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INTERNATIONAL ATOMIC WEIGHTS FOR 19431

Element

Sym-
bol

Atomic
number

Atomic
weight

Element

8>m-
bol

Atomic
number

Atomic
weight

Aluminum

26 97

Moh bdenum

95 95

Antimony.

121.76

Neodymium

144 27

39 944

Neon . .

20 183

Arsenic

74.91

Niekel

58.69

Barium

137 36

Nitrogen

14 008

Beryllium

9 02

Omnium

190.2

Bismuth .

209.00

Oxygen

16.0000

Boron .

10 82

P,t adiu

106.7

Bromine .

7^> ' 6

F' .split

30 98

Cadmium

112 41

Platmun

195.23

Calcium.

40.08

Potassium

39.096

Carbon

12 010

Praseodymium

140.92

Cerium

140.13

Protactinium

231

Cesium .

132 91

Radium

226 05

Chlorine

35.457

Radon

222

Chromium

52 01

Rhenium .

186 31

Cobi ,

58.94

R1 !mm

102 91

Colunibium

92 91

Ruoidium

37 %

85*. 48

Copper

63 57

Ruthenium

101.7

Dyspi oeium

162. 4t»

Samarium

150 43

Erbium

167.2

Scandium .

45.10

Europium

152 0

Selenium

78.96

Fluorine . .

19.00

Silicon

•i

28.06

C •Holnuum

156 9

Silver

107.880

( mm ,

69.72

Sodium ...

22.997

Germanium

72 60

Strontium. .

87.63

Gold

197 2

Sulfur .

32 06

Hafnium .

178 6

Tantalum

180 88

Helium

4.003

Tellurium

127 61

H^'mium

164.94

Terbium

159.2

' arogen

1 0080

Thallium

204.39

Indiui

114 76

Thorium

232.12

lodi"r ..

126 92

Thulium . .

Tin

169.4

193 1

Tin

118.70

Iroj

55.85

Titanium

47.90

K i.

~3fi

83 7

Tungsten

183.92

La lum

138.92

Uranium

238.07

207.21

Vanadium

50.95

Lit .m . .

6 940

Xenon .

_Ce

131.3

Lu mm

174 99

Ytterbium

\T>

173.04

Mb lesium .

24 32

Yttrium . .

88.92

Manganese

54 93

Zinc

65.38

Mercury

200 61

Zirconium

91.22

J. Am. Chem. Soc., 65, 1446 (1943).

OUP-880— 5-8-74— 10,000.

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PHYSICAL CHEMISTRY
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PHYSICAL CHEMISTRY
FOR COLLEGES

A Course of Instruction
Based upon the Fundamental Laws of Chemistry

by E. B. MILLARD

Professor of Physical Chemistry
Massachusetts Institute of Technology

Sixth Edition

McGRAW-HILL BOOK COMPANY, INC.

NEW YORK AND LONDON
1946

PHYSICAL CHEMISTRY FOR COLLEGES

K H MlLLARD

PHINTLD IN THE UNITED STATES OF AMERICA

pa rtt, thereof, may not be reproduced

in any form without permission of

the author

THE MAPLE PRESS COMPANY, YORK, PA.

PREFACE TO THE SIXTH EDITION

The author has again attempted the difficult task of presenting
physical chemistry to beginners with such simplicity that they
can understand it after careful study, and yet with such pre-
cision that they will acquire a sound background with which to.
continue in the field beyond the introductory course.

Such an attempt requires compromises that may not be accept-
able to all in any representative group of teachers, regardless
of the level at which the text is written. Moderate changes
toward or away from pedantic accuracy will shift the boundaries
of the group that is pleased without greatly changing the number
in the group. An arbitrary choice among the many important
topics of physical chemistry as to which should be treated "thor-
oughly," which "briefly," and which omitted is a matter on
which there will be differences of opinion, as is the amount of
detail which constitutes thoroughness.

In this sixth edition the selection and order of topics are the
same as in the preceding edition, the level of treatment is some-
what more precise, and there is a moderate redistribution of
emphasis among the topics. The greater part of the text has
been written anew, with the inclusion of new experimental
material where it was available; the remainder of the text has
been carefully studied and brought up to date to the extent that
limitations of space and the author's ability permit. The first
treatment of thermodynamics has been amplified and is now a
separate chapter, and more emphasis is placed on thermo-
dynamics in some of the other chapters as well. Opportunities
for introducing additional thermodynamics at several points
in the text will be evident to teachers who prefer to do so, but
these opportunities are not thrust upon teachers who prefer to
reserve most of the thermodynamics until the student has
acquired some background in physical chemistry. In the last
two chapters most of the important topics of the text are brought
together through the use of free-energy calculations in order to
provide a thorough review and summary with which to close

vi PREFACE TO THE SIXTH EDITION

the introduction to physical chemistry. Some of the problems
in the fifth edition have been retained, some amplified, some
omitted, and new problems have been added.

It is a pleasure to acknowledge the helpful suggestions received
from teachers of the text, especially those from associates at the
Massachusetts Institute of Technology.

E. B. MILLARD
CAMBRIDGE, MASS.,
May, 1946.

PREFACE TO THE FIRST EDITION

This book has been prepared to bring before college students
certain of the more important aspects of physical chemistry,
together with accurate modern data which illustrate the appli-
cability of its laws to the phenomena observed in the laboratory.
It has been assumed that the student is familiar with inorganic
and analytical chemistry, that he has had an adequate course in
college physics, and that the simple processes of calculus are
familiar to him.

No attempt has been made to cover the whole of physical
chemistry in a single volume; its most important topics have
been treated at such length as the size of the volume allows, and
numerous references to recent periodical literature are included
for those who would pursue any given topic further.

The limitations of the orthodox laws of physical chemistry
have been emphasized more than is commonly done in beginning
courses of physical chemistry. To this end the illustrative data
have been carefully chosen from modern experimental work, in
order to minimize the " experimental errors7' which are so often
used to conceal real deviations of a law from the facts it is
intended to express. A trusting belief in inadequate physical
laws will only retard the scientific progress of a student, and
weaken his faith in adequate laws ; whereas a wholesome apprecia-
tion that physical chemistry is an unfinished and growing science
may stimulate thoughtfulness and research. The proper time to
encourage a research attitude is in the very beginning of a
student's chemical career.

A qualitative treatment of the subject, so-called descriptive
physical chemistry, may be obtained from the text alone; but
careful consideration and study of the numerous tables of data
will be required if anything approaching an appreciation of quan-
titative physical chemistry is desired. A quantitative point of
view has been maintained as far as possible, even in the descrip-
tive material.

Rather more tabulated data have been presented than might
seem necessary in a beginning text. This is done to furnish the

vii

viii PREFACE TO THE FIRST EDITION

basis for numerous problems of a quantitative character. Many
such problems should be solved in the course of a term, since
they stimulate interest and increase the usefulness of the material
taught m the class room. The problems at the end of each chap-
ter will riot be sufficient to test the ability of general classes;
they are type problems in many instances, and should be supple-
mented by others designed by the teacher to suit the needs and
ability of his particular class Problems should often be assigned
for which data must be obtained directly from tables in the text.
Much of the value of problem working is lost when a student
knows he must use all of the data given him arid no other; this
too often results in suggesting the entire solution of the problem.
Moreover, fresh problems must be given every year if iresh
interest in physical chemistry is maintained, this can be done
only when ample data are at hand Some of the more difficult
problems at the end of a chapter should be worked by the instruc-
tor in class.

References to original sources are not intended primarily as
citations of authority for statements made; they are first of all
suggestions for further reading. With this in mind, most of the
references are to periodicals in English, and to those which are
available in small libraries The author has not hesitated to
draw upon little known periodicals when the material to be
obtained in them suited the needs of the text ; he has not ignored
foreign publications in the search for material; but for obvious
reasons much of the best data is published in the widely circulated
journals, and to these most of the reference work is confined.

The author is greatly indebted to Prof. James F. Norris and
to Mr. Charles R. Park for reading the manuscript and mak-
ing many helpful suggestions and criticisms based on their
teaching experience. Their assistance has aided materially in
detecting errors. He is also indebted to many other friends for
kindly interest and encouragement during the preparation of the
manuscript. It will be considered a favor if those who find
errors of any kind in the text will communicate them to the
author.

E. B. MILLARD.

CAMBRIDGE, MASS.,
Majfl, 1921.

CONTENTS

PAGE

PREFACE TO THE SIXTH EDITION v

PREFACE TO THE FIRST EDITION vii

CHAPTFK

1. Introduction. Determination of Atomic Weights 1

IT. Elementary Thermodynamics 29

III. Properties of Substances in the Gaseous State 51^

IV. Properties of Substances in the Liquid State 102

V. Crystalline Solids . 144

VI. Solutions . 179

VII. Solutions of Ionized Solutes 231

VIII. 'Thermochemistry 292

IX. Equilibrium in Homogeneous Systems 332

X. Heterogeneous Equilibrium 392

XI. Phase Diagrams . 427

XII. Kinetics of Homogeneous Reactions 464

XIII. Radiation and Chemical Change. 502

XI V.i Periodic Law of the Elements 517

XV. Radioactive Changes 525

X VI J Atomic Structure. 538

XVII. Colloids. Surface Chemistry . 566

XVIII. Free Energy of 'Chemical Changes . 591

XIX. Potentials of Electrolytic Cells . ... 630

AUTHOR INDEX . 673

SUBJECT INDEX 679

PHYSICAL CHEMISTRY FOR
COLLEGES

CHAPTER I

INTRODUCTION
DETERMINATION OF ATOMIC WEIGHTS

The title " physical chemistry" is almost universally accepted
for the field of science that is concerned with the physical effects
that attend or alter chemical changes Important aspects of
physical chemistry are the influence of temperature, pressure,
concentration, and relative proportions upon the rate and com-
pleteness of chemical reactions; the heat or work which they
may produce; the structure of atoms or molecules as revealed by
spectroscopy; the prediction from the properties of individual
substances of the extent to which they will react if conditions are
favorable; and, in general, the scope and limitations of the laws
or theories that apply to chemical systems. Even this long list
is not a complete description of the field; other topics of almost
equal importance could well be added

As a preliminary to the study of mixtures or of reacting sys-
tems, it is convenient to know how the properties of single sub-
stances change with pressure or temperature, the conditions
under which they are gaseous or liquid or crystalline, the condi-
tions under which they exist at equilibrium in two or more states
of aggregation, their heats of formation, and other properties.

Since it is obviously impossible to study experimentally
every chemical system at every temperature, every pressure,
and every concentration, one of the main functions of physical
chemistry is the formulation of laws and theories that show the
relation among the properties of chemical systems and the test-
ing and revision of these theories and laws as experimental
studies reveal minor or serious faults in them. The laws of

2 PHYSICAL CHEMISTRY

thermodynamics, which are in no sense the " property" of
chemists, have been most helpful in developing physical chem-
istry, but they could not have given this help to the extent that
they have without accurate experimental data of the most
varied kind. Notwithstanding the diligent work of thousands
of chemists for many years, the supply of data is still inadequate;
and notwithstanding the diligent work of many theorists for
many years the theoretical foundation of physical chemistry
is still inadequate. But research and study are continuing at
an increasing rate, and while the prospect of complete theory
or of complete experimental solution of the problems is remote,
the progress already made is both impressive and useful.

Physical chemistry correlates mathematics, physics, and
chemistry, using general methods of treating specific cases and
thus providing a classification that puts less stress on memory.
What is said of a selected system may be said of hundreds of
particularized systems, almost without modification It is only
for this purpose that "principles" are important and only in
this sense that the principles or generalizations of chemistry have
come to be called physical chemistry Thus physical chemistry
is not a subdivision of chemistry like inorganic or organic chem-
istry, but a theoretical foundation for all of chemistry.

The following pages are intended to be a first survey of its vast
field, with emphasis upon what has been accomplished and with
some indication of what yet remains to be done Of necessity
many topics have been treated briefly and others have been
omitted entirely in order to keep the length of text within reason-
able limits; but numerous references to the original sources are
given in the footnotes, and suggestions for further reading are
given at the ends of the chapters. Since the experimental facts
are more important than theories, we shall speak of the devia-
tions of theories or laws from the facts, rather than the deviations
of the facts from the theory.

Laws of Nature (Results of Experience). — Some of the general
laws of chemistry appear to be absolutely exact; they describe
faithfully the results of most carefully conducted experiments,
and the apparent deviations of these laws become less and less
as the manipulative skill employed in testing them increases.
Among these laws may be mentioned the law of the indestructi-
bility of matter (conservation of matter), the law of definite

INTRODUCTION 3

proportions, Faraday's law of electrolysis, and the laws of
thermodynamics. Other so-called "laws" fail to describe actual
conditions, and the deviations are not due to experimental errors
in the data. The deviations may be small under certain condi-
tions and larger under other conditions. Such "laws" are useful
approximations, which show the properties of substances in a
qualitative way and which more or less accurately show their
quantitative behavior. Thus, no simple law is known that shows
exactly how the pressure of a quantity of gas changes as the
volume or temperature changes. An approximation is known that
shows these changes for most gases at moderate pressures within
1 or 2 per cent but is seriously in error at high pressures. Hence,
it is as necessary to have a wholesome appreciation of the limited
applicability of this "law" as it is to know the law itself.

As the Y_arious_Law_s are stated*, we shall state the experi-
mental facts which confirm them or which show the extent of
their errors and so endeavor to learn whether judgment is
required in the use of a "law" or whether it is rigidly accurate
under all circumstances. For this purpose a sense of proportion
is essential. If a law appears to be exact in all but one case from
a hundred thousand, as is true oi the law of definite proportions,
this "exception" may point to a new fact. In these circum-
stances one must examine the data more carefully or reconsider
the fundamental assumptions or look for an unjustified interpre-
tation. If the "exception" reveals a new fundamental fact, as
is true here, much detailed study may be required before its
full significance is appreciated. But while this study is in prog-
ress, it would be^absurd to allow this "exception" to divert one's
attention from the practically universal validity of this law. On
the other hand, Avogadro's law and Boyle's law are "limiting
laws," which become more nearly exact as the pressure of the gas
is i educed but which are not strictly true at atmospheric pressure.
They may be quite inaccurate at high pressures, and due account
of the deviations must be taken in considering compressed gases.
Some approximate laws describe the results of experiment quite
accurately under certain conditions but deviate to a larger extent
under other conditions. When this is so, the limiting conditions
under which the law is accurate to within 1 or 2 per cent will be
stated.

The statements put forward as laws of nature are sometimes

4 PHYSICAL CHEMISTRY

the result of experience alone (empirical laws). There is always
a possibility that some future experiment will demonstrate the
untruth of what we have considered as a law, but the proba-
bility of this becomes less and less as the number of experiments
increases. No change has ever been observed in the total mass
of the substances involved in a chemical reaction; z.e., no matter
is destroyed in being changed into other forms.1 As the methods
of experiment have become more and more refined, and as the
experimenters have become more skillful in their work, this law
remains unshaken as a statement of universal experience, and it
is now commonly accepted as an exact law of nature. Other
simple laws, such as Boyle's law and Charles's law for gases,
are also the result of experience ; but as the experimental methods
have become more refined, real deviations of these laws from the
facts observed have been discovered. These experiments point to
a failure of the supposed laws to explain completely the behavior
of substances and are not to be traced to errors of experiment.

Proper reserve should always be exercised in drawing general
conclusions from a set of experimental data. The phenomena of
nature are often more complex than we think, and what appears
to be a general law may be true only under restricted conditions.
To state such a law without mentioning the qualifying*circum-
stances under which it is applicable is to misrepresent the facts.

Theories are plausible beliefs advanced to explain observed
facts. They serve to guide further experiments in a given field.
Thus the theory that a gas consists of molecules, separated from
one another by considerable distances and in rapid motion, offers
a ready explanation of the compressibility of gases, of their
diffusion, of their ability to mix with other gases in all propor-
tions, and of practically all their properties. The evidence in
favor of the theory is abundant and convincing; no facts are
known that contradict it; and deductions based on this theory
are in accord with the results of experiment. It is therefore
universally accepted as a fact but is referred to as the kinetic
theory of gases.

Many such theories are found in chemistry. They are

1 The exception to this statement that became so conspicuous in the
summer of 1945 had been known for years. It was, and still is, so
exceptional as to leave the conservation of matter one of the most valuable,
if not the most valuable, guiding principle in chemistry.

INTRODUCTION

accepted so long as they are in accord with the facts; they may
be altered to fit new discoveries, but they should be discarded
in favor of newer and more satisfactory ones when they seriously
conflict with the results of experiments. Before proceeding to a
study of new laws and theories, it will be advantageous to review
some of those already studied in earlier courses in chemistry.

Indestructibility of Matter. — It is a familiar fact that matter
may be changed into various forms by combination and rearrange-
ment of the elements in various ways without any loss in the
total mass of material The many operations of analytical
chemistry depend on the validity of this fact; but since there is
no reason why there might not be a change of mass during chemi-
cal change, it has been necessary to test this belief experimentally
before accepting it. Perhaps the best
known tests are Landolt's experiments1
extending over a period of 20 years and
devoted to a careful study of 15 different
chemical reactions, which wrere examined
with great skill and patience. The react-
ing substances were enclosed in the separate
arms of sealed vessels, such as that of Fig.
1, to prevent the possibility of mechanical
loss of material. The tubes were wreighed
on a very sensitive balance, a counterpoise
of the same size and shape being used. Then, by tipping
the vessel, the two solutions were brought in contact a
little at a time. After the reaction had been completed, the
vessel was weighed again. The weighings were made several
times, and an average was taken. As a result of his work,
Landolt concluded that, if there was a change in mass during
chemical reaction, it was less than the error of weighing, which
was 1 part in about 10,000,000.

The later work of Manley2 on the reduction of silver nitrate
by ferrous sulfate was carried out with extreme care. His experi-
ments showed that the change in weight attending chemical
reaction was less than 1 part in 32,000,000, which is less than
the probable error in weighing. In another series of experiments
Manley showed that the reaction between barium chloride and

1 Z. physik. Chem, 12, 1(1893); Sitzber. preuss. Akad. Wise., 1908, 354.

2 Phil. Trans. Roy Soc. (London], (A) 212, 227 (1913).

6 PHYSICAL CHEMISTRY

sodium sulfate was attended by a change in mass of less than 1
part in 100,000,000. In the light of these experiments there is
no reasonable doubt that mass is conserved in chemical changes ;
we may, therefore, state that matter (anything which has mass)
does not change in mass during chemical change.

One of the postulates of Einstein's theory of relativity states
that matter is converted into energy under certain circumstances.
The extraordinary velocities of some of the particles produced
in nuclear reactions (which are briefly discussed in Chap. XVI)
confirm experimentally the conversion of minute quantities of
matter into energy. The astonishing amount of energy radiated
by the sun is also claimed to be due to the conversion of matter
into energy, with the loss of 3 0 X 1011 tons of mass per day
required. Recently a few grams of matter were converted
into an enormous amount of energy under circumstances that
attracted world- wide attention and set off an equally large
amount of speculation as to future developments; some of them
are very attractive indeed. For the purposes of this text, we
may well leave the future to the future and confine our atten-
tion to the chemical reactions with which we are likely to be
concerned. In these reactions, mass is conserved within our
limits of measuring it. The relation between the loss of mass
and the energy produced is AE = Arac2, where c is the velocity
of light, 3 X 1010 cm per sec. Hence, ii the total energy evolved
by the combustion of 12 grams of carbon to carbon dioxide came
from the destruction of matter the loss in mass would be about
jQ-s gram, which is far beyond the precision of any weighing
device yet discovered.

Elements and Compounds. — The number of kinds of matter is
very great indeed, but attempts to resolve matter into its ulti-
mate constituents by chemical means have brought to light about
92 substances that cannot be resolved, or at least that have not
yet been resolved, into simpler substances. These substances
are called elements. The number of experiments performed upon
most of the known elements is so great as to make it improbable
that they consist of two substances which may be separated later
by some chemical process.

The separation of elements into isotopes, which are atoms
of different mass and practically identical chemical properties,
will be discussed briefly in Chap. XVI. We may mention here

INTRODUCTION 7

that deuterium, or hydrogen of atomic weight 2, has been sepa-
rated in a practically pure state from natural hydrogen,1 that
neon (atomic weight 20.18) has been separated into portions of
atomic weight 20 and 22, and that lithium (atomic weight 6.94)
has been separated into portions of atomic weight 6.0 and 7.0;
there are other instances of more or less complete separation of
elements. The separation of chlorine (atomic weight 35.45) into
portions of which one contained 99 per cent of the isotope of
mass 37 0 has also been reported 2 The isotopes of hydrogen
are called protium and deuterium (symbol D). Deuterium oxide,
or " heavy water/' contains about 20 per cent of "heavy"
hydrogen, as compared with 11 per cent hydrogen in ordinary
water; it boils at 101.42°, freezes at 3 8°, has a density of about
1 1, and its surface tension, vapor pressure, latent heat, and other
properties differ from those of ordinary water.

Lead of atomic weight ranging from about 206.0 to 208.0 has
also been found in small quantities in some rare minerals, proba-
bly as the result of radioactive changes. These isotopes are the
result of "natural" processes, in the sense that they have not
been carried out in a laboratory for the purpose of making this
separation, and they are accordingly naturally occurring excep-
tions to the constancy of atomic mass.

As may be seen from the periodic table in Chap. XIV, it is
improbable that there are many undiscovered elements of atomic
weight less than uranium, and there is yet no evidence of natural
elements of higher atomic weights.8 The discovery of "element
93" or "element 94" would cause no change in the periodic
arrangement of the elements. It is customary to regard isotopes
as different forms of the same element and to assign them all to
a single place in the periodic table. But the discovery of another
alkali element having an atomic weight between those of sodium
and potassium, for instance, is most improbable, as is the dis-
covery of any new element for which no place is available in the
periodic table

Law of Definite Proportions. — This law states that the quantity
of an element which will combine with a given weight of another

1 UREY and TEAL, Rev. Modern Phys , 7, 34-94 (1935)

2 HiRscHBOLD-WiTTNER, Z anorg allgem. Chem , 242, 222 (1939).

3 "Synthetic" atoms of higher atomic weight have been prepared by
methods that will be discussed in ("hap XVT

8 PHYSICAL CHEMISTRY

element to form a pure chemical compound is a fixed quantity,
regardless of the method of preparation of the compound. In
other words, the percentage of each element in a pure compound
is always the same, and the presence of an excess of one element
does not result in the formation of a compound containing more
of it. The atomic theory was suggested to Dalton by this law,
and the theory furnishes a ready explanation of the law. Identi-
cal whole atoms of an element, by combining with identical
whole atoms of another element, must yield molecules of a fixed
composition.

Table 1 shows data1 on the synthesis of silver bromide from
carefully purified silver and bromine, together with the weight of
bromine combined with each grain of silver. Elaborate precau-
tions were taken to ensure the purity of the substances weighed
and to avoid mechanical loss during the synthesis.2

The synthesis was conducted by supplying ammonium bromide
to a weighed quantity of silver that had been converted into
nitrate, until no more bromine would combine with the silver,
after which the silver bromide was collected and weighed.

Other examples of the law of definite proportions are shown in
Table 1. A quantity of iron was converted into ferric, oxide and
heated with an excess of oxygen until no more would combine
with it.8 The ferric oxide was weighed, then heated in a current
of hydrogen until all the oxide had been completely reduced to
iron, which was then weighed. The synthesis of tin tetrabromide
is also shown in Table 1. It will be seen again that the composi-
tion of the product is constant, insofar as it is possible for the best
quantitative chemistry to determine it 4

Molecular Theory. — The theory that matter of all kinds con-
sists of very small particles or molecules is now commonly

1 BAXTER, J. Am. Chcm. Soc , 28, 1322 (1906)

2 Students will note that six significant figures are given in most of the
weights in Table 1. This is justified in view of the elaborate precautions
that atomic-weight work requires. All the reagents are purified with
great care, and manipulative precautions are taken with which students of
ordinary quantitative analysis are quite unfamiliar For an excellent
description of such work, see Baxter, Proc Am. Acad Arts Sci., 40, 419
(1904), and 41, 73 (1905), in connection with the atomic weight of iodine.
Students who read these papers with care will find themselves well repaid.

3 RICHARDS and BAXTER, Proc. Am. Acad. Arts Sci., 36, 253 (1900).

4 BONGART and CLASSEN, Ber., 21, 2900 (1888).

INTRODUCTION
TABLE 1 — DATA ILLUSTRATING DEFINITE PROPORTIONS

Weight

Weight of

Grams of bromine

silvei bromide

combined with each

silver

formed

gram of silver

5 01725

8 73393

0 74078

5 96818

10 38932

0 74079

5 G2992

9 80039

0 74077

8 13612

14 16334

0 74080

5 07238

8 82997

0 74079

4 80711

8 36827

0 74081

5 86115

10 20299

0 74078

6 38180

11 10930

0 74078

6 23696

10 85722

0 740V9

9 18778

15 99392

0 74078

8 01261

13 94826

0 74079

8 59260

14 95797

0 74079

8 97307

15 62022

0 74079

Average 0 74079

Weight of
iron

Weight Fe.Oa

Pei cent Fe in Fe2O3

2 78115

3 97557

69 956

3 42558

4 89655

69 959

3 04990

4 35955

69 959

4 99533

7 14115

69 951

4 49130

6 42021

69 956

Weight of
tin

Weight of
SnBr4 formed

Per cent tin in SnBr4

2 8445

10 4914

27*113

4 5735

16 8620

27 123

4 5236

16 6752

27 119

3 0125

11 1086

27 116

2 8840

10 6356

27 113

3 0060

11 0871

27 123

accepted as a fact. This theory is in complete accord with all
the known facts of chemistry; it explains in a simple way all our
chemical reactions; and it forms the basis of modern chemical
thinking. The molecules of which a substance consists cannot
be divided into smaller particles without a complete change in

10 PHYSICAL CHEMISTRY

,the properties of the resulting particles. They are the limit of
divisibility for a given kind of matter. When there are two or
more kinds of molecules or molecular species present, the mass of
matter is called a mixture The usual criterion of a mixture is
that it may be prepared in varying proportions, while a pure
substance always has the same composition.

If all the molecules of a pure substance are of the same species,
every molecule must have the same composition as the whole
mass of pure substance. Two matters at once claim interest:
the relative weights of the molecules of different substances, and
the way in which the molecules are formed from their constituent
elements

The relative molecular weights cannot be determined from
comparisons of single molecules on account of their small size,
but equally satisfactory results may be obtained li we have a way
of counting out the same number of molecules of each substance
for comparison We consider next, a procedure thai- accom-
plishes this purpose

Avogadro's Law. — A provisional statement of this important
law, which will require some modification to put it into exact
form, is that equal volumes of gases at the same temperature
and pressure contain the same number of molecules. If this law
is accepted, we may determine the relative weights of the mole-
cules of two gases by comparing the weights of equal volumes
at the same temperature and pressure. In order to put these
comparisons on a numerical scale, the next step is obviously to
select some substance as a reference standard, and chemists by
common consent have ^adopted 32 as the "molecular weight"
of oxygen. l Since they employ the gram as a unit of weight, 32
grams of oxygen is therefore accepted as a "gram-molecular
weight," or a gram molecule. On account of the extensive use
of this term, it has been abbreviated to "mole," which is written
without a period. It is not an abbreviation of the word "mole-
cule," but a separate newT word meaning gram-molecular weight
or formula weight. Molecular weights based on gas densities
are usually free from any uncertainty as to the formula of the
substance, but we shall call 18 grams of liquid water or 58.5

1 Strictly speaking, it is 16 00 as the atomic weight that was arbitrarily
accepted as the standard. Since oxygen is diatomic, its molecular weight
is 32.00.

INTRODUCTION 11

grams of sodium chloride a mole without commitment as to
whether a molecule of this composition actualty exists or not.

A molecular weight of a gas is that weight of it which occupies
the same volume as 32 grams of oxygen at the same temperature
and pressure. If we define a molecular volume of gas as the
volume occupied by 32 grams of oxygen, we may then define the
molecular weight of any gas as that weight which fills a molecular
volume. The facts (1) that most of the precise data on gas
densities are reported at 0°C. and (2) that most texts on elemen-
tary chemistry give the niolal volume for 0°C and 1 atm. pres-
sure as 22.4 liters often leave students with the unfortunate
misconception that Avogadro's law applies only to " standard
conditions/' even though the language in the texts correctly
states that only the same temperature and pressure are required
At 1 atm pressuie a molecular volume of gas is 24.4 liters at
25°C., 30 0 liters at 100°C , and 22.4 liters at 0°C.; but a molec-
ular volume is also 24 4 liters at 0°C. and 0.92 atm., or 30 6 liters
at 0°C. and 0 73 atm., or 22 4 liters at 100°C. and 1.37 atm., for
all these figures are the volumes of 32 grams of oxygen under the
conditions stated A molecular weight of any gas is the weight
required to fill any of these volumes at the corresponding tem-
perature and pressure, and we may of course compute the
weight of a molecular volume from the weight of any convenient
volume.

Avogardro's law is an example of a " limiting law" which
becomes more nearly exact as the pressure at which the gases
are compared is reduced but which may be largely in error at
high pressures or near the condensation point of a gas. For
11 permanent" gases at 1 atm. and ordinary temperatures the
number of molecules per unit volume is the same within about
1 per cent. But the fractional expansion for a given pressure
decrease at constant temperature is not quite the same for all
gases, and therefore precise molecular weights may be deter-
mined through Avogadro's law only at low pressures. (The pro-
cedure for accomplishing this comparison will be given in detail
presently.) We may now state Avogadro's law in a workable
and exact form : Equal volumes of gases at the same temperature
and the same very low pressure contain the same number of mole-
cules. The necessity for this form of statement may be illus-
trated by the ratio of the density of N20 to that of oxygen at 0°,

12 PHYSICAL CHEMISTRY

which is 1.9782/14289 = 1.3844 for the gases at 1 atm. and
0.9855/0.7142 = 1.380 at M atm. The limit that this ratio of
densities at equal pressures approaches as the pressure approaches
zero is 1 3765. Hence, 32.000 X 1.3765 = 44.020 is the molec-
ular weight of N2O at such a low pressure that Avogadro's
law is exact 1

Atomic Weights. — An atomic weight of an element is the
smallest weight of it found in a gram molecule of its compounds.
Since a molecule must contain a whole number of atoms of each
element , a gram-molecular weight must contain a whole number
of atomic weights of each element. Thus, the accepted values of
atomic weights represent the smallest quantity of each element
found in a gram molecule of any compound so far; the possibility
of discovering a compound with less of the element per molecular
weight always exists, but the probability of this discovery
becomes less as the number of compounds studied increases.
There are many ways in which the value of an atomic weight
may be checked, such as by its specific heat, its place in the
periodic system, and its characteristic X-ray spectrum It is
most improbable that any of the atomic weights now accepted will
need to be divided by a A\hole number. The currently accepted
atomic weights are given in Table 4 and repeated on the inside
front cover for convenient reference.

The molecules of helium, argon, the other gases of the zero
group in the periodic table, and most metals in the vapor state
consist of a single atom, so that atomic weights are identical with
molecular weights for these substances Oxygen, nitrogen,
chlorine, and hydrogen,*or, in general, any element whose condi-
tion in the vapor state is indicated by the symbol E^ have atomic
weights that are half the molecular weights.

Atomic weights, by which chemists always mean gram-atomic weights
of course, are not proportional to weights of atoms unless all the individual
atoms have the same mass. As will be explained in Chap XVI, there is
no particle in chlorine that weighs 35 46/(6 03 X 1023), even though 35 46
is the correct "atomic weight" of chlorine. The individual atoms have
weights corresponding to 35 00 and 37.00 on the oxygen scale, but the mix-
ture pf these particles in the proportion of about 3 : 1 bears the name of the
element chlorine. All the occurrences of chlorine in nature are of the
same composition within 1 part in 10,000 or more, and the mixture behaves

1 The data are by Moles and Toral, through the report of the International
Committee on Atomic Weights, /. Am. Chem Soc., 60, 739 (1938)

INTRODUCTION 13

like a single substance in every chemical process. Samples of chlorine
collected from widely separated sources, and from rocks that have probably
never been in contact with the ocean, show no detectable variation in
atomic weight from chlorides derived from the sea Samples of lead from
minerals not associated with radioactive materials have been collected from
sources all over the earth in a further test of the constancy of atomic weights.
The atomic weight of lead from these materials was most carefully deter-
mined [BAXTER and G ROVER, J Am Chem Soc , 37, 1027 (1915)] and found
to be 207 21 ± 0 01 (see Table 90) It will be shown later that common
lead consists mamlv of isotopes of mass 206 00, 207 00, and 208 00, but the
figure 207 21 in Table 4 is still the proper atomic weight of lead The
long series of radioactive changes, of which the decav of radium itself
is the best kno\\n, results in an isotope of lead of atomic weight 2060.
Anothei series of such changes of which thorium is the parent element ends
in an isotope of lead of atomic weight 208 0 Hence, in lead from radio-
active materials the atomic weight values vary from nearlv 208 0 to nearlv
206 0, depending upon the source Some data bearing upon these "radio-
genic" leads arc given in Table 91

The atomic weights of elements that do not form gaseous
compounds are determined from exact chemical analysis of their
compounds, together with supplementary data that show the
formula of the compound. For example, 63.57 grams of copper
combine with 10 00 grams of oxygen to form cupric oxide, and
03.57 is the combining weight of copper. This is also shown to
be the atomic weight of copper when it is established that these
elements combine in the atomic ratio 1:1. It is found that
50 708 grams of iodine combine with 10.00 grams of oxygen to
form a stable pure substance which is shown by supplementary
data to be iodine pentoxide, I2O5, whence the atomic weight of
iodine is 120.92. The analytical data alone show only that
50.708 grams is the weight of iodine combining with 10 00 grams
of oxygen; they furnish no way of deciding what multiple or
submultiple of this weight is the actual atomic weight.

Although the most direct method of determining precise atomic
weights would be the analysis or synthesis of oxides, very few
of the elementary oxides may be prepared in a sufficiently pure
form for this purpose. Metallic halides are more readily puri-
fied, and ratios such as EC1: Ag may be used to determine atomic
weights if the atomic weights of Cl and Ag are accepted. But
these atomic weights involve those of other elements. The basic
quantities are the atomic weights of 11 elements, which are
related to one another through 71 ratios that have been most

14 PHYSICAL CHEMISTRY

carefully determined. From these ratios, F. W. Clarke1 derives
43 estimates of the atomic weight of silver, 32 for chlorine, 16 for
bromine, 22 for nitrogen, etc , and finally determines the basic
values of the 11 atomic weights, H, C, N, S, Cl, Br, I, Li, Na, K,
and Ag.

In place of attempting to follow this rather involved calcula-
tion, we may illustrate the principle by some simpler calculations
involving three of the fundamental ratios, as follows:

I:O = 50.768.16.000
I205 2Ag = 100 64 623
Ag Cl - 100.32867

Accepting the formula of I^Os, the first ratio establishes

1 - 126.92

The second ratio then establishes Ag = 107. 88; and the third ratio
establishes Cl = 35.45. From the ratio Ag I = 100.1176433,
independently determined,2 these atomic weights are confirmed;
from the ratio AgCl.AgI = 100.163.8062, another confirmation
is obtained.3

Once these atomic weights are established, the ratio ECl:Ag
may be used on another substance, say KC1, for winch the ratios
are4Ag-KCl = 100. 69. 1085 and AgCl KC1 = 100 52.016. The
molecular weight of KC1 that is here established may be checked
from the ratio KC1O3:KC1 = 100.60.836, which goes back to
the fundamental standard of oxygen.

Some of the common procedures for determining atomic
weights will now be described

Atomic -weight Methods, a. From Gas Densities Alone —
When the number of atoms in a molecule of an elementary sub-
stance has been established, the atomic weight may be deter-
mined by dividing the molecular weight by the proper whole
number. Similarly, the atomic weight of bromine may be
determined by subtracting from the molecular weight of hydrogen
bromide the atomic weight of hydrogen, since its molecule is

1 Mem. Nat. Acad. Sci., 16 (3), part V, pp 1-418 (1922), this particular
operation is given on p 116 of the memoir.

2 BAXTER and LUNDSTEDT, /. Am Chem. Soc., 62, 1829 (1940).
8 BAXTER and TITUS, ilid , 62, 1826 (1940).

4 BAXTER and HARRINGTON, ibid., 62, 1836 (1940).

INTRODUCTION

known from combining volumes to contain one atom of each
element. We have seen above that Avogadro's law is a useful
approximation at atmospheric pressure and an exact law at very
low pressures and hence that, in order to determine precise molec-
ular weights, densities must be determined at low pressures.
The pioneer work of Guye and his students has been supple-
mented by that of several other groups to such an extent that gas
densities are among the most precise methods of determining
molecular weights. It has been found that the ratio of density
to pressure is a linear function of the pressure, and hence, by
plotting d/p against p and extrapolating to zero pressure (or at
least to very low pressures), one may determine the density of a
gas under conditions such that it is substantially an ideal gas.

TABLE 2 — DENSITY OF CARBON DIOXIDE AT 0°

Pressure,

Density, grams

Ratio

a tin

per liter

d/p

1 976711

I 97676

1 314823

1 97226

0 985018

1 97010

0 655922

1 96788

0 491678

1 96676

0 327606

1 96566

1 96346

As an example of the precision that may be attained we quote
some data1 for CO2 at 0° in Table 2. When these ratios of d/p
are plotted against the pressure, as is done in Fig. 2, they fall on
a straight line that may be extended to zero pressure to deter-
mine the limiting density. This limiting density d/p is 1.96346;
and when similar data for oxygen2 are treated in the same way,
the limiting ratio of d/p is found to be 1.42767. The ratio of
these limiting densities is the ratio of their molecular weights
according to Avogadro's law, which is exact at the limit; and

1 DIETBICHSON, MILLER, and WHITCHER, not yet published.

2 BAXTER and STARKWEATHER, Proc Nat Acad. Sci , 14, 50 '(1928);
see also 10, 479 (1924); 11, 231, 699 (1925), 15, 441 (1929) for data on other
gases. The data for oxygen at 0° are as follows:

Pressure, atm
Density, grams per liter

1 000
1 42896

H
1 07149

0 71415 0.35699

PHYSICAL CHEMISTRY

since the molecular weight of oxygen is 32.000 by definition, the
molecular weight of C02 is 32.000(1.96346/1.42767), or 44.010,
and C = 12.010.

This method is particularly suited to accurate atomic-weight
determinations on the gases of the "zero" group in the periodic
table, since these elements do not form compounds Thus for
neon and argon the limiting ratios d/p at 0° are 0.90043 and
1.78204, respectively;1 and the molecular weights are 20.183 ior

neon and 39 944 for argon
Since the molecules are mona-
tomic, these are also the
atomic weights.

b From Molecular Weights
and Compositions by Weight —
When large numbers of gase-
ous compounds of any ele-
ment are examined, the
smallest weight of an element
found in a mole of any of its
compounds is called the
atomic, weight of that ele-
ment Since determinations
oi molecular weight are not
usually performed with great
accuracy, careful analytical
data are used to supplement
this work. An example will
make the procedure clearer
In Table 3 are given some approximate data for gaseous nitrogen
compounds, using only whole numbers. It is seen from this
table that no nitrogen compound contains less than 14 grams of
nitrogen per molecular weight; this is, then, the approximate
atomic weight. But it is not an exact value, since it is derived
from rough data Accurate values may be determined from
gravimetric analysis of nitrous oxide, nitric oxide, or nitrosyl
chloride ; from the limiting densities of N2O, NO, or NH3; from the
ratios AgfAgN03 = 100:157.48, NaCl:NaNO3 = 100:145.418,
N2O6:K20 = 100:87.232, all of which indicate N = 14.008.
None of these gravimetric ratios would show whether N = 14 or
1 BAXTER and STARKWEATHER, ibid., 14, 50 (1928), 15, 441 (1929).

1.977
1976
1975
1974
1973
1972
1971
1970
1969
1.968
1967
1966
1.965
1.964
1963

[

/
/

f8*"

Limiting c

fens/fy '196.

146

0 02 04 06 0.8
Pressure in Atmospheres

FIG. 2 — Limiting density of COa at 0'

INTRODUCTION

N = 28, but the latter value is excluded by the densities of NO
and NH3, so that N = 14.008 is the proper atomic weight for
nitrogen

TABLE 3 — NITROGEN CONTENT OF COMPOUNDS

Weight

Per cent

Weight

Substance

of a molal
volume

nitrogen
in the

of nitrogen
in a molo

of gas

compound

of gas

Nitric oxide

Ammonia

Nitrous oxide

Nitric acid

Nitrosyl chloride.

Hydrazine. . . .

c From Analytical Data and Specific Heats — The metals ami
some other elements do not form gaseous compounds at tempera-
tures suited to accurate work, and determinations of their atomic
weights rest on other considerations But the weight of a metal
that combines with 16 grams of oxygen is either an atomic
weight, two atomic weights, half an atomic weight, or two-thirds
or three-fourths or two-fifths of an atomic weight, depending on
whether the formula of the oxide is EO, E2O, EO2, ~E>zOa, EsO^ or
E2O5. Analysis of the oxide will give, therefore, an exact value
of the atomic; weight or a simple fraction of it, and it requires
only a rough determination in some other way to indicate which
multiple of the weight combined with 16 grams of oxygen is the
true atomic weight. The law of Duiong and Petit furnishes such
a method of fixing the multiple for heavy elements. This law
states that for metals and the heavy elements the atomic heat
capacity at room temperature is about 6.2, that is, that the
quantity of heat required to raise an atomic weight of an element
through 1° is the same for all solid elements. From the data of
Table 1 we see that iron oxide is 09.956 per cent iron; hence the
weight of iron combined with 16 grams of oxygen is

a: 16 = 69:956: (100.00 - 69.956)

or x = 37.256. This is either the atomic weight of iron or a
simple fraction of it. The specific heat of iron is 0.115, and 6.2

18 PHYSICAL CHEMISTRY

divided by 0.115 is 54, which is approximately the true atomic
weight. It will be seen that 37.256 is about two-thirds of 54,
whence the true atomic weight is 1.5 X 37 256, or 55 88.

The atomic weight of bromine has been established by several
methods at 79.92. From this we may compute the atomic
weight of silver from the data of Table I, since an atomic weight
of silver combines with a whole number oi atomic weights of
bromine. Thus

Ag.Br - 1.000:0.74078 - x. 79.92

whence x — 107.88 The specific heat of silver is 0.056, and
6.2 divided by 0.056 is 110. Thus the true atomic weight of
silver is 107 88

An example ol a more complete set of experiments is the follow-
ing one, which serves to determine the atomic weights of silver,
(Jilorine, and lithium with reference to oxygen. By reducing
lithium perch! orate to chloride it was found1 that 100 grams oi
the former gave 39.845 grams of the latter. The formula of the
perch! orate is LiClO4, whence it follows that the molecular weight
of lithium chloride is x (x + 4 X 16.000) = 39 845.100.000, or
x — 42.393. The lithium chloride was then treated with silver
nitrate solution made from a weighed quantity of silver, from
which it was found that each gram of lithium chloride required
2.54460 grams of silver, giving the atomic weight of silver as
2.54460 times the moleculai weight of lithium chloride, or 107.871 .
This will be seen to be in accord with its atomic weight calcu-
lated above from the synthesis of silver bromide. Then, by
weighing the silver chloride formed, the ratio of silver to silver
chloride was found to be 1 . 1 3287, from which the atomic weight
of chlorine is given by 107 871 : (107.871 + y) = 1.000.1 3287,
or y — 35.454. Returning now to the ratio of lithium chloride
to perchlorate we see that the atomic weight of lithium may
be calculated from the atomic weight of chlorine just found,
since

LiC!O4:LiCl = (z + 35.454 + 4 X 16.000) : (z + 35 451)

= 100.000:39.845
whence z = 6.939.

1 RICHARDS and WILLARD, / Am. Chem Sac , 32, 4 (1914)

INTRODUCTION 19

d. From X-ray Spectra. — The frequency of the characteristic
X radiation emitted by an element when it is bombarded with
electrons serves to fix the position of the element in the periodic
table and so to determine its atomic weight from the combining
weight. We shall see in Chap. XIV that when the square root
of this frequency is plotted against the atomic number, which is
the order number of the elements in the table, a straight line
results. This discovery of Moseley's places beyond doubt the
positions of the elements m the table.

e. From the Mass Spectrograph. — Since a discussion of this
procedure is given in Chap. XVI, it will be mentioned only briefly
here. The method depends upon the fact that a charged par-
ticle is deflected, upon passing through an electric field, to an
extent depending upon the ratio of charge to mass, in addition
to other factors. By proper design of the apparatus, molecules
or atoms of equal charge are caused to record their deflections
upon a scale linear with respect to mass, from which precise
determinations of atomic mass are derived. The atomic masses
of hydrogen and iodine, as measured by this method, are 1.00813
and 120.933, respectively; the chemically determined atomic
weights are 1.0081 and 126 92, respectively. This method meas-
ures the masses of the individual isotopes rather than those of
the naturally occurring mixtures which chemists call the ele-
ments, but the comparison of the atomic weight of iodine from
the- two methods is justified by the fact that this element consists
of a single species of atom A similar comparison for neon, whose
atomic weight by the limiting density method is 20.183, shows
from mass spectrographic data that this element consists of 90
per cent of atoms of mass 19 997, 9 73 per cent of mass 21.995,
and 0.27 per cent of mass about 21 (the precise mass has not been
determined). These isotopic masses are referred to that of the
most abundant oxygen isotope as 16.000 and must be corrected
to the chemical scale by allowance for the small amount of the
heavier isotopes before being compared with the data based upon
natural oxygen. The multiplying factor for this correction is
1.00027.

/. From Lattice Constants and Crystal Densities. — The recent
precise determinations of lattice constants of crystals from X-
ray diffraction give another method of determining relative
'atomic weights or molecular weights that is most promising,

20 PHYSICAL CHEMISTRY

although it has not yet been applied to many substances. A
single illustration must suffice here Both LiF and NaCl have
the crystal structure shown in Fig 22, the edge of a cube con-
taining 4LJF1 is a = 4.0181 X 10~8 cm., and the edge of a cube
containing 4NaCl is a = 5.0301 X 10~8 cm. The densities at
25° are 26390 for LiF and 2.1623 for NaCl, and so the ratio
of the weight of 4LiF to 4NaCl is the ratio of azd for the two
substances :

owMuv = (40181 X 10-HV>. =
aNaci^Naa (56301 X 10-h)32.1623 U 6

If the atomic weights of Li, Na, and 01 are accepted, F = 18 994,
which is in close agreement with the accepted value, 19.00 in
Table 4 In these determinations the relative lattice constants
are readily determined to six significant figures and the densities
to about the same precision. Since atomic weights and molec-
ular weights are relative quantities, this method may develop
into the most precise one for atomic weights.

Atomic -weight Table. — The table of international atomic
weights published each year is based upon careful study of all
available data by a Committee on Atomic Weights of the Inter1
national Union of Chemistry.2 The importance assigned to each
determination in computing the weight for general use depends
upon the number of individual experiments m a scries and the
probable accuracy of the work. The most recent atomic weights
obtainable are given in Table 4 All the figures stated are sig-
nificant. Thus 14.008 for nitrogen indicates that this atomic
weight is accurate to a thousandth of a unit; 197.2 for gold indi-
cates that the second decimal place is still uncertain

Units and Standards. — It will be convenient to define and
record here the numerical values of some quantities for use
throughout the book. We shall use the centimeter-gram-second
(c.g.s.) system of units and the centigrade temperature scale in
our calculations, following the usual custom of physicists and

1 JOHNSTON and HUTCHINSON, Phys Rev , 62, 32 (1942) In the original
paper the data are given to six figures

2 Summaries of current work on which changes are based appear with the
report each year. This is reported in J Am Chem Soc. and other peri-
odicals. A summary of all of the atomic-weight work done prior to 1920 ,
is given in Mem. Nat Acad Set , 16 (3), part V, pp 1-418 (1922)

INTRODUCTION

chemists The chief advantages of this c.g.s. system are that (1)
each unit used is a decimal multiple of the smaller unit, (2) a
unit volume of water has unit weight, and, especially, (3) the
recorded data of physical chemistry are published in these units.

TABLE 4 — INTERNATIONAL ATOMIC WEIGHTS FOR 19431

Element

Sym-
bol

Atomic
number

Atomic
weight

Element

Sym-
bol

Atomic
number

Atomic
weight

Aluminum

26.97

Molybdenum

95 95

Antimony

121.76

Neodvmiuiu

144 27

Argon

39 944

Neon ...

20.183

Arsenic

74 91

NickeL

58.69

Barium

137 36

Nitrogen

14.008

Beryllium

9 02

Osmium

190.2

Bismuth

209 00

Oxygen .

16 0000

Boron

10 82

Palladium

106 7

Bromine

79 916

Phosphorus

30.98

Cadmium

112 41

Platinum

195 23

Calcium

40 08

Potassium

39 096

Carbon

12 010

Praseodymium

140 92

Cerium

140 13

Protactinium

231

Cesium

132 91

Radium

226.05

Chlorine

35 457

Radon

222

Chromium

52 01

Rhenium

186 31

Cobalt

58 94

Rhodium

102 91

Columbium

92 91

Rubidium

85.48

Copper

63 57

Ruthenium

101 7

Dysprosium

162 46

Samarium

150.43

Erbium

167 2

Scandium

45 10

Europium

152 0

Selenium

78.96

Fluorine

19.00

Silicon

28 06

Gadolinium

156 9

Silver

107 880

Gallium

69 72

Sodium .

22.997

Germanium

72 60

Strontium

87 63

Gold

197.2

Sulfur

32 06

Hafnium

178 6

Tantalum

180 88

Helium

4.003

Tellurium

127.61

Holmium

JIo

164 94

Terbium

159 2

Hydrogen

1.0080

Thallium

204.39

Indium

114 76

Thorium

232.12

Iodine

126.92

Thulium

169.4

Indium

1^3 1

Tin

118.70

Iron

55 85

Titanium

47.90

Krypton

83.7

Tungsten

183 92

Lanthanum

138 92

Uranium

238 07

Lead

207 21

Vanadium

50 95

Lithium

6 940

Xenon , .

131 3

Lutecium

174.99

Ytterbium

173 04

Magnesium

24 32

Yttrium

88 92

Manganese .

54 93

Zinc

'30

65.38

Mercury

200.61

Zirconium

91 22

1 J. Am. Chem Soc., 65, 1946 (1943).

22 PHYSICAL CHEMISTRY

But daily use of the English units may make it easier to under-
stand the first statement of a new law in familiar units and to
obtain sooner a sense of proportion. Students of engineering
may study applied mechanics in English units and physical
chemistry in metric units at the same time, and considerable
confusion of quantities is an inevitable result. When ratios or
relative quantities are concerned, one set of units will do as well
as another. The units and conversion factors stated below are
for the convenience of students in working problems, and they
are stated with sufficient precision for this purpose. It will be
of little use to know that a cubic centimeter is 1 000027 ml. or
that the density of water at 4°0. is not unity but 0 999973 in
this connection.

Mass or weight usually will be expressed in grams, though
milligrams (J/fooo Sram) and kilograms, or kilos (1000 grams),
are sometimes more convenient units

The acceleration of gravity is 980. GG cm. per sec.

Volume is to be stated in liters or millihters. A milliliter
of water at 4°C. has a mass of 1 gram.

Force is expressed in dynes, a dyne being the force that will
impart to 1 gram mass a velocity of 1 cm per sec in a second.

Pressure is defined as the force acting on a unit area The
absolute unit of pressure is 1 dyne per sq. cm ; a convenient
multiple is the bar, which is 1,000,000 dynes per sq. cm.1 In
spite of the convenient size of this unit, which is closer to the
average atmospheric pressure than the standard "atmosphere/7
the latter remains the common unit of pressure in scientific work.
The main obstacle to its adoption is that the "steam point "
is defined as the boikng point of water under a pressure of 1
atm. and established as 100° on the centigrade temperature
scale. A standard atmosphere is a pressure that will support a
column of mercury 7G.OO cm. high at 0° when g = 980.66; it is
1.01325 bars. This multiplicity of pressure units is frequently
a source of confusion, but custom has not sanctioned the elimi-
nation of any of them so far.

1 Occasionally the pressure of 1 dyne per sq. cm is called a bar, and the
quantity defined as a bar above is called a megabar. The c.g.s. unit of
pressure is also called a barye Since the quantities differ by 106, no con-
fusion will arise. The definition which we have given is used in the " Inter-
national Critical Tables" and by the U.S. Weather Bureau.

INTRODUCTION 23

Work or Energy. — Small quantities of work or energy will be
expressed in calories (abbreviated ca/.), and large quantities in
kilocalories (abbreviated kcat ). The quantities are, respec-
tively, the amount of heat required to raise one gram of water
one degree centigrade, and 1000 times this quantity. For our
purposes it will not be necessary to consider whether the quan-
tities are in " 15° ca]." or "mean calories/' for the ratio of one to
the other is 1.00017, and almost none of the experimental data
we shall consider are precise enough to raise the question of which
calorie has been used. Similarly, we shall use 4 18 joules as
equivalent to 1 cal. without considering whether we mean 1
"absolute joule" or 1 "international joule," for the ratio of one
to the other is 1.0004. The work done when a piston of 1 sq.
cm. area moves 1 cm. against a pressure of 1 atm is called " 1
ml.-atm." and is the work done for each milhliter increase in
volume during evaporation against the atmospheric pressure.
One calorie is 41 3 ml -atm., or 1 ml.-atm. is 0.0242 cal ,x or 1
liter- atm. is 24.2 cal.

Temperature will be given on the centigrade scale, which takes
the ice point as 0° and the steam point as 100° ; or on the Kelvin,
or absolute centigrade, scale, which takes 273 16° as the ice point
and 373 16° as the steam point. It will usually be sufficient to
take 273° as the quantity to add to centigrade temperatures to
convert them to absolute, or Kelvin, temperatures

A mole, or formula weight, of substance will ordinarily be used
to describe a quantity of reacting substance. For gases this is
the quantity that fills the same volume as 32 grams of oxygen
at the same temperature and pressure; for liquids or solids it
will be the quantity corresponding to the usual chemical formula.
We shall refer to 142 grams of Na2SO4 as a mole of sodium sulfate,
whether or not a molecule of this composition actually exists;
and we shall call 18 grams of water a mole in the liquid state

1 For those working in English units, the following conversion factors will
be useful:

1 foot = 30 480 centimeters 1 cubic foot = 28.317 liters

1 pound = 453 59 grams 1 atmosphere pressure = 14 69
1 pound per square inch = 68,947 pounds per square inch

dynes per square centimeter 1 atmosphere pressure = 29 92

1 British thermal unit (60°F ) = inches mercury

1054 6 joules 7^ - tv + 459 7
1 gallon = 3.785 liters

24 PHYSICAL CHEMISTRY

whether liquid water consists of H2O molecules or (H2O)n mole-
cules. The volume of 18 grams of water (or of 98 grams of
sulfuric acid) will be called a molal volume, and the heat capacity
of- 18 grams of water will be called its molal heat capacity.

The ideal gas constant will be explained in the next chapter,
but its numerical value is recorded here as

R = 8 315 joules/mole-°K

or 0.08206 liter-atm./mole-°K., or 1.987 cal./mole-°K. In most
of our calculations these figures may be used as 8.32, 0.082, and
1.99, respectively.

Concentration. — This word is used somewhat loosely in chem-
istry to designate several ways in which the composition of a
solution is expressed; it may mean moles or equivalents of a
solute in a unit weight or volume of solvent or of solution. For
the purposes of this book two ways of expressing concentration
will serve every ordinary need. We shall define the molanty of a
solute as the number of moles of solute per 1000 grams of solvent,
arid O.lrn. will thus indicate 0.1 mole of solute in 1000 grams of
solvent. Compositions so expressed do not vary with the tem-
perature, and they are readily convertible into mole fractions,
which will be defined later. Certain properties of solutions
depend upon the quantity of solute per unit volume of solution,
and the moles of solute per liter of solution will be called the
volume concentration or simply the concentration of the solu-
tion. Since solutions expand slightly when heated, it is necessary
in precise work to specify the temperature at which the concen-
tration is given. An equivalent of solute pei liter of solution
will be called a normal solution, as in volumetric analysis. In
dilute aqueous solutions the difference between molality and con-
centration is small, but it is not to be ignored in precise calcu-
lations; and for solvents other than water the difference is always
important. For example, a solution of 0.1 mole of dissolved
substance in 1000 grams of chloroform has a volume concen-
tration of 0.15.

To illustrate these definitions, a solution containing 5 per cent
Bulfuric acid by weight has a density of 1.0300 at 25°; it contains
52.63 grams of H2S04 per 1000 grams of water and is

0.537w.

INTRODUCTION 25

The volume of 1052.63 grams of this solution is 1.0219 liters, and
its concentration is 0.537/1.0219 = 0.525 moles per liter of solu-
tion, or 1.050 equivalents per liter of solution. In the notation
that we shall use, fa = 0.537, C = 0.525, and N = 1.050.

Ionic Strength. — For certain purposes in connection with
ionized solutes the composition is expressed as the ionic strength
/z, which is half of the sum of each ion concentration multiplied
by the square of the valence of the ion. Thus, O.lm. BaCU has
an ionic strength /i = J^(0.1 X 22 + 0.2 X I2) = 0.3; in 0.12m.
CuS04, M = H(0.12 X 22 + 0.12 X 22) = 0.48; in 0.3m. HC1,
M = 1^(0.3 X I2 + 0.3 X I2) = 0.3.

Problems

Numerical data for some of the problems must be sought in the text. A table
of logarithms will be found in the back cover of the book

1. (a) Calculate the molecular weight of KBr from the following series
of weighings .

Wt KBrO3 7 44818 10 69361 10 36524 9 78481

Wt KBr 5 30753 7 62021 7 38620 6 97233

(6) Calculate the atomic weight of silver from the following series:

Wt. KBr 6 93122 7 62092 7 38622 6 97265

Wt. Ag 6 28281 6 90813 6 69531 6 32040

(c) Calculate the atomic weights of K and Br from these data and the
ratio in Table 1 [McALPiNE and BIRD, / Am. Chem Soc , 63, 2960
(1941) ]

2. The average of nine determinations of the ratio of carbon to oxygen
is 0 375262 Calculate the atomic weight of carbon corresponding to this •
ratio, and compare it with the atomic weight from the limiting density on
page 16 [BAXTER and HALE, / Am Chcm Soc , 58, 510 (1936).]

3. The ratio AsCl3:3Ag is given as 056022 in J. Am. Chem. Soc., 53,
1629 (1931), and as 0 56012 in ibid , 55, 1054 (1933); the ratio AsCl3:I2 is
given as 0714200 in ibid, 67, 851 (1935) Should the atomic weight of
arsenic be revised? (The atomic weights of silver and iodine have been
unchanged for many years.)

4. Potassium chlorate contains 39 154 per cent of oxygen, and a gram of
silver when converted into silver nitrate will react with 0.691085 gram of
potassium chloride (a) Calculate the molecular weight of potassium chlo-
ride and the atomic weight of silver from these data. (6) Calculate the
atomic weight of chlorine from that of silver just found and the ratio of
silver to silver chloride given m the text, (c) Calculate the atomic weight
of potassium from the composition of potassium chlorate and this atomic
weight of chlorine.

PHYSICAL CHEMISTRY

6. Pure silicon tetrachloride was decomposed with sodium hydroxide
solution, and the chloride was precipitated with silver nitrate made from
weighed portions of silver. [J. Am Chem. Soc , 42, 1194 (1920) ]

Weight SiCl4

Weight silver

Ratio SiCl4:4 Ag

10 4353

26 4952

0 39386

5 9785

15 1830

0 39376

8 7905

22 3213

0 39381

6 8352

17 3562

0 39383

Calculate from each experiment the molecular weight of silicon tetra-
chloride, and calculate an average value of the atomic weight of silicon, using
as the atomic weights of silver and chlorine 107 880 and 35 457 Calculate
the percentage deviation of this value from that for silicon in Table 4

6. The ratio of density in grams per liter to pressure in atmospheres at
0° for silicon tetrafluonde is

d/p

1 00 0 750 0 500
4 69049 4 67877 4 66705

(a] Determine the molecular weight of silicon tetrafluonde from these
data and such others as are required in the calculation (6) Calculate the
atomic weight of silicon, taking the value for fluorine from Table 4. [MOLES
and TOEAL, Z anorg allgem Chem , 236, 225 (1938) ]

7. (a) The chloride of an element reacts with silver to form silver chloride,
and in a certain experiment 3 418 grams of the chloride tequired 8 673 grams
of silver. From this fact, what is the lower limit for its atomic weight if
Ag = 107.88 and Cl = 35 457? (6) Given the further fact that at 1 atm.
and 140°C this (gaseous) chloride has a density of about 5 grams per htei,
what is the upper limit of its atomic weight? (c) What other facts would
be required to determine its atomic weight with certainty?

8. The density-pressure ratio of phosphinc gas at 0° is as follows:

Pressure, Atm.
1 0000
0 7500
0 5000
0.2500

d/p
1 5307
1 5272
1 5238
1 5205

Calculate the molecular weight of PH3 and the atomic weight of phos-
phorous, taking the value of hydrogen from Table 4 [RITCHIE, Proc Roy.
Soc (London), (A) 128, 55 (1930).]

9. The average of 15 determinations of the ratio POCl3:3Ag is given as
0 473833. Calculate the atomic weight of phosphorus from this ratio,
and compare with that of Problem 8. [HONIGSCHMID and MENN, Z. anorg.
allgem. Chem., 236, 129 (1937) ]

INTRODUCTION 27

10. Some of the gas-density data on nitrogen compounds arc as follows:

Pressure,

Density at 0°

ritm

NH. (1)

NH3 (2)

NH, (3)

N,O (4)

N2 (5)

1 000

0 77169

0 77143

0 77126

1 9804

1 25036

0 51182

0 51161

1 3164

0 83348

0 38293

0 38281

0 38282

0 9861

0 25461

0 25458

0 6565

0 41667

On the basis of these data, should a change be made in the atomic weight
of nitrogen, which foi many years has been given in the international tables
as 14008? [The sources of data are (1) MOLES and BATUECAS, Anales
soc espml fis qmm , 28, 871 (1930), (2) MOLES and SANTHO, ibid , 32, 931
(1934), (3) J Am Chcm Soc , 65, 1 (1933), (4) /. chim phys , 28, 572 (1931),
(5) BAXTER and STARKWEATHER, Proc Nat Acad Sc? , 14, 57 (1928) ]

11. The ratio 2Ag ZnBr2 is 100 104 380, and ZriBr* contains 29 030 per
cent zinc (a) Calculate the molecular \\eight of zinc bromide, using
107 880 as the atomic weight of silver (b) Calculate the atomic weights
of zinc and bromine

12. The specific heat of zinc is 0 092, and zinc oxide contains 80 311 per
cent zinc Calculate a new atomic weight of zinc, and compare with that
from Problem 11

13. The following data may be used to calculate values of the atomic
weight of phosphorus SAgCl PCI, = 100 31 951, and Ag3PO4 3AgCl =
100 102 704 Calculate atomic weights of phosphorus corresponding to
each of these data, and compare with the result from Problems 8 and 9.

14. (a) The chloride of a certain element E boils at 346°C under 1 atm
pi essure, and the density of the vapor is about 8 0 grams per liter under
these conditions What may be concluded as to the atomic weight of E
and the formula of its chlonde from these facts alone? (b) This chloride
contains 53 60 per cent chlorine With this additional fact what may be
said of the atomic weight of E and the formula of the chloride? (c) The
oxide of E contains 20 68 per cent oxvgen What additional information is
furnished by this fact? (d) The specific heat of E is 0 033 What is the
atomic weight of Et What are the formulas of its chloride and oxide?

15. (a) The chloride of an element requires 1 7853 grams of silver in solu-
tion to react with 1 0000 gram of it What is the lower limit of the atomic
weight of this element? (b) At 200°C and 1 atm the specific volume of this
gaseous chloride is 200 ml per gram. What is the upper limit for the atomic
weight of the element? (c) The specific heat of the clement is 0.09 cal. per
gram. What is the atomic weight of the element? What is the formula of
its chloride?

16. The density of chlorine (in grams per liter), the pressure (in atmos-
pheres), and the ratio of pressure to density at 50° are as follows:

28 PHYSICAL CHEMISTRY

p . 0 3134 0 6524 0 9893 1 605 2 0184

d 0 8410 1 756 2 673 4 361 5 509

d/p 2 683 2 692 2 702 2 717 2 789

Calculate a value for the atomic weight of chlorine from the limiting density
at 50°, assuming limiting densities proportional to absolute temperatures
for gases [Ros.s and MAAVSS, Can. J. Research, 18, B, 55 (1940) ]

CHAPTER II
ELEMENTARY THERMODYNAMICS

The purpose of this chapter is to outline very briefly the laws
of thermodynamics and the fundamental concepts on which they
are based, to derive a few therm odynamic equations that have
been found useful in physical chemistry, and to stimulate those
who are interested to read further.1 As the name implies,
thermodynamics relates to the flow of heat and the conversion
of heat into work or, in general, the conversion of energy from
one form to another form For our convenience we classify the
forms of energy to be considered as heat and work, heat being
that form of energy which flows under a temperature gradient,
and work including the action of a force through a distance,
expansion against an opposing pressure, production of electric
currents, etc., in short, all forms of energy other than heat. Foi
our further convenience we define heat as positive when it is
absorbed by a system arid work as positive when it is done by the
system. We measure heat and work in the same units of calories
or joules

By including a discussion of thermodynamics in a treatise
on physical chemistry we do not imply that thermodynamics is
an aspect of this field alone; for the laws apply in all fields,
whether physics, engineering, or some other science; they are as
general as the law of conservation of matter. But since the
physical aspects of chemical changes are our chief concern, most
of the applications of thermodynamics that we shall study will
be illustrated by chemical reactions.

The laws of thermodynamics are powerful tools with which

1 See for example, STEINER, ''Introduction to Chemical Thermody-
namics," McGraw-Hill Book Company, Inc., New York, 1941; WEBER,
"Thermodynamics for Chemical Engineers," John Wiley <fe Sons, Inc ,
New York, 1939; MACDOUGALL, " Thermodynamics and Chemistry,"
John Wiley & Sons, Inc., New York, 1939; LEWIS and RANDALL, " Thermo-
dynamics and the Free Energy of Chemical Substances," McGraw-Hill
Book Company, Inc , New York, 1923.

30 PHYSICAL CHEMISTRY

to show the relation of observed physical quantities to one
another, but they do not of themselves specify the properties of
material systems. In order to make them useful, we must
supplement them with adequate experimental data or with
suitable approximations when data are lacking.

Precise definitions of the terms used in thermodynamics must
be given as a necessary preliminary to this outline; these defini-
tions must be carefully read and the distinctions stated or
implied in them must be caiefully followed if the statements of
thermodynamics are to have any clear meaning. In order to
simplify these statements, certain quantities are designated by
letters, as p for pressure, v for volume, T for absolute tempera-
ture, E for energy content The notation used in this outline
is standard or as nearly standard as is possible4,1 and the defini-
tions and conventions as to signs are likewise those in common
use.

Definitions.— A system is defined as any combination of matter
that we wish to study; a closed system is one that is not exchanging
matter with any other system; an isolated system is one that
exchanges neither matter nor energy with any other system.
For convenience we usually give our attention to a single fixed
quantity of matter which we designate as "the system" and call
all other systems with which it may exchange energy "the
surroundings "

The state of a system is fixed when we specify so many of its
properties that all of them have definite values For example, if
we specify the pressure, temperature, quantity, composition,
and state of aggregation of a homogeneous (one phase) system,
all its other properties, such as volume, density, and energy con-
tent, are also fixed; and the system is in a definite state Its state
will also be fixed if we specify the volume in place of the pressure
or the density in place of the quantity of matter But the prop-
erties most readily measured are those first listed, and they are
the properties we shall ordinarily specify to fix the state of a
system. If the system is of more than one phase (partly solid,
partly liquid or vapor), one must specify the quantity and compo-
sition of each phase A change in one or more of the properties

1 The quantity which is called E in this text is sometimes designated
by U, and the quantity F given later is designated by G in some books In
Gihbs's notation, which is occasionally used, E — €, H — x, F = £ and A — $.

ELEMENTARY THERMODYNAMICS 31

of a system is a change in state, and of course all the properties of
a system are not independently variable.

A process is not completely described by a change in state but
is described by specifying the change in state and giving addi-
tional information as to the mechanism or how the pressure,
temperature, or other property varied as the change proceeded.
For illustration, a change in state is described by the following
scheme :

10 grams air) ( 10 grams air
20°, 5 atm J "^ ( 20°, 1 atrn

This is, moreover, an isothermal change in state, for the initial
and final temperatures are the same But in order to describe
the process we must also say whether the temperature remained
constantly at 20° (which would make it an isothermal process) or
whether the temperature varied as expansion took place and was
afterward brought to 20° (which would not be an isothermal
process). We must state whether the expansion was so conducted
that the pressure overcome was always infinitesimally less than
the pressure of the air (a reversible process) or whether the pressure
overcome was less than the maximum (an irreversible process)

An adiabahc process is one in which no heat is exchanged
between the system and the surroundings. The change in state
described in the preceding paragraph could not take place adia-
batically, since even during expansion into a vacuum (so that
no work was done) there would be a slight change in temperature.
This is not to say that air cannot expand adiabatically, but only
that the initial and final temperatures will not be the same when
it expands adiabatically.

Some further explanation of a reversible process in the thermo-
dynamic sense will not be out of place. The isothermal operation
of an electric cell against an opposing potential infmitesimally
less than its owrn is a reversible process, or one in which the
maximum amount of work is done. In general, a process is
reversible when the pressure or temperature or other intensive
property of the operating system differs infinitesimally from the
pressure or temperature or other property of the system against
which it operates. Thus an irreversible process is not one that
may not be reversed — it is one that may not be reversed by
infinitesimal changes in the variable properties of the system.

32 PHYSICAL CHEMISTRY

The transfer of heat from a body at T to a second body at
T — dT may be reversed by making the temperature of the
second body T + dT, and such a process is called a reversible
transfer of heat. Although, of course, no heat would pass
between bodies at exactly the same temperature, it is customary
to call the transfer reversible or isothermal when the temperature
difference is infinitesimal.

A cyclical process is one in which the system returns to its
initial state after completing a series of changes. Cycles, like
other processes, may be conducted reversibly or irreversibly. In
evaluating some quantities, such as heat absorbed or work done,
it will be important to state whether the cycle was reversible or
irreversible.

Temperature will usually be described on the absolute centi-
grade or Kelvin scale, on which the melting point of ice is 273. 1°K.
and the boiling point of water at 1 atm. is 373. 1°K., and such
temperatures will be denoted by T. Centigrade temperatures,
based on 0° as the melting point of ice and 100° as the boiling
point of water, will be denoted by t so that the boiling point of
oxygen will be written t = — 183°C. or, more commonly, T =
90°K. Thus the relation between the two temperatures is
/ + 273.1 = T. The means of determining this quantity 273.1°
will be given in the next chapter.

Laws of Thermodynamics. — The " first law" of thermo-
dynamics asserts the conservation of energy and denies the pos-
sibility of obtaining work without the expenditure of energy of
some kind, the "second law" imposes some limitations on the
conversion of heat into work, and the " third law" specifies the
limit that one particular thermodynamic quantity approaches
as the temperature approaches absolute zero. No " fourth law"
has so far been suggested. We now consider the three laws in
order.

The first law of thermodynamics is already familiar under the
name " conservation of energy." It may be stated in a variety
of ways. For example, the energy content of an isolated sys-
tem is a constant, or energy is not created or destroyed in any
process, or the energy content is a point function of the state of
a system. If we denote by E the total energy in all forms asso-
ciated with a system, any increase in the energy content of this
system requires a corresponding decrease in the energy content

ELEMENTARY THERMODYNAMICS 33

of some other system. A fixed quantity of matter does not have
a definite quantity of energy associated with it under every condi-
tion, of course, for its energy content varies with the state of the
system.

A system in a specified state has a definite energy content; and
when the system changes from state 1 to state 2, its energy con-
tent changes from E* to E* by exchange of energy with its sur-
roundings. This may be written

AE = E2 - Ei (1)

Upon restoring the system to state 1 its energy content again
becomes E\ by another exchange of energy, which is quantita-
tively the reverse of the first one. In other words, the energy
content of a system in a specified state is a property of the system.
Hence, one form in which we may express the first law is. that, in
any cycle of changes whereby a system is restored to its initial
state, the summation of the energy exchanges with the surround-
ings is zero. In mathematical language

f dE = 0 (2)

and dE is an exact differential. We may also say that the energy
content E is a point function of the state of a system, since AE
depends only upon the change in state, not upon the path fol-
lowed or the mechanism by which the change takes place. We
have classified the several forms of energy as heat and work,
and we have defined heat absorbed by the system as posi-
tive and work done by the system as positive. If we express
heat and work in the same units,1 the equations for the first law
are

/ dE = f (dq - dw) (3)

dE = dq - dw (4)

A# = q - w (5)

Although it is true that AE and hence (q — w) depend upon
the change in state taking place and not upon the manner in
which this change is brought about, it is not true that q and w
individually are independent of the manner in which the change
is brought about. For example, a quantity of compressed air
might expand and do useful work, or it might expand without the

1 The necessary conversion factors are given on p. 23.

34 PHYSICAL CHEMISTRY

performance of any work; but work would be required from an
outside source to compress the air again, regardless of the manner
of its expansion. Let the change in state be

10 grams airl f 10 grams air
5 atm., 20° } ~* { 1 atm., 20°

The first law states that AE — q — w regardless of the path;
hence more heat would be absorbed by the air during the expan-
sion in which work was produced than in the expansion in which
no work was done. The first law does not state how much work
such an expansion could do, nor does it give a numerical value
to AE for this change in state; but it does state that the heat
absorbed must be equal to &E for the process doing rio work and
to AE plus heat equivalent to whatever work is done in an expan-
sion doing work. The system may remain at 20° during the
expansion; or its temperature may change during the process; but
if its final temperature is 20°, &E will have the same value for any
path, while q and w are indefinite quantities until we specify
exactly how the change occurs. It should be understood that
while AE has a definite value for this change in state, we are
unable to calculate its value from thermodynamics without the
help of experimental data or suitable approximations, and we
are unable to calculate q or w without information as to the exact
mechanism of the expansion, whether it took place into a vacuum,
reversibly against the maximum pressure that could be overcome,
against the atmosphere, or in some other way, and whether the
temperature remained constant throughout or varied during
the expansion.

We shall see in the next chapter that for an ideal gas

dv ) T

and since the deviation of air from the ideal gas law is slight in
this pressure range, AE is approximately zero. From experiments
on the expansion of air we fftid AE is slightly more than 2 cal.
for this change in state. But q and w, while almost equal for
the specified change in state, are not even roughly determined
when AE is determined. If the vessel containing 10 grams of air
at 20° and 5 atm. (about 1.65 liters) is connected to an evacuated
vessel of such volume (about 6.6 liters) that the final pressure

ELEMENTARY THERMODYNAMICS 35

after isothermal expansion is 1 atm., w = 0 and q = AE = 2 cal. ;
if the expansion takes place reversibly at constant temperature,
w = fp dv = 325 cal., and q = 327 cal.; if the expansion takes
place isothermally against the atmospheric pressure, w = p9Av
= 162 cal., and q = 164 cal.

Since the minimum amount of work that must be done upon
the system to produce the change in state

10 grams air) 1 10 grams air
1 atm., 20° j I 5 atm., 20°

by an isothermal process is 325 cal. and the actual requirement
exceeds this amount, the work done by the system for this change
in state is —(325 + x) cal., and we may set no value for x with-
out exact knowledge of the process. For this change in state
AE = —2 cal., and therefore q will be equal to or greater than
— 327 cal. Thus, while E is a point function, a property of the sys-
tem in a specified state, and dE is an exact differential, the quan-
tities q and w depend upon the mechanism whereby the change
takes place, and not alone upon the change in state.

We consider next another therm odynamic quantity called the
enthalpy or heat content,1 designated by // and defined by the
equation

H = E + pv (6)

Since E, p, and v are all properties of a specified system, // is also
a property of a system, a quantity whose value is a point function
of the state of the system. The change in enthalpy attending
any change in the state of a system depends only upon the change

1 The word "enthalpy," rather than heat content, has long been used
abroad for H, and its use in the United States is increasing. The term
"heat content" has the unfortunate implication that a change in H requires
the absorption of an equivalent amount of heat, and this is true only under
certain conditions. For illustration, the isothermal expansion of a gas
with the performance of work absorbs a quantity of heat nearly equivalent
to the work done when the pressures involved are moderate, so that both
q and w are much larger than AH. Some objection is raised against the
word enthalpy because of its similarity in sound to entropy, which has
an entirely different meaning. This is readily met by a little care in speak-
ing. If enthalpy is accented on the second syllable (entropy being usually
accented on the first syllable), no serious difficulty will arise. We shall
use the terms enthalpy and heat content interchangeably in this book, but
with enthalpy as the preferred word.

36 PHYSICAL CHEMISTRY

in state, not upon the path Such changes may be shown by
the equation

AH = AE + A(pv) (7)

•

and for a cycle of changes f dH = 0, as was true of the energy
content.

When a change in state takes place at constant pressure and
without the performance of any work other than mechanical
work, w = p(vz — Vi), and A(pv) = p(v% — vi), whence it will
be seen that AH — q — w + A(pv) = q for a constant-pressure
change in state. Thus, in a constant-pressure reaction, for
example, the heat absorbed by a chemical change is equal to AH.
In succeeding chapters in this book, and especially in Chap. VIII,
where the heat effects of chemical reactions are considered in
detail, we shall use AH to describe the heat effects attending
constant-pressure processes.

Heat Capacity. — The heat capacity of a system is the ratio of
the heat absorbed to the rise in temperature attending the heat
absorption, but two facts make it necessary to be more specific
in the definition: (1) A given quantity of heat will not produce
the same temperature rise in a system for all initial tempera-
tures; in other words, the heat capacity is a function of the
temperature. (2) For a given initial temperature the tempera-
ture rise produced by a definite quantity of heat depends upon
the manner of heating, whether at constant pressure or constant
volume. We define the heat capacity at constant volume as

The heat capacity at constant pressure is denned by the equa-
tions

or '

of which the second follows from the first and the definition

H = E + pv

In the second definition (dE/dT)p is of course not Cv, which is
(dE/8T)v. In order to find its value we note that the energy

ELEMENTARY THERMODYNAMICS 37

content of a system of constant composition is a function of two
independent variables, and we may take them as T and v,

E = f(T, v)
for which the total differential is

- 0? X* + (f X* <io>

arid

dTp - \dTr \dvdTp

Upon substituting this relation in equation (9) denning Cp, we
have

Since the first term is equal to C,, from equations (8) and (11)
we find

The second law of thermodynamics imposes certain limitations
upon the flow of heat from one system to another and upon the
conversion of heat into work. The limitations do not appear
from the first law, which says nothing about any such restric-
tions so long as the quantities of energy as heat or work exchanged
between systems are equivalent. As an illustration of such a
restriction, if a given quantity of water at 25° be divided into
two nearly equal parts, one part might be heated to 50° by cool-
ing^ the other part to 0° (the slight inequality of the parts being
required by the variation in heat capacity of water with tem-
perature), and this process might occur spontaneously for any-
thing the first law of thermodynamics has to say. A heat engine
and generator immersed in a lake might deliver large amounts
of electric energy at the expense of the heat energy of the water,
and so long as the cooling of the lake gave a calorie to the heat
engine for each 4.18 joules -of electrical energy produced, the
requirements of the first law would be met in the process. But
these processes and numerous others are declared impossible by
the second law and found to be impossible by experience.

38 PHYSICAL CHEMISTRY

In place of attempting to state the second law of thermo-
dynamics in a form that will be applicable to all circumstances,
we shall state several facts that together will constitute a suffi-
cient formulation of it for the purposes of this text. (1) No work
may be produced by a complete cycle operating in surroundings
of constant temperature. (2) Heat will not flow spontaneously
from an object of lower temperature to one of higher temperature.
(3) When work is produced by a cycle operating between two
absolute temperatures 7\ and Tz, the maximum amount of work
to be derived from the cycle is

tiw = qi -7-1--2 (13)

where q\ is the heat absorbed at the higher temperature T\. It
will be observed that only a fraction of the heat absorbed at T\
may be converted into work, and that the remainder is rejected
at a lower temperature T% (4) No process is possible by which
heat is changed into work without some other attending process.
This attending process may be a change in the state of the system
when the process is isothermal, which excludes the cyclical
isothermal conversion of heat into work as was stated in (1)
above. It may be the transfer of heat to a lower temperature,
as in that fraction of the heat not converted into work in illus-
tration (3).

Carnot's Cycle. — In order to derive the equation that limits
the fraction of the heat convertible into work, let us assume that
we have two very large heat reservoirs from which heat may be
withdrawn or to which Jieat may be given. One of these reser-
voirs is maintained at the higher temperature t\ and the other
at the lower temperature t%. W,e may assume also a working
system called a "Carnot engine/' i.e., some system that can
absorb heat and produce work or evolve heat when work is done
upon it. In order to make the processes described seem real, we
may suppose this engine to consist of a quantity of gas or other
compressible fluid confined in a cylinder fitted with a frictionless
piston, but we need make no assumptions as to the properties of
the substance contained in the engine. In the " Carnot cycle/'
the engine passes through a series of reversible changes consti-
tuting a complete cycle, i.e., such a series that at its completion
the original state of the engine is restored in every particular.

ELEMENTARY THERMODYNAMICS 39

During this cycle, a quantity of heat is absorbed from the reser-
voir at ti, a portion of the heat is converted into work, and the
remainder of the heat is rejected to the reservoir at t%. Since in
a cycle of changes <f> dE = 0 for the operating system by the
first law of thermodynamics, the summation of the work done in
all the steps of the cycle must be equal to the difference between
the heat absorbed and the heat rejected. The steps in the cycle
are as follows:

1. Let the working system be put into thermal contact with
the heat reservoir at t\ and withdraw a quantity of heat qi by a
reversible isothermal expansion.

2. Let the system expand reversibly and adiabatically until its
temperature falls to t%.

3. Let the system be put into thermal contact with the heat
reservoir at t% and give to the reservoir a quantity of heat — #2, by
a reversible isothermal compression. Note that, according to
our standard convention, q is always the heat absorbed by the
system, so that giving —qz cal. to the reservoir corresponds to
+qz cal. absorbed by the system at Z2. It is inherent in the opera-
tion of a heat engine which produces work that some of the heat
is rejected at the lower temperature, and q% is thus a negative
quantity of heat absorbed by the system at the lower temperature.

4. Let the system be compressed reversibly and adiabatically
until its temperature rises to t\ and the system is restored to its
initial state.

Since every stage of the cycle took place reversibly, the work
obtained is the maximum obtainable by such a series of changes.
The system has undergone a complete cycle, for which j> dE = 0,
and so by the first law,

Wma* = qi + q*

Upon dividing both sides by gi, we obtain as a measure of the
fraction of the heat absorbed at ti that was converted into work

This measures the efficiency of the process, if we define efficiency
as the fraction of the total heat convertible into work.

We now show that the efficiency of a reversible engine operating
on a Carnot cycle depends only on t\ and /2. Let us suppose that,

40 PHYSICAL CHEMISTRY

of two Carnot engines A and B operating reversibly between t\
and /2, the first, A, is more efficient. Let A perform a Carnot
cycle and B a reversed Carnot cycle. We can adjust the engines
so that the amount of heat — q^A given to the heat reservoir at /2
by the engine A equals numerically the heat +gzB withdrawn
from the reservoir at fa by the engine 5, and, by so doing, we can
restrict the heat effects to the reservoir at t\. Since A is supposed
to be more efficient, WA will be greater than WB, and hence by the
first. law q\A is greater than q\B. If these engines are coupled
together and considered as a single heat engine, the net result of
one cycle will be the production of a quantity of work WA — WB
and the absorption of a quantity of heat qiA — qiB from the heat
reservoir at t\. But this would constitute the conversion of heat
into work by an isothermal cycle, which is declared impossible
by the second law. Hence, A cannot be more efficient than B,
and the efficiencies of all reversible engines operating between ti
and t% are functions of t\ and t% only. That is,

Wma* _ qi + qz _ .,. . ^ qi _ ,,. . ^ n -.

-—— - — - - - J(ti, h) or — - A*i> W (15)
q\ qi qi

In this equation q\ + qz is less than <?i, for we have already
seen that q% is negative, since heat is always rejected at the lower
temperature.

In order to make this relation quantitative, it is necessary to
show what function of the temperature governs the fraction of
heat converted into work and to select a scale on which to express
the temperature. The simplest relation would be obtained from
a temperature scale on -which the fraction of the heat converted
into work would also be the fractional decrease in temperature.
Such a thermodynamic temperature scale would be defined by
the equation

<?1 I 1

The temperature scale so defined is identical with the absolute
temperature scale derived from the expansion of an ideal gas at
constant pressure and already familiar from earlier work in
chemistry.

The form in which this equation appears is not the usual one,
but it is consistent with the conventions regarding q. The more

ELEMENTARY THERMODYNAMICS 41

common form designates q\ as the heat absorbed at T\ and #2
as the heat evolved at T^ so that the law then appears in one of
the forms

gi ~ 92 _ T±^_T* or ?I = i*
51 3Ti Ti r«

It was in this form that Clausius stated it. This form emphasizes
the fact that only a portion of the heat is converted into work;
but its notation is inconsistent with respect to g, and it is not well
adapted to a derivation of the entropy concept to which we
shall come in a moment.

By combining equations (15) and (16) we obtain the desired
statement of the law limiting the conversion of heat into work
through a reversible cycle, namely,

(13)

"'max ^1 m

1 I

This equation shows that the complete conversion of a quantity
of heat into work by a cycle of changes is impossible, since this
would require absolute zero for a rejection temperature. The
lower temperature 7\ for the practical operation of a cyclical heat
engine is the prevailing climatic temperature, which will ordi-
narily be between 275 and 300°K., and therefore the fraction of
the heat that may be converted into work by a cycle of changes
may be increased only by using higher initial temperatures.
While there is almost no difficulty in obtaining temperatures
much higher than the effective Ti in the operation of a steam boiler
(for example), there are practical difficulties in finding a suitable
working material for use in the "engine" and suitable structural
materials with which to build boilers and engines.

Entropy. — For the purpose of defining another useful thermo-
dynamic function, we may put equation (16) into the form

tfi , #2 A ,*,-.

— + — = o (17)

1 \ 1 2

which shows that the summation of q/T for the reversible cycle is
zero, or, in mathematical language,

'%r = o (is)

42 PHYSICAL CHEMISTRY

This relation defines a function, a property of the system, which
is called entropy and usually designated by S, such that

S = / qp + const. or dS = ^=

whence, for a change in state,

f2 3n

^ (19)

It should be noted that for irreversible processes / dq/T is not
the entropy, or any definite quantity. This is not to say that
AS for a system undergoing a change irreversibly is different
from AS for the reversible process; for the value of S is a point
function of the state of the system, and AS is independent of the
path. But / dq/T is not a measure of the entropy change, and
/ dqr^/T is a measure of the entropy change.
As an illustration, consider the change in state:

Us) \ I I8(Z)

386°K, 1 atm.J (38(>°K., 1 atm.

for which AH is 3650 cal. Since the stated temperature is the
melting point of iodine, the change in state takes place reversibly
when iodine crystals are heated, and AS = 3650/386 = 9.54
entropy units (usually written 9.54 e.u., meaning 9.54 cal. per
mole per deg.), and for the reverse change AS = —9.54 e.u.
But if liquid iodine is undercooled to 376°K. and crystallization
occurs at this temperature, the value of AS is not AH for the
irreversible change divided by 376°. The difference between the
entropies of liquid iodine and crystalline iodine at 376°K. may
be obtained by (1) calculating AS for the reversible heating of
crystalline iodine from 376 to 386°K. from equation (19), (2)
reversible melting of the iodine at 386°K., for which AS has been
calculated above, (3) calculating AS for the reversible cooling of
liquid iodine from 386 to 376°K. from equation (19), and adding
these three quantities. The calorimetric effect observed when
undercooled iodine crystallizes, divided by 376, would not be
equal to the AS calculated above; moreover, the temperature
could not be maintained at 376°K. during the irreversible change.
The quantity S is a very important one in thermodynamics.

ELEMENTARY THERMODYNAMICS 43

Although a clear concept of entropy is not to be obtained by a
slight acquaintance with it, time is probably gained if cultivation
of this acquaintance is begun early and continued throughout
physical chemistry. Accordingly, we shall make occasional use
of entropy in the calculations of this book, and the student will
find many others in more advanced courses. But most beginners
find it easier to understand derivations in which reversible expan-
sion against a pressure, heat absorption, electric potential, and
other familiar quantities are involved than derivations based
upon the more elusive concept of entropy. Since this book is
addressed to beginners in physical chemistry, it will be our usual
custom to derive the equations without the use of entropy and
to repeat the derivations of some of the equations using the
entropy concept.

The Third Law of Thermodynamics. — A simple and almost
accurate statement of this law is that the entropy of any pure
crystal is zero at the absolute zero of temperature.1 If this
theorem is accepted, one may determine the entropy of a sub-
stance at any temperature b}^ integrating equation (19) with
absolute zero as the lower limit and taking S at 0°K. as zero.
Through equations that will be developed in a later chapter, the
entropies so obtained enable one to calculate chemical equilibrium
from thermal data alone. In order to integrate equation (19)
we may write it in the form

for constant-pressure changes. If the lower limit it taken as T\
= 0°K., the heat capacity must be known as a function of the
temperature to within a few degrees of absolute zero and up to the

1 An exact statement of the third law given by Eastman [Chem. Rev., 18,
272 (1936)] is: Any phase cooled to the neighborhood of the absolute zero,
under conditions such that unconstrained thermodynamic equilibrium is
attained at all stages of the process, approaches a state of zero entropy.
He follows this statement with an admission that it is unnecessarily restric-
tive, since many constrained systems also approach zero entropy. The
inaccuracy of the simple statement given above may be removed by a
sufficiently stringent definition of the term "pure crystal." The definition
excludes only a few substances in which we shall have no interest in this
simple discussion. See Kelley, Bulletin US. Bur. Mines, 434, 3 (1941),
for a discussion of these exclusions.

44 PHYSICAL CHEMISTRY

desired temperature. Graphical integration from a plot of CP/T
against T or of Cp against 2.3 log T over the temperature range
of the data gives the entropy increase in this range. The small
entropy increase in the range from 0°K to the lowest temper-
ature at which CP has been measured is calculated from an equa-
tion that need not concern us here, 1 since the quantity is usually
not more than 0.1 e.u. Cp not only approaches zero at 0°K., but
Cp/T also approaches zero at 0°K. ; therefore, the molal entropies
are all finite.

For a substance that has no phase transitions and does not
melt below the temperature at which S is desired, the entropy is
given by the equation

and for one that has no phase transitions other than fusion at T '/,
the entropy of the liquid at T is

in which Cs is the heat capacity of the solid and Ci the heat
capacity of the liquid. For substances undergoing solid-solid
transitions or that evaporate below the desired temperature,
additional terms such as AHtnaut/Ttnaa or A//evap/Tovap must be
included, and separate integrations of (Cp/T)dT must be per-
formed over the temperature ranges between transitions. It
must be remembered that S = J dqrev/T, not / dq/T, when the
heating takes place irreversibly. This restriction makes it neces-
sary to conduct the heating so slowly that no irreversible heat
effects are included.

The necessary low-temperature heat capacities have now been
measured for many substances, and standard entropies at 298°K.
are available in sufficient quantity for calculations of numerous
equilibriums through equations that will be given later.2

It may be profitable to repeat with emphasis a statement made
at the beginning of this brief discussion: The laws of thermo-

1 See, for example, Steiner, op. cit., Chap. XV. ,

2 See, for example, Kelley, U.S. Bur. Mines Bull , 434 (1941), for the
low-temperature heat capacities and 298* entropies of inorganic substances.

ELEMENTARY THERMODYNAMICS 45

dynamics are among the most useful tools that the chemist has
available. But one cannot build with tools alone, he requires
materials as well, and for chemists the materials are the accumu-
lated experimental data of physics and chemistry. For illustra-
tion, the change of entropy of a substance at constant pressure
is related to the heat capacity of the substance by the equation
dS = Cp dT/Tj but if we have no data expressing Cp as a function
of the temperature we may not integrate the equation.

Thermodynamic Properties. — The properties of a system that
we have considered so far are the intensive properties, pressure p
and temperature 77, and the extensive* properties, volume v,
energy content E, enthalpy //, and entropy S. They are not,
of course, the only properties of a system in a specified state, nor
are they independently variable We have already had some
equations that express relations among them, and presently we
shall define two more quantities in terms of those listed above.
In giving a definition, the usefulness of the property alone justi-
fied doing so; for example, a thermodynamic property might be
defined as X = E — pv, in place of the enthalpy, which is defined
as E + pv But E — pv is not a useful property for many calcu-
lations, and E + pv = II is a property, independent of the path
followed during a change in state, that measures the heat absorbed
at constant pressure. Since most processes are conducted at
substantially constant pressure, // is a useful property to define,
and changes in // attending chemical reactions or other changes
are useful quantities for tabulation. If the common procedure
were to conduct changes at constant volume, there would be
little use for the quantity H] and since there is no apparent use
for a quantity defined by E — pv, there is no need to define it.

Two useful quantities will now be defined, the first by the
equation1

A = E - TS (22)

and a second property F, which is related to A in the same way
that H is related to E, by the equation

F = // - TS (23)

1 This A is the property that Helmholtz calls the free energy, but most
American publications call the quantity F, defined by equation (23), the
free energy, following Lewis, in J. Am Chem Soc., 36, 1 (1913).

46 PHYSICAL CHEMISTRY

which is equivalent to F = A + pv, since H — E + pv. This
quantity F is the "Gibbs's free energy" and is written G in some
books to emphasize this fact. We shall call it simply the "free
"energy," following Lewis and most American writers. These
two definitions complete the list of thermodynamic properties
that we shall have to use in this text, the full list being p, v, T,
E, H, S, A, and F. Each of the two new definitions applies to a
quantity that experience has shown to be useful. For reasons
that will appear as we proceed, F is more convenient than A in
most of the calculations of physical chemistry, and hence F is
the quantity we shall use. If constant volume were a common
procedure, A would be a more useful quantity than F. We turn
now to some equations involving these quantities.

Some Thermodynamic Equations. — In specifying a few restric-
tions which we wish to impose upon the first law of thermo-
dynamics in deriving equations applicable to reversible processes,
we imply, not that there are any restrictions to the applicability
of the first law itself, but only that we wish to impose some for
our present convenience. We confine our attention to reversible
changes in state taking place in closed systems in which gravita-
tional effects are negligible, in which there are no distortional
effects or electric fields large enough to be important, and in
which the only form of work considered is reversible expansion
at a single piston. Under these conditions dq = dqrev = T dS
and dw = p dvt so that the equation for the first law becomes

dE = TdS - pdv (24)

Another equation, subject to the same restrictions, is obtained by
differentiating the enthalpy equation H = E + pv,

dH = dE + p dv + v dp
and substituting the value of dE from (24),

dH - T dS + v dp (25)

In the previous section we defined the quantity A by the equa-
tion

A = E - TS (22)

Differentiating,

dA = dE - TdS - SdT (26)

ELEMENTARY THERMODYNAMICS' 47

and, by substituting the value of dE from (24),

dA = -SdT - pdv (27)

For an isothermal process the first term on the right side of this
equation is zero, and dA is seen to be the negative of the iso-
thermal work,

dA = -dwm« (t const.) (280 l

The quantity A is sometimes called the isothermal work content,
and an equation is written

AA = A* - A! = -wmax (290

which is a correct statement, subject to the condition that the
process is isothermal. But it must be kept in mind that when the
process is not isothermal the maximum work is not measured by
A-A, even though A is a property of a system and AA depends
upon the change in state regardless of the path.

As has been said before, the equations involving the free energy
F are more useful in physical chemistry than the equations involv-
ing A, or at least they are more commonly used. The definition
of F has already been given, namely,

F = H - TS (23)

which gives upon differentiation

dF = dH - TdS - SdT (30)

Substituting the value of dH from equation (25) and canceling
terms that are equal and of opposite sign,

dF = -SdT + vdp (31)

For isothermal expansion or compression in a system of constant
composition, the first term on the right is zero, and the relation
is

AF = fv dp (t const.) (320

Two other equations applicable to isothermal changes in state
for which we shall have frequent use in later chapters follow from
the equation defining F :

AF = AH - T AS (t const.) (330

1 The letter t included with the number of an equation indicates the
restriction of the equation to changes at constant temperature.

48 PHYSICAL CHEMISTRY

and

AF = A A + AO) (/ const.) (340

Most of the partial derivatives that can be formed from the
thermodynamic quantities have no practical interest, but a few
of them are very useful indeed. For example, the relations

(dE\

(as). =

and

follow at once from equation (24) above. Relations involving
four of the thermodynamic quantities may be derived almost
without limit, but again very few of them are interesting. The
following are some of Maxwell's relations, and they will fre-
quently be useful:

dvs \dS

(8p\ = (dS

\dT/v \dv
_(dS\ =(dv\
\dpjr \dTjf

Most of the equations that have now been given will appear
later as the need for them arises, and a few more will be derived
in later chapters.

All the equations of thermodynamics are exact, but many of
the useful ones are differential equations. Before integrating
those containing more than two variables, it will be necessary to
express all but two in terms of the selected two variables and to
be sure that the quantities assumed constant remain constant.
The necessary data for expressing the volume as a function of
temperature and pressure (for example) are sometimes lacking,
and an approximation must therefore be used. This is a per-
fectly legitimate procedure whenever one is willing to accept the
errors inherent in the approximation, but the " equation" that
results from combining an exact thermodynamic equation with an
approximation is not strictly an equality at all. It may (and
usually does) give a result that is all that is required. As an

ELEMENTARY THERMODYNAMICS 49

illustration, the volume of a gas at moderate pressure is very
nearly v = nRT/p; and if one substitutes this relation into equa-
tion (32t) to calculate AF for the expansion of a gas from pi to p%
at T, the result is

AF = nRT In ^ (t const.) (350

If both pi and pz are moderate or low pressures and if T is far
from the condensation temperature, the use of this equation will
give a definite value to AF for the specified change in state, which
is all that would ordinarily be required. But equation (35£) is
not a " thermodynamic equation"; it is a satisfactory approxi-
mation based upon a thermodynamic equation and the ideal gas
law.

References

STEINER. "Introduction to Chemical Thermodynamics," McGraw-Hill

Book Company, Inc , New Yoik, 1941
DODGE- "Chemical Engineering Thermodynamics," McGraw-Hiil Book

Co , Inc , New York, 1944.
MAcDouGALL: "Thermodynamics and Chemistry," John Wiley & Sons,

Inc , New York, 1939.
NOTES and SHERRILL: "Chemical Principles," The Macmillan Company,

New York, 1938
WEBER: "Thermodynamics for Chemical Engineers," John Wiley & Sons,

Inc , New York, 1939

Problems

1. The entropy of oxygen gas at 298°K and 1 atm is 49.0 cal. per mole
per deg Calculate its entropy at 373°K and 1 atm , taking

Cp = 65 -f 0 001 T cal per mole per deg.

foi the heat capacity in this temperature range

2. The standard entropy of COt(g) at 298°K arid 1 atrn. pressure is
51 08 and CP = 7 70 + 0 00537 - 0 83 X 10~KT2 Calculate the entropy
of CO2(0) at 798°K and 1 atm pressure

3. The volume of a mole of liquid water at 373°K and 1 atm pressure is
18.8 ml , that of a mole of water vapor under the same conditions is 30,200
ml , and the latent heat of evaporation at 373°K. is 9700 cal. per mole
Calculate AH and AE for the change in state

H2O(Z, 373°K , 1 atm ) * H2O(0, 373°K., 1 atm.)

4. Calculate AF, AA, and A/S for the change in state described in Problem 3.

5. For the isothermal change in state

O2(0, 298°K, 1 atm.) - Ot(g, 298°K, 0.1 atm.)

50 PHYSICAL CHEMISTRY

A// and A(pv) are negligible, pv = RT for the gas, and R = 1.99 cal. /mole- °K.
Calculate AF, AA, and A$ for this change in state.

6. The heat capacity of solid bromine, in calories per mole per degree,
changes with the Kelvin temperature as follows:

T 15 25 30 50 75 100 150 200 245 266

6 10 54 11 75 12 87 13.92 15 12

The molal latent heat of fusion of bromine is 2580 cal per mole at 266°K ,
and the heat capacity of liquid bromine is 17 cal per mole per deg. (a) Plot
Cp/T against T for the solid, join the points with straight lines (as a suffi-
cient approximation for illustrating the integration), and determine $266 for
Br2(s) (b) Determine $298 for Br2(0 [Data from LATIMER and HOEN-
BHEL, J. Am. Chem Soc., 48, 19 (1926) ]

7. The atomic heat capacity of silver changes with the Kelvin temperature
as follows:

T ... 15 40 50 60 80 100 150 200 250 298
Cp 0 160 2 005 2.784 3 420 4 277 4 820 5 490 5 800 5 989 6.092

Plot Cp/T against T for each of these temperatures, join the points by
straight lines (as a sufficient approximation to illustrate the method of
integration), and determine the entropy of silver at 298°K. [Meads, For-
sythe, and Giauque, / Am Chem Soc , 63, 1902 (1941), find 10 21 from an
exact treatment of this and other data ]

8. Calculate the entropy of diamond at 298°K from the heat-capacity
data in Table 23.

CHAPTER III
PROPERTIES OF SUBSTANCES IN THE GASEOUS STATE

This chapter will present .first the simple equations that
approximately describe the behavior of gases and gas mixtures
at low or moderate pressures and then the more complex equa-
tions that apply at higher pressures. Through these laws we
establish the temperature scale, determine molecular weights,
estimate heat capacities, measure chemical equilibrium and the
rates of reactions, and obtain other important information. The
" ideal gas" will receive due attention, and we shall emphasize
the important fact that "ideal" gas behavior is approached but
not attained, as is true of almost any ideal ; that the concept of an
ideal gas is useful under certain conditions and a source of hazard
if carried outside the bounds of its applicability. The ideal gas
law usually does well enough when applied at pressures near or
below atmospheric pressure, it may do well enough at higher
pressures, but it may also be in error by 50 per cent or more at
50 atm. pressure.

A gas is a fluid that distributes itself uniformly throughout any
space in which it is placed, regardless of the amount of gas or
space so long as the space is large enough to prevent partial
condensation to liquid. Thus a substance may or may not be
a gas, according to the conditions of temperature and pressure;
a more accurate expression would be "a substance in the gaseous
state. 'r It is this phrase that is to be understood when the word
gas is used. All the substances that we ordinarily call gases
have been liquefied and solidified by suitable reduction in the
temperature. Many of the common liquids and solids may be
changed to the gaseous state at high temperature and at low
pressures. The common metals, most metallic halides, and
many simple organic compounds are readily changed to gases by
heating; but salts of oxygenated acids, complex organic com-
pounds and metallo-organic compounds usually decompose before
their vapor pressures reach 1 atm.

52 PHYSICAL CHEMISTRY

Mixtures of two or more kinds of molecules exhibit in the
gaseous state most of the physical properties of a gas containing
only one kind of molecules; they follow the laws that describe
the behavior of single gases and may usually be treated as a
single gas. For example, in its physical properties dry air at low
pressures acts as if it were a single substance of molecular weight
29 at all temperatures above 100°K.

Structure of a Gas. — The fact that a small quantity of liquid
yields a very much larger volume of vapor at the same tempera-
ture and pressure is evidence that the molecules in the vapor are
separated from one another by distances that arc large compared
with the diameters of the molecules Eighteen grams of liquid
water occupies 18 8 ml in the liquid state at 100° and 1 atm.
pressure, but these same molecules occupy about 30,200 ml when
changed to a gas at this temperature and pressure. Thus in the
gaseous phase1 the volume available for the use of each molecule
is about 1600 times what it was in the liquid state. We do not
believe that the volume of the molecules has changed to any
great extent during evaporation, but only that the free
space around them is larger. This will be taken up in more
detail in connection with the kinetic theory of gases later in this
chapter.

The molecules of a gas are not stationary but are moving
about in space with very high velocities. They collide with
each other frequently and strike the walls of the containing vessel,
giving rise to the pressure exerted by the gas If the volume of
a gas is increased, the number of molecular impacts on a given
area is decreased, a smaller number of molecules strikes any area

1 The homogeneous parts of any sj^stem that are separated from one
another by definite physical boundaries are often called its phases For
example, ice, 'liquid water, and water vapor are the phases, or states of
aggregation, common to water A solution is a single phase because there
are no visible boundaries between solvent and dissolved substance. A
mixture of several gases constitutes a single phase; for gases mix in all pro-
portions, and there is no physical boundary between one gas and another
A mixture of crystals forms as many phases as there are kinds of crystal
present, since each is divided from the others by definite boundaries. When
a single solid substance is capable of existing in two different crystalline
modifications, each of these is considered a separate phase. Rhombic and
monoclmic sulfur, red phosphorus and yellow phosphorus, gray tin and
white tin are familiar examples of pure substances forming two definite solid
phases, though many others also exhibit this property.

PROPERTIES OF SUBSTANCES IN THE GASEOUS STATE 53

of the wall in a given time, and the pressure decreases. The
pressure of a gas at constant volume increases as the tempera-
ture rises, which means that there are more collisions of the mole-
cules with the walls in a unit of time and hence that the velocity
of the molecules increases at higher temperatures. The pressure
exerted by a gas does not decrease with time; therefore, the
collisions between molecules are perfectly elastic, and no decrease
in average velocity results from a collision. The "empty space"
between molecules bears some resemblance to that between the
spokes of a rapidly revolving wheel. The spokes do not fill all
the space in which they revolve, but the whole of this space is
effectively occupied, so that nothing else can be kept in it. In
the same manner, other molecules cannot be inserted into the
"empty " space between molecules without increasing the number
of collisions and hence the pressure of the gas.*

The treatment of gases at moderate pressures and at tempera-
tures well removed from their condensation points is compara-
tively simple, for all of them have properties in common, which
are expressed approximately by a few simple laws.

Boyle's law states that at any constant temperature the
volume occupied by a quantity of gas is^ inversely proportional

TABLE 5 — PRESSURE-VOLUME RELATIONS OF HELIUM1 AT 0°

Pressure,
mm. of Hg

Volume,
cc.

pv product

Per cent deviation
from average
(56,580)

837 63

67 547

56,579

-0 0018

794 81

71 191

56,583

+0 0056

761 56

74 293

56,579

-0 .0018

*732 17

77 278

56,581

+0 0018

613 09

92 279

56,575

-0 0087

561 40

100 777

56,576

-0 0071

520 37

108 720

56,575

-0 0087

462 54

122 320

56,576

-0 0071

310 31

182 341

56,582

+0 0036

237 84

237 895

56,581

+0 0018

169 48

333 881

56,586

+0 0105

147 16

384 539

56,589

+0 0159

1 BURT, Trans Faraday Soc., 6, 19 (1910) Baxter and Starkweather con-
firm Boyle's law for helium at 0° from their densities, 0.17845 gram per
liter at 1 atm., and 0 08923 at 0 50 atm. [Proc. Nat. Acad. Sci., 12, 20 (1926).]

PHYSICAL CHEMISTRY

to the pressure exerted upon it, provided that the composition of
the gas does not change through dissociation or polymerization
when the pressure changes. Very careful experiments have
shown that the law is not exact but is a limiting law that describes
the behavior of a gas more closely as the pressure decreases. At
pressures near or below atmospheric, the deviations from Boyle's
law are quite small for most gases, as may be seen from Table 5

FIG. 3.-

100 150 200 250
Pressure in Atmospheres
-Deviations from Boyle's law at high pressures.

and the limiting-density data in Chap. I. At 0°C. the pv product
for C02 at Y<i atm. is 1.0033 times that for 1 atm., and the pv
product for oxygen at Y% atm. is 1.00047 times the value for 1
atm. The pv products for some other gases are shown in Fig. 3
And Table 6 in both of which pv is taken as linity at 0°C. and 1
atm. pressure.

The pv product for most gases at constant temperature at first
decreases with increasing pressure, then passes through a mini-

PROPERTIES OF SUBSTANCES IN THE GASEOUS STATE 55

mum, and finally increases with increasing pressure; but for
hydrogen and helium the pv product increases with increasing
pressure Without passing through a minimum when the con-
stant temperature is above room temperature. At low tem-
peratures the pv products for these gases also pass through

minima.

TABLE 6. — CHANGE OF pv PRODUCT WITH -PRESSURE1
(pv * 1 000 at 0° and 1 atm )

Car-

Oxygen

bon di-

Hydrogen

Ethylene

Nitrogen

p, atm.

oxide

0°

100°

0°

100°

20°

100°

0°

100°

1 000

1 368

1 372

1 000

1 366

1 082

1 000

1 367

0 959

1 206

1 033

1 403

0 629

1 192

0 985

1 389

100

0 926

1 375

1 030

1 064

1 436

0 360

1 005

0 985

1 411

150

0 878

0 485

0 924

200

0 914

1 400

0 815

1 134

1 511

0 610

0 946

1.036

1 496

300

0 963

1 453

0 890

1 205

1 584

0 852

1 133

1 136

1 597

400

1 051

1 532

1 039

1 276

1 656

1 084

1 356

1 256

1 711

There is for every gas a temperature, called the Boyle tem-
perature, above which [d(pv)/dp]T is positive and below which
it is negative, as the pressure approaches zero. Thus, at the
Boyle temperature the plot of pv against p for constant tempera-
ture is horizontal at its lowest pressures, but this is not to say
that it is horizontal at high pressures.

When gases with more complex molecules are studied, the
deviations from Boyle's law become much larger. We quote the
data for ethyl ether2 at 300°C. as an illustration of this fact.

1 Quoted from " International Critical Tables/' Vol III, pp. 9/.

2 BEATTIE, J. Am. Chem Soc , 49, 1128 (1927) Data for other substance*-
may be found in the "Landolt-Bornstem Tables"; in the Communication*
of the Physical Laboratory of the University of Leiden in Holland (available in
English); in the tables published by the Smithsonian Institution; in Vol,
III of the " International Critical Tables"; and inProc. Am. Acad. Arts Sci.^
63, 229-308 (1928). Bartlett [/ Am. Chem. Soc , 62, 1363 (1930)] carries
experiments on N2, H2, and the mixture N2 + 3H2 to 1000 atm Data w^
sometimes presented in "Amagat units," in which the unit is the mail^fl
1 liter at 0° and 1 atm. pressure, or in "Berlin units," in which ^ffv$$
volume is at 0° and a pressure of 1 meter of mercury.

56 PHYSICAL CHEMISTRY

(It should be noted that the pv product for a mole of ideal gas
at 300°C. is 47.0, for comparison with the pv product in the last
line of the table.)

Pressure, atm . . 16 732 19 276 22 708 27 601 35 194 48 430

Molal volume, liters 2 593 2 222 1 852 1 482 1 111 0 741

Product 44 38 42 83 42 05 40 89 39 10 35 87

At 300°C. and pressures of 1 atm or less, ether vapor conforms
to Boyle's law within 1 per cent; at lower temperatures and these
same high pressures, its deviations are greater than those shown
above.

Law of Gay-Lussac (or Charles).— When a quantity of gas at
an initial low pressure is heated at constant volume, the pres-
sure is a linear function of the temperature. For example, if
the pressure were 0.100 atm. at 0°, it would be 0.1366 atm. at 100°
for a gas that was ideal and very nearly this pressure for all gases
that are chemically stable. The increase of pressure per degree
is 0.00366 of the pressure at 0°, for any low pressure; and since
the reciprocal of this quantity is 273, the pressure at any tem-
perature I is (273 + 0/273 times the pressure at 0°. This law,
like Boyle's law, is a limiting law that becomes exact as the gas
pressure becomes very small For " permanent " gases near
atmospheric pressure, it is in error by less than 1 per cent, but it
may be largely in error at high pressures. Some data are given
in Table 6.

The important point to be noted is that this same coefficient
applies to all gases that are chemically stable. Other materials
such as solids also have nearly linear temperature coefficients of
expansion, but they are. different for different substances. But
nitrogen, hydrogen, helium, ammonia, every gas increases its
pressure at constant volume and a low pressure by 36.6 per cent
of the pressure at 0° when heated to 100°. Since the pressuie
increase is due to an increased energy of a fixed number of mole-
cules with increasing temperature, the convergence of all the
energies toward zero at the same temperature ( — 273°C.) indi-
cates that this is a temperature of "absolute" zero in the sense
that no temperature can be lower. Since the temperature scale
based on gas behavior, as defined in the next section, coincides
W!$L the "thermodynamic" temperature scale defined from
Carn&t's cycle on page 40, it is necessary to fix the position of
|he ice point on this scale with precision.

PROPERTIES OF SUBSTANCES IN THE GASEOUS STATE 57

Determination of the "Ice Point" on the Absolute Scale. — The

ice point is defined as the temperature at which ice and water
saturated with air are in equilibrium at 1 atm. pressure, and the
steam point is defined as the temperature at which liquid water
and water vapor are in equilibrium at 1 atm. pressure. On the
centigrade scale the interval between them is denned as 100°.
In order to fix these points on an absolute scale through the
properties of an ideal gas, we define av as (pBteftm — plcfl)/pice,
which is the fractional increase of pressure at constant volume
for the fundamental interval of 100°. Since this quantity varies
with the pressure at 0° for an actual gas, the expansion coefficient
is plotted against the pressure at the ice point and extrapolated
to zero pressure. The following figures1 are for nitrogen, with
the pressure in meters of mercury:

p 0 90959 0 75117 0 60020 0 45032 0 33409 zero

«„ 0 3674118 0 3670689 0 3668750 0 3666780 0 3665327 (0 3660852)

The reciprocal of the extrapolated value of av is 2.7316; therefore,
iOO/ar is 273.16, which is the temperature of the ice point on the
gas scale. It is the figure that is added to centigrade tempera-
tures to convert them into absolute temperatures. Although
this is sometimes called the value of absolute zero, there is no
implication that such a temperature has been reached; and the
experiments on which the value is based were performed at 0
and 100°C. The mean value of all experiments made since 1900
to determine the ice point is 273.16.

Since the fundamental interval between the ice point and the
steam point is 180° on the Fahrenheit scale, absolute zero on this
scale is 180/0.366085 = 491.69° below the ice point; and since
the ice point is 32°, Fahrenheit temperatures are converted to
absolute or "Rankine" temperatures by adding 459.69° to the
Fahrenheit reading.

The absolute centigrade temperature scale, which is denned as
proportional to the pv product of an ideal gas and which is very
nearly proportional to the pv product for actual gases at low

1 BEATTIE, " Symposium on Temperature of the American Institute of
Physics/' p 74, 1940. Other less precise data for other gases support these
figures at the limit; for example,

Pressure, atm 10 5 1 Limit

a „ for helium 03635 ... 0.3658 03661

av for oxygen 0 3842 0 3752 0 3679 0 3660

58 PHYSICAL CHEMISTRY

pressures, is often called the Kelvin scale in honor of the cele-
brated physicist. Although Kelvin's originally defined scale was
the thermodynamic scale, which is proportional to the fraction
of heat convertible into work in a reversible cycle, these scales
are identical. We shall use the terms 273.16° abs. and 273.1G°K.
interchangeably in the text to indicate the temperature at which
ice and water satuiated with air are in equilibrium at 1 atm.
pressure. -In this book the usual custom of denoting tempera-
tures is followed, centigrade temperatures by t, and absolute
temperatures by T. Thus T = 273.16 + *; and unless the
highest precision is required, we shall be content to write T =
t + 273 as an adequate figure.

Measurement of Temperature. — If a quantity of gas at con-
stant volume has a pressure p0 in melting ice, a pressure pioo when
surrounded by water boiling at 1 atm., and a pressure pt at some
other temperature t, then this temperature may be determined
from the equation

t = 100 Pt ~ P° (v const ) (la)

Pioo — Po

A corresponding set of measurements of the volume of a quan-
tity of gas at constant pressure at the two standard temperature
points and at temperature t leads to the expression

t = 100 Vi ~ Vo (p const.) (16)

VWQ ~ #o

If the actual gases were ideal gases, these scales would be iden-
tical and each would give exact temperatures. But pressure
measurements on actual gases at constant volume do not yield
exact absolute temperatures, nor do they give quite the same
temperatures as the constant-pressure scale. Adequate, but
rather complex, means are available for correcting these measure-
ments so that their readings yield correct temperatures. On the
absolute scale, these relations may be written

7=r = ^ (p const.) or — = ^ («; const.) (Ic)

1 Q VQ J o PO

These scales are known, respectively, as the constant-pressure
gas scale and the constant-volume gas scale. They both give
true absolute temperatures to within small fractions of a degree.
It should be noted that equation (Ic) is true only if the expansion

PROPERTIES OF SUBSTANCES IN THE GASEOUS STATE 59

per degree is H?3 °f the volume at zero. This is not true of
equation (la), in which it is necessary only that the temperature
coefficient of pressure increase at constant volume is linear
throughout the temperature range; i.e., it is necessary only that
p = kt + a; equation (Ic) requires that a is 273k.

If 777 is the measure of any property of a substance that changes
linearly with temperature, its value is ra0 at the ice point, raioo at
the steam point, and mt at any temperature. Then the tem-
perature is defined by an equation similar to (la) above, namely,

L (id)

But since such a property is hard to find (actually none is known
that is exactly linear), all thcrmometric scales require slight cor-
rections when high precision is desired. The corrections are
smaller for the gas scale over a wide range than for most other
thermometric substances. For illustration, if nitrogen gas at
1000 mm, and 0°C. is used to measure temperatures through
equation (la), when the thermometer indicates 473.00° the
Kelvin temperature is 472.975°; when the thermometer indicates
873 00° the Kelvin temperature is 872 75°. The correction at
473°K. for a platinum resistance thermometer would be about
4.3° and for a mercury thermometer something like 2°, depending
upon the glass used in its construction.

We shall see in the next chapter that the vapor pressure of a
pure liquid is a function of the temperature alone, and thus a
vapor-pressure thermometer is another means of measuring tem-
peratures. But since the vapor pressure is very far from a
linear function of temperature, the scale will not be linear. For
example, the vapor pressure of water changes more between 99
and 100° than it does between 0 and 25°; and so an equation of
the form given in (la1) would be quite unsuitable. (The actual
relation is nearly log p = A/T + const.)

Certain other " fixed points " on the thermometric scale have
been established by international agreement for the purpose of
calibration, such as 90.19°K. for the boiling point of oxygen;
32.38°C., or 305.54°K, for the transition point of Na2S04.10H20;
444.60°C., or 717.76°K., for the boiling point of sulfur.1

1 See Burgess, J. Research Nail. Bur. Standards, 1, 635 (1928), for other
fixed points and a discussion of the international thermometric scale.

60 PHYSICAL CHEMISTRY

Ideal Gas Law. — By combining the two laws just given we
obtain the equation

Pv x

^777 = const.

in which the numerical value of the constant depends on the
units chosen for expressing p and r and on the quantity of gas
under consideration If we take a mole of gas as the standard
quantity, then the numerical value of the constant in a given set
of units is independent of the nature of the gas and is usually
denoted by R. The equation then becomes, for one mole of ideal
gas,

pvm = RT (2)

This equation is part of the definition of an ideal gas, and it is
also an approximate relation for actual gases. Equation (2)
alone is not a full definition of the ideal gas, and therefore we
give here for the sake of completeness the remaining equations
that complete the definition

(f) =0 or (f) = 0

\dv/T \dP/i'

(3)

Our discussion of this part of the definition will come later in
this chapter after we have considered equation (2) further. A
mole of gas is chosen as a unit in preference to a gram, since the
molecular weight of any gas occupies the same volume as the
molecular weight of any other gas. Engineers commonly use
1 Ib. of gas as the unit quantity in their calculations and employ
a different constant for each gas. This is less convenient than
the use of molal quantities, which require the same constant for
all gases.

Since the volume of n moles of gas is obviously n times the
volume of one mole, the equation may be written to describe the
behavior of any quantity of gas in terms of the one constant R.

pv = nRT (4)

The numerical value of the ideal gas constant R depends only
on the units chosen to express p and v. It should be noted that
R has the dimensions of work, since the product pv is force pel-
unit area X volume, or force X distance; and the quantities n

PROPERTIES OF SUBSTANCES IN THE GASEOUS STATE 61

and T are numbers. l Suppose a cylinder of area a, is fitted with a
tight piston. When this piston moves through a distance h
against a pressure of p on each square centimeter of the piston,
the force exerted is pa and it acts through the distance A; but
since ah is the volume of the cylinder, pah is pv and this has the
dimensions of work.

The limiting density (ratio of density to pressure at very low
pressure) for oxygen was given as 1.42707 grams per liter at 0°
on page 15, and from this value the molal volume of oxygen in
the state of an ideal gas at 0° and 1 atm. is

32.000/1.42767 = 22.414 liters

Hence, the pv product of an ideal gas is 22 414 liter-atm. per mole
at 0°, and this is equal to RT, whence R = 22.414/273.16, or

R = 0.0820G liter-atm. /mole-°K.

The actual density of oxygen at 0° and 1 atm. corresponds to a
molal volume of 22 394 liters, and upon dividing this pressure-
volume product by 273.16 we obtain R = 0.08198 by applying
the ideal gas law to a gas that deviates slightly from the ideal.
For most calculations R may be rounded off to 0.082 liter-atm.
per mole per deg. When pressure is expressed in dynes per
square centimeter, the ideal constant is

7? = 8.315 X 107 ergs/mole-°K.

When the pressure is in atmospheres and the volume is in milli-
liters per mole,2

R = 82.06 ml.-atm /mole-°K.
We record for later use two other values,

R = 8.315 joules/mole-°K

1 The usefulness of equation (4) is not confined to the c g s. system of units.
Pressure may be expressed in pounds per square foot and the quantity of
gas in pound-moles As explained on p 57, tf -f 460 = TR, where the sub-
script R indicates the Rankme, or Fahrenheit absolute, scale. Using this
absolute scale, with pressure in pounds per square foot, volume in cubic feet,
Mid quantity of gas in pound-moles, the value of the constant R in equation
(4) is 1544 ft.-lb./lb.-mole-°R

2 A milliliter-atmosphere is the work necessary to move a piston of 1 sq.
cm. area through a distance of 1 cm. against a pressure of 1 atm. One
small calorie is equivalent t^41.3 ml -atm.

62 PHYSICAL CHEMISTRY

and

R = 1.987 cal./mole-°K.

Equation (4) describes the behavior of most gases under
moderate variations in pressure and temperature with an accu-
racy of about 1 or 2 per cent. An ideal gas is one the behavior
of which would be exactly in accordance with this equation. No
such substance is known, but all actual gases approach the
condition of the ideal gas more closely as the pressure decreases
and as the temperature increases. The "ideal gas" is thus the
limiting condition for all gases, and equation (4) is called the
ideal gas law or idea] gas equation. The term "perfect gas"
is also commonly employed in this connection, but "ideal"
serves to keep constantly before us the imaginary character of
such a substance. In a later section we shall consider gases
under conditions of high pressures and at temperatures near the
condensation point, where the ideal gas law applies only roughly.
But for calculations at temperatures well removed from con-
densation points and at moderate pressures (up to 5 atm , for
example) the deviations of gases from the equation pv = nRT
are commonly less than 2 per cent, though they may be greater
for some gases.

The fact that conformity to the ideal gas law improves with
increasing temperature is well illustrated by the data for pro-
pane,1 which are plotted in Fig. 4. Propane (CH3CH2CH3)
boils at about — 42°C., so that all the curves are for temperatures
above the boiling point but not above the condensation tem-
perature for some of tho. pressures. For example, propane con-
denses to a liquid at 28 atm. and 80°C., and the sharp minimum
in the curve for 100°C. is very close to the critical temperature
and pressure above which no condensation is possible. At 60
atm. and 100°C. the value of pv/RT for a mole of propane is only
0.25; at 60 atm. and 325°C. it is about 0.92. Since thermal
decomposition of propane is observed at about 350°, the experi-
ments could not be carried to higher temperatures.

Large deviations from the ideal gas law at low pressures usu-
ally indicate a change in the number of moles present. Of
course, equation (4) cannot be expected to describe the changes
of p or v with T if the number of moles present is also changing.

1 DESCHNER and BROWN, Ind. Eng. Chem., ft, 836 (1940).

PROPERTIES OF SUBSTANCES IN THE GASEOUS STATE 63

For example, phosphorus pentachloride vapor partially decom-
poses according to the equation PC16 = PC13 + C12, and the
extent of the dissociation depends upon the temperature and
pressure. On the other hand, simultaneous measurements of p,
r, and T for a weighed quantity of such a material afford a means
of determining the number of moles present for these conditions.
A numerical example is given in a later paragraph. Such
apparent " deviations" are only examples of the misapplication
of a law to conditions for which it was not derived and to which
there is no reason to expect it to apply.

20 40 60 80 100 120
Pressure, Atm

FIG 4 — p-v-T relations of propane.

140

Mole Fraction. — A common method of expressing the com-
position of a mixture is in terms of the number of moles of each
substance present, divided by the total number of moles of all
substances present. As an example, the composition of the
earth's atmosphere may be computed in terms of the mole frac-
tions of the constituents. Analysis shows that 100 grams of dry
air contains 23.25 grams of oxygen, 75.5 grams of nitrogen, and
1.24 grams of argon. On dividing each of these weights by the
molecular weight of the substance we find 0.727 mole of oxygen,
2.70 moles of nitrogen, and 0.032 mole of argon, a total of 3.459
moles in 100 grams of -air. The mole fraction of oxygen is 0.727 '/
3.459 = 0.210, that of nitrogen is 2.70/3.459 = 0.781, and that
of argon is 0,032/3.459 = 0.009.

04 PHYSICAL U

At 20°C. and 1 atm. pressure the volume of 32 grams of oxygen
is 24 liters. By mixing 6.76 grams of oxygen, 21.88 grams of
nitrogen, and 0.36 gram of argon a total volume of 24 liters at 20°
and 1 atm. is obtained, and the mixture has properties identical
with air. The mixture contains 0.210 mole of oxygen, 0.781
mole of nitrogen, and 0.009 mole of argon, a total, therefore, of
1 mole. We may thus properly speak of this 24 liters of air as a
mole of air, though it contains less than a mole of any one sub-
stance. By multiplying the number of moles of each substance
in a mole of air by its molecular weight and adding, we find that
a mole of air weighs 29.0 grams. This "molecular weight of air"
is useful in applying the ideal gas law to air and in calculating
molecular weights of gases from the densities expressed as multi-
ples of the density of air under the same conditions. For most
approximate calculations it is sufficient to assign air the compo-
sition 0.21 mole of oxygen and 0.79 mole of nitrogen, since both
nitrogen and argon are chemically inert.

As one more illustration, we shall consider a mixture of 0.18
mole of hydrogen, 0.31 mole of iodine vapor, and 1.76 mole of
hydrogen iodine, a total of 2 25 moles of gas. In this mixture
the mole fraction of hydrogen is 0.18/2.25 = 0.080, that of
hydrogen iodide is 1.76/2,25 = 0.782, and that of iodine vapor
is 0.31/2.25 = 0.138.

Gas Dissociation. — The extent of dissociation (or of polymeri-
zation, or of reaction in general) in a gas mixture at moderate
or low pressure may often be determined from the pressure,
volume, and temperature of a known quantity of mixture of
known initial composition. For example, the density of phosgene
and its dissociation products at 823°K. and 1 atm. total pressure
is 0.820 gram per liter, and the calculated density of undissociated
phosgene is 1.475 grams per liter under these conditions. This
actual density is sometimes called an " abnormal" density or a
" deviation " from the ideal gas law. It is neither an abnormality
nor a deviation, but the density of a mixture formed through the
incomplete chemical reaction

COC12(0) = C0(0) + Cl,(flf)

which increases the number of moles for a given weight and so
leads to an increase in volume and a decrease in density for a

PROPERTIES OF SUBSTANCES IN THE GASEOUS STATE 65

given pressure and temperature. The measured density affords
a means of determining the extent of the dissociation. Con-
sider one mole of undissociated COC12, which is 99 grams, as a
working basis, and let x be the moles of CO formed. From the
chemical equation we see that x is also the moles of C12 formed
and the number of moles of COC12 decomposed, and so the
composition of the mixture is

x = moles CO
x — moles C12
1 - x = moles COC12

1 + x = total moles from 99 grams

The volume of 99 grams of mixture of density 0.82 gram per liter
is 99/0.082 = 120.7 liters, and upon substituting into pv = nR T
we have

1 X 120.7 - (1 + :r)0.082 X 823

whence x — 0.80, and this is the fraction of phosgene dissociated
at this temperature and pressure.

Any material basis for the calculation will serve as well as any
other, and we might have used 0.82 gram or 0 82/99 = 0.0083
mole of COC12 in 1 liter for the calculation. The composition
of the mixture is

y = moles CO
y = moles C12
0.0083 - y = moles COC12

0.0083 + y = total moles per liter

From the ideal gas law we find 0.0149 mole per liter at 823°K.
and 1 atm., whence y = 0.0066 and the fraction dissociated is
0.0066/0.0083 = 0.80 as before.

One more illustration will serve to show that the choice of a
material basis for calculation is merely one of convenience. At
823°K. and 1 atm. a molal volume of gas is 67.4 liters, and
67.4 X 0.82 = 55.2 grams per molal volume. In this volume we
have

z = moles CO

z = moles CU
1 - 2z = moles COC12

66 PHYSICAL CHEMISTRY

Upon multiplying each of these quantities by the appropriate
molecular weight, we obtain as the weight of a mole of mixture
282 + 712 + 99(1 - 2z) = 55.2 from which we find z = 0.445
mole CO and C12 and 1 - 2z = 0.11 mole COC12; and the frac-
tion dissociated is 0.455/(0.455 + 0.11) = 0.80 at 823°K. and
1 atm. total pressure. At some other temperature and pressure
the method would be the same, though the fraction dissociated
would not be 0.80, but another value.

Since this method in any of its forms depends upon measuring
the total moles of gas in a mixture through the ideal gas law, it
is obviously not applicable to reactions in which there is no
change in the number of moles. Dissociations such as 2HI =
H2 + I2 and 2NO = N2 + 02 must be measured in some other
way.

Partial Pressures. — The partial pressure of a gas in a mixture
is defined as the product of its mole fraction and the total pres-
sure of the mixture. If p is the total pressure on a mixture of
several components, a, b, c, . . . , whose mole fractions are

%aj •£(>) Xo • • •

Pa = pXa Pb = pXb pc = pXc (5)

In the dissociation problem treated at the top of page 65,
the partial pressure of phosgene was p(l — x)/(l + x), for
example. In any mixture of gases the ratio of the partial pres-
sures is thus the ratio of the number of moles of each in the
mixture, or

pi = wi

p2 n2

Dalton's law states that the total pressure of a mixture of gases
is equal to the sum of the pressures of the separate component
gases when each is at the temperature and each occupies the total
volume of the mixture. The pressures of the separate pure gases
are called the Dalton pressures.

Suppose the three bulbs A, By and C of Fig. 5 to be of equal
volume v and filled with HO moles of oxygen, nN moles of nitrogen,
and nH moles of hydrogen, respectively, at the temperature T.
Now let the stopcocks a and 6 be opened and the whole mixture
be forced into the bulb A. The pressures p0, PN, p& of the un-
mixed gases can be computed by the ideal gas law to be

PROPERTIES OF SUBSTANCES IN THE GASEOUS STATE 0Y
noRT

po =

and

5y Dalton's law, the pressure p of the mixture in the bulb A is
he sum of the pressures of the unmixed gases.

r»/T»

p = Po + PN + PH = (no + nN + nH)

Thus the equation for a mixture of ideal gases has exactly the
ame form as that for a pure gas. From the relations given, we
ee that each partial pressure is the product of mole
raction and total pressure,

FIG. 5.

?or ideal gases the Dalton pressure of a gas in a
nixture is equal to its partial pressure; for mixtures
>f real gases at low pressure they are approximately
jqual.1

It has been possible to find a few materials that
illow the free passage of the molecules of one gas
>ut not of other gases and so to measure partial
>ressures directly. Thus Ramsay2 found that when
i palladium bulb filled with nitrogen at 280° was
uirrounded by a stream of hydrogen the pressure
vithin the bulb increased almost as much as the
,otal pressure of hydrogen outside of the bulb.
Sis experiments were not continued until equi-
ibrium was attained, and the partial pressure of hydrogen
vithin the palladium bulb never reached the total hydrogen
)ressure outside. In a series of rather hasty experiments, he
bund that the partial pressure of hydrogen inside the bulb varied
rom 87 to 98 per cent of the pressure outside and that the
ictual figure depended somewhat upon the condition of the
palladium.

1 For a thermodynamic treatment of gas mixtures we are interested in the
squilibrium pressure of a gas in a mixture [Gillespie, /. Am. Chem. Soc.,
17, 305 (1925)]. The equilibrium pressure of a gas is the pressure that it
vould exert through a membrane permeable to it alone. For mixtures of
deal gases the equilibrium pressure is equal to the partial pressure; for
nixtures of real gases at low pressure they are approximately equal.

*Phil.M aa.. 38.206 (1894).

68 PHYSICAL CHEMISTRY

Lowenstein1 made use of the permeability of platinum to
hydrogen at higher temperatures in studying the extent of disso-
ciation of water vapor. A platinum tube connected to an oil
manometer was surrounded by water vapor contained in an
electrically heated furnace. As platinum allows the free passage
of hydrogen molecules through it, but not of oxygen or water
vapor, the manometer should show the partial pressure of hydro-
gen. By means of this method it was found that, at 1500°, water
vapor is about 0.1 per cent dissociated into hydrogen and oxygen,
which agrees with other methods of measuring the dissociation

With the exception of these experiments at high temperatures
upon mixtures containing hydrogen, there are no direct measure-
ments of partial pressures, because of the lack of suitable semi-
permeable membranes. The chief support for the belief that
correct equilibrium pressures or partial pressures are calculated
from the product of total pressure and mole fraction comes from
the study of chemical equilibrium itself. This topic will be
discussed fully in later chapters; here we need say only that
equilibrium compositions calculated from Dalton's law in gas
mixtures at moderate pressures are in agreement with measured
equilibrium compositions based upon analytical chemistry or
other means.

It is not to be expected that Dalton pressures will be additive
at high pressures, for the individual gases are not ideal at high
pressures; and such data as we have confirm this idea. For
example, in mixtures of argon and ethylene at 30 atm. total pres-
sure the actual pressures are less than the sum of the Dalton
pressures by 0.75, 0.85, |ind 0.45 per cent, respectively, when the
mole fractions of ethylene in the mixture are 0.25, 0.50, and 0.90.

Mixtures of nitrogen and ammonia at total pressures of 10 to
60 atm. also show that Dalton's law is inaccurate at high pres-
sures. In a steel bomb the pressure of NHa developed by the
dissociation of solid BaCl2.8NH3 is 7.123 atm. at 45°, and this
ammonia pressure remains almost constant when nitrogen is
added to the bomb.2

1 Z. physik. Chem., 64, 715 (1906).

* Data from Lurie and Gillespie, J. Am. Chem. Soc., 49, 1146 (1927),
53, 2978 (1931); the increase of dissociation pressure with total pressure is
calculated by a method similar to that on p. 109 for the vapor pressure of
water.

PROPERTIES OF SUBSTANCES IN THE GASEOUS STATE 69

In the following table the first line gives observed total pres-
sure of N2 + NH3 in atmospheres, the second line gives the
pressure of NH3 in equilibrium with BaCl2 8NH3 and BaCU at
45°, and the third is the product of total pressure and mole frac-
tion of NH3. Ammonia itself deviates from the behavior of an

Total pressure 10 13 13 27 23 70 32 82 60 86

7>(NH<) 7 14 7 16 7 22 7 27 7 44

prr(NH,) 7 28 7 51 7 85 8 13 9 03

Per cent difference 19 48 87 118 21 4

ideal gas by about 7 per cent at 45° and 7 atm., and larger devia-
tions are shown in the presence of nitrogen, which increases the
total pressure.

These data are quoted to show that while the ideal gas law is
a useful and convenient simplification at low pressures, it is not
to be used outside of certain limits without appreciable error.
It does not apply exactly to any gaseous system, but it ordinarily
yields calculations within 1 per cent of the truth with gases or
gas mixtures at pressures not much above 1 atm.

Change of Barometric Pressure with Altitude. — The decrease
of pressure in any "fluid of density p with increase in height above
a chosen reference point is shown by the equation

— dp = pg dh

in which p dh is the mass of a layer of unit cross section and
thickness dh and g is the acceleration of gravity. For an ideal
gas p = m/v = Mp/RT, whence, for changing barometric pres-
sure with altitude ft, we have

If a uniform temperature is assumed for the air column, we may
integrate the equation between the limits po at fto and p at an
altitude ft, as follows:

2.3 log 2! = (h - *.)

Substituting R = 8.32 X 107, T = 293 for an assumed tempera-
ture of 20°C., ft = 160,900 cm. for 1 mile, M = 29 for air, and
g = 980 cm. per sec.2, we find the pressure to be 0.83 atm. 1 mile

70 PHYSICAL CHEMISTRY

above sea level. Similarly, the pressure is found to be 1.019
atm. at the bottom of a 500-ft. shaft by taking h = --15,000 cm.

Avogadro's Law. — We have already seen in the previous
chapter that equal volumes of gases at atmospheric pressure and
at the same temperature contain almost the same number of
molecules. At very low pressures equal volumes at the same
temperature contain exactly the same number of molecules, as
shown by the agreement of atomic weights derived from gas
densities with those based on other methods. The fact that
the volumes of gases entering into chemical reactions are equal
or simple whole multiples of one another and of the volume of
the gaseous products is also evidence of the correctness of the
law. These volume ratios alone led Avogadro to propose the
law in the first place. But convincing confirmation of the law
came from determinations of the actual number of molecules
in a gram molecule. We turn now to some of the methods by
which this was accomplished.

Avogadro's Number. — The early experiments upon the
behavior of colloidal particles, which showed that if they approxi-
mated molecules in their properties the number of molecules in
a gram molecule was 6 X 1023 or 7 X 1023, are now of historical
interest only. But the scattering of solar radiation in the upper
atmosphere, the energy of the products of radioactive decompo-
sition, the radiation laws, and other data also pointed to these
figures, 6 X 1023 being nearer the probable number than 7 X 1023.
We may review briefly three methods of obtaining this number, 1
of which the first is so convincingly direct as to leave no room
for doubt of its validity^ and the second and third yield the most
precise values available.

The radioactive decay of radium expels charged helium atoms
(alpha particles) of such high velocity that the impact of a single
atom upon a screen of zinc sulfide produces a flash of light that
is visible in a microscope. There are other ways in which the
effect may be observed. By adjusting the quantity of radium
and the distance to the counting mechanism so that an actual
count could be made, it was found that the enormous number

1 A review of the early experiments which led to estimates of Avogadro's
number is given by Dushman in Gen. Elec. Rev., 18, 1159 (1915); the more
precise modern values are reviewed by Birge in Phys. Rev. Suppl., 1, 61
(1929); and by Virgo in Science Progress, 27, 634 (1933).

PROPERTIES OF SUBSTANCES IK THE GASEOUS STATE 71

1.36 X 1011 alpha particles were emitted each second from a
gram of radium.1 In other experiments it was found that
0.156 ml. of helium (0° and 1 atm.) was produced per gram of
radium per year. Upon multiplying 1.36 X 1011 by the number
of seconds in a year, one obtains the number of atoms of helium
in 0.156 ml. and, by proportion, the number in 22.4 liters, which
is 6.16 X 1023 atoms per molal volume of this monatomic gas.

A second method involves Faraday's law of electrolysis, the
important aspect of which for this purpose is the deposition of
silver from silver nitrate by electrolysis. This reaction is

Ag+ + e~ = Ag

and careful experiments have shown that 96,489 coulombs of
electricity deposit one atomic weight of silver. The charge of
an electron is 1.598 X 10~19 coulomb.2 The number of electron
charges in a faraday of electricity is the number of atoms of
silver in an atomic weight, or Avogadro's number, which is thus
96,489/1.598 X 10~19 = 6.03 X 1023.

The third method involves determining (1) the wave length of
X rays from a ruled grating, (2) the spacing of atomic planes in
a crystal by using these planes as a diffraction grating for the
X rays, (3) the density of the crystal, from which, together with
the atomic weights of the elements, one determines (4) the gram-
molecular volume. For sodium chloride, the edge of a cube
containing 4 atoms of sodium and 4 atoms of chlorine is
5.638 X 10~8 cm., the density is 2.163, the molecular weight is
58.454, and 4 X 58.454/2.163 = 108.10 cm.3 is the volume of 4
molecular weights of sodium chloride. Avogadro's number is
then found by dividing 108.10 by the cube of 5.638 X 10~8, which
gives* 6.032 X 1023. A more recent determination based on the
spacing of calcite3 gives 6.0245 X 1023.

Viewed in the light of this number the attainment of a " vac-
uum" seems quite hopeless; for the lowest pressures ever meas-
ured, after the most efficient removal of gas from a container,

1 The figures are quoted from Sir Ernest Rutherford's lecture printed in
the annual report of the Smithsonian Institution, 1915, p. 167.

2 Milhkan, Ann. Physik, 32, 34, 520 (1938), gives the electronic charge
as 4 796 X lO"10 e.s.u , which is 1.598 X lO"20 abs. coulomb or 1.598 X 10~19
int. coulomb, since the absolute ampere is 10 int. amp.

8 BEAKDEN, /. Applied Phys., 12, 395 (1941).

72 PHYSICAL CHEMISTRY

are about 10~6 dyne per sq. cm. (this is approximately 1/1,000,-
000,000,000 atm.), and in this "vacuum" the number of mole-
cules per milliliter is greater than the population of the earth.
As a further illustration of the astonishingly large number of
molecules in a weighable quantity of matter, it may be observed
that, if 1 gram of water were spread uniformly over the surface
of the entire earth, there would be 3500 molecules per sq cm.

Molecular -weight Determinations — Direct Method. — When
the ideal gas equation is written pv = (m/M)R7\ it will be seen
that the molecular weight M of a gas may be determined from
the weight m of a known volume at some definite temperature
and pressure. A glass bulb of 300 to 500 ml. capacity is evacu-
ated and carefully weighed, then filled at a fixed temperature and
pressure with the gas under consideration, and weighed again
The precise data on pages 15 and 27 illustrate an extension of
this method, which has been useful for many other gases. But
it should not be concluded that its application to all substances
is free from complications. We record here for illustration the
observed temperature and pressure (in millimeters of mercury)
for 0.2429 gram of formic acid vapor in a bulb of 521 8 ml capacity
and the "molecular weight" obtained from the data for each
temperature.

t°C... 10 20 30 40 50 60 70

p, mm. 10 1 11 02 12 13 13 42 14 90 16 50 18 10

M.. 814 773 724 676 629 585 549

The vapor of formic acid is a mixture of HCOOH and
(HCOOH)2 molecules jn proportions varying with the tem-
perature, and each of the figures in the third line above gives
the weight of a molal volume of the mixture under the stated
temperature and pressure. The data do not illustrate a failure
of the ideal gas law; they provide a means of determining the
composition of the vapor. If all the molecules were HCOOH,
the pressure would be 17.9 mm. at 10° and 21.7 mm. at 70°.

Dumas's Method. — If the substance whose vapor density is
desired is a liquid at room temperature, about 10 ml. of it may
be placed in a weighed bulb with a long capillary stem. All the
bulb except its tip is then immersed in a constant-temperature
bath (usually boiling water), and the air and excess liquid are
expelled from the bulb. When all the liquid has been vapor-

PROPERTIES OF SUBSTANCES IN THE GASEOUS STATE 73

ized, the bulb is sealed and the barometer is read. At the
moment of sealing, the bulb is filled with vapor at the barometric
pressure and at the temperature of the bath. Thus T and p are
known, and m is determined by weighing the sealed bulb again
and v by filling the bulb with water arid weighing again. As the
bulb when first weighed is filled with air that is expelled by the
boiling liquid, it is necessary to compute the weight of air expelled
and subtract it from the first weighing in order to obtain the
weight of the empty bulb.

Actual data on carbon tetrachloride may be used to illustrate
the method of calculation

Bulb (filled with air) 51 43 grams

Bulb with C014 vapor 52 86 grams

Bulb filled with water 411 grams

The difference between the weight of the bulb when filled with
water and the weight filled with air is 360 grams, and this is sub-
stantially the volume of the bulb in millihters. The weight of
air contained in the bulb at its first weighing was not present
at the second weighing. Its weight may be obtained by sub-
stituting in the equation pv = nRT, from which it will follow
that 0.015 mole of air, or 0.43 gram, was present. The empty
bulb, therefore, weighed 51.00, and hence 1.86 grams of carbon
tetrachloride vapor filled the volume of 360 ml. and exerted a
pressure of 1 atm. at 100°C when the bulb was sealed. Upon
substituting these values in pv = (m/M)RT, M is found to be
160, which should be compared with 154, the formula weight.
The difference is mostly due to the fact that the vapor of CCU
is not ideal under the experimental conditions, and closer agree-
ment* is not to be obtained by more careful experimentation.
The method of limiting densities would give 154 if correctly
applied to CC14 at 100°C.

Victor Meyer's Method. — This procedure is adapted to sub-
stances that vaporize at somewhat higher temperatures than
those suited to Dumas's method; indeed, it can be applied at
temperatures up to the softening point of porcelain or quartz.
In principle, the method consists in vaporizing a weighed quan-
tity of the liquid or solid substance in a vessel filled with hot air
or nitrogen at such a temperature that the substance vaporizes
readily. The hot bulb is made much larger thato the volume that

74 PHYSICAL CHEMISTRY

the substance will occupy as a vapor, and when vaporization
takes place a mole of air is expelled for each mole of vapor
formed. For convenient measurement, the expelled air is col-
lected in a burette over water. From the barometric pressure,1
volume, and temperature of the air in the burette the number
of moles of air expelled is calculated from pv = nRT, and since
this is also the number of moles formed by a known weight of
substance vaporized in the hot tu.be, M = m/n. The method
may not be applied to dissociating substances; for the vapor
mixes with the hot nitrogen in the tube, and the extent of
dissociation is altered by dilution at constant temperature.
Dumas 's method and the direct method are free from this
restriction.

KINETIC THEORY OF GASES

The purpose of the paragraphs that follow is to consider the
properties of the molecules in a gas and to develop equations in
terms of the mass and velocity of the molecules that apply to
the behavior of gases and that can be tested by experiment.
Since the number of molecules in any quantity of gas upon which
experiments can be performed is exceedingly large, we are to be
concerned with average velocities or average kinetic energies
rather than with those of individual molecules.

Fundamental Equation. — The molecules of a gas are not at
rest but move about through the confining space with great
rapidity,2 colliding frequently with each other and with the walls
of the vessel surrounding them. This statement is supported
by the fact that when-two gases are brought in contact and the
mixture is allowed to stand it finally becomes homogeneous
throughout. If a quantity of chlorine be placed in the bottom
of a vessel by displacing part of the air in it, a distinctly greenish
layer will be seen. When this is allowed to stand for some
time, the green layer diffuses upward throughout the whole

1 The* partial pressure of the air is of course the barometric pressure less
the vapor pressure of water, which is given in Table 14.

2 The average velocity of molecules in air is about H mil6 per sec., but
the average straight-line distance traveled between collisions is only about
0.0001 mm., the number of hits per second for each molecule being thus
about 5,000,000,000. Actual velocities of molecules were determined by
Stern [Z. Physik, 2, 49 (1920)] and found to agree with those expected from
the kinetic theory.

PROPERTIES OF SUBSTANCES IN THE GASEOUS STATE 75

vessel and there is no longer any visible boundary between the
two gases. This mixing is not dependent on stirring but will
take place if the vessel is kept absolutely quiet and at a constant
temperature, in spite of the different densities of the gases.

The pressure exerted on the walls of a container by a gas is
entirely due to collisions that take place between the moving
molecules and the walls. It is known that the pressure does not
decrease if a gas is allowed to stand indefinitely in a closed space
at constant temperature and that a gas does not continuously
absorb heat from the surroundings to supply the energy of
motion of its molecules. This can be true only if the molecules
are perfectly elastic as regards their collisions with one another;
for otherwise the collisions would absorb energy, and the intensity
of motion would gradually decrease and cause the pressure to
fall off. The pressure is perfectly constant on all the walls at
all times, and therefore the bombardment of the walls must be
uniformly distributed.

Within a gas the molecules move about in the utmost chaos,
with no regularity whatever, and at widely different velocities.
A molecule that has a high velocity at one instant may suffer a
collision that changes its direction and velocity at any moment.
Indeed, the path of each molecule is absolutely haphazard, and
the state of a gas must be thought of as absolute confusion.
But it is convenient in visualizing the behavior of molecules, as
regards pressure exerted on the* surrounding walls, to consider
their motions along three axes perpendicular to the faces of a
confining cube and to consider the mean1 velocity of all the

1 By applying the laws of probability Maxwell has shown that the dis-
tribution of velocities among a
large number of molecules which
have a given mean velocity is
shown by the equation

o.c>

0,4
0.2
0

!!\

j[ \

•

) 1.0 2.0 3.0 4.0

Velocity

where y denotes the probability of
a velocity whose magnitude is re,
the most probable velocity being
taken as unity. Figure 6 shows
this curve graphically. The arith-
metic average velocity is 1.13 times the most probable velocity; and the
"mean" velocity is 1.22 times the most probable one. By "mean" is

FIG. 6.

76 PHYSICAL CHEMISTRY

molecules, in place of the rapidly changing velocity of a single
molecule.

For convenience in deriving the desired equation, we may
assume a cubical container of edge /, of which one corner is the
"origin," and resolve the motions of the molecules along the
rectangular x, y, and z axes meeting at this corner. The root-
mean-square velocity \/2^2/n, which we shall call the mean
velocity u, or the velocity from which to compute the average
kinetic energy of a molecule, is evidently related to the velocities
resolved along these axes by the equation

Let n be the number of molecules in the container, and let m
be the mass of one molecule Consider one face of the cube,
perpendicular to the x axis, which a molecule approaches with a
velocity whose ^-component is ux and from which it recedes with
a velocity whose ^-component is —ux after colliding with the
wall. The change in momentum caused by this impact is 2mux,
and this momentum will be imparted to the wall by every mole-
cule striking it. Before the molecule can strike this wall again,
it must travel the distance 21 to the opposite face and back,
which will require 21 /ux sec. In other words, the number of im-
pacts on this wall by one molecule will be ux/2I per second or
nux/2l impacts per second for all the n molecules.

The total momentum imparted to the wall per second will
be the product of the change in momentum per hit and the
number of hits per second, which is 2mux(nux/2l). Since the
force / exerted on the* wall is the rate at which momentum is
imparted to it,1 we have as a measure of this force

. 2mux X nux

21 I

meant that velocity which would give the average probable kinetic energy.
This is the square root of the average of the squares, or root-mean-square
(r.m.s ) velocity, arid is denoted by u in the above text.

Since the area under the curve in Fig 6 is unity, the fraction of all of the
molecules which have velocities between OA and QB is denoted by the shaded
area. For a further discussion of these matters see Dushman, Gen. Elec.
Rev., 18, 952 (1915).

1 Force has the dimensions ml/t2, and momentum is ml/t; hence ml/t X 1 A
is the rate of imparting momentum to a surface.

PROPERTIES OF SUBSTANCES IN THE GASEOUS STATE 77

Since experiment shows that the force acting upon the walls
of the container is the same for all walls, it follows that the
velocities resolved along the three axes must be equal, so that
ux2 + uy2 + uz2 = u2 = 3ux2 Upon making this substitution,
and dividing both sides of the equation by I2, we have

/ _ run a2
T2 ~ W~

Note now that the left side of the equation, //72, is the pressure
and that /3 is the volume v of the container, so that the equation
becomes

P =

1 3 mnu'2

Since for a mole of gas pv is equal to RT [equation (4)], we may
write

prm = y3mnu2 = RT (7)

which is the fundamental equation of the simple kinetic theory.
If p is in dynes per square centimeter, v is in milliliters, m in
grams, u in centimeters per second, n is Avogadro's number
(6.03 X ;()28), and R has the value 8315 X 107 ergs/mole-°K.
Since mn equals M, the molecular weight, this equation may also
be written

pvm = l£Mu2 = RT (la)

If we write equation (7) in the form

%n XlAmu2 = RT

it will be seen that %n is constant for a molal volume (Avogadro's
law) ; and hence %mu2 must be the same for all gases when T is
constant, since nothing has been assumed as to the kind of gas
molecules. Thus %mu2 = f(T).

When two different gases at the same temperature are mixed
there is almost no change in temperature; consequently, the
average kinetic energy of the molecules (y^mu2) must be prac-
tically the same for all gases at the same temperature and must
increase at the same rate for all gases. If the kinetic energy of a
gas molecule depends only on its temperature and is independent
of the nature of the gas,

pv = % X %miui2ni = % X

PHYSICAL CHEMISTRY

If p\Vi = p&<i at a given temperature, the same volume of the
two gases must contain the same number of molecules, that is,
HI = rc2, since J^WiUi2 = ^m^u^, and this is the law of
Avogadro.

Thus we see that our fundamental equation (7) is in substantial
agreement with the known facts concerning gaseous substances
at moderate pressures.

Rate of Effusion of Gases. — At any given temperature the
kinetic energies of two kinds of molecules should be the same
according to our equation; i.e ,

HI _ Im2 _
u, ~ \^ "

(8)

since the masses of the molecules are proportional to the molecular
weights Mi and M2 and to the densities di and d2. This equation
states that the velocity of the molecules should be inversely pro-
portional to the square root of the density of the gas. Since
effusion through a small hole is a manifestation of molecular
motion, the correctness of this equation may be tested by com-
paring the rates of effusion of gases through a given opening.
The statement in equation (8) is Graham's law of effusion of
gases. Some of his data are quoted in Table 7 to show that
this consequence of equation (7) is proved by experiment.
TABLE 7. — RATE OF EFFUSION OF GASES1 4

Gas

Density
relative
to air

Time of
effusion
relative
to air

Square root
of density

Velocity of
effusion ,
relativ'ijf^
to airlJJf

Velocity calcu-
lated from
square root of
density

Air

1.0000

1.000

1.0000

1.000

1 000

Oxygen
N»

1.1056
0.9714

1.053
0.984

1.0515
0.9856

0.950
1.016

0 951
1.015

CO.

0 9678

0.987

0.9838

1.012

1 016

CH4. . . .
CO,
N,0....

0.5549
1.5290
1.5290

0.765
1.218
1.199

0.7449
1.2350
1.2350

1.322
0 821
0.834

1 342
0 809
0.809

1 GHAHAM, Phil Trans. Roy. Soc. (London), 136, 573 (1846). See Edwards,
Natl. Bur. Standards Tech. Paper, 94 (1917), for a description of an improved
experimental method; also Kemp, Collins, and Kuhn, Ind. Eng. Chem.,
Anal. Ed., 7, 338 (1935).

PROPERTIES OF SUBSTANCES IN THE GASEOUS STATE 79

The following calculations will illustrate the method of apply-
ing equation (8) under a small (constant) driving pressure.
Suppose that 100 ml of air will effuse through a pinhole in a thin
plate in 75 sec. and that under the same conditions 100 ml. of
another gas escape in 92 sec. Since the faster moving molecules
will escape at a higher rate, the velocities are inversely propor-
tional to the relative times of escape and equation (8) becomes

Upon substituting 75 sec. for ti, 92 sec. for Uy and 29 for M\ it
is found that Mz is 44.

Heat Capacity of Monatomic Gases at Constant Volume. —
Since a quantity of gas is usually described by the number of
moles in the calculations of physical chemistry, we shall be con-
cerned with the molal heat capacity, which is the ratio of the
heat absorbed by & mole of gas to the rise in temperature pro-
duced, C = dq/dT. But since q depends upon the manner of
heating, some further specification is required to make the heat
capacities definite. The only processes that concern us are heat-
ing at constant volume and heating at constant pressure, for
which the definitions are

and

dg\ _ (dE\

A WA

For gases at moderate pressures the equations
9E\ A j

* A = ° and

are substantially true; therefore, Cp is the same for any constant
pressure, and Cv is the same for any constant volume.

An increase of temperature increases the kinetic energy of
translation of the molecules by an amount that may be calcu-
lated from equation (7). This will not be equal to the increase
in the energy content E unless the other forms of energy do not
change. The total energy content of a gas includes kinetic,

80 PHYSICAL CHEMISTRY

rotational, vibrational, electronic, and all other forms; and since
Cv = dE/dT, this will not be equal to dEkm/dT unless the energy
absorbed in other forms is zero. Thus dEkin/dT is the minimum
value that Cv may have. For monatomic gases this is the actual
value of Cv, but for all other gases the rotational energy is impor-
tant even at room temperatures. For all gases the other forms
become important at high temperatures. Similarly,

by definition, and thus the heat capacities of all gases at constant
pressure will be greater than those for constant volume. The
calculation for a monatomic gas will now be given.

Let HI be the mean velocity of the molecules at the absolute
temperature T\ and i/2 the mean velocity at the higher tempera-
ture T% after the quantity of energy AE has been absorbed by
a mole of the gas. The increase in kinetic energy of all the
molecules is

where n is Avogadro's number of molecules in a mole of gas;
and this increase in kinetic energy is equal to the heat added
Since we are concerned with a molecular weight of gas, the
product nm is equal to the molecular weight of the gas M. From
equation (7a) we obtain

and

P*>~ = hMuj = m\

By multiplying each of these equations by % and subtracting
the first from the second, we obtain

y2Mu^ - YiMuJ = %R(TZ - T,) (9)

as the difference between the kinetic energies of the molecules
at the temperatures r2 and T\. This is equal to the heat
absorbed, which is equal to the molal heat capacity of the
gas multiplied by the increase in temperature ; that is,

PROPERTIES OF SUBSTANCES IN THE GASEOUS STATE 81
On substituting these quantities in equation (9), we have

AE = CV(T, - 7*0 = Y2M(uf - t/!2) = HR(T* - TJ (10)
whence the molal heat capacity at constant volume is

Cv = %R = 2.98 cal. per deg. (11)

By multiplying both sides of equation (la) by %, we obtain
an expression for the kinetic energy of the molecules,

and if increase in kinetic energy is the only effect of energy
absorption upon heating at constant volume, the value of Cv is
obtained by differentiating this equation,

irri

Since the relation (dE/dv)T — 0 is part of the definition of an
ideal gas, it will be seen that (6E/dT)v = Cv is also independent
of the volume. This relation is also nearly true for actual gases
at pressures of a few atmospheres; we may therefore write that
dCv/dv = 0.

Heat Capacity of Monatomic Gases at Constant Pressure. — If
the gas is heated from TI to Tz at constant pressure, expansion
attends the heating and work is done against the external pres-
sure. Since the increase of kinetic energy is the same whether
heating occurs at constant volume or constant pressure, the latter
process requires the absorption of additional heat equivalent to
the work done. This work is p(v% — Vi), which for a mole of gas
is R(T2 - Ti), whence

Cp = %R = 4.97 cal. per deg. (12)

This equation, like equation (11), is applicable only to gases in
which none of the energy absorbed in heating is used to increase
the rotational or vibrational energy of the molecules or to over-
come attractive forces between molecules; and only monatomic
gases meet these requirements. The experimental data of Table
8 will be seen to agree with the heat capacities calculated in
equations (11) and (12).

82 PHYSICAL CHEMISTRY

TABLE 8 — MOLAL HEAT CAPACITIES OF MONATOMIC GASES

Substance

CP - R =

Experiments bv

Mercury vapor

4 97

2 98

Kundt and Warburg

Helium

5 10

3 11

Behn and Geiger

Argon

4 99

3 00

Niemeyer

Argon

4 97

2 98

Pier

Argon

5 07

3 07

Heuse1

Ratio of Cp to Cv for Monatomic Gases. — In addition to evi-
dence from experiments on the temperature change during expan-
sion into a vacuum (to be discussed presently), there is another
way in which the correctness of equations (11) and (12) may be
tested. It will be remembered that these equations were derived
on the assumption that all the energy added to the gas increased
the kinetic energy of the molecules or performed work in over-
coming the pressure of *the atmosphere during expansion. Let
us assume for the moment that there is some unknown absorp-
tion of energy in addition to those stated. The equation
Cp — Cv = R has been established by experiment ; and the quan-
tity of energy %R must be absorbed to increase the kinetic
energy of the molecules and account for the experimentally
proved increase in pressure with the temperature. Let x denote
the energy required for other purposes. Then the ratio of
specific heats at constant pressure and at constant volume is

5R + 2x
3R + 2x

= y

It is possible to determine the ratio of these two specific heats
from the velocity of sound in a gas,2 and the ratio for monatomic

1 Ann. Physik, 59, 86 (1919) *

2 Laplace has shown that the hydrodynamic equation for the velocity of
sound in a medium of density p is

<•*>•

where v9 is the specific volume of the medium For an adiabatic expansion,
such as attends the passage of sound through a gas, pv,y « const., or
In p -f y In v, = In const., where y is the ratio CP/CV for the gas in which
sound travels. Upon differentiating,

PROPERTIES OF SUBSTANCES IN THE GASEOUS STATE 83

gases is 1.667. Now this is ^3, and hence x in the above equation
must be zero. Thus the heat-capacity equations are supported
by the results of experiment.

Heat Capacity of Diatomic Gases. — The definitions of heat
capacity that have already been used for monatomic gases apply
to all gases, namely,

C - dE and r - —

C' ~ dT a p ~ dT

and of course the relation pvm — RT applies to them. By
combining this equation with the definition H = E + pv, we
find // = E + RT for a mole of gas; and, upon differentiating
with respect to T, we have

dT ~ dT
which gives the difference between Cp and Cv for any gas as

Cp - Cv = R

whether the gas is monatomic or polyatomic, so long as it con-
forms to the relation pv = nRT. This same relation follows
from equation (12) on page 37, which was

since (dE/dv)T = 0 for gases and p(dv/dT)p = R from the gas
law.

dv V8 vsz

In an ideal gas the specific volume is RT/pM, and the product of pressure
and specific volume is RT/M, whence (2) becomes

dp RT

dv Mvf*

and (1) becomes

PHYSICAL CHEMISTRY

If the molecules of a gas contain more than one atom, consid-
erable quantities of energy may he absorbed in increasing rota-
tion of the molecules or in increasing internal vibrations, i.e.,
displacement of one of the atoms relative to another. Experi-
ment shows that the pressure of the diatomic and triatomic
gases increases with the absolute temperature in the same way
as that of the monatomic gases, which could be true only if the

TABLE 9 — MOI-AL HEAT ("APATITY RATIO FOR OASES

Substance

p, a tin

7 = CP/C>

Air

6 95

1 40

Air

100

1 58

Air

- 79

100

2 20

6 94

1 40

100

1 66

6 97

1 40

-180

1 45

C12

8 15

1 36

HC1

7 07

1 41

SO2

9 71

1 29

CO2

8 75

1 30

C02

3 52

C02

- 75

8 08

1 37

CaHc

11 6

1 28

Ether

27 7

1 08

kinetic energy of the molecules increases with increasing tem-
perature in the same way. The heat absorbed and converted
into rotation or vibration of the molecules is in addition to that
required to increase the kinetic energy or to do work of expan-
sion; therefore, the heat capacities are higher for diatomic gases.
If we call the extra energy absorption during heating the "internal
heat capacity/' Cmt, the equations that apply are

and

cv = HR + cint
cp = %R + R + cmt

These equations show that CP/CV will be less than % if Cmt is
appreciable. Since CP/CV = 1.4 for diatomic gases, we estimate
Cat = R for them as a first approximation. A clue, though not
a complete explanation, is furnished by the law of equipartition

PROPERTIES OF SUBSTANCES IN THE GASEOUS STATE 85

of energy, which says that Cvi$%R for each " degree of freedom"
of the molecule. Monatomic gases have three degrees of trans-
lational freedom; and since Cv = %R f°r them, they have no
appreciable rotational energy; diatomic gases have three degrees
of translational freedom and two of rotational freedom, which
should give Cv = %R, Cp = %R, and CP/CV = 1.4 if no energy
is absorbed in other ways. These figures are close to the experi-
mentally determined heat capacities of H2, N2, 02, CO, NO, and
HC1 at ordinary temperatures, which is an indication that there
is no appreciable internal heat capacity other than rotation at
ordinary temperatures. The molal heat capacities of Br2(0) and
I2(0) at constant pressure are 9.0 at ordinary temperatures which
shows that these gases have " internal heat capacity " other than
rotation; the usual interpretation is vibration of the atoms in the
molecule. For the other diatomic gases CP increases at higher
temperatures, which is an indication that vibrational effects
become more important as the temperature rises. The increase
for chlorine is conspicuous, Cp changing from 8.1 at 300°K. to 8.6
at 500°K. and to 8.9 at 2000°K., probably because the vibrational
heat capacity changes rapidly with rising temperature. Equa-
tions for the change of heat capacity with temperature are given
in Table 56 and some data for CP/CV are given in Table 9.

Mass of Gas Striking a Unit of Surface. — As shown on page
76, pressure is the momentum imparted to a unit area in unit
time. If w is the mass of gas striking a unit of surface in unit
time and ux is the velocity resolved on the x axis perpendicular
to this surface, the pressure is p = 2wux. In a gas the velocities
resolved upon the three axes are equal, for the pressure is the
same on all walls of the vessel; therefore,

From equation (7a),

RT =

and by combining these relations, the mass of gas striking unit
surface each second is given by the equation

where w is the mass of gas in grams per second per unit surface,

86 PHYSICAL CHEMISTRY

p is the pressure of the gas in dynes per square centimeter, M
is the molecular weight of the gas, T is the absolute temperature,
and R has the value 8.315 X 107 ergs/mole-°K. Langmuir1 has
derived a more exact expression for the mass of gas striking a
unit area during each second, by taking into account the dis-
tribution of velocities around the most probable one. His equa-
tion differs from the one above only by a numerical constant.
The more exact equation is

We may illustrate the application of this equation by calculating
the mass of oxygen striking each square centimeter of a surface
exposed to air under ordinary conditions. The partial pressure
of oxygen is 0.21 atm., or 21.2 X 104 dynes per sq. cm., T is 293,
and the other quantities have been given above. By substituting
these quantities into equation (14), we find w is 3.1 grams per sec.
Energy Absorbed in Expansion, Joule Effect. — The fact that
the pressure-volume product of gases at constant temperature
is nearly constant for moderate pressure changes indicates that
the attraction between molecules is relatively small under these
conditions. But if during an expansion the molecules exert
considerable attractive (or repulsive) forces on one another,
these forces will resist (or assist) the expansion. In the expan-
sion of a compressed gas taking place in an isolated system and
arranged so that no work is done (a " Joule expansion"), the
attractive forces of the molecules for one another must be over-
come at the expense of the kinetic energy of the molecules and
the temperature will not remain constant if these forces are
Appreciable. Consider a vessel of 6 liters capacity containing a
mole of gas at 20°C. and connected by a tube, containing a closed
stopcock, to an evacuated vessel of 18 liters capacity, and assume
the whole system isolated so that no heat can enter or leave it.
When the stopcock is opened, gas passes into the empty vessel
until the pressure is the same (about 1 atm.) in both. No heat
is absorbed, and no work is done by the system, so that AJ57 is
zero; and if no " internal' ' work is done 'against the attractive
forces, the temperature will still be 20°. These facts may be

. Rev., 2, 329 (1914).

PROPERTIES OF SUBSTANCES IN THE GASEOUS STATE 87

expressed by the equation (dE/dv)T = 0, which is part of the
definition of an ideal gas.

Experiments on the Joule expansion of actual gases show that
the temperature changes during the expansion. For these expan-
sions A^ is zero, but the temperature is not constant; therefore,
AE for the isothermal expansion is AE for heating the gas
to the original temperature at constant volume, or JCV dT.
Attempts to measure the temperature changes during Joule
expansions have been unsuccessful because of heat transfer from
the container to the expanded gas, heats of adsorption and
desorption, and other difficulties. One may calculate what the
temperature change would be if these effects were absent from
other experiments on actual gases, but the observed temperature
changes differ from the calculated ones. Even so, the experi-
ments show that for isothermal expansion of a real gas (dE/dv)T
is not zero, and they indicate that molecular attraction is one of
the main causes. The calculated temperature change for carbon
dioxide expanding as indicated above is about 1°.

Joule-Thomson Effect. — One of the best means of showing
the change of internal energy of a gas upon expansion consists
in passing it through a tube thermally insulated from its sur-
roundings and obstructed by a
porous plug, as shown in Fig. 7.
There will thus be a pressure

difference on the two sides of the ^ [

plug; and if the expansion is at-

tended by an energy change, the (p^pxv^AV)-"'1 P|v,

° °

temperature on the, two sides of * FlG 7

the plug will not be the same.

The change in temperature, called the " Joule-Thomson effect"

after its discoverers,1 depends upon the initial temperature and

for a given temperature varies with the initial pressure.

The gas in its passage through the plug will come to a steady
condition, provided that the pressure and temperature before
the plug remain constant and the pressure on the far side is
constant. To secure the steady state the tube and its plug must
be nonconductive for heat, or corrections will be required to
allow for flow of heat along the tube or plug. Assuming the ideal
conditions, an examination can be made of the physical change

lPhil. Trans.. 149, 321 (1854).

88 PHYSICAL CHEMISTRY

of state in the gas as it passes at a slow constant rate through the
uniformly porous plug. Referring to Fig. 7, consider sections
through the plug, and fix attention on one where the pressure
on the right side is pi and the volume v\. As the gas flows, the
pressure changes to p\ + Ap and the volume to Vi + Ay. The
gas in each thin section does work on the section ahead, and we
have the following difference for the work done upon the gas:

P&i — (Pi + Ap)(yi + Ay)

or, since we consider work done by the system as positive,
w = p Ay + v Ap + Ap • Ay

In the limit of infinitely thin sections, there is obtained for the
element of work, products of small quantities being dropped, the
expression

dw = d(pv)

This equation applies to a process where heat has no access to
the system, and hence — dw must equal the energy change in the
gas, dE. We obtain therefore the special thermodynamic equa-
tion for the Joule-Thomson effect,

dE = ~d(pv)

This equation may be integrated, and the following relation is
obtained for the conditions before and after the plug, as repre-
sented in Fig. 7 :

Eg + (pv)v = Ef + (pv)f

The quantity that it is desired to obtain from the Joule-
Thomson experiment is the change of temperature in relation to
the corresponding change in pressure, that is, dT/dp under the
condition that H or (E + pv) is constant. The following exact
equation1 is valid:

_ T(dv/dT}p - v

We see that a qualitative statement about the effect may be made
at once, since the heat capacity at constant pressure, CPJ is always
positive and (dv/dT)p is .positive. The sign of (dT/dp)H will

1 For its derivation, see Glasstone, "Physical Chemistry," p. 279.

PROPERTIES OF SUBSTANCES IN THE GASEOUS STATE 89

therefore be positive or negative according as T(dv/dT)p is
greater than v or less than v, and (dT/dp)H will be zero when
T(dv/dT)p = v. The temperature at which these quantities are
equal is the inversion temperature; and unless a compressed gas
is cooled below this temperature, its expansion through a porous
plug will not produce further cooling.

The Joule-Thomson coefficient for various gases is commonly
recorded1 in degrees centigrade per atmosphere of pressure
change, dT/dp, and is positive when cooling takes place, since
the pressure always decreases in these experiments. The coeffi-
cient depends upon the initial pressure and the initial tempera-
ture of the expanding gas, as shown by the data for carbon dioxide
in Table 10.

TABLE 10. — JOULE-THOMSON COEFFICIENTS, (dT/dp)a, FOR CARBON

DIOXIDE 2

\atrn

100

140

300
200

0 2650
0 3770

0 2425
0 3575

0 2080
0 3400

0 1872
0 3150

0 1700
0 2890

100

0 6490

0 6375

0 6080

0 5405

0 4320

0 7350

0 7240

0 6955

0 5973

0 4050

0 8375

0 8325

0 8060

0 6250

0 2625

0 9575

0 9655

0 9705

0 2620

0 1075

1 1050

1 1355

0 1435

0.0700

0 0420

1 2900

1 4020

0.0370

0.0215

0 0115

Liquefaction of Gases. — At high pressures, and especially at
low temperatures, the cooling effect available from a Joule-Thom-
son expansion may be quite large. By employing an insulated
expansion apparatus in which efficient heat interchange takes
place between the outgoing expanded gas and the entering high-
pressure gas, sufficient cooling may occur to cause liquefaction.
Since the gases are warmed by compression, it is advantageous
to cool the compressed gas by passing it through refrigerated
tubes before the cooling effect of expansion takes place. There

1 See ''International Critical Tables," Vol. V, p. 144, for data.

2 ROEBUCK, MURRELL, and MILLER, J. Am. Chem. /Soc., 64, 400 (1942).

PHYSICAL CHEMISTRY

is a "critical temperature" for each gas, above which no liquid
forms under any pressure, and for ordinary gases thi§ iff ar below
room temperature. For example, the Critical temperature of
oxygen is — 118°C., and even at this low temperature the pressure
required for condensation is about 50 atm. In the manufacture
of liquid air, if the compressed air enters the expansion chamber
at about 200 atm. and 0°, during its expansion to atmospheric
pressure the temperature falls to —182° and about 11 per cent
of the air liquefies. By cooling the compressed air to —50°
before expansion takes place, the yield of liquid is approximately

doubled.

The liquefying apparatus is
in principle a special porous-plug
apparatus (see Fig. 8) in which
heat interchange is brought
about between the expanded gas
and the incoming high-pressure
gas. It will be assumed that the
apparatus is so insulated as to
prevent heat flow and that the
low-pressure outgoing gas is
brought to exactly the same
temperature as the incoming
high-pressure gas by the heat interchanges Under these condi-
tions we are dealing with a constant-enthalpy process, but we
must consider the fluid in the three states, high-pressure gas at
Pstart and Tatart, liquid at pua and T7^, and exit gas at pe*it,
T^i = 778tart. Let x represent the fraction of the incoming gas
that becomes liquefied. We may then write the enthalpy-balance
equation as follows:

Lowpressurei
9™

Heat
fnterchcinger

High pressure

-Insulation

FIG. 8.

#start = H^X + jff«t(l - X)

Solving for x, the simple equation is obtained,

Hex* - #]

liq

A larger cooling effect may be obtained in the production of
liquid air by expanding the cold compressed air in an engine,
and so decreasing the energy content of the gas through the
performance of work. The Claude method employs this pro-

PROPERTIES OF SUBSTANCES IN THE GASEOUS STATE 91

cedure, which has some theoretical advantages over the method
based on overcoming molecular attraction alone, such as opera-
tion at a lower pressure and greater efficiency in operation.
There are practical difficulties in its operation, of which the design
and proper lubrication of an engine running at a low temperature
may be mentioned. In actual practice there is not much differ-
ence between the efficiencies of the two methods.

Mixtures of gases that have been liquefied may be separated
by fractional distillation in the same way as other liquids are
fractionated. The operation requires careful control of tempera-
tures, but it is in common use for the preparation of industrial
oxygen, nitrogen, argon, and neon. One other striking example
of its application is in the separation of helium from natural gas,
most of which contains not more than 1 per cent of helium.

Deviations from the Ideal Gas Law.— The simple equation for
ideal gases, pv = nRT, is not valid at high pressures, and many
expedients have been suggested for taking the variations into
account. One common procedure is to add terms in increasing
powers of the pressure and determine empirically the numerical
values of the coefficients from the measured pressure of the gas.
Such equations contain parameters that are coefficients of the
pressure terms valid for a given temperature but are different at
different temperatures. The equations for oxygen will be a
sufficient illustration. Upon taking pv = 1.000 at 0° and 1 atm.?
the pv product at 0° for any other pressure (in atmospheres) is

2^273 = 1.0010 - 0.000994p + 0.00000219?2

and at 20°C. the pv product for any pressure, again upon taking
pv — 1.000 at 0° and 1 atm., is

2^293 = 1.07425 - 0.000753? + 0.00000150?2

Another common procedure is to include terms that allow for
molecular attraction and " incompressible " volume, or that part
of the volume which is not reduced by increased pressure. Of
the many such equations proposed (probably more than a hun-
dred), we consider a few that are typical of them all.

van der Waals' Equation. — Since liquids, in which the molecules
are much closer together than in gases, are very slightly com-
pressible, it seems reasonable that compression of a gas changes
only the volume of free space between the molecules, At high

92 PHYSICAL CHEMISTRY

pressures this " volume of the molecules," or "incompressible
volume/' becomes a considerable portion of the total volume;
therefore, a better representation of the observed compressi-
bility of a gas is obtained by writing

p(vm - b) = RT

in which 6 is understood to be a volume correction, not the volume
that the molecules would have in the liquid state

The Joule-Thomson coefficients indicate a " cohesive pressure'7
that is overcome during expansion at the expense of energy, and
thus a correction for attractive forces is evidently required. It
would have the same effect qualitatively as an increase in pres-
sure, which may be indicated by writing an equation of the form

(p + A)(vm- 6) = RT

The Joule-Thomson coefficients in Table 10 show that the cohe-
sive pressure decreases with rising temperature for a given pres-
sure, which indicates that the cohesive pressure is a function of
the volume. This is supported by the known fact that the
deviations of actual gases from the ideal law become smaller as
the pressure becomes smaller, whereas, if A is a constant, its
importance would become greater relative to p at lower pressures.
If we consider the layer of molecules about to strike a given wall
at any instant of time, we see that the attraction holding them
back will be proportional to the number of molecules attracting
them. Since the number about to strike at any instant is also
proportional to the number present, it follows that this attractive
force is proportional jfco the square of the density of the gas or
inversely proportional to the square of the volume occupied by
a mole of gas. Our equation may then be written with a/vz in
place of A9 when we have1

+ £ij ("- ~b)=RT (15)

This is van der Waalb' equation for the behavior of a mole of
gas, though the argument on which it is based is not the same as
that used in its original derivation.

1 The equation will usually be required in this form. When any quantity
other than a mole is involved, the equation for n moles of gas is

[P + ° firYl (»-«&)= nRT

PROPERTIES OF SUBSTANCES IN THE GASEOUS STATE 93

In order to show the meaning of the equation more clearly, it*
is sometimes written in the form

P =

Vm —

in which the first term is the "thermal" pressure and the second
is the "cohesive" pressure. An increase in b relative to vm at a
given temperature would obviously increase the thermal pressure
above RT/vm, and a decrease in vm would increase the value of
a/vw2.

The equation of van der Waals is more difficult to handle than
is the ideal gas lawT; it is a cubic in v, and it contains character-
istics of the particular substance. The best means of determining
the numerical values of a and b is through two measurements of
pressure and volume for a substance at a known temperature.

TABLE 11. — VAN DEK WAALS' CONSTANTS a AND b
(For pressures in atmospheres and molal volumes in milliliters)1

a/vmz, atm., when

vm is

Substance

Ml per

mole

500 ml.

5000 ml

0 19 X 106

23 0

0 76

0 008

1 36 X 106

31 6

5 44

0 054

1 31 X 106

37 3

5 24

0 052

C02

3 61 X 106

42 8

14 4

0 14

1 43 X 106

38 6

5 72

0.057

S02

6 69 X 106

56.5

26.8

0.27

C2He

60 X 10G

69.9

24 0

0.24

H20

5 87 X 106

33 2

23.5

0.23

NH3 . .

4 05 X 106

36.4

16.2

0.16

For example, when the molal volume of C02 is 1320 ml., the
pressure is 15.07 atm. at 273°K. and 18.40 atm. at 321°K. Upon
substituting these measured quantities into van der Waals' equa-
tion and solving for values of a and b that satisfy these condi-

1 For other data see Z. physik. Chem , 69, 52 (1910), and "Landolt-Born-
stein's Tables/' pp. 253-263, 1923. Since the unit of volume used in these
tables is a molal volume at 0° and 1 atm., the values of a given there should
be multiplied by (22,400) 2 and those for b by 22,400, if they are to correspond
to the units used in this table.

94 PHYSICAL CHEMISTRY

.tions, we find a = 4.6 X 106 and b = 47 ml. per mole. But if
this process is repeated with other data for C02, somewhat dif-
ferent values of a and b are obtained, which shows that van der
Waals' equation is not a complete representation of the prop-
erties of gases. It will readily be seen that the values of a and b
that apply to C02 do not apply to some other gas, such as NH3
or SO2, since the volume and attractive force depend upon the
substance. Data for various gases will be found in Table 11

Many of the recorded data for a and b are derived from the
critical constants through a "reduced" equation of state that
will be given in the next chapter. The quantities so derived are
less suitable for pressure calculations at temperatures and pres-
sures far removed from critical conditions than are a and b based
on actual gas densities, since van der Waals' equation is not valid
in the critical region.

The experimental facts (1) that a is not zero or negative for
hydrogen and (2) that the Joule- Thomson expansion of hydrogen
is attended by a rise in temperature show that b is not alone a
volume correction but that repulsive forces of some kind are
involved.1

For a constant molal volume the cohesive pressure a/vm~ in
van der Waals' equation has the same value for a given gas at all
temperatures, and for all temperatures and pressures the " incom-
pressible volume" correction has the same value for a given
substance. It seems more probable that these corrections are
temperature functions, rather than constants, and the devia-
tions of calculated pressures from observed pressures also show
that some further corrections are required. The equation is a
second approximation that indicates the type of correction
needed but furnishes inadequate correction. If the same a and
b are used over wide ranges of temperature and pressure, van der
Waals pressures are sometimes in error more than ideal gas
pressures; but in general a pressure calculated from van der

1 For a change of pressure from pi atm. to p% atm., the temperature change
in a Joule-Thomson expansion is, nearly, AT7 = ( j£jz — b J ( — ^-^) if the

van der Waals equation is accepted. It will be seen that AT7 is zero only
when 2a/RT — b. The " inversion" temperature for hydrogen is about
— 80°C., while that for most other gases is above room temperature. Thus,
at temperatures below — 80°C. hydrogen is cooled by expansion as is true
of other gases.

PROPERTIES OF SUBSTANCES IN THE GASEOUS STATE 95

Waals' equation will be more nearly correct than a pressure calcu-
lated from the ideal gas law.

As an illustration, we calculate the pressure for ethyl ether at
303°C. when the volume is 2120 ml. per mole and for which the
measured pressure is 20.4 atm. Using the values of a and b in
Table 11, we calculate from van der Waals' equation that the
pressure is 20.3 atmv and from p = RT/vm we calculate the pres-
sure to be 22.3 atm. Thus the ideal gas law pressure is 9 per
cent above the actual pressure, and the van der Waals pressure
is 0.5 per cent less than the actual pressure. Some other calcu-
lations involving these equations are shown in Table 13.

When the pressure becomes small and the volume of a mole
of gas correspondingly large, the term a/vm2 becomes so small in
comparison with p that it may be neglected; also, the volume 6
is negligible in comparison with the molal volume vm, and it may
be neglected. The equation of van der Waals thus reduces to
the simple gas law at large molal volumes

Key^s's Equation.1 — This equation, which agrees quite well
with observed experimental data, may be written

(16)

where a and I are constants characteristic of each substance and
the logarithm of 6 is a function of the volume, 6 = /3e~a/v. Cal-
culations based on this equation are rather difficult to carry out,
but the agreement between observed and calculated pressures is
excellent.

The Beattie-Bridgeman Equation of State. — When it is neces-
sary to calculate pressures to within a few tenths of 1 per cent,
the Beattie-Bridgeman equation2 is recommended. It is

„ Brj- 0 -L ? jl 8 m\

P = 7- + ri + — 3 + — 4 (17)

Vm Vm Vm Vm

1 KEYES, F. G., Proc. Nat. Acad Sci , 3, 323 (1917)

2 J. Am. Chem. Soc , 49, 1665 (1927); 60, 3133 (1928); Proc. Am. Acad.
Arts SCL, 63, 229 (1928). A close approximation when volumes are to be
calculated is

in which the Greek letters have the same significance as in the otherform of

thp prmntirvn

PHYSICAL CHEMISTRY

in which the Greek letters represent constants and temperature
functions as follows :

R KTR A R°

p — til £>o — ^o — TffZ

y = -RTBob + A0a -
_ RB0bc

In this equation R is 0.08206 liter-atm./mole-°K, vm is the molal
volume in liters, and the quantities A<», a, BQ, 6, and c are con-
stants for a given gas but different for each gas. The values of
these constants are given in Table 12.

Calculations made to check the validity of this equation show
that it agrees with measured pressures up to 100 atm., and at
temperatures of — 150°C. or above, to within 0.3 per cent or less
except near the condensation pressures for the temperatures used.
Some of the calculated pressures for carbon dioxide are given in
Table 13.

TABLE 12 — CONSTANTS OF THE BEATTIE-BRIDGEMAN EQUATION

Gas

10-<c

0.0216

0.059 84

0.014 00

0.0

0.0040

0.2125

0.021 96

0.020 60

0.0

0.101

1.2907

0.023 28

0.039 31

0.0

5.99

0.1975

-0.005 06

0.020 96

—0.043 59

0.0504

1.3445

0.026 17

0.050 46

-0.006 91

4.20

1.4911

0.025 62

0.046 24

0.004 208

4.80

Air

1.3012

0.019 31

0.046 11

-0.011 01

4.34

CO2

5.0065

0.071 32

0.104 76

0.072 35

66 00

CH4

2.2769

0.018 55

0.055 87

—0.015 87

12 83

(C2H6)2O

31.278

0.124 26

0.454 46

0.119 54

33 33

C2H4 •

6.1520

0.049 64

0.121 56

0.035 97

22 68

NH,

2 3930

0 170 31

0 034 15

0 191 12

476 87

1.3445

0 026 17

0 050 46

— 0 006 91

4 20

N2O

5.0065

0.071 32

0.104 76

0.072 35

66 00

Other Equations for Gases. — The equations that have been
given above are not the only ones that have been proposed to
represent the changes of pressure and temperature of a gas with
volume; many others have been suggested, and new ones are being

PROPERTIES OF SUBSTANCES IN THE GASEOUS STATE 97

proposed from time to time.1 A compressed gas is a complex
system in which attractive and repulsive forces operate between
the molecules and in which the "volume of the molecules " is a
function of temperature and total volume. In a dilute gas these
effects are not as important as they are in the compressed gas, of
course, but they are not negligible if high precision is desired.

TABLE 13. — OBSERVED AND CALCULATED PRESSURES FOR CARBON DIOXIDE 2
(Density in moles per liter, pressure in atmospheres)

«,°c.

Density

Actual p

14 75

28 47

53 30

75 06

94 45

100°

p = RT/v

15 3

30 6

61 2

91 8

122 4

Eq. (15)

14 74

28 3

52 5

72 8

89 5

Eq (17)

14 77

28 42

53 21

74 68

93 62

Actual p

13 45

25 69

47 01

64 77

79 50

70°

p = RT/v

14 45

28 9

57 8

86.7

115 6

Eq (15)

13 85

26 6

48 8

66 9

81 8

Eq (17)

13 46

25 74

47 07

64 68

79 37

Actual p

12 15

22 94

40 86

54 58

64 79

40°

p = RT/v

12 8

25 6

51 3

76 9

102 6

Eq. (15)

12 2

23 2

41 7

55 9

66 1

Eq (17)

12 14

22 94

40 83

54 44

64.67

Actual p

11 26

21 03

36.56

47 49

54 57

20°

p = RT/v

12 0

24 0

36.0

48.0

60.0

Eq. (15)

11 35

21 5

38 0

50 5

58 0

Eq. (17)

11 26

21 05

36.59

47.43

54.53

For engineering purposes, one may use an empirical treatment
of the data by defining a quantity

JLt =

(19)

which may be plotted against the pressure or some function of
the pressure, as was done in Fig. 4 for prdpane. Another com-
mon device i& to plot p, against the " reduced pressure/' which is
the ratio of the actual pressure to the critical pressure (the vapor

1 A review of some of these equations, with historical notes, is given in
/. Chem. Education, 16, 60 (1939).

2BEATTiE and BRIDGEMAN, Proc. Am. Acad. Arts Sd.j 63, 229 (1928).

98 PHYSICAL CHEMISTRY

pressure for the highest temperature at which condensation is
possible), for such a plot is linear for many gases. Whichever
device is used, a separate line is drawn for eacli temperature or
for temperatures at convenient intervals for interpolation.

References

Current research, on gases frequently appears in the Philosophical Maga-
zine, Proceedings of the Royal Society of London, Communications of the Physi-
cal Laboratory of the University of Leiden, \Visscnschaftlichc Abhandlungcn der
Physikahschen-Technisclien Reichscnistalt, Journal of the American Chemical
Society, Zeitschnft fur Pkysik, and Physical Review

Further treatment of the topics in this chapter may be found in books by
Glasstorie, "Text Book of Physical Chemistry," New Voik, 1940, arid
Kennard, "Kinetic Theory of Gases," McGraw-Hill Book Company, Inc ,
1938.

Problems

Numerical data for solving some of the problems must be sought in tables in
the text.

1. (a) Calculate the volume of a balloon with a lifting power of 400 kg
at 20° and 1 atm , if the balloon is filled with hydrogen (6) Repeat the
calculation for helium as the gas filling the balloon (c) Calculate the vol-
ume of the helium balloon in the stratosphere at — 60° C and 0 1 atm.

2. When air is passed through a bed of iuel, part of the oxygen reacts to
form CO and 1he remainder to form CO.., and a molal volume of the emerging
gas weighs 29 grams. Assume air to contain 21 mole per cent oxygen and
79 mole per cent nitrogen, arid calculate the composition of the emerging
gas.

3. When 0 00413 mole of bromine is introduced into a flask of 1050 ml.
volume at 300°K. containing NO at an initial pressure of 0 229 atm., a
chemical reaction as shown by the equation 2NO + Br2 = 2NOBr takes
place incompletely, arid the final pressure becomes 0 254 atm (a) What
fraction of the NO originally present has formed NOBr? (b) What is the
partial pressure of the residual bromine vapor? (c) When this same mixture
is heated to 500°K. in the same flask, the total pressure becomes 0 529 atm.
Under these conditions what fraction of the original NO is combined with
bromine?

4. (a) Calculate the weight of air in a 200-ml. incandescent light bulb if
the pressure at 20°C. is 1 dyne per sq cm. (6) Calculate the number of mole-
cules in the bulb.

6. (a) Calculate the velocity of oxygen molecules in air at 25°C. (b)
Calculate the velocity of nitrogen molecules in air at 25°C. (c) At what
temperature would the velocity of oxygen molecules be 1 mile per sec. (1610
meters per sec.) ? (d) At what temperature would the velocity of hydrogen
molecules be 1 mile per sec ?

6. From the data on page 72 calculate what fraction of the formic acid
vapor has reacted according to the equation 2HCOOH = (HCOOH)2 at
each of the temperatures.

PROPERTIES OF SUBSTANCES IN THE GASEOUS STATE 99

7. (a) If 100 ml. of nitrogen under a constant pressure will flow through a
given orifice in 155 sec., what is the molecular weight of a gas of which 100
ml. under the same pressure will flow through the same orifice in 175 sec.?
(b) Assuming the gas to be a mixture of nitrogen and argon, calculate the
mole fraction of argon in it.

8. Calculate the mass of 002 striking each square centimeter of a leaf
in an containing C02 at a partial pressure of 0 0010 atm at 25°C.

9. (a) Calculate the value of the gas constant R from the limiting density
data for CO2 on page 15 (b) Calculate another value of R from the
density of helium given in the footnote on page 53.

10. The ratio CP/CV for CO2 at 293°K is 1 30 for 1 atm. pressure. Cal-
culate the "internal heat capacity," the energy absorbed on heating and
not used for increasing the trarislatioiial kinetic energy of the molecules or
for doing work

11. In the manufacture of SOs by the contact process 8 0 moles of air
(assumed 21 mole per cent oxygen and 79 mole per cent nitrogen) enter a sul-
iur burner for each atomic weight of sulfur burned, and the density of the
emerging mixture of SO;, SOj, O2, and N2 is 0 605 gram per liter at 700°K
and 1 atm total pressure. Calculate the partial pressures of 80s, S02, and
O2 m the mixture

12. A capsule containing 0 356 gram of a solid was dropped into a Victor
Meyer bulb at 400°C , expelling 33 2 ml of air, measured over water at 20°
and 1 atm total pressure Calculate the molecular weight of the substance
at 400°

13. When 6 40 grams of S02 and 4 26 grams of chlorine are introduced into
a 3-hter flask, partial union as shown by the equation SO2 + C12 = SO2C12
takes place, and the total pressure at 463°K becomes 1.69 atm Calculate
the partial pressure of each gas m the mixture.

14. The vapor of acetic acid contains single and double molecules in
equilibrium as shown by the reaction (CH3COOH)2 ?=» 2CH3COOH. At
25° and 0 020 atm pressure the pv product for 60 grams of acetic acid vapor
is 0 541/("F, and at 40° and 0 020 atm. it is 0.593ft!7. Calculate the fraction
of the vapoi forming single molecules at each temperature. (Ans.: 0.186
at 40°) [MAcIJouGALL, / Am Chem. Soc., 58, 2585 (1936).]

16. A glass bulb of 373 ml. volume, with a long capillary stem, weighs
29 450 grains when open to the air at 20° and 1 atm. In a molecular-weight
determination by Dumas's method an excess of a volatile liquid is placed in
the bulb, which is then heated m boiling water until the air and the excess of
substance are expelled. The bulb is sealed and after cooling is found to
weight 30 953 grams. Calculate the molecular weight of the substance.

16. One rnole of ethane (C2He) is exploded with 15 moles of air, and the
products are cooled to 320°K. and 1 atm. total pressure. Assume that air
is 21 mole per cent oxygen and 79 mole per cent nitrogen, that the only sub-
stances present are CO(0), CO2(0), N2(0), H2O(gr), and ELO(Z), that all the
gases are ideal, that the volume of condensed water is negligible, and that
the vapor pressure of water at 320°K. is 0 10 atm. Calculate the volume
of the mixture, the weight of condensed water, and the partial pressure of
each of the gases.

100 PHYSICAL CHEMISTRY

17. When 0 296 mole of iodine is added to a space of 34.6 liters at 422°K.
containing 0 413 mole of NOC1, partial reaction as shown by the equation

2NOCl(flO + I2(<7) - 2NO(0) + 2101(0)

takes place and the final pressure becomes 0.866 atm. Calculate the partial
pressure of NO in the final mixture.

18. Lead nitrate decomposes on heating according to the chemical equa-
tion Pb(NO3)2(s) = PbOO) -f N2O4(0) + MO2(0). When the gaseous
products are brought to 323°K , 45 per cent of the N2O4 is decomposed into
NO2 and the partial pressure of oxygen in the mixture is 0 184 atm. (a)
Calculate the partial pressures of NO; and N204. (&) Calculate the weight
of a liter of the gaseous mixture at 323°K.

19. When 3 atomic weights of phosphorus and 7 moles of chlorine are
brought together at 523°K. the phosphorus is completely converted to a
mixture of PCls and PC10. At a final total pressure of 5 atm. 55 per cent of
the phosphorus is in the form of PC13. (a) Calculate the density of the
mixture in grams per liter at 523°K. and 5 atm. pressure. (6) Calculate
the partial pressure of chlorine in the mixture.

20. When 1 mole of N2 and 1 mole of H2 react to equilibrium at 623°K ,
the chemical reaction N2 + 3H2 = 2NHS takes place incompletely and the
density of the mixture is 3.10 grams per liter at a final total pressure of
10 atm. (a) What is the partial pressure of ammonia? (6) What fraction
of the hydrogen reacted?

21. (a) Calculate the molecular volume of carbon dioxide at 70°C. and
23.56 atm and from this the specific volume in milliliters per gram, assum-
ing it to be an ideal gas. (b) Calculate the molecular volume under these
conditions by means of van der Waals' equation, solving the cubic by trial
and using the measured specific volume, 25 ml. per gram, as a first estimate

22. The pressure in a liter flask containing 0.500 gram of NO2 changes
with the temperature as follows:

T, °K - 521 615 658 714 795 820

p, atm. . .0488 0628 0.705 0.81(1 0.965 1.000

The deviation of this pressure from that to be expected of NO2 as an ideal
gas is due to the incomplete chemical reaction 2NO2 = 2NO + O2. (a)
Plot the observed pressure against the absolute temperature, and show the
pressure to be expected of undissociated NO2 by a dotted line on the same
diagram. (6) Derive a relation between the pressure to be expected of the
undissociated gas, the increase over this pressure, and the fractional dissoci-
ation; and apply this relation to the diagram to determine the fraction dis-
sociated at each temperature, (c) Calculate the partial pressure of each
substance in the mixture at 820°K.

23. When a mixture of 2CS2 and 5C12 is heated, 90 per cent of the chlorine
reacts as shown by the equation CS2(gr) -f 3C12(0) - CC14(00 -f S2C12(0).
Calculate the volume of the resulting mixture at 373°K. and 1 atm. total
pressure, and the partial pressure of each gas in the mixture.

PROPERTIES OF SUBSTANCES IN THE GASEOUS STATE 101

24. When a mixture of 2CBU and 1H2S is heated, the reaction

CH4(0) + 2H2S(<7) = CS2(0) + 4H2(0)

takes place incompletely, and the final volume is 259 liters at 973°K. and
1 atm. Calculate the partial pressure of each gas, in the mixture.

25. (a) Calculate the pressure in atmospheres at which ammonia has a
specific volume of 50 0 ml per gram at 200°C., assuming it an ideal gas.
(b) Recalculate the pressure from van der Waals' equation. The meas-
ured pressure under these conditions is 41 9 atm.

26. The total pressure in a liter flask containing 1.159 grams of N2O4 is
0 394 atm at 25°, 0 439 atm at 35°, and 0 489 atm at 45°. Practically
all the deviation from the ideal gas law is due to the incomplete dissociation
of N2O4 into NO2 Calculate the extent of this dissociation at 25° and at
45°. [VERHOEK and DANIELS, /. Am. Chem. Soc., 53, 1250 (1931).]

CHAPTER I\
PROPERTIES OF SUBSTANCES IN THE LIQUID STATE

The purpose of this chapter is to consider the vapor pressures,
surface tensions, latent heats, viscosities, critical constants, and
other properties of liquids. Since all gases may be changed to
liquids by suitable changes in temperature and pressure and
many liquids may be changed to gases or solids, it is evident
that a liquid is only a substance in the liquid state under certain
conditions. Under other conditions it may be a solid or a gas,
and under suitable conditions a liquid may exist in equilibrium
with both solid and vapor of the same composition or with either
phase in the absence of the other. There is for every vapor a
certain " critical temperature" above which it may not be con-
densed to liquid under any pressure. This critical temperature
is 374.2°C. for water vapor, 31.1°C. for carbon dioxide, - 118.7°C.
for oxygen, and some characteristic temperature for every vapor.
Below this critical temperature and above the "triple point" at
which solid, liquid, and vapor are in equilibrium, there is for each
temperature a single pressure at which liquid and vapor may be
in equilibrium. This " vapor pressure" is also different for each
substance at a given temperature; it is 57.0 atm. for carbon
dioxide at 20° and 0.0231 atm. for water at 20°; and since 20° is
above the critical temperature of oxygen, no pressure, however
great, will cause oxygen to liquefy at 20°C.

Substances in the liquid state have greater densities, greater
internal friction, larger cohesive pressures, and much smaller
compressibilities than they have in the gaseous state. Many
of the changed properties are due to greater attractive forces
acting between the molecules. The molecules probably have
the same kinetic energies as those characteristic of the gaseous
state at the same temperature ; but they have much shorter paths
between collisions, much less freedom of motion, and much
greater damping effects upon their motion. In contrast to the
crystalline state that most liquids assume at still lower tem-

102

PROPERTIES OF SUBSTANCES IN THE LIQUID STATE 103

peratures, liquids have no shape, no form elasticity but only
internal friction. They usually have larger compressibilities,
larger temperature coefficients of expansion, higher specific heats,
and smaller densities than the substances in the crystalline form.

Our knowledge of the liquid state is much less complete than
that of the gaseous state or the crystalline state, in spite of
diligent study by competent physicists and chemists for many
years. No experimental measurements yet enable us to calculate
directly the attractive forces that cause condensation of a vapor
to liquid or the cohesive forces between molecules in a liquid,
but experiments on the angle and intensity of scattering at a
given angle for molecular or atomic beams appear to be promising.
It is estimated that the attractive force between molecules varies
inversely as the seventh power of the distance between nonpolar
molecules, but at close approach there are also repulsive forces
acting between them. Such " calculated" attractions as we have
rest upon assumptions of uncertain validity. From a review of
numerous papers attempting to correlate the properties of liquids
or to calculate some of their properties, Herzfeld1 finds that cal-
culations often disagree with measured properties by 50 per cent
to fourfold. Evidently new experimental methods are urgently
needed. We turn to a brief consideration of some of the experi-
mental facts and such interpretations as are available

Liquid Solubilities. — Although gases mix with one another in
all proportions without seriously influencing the properties of
each gas (such as partial pressure), this is not true of all liquids.
Some pairs of liquids, such as alcohol and water, chloroform and
carbon tetrachlonde, benzene and xylene, do mix in all propor-
tions; other pairs, such as aniline and water or ether and water,
mix only to a limited degree; still others, such as benzene and
water or alcohol arid mercury, do not dissolve in each other to an
appreciable extent In the gaseous state all these substances
mix in all proportions, but this is doubtless because of the greater
separation of the molecules and the consequent lack of strong
forces acting between them. In the liquid state, where molecules
are very close to each other, specific attractive forces act between
them, and these forces seem to govern the extent to which one
liquid will dissolve in another. No general rules for solubility
of liquids are free from exceptions, but it is usually true that

1 /. Applied Phys , 8, 319 (1937); 43 references to recent work.

104 PHYSICAL CHEMISTRY

liquids of the same chemical type (two hydrocarbons, two liquid
metals, or water and alcohols) are soluble in each other, while
liquids of quite different natures exhibit slight attractions for
each other. Thus when benzene (Celle) dissolves in toluene
(CrHs), the attractive forces between molecules are probably
changed but little, because of the chemical similarity of the
substances.

"Slightly soluble " liquids usually increase in solubility as the
temperature rises; they often become completely soluble in one
another at a sufficiently high temperature, but this temperature
may be above the boiling point of the mixture for 1 atm. pres-
sure. At 20°C. a saturated solution of phenol in water contains
about 8 per cent phenol; when a larger percentage is present, a
second liquid layer containing 72 per cent phenol and 28 per cent
water is in equilibrium with the solution containing 8 per cent
phenol and 92 per cent water. With rising temperature the
compositions of the two layers approach one another, and above
66.8°C. the liquids mix in all proportions to form a single solu-
tion. Water and aniline also form two layers, which at 100°C.
contain, respectively, 7 2 and 90 per cent aniline by weight.
Complete solubility of each in the other is reached at 167°C.
with the application of sufficient pressure to prevent evaporation.

Liquid solubilities also change slightly with pressure at con-
stant temperature, but climatic variations in atmospheric pres-
sure produce only negligible changes. Application of 100 atm.
pressure raises the critical solution pressure of phenol in water
by about 4.6°, and the effect of pressure upon other systems is
likewise small.

Vapor Pressure. — The v»por pressure of a pure liquid is that
pressure at which the liquid and vapor are in equilibrium. This
equilibrium pressure, or saturation pressure, is a function of the
temperature alone and is independent of the relative quantities
of liquid and vapor present. Different liquids have different
vapor pressures at a given temperature, and the vapor pressures
change with temperature at different rates; but for a given pure
substance at a given temperature there is only one pressure at
which liquid and vapor are in equilibrium. If the volume of a
vapor is gradually decreased at a constant temperature that is
below the critical temperature, the pressure increases until the
vapor pressure for that temperature is reached; after this further

PROPERTIES OF SUBSTANCES IN THE LIQUID STATE 105

decrease in volume at constant temperature causes more conden-
sation to liquid, and no increase in pressure is observed until
condensation is complete.

Mixtures of liquids, and solutions in general, also have vapor
pressures; but they depend upon the nature and relative propor-
tions of the substances in the solution at a given temperature.
We shall consider these vapor pressures in a later chapter, but
we are now considering only the vapor pressures of pure liquids —
one-component systems in which liquid and vapor have the same
composition and exist together at a single pressure for a fixed
temperature.

Vapor pressures of readily purified substances may be used to
calibrate pressure gauges. For illustration, the equilibrium
pressure between liquid C02 and its vapor at 0°C. is 34.041 atm.;
and since 0°C. is the most readily reproduced standard tempera-
ture, a pressure gauge that does not read 34.041 atm. for the
vapor pressure of CO2 packed in a mixture of ice and water is in
error by the amount its reading deviates from this pressure.

In the absence of liquid, the pressure of a vapor may be any-
thing less than the vapor pressure for the prevailing temperature
and retain this value indefinitely. Thus the difference between
a vapor pressure and the pressure of a vapor is neither a pedantic
distinction nor a play upon words; it is an important difference
that must be clearly understood. An illustration or two may be
helpful. Consider a flask of 12.045 liters at 50°C. containing a
gram of water, which exerts a pressure of 0.1217 atm. Since
this pressure is the vapor pressure of water at 50° and this volume
is the specific volume of saturated water vapor at 50°, we have a
pressure of water vapor equal to the vapor pressure. If we
double the volume occupied by a gram of water vapor, the pres-
sure of water vapor will become 0.0607 atm. ; we no longer have
saturated vapor, but the vapor pressure of water at 50° is still
0.1217 atm. If we increase the temperature to 70° and keep the
volume 12.045 liters, the pressure will become 0.129 atm. but
this pressure is not the vapor pressure of water at 70°C. or a
quantity from which it may be calculated. The vapor pressure
of water at 70°C. is 0.3075 atm., a pressure found by experi-
ment upon water in equilibrium with its vapor at 70°C. If we
ccfcl the flask to 20°C., part of the water vapor will condense and
tjfre pressure of water vapor at equilibrium is 0.02307 atm.,

106 PHYSICAL CHEMISTRY

which is also the vapor pressure of water for this temperature.
Doubling the volume at 20° would not evaporate all the gram of
water, and therefore the pressure of water vapor and the vapor
pressure would still be the same. But if the volume were
increased beyond 57.87 liters (the specific volume of saturated
vapor at 20°C.), the pressure of water vapor would decrease as
indicated by the gas laws and would no longer be equal to the
vapor pressure.

In the presence of air or of any inert slightly soluble gas at
low pressure, the equilibrium pressure or saturation pressure of
a liquid is substantially the same as its vapor pressure in the
absence of the gas. Thus, in a mixture of 0.023 mole of water
vapor and 0.977 mole of air at 20° and 1 atm total pressure, the
partial pressure of water vapor is the same as its vapoi pressure.
If this mixture is heated to 30° at 1 atm., the partial pressures
are unchanged; but since the vapor pressure of water at 30° is
0.0419 atm., the air at this temperature is 55 per cent saturated.
Two other common expressions for the moisture content of the
mixture at 30° are that the relative humidity is 55 per cent and
that the dew point is 20°C.

Equilibrium between a liquid and its vapor, like any other
condition of equilibrium, is not a stationary state but a condi-
tion of reactions at equal rates in opposite directions. Thus at
20° the pressure exerted by a gram of water in a volume of 1
liter, or 10 liters, or 50 liters is 0.02307 atm., but at each volume
we must suppose that water is evaporating and water vapor is
condensing at the same rate to keep this pressure constant. If
the volume is quickly 3ecreased, there is a temporary increase
in pressure, which increases the rate of condensation while the
rate of evaporation remains constant; and with the passage of
time the pressure returns to 0.02307 atm. after the removal of
enough heat to restore the temperature to 20°.

Measurement of Vapor Pressures. — In theory the measure-
ment of a vapor pressure over a range of temperatures is a very
simple operation; namely, one measures on a gauge the pressure
under which liquid and vapor exist at equilibrium for each tem-
perature. But there are many experimental difficulties in carry-
ing out this simple operation in such a way as to yield precise
data. Removal of the last traces of dissolved air from a liquid
(which requires prolonged shaking with periodical pumping out

PROPERTIES OF SUBSTANCES IN THE LIQUID STATE 107

of air, followed by repeated distillation under very low pressure)
is necessary if the gauge is to show the pressure of vapor alone
and riot the pressure of vapor plus air. Containers and precise
gauges that will withstand high pressures, be inert to the liquid,
and possess the requisite mechanical, thermal, and elastic prop-
erties are difficult to design and construct. All these problems
have been solved, and reliable vapor-pressure data are available
for water and most of the fluids used in refrigeration. Vapor
pressures of most of the common liquids at temperatures below
their boiling points have also been measured, but one must
exercise some judgment in selecting data, for some of the pres-
sures were measured before the experimental difficulties involved
were fully appreciated.

Vapor pressures for several substances are given in Table 14.

Air -bubbling Method. — Vapor pressures of liquids at tempera-
tures well below their boiling points may be measured with fair
precision by saturating a known quantity of air or nitrogen
with the liquid, passing the mixture of air and vapor through
an absorbing agent, and weighing the absorbed vapor. For
example, if 10 liters of air at 2()°C. and 1 atm. are bubbled through
several tubes of water at 20 °C and the water in the saturated
air is absorbed in sulfuric acid and weighed, it will be found that
0.178 gram of water saturated the air. Reducing these figures to
moles, 0.416 mole of air and 0.00984 mole of water vapor emerged
from the saturating vessel at 20 °C. and 1 atm. The partial
pressure of water vapor in the mixture, on the basis of Dalton's
law of partial pressures, is 0.00984/0.4258 = 0.023 atm., which
is also the vapor pressure of water at 20°. If the experiment is
repeated at 25°, 0.0132 mole of water vapor will saturate 0.416
mole of air and the vapor pressure will be found to be 0.031 atm.
But one may not find by this method that the vapor pressure of
water is 0.02307 atm. at 20° and 0.031254 atm. at 25°, no matter
how carefully the experiments are performed, for the ideal gas
law does not apply to this mixture of gases with the requisite
precision.

Change of Vapor Pressure with Total Pressure. — In the air-
saturation method of measuring vapor pressure, the total pres-
sure acting on the liquid phase is 1 atm., while at equilibrium in
an evacuated space the pressure on the liquid is only 0.023 atm.
at 20°. There is a very slight increase of vapor pres&ure caused

108

PHYSICAL CHEMISTRY

TABLE 14 — VAPOK PRESSURES OF LIQUIDS
(In millimeters)1

t,°c.

H20

CC14

C2H6OH

Ethyl
ether

CeH6

n-C8H18

SO2

9 21

23 6

291 7

5 62

2 256a

12 79

32 2

360 7

17 53

43 9

442 2

10 45

3 288a

23 75

113 8

59 0

537 0

31 82

141.5

78 8

647 3

119 6

18 40

4 498a

42 17

174 4

103 7

775 5

148 2

55 31

213.3

135 3

1.212a

182 7

30 85

6 125a

71.86

258 9

174 0

223 2

92.50

312 0

222 2

1.680a

271 3

49 35

8.176a

118 04

373 6

280.6

340 7

149.38

444 3

352.7

2 275a

391 66

77 55

10 73a

233.69

617 43

542 5

3.021a

551 0

117 9

13 87a

355.18

l.lOa

812.6

3.939a

757 6

174 8

17 68a

525 82

1 46a

1.562a

5 054a

1 42a

253 4

22 27a

100

l.OOOa

1.92a

2.228a

6 394a

1 76a

353 6

27 71a

110

1 414a

2 47a

3 107a

7 987a

2 29a

34 09a

120

1 959a

3 20a

4 243a

9 861a

2 93a

41 43a

130

2.666a

3 95a

5.685a

12.05a

3.71a

49 70a

TABLE 15. — VAPOR PRESSURE, VOLUME, AND AH FOR WATER

Vapor

Specific volume of

dp/dT,

pressure,

T, abs.

atm. per

A#, cal.

atm.

liquid

vapor

degree

per gram

0 1217

323

1 0121

12045.0

0 006039

568.9

1.0000

373

1 0434

1673 2

0 0357

539.0

4.6977

423

1.0906

392 46

0 1260

504 9

15 352

473

1.1565

127.18

0 3211

463.3

39 256

523

1 2512

50 06

0 6629

409.6

84 776

573

1 4036

21.62

1.1942

334.9

163.164

623

1.7468

8.802

2.0031

213.2

218 5

647.3

3 15

3.15

1 Pressures marked a are in atmospheres. The data for water are from
Smith, Keyes, and Gerry, Proc. Am. Acad. Arts Sri., 69, 137 (1934); for
CCU and CeH6 below 1 atm. from Scatchard, Wood, and Mochel, /. Am.
Chem. Soc , 61, 3206 (1939); for other substances from " International
Critical Tables. " Some additional data for water are

p, mm . ,

16
13.63

17 18
14.53 15 48

19 21
16.48 18 65

22 23 24
19 83 21 07 22 38

PROPERTIES OF SUBSTANCES IN THE LIQUID STATE 109

by this increase of pressure on the liquid. The equation for this
increase is1

a/Vr ~ T, (1)

where p is the vapor pressure, P the total pressure, Vi the molal
volume of the liquid, and vg the molal volume of saturated vapor
at T. The equation may be integrated between limits, after
separating the variables, by neglecting the slight compressibility
of the liquid and assuming vg = RT/p for the vapor. Then the
equations are

and

2.3 log £-2 = ^ (P* ~

If air at 100 atm. presses upon liquid water at 25°, the partial
pressure of water vapor in the air at equilibrium will be about
1.07 times the vapor pressure when no air is present, as will be
found when the appropriate quantities are substituted in this
equation. For air at 1 atm. in contact with water, the increase in
vapor pressure with the total pressure (about 0.07 per cent) is
commonly neglected.

, Change of Vapor Pressure with Temperature. — The vapor
pressures of liquids increase with increasing temperature, and
the increase per degree also increases as the temperature rises.
Data showing the vapor pressures of some common liquids are
given in Table 14. 2 The rate at which the vapor pressure
changes with the absolute temperature is given by the following
exact equation, called the Clapeyron equation:

1 The equation follows from equation (32) on p. 47 in view of the fact that
AF is zero for any phase change taking place isothermally at equilibrium,
since dFi then equals dFff when the pressure changes and

vp dp = vi dP (t const.)

which rearranges to give (1) above.

2 For the vapor pressures of most substances that have been studied, see
"International Critical Tables," Vol III, pp. 201-249; a review of the data
on vapor pressures of inorganic substances is given by Kelley in U.S. Bur.
Mines Bull., 383 (1935).

110 PHYSICAL CHEMISTRY

dp_ AH AH_

dT (vg-vi)T TAv w

In this equation AH is the quantity of heat absorbed in vaporizing
vi ml. of liquid to form vg ml. of saturated vapor, dp/dT is the
rate at which the vapor pressure increases with the temperature,
and Av is the increase in volume attending evaporation.

The Clapeyron equation follows from equation (31) on page
47, which was

d¥ = -SdT + vdp

We note that for the isothermal evaporation of a liquid under its
vapor pressure AF = AH — T AS = 0 from equation (33£) on
page 47 ; therefore, the free energies of liquid and vapor change
with temperature by the same amount. The equations for each
phase are

dFg = -S0dT + v(,dp
d¥i = -StdT + Vidp

and upon equating them and rearranging , we have

dp _ Sg — Si
dT ~ vg - vi

But Sg — Si = AS, which is AH/T when evaporation takes place
isothermally and reversibly; and, upon making this substitution
above, we obtain

dT TAv

Clapeyron's equation follows from the third "Maxwell rela-
tion" given on page 48; but since the system i& monovariant
when a liquid and its vapor are at equilibrium, there is only one
independent variable and the equation becomes

dp = dS = AS
dT dv Av

Upon multiplying numerator and denominator of the right side
by T and noting that T AS = A/7, the equation is then

/ON
dT TAv (6)

PROPERTIES OF SUBSTANCES IN THE LIQUID STATE 111

The Clapeyron equation may also be derived from a cycle of
changes whereby heat is transferred from one temperature to
another by a reversible cycle involving the phase change and for
which the maximum work is given by equation (13) on page 38,

, dT

where q is the heat absorbed at the higher temperature. Let
the cycle consist of the following steps: (1) Evaporate a quantity
of liquid reversibly under its vapor pressure p at T, for which the
work done is w\ = p(vg — Vi) and the heat absorbed is AIL (2;
Cool the vapor to T — dT} by this means the pressure becomes
p — dp and the volume of saturated vapor becomes vg — dvg, for
which w<i = — (p — dp)dvg or —pdvg if the second-order quan-
tity is neglected. (3) Condense the vapor to liquid reversibly
under its vapor pressure p — dp at T — dT, for which

wa = (p - dp)[(vt - dvi) - (va - dvg)]

= pvi — p dvi — pvg + p dvg — Vi dp + vg dp

if the second-order quantities are neglected. (4) Heat the liquid
to T, for which w* = p dvj. The summation of these work quan-
tities is (vg — vi)dp; and, upon substituting this quantity for dw
and AH for q in the equation above, we have

dp _ AH = AH

dT ~ (va - Vi) ~ T Av

which is again equation (3).

This equation, while derived for the change of vapor pressure
of a liquid with changing temperature, came from fundamental
equations of the second law of thermodynamics applying to any
equilibrium phase change in a system of constant composition.
We shall also use it later for the change of melting point of a
solid with pressure, for the vapor pressures of solids, and for any
change for which the pressure is a function of temperature alone.

The Clapeyron equation does not apply when the pressure is a
function of some quantity other than temperature. For example,
the pressure at which Na2C03(s)? NaHC03(s), H20(^), and
C02(0) are at equilibrium depends upon the composition of the

112 PHYSICAL CHEMISTRY

gas phase as well as upon the temperature, and tnus the Clapeyron
equation does not apply to this system at every composition.

For calculations involving equilibrium between a liquid and
its vapor at pressures near or below 1 atm. and over small ranges
of temperature, the Clapeyron equation may be put into a more
convenient form by the use of some approximations. The
derived equation is, of course, valid only to the extent that the
approximations are valid. If we assume that Vi is negligible in
comparison with v0, that va = RT/p, and that AH is a constant,
the equation becomes

dp p
-Z

,
*

in which AHm is now the molal latent heat, since &v is taken as
RT/p and not nRT/p.

A plot of In p against 1/77 for the vapor pressure of water
between 323 and 373°K. is substantially linear, and equation (4)
shows that its slope should be —AHm/R, from which we find
A#m = 10,100 cal. The true value of Affm is 10,250 cal. at
323°K. and 9700 cal. at 373°K. At higher pressures the curva-
ture of the plot becomes apparent, and larger deviations are
found. Between 473 and 573°K. saturated water vapor deviates
widely from ideal gas behavior, and A/fm changes 25 per cent.
In this range a plot of In p against \/T shows some curvature,
and the slope at 523 °K. gives AHm = 9100 cal., while the correct
&Hm at this temperature is 7370 cal. Thus the fact that the
curvature is small is not proof of the validity of the simplifying
assumptions. At thes^e high pressures the decrease in AHm is
somewhat compensated by the fact that &v is less than RT/p,
so that a plot of In p against l/T is nearly straight but of the
wrong slope. We must understand that these deviations are
due to the assumptions made in obtaining equation (4) from (3),
and not to any defect in equation (3), which is exact. If meas-
ured volumes of liquid and vapor and the correct slope of the
vapor-pressure-temperature curve at 523°K. are substituted into
equation (3), the correct AHm will be found, namely, 7370 cal.

Over moderate ranges of temperature in which the vapor pres-
sure is near or below 1 atm. the vapor pressure may be expressed
as a function of the temperature with reasonable approximation
by the integral of equation (4),

PROPERTIES OF SUBSTANCES IN THE LIQUID STATE 113

log p = - 2ZRT + C°nSt* ^

The change of Aff with temperature is usually expressed by an
equation in ascending powers of T\ therefore, for higher precision,
vapor pressures are expressed by equations of the form

log p = ^ + BT + CT* + D + - - - (6)

and the coefficients A, B, C, D are adjusted to fit the data for
the chosen units of pressure. As an illustration, the vapor pres-
sure of S02 below 273°K., in centimeters of mercury, is given by
the equation1

log p = - 186?-52 - 0.015865 17 + 0.000015574772 + 12.0754

For another substance the equation would have a different set
of numerical quantities but would be of the same form.

For some purposes the integral between limits of the approxi-
mate equation (4) is convenient. If A// is sufficiently constant
over the interval involved,2 the integral is

As the pressures appear in a ratio, PZ/PI, they may be expressed
in any units, but R = 1.99 cal. when AHm is in calories per mole.
By substituting p = 0.0946 atm. for 45°C. and p = 0.1553 for
55° in this equation, AHm for water at 50° is calculated as 10,300
cal., which is satisfactory. As another example, the vapor pres-
sure of benzene is 700 mm. at 77.43° and 777.2 mm. at 80.82°,
whence A#m = 7600 cal., which should be compared with 7600 by
direct experiment at 80.1°.

-Boiling Point. — The boiling point of a liquid is defined as the
temperature at which its vapor pressure is 1 atm. The tempera-
ture at which a liquid is observed to boil in the laboratory is a

1 GIAUQUE and STEPHENSON, /. Am. Chem. Soc , 60, 1389 (1938).

2 The change of AHm (in calories) with the temperature for water is as
follows:

t ............... 0° 50° 95° 100° 105° 200° 300°

AHm ...... 10,760 10,250 9760 9700 9640 8360 6030

114 PHYSICAL CHEMISTRY

variable quantity depending upon the existing barometric pres-
sure, and it is often necessary to apply a correction to such
observed boiling temperatures in order* to change them to
standard boiling points. This correction is usually small, but
in places of high altitude it may be several degrees; failure to
make such corrections in reporting boiling points has led to small
errors in recorded data. It is partly for this reason that the
melting point of an organic substance (which is not appreciably
affected by moderate changes of pressure) is a better guide to
its purity than the boiling point.

The rise in boiling point of a pure liquid per millimeter increase
in external pressure is nearly the same fractional amount of the
absolute boiling point for all substances, about 0.00010 In
using this approximate rule to compute a boiling point at 1 atm.
from that observed at some other pressure, one should subtract
0.00010T (p - 760) from the observed temperature. For illus-
trations, water1 boils at 100.73° under a pressure of 780 mm.,
and 20 X 373 X 0.00010 is 0.75°; benzene2 boils at 79.80° under
a pressure of 753.1 mm., and 6.9 X 354 X 0.00010 is 0.25°,
whence the calculated boiling point at 760 mm. is 80.05° and the
observed one is 80.09°.

For pressures far removed from atmospheric, this simple rule
will not give the proper correction. Thus at 525 mm. pres-
sure the boiling point of water calculated according to this
rule is 91.2°; the experimental boiling point under this pressure
is 90.0°. When it is desired to calculate boiling points at pres-
sures considerably removed from 1 atm., the approximate form
of the Clapeyron equation (7) will give results of reasonable
accuracy; thus in the example just considered, by substituting
9700 cal. for A#m, 1.99 cal. for R, 373 and 760 for T2 and p2,
525 for pi, and solving for T\, we find TI = 362.8, whence t is

1 The change in boiling temperature of water with changing barometric
pressure is as follows:

p, mm . . . 700 720 740 780 800

<, °C. .. . . 97.712 98 492 99 255 100 729 101 443

2 SMITH and MATHESON, J. Research Natl. Bur. Standards, 20, 641 (1938),
give the boiling temperature of benzene at various pressures as follows :

p, mm . . 674 4 699.6 712.6 739 4 753 1 764 8 777 2
*, °C . 76.26 77 43 78.02 79 20 79 80 80 29 80 82

PROPERTIES OF SUBSTANCES IN THE LIQUID STATE 115

calculated to be 89.7°, compared with 90° by experiment. These
calculations will illustrate the errors to be expected from the use
of these two approximate rules.

An equation giving the boiling point of water to within about
0.001° in the pressure range 700 to 830 mm. is1

t = 100 + 0.03697(p - 760) - 1.959 X 10~6(p - 760) 2

Latent Heat of Evaporation. — Recorded latent heats are usually
for evaporation at 1 atm. pressure and are written A//, so that
enthalpy increase for evaporation would be a more precise term.
For evaporation into an evacuated space qv = AJ^ = A// — A(pv),
and at moderate pressures At> is nearly the volume of vapor
formed, which is nRT/p. We shall confine our discussion to
evaporation at constant pressure, for which the latent heat is
A//.

The experimental determination of latent heats is very simple
in theory and very difficult in practice. One need only measure
the quantity of heat added to a liquid at its boiling point and
the quantity of vapor formed. But, in order to be sure that
all the added heat is used in evaporation, one must prevent heat
flow through apparatus in which temperature gradients exist or
apply corrections for them, prevent reflux of condensed vapor to
the evaporator, prevent entrainment of spray in the escaping
vapor, and prevent superheating of the vapor. If the calorimeter
is run as a condenser, one must eliminate spray without super-
heating the vapor, avoid incomplete condensation, prevent or
correct for heat flow along the condenser coil, and meet other
difficulties. All these problems have been solved2 and accurate
latent heats of evaporation for water have been measured over
a wjde temperature range, but a glance at the reference quoted
will show that much skill and patience were required.

Exact latent heats may also be obtained from the Clapeyron
equation through the use of measured volumes of liquid and
saturated vapor and from dp/dT obtained by differentiating
the vapor-pressure equation with respect to temperature. The
many experimental difficulties were troublesome in this method

1 MICHELS, BLAISSE, SELDHAM, and WOUTERS, Physica, 10, 613 (1943).

2 See for example, OSBORNE, STIMSON, "and FLOCK, /. Research Nail Bur.
Standards, 5, 411 (1930); OSBORNE, STIMSON, and GIDDINGS, ibid., 18, 389
(1937); 23, 197 (1939).

116 PHYSICAL CHEMISTRY

as well, but they have been solved;1 and the method has been
used to determine latent heats for water that agree with those
based on direct calorimetry to within 1 part in 3000. Data of
nearly as good quality are available for a few other substances
used in refrigeration over suitable temperature ranges, but most
of the recorded latent heats are for 1 atm. pressure and the
normal boiling point.2 Those based on vapor pressures or from
direct calorimetry are usually reliable to 2 or 3 per cent, but many
of the latent heats of evaporation in tables have been derived
from boiling-point changes for solutions through equations that
will be derived in Chap. VI. Some of these are also reliable to
2 or 3 per cent, but many of them are in error by something like
10 per cent, and tables do not usually indicate sources of data or
probable errors. For example, the latent heat of evaporation
for a mole of bromine at 59°C. is given in the common reference
books as 7280, 7410, 7000, 7200, and 7520 cal., with no means
of deciding which value is best.

Latent heats of evaporation decrease with rising temperature
and become zero at the critical temperature. The rate at which
the latent heat decreases also becomes greater at higher tempera-
ture, as may be seen from the data for water in Table 15 and for
alcohol on page 140.

Molal latent heats are roughly the same for liquids of the same
boiling point and are higher for liquids of higher boiling point.
This fact is expressed in the so-called "Trouton's rule," which
states that the molal latent heat in calories is 22 times the abso-
lute boiling point of the liquid. This approximation may be
written

= 22 or A&vap = 22 (8)

It is at best only a rough estimate, as shown by the fact that in
a tabulation for 153 liquids the average Trouton " constant "
was 22.1 and 4Cf of the liquids deviated from this average by
more than 10 per cent. From this rule the estimated Affm for
water is 8200 cal., compared with 9700 by experiment; the esti-

1 See, for example, SMITH, KEYES, and GERRY, Proc. Am. Acad Arts Sci.,
69, 137, 285, (1934), 70, 319 (1936).

2 The best compilation of latent heats is by Kelley, U.S. Bur. Mines Bull,
383 (1935).

PROPERTIES OF SUBSTANCES IN THE LIQUID STATE 117

mated A//m for benzene is 7800, compared with 7600 by experi-
ment. Large deviations are usually found for liquids in which
the dipole moments are capable of associating the hydrogen
bonds.1

Critical Conditions. — There is a critical temperature for each
substance above which it cannot be condensed to a liquid phase
at any pressure. At any temperature below the critical tem-
perature a vapor condenses when the applied pressure reaches
the vapor pressure for that temperature. Since the vapor pres-
sures of most substances at the critical temperature are less than
100 atm. and much higher pressures are readily reached, it seems
suprising that higher pressures cannot cause condensation above
a sharply defined temperature that is usually about 1.5 times
the boiling point on the absolute scale for 1 atm. pressure. Yet
there is ample experimental evidence that no condensation to
liquid occurs above the critical temperature, even at extreme
pressures. There are additional facts showing that there is a
temperature above which liquid does not exist, such as that (1)
the densities of liquid and saturated vapor become identical at
the critical temperature, (2) the surface tension approaches zero
at this temperature, (3) the latent heat of evaporation becomes
zero at this temperature, and (4) the isotherms near the critical
volume have different characteristics above and below this tem-
perature (see Figs. 10 and 11).

The critical pressure is the last point on the vapor-pressure
curve, the critical density is the density of both liquid and- satu-
rated vapor at the critical temperature, and the critical volume is
the volume of a gram of liquid (or vapor) at the critical tem-
perature and pressure.

The so-called "law of Guldberg-Guye " states that the critical
temperature is 1.5 times the boiling point, both temperatures
being on the absolute scale; that is, Tc/Ti = 1.5. The ratio is
between 1.45 and 1.55 for many liquids, but wider deviations are
not uncommon; for example, the ratios are 1.72 for water, 1.88
for oxygen, and 1.69 for ammonia, so that the "law" is only a
rough approximation. Some data for liquids are given in Table
16.

Law of Average Densities. — As the temperature rises, the
density of saturated vapor increases rapidly, owing to the increase

1 HILDEBRAND, Proc. Phys. Soc. (London), 56, 221 (1944).

118

PHYSICAL CHEMISTRY
TABLE 16. — DATA FOR LIQUIDS

Substance

Absolute
boiling
point

Absolute
critical
temper-
ature

Atfm,
cal per
mole at
1 atm.

Critical
pressure,
atm

( 'ritical
density,
g per ml

Acetic acid

391 4

594 8

5800

57 2

0 351

Acetylene

189 5

309

0 231

Ammonia

239 7

405 6

5560

111 5

0 235

Argon

87 4

151

1500

0 531

Benzene

353 3

561 7

7600

47 7

0 304

Butane (ri)

273 7

425 2

5320

37 5

0 225

Carbon dioxide

304 3

73 0

0 460

Carbon monoxide

81 1

133 0

1480

34 5

0 301

CC14

349 8

556 3

7290

45 0

0 558

Chlorine

240

417 2

4410

76 1

0 573

Ethane

184 8

305 4

7800

48 2

0 203

Ethanol

351 4

516 2

9400

63 1

0 275

Ethyl chloride

285 3

460 4

5960

0 33

Ethyl ether

307 7

466 0

6220

35 5

0 263

Ethylene

169 3

282 8

50 9

0 22

Helium

4 2

5 2

2 3

0 069

Heptane (ri)

371 5

540 2

7650

27 0

0 243

Hexaiie (ri)

342 1

507 9

6830

29 6

0 234

Hydrogen

20 5

33 3

215

12 8

0 031

Methane

111 7

190 7

2040

45 8

0 162

Methanol

337 8

513 2

8420

98 7

0 272

Methyl chloride

249 3

416 3

5170

65 8

0 37

Neon

27 2

44 5

415

25 9

0 484

Nitrogen

77 3

126 1

1330

33 5

0 311

Octane (ri)

397 7

569 4

8100

24 7

0 233

Oxygen

90 1

154 4

1595

49 I

0 430

Pentane (ri) .

309 3

470 3

33 0

0 232

Propane

228 6

377 4

42 0

0 226

Sulfur dioxide

263 0

430 4

6070

77 7

0 52

Sulfur trioxide

317 7

491 5

9500

83 6

0 630

Toluene

383 6

593 8

7980

41 6

0-292

Water.

373 1

647 3

9700

218 5

0 318

of vapor pressure of the liquid. The density of the liquid phase
decreases as the temperature rises, at first slowly, then more
rapidly as the critical temperature is approached. At the critical
temperature the density of liquid becomes the same as that of
the saturated vapor. In this region there is considerable diffi-

1 Vapor and liquid not in equilibrium at 1 atm pressure. .

PROPERTIES OF SUBSTANCES IN THE LIQUID STATE 119

culty in distinguishing the separate phases, and an exact deter-
mination of the critical density is difficult. It has been found
that, as the critical temperature is approached, the average of
the density of the liquid and its saturated vapor is a linear func-
tion of the temperature. This statement will be clearer from
Fig. 9, which shows the density of liquid argon and its coexisting
saturated vapor. By plotting this average density against the
temperature and drawing a straight line through the points it is
easy to determine the point at which this line intersects the
curve showing the density of each phase and thus to read the
critical density. This statement is known as the law of Cailletet

u-190 -180 "170 -160 -150 ~140 ~I30
Temperature

-20

FIG. 9. — Densities of liquid argon and its saturated vapor.

and Mathias, after its discoverers, or as the law of rectilinear
diameters, since the diameter of the density curve is a straight
line.

Isotherms in the Region of Condensation. — When pressure is
plotted against molal volume at a series of constant temperatures,
a diagram such as Fig. 10 results. For a temperature T7!, which
is below the critical temperature, the pressure increases with
decreasing volume along CD until the vapor pressure for T\ is
reached at the point C. Under this constant pressure the
volume decreases from C to B while condensation takes place.
In the region between C and B the " molal volume " is governed
by the fraction condensed and is thus not a function of p and T
alone. At a higher temperature such as TZj condensation occurs
at a higher pressure, the molal volumes of saturated liquid and
saturated vappr are more nearly equal, and AHm is smaller.

120

PHYSICAL CHEMISTRY

These changes all continue until the critical temperature 2% is
reached, and at this point vlla = vv&l)or and AHm = 0. The
behavior of a fluid in this region is shown by measurements on
ethane,1 which are plotted to scale in Fig. 11 for temperatures
very close to the critical temperature. It will be noted that the

'012 014 016

Volume, Liters per Mole

0.18

Volume

FIG. 10. — Isotherms on a pressure- FIG. 11. — Isotherms of ethane in the
volume plane critical region.

critical isotherm at 32.27°C., which is tangent to the two-phase
area, is horizontal at the critical temperature. At this point

(?) -o

\dv/T

and

dv*

Reduced Equation of van der Waals. — Since at the critical
temperature (v — vc) = 0, one may expand the equation
(v — vcy = 0, whick gives

write van der Waals' equation in the expanded form

3 /, , RT\ 2 , /a\ ab A

v 3 - ( b H h;2 + I — ) v = 0

\ PC / \Pc/ PC

and, by equating the coefficients of the various powers of v,
derive the relations

a =* 3vc2pc and b = ~
1 BEATTIE, Su, and SIMARD, /. Am. Chem. Soc., 16, 924 (1939).

PROPERTIES OF SUBSTANCES IN THE LIQUID STATE 121

Such a procedure is often used to obtain numerical values of
these "constants" a and 6; but since van der Waals' equation
near the critical region is not reliable, the values of a and b so
derived will not be suitable for calculations involving this equa-
tion at temperatures and pressures far removed from the critical
region.

The critical data for carbon dioxide in Table 16 lead to the
values a = 8.4 X 106 and b = 32 ml. per mole; and if the con-
stants so derived are used to calculate the pressure at which
carbon dioxide has a molal volume of 880 ml. at 323°K., the cal-
culated pressure is 21.5 atm. compared with the observed pressure
of 26 4 atm. The values of a and b in Table 11 lead to a calcu-
lated pressure of 26.9 atm. for these conditions.

If the "reduced" temperature is defined as T/TC = 0, the
"reduced" volume as v/vc = <p, and the "reduced" pressure as
P/Pc = K, so that the quantities are expressed as fractions of the
critical quantities for each substance in place of being in the same
units for all substances, one obtains van der Waals' reduced
equation of state,

/ Q\

fy ~ i) = se (9)

It will be observed that there are no quantities appearing directly
in the equation which are properties of any particular substance ;
but, of course, the reduced quantities themselves have the char-
acteristic constants pc, vcj and Tc in them. Thus a reduced
pressure of unity is 73 atm. for carbon dioxide, 52 atm. for ethyl
chloride, 218 atm. for water, etc. These reduced quantities lead
to certain simple relations more suited for plotting than the
actual data; for example, reduced isometrics (plots of TT against 6
for constant <p) fall on the same straight line for all the hydro-
carbons, CH4, C2H6, C2H4, C8H8, C5H12, and C7H16.

Plots of \i = pvm/RT against the reduced pressure for a range
of reduced temperatures for all saturated hydrocarbons above
methane are identical for each reduced temperature. This fact
indicates that reduced temperatures, pressures, and volumes are
"corresponding states" and thus indicates that there is some
fundamental "law of corresponding states." But since the
reduced equation of van der Waals does not yield exact pressures
or volumes, it will be clear that some further modification or

122 PHYSICAL CHEMISTRY

some other equation of state is required to show fully what this
law is.

Surface Tension.1 — The familiar fact that drops of liquid are
nearly spherical indicates some kind of tension within the sur-
face that acts to reduce the surface area to the smallest value
consistent with existing conditions. This force is due to molec-
ular attraction. A molecule in the bulk of a liquid is attracted
equally in all directions by surrounding molecules in a region of
equal density, but a molecule in the surface is attracted toward
the liquid phase more than toward the vapor phase of smaller
density, and there is a resultant force acting upon it in the inter-
face. Surface tension is measured in dynes per centimeter of
film edge, and the surface free energy is the work required to
increase the surface area 1 sq. cm.; i.e., we measure surface
tension in dynes per centimeter and surface free energy in dynes
per square centimeter.

For any pure liquid the surface tension has a fixed value at a
fixed temperature. It may be measured by the height to which
a liquid rises in a capillary tube of known radius, by the maximum
weight of a drop that will hang from a circular tip, by the pres-
sure required to form bubbles at the end of a submerged tube,
by the force required to pull a submerged ring out of a surface,
and by other methods. Surface tensions of solutions may also
be measured~by these methods, but they depend upon the nature
and concentration of dissolved substance as well as upon the
temperature. We shall see later that the composition of a surface
layer may be quite different from that of the bulk of the solu-
tion; therefore, when^the surface is extended, sufficient time must
be allowed for the new surface to come to equilibrium with the
underlying liquid before the surface tension is measured. Serious
errors in some of the recorded data are due to failure to allow
sufficient time, which may be hours rather than minutes for some
solutions.

, The rise of a liquid in a capillary tube that is wet by the liquid
may be used to measure its surface tension. If 7 is the surface
tension in dynes per centimeter and h is the height to which a

1 For a detailed treatment of this subject see Rideal, " An Introduction to
Surface Chemistry," Cambridge University Press, London, 1926; for a gen-
eral survey of experimental methods, see Dorsey, Nail. Bur. Standards Sci.
Paper, 21, 563 (1936).

PROPERTIES OF SUBSTANCES IN THE LIQUID STATE 123

liquid of density d rises above the horizontal surface in a tube of
radius r, the equation connecting these quantities is

7 = lirhdg (10)

This equation results from equating the surface tension to the
weight of liquid supported by it when equilibrium of forces is
reached. The length of film edge is the circumference of the
tube, 2irr-} hence the upward force is 2irrj, and this is balanced
by a volume of liquid irr2h of density d acted upon by the force of
gravity g. It follows from this that 2irry = Trr2hdg] and, upon
solving for 7, equation (10) results. It has been assumed in this
derivation that the angle of contact between the liquid and the
surface it wets is zero, or otherwise the upward force would be
2irry times the cosine of this contact angle. The fact that the
surface tensions of water and most liquids as determined by the
capillary-rise method without correction for an appreciable angle
of contact are in agreement with those from other methods
indicates that the angle is zero for these liquids.1 But, for
liquids that do not wet the material of the capillary tube, equa-
tion (10) without correction for the contact angle will give
incorrect results.

The chief error in capillary-rise measurements comes from
uncertainty of the radius r, owing to irregular diameters of the
capillary tubes. In the method as modified by Jones and Ray2
and shown diagrammatically in Fig. 12, the meniscus is brought
to the same part of the capillary tube for each measurement by
adjusting the level of liquid in the large tube. Thus a capillary
rise h0 for a liquid of known surface tension (such as water) with
the meniscus in the capillary at the index point serves to deter-
mine the radius at this point, whereas the length of a weighed
mercury thread in the capillary would yield only the average
radius of that part of the tube which it occupfes.

When a liquid of smaller surface tension than water is put in
the apparatus in such quantity that the meniscus in the capillary*
rests at the index point, a larger quantity of liquid is required
to bring the level in the large tube to the position shown by the

1 For methods of measuring the contact angle, see Ferguson, "Fifth
Report on Colloid Chemistry," Brit. Assoc. Advancement Sei. Rep,, 1923,
1-13.

2 JONES and RAY, /. Am. Chem. Soc., 59, 187 (1937).

124

PHYSICAL CHEMISTRY

dotted line in Fig. 12. In precise work the density d in equation
(10) should be written (d — ft), where ft is the density of the
vapor. Then, indicating the quantities for water with sub-
scripts w and omitting subscripts from the corresponding quan-
tities for the unknown liquid, we have from equation (10)

rg(d -

rg(dw — ftv)hw

(11)

When capillary rise is determined for a liquid-liquid interface
bet ween two insoluble liquids, the term (d — ft) becomes (d\ — ^2),
the difference in density of the two liquids. Such interfacial

Index
point

FIG. 12. — Apparatus for measuring capillary rise. (Jones and Ray.)

tensions are important factors in determining the stability of
emulsions. The surface tensions of some common liquids are
given in Table 17.

TABLE 17. — SURFACE TENSION OF SOME PURE LIQUIDS
(Measured in dynes per centimeter at 20° by the capillary-rise method)

Water.

72 62

Toluene

28 58

Benzene . . .

28 88

Isobutyl alcohol

22 85

Methyl alcohol

22 61

Ethyl butyrate

24 54

Ethyl alcohol

22 27

PROPERTIES OF SUBSTANCES IN THE LIQUID STATE 125

Surface Tension and Drop Weight.1 — The maximum weight
of drop that will hang from the end of a rod or other tip is deter-
mined by the surface tension of the liquid. But the simple
theory that equates the product of the tip circumference and
surface tension to mg, the product of the mass of the drop that
falls and gravity, is incorrect; for a considerable portion of the
hanging drop adheres to the tip when the remainder falls. The
mass of the ideal drop, rat, which gives the correct surface tension
through the relation

•2 0750r

2irry = mlg

(12)

^ 0125

£ 0,100
I o.ei5

o 0.650

4 05)25

£ 0600

§ 0515

t 0.550

rtf*

is a function of the mass of
the actual falling drop, the
diameter of the tip, and the
cube root of the volume of
the drop;2 that is, mt (ideal)
= m/f(r/V*). In Fig. 13
the fraction of the ideal drop
that falls is plotted against

(r/F**). Once the volume of the actual falling drop has been
determined by experiment, the ratio of the tip radius to the cube
root of its volume is calculated; and then the fraction by which
the volume of the actual drop must be divided to give the vol-
ume of the ideal drop is read from the curve. Its mass when
substituted in the equation

fl04 05 0,6 01 08 09 1 6 I.I U 1,3 1.4 1 5 1.6

' Cube Root of Volume tf Qrop* V^
FIG 13.

7 =

&irr

gives the correct surface tension. In order to emphasize more
fully the imaginary character of the ideal drop whose mass is m»,
it is better to write this surface tension equation in the form

(13)

mg = 2irryf ( -™

1 For a discussion of the method and a survey of the literature upon it, see
Harkins and Brown, ibid, 41, 499 (1919)

2 HARKINS, ibid., 38, 228-*253, 39, 354-364, 541-596 (1917), 41, 499 (1919).
Cf. Lohnstein, Z. phyaik. Chem., 84,410 (1913), for a criticism of the method.
Tabulated values of a function F, such that 7 = (mg/r}F, are given "for
various values of V/r8 in "International Critical Tables," Vol. IV, p. 435.
For recent studies see Hauser, /. Phy*. Chem., 40, 973 (1936), 41, 1017 (1937).

126 PHYSICAL CHEMISTRY

where m is the mass of the actual falling drop determined by
experiment.

The drop-weight method has the advantage of employing a
much larger liquid surface than the capillary method When
suitable precautions are taken in the experiments and when the
drop volumes or drop weights are properly used in the calcula-
tions, the method gives surface tensions that are comparable in
precision with those derived from capillary rise. The details of
manipulation allow of less latitude than has commonly been
supposed;1 for example, a tip diameter should be chosen such that
(r/V^) is between 0.7 and 1.0; and adequate time must be allowed
for orientation of the molecules at the interface and adjustment
of the molecular forces before the drop falls. This time is seldom
less than 5 min. per drop, and for soap solutions falling into oil it
may exceed 20 min. per drop, as shown by variation in the drop
size with time when drops are allowed to form too fast.

Other experimental methods include measuring the force
required to draw a straight wire or a wire ring vertically out of a
horizontal surface (du Nuoy method)2 and the pressure required
to initiate bubble formation on a submerged tip. Both these
methods, like the drop-weight method, require the formation
of new interfacial surface so slowly that orientation reaches
equilibrium.

Surface tensions of mixtures of liquids are not linear functions
of the concentration at constant temperature. In mixtures of
benzene and cyclohexane, for example, the plot of surf ace 'tension
against concentration passes through a minimum ; other mixtures
do not show such minima but are not linear.

Surface Tension and Temperature. — The empirical equation
of Ramsay and Shields shows the change of molecular surface
energy with temperature, where the molecular surface energy is
proportional to the product of surface tension and the molecular
volume to the two-thirds power. The equation is

y(Mv)% = k(tc - t- d)

1 HARKINS and BROWN, /. Am. Chem. Soc., 41, 499 (1919); HAUSER,
EDGERTON, HOLT, and Cox, /. Phys. Chem., 40, 973 (1936); AUBRY, Compt.
rend., 208, 2062 (1939)

2 See Dale and Swartout, J. Am. Chem. Soc., 62, 3039 (1940), for a "twin-
ring" modification of this method into a means of precise measurement.

PROPERTIES OF SUBSTANCES IN THE LIQUID STATE 127

where tc is the critical temperature, t is the temperature at which
7 is measured, d is a "correction factor" of 6° (required for
unexplained reasons), & is a constant, and (Mv)** is proportional
to the surface of a mole of liquid. Some results of applying the
equation are shown in Table 18.

TABLE 18. — CRITICAL TEMPERATURES CALCULATED FROM DROP WEIGHTS
AT DIFFERENT TEMPERATURES1

Substance

Milligrams
Weight per drop at

ww(Jlf») & at

Calcu-
lated

Experi-
ment

18° ,

60°

18° | 60°

Benzene

30.96
48 10
28.85
35.87
40 37

25.10
43.09
23.55
30.48
34.14

614.99
1,156.42
607.14
780.31
751.71

516.44
1,057.23
512.89
682.00
654.35

2.326
2.329
2.332
2.328
2.330

288

520
285
360
347

288

283
359

Quinolme

ecu

C6H6CL ...
Pyridine . .

From the equation it is seen that the surface tension 7 decreases
as the critical temperature is approached and that it becomes
zero when (tc — t) is equal to d. The quantity d is a correction
factor of 6°, introduced into the equation to show that 7 becomes
zero at 6° below the critical temperature, instead of at the critical
temperature as might be expected. The value of fc in the equa-
tion is obtained from measurements of 7 at different temperatures
for one substance; it has the same value for all "normal"
substances.2

Molecular Attraction. — In the simple gas law, it was assumed
that there were no attractive forces acting between the molecules;
and it was found that at high pressures this assumption was not
correct. To allow for it, a term (a/vm2) was introduced into van
der Waals" equation. Without the attractive forces that cause
the vapor to condense (i.e., an ideal gas being assumed), the
pressure necessary to bring a mole of water vapor into a space of
18 ml. at 20°C. is given by p X 18 = 82 X 293, whence the
pressure is 1340 atm., but the pressure exerted by water at 20°

1 MORGAN and THOMSSEN, ibid , 33, 657 (1911).

2 Research upon surface tension near the critical temperature involves
difficulties that have not been fully realized by all who have experimented
in this region. For a consideration of these matters, see Winkler and Maass,
Can. J. Research, 9, 65 (1933).

128

PHYSICAL CHEMISTRY

'(its vapor pressure) is only about 0.02 atm. It appears that the
attractive pressure, or cohesive pressure, must, therefore, be very
large. The cause of molecular attraction is not at all understood
but is believed to be due to stray electric fields caused by the
electrons within the atoms.

TABLE 19. — INTERNAL PRESSURE (IN ATMOSPHERES) OF VARIOUS LIQUIDS

Wmther1

Traube2

Walden2

Lewis3

Mathews4

Ether

1220

990

1360

1930

1970

Ethyl acetate

1490

1140

1730

2640

2460

CC14

1820

1305

1680

2520

2660

Benzene

1790

1380

1920

2640

2940

Chloroform

1680

1410

1950

2780

2910

CS2

2200

1980

2400

2920

3950

Ethyl alcohol

2030

2160

4000

3600

Many equations have been put forward by various investiga-
tors for calculating the internal pressure of a liquid from a/rm2,
from the latent heat of evaporation, and from other data. But
the internal pressures so calculated are not in good agreement
with one another. Table 19 shows the internal pressure6 in
atmospheres according to the calculations of various workers.
It should be borne in mind that all these calculations are based
on certain assumptions and that the actual internal pressure has
not been measured directly. The deviations among the values
for any one liquid will indicate the uncertainty of the assump-
tions made as to the way in which the attractive forces act ; but
all the calculations agree in showing that there is an internal
pressure and that it is very great.

When van der Waals' equation of state is written in the form
p = [RT/(vm — b)] — (a/vm*) it will be seen that the measured
pressure p is the difference between two terms, of which the first
may be called the thermal pressure and the second the cohesive
pressure. For small molecular volumes both these terms are
large compared with the difference between them, and under

1 From optical properties.

2 From surface tension and van der Waals' a and 6.
8 From thermal data.

4 From latent heats and surface tension.
6HiLDEBRAND, /. Am. Chem. Soc., 38, 1459 (1916).

PROPERTIES OF SUBSTANCES IN THE LIQUID STATE 137

themselves so that the change of properties attending passage
from one phase to the other will be as gradual as possible. For
example, the hydrocarbon part of a molecule in an oil-water
interface is probably directed toward the oil layer. Of course,
there is no evidence that this arrangement persists for more
than one molecular length. The " water-soluble " portion or
active group (— COOH, —OH, = CO, — CN, — CONH2, or
inorganic radical) will be directed toward the water layer. This
tendency may result in a preferential solubility of a dissolved
substance in the interface. For example, the surface tension at
a benzene-water interface is greatly decreased by very small
amounts of soap, of which the composition may be represented
roughly by the sodium salt of palmitic acid (Ci5H3iCOONa).
The interfacial tension decreases from that of benzene-water
(35 dynes) almost in proportion to the concentration of soap,
falling to about 2 dynes for 0 01 N soap solution, after which
further additions of soap cause only a slight decrease (to 1.8
dynes for 0.1 N soap, for example). A probable explanation is
that the interface becomes nearly saturated with soap molecules
oriented in such a way as to give the minimum surface tension
through preferential solution in the interface long before the
water layer as a whole is saturated. When this surface satura-
tion is attained, the addition of more soap to the water layer
causes only a little increase in the soap concentration in the inter-
facial layer and hence only a slight change in the surface tension.
The behavior of soap solutions is complicated by other factors
such as the alkalinity of the aqueous layer and the nature of the
nonaqueous layer, which are best omitted from a preliminary
discussion, but the most important properties of soaps are those
which result from the formation of surface layers much richer in
soap than the body of the solution.

Dr. Katherine Blodgett1 has modified the monolayer technique
so that parallel layers of barium stearate and other insoluble
substances may be deposited one upon another to a total thick-
ness of some 300 molecules.

Monolayers such as these are probably the most important
single factor in determining the structure and properties of the
water shells around the particles in Bydrophyllic colloids, as
we shall see in a later chapter. Similar layers are probably

1 See Science, 87, 493 (1938), for references to papers upon this topic.

138 PHYSICAL CHEMISTRY

present upon most solid surfaces in contact with liquids or
solutions. While there are many complications, such as changes
produced by pressure or minute amounts of solutes, the fact
that oriented monolayers form is the prime fact to be kept in
mind.

X-ray Diffraction in Liquids. — In the next chapter a method
for determining the distance between atomic centers in a crystal
is described. We may anticipate this treatment here by a brief
statement of the results of its application to liquids. The " pat-
tern'7 shown by X-ray diffraction of liquids consists of one or
more broad diffuse rings, differing markedly from the sharp rings
so typical of a crystalline material. By making a Fourier
analysis of the X-ray pattern of a liquid, a radial distribution
curve is obtained that gives the distribution of atoms with
respect to any average atom in the liquid. In such a distribu-
tion curve for liquid sodium, the first peak occurs at about
4 X 10~~8 cm., and this distance corresponds approximately to
the diameter of the sodium atom.1 Measuring from the center
of any sodium atom we should not expect to find the center of
any other atom at a distance less than the " diameter " of the
atom. At this distance we should expect to find several atoms,
since any atom in a liquid will always be in approximate contact
with several neighboring atoms.

A similar study of water shows an average distance 2.9 to
3.0 X 10~8 cm. between oxygens (the X-ray diffraction of
hydrogen atoms is too feeble to indicate their positions), which is
greater than the O-O distance in ice (2.76 X 10~8 cm.) in spite of
the smaller density of -ice. The interpretation is that in liquids
a molecule has no permanent neighbors, but at any instant a few
molecules are in approximate contact and others at greater
distances are either approaching or receding.

The patterns obtained in a vitreous " liquid" such as a simple
glass are also diffuse rings. In fused quartz or vitreous silica,
which is an example of a simple glass, the X-ray results show
that each silicon atom is tetrahedrally surrounded by four
oxygens at a distance of 1.62 X 10~8 cm. Each oxygen is bonded
to two silicons, the two bonds being roughly diametrically oppo-
site. As far as nearest neighbors are concerned, the structure in
the glassy form of silica is exactly the same as the crystalline

1 WARREN, /. Applied Phys., 8, 645 (1937).

PROPERTIES OF SUBSTANCES IN THE LIQUID STATE 139

forms. The glass differs from the crystal only in the fact that
no definite scheme of structure repeats itself identically at regular
intervals.

Application of X-ray diffraction to alcohols shows that the
C-C distance1 is 1.54 X 10~8 cm., which agrees with the length
of hydrocarbon chain per atom of carbon determined in the oil-
film experiments described in the previous section.

Problems

Numerical data for some of the problems must be sought in tables in the text

1. (a) Calculate the latent heat of evaporation per mole of water at 80°
from the vapor pressures in Table 14, using the approximate Clapeyroii
equation, (b) Calculate this quantity from the slope of the vapor-pressure
curve, 0 01893 atm per dcg. at 80°, and the specific volumes of liquid and
saturated vapor, 1 029 and 3409 2 ml per gram, respectively.

2. A cylinder fitted with a movable piston contains 5 4 grams of a satur-
ated vapor, which occupies 1 liter at 350°K. (=77°C ) and 1 atm pressure,
(a) When the temperature is reduced to 323 °K arid the volume remains
1 liter, part of the vapor condenses to liquid and the pressure becomes 0.41
atm. Calculate the weight of condensed liquid, assuming the vapor an
ideal gas and neglecting the volume of the condensed liquid, (b) The latent
heat of evaporation of the liquid is substantially constant m this temper-
ature range. Calculate the quantity of heat that must be added to the
vessel at 323°K. to evaporate the condensed liquid if the pressure is kept at
0 41 atm through the motion of the piston

3. Calculate the area covered by a monolayer of stoaric acid spread upon
water for each milligram of acid, whose formula is CnH^COOH

4. The slope of the vapor-pressui e curve of liquid nitrogen tetroxide at
294°K (the boiling point) is 0 0467 atm per deg. (a) Calculate the latent
heat of evaporation per mole of vapor formed at the boiling point (b) The
vapor consists of N«O4 and NO2 molecules, and the measured latent heat of
evaporation of 92 grams of liquid is 9110 cal. at the boiling point Calculate
the degree of dissociation of N2O4 into NC>2 at 294°K [GIAUQUE and KEMP,
/. Chem Phys , 6, 40 (1938) ]

5. In the experiment described in Problem 15 (page 99) assume that the
sealing was imperfectly performed, so that, when the bulb is cooled to 20°
for weighing, some air enters the bulb, part of the substance condenses to a
liquid, but none is lost. Under these conditions the bulb weighs 31 300
grams. Assume the vapor pressure of the substance to be 0.227 atm , and
calculate the molecular weight of the vapor.

6. One mole of CH4 is exploded with 9 moles of air (assumed 21 mole per
cent oxygen and 79 mole per cent nitrogen), and the resulting mixture is
assumed to contain only H^O, CO, CO2, and N2. (a) Find the temperature
at which this mixture is just saturated with water vapor (see Table 14).

1 HARVEY, /. Chem. Phys., 7, 878 (1939).

140

PHYSICAL CHEMISTRY

(b) The mixture is cooled to 25°C and 1 atm. total pressure Calculate the
weight of condensed water and the partial pressures of CO, CO2, and Na.

7. The volume of a quantity of air saturated with water vapor at 50° is
2.50 liters when the total pressure is 5 0 atm (a) Calculate the final total
pressure if this air is expanded over water at 50° until the total volume
becomes 46 liters. (6) How many moles of water evaporate to establish
equilibrium?

8. Benzene has a surface tension of 28 88 dynes at 20°, and its density is
0 879. What is the radius of a capillary tube in which benzene rises 1 cm ?
How high would water rise in the same lube?

9. (a) Calculate the total pressure in a 10-liter flask containing 0 1 mole
of CC14 and 0 3 mole of air when the temperature is 50, 40, 30, and 20°
(b) Determine from a suitable plot the temperature at which the mixture is
just saturated with CCh

10. The critical temperature of ethanol (C2H&OH) is 243°C , the critical
pressure is 63 1 1 atm , and the following data apply at lower temperatures.

t, °C.

Vapor
pressure,
atm

Surface
tension,
dynes
per cm

Liquid
density,
grams per
ml

Saturated
vapor den-
sity, grams
per ml.

AH
evaporation,
cal. per mole

0 0577

22 75

0 7895

10,000

0 0776

22 32

0 7852

0 2925

20 14

9,800

78.3

1 000

0 7365

0 00165

9,400

100

2 228

15 47

0 7157

0 00351

8,900

150

9 70

10 16

0 6489

0 0192

7,490

200

29 20

4 26

0 5568

0 0508

5,280

220

42 38

0 4958

0 0854

3,950

240

59.92

0 3825

0 1715

1,760

(a) Estimate the critical density from a suitable plot of the above data.
(6) Calculate AHm at 220° from the vapor-pressure data, (c) Calculate
AHm at 220° from the slope of the vapor-pressure curve, which is 0 750 atm.
per deg. at 220°.

11. (a) Calculate the weight of ethanol evaporated when 100 liters of air
at 50° and 1 atm. are bubbled through ethanol at 50° so slowly that equi-
librium is reached and the mixture of air and ethanol emerges at 50° and
1 atm. total pressure, (b) Calculate the weight of ethanol condensed when
this mixture is cooled to 25° and 1 atm. total pressure.

12. The vapor pressure of phenylhydrazine in atmospheres is given by
the equation log p = 5.0238 - 2810/77 in the range 365 to 415°K. Cal-
culate AHm, assuming the vapor an ideal gas, [WILLIAMS and GILBERT, /.
Am. Chem. Soc., 64, 2776 (1942).]

13. Drop-weight experiments were made at 20° for water and for benzene
with the following results :

PROPERTIES OF SUBSTANCES IN THE LIQUID STATE 141

Tip radius,
centimeters

Drop weight for

Water, grams

Benzene, grams

0 1477
0 2680
0 3419

0 0469
0 0775
0 0964

0 0175
0 0297
0 0383

Calculate from these drop weights the surface tension of benzene and of
water at 20°, and compare them with the measured results given in Table 17.
[HARKINS and BROWN, /. Am. Chcm. Soc , 41, 449 (1919).]

14. The slope of the vapor-pressure curve for formic acid is 6 3 mm. per
deg at 50°C and 25 mm. per deg. at 100°C.; the vapor pressure is 130 mm.
at 50°C. and 748 mm. at 100°C. (a) Calculate AHm at each temperature,
assuming the vapor to be an ideal gas. (b) The recorded latent heat of
evaporation of formic acid at 100° and 1 atm. pressure is 120 cal. per gram.
See page 72 for other data on formic acid, and suggest an explanation of the
values obtained in (a).

16. (a) Calculate the molal latent heat of evaporation for water at 120°
from the data in Table 14, assuming the vapor to be an ideal gas. (6) The
slope of the vapor-pressure curve at 120° is 0.0621 atm. per deg , the specific
volume of the vapor is 891 8 ml per gram, that of the liquid is 1.06 ml. per
gram. Calculate a more accurate value of AH for the evaporation of a mole
of water at 120°.

16. The following data refer to ammonia:

Vapor

Specific volume, ml.

AH,

dp/dT,

pressure,

cal. per g.

atm per

atm.

Liquid

Vapor

deg.

233 1

0 708

1 45

1551

331 7

0 0378

238 1

0 920

1 46

1215

293 1

8 459

1 64

149 5

283.8

0 270

298 1

9 986

1 66

128 4

313 1

15 34

1 73

83 3

263 1

0 426

318 1

17 58

1 75

72 6

(a) Calculate A// over the three 5° temperature intervals from the
approximate Clapeyron equation. (6) Calculate AH at 233.1, 293.1, and
313.1°K. from the exact equation, and compare with the experimental
values.

17. The slope of the vapor-pressure curve for acetic acid is 0.0187 atm.
per deg. at 100°C., and the vapor pressure at this temperature is 0.548 atm,
(a) Calculate the heat absorbed per molal volume of vapor formed at 100°C

142 PHYSICAL CHEMISTRY

(b) The vapor consists of (CH8COOH)2 and CH8COOH molecules, and the
heat absorbed by the evaporation of 120 grams of acetic acid at 100°C. is
11,800 cal. Calculate the degree of dissociation of the dimer into the
monomer at 100°C. and 0.548 atm., assuming this to be the only cause of the
deviation.

18. (a) Air at 17° and 1 atm. pressure, 70 per cent saturated with water
vapor, is pumped into a 1000-liter tank until the pressure becomes 6 0 atm.
and the temperature rises to 27° The vapor pressure of water is 14 5 mm
at 17° and 26.7 mrn. at 27°. Assume the ideal gas law to apply, neglect the
volume of condensed water in comparison with the volume of the tank, and
calculate the weight of liquid water in the tank at 27° (b) Determine from
a suitable plot the lowest temperature at which all the water in the tank
would be in the form of vapor.

19. (a) Calculate the values of RT/vm, and of [RT/(vm - b)] - (a/vm2)
for water at 100° and a molal volume of 18 8 ml. (b) Calculate the pressure
at which water vapor would have a molal volume of 30.16 liters at 100° from
the ideal gas law and from van der Waals' equation. (The measured molal
volume of water vapor at 100° and 1 atm. is 30 16 liters )

20. (a) One step in the manufacture of nitrocellulose cakes involves the
removal of ethanol (C2H5OH) from the cakes by evaporation in a current
of dry air If the air-ethanol mixture emerges from the drier at 35° and
1 atm. total pressure, 73 per cent saturated with ethanol, what weight of
ethanol is evaporated for each 90 moles of air entering? (b) This mixture
is passed over brine pipes at —15° to recover the ethanol, and air emerges
from the cooler at —15° and 1 atm. 100 per cent saturated with ethanol
What weight of ethanol is condensed from each 90 moles of air? (The latent
heat of evaporation of ethanol is 10,000 cal per mole in this temperature
range )

21. What volume of dry air at 20° and 1 atm. must be bubbled through
n-octane at 50° in order to evaporate 10 grams of it, assuming the mixture
of air and n-octane to emerge from the evaporator at 50° saturated with
n-octane?

22. Hot air is passed over a product to remove CC14 from it, and the air
emerges from the drier af45°, 59 per cent saturated with CC14, and at 1 atm
total pressure, (a) How many moles of air enter the drier for each mole of
CCU evaporated? (6) This mixture of air and CC14 is passed over refriger-
ated coils, which cool it to 0° to recover the CC14. Calculate the vapor pres-
sure of CCh at 0° and the fraction of CC14 recovered, assuming the total
pressure to remain at 1 atm.

23. A refrigerator derives its cooling effect from the reaction

NH3(Z) - NH8(<7)

which takes place at -10°C. (263°K.). Assuming NH3 to be an ideal gas
and that AH is independent of temperature, calculate the heat absorbed in
the refrigerator for each 100 liters of saturated vapor formed at — 10°C.

24. The following data apply to equilibrium between liquid and vapor of
carbon dioxide:

PROPERTIES OF SUBSTANCES IN THE LIQUID STATE 143

r, °K.

dp/dT,
atm /deg

Molal volume, nil

Liquid

Vapor

223

0 273

2520

293

1 35

234

(a) Calculate AHm at each temperature. (6) The vapor pressure of COa
at 293°K is 57 atm. Calculate the per cent error in taking Av — RT/p at
293°K.

26. Air 50 per cent saturated with ethanol at 20°C and 1 atm total pres-
sure is pumped into a 100-liter tank until the total pressure becomes 10 0 atm .
and the temperature rises to 30° (a) Calculate the moles of liquid C2H6OH
in the tank, (b) Find the lowest temperature at which the ethanol in the
tank will be completely evaporated, (c) Find the total pressure in the
tank at this temperature.

26. The vapor pressure of sulfur dioxide (in millimeters of mercury)
changes with the absolute temperature as follows:

T
P

197.6 205 1 214 1 228 3 238 3 249 6 256 4 263 5
12 56 23 58 46 77 121 57 217.62 402 27 558 97 773 82

(a) Plot log p against l/T over the entire temperature range, draw a
smooth curve through the points, and state whether AH is a constant over
the range (b) Calculate A// from the vapor pressures at the two highest
temperatures, assuming the vapor an ideal gas, and compare with the meas-
ured A//, which is 5960 cal per mole at 263 08°K , the boiling point. [GiAU-
QUE and STEPHENSON, /. Am. Chem. Soc., 60, 1389 (1938).]

CHAPTER V
CRYSTALLINE SOLIDS

The purpose of this chapter is to present very briefly such
experimental facts on the properties of crystalline solids as we
shall need in later chapters — their vapor pressures, thermal
properties, and the arrangement of atoms in their crystals.
Crystalline solids result when pure liquids are cooled to tem-
peratures characteristic of the substances, when solutions of
these substances are cooled or evaporated, or when vapors con-
dense under such conditions that the liquid does not form.
Iodine crystals, for example, may be formed in any of these ways :
by cooling liquid iodine to 114.15°, by evaporating a solution
of iodine in CCU, or by cooling iodine vapor that has a partial
pressure of less than 94 mm., which is to specify that the tem-
perature is below 114.15° when condensation begins. It is not
definitely known that there are any noncrystallirie solids that
are stable over long intervals of time. But whether these exist
or not, there are some substances that are evidently solid and
not demonstrably crystalline. Since we are to consider the
equilibrium properties of solids and since the noricrystallme
solids are probably not in equilibrium states, we shall not con-
sider them.

The change from liquid to solid at the melting p'oint is attended
by a moderate change in volume, by a decrease in energy content,
and by the assumption of rigidity. Although the shape of the
mass of crystals obtained from complete solidification of a liquid
is usually that of the container in which it occurred, if partial
solidification occurs, the crystals formed will have characteristic
geometric forms. Under either circumstance the internal
arrangement of atoms or molecules in the crystal conforms to a
definite pattern. In crystals the molecules or atoms are held in
fixed positions; and though they probably vibrate about these
positions, they have no net motion in one direction, no mobility
at ordinary temperatures. There is, however, abundant evi-

144

CRYSTALLINE SOLIDS 145

dence of intercrystalline diffusion at higher temperatures, which
are still far below those at which liquid forms. The viscosity of
a crystal is substantially infinite; it may be crushed or sheared,
but by the application of a reasonable force it may not be changed
into another shape that it will retain when the force is removed.

If a crystal of a pure substance is heated at atmospheric pres-
sure, it changes to a liquid sharply at its melting point and when
cooled it assumes again its characteristic external shape and
internal symmetry as it crystallizes at the melting point.

Since all pure liquids become crystalline when sufficiently
cooled and most crystals become liquid when sufficiently heated
(except those which decompose before reaching the melting
point), we must understand that by a crystal we usually mean
a state of aggregation rather than a chemical substance capable
of existence only in solid form. The changes in volume and in
energy content that attend the formation of solids from liquids
are much smaller than those attending the condensation of vapors
to liquids. These phase changes for pure substances t)ccur at
constant temperatures for any specified pressure, and the effect
of pressure upon the temperature of the phase change from
liquid to solid is much smaller than that for vapor to liquid.
The density of a crystalline phase is commonly within 10 per
cent of that of the liquid from which it forms, while the density
of a liquid may be a thousand times that of the vapor from which
it condenses at atmospheric pressure. Solids have characteristic
vapor pressures that change with the temperature, as was true
of liquids; and, of course, the vapor pressure of the solid is equal
to that of the liquid at the triple point where all three phases,
solid, liquid, and vapor, are in equilibrium.

Vapor Pressures of Crystalline Substances. — A solid phase
in equilibrium with its saturated vapor is a monovariant system,
one in which the equilibrium pressure is a function of the tem-
perature alone, and hence the change of vapor pressure or
" sublimation" pressure with changing temperature is shown by
the Clapeyron equation

dT T At; '

in which A#8 is the heat absorbed by the phase change from
solid to vapor and Ay is the increase in the volume of the vapor

146 PHYSICAL CHEMISTRY

over that of the solid. By making the same assumptions as were
used for the liquid- vapor change in the previous chapter, we may
derive an approximate form of this equation suitable for low
pressures. These assumptions are that the volume of solid is
negligible compared with that of the vapor, that the volume of
the vapor is RT/p, and that AHS is constant over the range in
which the equation is used. Since RT/p is the volume of 1 mole
of vapor, AH a must now be the heat absorbed in the formation
of 1 mole of vapor. The equation and its integral between limits
then become

j ni A rr / rji rn

amp — —,5— 7™- and 2.3 log — = —7; *
ri 1 pi n

As an illustration of the change of vapor pressure of a solid
with changing temperature we quote the data for iodine.1

/. . . . 20° 25° 30° 40° 60° 80° 100° 114 15°(m. pt )
p, mm v 0 201 0 309 0 467 1 027 4 276 15 04 45 97 94 18

By using the vapor pressures for 20 and 30° one may calculate
AJf7s per mole of iodine vapor formed at 25° from the approximate
equation to be 14,960 cal. From more precise treatment of the
data, the authors calculate &H8 — 14,880 cal. per mole of vapor
formed.

Carbon dioxide is one of the few substances of which the solid
phases have vapor pressures greater than 1 atm., as the following
data show:2

r, °K 174 7 182 3 192 66 194 6 195 83 203 213 216

p, atm . 0 160 0 339 0 845 1 000 1 100 2 02 4 18 5 13

Since 216°K. is the triple-point temperature, 5.13 atm. is the
last point on the vapor-pressure curve for the solid and the first
point on the vapor-pressure curve for the liquid. Liquid carbon
dioxide has no boiling point, since its liquid and vapor phases
are not in equilibrium at 1 atm. pressure for any temperature.
These vapor pressures afford a means of calculating the heat of
sublimation from the exact Clapeyron equation (1), but they do
not give a correct heat of sublimation when substituted into equa-
tion (2), since carbon dioxide deviates from ideal gas behavior

1 GILLESPIE and FBASER, J. Am. Chem. Soc., 68, 2260 (1936).

2 GIAUQUE and EGAN, /. Chem. Phys., 5, 45 (1937).

CRYSTALLINE SOLIDS 147

at these temperatures and pressures. Thus, substitution of the
pressures 1.10 atm. and 0.845 atm., with the appropriate tem-
peratures, into equation (2) gives A#B = 6400 cal. per mole,
while equation (1) gives &H8 = 6030 cal. per mole at 194. 6°K.
As has been said before, an approximate equation is useful only
to the extent that the assumptions inherent in it are valid. In
this instance the assumption of ideal gas behavior is not valid,
but in the illustration at the end of the preceding paragraph the
same equation gave A#s for iodine vapor within 0.5 per cent
because at the higher temperatures and lower pressures involved
the assumptions were closer to the truth.

Melting Point. — The temperature at which the liquid and solid
phases of a pure substance are in equilibrium under a pressure
of 1 atm. is defined as the melting point. Since the presence of a
foreign substance in a liquid lowers the temperature at which
equilibrium with the solid phase is reached, melting points are a
useful indication of the purity of a preparation. Under the
procedure usually followed the liquid is saturated with air, which
is an "impurity" affecting the melting point slightly; but unless
the very highest precision is required, the change produced by
air may be neglected. For example, centigrade zero is defined
as the temperature at which ice and water saturated with air are
in equilibrium under 1 atm. pressure. Removal of the air
would raise the equilibrium temperature to +0.0023°, which is
thus the true melting point of ice. The effect of dissolved air
on other substances is also of this order of magnitude.

Changes in barometric pressure produce only negligible changes
in the melting point, but high pressures cause changes in melting
points that may be large; for example, under 2000 atm. pressure
ice^and liquid water are in equilibrium at —22°.

A solid phase in equilibrium with its liquid is also a mono-
variant system to which the Clapeyron equation

dp =

dT ~ T Av

may be applied. If the pressure effect is desired in atmospheres
per degree, Av should be expressed in milliliters and Aff in milliliter
atmospheres (calories X 41.3). For example, when a gram of
ice melts at 0° and 1 atm., there is a volume decrease of 0.09 ml.
and a heat absorption of 79 cal., or 3260 ml. -atm.; upon sub-

148

PHYSICAL CHEMISTRY

stituting these quantities in the Clapeyron equation, dp/dT is
found to be — 132 atm. per deg., which is a change of the melting
point of —0.0075° per atm. This is not to say that some very
high pressure would produce a change proportional to this figure.
For example, the application of 2000 atm. would not change the
melting point to 2000/(-132), or -15°, but to -22° as was
stated above. Such a calculation leaves out of account the
important facts (1) that ice and water have different compressi-
bilities so that Av is not —0.09 ml. over the range of 2000 atm.
and (2) that A// is not 79 cal. per gram over a 22° range. When
Av and AH are suitably expressed as functions of pressure and
temperature, the Clapeyron equation leads to the correct tem-
perature, as it always does when properly used.

Heats of Fusion. — The heat absorbed by the melting ot a
solid to a liquid at the melting point is called the heat of fusion
or the " latent heat" of fusion. It is best determined by direct
calorimetry but may be derived from the freezing-point depres-
sions of solutions through some of the equations to be given in
Chap. VI. Some of the recorded data based on the latter method
are unreliable because of incorrect use of the data or the use of
unreliable data, but such figures are often recorded in the same
tables with directly measured heats of fusion and properly calcu-
lated ones. Since no reliable rules are known for estimating
latent heats of fusion, one must select the sources of data with
care or be prepared for discrepancies. The ratio AHf/T of the

TABLE 21 — LATENT HEATS OF FUSION
(In calories per mole at the melting point)

Substance

A#,

Substance

A#/

932

2550

Acetic acid

289 7

2690

Cl,

238

1615

Ethylene dibromide

282 7

2570

H2 .

Ethyl alcohol

158 7

1145

Pb ...

600

1224

Carbon tetrachloride

249 1

644

Mg ...

923

2160

p-Dichlorbenzene

325 8

4360

234

557

Nitrobenzene

278 8

2770

505

1720

Benzene

278 5

2365

H2O

273

1436

Phenol

298 5

2720

LiCl

887

3200

Naphthalene

353 0

4550

NaCl

1073

7220

Diphenyl .

382 3

4020

KC1

1043

6410

Benz ophenone

321 6

4290

NH8

196

1426

Anthracene

489 7

7800

CRYSTALLINE SOLIDS 149

molal latent heat of fusion to the absolute temperature varies
widely for different substances, from 1.6 for cesium to 18.2 for
Aids, for example. Thus the ratio is not even a rough approxi-
mation, and it would be useless for checking the reliability of
recorded data. A few measured heats of fusion are given in
Table 21 '

Heat Capacities of Crystalline Solids. — We shall consider
only heat capacities at constant pressure, since virtually all the
data are taken at constant pressure; and, in conformity to the
common custom, we shall discuss the atomic heat capacity of
elements and the molal heat capacity of compounds. Thus
Cp = dH/dT = 5.82 for aluminum at 298°K. is the ratio of the
heat absorbed (in calories) by an atomic weight of aluminum
to the rise in temperature produced at or near 298°K , and
Cp = 4,80 + 0.0032 17 is an expression for the heat capacity of
an atomic weight of aluminum, valid to 2 per cent, in the tem-
perature range 273 to 932°K. Since Aff = JC3> dT between the
appropriate temperature limits, the heat required to raise the
temperature of an atomic weight of aluminum from 273 to
673°K. is the integral of the heat-capacity equation between
these temperature limits, or 2560 cal. Use of the "room-
temperature" heat capacity over this range of temperature would
give 400 X 5 82 = 2320 cal , which is obviously not correct; but
between 288 and 298°K. the equation gives 57.2 cal., and the
single heat capacity gives 58.2 cal., either of which would be
close enough in most calculations.

The restrictions as to temperature range and validity of a heat-
capacity equation are important. Thus substituting T = 298
in the equation Cp = 4.80 + 0.00327" gives 5.76, which is within
1 per cent of 582; but by substituting T = 50 in this equation
one obtains Cp — 4.96, while the correct atomic heat capacity of
aluminum at 50°K. is 0.92. The upper limit is set by the melting
of aluminum at 932°K. ; the lower limit is a conventional one
arising from the custom of discussing "low-temperature" heat
capacities and " high- temperature " heat capacities from different

1 The best critical summary of heats of fusion of inorganic substances
is by K. K. Kelley m U.S. Bur Mines Bull , 393 (1936), from which the data
in Table 21 were taken. Data for organic substances will be found in
"International Critical Tables/' Vol. V, pp. 132jf, in which the data are in
joules per gram or kilojoules per formula weight. One kilojoule is 238.9 cal.

150

PHYSICAL CHEMISTRY

points of view. There is no implication that heat capacities
change abruptly at the melting point of ice. One more illus-
tration will serve to emphasize the necessity of heeding the
restrictions stated with such equations For iron the equation
CP = 4.13 + 0.00638 T is valid to 3 per cent in the range 273 to
1041 °K. The melting point of iron is 1803°K ; but the equation
given is not to be used through the upper limit stated because of
a phase transition to another form of iron, which takes place at
1041°K. with the absorption of "heat of transition," and the
formation of a phase with a different heat capacity. Many
other substances undergo phase transitions, some at low tem-
peratures, some at high temperatures; some (including iron)
undergo more than one solid-solid transition; and for all of them
there is a constant-temperature absorption of heat at the transi-
tion temperature for which no allowance can be included in a
heat-capacity equation.

TABLE 22 — HEAT CAPACITIES OF SOME SOLID ELEMENTS
(In calories per atomic weight at 298°K and constant pressure)

Element

<"P

Element

Aluminum

5 82

Lead

6 39

Antimony

6 03

Lithium

5 65

Beryllium

4 26

Magnesium

5 71

Bismuth

6 10

Nickel

6 16

Cadmium

6 19

Potassium

6 97

Calcium

6 28

Silicon

4 73

Carbon (graphite)

2 06

Silver

6 10

Carbon (diamond)

1 45

Sodium

6 79

Copper

5 86

Sulfur (r)

5 41

Gold

6 03

Tm (white)

6 30

Iodine

6 57

Tungsten

5 97

Iron

6 03

Zinc

6 07

The atomic heat capacity of most of the solid elements at
ordinary temperatures is about 6.2, a fact that has long been
known as the "law of Dulong and Petit." As may be seen in
Table 22, carbon, beryllium, and silicon are conspicuous excep-
tions, and most of the elements of atomic weight below 39 deviate
by more than 10 per cent from this average figure. This "law"
is thus only a rough approximation. Another rough approxi-
mation, known as "Kopp's law," states that the heat capacity

CRYSTALLINE SOLIDS 151

of a solid compound is equal to the sum of the heat capacities of
the elements of which it is composed. The sum of the atomic
heat capacities of Cu and S is 11.17; the molal heat capacity of
CuS is 11.43; for FeS the corresponding figures are 11.44 and
13.06, which shows that considerable error may be involved in
accepting this "law." Fortunately, there is now little need for
either of these "laws," since abundant modern heat-capacity
data are available,1 especially at low temperatures, because of
the importance of standard entropies computed from them. It
will be recalled from Chap. II that the entropy of a substance at
(say) 298°K is obtained by integrating Cp dT/T from 0 to 298°K
and that the heat capacity must be known as a function of
temperature for this integration.

The heat capacities of all crystalline substances become zero
at 0°K., but the rates at which they decrease at temperatures
below 298°K. are quite different for different substances. For
example, Sb, Au, and Fe all have atomic heat capacities of 6.03
at 298°K , but at 50°K they are, respectively, 3.0, 3.5, and 0.71
Their standard entropies at 298°K., which are obtained by
integrating Cp dT/T from 0 to 298°K., also illustrate this dif-
ference; they are 10.5 for Sb, 11.4 for Au, and 6.47 for Fe. Some
low-temperature heat capacities are given in Table 23, and many
others will be found in the reference quoted with the table.

So-called "high-temperature" heat capacities are commonly
represented by equations such as Cp = a + bTorCp = a + bT + cT2.
Plots of heat capacity against temperature often have marked
curvature at ordinary temperatures and become nearly linear
(though not horizontal) at higher temperatures. Such varia-
tion is better shown by an equation of the form suggested by
Mftier and Kelley,2 Cp = a + bT - c/T2. Thus for zinc oxide
the molal heat capacity is given by the equations

Cp = 6.63 + 11.26 X 10-3r - 4.72 X IQ~«T2

(2 per cent, 273 to 1600°K)
Cp = 11.40 + 1.45 X 10-3?7 - 1.824 + W6/T2

(1 per cent, 273 to 1573°K.)

1 For " low-temperature" heat capacities (0 to 298°K.) see the excellent
compilation of Kelley in U.S. Bur. Mines Bull , 434 (1941); for "high-tem-
perature" heat capacities (273°K. to the highest temperatures for which
data are available) see Kelley, ibid., 371 (1934).

2 J. Am. Chem. Soc.t 54, 3243 (1932).

152 PHYSICAL CHEMISTRY

"TABLE 23 — ^LOW-TEMPERATURE HEAT CAPACITIES1

Substance

10°K

25°K

50°K

100°K

150°K

200° K

298°K

0 66

3 36

5 11

5 83

6 06

6 20

6 39

C (diamond)

0 00

0 06

0 25

0 58

1 45

C (graphite)

0 00

0 04

0 11

0 40

0 77

1 20

2 06

0 93

5 12

8 79

10 96

11 86

12 42

13 14

0 14

1 44

3 82

5 40

5 93

6 25

6 79

NaCl

0 04

0 58

3 82

8 44

10 15

11 09

12 14

KC1

0 10

1 30

5 04

9 38

10 89

11 58

12 31

AgCl

0 40

2 95

6 59

10 00

11 22

11 88

12 14

HgO (red)

0 19

1 94

4 31

6 89

8 39

9 46

10 93

In spite of the widely different coefficients, these equations are
both valid for the heat capacity within the limits stated.

A glance at Fig. 15 will show that for elementary solids the
change of heat capacity with temperature is not a simple matter-
governed by a universal rule. Yet qualitatively all these curves

are at first convex toward the
temperature axis, with the
heat capacities at the lowest
temperatures proportional to
T3; all have nearly straight
portions followed by portions
with concavity toward the
temperature axis as the tem-
perature increases; and at
higher temperatures the
curves become more nearly
horizontal. Hence, one might
expect to derive an equation
of the same algebraic form,
with one or two characteristic
constants for each substance,
showing this change. Upon the assumption that the atoms
of an elementary crystal vibrate about their mean positions
with a characteristic frequency, independent of T, and an
intensity varying with T7, Einstein derived an equation for

> From Kelley, U.S. Bur. Mines Bull, 434 (1941), in which the heat
capacities of hundreds of substances are given. This bulletin is the best
compilation of such data.

100 200

Absolute Temperature

FIG 15. — Change of atomic heat capac-
ity with absolute temperature

CRYSTALLINE SOLIDS 153

a curve of the right form. Nernst and Lindemann assumed two
characteristic frequencies ; Debye assumed a range of frequencies
from zero to a certain maximum; others took into account the
energy absorption of the electrons, changing " degrees of free-
dom " in vibration and other factors. All the equations were
quite complex, and we shall give only the Debye equation1
applicable at very low temperatures,

C9 = 77.94 X 3/2 [j] (3)

where 6 is proportional to the maximum vibration frequency of
the atoms. At high temperatures the equation approaches
Cv = 37?, which is in fair agreement with the horizontal portions
of the curves in Fig. 15.

Much remains to be done upon the problem of heat capacity.
Thus the atomic heats of sodium, potassium, and magnesium
tend toward higher values than the 3R predicted by Debye 's
equation;2 and the elements iron, nickel, cobalt, bismuth, tin,
and chromium do not approach 3R as an upper limit of their
atomic heat capacities;3 but aluminum, copper, silver, zinc, and
cadmium do approach such a limit. The excess heat capacity
above 3R is not due to the partial heat capacity of the electrons4
in the atoms, though no explanation is known for the excess
above 3R. By taking into account the decreasing " degrees of
freedom" at low temperatures and the corresponding loss in
thermal agitation of the atoms, A. H. Compton5 derived a rela-
tion that is in good agreement with measured heat capacities over
a wide range of temperature. Other suggestions, which need not
concern us here, have appeared more recently.

Forces Acting between Atoms or Molecules. — While it must
be said that our knowledge of these forces is inadequate, the
available theory in its incomplete form allows the calculation or
close approximation of the forces in some simple crystals. The

1 Ann. Physik, 39, 789 (1913). For an excellent treatment of Debye's
theory of specific heats, see Slater, "Introduction to Chemical Physics,"
McGraw-Hill Book Company, Inc , New York, 1939.

2 LEWIS, Proc. Nat Acad. Sci., 4, 25 (1918).

3 SCHUBEL, Z. anorg. Chem., 87, 89 (1914).

4 EASTMAN, J. Am. Chem. Soc., 48, 552 (1926).
*Phys. Rev., 6,377 (1915).

154 PHYSICAL CHEMISTRY

slight compressibilities of solids indicate that the molecules or
atoms are already under very high compressive forces, so that
the application of more pressure does not largely increase the
total. The tensile strength of solids, particularly of the metals,
is an indication of large forces holding the material together, but
the true cohesive strength of a metal is not measured by the
breaking tension of a standard test bar. The fact that crystals
have constant axial ratios and interfacial angles shows the precise
nature of the forces but does not enable us to calculate the forces.
The application of X rays to crystal analysis has greatly increased
our knowledge of crystalline solids, particularly of the regular
arrangement of atoms, ions, or molecules into space-lattices, but
these data have not yet led to calculations of the forces or indeed
to a clear understanding of their nature. The work is still being
pressed actively, both by experiment and by the application of
all known theoretical means, and the results achieved so fur are
most promising even in their incomplete form.

Arrangement of Atoms in Crystals. — Before discussing the
modern work on this subject, it will be instructive to consider
briefly what knowledge preceded this work and to speculate
upon the various possible arrangements that agree with this
knowledge. It is a familiar fact that the crystals of different
substances have different external forms. The cry/.tallographer
measures the angles between the faces of a crystal, and he refers
the planes forming these faces to imaginary axes placed wit hin the
crystal. He finds that the intercepts of these planes, when
the axes have been properly chosen, occur at distances from the
origin which are to one another as simple whole numbers. The
classification of crystals is more simple when their symmetry
is considered with reference to the proper axes than when the
faces are considered. For many crystals the axes are not at 90
deg. to one another, and often the axes are of unequal lengths.1

1 All crystals may be classified according to the following systems: (1)
cubic, with the three crystallographic axes of reference of equal length
and at right angles to one another; (2) tetragonal, with only two axes
equal, but all at right angles; (3) rhombic, with three unequal axes at right
angles; (4) monoclinic, with two axes at right angles and all of unequal
length; (5) trichnic, with three oblique unequal axes; (6) hexagonal, with
three axes in a plane intersecting at angles of 60 deg. and a fourth axis
through the intersection and perpendicular to the plane; (7) trigonal, with
three axes of equal length, at equal angles other than 90 deg. For a dis-

CRYSTALLINE SOLIDS 155

Some of the crystal faces may be parallel to one or two of the axes
and so have no intercept at all upon them.

The constancy of crystal form in a given substance, regardless
of the size of the crystals, suggests that a unit of packing is
repeated over and over throughout the crystal, corresponding to
some systematic arrangement of points or volume elements in
space In elementary substances the unit might contain only
a single atom, and single atoms or ions (rather than molecules)
of compounds sometimes make up the "points" that form the
basis of the "space-lattice/' as it is called The repetition of
this unit of packing in space constitutes the structure of the
crystal.

It is interesting to speculate upon what arrangement the
atoms may take. We have no information as to the shape of an
atom or molecule, 1Ji)ut in the absence of information it will be
instructive to assume that the atoms or other structural units
that make up the crystal are incompressible spheres. We shall
see later that certain metallic elements have the internal arrange-
ment which spheres assume under pressure and shaking; but the
internal arrangement of other elementary substances is not that
taken by spheres. In binary compounds we must imagine
spheres of different sizes for the two elements, and we may
abandon the sphere concept entirely in connection with other
compounds. Thus this useful concept, like any mechanical
analogy, must not be pressed too far just because it is useful in
a few simple instances.

The fact that a substance crystallizes in a cubic system does
not mean that its atoms are arranged at the corners of imaginary
cubes; but since all crystals may be described with reference to
axes which are straight lines and since the natural faces of crystals

cussion of the development of crystal faces referred to axes in the various
systems, reference should be made to texts on crystallography or to any
standard encyclopedia.

1 Measured dielectric constants of liquids may be used to calculate dipole
moments, which in turn yield some information as to the shape of the
molecules of liquids. Such experiments have shown, for example, that
H2O and H2S are triangular, by which we mean that the atomic centers are
arranged at the corners of a triangle and not that the exterior of the molecule
is a triangle with no third dimension. The atoms in CO2 are arranged
linearly, NHs is pyramidal, and chain hydrocarbons are linear, as has been
found from the spreading experiments.

156

PHYSICAL CHEMISTRY

are planes, it seems proper to assume that the arrangement is one
in which the constituent units lie in planes. It seems reasonable
to suppose, also, that some of these planes, perhaps the most
important, are parallel to the developed faces of the crystal.
For example, in the piles of spheres shown at the bottom of Fig
16, the external form of the "crystal" is not that of a cube. The

FIG 16 — Illustrating cubic close packing

arrangement of the spheres may be shown to possess cubic
symmetry in both of these arrangements, however, by removing
some of the spheres and noticing the "unit cube" of black balls,
which is the same in both arrangements. While the external
form of the two piles of spheres is different, the internal arrange-
ment is that of a face-centered cube for both pyramids. The
different external shapes result from developing different planes.
We shall return to a consideration of the problem in three dimen-
sions after a brief examination of a simpler one in two dimensions

CRYSTALLINE SOLIDS

157

to illustrate the method of attack, but it may be suggested here
that a determination of the relative spacings of these planes
would give some information regarding the method of packing
the atoms in a crystal.

It is a familiar fact that as one rides by an orchard1 planted
in some systematic way the confusion of tree trunks is resolved
into straight rows of trees when the orchard is viewed from
certain angles. As one rides on, confusion appears to replace
regularity until presently at some other angle straight rows are
seen again. It is probably a less familiar fact that the distance

Slant I :

'-0.447
0.316

SlanW

FIG 17.

between the straight rows would be different in viewing the
orchard at different angles, but a glance at Fig. 17 will show that
this must be so. Now suppose that one is given the distance
between these straight rows of trees as viewed from several
distant points and that it is required to draw a plan of the orchard
from these spacings. A set of such spacings is given in the first
column of Table 24, with the largest distance given first and the
others in order of decreasing distance; in the second column
the ratio of each of these spacings to the largest one has been
obtained by dividing each distance by 17.7 ft.

The next step is to assume some simple plan and see whether
the relative distances between straight rows are in agreement

1 The author is indebted to Dr. W. P. Davey for the illustration of the
orchard [see Gen. Elec. Rev., 28, 586 (1925)].

158

PHYSICAL CHEMISTRY

with it. Let us assume as a beginning that the orchard is
planted with trees at the corners of squares 17.7 ft. on a side and
that the angles of view are illustrated in Fig. 17. The third
column of Table 24 shows the ratios calculated for this simple
square arrangement for the various angles, and it is obvious at
once that some of the ratios correspond to such a plan and others
do not. This is, therefore, not the correct plan, for a correct one
must correspond to all the ratios observed; but it is probable
that a square enters into the plan, since the first four ratios agree
with the experimental ones. Incidentally, the table illustrates
the need of sufficient data before reaching a definite conclusion,
for had only the first four ratios been studied it would appear
that the correct plan corresponded to a simple square. Let us

TABLE 24. — DISTANCE BETWEEN Rows OF A SIMPLE SQUAKE AND A
CENTERED SQUARE

Distance between
rows (feet)

Ratio from
experiment

(17.7 = 1)

Ratio calculated
for simple square

Ratio calculated
for face-centered
square

17 7

1 00

12 5

0 71

0 707

7 9

0 45

0 447

0.447

5.6

0 32

0 316

3.4

0 19

0 277

0.195

3 0

0.17

0 242

0.171

next assume that the plan of the orchard consists of a tree at
each corner of an imaginary square and an additional tree in the
center of each square (a centered square such as the "five"
face on dice). The spacing of the straight rows of trees as viewed
from some of the points of observation would be changed, but it
would be unchanged when viewed from some other points, such
as the 1:1 ratio. Furthermore, the largest distance between
rows would be less than the side of the assumed square, for a view
directly at the side of the square would show a row corresponding
to the trees in the centers of the squares. That this set of
measurements corresponds to a "face-centered square" of 25 ft.
is shown by the figures in the last column of Table 24. Once a
method applicable to the spacing of planes of atoms in crystals
has been developed, the problem in three dimensions may be

CRYSTALLINE SOLIDS 159

attacked in the same way, by choosing some simple arrangement
as a working basis and discarding it in favor of another as soon
as it is found to be incorrect.1

To return now to the piles of spheres shown in Fig. 16, it
will be seen from the black spheres that the arrangement is a
face-centered cube, i e., that each sphere in the face of the "unit"
formed by black spheres is equidistant from four others in the
same plane with it. The "crystal," therefore, has the same
atomic plan as the orchard, if the proper planes are considered.

Application of X Rays to Crystal Structure. — This topic, like
so many others that we consider briefly, is one about which a
book should be read as an introduction to the fundamental
theory and an outline of some of the simpler results.2 Since
only a few pages are available for the topic, it is necessary to
omit entirely the historical development,3 the means of measuring
the wave lengths,4 and the procedures by which the X-ray diffrac-
tion of single crystals or of crystalline powders has revealed the
arrangement of atoms or ions or molecules in crystals.

The fascinating chain of scientific events that has so enriched
our knowledge of crystals started in 1912 from the application
of three fundamental facts to this problem: (1) X rays were
shown to possess properties similar to light, of a wave length
about 10~8 cm , and capable of penetrating matter that was
opaque to visible light. (2) Avogadro's number (6 X 1023)
showed that atomic spacing in a crystal was of the order 10~8 cm.
(3) The plane faces of crystals made it probable that there were
planes of atoms or molecules regularly spaced throughout the
crystal.

1 More general analytical methods have been developed which are appli-
cable to the problem in three-dimensional space. See R. W. G. WYCKOFF,
"The Structure of Crystals," 2d ed , Chemical Catalog Company, Inc.,
New York, 1931

2 There are several excellent books available, of which Bragg, "The
Crystalline State/' and Wyckoff, op cit , are worthy of special mention.

3 See Richtmyer, " Introduction to Modern Physics," 1934, Chap. XIII,
for a brief but most excellent historical outline

4 The wave length was at first derived from the relative spacings of planes
parallel to the cube face, face diagonal, and cube diagonal [BRAGG, J. Chem
Soc. (London) , 109, 252 (1916)] and later by diffraction from a ruled grating
[COMPTON and DOAN, Proc Nat. Acad. Sci., 11, 598 (1925)]; see also RUAEK,
Phys. Rev., 45, 827 (1934); GOTTLING and BEAKDEN, Phys. Rev., 46, 435
(1934).

160 PHYSICAL CHEMISTRY

These facts led von Lauc to suggest to Friedrich and Knipping1
that a crystal with its three-dimensional symmetry should be
able to serve as a diffraction grating for X rays in the same way
that a ruled grating may be used to diffract visible light. By
passing a pencil of general X radiation for some hours through a
crystal mounted in front of a photographic plate, they obtained
on the plate a symmetrical pattern of spots about the image
of the transmitted beam, from which they confirmed the wave-
like properties of X rays, demonstrated the three-dimensional
space-lattice of the crystal, and showed that the wave lengths
in the beam were about 10~8 cm.

Following this discovery, means were developed for providing
nearly " monochromatic " X rays, for precise measurement of the
wave lengths, and for precise determination of atomic plane
spacing in crystals. We may assume that the internal structure
of crystals in three dimensions wa» then inferred in a way similar
to that used in the "orchard" example, though, of course, other
procedures have also been used. The fundamental equation
relating the distance d between atomic planes, the wave length
X of the X rays, and the angle 6 at which the "reflected" X-ray
beam has its maximum intensity is

X - 2d sin 8 (4)

which is known as Bragg's law.2

In order to derive the equation let the parallel dash lines of Fig. 18 repre-
sent the advancing wave front of a beam of X rays of a single wave length X
and the horizontal lines correspond to the planes of atoms in a crystal
separated by the distance d If the beam is striking at such an angle 6 that
the " reflected" beam along the line hcg is not in phase, destructive inter-
ference results and the intensity of the reflected beam is very low Only
when the angle 0 is such that the difference in the paths ecg, mhg, akg, etc ,
is a whole number of wave lengths will the reflections from different planes

1 See Sitzber. kgl. bayer. Akad. Wiss (1912) ; Jahrb. Radioakt Elektronik, 11,
308 (1914), for the first papers on the topic. An excellent account of the
later developments, experimental technique, and interpretation of the
photographs is given in Wyckoff op. cit., and especially in Bragg, op cit.
For briefer accounts see Ruark and Urey, "Atoms, Molecules, and Quanta,"
pp. 209-236 (1930), or Richtmyer, op cit. Chap. XIII.

*Proc. Cambridge Phil Soc., 17, 43 (1912). The usual form of the law
is nX = 2d sin 6, where n is a whole number called the "order " and signifying
the number of wave lengths by which the paths of the X-ray beam differ
when there is constructive interference.

CRYSTALLINE SOLIDS 161

reinforce one another and give rise to an intense reflected beam, for the
apparent reflection of X rays differs from ordinary reflection of light in that
the beam penetrates into the crystal and gives rise to reflection from many
planes. It will be seen also that, unless the planes of atoms are accurately
spaced at the distance d, destructive interference would take place for all
incident angles of the beam and there would not be any reflected beam of
marked intensity. Suppose 6 is so chosen that the reflected beam has

•*4T) /^x

/ \ J-''

FIG IS

maximum intensity, the difference in the paths ccg and bhg is a whole num-
ber of wave lengths n\. As ec IK equal to bj and hf is equal to he, the differ-
ence in path bhg — ccg = he - hj and this is equal to 7i/ — hj, or jf Now
jf divided by cf is the sine of the angle 6, and cf is twice the interplanar dis-
tance, then it follows that

n\ = Id sin 0

The Unit Cell. — In considering the internal structure of
crystals it is convenient to imagine that the space is divided
into identical unit cells of suitable dimensions such that each
cell contains a unit of the pattern. The points at which atoms
occur in this cell form the space-lattice which shows how the
"design " is repeated. The cell is made as email as it may be and
still be identical with every other cell. In the cubic system to
which we shall confine most of our attention in this brief treat-
ment, the unit cell is a cube; but in other types of crystals the
planes bounding the cells may meet at angles other than 90 deg.,
and the lengths of the edges of the cells may not be equal. If
one corner of a cell is taken as the origin, the edges of the cell
along the x, yy and z axes are a, b, and c. In place of giving the
actual lengths of these edges, it is usually sufficient to express them
in terms of b as unity, but in the cubic system a = b = c = 1.

Types of Unit Cells. — The sketches1 in Fig. 19 show the types
of cell in the cubic system and the hexagonal close-packed cell.

1 From the Department of Metallurgy at Massachusetts Institute of
Technology. The dimensions of the cells are in angstrom units, of which
lA - 10~8 cm.

162

PHYSICAL CHEMISTRY

These cells have been drawn in the conventional manner, but
it should be understood that in all of them the "corner" atoms
are also the " corner" atoms of other cubes formed by extending
the plane faces beyond the distances shown and that those
in the faces of the cubes lie in the faces of the adjoining cubes.
An element of space such as that shown for the face-centered
cube contains one-eighth of each of the atoms shown at the
eight corners and one-half of each of the atoms shown in the
cube faces, or a total of 4 whole a-toms. Similarly, the body-
centered cube contains one-eighth of each of the eight corner

Face-Centered Cubic Body-Ceniered Cubic Diamond Cubic Hexagonal Close Packed

ELEMENT

404

556

Fe (y)

361

355

354

360

382

395

406

512

380

393

408

491

504

^>*~~

f^<

£&

/ \

- " '"*•"-"- i

ELEMENT

c/a

22V

Ibtt
762

12
30

322

267

186

296

IB9

297

159

323

159

365

163

251

163

269

159

271

159

FIG. 19. — Crystal structures of elements

atoms and all of the Center atom, or a total of 2 atoms; the
hexagonal cell contains 3 atoms entire, one-half of each of 2,
and one-sixth of each of 12, or 6 altogether.

The face-centered cubic arrangement is obtained by dividing
the space in a crystal into closely packed cubes and placing an
atom at each cube corner and at the center of each cube face.
This arrangement is also called cubic close packing and is
one of the two alternative arrangements that hard spheres
of equal size assume when closely packed by pressure and shak-
ing. The body-centered cubic arrangement has an atom at
each cube corner and at the center of each elementary cube.
Spheres so arranged are not so closely packed as in the face-
centered cubic arrangement, and this arrangement is not stable

CRYSTALLINE SOLIDS 163

for spheres. Hexagon close packing is obtained by dividing
the space into equal, closely packed, right-triangular prisms, the
bases of which are equilateral triangles and the altitudes 1.63
times the side of the triangles. An atom is located at each prism
corner and at half of the prism centers. This is the second
alternative arrangement assumed by equal spheres under pressure
and shaking. As has been said before, the concept of a spherical
unit is not necessarily the correct one, but the arrangements
that have beon described are those actually assumed by the atoms
in a considerable number of crystals of elements and compounds.

The simplest arrangement of all would appear to be that
obtained by dividing the space into equal elementary cubes
with the center of a sphere at each cube corner. Such an
arrangement is not stable for equal spheres that are pressed
and shaken, and no elementary substance has this arrangement,
though some compounds have structures of this kind, involving
spheres of two different sizes, as we shall see later.

The Coordination Number. — In any symmetrical arrangement
of spheres or points repeated in space of three dimensions, each
sphere or point would have a certain number of " nearest neigh-
bors/' and this number is defined as the coordination number.
For example, if a rectangular box of which the dimensions are
whole multiples of 1 in. is filled with uniform spheres 1 in. in
diameter in such a way that all the edge members of each layer
touch the sides of the box, the arrangement has simple cubic
symmetry, for each sphere is in contact with six others, its nearest
neighbors.

In the body-centered cubic arrangement shown in Fig. 19,
which could be produced in the box of spheres by shifting every
other layer half the radius in two directions and decreasing the
vertical spacing of the layers, the coordination number is 8.
Each sphere in the second layer, for example, is in contact with
four in the first layer and four in the third layer, these eight
forming the "unit cube." Of course, the spheres in the second
and fourth layers form "unit cubes" in which the spheres in
the third layer are the center spheres, so that, except for the
outside spheres touching the box, each one has eight nearest
neighbors.

In the face-centered cube (Fig. 20) the coordination number is
12. Consider for a moment the spot in the front face of the cube

164

PHYSICAL CHEMISTRY

from which the 4 spots F, G, B, and E are separated by half the
diagonal of the cube face. Four others in the plane a/2 behind
this front face are also half the diagonal of a cube face from it;
and if we imagine another plane a/2 in front of the plane contain-
ing F, (7, B, and E, it will also contain 4 spots at this distance
from the one in the center of the face FGBE, making a total of
12 at the distance a/\/2, or 0.707a, from center to center.

In hexagonal close packing the coordination number is also
12, as may be seen from Fig. 19. The spot m the upper face, for
example, has six spots in the plane of this face, three in the plane
c/2 below it, and, of course, another three in the plane c/2 above
this plane. Since hexagonal close packing has the same coordi-
n nation number as that of the face-
centered cube and both arrangements
are stable for spheres, it might seem
at first thought that the arrangements
were identical and made to appear
different by an artificial choice of
volume element ; but this is not true.
Hexagonal close packing could be
changed to face-centered cubic pack-
ing by moving the three "inside"
spots of the hexagonal unit cells packed
above and below the ones shown in
Fig. 19 around the vertical axis 60 deg., but keeping them in the
same horizontal plane. This may readily be seen by packing at
least four layers of spheres in a glass box or frame; but it is some-
what difficult to imagine from the single cells sketched, and plane
drawings of several cells are too confusing to be useful. The
two arrangements give slightly different densities, which are
nevertheless real, again confirming the fact that the arrange-
ments are not quite the same.

Other coordination numbers are also found in crystals. The
lowest possible coordination number would, of course, be 1,
corresponding to two spheres in contact, with these pairs arranged
in a symmetrical lattice spaced at a distance greater than a
sphere diameter. Another possible arrangement would be in
linear chains, in which 2 would be the coordination number.
In the diamond cubic arrangement sketched in Fig. 19, each
sphere has four nearest neighbors arranged with the centers

FIG 20 — Face-centered cubic
unit

CRYSTALLINE SOLIDS 165

forming a tetrahedron around it. This will be clearer from Fig.
21, in which spheres are arranged in this same way.

From the distance between atomic centers in an elementary
crystal and the coordination number, we may calculate the radii
of equal spheres which will just be in contact when packed in
this way. This calculated quantity is commonly called the
atomic radius or distance of closest approach, though, of course,
we have no knowledge that the atoms are actually spheres or
actually of any recognizable "shape "

In the discussion of chemical compounds
later m the chapter, especially of compounds
in which the lattice unit is an ion, we shall
consider ionic radii as well, and these will not
in general be equal for the two ions in a
crystal. The point which should be made

here is that the atomic radius of sodium in FIG 21 Ar-

sodium metal, for example, will not be the laiiRementof

. &pheio& in totra-

sarne as the radius ot sodium ion in a sodium hodial symmetry

chloride crystal for several reasons, of which (diamond-type

•ii i • 14 lattice),

some will be given later.

Arrangement of Atoms in Elementary Crystals. — Crystalline
structures of the true metals are characterized by their extreme
simplicity and by the closeness of packing. The common
arrangements are face-centered cubic and hexagonal close packed,
in each of which the coordination number is 12, representing the
closest packing of spheres; and body-centered cubic with a
coordination number of 8 and ih which the packing is not
quite as close as in the first two types. Some correlation of
arrangement with physical properties has been observed; for
example, the metals that are ductile and good conductors of heat
and electricity (Cu, Ag, Au, Al) are face-centered cubic. But it
is not safe to generalize that all face-centered cubic metals will
have these properties to an exceptional degree compared with
those of some other symmetry.

There is a tendency for members of the same group in the
periodic table to show the same symmetry (for example, Li, Na,
K; Cr, Mo, W; Cu, Ag, Au), but exceptions are found. It
should be noted that the tetrahedral arrangement shown by C,
Si, Ge, and gray Sn, all in the fourth column of the periodic
table, is not shown by Pb, which is also in the triad with Ge and

166 PHYSICAL CHEMISTRY

Sn. The high melting point of carbon is less marked in the
succeeding elements (Si melts at 1420°, Ge at 958°), though the
hardness persists in Si to some extent and is especially con-
spicuous in SiC, which is of the same structure.

The nonmetallic elements N2, 02, Br2, and I2 have these
diatomic molecules as the unit in the crystal, rather than atoms,
which is to be expected from the stability of the molecules in the
vapor. Chlorine has a different arrangement of molecules from
bromine and iodine, which shows again that not all elements in
one column of the periodic table have the same structure. The
structure of crystalline fluorine has not yet been determined.

Only ftie simpler structures for elements are discussed here,
but it will be understood that not all elements c^stallize in the
cubic system, and hence the structures of some of them are more
complicated than one would infer from the examples given. The
atoms in most of the elementary structures outside of the cubic
system are arranged symmetrically with coordination numbers
of 2, 4, 8, 12, etc., as is true of cubic crystals, but of course the
axes are unequal or inclined at angles other than 90 deg., so that
the "unit cell" is not a cube, but another geometric unit.

As has already been suggested, attempts to explain hardness,
melting point, thermal or electrical conductivity, color, ductility,
or other physical properties of crystalline elementary solids in
terms of the arrangement of atoms in crystals have been only
partly successful. Some of these properties depend upon the
nature of the bonds between atoms and the part taken by the
electrons in these bonds — doubtless upon other factors as well.
Much experimental work is still being done, and many of the
facts already known await satisfactory interpretation. The
bare outline of some of the work given here will suffice to show its
general nature; full accounts are available to those who wish
to study further.1

Arrangement of Atoms in Binary Compounds. — When the
elements forming a binary compound come from widely sepa-
rated columns of the periodic table, the chemical bond is usually
due to a complete (or nearly complete) transfer of an electron

^TILLWELL, " Crystal Chemistry," McGraw-Hill Book Company, Inc.,
New York, 1938, and EVANS, " Introduction to Crystal Chemistry/'
Cambridge University Press, London, 1939, are suitable texts in which to
read further on the correlation of properties to internal arrangement.

CRYSTALLINE SOLIDS

167

from one atom to another.1 An alkali metal readily loses its
one valence electron to chlorine or other halogen, which has
seven valence electrons, so that the outer shell of eight is com-
pleted in the halogen. The crystals of such substances are
presumably formed of ions and are termed ionic crystals. X-ray
diffraction shows the positions of the atomic centers; but since
the ions do not have equal " atomic radii/' the conventional rep-
resentation of the structure is by spheres of unequal size repre-
senting the two elements. It does not follow that the radius of
sodium ion in sodium chloride is the same as that in sodium

FIG. 22. — Sodium chlonde FIG 23 — Arrangement of atoms of

structure sodium (small spheies) and chloime iri

sodium chloride

bromide, for the different atomic volume and the larger number
of electrons in bromine alter the volume available to the sodium.
It should not be assumed that NaCl and CsCl have the same
internal arrangement (for they do not), nor does it follow that
another compound of the type XY will have the arrangement of
either NaCl or CsCl We consider briefly some simple examples.
In sodium chloride the ion centers of sodium and chloride
ions alternate at the corners of equal cubes, as sketched in Fig.

1 The two types of bond which we need to consider are the so-called " polar
bond," which results from a complete transfer of an electron from one atom
or group to another, and the covaleiit or homopolar bond, which results from
the sharing of a pair of electrons by two atoms, as in the compound C12. In
general, an atom of the nth group may share no more than (8 — ri) electrons.
Thus in Oa the atoms share two pairs of electrons, corresponding to a chem-
ical valence of 2

168 PHYSICAL CHEMISTRY

22, which shows the conventional unit cube. A photograph of
an arrangement of large dark spheres, representing chloride ions,
and smaller white ones representing sodium ions is shown in
Fig. 23, which is eight "unit cubes.7' It should be understood
that the corner ions in Fig. 22 differ in no way from those in the
face centers, for this pattern is repeated over and over again in
space. One-eighth of each corner ion, one-half of each face-
centered ion, one-fourth of each ion in the cube edge lies within
the cube shown. The coordination number is 6, each ion of
sodium having six neighboring chloride ions and each ion of
chloride six neighboring sodium ions. Although the structure
io apparently a simple cubic one, it is not commonly so called;
for a cube having half the edge of that sketched in Fig 22 would
not show the correct structure by repetition in space This
is the structure of most alkali hahdes (though not of all of them)
and* of many oxides and sulhdes. It is commonly called the
"sodium chloride structure/7

The radius assigned to sodium ion in sodium chloride is 0.96 A,
and that assigned to chloride ion is 1 83A It may be noted
for comparison that the atomic radius of sodium atoms in sodium
metal is 1.86 A.

Cesium chloride is a body-centered cubic structure in which
* half the atoms are different from the other half. It is also
an ionic crystal, and one in which the assigned ionic radii are not
equal. For each ion in this structure the coordination number
is 8. The hahdes of cesium, thallium, and ammonium are
other examples of ionic crystals of this type, but nonionic crys-
tals of compounds are* known that are of this type also. In
the ammonium halides we have an example of an ionic group
(NH4+) forming the unit of structure, and this is true of the
structure of other compounds such as nitrates and carbonates.

As another illustration of a binary compound having an
arrangement similar to that of an elementary substance, ZnS
has the diamond structure with half the atoms unlike the other
half. This structure is shown by many less polar binary solids.
Presumably, but not certainly, the units in this structure are
atoms rather than ions. As has been said before, one must not
assume that chemically analogous compounds have the same
structure; for example, ZnO does not have the same structure
as ZnS, but SiC has the same structure as ZnS.

CRYSTALLINE SOLIDS 169

These examples will show the general nature of the arrange-
ment in simple crystals, though not all binary compounds
crystallize in the cubic system, of course, and not all the types
have been listed. Further complications arise in more complex
crystals, as would be expected, but the structures of many
hundreds of crystalline solids have been worked out1 by the
application of X-ray diffraction.

Though correlation of crystal structure with physical properties
is not a simple matter, since several different factors are involved,
it is generally true that increasing distance between atomic
centers in ionic crystals is attended by decreased hardness and
lower melting point. In crystals of substances joined by homo-
polar bonds (shared electrons), these forces hold together the
two atoms in the molecule, and the crystal structure derives its
strength from other less intense forces that are described by
the vague term "residual." Such crystals will usually be of
much less strength and of lower melting point, though the cor-
relation of properties to structure is more difficult for these
substances.

Many inorganic crystals are probably not of the ionic type
but consist of atoms. This is particularly true of crystals of
intermetallic compounds, most of which have bonds similar
to those in crystals of a single metal. Crystals of organic
compounds usually consist of molecules arranged in space-
lattices. The chemical bond is probably "covalent" in these
substances, which is to say that two elements share one or more
electron pairs, rather than transferring electrons more or less
completely from one atom to another as in "polar" compounds
such as sodium chloride. Since hydrogen atoms diffract X rays
to a comparatively slight extent, the crystal study by this
method usually locates the other atoms in an organic compound
and leaves the position of hydrogen to be inferred.

Determination of Avogadro's Number. — Since wave lengths
of X rays may be determined from ruled gratings, their diffrac-
tion by crystals furnishes a means of calculating Avogadro's
number from the size of the "unit cell" in a crystal of known
structure. For example, the "unit cube" shown in Fig. 22
contains 4 atoms of sodium and 4 atoms of chlorine. The edge

1 Most of them are described in Wyckoff, op. cit., and in the 1935 supple-
ment; nearly all of them are given in the six volumes of "Strukturbericht."

170 PHYSICAL CHEMISTRY

of this cube is 5.638 X 10~8 cm., or its volume is (5.638 X 10~8)3
cm3. The density of NaCl is 2.163, whence the volume occupied
by 4 gram atoms of each element in the compound is

4(23 0 + 35.45)
2. 103

or 108.1 cm3. The ratio of the volume of 4 "gram molecules " of
NaCl to the ^olume of 4 "molecules" of NaCl is

108 ] = 6.03 X 1023

(5638 X 10-*) 8

which is the number of molecules per mole.

Structure of Surfaces. — We have seen that in crystals the
atoms or other structural unite are held together in symmetrical
patterns by something which may be called "bonds " These
atomic or molecular forces, or "bonds," are exerted in all direc-
tions within the body of the crystal, no doubt chiefly upon the
immediate neighbors, but possibly upon a second or third "layer"
as well. Molecules or atoms in the surface of a crystal may
be presumed to have these forces unsatisfied outside the crystal.
If the crystal is in contact with its vapor at a sufficient pressure
or with a solution of the substance at a sufficient concentration,
it will add on other layers and grow in size. This growth of
crystals, which may be readily observed in the laboratory, is
evidence of the existence of the residual forces.

Lacking an opportunity to attach molecules of its own kind,
the crystal may attach molecules of some other substance.
The "bond" hoi ding -such molecules is possibly of a different
character and less intense than a "bond" to a molecule that
may fit into the crystal lattice, though we have no means of
showing how the molecule may be held. Experimental evidence
is available for the formation of attached layers of nitrogen
upon mica,1 of water vapor and other gases upon glass or silica,
of many gases upon charcoal,2 and of many solutes upon charcoal
or other solids.

An initial monolayer might be held by the residual forces at
the face of the crystal. The formation of a second layer could
result only if the crystal forces reached out into space more than

IL,ANGMUIR, /. Am. Chem. Soc., 40, 1361 (1918).

2 For example, see COOLIDGE and FOKNWALT, ibid , 66, 561 (1934).

CRYSTALLINE SOLIDS 171

molecular distances (which is considered improbable) or by the
forces acting between the molecules of the attached substance.
The latter effect would resemble condensation to a liquid phase,
and adsorbed layers form upon surfaces when the pressure of
the gas supplying the attached layers is a very small fraction
of that necessary for true condensation.

Adsorption. — This term is commonly used to signify an
attached layer upon a solid or liquid surface such as is discussed
in the previous section. The mechanism of adsorption is
described by Langmuir1 as follows:

. . . when gas molecules impinge against any solid or liquid surface
they do not in general rebound elastically, but condense on the surface,
being held by the field of force of the surface atoms. These molecules
may subsequently evaporate from the surface. The length of time
that elapses between the condensation of a molecule and its subsequent
evaporation depends on the intensity of the surface forces. Adsorption
is a direct result of this time lag. If the surface forces are relatively
intense, evaporation will take place at only a negligible rate, so that
the surface of the solid becomes completely covered with a layer of
molecules. In cases of true adsorption this layer will usually be not
more than one molecule deep, for as soon as the surface becomes covered
by a single layer the surface forces are chemically saturated. When, on
the other hand, the surface forces are weak the evaporation may occur
so soon after condensation that only a small fraction of the surface
becomes covered with a single layer of adsorbed molecules.

In agreement with the chemical nature of the surface forces, the
range of these forces has been found to be extremely small, of the order
of 10~8 cm. That is, the effective range of the forces is usually much
less than the diameter of the molecules. The molecules thus orient
themselves in definite ways in the surface layer since they are held to
the- surf ace by forces acting between the surface and particular atoms
or groups of atoms in the adsorbed molecule.

The atoms in the space-lattice may be thought of as resembling
a "checkerboard" on which adsorbed molecules take up definite
positions. Since not all the atoms in the crystal face are alike,
not all the spaces will necessarily hold an adsorbed atom or
molecule. Large molecules might occupy several spaces or at
least prevent the occupation of adjoining spaces by other mole-
cules. If nearly all the gas molecules striking a solid surface
condense and if a molecule of gas striking another molecule of

1 Ibid., 40, 1361 (1918).

172 PHYSICAL CHEMISTRY

gas already adsorbed evaporates immediately (or rebounds
elastically), the rate of condensation will be proportional to
the pressure of the gas and to the fraction of the surface that is
bare. The rate of evaporation will be the product of the rate
for a saturated surface and the fraction of the surface covered;
and at equilibrium the two rates will, of course, be equal.

At low gas pressures the amount of adsorbed gas usually
decreases rapidly as the temperature is raised, since this greatly
increases the rate of evaporation At high pressures the surface
may be nearly covered with a monolayer, so that the adsorption
varies only slightly with increasing temperatures.

Much of the experimental work tending to show that adsorbed
layers are or are not monomolecular is difficult to interpret,
owing to the uncertainty as to the actual area of adsorbing
surface available. For the area of a square centimeter of
"rough" surface has no meaning, and when molecular dimensions
are considered smoothness may be an ideal beyond attainment.

Langmuir has derived an expressionjfor the fraction of a solid
surface covered by an adsorbed layer of molecules of gas at
equilibrium, in terms of na, the number of molecules striking a
square centimeter of surface each second [which may be com-
puted from equation (14), page 86], the fraction x of these
molecules that condenses upon the surface (usually near unity),
and ne, the number evaporating each second from a square
centimeter of completely covered surface This relation is

TL 7*

Fraction covered = **

He + USX

Experiments show that this relation is valid insofar as one is
able to determine the quantities appearing in it. The chief
difficulty lies in determining the actual area of the solid surface.
A more common but less accurate relation, the Freundlich
equation, gives the quantity of adsorbed substance 'as

q = apl/n (5)

where q is the quantity of adsorbed substance per unit area of
surface, p is the pressure, and a and n are constants. Over
narrow ranges of pressure the equation fits experimental data
fairly well, though the term n is not a constant but a function of

CRYSTALLINE SOLIDS

173

the pressure. This may be seen in Fig. 24, which is a plot of the
data in Table 25. At low pressures the adsorption might well
be expected to be proportional to the pressure (i.e., to the num-
ber of molecules striking the surface), while as the pressure is
increased the surface layer approaches saturation and there is
no further increase of adsorption because there is no more
uncovered surface at which the residual attraction of the surface
atoms can act

TABLE 25 — ADSORPTION OF NITROGEN ON MICA AT 90° ABSOLUTE

Pressure (dynes
per squaie

Moles adsorbed
X 106

( Calculated from
Freiindhch

Per cent deviation
of Freundhch

centimeter)

equation

34 0

1 37

1 54

+ 11

23 8

1 28

1 31

+ 3

17 3

1 17

1 14

13 0

1 06

1 01

- 5

9 5

0 995

0 883

-12

7 4

0 90

0 795

-11

6 1

0 79

0 726

- 7

5 0

0 707

0 68

- 4

4 0

0 628

0 62

- 1

3 4

0 556

0 58

+ 4

2 8

0 500

0 536

+ 7

The calculated values were obtained from the equation qp = 8 4p(} 417
At the lowest pressure the slope of the plot (log p against log q) corre-
sponded to l/n = 0.68; at higher pressures it decreases to 1/n = 0 20.

LangminVs adsorption data for nitrogen are given in Table 25
as typical of modern work.1 These results were obtained by a
simple and ingenious method. A quantity of mica whose
area was 5750 sq. cm. was placed in one of two connecting
bulbs of nearly equal volume, and both bulbs were very care-
fully and completely exhausted. A small quantity of nitrogen
was admitted to the empty bulb, and its pressure was deter-
mined. Then connection was established between this bulb
and the one containing mica, and the pressure was measured
again. The difference between the pressure to be expected from
the relative volumes of the two bulbs and the pressure actually

1 A summary of the numerous papers of Langmuir and his associates dur-
ing the last 20 years is given in Science, 87, 493 (1938).

174

PHYSICAL CHEMISTRY

measured gave the quantity of nitrogen that had been adsorbed.
Next, the tube connecting the two bulbs was closed, and the one
containing no mica was carefully pumped out again. When the
connecting tube was opened a second time, the difference between
the expected and observed pressures was a measure of the amount
of nitrogen adsorbed on the mica at the lower pressure.

In order to evaluate the constants of the Freundlich equation,
log q was plotted against log p (solid line), and n was so chosen
as to give a " straight line" through these points. As will be seen
from Fig. 24, the Freundlich equation (represented by a dotted

l.U

1.6
1.2

PL,

-^0.8
0,4

FIG. 2^

/ ^

/-''

A -6.3 -fc.2 -6.1 -6.0 -5.9 -5.fi
1°9<1

t — Adsorption as a function of pressure.

line) is not a very satisfactory one for expressing adsorption as
a function of the pressure.

Adsorption decreases as the temperature is raised. Therefore,
when it is desired to remove an adsorbed film of gas from a solid
surface, this is usually done by pumping out at a high tempera-
ture. Thus the evacuation of double-walled flasks for the
storage of liquid air is usually carried out at a temperature just
below the softening point of the glass. Since adsorption increases
at lower temperatures, the evacuation of a flask may be made
fairly complete by attaching it to a bulb filled with charcoal
and immersing the charcoal bulb in liquid air while gently warm-
ing the flask to be evacuated.

Experiments on adsorption of gases at high pressures and
with materials of large surface for a given weight are more
difficult to interpret, and the quantity of gas adsorbed by a unit

CRYSTALLINE SOLIDS

175

weight of adsorbent is not a simple function of the pressure,
as may be seen from the data expressed in Fig. 25 for nitrous
oxide adsorbed on charcoal.1

While the formation of monolayers on solids is greatly influ-
enced by the surface lattice of the solid, such layers forming on
liquids are probably not dependent upon the structure of the
underlying liquid. Oriented monolayers of solutes also form at
liquid-liquid interfaces and at liquid-solid interfaces. These
layers are of the greatest importance in determining the stability
of emulsions and suspensions, in the concentration of minerals

R
FIG. 25 — Adsorption isotherms for nitrous oxide on charcoal.

by froth flotation, and other processes. Some of these matters
will -be considered in a later chapter.

Liquid Crystals. — Certain substances of complex organic
nature melt to turbid liquids having quite different properties
from those of ordinary liquids. As the temperature is further
raised, a point is reached at which each liquid changes sharply to
a clear liquid of ordinary properties. The substance thus shows,
in addition to its usual melting point, another transition tempera-
ture at which it assumes the properties of liquids. While in
this intermediate state, the liquid exhibits double refraction,
a property characteristic of crystalline substances. When a

1 COOLIDGE and FOBNWALT, J. Am. Chem. Soc., 66, 561 (1934).

176 PHYSICAL CHEMISTRY

beam of light passes through a doubly refracting substance, there
are two emerging beams, only one of which follows the ordinary
laws of refraction, and the rays are polarized. This occurrence
is characteristic of substances which are not isotropic, i.e.,
whose properties are not the same when measured in different
directions. It follows that the intermediate "liquid" state is
one in which the properties of the liquid are not the same in all
directions. Lehmann1 calls this intermediate condition the
"liquid-crystalline" state; perhaps a better name would be
doubly refracting liquids. Apparently weak forces such as
those acting in crystals are at work arranging the molecules in a
kind of space-lattice similar to that of crystals, but less definite in
character. The sharp disappearance of this double refraction at
a definite temperature bears a resemblance to the melting point
of crystals, except that in this case the substance is already fluid.

An early explanation of liquid crystals (Nernst, Bose) was that
there were "molecular swarms," but this idea has been found
inadequate to explain the observations. Born2 and Voigt3 both
consider that in liquid crystals there is an arrangement of the
molecules in some particular way, perhaps parallel to one another
with respect to some one axis, and that this is responsible for
the behavior of liquids in this peculiar state. If there is a space-
lattice, it differs sharply from the one found in solids. At the
second transition point, or clearing point, this molecular lattice
is lost, and with it the double refraction characteristic of aniso-
tropic substances.

Over 170 substances showing two transition points4 have been
prepared. A study of "them has shown no space-lattice detecta-
ble by the usual X-ray methods applicable to solid substances.
These liquid crystals have optical rotatory powers as high as
4000 deg. for a film 1 mm. thick; a quartz plate of this thickness
has a rotation of only about 25 deg. There is apparently no
relation between the constitution of the compounds and their
capacity for producing liquid crystals.6 It may be that all

1 A review of his very numerous papers on this subject is given in Physik.
Z., 19, 73 (1918).

*Sitzber. kgl. preuss. Akad. Wiss., 1916, 614.

8 Physik. Z., 17, 76, 152 (1917).

4 Engineering, 106, 349 (1918); a review of the subject.

6 CHAUDHARI, Chem. News, 117, 269 (1918).

CRYSTALLINE SOLIDS

177

organic substances are capable of forming liquid crystals, but
the temperature ranges of their existence are so small that they
have escaped detection. This is rendered unlikely by the fact
that some of the substances exhibit their peculiar properties
through a range of 35°. A few examples are mentioned in Table
26.

TABLE 26 — SUBSTANCES FORMING LIQUID CRYSTALS 1

Transition tem-

Range of ex-

Substance

peratures,
degrees

istence of
liquid crys-
tals, degrees

Oholesterin benzoato

145

179

p-Azoxyamsole

118

136

p-Azoxyphenetole

134

169

Pyndme nitrate

105

Qumolme nitiate

102

119

p-Methylaminobenzaldehyde phenyl hv-

drazoiie

170

190

p-Ethylammobenzaldehyde phenyl hydra-

zone

160

181

Problems

1. The beat of fusion of monoclmic sulfur is 13 cal per gram, the melting
point is 119°, tbe density of the solid is 1 960, arid that of the liquid is 1,80
Calculate the melting point at 50 atm pressure

2. The vapor pressure of ice is 4 58 mm. at 0° and 3 28 mm. at —4°.
Calculate the heat of sublimation of ice

3. Calculate the heat of sublimation of iodine at 110° from the vapor
pressures on page 146

4. "The unit cell of chromium is a cube of edge 2.89A, its density is 7.0
Calculate this density upon the assumptions of (a) face-centered and (b)
body-centered structure

5. (a) Calculate the weight of nitrogen gas necessary to cover the surface
of a cube of 1-liter capacity with a layer one molecule deep, making a
reasonable assumption as to the diameter of an atom, and assuming both
atoms of the molecule in contact with the adsorbing surface, (b) How
large an error would the loss of these molecules produce in the calculated
pressure at 20° and 1 atm.?

6. MgO has been shown to have the sodium chloride arrangement, and
the edge of a cube containing 4MgO is 4. 20 A Calculate its density.

1 ROTABSKI, Ber., 41, 1994 (1908).

178 PHYSICAL CHEMISTRY

7. Given the density of KI as 3. 1 1, calculate the edge of a cube containing
4KI, assuming the sodium chloride arrangement. The measured edge is
7.1JL

8. Copper crystallizes in the face-centered cubic arrangement, and the
edge of the unit cube is 3 6A. Show that the density calculated upon the
assumption of this arrangement is in agreement with the measured density,
which is 8.93.

9. Cesium chloride forms a body-centered cube arrangement, and the
cube containing ICsCl has an edge of 4 12A Show that this anangement
is in conformity with its measured density and not in conformity with the
arrangement in which most of the alkali halides crystallize The density of
CsCl is 3 97.

10. The edge of the unit cell of lead is 4 92A, and the stiucture is face-
centered. Calculate the sine of the smallest angle at which constructive
interference of X rays of wave length 0 708A would occur for planes of
atoms parallel to the cube face and to the face diagonal

CHAPTER VI
SOLUTIONS

The solutions that are to be studied in this chapter are liquid
phases in which a gas, liquid, or solid solute is molecularly
dispersed. Solutions in which the solute is ionized are con-
sidered in the next chapter; " solid solutions" are discussed
briefly in Chap. XI; colloidal " solutions," in Chap. XVII. Such
a subdivision of the general topic of solutions brings us to the
simpler systems first. The experimental quantities used in study-
ing solutions are solubility, partial pressure of solvent vapor
above the solution, partial pressure of solute vapor, boiling point,
freezing point, and osmotic pressure and the changes in these
properties with changing temperature or pressure or composition.
We shall develop equations relating some of these properties to
others that are exact for very dilute solutions and useful approxi-
mations for stronger solutions; and it will be necessary to exercise
some judgment in applying them to solutions that are not dilute,
as it was necessary to use the ideal gas law with discretion at
high pressures or low temperatures.

Solubility. — There are no fixed rules by which to predict
whether a substance will dissolve in a given liquid or not or to
what extent. The probability that a solution can be formed
increases with the resemblance of the solvent to the dissolved
substance; hence most closely related liquids mix with one
another in all proportions. Chemically unlike substances, such
as water and silver nitrate or water and sodium chloride, also
form solutions over a wide range of compositions; yet silver
chloride dissolves in water scarcely at all. Carbon bisulfide is
soluble in all proportions in alcohol, but very slightly soluble in
water, though water and alcohol are soluble in one another in
all proportions. Hence direct experiment is the only method of
determining solubility. The solubility of a substance in a given
liquid is a function of the temperature and the pressure, though
variations in atmospheric pressure produce only negligible

179

180 PHYSICAL CHEMISTRY

changes in the solubilities of liquids and solids. Large varia-
tions in pressure may cause large changes in solubility even in
these " condensed " systems, and the solubilities of gases change
in direct proportion to the partial pressure at low pressures
Most solubilities at constant pressure increase with increasing
temperature, some decrease with increasing temperature, and a
few do first one and then the other. Plots of solubility against
temperature for a single crystalline form of solute are smooth
curves. Sudden breaks in a solubility-temperature curve indi-
cate a change in crystalline form or crystalline composition; for
example, Na2SO4.10H2O changes to rhombic Na2S04 without
water of crystallization at 32.38°, and at this temperature there
is an abrupt change in the curve showing the solubility of
" sodium sulfate" as a function of temperature.

Concentration in Solutions. — The composition of a solution
may be expressed in a great many ways, such as the number of
moles or equivalents of dissolved substance (called the solute)
per liter or per 1000 grams of dissolving liquid (called the
solvent) or per liter or 1000 grams of solution. Unfortunately
for clearness, each of these quantities is sometimes called a con-
centration; and since each such " concentration " is a convenient
quantity in some kinds of work, no one of them has a greater
claim to the term than any other. For our purposes two of
these " concentrations " will fill almost every need. The molality
of a solution is defined as the moles of solute per 1000 grams of
solvent, and it will be better to form the habit of calling it the
molality rather than the molal concentration. The volume con-
centration is defined as the moles of solute per liter of solution.
Of course, the equivalent concentration is defined as the number
of equivalents per liter of solution, as is customary in analytical
chemistry and as will be requisite in considering some of the
electrical properties of solutions. The molality of a solution has
the advantage that it does not change with the temperature,
whereas volume concentrations change with temperature owing
to thermal expansion.

For many purposes the mole fraction of solvent or solute in a
solution is a convenient method of expressing composition. This
quantity is defined for any component as was the mole fraction
in a gaseous mixture, namely, as the number of moles of it in a
mixture, divided by the sum of the moles of all substances present.

SOLUTIONS 181

An example will make these definitions clearer. A solution
containing 10 per cent by weight of ethanol (C2H6OH = 46.0)
has a density of 0.9839 grams per milliliter at 15.5°. A
liter of this solution contains 98.39 grams, or 98.39/46.0 = 2.14
moles of ethanol; and it contains 885.5 grams, or 49.4 moles of
water. In this solution the volume concentration of ethanol
is 2.14; its molahty is 2.14/0.8855 = 2.42; its mole fraction is
2.14/(2.14 + 49.4) = 0.0416. Our standard notation for these
quantities is C = 2.14, m = 2.42, and x = 0.0416.

Ideal Solutions. — The ideal solution, like the ideal gas, is a
convenient fiction that is closely approached by some actual
solutions at moderate or high concentrations and by most
solutions at low concentrations of solute. There is no solution
that conforms strictly to the laws of ideal solutions, just as there
is no gas that conforms strictly to the equation pv = nRT. Yet
each serves the same useful purpose ; namely, it provides an ideal
that is approached by actual system at low concentrations and
a means of obtaining approximations when data are lacking.
There are many solutions of which the actual properties are
within 1 or 2 per cent of those calculated for an ideal solution
and many circumstances in which a knowledge of the properties
of the solution within this accuracy is desirable. There are also
many solutions for which this is not true, and of which the
properties must be determined by. experiment. We shall con-
sider both types in this chapter.

In an ideal solution of two liquids, the components dissolve
in one another in all proportions, without the evolution or
absorption of heat, to form a mixture the volume of which is the
sum of the volumes of the components. In ideal solutions there
is no distinction necessary between solvent and solute, but in
actual solution it will be necessary to distinguish carefully
between the "solvent," which is the component present in excess,
and the "solute," which is the component present in small quan-
tity. In mixtures such that "excess" and "small quantity"
do not apply, it is usually necessary to determine the properties
experimentally. The properties of ideal solutions may be cal-
culated from those of the components through simple laws called
the laws of ideal solutions. But the properties of many solu-
tions of gaseous or solid solutes in liquid solvents at moderate
concentrations may also be calculated from these simple laws,

182 PHYSICAL CHEMISTRY

within the limitations of a few per cent. At low concentrations,
or in "dilute" solutions, the agreement between calculation and
experiment is even better. These laws are thus " limiting"
laws from which we may calculate the properties of very dilute
solutions but from which the deviations are important in some
concentrated solutions and small in other concentrated solu-
tions. The experimental data in the sections that follow will be
chosen so as to represent both classes of solutions. As the laws
are stated, their limitations will also be stated. Failure to
appreciate the fact that many solutions do not conform to these
ideal laws may lead to serious errors. Thus, the measured vapor
pressures of solutions of CCU in SiCl4 agree with the calculated
pressures within less than 5 per cent; but the measured vapor
pressure of a solution of a mole of alcohol in a mole of water at
80° is 30 per cent greater than the one calculated for an ideal
solution.

Vapor Pressure of the Solvent from Solutions. Raoult's Law.
The partial pressure of solvent vapor at equilibrium with a solution
at a fixed temperature is proportional to the mole fraction of the
solvent in the solution. Stated in other words, the partial
pressure of the solvent vapor decreases as the mole fraction of
the solute increases, and the fractional decrease in solvent vapor
pressure at a fixed temperature is equal to the solute mole fraction.
If PQ denotes the vapor pressure of the pure solvent and p the
equilibrium pressure of solvent vapor above the solution, these
statements of Raoult's law may be written as equations

P = /^solvent (t COnSt.) (1)

and

~ £- = x.oiute (t const.) (2)

These equations are only different algebraic forms of the same
law, as may be seen by substituting (1 — xBOivmi) for z80iute in (2)
and solving for p, whereupon equation (1) will result.

It should be clearly understood that, in these equations for
Raoult's law, p is the partial pressure of solvent vapor, and this
will not be the total pressure of vapor in equilibrium with the
solution if the dissolved substance is volatile. The partial pres-
sure of solute vapor as described by Henry's law is given in the
next section, and the total vapor pressure of a solution is the sum

SOLUTIONS

183

of the partial pressures of solvent plus solute. But the solvent
vapor pressure at a fixed temperature is decreased by the addition
of a solute whether or not the solute has an appreciable pressure.
It may be seen from Table 27 that Raoult's law gives correctly
the lowering of vapor pressure of the solvent for solvent mole
fractions from 1.00 to 0.983, which is to say for solute mole frac-
tions from zero to 0.0176 or solute molalities up to unity. Some
other aqueous solutions of nonionized solutes in water over this
range show similar conformity within the experimental error.
The largest deviation shown in Table 27 is 0.002 mm., which
probably exceeds the experimental error of these measurements;
but vapor pressures are difficult to determine experimentally and
are only rarely accurate to this extent.

TABLE 27. — AQUEOUS SOLUTIONS OF MANNITOL AT 2001

Vapor-pressure lowering, mm.

Molality

po — p observed

Calculated from
pQin/(m -f 55.54)

Per cent
deviation

0 0984

0 0307

0.0310

+1 0

0 1977

0 0614

0.0622

+1 3

0 2962

0 0922

0.0930

+0 9

0 3945

0 1227

0 1236

+0 7

0 4938

0 1536

0.1545

+0 6

0 5944

0 1860

0.1857

-0 2

0.6934

0 2162

0 0

0.7927

0 2478

0.2467

' -0 4

0 8922

0 2792

0 2772

-0 7

0.9908

0 3096

0 3073

-0.7

The vapor pressures of aqueous solutions of sucrose calculated
from Raoult's law are not in close agreement with experiment, as
may be seen in Table 28. These experiments are probably as
reliable as those quoted in Table 27, so that the differences are
real deviations of Raoult's law. But Table 27 is more nearly
typical of dilute aqueous solutions in general, and such vapor
pressures as have been determined at molalities below unity
usually agree with Raoult's law within the experimental error.

In nonaqueous solutions of nonvolatile solutes, RaoultVlaw

1 FRAZEB, LOVELACE, and ROGERS, /. Am. Chem, Soc., 42, 1793 (1920).

184

PHYSICAL CHEMISTRY

is usually reliable for solute mole fractions below 0.05, and
occasionally over wider ranges. The following vapor pressures
for benzene solutions of biphenyl (C^Hio) at 70° are probably
accurate to 1 per cent and so the data show conformity to
Raoult's law within this range 1

I 000 0 930 0 890 0.848 0 786 0 699

550 511 492 472 435 386

511 490 466 432 385

.0 -02 -13 -07 -03

Mole fraction
p(C6H6), mm

P (^solvent

Per cent deviation

TABLE 28. — VAPOR PRESSURES OF AQUEOUS SOLUTIONS OF SUCROSE AT 30° 2

Molahty

Vapor
pressure, mm

Mole fraction
of solute

Po ~ P

Per cent
deviation

0 993

31 22

0.0175

0 0194

1 65

30 76

0 0288

0 0338

2 38

30 21

0 0410

0 0520

3 27

29 43

0 0555

0 0746

4 12

28 72

0 0690

0 0980

5 35

27 55

0.0877

0 1326

6 36

26 70

0 1025

0 1612

Raoult's law may be written in terms of the weights of solvent
and solute in a solution,

9o — p __ __ m/M

PO m/M -

(t const.) (3)

in which ra0 denotes the grams of solvent, M0 its molecular
weight, ra the gram§ of solute, and M its molecular weight in
the solution. This relation allows one to calculate molecular
weights from vapor-pressure data; but since vapor pressures are
more difficult to measure experimentally than are other properties
related to them (boiling points arid freezing points), vapor
pressures are not ordinarily available for such calculations.
The vapor pressure of benzene is 639.8 mm. at 75°, and the
equilibrium pressure above a solution of 8.84 grams of naph-
thalene (CioH8 = 128) in 100 grams of benzene (C6H6 = 78) at
75° is 607.4 mm., whence M = 129 from equation (3). This
calculation indicates that solutions of naphthalene in benzene

1 GILMAN and GROSS, ibid., 60, 1525 (1938).

2 BERKLEY and HARTLEY, Trans. Roy. Soc. (London), (A) 218, 295 (1919)

SOLUTIONS 185

are nearly ideal, and other data1 upon this system support this
conclusion.

It should not be assumed that all solutions of nonvolatile
organic solutes in benzene at low molalities will be ideal, for this
is not true. The tendency of hydroxylated compounds to form
double molecules (dimers) or higher complexes (polymers) in
benzene has long been known, and is a reasonable explanation of
vapor pressures higher than would be calculated from Raoult's
law. If the molalities of phenol (CeHsOH) solutions in benzene
are computed upon the assumption that the molecular weight
of the solute is 94 and the molalities of solutions of naphthalene
(CioH8) in benzene are computed upon the assumption that the
molecular weight of the solute is 128, the molalities for solutions
of equal vapor pressure will not be equal. Some of these
" molalities" for equal vapor pressures at 25° are2

m (phenol) 0 2221 0 4014 0 6634 0 7369 1 036 1.368

TO (naphthalene) 0 1989 0 3344 0 5070 0 5608 0 7314 0 9061

Ratio 1 117 1 199 1 307 1 313 1 416. 1 509

Since these ratios are not whole numbers and since they
increase with increasing molality, a reasonable interpretation is
partial association of phenol into dimers to an extent that
increases with the molality.

Vapor Pressure of the Solute. Henry's Law. — This law states
that the solubility of a gas at a given temperature is propor-
tional to the equilibrium pressure of the gas above the solution
Expressed as a vapor-pressure law, it states that the partial
pressure of a volatile solute in equilibrium with a dilute solution
is proportional to its mole fraction in the solution. In the form
of- an equation Henry's law is

Psolute = &ZBOlute (t COttSt.) (4)

where p is the partial pressure of the solute vapor in equilibrium
with a solution in which x is the mole fraction. It will be seen
that this law resembles Raoult's law for the vapor pressure of
solvent from a solution, the important difference being that the
proportionality constant k is not the vapor pressure of the pure

1 WASHBUHN, Proc. Nat. Acad. Sd., 1, 191 (1915) ; ROSANOFF and DUNPHY,
/. Am. Chem Soc , 36, 1416 (1914).

2LASSETTRE and DICKINSON, ibid., 61, 54 (1939).

186 PHYSICAL CHEMISTRY

solute. The value of k is a joint property of the solvent and
solute; it must be determined by experiment for each solute in a
chosen solvent at each temperature. Henry's law affords only a
means of calculating a solubility at some new pressure or a
solute pressure at a new mole fraction, when k is known for
the system involved at the required temperature. The total
vapor pressure above a solution will be the sum of the partial
pressures of solvent vapor and solute vapor, and Henry's law
applies only to the solute, as Raoult's law applies only to the
solvent.

In dilute solutions the mole fraction of solute will be nearly pro-
portional to its molality or its concentration, since n\/(n\ + 712)
is nearly equal to ni/n2 when HI is small. For dilute solutions
the pressure of solute will be proportional to the molality or the
concentration if the solute conforms to Henry's law; this may be
stated in equations such as

m = k"p or p = k'm or p = k'"C (t const.) (5)

but of course none of these constants will be equal to k in equa-
tion (4) above. The point is that in a dilute solution the equi-
librium pressure of solute (in any units) is proportional to the
quantity of solute in the solution (in any units).

This law applies to the distribution of a single molecular species
between the vapor phase and the solution at moderate pressures
and concentrations. It is not valid at high pressures, or for solu-
tions in which the solute forms a compound with the solvent or
is polymerized or ionized, without allowance for these effects.
Solutions of S02 in GHC13, HC1 in C6H6, H2S in water, and C02
in water, for example, conform to Henry's law at moderate pres-
sures; but aqueous solutions of HC1 and S02 do not conform.
Any convenient units may be employed to express the solubilities
and pressures; but since there is no standard way of reporting
such data, it will be necessary in consulting the literature to give
careful attention to the units employed. The "Bunsen coeffi-
cient" a is the milliliters of gas, reduced to 0° and 1 atm., that
dissolve in 1 ml. of solvent when the partial pressure of the
solute is 1 atm. ; hence, ap/22.4 gives concentrations of solute in
moles per liter of solvent, which will be substantially moles per
liter of solution, and molalities of solute are given by ap/22Ad if
d is the density of the solvent. In some tables of data the total

SOLUTIONS 187

pressure of solvent plus solute is given, and from these tables
concentrations are calculated after subtracting the vapor pres-
sure of the solvent from the total pressure. Such coefficients in
terms of total pressure are frequently designated ft. Equilibrium
pressures may be in atmospheres, millimeters of mercury, or other
units; liquid-phase compositions may be given in any one of a
dozen ways. Some illustrative data will now be given.

The solubility of C02 in water1 at 50° and at 100° is given in
milliliters of gas (reduced to 0° and 1 atm.) per gram of water
under the following total pressures :

Total pressure, atm . . 25 50 75 100

ft = solubility at 50° . . 9 71 17 25 22 53 25 63

0 = solubility at 100° .... . 5 37 10 18 14 29 17 67

Upon dividing these solubilities by 22.4, they become molali-
ties, and pco2 is obtained by subtracting 0.13 atm. from the total
pressure at 50° and 1.05 atm.2 at 100°. Since C02 is not an ideal
gas at such pressures, it is not to be expected that Henry's law
will apply exactly. The ratio of pressure to molality is

pco2 25 50 75 100

k' = p/m, 50° . 57 65 75 87

k' = p/m, 100° 104 110 117 127

Aqueous solutions of H2S in water conform -to Henry's law,
as shown by the data3 for 25°:

p, atm . 1 00 2 00 3.00

molality . . 0 102 0 204 0 305

m/p = k" 0 102 0 102 0. 102

-This ratio ra/p, which is constant for a given temperature
according to Henry's law, changes with changing temperature, as
is true of almost every equilibrium ratio. In this system the
ratio m/p changes with the temperature as follows:

t 10° 20° 30° 40° 50°

k" - m/p 0.153 0116 0.092 0.075 0.064

1 WEIBE and GADDY, ibid., 61, 315 (1939).

2 The vapor pressure increases with the applied total pressure and becomes

1 08 atm. at 100° for a total pressure of 100 atm. We subtract 1.05 as a
sufficient correction at all pressures in this table.

3 WEIGHT and MA ASS, Can. J. Research, 6, 94 (1932).

188 PHYSICAL CHEMISTRY

The molality of HC1 is proportional to the pressure of HC1
above the solution when the solvent is nitrobenzene, CHCls,
CCU, chlorobenzene, benzene, or toluene. We quote the data
for HC1 in toluene1 at 25°:

p, aim . 0 282 0 250 0 158 0 0960 0 0338

m 0 137 0 119 0 0762 0 0468 0 0167

p/m = k' 2 05 2 11 2 07 2 06 2 16 av 2 09

It should be understood that this ratio is for a given solute
and a given solvent, a joint property of both, for a single tem-
perature. For example, the ratio p/m in the same units at the
same temperature is 6.4 for HC1 in carbon tetrachloride2 and 1.02
for HBr in toluene.

Hydrogen chloride is largely ionized in aqueous solution, and
there is no reason to expect proportionality between the partial
pressure of HC1 molecules and a molality of ions in a solution.
Since there is no reliable way of measuring what fraction of the
total dissolved hydrogen chloride is in the form of un-ionized
molecules, it is impossible to say whether Henry's law applies to
the HC1 molecules or not. The data for 25° are as follows:3

Molality HC1 4 5 6 7 8 10

104p, atm. 0 24 0 70 1 84 4 58 11 1 55 2

Ratio 16 7 71 33 1 5 0 72 0 18

When sulfur dioxide dissolves in water, both ionization and
hydration occur, so that one would not expect the ratio m/p to
be constant. If a fixed fraction of nonionized solute is hydrated,
which is a reasonable expectation from the laws of chemical
equilibrium, the ratio of p(S02) to the molality of (H2S03 + S02)
should be constant. The following table4 gives for 25° total
SO2 in all forms as the molality, p the pressure of S02 in atmos-
pheres above the solution, a the fraction of the solute which is
ionized, so that m(l — a) is the molality of un-ionized solute,
and K = m(l — a) /p. It will be seen that this ratio is sub-
stantially constant, but that m/p is not constant.

1 O'BRIEN and BOBALEK, J. Am. Chem. Soc., 62, 3227 (1940).

2 ROWLAND, MILLER, and WILLARD, ibid., 63, 2807 (1941).

3 BATES and KIRSCHMAN, find., 41, 1991 (1919).

4 JOHNSTONE and LEPPLA, ibid., 56, 2233 (1934).

SOLUTIONS 189

Molality

0271

0854

166

0 287

.501

0 764

1 027

?>(S02)

0104

0450

097

0 179

333

0 526

0 723

524

363

285

0 230

184

0 153

0 134

m(l - a)

0129

0544

119

0 221

409

0 647

0 890

m/p

1 61

1 45

1.42

K = m(l - a)/p

1 23

It should be understood that this constant K is for a single
temperature; the ratio of m(l — a) to p changes with changing
temperature as follows:

/ 0° 10° 18° 25° 35° 50°

K = m(l - a)/p . 3 28 2 20 1.55 1 23 0 89 0 56

The pressure of chlorine above an aqueous solution would be
proportional to the molality of dissolved chlorine as such, but
not proportional to the total chlorine that dissolves, since a con-
siderable proportion of it reacts with water to form hypochlorous
acid and hydrochloric acid. No corrections were required on
page 187 for the very small fraction of carbonic acid or of
H2S ionized, and therefore Henry's law applies directly to these
solubilities.

Distribution of a Solute between Liquid Phases. — Consider
two mutually insoluble liquids in each of which a third substance
is soluble, the molecular condition of the solute being the same in
both solvents. The distribution law states that at equilibrium
the ratio Ci/Cz of the concentrations in the two solvents is a
constant for a given temperature, whatever (small) quantity of
solute is used. Like Raoult's law and Henry's law, the distri-
bution law applies only to a single molecular species. The ratio
Ci/C* will not be constant when the solute is ionized or poly-
merized or solvated in one solvent and not in the other, without
allowance for these effects. Even when these effects are not
known to be responsible, variations in the ratio Ci/Cz are often
found at high concentrations, so that the law is strictly appli-
cable only in dilute solutions. When the distribution ratio varies
with the concentration, a plot of Ci/C* against Ci is a useful
device for determining C2.

In dilute solutions the distribution ratio at constant tempera-
ture may be expressed in several ways, such as molalities, mole
fractions, or volume concentrations:

— = const. ~ = const. or — = const. (6)

190

PHYSICAL CHEMISTRY

The numerical values of rai/ra2 and Ci/C2 will not be the same,
of course, and it is important to know in what units a distribution
ratio has been stated when it is used in calculations. There is no
standard form for recording these ratios.1

Some illustrations are quoted in Tables 29 and 30, from which
it will be seen that the ratios are substantially constant at low
concentrations. Table 29 shows that the equilibrium ratio is a
function of the temperature, as is true of all equilibrium ratios
that are constant for constant temperatures.

If a gaseous solute at some fixed pressure is in equilibrium
with two mutually insoluble solvents, the concentrations in

TABLE 29 — DISTRIBUTION OF SUCCINIC ACID BETWEEN WATER AND ETHER2
(Concentrations are in moles of acid per 100 moles of solution)

15°

20°

25°

Water
layer

Ether
layer

Water
layer

Ether
layer

Water
layer

Ether
layer

0.372
0.440
0.575
0.880
0.963

0 305
0.358
0.468
0.714

0.778

1.223
1.229
1 228
1.233
1.237

0.2025
0.431
0.495
0.629
0.936
1.211

0.1535
0.319
0.366
0.465
0.686
0.889

1.322
1.351
1.353
1.355
1.364
1.363

0.364
0.720
1.088
1.513

0 248
0.485
0.727
1.014

1.471
1.485
1.493
1.489

each will be determined by the constant of Henry's law for
each solvent. The distribution ratio is then the ratio of these
constants, for the two liquid phases are in equilibrium with the
same gas phase and so must be in equilibrium with each other.
When two phases are in equilibrium with one another as regards
some particular component and one of these is in equilibrium
with a third phase, the other is also in equilibrium with this third
phase. If the third phase is the solid solute itself, then when one
liquid is saturated with the solid and in equilibrium with another
liquid this second liquid must also be a saturated solution of the
solute. Thus the distribution constant for a given substance
between two solvents is the ratio of the solubilities of that sub-

1 Distribution ratios of many systems for volume concentrations are given
in "International Critical Tables," Vol. IV, pp. 418/.

2 FORBES and COOLIDGE, /. Am. Chem. Soc., 41, 140 (1919).

SOLUTIONS 191

TABLE 30. — DISTRIBUTION OF AMMONIA BETWEEN WATER AND CHLOROFORM l

At low concentration

At high concentration

Concentra-
tion in water

Concentra-
tion in
chloroform

CW/C0

Concentra-
tion in water

Concentra-
tion in
chloroform

CV/C0

0.0443

0.00165

26.2

1.02

0.045

22.7

0 0220

0 00091

24.1

3.13

0.146

21.4

0 0110

0 00044

24.7

5.24

0.283

18.5

0.00572

0.00021

25.7

£.29

0.457

15.9

0.00275

0.00011

24.6

9.35

0.710

13.2

12.25

1.227

10.0

stance in the two phases, provided that the distribution law holds
for such concentrated solutions.

In order to emphasize the fact that distribution ratios are
not constant when the solute is in a different molecular condition
in the two solvents, we quote the data for acetic acid distributed
between water and benzene at 25°. The acid is largely in the
form (CH3COOH)2 in benzene and largely in the form CH3COOH
in water, and thus the distribution ratio in terms of total con-
centrations is not constant.

CB . 0.0159 0 0554 0 2250 0 9053

CV-... .. 0 579 1 382 3 299 6 997

CB/CW . . 0 0274 0.0401 0 0776 0.1290

As has been said before, the distribution law applies strictly in
dilute solutions only. The addition of large quantities of the
distributed substance usually increases the mutual solubilities
of the " insoluble " solvents and may so increase them as to form
a single three-component liquid. For example, the distribution
ratio of acetone between chloroform and water, which are sub-
stantially insoluble in one another, is 2.25 at 0°C., but the addi-
tion of 62 grams of acetone to 18 grams of chloroform and 20
grams of water forms a single liquid. Similar behavior is
observed in the addition of pyridine to water and benzene and
in the addition of alcohols to water and ethers.

Summary of Three Distribution Laws. — Raoult's law, Henry's
law, and the " distribution law" are all distribution laws, each
for a single species of molecule between two phases at the same

1 Z. physik, Chem., 30, 258 (1899). J. Am. Chem. Soc., 33, 940 (1911).

192 PHYSICAL CHEMISTRY

temperature. In a vapor the concentration in moles per liter
is C = n/v = p/RT] and, by combining R T with the constant,
Raoult's law becomes

C (solvent in vapor)

„ , . --. , .. \ = const.

C (solvent in solution)

Henry's law as stated in equation (5) may be put into the same
form by the same device, namely,

C (solute in vapor) ,

>> / i , — i — T-TV--S = const.
C (solute m solution)

and equation (6) is already in the form

C (solute in LI) _
C (solute in L2)

Vapor Pressures of Binary Liquid Mixtures at Constant
Temperature. — When two liquids A and B form an ideal solu-
tion, the partial pressure of each component in the vapor in
equilibrium with the solution at constant temperature is pro-
portional to its mole fraction in the solution,

PA = POAXA and pB = POBXB (t const.)

where PQA and pOB are the vapor pressures of the pure components
and XA and XB are their mole fractions in the solution. These
partial pressures and the total vapor pressure, which is their sum,
are shown in Fig. 26 for an ideal system.

When the components are present in the liquid phase mole for
mole, the partial pressures in the equilibrium vapor will be 3^Po4
and HPOB, or cd and ce in Fig. 26. The total pressure is the sum
of these partial pressures, or cf, and the equilibrium mole frac-
tions in the vapor (which are denoted by y) are yA = cd/cf and
yji = ce/cf. As cd and ce are not equal, it will be evident that the
vapor in equilibrium with an ideal solution at a given temperature
does not have the same composition as the liquid. In general,
the greater the difference between the vapor pressures of the two
components, the greater the difference in composition between a
liquid and a vapor in equilibrium with it.

Ideal solutions of this kind are formed only when the two com-
ponents are chemically similar. For most pairs of liquids that
mix in all proportions the deviations from ideal solutions are

SOLUTIONS

193

considerable when both constituents are present in large propor-
tion, for example, when the mole fractions are between 0.1 and
0.9 for both. This may be due to the formation of complexes
between solvent and dissolved substances, or to the dissociation
of double molecules of solvent, either of which would render the
mole fractions calculated from the composition by weight in error,
or to other factors.

^200

SI50

| ccU
is 100

Mol Fraction of B

26 — Vapor piessuies m
ideal solution.

02 04 06 08
Mol Fraction Si Cl 4

FIG 27. — Vapor pressures of CCU
and SiCU solutions at 25°.

The experimental procedure by which solutions and their
vapors are studied consists in establishing equilibrium between
the liquid and vapor phases at a fixed temperature, measuring the
total vapor pressure, and analyzing the vapor. Although we
cannot measure directly a partial pressure, the product of total
pressure and mole fraction in the vapor is usually a sufficient
measure of the partial pressure. We designate the mole fraction
oi a component in a liquid by* x and the mole fraction of it in the

TABLE 31 — VAPOR PRESSURES OF MIXTURES OF SiCU AND CCU AT 25

Mole fraction SiCl4 in

Total vapor

f<3*Pi ^

.~ p. .

Per cent

pressure, mm

Liquid

Vapor

deviation

114 9

153 0

0 266

0.436

63.4

66 7

5 0

179.1

0 472

0.648

112.4

116 1

3.3

198.5

0 632

0 773

150 5

153 4

1 9

238.3

1 00

1.00

194 PHYSICAL CHEMISTRY

vapor by y. If the solution is ideal, the partial pressure of the A
component is P^XA, and this is equal to pyA when the total vapor
pressure is p and the mole fraction of A in the vapor is yA. When
the solution deviates from ideal behavior, we shall take PA = pyA
as a measure of the partial pressure of A and call the difference
between this quantity and POXA the deviation of PA from that for
an ideal solution.

Mixtures of CC14 and SiCl4 conform to the simple laws of
ideal solutions quite closely, as may be seen from the data1 in
Table 31. These data are plotted in Fig 27, in which the
solid lines show measured total pressures, and the products of
these pressures and the mole fractions of SiCl4 in the equilibrium
vapors. The dotted lines show calculated total pressures and
calculated partial pressures from Raoult's law for an ideal
solution.

Mixtures of benzene and toluene2 have vapor pressures from
which the calculated ones deviate 6 per cent or less. Mixtures
of benzene and cyclohexane3 show closer conformity to the ideal
laws. In all these systems the deviations are real ones, far
outside of the experimental error; in all these mixtures the com-
ponents are chemically similar, which is the favorable condition
for ideal conformity.

We turn now to some systems which are more typical of solu-
tions in general and in which large deviations are found at high
mole fractions. Even in these systems we shall frequently find
close conformity to Raoult's law when the mole fractions of
solute are below 0.05, considering first one component and then
the other as solvent According as it is present in a large mole
fraction.

Mixtures of chloroform (CHGU) and ethanol (C2H2OH) are
more nearly typical of solutions in general. Raoult's law yields
nearly correct vapor pressures of ethanol when its mole fraction
is between 0.8 and 1.0, but the pressures of chloroform from these
mixtures deviate largely from the ideal. In such mixtures the
chloroform pressures are nearly proportional to the mole fractions
of chloroform, so that Henry's law applies, but the proportional-
ity constant is not the vapor pressure of pure chloroform. Let

1 WOOD, ibid., 59, 1510 (1937).

2 SCHULZE, Ann. Physik, 69, 82 (1919).

8 SCATCHARD, WOOD, and MOCHEL, J. Phys. Chem.y 43, 119 (1939).

SOLUTIONS 195

xe denote the mole fraction of ethanol in the liquid phase, ye
the mole fraction in the equilibrium vapor, and p the measured
total pressure. Then p^xe should be equal to pyf if Raoult's law
applies and if the vapor is an ideal gas. The data1 for that part
of the system rich in ethanol are as follows for 45°, with pressures
in millimeters:

xt 1 000 0 9900 0 9800 0 9500 0.9000 0 8000

ye \ 000 0 9610 0.9242 0 8202 0 6688 0 4640

p. 172 76 177 95 183 38 200 81 232 58 298 18

poxe 171 03 169 30 164 12 155 48 138 20

pye % 171 01 169 57 164 70 155.54 137.90

Considering only this part of the data, one might conclude
that since Raoult's law applies over a wide range the solu-
tion was ideal. But the partial pressure of chloroform in
equilibrium with the solution in which xe is 0.8 is 298.18 — pye,
or 160.3 mm., and PQCXC is 86.7 mm.

Turning now to mixtures rich in chloroform, we find that in
the corresponding range of composition p^cxc deviates some-
what more from pyc, as these figures for 45° show:

xc. 1 000 0 990 0 980 0.950 0.900 0.800

yc I 000 0 9793 0.9626 0.9254 0 8868 0.8448

p, mm 433 54 438 59 442 16 449 38 455 06 454.53

p&c . 429 18 424 87 411 86 390 19 346.83

pyc . . 429.52 425 63 415 87 394.46 383.98

The difference between pGcxc and pyc exceeds 1 per cent when
xc is 0.95; the corresponding difference between p§exe and pye
is below 0.4 per cent when xe is 0.95. In the solution in which
xc is 0.8, PQCXC deviates from pyc by about 10 per cent, but p^e
and pye still agree within 0.4 per cent when xe is 0.8.

^Similar behavior is shown by many mixtures, with smaller2
or even larger3 deviations from the ideal. Without experiment-
ing upon the mixture there is n<j way to decide whether or not a
given mixture will form an ideal solution over a wide range of
composition. There are only the general rules (1) that chem-
ically similar components usually yield solutions that are approxi-

1 SCATCHARD and RAYMOND, /. Am. Chem. Soc., 60, 1278 (1938).

2 Benzene and acetic acid, HOVORKA and DRIESBACH, ibid., 56, 1664
(1934) ; benzene and CS2, SAMESHIMA, ibid., 40, 1503 (1918); CC14 and CeHe,
SCATCHARD, WOOD, and MOCHEL, ibid., 62, 712 (1940).

8 Acetone and CS2, ZAWIDSKI, Z. physik. Chem., 35, 172 (1900).

196

PHYSICAL CHEMISTRY

mately ideal and (2) that " dilute7' solutions have vapor pressures
which conform to Raoult's law and Henry's law.

Constant-temperature Distillation. — We have already quoted
the equilibrium mole fractions of liquid and vapor for mixtures
of ethanol and chloroform at 45° for " dilute" solutions. For
the purposes of this section we quote the remaining data applying
at 45° for mole fractions of ethanol between 0.2 and 0 8:

p, mm

0 300 0 400
0 1850 0 2126
446 74 435.19

0 500
0 2440
417 71

0 600 0 700
0 2862 0.3530
391.04 353 18

These data, together with the other equilibrium mole fractions
already quoted, are plotted in Fig. 28 in which the total vapor

J"200

02. 04 06 08 10

Mole Fraction of Ethanol

FIG. 28. — Constant-temperature distillation of mixtures of chloroform and ethanol

at 45°

pressure is plotted against the mole fraction of ethanol in the
liquid phase as a solid line and the dotted line shows the equi-
librium mole fraction of ethanol in the vapor for each total pres-
sure at 45° on the same composition scale. For example, at 45°
and 380 mm. total pressure, liquid of composition x\ is in equi-
librium with vapor of composition y^\ liquid of composition x2 is
in equilibrium with vapor of composition t/2 at 325 mm. and 45°.
Such lines as x\y\ and x^y* are called "equilibrium tie lines"
or, more briefly, "tie lines/' since they tie together the composi-
tions of two phases at equilibrium. These lines apply to con-
stant-temperature diagrams, each for a given pressure; but in a
later section we shall also use tie lines on constant-pressure
diagrams, each applying to a single temperature.

SOLUTIONS 197

Diagrams such as Fig. 28 may be used to show approximately
the composition of each phase when a moderate fraction of the
total liquid is distilled at constant temperature. Starting with a
liquid of composition 0*1, which would yield & first vapor of compo-
sition 2/1, suppose the distillation is continued at constant tem-
perature and decreasing pressure until the liquid composition
becomes x2. The last portion of vapor leaving the liquid would
have the composition 2/2, and when the distillation range is not
too great, } -2(2/1 + 2/2) will nearly represent the composition of
the whole distillate It should be noted that the composition of
the liquid residue is x2 and not %(xi + ^2) and that a line join-
ing the compositions x2 and 3-2(2/1 + 2/2) is not an equilibrium tie
line Fractional distillation for the purpose of separating a mix-
ture into portions of different composition is more commonly
carried out at atmospheric pressure and changing temperature,
rather than at constant temperature, as we have done here, since
the former procedure is more convenient and the latter is experi-
mentally difficult. We shall consider this process in a later sec-
tion, after discussing boiling solutions in which only the solvent
is volatile from the solution.

Boiling Points of Solutions of Nonvolatile Solutes. — The boil-
ing point of a solution is the temperature at which its total vapor
pressure is 1 atm. Solutions from which both solute and sol-
vent are volatile are discussed in the next section; and solu-
tions from which only the solvent has an appreciable vapor
pressure are discussed in this section. At any given tempera-
ture, such as the boiling point of the pure solvent, the vapor
pressure of solvent from a solution will be less than p0 for this
temperature. It is thus necessary to heat a solution contain-
ing " a nonvolatile solute to a temperature above the boiling
point of the pure solvent before the solution will boil.

We have seen in previous sections that the vapor pressure is
not a linear function of the temperature and that for ideal solu-
tions the fractional decrease in solvent vapor pressure produced
by a fixed mole fraction of solute is the same at all temperatures.
Hence, plots of vapor pressures against a considerable range of
temperature for a pure solvent and for a solution of a nonvolatile
solute will yield lines that are neither straight nor parallel. Yet
when such a plot is made over a range of 2° or so near the boiling
point for the pure solvent and a solution in which the mole frac-

198

PHYSICAL CHEMISTRY

tion of solute is 0.02, the lines are so nearly straight and parallel
that a diagram similar to Fig. 29 results.

We shall use this diagram to determine the relation between
T — To, the boiling-point elevation caused by the addition of a
nonvolatile solute to a solvent of which the boiling point is jT0,
and x, the mole fraction of solute. At T0 the vapor pressure of
the solution is less than 1 atm. by the distance ab. In order to
bring the solution to its boiling point, it must be heated while
the vapor pressure increases along the line be until the point c
is reached at the temperature T. The relation between the

lowering of the vapor pressure
and the boiling-point raising is
ab/ac = (po - p)/(T - TQ).
But for small temperature
changes, ab/ac is the slope of
the dotted line, i.e., it is the
rate of change of the vapor
pressure of the solution with
the temperature. The dotted
and solid lines are nearly par-
allel for the short distances
involved in a small change of
boiling point, and hence we
may write dp0/dT for ab/ac,
in place of dp/dT, employing
the change of vapor pressure of the pure solvent with the tem-
perature in place of the change in vapor pressure of the solution
with the temperature.- Then we may write

Temperature

FIG 29 — Vapor-piessure relations
near the boiling point.

Po — p = dpv
T - To dT

pox

T - To

(7)

since p0 — p is equal to p^x from Raoult's law. On solving the
equation for the elevation of the boiling point, which is A!T6, we
have

T — T -
1 1 °

dpo/dT
= kx (p const.)

(8)

Since po and dp^/dT are characteristics of the solvent, the
change in boiling point depends upon the mole fraction of solute

SOLUTIONS

199

but not upon its nature, provided that its vapor pressure from
the solution is negligible. The relation provides a means of
determining molecular weights of solutes in solvents for which
k is known. The value of k will not be the same for all solvents
but must be determined in one of the ways explained below. The
validity of this equation is illustrated by the data of Table 32
for solutions of biphenyl in benzene.

Table 32 shows better than average conformity of a system
to the ideal equation, though it is not unique. A more typical
set of data, so far as usual deviations are concerned, is the fol-
lowing for salicylic acid in ethanol :

Molahty

03
25 5

05
26 3

07
26.9

26 9

1.5
28.0

Wide deviations may be found when association, dissociation,
solvation, or reaction of solute with solvent occurs, but these are
misapplications of the equation rather than deviations. Yet
variations in AT/x with increasing x are sometimes found when
none of these factors is known to be responsible, and no expla-
nations have yet been found

TABLE 32 — BOILING POINTS OF SOLUTIONS OF BIPHENYL IN BENZENEX

Mole fraction solute

^=fc

0 0380

1.333

35 7

0.0490

1.709

35 0

0.0613

2.152

35.1

0.0718

2.521

35.0

0.0890

3 142

35.3

In laboratory practice, the composition of a solution may be
expressed in terms of the moles of solute per 1000 grams of
solvent, and the elevation of the boiling point produced by a

1 WASHBUKN and READ, /. Am Chem £oc^41, 729 (1919). The vapor
pressures of benzene solutions of biphenyl are given very closely by Raoult's
law, and therefore conformity to equation (8) is to be expected in these solu-
tions. Data for 70° [by Gillman and Gross, ibid., 60, 1525 (1938)] are given
on p 184. Their data for 50° are

xs. . .
p, mm.

1 00
270

0.930
249
251

0.890
240
241

0.848
228
229

0.786
215
212

200

PHYSICAL CHEMISTRY

mole of solute in 1000 grams of solvent is called the molal eleva-
tion of the boiling point, B For example, 1000 grams of water is
1000/18, or 55.5, moles; and when a mole of solute is dissolved in
1000 grams of water its mole fraction is x = 1/(1 + 55.5) = 0.0177.

The vapor pressure of water at
its boiling point changes at the
rate of 0.0357 atm per deg , and
by substituting these quantities
into equation (8) we find

An =

00177
00357

- 0.50°

Cooling Water -w-

for a solution of a mole of solute
in 1000 grams of water. Then
the boiling-point elevation of
any dilute aqueous solution of a
nonvolatile substance in water is

An = 0.50m = Bm (9)

This equation furnishes a con-
venient means of determining
approximate molecular weights
of dissolved substances, since
the moles per 1000 grams of sol-
vent is given by equation (9)

k from the boiling-point elevation

Jj - and the grams of solute per 1000

grams of solvent is known from
analysis.

It will be noted that the
boiling-point elevation has been
expressed in two ways, An = kx
and An = Bm. Both these
equations state the same fact,
namely, that the boiling-point

elevation for a dilute solution of a nonvolatile solute is propor-
tional to the quantity of solute in a given quantity of solvent.
If ra is the molality of a solute and M0 is the molecular weight
of the solvent, the mole fraction of solute is

FIG. 30 — Boiling-point apparatus

The narrow tube a serves to pump
an intimate mixture of solution and
vapor over the thermometer Weighed
pellets of solute are introduced thiough
c, or the solution may be analyzed after
a determination The condenser is so
arranged that cold solvent returning to
the solution from' it does not touch the
thermometer, but runs down the pump
tube. [Cottrell, J. Am. Chem. Soc , 41,
721 (1919).]

X =

SOLUTIONS 201

m + (1000/Afo)

In a dilute solution m is small compared with 1000/Mo, and the
mole fraction is nearly m/(1000/M0). Thus the mole fraction
and molality are almost proportional to one another; but since
_!/[! + (lOOO/3/o)] is not a unit mole fraction when the molality
is unity, it will be evident that the numerical values of k and B
are not the same for any solvent or proportional to one another
for different solvents. Some values of these constants are given
in Table 33.

It will be recalled that the approximate Clapeyron equation
expresses the change of vapor pressure in terms of the molal
latent heat of evaporation. By substituting

dpn _ poA7
dT ~ ~RT

in equation (7) we have

= = _

dT ~ 7JZV T - 770

and on solving for AT& we have

7?T 2

An = ^- x = kx (p const.) (10)

A//m

For comparison we calculate a value of B for water from this
equation. The heat of evaporation of water is 9700 cal. per mole,
whence for one mole of solute per 1000 grams of water

Ay 1.99 X (373)2
A7fe- X

9700 1 + 55.5 ~ '

It will be observed that k may be obtained from RTQ2/AHm as
shown in equation (10), from p0/(dpo/dT) as shown in equation
(8), or directly from boiling-point measurements as shown in
Table 32. Yet when all three of these procedures are used,
slightly discordant values of k result, and the disagreement
seems to lie outside the probable errors of the experiments even
when very dilute solutions are concerned. No definite explana-
tion of the discordance is known.

202

PHYSICAL CHEMISTRY
TABLE 33. — BOILING-POINT CONSTANTS

Solvent

Boiling point

Benzene

80 09

2 6

Carbon bisulfide

46 0

2 4

Carbon tetrachlonde

76 5

33 4

5 05

Chloroform

61 2

32 0

3 4

Ethyl alcohol

78 26

1 24

Ethyl ether

34 5

30 2

2 21

Hexane

68 6

34 1

2 9

n-Octane

125 8

38 9

4 4

Water

100 0

28 9

0 51

Fractional Distillation at Constant Pressure. — Liquid mixtures
of two volatile components are in equilibrium with vapors in
which the mole fractions usually differ from those in the liquid
phase, as we have seen in Fig. 28. In place of considering these
quantities for constant temperature, we now consider the equi-
librium mole fractions in the two phases at a constant pressure
of 1 atm. and bring the mixture to this pressure by adjusting
the temperature. When heat is applied to these mixtures, vapor
is expelled and may be condensed, as in the familiar process of
distillation. The first portion of distillate represents the composi-
tion of vapor in equilibrium with the liquid from which it was
expelled, provided that the quantity of distillate is very small
compared with the quantity of liquid remaining. It will be
assumed that distillation is conducted so slowly as to maintain
equilibrium in the distilling vessel and that condensation of the
vapor is complete so^that the composition of condensate is the
same as that of the equilibrium vapor.

We take up first the temperature-composition diagrams for
equilibrium between liquid and vapor at 1 atm. total pressure,1
next the compositions of residue and distillate obtained when a
single portion of distillate is collected from a fixed quantity of
liquid by distillation over a moderate temperature range (in

1 These diagrams are usually applicable at any constant pressure near
1 atm., without correcting for geographical or climatic variations in atmos-
pheric pressure; but, of course, the experimental data must all be taken for a
single pressure. Daily variations of atmospheric pressure in a given local-
ity may produce changes in observed boiling points of as much as 1° above
or below the normal, and for precise work these observed temperatures must
be corrected to 1 atm.

SOLUTIONS 203

which the compositions of both phases change continuously as
distillation progresses), and finally complete fractionation by
which through repetition of partial distillation and partial
condensation the mixture is separated into its components or
into one component and a constant-boiling mixture. This third
procedure will yield the pure substances when the boiling points
of all mixtures lie between those of the components. If some of
the mixtures boil outside of this temperature range, separation
by repeated fractionation may be carried only to the formation
of a maximum (or minimum) boiling mixture as a final residue
(or distillate) and one pure component as a final distillate (or
residue). In discussing fractional distillation, it will be impor-
tant to make clear whether equilibrium compositions, single
distillates, or complete fractionation is being discussed.

As is common practice, we shall designate mole fractions in the
liquid mixture by x with a suitable subscript and mole fractions
in the vapor or distillate by y with a suitable subscript. The
partial pressure of any component will be understood to be the
product of total pressure and its mole fraction in the vapor, for
there is no way of measuring partial pressures directly.

a. Equilibrium Compositions. — Toluene (CyHg = 92) and ace-
tone (CsHeO = 58) mix in all proportions, and the boiling points
of all mixtures of them lie between those of the components *
Table 34 gives the boiling points and equilibrium mole fractions
of liquid and vapor for several mixtures. These data are plotted
in Fig. 31, in which liquid composition is shown by a solid line
and vapor composition by a dotted line. "Tie lines " such as
Xiyi and X2y2 show the equilibrium compositions for selected
temperatures.

Consider a vessel closed by a movable piston, in which a mix-
ture of 0.2 mole of acetone and 0.8 mole of toluene is heated while
the pressure remains 1 atm., but no vapor escapes from the con-
tainer. At 84° the solution reaches its boiling point and expels
a first vapor of composition y\. If the heating is continued, say
tp 87°, the liquid composition changes along the solid line from
x\ to #2 while the vapor composition changes from y\ to yz along

1 Other systems in which this simplicity is observed are ethanol-n butanol
[for which data are given by Brunjes and Bogart in Ind. Eng. Chem.j 35,
255 (1943)] and CC14-C2C14 [for which data are given by McDonald and
McMillan in Ind. Eng. Chem., 36, 1175 (1944)].

204

PHYSICAL CHEMISTRY

110

too

°90

CD 70

the dotted line. Upon further heating the compositions change,
the quantity of vapor increases, the quantity of liquid decreases
until evaporation becomes complete at 104°, and in the last drop
of liquid to evaporate XA is about 0.02. This imaginary process
has been described to illustrate the meaning of Fig. 31, but it
would be inconvenient, since it would require a vessel of some 35
liters capacity to carry it out.

For all ranges of temperature and composition within the field
below the solid line in Fig 31, a liquid phase alone results at 1

atm. pressure; for all ranges
of temperature and composi-
tion above the dotted line
only vapor exists at 1 atm.
Between these lines a liquid
phase of composition x and a
vapor phase of composition y
are at equilibrium, so that
this is a two-phase area. For
illustration, a vapor contain-
ing 60 mole per cent of ace-
tone begins to condense at
about 87° and when cooled to
70° without the escape of con-
densate consists of a liquid
phase in which XA is 0.48 and a vapor in which yA = 0.86.
Partial condensation serves to separate a vapor mixture into two
portions of different composition, just as partial evaporation does.
b. Fractional Distillation. — The usual procedure in distillation
is to remove the vapor as fast as it forms by passing it into a
condenser. If the mixture in which XA = 0.20 were distilled until
the boiling point rose from 84 to 87°, the vapor (or distillate)
composition would vary from yl to yz (Fig. 31), say from 0.64 to
0.60, so that in the whole distillate yA would be 0.62. The compo-
sition of the residue in the flask would be xz, or about 0.16, and
not the average of x\ and x2. It should be noted that y = 0.62
and x = 0.16 are not on a horizontal tie line and should not be,
since the whole distillate was not in equilibrium with (or expelled
from) a liquid of composition XA = 0.16. By a continuation of
this process, with fresh receivers under the condenser, the entire
mixture could be separated into fractions passing over in 3°

02 04 06 08
Mol Fraction of Acetone
FIG 31 — Boiling-point composition
diagram for toluene and acetone at
1 atm pressure

SOLUTIONS

205

TABLE 34 — BOILING POINTS AND COMPOSITIONS OF TOLUENE-ACETONE

MIXTURES1

Mole fraction of acetone in

Boiling point

PA = pyA

Liquid

Vapor

109 4

93 5

0 108

0 449

341 mm.

85 0

0 187

0 636

484

72 8

0 383

0 811

616

67 0

0 572

0 883

671

64 0

0 686

0 916

696

61 2

0 790

0 941

715

59 5

0 871

0 964

732

58 0

0 938

0 981

746

56 5

1 00

760

ranges. Each succeeding distillate would be richer in toluene;
the 102 to 105° distillate would have about the composition of the
original mixture, for example; and the residue after one more
distillation would be almost pure toluene. We shall come in a
moment to a method by which the distillates are distilled again
and the residues suitably combined for further distillation until
substantially complete separation into the pure components is
obtained for this type of mixture.

A material balance enables us to compute the weights of dis-
tillate and residue obtained in a single fraction. In the illustra-
tion given above, a mixture of 0.2 mole of acetone and 0.8 mole
of toluene was distilled until a 3° fraction of distillate resulted, yA
being 0.62 in the distillate and XA being 0.16 in the residue. If
d moles of distillate resulted, 0.62d mole of acetone were in the
distillate and 0.16(1 — d) mole of acetone remained in the flask.
The total acetone in the original mixture was 0.2 mole, so that
0.62d + 0.16(1 - d) = 0.2 and d = 0.087. The quantity of
acetone in the distillate is 0.087?/,i, or 0.054 mole; the quantity
of toluene is 0.087(1 — J/A), or 0.033 mole. These quantities are,
respectively, 3.13 and 3.04 grams, or a total of 6.17 grams of dis-
tillate. The original mixture weighed 85.2 grams, this being
0.2 X 58 + 0.8 X 92, and therefore 79.0 grams remained in the
flask.

, BACON, and WHITE, J. Am Chem. Soc.t 36, 1803 (1914).

206 PHYSICAL CHEMISTRY

This 6.17 grams of distillate in which XA is 0.62 would boil at
about 66°, as shown in Fig. 31, and yield a new distillate in the
first portion of which yA would be 0.90; a 3° fraction would be
about 88 mole per cent acetone and much smaller in quantity
than 6.17 grams. A third distillation of this second distillate
would yield a very small amount of third distillate in which yA
would be about 0.99.

c. Complete Fractionation. — In order to illustrate the principle
of the procedure for obtaining larger quantities of nearly pure
toluene and acetone from a mixture in which XA is 0.20, for
example, consider a simple (but experimentally inadequate)
arrangement of four vessels containing mixtures of these sub-
stances at their boiling points, as follows:

(1)

(2)

(3)

(4)

XA = 0 02

XA = 0 20

xA = 0 50

XA = 0 85

t = 105°

t = 84°

i = 68°

t = 60°

I/A = 0 20

yA = 0 64

yA = 0 86

yA = 0 96

Each vessel has a long and short exit tube so arranged that the
vapor from (1) is discharged under the liquid in (2), the vapor
from (2) discharges under liquid (3), etc., and finally, the vapor
from (4) passes into a condenser. Vessel (1) is heated; the others
are thermally insulated and not heated. (Note the location of
the tie lines corresponding to these four liquids and vapors on
Fig. 31 before reading the next paragraph.)

The vapor expelled from (1) at 105° is cooled to 84° in (2),
causing partial condensation ; the latent heat of this condensation
is used to form a vapor in which yA is 0.64, while the liquid in
(2) is enriched in toluene. Vapor expelled from (2) at 84° is
cooled to 68° in (3), causing some enrichment of this liquid in
toluene and the formation of a vapor in which yA is 0.86. This
vapor is cooled to 60° in (4), where partial condensation yields
the heat required to expel a vapor in which yA is 0.96. As these
operations continue, the liquid in (1) approaches pure toluene,
since XA is only 0.02 and yA is 0.25. Use of one or two more
vessels on the toluene side would yield a final liquid residue
that is nearly pure toluene; addition of one or two more on the
acetone side would yield nearly pure acetone vapor for the final
condenser. If to such a plan we add means of keeping the liquid
compositions constant by flowing liquid from (4) to (3), from

SOLUTIONS

207

• Vapor ou fief'

(3) to (2), from (2) to (1) and if the losses of the pure com-
ponents from this multiple distilling arrangement are made up
by adding more boiling liquid 20 mole per cent solution to vessel
(2), a continuous yield of both components results.

Such an arrangement would, of course, be too crude for actual
use. In practice, the vessels are called " plates" or trays; they
are arranged one over another
in a " fractionating column"
with "bubble caps" to pro-
mote contact between liquid
and vapor and with down-
takes for the liquid to flow
toward the high-boiling por-
tion of the column, as illus-
trated in Fig. 32. Heat is
supplied at the bottom of
this fractionating column, the
high-boiling component is
withdrawn as a liquid at the
bottom, and the low-boiling
component leaves the top of
the column as a vapor, which
is condensed in a ^separate
condenser. The liquid to be
fractionated is heated to its
boiling point and fed in on

Feed pipe

FIG. 32.

Heating coil

Liquid outlet'
Idealized fractionating column

the plate of which the liquid phase has the same composition.

In the laboratory, a flask containing the boiling mixture
serves as the bottom "tray," and a glass tube containing "pack-
ing" or supplied with baffles and depressions for the liquid
constitutes the "column," to the top of which a condenser is
attached. When the boiling points of the components to be
separated differ by 5°, complete fractionation may be accom-
plished with as little as 20 mg. of liquid. Fractionating towers
in industry may be 32 ft. or more in diameter and 60 to 115 ft.
in height and may contain 30 to 80 plates with 1000 or more
bubble caps to each plate. A single tower may handle as much
as 100,000 barrels of oil per day.

When binary mixtures are to be separated, operation is usually
at atmospheric pressure ; but petroleum fractionating towers some-

208

PHYSICAL CHEMISTRY

times operate under pressures of 300 Ib or more, and in other
industrial distillation the stills operate under reduced pressures.
Side streams are sometimes withdrawn from a plate and passed
through stripping towers or otherwise treated

Mixtures of three or more components are sometimes separable
by fractional distillation as well, but they require special pro-
cedures that we cannot consider here The design of efficient
fractionating columns is a complex problem for a chemical engi-
neer, but the fundamental data that he requires for this purpose
are the equilibrium mole fractions of liquid and vapor, such as
are shown in Fig 31.

Constant -boiling Mixtures (Azeotropes). — Many pairs of
liquids form certain mixtures boiling higher than either com-
ponent or lower than either component and of course one mixture
with a maximum (or minimum) boiling point. Such mixtures
are called azeotropic mixtures, and the pairs of liquids forming
them are called azeotropes. The maximum (or minimum) boil-
ing mixture cannot be further separated by fractional distillation
at constant pressure. A few illustrations at 1 atm pressure are
quoted here, and thousands of others are known.

Components and boiling points

Constant-boiling mixture

Water, 100° Ethyl alcohol, 78 26°
CC14, 76 5° Ethyl alcohol, 78 26°
Water, 100° Nitric acid, 86°
Water, 100° Ethyl acetate, 77 1°

89 4 mole per cent alcohol, 78 15°
39 7 mole per cent alcohol, 64 95°
62 mole per cent water, 122°
24 mole per cent water, 70 . 4°

These constant-boiling mixtures are not compounds, for the
mole ratios in them are seldom whole numbers, and they change
materially when the distillations are carried out at pressures
other than 1 atm. For example, the mole per cent of ethanol
in the azeotropic mixture with water changes with the pressure
at which the distillation is conducted, as follows:1

Pressure, atm .

Mole per cent ethanol

In some industrial alcohol fractionating columns the pressure
is as low as 0.125 atm., at which pressure the boiling temperature

1 BEEBE, COULTER, LINDSAY, and BAKER tlnd. Eng. Chem., 34, 1501 (1942).

10 0 50 0 25 0.125
89.4 91 5 94 1 99.7

SOLUTIONS 209

of water is about 50°C. and the boiling temperature of the azeo-
trope is about 35°C., but there are important reasons other than
the enriched azeotrope for conducting the distillation at such
a low pressure. Azeotropes are not ordinarily " broken " by this
means, since more economical methods are available.

Another example is constant-boiling hydrochloric acid, for
which the azeotropic mixture boiling at 1 atm. pressure contains
20 22 per cent HC1 by weight, which is very nearly the compo-
sition HC1.8H2O. A solution containing more water than
this mixture expels water in a higher mole ratio than 1:8 and
approaches this composition; one containing less water expels
more than 1HC1 to 8H2O and likewise approaches 20.22 per cent
HC1 by weight. But when the distillation is conducted at some
pressure other than 1 atm., the ratio of HC1 to water in the con-
stant-boiling mixture changes, so that the evidence for compound
formation is not convincing. Since the preparation of " constant-
boiling hydrochloric acid" is a convenient means of obtaining a
solution of accurately known composition, we quote the data
applicable to climatic changes in pressure.1

Pressure, mm 770 760 750 740 730

Weight per cent HC1 20 197 20 221 20 245 20 269 20 293

Mimimum-boiling mixtures are somewhat more common than
maximum-boiling mixtures. The only difference in their treat-
ment is that the minimum-boiling mixture is the ultimate
distillate in complete fractionation, while the maximum-boiling
mixture is the ultimate residue in this process.

Systems of two components in which maximum-boiling or
minimum-boiling mixtures form may be separated into portions
of different compositions by fractional distillation except when
the system has the composition of the azeotropic mixture. ( A
single fraction of distillate may be collected, or repeated fraction
may be performed; but this latter operation will not yield the
two pure components. Equilibrium mole fractions of liquid and
vapor for 1 atm. and varying temperature may be shown on dia-
grams such as Fig. 34, which is read in the same way as Fig. 31.
Another common way of plotting the data is shown in Fig. 33,

1 Foulk and Hollmgsworth, J. Am. Chem. Soc., 46, 1227 (1923); for other
examples of azeotropic ratios changing with pressure see J. Phys. Chem.,
36, 658 (1932).

210

PHYSICAL CHEMISTRY

which is easier to read for compositions but which does not show
the boiling temperatures.

Equilibrium data for ethanol (ethyl alcohol) and water1 at
1 atm. pressure are given in Table 35 and plotted in Fig. 34

0 02 04 06 Q8
Mol Fraction Ethanol in Liquid

20
Mol

FIG. 33. — Equilibrium mole frac-
tions in liquid and vapor at 1 atm.
pressure for ethanol and water.

40 60 80
Per Cent Ethanol

FIG. 34. — Temperature-composition
diagram for water and ethanol at
1 atm. pressure.

Liquid composition is shown by a solid line and vapor composi-
tion by a dotted line, as was done in earlier diagrams.

TABLE 35. — EQUILIBKIUM MOLE FRACTIONS OF ETHANOL AND WATER

B. pt.

2/e

86 4

0 100

0 442

79 1

0 600

0.699

83 3

0 200-

0 531

78 6

0.700

0 753

81 8

0 300,

0 576

78 3

0 800

0 818

80 7

0 400

0 614

78 2

0 894

79 8

0 500

0 654

78 3

1.000

1 000

A solution of 20 mole per cent ethanol would boil at 83.3° and
yield a first vapor in which ye was 0.53; a fraction collected
between 83.3° and 84.3° would be about 50 mole per cent ethanol.

1 From Cornell and Montonna, Ind. Eng. Chem., 25, 1331 (1933); data for
methanol and water, and for acetic acid and water are given in the same
paper. Data for ethanol and water in substantial agreement with those
above are given by Baker, Hubbard, Huguet, and Michalowski, ibid., 31,
1260 (1939).

SOLUTIONS 211

The first vapor from redistillation of this small distillate would be
about 65 mole per cent ethanol. Repeated fractionation in a
column such as that shown in Fig. 32 would separate the mixture
into a final residue of pure water, and a final distillate containing
89 mole per cent (or 96 weight per cent) ethanol, the minimum-
boiling mixture. This would be true of any mixture containing
less than 89 mole per cent of ethanol. Although it is true that
azeotropic compositions change with the pressure under which
distillation is conducted, it is usually not practical to apply this
fact to the further enrichment of the distillate, since other means
better suited to the preparation of anhydrous ethanol from the
89 mole per cent mixture are known. Any mixture containing
more than 89 mole per cent ethanol would also yield the azeo-
tropic mixture as a final distillate upon complete fractionation
and pure ethanol as a final residue.

Similar statements would apply to any system in which one
mixture has a minimum boiling point ; this mixture would be the
final distillate upon complete fractionation, and the final residue
would be whichever pure component has to be removed to pro-
duce this composition. Maximum-boiling mixtures form the
final residue upon complete fractionation, and one pure com-
ponent forms the final distillate. Through the use of material
balances the quantities of distillate and residue may be computed,
as was done in an earlier section. For example, 1000 grams of
20 mole per cent ethanol is 8.48 moles of ethanol and 33.92 moles
of water; the distillate resulting from complete fractionation
would contain all the ethanol, making 8.48/0.89 = 9.53 moles of
distillate, 1.05 moles of water, and 8.48 moles of ethanol. The
residue would be pure water, 33.92 - 1.05 = 32.87 moles, or 592
grams of water.

While it is true, as suggested above, that azeotropes may not
be separated by fractional distillation in a two-component
system at constant pressure and that their separation by changing
the pressure is tedious or at least uneconomical, it is not true that
such mixtures are incapable of separation by distillation, for they
are% " broken " industrially in many processes. The usual expe-
dient is to add a third substance called an "entrainer," which
may or may not form an azeotrope with one or the other com-
ponent of the original mixture, and to fractionate the three-
component system. The addition agent, or "entrainer," cycles

212 PHYSICAL CHEMISTRY

through the process with little loss, and the end products are the
two components of the original azeotrope. A common entrainer
is benzene for the preparation of anhydrous alcohol from the
azeotrope with water, and many others are known.1

Distillation of Insoluble Liquids with Steam. — If two liquids
are mutually insoluble, neither one lowers the vapor pressure of
the other and the total vapor pressure of a mixture of them is the
sum of their vapor pressures. When such a mixture is heated
in a distilling flask until this sum reaches atmospheric pressure,
the mixture boils and the substances pass out of the flask in the
mole ratio of their vapor pressures. Liquids insoluble in water
may thus be distilled with steam at temperatures that are not
only below the boiling points of the liquids, but below the boiling
point of water as well. For substances of high boiling point
that do not react with water, steam distillation is a convenient
expedient for effecting distillation at low partial pressures without
the use of vacuum equipment.

Consider, for example, a mixture of water with terpinene
(CioHie, boiling point 182°), whose vapor pressures are

t 90° 95° 100°

p, mm for terpinene 91 110 131

p, mm for water 526 634 760

The liquids are substantially insoluble in one another; the
total vapor pressure is 744 mm at 95° and 891 mm. at 100°.
While the vapor-pressure curves are not quite linear functions
of the temperature over a range of 5°, it will be evident that at
about 95 5° the total vapor pressure will be 1 atm. from this
mixture (Actually dp/dT is 24 mm. per deg. for water at 95°
and 4 mm. per deg. for terpinene, or 28 mm. for the two together,
and l%8° is sufficiently near to 0.5°.) In the vapor expelled
from the flask, pw will be 648 mm., pt will be 112 mm., and the
mole ratio in the distillate will be the ratio of these pressures.
Each mole of distillate will thus contain 0.147 mole of terpinene
and 0.853 mole of water, or 57 per cent terpinene by weight; §nd
distillation will take place 87° below the boiling point of pure
terpinene.

1 For a discussion of azeotropic distillation, see Ewell, Harrison, and Berg,
ibid., 36, 871 (1944).

SOLUTIONS 213

Substances of higher boiling point will have lower vapor pres-
sures near 100°, and thus the yield in moles per mole of distillate
will be smaller; but against the small yield must be set the advan-
tage of convenient distillation at low partial pressures.

So long as both substances are present at equilibrium in the
distilling flask, the temperature will remain constant and the
composition of distillate will be independent of the relative
quantities in the flask, since each substance exerts a vapor pres-
sure dependent upon temperature alone and independent of the
quantity of liquid present.

An accurate measurement of the temperature of a steam
distillation and of the weight composition of the distillate serves
to determine the molecular weight of the vapor of an insoluble
substance, as well as its vapor pressure at this temperature.
For illustration,, suppose a substance A distills with steam at
99.0° under an observed barometric pressure of 752.2 mm., yield-
ing a distillate that is 25 per cent A by weight. The vapor
pressure of water at 99° is 733.2 mm., and that of the substancifc,
A is thus 19.0 mm. The mole ratio in the distillate is 733.2/752.2
to 19/752.2, or 0.975 mole of water to 0.0252 mole of A. In 100
grams of distillate there are 7%g = 4.17 moles of water to 25/M
moles of A , and these quantities must be in the ratio of the partial
pressures. Then 0.975 :0.0252 = 4.17 : (25/M), whence M = 232
for the substance.

Liquids which are slightly soluble in water and in which water
is slightly soluble may also be distilled with steam, but the mole
ratio in the distillate is not to be computed from the vapor pres-
sures of the pure substances or from them and Raoult's law, for
such solutions are far from ideal. For illustration, when aniline
(CeHyN) and water are shaken to equilibrium at 100°, there are
two phases, containing 7.2 and 89.7 per cent aniline by weight,
respectively. In these mutually saturated solutions the mole
fractions of aniline are 0.015 and 0.63; but since the solutions are
in equilibrium with one another, they are both in equilibrium
with the same vapor. The value of y*. in this vapor is about
0.045, as determined by analysis of the distillate, and this value
could not be obtained from calculations assuming either layer to
be an ideal solution. The vapor pressure of aniline at 100° is
0.060 atm.; and from Raoult's law pA would be 0.63 X 0.060, or
0.038, and pHzo would be 0.985, whence y* is calculated to be 0.037

214

PHYSICAL CHEMISTRY

in place of 0.045 found by experiment It will always be true
that the calculated partial pressures are below the observed ones
for liquids of limited solubility.

Freezing Points of Solutions. — The freezing point of a solution
is denned as the temperature at which the solution is in equi-
librium with the pure crystalline solvent. Since solutions when
cooled usually deposit one component as a solid before the other,
the freezing point of a solution is not the temperature at which
the solution as a whole becomes solid but the temperature at
which it begins to deposit solid solvent if cooled so slowly that
equilibrium is maintained. Equilibrium is more readily attained
when the cooled solution is poured over a liberal excess of crystal-
line solvent; the composition
of the solution may be deter-
mined by analyzing a portion
of it withdrawn after the equi-
librium temperature has been
measured.

Addition of a solute will
lower the equilibrium pressure
of solvent vapor at any given
temperature, and for ideal
solutions the decrease in sol-

FlG

T2 T, T0

Temperature

35 — Vapor-pressure relations
near the freezing point

vent vapor pressure is shown
by Raoult's law. At r0, the
freezing point of the pure solvent, the vapor pressures of the
solid and liquid phases are equal, but at this temperature
the vapor pressure's of crystalline solvent and a solution will not
be equal. Since the vapor pressure of the solid decreases with
falling temperature more rapidly than does the vapor pressure
of the solution, cooling will bring the two vapor pressures to
equality at the freezing point of the solution.

A diagram of these conditions is given in Fig. 35, in which ae
shows the change in vapor pressure of liquid solvent with tem-
perature, hda this change for solid solvent, and dbf the change of
solvent vapor pressure for a solution in which x\ is the mole frac-
tion of solute. Since hda and dbf intersect at d} Ti is the freezing
point of this solution and cd or TQ — Ti is the freezing-point
depression AT7/. The relation between cd and the mole fraction
of solute Xi is desired. It will be seen from the figure that ab

SOLUTIONS 215

is po — p, which is connected through Raoult's law with x\.
From the relations in Fig. 35 it will be evident that

ab ac be
cd cd cd

For small temperature intervals such as are involved in freezing-
point changes in dilute solutions, ac/cd is substantially the slope
of the vapor-pressure curve for the solid solvent at the freezing
point; and bc/cd, which is the slope of the vapor-pressure curve
for the solution, is substantially equal to the slope of the vapor-
pressure curve for the pure solvent at the freezing point.

These slopes are given by the approximate form of the Clapey-
ron equation as

ac dp^u po Ag.^ , be dpo p0 A//ev»p

cd ~ dT ~ 72ZV cd dT ~ RT<? ( }

Upon subtracting the second of these relations from the first,
(noting that A//8Ubi — A//eVaP = A//fU81on for a mole of solvent ai»,
the freezing point), we obtain the relation between ab and cd,
which is

ab _ AfffuMoapo _ PO — p
~cd ~ ~~ ~ '~

Upon rearranging and putting x\ for (p0 — p)/po from Raoult's
law, we obtain the freezing-point equation

Since the triangles abd and ajh in Fig. 35 are similar, the relation
between aj and hk is

These equations show that the freezing-point depression, which
we shall write AT7/, is proportional to the mole fraction of solute
and that the proportionality constant is .RTV/A///, which may be
calculated from the properties of the pure solvent. For a dilute
solution the relation stands

AT1/ = = Kx • (12)

216 PHYSICAL CHEMISTRY

The proportionality constant K is the factor by which the mole
fraction of the solute in a dilute solution must be multiplied to
give the freezing-point depression. This quantity is 104° for
water, but there is no aqueous solution that freezes at —104°;
even if the mole fraction of solute were 0.10, it would not follow
that AT7/ is 10.4° except by accidental compensation, for such a
solution does not meet the assumptions made in deriving the
equation.

Since this equation closely resembles equation (8) for the
boiling-point elevation, we should note that the same approxi-
mations regarding slopes were made in deriving both equations,
and hence each one is valid only so long as these approximations
are justified.1 For solutions in which ionization or polymeriza-
tion of the solute does not occur, the equation will give AT7/ in
substantial agreement with experiment when the mole fraction of
solute is not greater than 0.02. It should be noted that this
equation does not require the solute to be nonvolatile. The
curves adh and dbf of Fig. 35 intersect when the solvent vapor
pressure is the same above the pure solid solvent and the solution,
whether or not the solute has a vapor pressure.

Climatic variations in barometric pressure produce negligible
changes in freezing points except when the highest precision is
necessary, which is not true of the boiling-point equation 2 But
the freezing-point equation does require that the crystalline
phase at equilibrium be the pure solvent, just as the boiling-point
equation requires that the vapor be pure solvent. We shall see
in a later chapter that some solutions deposit crystalline phases
that are not pure solid solvent; of course, equation (12) does not
apply in these systems.

1 A more exact equation relating the freezing-point depression to x, the
mole fraction of solute, is

dAu^-j;) _ A/7,

dT ~ RT* (i6)

If A#/ is constant over the temperature interval involved, the integral of
this equation between 770 and T is

This relation will give better agreement for high mole fractions of solute;
for dilute solutions it reduces to equation (12) above

2 If the barometer changed from 760 to 740 mm , the freezing point of an
aqueous solution would rise about 0.0002°, its boiling point would fall 0.74°,

SOLUTIONS

217

In precise work it is necessary to remove air from the solutions,
since air would act as a solute with the usual effect upon the
freezing point. We have already seen that the definition of
centigrade zero includes a provision that ice be in equilibrium
with water saturated with air at 1 atm. and that complete
removal of the air raises the freezing point 0.0023°.

In dilute solutions Ni/(Ni + Nz)9 the mole fraction of solute,
is close to Ni/Nz, and therefore the freezing-point equation may
be written

AT7/ = Fm (14)

in which F is the lowering produced per mole of solute in 1000
grams of solvent. The freezing-point depression for 1 mole of
solute in 1000 grams of benzene may be calculated from equation
(12), since the fusion of benzene at 5.4° absorbs 30.3 cal. per
gram, as follows:

AT7/ =

1.99(278.5)2

30.3 X 78 1 + 100^78

= 4.7°

It should be noted that 1000 grams of benzene is 12.8 moles and
that the mole fraction of solute is hence 0.072, which is scarcely
a " dilute solution." If the above calculation is repeated for 0.1
mole of solute in 1000 grams of benzene, AT7/ will be 0.51 and
accordingly F = 5 1 is obtained. In the limit F = JT/(1000/M),

TABLE 36 — FREEZING-POINT CONSTANTS1

Substance

M
Pt

Substance

M.
Pt.

Acetic acid

3 9

Ethylenc bromide

12 5

Benzene

5 5

5 1

Naphthalene

7 0

Benzophenorie

47 7

9 8

Nitrobenzene

5 7

7 0

Camphor

179

Stannic bromide

26 4

24 3

Diphenyl

8 2

Stearic acid

69 3

4 5

p-Dichlorberizene

52 9

7 5

Water

104

1 86

1 For F in other solvents see "International Critical Tables," Vol IV, p
183, additional values of K may be computed from the latent heats of fusion
in Table 21 The recorded values of K and F are not among the most satis-
factory data in physical chemistry. Values which are stated to 0.1° are
frequently in error by 1° or more, and there is no simple way of sorting the
good data from the poor.

218

PHYSICAL CHEMISTRY

where M is the molecular weight of solvent, and since K is 65
for benzene and 1000/M is 12.8, F = 65/12.8 = 5.1, which is the
value given in Table 36.

These calculations have been given to show that equation (14)
is a suitable approximation for dilute solutions, and not appli-
cable to solutions of high molality. Freezing-point data are fre-
quently recorded in tables of ra and A!T//ra, which is a useful
device, but it will be found that ATf/m is not constant in these
tables For most calculations in which the solute concentration
is high, equation (12) will be a better choice than equation (14).

Some experimental values of F = AT//W for water are given
in Table 37. Values of both K and F for some common solvents
are given in Table 36.

TABLE 37 — FREEZING POINTS OF SOLUTIONS OF MANNITOL IN WATER1

Molal
concentration

Freezing-point
depression

F = LTf/m

0 006869

0 01274

1 853

0 01006

0 01846

1 847

0 01041

0 01930

855

0 02039

0 03790

859

0 02249

0 04171

854

0 05061

0 09460

868

0 06062

0 11265

858

0.09574

0 1790

870

0 1197

0 2225

858

Freezing-point depressions furnish a convenient means of
determining the molecular weights of solutes when such effects
as ionization or polymerization or solvation of the solute are
absent. For example, the molecular weight of triphenylmethane
[(CeHs^CH = 244.1] in benzene as derived from freezing points
is shown in Table 38.

By means of thermocouples it has been possible to measure
very accurately the freezing points of quite dilute solutions.
Usually the solution is made up somewhat stronger than needed
and poured over an excess of crystalline solvent. The mixture
is stirred until equilibrium is established, the freezing tempera-

1 FLUGBL and ROTH, Z physik. Chem., 79, 577 (1912).

SOLUTIONS

219

ture is accurately determined, and a sample of the solution is
withdrawn through a chilled filter and analyzed. This procedure
is more accurate than that of chilling a solution of known con-
centration until solid begins to separate, for a correction must
then be applied to allow for the solid that has separated. When
a large quantity of solid is used, equilibrium is more readily and
more certainly established, and the added labor of analyzing the
solution actually at equilibrium is well justified. If a solution
of known strength is cooled until solid separates, undercooling is
almost unavoidable, equilibrium is established slowly, and the
correction for the quantity of solid deposited is uncertain.

TABLE 38. — FREEZING POINTS OF TRIPHENYLMETHANE IN BENZENE1

Molahty

Freezing-point
depression

Molecular
weight

0 000313

0 00158

244 5

0 000634

0 00322

243 5

0 000986

0 00497

245 4

0 004096

0 02082

243 5

0 0248

0 1263

243 1

0 04375

0 2214

244 6

The molal freezing-point depressions AT7//??! calculated for
dilute solutions of inorganic salts in water will not be constant
or close to 1 86°, because of ionization of the solutes. But the
extent of ionization in these solutions is not to be calculated
simply by assuming that A77//m divided by 1.86 gives the total
number of solute moles (molecules plus ions) per formula weight
of salt. This topic is discussed in the next chapter.

Molecular weights derived from freezing-point determinations
in nonaqueous solvents frequently require interpretation as well,
for effects such as ionization or polymerization into double
molecules or solvation sometimes occur. The figures for tetra-
butyl ammonium perchlorate (formula weight 341.8) in benzene2
are an extreme example in which the interpretation is made more
difficult by an appreciable conductance of the solutions. The
data are shown in Table 39.

1 BATSON and KRAUS, J Am Chem Soc , 56, 2017 (1934).

2 ROTHROCK and KRAUS, ibid., 59, 1699 (1937),

220

PHYSICAL CHEMISTRY
TABLE 39 — MOLECULAR WEIGHTS

Moles per 1000
grams benzene

AT,

A7V

AT^deal

Apparent molec-
ular weight

0 00109

0 00184

0 333

1029

0 00434

0 00535

0 243

1404

0 00962

0 00982

0 202

1692

0 01423

0 0120

0 166

2052

Solutions of urea in water are more nearly typical of solutions
in general than are the examples of close conformity to the ideal
laws or the extreme deviations from them that have been quoted.
They conform fairly closely at moderate concentrations, more
closely at low concentrations, and deviate at high concentrations.
The freezing-point depressions1 illustrate this fact.

m
AT,

0 3241 0 646

0 5953 1 170

1 837 1 811

521
673

757

3 360
5 490
1 660

5 285
8 082
1 529

8 083
11 414

1 412

Solutions of ethanol (ethyl alcohol) in water also conform to
the ideal equation for freezing-point depression in dilute solution
and deviate at higher molalities. In these solutions the ratio
A!T//ra increases with the molality, while the same ratio decreased
with increasing molality for the urea solutions above. There is
no way of predicting whether the deviations will be in one way or
the other. The data for ethanol are as follows:

m
AT,/m

0 1
t 83

1 0
1 83

2 0

1 84

4 0
1 93

6 0
2 05

7 0
2 12

10 0
2 2

15 0
2 0

Osmotic Pressure. — The molecules of a solute in a dilute solu-
tion are separated from one another by distances that are large
compared with the diameters of the -molecules, and they have a
certain freedom of motion. This condition is similar to that of
the molecules of a gas, the main difference being that the space
between the molecules in a solution is filled with solvent. Early
experiments showed that the pressure necessary to prevent the
flow of water through an animal membrane into a solution was
proportional to the concentration of solute and that this pressure
increased nearly in proportion to the absolute temperature.

1 CHADWELL and POLITI, ibid., 60, 1291 (1938)

SOLUTIONS 221

These facts led van't Hoff to suggest that the solute exerts an
" osmotic pressure " corresponding to the pressure that it would
exert in the form of gas in the same volume if the solvent were
removed. To test this supposition it would be necessary to
devise a membrane that was impermeable to solute molecules
and allowed free passage of solvent.

Consider a cylinder closed at one end by such a membrane,
filled with a solution, fitted with a movable piston, and immersed
in pure solvent. If "the pressure exerted by the piston exceeds
the osmotic pressure, solvent will be forced out of the solution
through the membrane; if the pressure is less than the osmotic
pressure, solvent will enter the solution through the membrane;
and when the pressure on the piston is equal to the osmotic
pressure, no solvent will pass through the membrane in either
direction. To the extent that this conception of osmotic pres-
sure is correct, the osmotic pressure in a dilute solution will be
equal to that calculated on trie assumption that the solute is an
ideal gas in the same volume at the same temperature.

In spite of experimental difficulties, which were many and
troublesome,1 suitable membranes have been devised, and some
osmotic pressures have been obtained. They confirm the
assumption that in a dilute solution an osmotic pressure exists
which is given by the equation

TTV = nRT or T = CRT (15)

in which C is the volume concentration, R has the same value as
in the ideal gas law, and ir is the osmotic pressure. Osmotic pres-
sures of sugar are shown in Table 40; a membrane of copper
ferrocyanide embedded in the walls of a clay vessel was used.
The columns headed "Ratio" show the ratio of the measured
osmotic pressure to the pressure calculated on the assumption
that the solute is an ideal gas occupying the volume of the solu-
tion. ' The deviations of these numbers from unity are no greater
than one might expect of a gas of molecular weight 342 at these
pressures and temperatures. Osmotic pressures of mannite at

1 See Morse, Carnegie Inst. Wash. Pub., 198 (1914); Berkley and Hartley,
Phil Trans Roy. Soc (London), (A) 209, 177 (1909); (A) 218, 295 (1919) for
the method and experimental data. The pressures in Table 40 are taken
from the paper by Morse.

222 PHYSICAL CHEMISTRY

molalities below 0.5, or at osmotic pressures below 12 atm , differ
from the calculated ideal gas pressures by less than 1 per cent.
If the osmotic-pressure equation is written in the form

TV = -^ RT
M

it will be evident that these experiments could be used to deter-
mine molecular weights of solutes Osmotic-pressure meas-
urements are experimentally difficult for solutes of moderate
molecular weight, chiefly because of the preparation of semiper-
meable membranes that will not "leak" solute ; therefore, molec-
ular weights are usually determined from freezing points or boiling
points. But the recent interest in high polymers, which may
have molecular weights of 100,000 or more, has directed atten-
tion to osmotic pressures as a means of studying them. A solu-
tion of 10 grams of such a substance in 1000 grams of water
would have a freezing-point depression of only 0.00018°, and the
presence of the slightest impurity would render the measured
depression uncertain. Such a solution would have an osmotic
pressure of 0.0025 atm , which is 26 mm. of water. Membranes
that are impermeable to such large molecules and capable of
withstanding this small pressure are comparatively easy to make,
but deviations from the laws of ideal solutions are quite high for
solutes of such high molecular weights, even at low mclalities^
To correct for them the common expedient is to plot the ratio of
osmotic pressure to concentration, extrapolate to zero concen-
tration, and determine the molecular weight from the limiting
ratio of TT to C, as "was done in determining precise molecular
weights of gases from densities in Chap. I.

For example, the ratio n/C for polyisobutylene in benzene1
is nearly independent of the concentration, but the ratio w/C
for the same preparation in cyclohexane changes rapidly with C.
Plots of TT/C against C are nearly linear for both solvents and
when extrapolated to zero concentration give the same limit of
TT/C, as shown in Fig. 36. The extrapolated value of nearly
0.001 atm. gives for a concentration of 10 grams per liter of
solution at 25°C. an average molecular weight of 250,000. In
such preparations the presence of larger and smaller molecules is

1 FLOKY, J. Am. Chem Soc , 66, 372 (1943).

SOLUTIONS

223

not excluded, and indeed their presence is probable. The freez-
ing-point depression of this solution in benzene would be about
0.0002°, and, while such a temperature difference can be meas-
ured, the presence of a slight impurity of reasonable molecular
weight would render the measured freezing point uncertain.

0.006

Ol
\

£ 0004
.£
o

0.002

Benzent

9 SO/^/^/O

0 5 10 15 20

Concentration in Grams per lOOOcc.

of Solution
FIG. 36.-— Osmotic pressures of polyisobutylene solutions at 25°.

The osmotic membranes are probably permeable to ordinary
solutes, and thus they correct for the presence of these solutes
and yield the average molecular weight of the polymer.

Similar wide deviations from ideal solutions are shown by other
systems, for example, polymethylmethacrylates in chloroform.1
TABLE 40 — OSMOTIC PRESSUKES OF SUGAR SOLUTIONS, IN ATMOSPHERES

0°

20°

40°

60°

.Molal

Aver-

concen-

Osmotic

age

tration

pres-

Ratio

pres-

Ratio

pres-

Ratio

pres-

Ratio

ratio

sure

0 1

2 462

1 106

2 590

1 130

2 560

0 998

2 717

1 000

1.06

0 2

4 723

1 065

5 064

1 060

5 163

1 012

5 438

1 001

1.03

0 4

9 443

1.060

10 137

1 060

10 599

1 037

10.866

1 000

1.04

0 6

14 381

1 077

15 388

1 071

16 146

1 053

16 535

1.015

1.05

0 8

19 476

1 091

20 905

1 093

21 806

1.068

22 327

1.025

1.07

1 0

24 826

1 130

26.638

1 130

27.701

1.085

28.367

1.045

1.10

1 HOFF, Trans. Faraday Soc., 40, 233 (1944).

224

PHYSICAL CHEMISTRY

Plots of TT/C against C are nearly linear but not horizontal, and
extrapolation to zero concentration gives acceptable molecular
weights.

The osmotic pressure is related to po/p, the ratio of the vapor
pressure of the pure solvent to that of the solvent from solution,
by the equation

(16)

in which vi is the volume of a mole of liquid solvent This
equation may be derived from an isothermal reversible cycle
of changes in which (1) a mole of solvent is expressed reversibly
from a large quantity of solution through a semipermeable
membrane, (2) the solvent is vaporized reversibly under its vapor
pressure po, (3) the solvent vapor is expanded reversibly to p,
and (4) the vapor is condensed reversibly into the solution. The
work done in these stops is

wz = PQ(VV — vj) = RT

ti>, = RT In -2- = RT In ^
vi P

Uh = P(VL - Vr) = -RT
TABLE 41. — CANE-SUGAR SOLUTIONS AT 30° '

TVTnlol

Measured

Calculated osmotic pressures

concen-
tration

pressure1
(atmos-

Equation

Per cent

Equation2

Per cent

pheres)

(15)

error

(16)

error

0 10

2 47

.2 47

2 47

0 0

1 00

27 22

24 72

27 0

1 0

2 00

58 37

49 40

58 4

0 0

3 00

95 16

74 20

96 2

0 0

4 00

138 96

98 90

138 3

0 5

5 00

187 3

123 60

182 5

2 5

6 00

232 3

148.30

230 9

0 6

1 ERASER and MYRICK, J Am. Chem Soc , 38, 1907 (1916)

2 Vapor pressures from BERKLEY, HARTLEY, and BURTON, Phil Trans.
Roy. Soc. (London), 218, 295 (1919) The data are as follows:

Molal concentration 1 00 2 00 3 00 4 00 5 00 6 00

Ratio pQ/p ,1.020 1,044 1072 1.104 1,140 1.17

SOLUTIONS 225

According to the second law of thermodynamics the summation
of work in a reversible isothermal cycle is zero, and this is such
a cycle, so that

-™i + RT + RT In 22 - RT = 0

and, upon solving this equation for TT, equation (16) results. As
may be seen from Table 41, this equation gives calculated osmotic
pressures that agree with experimental pressures within the
error of the data, while equation (15) deviates seriously from the
measured pressures.

Problems

Numerical data for some of the problems must be sought in tables in the text.

1. When the concentration of SOz is 1 mole per liter of CHClj at 25°, the
equilibrium pressure of SO2 above the solution is 0 53 atm. When the total
SO2 is 1 mole per liter of water at 25°, the equilibrium pressure of SO2 above
the solution is 0 70 atm , and 13 per cent of the, solute is ionized into H+ and
HSO3~. Sulfur dioxide is passed into a 5-liter bottle containing a liter of
water and a liter of CHOI* (but no air) until the total moles of SO2 per liter
of water at equilibrium is 0 20 at 25° Under these conditions 25 per cent
of the SO 2 in the water layer is ionized Henry's law applies to the non-
ionized portion (SO2 + H2SOs) in water and to SO2 in CHC13 (a) How
many moles of SO2 were passed into the bottle? (b) More SO2 is passed
into the bottle until the total quantity is 1 mole Estimate the fraction
ionized in the water layer under these conditions by interpolation from the
data on page 189, and calculate the moles of SO2 in each of the three phases

2. The boiling point of methanol (CH3OH = 32) is 65°, its molal latent
heat is 8400 cal at 65° and may be assumed constant over the temperature
range of this problem A solution of 0 5 mole of CHC13 in 9 5 moles of
CH3OH boils at 62.5° Calculate the total vapor pressure and the compo-
sition of the vapor in equilibrium with a solution containing 1 mole of CHC13
and 9 moles of CH3OH at 62 5°.

3. (a) The ratio of the pressure of CO2 in atmospheres to the molahty
of the saturated solution is p/m = 29 at 25°C. Calculate the total pressure
at equilbrmm in a 2-liter bottle containing 0 10 mole of CO2 and 1000 grams
of water (but no air) at 25° (6) The ratio p/m = 100 for CO2 in water at
100°. Calculate the total pressure in the bottle at 100°C , neglecting small
corrections (c) List the factors neglected in the calculation of part (b).

4. The latent heat of evaporation of toluene (CyHg = 92) is 85 cal. per
gram at 110°C (the boiling point). When toluene is distilled with steam
at 1 atm. total pressure, the distillation temperature is 84°C. Toluene
and water are mutually insoluble. How many grams of toluene will be in
100 grams of distillate?

226 PHYSICAL CHEMISTRY

5. Calculate the boiling-point constants k and B and the freezing-point
constants K and F for benzene from the physical constants of benzene in
Tables 16 and 21, and compare with the values in Tables 33 and 36

6. Ethanol (C2H6OH) boils at 78 3°, and its molal latent heat is 9400 cal.
A solution of 0.07 mole of benzene in 0 93 mole of ethanol boils at 75° and
1 atm. (a) Calculate the partial pressures of ethanol and benzene in the
vapor, (b) Calculate the partial pressure of each substance above a solu-
tion of 0 1 mole of benzene and 0 9 mole of ethanol at 75°

7. The vapor pressure of a solution of 2 38 moles of cane sugar (C^H^On)
in 1000 grams of water at 30° is 94 88 per cent that of pure water Calculate
the osmotic pressure of this solution from the vapor pressure Calculate
also its osmotic pressure, assuming that it behaves as an ideal gas at this
concentration The measured osmotic pressure is 73 atm

8. The change of vapor pressure of benzene (Cell 6 = 78) with temper-
ature is given in a footnote on page 114, and its boiling point for 1 atm is
80 09°. (a) Calculate the vapor pressure at 80 09° of a solution containing
0 20 mole of nonvolatile solute in 1000 grams of benzene (b) Calculate
the boiling point of this solution from the vapor-pressure data (r) Cal-
culate the boiling-point constants k and B for benzene from the vapor-pres-
sure data (d) Calculate another value of k, taking A//w = 7600 cal for
benzene.

9. Calculate the weights of ethyl alcohol, of ethylene glycol, and of
glycerol required for 25 kg of solution that would not deposit ice at 0°F

10. Equilibrium mole fractions for ethanol-water mixtures are given in
Table 35 and Fig 34 (a) If 1000 grams of a mixture that boils at 83 3°
are distilled until the boiling point rises to 86 4°, what weight of distillate
will be obtained? (b) Calculate the temperature at which the original
mixture would begin to deposit ice, assuming it an ideal solution Recal-
culate this temperature from the data on page 220 (c) Calculate the
weight of ice deposited per kilogram for the residue obtained in part (a)
if this residue were cooled to the actual freezing point of the original mixture
(d) A vapor mixture of 0 A mole of ethanol and 0 6 mole of water is cooled to
81.8° and 1 atm. without removing the condensate from the vessel. What
are the equilibrium mole fractions in this system ? What weight of vapor
remains uncondensed?

11. Ethyl iodide is an ideal solute when dissolved in p-chlorotoluene (mol
wt. 126 5, m. pt. 7.80°). The freezing-point depression for this solution
changes with m, the moles of solute per 1000 grams of solvent, as follows :

AT7/ 0 263° 0 487° 0 708° 1.262°

m 0 0468 0 0875 0 128 0.227

For chloroacetic acid (C2H3O2C1 = 94 5) in p-chlorotoluene, A71/ changes
with the grams of solute (g) per 1000 grams of solvent as follows.

. . 0.27 0 42 0 52 0.62

. 9.2 13 6 18.0 20.8

SOLUTIONS 227

Calculate A/// and F for p-chlorotoluene and the molecular weight
of chloroacetic acid in p-chlorotoluene [/ Chem Soc (London), 1934,
1971.]

12. The distribution ratio Cw/Cf for formaldehyde (HCHO) between
water and ether at 25° is 9 2 for volume concentrations (a) How many
liters of water will be required to remove in one extraction 95 per cent of the
formaldehyde from a liter of ether containing one mole of formaldehyde?
(b) How much formaldehyde would remain in a liter of ether containing
initially one mole of formaldehyde after eight successive extractions with
50 ml of water?

13*. The freezing-point depression for 0 05 mole of bromine in 1000 grams
of water is 0 0938° , that for 0 05 mole of chlorine in 1000 grams of water is
0 157° What is the chemical explanation of the difference m A7"//m for
the two solutions?

14. Calculate the Bunsen coefficient a for H2S in water at 25° and for
SO2 in water at 25°, from the data on pages 187 and 189

16. Calculate the boiling-point constant k for ethyl ether from the data
in Table 14, calculate another value from Table 16, compare with the entry
in Table 33

16. (or) Calculate the temperature at which n-CgHis will distill with steam
at 1 atm total pressure and the composition of the distillate in per cent by
weight (6) Repeat the calculation for a total pressure of 0 84 atm , which
would prevail 1 mile above sea level (See Table 14 for data )

17. The solubility of H2S in water at 25° is 0 102 mole per liter of solu-
tion when />(Il2S) is 1 atm , and the distribution ratio between benzene
(Cr,H<5 = 78) and water is CW/C\ = 5 72 for volume concentrations. The
vapor pressures of the pure liquids at 25° are 0 12 atm for benzene and
0 03 atm for water Hydrogen sulfide is passed into a 5-liter bottle contain-
ing a liter of benzene and 400 ml of water (and no air) until the total pressure
at 25° is 5 atm (a) Neglecting effects calculable from Raoult's law, calcu-
late how many moles of H2S are in each of the three phases at equilibrium
(b) Show that the neglected pressure changes are negligible compared with
the total pressure (r) Calculate the total pressure at equilibrium if 2 more
liters of benzene are forced into the bottle and no gas escapes

18. Equilibrium mole fractions at 1 atm total pressure for nitric acid
(ilNOs = 63, b pt 86°, symbol N) and water (symbol W) change with the
temperature as follows.

Temperature 110° 120° 122° 120° 115° 110° 100°

Mole per cent N m liquid 11 26 38 45 50 54 62

Mole per cent N in vapor 1 10 38 70 84 90 93

(a) Draw a complete temperature-composition diagram for this system.
(b) A vapor mixture of 3N 4- 2W is cooled to 114°, and no condensate is with-
drawn from the system What are the equilibrium compositions in the
liquid and vapor phases? (c) A liquid mixture of 3N -f- 2W is completely
fractionated by repeated fractional distillation. State the composition and
calculate the weight of distillate and of residue obtained, (d) A liquid
mixture of 3N + 2W is distilled until the boiling point rises 4°, and the

228 PHYSICAL CHEMISTRY

distillate is collected in a single portion Calculate the weight and com-
position of the distillate

19. Equilibrium mole fractions at 1 atm total pressure for carbon tetra-
chloride (CC14, rriol wt. 154, b. pt 1208°, symbol C) and tetrachloro-
ethylene (C2C14, mol wt 166, b. pt 76 9°, symbol T) change with
temperature as shown in the following table

t ... 108 5 100 8 93 0 89 3 86 0 83 5 81 5 79 9 77 5

Xc 0 100 0 200 0 300 0 400 0 500 0 600 0 700 0 800 0 900

ye 0 469 0 670 0 800 0.861 0 881 0 918 0 930 0 958 0 980

(a) Draw a complete temperature-composition diagram for this system
(b) A vapor mixture of 3O and 7T is cooled to 100 8°, the pressure remains
1 atm , and no condonsate escapes from the vessel What arc the quanti-
ties of C arid T in the vapor? (c) A liquid mixture of 3(1 and 7T is distilled
until the boiling point rises to 100 8°, and the distillate is collected in a single
portion. What are the quantities of C and T in the distillate? (d) This
distillate is distilled until the boiling point rises 5° Calculate the composi-
tion and weight of distillate obtained [Data from McDonald and McMil-
lan, Ind Eng Chem , 36, 1175 (1944) ]

20. The steam distillation of an insoluble liquid takes place at 90°C.
and 1 atm total pressure, and the distillate contains 24 per cent by weight
of water (a) Calculate the molecular weight of the substance distilled
(b) This substance boils at 130° Calculate its molal heat of evaporation

21. The distribution ratio Cw/Ct for acetone between water and toluene
is 2 05 The constant (\,/p = k} is 2 8 for acetone in water, when the con-
centrations are m moles per liter and the pressures in millimeters of mercury
(a) Calculate the moles of acetone extracted from 650 ml ol water containing
0 25 mole of acetone, il it is shaken three successive times with 50- ml por-
tions of pure toluene. (b) Calculate the constant Ct/p = k^ for acetone
dissolved in toluene

22. (a) If 1 gal of glycerol and 3 gal of water form the solution in an
automobile radiator, at what temperature will ice begin to separate out of
the solution? (b) What weight of ice will deposit from this solution at
0°F. (= -178°C.)? A gallon of water weighs 3785 grams; glycerol
(CaHgOa = 92) has a density of 1 26, the latent heat of fusion of ice is
79 cal per gram, (c) Repeat the calculation of the weight of ice deposited
at 0° from a solution of 1 gal. of alcohol (C2H5OH = 46, density 0 79) in 3
gal of water.

23. Naphthalene (Ci0H8 = 128) is soluble in benzene and not volatile
from the solution. The vapor pressure of a solution of 90 grams of naph-
thalene in 1000 grams of benzene (C6Hfi = 78, b pt 80 1°) is 0 80 atm at
75°, the latent heat of evaporation of benzene is 7600 cal per mole (a)
Calculate the vapor pressure of pure benzene at 75°. (6) Calculate the
boiling point of the solution.

24. Mixtures of carbon tetrachloride and ethylene dichlonde are partly
distilled, and the equilibrium vapor compositions are determined from the
"boiling points of the first portion of each distillate. The data are as follows:

SOLUTIONS 229

Mole per cent C,H4C12 111 liquid 0 10 30 60 80 90 100

Boiling point of liquid 76 5 75 7 75 3 76 5 78 5 80 2 82 7

Boiling point of distillate 76 5 75 5 75 3 75 7 77 0 78.5 82 7

Sketch the distillation diagram, showing vapor composition by a dotted
line Estimate from the diagram the quantity and composition of distillate
and residue resulting if 1000 grams of a liquid mixture of 70 mole per cent
C2H4C12 was distilled until the boiling point rose 2°.

26. Beiizophenone (C6H5( 'OO6H6 = 182, m pt 47 7°) and diphenyl
(Ci2Hio = 154, m pt 69°) mix m all proportions in the liquid phase A
solution containing 22 8 mole per cent diphenyl begins to deposit solid
benzophenone at 35 0°, and a solution containing 78 0 mole per cent diphenyl
begins to deposit solid diphenyl at 56 2° (a) Calculate the freezing-point
constants RT^/\Hm for these substances (b) Considering first one and
then the other as the solvent, calculate the composition of a mixture of
these substances that would freeze at 25 2° The freezing-point curves
are found by experiment to intersect a 39 3 mole per cerit»diphenyl and at
25 2°. [LEE and WARNER, ,1 Am Chein Soc , 65, 209 (1933) ]

26. Nitrobenzene ((yCH5NO2 = 123) is only slightly soluble m water At
99 3° the two solutions contain 012 mole per cent nitrobenzene and 91 2
mole per cent nitrobenzene, respectively The vapor pressure of each of the
solutions is 1 00 atm at 99 3°, and at 99 3° the vapor pressure of pure
nitrobenzene is 0 0275 atm (a) What is the composition of the vapor in
equilibrium with the solutions, in mole fraction, and in weight fraction?
(b) Calculate the vapor pressure of water at 99 3° from the data in this
problem.

27. Equilibrium mole fractions of acetone in the liquid (xa) and vapor (ya)
for mixtures of acetone and chloroform at 1 atm total pressure change with
temperature as follows

/°C 56° 59° 62 5° 65° 63 5° 61°

xa 0 0 20 0 40 0 65 0 80 1 0

ya 0 0 11 0 31 0 65 0 88 1 0

(a) Draw a temperature-composition diagram for this system (b) A
liquid mixture of 1 mole of chloroform and 4 moles of acetone is distilled
until the boiling point rises to 60°, and the distillate is collected in a single
portion Calculate the weight of distillate obtained (c) A mixture of
1 mole of chloroform arid 4 moles of acetone is completely fractionated by
repeated distillation. Calculate the weight of distillate and weight of
residue obtained

28. The atomic heat of fusion of cadmium at its melting point (596°K.) is
1460 cal., the atomic heat of fusion of bismuth at its melting point (546°K.)
is 2500 cal., the liquids mix in all proportions, and both have monatomic
molecules, (a) Calculate the temperature at which a solution containing
10 atomic per cent bismuth would be in equilibrium with solid cadmium
and the temperature at which a solution containing 10 atomic per cent
cadmium would be in equilibrium with solid bismuth, (b) Calculate the
freezing point of a solution containing 40 weight per cent cadmium, assum-

230 PHYSICAL CHEMISTRY

ing first cadmium and then bismuth to be the solvent. (Experiment shows
that a solution containing 40 weight per cent cadmium is in equilibrium
with both solid metals at 413°K.)

29. The ratio ir/C of the osmotic pressure (in millimeters of Hg) to con-
centration (in grams per liter) for a solution of serum albumin in water at
25° changes with concentration C as follows.

TT/C 0 430 0 385 0 335 0 315

C 73 50 30 18

(a) Calculate the molecular weight of the solute from the limiting v/C
(b) Calculate the freezing-point depression for the solution containing 30
grams per liter

30. The following table gives p, the partial pressure of HC1 in atmospheres,
and x, the mole fraction of HC1 in CC14 at 25°:

p 0 235 0 500 0 559 0 721 0 872

x 0 00379 0 00803 0 00922 0 01190 0 01415

(a) Calculate the Henry's law constant k as defined m equation (4) for
this system. (6) From the average value of k calculate the constants k'
and k"' as defined in equation (5) for this system, taking the density of
CC14 as 1 498 at 25° (c) Calculate the Bunsen coefficient a as defined on
page 186 for this system at 25°. [HOWLAND, MILLER, and WILLARD, J
Am. Chem. Soc., 63, 2807 (1943) ]

CHAPTER VII
SOLUTIONS OF IONIZED SOLUTES

This chapter presents some experimental facts relating to
vapor pressures, freezing points, conductances, and other prop-
erties of solutions in which ions rather than molecules are the
important solutes; it considers the products formed when an
electric current passes between electrodes in these solutions, the
changes in the quantity of solutes near the electrodes, and the
interpretation of these effects in terms of the velocities and other
properties of the ions. After the necessary facts have been
presented, the underlying theory will be considered.

The standard methods for determining molecular weights of
solutes, such as were described in the previous chapter, lead to
impossible values when applied to solutions of inorganic salts in
water. For example, the freezing-point depression of a solution
of 30 grams of sodium chloride in 1000 grams of water is about
1 7°, which would indicate a molecular weight of 32, while 58.5 is
the sum of the atomic weights of sodium and chlorine. The
vapor density of hydrogen chloride agrees with the common
formula HC1, but the freezing-point depression for 3.65 grams of
HC1 in 1000 grams of water is 0.35° in place of 0.186°, which
would be expected of 0 1 mole of " ideal" solute. Similar effects
are found for almost all inorganic solutes in water.

'Such solutions conduct electricity to a moderate extent, while
the solutions studied in the previous chapter have only negligible
conductances.1 From a study of the properties of these solu-
tions, Arrhenius suggested that the solutes in conducting solu-
tions are dissociated into charged particles called ions; and since

1 Even the best conducting solutions are poor conductors compared with
metals. For example, the resistance of a centimeter cube of molal potas-
sium chloride solution at 20° is about 10 ohms A copper wire of 1 sq. cm.
cross section and of this resistance would be about 35 miles long. A centi-
meter cube of molal sugar solution would have a resistance of about 10 meg-
ohms. Thus the conductances of the three types of systems are of different
orders of magnitude.

231

232 PHYSICAL CHEMISTRY

this results in the formation of two effective moles of solute ion
for each formula weight of sodium chloride (for example) that
ionized, a partial explanation of the small molecular weights was
at hand. The anomalous molecular weights were always less
than the formula weight but greater than half of it for solutes of
this type, and they decreased with decreasing concentration. He
therefore assumed that ionization was incomplete, that it was a
dissociation equilibrium that changed with concentration, as
would be true of any dissociation. The original teim was
" electrolytic dissociation'7 rather than ionization.

Experimental work upon 'the properties of aqueous solutions
was begun about 1890 by Arrhenius, Kohlrausch, Ostwald, van't
Hoff, and Hittorf and continued by many others until sufficient
data were available for a fairly comprehensive theory that
explained the behavior of these solutions "within the experi-
mental error." But as experimental errors were largely elimi-
nated, it became evident that the theory was unable to explain
many of the experimental facts. For example, the "fractional
ionization" as derived from mole numbers (page 237) or from
the conductance ratio (page 276) did not change with concen-
tration in the way to be expected from the laws of chemical
equilibrium Moreover, the extent of ionization in a given solu-
tion as measured by the two methods was not the same. There
was much discussion of the "abnormality of strong electrolytes"
but no clear definition of the term "extent of ionization." If
ionization meant the transfer of an electron from sodium to
chlorine, ionization was complete m any solution, and we now
believe that this effect attends the formation of sodium chloride
from its elements. If complete ionization meant the separation
of the ions by dilution to such an extent that they were "normal
solutes" completely freed from influence upon one another, there
was no evidence that this condition was attained in the most
dilute solutions that could be studied experimentally.

Suggestions of "complete ionization" were occasionally heard
before 1910, and between 1915 and 1925 most physical chemists
accepted the idea that "strong" (highly ionized) electrolytes
were completely ionized. Of course, this idea was not applied
to "weak," or slightly ionized, solutes such as ammonium
hydroxide or acetic acid, for there is no evidence that they are
ionized more than a few per cent in solutions of moderate con-

SOLUTIONS OF IONIZED SOLUTES 233

centration. The assumption of complete ionization for "strong"
electrolytes meant only that an effort would be made to explain
the properties of these solutions on grounds other than a sup-
posed fractional ionization, namely, an interionic attraction
existing between the ions of opposite charge.

A large amount of experimental work on solutions is still in
progress in many laboratories ; extensions and revisions of theories
are still under way ; and while a fairly satisfactory general theory
has been developed, much still remains to be done. Under these
circumstances it seems best to present the bulk of the experi-
mental evidence first, then the interpretation that is beyond
question, then a summary of the older theory and its short-
comings, and finally a brief review of the newer theory.

Types of Electrolytes. — Ionizing solutes may be divided into
classes according to their products upon ionization. Simple
binary (or uni-univalent) electrolytes, such as hydrochloric acid,
sodium nitrate, and potassium acetate, yield a single positive
ion bearing a unit positive charge, or having lost 1 electron, and
a single negative ion bearing a unit negative charge, or having
acquired 1 electron. Solutes of this type exhibit the simplest
phenomena in solution and have been more extensively studied
than salts of other types. Another simple type of ionization is
shown by copper sulf ate and other salts ; each ion bears two units
of electricity, but a molecule forms only two ions. The remain-
ing types of ionized solutes are more puzzling in their behavior
and more difficult to study experimentally, because of the possi-
bility of ionization in different ways or in different steps. Thus
sulfuric acid ionizes according to the reaction

H2S04 = H+ + HSO4~
and the negative ion may ionize further.

HSOr = H+ + SO4—

The formation of intermediate ions of the HSO4~ type is very
common in the ionization of weak acids, which form ions such as
HS-, HCOr, HSO3-, and HPO4— . These are the important
negative ions in solutions of NallS, NaHCOs, NaHSOs, and
Na2HPO4, respectively. The presence of ions such as ZnCl+ in
zinc chloride solutions is also a possibility, and the evidence
for ion.s of the composition Fed4"1" and FeCl2+ in ferric chloride

234 PHYSICAL CHEMISTRY

solution is convincing. No satisfactory general methods have
been devised for establishing definitely the presence or absence
of these intermediate ions.1

Mole Numbers for Ionized Solutes. — We have defined a molal
solution as one containing a mole or formula weight of solute per
1000 grams of solvent and a normal solution as one containing
a chemical equivalent of solute per liter of solution. In this
chapter we adhere to these definitions, of course, but we do
not find by experiment that a mole of a salt produces the effect
upon vapor pressure or freezing point that would be expected
of a nonionized solute. For our convenience in studying the
results of experiment, we define a quantity called the mole
number, which van't Hoff designated by z, and which is the ratio
of the moles of solute as calculated from a vapor-pressure lower-
ing (or other change) to the moles of solute as indicated by the
conventional formula weight. Thus, the vapor-pressure lower-
ing produced by 58.5 grams of sodium chloride in 1000 grams of
water at 18° is 0.475 mm , and Raoult's law indicates that 1.75
moles of ideal solute in 1000 grams of water produces this effect.
Hence 1.75 is the mole number for Im sodium chloride at 18°.
The freezing-point depression of a solution of 40.8 grams of
sodium chloride in 1000 grams of water is 2 705°, and the ratio
2.705/1.86 = 1.455 indicates 1.455 moles of solute per 1000 grams
of water. From the weight composition of the solution,

46.8/58.5 = 0.80

mole of solute, and 1.455/0.80 = 1.82 is thus the mole number
for 0.80m. NaCl at the freezing point.

It was formerly supposed that the change of i with the con-
centration was due to changing fractional ionization and that for
an electrolyte of the A+B~ type, a = i — 1 measured the extent
of ionization. From the fact that i = 2.15 for 1m. LiBr it is

1 Experiments in which solutions of the chlorides of Ba, Sr, Ca, Zn, Cd,
Co, Mg, Ni, or Cu were shaken with ammonium permutite to equilibrium
indicate that no ions of the type MC1+ exist below normal concentrations.
[GfrNTHER-ScHULZE, Z Elektrochem., 28, 387 (1922) ] On the other hand,
transference data for concentrated solutions of cadmium chloride are
difficult to interpret unless CdCl+ ions exist in solution, and experiments
upon the behavior of sulfuric acid indicate definitely the presence of HSO4~
ions in solution. There is also good evidence for the existence of PbCl+ in
lead chloride solutions and for PbOH"1" as the hydrolysis product for lead ion

SOLUTIONS OF IONIZED SOLUTES

235

evident that mole numbers do not measure the extent of ioniza-
tion. Other univalent electrolytes also have mole numbers
greater than 2 at high concentrations, though all these mole
numbers fall below 2 at lower concentrations and again approach
2 at the limit of dilution.

Another quantity sometimes used in discussing freezing-point
or vapor-pressure data of solutions of electrolytes is the "osmotic
coefficient" v, which is ^ divided by the number of ions formed by
the dissociation of 1 mole Thus, <p = t'/2 for NaCl or MgSO4
and (p = i/S for MgCl2 or H2S04.

Vapor-pressure Lowering for Ionized Solutes. — Table 42 gives
the vapor pressures of some solutions of electrolytes in water at
18°. The data show that solutions of the same molality do not
have the same vapor pressure, and hence i depends upon the
particular solute as well as upon the ionic type. Because oT the
serious experimental difficulties, few precise measurements of

vapor pressures below 1m. have been made.

TABLE 42 — VAPOR PRESSURES OF AQUEOUS SOLUTIONS1 AT 18°
(po = 15 48 mm at 18°)

Vapor pressure, mm. Hg

LiBr

NaCl

LiCl

KC1

1 00

14 90

15 02

14 94

15 01

2 00

14 18

14 46

14 27

14 52

3 00

13 34

13 88

13 46

14 00

4 00

12 32

13 19

12 57

13 48

5 00

11 29

12 46

11 55

Freezing Points of Ionized Solutes. — Data are available in
much larger quantities for the freezing-point depressions pro-
duced by salts; the data in Table 43 may be taken as typical of
modern work of high quality. One form of apparatus for such
work is shown in Fig. 37. It will be noted that the mole num-
ber, whicti is obtained by dividing AT7/ by 1.86m. in Table 43,

1 The data of A. Lannung, Z. physik. Chem., (A) 170, 139 (1934), were
plotted on a large graph from which vapor pressures at these concentrations
have been read. He gives data at irregular concentrations up to saturation
for all of the alkali hahdes in aqueous solution at 18°. Other data on vapor
pressures of salt solutions will be found in Table 53.

236

PHYSICAL CHEMISTRY

is very far from unity and that it varies with the concentra-
tion. Like the mole numbers based on vapor pressures, the}'
are not the same for different salts at the same concentration.

FIG 37 — Freezing-point apparatus

This difference io particularly noticeable when salts of different
types, such as potassium nitrate and magnesium sulfate, are
compared.

TABLE 43 — FREEZING POINTS OF AQUEOUS SALT SOLUTIONS'

Freezing-point depression

Molalitv

KNO3

LiNO3

NaCl

MgSO4

0 01

0 03587

0 03607

0.03606

0 0300

0.02

0 07072

0 07159

0.07144

0 0565

0.05

0 1719

0 1769

0 1758

0 1294

0.10

0 3331

0 3762

0 3470

0 2420

0.20

0 6370

0 7015

0.6849

0 4504

0.50

1 414

1 786

1 692

1 0180

0 80

2 144

2 928

2.705

«r

An examination of the available data upon freezing points of
salts in dilute aqueous solution shows that salts of the same type

1 SCATCHARD, PRENTiss, and JONES, /. Am Chem. Soc., 64, 2690 (1932),
66, 4335 (1933) The data for MgSO4 are by Hall and Harkins, ibid., 38,
2672 (1916).

SOLUTIONS OF IONIZED SOLUTES

237

have roughly the same mole numbers at a given concentration.
Thus for salts of the KC1 type the maximum and minimum mole
numbers for O.lm. were 1.90 and 1.78. Some of the data for
other salts are shown in Table 44.

TABLE 44 — MOLE NUMBERS DERIVED FROM FREEZING-POINT LOWERING*?*

Solute

Molal concentration

0 005

0 010

0 020

0 050

0 10

0 20

0 50

1.00

2 00

HC1
AgN03
NaCl
KC1
KNO3
NH4NO3

1 96

1 95
1 96
1 96

1 94
1.94
1 94
1 94
1.93
1 92
2 64
2 77
2 72
1 57
1 45
2 47

1.92
1 90
1.91
1.92
1.90
1.90
2.51
2.71

1.90
1.84
1 90
1.89
1.85
1.87
2 30
2.66
2.68
1.30
1.22
2.21

1.89
1.79
1.87
1.86
1.78
1.83
2.13
2.66
2.66
1.21
1.12
2.17

1 90
1 72
1.83
1.83
1.70
1.77
1.93
2.67
2.68
1.13
1.03
2.04

1.98
1.59
1 81
1.78
1.55
1.68
1.57
2 70
2.90
1.07
0.93
1.99

2 12
1 42
1.81
1.75
1.38
1.57
1 31
2.80
3 42
1.09
0.92
2.18

2 38
1 17
1 86
1.73

1.43

2.95

4.8

2 74

Pb(N08)2
ZnCl2
MgCl2
MgSO4

2 74
2.84

1 62
1 55
2 59

CuSO4
H2SO4

Boiling-point elevations, like freezing-point depressions, meas-
ure the change in vapor pressure of solvent caused by decreased
mole fraction of solvent and thus furnish a measure of the mole
number. Mole numbers change but little with temperature, and
the freezing-point depressions are easier to measure precisely, so
that there are few data based on boiling points. The following
data for silver nitrate are typical :

Molality
Mole number

0 05

1 82

0.20
1 70

0 50
1.69

0
59

It will be evident from the mole numbers based on any of these
methods that something fundamentally different in the properties
of the solute is indicated. No slight deviation from the laws of
ideal solutions can explain them. The fourth line of Table 44
does not mean that potassium chloride molecules deviate from
the behavior of an ideal solute 96 to 73 per cent, depending upon
the concentration, and it is improbable that dissociation or

1 Based upon freezing points from "International Critical Tables," Vol.
IV, pp. 254-263.

238 PHYSICAL CHEMISTRY

ionization to this extent is alone responsible. We shall postpone
a discussion of the mole numbers until other important experi-
mental facts have been presented.

Conduction of Electricity. — Aqueous solutions which have the
properties given in the preceding paragraphs also conduct elec-
tricity, while those which do not show these deviations from the
molal properties of ideal solutions have negligible conductances.
Because of this property of conducting electricity, substances
that ionize in solution are often called " electrolytes." There is
one fundamental difference between the conduction of these
solutions and that of the metals. Metallic conduction is not
accompanied by the movement of matter, while electrolytic con-
duction is always attended by chemical reactions at the elec-
trodes, in which electricity is given to uncharged atoms (or
atom groups) or is accepted from them, and by the motion of
charged particles through the solution. For example, when an
electric current is passed through an aqueous solution* of copper
chloride between chemically inert electrodes, metallic copper is
plated on the negative electrode, chlorine gas is evolved at the
positive electrode, and concentration changes occur near both
electrodes which indicate that both cupric ions and chloride ions
have taken part in carrying electricity through the solution.
Corresponding effects are observed when electricity is passed
through any conducting solution, though as we shall see presently
it is not necessarily true that the ions which form or discharge
at the electrodes during electrolysis are those which carry most
of the electricity through the solution. The products of elec-
trolysis depend on the material of the electrodes, the current
density, and the concentration of solute, as well as on the nature
of the solute.

The decomposition that results when electricity passes through
a solution is called electrolysis; the metallic conductors through
which electricity enters or leaves the solution are called the anode
and cathode, or the electrodes. At the anode, or positive
electrode, a chemical reaction takes place by which electrons are
given to the metal and oxidation takes place. At the cathode,
electrons are received from the metal, and chemical reduction
takes place. During these reactions charged ions move through
the solution in opposite directions at characteristic velocities and
in such quantities that the sum of the equivalents of positive ion

SOLUTIONS OF IONIZED SOLUTES 239

crossing any boundary in their motion toward the cathode and the
equivalents of negative ion crossing this boundary in their motion
toward the anode is equal to the total quantity of electricity
passed through the solution. These processes occur simultane-
ously of course, but we shall consider the electrode reaction first
and then the motion of the ions through the solution.

Faraday's law states that when electricity passes through a
solution the total quantity of chemical change produced at each
electrode is strictly proportional to the quantity of electricity and
dependent on that alone and that in electrolysis chemically
equivalent quantities of substances are produced or destroyed
at the electrodes. The nature of these chemical changes depends
on the ions in solution and the material of the electrodes, but
the total quantity of chemical change, measured in equivalents,
is independent of every factor except the quantity of electricity.
The electromotive series or potential series, which is given in
Table 99, gives the anode potentials for electrode reactions.
Of all possible anode reactions, the one of highest potential tends
to take place first. Electrode potentials are given for anode
reactions or oxidations, and since cathode reactions are all
reductions the one of lowest anode potential has the greatest
tendency to act as a cathode; hence, of all possible cathode
reactions, that of the lowest potential in the electromotive series
tends to take place first. These potentials vary with the con-
centration of the solute in a way we are to consider in Chap.
XIX, but in the examples considered here the differences are
great enough for changing concentration not to change the order
in which the reactions occur.

Jn order to illustrate the application of Faraday's law, suppose
four vessels, each containing a solution and a pair of electrodes,
to be arranged as shown in Fig. 38 and connected in series so
that the different chemical effects of a fixed quantity of electricity
may be observed. The anode is defined as the electrode at which
electrons are given to the electrode, and therefore the left-hand
electrode is the anode in each vessel ; it is the electrode at which
oxidation takes place. If a current is passed through these cells
in the direction indicated, the products of electrolysis will appear
as deposits on the electrodes, as gases evolved from solution, or
as new solutes in solution near the electrodes, as follows: (a)
chlorine is evolved from the carbon anode, hydrogen is evolved

240

PHYSICAL CHEMISTRY

from the platinum cathode, and sodium hydroxide is formed in
the solution around it; (6) zinc chloride is formed in solution
around the zinc anode, silver chloride is reduced to silver at the
cathode, and sodium chloride is formed in the solution around
this electrode; (c) oxygen is evolved from the platinum anode,
nitric acid forms in the solution around it, and silver is deposited
upon the platinum cathode; (d) oxygen is evolved from the anode,
sulfuric acid is formed in solution near it, and copper is deposited
upon the cathode.

The solutions are not assumed to be of the same strength or
at the same temperature or of the same resistance. The only
conditions imposed are that all the electricity which passes
through one cell must pass through the others and that the cur-

- Source of current^

\ "1

|Zn AgCli

p* pt

pt pt]

yir:

-r-i

"ZJr""!!:1

_—_-_•

— — —

— —

— — —I

(a) Na Cl (b) Na Cl (c) Ag N03 (d) Cu S04

FIG 38 — Electrolysis diagiam for Faraday's law

rent density at the electrodes is such that the reacting ions reach
the electrodes by migration, diffusion, or convection fast enough
to produce clean chemical reactions free from "side reactions/'
This condition is imposed here because Faraday's law governs
the total quantity of chemical reaction produced by a given
quantity of electricity even when several reactions occur at an
electrode, but it does not say what ions react. At the cathode
in the copper sulfate solution, for example, if the current density
is too high, both hydrogen and copper plate out, since copper
ions cannot reach the electrode and discharge fast enough to
carry the total current. Under these conditions Faraday's law
accurately describes the total number of equivalents of hydrogen
plus copper discharged, but the weight of copper deposited will
not correspond to the total quantity of electricity.

A quantitative examination of the products of electrolysis will
show that the sulfuric acid formed at the anode in d is just suffi-

SOLUTIONS OF IONIZED SOLUTES 241

cient to neutralize the sodium hydroxide formed at the cathode in
a; that the chlorine evolved from the anode in a will convert all
the silver deposited on the cathode in c into silver chloride or
all the copper on the cathode from d into copper chloride; that
the silver formed from silver chloride in b is equal in weight
to that deposited in c; that the sodium hydroxide of a will
precipitate all the zinc ion formed at the anode in b as zinc
hydroxide; arid that the zinc hydroxide so formed is just sufficient
to react with the sulfunc acid of d.

All these chemical details may be summarized in the single
statement that a fixed quantity of electricity passing through
a solution produces chemical substances at the electrodes which
are equivalent to one another. Special experimental conditions,
such as control of current density and concentration, are often
required to restrict each electrode reaction to a single chemical
change, as has been said before; and when these precautions are
observed, the quantity of chemical change as shown by a single
electrode reaction is proportional to the quantity of electricity
and independent of every other influence.

Since the ampere is defined as a uniform current that deposits
00011180 gram of silver from silver nitrate solution each sec-
ond and since the atomic weight of silver is 107.880, the ratio
107 880/0.0011180 gives the number of ampere-seconds or
coulombs of electricity required to deposit a chemical equivalent
of silver. This quantity is 96,494 amp -sec. (usually rounded off
to 90,500 except in the most precise calculations), and it is called
1 faraday of electricity

Faraday's law may be restated in terms of this constant as
follows: The passage of 1 faraday of electricity through an elec-
trolytic solution produces one chemical equivalent of some
chemical change at each electrode. Faraday's law is an exact
law to which there are no known exceptions; it has been con-
firmed by experiments upon the widest variety of solutes in
water and for solutions of silver nitrate in fused potassium
nitrate1 and in pyridine2 and other nonaqueous solutions. As

1 RICHARDS and STULL, Proc. Am Acad Arts Sci , 38, 409 (1902). Silver
was deposited from an aqueous solution of silver nitrate at 20° and in the
same circuit from silver nitrate dissolved in fused sodium nitrate and potas-
sium nitrate at 250°. The weights of silver deposited agreed within 1 part
in 20,000.

2 KAHJJSNBERG, / Phys. Chem., 4, 349 (1900).

242 PHYSICAL CHEMISTRY

the precision of the experiments is increased, the equivalence
of the chemical changes becomes closer.

Calculation of Avogadro's Number. — A univalent positive ion
is an atom or group of atoms that has lost an electron, and its
discharge at a cathode takes place when it acquires the electron.
The ratio of Faraday's constant to the electronic charge is thus
the number of electrons in a faraday, which is the number of
atoms in a gram atom, or Avogadro's number. In the absolute
electromagnetic system of units, 1 faraday is 9G49.4 absolute
coulombs, and in the same units the charge of an electron is
1 598 X 10~20, whence Avogadro's number is

9649 4

JO*J* = 6.03 X 1023

1 598 X 10-

It will be seen that this value is in agreement with determinations
by the other methods given on pages 71 and 170. It is one of
the most accurate values for Avogadro's number that we have
at the present time.

Electrode Reactions. — It has been stated above that Faraday's
law says nothing as to which of several possible reactions will
occur at an electrode; it describes only the total quantity of
chemical change produced. The electric potential determines
which reaction occurs; if the current density is not too high,
only the reaction of lowest required potential takes place
Electrode potentials such as those listed in Table 99 are
expressed in volts for changes in state by which the ions con-
cerned are used reversibly at unit activity or formed reversibly
at unit activity These potentials change with the concentra-
tions of ion solute in a way that is explained in Chap. XIX,
but we may note here that for univalent ions the potential
changes about 0.06 volt for a tenfold change in ion concentration.
For example, in the first cell in Fig. 38, hydrogen was evolved
at the cathode and no sodium was deposited. It may be seen
in Table 99 that sodium is near the top of the list of anodic
potentials and thus that it would require a high opposing poten-
tial to cause the deposition of sodium at the cathode, whereas
hydrogen is lower in the list and would require a smaller potential
for its evolution. Quantitatively, the potential required to
deposit sodium is about 3.0 volts higher than that required to
discharge hydrogen ions under these conditions. As hydrogen

SOLUTIONS OF IONIZED SOLUTES 243

ions are present from the slight ionization of water, these are
discharged and the required potential for sodium is never reached.
Similarly in 6, the potential required to discharge chlorine at the
zinc anode is 2 volts higher than that required for zinc to pass
into solution. The reaction requiring the lowest potential always
takes place first. There are, of course, hydrogen ions from water
in the silver nitrate solution of c, but the potential required to
discharge them is higher than that required for silver by about
1.2 volts; therefore, the metal deposits. In a, chlorine is evolved
at the anode in place of oxygen from the hydroxide ions of water,
for chlorine has a lower discharge potential than oxygen under
these conditions.

If we denote a faraday of negative electricity, or Avogadro's
number of electrons, by the symbol e~~, electrode reactions are
readily described by chemical equations in which this quantity
is written as if it were a reacting substance or a reaction product.
Thus, the electrode reactions described on page 240 are

ci- = 2

H20 + e- = Oil- + (a)

rr + e-

AgCl + e- = Ag + Cl-

i^H20 = H+ + 3^O2 + c~ , ,

A 4- I 4 (C)

Ag+ + e~ = Ag

_ TT+ 4- 3^O« 4- *>-'

— J-J- i^ /A\J% l^ * / -i\

H + e-JiJcu (d)

It was stated on page 240 that the products of the anode
reaction for nitrate ions on platinum and for sulfate ions on
platinum are the same, and we have shown above that neither
electrode reaction mentions the negative ion. In each reaction,
oxygen is evolved, and hydrogen ions form in solution at the
expense of decomposed water. While it is sometimes stated that
the sulfate ion or nitrate ion discharges and then reacts with
water to form sulfuric acid or nitric acid and oxygen, there is no
experimental evidence for these statements. Even if this
peculiar mechanism were true, it is a fact that no change in the
number of sulfate ions finally results from the electrode reaction.
The equations as written express the observed facts, and nothing
is to be gained by combining these facts with fanciful assump-
tions such as the deposition of nitrate ions or sulfate ions or the

244 PHYSICAL CHEMISTRY

plating out of sodium metal on the cathode from an aqueous
solution, followed by a reaction between sodium and water to
produce hydrogen and sodium hydroxide.

The common effects observed at an anode are (1) the dis-
charge of a negative ion when it is not an oxygenated ion and
when the anode metal is inert, (2) the formation of a positive
ion when the metal of the electrode forms ions that do not precipi-
tate with those of opposite charge in the solution, (3) the forma-
tion of an insoluble salt when precipitation takes place between
the ion of the anode metal and the negative ion in solution, and
(4) the evolution of oxygen gas. This evolution of oxygen is
attended by the loss of hydroxyl ions and the formation of water
in alkaline solutions, as shown by the reaction

OH- = i£H20 + M02 + cr

and by the decomposition of water with the formation of hydro-
gen ions when the solution is neutral or acid, as shown by the
equation

^H20 = H+ + }±0* + r

Although this is not a full list of the chemical effects observed
at anodes, it will suffice for the purpose in this chapter and we
shall return to the topic later.

The common effects observed at a cathode are (1) the discharge
of a positive ion when the ion is below hydrogen in the electro-
motive series, (2) the formation of a negative ion from a reducible
material such as chlorine gas, (3) the reduction of an insoluble
salt with the formation of a negative ion into solution, and (4)
the evolution of hydrogen gas when the positive ion lies above
hydrogen in the electromotive series. In acid solutions this
is attended by the loss of hydrogen ions from solution as shown
by the reaction

H+ + <r = MH2

and in neutral or alkaline solutions it is attended by the forma-
tion of hydroxyl ions and the decomposition of water as shown
by the reaction

H20 + e- = HH2 + OH-

SOLUTIONS OF IONIZED SOLUTES 245

Measurement of the Quantity of Electricity. — The number of
coulombs of electricity passing through an electric circuit is best
measured through an application of Faraday's law, by weighing
the silver deposited upon a platinum cathode from silver nitrate.
Since this reaction is the basis of the definition of the inter-
national ampere, it has been most carefully studied to devise
apparatus and procedures for limiting the cathode reaction to
this single effect.

The standard coulometer in which this is done is shown in Fig 39, 1 in
which a porous cup of unglazed porcelain surrounds an anode of pure silver
and is suspended above a platinum dish
serving as a cathode on which silver deposits
Both dish and cup are filled with silver
nitrate solution After electrolysis the silver
deposit is carefully washed free of silver nitrate
and dried and weighed.2

Unless precautions are taken to prevent
the electrolyte around the anode from reach-
ing the cathode, deposits are obtained that
are too heavy, owing to the formation of
some unknown substance at the anode (possi-
bly colloidal charged silver), which deposits
and which is not removed by washing

When commercial quantities of electricity

are involved, the use of silver is out of the FlG' 39 —Porous-cup type

; „ , . „ of silver coulometer.

question, and copper is usually deposited from

copper sulfate for this purpose Lead from solutions of lead silico fluoride
may also be used, or the volume of hydrogen evolved from a cathode in acid
solution may be measured The commercial processes of copper refining and
electroplating are everyday confirmations of the law of electrolysis. It is the
universal experience in such processes that the weight of metal deposited is
strictly proportional to the quantity of electricity passed through the elec-
troRjating cell when the current is not allowed to cause other reactions, such
as the evolution of gas from the electrodes. The character and adherence of
the metal film depend on current density, the concentration of electrolyte,
efficient stirring, and temperature control, but the weight of metal deposited
is independent of these factors.

1 Bull. U.S. Bur. Standards, 1, 3 (1904).

2 Special reference should be made to the work of the U.S. Bureau of
Standards and England's Najbional Physical Laboratory. Important papers
will be found in Bull. U.S. Bur. Standards, 1, 1 (1904) ; 9, 494 (1912) ; 10, 425;
11, 220, 555 (1914); Sci. Paper, 283 (1916); Richards, Proc. Am. Acad. Arts
Sti., 37, 415 (1902); 44, 91 (1908); J. Am. Chem. Soc., 37, 692 (1915); Smith,
Mather, and Lowry, Nat. Phys. Lab. Researches, 4, 125.

246

PHYSICAL CHEMISTRY

Atomic-weight Ratios from Electrolysis. — Since Faraday's law
is an exact law, an electric current passing through solutions may
be used to liberate or deposit chemically equivalent quantities of
substances from solutions in the same circuit, whether or not the
reactions take place in the same solution. It is necessary only
that the chosen electrode reactions involve a single ion solute
and a single product. Two electrode reactions that meet this
requirement are

Ag+ + er = Ag
and

I- = y2i2 + e-

Since silver iodide is an insoluble salt, these reactions may not
be carried out in the same solution, but a suitable experimental
arrangement is a silver coulometer in series with electrodes dip-
ping into potassium iodide solution. Data that show the experi-
mental results of electrolysis with this arrangement are given
in Table 45. Upon dividing the weights of iodine in the second

TABLE 45. — DATA ILLUSTRATING FARADAY'S LAW1

Calculated coulombs

Milli-

Weight of
silver de-
posited

Weight
of iodine
deposited

From sil-
ver cou-
lometer

From po-
tential
and resist-

Differ-
ence in
per cent

grams of
iodine
per cou-
lomb

the fara-
day (I =
126.92)

ance

4 10469

4 82862

3,671 45

3,671 53

0 002

1 31518

96,504

4 09903

4 82224

3,666 39

3,666 55

0 004

1 31526

96,498

4 10523

4 82851

*3,671 94

3,671 84

0 003

1 31498

96,518

4 10475

4 82860

3,671 51

3,671 61

0 003

1 31515

96,506

4 10027

4 82247

3,667 50

3,667 65

0 004

1 31492

96,523

4.10516

4 82844

3,671 88

3,671.82

0 001

1 31498

96,519

Average value of the faraday: 96,515

column by 'the corresponding weights of silver in the first column,
the ratio Ag:I will be found to be 1:1.1762. A careful deter-
mination of the combining ratio of silver and iodine by gravi-
metric analysis2 gave the ratio 1:1.17643, which shows that

1 BATES and VINAL, ibid., 36, 916 (1914); Bull. U.S. Bur. Standards, 10,
425 (1914).

2 BAXTER and LUNDSTEDT, J. Am. Chem. Soc., 62, 1829 (1940).

SOLUTIONS OF IONIZED SOLUTES 247

Faraday's law is accurate within 2 parts in 10,000. There is of
course no implication that this small difference is due to any
failure of Faraday's law, for the limit of accuracy of the experi-
ments is about 1 part in 10,000.

In the experiments recorded in Table 45 the quantity of elec-
tricity was measured from the potential drop across a known
resistance and the time of the electrolysis; thus the experiments
also yielded a determination of the faraday. If 126.92 is accepted
as the atomic weight of iodine, these calculated values of the
faraday are shown in the last column of the table.

Resistance and Conductance. — The familiar law of Ohm that
the current flowing in a conductor is equal to. the applied elec-
tromotive force1 divided by the resistance of the conductor
applies also to solutions that conduct electrolytically. This
law, / = E/R, is often used in a form in which the resistance R
is replaced by its reciprocal 1/jR, which is called the conductance.
Ohm's law is then

/ = E X conductance (1)

The specific resistance of a substance is the resistance of a centi-
meter cube of it; the reciprocal of this is the specific conductance,
L. The conductance of any substance increases with its cross
section and decreases in proportion to its length. If the specific
conductance is L, the conductance of a quantity of material in
a form other than a centimeter cube is

Conductance = L j (2)

where q is the cross section and I the length of the conductor.
Conductance is expressed in reciprocal ohms; thus, if the resist-
ance is 175 ohms, the conductance is 1/175 = 0.00572 reciprocal
ohm. The specific conductance of a given salt solution increases
almost in proportion to the concentration up to about 0.1 N9
and it increases almost linearly with increasing temperature; but
the specific conductances of different " strong" electrolytes at the

1 If resistance measurements are made with direct current, the applied
electromotive force must be corrected for that of the electrolytic cell formed
by the products of electrolysis. Measurements of resistance are usually
made with alternating current to avoid this correction. The method will
be described on p. 254.

248 PHYSICAL CHEMISTRY

same temperature and same moderate concentration may differ
from one another by fivefold or more. Exact relations of specific
conductance to temperature and concentration are determined
by experiment only.

Equivalent Conductance. — The equivalent conductance of a
solution at a given concentration is defined as the product of
its specific conductance and the volume of solution containing
one equivalent of electrolyte. Thus it is the conductance of a
sufficient number of centimeter cubes in parallel to contain one
equivalent of solute. Denoting the equivalent conductance by
the Greek letter lambda, A, as is the usual custom, and the con-
centration as C equivalents per liter of solution, the relations
between these two quantities are

or L==

Tooo

As a means of visualizing the equivalent conductance, consider
two parallel electrodes of indefinite extent, 1 cm. apart, between
which 1 liter of normal solution is placed. The cross section of
the solution is 1000 sq. cm., and the length of the conducting
column is 1 cm. Thus, from equation (3), we have

1000

If this solution is diluted to some lower concentration C, the
volume becomes 1000/C ml., which is the cross section since the
length is still 1 cm. Experimentally the quantities measured are
L and C, but the data commonly recorded are A and C, for con-
venient interpolation and for other purposes that will be explained
later in the chapter.

Table 46 shows the change of equivalent conductance with
concentration for a few electrolytes.1 It will be noted that for
salts of the same ionic type the equivalent conductance increases
with decreasing concentration to about the same fractional

1 Data for almost all aqueous solutions will be found in " International
Critical Tables/' Vol. VI, pp. 230-258. Recent work is reported in the
current literature of chemistry. See also KRAUS, "The Properties of Elec-
trically Conducting Systems," Chemical Catalog Company, New York, 1922,
and HARKED and OWEN, "The Physical Chemistry of Electrolytic Solutions/1
1943.

SOLUTIONS OF IONIZED SOLUTES
TABLE 46. — EQUIVALENT CONDUCTANCES AT 25°

249

NaCl

KC1

HC1

LiCl

HNO3

KNO8

HI03

0 0005

124 5

147 8

422 7

113 2

416 2

142.8

386 3

0 0010

123 7

147 0

421 4

112 4

414 6

141 8

383.9

0 0020

122 7

145 8

419 2

111 1

412 9

140 5

379 9

0 0050

120 7

143 6

415 8

109 4

409 0

138 5

370 9

0 010

118 5

141.3

412 0

107 3

405 2

135 8

359 7

0 020

115 8

138.3

407 2

104 6

400 8

132 4

343 0

0 050

111 1

133 7

399 1

100 1

392 5

126 3

310 7

0 100

106 7

129 0

391 3

95 9

384 2

120 2

278 3

0 200

101 6

123 9

379 6

89 9

374 4

113 3

242 2

0 500

93 3

117.2

359 2

81 0

356 6

101.4

219 5

1 000

111.9

332.8

73 1

333 2

NaOH

AgN03

^H2S04

MCuS04

MBaCU

0 0005

245 6

131 6

413 1

121.6

0 0010

244 7

130 5

399 5

115.2

134 5

0 0020

142

128 7

390 3

110 3

131 7

0 0050

240 8

127 2

364 9

94 1

127 7

0 0100

238

124 8

336 4

83.1

123 7

0 0200

227

121 4

308 0

72 2

119 2

0 0500

221

115 2

272 6

59 0

111 7

0 100

109 1

250 8

50 6

105.3

0 200

101 8

234 3

43 5

98.6

0 500

222 5

35 1

88 8

1 00

29 3

80 5

extent over a given concentration range. At low concentrations
a plot of equivalent conductance against the square root of the
equivalent concentration is almost a straight line for all "strong"
electrolytes, as may be seen in Fig. 40. It will also be evident
from this figure that slightly ionized solutes such as acetic acid
or hydrofluoric acid change equivalent conductance with con-
centration in an entirely different way.

Limiting Equivalent Conductance. — The equivalent conduct-
ance continues to increase with dilution down to the lowest
concentrations at which experiments are possible for all sub-
stances. For salts of the KC1 type, A at 0.001 N is about 98
per cent of the limiting value determined in the way to be
explained below. For so-called "weak" electrolytes, which are

250

PHYSICAL CHEMISTRY

slightly ionized at moderate concentrations, the equivalent con-
ductance is still increasing rapidly with decreasing concentration
in the most dilute solution that can be measured The data for

FIG. 40 — Change of equivalent conductance with concentration

very dilute acetic acid1 and HC1 at 25° will illustrate the great
difference in these changes with concentration :

C...

A for HC1

A for HAc

0 001028 0 0001532 0 0001113 0 0000280
421.4 424 4 424 6 425 13

48 13 112 0 127 7 210 3

There are reasons that will appear below for expecting the
equivalent conductance to reach a limit of 426 0 for HC1 and a
limit of 390.6 for acetic acid, but the limit 390.6 may not be
determined from the data quoted above or from measurements
on more dilute solutions. The data for other "weak" electrolytes

1 MAC!NNES and SHEDLOVSKY, J. Am Chem. Soc , 64, 1429 (1932).

SOLUTIONS OF IONIZED SOLUTES 251

show similar behavior, but the ratio of the conductance at 0.001
N to that at limiting dilution for different weak electrolytes
shows no regularity such as that found for strong electrolytes;
it may vary a thousandfold. Thus it is evident that the small
change of A with C for strong electrolytes may not be explained
in the same way as the very large change of A with C for weak
electrolytes. We shall see later that the change of A with C for
salts is due mainly to decreasing attractions between the charged
ions at lower concentrations, while the increase in A with decreas-
ing C for weak electrolytes is due mainly to an increased frac-
tional iomzation of the solute, which produces an increase in the
number of ions available for carrying electricity.

For salts and other highly ionized solutes the limiting conduct-
ance may be obtained by plotting the equivalent conductance
against some function of the concentration, extrapolating the
curve to zero concentration, and reading the intercept It
should be understood that this limiting value of the equivalent
conductance, which is written A0, is not the conductance ot pure
water, for in these dilute solutions the slight conductance of the
water is subtracted from the measured conductance of the solu-
tion to give that due to the solute.

More than 30 functions suitable for this extrapolation have
been proposed1 at one time or another Kohlrausch observed
empirically that the relation

A = Ao - A VC

was valid in dilute solutions of strong electrolytes, where the
constant A applied only to a single solute at a single temperature.
The equation of Onsager2 is also of this form, but he is able to
calculate the quantity A from the valencies of the ions, the vis-
cosity and dielectric constant of the solvent, and other constants.

1 KOHLRAUSCH, Wiss. Abhandl. phys -tech Reichsanstalt, 3, 219 (1900),
NOTES and FALK, /. Am Chem Soc , 34, 454 (1912); ONSAGER, Physik, Z.,
27, 388 (1926), 28, 277 (1927), SHEDLOVSKY, J. Am. Chem. Soc., 64, 1405
(1932); JONES and BICKFORD, ibid , 66, 602 (1934)

2 Physik. Z , 27, 388 (1926); 28, 277 (1927). A discussion of this some-
what complex equation, of the factors that are taken into account in its
derivation, and of its applicability and limitations and some illustrations of
its use m obtaining limiting conductances are given by Machines in ./
Franklin Inst , 226, 661 (1938).

252 PHYSICAL CHEMISTRY

Recently the conductances of very dilute solutions have been
intensively studied and measured The theory of Debye and
Hiickel, which will be discussed briefly later, was mainly respon-
sible for this renewed interest, but it is beyond the scope of this
book to consider the experimental technique or interpretation
of the work. Students should consult references such as those
below for the details.1 The limiting conductances for salts and
other highly ionized substances, as estimated by the various
methods, usually agree within a few tenths of a unit

The limiting equivalent conductance for weak acids and bases
may not be obtained from extrapolation oi conductance data for
the acid itself but is available through a simple procedure. The
difference between the limits for HC1 and NaCl is the same as the
difference between the limits for HX and NaA", whatever um-
valent ion we denote by X, namely, the difference between A0
for H+ and Na+. This difference at 25° is 299 7. To obtain A0
for lactic acid at 25°, one need only determine the limit AQ for
sodium lactate and add to it 299 7. This limit for sodium lactate
is 88.8, and therefore the limiting equivalent conductance of
lactic acid at 25° is 388.5. Since lactic acid is a weak acid, the
limit cannot be obtained by direct measurement of lactic acid,
as has been said before A similar procedure serves for calcu-
lating the limiting equivalent conductance of any weak acid.
For weak bases, it should be noted that the difference between
the limiting equivalent conductances of NaOH and NaCl would
be the same as the difference between the limits for BOH and
BCl, whatever univalent positive ion we denote by J5, namely,
the difference between A0 for OH~ and Cl"~, which is 120.7 at 25°.
The limiting equivalent conductance of NH4C1 at 25° is 149.7,
and, by adding 120.7 to this quantity, we have 270.4 as the
limiting equivalent conductance for NH4OH at 25°. This
limit could not be obtained by direct experiment on dilute
NH4OH. For any solute the limiting equivalent conductance
is evidently the sum of the limits for its individual ions, and a
method of obtaining these individual conductances is to be given
on page 266.

1 DAVIES, "The Conductivity of Solutions, " John Wiley & Sons, Inc ,
New York, 1933, JONES and DOLE, / Am Chern. Soc , 52, 2245 (1930), 56,
602 (1934); SHEDLOVSKY, BKOWN, and MAC!NNES, Trans. Electrochem Soc ,
66, 237 (1934); KRAUS ET AL , /. Am Chem. /Soc., 55, 21 (1933); and earlier
papers, HARNED and OWEN, op. cit.

SOLUTIONS OF IONIZED SOLUTES 253

Conductance Ratio. — The ratio of the equivalent conductance
of a solution at some concentration C to its limiting value at the
same temperature, which is called the conductance ratio Af/A0,
was at first assumed to measure the fractional ionization of tliQ
solute at the concentration C. This ratio would be a measufe
of the extent of ionization if the change in equivalent conductance
with concentration were due only to an increasing concentration
of ions of constant mobility with increasing dilution. But the
experiments to be discussed on page 256 show that the ratio of
the ionic velocities changes with the concentration, and hence at
least one ion changes its velocity with changing concentration.
Since the motions of the positive and negative ions in opposite
directions through the solution would be influenced by those of
opposite charge to an extent that depends on the concentra-
tion, it is improbable that the velocity of either ion under a fixed
potential gradient is constant There is probably little relation
between the conductance ratio and the fractional ionization of
any highly ionized solute.

The data quoted for salts show that the equivalent conductance
increases about 2 per cent below 0 001 Ar, while for acetic acid
the increase below this concentration is about sevenfold

In 0.001 N acetic acid the ionized fraction is not much over
10 per cent; thus the ion concentration is about 0 0001 N, and
at this ion concentration the ionic attractions, insofar as they
interfere with the conductance, are small. Hence A/A0 is almost
a measure of the fraction ionized in solutions 6f weak electrolytes.
We shall see in Chap IX that for slightly ionized solutes the
fraction ionized changes with the concentration in the way to
be expected from the laws of chemical equilibrium when A/A0
is taken as a measure of this fraction. But we shall also see
in the same place that A/A0 is not a measure of the fraction
ionized for strong electrolytes

Measurement of Conductance. — In laboratory practice the
resistance of a solution is measured by means of a Wheatstone
bridge, using an alternating current of fairly high frequency
from a suitable generator E (Fig 4 la), with a telephone receiver
T, or other convenient apparatus, in place of a galvanometer.
A resistance R is chosen for the box of such size that there is a
point b near the middle of the bridge wire abc at which there is
no audible sound in the telephone receiver. Then the resistance

254

PHYSICAL CHEMISTRY

of the box is to that of the cell as the corresponding lengths of
the uniform resistance wire abc] that is, R^^'.R^n = ab:bc. The
reciprocal of this resistance is the conductance of the cell, and
from its dimensions the specific conductance can be calculated
by means of equation (2), then the equivalent conductance from
equation (3).

For the purpose of reducing electrolysis effects at the electrodes to a
minimum, alternating current of low potential is employed at frequencies
of 1000 to 5000 cycles, the electrodes of the conductance cell are coated
with " platinum black" to increase their effective surface, and the current

FIG. 4la — Ariangement of Wheat-
stone budge for measuring conductivity
of a solution

FIG. 41&. — Conductance cell,
pipet type.

passing through the cell is made as small as the detector permits. A con-
venient form of conductance cell with electrodes sealed inside a glass cham-
ber is shown in Fig 416 Since the distance between electrodes in such a
cell is more difficult to measure than the conductance of a solution between
the electrodes, it is customary to determine the "cell constant7' L/L' from
L', the actual conductance of a standard solution It is more convenient
to weigh the salt and the solution than to weigh salt and water, since in the
former procedure some of the water may be used in effecting transfer of the
salt Suitable conductances, corrected to vacuum weights for both salt
and solution, and corrected -^or conductance of the water (about 10 ~6), are
as follows*1

Grams KC1 per
1000 grams
of solution

Specific conductance

0°

18°

25°

71.1352
7.41913
0.745263

0.065176
0.0071379
0.00077364

0 097838
0.0111667
0.00122052

0.111342
0.0128560
0.00140877

1 JONES and BRADSHAW, J. Am Chem Soc , 65, 1780 (1933).

SOLUTIONS OF IONIZED SOLUTES 255

Through the use of vacuum-tube generators for the alternating current,
new amplifiers, and high-precision bridges, the method of measuring the
conductances of solutions has been brought to a high state of perfection
Some of these improvements are described by Jones and Josephs, / Am
Chem Soc , 60, 1049 (1928) [see also Jones and Bellinger, ibid , 61, 2407
(1929), Jones and Bradshaw, ibid , 65, 1780 (1933)]. A new type of cell,
and a screened bridge are described by Shedlovsky, ibid , 62, 1793 (1930),
a simpler bridge is described by Luder, ibid , 62, 89 (1940), a cathode-ray
oscillograph detector is described by Jones, Mysels, and Juda in ibid , 62,
2919 (1940). The preparation and storage of water of sufficient purity for
accurate work on dilute solutions involve repeated distillation and elaborate
precautions against contamination [see Kendall, ibid , 38, 2460 (1916), 39,
9 (1917); Weilaiid, ibid, 40, 131 (1918)] Apparatus, procedures, errors,
calibrations, and an ample bibliography are given in catalog EN-95 of the
Leeds & Northrup Co (1938) The electrical characteristics of the bridge
assembly are discussed by Acree, Bennett, Gray, and Goldberg in J Phys.
Chem , 4=2, 871 (1938)

Conductance of Pure Water. — As stated above, the conduct-
ance oi the water used in preparing a solution is subtracted from
the measured conductance in determining that due to the salt.
Careful experiments have shown that water itself is ionized to
a slight extent, such that at 25° the concentration of hydrogen
ion (and of hydroxide ion as well) is 0.0000001 N. It is not
from this source that most of the error in measuring conductivities
of dilute solutions arises, but from the presence of dissolved
impurities. Even after careful distillation, water may contain
ammonia and carbon dioxide; and it will dissolve sodium and
calcium salts from glass in a very short time. Perfectly pure
water has a specific resistance of 20,000,000 ohms, ordinary
distilled water a specific resistance of perhaps 100,000 ohms, and
a good quality of "conductivity water" from 1,000,000 to 10,-
000,000 ohms. Water of a resistance greater than 1,000,000
ohms per centimeter cube can be preserved in glass for not more ?
than *a very few hours — perhaps for a day or so in quartz vessels.
For this reason, conductivity water is freshly prepared for a set
of measurements, first by distillation in the usual way, then by
a second distillation (often directly into the conductivity appa-
ratus) from alkaline permanganate solution, the first third of
the distillate being rejected.

Change of Conductance with Temperature. — The equivalent
conductance for a given salt at a given concentration increases
rapidly with temperature. The data for NaCl in Table 47 are

256 PHYSICAL CHEMISTRY

TABLE 47 — EQUIVALENT CONDUCTANCE OF SODIUM CHLORIDE!

Concentration

0 0005

0 0010

0 0020

0 0050

0 0100

15°

101 20

99 64

99 00

98 12

96 49

94 88

25°

126 48

124 54

123 77

122 69

120 67

118 55

35°

153 85

151 43

150 47

149 14

146 64

144 03

45°

182 73

179 79

178 62

177 00

173 96

170 78

typical of the behavior of most salts. The change of A0 with
changing temperature almost parallels the change in fluidity of
water with temperature; namely, each increases about 2 per cent
of the value at 0°C for every degree rise in temperature. At
moderate concentrations the fractional change in equivalent
conductance is slightly less than the change for A0 Since the
temperature coefficient of A0 for IIC12 is less than that for NaCl
and the change for NaCl is less than the temperature coefficient
of fluidity, it will be evident that factors other than fluidity of
the water affect the change of conductance with temperature.
In order to show the relative changes, the ratios <pt/<pzb° for water,
A,°/A25° for HC1, and V/A250 for NaCl are plotted against the
temperature in Fig. 42.

Transference Numbers. — Faraday's law states that the
quantity of electricity passing through a solution is strictly
proportional to the quantity of chemical change at each electrode,
that 96,500 amp -sec., or 1 faraday, of electricity produces or
destroys one chemical equivalent of chemical substance at each
electrode, and that the total changes at the electrodes may be
shown by a pair of electrode reactions which add to an ordinary
chemical equation But each electrode reaction involves an
equivalent of one ion a"hd none of the opposite chaige, and elec-
trical neutrality must be maintained at all times in all parts of
the solution. The loss of an equivalent of negative ion from the
solution near the anode by electrolysis is partly compensated by

1 GUNNING and GORDON, ibid., 10, 126 (1942) Data for potassium chlor-
ide are given in the same paper.

2 The limiting equivalent conductance of HC1 at various temperatures is

t 5° 15° 25° 35° 45° 55° 65°

Ao 297 6 362 0 426 2 489 2 550.3 609 5 666 8

OWEN AND SWEETON, /. Am. Chem. Soc., 63, 2811 (1941),

SOLUTIONS OF IONIZED SOLUTES

257

the movement of negative ions into this portion of the solution
and partly by the movement of positive ions out of this part of
the solution, the sum of these effects being equal to the loss by
electrolysis.

If N faradays pass through the solution and N equivalents of
negative ion are lost in the anode reaction, Nc equivalents of

1.8

I 10

kHCl

20 30 40 50

Temperature
FIG 42 — Change of fluidity and limiting conductance with* temperature.

positive ion leave the anode portion and Na equivalents of nega-
tive ion enter it. The relation between these quantities is

= Nc

but.it does not follow that Nc and Na are equal. In the elec-
trolysis of HC1 with a silver anode, for example, each faraday
passed through the solution causes the loss of an equivalent of
chloride ion by electrolysis from solution around the anode, and
electrical neutrality is maintained by the loss of 0.83 equivalent
of hydrogen ion from the solution near the electrode and the entry
of 0.17 equivalent of chloride ion. Thus Nc is 0.83N and Na is
Q.17N. In the electrolysis of sodium chloride under the same
conditions there is the same loss of negative ion by electrolysis,
and analysis of the solution near the anode shows the loss of 0.38

258 PHYSICAL CHEMISTRY

equivalent of sodium chloride. For this solution Nc is 0.38Af
and Na is 0.627V; the changed fractions are due to the fact that
sodium ion has a much smaller velocity than hydrogen ion.

We now define a quantity called the transference number, which
for the positive ion in a solution of a single electrolyte is given by
the equation

and for the negative ion by the equation

rr\ _ * * a ___

la ~ ~

The transference number of an iori is thus the fraction of the total
electricity carried in the solution by that ion. It is the fraction
of an equivalent of ion transferred across any boundary in the
solution per faraday of electricity carried through the boundary.
But ions move with different velocities in a solution of the same
concentration under the same potential gradient, and thus these
fractions are not one-half.

The actual velocities of ions in solutions of the same concentra-
tion, at the same temperature, and under the same potential
gradient are characteristic properties of the ions; thus the
transference number of chloride ion (for example) will depend
upon the velocity of the positive ion with which it is associated.
If Vc and Va are the ionic velocities, the transference numbers
may also be defined by the relations

V V

W — _ 1 <± _ _ Qnr] rn _ v «

~ ~

Transference numbers may be derived from several types of
experiment, of which two will be described in this chapter and
another in a later one.

In the gravimetric method a measured quantity of electricity
is passed through a solution, and separate portions of it are
analyzed after the electrolysis to determine the gains and losses
in the portions of the solution near the electrodes. (A descrip-
tion of the apparatus and the means of withdrawing the portions
without mixing will be given presently.) It is an essential

SOLUTIONS OF IONIZED SOLUTES

259

characteristic of these experiments that a "middle" portion of
the solution be unchanged at the end of an experiment, which is
accomplished by using a long tube for the electrolysis and so
adjusting the experimental conditions that the changes are con-
fined to the region near the electrodes. We shall first describe
an idealized experiment in which a large tube is filled with O.lm.
sodium chloride and fitted at its ends with a platinum cathode
and a silver anode. The tube is so fitted that after the passage
of a faraday of electricity the solution may be withdrawn from
the cathode region, then from two separate "middle portions"
(for check analysis), and finally from an anode portion The
electrode portions will have changed composition, but there must
be no change in the ratio of salt to water in the middle portions
Diagrammatically this arrangement is as follows*

1 °

Anode

Anode-middle

Cathode-middle

Cathode

t *

portion

portion of

portion

v (j

of NaCl

NaCl

of NaCl

i
o

e
e

n d

When 1 faraday of electricity is passed through the solution,
the anode reaction is

Ag(*) + Cl- = AgCl(s) + <r

by which one equivalent of chloride ion is lost from the solution
near the anode. At the same time the cathode reaction

H20

OH-

forms an equivalent of negative ion at the cathode. Thus the
electrical neutrality of the whole solution is maintained as the
result of these reactions. To maintain the electrical neutrality
of the anode portion, part of an equivalent of chloride ion moves
into this portion and part of an equivalent of sodium ion moves
out of it into the middle portion. To maintain electrical neu-
trality in the cathode portion, some chloride ions move out of it
and some sodium ions move into it,

260 PHYSICAL CHEMISTRY

Analysis of the separate portions of the solution after passing
I faraday through the whole cell shows that the anode portion
contains 0.38 mole less sodium chloride than was associated with
the amount of water in this portion before the electrolysis took
place In the cathode portion there is 0.62 mole less sodium
chloride than before, and 1 00 mole of sodium hydroxide. In
both middle portions the ratio of salt to water is still 0 1 mole
of NaCl to 1000 grams of water Sodium ions have been trans-
ferred from the anode portion through the middle portion to the
cathode portion, and chloride ions have been transferred from the
cathode portion through the middle portion to the anode portion

In the anode portion, where the electrode reaction is

Ag(s) +C1- - AgCl(*) +c~

the solution has lost one equivalent of chloride ion by this
electrolytic reaction, with no positive ion involved. The ana-
lytical data showed the net loss of 0 38 equivalent of Na+ arid
Cl~ from this portion, and these results may be explained simply
if 0.38 equivalent of Na+ left the anode portion by transfer into
the middle portion, while 0 02 equivalent of 01~ entered the
anode portion from this middle portion. Since there was no
change in the ratio of salt to water in the middle portions, these
effects of transfer evidently took place across the other boundaries
in the solution as well. Since 38 per cent of the faraday of elec-
tricity was carried by sodium ions, the transference numbers
for O.lm. are

TN& = 0.38 and TC] = 0 02

The same fractions are obtained from the results in the cathode
portion, where the electrode reaction was

H20 + <r = MH2(0) + OH-

Electncal neutrality is maintained in this portion of the solution
by the loss of O.G2 equivalent of chloride ion into the middle
portion and the gain of 0.38 equivalent of sodium ion from it.
These transfers to or from the middle portion are compensated
by corresponding transfers from or to the anode portion, so that
the total sodium chloride in the middle portion is unchanged.
The transfer of electricity through this solution was by 0 38
equivalent of Na+ and 0.62 equivalent of Cl~, and these frac-

SOLUTIONS OF IONIZED SOLUTES

261

tions are the transference numbers, namely, 77Na = 0.38 and
!Fci = O.G2

It will be understood, of course, that the OH~ ions formed at
the cathode take part in the conductance of the solution near the
cathode; these ions move toward the anode faster than do chloride
ions. But it is a necessary characteristic of transference experi-
ments such as this that the electrolysis must be interrupted
before any hydroxyl ions reach the middle portion This illus-
tration does not show what fraction of the total electricity is
carried by sodium ions in a mixture of NaOH and NaCl; it shows
only the relative velocities of the ions of sodium chloride in the
unchanged middle portions of the solution by considering the
changes in the electrode portions

The results of transference experiments may, for the sake of
clearness, be summarized in gam-and-loss tables like the fol-
lowing:

ANODE PORTION
+ Cl- = AgCl(s)

CAT-HOD K PORTION

HA) + c~ = l>2II,(g) + OH"

Gain

Loss

1 0 Cl-

0 62 Cl-

0 38 NLI^

0 38Na-|(ir

Gain

Loss

1 OOII-

0 38 Na+

0 62 Cl-

1 ONa+OH-

0 62 Na+01-

Electrolysis
Transference

In order to show that the transference effects are independent
of the nature of the electrodes, while the net changes in the
electrode portions of solutions are not, assume this experiment
to be repeated with a silver chloride cathode, but with the silver
anode retained. Analysis of the anode portion still shows the
net loss of 0.38 NaCl from it. The gain-and-loss tables would
then be as follows.

ANODE PORTION

CATHODE PORTION

AgCl(s) + e- = Ag(s) + 01-

Gain

0 62 Cl-

Loss

Gain

1 0 Cl-

0 38 Na+

Electrolysis
Transference

1 0 Cl~

0 38 Na+

0 38 Na+Cl- Net changes 0 38

Loss

0 62 Cl-

262 PHYSICAL CHEMISTRY

The net effects have been changed, but the interpretation
of them in connection with the gain or loss required by Faraday's
law is still the transfer of 0.38 equivalent of sodium ion out of the
anode portion and into the cathode portion per faraday of elec-
tricity passing, arid thus 0.38 is transference number of sodium
ion in this solution.

When the solute is changed, the transference effects are also
changed. Assume the electrodes to be a silver anode and a
platinum cathode, and the apparatus to be filled with 0 1 N
HOI in place of NaOl The results of passing a faraday through
this solution are shown by new tables, as follows:

PORTION CATHODE PORTION

Ag(«) + (T- = AgClW + e~ H+ + e~ = 12H2(<7)

Gain

Loss

Gam

Loss

0 172(1"

1 001-

0 828 H+

Electrolysis
Transference 0 828 H+

1 OH*"
0 172 CI-

0 828 H+C1-

Net changes

o mii-'ci-

With the same loss by electrolysis in the anode portion the
net loss from it is much higher, which is explained by the fact
that hydrogen ions move faster than sodium ions and therefore
carry a larger fraction of the electricity through the solution.
Thus the transference number of chloride ion in O.lm 1101 is
0.172, while in the NaOl solution the transference number of
chloride ion is O.G2.

An apparatus large enough to conduct such an experiment
with a whole faraday would be cumbersome and is unnecessary.
Since both electrolysis and transference are proportional to the
quantity of electricity, an apparatus such as that shown in Fig.
431 and about 0.01 faraday are used Two silver coulometers
S measure the total quantity of electricity passing; C is a cathode;
A, an anode. The two parts of the apparatus are joined at Z), the
whole is filled with solution, and the experiment is run with both
large stopcocks open, while the apparatus is immersed in a
thermostat. The stopcock keys must have a bore as large as the
diameter of the tubing (about 3 cm.) to prevent local heating and

1 WASHBURN, ibid , 31, 332 (1909).

SOLUTIONS OF IONIZED SOLUTES

263

convection. When the experiment is finished, these stopcocks
are closed to isolate the anode and cathode portions, and two or
three middle portions may be withdrawn through the side tubes
a and c for check analysis. The apparatus is divided at D, and
the parts containing the anode and cathode portions are weighed
and opened for analysis of the solution.

The actual data on which our first illustration of transference
was based were obtained from an experiment in such a piece of
apparatus and are as follows. The anode portion of solution

FIG. 43. — Diagram of transference apparatus and connections.

weighed 176 15 grains and contained 0.852 gram of sodium
chloride and 175.3 grams of water. The original solution put
into -the apparatus contained 5.485 grains of sodium chloride per
1000 grams of water, or 1.025 grams in 175.3 grams of water.
Thus the loss of sodium chloride from the anode portion was
0.173 gram, or 0.00295 equivalent. The silver coulometer in
series with the electrolysis apparatus to measure the quantity
of electricity deposited 0.842 gram of silver, which required
0.842/107.88 = 0.00780 faraday. The middle portions must be
shown to have the same ratio of sodium chloride to water at
the end of the experiment as at the beginning. This condition
was met in this experiment, arid therefore all the changes are

264

PHYSICAL CHEMISTRY

shown by the analysis of the anode portion of solution. These
1 changes and a corresponding set of figures for the cathode por-
tion are shown in the following table :

ANODE PORTION

Ag(«) + Cl- = AgCl(s) + e~
(Basis 0.00780 faraday)

CATHODE PORTION

H20 + c- = ^H2(0) + OH-

(Basis 0 00780 faraday)

Gain

Loss

Gain

Loss

0 00485 CJ-

0 00780 01-
0 00295 Na+

Flpp
, , 0 00780 OH-

trolysis

Trans- N&+
ferenoe

\T i

0 00485 Cl-

0 00295 Na+Cl-

,^ct 0 00780 Na+OII-
change

0 00485 Na+Ol-

A loss of 0 00295 equivalent of sodium ion from the anode
portion for 0.00780 faraday of electricity gives

TN&+ = 0 00295/0 00780- 0.38

as before. It is suggested that gain-and-loss tables be set up on
the actual basis of the experiment in the solution of problems at
the end of the chapter.

Moving Boundary Method for Transference Numbers. — We
have seen on page 261 that the changes due to transference are
independent of electrode reactions and that the ratio of the
transference numbers is the ratio of the equivalents of each ion
moving through the middle portion. The number of equivalents
of ion passing through any cross section of solution is the product
of concentration, cross section, and distance moved. Since dis-
tance and cross section are commonly expressed in centimeters
and square centimeters, respectively, the concentration of ions
must be in equivalents per centimeter cube, or CyiOOO if C is
the normality of the solution. For the positive and negative
ions the expressions for equivalents passing are

1000

qd+ and N- =

C
1000

qd-

and for a single solute C and q are common to both ions. Hence
T+/T- = d+/d-, and any method of determining these distances

SOLUTIONS OF IONIZED SOLUTES

265

moved by the ions would yield values of the transference number
through the relation

rr _ d+

Cathode

Sodium
acetate

Sodium
chloride

An idealized diagram of the moving boundary method1 is shown
in Fig 44, wrhich assumes a layer of sodium chloride over one of
lithium chloride and beneath one of sodium
acetate, with the boundaries before elec-
trolysis shown by the solid lines. When
electricity passes through the cell, the
boundaries move as indicated; and since
chloride ions are followed by the slower
acetate ions and sodium ions by the slower
lithium ions, the boundaries remain sharp and
may be located by the different indexes of
refraction of the solutions. After electricity
has passed for a suitable length of time, the
boundaries move to the positions indicated
by the dotted lines, and it is found that, in
0 1m sodium chloride solution at 25°, the ratio
of the distances moved is dn*'.dc\ = 38:62,
and therefore TNa+ is 0.38, as was found in the
gravimetric method.

The experimental difficulties of the
method, which are many, have been overcome
so completely that transference numbers
from this source are probably the most reliable of any now
available. Agreement between this method, the gravimetric
one, and a third method based on the potentials of concentration
cells (to be given in Chap. XIX) is satisfactory. Some data are
given in Table 48, from which it will be seen that there is a small
but unmistakable change of transference number with concen-
tration. At higher concentrations the changes are much greater.

Transference numbers also change with temperature, the gen-
eral effect of rising temperature being to bring the transference

Sodium
chloride

Lithium
chloride

<?NO

Anode
FIG. 44.

1 The actual apparatus and method for obtaining precise transference data
from moving boundaries are given by Maclnnes and Longsworth in Chem.
Rev , 11, 171 (1932); J Am. Chem. Soc., 60, 3070 (1938).

266 PHYSICAL CHEMISTRY

TABLE 48 — TRANSFERENCE NUMBERS OF POSITIVE loNs1 AT 25°

Electrolyte

Equivalent concentration

0 01

0 02

0 05

0 10

0 20

KC1

0 490
0 392
0 329

0 825
0 554
0 488
0 483
0 491
0 465

0 490
0 390
0 326

0 827
0 555
0 488
0 483
0 491
0 465

0 490
0 388
0 321
0 829
0 557
0 488
0 483
0 491
0 466

0 490
0 385
0 317
0 831
0 559
0 488
0 483
0 491
0 468

0 489
0 382
0 311
0 834
0 561
0 489
0 484
0.491

NaCl

LiCl

HC1

NaAc . .

KBr

NH4C1

AgNOs ....

numbers closer to 0.5 for all ions. The slower ions thus have
larger temperature coefficients oi velocity than the faster ions.
The following data2 for the transference number of sodium ion
in sodium chloride are typical of salts in general:

Concentration
7W at 15°
TN.' at 25°
7W at 35°

JW at 45°

0 001
0 3914
0 3947
0 3987
0 4023

0 010
0 3885
0 3918
0 3958
0 3996

0 100
0 3820
0 3853
0 3892
0 3932

The transference number of hydrogen ion in 0.01 N HC1
changes somewhat more rapidly with temperature, as shown
by the following data 3

f°C

0°
0 846

18°
0 833

30°

0 822

50°

0 801

96°
0 748

Limiting Conductances of the Separate Ions. — It has already
been explained that each salt approaches a limiting equivalent
conductance as the concentration decreases. In very dilute
solutions, where this limit is essentially reached, each ion is free
to move almost as if no other ions were present. From the
transference number obtained in dilute solutions and from the

1 Longsworth, ibid., 67, 1185 (1935), using the method of moving bounda-
ries. Transference numbers by the gravimetric method are given in " Inter-
national Critical Tables/' Vol. VI, p. 309, for these and other electrolytes.

2 ALLGOOD and GORDON, /. Chem. Phys , 10, 124 (1942).
3 LONGSWORTH, Chem. Rev., 11, 171 (1935).

SOLUTIONS OF IONIZED SOLUTES

267

limiting equivalent conductance, the limiting equivalent conduct-
ance of each ion in a solution may be calculated. Thus the
limiting equivalent conductance for sodium chloride at 18° is
108 9 reciprocal ohms; and if this is multiplied by the transfer-
ence number for sodium ion, 0.398, the icsult is 43 4 for the limit-
ing equivalent conductance of the sodium ion. Since at limiting
dilution the ions are substantially without influence upon one
another, the A0 values of the ions are additive, and 108 9 — 43.4
gives 65 5 as the limiting equivalent conductance of chloride ion.
The limiting equivalent conductance for sodium nitrate is 105.2
reciprocal ohms, 43 4 of which is due to sodium ion. Hence the
limiting equivalent conductance of nitrate ion is obtained by
subtraction, 105.2 — 43.4 = 61.8. For potassium nitrate the

TABLE 49 — LIMITING CONDUCTANCES OF

Temporaturo

Ion

100

128

156

K +

40 4

64 2

73 5

115

159

206

263

317

Na+

26 0

43 2

50 1

116

155

203

249

NH4+

40 2

64 3

73 4

115

159

207

264

319

Ag+

32 9

53 8

61 9

101

143

188

245

299

1-2BU++

63 6

104

149

200

262

322

]2Oa++

59 5

142

191

252

312

41 1

65 2

76 3

116

160

207

264

318

NOr

40 4

61 6

71 4

104

140

178

222

263

C2H,02-

20 3

34 6

40 9

130

171

211

^scv -

79 8

125

177

234

303

370

H+

240

315

349 8

465

5(55

644

722

777

OH-

105

174

197 6

284

360

439

525

592

limiting value is 126.3; hence 126.3 — 61 8 = 64.5 is the equiva-
lent' conductance of potassium ion. The limiting value for
potassium chloride should then be the sum of the values for
potassium ion arid chloride ion found above, 64.5 + 65.5 = 130.
Proceeding in this way, one may calculate limiting conductances
for all the ions. A few values for tho common ions at a series

1 Some other limiting conductances at 25° are Li+ 38 7, ^2^++ 59 5,
MMg++ 53 1, Bi- 78.4, I~ 76 9, and HCOr 44 5 The values for 25° in the
table are from Maclnnes, J Franklin Inst , 225, 661 (1938) Other data
will be found in " International Critical Tables," Vol VI, p 230.

268 PHYSICAL* CHEMISTRY

of temperatures are given in Table 49. Since the temperature
coefficients are almost linear, values for temperatures other than
those given in the table may be obtained by interpolation.

The limiting equivalent conductance for sodium lactate at 25°
was given on page 252 as 88.8; and by subtracting 50.1, which
is the limit for sodium ion, the limiting equivalent conductance
of lactate ion is 38 7. Upon adding 349 8 for hydrogen ion to
this we obtain 388 5 for the limiting equivalent conductance of
lactic acid, which is in agreement with the value given on page
252

Calculation of Conductances. — The conductance of any dilute
aqueous solution of a "strong" electrolyte involving the A()
values of the ions listed in the tables may be calculated from these
values and an estimate of the conductance ratio This ratio
A/Ao is about the same for all strong electrolytes of a given ionic,
type at the same concentration, but it is not the same for salts
of different ionic types at the same concentration. Some typical
values are given in Table 52 on page 270

For illustration, we may calculate the specific conductance of
0.1 N KNO3 at 25°. The conductance ratio at 0.1 N is 083
for NaNOs and KC108, so that we may take 0 83 for KNO3, and
Ao i = 0.83(73 5 + 71.4) - 120 2, whence L = 0 01202 by cal-
culation and 0.01203 by experiment But Nad is also of the
same ionic type, and A<» i/A0 = 0 85 for this salt, and from this
ratio the computed L ior KNOs would have been 0.0123 It
will usually be true that the calculations agree better with experi-
ment when the conductance ratio is taken for a salt resembling
as closely as possible that for which the calculation is being made
Over small ranges of concentration the equivalent conductance
changes only slightly, and therefore the specific conductance is
almost proportional to the concentration For illustration, the
specific conductance of 0.12 Ar KN03 is very close to 1 2 X 0.0120,
or 1 .2 times the specific conductance *f or 0.10 N. But the specific
conductance of 0.5 N KNO3 would not be 5 times that for 0.1 TV,
for in this concentration range the equivalent conductance
changes 18 per cent.

Conductance of Mixtures of Electrolytes. — The specific con-
ductance of a mixture of electrolytes of the same ionic type is
almost the sum of the specific conductances of the individual
salts present, calculated in the way shown above; but in calculat-

SOLUTIONS OF IONIZED SOLUTES 269

ing each one the ratio A/A0 applicable at the total concentration
should be used. For example, in a solution 0.05 N in HC1
and 0.10 N in NaCl the ratio A/A0 for 0.15 N should be used
on both solutes, and for this ionic type the value is about 0.85
At 25°, Ao = 426 for IIC1 and A0 = 127 for NaCl. The respec-
tive specific conductances are then computed and added together
to give that of the mixture, as follows*

0 85 X 42(3 X 0 05

-Luci — -.TT/OX = IJ.Ulol

and

_()85X 127X0 10 _0010S

-'-'NaC 1 — " T7\~f\r\ — vjo

The calculated specific conductance of the mixture is the sum
of these two quantities, or 0.0289; the measured specific conduct-
ance is 0 0291. The chief error in such calculations is of course
the estimate of the conductance ratio, which differs at 0.1 N by
1 per cent for KC1 and NaCl, by 3 per cent for HC1 and HNO3,
and by 7 per cent for HC1 and NaCl. Use of 0 88 in the above
calculation in place of 0.85 would change the calculated specific
conductance to 0.0299, for example.

It will be seen from these equations that the conductances are
computed on the assumption that the solutes act as conductors
in parallel when they are in the same solution. The actual
conductance of the mixture is slightly less than would be that of
the two solutions connected in parallel, since both solutes move
in the same solvent and hence influence the motion of one another.
This effect is taken into account by using the conductance ratio
corresponding to the total concentration, since the main effect
of the ions on one another arises from interaction between
their charges, which is a function of the total concentration.

These equations may not be applied to mixtures such as acetic
acid and sodium acetate without alteration, for the presence of
acetate ions changes the fractional ionization of acetic acid in a
way that cannot be estimated from the conductance ratio. (We
shall see in Chap. IX how to calculate the ionization of the acid
in the presence of the salt.) In mixtures of salts of different
ionic type, such as NaCl and BaCl2 or CuS04 and H2S04, the
estimate of a conductance ratio applicable to the mixture is so
uncertain as to make calculated conductances of little value.

270 PHYSICAL CHEMISTRY

Conductimetric Titration of Acid or Base. — It will be observed
that the limiting conductances of hydrogen ion and hydroxyl
ion are much larger than those of other ions. During a titration
of acid with a standard base the specific conductance decreases
rapidly as hydrogen ion is removed from solution by neutraliza-
tion and replaced by some slower positive ion; but after passing
the end point the conductance increases because of the pres-
ence of rapidly moving hydroxyl ions. If the acid solution is
diluted to about 0 01 JV, and titration is carried out with 0.10 N
NaOH, a sharp minimum will be obtained in a plot of conduct-
ance against burette readings. If a set of dipping electrodes is
used, a conductimetric titration will not require much more time
than the usual type, and it may be employed under some cir-
cumstances when the use of an indicator is not permissible, as in
a colored or a strongly oxidizing solution

Hydration of Ions. — There is no completely decisive method
for determining the quantity of water combined with solute in a
solution, though much work has been done in connection with
freezing-point deviations1 and in other ways. There is, however,
a method for determining whether or not the two ions of a solute
carry different quantities of water, and this may be used to calcu-
late the quantity of water combined with one ion if the other is
assumed to carry none. Thus, suppose a transference experi-
ment to be conducted on a solution of sodium chloride to which
a little sugar has been added. Sugar is not an ionized substance
and it does not move through a solution when electricity is passed;
thus, if at the end of an experiment the ratio of sugar to water in
the anode portion has changed, water must have come into this
portion on the anion or Jiave been carried out ot it on the cation.

When electricity passes through a normal solution of sodium
chloride, 0.76 mole of water per faraday is lost from the anode
portion, as is shown by a change in the ratio of sugar to water.
Assuming that the chloride ion does not carry any water, this 0 76
mole of water must have been carried out by 0.38 equivalent of
sodium ion, this being the quantity of sodium ion leaving the

1 A review of hydration in general is given by Washburn, Tech. Quart ,
21, 360 (1908), together with a criticism of each method. More recent
work on cryoscopic determination of hydration [BouRioN and ROUYER,
Compt. rend., 196, 1111 (1933)] seems to show that about 25 per cent of the
water in molal sodium chloride solution is combined with the solute.

SOLUTIONS OF IONIZED SOLUTES

271

anode portion per faraday passed. That is, each sodium ion is
associated with two molecules of water, since 0.38 equivalent
carried away 0.76 mole of water. If it be assumed that chloride
ions also carry water in these experiments, the hydration of the
positive ions is correspondingly greater. For example, if a
chloride ion carries four molecules of water, 0.62 equivalent of
chloride ion would carry 0.62 X 4 = 2.48 moles of water into
the anode portion. The net loss of water from the anode portion
requires that 0 38 equivalent of sodium ion carry out this 2.48
moles of water and 0.76 mole in addition, making 3.24 moles
of water on 0 38 equivalent of sodium ion, or a hydration of
3 24/0 38 = 8.5 moles of water per equivalent of sodium ion.
This particular method gives only the difference between the
hydration of one ion and another; but since the chloride ions
cannot carry less than no water at all, the lower limit of hydra-
tion for the sodium ion under these conditions, and as shown by
this method, is 2 0 moles per equivalent. Other " reference
substances," such as resorcinol, arsenious acid, alcohol, and
raffinose, in place of sugar, show the same hydrations and thus
show that the effect is produced, not by the reference substance,
but by actual motion of water with the ion. Some ionic hydra-
tions by this method are given in Table 50.

TABLE 50 — HYDRATION OF loNS1 IN NORMAL SOLUTION

Moles water on positive

Moles water carried

Transfer-

ion when chloride ion

Salt

from anode to cathode

ence num-

is assumed to have

per faraday of elec-

ber of posi-

tricity

tive ion

HC1

0 24

0 844

0 3

1 0

2 1

OsOl *

0 33

0 491

0 7

4 7

11 0

KC1

0 60

0 495

1 3

5 4

11 5

Nad

0 76

0 383

2 0

8 5

18 0

LiCl

1 50

0 304

4 7

14 0

28 0

Diffusion experiments in the presence of electrolytes are said2
to show that hydration of strong electrolytes does not change

1 WASHBUBN and MILLARD, J. Am. Chem. Soc., 37, 694 (1915).

2 GOTZ and PAMFIL, Bull. sect. sci. acad. roumaine, 8, 266 (1923); Chem.
Abst., 18, 3132 (1924).

272

PHYSICAL CHEMISTRY

with the concentration of the solution, which is a direct contra-
diction of the law of chemical equilibrium. Sugden1 states that
only cations are hydrated and that hydration is independent of
the concentration of the electrolyte. If these statements are
correct, the hydration of each positive ion in Table 50 is that
shown under 0. Somewhat different quantities of water trans-
ported per faraday are reported by Baborovsky,2 as follows:

HC1
0 43

KC1

0 47

NaCl
0 90

LiCl
1 02

KBr

0 89

NaBr

1 58

LiBr
2 10

Conductance in Solvents Other than Water.3 — Inorganic solutes
in solvents such as formic acid, liquid ammonia, organic amines,

TABLE 51. — CONDUCTANCE OF SODIUM IODIDE IN AcETONE4 AND ISOAMYL

ALCOHOL5

Equivalent
concentra-
tion

Equivalent conductance in acetone

Equivalent con-
ductance in iso-
amyl alcohol at
25°

0°

25°

40°

1.0000

26 4

28.65

0.5000

32 5

38 4

41 00

1.396

0.2000

44 9

52 7

56 60

1.339

0.1000

53 7

64.1

68 90

1.294

0.0500

63.1

76 1

82 70

0.0200

77.2

95 0

103 80

1.649

0.0100

89 0

109 7

121 40

2.024 -

0.0050

99 0

124 5

139 40

2.560

0.0020

111 0

143 2

163 00

3.394

0.0010

118 5

155 0

178 00

4.184

0.0005

125 0

164 6

188 90

•

0.0002

129 4

171 7

197 20

6.115

0.0001

129 9 -

173 6

199 90

6.636

0.0000

(131.4)

(176.2)

(204 00)

(7 790)

acetone, alcohols, dioxane, and other liquids have appreciable
conductances, some of which approach those of aqueous solu-
tions at the same temperature and concentration. For example,

1 J Chem. Soc. London, 129, 174 (1926).

2 J. chim. phys., 25, 452 (1928)

8 The data on electrically conducting systems have been brought together
in a single volume by C A Kraus, op cit , to which reference may be made
for detailed information concerning nonaqueous solvents.

4 McBAiN and COLEMAN, Trans. Faraday Soc., 15, 27 (1919).

6 KEYES and WINNINGHOFF, Proc. Nat. Acad. Sci , 2, 342 (1916)

SOLUTIONS OF IONIZED SOLUTES 273

Ao for Nal at 25° is 7.8 in isoamyl alcohol, 167 in acetone, 61 in
pyridine, 301 in liquid ammonia, compared with 126.94 in water.
The ions in these solutions are the same as in water solutions,
and Faraday's law applies, though the relative velocities of the
ions are not the same. Mole numbers for a given solute at a
given concentration vary widely with the nature of the solvent.
The conductance data in Table 51 are typical of nonaqueous
solutions

Conductances of Pure Liquids. — Most common liquids have
very slight electrical conductances at ordinary temperature.
The specific conductance of pure water is 1.0 X 10~8 reciprocal
ohm at 0°C., 4.5 X 10~8 at 25°, and 50 X 10"8 at 100°; and most
liquids have even smaller conductances. Assuming that the
conductance of water is due to H+ and OH~ ions, we may calcu-
late the concentration of these ions from the conductance by
means of equation (3) and the data in Table 49. At 25°, for
example,

4.5 X 10-8 = (349.G + 197)

whence C = 1.0 X 10~7 mole per liter of H+ and OH~~; the con-
centrations at the other temperatures are found through the
same relation to be 0 1 X 10~7 at 0° and 7 X 10~7 at 100°. These
ionic concentrations in water have been confirmed by several
other methods, some of which will be given in later chapters.

Fused salts, on the other hand, are very good conductors of
electricity The conductance is undoubtedly due to ions, just
as that of their aqueous solutions is due to ions; the products
of electrolysis are often the same in aqueous solutions, except
where these would react with water. Fused lead chloride
yields, upon electrolysis, lead at the cathode and chlorine at the
anode; the same products result when aqueous solutions of it are
electrolyzed

Fused sodium hydroxide yields metallic sodium at the cathode
and oxygen at the anode when it is electrolyzed; this same effect
is produced by electrolysis of sodium hydroxide solution with
a mercury cathode in which sodium can dissolve and be protected
from the action of water;1 Faraday's law describes quantitatively

1 When sodium hydroxide solution is electrolyzed with a platinum cathode,
sodium does not deposit and then react with water to produce sodium

274 PHYSICAL CHEMISTRY

the yield in both cases. But since salts in the fused condition
are, acting as both solvent and solute, ionic velocities have
not yet been determined, and transference experiments are
impossible.

The industrial importance of electrolysis of fused salts is very
great. Metallic sodium is produced almost entirely by elec-
trolysis of fused sodium chloride, magnesium metal from the
electrolysis of fused magnesium chloride, and aluminum from
the electrolysis of a solution of aluminum oxide in fused cryolite,
a fluoride of sodium and aluminum. Attempts have been made
to develop the theory of fused salts,1 but an adequate treatment
of them has not yet been accomplished.

IONIC THEORY

Most of the important experimental facts that we shall need
for a brief discussion of the ionic theory and for use in later work
in this book have now been given To account for these facts
the ionic theory has been built upon the following assumptions,
about which there seems to be no serious doubt at the present
time :

1 Inorganic salts and strong acids and bases dissolved in
water (and some other solvents) are dissociated into two or more
parts bearing charges of positive or negative electricity and
called ions

2. The conduction of electricity through these solutions is due
wholly to the movement of ions. Positively charged ions move
toward the negative pole; negatively charged ions move toward
the positive pole.

3. Ions have charges that are whole multiples of the charge
of the electron. Chloride ions carry 1 electron per atom; nitrate,
acetate, bicarbonate, and other univalent ions carry 1 electron

hydroxide and hydrogen, as is sometimes stated Hydrogen is evolved at
the cathode and oxygen at the anode during this electrolysis, when the
applied electric potential is insufficient to cause the deposition of sodium
Metallic sodium is deposited in a mercury cathode as an amalgam only
upon application of a much higher potential than is required to discharge
hydrogen at a platinum electrode

1 See Kraus, op. cit , Chap XIII, for a discussion of these systems The
data referring to fused salts are collected in Vol. Ill of the "International
Critical Tables."

SOLUTIONS OF IONIZED SOLUTES 275

per atom group. Corresponding positive ions are atoms or atom
groups that have lost one or more electrons, and thus become
positively charged. The unit charge is 1 598 X 10~20 absolute
electromagnetic unit, or 1.598 X 10~19 coulomb.

The older ionic theory as developed by Arrhenius and others
also contained the following assumptions, which are now believed
to be incorrect:

4. The dissociation of salts into ions is incomplete. Frac-
tional ionization increases with decreasing concentration and
approaches complete ionization as the concentration approaches
zeio The increase in equivalent conductance upon dilution is
due to an increase in the number of charged ions of constant
mobility.

5. Ions behave like independent molecules of solute as regards
the properties of solutions that are determined by the mole
fraction of the solute, such as vapor pressure, freezing point,
boiling point, and osmotic pressure. Each of these particles
exerts the same effect upon the freezing point as a whole molecule
Nomomzed molecules exert their usual effects.

Granting the last two assumptions lor the moment, two
methods become available for calculating the extent of ioniza-
tion, and the agreement between the methods, faulty as it was
in many solutions, was thought for many years to prove that the
extent of ionization changed with the concentration. If C is the
concentration and a the fractional ionization of a salt of the KC1
type, Ca gave the concentration of each ion and C(l — a) the
concentration of un-ionized molecules, whence ^ = 1 + a or
a = i — 1 Freezing points of aqueous solutions furnished the
best means of measuring i\ some values of 100(t — 1) from this
source are given in Table 52, marked F.P.

On the assumption that the change of equivalent conductance
with* concentration is owing only to a change in the number of
ions per equivalent of solute, the fraction of the solute ionized
is given by A/A0 = a. Table 52 shows some values of 100(A/A0),
marked C.R.

These two quantities i — 1 and A/A0 were accepted as meas-
ures of the fraction ionized long after it had been shown that
their values were not the same in a given solution and that
experimental error was not the cause of the variation. More-
over, the change of transference numbers with concentration

276

PHYSICAL CHEMISTRY

showed that some of the ion mobilities were not constant, and
this should have raised the question regarding the others.

The most serious objection to these "fractional ionizations"
was the fact that the change with concentration did not follow
that calculated from the laws of chemical equilibrium, which

TABLE 52 — COMPARISON or "PER CENT IONIZATION" FROM MOLE NUMBERS
BASED ON FREEZING POINTS (F P }l AND FROM CONDUCTANCE RATIO (C R )2

Solute

Method

Equivalent concentration

0 01

0 Of>

0 1

0 5

1 0

2 0

KC1

FP
CR

94 3
94 1

88 5
88 9

86 1
86 0

80 0

77 9

75 0
75 8

71 3

NaCl

FP
CR

93 8
93 6

89 2
88 2

87 5
85 2

81
68

85
59

LiCl

FP
CR

93
92 5

87 4

89
84 3

71 8

104
64 3

137
54

NaNO3

FP.
CR

91
93 3

86
87 0

83
83 1

70 5

62
62 7

50
52 1

KN03

93 3

93 8

84 7
87 3

78 4
83 1

55 2
70 8

37 8
63 9

54 9

HC1

93 5

97 2

90
94 0

89 0
92 0

97 6

85 8

112
79 0

66 7

HNO3

F.P
CR

95 5
94 2

90 8
91 4

88 6
89 4

86
83 2

92 3
79 0

103 5
67 I

MgSO4

FP ~
CR.

62
67 0

39
50 2

30
43 9

9 5
30 9

7 0
25 6

8 5
18 9

requires Ca2/(l — a) to be a constant. Thus, if KC1 were 86
per cent ionized in 0.1 N solution, it should be 51 per cent ionized
at 1 N, but the fractional ionization from freezing point and
conductance ratio showed about 75 per cent ionization.

These contradictory interpretations were grouped under the
inclusive heading " anomaly of strong electrolytes " rather than

1 Ibid., Vol. IV, pp. 254^.

2 Ibid., Vol VI, pp. 230JF.

SOLUTIONS OF IONIZED SOLUTES 277

under a more descriptive one such as "need of revision of the
theory/' and the term " extent of ionization" was, never clearly
defined. It will not be profitable to study the early stages by
which a new theory evolved and gained ground and " complete
ionization" was gradually accepted; for some of the first conse-
quences of its acceptance were mildly absurd. We turn now
to some aspects of this theory based upon complete ionization
as one working hypothesis

ASSUMPTION or COMPLETE IONIZATION

In assuming complete ionization of salts in dilute aqueous
solution, we assume that no neutral solute molecule such as
KC1 or HC1 exists, but we do not assume that a mole of hydrogen
chloride yields two moles of ideal solute, for this w^ould require
mole numbers of 2.0 at all concentrations, which would be con-
trary to experimental knowledge. We decide only that the
properties of solutions of salts in water and other ionizing solvents
will be considered in terms of properties other than a supposed
fractional ionization. In an address in 1908, Lewis1 pointed out
that many of the properties of electrolytic solutions were additive
properties of the ions up to concentrations approaching 1 N, in
which the degree of dissociation was currently supposed to be
about 75 per cent. He said of this additivity: "If it is an argu-
ment for the dissociation of electrolytes, it seems to be an argu-
ment for complete dissociation." Chemists were not at that
time prepared to accept a theory of complete ionization; in the
paper we have just quoted, Lewis himself makes the statement:
"I believe we shall make no great error in assuming that the
degree of dissociation as calculated from conductivities is in
most cases substantially correct . . ."

Thus the data available over 35 years ago showed evidence
for coftiplete dissociation of strong electrolytes in aqueous solution
to which the scientists were not blind and evidence of incomplete
dissociation, which was then thought to be more probable. Sub-
stances such as H2S03 and H2C03 are certainly not completely
ionized, the possibility of solutes such as T1C1 or Bad4" still
exists, and the ions PbCl+, FeCl++, and FeCl2+ have almost
certainly been shown to exist; but the change of transference

1 LEWIS, "The Use and Abuse of the Ionic Theory/' Z physik Chem., 70,
215 (1909) (in English).

278 PHYSICAL CHEMISTRY

number with concentration and the interionic attraction theory
alike point tcwthe impossibility of measuring the fractional ioniza-
tion of a highly ionized solute from the conductance ratio One
should not, however, lose sight of the fact that we still have no
conclusive evidence that ionization of salts is complete; we still
have the intermediate ion (such as HSO4~ or HSO3~ or HC03~)
to explain; we still have weak acids and bases that no one sup-
poses completely ionized; we still have acids that are neither
decisively "weak" acids nor yet completely ionized acids; and
we still have no property of a solution of a salt or other " strong"
electrolyte that is unquestionably connected with salt molecules
in such a way as to demonstrate their presence at concentrations
below 1m.

In the discussion of some aspects of modern work on ionized
solutes, we shall still accept the first three assumptions of the
ionic theory given on page 274, but in place of those numbered
(4) and (5) we shall now assume that

6. Ionization is complete in dilute aqueous solutions of salts
and "strong" acids and bases, and un-ionized molecules are not
present in these solutions.

7. The activity of an ionic solute, which is its effective con-
centration in influencing a chemical equilibrium or a potential
or a reaction rate, is equal to its concentration only in extremely
dilute solutions; at other concentrations the activity is a = my,
where m is its molality and 7 is the "activity coefficient." Thus
the activity has the dimensions of a concentration, and the
activity coefficient 7 = a/m is a number.

8. The change of equivalent conductance with concentration
is due mainly to the interionic attraction between the charged
ions for strong electrolytes; but the change of equivalent con-
ductance with concentration for weak electrolytes is due mainly
to increased ionization.

A brief discussion of the consequences of these assumptions
will now be given.

Conductance and Ion Velocities. — If only one charged "par-
ticle" were concerned in the conduction of electricity, as is true
in metallic conduction, the total quantity of electricity passing
would be given by the equation

N = cqd (4)

SOLUTIONS OF IONIZED SOLUTES 279

in which N is the number of faradays passed, c is the concentra-
tion of moving particles in equivalents per centimeter cube, q the
cross section of the conductor, and d the distance moved by the
particles. But in electrolytic solutions all the ions present take
part in the conduction in proportion to their concentrations
and velocities, as is shown by transference experiments. For a
single ionized solute yielding one negative and one positive ion
of unit charge,

N = Nc + Na

and the relation N = cqd may be applied separately to each ion.
The positive and negative ions move in opposite directions, of
course, but the motion of positive charges in one direction
produces the same electrical effect as the motion of negative
ions in the opposite direction. In the relation N — cqd, the
product qd is a volume and c is the quantity of material in a
unit volume; thus if q is in square centimeters and d is in centi-
meters, c will be in equivalents per centimeter cube, which is
0/1000 if we express concentrations in equivalents per liter of
solution. Writing the equation for the positive ion only, we have

in which Vc is the velocity of the ion in centimeters per second
and t is the time in seconds. These solutions obey Ohm's law,
which requires that the velocity of the ion be proportional to the
applied voltage, since C and q are constant. The mobility U
of an ion may be defined as the velocity under unit potential
gradient, and the quantity of electricity carried by the cation
is then

A similar expression containing Ua, the mobility of the negative
ion, shows the quantity of negative electricity passing, and the
total quantity is given by

N = Nc + Na = qt(Ue + Ua) ~ (7)

280 PHYSICAL CHEMISTRY

The current I is measured in coulombs per second, or NF/t, and
is by Ohm's law equal to E/R, which from equations (1) and (2)
is

I-EL2 (8)

Upon multiplying both sides of equation (7) byF/t and combining
with equation (8), we have

, NF C „ ... . 7T .E ELq

- -

After canceling E and q/l, we obtain the relation of the specific
conductance to the ion mobility, which is

L = (Ue + Ua)F

and the relation of equivalent conductance to mobility follows
by combining this equation with equation (3).

A = (Uc + Ua)F = Ar + Aa (11)

This relation implies that the equivalent conductance of a given
ion is independent of the other with which it is associated. As a
test of this implication, we may calculate the equivalent con-
ductance of chloride ion at 25° and 0.01 N in several solutions
by multiplying the equivalent conductance of the salt by the
transference number of chloride ion.1

TciAxci = 72.07 = Ao.oi for Cl~ at 25°
TciANaci = 72.05 = Ao.oi for Cl" at 25°
rCiAHci = 72.06 = Ao.oi for Cl" at 25°
7ciALlCi *= 72.02 = Ao.oi for Cl" at 25°

The corresponding figures for 0.10 N are, respectively, 65.79,
65.58, 65.98, and 65.49; and at higher concentrations the dif-
ferences are somewhat larger. It seems proven that at low con-
centrations the ions have independent mobilities, as was first
suggested by Kohlrausch many years ago. For chloride ion at
25° and 0.01 N this mobility under unit potential gradient is

72 0^
Ucl- = —- = 0.000746 cm. per sec.

1 MAC!NNES, J. Franklin Institute, 225, 661 (1938).

SOLUTIONS OF IONIZED SOLUTES 281

Limiting mobilities may be calculated from the limiting equiva-
lent conductances in Table 49 through the same relation; for
example, the limiting mobility of chloride ion at 25° is

76 3

= 0.00079 cm. per sec.

96,500

The Activity Function. — The activity of any constituent of a
solution is denned by Lewis1 as its "effective" concentration (its
effect in changing a chemical system at equilibrium). In an
ideal solution the activity and the actual concentration are
equal; in aqueous solutions of ions the activity and the ion
concentration are not equal, but they approach equality as the
concentration approaches zero.

Following the notation of Lewis and Randall,2 the activity
of a solvent is designated by ai and of a solute by a2. Thus
the vapor pressure of a solvent over a solution would be propor-
tional to «i, and for an ideal solution this could be computed
from Raoult's law No simple law for calculating the activity
of an ionized solute has yet been discovered. We may, however,
designate by a_j. and a_ the activities of the positive and negative
ions, respectively, and by a^ the activity of the noniomzed
molecules. Then by definition

In the absence of definite information regarding the concentra-
tion of nonionized solute in an electrolytic solution, Lewis defines
K as unity so that

Since at finite concentrations the two ions of a solute may not
have the same activity, it is often expedient to consider the
geometric mean of the two ion activities, which may be defined

a± =

lProc. Am. Acad. Arts Sa.t 43, 259 (1907).
2/. Am Chem Soc., 43, 1112 (1921).

282

PHYSICAL CHEMISTRY

The Activity Coefficient. — Lewis defines the activity coefficient
as the activity divided by the molality, i.e.,

7 =

(12)

This coefficient is not, and should not be confused with, a frac-
tional ionization. It is a factor, sometimes greater than unity,
by which the molality must be multiplied to give the effect that
a solute produces upon a chemical equilibrium or electrode poten-
tial or other property. Some of the methods by which activity
coefficients are obtained will be given in the next section, and
others later in the text.

Methods of Determining an Activity Coefficient. — The activity
coefficients of solutes may be determined from their vapor pres-
sures when the solute is sufficiently volatile; from freezing points
of their solutions (but not from equating i — 1 toy), or from the
potentials of concentration cells in a way which will be explained
in Chap. XIX. The activity of the solute may be calculated
from the vapor pressure of the solvent by means of the equation

d In a\ = — •— d In

(13)

in which NI and N2 are the moles of solvent and solute and a\
and a2 are the corresponding activities. For convenience in

TABLE 53 — COMPARISON OF ACTIVITY COEFFICIENTS AT 25°

NaCl

KOI

H2SO4

Vapor

Cell

Vapor

Cell

Vapor

Cell

pressure1

potential2

pressure1

potential3

pressure1

potential4

0 10

0 781

0 778

0 770

0.769

0 265

0 20

0 737

0 732

0 719

0 209

0 50

0 685

0 679

0 651

0.651

0 156

0 154

1 00

0 661

0 656

0 606

0 131

0 130

2 00

0 667

0 670

0 571

0 576

0 127

0 124

3 00

0 713

0 719

0 567

0 571

0 142

0 141

HAMER, and WOOD, J Am Chem Soc , 60, 3061 (1938)

2 HARNED and NIMS, ibid., 64, 423 (1932).

3 EARNED and COOK, ibid., 69, 1290 (1937).

4 HARNED and HAMER, ibid , 67, 27 (1935)

SOLUTIONS OF IONIZED SOLUTES 283

integrating, the equation is often transformed into terms of
molalities and activity coefficients, by methods which need not
concern us here. Table 53 shows some activity coefficients for
25° at several molalities derived from vapor-pressure measure-
ments and for comparison the coefficients derived from cell
potentials.

Agreement between the two methods is as close as that among
various experimenters using the same method. Activity coeffi-
cients may also be calculated from freezing-point depressions,
provided that the data cover a range of molalities extending
below 0 Olm. In discussing the freezing points of dilute aqueous
solutions, it has become common practice to use another func-
tion in place of the actual freezing-point depression, called the
j function, and denned by the equation

~ 1.858m

where A/ is the freezing-point depression, m the molality of the
solution, and v the number of ions produced by a mole of salt,
In terms of j, the relation between the activity a± of a solute, its
freezing-point change A£, and the molality m is1

d In — = d In 7 = —dj — j din m (15)

Tli

Since the activity coefficients change with temperature, values
derived from freezing points should not be compared with those
from vapor pressures or electromotive forces of concentration
cells without first correcting them to the same temperature.

Some activity coefficients for 25° are given in Table 54, and
others will be found in Table 98.

The mean activity coefficient for simple electrolytes in a
mixture of two salts at a total concentration of c\ + C2 is about
the same as that for each salt when it is alone present at the
concentration c\ + C<L. Accurate data on the activity coefficients
in mixtures have shown that this simple rule is not strictly true,
but so far no accurate general law has been discovered.

1 LEWIS and RANDALL, " Thermodynamics," Chap. XXVII, equation (3).
Methods of integrating the equation are also discussed in Chap. XXIII of
this excellent text.

284 PHYSICAL CHEMISTRY

TABLE 54. — MEAN ACTIVITY COEFFICIENTS OF IONS AT 25°C.

0 10

0 20

0 50

1 00

2 00

3 00

LiCl

0 792

0 761

0 742

0 781

0.931

1 174

NaBr

0 781

0 739

0 695

0 687

0 732

0 817

NaNO3

0 758

0 702

0 615

0 548

0 481

0 438

MgCh

0 565

0 520

0 514

0 613

1 143

CaCl2

0 531

0 482

0 457

0 509

0 807

Na2SO4

0 45

0 36

0 27

0 20

ZnSO4

0 15

0 11

0 065

0 045

0 036

0 04

Change of Activity Coefficient with Temperature. — Activity
coefficients change somewhat with temperature, so that those
based on freezing points require correction before being com-
pared with coefficients derived from cell potentials at 25°. The
following data are typical:

0°

10°

20°

30°

40°

50°

60°

0 1m. HC1

0 803

0 802

0 799

0 794

0 789

0 785

0 781

0 1m. NaCl

0 781

0 779

0 777

0 774

0 770

0 766

1 Om. HC1

0 842

0 830

0 816

0 802

0 787

0 770

0 754

1 Om. NaCl

0 638

0 649

0 654

0 657

0 656

0 654

INTERIONIC-ATTRACTION THEORY1

The most important recent event in theoretical electro-
chemistry is certainly the publication of papers on the interionic
attraction theory of electrolytes by Debye and Hiickel2 and by
Onsager.3 Although the picture these authors give of the
phenomena occurring fn solutions of electrolytes has none of the
engaging simplicity of the electrolytic-dissociation theory as
advanced by Arrhenius, there is little doubt that the later theory,
incomplete as it must be granted to be in details, is remarkably
successful in organizing and predicting the results of measure-

1 These paragraphs are condensed from the excellent paper of Shedlovsky,
Brown, and Maclnnes in Trans. Electrochem. Soc., 66, 237 (1934). For an
extensive bibliography and further discussion of this material, see Scatchard,
Chem Reviews, 13, 7, (1933), Maclnnes, ''Principles of Electrochemistry,"
Chap. VII, 1939, or Earned and Owen, op. at., 1943.

2 DEBYE and HUCKEL, Physik Z., 24, 305 (1923), 26, 93 (1925)
8 ONSAGER, ibid., 27, 338 (1926), 28, 277 (1927).

SOLUTIONS OF IONIZED SOLUTES 285

ments. In the interionic-attraction theory of electrolytes the
properties of the solutions are considered to be due to the inter-
play of electrostatic forces and thermal vibrations The first
of these tends to give the ions a definite arrangement, and the
second acts to produce a random distribution.

The methods of Debye and Huckel are still the subject of
discussion and occasionally of acrimonious dispute, but they
have led to equations that could be tested experimentally. It
appears to be a safe statement that, in dealing with the thermo-
dynamic properties of dilute solutions of electrolytes in solvents
of high dielectric constant, the more accurate the experimental
data the more surely they can be fitted by equations obtained by
Debye and Huckel or by extensions devised to make them
mathematically more adequate.

These equations take account^ of the fact that the ions are
not fully independent but must attract and repel each other in
accordance with Coulomb's law. If these electrical forces were
the only ones acting on the ions, they would tend to arrange them-
selves in a space-lattice, as in a salt crystal. However, the
ions are also subject to thermal vibration of increasing intensity
as the temperature is raised. The properties of an ionic solution
are thus due largely to the interplay of these two effects. Since
the electrostatic forces increase as ions approach each other, it
follows that these properties must change as the concentration
changes, and that the ions cannot have the same mobilities and
osmotic (thermodynamic) properties in concentrated and dilute
solutions, as postulated by the Arrhenius theory. It is a real
triumph for the modern theory that the changes of these prop-
erties, at least in dilute solutions, are quantitatively as predicted.

It is a result of the presence of electrostatic forces that any
selected ion, a positively charged one, for instance, will, on the
average, have more negative ions near it than if the distribution
were purely random. This is known as the "ion atmosphere" ol
the selected ion. This distribution gives rise to a potential
around the ion that may be computed from the Debye-Hiickel
equation.

From the thermodynamic point of view the effect of the
presence of the ionic atmosphere is to reduce the activity coeffi-
cients of the ions. The presence of the ionic atmosphere has
at least two results on electrolytic conductance, both of which

280 PHYSICAL CHEMISTRY

tend to decrease the ion mobilities with increasing ion concentra-
tion. These are known as (1) the electrophoretic effect and (2)
the time of relaxation effect. Both these were considered by
Debye and Huckel. However, the theory of conductance of
electrolytes in its present form is to a large extent the work of
Onsager.

We have space hero only for the original equation of Debye
and Huckel, which is

In this expression z is the valence of the ion, R is the gas con-
stant, K is the dielectric constant of the solvent, T is the absolute
temperature, e is the electronic charge, c is the ion concentration
per centimeter cube, and N is Avogadro's number. For an
aqueous solution of a salt of the KC1 type at 25° this equation
may be reduced to the following one in which all the constants
are combined into a single term,

- log 7 = 0 50 Vm (17)

where ra is now the molality of the solution. It will be noted
that this equation contains no term which is characteristic of
the solute. This relation is valid only in very dilute solutions;
a better approximation is

- -Q50 Vm
1 + \/m

When ions of valence other than unity are present in solution,
this relation is best given in terms of the valences z+ and Z- of
the ions and the ionic strength /x, which is defined as \i =
The relation is

V M

Comparisons of measured activity coefficients with those calcu-
lated from these equations showr a remarkable agreement at low
concentrations, but the agreement is much less satisfactory at

SOLUTIONS OF IONIZED SOLUTES 287

moderate concentrations.1 Among the additional effects that
required consideration were the size of the ion, the variation in
dielectric constant of the solvent produced by the presence of the
solute, attraction between ions and solvent molecules, alteration
of the forces acting between solvent molecules produced by the
solute, changes in the hydration of solute ions at higher concen-
tration, and possible ionic association. To allow for some of
these effects, additional terms involving higher powers of the
molality than its square root have been added to the equation
above, but a consideration of the more complex equation would
be out of place in an elementary text

According to the original treatment of Debye and Htickel
or to the correction and extension of Onsager, the equivalent
conductance decreases with increasing concentration for two
reasons. The first, called the time of relaxation effect, comes
from the fact that the ion atmosphere of a moving ion always lags
behind; thus ahead there is always too little of the opposite
charge for equilibrium, and behind there is always too much.
The second, called the cataphoresis effect, arises from the fact
that the ion must move through a medium bearing the opposite
charge and therefore moving in the opposite direction.

As has been pointed out recently, the behavior of solutions*
containing "ionic atmospheres" is much more complex than any
theory yet proposed assumes. When changes of hydration,
Debye-Huckel electric effects, ionic association, dielectric con-
stant of the medium, etc , unite in influencing the behavior of
ions, any theory that pretends to explain the observations on
the basis of one or a few of these variables cannot possibly be
trusted as a sound solution of the problem. It should not be
overlooked that ionized solutes exert a very marked effect also
on molecules having no electric net charge.

Procedure to Be Followed in This Book. — In the present state
of our knowledge the calculation of an activity coefficient is
difficult and somewhat uncertain except m a dilute solution con-
taining one salt of the simplest type. Comparatively little work
has been done on the activity coefficients for ions in mixtures

1 An empirical extension of this equation suggested by Davies [/. Chem.
Soc (London), 1938, 2093] is obtained by subtracting 0.2/z from the one just
given. It is claimed that the usual deviations from this equation are about
2 per cent in O.lm solutions and proportionately less in more dilute solutions.

288 PHYSICAL CHEMISTRY

of salts. In the treatment of chemical equilibrium in the fol-
lowing chapters it would be very desirable to multiply the con-
centration of each ion by the appropriate activity coefficient if
this were known. It is, however, unknown and we shall there-
fore make most of the calculations by using the ion concentration
itself without an activity coefficient as a rough measure of the
activity. We shall do so with the understanding that this pro-
cedure is not correct but that under present circumstances
it is inexpedient for beginners to attempt exact calculations.
When there is reason to believe that the solute is substantially
un-ionized, we shall treat it as if it were not ionized. Problems
involving solutes that are not "largely ionized" but that are
not substantially un-ionized will not be treated in this text.

Problems

Numerical data for solving problem* should be sought ^n the tables

1. Write electrode reactions that illustrate each oi the effects listed for
anodes and cathodes on page 244

2. The limiting equivalent conductance and the equivalent conductance
at 0 01 N for potassium chloride change with temperature as follows:

t 15° 25° 35° 45°

Ao 121 1 149 9 180 5 212 5

Ao 01 114 3 141 3 169 9 199 7

(a) Plot these conductances against the temperature on a §cale wide
enough to allow extrapolation to 0° and 50°, arid compute the conductance
ratio A/A 0 for 0° and 50°. (6) The fluidity of water is 55 8 at 0°, 111 6 at 25°,
and 182 at 50° Recalculate A0 for KC1 at 0° and at 50° from the stated
value for 25°, on the basis that all the change of conductance is caused by
the changing fluidity of water, and draw on the same plot a line through
these computed conductances and the actual conductance for 25° [GUN-
NING and GORDON, ,7 Chem Phys., 10, 126 (1942) ]

3. Calculate the mole numbers for LiCl from the vapoi-piessure data in
Table 42.

4. Calculate the current required to deposit an atomic weight of chro-
mium in 10 hr , on the assumption that the electrolyte is a solution of chromic
acid and that 90 per cent of the electricity is used in the evolution of hydrogen
gas at the cathode and 10 per cent is used in reducing chromic acid to
chromium.

6. A transference experiment is run on a solution containing 8 00 grams
of NaOH per 1000 grams of water, with a platinum anode and a silver
chloride cathode, until 122 ml of oxygen (25°, 1 atm ) is evolved. The
cathode portion weighs 252 53 grams and contains 1 36 grams of NaOH.
(a) Write the electrode reactions and complete gam-arid-loss tables for the
anode and cathode portions, and calculate the transference number of

SOLUTIONS OF IONIZED SOLUTES 289

hydroxide ion. (6) Assume that the cathode portion is thoroughly mixed
after its removal from the apparatus, that the conductance ratio is 0.85,
and calculate its specific conductance at 25°. (c) The transference tube was
18 sq. cm in cross section, and the experiment ran for 4 hr. How far did
the hydroxide ions in the middle portion move during the experiment?

6. The freezing-point lowenngs of solutions of MgSO4 at several molah-
ties are given in Table 43. Calculate the mole number corresponding to
each of the concentrations.

7. A 0 1m. solution of lithium iodide is electrolyzed in a transference
experiment The electrodes consist of a platinum anode and a silver
iodide cathode. By titration with thiosuliate solution, it was found that
the anode portion contained 1 27 grams ol free iodine The net gain of
lithium iodine in the cathode portion is 0 445 gram, (a) Construct gam-and-
loss tables for both anode and cathode portions (6) Calculate the trans-
ference number of iodide ion in Lil (r) Assume the experiment repeated
with a solution 0 Ira in HI in the same apparatus with the same quantity
of electricity used. Write new gam-and-loss tables for the experiment,
taking 0 18 as the transference number of iodide ion m HI.

8. A solution of 10.00 grams of perchloric acid per 1000 grams of water is
electrolyzed at 25° in a tube of 20 sq cm. cross section between a silver
anode and a platinum cathode with a current of 0 134 amp for 2 hr AgClO4
is a soluble salt. The anode portion after electrolysis weighed 405 2 grams
and contained 3 16 grams of HC104 (a) Write the electrode reactions and
complete gain-and-loss tables for both portions, and calculate the transfer-
ence number of perchlorate ion in the solution (6) Calculate the distance
moved by perchlorate ions in the middle portion (r) Calculate the specific
conductance of the anode portion. Assume that normality is equal to
molality and that the conductance ratio is 0 90

9. A solution of 1 gram of HF per 1000 grams of water was electrolyzed
between silver electrodes for 10 hr with a current of 0 01 arnp. An anode
portion weighing 480 2 grams contains 0 415 gram of HF AgF is a soluble
salt Write electrode reactions and gam-and-loss tables for the anode and
cathode portions of solution, and calculate the transference number of
fluoride ion in HF.

10. A solution containing 3 65 grams of HC1 per 1000 grams of water
is electrolyzed for 10 hr at 25° with a uniform current in a tube 10 sq cm
in cross section, between silver electrodes. The anode increases in weight
LOOT gram, and the anode portion after electrolysis weighs 601 3 grams and
contains 0 0364 equivalent of HC1. (a) Write the electrode reactions, and
show the gams and losses of each ion in the anode portion (b) What is the
transference number of chloride ion in this solution? (c) How far did the
chloride ions move in 10 hr ? (d) What was the current? (e) The limiting
equivalent conductance of chloride ion at 25° is 76 reciprocal ohms. Esti-
mate the specific conductance of the middle portion, and state within about
what limits the estimate is reliable.

11. A current of 0.0193 amp. passes for 2.78 hr. through a solution of
6.3 grams of nitric acid per 1000 grams of water at 25° in a long tube fitted
with a silver anode and a platinum cathode. After the electrolysis the

290 PHYSICAL CHEMISTRY

anode portion weighs 40275 grams and contains 2415 grams of HNO<
(a) Compute the weight of AgNO3 in this portion arid the change in the
weight of HNO3 in it (6) Write gam-and-loss tables for the anode and
cathode portions, with the electrode reactions at the head of each table, and
compute the transference number of hydrogen ion in 0 1 N HNO.< at 25°

12. The resistance of a centimeter cube of 0 1 Ar HNOa at 25° is 26 0 ohms
A potential of 10 volts is applied to a tube of 0 1 A* HNOj 15 cm long and
5 sq cm. in cross section for 1 mm Neglect concentration changes near the
electrodes, and calculate the number of faradays carried by the nitrate
ion. How far did these ions move?

13. A sample of "hard water" known to contain only calcium sulfate and
calcium bicarbonate in appreciable quantities is submitted for analysis
At 18° the specific conductance of the hard water is 0 00100 It is boiled
(without loss of water) and cooled to 18°, when its specific conductance is
found to be 0 000757 Assume that A /A,, is 0 85 for each salt, that boiling
completely changes the calcium bicarbonate to insoluble OaCOj, and
calculate the concentration of calcium sulfate ("permanent hardness")
and of calcium bicarbonate ("temporary hardness") Express the results
as molecular weights per liter The limiting equivalent conductances are
Ca = 51, SO4 = 68, and HC(X< = about 35

14. A solution of 65 60 grams oi NaCl in 1000 grams of water is elec-
trolyzed in a transference apparatus at 25° with a silver anode and a silver
chloride cathode A coulometer in the circuit deposited 5 670 grams of
silver. The anode portion after the experiment weighed 120 23 grams and
contained 6 409 grams of Nad Write the electrode reactions and com-
plete gam-and-loss tables for both electrode poitions, and calculate the
transference number of sodium 1011 in the solution

15. A transference experiment is made with a solution containing 7 39
grams of AgNO-? per 1000 grams of water and using two silver electrodes
A coulometer in the circuit deposited 0 0780 gram of silver At the end
of the experiment the anode portion weighed 23 38 grams and contained
0 2361 gram of AgNO? (a) Write complete gam-and-loss tables for both
electrode portions, and calculate the transference number of silver ion (&)
The cathode portion weighed 25 00 grams. How much silver nitrate did
it contain?

16. (a) Show by a diagram approximately how the specific conductance
would change as 0 1 N HC1 is added to Q.I N sodium acetate in the following
proportion :

HC1, ml 90 99 100 101 110

NaAc, ml 100 100 100 100 100

Bear in mind that acetic acid is only very slightly ionized in the presence
of HC1 or NaAc (b) Calculate the specific conductance of the solution
containing 110 ml. of HC1.

17. A 10-ml. sample of commercial liquid bleach, containing sodium
hydroxide, sodium chloride, and sodium hypochlonte, is diluted to about
500 ml. and titrated with 0.5 N hydrochloric acid, using the electrical
conductance of the mixture as an indicator (since color indicators are

SOLUTIONS OF IONIZED SOLUTES

bleached by hypochlorites) Draw a plot roughly to scale showing burette
reading against conductance, which is taken after every 1.0-ml addition of
acid Indicate how the plot should be read to determine the quantities of
sodium hydroxide arid sodium hypochlonte present. (A typical analysis
might show about 0 5 N sodium hydroxide, 2 TV sodium chloride, and "125
grams per liter available chlorine ")

18. The equivalent conductance at 25° for monoethariolammomum
hydroxide changes with the concentration as follows

10(C 0 228 0 385 0 490 1 018 2 687 5 347

A 74 87 60 12 54 14 39 07 24.93 17 95

The limiting equivalent conductance of the chloride of this base is 118 58
at 25° Calculate the fractional lomzation of the base in these solutions
[SrvKRTZ, REITMEIKR, and TARTAR, J Am. Chcm Soc , 62, 1379 (1940) ]

19. A transference experiment is made at 25° with a solution containing
185 2 grams of CsOl per 1000 grams of water and with a silver anode and a
silver chloride cathode A silver coulometer in the circuit deposited 5 48
grams of silver The cathode portion weighed 117 22 grams and contained
21 88 grams of CsCl. (a) Write the electrode reactions and complete
gam-and-loss tables for the anode and cathode portions, and calculate the
transference number of cesium ion in this solution (fo) Assume that the
experiment was made with a platinum cathode arid the same quantity of
electricity and that the cathode portion after electrolysis contained the
same weight of water as in part (a), arid write a new gam-and-loss table
for the cathode portion.

CHAPTER VIII
THERMOCHEMISTRY

The purpose of this chapter is to show how the recorded
calorimetnc data and the first law of thermodynamics may be
combined with certain useful approximations to calculate the
heat effects attending chemical reactions Measured heat
effects are available for many reactions, and therefore calcula-
tions are not always required; but the obvious impossibility of
measuring the heat effect attending every chemical reaction at
every concentration and every temperature makes calculations
from the available data a most important matter for chemists
and engineers. The available materials for these calculations
are (1) an adequate theory, (2) experimental data, and (3) useful
approximations with which to supplement the data when neces-
sary. Since the enthalpy increase attending an isothermal
chemical change vanes with the temperature and concentration
of the reacting substances, it is necessary to specify carefully the
composition of the systems involved if the enthalpy change is
to have an exact meaning. Before proceeding with the actual
calculations we review^ briefly the factors that determine the
"state" of a system, we review the first law of thermodynamics,
and we consider the experimental methods by which the data
are obtained. It is suggested that pages 33 to 36 be read again
in this connection.

Since the changes involved in this chapter are taking place
either at constant volume or at constant pressure, the work done
will be either zero or p(vz — vj. When only liquids and solids
are involved, the work corresponding to changes in volume
against atmospheric pressure is negligible ; and for systems involv-
ing gases Ay will be substantially equal to AnRT, An being the
change in the number of moles of gas in the chemical reaction.

For the purposes of this chapter it will be sufficient to consider
a calorie as the heat required to raise the temperature of a gram
of water 1° and a kilocalorie (written kcal.) as 1000 times this

292

THERMOCHEMISTR Y 293

quantity, without specifying whether it is a "15° calorie," a "20°
calorie," a "mean" calorie, or a "defined" calorie; and it will
be sufficient to assume 4.18 joules per cal. We may leave until
the need arises the definition that a "15° calorie" is 4.185 abs.
joules and 4.1833 "international" joules or that a "20° calorie"
is 4.1793 international joules. These distinctions are important
for exact work but are not required for a first consideration of
thermochemistry.

It should be recalled that the definition of an ideal gas is con-
tained in two equations

pv = nRT and . ^
\ dv

The second of these relations, combined with the definition
H = E + pv, gives

(T) -»

\dp/T

which means that the enthalpy of an ideal gas at constant
temperature is independent of pressure. Thus this equation and
pv = nRT also define the ideal gas.

Changes in the State of a System. — When the state of a
system is fully specified, every property of it is uniquely deter-
mined, though, of course, it is not necessary to specify every
property of a system in order to fix its state. We need specify
only so many properties that the others are fixed; for example,
if (1) quantity, (2) composition, (3) state of aggregation, (4) pres-
sure, and (5) temperature of a system (or of each of its parts if it
consists of more than one phase) are stated, all the properties are
determined, and the system is in a definite "state." A change
in any property of the system constitutes a change in the state
of the system. It is commonly true that the properties listed
above are those observed experimentally, and they are the ones
we shall ordinarily use in this book, though others may be used in
place of them. For example, we may specify the volume of a
homogeneous system in place of the temperature or pressure
In the calculations that follow, a pressure of 1 atm. is assumed
to prevail unless some other pressure is specified.

294 PHYSICAL CHEMISTRY

It will be recalled that the energy content E and the enthalpy
H are properties of a system in a specified state, that changes in
them are dependent on the change in state and fully determined
by the initial and final states of the system undergoing change
without regard to the mechanism or process of the change. This
is not true of the heat absorbed during a change in state or of
the work done by the system during a change in state. It is for
this reason that the heat effects are described by AE and AH in
this chapter, as is the common custom in physical chemistry.

The " surroundings" of a system may be defined as any matter
or space with which the system exchanges energy.

First Law of Thermodynamics. — The relations by which the
first law of heat, or the first law of thermodynamics, are expressed
were given on page 33. They are

£ dE = 0 A# = #2 - #1 A# = q - w (\)

In these expressions E denotes the energy content of a system in
a specified state, A£J the increase in energy content attending
a change in state, q the heat absorbed by the system in such a
change of state, and w the work done by the system. Since the
integral of dE around a complete cycle is zero, it follows that
AE for any part of a cycle is equal to — &E for the remainder
of the cycle. Hence, if AE is the increase in energy content
attending a change in state by any path, AJ5/ for the reverse
change in state by any path has the same numerical value and the
opposite sign; for only so may the energy content of the system
return to its initial value when the system returns to its initial
state.

The relation in the form &E = E2 — EI emphasizes the fact
that AE has the same value for a specified change in state by all
paths. Hence, if by a series of reactions the same change in state
is produced as by a single reaction, AJ£ for the over-all change
in state is the sum of the separate &E values of the individual
steps. This important fact allows the calculation of AE for
reactions that are inconvenient to measure calorimetrically but
that are the sums of readily measurable steps or the differences
between readily measurable steps. The fact that &E for a
given change is equal to the sum of the &E values for a series of
changes producing the same net effect was proved experimentally
about a hundred years ago by the experiments of Hess and was

THERMOCHEMISTR Y 295

long known as the law of Hess. This law has been of the greatest
service in thermochemistry, but it is only a special statement
of the first law of thermodynamics.

Another quantity called the enthalpy, which is a* property of
a system in a specified state, is defined by the equations t

PI = E + pv
A// = A# + AO)
A// = //2 _ //, (2)

Since E, p, and v are all properties of a system, it follows that
// is a property of a system, that <f> dH = 0, and that dH is
an exact differential.

The relation A// = 7/2 — Hi shows that AH for a given change
in state produced in a single step is equal to the sum of the A/f
values for a series of changes which produce the same over-all
change in state. This fact will be of great service in the calcu-
lations that are outlined in this chapter. Although A// is not
restricted to changes at constant pressure or restricted in any
way, it will be the convenient quantity to sum for constant-
pressure processes, since it is then equal to g, as we shall see
below. AE will be the convenient quantity to sum for constant-
volume processes, since in these changes AE = q.

For the special condition of changes in state at constant pres-
sure, during which no work is done other than changes in volume
against constant pressure, the A// relation is

AH = q — w + p(v2 — Vi) = qp (3)

and AH is a measure of the heat absorbed. Similarly, for changes
in state at constant volume, AE = q — w = qv, since w = 0 when
the volume is constant. When both pressure and volume change,
the general relations

&E = q - w and A// = &E + A(»

may still be used, since they imply no restrictions as to the
mechanism of the process. It should be noted that w is the
work actually done and not the work that might have been
done in a more efficient process. This work is p(vz — Vi) when
the process takes place at constant pressure. When the pressure
varies as the process takes place, it is necessary to express p as
a function of v before integrating p dv.

296 PHYSICAL CHEMISTRY

Thermochemical Equations. — Chemical equations are incom-
plete descriptions of changes in state, and they may be made into
complete statements by specifying the pressure and temperature,
together with the state of aggregation when this is not obvious.
For example, the equation

CH4 + 202 = C02 + 2H2O

does not constitute a complete formulation of a change in state,
though it states the quantities and compositions of the substances
undergoing change. In order to specify definitely the change in
state we should write

CH4(<7, 25°, 1 atm.) + 202(0, 25°, 1 atm ) = C02(<7, 25°, 1 atm.)
+ 2H2O(/, 25°, 1 atm.) A// = -212.79 kcal.

Since at low pressures (dH /dp)T is zero or very small for gases,
this change in state may be formulated more briefly, and yet so
fully as to be completely understood, as follows:

OH4(<7) + 202(0) = C02(<7) + 2H20(Z)
A#298 = -212.79 kcal.

An example of a change in state in which no chemical change
occurs is

H20(Z) = H20(gf, 1 atm.) AF373 = 9700 cal.
A#373 = 8950 cal.

The subscript attached to A//" is always understood to mean A#
for the isothermal change in state. Later we shall see that A#
for any change in state is a function of the temperature and that
means are available for calculating its change with changing
temperature, but we may give as a simple example to show the
necessity of specifying the temperature

H,O(Z) = H2O(<7, 0.1 atm ) A#323 = 10,250 cal.

It must be clearly understood that the changes in state formu-
lated are complete changes. For illustration, the expression

H2(0, 1 atm.) + I2(p, 1 atm.) = 2HI(0, 1 atm.)
= -3070 cal.

means that this increase in enthalpy attends the formation of
2 moles of HI at 573°K. It does not mean that when 1 mole of

THERMOCHEMISTR Y 297

hydrogen and 1 mole of iodine vapor are brought together at
573°K. this effect will be observed; for the reaction is incomplete,
and substantial quantities of both hydrogen and iodine remain
at equilibrium with less than 2 moles of HI in this system.

Much confusion has been brought into thermochemistry by
using the term "heat of reaction," which some writers define as
the heat absorbed by a reaction and others as the heat evolved.
It is partly to avoid this confusion (but chiefly because the terms
depend on the change in state and are independent of the path
followed) that we use AH arid AE. Students should form the
habit of saying "heat absorbed by a reaction" or "AH for a
reaction," rather than using the ambiguous "heat of a reaction,"
which may be misunderstood. Some tables of thermochemical
data record the heat evolved by chemical changes; others give
heat absorbed. Data are given in small calories (usually abbre-
viated cal.) or in large calories (written kg.-cal. or kcal. or Cal.
for 1000 cal ) or in kilojoules (written kj. for 238.9 cal.). When-
ever reference books are consulted, it will be necessary to give
careful attention to this difference in notation and usage.

We shall not follow the older custom of writing a thermo-
chemical equation in the form

H2(0) + M02(0) = HaO(Z) + 68.32 kcal. at 25°

in which a positive sign attached to a heat quantity signifies heat
lost from the system. We shall follow the practice, which is now
almost standard, of writing this same fact in the form

H2(flf) + M02(0) = H,0(/) A//298 - -68.32 kcal.

since it is the enthalpy increase attending a change in the state
of a system that is used in the thermodynamic calculations of
physical chemistry, and it is best to become accustomed to this
usage at the start.

Thermochemical Methods. — Heat effects attending changes
in state are measured in a calorimeter, which is a reaction vessel
immersed in a tank of water isolated from its surroundings. The
change in temperature of the calorimeter and its heat capacity
furnish the quantities for computing the heat effect for an iso-
thermal change in state. This is equal to AE if the change in
state takes place at constant volume and to AH if it occurs at
constant pressure. But in the calorimetric process itself AE is

298 PHYSICAL CHEMISTRY

zero and the temperature is not constant. In order to make this
clear, consider the change in state

C0(25°, 1 atm.) + ^02(25°, 1 atm.) = CO2(25°, 1 atm.)

for which AE at 25°C. is desired. Imagine a calorimeter large
enough to contain ICO + J'2^2 at 25° and 1.5 atm. totaj pres-
sure. The change in state taking place in the calorimeter when
these substances react is

l(X)(p = 1 atm.) + }4O2(p = Ji atm.) = ]CO2(p above 1 atm )
t = 25° t = 25° + At

and for this change in state AE = 0. By removing a quantity
of heat equal to At times the heat capacity of the calorimetric
system in its final state (this system is a mole of C02, a quantity
of water, the container, and the temperature-measuring devices),
the final system is restored to 25°. If — q is the heat removed
from the system, then — q will be the heat evolved by the iso-
thermal change in state

1CO(25°, 1 atm ) + MO2(25°, Y2 atm.) = C02(25°, 1 atm.)

for this change is the sum of the calorimetric process for which
AE was zero and the cooling process for which AE = —Cv At.
Since the volume was constant, no work was done and AE will
be equal to the heat absorbed by the system, which is +g. The
heat of mixing the gases is negligible at these pressures, and
(dE/dv)T is also negligible for gases at low pressures; therefore
AE for the process occurring in the calorimeter is substantially
equal to AE for the initial change of state formulated. We may
write for this change in state

100(25°, 1 atm ) + MO2(25°, 1 atm.) = 1CO2(25°, 1 atm.)
AEW = -67.64 kcal.

In this book we shall express the quantities AH and AE in
small calories when they .are small ^and usually in kilogram-
calories when they are large, and AH will be positive when heat
is absorbed .by, the. gystem atjconstant pressure. LikewisQ.^A^
will be positive when heat is absorbed by the system at constant
volume,

THERMOCHEMIS TRY

299

In order to prevent any loss of heat by exchange between the calorimeter
arid it>s surroundings, the latter are often maintained at the same temperature
as the calorimeter itself As the temperature of the calorimeter rises during
a reaction produced in it, a parallel rise is produced in the surroundings,
usually by adding sulfunc acid to a solution of sodium hydroxide or by
electric heating A diagram of such a piece of apparatus1 is shown in Fig 45,
which shows the bomb type of calorimeter arranged for burning a volatile
liquid.

FIG 45 — Calorimeter, arranged foi combubtion of a volatile liquid

The^matenal to be burned is placed in a glass receptacle of very thin
walls in a platinum crucible suspended in a heavy steel bomb lined with
gold, which is then filled with oxygen under considerable pressure The
bomb is placed m the calorimeter (the inner vessel of water), and the sub-
stance is burned completely by means of the excess oxygen present. The
heat liberated causes a rise in temperature that is indicated on the ther-
mometer reaching to the inner vessel, and a parallel rise in temperature
of the outside vessel of sodium hydroxide solution is produced by adding
strong sulfuric acid from a burette at the required rate. Since the outer bath
is always kept at the same temperature as the calorimeter within it, there is

1 RICHARDS and BARRY, J. Am. Chem. Soc , 37, 993 (1915).

300 PHYSICAL CHEMISTRY

no exchange of heat between them, and all the heat of reaction is used to
change the temperature of the calorimeter itself The total heat evolved
hv an isothermal change is then the product of temperature change and heat
capacity of the calorimeter system A convenient means of measuring the
heat capacity of a combustion bomb, the water surrounding it, and its con-
tamer is by burning benzoic acid, which evolves 6324 cal per gram (weighed
in air or 6319 cal per gram weighed in vacuo] in the same vessel. Thus all
the data needed for the calculation are at hand The heat evolved per gram
of unknown substance is to 6324 cal as the temperature change produced per
gram of unknown is to the temperature change produced per gram of benzoic
acid in the same apparatus

An outline of the computation arid of the necessary corrections (for
incompletely condensed water in the bomb, the formation of traces of nitric
acid, heat of combustion of the ignition wire, etc ) to the observed temper-
ature rise in a calorimeter is given by Washburn.1

Heat effects for reactions taking place in solutions may be determined in
the same way, a thin platinum vessel containing one solution being substi-
tuted for the bomb The other solution is discharged into this vessel from
a pipette immersed in the calorimeter, in order that the solutions may be at
the same temperature when they are mixed for the reaction.

Heat Capacity and Specific Heat. — The heat absorbed by a
substance during a change in temperature is a quantity that
must frequently be calculated. While it is true in general that
a heat capacity is defined by the relation c = dq/dT, it is neces-
sary to specify the conditions under which the heating occurs
before this relation has an exact meaning. We define the heat
capacity at constant volume by the relation

c -
v \d

and the heat capacity at constant pressure by the relation

_(dE\ + n(dv

~ \ar)p + p \dr

The' specific heat is defined as the quantity of heat absorbed per
gram per degree, and the heat capacity of any quantity other
than a gram is the product of specific heat and mass. All the
data given in this chapter, and in general in the chemical litera-
ture, refer to atomic heat capacity or molal heat capacity. Since
A# for a change in state which involves heating a system at
constant pressure through a range of temperature is JCpdT

1 J. Research Nat. Bur. Standards, 10, 525-558 (1933).

THERMOCHEMISTR Y 301

between the initial and final temperatures, it is necessary to
express Cp as a function of temperature before performing the
integration except for the comparatively few substances of which
the heat capacities do not change with temperature. For
monatomic gases we have already seen on page 81 that the molal
heat capacities are Cv = %R and Cp = %R, both independent
of temperature For any gas that conforms to the ideal gas law
pvm = RT, the relation

Cp — Cv = R

gives the difference between the molal heat capacities, whether
the molecule has one or several atoms.

The molal heat capacities of diatomic gases are higher than
those of monatomic gases, and they increase with rising tempera-
ture As a sufficient approximation for the solution of problems
at the end of the chapter we may take the molal heat capacity
as

Cp = 6.5 + 0.001 T

for 02, N2, H2, CO, HC1, HBr, HI, NO, and any diatomic gas
or mixture of diatomic gases (except the halogens) at any mod-
erate pressure and m the temperature range 300 to 2000°K.
This equation will give the heat absorbed within 2 or 3 per cent;
more accurate heat-capacity equations are given in Table 56.

Some other convenient approximations for use in the problems,
which are intended to illustrate the methods rather than to pro-
vide precise answers, are Cp = 8.5 cal. per mole per deg. for
water vapor below 800°K., Cp = 2.0 + 0.005T for carbon (300
to 1000°K ), Cp = 7.0 + 0.00777 for C02 or S02 in the same
temperature range.

The entries in Table 55 will be useful in calculating the heat
absorbed by some common gases when heated. They show the
heat absorbed upon heating a mole of gas through 100° intervals.
For example, the integral of Bryant 's equation for Cp of carbon
dioxide between 273 and 373°K. is 935 cal., the integral between
273 and 473°K. is 1936 cal., and these are the first two entries in
the column headed H - ff273 for C02 in Table 55. The dif-
ference between these quantities is given under A and is obviously
AH for the interval 373 to 473°K. Linear interpolation is of
course permitted, and the heat absorbed in the interval 273 to

302

PHYSICAL CHEMISTRY

TABLE 55. — INTEGRALS OF HEAT-CAPACITY EQUATIONS FOR GASESI
(In calories per mole from 273°K. at constant pressure)

Temp ,
°K.

CO2

H2O

77 -

Hz7Z

// -

//273

77-

7/27.

77 -
77273

273

693

707

688

935

791

373

693

707

688

935

791

695

728

705

1001

822

473

1388

1435

1393

1936

1613

700

749

722

1064

852

573

2088

2184

2115

3000

2465

703

767

737

1120

880

673

2791

2951

2852

4120

3345

707

784

752

1172

908

773

3498

3735

3604

5292

4253

712

800

766

1220

936

873

4210

4535

4370

6512

5189

718

814

779

1261

963

973

4928

5349

5149

7773

6152

724

826

791

1299

989

1073

5652

6175

5940

9072

7141

731

837

802

1331

1014

1173

6383

7012

6742

10403

8155

738

847

812

1358

1039

1273

7121

7859

7554

11761

9194

745

854

821

1381

1064

1373

7866

8713

8375

13142

10258

754

861

830

1397

1086

1473

8620

9574

9205

14539

11344

763

865

836

1411

1109

1573

9383

10439

10041

15950

12453

772

869

844

1418

1132

1673

10155

11308

10885

17368

13585

782

870

849

1420

1153

1773

10937

12178

11734

18788

14738

793

871

854

1418

1173

1873

11730

13049

12588

20206

15911

804

869

857

1410

1194

1973

12534

13918

13445

21616

17105

815

867

860

1399

1213

2073

13349

14785

14305

23015

18318

828

861

862

1380

1231

2173

14177

15646

15167

24395

19549

1 G. B TAYLOR, Ind. Eng. Chem., 26, 470 (1934), based on the heat-capac-
ity equations of Bryant, ibid , 26, 820 (1933).

THERMOCIIEMISTR Y 303

873° will differ but little from the heat absorbed in the interval
293 to 893°. These data may of course be used in the solution
of problems. The column headed CO may also be used for N2,
the column headed H2O may also be used for H2S, and the
column headed C02 may also be used for SCV

One should not conclude too hastily that apparently different
heat-capacity equations for a given substance are discrepant
when the constants in them are not the same. As a single illus-
tration, we quote four equations for the heat capacity of carbon
dioxide at constant pressure and give after each one its integral
between 400 and 500°K., which is the calculated heat absorption
when a mole of CO2 is heated through this range:1

(1) Cp = 10.34 + 0 0027477 - 1.955 X IW/T2 AH = 1060 cal.

(2) CP = 6 85 + 0.00853 T - 0.00000247772 AH = 1030 cal.

(3) Cp = 0 37 + 0.01 01 T - 0.0000034772 AH = 1020 cal.

(4) Cp = 5.17 + 0.015277 - 0.00000958 772

+ 2.26 X 10-9?73 AH = 1030 cal.

But one must also be prepared to find heat-capacity equations
which do not give the same heat absorption and between which
it is difficult to choose For example, the integral of another
heat-capacity equation for CO2,

Cp = 7 7 + 0.0053 T - 0.00000083 T2

from 400 to 500°K., is 900 cal ; yet this equation at temperatures
above 1200°K. gives the heat absorption for CO2 as well as any
and is probably the best one for high temperatures.2

The data quoted have been for heating at constant pressure.
Since (dH/dp)r is zero for ideal gases and very small for real
gases at low pressures, these equations may be used at any
constant pressure below 3 to 5 atm. unless high precision is
required. Under these conditions pv = nRT will also apply,
and hence heat capacities of gases at constant volume may be
obtained by subtracting R cal. per mole from the constant-

1 The equations are from (1) Gordon and Barnes in Kelley's compilation,
U.S. Bur. Mines Bull , 371, 18 (1934), (2) Bryant, Ind. Eng. Chem , 25, 820
(1933), (3) and (4) Spencer and Flannagan, J. Am. Chem. Soc., 64, 2511
(1942). -

2 EASTMAN, U.8. Bur. Mines Tech. Paper, 445 (1929).

304 PHYSICAL CHEMISTRY

TABLE 56. — SOME HEAT CAPACITIES AT CONSTANT PRESSURE1

Sub-
stance

Molal heat capacity at constant pressure

Per

cent
error

Temperature
range

6 85 + 0. 00028 T + 0 22 X ICT6?72

1.5

300-2500

02, N2

6 76 + 0.00060677 + 0 13 X IQ-'T2

1.5

300-2500

C02

7 70 + 0 005377 - 0 83 X W~«T2

2.5

300-2500

NH8

67+0 006371

1.5

300- 800

H20

8 22 + 0 0001577 + 1 34 X 1Q-«T2

1 5

300-2500

H2S

72+0 003677

5-10

300- 600

S02

7 70 + 0 005377 - 0.83 X 10-6T2

2 5

300-2500

C12

8 28 + 0 0005677

1 5

27&-2000

2 673 + 0 0026277 - 1.17 X 106/^2

273-1373

HC1

6 70 + 0 0008477

1 5

273-2000

HBr

6 80 + 0 0008477

273-2000

pressure equations; for example, Cv = 4 5 + 0.001 T is a suitable
approximation for the diatomic gases.

No general expressions are known for the heat capacities of
liquids; they are usually larger than those for the corresponding
solids.

The heat capacities of most of the solid elements approach
about 6 cal. per atomic weight per degree near room tempera-
ture; they fall off rapidly at lower temperatures in a way that is
not expressible by a simple equation such as that used for gases,
as shown in Fig. 15 on page 152 Above ordinary temperatures
the atomic heats of most solid elements increase slightly.

• The heat capacities of solid compounds are roughly equal to
the sum of the heat capacities of the elements in them (Kopp's
law). Thus the molecular heat capacity of lead iodide is about
equal to that of an atomic weight of lead plus that of two atomic
weights of iodine, or about 18.6 cal. per mole per deg.; but large
deviations from this "law" are so common as to make it of
little value except as a rough guide when data are unavailable.

Kelly, U.S. Bur. Mines Bull, 371 (1934), who gives a critical
review of the heat capacities of inorganic substances together with equa-
tions expressing the "best values" as functions of the temperature These
equations are in the conventional form Cp = a + bT + cTz and also in the
form Cp = a + bT — c/T* for some hundreds of substances Equations
for 59 gases, in both these forms, are given by Spencer and Flannagan in J
Am. Chem. Soc., 64, 2511 (1942).

THERMOCHEMISTRY 305

An aqueous solution usually has a heat capacity less than that
to be expected from a mixture rule, and for calculations involving
temperature changes in solutions it is necessary to measure
heat capacities experimentally.1 In approximate calculations
a fair assumption is that the heat capacity of a solution
is equal to that of the water it contains. For example, one
may assume that a 10 per cent aqueous solution has a heat
capacity of 0.90 cal. per gram per deg., a 20 per cent solution
0.8 cal. per gram per deg., etc. In general, the actual heat
capacities are even less than such estimates; for example, the
heat capacity of a 10 per cent solution of MgBr2 is 0.79 cal. per
gram per deg. The effective heat capacity of dissolved KC1 is
shown in the following table, in which m is the moles of KC1
added to 1000 grams of water and Cp is the heat capacity of the
resulting solution. It will be seen that the heat capacities of
solutions must be measured rather than estimated, since the
addition of KC1 to water decreases the heat capacity of the
solution to less than that of the water alone.2

m 0 55 1.11 2 22 3 33 4 44

Cp 986 975 968 968 966

AC/m -26 -23 -15 -10 -8

Changes in State of Aggregation. — Heats of evaporation have
already been considered in Chap. IV. The heat absorbed in
small calories per mole of liquid evaporated at constant pressure
is approximately 22 times the absolute boiling point (Trouton's
rule) for many liquids, but large deviations from this rule are
often found, and recourse to experiment is necessary when
reliable data are required. Some latent heats of evaporation
at atmospheric pressure are given in Table 16.

No general rule similar to Trouton's rule is applicable to latent
heats t)f fusion. The ratio AHf/T for a mole of substance varies

1 Data are recorded in the " International Critical Tables," Vol. V, p. 122,
and by Rossini in J Research Nat. Bur Standards, 4, 313 (1930).

2 In more dilute solutions the " partial molal heat capacity" approaches
a definite limit. The heat capacities, in calories per gram of solution at 25°,
for KC1 and NaCl are given by Hess and Gramkee in J. Phys. Chem., 44,
483 (1940), as follows:

m 0 010 0.050 0 070 0 100 0 300 0 700 1 03

Cp(KCl) 0.9968 0.9929 0.9908 0 9881 0.9695 0.9342 0.9090

cp(NaCl) . , . 0.9971 0,9943 0.9928 0.9903 0.9762 0,9501 0,9319

306 PHYSICAL CHEMISTRY

from 1.6 to 18.2; it has no constant value that may be used in
estimating heats of fusion. Some molal latent heats of fusion
are given in Table 21. 1 Many of the heats of fusion given in
tables are derived from freezing points of solutions through the
equations on page 215, which is permissible, of course, if the
data are reliable. Unfortunately, not all the freezing-point data
represent true equilibrium between a solution and the crystalline
solvent, and therefore not all the recorded heats of fusion from
this source are reliable. For example, the molal heat of fusion
of bromine is 2580 cal. by direct calorimetry; and two values
said to be based on the freezing-point constant are 2380 and
2780. Even wider variations are not uncommon.

Transitions from one crystalline form to another also absorb
small quantities of heat, for example, Srhoni = Smonoci; A//368 = 95
cal. and Cd,am = CBraph; A//298 = —454 cal , of which the first has
been measured both directly and by several indirect methods and
the second is the difference between the heats absorbed by the
combustion of diamond and /3-graphite.

Heat Absorbed by Reactions at Constant Pressure and at Con-
stant Volume. — Two methods of procedure are followed m calori-
metric work, and it is convenient to correct the values obtained
by one procedure to those which would have been obtained had
the other procedure been employed. Thus when iron is dissolved
in acid in an open vessel, the hydrogen formed must force back
the atmosphere to make room for itself, thus doing work. If
the reaction had been carried out in a closed bomb, a pressure of
hydrogen would have been built up and no work would have
been performed. The work done is p(vz — Vi) in the first process
in which hydrogen was evolved at 1 atm. and is zero for the con-
stant-volume process. An amount of heat equivalent to this
work is absorbed in the constant-pressure process but not in the
constant-volume process. The heat absorbed during the reac-
tion at 1 atm. pressure is A//; but since no work is done by the
reaction that takes place at constant volume, the heat absorbed
is AE. By definition these quantities differ from one another
by A(TW), that is,

AH = AE + p(v2 - vi) (4)

1 For the best compilation of heats of fusion, see Kelley, U S. Bur. Mines
Bull, 393 (1936).

THERMOCHEMISTH Y 307

In this expression v2 is the volume of a mole of hydrogen plus that
of a mole of dissolved ferrous chloride, and Vi is the volume oi the
iron and acid from which it was formed. There is only a slight
change in the volume of the solution, and the volume of the
iron may be neglected in comparison with that of the gas. The
work term then becomes practically pv^ which from the ideal
gas equation is RT. Since the value of R is 1.99 cal., the correc-
tion term is at once available in calories, and the difference
between AH and AE for this reaction at 20° is 1.99 X 293 = 580
cal. of heat absorbed per mole of gas generated. This should bo
rounded to 600 cal , since otherwise upon addition we should
write down a larger number of significant figures than the experi-
mental work justifies.

For reactions involving only solids and liquids, the difference
between heats of reaction at constant volume and at constant
pressure usually need not be taken into account. For reactions
in which gases are involved, the increase in volume is due to the
increase in the number of motes of gas during the reaction. In
general,

AH = AE + AnRT (5)

where An is the increase that takes place in the number of moles
of gas when the reaction occurs. For the combustion of methane
at 20°, for example, the change in state may be written

CH4(0) + 202(<7) = C02(0) + 2IIaO(Z)
A//291 = - 212.79 kcal

Since An = —2, AnRT = —1.17 kcal., and AE for this change
instate is -211.62 kcal.

All the data quoted in this chapter are for constant pressure.

Melal Enthalpy of Combustion (Heat of Combustion). —
Enthalpy increases for combustion have been determined pre-
cisely for almost all combustible substances, and they are used
to calculate enthalpies of formation, as will be explained in the
next section. Some illustrations are quoted in Table 57, and
many others are known.1

1 Data for about 1500 substances are given by Kharasch in /. Research
Nat Bur. Standards, 2, 359 (1929); see also " International Critical Tables/'
Vol. V, pp. 163-169.

308

PHYSICAL CHEMISTRY

The combustion of organic compounds severs C — C, C — H,
C=C, and C=C bonds and changes C — O to C=O, and to
each of these changes a fixed enthalpy increase may be assigned.
These assigned quantities enable one to estimate the enthalpy
of combustion when data are lacking and are hence useful
approximations. The assigned enthalpies of combustion are1

TABLE 57. — MOLAL ENTHALPY OF COMBUSTION

[In kilogram-calories absorbed per mole of substance oxidized to CQs(g) and
H2O(Z) at 25° and constant pressure]

Substance

A/7 298

Substance

A//298

Methane (g)

- 212 79

Methyl alcohol (/)

- 173 64

Ethane (g)

- 372 81

Ethyl alcohol (/)

- 326 66

Propane (g)

- 530 57

n-Propyl alcohol (/)

- 482 15

n-Butane (g)

- 687 94

n-Butyl alcohol (I)

- 638 10

n-Pentane (gr)

- 845 3

iso-Butyl alcohol (I)

- 638 2

Acetylene (g)

- 310 5

Benzoic acid (s)

- 771 85

Ethylene (g)

- 337 3

Salicylic acid (&)

- 722 0

Naphthalene (&)

-1231 0

Formic acid (I)

- 63 0

Benzene (/)

- 781 0

Acetic acid (7)

- 206 7

Toluene (I)

- 934 6

Sucrose (&)

-1349 6

- 52.25 kcal. for each C— II bond, -53. 72 for each C— C, -121.8
for each C=C, -203.2 for each C=C, and -15.0 for each
C — O bond in the compound. For illustration, ethane contains
six C — H bonds and one C — C bond, whence AH for its combus-
tion is estimated to be —367.3 kcal., compared with —372.81
by experiment. For saturated hydrocarbons2 above C&Hi2 the
addition of each CH2 group increases the molal AH of combus-
tion by —157.0 kcal. For all saturated hydrocarbons the molal
enthalpy of combustion is almost —52.7 kcal. for each atomic
weight of oxygen used;3 this approximation gives —368.9 kcal.
for ethane. As is always true, experimental data are better than
approximations, but data are not always available.

Molal Enthalpy of Formation (Heat of Formation). — The
enthalpy increase for the formation of compounds that may be
synthesized in pure form from the elements is determined directly
in a calorimeter. For example,

1 SWIETOSLAWSKI, J. Am. Chem. Soc., 24, 1312 (1920).

2 ROSSINI, Ind Eng. Chem., 12, 1424 (1937).
8 THORNTON, Phil. Mag., 33, 196 (1917).

THERMOCHEMISTRY 309

(g) = HCl(flf) A#298 = -22.06 kcal.
S(«) + 0,(0) = S02(0) A#298 = -70.94 kcal.

Data for compounds that are not readily formed in a calorim-
eter may be obtained through the fact that AH for a specified
change in state is the same by all paths. It is necessary only
to devise paths by which the desired change may be brought
about, one of which includes the reaction whose enthalpy increase
is desired. As an illustration, suppose AH were required for the
synthesis of benzoic acid, which may not be made directly from
the elements by procedures adapted to calorimetry. One path,
by which benzoic acid is formed and then burned to CO2 and
H2O, is shown by the equations

7C(s) + 3H2(0) + 02(0) = C6H5COOH(s) A//291 = x kcal.

C6H&COOH(s) + 7^02(g) = 7C02(g) + 3H,0(0

A7/291 = -771.85 kcal.

The sum of these equations gives another path by which 7C02
and 3H20 may form from the elements,

3H2(</) + 7C(«) + 8HO,(flf) = 7C02(<7) + 3H,O(Z)
A//291 = -8(53.17 kcal.

and for which AH is 7(-94.03) + 3(-68.32) = -863.17 kcal.,
from Table 58. Since the enthalpy increase —771.85 + x must
be equal to -863.17, it follows that A//29i is -91.32 kcal. for the
formation of solid benzoic acid from its elements.

The enthalpy of benzoic acid determined in the preceding
paragraph is probably reliable to within ±0.5 kcal., for its heat
of combustion is a calorimetric standard that has been meas-
ured with care. Less reliable data and smaller differences
between large quantities often yield small enthalpies of high
percentage error. For example, the enthalpy of liquid toluene
may be calculated from the following equations:

7C(«) + 4H2(0) = C6HBCH8(/) AH = x cal.
C,H5CH,(Z) + 902(<7) = 7C02(0) + 4H2O(Z)
_ AH = -934.6 kcal.
7C(«) + 4H2(<7) + 902(<7) = 7C02(0) + 4H20(/)

AH = -931. 5 kcal.

310 PHYSICAL CHEMISTRY

Thus the enthalpy of liquid toluene is 3.1 ± 0.5 kcal., since at
least this error is possible in the heat of combustion (though not
in the data for C02 and H2O) and a larger error is not excluded.
Since the heat of evaporation of toluene is 8.0 kcal., the enthalpy
of CeHsCHs^) is 11.1 kcal., and this may be used to calculate
for a reaction such as

*-C7H16(<7) = C6H5CH3(<7) + 4H2(0)

to be 59.4 kcal. with an uncertainty of less than 1 kcal. by a
method that is explained in the next section.

Molal Enthalpy of Compounds (Molal Heat Contents). — It has
become customary to define the molal enthalpy of an elementary
substance in its stable state at 1 atm. pressure and a standard
temperature as zero. Under this convention the enthalpy
increase for the formation of a compound from its elements
becomes its molal enthalpy. For example, the formation of
Na2S from its elements at 18° is shown by the equation

2Na(s) + S(s) = Na2S(s) A#29i = -89.8 kcal.

and since the enthalpies of the elements are defined as zero, the
molal enthalpy, or molal heat content, of Na2S is —89.8 kcal.

This definition of enthalpies of elements as zero is an arbitrary
one, and thus the enthalpies of the compounds are relative to
this standard. If the enthalpy of solid sodium is taken as zero
at 18°, its enthalpy at 25° is obviously not zero relative to this
standard. The point is that, if sodium and sulfur are zero at
18°, Na2S is —89.8 kcal. at 18°; if sodium and suliur are zero at
25°, Na2S is -89.8 at 25°. (The change in heat capacity
attending the reaction, which is neglected in this calculation and
which we shall consider~later in the chapter, would influence the
third figure after the decimal point in this calculation and is
outside of the precision of the data.)

Enthalpy tables sometimes contain entries such as

Br2(g) = 7.4 kcal.

If the enthalpy of liquid bromine is zero at 25° and the latent
heat of evaporation to form saturated vapor at 25° and 0.28 atm.
is 7.4 kcal., the enthalpy of the vapor is thus 7.4 kcal. at 25° and
0.28 atm. For the imaginary state of bromine vapor at 25° and
1 atm. the same figure is used, since (dH/dp)T is substantially

THERMOCHEMISTR Y 311

zero for gases at moderate pressures. The value I2(0) = 14.88
kcal. is similarly obtained from the heat of sublimation of the
solid.

If we denote the enthalpy of any chemical system in state 1
by Hi, which is the sum of the enthalpies of all the substances
in the system, and the enthalpy of the system in state 2 by Hz,
the change in enthalpy attending the change from state 1 to
state 2 is the difference between HI and #2, which may be
written

Hi + A// = H* or Hi = #2 - A# (6)

We may thus calculate A# for any isothermal chemical reaction
when the molal enthalpies of the reacting substances and reaction
products are known, whether or not the reaction is adapted to
calorimetry. The recorded enthalpies are for the formation of
compounds at 1 atm. in their stable states at the standard
temperature from the elements in their stable states at 1 atm.
and the standard temperature.

Unfortunately the standard temperature selected by various
writers is not the same,1 but it is usually 18° or 25°. When the
enthalpies of compounds at 18° are referred to the elements at
18°, the quantities are almost equal to those for compounds
at 25° referred to the elements at 25°; and the differences are
usually less than the experimental errors in the fundamental
data. An exception to this statement is required for reactions
involving the formation or use of ions; for in these reactions AJEf
is often small, and the change with temperature is usually large.
The actual calculation, with the underlying theory, will be given
later in the chapter, but we may note here for illustration

1 Bichowsky and Rossini use 18° in their " Thermochemistry of Chemical
Substances," and record (?/, which is our —AH, for the formation, transition,
fusion, and evaporation of all substances for which data exist (5840 values of
Qf for formation) except for organic compounds containing more than two
atoms of carbon. The data in " International Critical Tables," Vol. V, are
for 18°. Lewis and Randall choose 25° as the standard temperature in
their " Thermodynamics," McGraw-Hill Book Company, Inc., New York,
1923. They record also molal free energies at this temperature, since it is
desirable to have both quantities at the same temperature to avoid laborious
corrections. Latimer;s extensive compilation of data in "Oxidation Poten-
tials," Prentice-Hall, Inc., New York, 1938, also uses 25° as the standard
temperature for molal enthalpies and molal free energies.

312 PHYSICAL CHEMISTRY

g = NH4+ 4g + OH~ Aq A#293 = 1070 cal.
g = LNH4+.A$ + OH- Aq A#298 = 865 cal.

The data for formic acid give an even more striking illustration.

HCOOH.Ag = H+.Ag + COOH~.Aq A#293 = +192 cal.
HCOOH.^ig = H+.Aq + COOTS'. Aq A#298 = - 13 cal.

With the exception of such reactions, the enthalpy data for 18°
may be used at 18°, 20°, 25°, or 27° (= 300°K.) without intro-
ducing errors that exceed the discrepancies in the available data.
Some molal enthalpies are given in Table 58 for use in the
problems at the end of the chapter. We may illustrate the use
of such data by calculating AH for the reaction at 25°.

Na2S(s) + 2HC%) = 2NaCl(s) + H2S(0)

The enthalpy of the system in its first state is -89.8 + 2(- 22.06)
kcal, whence H± = -133.92 kcal.; and #2 is 2(- 98.36) -4.8, or
— 201.52. Since AH = HZ — Hi, its value for the reaction is
A#298 = —67.60 kcal. A simple procedure for carrying out such
a calculation, applicable to any reaction, is to write under
each chemical formula in an equation the molal enthalpy of that
substance, multiplied by the coefficient for the substance in the
equation, as follows:

Na2S(s) + 2HC%) = 2NaCl(s) + H2S(0)
-89.8 + 2(- 22.06) = 2(- 98.36) + -4.8 -AH
Hi = Hz -AH

It will be seen that we may substitute the molal enthalpy of a
chemical substance for its formula in a chemical equation and
by adding — AH as a term at the end of the equation obtain an
equation which is readily solved for AH. Thus the sign of the
quantity and its numerical value are obtained by a procedure
that is not likely to cause an error.

This same result might have been obtained, somewhat more
laboriously, by adding together in an appropriate way the
expressions for the individual molal enthalpies, as follows:

Na2S = 2Na + S AH = 89.8 kcal.

2HC1 = H2 + Clt AH = 44.12 kcal.

2Na + Cla = 2NaCl AH = -196.72 kcal.

H2 + S = H2S _ AH = -4.8 kcal.

Na2S + 2HC1 = H2S + 2NaCl AH = -67.60 kcal,

THERMOCHEMIS TRY 313

This calculation has been made for a change in state involving
only the solids and gases. Since all these substances dissolve
in water with appreciable enthalpy changes, AH for this reaction
in water would not be —67.60 kcal., but another value. In
order to calculate AH for the reaction in solution we require
the molal enthalpies of the substances as solutes. This calcu-
lation will be given later in the chapter, after we have discussed
enthalpy changes attending solution in water.

A word of explanation regarding quantities such as

H2C%) = -57.82

and NH3.4<7 or NH4OH.w4g may be helpful. The enthalpy for
water vapor comes from —68 32, which is the enthalpy of liquid
water at 25° and 1 atm. referred to hydrogen and oxygen, and
from AH = 10.50 kcal. for the evaporation of water to form
saturated vapor at 25° and 0.03 atm., which would give —57.82
kcal for the enthalpy of water vapor at 0.03 atm. For the
compression to the unstable condition of vapor at 25° and 1 atm
we assume A# = 0, since (dH/dp)T = 0 for the compression of
an ideal gas. There is no implication that water vapor has been
observed at 25° and 1 atm. pressure; we accept the value as a
convenience in making calculations that involve water vapor
Since AH is independent of path, the same result is obtained
with less labor than is required for calculations along the actual
path. This will be clear if the student will calculate A// for the
two following paths, which accomplish the same net change in
state :

(1) H2O(/, 25°, 1 atm.) = H20(Z, 100°, 1 atm.)

= H2O(g, 100°, 1 atm.)
and .

(2) H20(/, 25°, 1 atm.) = H2O(0, 25°, 0.03 atm.)

= H2O(0, 100°, 0.03 atm.) = H2O(0, 100°, 1 atm.)

The difference between NH3.^(? and NH^OH.Aq will of course
be —68.32 kcal., which is the standard enthalpy of a mole of
liquid wa'ter. We have no information as to what fraction of
the dissolved ammonia is NH3 and what fraction is hydrated
ammonia (NH4OH); and since AH must be the same quantity
whether a neutralization equation, for example, is written

314 PHYSICAL CHEMISTRY

NH4OH.ylg + tt+.Aq

it is evident that —68.32 kcal. must be the difference between
NH3.^4g and NH4OH.^4g. The same statement applies, of
course, to COz.Aq and H2C03.Ag or to SO^.Aq and H2S03.^4g.
Since these differences are conventional ones to serve the purpose
illustrated, it will be evident that the quantities may not be
used to show that Aff = 0 for "reactions" such as NH3.Ag +
H20(7) = NB^OH.Ag, for we have no evidence that this is a
reaction. *

Enthalpy of Solution (Heat of Solution). — A few substances
dissolve in water with the absorption of heat, but the majority
of solids, liquids, and gases dissolve with the evolution of heat.
The molal enthalpj^ change upon solution varies with the tem-
perature and the quantity of solvent per mole of solute, up to a
certain limit characteristic of the solute. For example, AT/ for
the solution of 1 HC1(0) in water at 18° varies with N, the moles
of solvent water as foljows:1

NAq 1 2 3 5 10 20

A//, kcal -6 24 -11 5 -13 37 -15 00 -16 29 -16 86

NAq 50 100 200 500 1000 Limit

A//, kcal . -17 20 -17 32 -17 41 -17 50 -17 52 -17.63

We should thus write -22.06 - 11.5, or -33.56, kcal. for
the molal enthalpy of HCl2Aq and -39.26 for UClSOAq or
HC1 (1m.). A solution of a mole of HC1 in so much water that
the addition of more water caused no appreciable heat effect
will be written HCl.^lg in thermochemical equations to indicate
indefinite dilution. This is sometimes written HC1. oo Aq} but
the notation HCl.Aq is preferred, since there are many prop-
erties of HC1 that change at lower concentrations and therefore
the solution is not " infinitely" dilute except in the thermo-
chemical sense. We use the symbol Aq, as in HC1.25^4(?, to
denote 25 moles of solvent water and reserve the formula H2O

1 Data for the dilution of HC1 at lower molalities are given by Sturtevant,
/. Am. Chem. Soc.y 62, 584 (1940). His heats of dilution are in agreement
with those above.

THERMOCHEMISTR Y 315

TABLE 58 — MOLAL ENTHALPIES OF COMPOUNDS1 AT 298°K.

Substance

A//298

Substance

AH29g

Substance

A//298

HC1(0)

- 22 06

NOC1(<7)

12 8

KClOi(s)

- 91.33

HBr(?)

- 8 65

NOBr(0)

17 7

KC104(s)

-112 71

Hlfo)

5 91

S02(0)

- 70 94

AgCl«

- 30 40

H20(</)

- 57 82

S02 Aq

- 77 20

AgBr(s)

- 23 81

H20(0

- 68 32

S0,(«)

- 105 2

Agl(«)

- 15 17

H2S(0)

- 4 80

S2C12(Z)

- 14 3

CuCl(s)

- 34 3

H2S Aq

- 9 27

Ag20(s)

- 7 30

CuCl2(«)

- 53 4

H2S04(Z)

-193 75

CaO(s)

- 152 2

NH4Cl(s)

- 75 20

HN03(0

- 41 66

PbO(s)

- 52 4

Hg2Cl2(s)

- 63 15

CS2(/)

15 4

PbO2(s)

- 65 9

PbCl2(s)

- 85 71

CC14(/)

- 33 8

HgO«

- 21 6

A1C1,(8)

-166 8

CH4(<7)

- 17 87

Cu20(s)

- 42 5

MgO(s)

-143 84

02H2(<?)

54 23

CuO(s)

- 38 5

Mg(OH)2(s)

-221 48

C,H4(0)

12 56

ZnO(s)

- 83 17

MgC03(s)

-265 4

CjHeQr)

- 20 19

Al203(s)

- 399 0

H+Aq

C3Hr,(0)

4 96

Ca(OH2)(s)

- 236 1

Na+ Aq

- 57 48

C,H8fo)

- 24 75

CaC03(s)

- 288 6

K+Aq

- 60.10

00(0)

- 26 39

NaCl(s)

- 98 36

NH4+Ag

- 31 71

C02(0)

- 94 03

NaOH(s)

- 101 96

Ag+Ag

+ 25 29

CH3OH(/)

- 57 45

Na2S(s)

- 89 &

Ca++ Aq

-129 87

C2H5OH(0)

- 56 95

Na2S04(s)

- 330 20

Mg++ Aq

-111 52

C2H6OH(/)

- 67 14

Na2S04-

Zn++.Aq

- 36 43

C2H6OH^9

- 69 82

10H20(s)

-1032 78

Cl- Aq

- 39 94

CH8COOH(0

-117 7

NaNOa(«)

- Ill 60

Er-.Aq

- 28 83

CH3COOH.Ag

-118 06

Na2C03(s)

- 270 97

l-Aq

- 13 61

H2CO,40

-167 06

NaHCO,(8)

- 226 97

OR-.Aq

- 54 95

NH,(?)

- 10 93

KCl(s)

- 104 19

RCOr-Aq

-165 22

NH4OH.4g

- 87 53

KOH(,s)

- 102 02

NOr.Atf

- 25 60

NOfo)

21 53

KI(8)

- 78 87

NO,- Aq

- 49 32

N02(0)

7 96

K2S(s)

- 121 5

S04— Aq

-215 8

NzOsto)

0 6

K2S04(s)

- 341 68

C03" Aq

-161 72

1 Additional data, in kilogram-calories evolved per mole of substance at 18°,
will be found in Bichowsky and Rossini, " Thermochemistry of Chemical
Substances/' where over 5000 entries are given, g = gas, I — liquid,
s = solid, and Aq — dilute solution Ionic enthalpies are all based on the
arbitrary assignment of zero to H+ Ag, and therefore Cl~ Aq has the same
assigned enthalpy as H+C1~. Aq We shall see in later chapters that hydrogen
ion has other assigned values of zero, such as its free energy. Since in all
calculations it is the difference in enthalpy between products and reacting
substances that is useful, zero for H+ is as good as any other quantity.

316 PHYSICAL CHEMISTRY

for water that is a reacting substance or reaction product, as in
the thermochemical equation

= Na+CJ- 5(Ug + H2O
A//298 = - 14.00 kcal.

As enthalpy equations, solution and dilution may be written

HC%) + 5Aq = TLCLdAq A#291 = -15.0 kcal.
HC%) + Aq = RCLAq A#29i = -17.03 kcal.

The difference between these two equations is obviously the
enthalpy of dilution of HCl.SAq to its limit, and it would be
written

HCl5Aq + Aq = HCl.Aq A7/291 - -2630 cal.

Since the symbol Aq without a figure attached means so large
an amount of solvent water that the addition of more water pro-
duces no heat effect, this last equation and that on page 314 for
neutralization are " balanced, " even though there appears to be
an excess of water on the left side. The solution indicated by
HCl.^lg is "indefinitely" dilute rather than "infinitely" dilute.
For dilutions with smaller quantities of water, the equations
may be written

HC1.20Ag + SQAq = HCl.lOOAg A//29i = -460 cal

Partial Molal Enthalpy of Solution or Dilution. — By plotting
AH per mole of solute against the moles of water added to 1
mole of a solute S, one obtains curves that usually rise steeply
at first and become horizontal for such large quantities of water
that the heat of further dilution is negligible. The tangent to
such a curve at TV = 5 moles of water, for example, is the "partial
molal AH of dilution" of a solution of composition S.SAq. It is
the heat absorbed upon adding a mole of water to so large a
quantity of solution that there is a negligible change in compo-
sition of the solution, and it is usually written dH/dNAq. There
is also a partial molal enthalpy for solution of the solute in a
solution, which is the heat absorption attending the solution of
a mole of solute in so large a quantity of &.5Aq that the change in
composition of the solution is negligible. These partial molal
heat quantities are important ones in many chemical calcula-
tions, such as the temperature coefficients of electromotive force

THERMOCHEMISTR Y 317

in cells that form a solute into a solution, which we shall con-
sider in Chap. XIX.

The relation of one to the other is shown by the following two
procedures for introducing 1 mole of S and 5 moles of water into
a large amount of solution of composition S.5Aq.:

1. Mix IS with 5Ag, cool (or heat) to the original temperature,
and add this solution of S.5Ag to the main body of S.5Ag. The
enthalpy increase for the first step is the " integral heat of
solution"

S + 5Ag = S.5Ag A// = a cal

For the second step, by which a solution is mixed with more
solution of the same composition, AH = 0.

2. Add a mole of S to the main quantity of solution; then
add 5Ag to this solution, for which the enthalpy increases are
(dH/dNs) and 5(dII/dNAq), respectively. Since the sum of
these steps produces the same net change as those in the first
procedure, the relation between the quantities is

dH\ . _

r — J = a

For example, when 1HC1 is added to 5Ag at 18°, AH is - 15,000
cal , but the heat effect upon adding 1HC1 to a large quantity of
HCl.SAg has another value. The slope of the curve obtained
by plotting the data for the dilution of HC1 on page 314 is
(d//)/(dNH2o) = -440 cal. at N = 5. The partial molal heat
of solution of HC1 is then obtained through the above equation
from these quantities, since

-5(-440) = -15,000

cWH

and (dH)/(dNHCi) is -12,800 cal. At N = 40Ag, (dH/dNAq)
is only —9 cal , and (dH/dNnci) is —16,780, since the integral
heat of solution, which is a in the above equation, is —17,140
cal. when HC1 dissolves in 40Ag

Reactions in Solution. — The method of calculating AH for
reactions in solution is the same as that for other reactions; one
writes the chemical reaction and under each chemical formula
the enthalpy of the solute ion or molecule or liquid or solid, then

318 PHYSICAL CHEMISTRY

uses the relation Hi = Hz — AH as before. As an illustration
we may calculate AH for the reaction of sodium sulfide with
dilute hydrochloric acid, using the data in Table 58

Na2S(s) + 2H+C1- Aq =
-89.8 + 2(- 39.94) = 2(- 97.42) - 9.27 - AH
Hi = Hi -AH

whence AH = —34.43 kcal. It will be recalled that for the
reaction involving only gases and solids (page 312) AH was
-67.60 kcal.

Enthalpy of Neutralization (Heat of Neutralization). — Neutral-
ization of a "dilute" highly iontzed acid by a " dilute" highly
ionized base causes an enthalpy increase of —13,610 cal. at 20°,
almost independent of the nature of the acid and base. The
chemical effect common to all such neutralizations, which is
substantially the only change responsible for the heat effect
observed, is the union of hydrogen ion and hydroxyl ion to form
water. This effect in a "dilute" solution may be shown by
thermochemical equations of the usual form:1

II+.Aq + OH-.A0 = H20 A7/293 = -13,610 cal.
H+.Aq + Oft-.Aq = H20 A//298 = -13,360 cal.

These values apply only for dilute solutions of acid and base
and only when both acid and base are highly ionized in solution.
Because of the different heats of dilution of acid, base, and salt,
AH for neutralization will have a different value for each molality
of salt formed when the concentrations are moderate or high.
For illustration, we quote the variation of AH2g& for the neutral-
ization of sodium hydroxide with hydrochloric acid of equal
strength as a function of the molality of the salt solution formed:2

m(NaCl) 05 1.0 2.0 3.0 40 50 60

-AH.. 13,750 14,000 14,600 15,500 16,500 17,700 18,950

When moderately dilute solutions of slightly ionized acids are
neutralized by dilute highly ionized bases (or when the acid is
highly ionized and the base slightly ionized), enthalpy increases
are observed that differ materially from —13,610 cal. per mole
of acid at 20° or — 13,360 cal. at 25°, since under these circum-

1 The data in this section are quoted from Pitzer, ibid., 69, 2365 (1937).
2KEGLES, ibid., 62, 3230 (1940).

THERMOCHEMISTR Y 319

stances the formation of water from its ions is not the only
thermal process attending neutralization. For example, the
neutralization of boric acid may be imagined to take place in
two steps as follows:

q = H+.Aq + BO2~ Aq

Aff298 = 3360 cal.

H+.Aq + OH-.Aq = H2Q _ Aff298 = - 13,360 cal.

^g + Na+OH- Aq = Na+B02- 4g + H20

A#298 = - 10,000 cal.

The experimentally determined quantities are —10,000 cal. for
neutralizing boric acid and — 13,360 cal. for the union of hydrogen
and hydrogen ions; and since AH is independent of path, the
heat absorbed by ionization is determined by difference. This
could not be determined by dilution of the acid solution with
water, since ionization is far from complete at any dilution for
which calorimetry is possible.

Enthalpy of Ionization. — Neutralization experiments such as
the one just given are not the best method of determining
enthalpies of ionization, since they are the differences between
comparatively large quantities. An attempt to determine AH
by this method for the reaction

CH3COOH.Ag = H+ Aq + CH,COO- Ag Atf 298 = - 112 cal.

would yield a value of little precision, and therefore AH for the
union of the ions is measured instead. When a dilute solution
of sodium acetate is mixed with a slight excess of dilute hydro-
chloric acid, the reaction is

Ag + H+Cl-.Ag = Na+Q-.Ag +
AH = 112 cal.

and thus AH for the ionization has the same value and the
opposite sign. Some other enthalpies of ionization at 25° are
— 691 cal. for butyric acid, 2075 cal. for carbonic acid, 3600 for
bicarbonate ion, —13 cal. for formic acid, —168 cal. for propionic
acid, and 4000 cal. for sulfurous acid.1

The quantities A#298 = -13,610 cal. and AHW = -13,360
cal. do not apply to the formation of water from its ions at other

1 PITZEB, ibid., 59, 2365 (1937).

320

PHYSICAL CHEMISTRY

temperatures, for the change of AH with temperature is excep-
tionally high for this reaction. The general method of calcu-
lating AH as a function of the temperature is given later in this
chapter, but we give here the final result,

R+.Aq + OH-.Aq = H20 AH = -28,260 + 5071

This equation is valid only in the range 273 to 313°K , since the
data from which it was derived lie in this range. Substituting
T = 373 into the equation, one obtains A//373 = 9610 cal , but
the result is unreliable and should at most be accepted as an
indication that AH is about 9 or 10 kcal. at 373°K.

Change of Enthalpy with Temperature. — When AH is required
at some single temperature other than that for which standard
enthalpy tables are available, it may be calculated by specifying
two paths for producing the same change in state and equating
the summation of AH for the two paths, selecting one of them so
that it involves the desired isothermal change at the desired
temperature. A convenient procedure is to combine two iso-
thermal steps for producing the chemical change with two con-
stant-composition steps involving *the temperature change, as
illustrated in the following diagram :

C0(0) + ^02(gr)A#4 = A#i + AHZ - AH, C02(0)
1473°K >1473°K.

AH, =

13,780 cal.

AH* =

C0(g)
298°K.

= -67,640 cal.

14,350 cal

C02(<7)
298°K

In this scheme the value of AHi is obtained from Table 58, and
AH* and AH$ have been obtained from Table 55, though they
could also have been calculated by integrating the heat-capacity
equations through the temperature range. Upon making the
summation indicated for AH MS, we find —67,070 cal. from these
data. The final AH should be rounded to Allies = —67.0 kcal ,
since the uncertainties in the basic data may exceed 0, 1 kcal. in
almost any such calculation.

THERMOCHEMISTR Y 321

When it is desired to express AH as a function of T for use
in some other equation or when many calculations are to be made
on the same change in state, a more general procedure is con-
venient. Consider any change in state in a homogene-
ous system for which the enthalpy increase is AH at T and
AH + d AH at T + d T. The enthalpy increase may be obtained
as a function of the temperature by equating S AH for two paths
by which the system passes from state 1 at T to a second state
2 at T + dT If the change in state occurs isothermally at T
and the products of the reaction are heated to T + dT at con-
stant pressure, the enthalpy increase is AH for the first step and
C3,2 dT for the second step, where Cpt is the heat capacity of the
system in state 2. If the reacting substances are heated to
T + dT and the change in state then occurs isothermally at
this temperature, the enthalpy change is CPl dT + (AH + d A7/),
where CPI is the heat capacity of the system in state 1. Upon
equating the enthalpy changes for these two paths producing the
same net change in state, we have

AH + Cp.2 dT = CPl dT + AH + dAH

d AH = (CP2 - CPl)dT = ACP dT (7)

Since heat capacities are usually functions of the temperature,
it is necessary to express them in powers of T before integrating
equation (7).

We may illustrate the use of this general equation by the
combustion of carbon monoxide,

+ H02(0) = C02(g) A#298 = -67,640 cal.

The heat capacity of a mole of CO and 0.5 mole of oxygen at
constant pressure is found from Table 56, namely,

CPl = 10.14 + 0.000977 + 0.19 X 10-6!F2
and the heat capacity of a mole of C02 at constant pressure is
CPl = 7.70 + 0.005377 - 0.83 X

322 PHYSICAL CHEMISTRY

whence ACP for this reaction is obtained by subtracting the first
of these equations from the second. Then we have

= (-244 + 0.004477 - 1.02 X
A# = -2.44T7 + 0.0022772 - 0.34 X lO-T3 + AH0

The integration constant, which is usually written A#0, is shown
to be —67,100 cal. from the A#298 value given above; therefore,
the complete expression is

A# = -67,100 - 2.44T7 + 0.0022772 - 0.34 X 10-<T8

The equation may, of course, be integrated between limits when
only a single new value of A# is desired. For example, if only
A//H73 is desired, integration between limits gives

A//1473 - A#298 = 600 cal.

Whence AHun = —67,040 cal. for this change in state, in agree-
ment with the value of page 320.

The integration constant A//0 is only an integration constant;
we do not imply that it is A# for the change in state at absolute
zero. The heat-capacity equations are valid in certain temper-
ature ranges only, and A/f 0 is a valid integration constant only
in these ranges as well.

When any substance in a system undergoes phase transition
(change of crystal form, fusion, or evaporation) in the tem-
perature interval involved in a calculation, A// may not be
expressed as a function of temperature by the relation „

-d A# = ACP dT

for phase changes involve heat absorption at constant tempera-
ture that cannot be included in heat-capacity equations. The
general method first given is of course applicable in these cir-
cumstances. For example, if liquid water is formed at the lower
temperature and water vapor at the higher one, there is a large
absorption of heat when evaporation takes place, with no
attendant change of temperature, and the temperature function
for Cp changes abruptly with the change in state of aggregation.
A calculation for the union of hydrogen and oxygen to form water
vapor at 150° will illustrate this point.

THERMOCHEMISTRY 323

2H2 + O2 A//2 2H,0(0)

150°, 1 atm. *150°, 1 atm.

A//3

2H2 + 02 Affi 2H,0(Z)

25°, 1 atm; *25°, 1 atm.

As before, AHi + A/f4 must be equal to A//3 + A//2, and A//I is
— 136,640 cal. from Table 58. A#4 is the sum of three steps by
which liquid water at 25° is changed to water vapor at 150°
through paths involving known data: First heat 36 grams of
water from 25° to 100°, absorbing 36 X 75 = 2700 cal.; then
evaporate the water, absorbing 2 X 9700 = 19,400 cal.; then
heat 2 moles of water vapor to 150°, absorbing 820 cal. The
sum of these quantities gives A//4 = 22,920 cal., and AH3 is
2710 cal. from Table 55. Thus

2710 + A//2 = -136,640 + 22,920 A#2 = -116.43 kcal.

The heat absorbed per mole of water vapor formed at 150°C. is
— 58.2 kcal. Final results of calculations should be rounded off
in this way, since writing —58.215 kcal. indicates a more exact
result than the data justify.

When enthalpies such as H2C%) = -57.82 kcal. at 298°K. are
available, the change in state may be set up in terms of water
vapor at both temperatures, the value of A#4 taken from Table
55, and the calculation involving two heat capacities and A# for
the phase change avoided. But there are many reactions for
which such entries are not available and which require the longer
procedure. All reactions involving solid-solid transitions or
chemical decompositions necessarily fall into this class.

Application of the equation d(&H)/dT = ACP to the data for
the union of hydrogen and hydroxyl ions leads to the sur-
prising value A(7P = 50 cal., as will be evident from the equation
AH = -28,260 + 5077 given on page 320. Since the heat
capacity of water is 18 cal. per mole, this means that the apparent

324 PHYSICAL CHEMISTRY

heat capacity of the ions is negative, — 32 cal. for the sum of the
apparent molal heat capacities of H+ and OH~ in a dilute solu-
tion. Other ions also have this strange property; for example,
the apparent ionic heat capacities in dilute solution are1 — 14 cal.
for K+, -14 for Oh, -7.5 for Na+, -16.1 for OH- and -15.9
for H+. At higher molalities these heat capacities change in
value but are still negative, for example, — 18 cal per mole for
KC1 at unit molality. This means that the addition of 74.5
grams of KC1 to a large quantity of 1m. KC1 solution decreases
the heat capacity of the system 18 cal. The quantities are thus
partial molal heat capacities, so that, at 1m., dCp/dNKC\ = — 18
cal and, in a very dilute solution, dCp/dNKCi = — 28 cal

Heats of ionization for weak electrolytes are commonly small,
with large changes in heat capacity, so that AH often changes
sign within a moderate temperature range. For example, AH
for the ionization of lactic acid is 768 cal. at 0°C., zero at 22 5°C.,
and —1313 cal. at 50°C., and the equation that expresses AH as a
function of the temperature within this range is

AH = 0.1355772 - 4.58 X
For the first ionization of carbonic acid,

AH = 78,011 - 427.6T7 + 0.58I'2

in the range 273 to 323°K. ; and in the smaller range 273 to 298°K
it is approximately 27,400 — 85 T. As one other illustration, for
the ionization of bicarbonate ion, AH = 13,278 - 0.108847'2 in
the range 273 to 323°K and from 273 to 298°K. it is approxi-
mately" 20,500 — 57 T. We shall return to a consideration of
these equations near the" end of the next chapter.

All these examples are only illustrations of the fact that AH for
a given change is the same by all paths, and there is of course
no implied limitation of the calculations to two isothermal paths
and two constant-composition paths. In all commercial com-
bustions air and fuel enter at about 20°, and stack gases and
ashes emerge at higher temperatures; and for such processes the
net heat available is AH for an assumed isothermal combustion
less the heat required to raise the products of combustion to their
emergent temperatures. For illustration, we may calculate the

1 PITZER, ibid., 69, 2365 (1937).

THERMOCHEMISTRY 325

theoretical maximum temperature attainable by burning carbon
monoxide with air for which we assume AH — 0. Since this
calculation assumes no loss of heat to the surroundings, which
would be impossible with the temperature differences involved,
a rough calculation will suffice. We have for the basis of the
calculation

CO + M02 + 2N2 = CO2 + 2N2 A#3oo = -67.6 kcal.

and this quantity of heat is available for heating 1 mole of CO2
and 2 moles of nitrogen to the maximum temperature T. We
may assume Cp = 7 + 0 007 T for C02 and Cp = 6.5 + 0.001 T7
per mole ol N2, which gives 20 + 0.00971 for the heat capacity of
the system. Then

+ 07,600 = |Jc (20 + 0.009 T)dT

and, upon integrating between T and 300° and rearranging the
equation, \vre have

74,000 = 2077 + 0.0045272

Solution of this equation yields an absolute temperature higher
than 2500°K But since at any such temperature some heat
would be lost to the surroundings and some heat would be
absorbed by the appreciable dissociation of C02, it is evident that
this temperature would not be reached. By expressing the extent
of dissociation of C02 as a function of the temperature, one may
obtain a more complex equation allowing for the dissociation
and thus may calculate the theoretical maximum temperature to
a closer approximation By employing well-insulated furnaces
one may almost reach this theoretical maximum temperature,
which is about 2100°K. for the reaction we have been considering.
Heat balances for flow processes, whether isothermal or not,
may be computed from enthalpy data and heat-capacity data in
the same way. For example, if equal volumes of 2m. NaOH
and 2m. HC1 enter a flow calorimeter at 298°K., A7/29g will be
— 14.00 kcal. for each mole of water formed isothermally ; and
since AH within the calorimetric system is always zero, 14.00
kcal. is available to heat the resulting sodium chloride solution,
which will be 1018 grams of water containing 58.5 grams of
sodium chloride. The specific heat capacity of the solution is

326 PHYSICAL CHEMISTRY

0.932 cal. per gram, or the heat capacity of the solution to be
heated is 0.932(1018 + 58.5) = 989.7 cal. per deg. Hence the
temperature of the effluent will be 14,000/989.7 = 14.15° above
that of the entering solutions.

Acetaldehyde may be made industrially by passing acetylene
into dilute sulfuric acid containing mercuric sulfate as a catalyzer
for the reaction. The over-all change in state in the reaction
vessel is

C2H2(<7) + H2O(/) = CH3CHO(0) A#298 = -29.5 kcal.

In order to keep the temperature in the reaction vessel constant,
cooling water is passed through a coil immersed in it. If this
water enters at 10°C. and emerges at 25°C., substantially 2000
grams of water will thus be required for each 44 grams of acetalde-
hyde vapor formed.

In discussing the temperature coefficients of the heat effects
attending reactions we have assumed a constant pressure,
summed A# values, and used heat capacities at constant pres-
sure. But the corresponding calculations for constant-volume
processes are carried out in the same way; one sums AE values
and uses heat capacities at constant volume for two paths for
a change from an initial state at T\ to a final state at TV We
have already seen that for gases at moderate pressures the differ-
ence in molal heat capacity is Cp — Cv = R] for liquids and
solids at moderate pressures the difference between Cp and Cv
may usually be neglected.1

Problems

Numerical data for the problems should be sought in tables in the text.

1. Calculate A# for each of the following changes in state, given Cp — 30
cal per mole for liquid C6H6, Cp = 6 5 -f- 0.0527" for C6H6(0), and the latent
heat of evaporation of CeH6 is 7600 cal per mole at 353°K.:

(a) C6H6a, 293°K.) = C6H6(/, 353°K.)

(6) C6H6(Z, 353°K ) = C6H6(0, 353°K) = C6H6(0, 453°K.), all at 1 atm.

(d) C6H6(Z, 293°K, 1 atm ) = C6H6(g, 453°K., 0.1 atm.)

2. Calculate the heat absorbed per mole of ethane formed when a mix-
ture of ethylene and hydrogen is passed at 25° over a suitable catalyst.

3. Calculate A# for the reaction CaCO3(s) - CaO(s) + COz(g) at
1100°K, taking CP = 12.0 for CaO and 23.5 for CaCO3.

lThe difference is Cp — Cv = a*vT/p, in which a = (l/v)(dv/dT)p and
|3 - ~(l/v)(av/dp)T.

THERMOCHEMISTR Y 327

4. Calculate AH at 385°K. for the change in state
2NaHC03(s) - Na2CO3« + H2O(0) -f

taking 29 as the molal Cp for NaHCO3 arid 30 for Na2CO.

5. For the solution of aluminum in HC1 200A</, AH 291 =* —127 kcal. per
atomic weight; for solution in HC1 20Ag, AH 291 = —126 kcal Refer to
page 314, and calculate A//29i for the dilution of AlCl3.60Ag to A1C18 600A</.

6. A#29i for the solution of magnesium in HC1.200A# is —110.2 kcal. per
atomic weight. Calculate A//29j for the reaction

3Mg + 2A1CU.A? = 2A1 + 3MgCl2 Aq

7. (a) Calculate A//, AE, A(pv), g, and w for the evaporation of a gram
of water at 100° and constant pressure. (6) Calculate these quantities
for the evaporation of a gram of water at 100° into an exhausted space of
such volume that the final pressure of the resulting vapor is 1 0 atm

8. Calculate AH for the evaporation of a mole of water at 150°, at this
temperature the vapor pressure of water is 4. 69 atm. Takes AH — 135 cal.
at 150° for H2O(0, 4 69 atm ) -» H20(?, 1 atm.).

9. (a) Given the heat of sublimation of S03 is 10,800 cal per mole at
25°, calculate A/7 for the change in state at 25°: S03(0) + H20(Z) = H2SO4(/).
(b) A gas mixture containing 5 moles of air and 1 mole of SO2 enters a cataly-
tic chamber, where practically all the SO2 is converted into SOa(flO The
resulting gas mixture is cooled to 200°C and then enters a tank containing
100 per cent H2S04 at 25°C. and atmospheric pressure Sufficient water
at 25°C is introduced into the tank to maintain the concentration of H2S04
constant. Formulate the change in state taking place in the tank, and
calculate the amount of heat that must be removed from the tank for each
5.5 moles of entering gas mixture so that the temperature in the tank will
remain constant at 25°C. Cp for SO3(<7) = 14

10. (a) Formulate carefully the change in state that occurs when a
mixture of lCaHo(fir) + 5O2(0) in a 25-liter vessel at 25° is exploded and the
vessel is brought back to 25° by the removal of heat. (Note that 0 032 mole
of water vapor remains uncondensed.) (6) Calculate A#, AE, q, and w for
this process, (c) Calculate AH, AE, q, and w for the ideal combustion
process at constant pressure, C2H6(0) + 3>£O2(gO = 2CO2(0) + 3H2O(0,
and compare them with the corresponding quantities for the actual process
described in part (a).

11. Calculate AH for the change in state

S0,(0, 1 atm., 400°K.) - S02(0r, 1 atm., 500°K.)
from the heat-capacity equation in Table 56 and from

CP - 11.9 + 0.0011 r - 2.64 X lOV^1.

[SPENCER and FLANNAGAN, /. Am. Chem. Soc., 64, 2511 (1942).]

12. Calculate AH for the reaction ZnO(s) + C(«) = Zn(0) + CO(0) at
1193°K. The atomic heat of fusion of Zn at 693°K. is 1:58 kcal., its heat of

328 PHYSICAL CHEMISTRY

evaporation at 1193°K. is 31.1 kcal., Cp is 10 cal per atomic weight of Zn(/)
and 13 cal per mole of ZnO(s).

13. The latent heat of evaporation of toluene (CrHg = 92) is 85 cal per
gram at 110°C. (the boiling point) when evaporation takes place against the
atmosphere, and the vapor pressure of toluene is 0 44 atm at 84°C Assume
AH independent of the temperature and that toluene vapor is an ideal gas
A small flask of such volume that it is filled bv 0 10 mole of liquid toluene
at 84° and 0 44 atm is connected through a stopcock to a 3-liter evacuated
flask. The stopcock is opened, and heat is added until the temperatuic
returns to 84°. (a) Formulate completely the change in state that occurs
(6) What weight of toluene evaporates ? (c) Calculate A/7, AT?, </, and w foi
this process.

14. The steam distillation of toluene occurs at 84°C and 1 atm total pres-
sure. At 84°C the vapor pressure of water is 0 56 atm , and the liquids are
mutually insoluble See Problem 13 for data on toluene (a) How many
grams of toluene will be in the first 100 grams of total distillate if steam
at 100° and 1 atm is passed into a flask containing a mixture of toluene and
water at 84° ? (6) How many grams of steam must be passed into the flask
to yield this 100 grams of distillate? (Assume the flask to be thermally
insulated and that steam entering at 100° is the only source of heat )

16. Carbon monoxide mav be manufactured by passing a mixture of
oxygen and carbon dioxide over hot carbon Since the oxidation of carbon
evolves heat and the reduction of carbon dioxide by carbon absorbs heat,
there is a mixture of oxygen and CO2 that can be passed over carbon at
1200°K , where it will be changed to practically pure carbon monoxide with-
out changing the temperature of the carbon bed (a) Calculate AH for each
reaction at 1200°K. , and the moles of oxygen per mole of CO« in a mixture
that would cause no change in the temperature of the carbon bed, assuming
that the gases enter at 1200°K and leave at 1200°K (6) Recalculate this
ratio, assuming that the reacting substances enter at 300°K and leave at
1200°K

16. Calculate the heat absorbed by the reaction H2(<7) + 12(17) = 2HI(gr)
at 600°K.

17. Estimate the heat of formation of HBr from its elements at 700°K.,
using the data in Problem 22*

18. Calculate AH for adding a mole of calcium oxide to a large quantity
of HI.100.Aff.

19. When a mole of 0 1m H3PO4 is neutralized with a mole of sodium
hydroxide m dilute solution, A//298 = —14,800 cal Phosphoric acid at
0 1m. is about 30 per cent ionized into H+ and H2PO4~ Calculate the heat
absorbed per mole of H^PO4 ionized into H+ and H2PO4~.

20. (a) Calculate AH for the reaction

Ci,HMOii(«) + H20(Z) = 4C2H6OH(/) + 4CO2(0)

at 25°C. (b) What additional data would be required for calculating AH for
the production of dilute alcohol from sugar solution?

21. Calculate AH for the gaseous reaction CO2 + H2 = CO + H2O at
1100°K.

THERMOCHEMISTR Y 329

22. The formation of iodine bromide is shown by the equation
I2(s) + Br2(Z) = 2IBr(0), A77298 = +19,720 cal

The molal heat capacities may be taken as 13 3 for 1 2(s), 17 2 for Br2(7), and
9 0 for lafe), Br2(7), and IBrO/), the molal heat of evaporation of bromine
is 7400 cal at 332°K , the molal heat of sublimation of iodine is 14,900 cal
at 387°K (a) Calculate A77 at 387°K for the reaction

(I)} Calculate A77487 for this reaction

23. Calculate the ratio of air to steam in a mixture that can be blown
through a fuel bed at 1000°K if the temperature of the fuel is to remain
constant Assume (a) that no water or oxygen passes through unchanged,
(b] that air is 4N2 -f- O2, (c) that the gases enter and leave the fuel bed at
1000°K through the use of a suitable heat interchange!1, and (d) that there is
no COs in the emerging gas

24. (a) Calculate A772<,8 for the reaction

A120,(«) + 3CW + 3C12(?) = Al2Clc(s) + 3CO(0)

(6) Calculate A/7 for the change in state Al2O3(s) + 3C(s) + 3C12(0) at
298°K = A12C1,,(0) ~h 3OO(0) at 435°K The (calculated) A// of sublima-
tion of A12C1<, at 298°K is 28 85 kcal , and Cr for Al2O6(flO is 34 cal per mole
26. The steam distillation of cblorobenzerie takes place at 90°C under a
total pressure of 1 atm , and the liquids are substantially insoluble in one
another Calculate the weight of chlorobenzene in the distillate and the
weight of condensed water in the flask after 100 grams of steam at 100° and
1 atm is passed into a thermally insulated flask containing 1120 grams of
chlorobenzene ( = 10 moles of Cf,Hr,Cl) at 20° Neglect the heat capacity
of the flask, arid use the following molal quantities. Cp for liquid CeH6Cl = 34,
arid AH for evaporation of C6H5C1 is 8800 cal at 90°

26. When 0 0340 mole of NaOH in 35 grams of water is added to 950 grams
of 0 050m NH4C1 at 25°, there is a heat absorption of -29.4 cal. Calculate
A/7 for the lomzation

KH4OH Aq = NH4+ Aq + OH~ Aq

assuming the heats of dilution are negligible and neglecting the small ioniza-
tion of NH4OH in the final solution

27. (a) Calculate A77 at 25° for the complete change in state

HBO2.Ag + NH4OH.Atf = NH4+BO2- Aq + H2O

(6) When 0 1 mole of HBO2 m 1000 grams of water is added to 0 1 mole of
NH4OH in 1000 grams of water at 25°, there is a heat absorption of —600 cal.
What fraction of the base has combined?

28. When a mixture of 1 mole of HC1 and 5 moles of air (02 + 4N2) passes
over a catalyzer at 386°, 80 per cent of the HC1 is oxidized to chlorine, (a)
Assuming that the gases enter the reaction vessel at 20° and 1 atm, total

330 PHYSICAL CHEMISTRY

pressure and leave it at 386° and 1 atm , formulate the change in state
taking place in the vessel (6) What quantity of heat must be removed
from, or added to, the vessel to keep its temperature 386°?

29. An important reaction for the recovery of sulfur from H2S is

S02(f7) - 2H20(g) + 3S«

Calculate AH for this reaction at 100°C.

30. One step in the manufacture of CCU involves the reaction

CS2(Z) + 3C1,(0) = CC14(/) + S2C12(Z)

which takes place in a water-cooled reaction vessel at 25°. How many
kilograms of cooling water at 10° must pass through coils in the reaction
vessel for each kilogram of chlorine reacting to keep the temperature at 25° ?

31. (a) Calculate A/7 at 20° for the reaction

4NH,(0) + 50,fo) + 20N2(0) = 4NO(<7) + 6H2O(0) + 20N2(g)

(6) Make the additional assumptions that (1) this mixture of 4NH3 + 5O2
-h 20N2 enters a vessel at 20° in which complete oxidation of the NH3
takes place upon a suitable catalyst, (2) constant pressure of 1 atm prevails,
(3) the mixture emerges at 200°, and (4) cooling water enters a coil in the
reaction vessel at 20° and leaves at 70° Calculate the kilograms of water
passing through the coil for each 29 moles of entering gas

32. In a flow type of water heater a flame of methane burns on a coil
through which water passes Assume that (1) 30 moles of methane and
300 moles of air (60O2 + 240N2) are used each minute, (2) complete oxida-
tion to CO2 and water vapor, (3) the gases enter the heater at 20° and leave
at 220°, and (4) the water enters at 15° and lea,ves at 65°. (a) How many
liters of water flow through the coil each minute if heat interchange is com-
plete? (6) What is the "dew point" of the stack gas?

33. (a) Calculate A/f 298 for the reaction Ca(s) -f- 2C(«) = CaC2(s), given
A#298 = -31.3 kcal for CaC2(«) 4- 2H2O(0 = Ca(OH)2(s) -f C2H2(0).
to) Calculate A#298 for the>eaction 3C(s) -f CaO(s) = CaC2(«) + CO(g)

34. In the manufacture of hydrochloric acid, HC1 gas at 100°C and water
at 10°C flow countercurrent through a vessel, and a solution of the com-
position HC1 5H2O leaves the vessel at 50°. Cooling water enters a coil in
the vessel at 10° and leaves at 50°. How many kilograms of water must
flow through the coil for each kilogram of HC1.5H20 leaving it? [The molal
heat capacity of HC1.5H2O is 85 cal per deg., and that of HC1(0) may be
taken as 7.0 cal. per deg.]

35. Derive an expression for A/f as a function of temperature for the reac-
tion C2H4(0) + H2C%) » C2H6OH(0) that will be valid in the range 290 to
500°K. In this range the following heat-capacity equations at constant
pressure are valid: Cp - 6.0 + 0.01577 for C2H4(0), 4.5 + 0.03827 for
C2H6OH(0), and 7a-f-0.003T for H2O(gr). [The first two heat-capacity
equations are from Pardee and Dodge, Ind. Eng. Chem., 35, 273 (1943).]

THERMOCHEMISTR Y 331

36. Plot AH against N from the table of heats of solution of HC1 on page
314, and draw tangents to the curve at N = 5 55 (corresponding to 10m.
HC1) and N — 13.9 (corresponding to 4m. HC1) Determine for each
molahty (dH/dNnzo), and from this slope compute (dH/dN&ci), the partial
molal heat of solution of HCl(^) in 4m and 10m* HC1

37. (a) Calculate A//29g for the dehydrogenation of n-heptane to toluene
as shown by the chemical equation n-C^Hi^l) = C6H5CH3(0 + 4H2(0),
using the data below and other data from the text. (6) Calculate AH for
the change in state n-C7Hi6(0, 298°K. = C6H5CH3(0) + 4H2(0) at 573°K
(c) Calculate AH for the reaction at 573°K : n-C7Hi6(0) = C6H6CH3(0) +
4H2(0). Data for n-heptane: A jf/2 98 (combustion) = — 11499kcal per mole,
A//371 (evap ) = 12.5 kcal. per mole, Cp(l) - 53 cal per mole, Cp(g) = 6.4
-f 0 0957". Data for toluene: A//383 (evap ) = 8.09 kcal. per mole,
Cp(l) = 36 cal. per mole, Cp(g) =50 + 0.07027.

CHAPTER IX
EQUILIBRIUM IN HOMOGENEOUS SYSTEMS

One of the most important problems of physical chemistry is
the extent to which chemical reactions take place. Most of the
familiar reactions of inorganic chemistry, especially those of
analytical chemistry, go forward until one of the reacting sub-
stances (the " limiting reagent ") is exhausted. In addition to
such complete reactions, there are many others in which sub-
stantial fractions of all the reacting substances remain unchanged,
even when a "stationary"1 state is reached. These fractions
vary with the proportions in which the reacting substances are
put together (though the proportions in which they react are
governed by the chemical equation) and with the pressure and
temperature

Equilibrium in gaseous mixtures at constant temperature and
moderate pressures will be considered first, then equilibrium
in dilute aqueous solutions at constant temperature, and finally
the effect of changing temperature upon equilibrium conditions.
Occasionally we shall use an excess of a liquid phase or a solid
phase of constant vapor pressure to control one partial pressure
in a gaseous mixture, or the partial pressure of a gas to control
its molality in a solution, or an excess of a solid phase to keep its
molality constant in a solution; but the main topic of this chapter
is chemical equilibrium in a single phase Equilibrium in sys-
tems of more than one phase will be presented in the next chapter.
In gaseous systems at moderate pressures, the experimental data
will be combined with the ideal gas law, Dalton's law of partial
pressures, and other general principles in order to calculate the
partial pressures in equilibrium mixtures. Gaseous systems at
such high pressures as to render the ideal gas law invalid require

1 At equilibrium, or an apparently stationary state, the initial reaction
and the reverse reaction are proceeding at equal rates in the opposite direc-
tions. The relation of these rates to the equilibrium concentration is dis-
cussed in Chap. XII.

332

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 333

special methods of treatment, to which we shall refer briefly
after the simpler systems have been studied.

The law of chemical equilibrium applying to a gaseous system
may be expressed in terms of molal concentrations or in terms
of the partial pressures of the substances. Partial pressures
are more commonly used, and their use leads to simpler calcula-
tions in constant-pressure processes. But partial pressures are
not directly measurable quantities ; they may be calculated from
the chemical composition of the equilibrium system and the total
pressure through Dalton's law when the ideal gas law applies.
They may be calculated in a system of nearly ideal gases at con-
stant volume from the difference between the observed total
equilibrium pressure and the initial pressure of the system before
reaction through the stoichiometry involved.

Since the composition of an equilibrium mixture changes with
temperature and pressure, it is usually not permissible to with-
draw a sample and cool it for analysis. Physical measurements,
such as density, pressure, color, volume change, heat evolution,
electrical conductance, spectrographic analysis, or the control
of a partial pressure through the presence of a liquid or solid
phase, must be applied to the system at equilibrium.

General Law of Chemical Equilibrium. — This law may be
stated as follows: At equilibrium the product of the partial
pressures of the substances formed in a reaction, each raised to
a power that is the coefficient of its formula in the balanced
chemical equation, divided by the product of the partial pressures
of the reacting substances, each raised to a power that is the
coefficient of its formula in the chemical equation, is a constant
for a given temperature. Thus for the general reaction

aA + 6B + - • • = dD + eE + • • •
the condition of equilibrium at any specified temperature is

PD'PE* = Kp (t const.) (1)

p^apBb • • • p ^ ' v '

Partial pressures are usually, though not always, expressed in
atmospheres, and it should be noted that the numerical value
of K will depend on the units used unless

a + b + - • - = d + e + • - •

334 PHYSICAL CHEMISTRY

that is, K will depend on the units in which pressures are
expressed for every reaction in which there is a change in the
number of molecules as the reaction proceeds. The equation
may be derived from the laws of thermodynamics for a system
of ideal gases.1

If the equilibrium expression is written in terms of the molal
concentrations of the substances involved, the relation becomes

c dr * • • •

v'D V'E rr t. . N /rtx

r *r * .^~T = Ac (l const.) (2)

v'A ^'B

These equilibrium constants both express the fundamental rela-
tion applying to a selected system at a definite temperature, but
the numerical values of Kp and Kc are not the same. If we
write the ideal gas law as applying to constituent A, for example,
PA = (n±/v)RT = CARTj and a similar expression for the other
constituents, substitution of CRT for these partial pressures in
equation (1) shows that Kp — Kc(RT)^n, where

An = d + e — a — b

is the increase in the number of moles of gas attending the
complete chemical reaction, pA is the partial pressure in atmos-
pheres, and R has the value 0.082 hter-atm./mole-°K.

In writing the equilibrium expressions above, we have observed
a custom to which we shall adhere throughout the book and
which is standard practice in physical chemistnr, namely, that
of writing the partial pressures or concentrations of the reaction
products in the numerator of the equilibrium expression. These
relations are independent of the mechanism by which equi-
librium is reached; and, of course, one may write the chemical
reaction in any desired way. But the equilibrium expression
should always be written with the products of the chemical
reaction as written in the numerator. For illustration, all the
following expressions are equally satisfactory representations of
the equilibrium between 80s, 862, and 02 at a given temperature:

(1) S02 + V&i = S08 #1 = P&0t „> = 1.85 at 1000°K.

Pao&oS*

(2) 2S02 + 02 = 2S08 K2 = —~- = 3.42 at 1000°K.

psoSpo*

1 LEWIS, Proc. Am. Acad. Arts Sci., 43, 259 (1907)

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 335

(3) S03 = SO2 + HO2 K3 = 80 = 0.540 at 1000°K.

pao,

(4) 2S03 = 2SOZ + 02 Kt = &°L£p = 0.282 at 1000°K.

Paos

The partial pressures are in atmospheres for the values of K
given. In each equilibrium expression the partial pressures in
the numerator are those of reaction products for the correspond-
ing chemical reaction; in each the pressure is raised to the power
that is the coefficient of its formula in the equation as written.
It will be clear that it is often necessary and always desirable
to write the chemical reaction for which an equilibrium constant
is evaluated, to state the units in which the quantities are
expressed, and to state the temperature. K% and K±, for exam-
ple, are both " dissociation constants" for SO3, but K± is the
square of K3, and without the attending chemical equation it
would be uncertain which one was meant.

Another fact about equilibrium constants is of the greatest
importance, namely, that they give the relation among the partial
pressures involved regardless of the quantities of the substances
present, regardless of the direction from which equilibrium is
approached, and regardless of the presence of other gases.
In systems consisting of 1 mole of SO2 and 2.3 of oxygen, or
SO2 + 2S03, or 041S02 and 0.21 oxygen and 0.79 nitrogen, or
in the flue gas from sulfur-bearing fuel, the equilibrium expres-
sions give the relation of the partial pressures of SO3, SO2, and 02
at equilibrium. Of course, the sum of these three partial pres-
sures is not the total pressure in two of these systems; and, in
computing xso2 from the composition of the equilibrium mixture,
the moles of SO2 divided by the total moles of all substances
present gives the mole fraction.

The law of chemical equilibrium has been tested and con-
firmed by experiments on a large number of chemical systems
of the most varied kind. Deviations from its predictions are
no greater than those found between the measured and ideal
properties of solutions or gases already considered. It is proba-
bly the most important law of physical chemistry; its proper
application will show what procedure is necessary to increase the
yield of a desired product in a chemical reaction or what should
be done to decrease the yield of an undesirable product. It
indicates the precautions to be observed in analytical chemistry;

336 PHYSICAL CHEMISTRY

it enables us to calculate the extent to which a reaction will
take place in solutions, the fraction of a substance hydrolyzed,
the quantity of reagent necessary to convert one solid completely
into another, and many other similar quantities.

The formulation of equilibrium expressions requires complete
knowledge of the chemistry of the reacting systems. The
chemical substances involved in a single equilibrium expression
must be those, and only those, shown in the chemical equation.
This is not to say that the methods are inapplicable in systems
in which more than one chemical reaction is taking place, for we
shall encounter many such systems and apply the laws of chem-
ical equilibrium to them. In treating them we shall write a
sufficient number of equations to describe all the reactions
taking place, and we shall formulate a corresponding number of
equilibrium expressions. Through stoichiometry, material bal-
ances, energy balances, a sufficient number of measured quanti-
ties, and suitable approximations, we shall be able to calculate
the pressures or concentrations of all the substances present at
equilibrium. When the pressure of a given substance appears
in more than one equilibrium expression, it will be understood
that it has the same value in every one, for there can be only one
equilibrium pressure of a given substance in a given system.

In gaseous systems for which the ideal gas law and Dalton's
law of partial pressures are inadequate approximations, the
equilibrium law is expressed in terms of the fugacities of the sub-
stances. For the general reaction above, the equilibrium law is

. ££i = ff/ (I const.) (3)

This expression is constant by definition, since the fugacity of a
substance is a quantity with the dimensions of a partial pressure
that represents its actual effect in a chemical system In a
system of ideal gases, the fugacities are equal to the partial
pressures; in any other system, they must be evaluated from the
equation of state for the gas. Since these calculations are some-
what difficult for beginners, they are best reserved for more
advanced courses.1 In this brief treatment we shall confine our

1 See Lewis and Randall, " Thermodynamics," pp. 190-201, for the meth-
ods and some illustrations.

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 337

discussion to systems at such moderate pressures that pA = ptot»i^A
is an adequate measure of the fugacity /A.

It cannot be emphasized too strongly that equilibrium expres-
sions do not apply to systems that are not at equilibrium. In
experimental work it is necessary to show that equilibrium has
been reached through the use of suitable methods. Chemical
systems react toward equilibrium at rates that decrease as
equilibrium is approached; and in some systems the rates are very
slow. One common procedure is to approach equilibrium from
both sides , by mixing A and B in one series of experiments and
by mixing D and E in another. If the same relation among the
partial pressures is obtained in both series, the system has
reached equilibrium. If different relations are found, the
system is not at equilibrium and more time must be allowed or
means of accelerating the reaction must be found. Another
common test consists in heating one chemical system up to the
desired temperature and in cooling another system to this tem-
perature after it has been kept at a higher temperature for a
sufficient time. Since equilibrium conditions change with the
temperature, this is another means of approaching equilibrium
from both sides These or other proofs of an equilibrium state
are absolutely essential in experimental work and are rou-
tinely carried out by competent workers. We turn now to the
application of these principles to some chemical systems.

1. Formation of Sulfur Trioxide. — When a mixture of 1 mole
of sulfur dioxide and ^ mole of oxygen is heated to 1000°K. in
the presence of a suitable accelerator for the reaction, 46 per cent
of the sulfur dioxide is converted to sulfur trioxide when the equi-
librium total pressure is 1 atm. The chemical reaction and its
equilibrium expression are

S02(g)

With 1SC>2 + J^02 as a working basis or a material basis for
the calculation, we see from the chemical reaction that 0.46 mole
of S03 requires 0.23 mole of 02 for its oxidation, leaving 0.54
mole of S02 and 0.27 mole of 02 in equilibrium with 0.46 mole of
S03. The equilibrium system has the composition, at 1000°K,

338 PHYSICAL CHEMISTRY

and 1 aim. total pressure,

0.46 mole S03
0.54 mole S02
0.27 mole O2

1.27 total moles

The partial pressures are 0.46/1.27 = 0.362 atm. for S03,
0.54/1.27 = 0.425 atm. for S02, and 0.27/1.27 = 0.213 atm. for
O2,' and, upon substituting these quantities in the equilibrium
expression, we have

0.362
0.425(0.213)^

Kp = ^^^ = 1.85 at 1000°K.

In the use of recorded equilibrium constants from the chemical
literature, attention must be paid to the conventions used and
to the units in which the equilibrium compositions are expressed.
For example, in the original paper from which these figures come,1
equilibrium compositions are given in moles of gas per liter,
partial pressures are given in millimeters of mercury, and the
equilibrium constant is Kc = 3.54 X 10~3 for the dissociation of
2 moles of SO3 with concentrations in moles per liter. One pro-
cedure is as good as any other so far as equilibrium is concerned;
we have used Kp for the formation of SO 3 to simplify calculations
that are to be made in later chapters, and in conformity with
the conventions used in tabulating free energies. If Kp were
given for partial pressures in millimeters of mercury for the forma-
tion of ISO*, its numerical value would be 1.85/\/760 = 0.067.

As has been said before, the relation among the partial pres-
sures given in equation (4) applies at 1000°K. to any mixture
containing S02, 02, and S03 at equilibrium, at any moderate
pressure, in any proportions, and in the presence of other sub-
stances. Some illustrations may not be out of place. Let the
original mixture of 1S02 + %0Z be compressed until the total
pressure at equilibrium is 2 atm., and let

x = moles S03
1 — x = moles S02
0.5 — 0.5# = moles O2
1.5 — 0.5x total moles

1 BODENSTEIN and POHL, Z. Elektrochem., 11, 373 (1905).

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 339

Upon substituting the product of total pressure and mole frac-
tion in terms of x into equation (4) we have

.5 - 0 5x)] V2[(0" - 0.5z)/(1.5 - 0.5z)]

While this equation is a cubic in x, the complete algebraic solu-
tion of a cubic equation is not required for practical purposes.
There is only one real root, it must be positive, the chemical
composition of the system places its value between 0 and 1.0, and
the equilibrium data above restrict it to still narrower limits.
Any compressed system reacts in the direction that relieves the
compression, and this particular system must react to form more
S03 in order to reach equilibrium under a higher total pressure.
The momentary effect of doubling the total pressure would
be to double each partial pressure and so bring the relation
Pso,/7>so2po2^ to a lower value than 1.85 which is required for
equilibrium. In order to restore the required relation among
the partial pressures, pa0a must increase, and both pSo2 and p02
must decrease, which requires the formation of SO3 by the
chemical reaction. The value of x in the new system at equi-
librium is thus greater than 0.46, less than 1, and nearer the
former value than the latter. Successive trials of 0.7, 0.6, 0.55,
and 0.53 for x will show that 0.53 satisfies the equation and hence
that 0.53 mole of S03 exists in this system when equilibrium is
reached at 1000°K. and 2 atm. total pressure.

The equilibrium mixture of 0.46SO3, 0.54S02, and 0.2702 had
a volume of 104 liters at 1000°K. and 1 atm. total pressure, as
shown by the equation pv = 1.27RT.

Suppose this mixture to be expanded to 208 liters, and, as
before, let

x = moles S03
1 — x = moles SC>2
0.5 — 0.5# = moles O2
1.5 — 0.5x total moles

Since expansion at constant temperature is attended by the
decomposition of S03, the new equilibrium pressure when the
volume is doubled will not be 0.5 atm., but a higher value, namely,
one that satisfies the ideal gas law in the form

p208 = (1.5 - Q.5x)RT

340 PHYSICAL CHEMISTRY

or p = 0.395(1.5 — 0.5x). Upon substituting the product of
each mole fraction times this total pressure into equation (4) and
simplifying, we have

1.85 =

(1 - x) V6.395(0.5 - 0

Again solving by trial, observing that .r is positive and must be
less than 0.46, we find x = 0.39 and

p = 0 395(1.5 - 0.5J-) = 0.52 atm.

Returning to the mixture of 0.46S08, 0 54SO2, and 0.2702 in
104 liters at 1000°K. and 1 atm pressure, suppose oxygen were
added to the mixture at constant volume until the total pressure
becomes 2 atm. at equilibrium. The ideal gas law shows 2 54
total moles of gas, a sulfur balance shows 1 mole of SO 2 + SO 3,
and therefore the moles of oxygen at equilibrium is 1.54 moles.
(This is not the quantity of oxygen added to the system, as we
shall see in a moment.) Let

x = moles SO3
1 — x = moles SO2
1.54 = moles 02
2 54 total moles

Upon substituting partial pressures based on these values into
equation (4), we have

2(*/254) _ x

1.10(1 - *) * —

whence x = 0.68 mole of SOs. The formation of this quantity
of 80s required 0.68 mole of S02 and 0.34 mole of O2; and since
the oxygen present at equilibrium was 1 .54 moles, the total oxygen
in the system (other than that in the original SO 2) is 1.54 + 0.34,
or 1.88 moles. The original system contained 0.50 mole of
oxygen and thus the oxygen added to bring the equilibrium
pressure to 2 atm. was 1.88 — 0.50 = 1.38 moles.

A balance for total oxygen gives the same result, namely,
1.5 X 0.68 = 1.02 moles of oxygen in SO3, 0.32 mole of oxygen
in 0.32 mole of S02, and 1.54 moles of oxygen uncombined, total
2.88 moles. Of this oxygen 1 mole was in the original S02, and

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 341

0.50 mole as free oxygen in the original mixture, which leaves
J .38 moles of added oxygen to make up the total.

As one more illustration, consider an initial mixture of a mole
of SO2 and 4 moles of air at 1000°K. and 1 atm. total pressure,
that is, 1S02 + 0.8402 + 3.16N2, and let the composition after
equilibrium is established be

x = moles SO 3
1 — x = moles S02
0.84 - 0 5z = moles 02
3.16 = moles N2
5 — Q.5x total moles

One may substitute mole fractions based on this table in equa-
tion (4) and find x = 0.41 mole of 80s at equilibrium in this
mixture at 1000°K. and 1 atm. total pressure.

It should be clearly understood that, while the equilibrium
relations which we have been discussing apply at any temperature
in any mixture containing these substances, the constant 1.85
applies only at 1000°K. At some other temperature a different
constant applies; for in this system and in every system the
equilibrium constant is for a given temperature. In this system
Kp changes with the Kelvin temperature T as follows :

Kp ... 31 3 13 7 6 56 3 24 1 85 0 95 0 63 0 36

T ... 801 852 900 953 1000 1062 1105 1170

Further discussion of these constants will be found at the end
of the chapter, where the equation governing the change of Kp
with temperature will be given.

The procedure that has been followed above is of such general
application in chemical equilibrium that it is worth while to sum-
marize the steps as routine for problem work. They are

1 Write a balanced chemical equation describing the chemical
change involved. This step should never be omitted, no matter
how simple or familiar the equation may be.

2. State the working basis of the calculation, the quantity of
each substance at the start, and the pressure, volume, and tem-
perature to be used in the problem.

3. Formulate the equilibrium expression in the standard way,
and note that the pressures of reaction products always appear

342

PHYSICAL CHEMISTRY

in the numerator of the equilibrium expression. When sufficient
data are- at hand for evaluating K, insert its value and note the
units employed in expressing it.

4. Set up a "mole table77 describing the quantity of each sub-
stance in the equilibrium mixture in terms of a suitable unknown.
The use of two or more unknowns is not excluded, but it will
usually be advantageous to restrict the number of unknowns to
one. Care in the choice of the unknown often yields a simpler
solution of the problem.

5. Solve for this unknown by appropriate use of the data.
This may be through a material'balance, or an expression for
total moles of gas from the ideal gas law, or a density expression
in terms of the fraction reacting, or direct substitution into Kpy
or any other procedure for which data are available.

2. The Synthesis of Ammonia. — As our second example of
equilibrium in gaseous systems we consider the data on synthetic
ammonia in a range of pressures in which deviations from the
ideal gas law become important. Table 59 shows how the mole
per cent of ammonia in equilibrium with a mixture of N2 + 3H2
varies with temperature and pressure. If we base our calcu-
lations upon the equation

TABLE 59. — MOLE PER CENT NEL IN EQUILIBRIUM WITH N2 + 3H2l

Temperature

Total pressure, atm.

°K.

°C.

100

623

350

7 35

17 80

25 11

648

375

5 25

13 35

19 44

30 95

673

400

3.85

10 09

15 11

24 91

698

425

2.80

7 59

11 71

20 23

723

450

2.04

5.80

9.17

16.36

748

475

1 61

4.53

7.13

12 98

773

500

1 20

3 48

5 58

10 40

and formulate the equilibrium constant in the standard way,
with the product of total pressure and mole fraction taken as the
partial pressure for each constituent, we have

and DODGE, /. Am. Chem. Soc., 46, 367 (1924).

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 343

rv (t const.) (5)

Of course, the chemical equation might have been written as a
dissociation rather than for synthesis or for 2 moles of NH3
rather than 1. But Kp for the reaction 2NH3 = N2 + 3H2
would be the square of the reciprocal of the Kp in equation (5).
The data of Table 59 yield at once the quantities required in
this expression, since one quarter of the difference between 100
and the mole per cent of NH3 is the mole per cent of nitrogen and
three quarters of this difference is the mole per cent of hydrogen
in the equilibrium mixture. If the ideal gas law were valid in
the equilibrium mixtures up to 100 atm. total pressure, all the
Kp values for a single temperature should be the same, but
Table 60, which records the value of this Kp X 1000 for partial
pressures in atmospheres, as calculated from the compositions
given in Table 59, shows that Kp changes with the total pressure.

TABLE 60 — CALCULATED

, IN ATM FOR >^N2 + 3^H2 = NH3

Temperature

Equilibrium pressure, atm.

°K

°C

100

623

350

26 59

27 34

27 94

648

375

18 15

18 43

18 66

20 30

673

400

12 92

12 93

13 05

13 78

698

425

9 20

9 34

9 90

723

450

6 60

6 76

6 91

7 27

748

475

5 16

5 14

5 13

5 33

773

500

3 81

3 86

3 89

4 03

The Kp values in Table 60 for pressures of 10 atm. may be used
to calculate the composition of any equilibrium system contain-
ing nitrogen, hydrogen, and ammonia in any proportions for
pressures near or below 10 atm. without large error. Consider,
for example, a mixture of 1 mole of nitrogen and 2 moles of
hydrogen that reacts to equilibrium at 623°K. and a total
pressure of 5 atm. If we let x be the moles of ammonia at
equilibrium, the "mole table" through which we express the
equilibrium composition becomes

344 PHYSICAL CHEMISTRY

x = moles

1 — 0.5z = moles N2

2 — I.5x = moles H2

3 — x — total moles

At equilibrium the partial pressures are 5z/(3 — x) for NH3,
5(1 - 0.5z)/(3 - x) for N2, and 5(2 - 1.5x)/(3 - x) for H2.
Substituting-these partial pressures into equation (5) and taking
#p for 623°K. from Table 60, one may solve by trial for the
moles of ammonia at equilibrium.

The values of Kp in Table 60 for higher pressures may also be
used for approximate calculations, by taking a value of Kp
adjusted for total pressure. But exact calculations, which are
required for ammonia synthesis in industry, are too difficult for
beginners.1

3. Dissociation of Nitrogen Tetroxide. — The experimental
method applied to this system consisted in measuring the total
pressure at equilibrium in a flask of known volume containing
a known weight of N2C>4. If we denote by m the initial weight
of N2C>4 added to a liter flask and by p the equilibrium total
pressure, the data for a series of experiments2 at 35°C. (= 308°K )
are

m, grams, 0 578 0 933 1 16 1 31 1 99

p, atm 0 238 0 365 0 439 0 487 0 706

Kp 0 317 0 316 0 300 0 287 0 264

The only important chemical reaction at this temperature is

N,04fo) = 2N02(</)
for which the equilibrium expression is

If we determine the total moles present at equilibrium from
pv = nRT and the moles of N2(>4 before dissociation by ra/92,
we may set up a mole table, calculate partial pressures, substitute

1 For an exact treatment of the system N2 + 3H2 up to 1000 atm., see
GiUespie and Seattle, Phys. Rev., 36, 743, (1930); /. Am. Chem. Soc., 52,
4239 (1930)

2VERHOEK and DANIELS, ibid., 53, 1250 (1931). The derived values
when Kp is plotted against the pressure and extrapolated to zero pressure
are 0.14 at 25°, 0.32 at 35°, and 0.68 at 45°

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 345

them in the equilibrium expression, and calculate the equilibrium
constant for this temperature. The results of this calculation,
for partial pressures in atmospheres, are given in the table above.
The "drift" in a quantity supposedly constant probably indi-
cates increasing deviation from ideal gas behavior on the part of
some component, and it should be noted that the mixture of
N204 and N02 condenses to a liquid at about 21°C. for a total
pressure of 1 atm. The common expedient is to plot the derived
Kp against the total pressure and extrapolate the curve to zero
pressure to determine the constant applicable to the state of an
ideal gas.

Some attention should be given to the qualifying statement
that at 35°C the only important chemical reaction is the dissocia-
tion of N204, for at higher temperatures another dissociation
becomes important, namely, 2N02 = 2NO + 02. The experi-
mental study of these systems would have been more difficult if
the second dissociation became appreciable before the first one
was substantially complete. From a study of the dissociation
of N02 at higher temperatures (500 to 900°K.) we may calculate
the extent of its dissociation at 308°K. through a relation to be
given later in this chapter. Such a calculation shows that the
partial pressures of NO and O2 are inappreciable in comparison
with the pressures of N02 and N204 at 308°K. They are below
0.0001 atm. in the systems given in the above table and thus
could not be detected experimentally by the method used in
studying the dissociation of N204. At temperatures higher than
about 425°K. the dissociation of N204 into N02 is substantially
complete, and the only important chemical reaction in the sys-
tem is 2N02 = 2NO + 02. Table 61 shows, for a total pressure
of 1 atm., how the various partial pressures at equilibrium change
with* the Kelvin temperature.

4. The "Water Gas" Equilibrium. — In some reactions involv-
ing hydrogen gas at high temperatures, advantage may be taken
of the fact that platinum is permeable to this gas and not to other
gases. Thus a platinum tube inserted into a reaction chamber
allows free penetration of hydrogen, and its partial pressure is
measured by a manometer attached to the platinum tube. This
method has been applied to the equilibrium

C02 + H2 = CO + H20

346

PHYSICAL CHEMISTRY

at high temperatures.1 A gaseous mixture containing known
proportions of carbon dioxide and hydrogen was brought to
equilibrium at a total pressure of 1 atm. As the total number of
moles of gas does not change during the chemical reaction, no
change of pressure is observed. But a decrease in hydrogen
pressure takes place when water is formed; hence the difference
TABLE 61 — PARTIAL PRESSURES IN AN EQUILIBRIUM MIXTURE

T, °K

PN204

PNO,

i
PNO

PO,

300

0 330

0 670

0 000

350

0 175

0 825

0 000

400

0 020

0 980

0 000

450

0 000

0 976

0 016

0 008

500

0 928

0 048

0 024

550

0 844

0 104

0 052

600

0 718

0 188

0 094

700

0 412

0 392

0 196

800

0 191

0 540

0 270

900

0 085

0 610

0 305

between the starting pressure of hydrogen (calculated from its
mole fraction in the original mixture) and the equilibrium pres-
sure of hydrogen represents water-vapor pressure From the
chemical reaction it follows that there is a mole of carbon monox-
ide formed for each mole of hydrogen used, i.e., for each mole of
water formed, and that a mole of carbon dioxide is used for every
mole of carbon monoxide formed. Thus a measurement of
hydrogen pressure gives (1) the partial pressure of hydrogen, (2)
the partial pressure of carbon monoxide, (3) the partial pressure
of water vapor (each of these last two being equal to the decrease
in hydrogen pressure during reaction), and by difference (4) the
pressure of carbon dioxide. Table 62 shows the mole per cent
of each substance in experiments at 1259°K., together with values
of the constant

* KP (6)

1 HAHN, Z. physik. Chem , 44, 513 (1903), NEUMANN and KOHLER,
Z. Elektrochem., 34, 281 (1928). BRYANT, Ind Eng. Chem , 23, 1019 (1931),
24, 592 (1932), and KASSEL, J Am. Chem Soc , 66, 1841 (1934), have
studied this equilibrium "system by quite different experimental methods
and have obtained results in substantial agreement with those reported in
Table 62.

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 347

TABLE 62 — EQUILIBRIUM DATA FOR CO2 + H2 = CO -f H20 AT 1259°K.

Original mixture

Equilibrium mixture

Mole

per cent
C02

per cent
H2

per cent
C02

per cent
CO = H2O

per cent
H2

10 1

89 9

0 69

9 4

80 5

1 60

30 1

69 9

7 15

22 96

46 93

1 58

49 1

51 9

21 22

27 90

22 95

1 60

60 9

39 1

34 43

26 45

12 67

1 60

70 3

29 7

47 50

22 82
Average

6 85

1 60

As has been said before, equilibrium conditions change mate-
rially with the temperature. Thus, the constant, which is 1.60
at 1259°K , changes with the temperature as follows.1

T, °K

0 219
800

0 412
900

0 675
1000

0 999
1100

1 37
1200

2 21
1400

3 11
1600

It is a matter of the first importance to bear this in mind when
comparing data from different sources.

An interesting feature of this equilibrium is the calculation of
dissociation constants from it. So far nothing has been said
about the presence of oxygen in this mixture, and there is in fact
only an insignificant quantity present. Its partial pressure
would have no effect upon the total pressure that could be
detected by experimental means. (From relations to be given
later in the chapter we may calculate the partial pressure of
oxygen in the equilibrium mixtures at 1259°K. to be about 10~14
atm., but there are no experimental means of finding such pres-
sures.) But the small quantity of 02 present must satisfy the
dissociations

2H20 = 2H2 + 02 and 2CO2 = 2CO
for which the equilibrium equations are

— A]

Hao

and

= J\C02

1 BRYANT, Ind. Eng. Chem., 24, 592 (1932). These data were not obtained
by measuring the partial pressure of hydrogen through platinum, but by
another procedure which is given in the next chapter.

348 PHYSICAL CHEMISTRY

Upon dividing the second of these dissociation expressions by
the first and extracting the square root, we obtain

which is the equilibrium expression of equation (6) for the reac-
tion CO2 + H2 = CO + H2O. This furnishes a means of calcu-
lating equilibrium constants from dissociation constants or of
calculating dissociation constants from measurements of equi-
libriums. It is an expedient that we shall often use.

5. Synthesis of Iodine Chloride. — For chemical reactions in
which no change in total moles attends the reaction, such as

pressure or density measurements afford no information, and
another expedient must be used. For this reaction we take
advantage of the fact that the chemical reaction

BaPtCleO) = BaCl2(s) + Pt(«) + 2Cl2(g)

maintains a constant pressure of chlorine at a given temperature
so long as all three of the solid phases are present Thus, the
use of a sufficient excess of solid BaPtCU serves to control the
partial pressure of one of the substances involved in the first
reaction. At 736. 5°K. the equilibrium pressure of chlorine is
9.5 mm. Consider a vessel containing an excess of BaPtCl6 at
736. 5°K., and into which enough iodine is introduced to give an
initial pressure of 174.7 mm. of iodine vapor. At equilibrium
the total pressure was found to be 342.3 mm., and therefore
342.3 — 9.5 = £>i2 + pici- It may be seen from the chemical
reaction that each IC1 requires J^I2, whence the pressure of IC1
is twice the decrease in iodine pressure. This gives an equation

342.3 - 9.5 = 2(174.7 - PI.) + plz

from which pi2 = 16.6 mm. and pic\ = 316.2 mm. Thus all
the partial pressures are known, and

Kp = ICI = ..- = 25.4 at 736.5°K.

16.6 -v/9.5

It is, of course, permitted to write the chemical reaction
I»(0) + CUfo) = 2IC%)

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 349

provided that the equilibrium constant is written
Kp = -^- = 640 at 736.5°K.

6. Dissociation of Nitrosyl Bromide. — Equilibrium between
nitric oxide, bromine, and nitrosyl bromide has been studied1
through the change in total pressure that attends the reaction

2NO(0) + Br2(0) = 2NOBr(0)

at constant volume. The equilibrium expression in terms of
partial pressures, formulated in the standard way, is

o
•rr 7?NOBr /o\

KP = Z — 27T" (°)

In one series of experiments a glass bulb of 1055 ml. volume con-
tained 0 0103 mole of NO and 0.0044 mole of Br2. The equilib-
rium pressure (in atmospheres) changed with the absolute
temperature as follows:

T 273 290 324 350 - 373 442 477

p 0 232 0 254 0 304 0 345 0 384 0 481 0 528

We may use the pressure at 350°K. to calculate the equilibrium
constant for this temperature. By substituting the observed
pressure, volume, and temperature in the ideal gas equation, we
find 0.0127 total mole present at equilibrium. In order to express
the composition of the equilibrium mixture in terms of a single
unknown, we may set up a "mole table" in terms of the original
quantities present, taking x as the number of moles of bromine
reacting. Then 0.0044 — x moles of bromine remain, and 2x
moles of NOBr have formed at the expense of 2x moles of NO,
as may be seen from the chemical equation. Thus the "mole
table" becomes

0.0044 — x = moles Br2 at equilibrium

2x = moles NOBr at equilibrium
0.0103 — 2x = moles NO at equilibrium
0.0147 — x = total moles at equilibrium

Since this total is 0.0127, x = 0.0020 and the mixture consists of
0.0040 mole of NOBr, 0.0024 mole of Br2, and 0.0063 mole of NO.

1 BLAIR, BRASS, and YOST, ibid., 66, 1916 (1934).

350 PHYSICAL CHEMISTRY

We divide each of these quantities by 0.0127 to obtain the respec-
tive mole fractions; multiply each one by 0.345 to obtain partial
pressures; and insert them in the equilibrium expression:

(0.11Q)2 ft A o

p -—^ - /rv i^r\\9/n /w»g\

p ?>No2pBr2 (0.170)2(0.065)

The same data may be used in a slightly different way to
obtain the equilibrium constant lor this system, though this
procedure is applicable only in systems reacting at constant
volume and constant temperature From the quantities of NO
and bromine present we may calculate that the initial pressures
would have been pQ = 0.280 atm. for NO and p0 = 0 120 aim
for Br2 at 350°K. if no reaction took place. It will be seen from
the chemical equation that each mole of NO which reacts forms
a mole of NOBr, and hence the sum of the partial pressures
PNO + ?>NOBr will remain constant at 0.280 atm But each mole
of NOBr formed required J^ mole of Br2, and the progress of the
reaction is attended by a decrease in pressure that measures the
bromine reacting. The difference between the sum of the initial
pressures (0.280 + 0 120 atm ) and the equilibrium pressure
(0.345 atm ) is 0.055 atm., which is the decrease in the bromine
pressure. Since each Br2 yields 2NOBr, 2 X 0.055 is the equi-
librium pressure of NOBr; 0.280 - 0.110 = 0.170 is the partial
pressure of the remaining NO, and 0.120 — 0.055 = 0.065 is the
partial pressure of the remaining Br2 These are the partial pres-
sures that appear in the equilibrium constant Kp given above.

The equilibrium relation among the partial pressures is valid
in any chemical system containing these substances in any
proportions, and in the presence of other gases, so ]ong as the
equilibrium pressure is low enough for reasonable conformity to
the ideal gas laws. For example, 0.0550 mole of NO and 0.0816
mole of Br2 in a 10-liter space at 350°K. will react to produce at
equilibrium a total pressure of 0.350 atm., and treatment of these
data in the way outlined above will yield a value of Kp in sub-
stantial agreement with that given above, namely, 6.4 for partial
pressures in atmospheres.

If the chemical reaction is written for the dissociation of nitrosyl
bromide,

2NOBr(0) =

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 351

the equilibrium relation is written as the reciprocal of the one
given above, and Kp is 0.16 for partial pressures in atmospheres.
Equilibrium in Aqueous Solutions. — The general law of equi-
librium that we have been considering in gaseous systems is
applicable in aqueous or other solutions. In dilute solutions of
nonionized solutes the law may be used for solutions in the
approximate form already given as equation (2),

cvcy

C^c^b = K<- (t const.)

The exact law for equilibrium in solutions is stated in terms of
activities, corresponding to the exact equilibrium law for gases
in terms of fugacities. It will be recalled that the activity a of a
solute is a quantity with the dimensions of concentration, so
defined that it is a measure of the "effective concentration/7
which is the effect of a solute upon the equilibrium. The equi-
librium expression in terms of activities is

and this expression is constant by definition

An activity coefficient is a factoi by which the molality or the
concentration must be multiplied in order to give the activity of
a solute. Since molality (moles of solute per 1000 grams of
solvent) and molal concentration (moles per liter of solution)
are somewhat different in aqueous solutions and largely different
in nonaqueous solutions, it will be evident that an activity is
defined in two different ways. The product of molahty and
activity coefficient 7 gives an activity a = my in moles per 1000
grams of solvent, and the product of molal concentration and
activity coefficient 7 gives an activity Cy in moles per liter of
solution. In this brief treatment of chemical equilibrium we
shall consider only dilute aqueous solutions, in which the dif-
ference between molality and concentration is slight, and we shall
use my and Cy interchangeably for an activity. In more con-
centrated solutions this difference must be considered, of course.

In ideal solutions y is unity at all molalities, in any solution y
approaches unity as the molality approaches zero, and in dilute
solutions of nonionized solutes 7 is very nearly unity and will be

352 PHYSICAL CHEMISTRY

assumed unity in this book. In aqueous solutions containing
ions the activity coefficient 7 is a function of the molality, the
valences of the ions, the effective " diameter " of the ions, and
some other quantities. It is so defined that it approaches unity
as the molality approaches zero, but in moderately dilute solu-
tions of ionized solutes 7 differs materially from unity. Some
activity coefficients at 25° are quoted for illustration, and others
are given in Tables 53, 54, and 98.

m 0 001 0 002 0 005 0 010 0 020 0 050

7 for HC1 0 966 0 952 0 928 0 904 0 875 0 830

7 for KC1 0 965 0 952 0 927 0 901 0 870 0 815

(The general equations showing the relation of m to 7 are given
on page 282.)

Upon substituting a = Cy in equation (9) and rearranging, we
obtain the equation

C1-n?CtTre 'VA°'VT»&

X5_^;E_ = Ka — d— e (t const.) (10)

Since the activity coefficients for each solute depend on the total
solute concentration and not alone on that of the individual
solute, it will be evident that the right-hand side of equation (10)
will often be nearly constant; we shall frequently be content to
assume it constant and write the expression

7—7^ = Kc (t const.) (11)

as a sufficient approximation for our purposes in a first treatment
of equilibrium in aqueous solutions.

In the use of this expression for solutions, calculated concen-
trations will differ from measured equilibrium concentrations
somewhat more than was true in gaseous systems. But such
calculations will not often be in error by 10 per cent and may be
within 1 or 2 per cent in many Instances. The expedient of
employing the approximate equation is commonly a necessary
one, for the use of the exact equation (9) is excluded by a lack
of sufficient data on activity coefficients in all but a few mixtures
at a single temperature. We shall see in some instances that
more exact calculations may be made by assuming that activity
coefficients which have been determined for one solute are appli-
cable to another solute of the same ionic type or to mixtures of

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 353

two solutes of the same type. We turn now to some calculations
that are useful.

Ionization of Weak Acids. — We have seen in Chap. VII that
the relatively small change of equivalent conductance with
changing concentration of a "largely ionized" solute is probably
not due to a changing ion concentration, but to other factors.
In solutions of slightly ionized solutes such as acetic acid or
lactic acid, on the other hand, the change of equivalent conduct-
ance with concentration is largely (though not entirely) due to a
change in the fractional ionization. Thus, the ratio A/A0 of the
equivalent conductance of lactic acid at a concentration C to the
limit that it approaches as C approaches zero is nearly a measure
of the fractional ionization. The chemical equation for the
ionization of lactic acid, which is CH3CHOHCOOH, may be
abbreviated

HLac = H+ + Lac-

and if we follow the common custom of denoting the concentra-
tion of a solute by its symbol in parenthesis, so that (H+) = CH+,
for example, the equilibrium expression is

(H+)(Lac-) =
(HLac)

Each of the ion concentrations in the numerator is CA/A0, and
the concentration of nonionized acid is the difference between
C and this quantity. The equivalent conductances at 25° are
as follows:1

C 00634 00374 00136 000741 0.00354

A 17.9 234 380 503 707

104KC 1.41 144 1.44 143 1.43

Tliey lead to values of Kc that are substantially constant over a
concentration range of twentyfold, and thus the value 1.43 X 10~4
is the ionization constant of lactic acid at 25°.

The equilibrium expression for this ionization in terms of
activities is

Ka =

1 MARTIN and TARTAR, ibid., 59, 2672 (1937). The limiting equivalent
conductance A o is 388 5, derived from extrapolation of the data for sodium
lactate as shown on p. 268.

354 PHYSICAL CHEMISTRY

If the concentrations of the ions are measured by the conductance
ratio, the activities are obtained by multiplying by the appro-
priate activity coefficients. In the strongest solution for which
conductance is given above, the ion concentration is about 0.003,
for which the activity coefficient would be 0 95; in the weaker
solutions, it would increase; and, for the weakest solution, it
would be about 0 97 Even though this factor appears in the
numerator raised to the second power, its effect upon the numeri-
cal value of Kc will not be great, and the variation in Kc within
this range because of assuming an activity coefficient of unity
will be about 4 per cent But in more highly ionized acids, such
as sulfurous acid or chloroacetic acid, larger variations in the
approximation Kc must be expected, and greater deviations of
Kc from Ka must also be expected.

Although the constant Kc was derived from measurements on
solutions containing only lactic acid, it, applies in solutions con-
taining other ions as well For illustration, (II4 ) is about 0.0013
in 0 013()m. lactic acid as shown by the data above. Addition of
0.01 mole of HC1 to 1 liter oi this solution would largely increase
(H+) and require a corresponding reduction in the lactate ion
concentration li the relation in equation (12) is to be maintained.
Let a be the fraction of IlLac ionized in a solution containing
0.0136 mole of HLac and 0010 mole of IIC1 per liter. Since
HC1 is substantially all ionized, (II4 ) = 0.01 + 0 ()136a;
(Lac~) = 0.013(>a, and by difference (HLac) = 00130(1 - a)
Upon substituting these quantities in equation (12), we find a:
is reduced from about 0 1 to 0.014, and (Lac~) becomes about
1.9 X 10~4. The addition of 0.01 mole of sodium lactate to the
acid solution in which G = 0 0130 would change (H+) in the
solution to 1 9 X 10~4; the addition of 0.10 mole of sodium
lactate would reduce (II4) to about 1.9 X 10~5.

Similar considerations would apply to the ionization of any
acid whose ionization was slight, though not to the ionization ol
HC1 in the presence of NaCl They would apply to a weak base
whose ionization was shown by BOH = B+ + OH~ and for
which

(BOH) c

This relation is valid in solutions containing BOH alone, and also

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS

355

for BOH in the presence of an added strong base like NaOH, or a
salt of the base, such as BC1

Some ionization constants for use in solving problems are
given in Table 63 for 25°. These constants have somewhat
different values at other temperatures, as is true of most equi-

TABLE 03 — -SOME IONIZ \TION CONSTANTS AT 2501

Solute

A'r ' Solute

Formic acid

1 7 X 10 4 1 Ammonium hydroxide

1 8 X 10~6

Acetic acid

1 8 X 10-*

Methylanmiomum

Propiomc ncid

1 3 X 1C)-6

hydroxide4

4 X 10"&

Acetoacetic acid *

1 5 X 10~4

1 )imet h vkimmonium

Chloroacetic acid

1 4 X 10~3

hydroxide

5 X 10- b

Phem lacetic acid

5 5 X 10 '6

Trimethvlairimomum

Nitrous acid

1 f> X 10 l

In dr oxide

0 5 X 10~6

Hvdiofluoric acid

7 X 10-*

Pvridinc hydroxide

2 3 X 10-°

Butyric acid

1 5 X 10~5

Aniline hydroxide

4 X 10~10

Valeric acid

1 5 X 10-*

Boric acid

0 6 X 10~10

Lactic acid

1 4 X 10~}

Hydrocyanic acid

4 x 10"10

Hypochloroub acid

5 G X 10-»

Cinnamic acid

3 5 X 10~6

Bcnzoic acid

6 2 X 10-6

Polvbasic Acids

A',

A' 2

Phosphoric acid

7 5 X J0~3

X 10-s

2 X 10~12

Carbonic acid

4 5 X 10~7

6 X 10"11

Sulfurous acid

1 7 X lO-2

X lO-8

Oxalic acid

5 X 10~2

X 10~5

Hydrogen sulnde ' 1 1 X 10~7

X 10~ll>

librium constants, but they may be used at 18° or 20° as well as
for 25° for most approximate calculations, since the change in
this small range of temperature is no greater than the possible
eiror in the values of the constants

Ionization of Polybasic Acids. — Weak acids, such as phosphonc
acid, carbonic acid, tartaric acid, and hydrogen sulfide, ionize in
steps, and an equilibrium expression may be written for each step.
For example, carbonic acid gives in its first-step ionization

1 For many more ionization constants, see Latimer and Hildebrarid,
" Reference Book of Inorganic Chemistry/' The Macmilian Company, New
York, 1940.

356 PHYSICAL CHEMISTRY

hydrogen ions and bicarbonate ions, as shown by the chemical
equation

H2CO3 = H+ + HC()3-

and its corresponding equilibrium expression1 is

(H+)(HCOr)
~~"

The denominator of this expression means (H2C()3 + CO2), of
course, since in all experiments it is this quantity that is meas-
ured; but we follow the usual custom of writing it simply (H2C03)
to indicate all the dissolved, nomomzed carbon dioxide.

The bicarbonate ion acts as a weaker acid than carbonic acid,
from which it came, and ionizes into hydrogen ions and carbonate
ions, HCOa~ = H+ + CO 3 , for which the equilibrium expres-
sion is

It should be noted that Kz is written for the ionization of an
ion into other ions. The expression (H+) in the numerator of
Kz indicates total hydrogen-ion concentration in solution, not
merely that part of it which came from the ionization of bicar-
bonate ions

The first step in the ionization of phosphoric acid is shown by
the equation

H3P()4 = 11+ + H2PO4-

for which the ionization expressions are

(ll3l U4J

Phosphoric acid is intermediate between " strong7' and "weak"
electrolytes in its ionization (about 25 per cent ionized into H+

ITT t U * • +• **U * *r (H+)(HCQ8T)yH+THCO.-

1 For a careful determination of the constant A 0 ~ — /o nr\ \ --

(11 2C U3) TH»CO«

see Maclnnes and Belcher, /. Am Chem Soc , 55, 2630 (1933), who find
Ka = 4.54 X 10~7 at 25°. The change of Ka with temperature is given by
Shedlovsky and Maclnnes [ibid , 57, 1705 (1935)] as follows-

t . 0° 15° 25° 38°

Ka X 107 2 61 3 72 4 31 4 82

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS

357

and H2P04~~ at O.lm.) ; the activity coefficients for the ions would
differ materially from unity, and thus Kc computed from con-
ductance data would not be a satisfactory constant. The second
and third steps in its ionization are shown by the equations

H2PO4- = H+ + HPO4— and HPO4— = H+ + PO4

and the corresponding equilibrium equations are

(li2P()r)

and

(HP04

= K*

(15)

(16)

These expressions are, of course, valid in the presence of phos-
phates and when acids other than phosphoric are present.

Ionization of Strong Electrolytes. — We have already stated
that there is no known measure of the fractional ionization of a
salt or strong acid or base in dilute aqueous solution and that
the bulk of the evidence points toward substantially complete
ionization of these solutes. The figures in Table 64 show that

TABLE 64 — "IONIZATION CONSTANTS" FOR SALTS OF DIFFERENT TYPES*

KC1

Ba(NO,)2

K4Fe(CN)6

Concen-

tration

0 0001

0 0075

0 001

0 000017

0 0005

0 7

0 001

0 035

0 005

0.00018

0 0020

18 0

0 01

0 132

0 01

0 00045

0 012

1,171

0 1

0 495

0 10

0 97

0 1

41,190

1 0

2 22

0 4

842,100

the conductance ratio is a most unsatisfactory measure of the
fraction ionized. The "constants" in this table result from
taking CA/A0 as a measure of the ion concentration and calcu-
lating the ionization "constant" from these ion concentrations.
Their wide variation from a constant value is no reflection upon
the law of chemical equilibrium but only an illustration of the

1 LEWIS, Z physik Chem , 70, 215 (1909).

358 PHYSICAL CHEMISTRY

fact that the fractional ionization of a salt is not to be measured
in this way. We shall assume that salts and strong acids and
bases are completely ionized in the calculations that follow.

Equilibriums Involving Ions. — There are many chemical
reactions involving ions with one another or ions with non-
ionized solutes, which lead to equilibriums that, may be calcu-
lated from the ionization constants oi the weak electrolytes.
For example, the reaction of the salt oi a weak acid and another
weak acid is represented by the chemical equation

Na+Ac" + HNO2 = Na+N(>2- + HAc

in which HAc is used lor acetic acid, which is CHsCOOH Since
complete ionization of the salts is assumed, we may write this
reaction

Ac- + HN<)2 - NO2- + HAc

and the corresponding equilibrium expression for the displace-
ment of one acid by the othei is

=
(Ac-)(HN()2)

The ratios (HAo)/(Ac~) and (NO2~)/(HN02) in this equi-
librium expression show that the ionization equilibriums of
nitrous acid and acetic acid must also be satisfied in the solution
Upon multiplying numerator and denominator of this expres-
sion by (11+) , we find a convenient means of evaluating Kc for
the acid displacement, namely,

(H+)(HAc)(N02-) XHNO,

(II+)(Ar-)(HN08)

(

( }

This means of evaluating equilibrium constants is one that we
shall use again and again. In any equilibrium expression in
which the concentration of a weak acid or weak base and the
concentration of a product of its ionization appear as a ratio,
this expedient should be considered.

In the presence of their salts, these weak acids are very slightly
ionized, so that in the expression for electrical neutrality, which is

(Na+) + (H+) = (NO.-) + (Ac-)
the sum (N02~) + (Ac") is very nearly equal to (Na+). We thus

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 359

neglect (H+) as an addition to (Na+), but, of course, we cannot
call (11+) negligible as a multiplying factor in such expressions as

(H+KNOr) _ 4 - Q_4 , (H+)(Ao-)

(HNO,)~ ~ " ("HA?) ' » X 10

Since all the molecular and ionic solutes appearing in these equa-
tions for the ionization of the acids are present in the solution
containing the two salts and the two acids, both these equi-
libriums must be satisfied Upon dividing the first of these
ionization constants by the second, we obtain as before

aj+)(N()a-)(HAc) = 4 5 X 10"4 = 0

(lT+)(Ac-)(HN02) 1 8 X 10"6

for the numerical value of Kc in equation (17) There can be
only one (H+) m the solution, of course; therefore, it may be
canceled from the expression

A numerical example will make the use of the equations
clearer Suppose a liter of solution containing 0 1 mole oi NaAc
is added to a liter of solution containing 0 2 mole of IINO2, and
let x be the equilibrium concentration of nitrite ion in the result-
ing solution The chemical equation shows that jc is also the
concentration of HAc, since they iorm in equal quantities The
total (HNO2) + (NO2~) is 0 2 mole in 2 liters, which gives
(HN02) = (0.1 - x)] also, (Ac") = (005 - x). [We have
neglected (H+) in setting (Ac~) + (NO2~) equal to (Xa4), and we
shall find in a moment that this assumption is justified by the very
small value of (11+) ] Upon making these substitutions in the
equilibrium equation,

(0 1 - a-) (005 - x)

we find x = 0.0482 and this is the equilibrium concentration of
N02- and of HAc.

If each of the original solutions had been 0.2m. the equilibrium
expression would have been

= 25

(0.1 - .r)2

from which x = 0.0833. Thus, in this second system XaAc was
present in larger quantity at the start, and therefore a larger
quantity of it reacted.

360 PHYSICAL CHEMISTRY

Returning now to the first system in which (HAc) = 0.0482,
(NOr) = 0.0482, (HNO2) = 0 0518, and (Ac-) = 0.0018, we
may insert these concentrations in either of the equilibrium
relations

(H^AC:) = 10_6

(HAc)
or

(H+.)(NOr)

/TTTVT/ \ \ - T.C* /\ 1 V/

(HM)2)

and solve for (H+), which is 4 8 X 10~4. Thus, in taking
(HNO2) as (0.1 — a*) in the calculation above, we have neglected
4.8 X 10~4 in comparison with 10"1, which is justified in view of
other assumptions that introduce a larger error.

The equilibrium in terms of activities for this system is

In dilute aqueous solutions the activity coefficients of nonionized
solutes are substantially unity, and the activity coefficients for
ions of the same valence in a mixture are determined largely by
the total ion concentration, which is unchanged in this system
as the reaction proceeds Hence Kc and Ka are substantially
equal in this system. But it must be understood that, in the
equilibriums shown in equations (12) to (10) and in many others
trO follow, there will be a real difference between Kc and Ka.

The lonization of Water. — The slight ionization of water into
hydrogen ions and hydroxyl ions is of the greatest importance
in some respects and of no consequence whatever in other
respects. Since the equilibrium hydrogen-ion concentration in
pure water at 25° is 1 mole in 10,000,000 liters of water, only
one molecule out of 550,000,000 is ionized at any given
moment, and it seems surprising that this could be of any conse-
quence or indeed that the dissociation could be measured. There
are several ways in which the ion product (H+)(OH~) may be
determined, and of these the conductance of pure water has
already been mentioned Other ways will appear later, some of
them in this chapter, and some in Chap. XIX. Following the
standard procedure, we write the chemical equation

H20 = H+ + OH-

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS

361

and formulate the equilibrium expression
(H+XOH-) = Kw

(t const.) (18)

It will be noted that the concentration of water has not been
included in the ionization equilibrium. In a dilute aqueous
solution the concentration of water is substantially constant
when a small fraction of the water ionizes; thus we are justified
in including this constant concentration in the value of Kc.
There would, of course, be no objection to inserting the concen-
tration of water wherever water is a reacting substance, but one
must consistently write the water concentration in the equi-
librium expression or consistently include its value in Kc. We
follow the more common custom of including the concentration
of water in Kc and of writing this special constant as Kw. In
the hydrolysis reactions considered in the next section and, in
general, whenever water is a reacting substance in dilute solutions,
we shall also include the water concentration in Kc.

TABLE 65 — ION PRODUCT FOR WATER1

*,°c

1014A^

/, °C

1014A'W

0 11

9 65

0 29

0 68

100

1 00

150

230

1 47

200

550

2 91

250

700

5 48

500

400

Not only does the ion product Ku change with temperature as
do other equilibrium constants — it is conspicuous for the rapidity
of this change. The value at several temperatures is given in
Table 65. Equilibrium between hydrogen and hydroxyl ions
prevails in every aqueous solution, whether acid, alkaline, or
neutral, and regardless of the presence of other solutes. This is
not to say that the ionization of water is important or even of

1 The values from 0 to 50° are by Harned and Mannweil«r, J. Am Chem.
Soc., 67, 1873 (1935) ; they are based upon the electromotive force of an acid-
alkali cell that is described in Chap. XIX See also Harned and Geary,
ibid , 59, 2032 (1937). The values for 60° and above are by Bjerrum in
"International Critical Tables," Vol VI, p. 152.

362 PHYSICAL CHEMISTRY

any consequence in every solution. In aqueous solutions of
strong acids, strong bases, and their salts and in solutions of all
but the weakest acids and bases in the absence of their salts,
water behaves as a nonionized, inert solvent. But the alkaline
reaction of sodium carbonate solution or potassium cyanide solu-
tion and the acid reaction of ammonium chloride solution are
connected with the ionization of water in a way that is explained
in the next section Since the product (H+)(OH~) is constant at
a given temperature, it will be clear that increase of one con-
centration requires a decrease in the other. If the hydrogen ions
in water are removed by union with some other ion, more water
ionizes to restore the equilibrium, and (H+) will no longer be
equal to (OH~). Even so, the product (H+)(OH~) will remain
constant at equilibrium.

Hydrolysis, a. Negative Ions — Since salts of weak acids
ionize in the same way and to the same extent as the salts of
strong acids, an aqueous solution of such a salt contains negative
ions of a weak acid from the ionization of the salt and hydrogen
ions from the ionization of water. Th^se ions require the
presence of nonionized weak acid at a concentration that satisfies
the ionization equilibrium for the weak acid. The chemical
reaction that supplies this acid is called hydrolysis. As an
illustration, the hydrolysis of cyanide ion is shown by the equa-
tion

CN- + H20 = OH- + HCN

and the equilibrium expression for the reaction is
(OH-)(HCN) „ u

^r^^pp-- = K* (t const- >

We combine the water concentration with Kc, as was done
for the ionization of water. In this system the equilibriums
(H+)(OH-) = Kw and (H+)(CN-)/(HCN) = #HCN must both
be satisfied; and, upon dividing the first of these by the second,
we obtain the numerical value of the hydrolysis constant

(H+)(OH-)(HCN) Kw .,

(H+XCN-) =^TN = Ac (/COnst') (19)

In the presence of cyanide ions the fractional ionization of HCN
is very small; thus its concentration is the salt concentration

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 363

multiplied by the fraction hydrolyzed. We may express the
concentrations of all of the solutes in terms of these quantities,
since, if h is the fraction hydrolyzed at the concentration C,
Ch = (OH-) = (HCN), and C(l - h) = (CN~). Substituting
also the numerical value of the constants in the expression for
Ke, we have

c/?2 i x io-14

1 - h 4 X

= 2 5 X 10~6 at 25° (20)

The fraction hydrolyzed may be reduced by adding KOH or
HCN to the solution; but if this is done, the special relation
C7/2/(l — h) is no longer a proper one, though equation (19) is
still valid. For instance, if to a liter of solution 0. Ira. in KCN we
add 0.01 mole of KOH, (OH-) becomes (0.1 A + 0.01), (CN~) is
0.1(1 - /?), and (HCN) is 0.1 fc.

It will be understood that as hydrolysis removes H"1" to form
HCN, more water ionizes to satisfy the equilibrium expression
(H+)(OH~) = Kw. Since the chemical reaction of the salt with
water forms a strong base and a weak acid in chemically equiva-
lent quantities, the solution at equilibrium is alkaline. In
O.Ira. KCN, h is about 0.016 and (OH~) = 0.0016, and the con-
centration of H+ is 6 X IO-12.

Similar behavior is shown by the negative ions of all weak
acids, with smaller fractions hydrolyzed when the ionization
constants are larger. Hydrolysis is not confined to ions of unit
valence and is indeed more likely for ions of higher valence. For
example,

S— + H2O = HS- + OH-

is a reaction that takes place when any sulfide dissolves in water.
Carbonate ions hydrolyze, as shown by the equation

C03— + H20 = HCOr + OH-

•

and form alkaline solutions when carbonates dissolve in water.

b. Positive Ions. — Salts of weak bases yield the positive ions of
the base when dissolved in water, and thus equilibrium between
these ions and hydroxyl ions from water is established. The
hydrolysis of ammonium ion is shown by the equation

NH4+ + H20 = NH4OH + H+
for which the equilibrium expression and its relation to Kw and

364 PHYSICAL CHEMISTRY

^NH4oH arc given by the equation

IT (NH4OH)(H+) Kw (. w,x ,on
Kc = /IVTT +\ = jr '* const.) (21)

(JN H 4 ) ANH4OII

This hydrolytic reaction forms a weak base and hydrogen ion in
chemically equivalent quantities; and since the slight ionization
of the weak base is greatly repressed by the relatively high con-
centration of ammonium ion from the salt, the resulting solution
is slightly acid at equilibrium. In the absence of added acid or
added NI^OH, the fraction hydrolyzed at any salt concen-
tration is given at 25° by an equation similar to (20), namely,

r/?2 1 v 10~14

c/i I X iu >5 x 10_ioat25°

(1 - h) 1 8 X 10-5

whence h = 7.4 X 1Q-6 at O.lm. and (H+) = 7.4 X 10~6. In
mixtures of NH4OH and NH4C1 the hydrogen-ion concentration
is shown by a rearrangement of equation (21),

CH+\ _ K K v H)-10
(H ) - 5.5 X 10 ~

__
(~NH40fl)

The fact that polyvalent positive ions hydrolyze in steps is
not as commonly realized as it should be. Ferric chloride solu-
tions are known to be acid, and ferric hydroxide is known to be
almost insoluble (about 10~9 mole per liter), and yet the common
explanation is the formation of hydrogen ions and ferric hydroxide.
The hydrolytic reactions are

H20 = FeOH-H- + H+
and

FeOH-H- + H20 - Fe(OH)2+ + H+

Both thf species FeOH++ and Fe(OH)2+ have been shown to
exist in ferric solutions,1 and the equilibrium constant for the
first reaction has been shown2 to be about 5 X 10~3.

c. Hydrolysis of Both Ions. — When salts derived from weak
acids and weak bases dissolve in water, hydrogen ions from water
combine with the negative ion of the salt to form the weak acid,
hydroxyl ions combine with the positive ion of the salt to form

JLAMB and. JACQUES, J. Am. Chem Soc., 60, 967 (1938)
2 RABINO WITCH and STOCKMAYER, ibid , 64, 335 (1942).

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 365

a weak base, and the extent of hydrolysis is much greater.
For an illustration consider ammonium lactate, for which
the hydrolytic reaction may be written as follows if Lac"" denotes
CHsCHOHCOO-

NH4+Lac- + H2O = NH4OH + HLac
and the equilibrium expression is

(NH4OH)IILac) v

TNH4+)(Lac-) = A'

,4 .. /OON

(t C°nst-} (22)

In order to evaluate Kc for this hydrolysis, we note that the
ionization constants of the acid, the base, and water must all
be satisfied in the equilibrium system By multiplying both
numerator and denominator of equation (22) by (H+) (OH~) we
see that Kc = K^/J^HLac-KNi^oH = 4 X 10~6 at 25° in this sys-
tem and that the fraction hydrolyzed in O.lm NHJ^ac is
h = 0.002 In this solution (H+) (OH~) = Kw, as is always
true in any aqueous solution; but since the weak acid and weak
base are not ionized to the same extent, (H"1") and (OH~) are not
equal. The equilibrium concentrations are (Lac~~) = 0.0998 and
(HLac) = 0.0002, whence, from the ionization constant of the
acid, we calculate (H+) = 2.8 X 10~7. Thus, it is shown that
in the above calculation the concentration of Lac" derived from
the ionization of the acid is an insignificant quantity compared
with that from the salt.

Experiment1 shows that ammonium phenolate is 84 per cent
hydrolyzed at 25°. If we abbreviate the equation

NH4+PH- + H20 = NH4OH + HPh

in which Ph stands for CeHsO, the hydrolysis equilibrium may be
written

(NH4OH)(HPh)_

(NH4+)(Ph-) ^°

If C is the original concentration and h the fraction hydrolyzed
(which we have stated to be 0.84), we see that

(NH4OH) = (HPh) = Ch (NH4+) = (Ph-) = C(l - fc)

1 This fraction is given by O'Brien and Kenny for 25° over the concen-
tration range 0 25 to 1 0, in J. Chem Education, 1939, p 140

366 PHYSICAL CHEMISTRY

and upon substituting these equalities in the equilibrium expres-
sion, C cancels out, leaving

" Jf — &w — OQ of OKO

JVc — 'irr TT — AO <*,l &O

a- 7-7-7, — Jc —

— rl)

For salts of weak bases and weak acids, in the absence of
added free base or free acid, the extent of hydrolysis is thus seen
to be independent of the salt concentration, to the extent that the
variation in the activity coefficients can be ignored. It will be
recalled that equation (20) contained the concentration of the
salt ; thus for the hydrolysis of positive ions alone or negative ions
alone the extent oi hydrolysis varies with the concentration.
The measured fractional hydrolysis and the known values of Kw
and Kc for the ioiiization oi NH4OH enable us to calculate from
equation (23) that Kc for the ioiiization of phenol as an acid is
2 X 10~n. Other experiments upon phenolates lead to a some-
what larger ioiiization constant for phenol at 25°, namely, about
1.3 X 10~10, which would correspond to about £3 for the fractional
hydrolysis of ammonium phenolate. In this instance, as in so
many in physical chemistry, it is difficult to choose between
conflicting determinations of a given physical quantity, and one
should remember that the experimental difficulties of measuring a
quantity as small as 10~10 or 10~n are great.

In general, the numerical values of Ka and Kc will be rather
close together for the hydrolysis of a positive ion or a negative
ion alone in a given solution, and the difference between Ka and
Kc will be much greater when both ions hydrolyze Denoting
by 7 the activity coefficient that applies to all the ions in a solu-
tion and remembering that the activity coefficients for nonionized
solutes in dilute solution are very close to unity, we find the
equilibrium expressions for the hydrolysis of a single ion and of
both ions to be

,., (Chy)Ch , ^ Ch-Ch

- rr— and

a CV(1 - h)*

It will be seen from these expressions that the activity coefficient
7 for the ions cancels from the expression for the hydrolysis of a
single ion and appears as the square in the denominator of the
expression for the hydrolysis of both ions.

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 367

Hydrolysis and lonization of Intermediate Ions. — The ions
formed in the first ionization of dibasic acids, such as HS~~ from
H2S, HC03- from H2C03, or HS03~ from H2SO3, may hydrolyze
and ionize in such quantity that both reactions must be consid-
ered in the same solution. For example, the reactions

HS- + H2O - OH- + H2S

HS- = H+ + S—

both occur in a solution of NaHS. The fact that the solution
is alkaline shows that the first reaction is more important than
the second. In such a solution there are six solutes at equilib-
rium, Na+, H+, HS-, 8 — , H2S, and OR-. If we consider a
solution O.lm in NaHS at 25°, (Na+) is 0.1 and the five other
equilibrium concentrations are fixed by five equations: an
equation for electrical neutrality, which always exists in any
ionic solution; a sulfur balance; and three ionization constants.
The equations are

(Na+) + (H+) = (HS-) + 2(S~) + (Oil-) (a)

0.1 = (HS-) + (S— ) + (H,SJ (6)

(H+XHS-)

(H2S)
H+XS— )

= 1.1 X 10-7 ^ (c)

~7n<Fr = l x 10~15 (d)

(H+)(OH~) = 1.0 X 10-14 (e)

A solution of five simultaneous equations is of course possible,
but tedious, and is unnecessary for the present purpose if some
suitable approximations are made. If we neglect (H+) in com-
parison with (Na+) and equate the right sides of equations (a)
and (6), we have

(S— ) + (OH-) = (H,S)

We may show that (S ) is small in comparison with (OH") by
dividing (d) by (e) and noting that (HS") is nearly 0.1, which
shows that (S )/(OH~) is approximately 0.01.

Thus we see that hydrolysis of the negative ion is the important
reaction in this solution, and that (OH") = (H2S) within 1 per
cent. The equilibrium relation for the hydrolysis and the value
of its constant are given by an equation like equation (19),

368 PHYSICAL CHEMISTRY

namely,

(H2S)(OH-) _ Ku 0

(HS-) ~ Xx " 9 X 1U at J5

Recalling that (H2S) and (OH~) are nearly equal and that (HS~~)
is about 0.1, we find (H2S) = (OH-) = 9.5 X 10~5. From the
value of Kw, (H+) = 1 X 10~10 ; and, upon substituting this in (d),
we find that (S — ) = 10~6 and (HS~) is between 0.0999 and 0.1.
Thus all the equilibrium concentrations are fixed within a per
cent or two, which is as close as the numerical values of the con-
stants will justify.

In a solution of O.lm. in NaHSO3 hydrolysis and the ioniza-
tion of HSOr are of nearly equal importance. The chemical
equations for the processes are

HSOr + H2O = OH- + H2SO3
HS03- = H+ + S08—

As before, we have five solutes in addition to sodium ion, H+,
OH", HSOr, SO3 , and H2S03, requiring five equations. They
are again an electrical balance, a sulfur balance, and three equi-
librium constants

(Na+) + (H+) = (HSOr) + 2(80,—) + (OH-) (a)

0.1 = (HSOr) + (S03— ) + (H2S03) (6)

(H+)(HSOr) 0017 K M

(H2S03) =0017 = A, (C)

_ 1Q_8 _ R

- b x 1U - Az W

(HSOr)
(H+)(OH-) = 10-14 = Kw (e)

For a first approximation we neglect (H+) and (OH~) in (a) and
equate the right sides of (a) and (6), though because of the smaller
ionization constant of H2SO3 the neglect of (H+) may not be
justified, and find

(S03— ) = (H2S03) (/)

Upon multiplying (c) by (d) and noting the equality in (/), we
find

(H+) = \/M~2 = 3.2 X 10~5

Now substitute this (H+) in (c), and note that (HSOr) is nearly
0.1, whence

(H2SO8) = 1.9 X 10-4 = (SO3— )

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 369

the last concentration following* from (/). Finally, from (e),
(OH-) = ^ = 3 X 10-10

The results of such a calculation should always be reviewed to
check the assumptions made. In this calculation we see that
(H+) is 17 per cent of the calculated (H2SO3) and hence not
negligible in equation (a), but we see also that (OH~~) as an addi-
tion quantity is negligible in this equation.

As a second approximation, transpose (H+ ) to the right side of
(a), then equate this to the right side of (b), which gives

(S08— ) = (H2S03) + (H+) (?)

If we take (HS(V) as 0.1 in (c), it follows that
(H+) = 0.17(H2S03)

whence, from (gr), (SO3 ) = 1.17(H2S03); this relation is now
substituted into (c) to obtain (H+), and then the other operations
are performed as before to obtain the other concentrations.

(H2SO3) = 2.1 X 10-4, (SO3— ) = 2.5 X 10~4, and

(Oil-) = 3.6 X 10-10

•

From (6), (HSO8") = 0.0995 in place of 0.10.

It should be noted that, while hydrolysis is more important
than ionization of HS~ in NaHS and while hydrolysis and ioniza-
tion are about equal in NaIIS03, both the effects are small.
Whenever hydrolysis involves only a negative ion or only a posi-
tive ion, the fraction hydrolyzed will usually be small at moderate
or high concentrations. But at extreme dilutions the fraction
hydrolyzed may be large, as, for example, in a saturated solu-
tion of CaC03 in which the molality is about 10~~4 and more than
half the solute is in the form of hydrolysis products.

Buffer Solutions. — In a mixture of a weak acid and one of its
salts the acid is very slightly ionized, and the salt is assumed to be
completely ionized, so that the very small hydrogen-ion concen-
tration is dependent on the ratio of salt concentration to acid
concentration at a given temperature. Such a solution will have
a hydrogen-ion concentration that is unchanged upon moderate
dilution and nearly unchanged by the addition of a relatively
small amount of acid or base. Thus in a solution containing 0.1
mole of acetic acid and 0.09 mole of sodium acetate per liter of

370 PHYSICAL CHEMISTRY

solution, the hydrogen-ion concentration is

(H+) = Kc ^j~~^ = 1.8 X 10~5 £~ = 2 X 10-6 at 25°

Dilution with a liter of water would leave the ratio (HAc)/(Ac~)
unchanged to the extent that Kc is unchanged, and thus (H+)
would also be unchanged within the same limitation. It will
be recalled that for the ionization of acetic acid Ka = Kcy2', and
since this dilution changes 7 from 0 82 to 0.87, Kc and (H+) will
change about 10 per cent. Hydrogen ions to yield 2 X 10~6 mole
per liter would come from the ionization of an amount of acetic
acid that is negligible in comparison with the total acid present,
and thus the solution is " buffered'7 to maintain a nearly constant
(H+).

The ionization constant for the second hydrogen ion of phos-
phoric acid is 6 X 10~8 at 25°, whence, by rearranging the expres-
sion for its ionization equilibrium, we have

(H+) = 6 X 10- at 25°

In a solution containing 0.10 mole of NaH2P04 and 0.06 mole of
Na2HPC)4 in any reasonable volume of water at 25° the ratio
(H2P04-)/(HP04— ) is 1%, (H+) is 1.0 X 10~7, and therefore
(OH-) is also 1.0 X 10~7. Addition of 0.001 mole of HC1 to
such a solution would cause the reaction

H+ + HPO4— = H2P04~

to take place, reducing the quantity of HPO4 from 0.060 to
0.059, increasing the quantity of H2PO4~ to 0.101, and changing
the ratio (H2P04~)/(HPO4 ) by about 2 per cent; accordingly
(H+) would be changed by this amount. Addition of 0.001 mole
of HC1 to a liter of water would change (H+) from 10~7 to 10~3.
This solution is a "buffer" that maintains a nearly constant
(H+), while water has no capacity to maintain a constant (H+)
against small amounts of acid or base. Of course, this phosphate
mixture is also a buffer against small amounts of alkali, and
against dilution as well.

Since the ionization constants of many weak acids are not
accurately known, the usual practice is to make up a series of
solutions of different ratios of (H2P04~~) to (HP04 — ), or other
salts and acids, and to determine the hydrogen-ion activity in

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 371

them from the potential of a cell composed of a hydrogen elec-
trode dipping into the solution and a reference electrode. The
electromotive force of such a cell is

E = 0.0592 log aH* + constant

as we shall see in Chap. XIX, and, for the approximation we have
been using, this becomes E = 0.0592 log (H+) + constant. For
illustration, solutions of 9.08 grams of KH2P04 per liter and
1188 grams of Na2HPO4 2H20 per liter, respectively, when
mixed in the quantities shown below, yield buffers in which (H+)
at 25° has the value given under each mixture, determined from
the potential of a cell

Ml Na2HPO4

1 0

2 0

3 ft

4 0

5 0

6 0

7 0

8 0

9 0

Ml KH2PO4

9 0

8 0

7 0

6 0

5 0

4 0

3 0

2 0

1 0

107(H+)

2 5

1 6

1 0

0.6

0 4

0 2

5 9

6 3

6 5

6 6

6 8

7 0

7 2

7 4

7 7

The hydrogen-ion concentrations of buffer solutions will
change with changing temperature, for Kw and the ionization
constants of the acids or acid ions change with temperature at
unequal rates. For illustration, Kw is 10~15 at 0°C , 10~u at
25°C., and 5 X 10~13 at 100°C ; and in this range of temperature
the ionization constant for acetic acid would change only a few
per cent.

Consider a solution containing 0 1 mole of K2HPO4 and 0.1
mole of KH2PO4 in 1000 grams of water at 25°. By titrating
a portion of this solution with bromophenol blue as indicator it
would appear to be about 0.1 N base, by testing it with nitrazine
yellow it would appear to be "neutral," and by titrating it with
phenolphthalein as indicator it would appear to be about 0.1 N
acid. The terms "acid," "neutral," and "alkaline" are all
inappropriate for describing this solution; the correct statement
applying to it is found in the sixth column of the table above,
namely, (H+) = 1.6 X 10~7.

The pH Scale. — In the range between "slightly acid" solutions
and "slightly alkaline" solutions the change of (H+) is so large
relatively, though (H+) is very small in all of them, that a
logarithmic scale is convenient. This scale was suggested by
S0rensen in 1909 and defined as

PH = - log (H+) or pH = log (24)

372 PHYSICAL CHEMISTRY

The pH values given for the phosphate mixtures above are
expressed in this way. For illustration, when (H+) = 5 X 10~7,
log(H~*) = — 6.3 and pH = 6.3. This is a reciprocal logarithmic
scale of acidities as defined and as commonly used; therefore, the
actual acidity of a solution in which pH is 7.3 is one-tenth of that
in a solution whose pH is 6.3. (Occasionally the alkalinity of a
solution is expressed as pOH, which is the logarithm of the
reciprocal of the hydroxide-ion concentration, but the use of pH
is more common.)

Such a definition is clear enough for most purposes when the
acidity is produced by an acid alone. But we have seen in the
preceding pages that pH is difficult to control without the use of
buffers when it lies between 4 and 10. The activity coefficient 7
in a mixture of an acid and a salt depends on the total ion con-
centration, its value is 0 8 to 0 9 when the salts added to the
weak acid in buffer solutions are 0.1 to 001m., and in these
mixtures aH+ = WH+TH+ Two other definitions, among the many
proposed for one reason or another, will suffice to show that con-
fusion results unless one states which definition of pH is being
used, namely, pH = — log aH+ and

nH = E - E*
p

2.3RT/F

in which F is Faradays' constant, E° is the potential of a constant
" reference electrode, " and E is the potential of an electrolytic
cell:

Pt, H2 (1 atm.), [unknown solution], KC1 (satd.), ref. elec.

There are valid objections, apparently, to any one definition of
pH and an obvious need for a single definition of pH that has
not yet been met. The distinctions are best reserved for a
second consideration of physical chemistry and omitted from a
first consideration,1 but beginners should realize that the con-
fusion exists. Admitting its existence, we postpone considera-
tion of the definition in terms of cell potentials until Chap. XIX,

1 See, for example, Maclnnes, Belcher, and Shedlovsky, J Am. Chem
Soc., 60, 1094 (1938), for a discussion of this topic and data on pH to be
assigned to acetate and phthalate buffers. Other buffer solutions and the
corresponding pH values in the range 2.27 to 11.68 are given by Bates,
Earner, Manov, and Acree, in /. Research Nat. Bur. Standards, 29, 183 (1942).

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 373

and consider pH = — log (H+) as a sufficient approximation for
the purpose of discussing indicators and titration errors.

Experimentally, the pH in a solution is measured cither from
the potential of a cell1 or by comparisons of indicator colors in the
solution with the colors in solutions of known pH. Each method
has certain advantages, and each has certain limitations that are
not as commonly appreciated as they should be. These limita-
tions will be given later in this chapter for the indicator method
arid in Chap. XIX for the potential method, but we may say
here that there is no method of determining pH applicable to
every kind of solution. " Interfering7' materials, especially
oxidizing or reducing agents, colloids, protein, and other organic
materials, and certain salts, may cause "measured" pH values
to be in error by 1 to 5 units, and many values recorded to 0 1
unit are in error by several times this amount.

Indicators. — An indicator is a substance that changes its color
with changing hydrogen-ion concentration. Most of the acid-
alkali indicators familiar in analytical chemistry change color
conspicuously within a pH range of 1.0 or less, and this rapid
change is desirable for such work Other indicators change over
ranges as wide as 2.0 pH, and they are useful for other purposes.
But an indicator is not in general a substance that changes color
at the true end point of a titration; it fulfills this desirable condi-
tion only when it is properly selected for the titration to be done.
The hydrogen-ion concentration at the end point should be that
in a solution of the pure salt formed from the acid and base, for
only "nder this condition will the acid (or base) added be equiva-
lent to the base (or acid) being titrated. Since the hydrogen-
ion concentration in 0.2m. ammonium chloride differs from that
in 0.2r?.\ sodium acetate by about 4 pH units, it will be evident
that an indicator suitable for ammonium hydroxide will not serve
for titrating acetic acid.

Most indicators behave as if they were weak acids that change
color when neutralized, though the color changes result from
structural changes that accompany the neutralization, rather
than from simple ionization. For our purposes we may consider
an indicator as a weak monobasic acid whose color changes upon
neutralization, and we define the "indicator constant " as

1 For descriptions of the apparatus and procedure see catalogues EN96
and EN96 (1) of Leeds and Northrup Co

374

PHYSICAL CHKMIRTRY

Kt = (H+)

(HIn)

(25)

in which expression the ratio (In~)/(HIn) is the ratio of the con-
centration of indicator ion to nonionized indicator. If we let
x be the fraction of the indicator showing its "alkaline" color and
(1 — x) be the fraction having its "acid" color, whether or not
these are actually ions and free acids, respectively, this equation
may be arranged in the form

(H+) = A'. -~

(26)

and used to determine hydrogen-ion concentration after K% is
known

When the indicator constant i» much smaller than the loniza-
tion constant of a weak acid being titrated, nearly all the acid
is neutralized before the indicator is neutralized, as may be seen
from an equation like (17). When the indicator is present in
much smaller quantity than the acid, as is commonly true, the
residual acid is negligible when the indicator is neutralized, if one
of the proper Kr has been chosen.

In the presence of relatively large amounts of neutral salts
such as KC1, the value oi Kt as defined in equation (25) changes
with the salt concentration,1 and change of color at constant (H+)
is observed upon the addition of KCL These changes in K% are
not much greater than those which would be observed in the
ionization constant Kc for any weak acid in the presence of KC1
at these concentrations. They arise from changing activity
coefficients and from other causes that are obscure In dilute

1 See CHASE and KILPATRICK, J Am Chem. Soc , 64, 2284 (1932). The
ratio of Ki in 0 Ira KC1 to its value in other molalities of KC1 to give correct
(H+) is as follows for three common indicators:

Ratio A\ to Kl m 0 Ira KC1

muicUJi v i\\'i

Bromocresol preen

( "hlorophenol red

Methyl red

0 5

1 31

1 22

0 89

1 0

1 18

1 48

0 75

2 0

0 95

1 20

0 51

3 0

0 78

0 72

0 28

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 375

solutions and when great precision in pH measurement is not
required, they may be ignored. But it should not be forgotten
that when (H+) is very small the other ion solutes are not ideal
at their much greater concentrations and that the activity
coefficient for hydrogen ion depends on the total ion concentra-
tion rather than on its concentration alone

Indicators give pH indications of reasonable accuracy when
they are used under proper conditions, and they are open to
moderate or serious errors when applied under improper condi-
tions. It is therefore important to realize that such conditions
exist as a limitation to the use of indicators, and we now turn
to some of them. (1) Proper temperature control is essential.
Some indicators change " range" by one pH unit or more for a
temperature change of 50°; and since ionization constants and
Kw are also temperature functions, the use of indicators at other
than the standard temperature yields uncertain pH determina-
tions (2) Organic liquids, such as alcohol, may shift pH indica-
tions by one unit or more, up or down, and no simple method of
estimating the shift is known. (3) Proteins shift pH indications
so seriously that indicators may not be used in their presence
except for rough measurements. (4) Colloids, soap, soil suspen-
sions, and colored solids in general render pH indications in error
by unpredictable amounts (5) Oxidizing or reducing agents
may bleach the color of an indicator and render its pH indication
wholly false. ((>) Insufficient buffering leads to false pll indica-
tions; for some of the indicators are acids, and others are made
up in dilute sodium hydroxide solution. The most-quoted illus-
tration is the shift of pH from 7 to 5 by the addition of a few
drops of methyl red to a test tube of pure water, but many other
less extreme examples are known.

While this list of restrictions to the use of color indicators is
discouragingly long, it is far better to realize that pH indications
are subject to these limitations than to make measurements in
ignorance of the conditions and rely on inaccurate results.

Most of the commercial indicators are described in terms of
the "pH range" within wrhich color changes are observed, as, for
example, bromothymol blue, yellow to blue, 6.0 to 7.6. Perma-
nent color standards for steps of 0.2 pH are available or may be
prepared in the laboratory from buffer solutions to which meas-
ured volumes of dilute indicator solutions are added. Table 66

376

PHYSICAL CHEMISTRY

shows the range of some common indicators.1 No satisfactory
indicators for solutions more alkaline than pH = 11 are known 2

TABLE 66. — SOME INDICATOR RANGES

Indicator

pH range

Color change

Metacresol purple

12-28

Red-yellow

Bromophenol blue

30-46

Yellow-blue

Methyl orange

28-40

Orange-yellow

Methyl red

42-63

Red-yellow

Bromoeresol green

40-60

Yellow-green

Bromocresol purple

52-68

Yellow-purple

Nitrazine yellow

64-68

Yellow-blue

Bromo thymol blue

60-76

Yellow-blue

Phenol red

68-84

Yellow-red

Cresol red

72-88

Yellow-red

Phenolphthalem

8 4-10 0

Colorless-pink

Thymol blue

80-96

Yellow-blue

Orthocresolphthalem

82-98

Colorless-red

Thymolphthalem

10 0-11 0

Colorless-red

Titration Errors. — While a perfect titration of an acid with a
base requires that the indicator change color at the (H+) of the
salt solution and not over a range of pH, this condition is neither
possible to meet nor necessary for an acceptable titration For
example, in the titration of lactic acid with NaOH, we may assume
that the sodium lactate concentration at the end point is about
O.lm. and calculate the fraction of the lactate ion hydro! yzed
and (H+) from equation (20),

(OH-)(HLac) _ O.U2 _ 30'14 . _0

— - - -

from which h = 8.5 X 10~6, (OH~) = 8.5 X 10"7, and
(H+) = 1.2 X 10~8

in the solution at the true end point But, in a solution in
which the end point is 0.1 per cent short, the ratio of free lactic
acid to lactate ion is 1 : 1000; and, upon substituting this ratio into

1 For other data see ''International Critical Tables/' Vol I, p 81 Dis-
cussion and procedures will be found in Clark, " Determination of Hydrogen
Ions," Britton, " Hydrogen Ions," and "The A B.C of Hydrogen Ion
Control," by the LaMotte Chemical Products Company.

2 See Ind. Eng Chem., Anal. Ed., 1, 45 (1929).

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS

377

the ionization expression Kc for lactic acid, (H+) = 1.4 X 10~7.
If the end point is 0 1 per cent overrun, hydrolysis of the lactate
ion is negligible in the presence of a slight excess of base, and
the titrated solution acts as a diluent for the excess standard
base. For each 100 ml. of base required for the titration, 0.1 ml.
in excess is diluted to the final volume of the titrated solution,

95 96 97 98 99 100 101

Per Cent of Theoretical Base Added
FIG 46 — Titratiori diagram for acids.

which might be about 200 ml. Thus, if the standard solution
were* 0.2 N, (OH~) = 0.2 X (0.1/200) = 10~4 and (H+) would
be 10~10. If an error of less than 0.1 per cent is acceptable, any
indicator that changes color between 1.4 X 10~~7 and 1 X 10~10 is
satisfactory, and all those listed between phenol red and ortho-
cresolphthalein in Table 66 (or any others of similar pH range)
will serve.

The " titration curves" that are familiar from analytical chem-
istry are only curves that show the fraction of a base or acid
titrated in terms of the pH of the solution. Points on these
curves are calculated in the way shown in the previous para-

378

PHYSICAL CHEMISTRY

graph. For the titration of a strong acid with a strong base, any
indicator that changes color between pH = 4 and pH = 10 will
serve; for weak acids or bases the range is narrower; and, for
extremely weak acids such as hydrocyanic acid or boric acid,
the range of pH for accurate titration is impossibly small.
Titration curves for a few acids are shown in Fig. 46. It may
be seen from this figure that an indicator which changes color at
pH = 7 (true neutrality) would cause an error of 0.5 per cent in
titrating acetic acid and an error of more than 10 per cent in
titrating carbonic acid Thymol blue would be excellent for
acetic acid but would cause an error of perhaps 1 per cent with
carbonic acid
TABLE 67 — PERCENTAGE DISSOCIATION OF GASES* AT 1 ATM. PRESSURE

Tabs

C02

II2

1,000

0 00002

1,200

0 00093

1,400

0 0146

1,600

0 110

0 005

1,800

0 546

0 029

1,900

1 04

2,000

1 84

0 112

2,200

5 0

0 392

2,500

15 6

1 61

3,000

48 5

9 03

3,400

24 5

4,000

62 5

Change of Chemical Equilibrium with Temperature.— Since an

increase in the temperature of a chemical system at equilibrium
requires the absorption of heat by the system, the qualitative
effect on equilibrium is seen to be a change of composition in
which the chemical reaction absorbing heat is favored. The
dissociations of S03, NH3, NOBr, and N2O4 are attended by the
absorption of heat, and the data quoted for these substances
earlier in the chapter show an increased 'extent of dissociation
at higher temperatures for all of them Hydrogen and CO 2 also

1 These figures are quoted from Langmuir, «/ Am. Chem Soc , 37, 417
(1915), Ind. Eng. Chem , 19, 667 (1927), slightly different extents of dissoci-
ation are given by Giauque, J. Am* Chem Soc , 52, 4816 (1930) ; by Gordon,
J. Chem. Phys., 1, 308 (1933); and by Kassel, /• Am Chem. Soc , 66, 1838
(1934).

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 379

dissociate with the absorption of heat, and the data of Table 67
show that they are more highly dissociated at higher tempera-
tures. The dissociation of NO into N2 and 02 evolves heat, and
the extent of its dissociation therefore decreases with increasing
temperature.

Before giving the quantitative relation between AH and the
equilibrium constant, we review the conventions followed in
writing chemical-equilibrium expressions and thermochemical
equations: Write a balanced chemical equation for the process
involved; append A// for the complete reaction as written, with
due regard to sign ; and formulate K, with the partial pressures or
concentrations of the reaction products in the numerator. One
may reverse the direction in which the reaction is written, change
the sign of AT/, and invert the expression for Kp or KCJ but one
may not perform some of these operations without performing
all of them.

The change of equilibrium constant with temperature for a
system of ideal gases or ideal solutes is shown by the differential
equation

dlnjt = A/7
dT KT* { }

This equation, which is usually called van't HorTs equation,
may be derived for a system of ideal gases from the second law
of thermodynamics through the use of a reversible cycle of opera-
tions involving the desired chemical reaction in one direction at
T and in the opposite direction at T — dT. (Another derivation
of the van't Hoff equation will be given in Chap. XVIII )

In such a reversible cycle operating between two temperatures
and absorbing q cal. at the higher temperature, the summation
of the work done is related to the fraction of the heat converted
into work during the cycle by the equation

dT
2w = q -^

in which dT IT is the fraction of the heat converted into work by
the cycle. (The equation is derived on page 39.)
We consider the general chemical reaction

aA + 6B + • • • = dD + eE + • • •

380 PHYSICAL CHEMISTRY

for which A# is the heat absorbed in the complete reaction and
to which the equilibrium relation

= Kp (t const.)

f A //B

applies. The derivation is accomplished through an "equi-
librium box" in which this reaction takes place and which serves
as the "engine" in the cycle. The equilibrium box is a chamber
containing these substances at equilibrium, it is fitted with four
cylinders, each containing one of the substances. Each cylinder
is closed by a frictionless piston; each connects with the equi-
librium mixture through a membrane permeable only to the sub-
stance in that cylinder, so that through motion of these pistons
the individual substances may be forced into or out of the equi-
librium chamber. The pressure of each substance in its cylinder
is thus equal to its equilibrium pressure in the mixture.

Each of the four steps in the cycle is conducted "reversibly,"
or so slowly that equilibrium is maintained at all stages of it.

1. In the first step at 77, a moles of A at pA and b moles of B at
pB are forced isothermally and reversibly into the equilibrium
mixture, and during these operations d moles of D at p-o and e
moles of E at pE are withdrawn from the mixture through theii
respective membranes into their cylinders isothermally and
reversibly. The change in state which is the sum of these opera-
tions is

aA (at PA) + kB (at pB) = dD (at pD) + eE (at pE)

Since the equilibrium box is unchanged in its contents by this
change, the work done is the sum of the p Av changes at each of
the cylinders. Denoting the volume of a moles of A by v A, b moles
of B by I>B, etc., this summation is

and A# is the heat absorbed by the change at T.

2. Each piston is clamped in a fixed position, and the whole
system is cooled to T — dT, by which the equilibrium pressures
become pA — dpA, PB — dpBj PD — dpv, and pE — dpE. The
volume remains constant during this change, and thus w2 = 0

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 381

3. In the third step d moles of D at pD — dp& and e moles of
E at PE — dpv are forced isothermally and reversibly into the
equilibrium chamber, and at the same time a moles of A at
PA. — dp A and b moles of B at pB — dp& are withdrawn into their
cylinders isothermally and reversibly The change in state that
is the sum of these operations is

dD (at p-D — dpi)) + eYj (at pE — dpE) = aA (at pA — dpA)

+ 6B (at pB — dp*)

The work done in this step is the summation of the (p — dp)Av
changes, which is

— dp*.)

4. Finally, the system is restored to its original state by heating
to T at constant volume, for which w± = 0

The work summation for the entire cycle of operations is

Sw = — VA. dp A — ?'B rfpB + Z'D ^?>D + ?'E dpv

and since each substance is assumed an ideal gas, rA = aKT/pA,
VB — bRT/p*,etc Upon making this substitution into the equa-
tion above and putting d In p for dp/p we have

Lw = HT (-d In pA° - d In pj> + d In p^ + d In pEf)
= RTdlnKp

By the second-law equation this summation is dT/T times the
heat absorbed at the higher temperature Tj which was AT/ for
the chemical reaction Equating these quantities,

RT dlnKp = AH (^-

which rearranges to give the van't Hoff equation

dluKp AH

dT RT2 ^ i}

It must be understood that AH in the van't Hoff equation is
for the complete change in state shown by the chemical reaction
on which Kp is based, and not for the incomplete reaction which
takes place when the substances on the left side of the equation

382 PHYSICAL CHEMISTRY

are mixed in the specified quantities. It is AH for the formation
of d moles of D and c moles of E. For gaseous reactions at
moderate pressure, AH calculated from equilibrium constants
through the van't Hoff equation will he in substantial agreement
with AH calculated for the same temperatures from enthalpy
tables and heat-capacity data. At high pressures, (dH/dp)r is
not zero for actual gases, and therefore A// calculated from equi-
librium constants unconnected for deviation from the ideal gas
law may not be the same as A// calculated for the reaction at
1 atrn. pressure.

If A// is sufficiently constant over the temperature interval
involved, equation (27) may be integrated between limits and
becomes

2.303 l»; = <2S)

In using this equation, R is expressed in calories if the heat of
reaction is so expressed Since the equilibrium constants appear
in this equation as a ratio, any units may be used in formulating
them, provided that the same units are employed at both tem-
peratures. Thus, if the partial pressures are in atmospheres in
K at one temperature, they must be at the other also.

By putting equation (27) in a form suitable for plotting,
namely,

(29)

it may be seen that a plot of In K against the. reciprocal of T
is a straight line of slope —AH/R if A// is independent of T
When the change of AH with T is slight, the plot will be almost a
straight line, and this condition is true of most of the data given
in this chapter. A plot of logic Kp for the reaction

MN2 + %H2 = NH,

at 10 atm. total pressure, as shown by the data of Table 60,
against 1000/7" yields a straight line of slope — A///2.37J, whence
AH = -12.7 kcal. between 350 arid 500°C. Since this chemical
system at 10 atm. pressure does not behave as a mixture of ideal
gases, which is required for the use of equation (29), one might
expect the derived AH value to be considerably in error. Yet a

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 383

precise calculation1 in which an accurate equation oT state is used,
together with an adjustment for ACP, yields AH = —12.66 kcal.
at 500°. Evidently some compensation yields a more precise
value in the A// calculated from equation (29) than one would
expect.

Application of the van't Hoff equation to the equilibrium data
on page 341 for the reaction SQ*(g) + %Oz(g) = SO3(0) yields a
straight lines of slope -AH/2.3R such that A// = -22.6 kcal.
when log Kp is plotted against lOOO/T. Calculations of this A//
from thermochemical data lie between -—21.9 and —23.0, which
is satisfactory agreement.

It is somewhat generally true that plots of log K against l/T
are more nearly straight and yield better values of A// than would
be indicated by the deviation of the systems from ideal gas
behavior. But when precise values are required, AH should be
expressed as a function of temperature, and pressures should be
low enough for ideal gas behavior, or an exact equation of state
should be employed to calculate the equilibrium partial pressures

The calculation involving A// as a function of the temperature
may be illustrated by the dissociation of carbon dioxide at
atmospheric pressure and high temperatures, which Table 67
shows to be 5 0 per cent at 2200°K. and 1 atm. We write the reac-
tion 2C02 = 2CO + O2 and calculate Kp = 67 X 1 0~6 for partial
pressure in atmospheres Since AH is given as a function of
temperature on page 322 for half of the reverse reaction, we
obtain A// for the dissociation by multiplying the equation there
given by —2, which gives

A// = 134,200 + 4.88 T7 - 0.0044712 + 0.68 X lO-6^3

for the reaction as written above. Upon substituting this A// in
equation (27) and integrating between T = 2000 and T = 2200,
we find the ratio ^2200/^2000 = 20, and X2ooo is 3.3 X 10~6.
This corresponds to 1.9 per cent dissociated, and Langmuir gives
1 .84 per cent for 1 atm. total pressure. More recently Kassel2
has calculated from other data that C02 is 1.55 per cent dis-
sociated at 2000°K.

Calculations such as the one just outlined are tedious rather
than difficult, and for many purposes it is sufficient to assume

'GiLLESPiE and BEATTIE, Phys. Rev., 36, 1008 (1930).
2 J. Am. Chem. Soc , 56, 1838 (1934).

384 PHYSiqAl, CHEMISTRY

AH constant unless the temperature interval is quite large. For
most of the data quoted in this chapter on change of Kp with
temperature, there are no reliable data on the heat capacities of
some of the substances involved, and for many systems the
equilibrium data and thermal data are not accurate enough to
justify calculations in which AH is assumed to vary with the
temperature.

Since AH may not be expressed by an equation in powers of T
over a range in which some substance changes its state of aggre-
gation, it is obvious that the van't Hoff equation may not be used
over such a temperature range. It is necessary to calculate up
to the temperature at which the change in state of aggregation
occurs, adjust AH for the new conditions, and compute it anew for
the heat capacities corresponding to the new states of aggregation.

The van't Hoff equation also applies to reactions in aqueous
or other solutions; but when these are attended by a change in
the number of ions, All is usually a temperature function for
which allowance must be made. Such data, as we have show that
ACP is not only large but a temperature function as well, though
there are comparatively few data at temperatures other than
25°C. For example, the change in the second iomzation constant
for carbonic acid with temperature requires an equation

A/7 = 13,278 - 0.1088772

for the heat absorbed in the ionization The data1 and the
derived quantities are

T 273 283 293 303 313 323

WnKz 2 36 3 24 4 20 5 13 6 03 6 73

A/A 5158 4565 3927 3278 2608 1915

ACP -59 3 -63 8 -64 9 -67.0 -69 3

The numbers in the last line of this table show that A77t is
not a function of the first power of temperature. From the
equation above, which is valid in a 50° range, one may calculate
that AHl will be zero at 349°K. and negative above this tempera-
ture. To the extent that an extrapolation of data taken over a
50° range is valid outside of that range, K would appear to pass
through a maximum at 76°C. and decrease with further rise in
temperature. Since the data for other weak acids often show

1 HARNED and SCHOLES, ibid., 63, 1706 (1941).

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 385

that K reaches a maximum at a characteristic temperature and
then decreases, it is probable that carbonic acid shows this effect,
but questionable whether the maximum is at 76°C.

The ionization constant of lactic acid reaches a maximum value
at 22 5°C., which requires that A//t be zero at this temperature,
positive below it, and negative above it. The data are as follows : '

t 0 10° 20° 22 5° 25° 30° 50°

104A' 1 317 1 356 1 388 1 389 1 387 1 378 1 274

A//, 768 458 98 0 -102 -315 -1313

It has been shown by Harned and Embree2 that the ionization
constant passes through a maximum with increasing temperature
for many weak acids. In the neighborhood of the temperature at
which the maximum occurs the change of K with temperature is
given by a single equation for all the weak acids studied, namely,

In -JF- = -1 15 X 10-4(!T - rmftx)2 (30)

J^max

Upon differentiating this equation with respect to jf, combining
with the van't Hoff equation, and solving for A//\, we find a
general equation for the heat of ionization of the acids,

A//t - -23 X IQ~4(T - Tmta)RT* (31)

Applying this general equation to the ionization of lactic acid,
for which 77max is 295.6, we find

AH, = 0 1355772 - 4 58 X IO~4T* (32)

which is the value given on page 324.

Since the temperature at which the maximum in K is observed
is different for different acids, the general equation (30) does not
require that all acids have the same AjfJt, even though there is
only a single constant in the equation.

Problems

Numerical data should be sought in the tables in the text

1. A constant bromine pressure of 0 107 atm. is maintained at 503°K
by the dissociation 2CuBr2(s) = 2CuBr(s) -f Br2(0), and at this temperature
the equilibrium constant for the gaseous reaction 2NO -f- Br2*= 2NOBr is

1 MARTIN and TARTAR, ibid., 59, 2672 (1937).

2 Ibid, 66, 1050 (1934).

386 PHYSICAL CHEMISTRY

0.050 for partial pressures in atmospheres. Calculate the final pressure at
equilibrium and the composition of the solid residue if 0.2 mole of CuBr2(s)
and 0 2 mole of CuBr(,s) are put into a 25-liter vessel containing 0.23 mole of
NO and 0 10 mole of Bra at 503°K.

2. When 0 090 mole of chlorine is dissolved in a liter of water at 25°C ,
36 per cent of the chlorine reacts with water to form un-ionized HC1O and
completely ionized HC1 (a) How many moles of HC1 must be added to this
solution to reduce the fraction of chlorine hydrolyzed to 0 20? (6) How
many moles of Nad would be required to produce the same result? (c)
The partial pressure of chlorine above the original solution containing
0 09 mole of chlorine is 0 96 atm. Calculate the total solubility of chlorine
when the chlorine pressure is increased to 2 0 atm

3. When 0 0060 mole of iodine is added to a liter flask containing 0 0140
mole of nitrosyl chloride, the gaseous reaction 2NOC1 + I2 = 2NO + 2IC1
takes place incompletely, and the equilibrium pressure at 452°K becomes
0 922 atm. (a) Calculate Ki for this reaction at 452 °K with partial pres-
sures in atmospheres (b) At 452°K the equilibrium constant K2 for the
reaction 2NOC1 = 2NO + C12 is 0 0026 atm Calculate K , for the reaction
2IC1 = I2 + Cl2 at 452°K (c) Show that the partial pressure of chlorine
in the equilibrium mixture of part (a) ih a negligible part of the total pressure.

4. The dissociation 2CuBr^(s) = 2CuBr(s) + Br-Xj?) maintains a con-
stant bromine pressure of 0 046 atm at 487°K when both solids are present.
Neither solid reacts with iodine, and when 0 10 mole of iodine is introduced
into a 10-hter space containing an excess of CuBr2(&) at 487°K , the reaction
Br2(0) + lato) = 2IBr(0) produces an equilibrium pressure of 0 746 atm
(a) Calculate the equilibrium constant for this reaction (b) The equi-
librium constant for this reaction at 387°K is 190 Calculate AH for the
reaction (The answer should check the answer to Problem 22, page 329.)

6. Calculate the upper and lower limits between which the hydrogen-
ion concentration must he for a titration of 0 02 N benzoic acid with 0 02 N
sodium hydroxide to be correct within 0 5 per cent

6. The apparent molecular weight of acetic acid vapor, as defined by
the equation M — dRT/p, changes with the total pressure at 100°C. as
follows :

p, atm 0 122 0 274 0 396

M 83 1 93.1 97.4

(a) Calculate the equilibrium constant for the reaction
(CH3COOH)2 - 2CH3COOH

at 100°C., assuming that this reaction is wholly responsible for the change of
M with p. (b) The apparent molecular weight of the vapor at 120°C and
0.396 atm. is 85.7, at 158°C. and 0 396 atm it is 70.9 Calculate AH for
the dissociation of the dimer [HITTER and SIMONS, J Am. Chem. Soc., 67,
757 (1945). There is said to be evidence of the formation of some tetramer
at temperatures below 140°C.]

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 387

7. Calculate Kp for the reaction 2NO2 = 2NO + O2 from the data on
page 346 for 600, 700, 800, and 900°K , plot log K against 1000/71, and
determine AH for the reaction from the slope of the curve

8. A small amount of phenol phthalem is added to a solution prepared by
mixing 20 ml of 0 1 TV NH4C1 with 3 ml of 0 1 TV NH,OH Calculate the
hydrogen-ion concentration in solution and the fraction of the indicator
transformed to the pink form if A"» = 10~10

9. A solution 0 Im in phosphoric acid is titrated with NaOH, using
methyl orange as indicator, and the end point is taken when pH = 43.
(a) Calculate what fraction of the acid has been converted into NaH2PO4.
(6) What fraction has been converted into Na2HPO4? (r) What per cent
error results from taking the end point at this pH?

10. The 0 1m H^PO4 is titrated with NaOH, using phenglphthalem
as indicator, and the end point is taken when pH ==87 Calculate the

ratio (HaPO4-)/(HPO4— ) and the ratio (HPO4— )/(PO4 ) corresponding

to this end point How much NaOH (0 10 TV) would be required for the
titration of 100 ml of 0 \m H^PO4 to this end point?

11. The solubility of cinnamic acid (Cf)H6CH:CHCOOH = HCm, mol.
wt 148) in water at 25° is 0 0038m Carbon dioxide is passed into a liter
of 0 1m sodium cmnamate at 25° in a 2-liter bottle (containing no air) until
0 010 mole of cinnamic acid is precipitated (a) Calculate the equilibrium
constant of the reaction NaCm -f- H2CO3 = HCm -f NaHCO. (6) Cal-
culate (H+) in the solution (r) Calculate the pressure of CO2 at equilibrium
and the quantity of CO2 required in the process The solubility of CO2 is
0034m at 1 atm. pressure (Note that the concentration of unionized
cinnamic acid is constant in the presence of the solid acid )

12. (a) Calculate Kp at 773°K for the gaseous reaction 2NH3 = N2 + 3H2
at 10, 30, and 50 atm from the data of Table 59, assuming the ideal gas law
to apply (6) Plot these values of Kp against the pressure, and extrapolate
the curve to 1 atm (r) Calculate from this Kp the equihbriurn quantity of
NH,} at 773°K and 1 atm in a system made from 1 mole of N2 and 1 mole
of H2. (Ans . about 0 002 mole of NH3 )

13. Calculate the total pressure at equilibrium after 0 030 mole of chlorine
has been pumped into the mixture described m Problem 13, page 99.

14. Calculate the equilibrium constant of the reaction

H2S -f NaHCOj = NaHS -f H2CO, at 25°

and the concentration of free H2S in a mixture of equal volumes of 0 02m.
H2S and 0 02m NaHCO3 No gases escape from solution

16. (a) If a liter of 0 1m ammonium formate is added to a liter of O.lm.
acetic acid, what fraction of the salt will be converted to NH4Ac? (6) What
will be the fraction converted to acetate when a liter of 0 1m. ammonium
formate is added to 0 5 liter of 0 3m acetic acid

16. A series of buffer solutions is to be prepared covering the range pH
4.0 to 5 4 in steps of 0.2 by mixing O.lm. acetic acid with 0 1m. sodium ace-
tate. What volume of sodium acetate solution must be added to 10 ml. of
acetic acid for each of these solutions?

388 PHYSICAL CHEMISTRY

17. The equilibrium constant for the reaction TLgBr^g) = Hg(0) + Br2(0)
is 0 040 at 1100°C for partial pressures in atmospheres. At what total pres-
sure would (partly dissociated) mercuric bromide vapor have a density of
1 gram per liter at 1100°?

18. When 0.20 mole of bromine and 0 30 mole of iodine reach equilibrium
in a 10-hter flask at 373°K, the reaction l'£\t(g) + KBr2(0) = IBr(0) takes
place incompletely, part of the iodine remains as a crystalline phase, and the
total pressure becomes 1 181 atm The vapor pressure of lodme at 373°K
is 0 0604 atm (a) Calculate the equilibrium constant for the reaction and
the quantity of solid iodine remaining (6) The equilibrium constant
changes with the Kelvin temperature as follows:

T 298 400 600 800 1000

KP 20 66 11 42 6 37 4 80 3 99

Determine A/7 for the reaction from a suitable plot

19. When a mixture of 1 mole of C2H4 and 1 mole of H2 is passed over n
suitable catalyst, part of the ethylene is converted into ethane, and the den-
sity oi the mixture at equilibrium LS 0267 gram per liter at 973°K and
1 atm (a) Calculate Kp for the reaction C2H4(0) + H2(0) = C2II6(^)
(6) For this reaction AH = — 32 6 kcal , arid ACP = 0 Calculate Kp foi
the reaction at 1173°K

20. Calculate the concentration of each important solute molecule or
ion in each of the following aqueous solutions at equilibrium at 25°. (a)
0034m H2CO3, (b) 0034m NaHGOj, (r) 0034/77 NaaCOj Note that of
the solutes H2CO8, HCO3~, COr~, H+, OH~, and Na+, some concentrations
are negligible in comparison with others in these solutions

21. (a) Calculate the hvdrogen-ion concentration at the correct end point
for 0 2 TV NH4OH titrated with 0 2N HC^l (b) Calculate also the hydrogen-
ion concentration when the end point is 0.1 per cent short of the true one
and when it is 0 1 per cent overstepped

22. At 100°C ammonium acetate m 0 Olm solution is 4 5 per cent
hydrolyzed, and at 100° the lomzation constant for ammonium hydroxide
is 1.3 X 10~6. Calculate the lomzatioii constant for acetic acid at 100°.

23. (a) Calculate the hydrogen-ion concentration in solutions formed
when 100 ml of 0.2 N acetic acid is titrated with 99, 99 8, 100, and 100 2 ml.
of 0.2 N sodium hydroxide, (b) What indicator would be suitable for this
determination?

24. The solubility of HaS in water at 25° is 0.10m, when the pressure of
H2S is 1 0 atm. H2S is passed into a liter of 0. 1m. NaBO2 in a 25 5-liter ves-
sel (containing no air) until (H+) — 10~8 in the solution, (a) What fraction
of the NaBOa is changed to NaHS? (6) How many moles of H2S are
required?

25. (a) Calculate the concentration of hydroxide ion in 0 1m. Na2CO3
solution at 100°C., assuming that hydrolysis- of carbonate ion is the only
important chemical reaction, and given the following data: for the lomzation
HCOr - H+ + COj— ' , A#298 « 3600 cal., and &CP = -60 cal; for the
ionization H2O = H+ -f- OH~ A//298 = 13,360 cal., and ACP = -50 cal.
(b) Calculate (H+) in the solution at 100°C.

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 389

26. Problem basis 27 1 grains of PC15 vapor at 523°K. and 1 atm. pres-
sure in a 10-liter vessel, (a) Calculate the dissociation constant of PCU for
partial pressures in atmospheres. (&) Chlorine is added to this 10-liter ves-
sel at 523°K until the total pressure at equilibrium becomes 2 atrn WKat
fraction of the PCI 5 is dissociated? What quantity of chlorine was required?
(r) The original 10 liters of vapor is expanded to 20 liters at 523°K. What
fraction of the PC1& is dissociated? (d) Chlorine is added to this 20-liter
vessel until the total pressure becomes 1 atm What fraction of the PC15
is dissociated at equilibrium? What quantity of chlorine was added?
[Note that the quantity is not the same as in (b) ]

27. When 46 grams of iodine and 1 gram of hydrogen are heated to equi-
librium at 723°K , the reaction mixture matches in color a similar vessel
containing 1 9 grams of iodine alone (a) Calculate the fraction of the
hydrogen convex ted to HI and the equilibrium constant of the reaction
H.J -f- I2 = 2HI. (b) Calculate the fractional dissociation of HI(p) at
723°K

28. Problem basis 0 30 mole C4H8 and 0 30 mole HI in 10 liters at 425°K

(a) The total pressure at equilibrium is 1 10 atm , and the only important
chemical reaction is C4H8 + HI = C-tHgl Calculate Kp for partial pres-
sures in atmospheres (b) If C4H8 is added to the mixture until the total
pressure becomes 1 50 atm , how many moles will be required? (r) If nitro-
gen is added to the original mixture until the total pressure becomes 1 5 atm ,
how many moles will be required? [(d) Students with sufficient curiosity
may calculate the partial pressures of H2 and I2 in the equilibrium mixture
of part (a) from the data in Problem 27 above and on page 296 or from that
in Problem 35 below 1

29. Calculate AH for the reaction CO2(0) + Hs(gr) = CO(0) + H2O(0)
at 1100°K fiom the equilibrium constants on page 347, and compare with
the result obtained in Problem 21, page 328.

30. From the data on page 185 calculate the equilibrium " constant" for
the reaction (CrJIbOH)^ = 2C6H5OH in benzene, assuming the whole
deviation from Raoult's law is due to this reaction

31. When a mixture of 2 moles of OH 4(0) and 1 mole of H2S(g) is heated
at 973°K and 1 atm total pressure, the reaction

CH4(flO + 2H2S(<7) - C8*(g) + 4H2(0)

takes ' place incompletely, and the partial pressure of hydrogen becomes

0 16 atm at equilibrium (a) Calculate the equilibrium constant for the
reaction, (b) Calculate the density of the equilibrium mixture m grams per
liter.

32. Hydrogen sulfide is passed into a 20-liter vessel at 25° containing

1 0 mole of NaHCO3 in 10,000 grams of water (and no air), until the total
pressure is 5 03 atm , of which water vapor is 0.03 atm The solubility
of H2S at 1 atm partial pressure is 0 102 mole per liter, and that of CQz is
0 034m. (a) Calculate the equilibrium constant of the reaction

NaHCO3 + H2S - NaHS + H2CO8

(b) What are the partial pressures of H2S and CO2 above the solution at
equilibrium? (c) How many moles of H2S were required?

390

PHYSICAL CHEMISTRY

33. At 1600°K the equilibrium constant of the reaction S02 -f %02 - S03
is 0 026 for partial pressures in atmospheres, and at 1 atm and this same
temperature C02 is 0 11 per cent dissociated into carbon monoxide and
oxygen. Calculate the equilibrium constant at 1600°K. for the reaction

S03 + CO - S02 + C02

34. For certain reactions the enthalpy change m calories at 298°K , the
increase in heat capacity, and the equilibrium constant at 298°K are as
follows:

Reaction

A//298

AC,

H20 = H+ Aq + OH~.Aq
NH4OH Aq = NH4+M<7 -f OH~ Aq
HB02.Aq = Jl+Aq + BOr Aq

13,360
865
3,360

-50
-30
-43

1 0 X 10~14
1 8 X 10~6
6 6 X 10-10

(a) From the above data express A/7 for the hydrolysis of ammonium
borate as a function of temperature (6) Calculate the fractional hydrolysis
of 0 1m ammonium borate at 75°C ( = 348°K ) (r) Calculate the fraction
hydrolyzed in a solution 0 Im in NH4HO2 arid 0.2w in NH4OH at 348°K.

35. The equilibrium constant for the reaction 2HI(<7) = H2(<7) + 12(0) is
1 84 at 700°K , A/7 = 3070 cal , and A£P is negligible. Calculate the frac-
tional dissociation of HI(#) at 800°K.

36. The equilibrium constants A^i for the reaction

2NO(0) + 2IC%) = 2NOCl(flf) + I*(g)

and K2 for the reaction 2NOC1(0) = 2NO(p) + C12(0) change with the
temperature as follows.

T, °K

409 422
0 255 0 159
3 09 6 08

437 452
0 090 0 055
13 1 25 7

(a) Calculate Kz for the reaction 2IC1 = I2 + C12 for each of these
temperatures (6) Calculate A77 for the dissociation of 2 moles of IC1
[McMoRRis and YOST, / Am. Chem Soc , 64, 2247 (1932) ]

37. Calculate the pH of the second, sixth, and tenth mixtures of Na2HP04
and KH2PO4 described on page 371 from the lomzation constants in Table 63,
assuming the salt-concentration ratio is equal to the ion-activity ratio

38. (a) What volumes of 0 1m. NH4C1 should be added to 10 0-ml portions
of O.lw. NEUOH to produce buffer solutions of pH 8.0 to 9.4 in steps of
0.2 pH? (b) If 1.0 ml of O.lw. HC1 were added to the buffer solution of
pH = 9, what would be the resulting pH?

39. Calculate the ionization constant of monoethanolammomum hydrox-
ide from the data in Problem 18, page 291.

EQUILIBRIUM IN HOMOGENEOUS SYSTEMS 391

40. Calculate the equilibrium constant for the reaction

(HCOOH)2(0) = 2HCOOH(0)

at 20°, 40°, and 60° from the data on page 72, and determine A// for the
dissociation.

41. The equilibrium constants for the reaction S02(0) + ^(MflO = S03(0)
at several temperatures are given on page 341. (a) Plot log K against
1000/7", allowing a sufficient range for extrapolating to 1600°K , and show
that the constant given in Problem 33 on page 390 is in reasonable conform-
ity with these data. (6) Determine AH for the dissociation of 1 mole of 803
from the slope of the plot.

42. In the following table p0 is the theoretical pressure of I2(0) calculated
on the assumption of no dissociation, p is the observed pressure after the
reaction 1 2(0) = 21(0) has reached equilibrium, and T is the temperature of
the experiment'

T 1273° 1173° 1073° 973°

po, atin 0 0736 0 0684 0 0631 0 0576

p, atm 0 1122 0 0918 0 0750 0 0624

(a) Calculate the equilibrium constant for the reaction at each temper-
ature, assuming all the pressure difference to be due to the dissociation.
(5) Calculate A# for the dissociation of a mole of iodine. [PERLMAN and
ROLLEFSON, J. Chem. Phy&., 9, 362 (1941) ]

CHAPTER X
HETEROGENEOUS EQUILIBRIUM

In many important chemical systems the equilibrium com-
position of a given phase is closely related to that of another
phase. The concentration or pressure of one substance in a
mixture may be fixed by the presence of its solid phase in excess,
and this in turn establishes the concentrations or pressures of
other substances through the equilibrium constant and a material
balance. We have already seen that the partial pressure of a
volatile solute controls its mole fraction in a solution (Henry's
law), that the partial pressure of solvent controls -its mole frac-
tion in solution (Raoult's law), that equilibrium between a solid
solvent and a solution is described by the freezing-point law,
and that the partial pressure of a single gaseous dissociation
product is kept constant by the presence of solid phases with
which it is in equilibrium Jn this chapter we shall consider
other aspects of chemical equilibrium in the gaseous phase
or in a solution at a constant temperature in the presence of solid
phases It will be true of these systems, as it was oi homo-
geneous systems, that the equilibrium constants apply ori]y to
the particular temperature at which the measurements were
made but to systems of all compositions at this temperature so
long as all the solids involved are present at equilibrium.

The value of the equilibrium constant K for a given tempera-
ture changes with changing temperature in these systems as
required by the van't Hoff equation; and in this equation A7/ is
for the complete change in state as used in formulating the
equilibrium constant. If the equilibrium involved is a change
of solubility with changing temperature, AH derived from the
van't Hoff equation is for dissolving the solid into the saturated
solution, a " partial mola!" heat of solution. The relation
between this partial quantity and the total heat of solution has
been given on page 317.

It should not be assumed without proof that a system is at
equilibrium just because the phases have been in contact for some

392

HETEROGENEOUS EQUILIBRIUM 393

time; they must be in contact for enough time for the attainment
of true equilibrium. For example, in measuring the equi-
librium pressure for the reaction Mg (OH) 2(s) = MgO(s) + H2O-
(g) at 485°K., the pressure of water vapor in a system reacting
toward the formation of water vapor wras 53 0 mm. after 2 days
and the pressure in a system reacting in the opposite direction
was 55 2 mm. After 6 days the pressures still differed by 1 mm.;
after 1 1 days they became identical at 54.4 mm and of course
remained identical after equilibrium was reached.1

In the study of heterogeneous equilibrium, as was true of
homogeneous equilibrium, it is first necessary to determine the
chemical changes involved. This includes correct identification
of the solid phases present as well as the composition of the gas
or solution in equilibrium with them. For illustrations, the
pressure of CO^(g) in equilibrium with PbO(s) and PbCOs^)
is not the same as that for the solid phases PbO.PbCO3 and
PbCOs; the pressure of water vapor in equilibrium with Na2-
HP04 7H2O and its saturated solution is not the same as that
between Na2HP()4 1211 2O and its saturated solution; the crystals
in equilibrium with a liquid mixture of bismuth and cadmium
are the pure crystals of the elements, but there is no liquid mix-
ture of bismuth and magnesium that is in equilibrium with
crystals of the two pure elements

We are first to consider systems in which only one important
chemical reaction takes place and then some systems in which two
or more reactions must be considered at the same time. Whether
there is one reaction or several, the necessary chemical equations
are written and balanced; a definite material basis is completely
defined (giving the quantities of all solids, liquids, gases, and
solutions); the equilibrium expression is formulated in the way
described in the next section; and the equilibrium composition
of the system is described in terms of the minimum number of
unknowns, before any calculations are begun.

Activities of Solid Phases. — The activity or equilibrium effect
of a pure solid phase at a given temperature remains constant
without regard to the quantity of solid present, since its partial
pressure or concentration is constant, and it is convenient to
define its activity as unity. Thus, for the equilibrium change

1 GIATTQTJE and ARCHBOLD, J. Am. Chem. Soc., 69, 561 (1937).

394 PHYSICAL CHEMISTRY

in state at 800°C.,

CaCO3(s) = CaO(s) + CO2(flf, 0.220 atm.)
the equilibrium expression might be written

, = Ka (t const }

but if the activities of the solids are unity and the fugacity of the
gas is equal to its pressure (as will be substantially true for
moderate pressures) this may be written more simply.

pco2 = KP (t const )

The expression in this form is in agreement with the experimental
fact that the pressure of CO 2 in equilibrium with CaO(s) and
CaCOs(s) is constant at a given temperature, regardless of the
relative quantities of the three substances present. There
would, of course, be no objection to writing Kp for this equilibrium
in the form

£c.D£ro, = K, (i }

7>CaC08

but we have no information on the vapor pressures of the solids,
only the belief that they are constant at a given temperature.
If this expression is rewritten with these two constant (though
inappreciable) pressures combined with K'

= pc02 = Kp (t const.)

PCaO

the same result is obtained as by defining the solid activities as
unity. For equilibriums in which the solid or liquid phases have
determinable vapor pressures either procedure may be followed,
but it is important to indicate clearly which one has been fol-
lowed, since this special definition that the activity of a solid
phase is unity at every temperature makes neither its vapor
pressure unity nor its solubility unity on a molality scale. One
may, of course, insert the partial pressure or the concentration
of any substance involved in a chemical reaction into the equi-
librium expression that governs it. The point is that, if the
pressure is not constant, it must^be included in the equilibrium
expression; if it is constant, it may be put in the equilibrium
expression or put in the value of the constant.

HETEROGENEOUS EQUILIBRIUM 395

Dissociation Pressures. — We have considered above the
dissociation

CaCO8(«) = CaO (a) + OO2(0)

for which the equilibrium was represented by

Kp = pco2 . (t const.)

This dissociation pressure (in atmospheres) changes with the
centigrade temperature as follows.1

t . 775° 800° 855° 894 4° 1000° 1100°

pCOz 0 144 0 220 0 556 1 000 3 87 11 50

It should be clearly understood that the expression Kp = pco2
is not applicable if only one of the solid phases is present; this
constant pressure for a given temperature requires that both
solid phases be present. In the absence of CaO the pressure of
CC>2 at 894 4°C. may be any pressure greater than 1 atm., and
in the absence of CaCOs the pressure of CO 2 may be any pressure
less than 1 atm. The implications attending Kp = pco2 might
be written pco2 = const, (t const., CaO and CaC03 present) ; and,
whether written or not, these conditions are essential for true
equilibrium in this system.

For any given temperature the equilibrium pressure is inde-
pendent of the direction of approach, whether by mixing CaO
and CO2 or by the direct dissociation of CaC03, and it is likewise
independent of the relative quantities of the solid phases present.
The same considerations apply in the dissociation of MnCOs,
FeCO3, ZnCO3, PbCO8, etc

In any of these systems the pressure is a function of tempera-
ture alone, which is the characteristic of a uni variant system.
The Clapeyron equation describes the change of equilibrium
pressure with temperature in such a system. For the dissocia-
tion CaCOs = CaO + CO2, Av is due to the formation of a mole
of gas and a mole of solid from a mole of another solid and is thus
substantially equal to the volume of the gas, since the solids have
very small volumes by comparison. Upon substituting RT/p
for Aw in the Clapeyron equation, we have

dp^ _ p AH
dT ~ ~

1 SMYTH and ADAMS, ibid., 46, 1167 (1923); SOUTHARD and ROYSTER,
J. Phys Chem., 40, 435 (1936)

396 PHYSICAL CHEMISTRY

which rearranges to give the van't Hoff equation

din p A//
dT ~ RT2

since p is equal to Kp for this system, by the equation above.

The system at equilibrium which is represented by the chemical
equation

BaCl2.8NH3(s) = BaCl2(s) + 8NH8(0)

is also a monovariant system in which the change of equilibrium
pressure with changing temperature is shown by the Clapeyron
equation. But it must be recalled that this equation is

dp^ = AH

dT ~ T Av

and hence, if AH is the heat absorbed by the chemical reaction
as written above, Av = SRT/p and not RT/p, since the dissocia-
tion of a mole of BaCl2.8NH3(s) yields 8 moles of gas.

As another illustration, the pressure of oxygen in equilibrium
with silver and silver oxide changes with the centigrade tempera-
ture as follows:1

/ 150° 173° 178° 183 1° 188 2° 190 0° 191 2° 200°

p0a, mm 182 422 509 605 717 760 790 1050

If the equation for this dissociation is written
Ag20(s) = 2Ag + M02(<7)
the equilibrium expression must be written

Kp = po^ (t const )

and AH taken for the dissociation of 1 mole of Ag2O. Of course,
the chemical reaction may be written for 2Ag20, when Kp = pQ2
and AH is for the dissociation of 2 moles of silver oxide.

The equilibrium expression for the dissociation of HgO into
oxygen and liquid mercury, 2HgO(s) = 2Hg(J) + Qz(g), may be
written

Kp = Po2 (t const.)

but, since the vapor pressure of liquid mercury is not negligible
in the temperature range in which this dissociation has been
studied, p0j is not equal to the total pressure. When the reaction is

2HgO(s) = 2Hg(0) + 02(<7)
1 BENTON and DRAKE, /. Am. Chem. Soc,, 64, 2186 (1932)

HETEROGENEOUS EQUILIBRIUM 397

in the absence of liquid mercury, the equilibrium constant must
be written

KP = PHK2po2 (t const.)

If mercuric oxide dissociates into an evacuated space, pHK is
two-thirds of the dissociation pressure (i.e., of the total pressure
developed by the dissociating oxide) and

KP = (hp)2(4p) = 0.148p8

The value oi Kp so determined will also apply when HgO dis-
sociates into a space containing excess oxygen or excess mercury
vapor, but (/3p)2(/^p) will not be applicable, since the mercury
vapor pressure is not twice the oxygen pressure under these
conditions. For example, the dissociation pressure is 90 mm.
at 360°C and Kp = (60)2(30) = 1.08 X 106 for partial pressures
in millimeters. If p is the total pressure at equilibrium when
HgO dissociates into a closed space containing oxygen at an
initial pressure of 25 mm., the equilibrium pressures in this system
are pHe = %(p - 25) and p0, = [H(p - 25) + 25], and by
substituting these quantities into the equilibrium expression one
may solve it for the equilibrium pressure.

The total pressure, in millimeters, developed by the dissocia-
tion of solid IlgO into oxygen and mercury vapor changes with
the centigrade temperature as follows:1

t 360° 380° 400° 420° 440° 460° 480°

p, mm 90 141 231 387 642 1017 1581

Further illustration of the equilibrium between solids and
gases is afforded by the evaporation and complete dissociation
of solid NH4HS. The equilibrium constant is Kp = pH2spNH8,
and each partial pressure is half the total pressure when this
substance evaporates into empty space. For its evaporation
into a space already containing ammonia at a pressure po, the
partial pressure of H2S would be lA(pt — PO), or half the differ-
ence between p0 and the equilibrium total pressure pt, and the
pressure of NH8 would be p0 + }4,(pt — Po).

The evaporation of ammonium carbamate is attended by
complete dissociation, with the formation of three moles of gas

1 TAYLOR and HTJLETT, /. Phys Chem , 17, 565 (1913).

398 PHYSICAL CHEMISTRY

for one of solid, as shown by the reaction

NH4C02NH2(s) = 2NH,(0) + CO.fo)
for which the equilibrium is written

Kp = ?>Nii82pco2

Other illustrations are the dissociation of salt hydrates into
water vapor and lower hydrates or anhydrous salts and the
dissociation of CuBr2 into CuBr and bromine vapor. The
dissociation of NaHCOs yields another solid and two moles of
gas, as shown by the reaction

2NaHC03(s) = Na2CO3(s) + H2O(g) + C0a(g)
for which the equilibrium constant is

Kp = pn&pcoi

The value of this constant for a given temperature is obtained
from the total pressure developed when NallCOs dissociates into
an evacuated space, for which the data are1

t 30° 50° 70° 90° 100° 110°

p, atm 0 00816 0 0395 0 158 0 545 0 962 1 645

In the absence of any other vapor each of the partial pressures
would be half the total pressure, and the numerical value of the
equilibrium constant would be p2/4 This constant also applies
when NaHCOs dissociates into a space containing excess COs or
excess water vapor; but it is the product Pco2pn2o which must be
used under these circumstances, and not p2/4, which applies only
when all the vapor comes from dissociation of the NaHCOa. It
may be worth saying again that the equilibrium which is repre-
sented by the product of the two pressures applies only when
both solid phases are present. It places no restriction on a
product that lies below the equilibrium value if NaHCOg is
absent or above the equilibrium product if Na2C03 is absent.
In the drying of moist NaHC03, for instance, it is desired to
prevent the formation of Na2CO3, and therefore the product
J>H2opco2 must be kept above the equilibrium value. So long as
the product of these partial pressures exceeds the equilibrium
value, no NaHCOs will decompose; and so long as the partial
pressure of water vapor in the equilibrium mixture is less than

^AVEN and SAND, /. Chem. Soc. (London), 99, 1359 (1911); 106, 2752
(1914).

HETEROGENEOUS EQUILIBRIUM 399

that of water from the saturated NaHC03 solution which is to
be dried, water may enter the vapor space. But if the total
pressure is constant, that of CO2 decreases as water evaporates;
and when the product of the two partial pressures falls below
the equilibrium value, dissociation becomes a possibility.

Reactions between Solids and Gases. — Equilibrium as shown
by the chemical equation Ag2S(s) + H2(0) = 2Ag(s) + H2S(0)
has been investigated over a range of temperature. The equi-
librium may be formulated

Ka = ^£^-1 (t const.)

as was done for the dissociation of CaCOs Since in the tempera-
ture range of these experiments the fugacities are substantially
equal to partial pressures and the activities of the solid phases
are defined as unity, we may use the simpler expression

Kp = 5* (t const.)

This ratio changes with the centigrade temperature as follows:1

f 476° 518° 617° 700°

pRzs/pH2 0 359 0 325 0 278 0 242

Although the equilibrium ratio in the gaseous phase is independ-
ent of the quantities of the solid phases present, the composition
and quantity of the gas phase provide data for calculating the
quantities of the solids through a material balance that takes
into account the quantities at the start (the " working basis").
If at 476° a mole of silver sulfide were put into contact with 10
moles of hydrogen, 1 mole of hydrogen sulfide would be formed
and 9 of hydrogen would remain, but no silver sulfide would be
present. This condition is one of true chemical equilibrium
between H2, H2S, and Ag, but it is not the equilibrium to which
the constant ratio of hydrogen sulfide to hydrogen applies, for
this requires the presence of solid Ag2S as well as solid Ag,

The least quantity of hydrogen that would reduce 1 mole of
silver sulfide at 476° is 1 mole for the actual chemical process
plus (1/0.359 = 2.79) moles to maintain the equilibrium ratio,
or a total of 3.79 moles. If a smaller quantity of hydrogen
reacts upon a mole of silver sulfide, equilibrium is established in

1 KEYES and FELSING, /. Am. Chem Soc., 42, 246 (1920).

400 PHYSICAL CHEMISTRY

the gaseous phase before all the sulfide is reduced; if a larger
quantity of hydrogen is employed, complete reaction taken place
without forming enough hydrogen sulfide to produce the equi-
librium ratio of partial pressures.

These figures illustrate the importance of equilibrium con-
siderations, for one who took no account of them and supposed
the substances involved to react completely as shown by the
chemical equation would calculate the efficiency of this reaction
at 476° to be 1/3.79, or about 26 per cent, whereas this yield is all
that could possibly be attained.

Reactions of similar type occur between metallic oxides and
carbon monoxide. The most common one in chemical industry
is

FeO(*) + 00(0) = Fe(«) + CO2(0)

for which data at various temperatures will be found in Problem 1
at the end of this chapter. As in the previous illustration, the
equilibrium constant is written for the gaseous substances only,
namely, Kp = pcojpco- Such an equilibrium constant is also
obtained when oxides of nickel or cobalt are the solid phases
reduced, but it should not be inferred that every metallic oxide
will reach equilibrium in the sense MO + CO = M + C02.
For example, aluminum oxide is not reduced at all by carbon
monoxide, and molybdenum dioxide is reduced to molybdenum
carbide by carbon monoxide. In these systems, as in all chemical
systems, the first requisite in studying chemical equilibrium is
a correct knowledge of the reactions taking place.
Another reaction of the same type is

FeO(s) + H,(0) = Fe(s) + H20(0)

for which equilibrium data at various temperatures are also given
in Problem 1 at the end of this chapter. Equilibrium between
hydrogen and water vapor in this sense is also observed for
other -metallic oxides, and experimental studies have been pub-
lished for the oxides of nickel, cobalt, and tin which show a
constant ratio of pH2o/pH2 &t a constant temperature, regardless
of the relative quantities of the solid phases, but provided both
solid phases are present at equilibrium.

The fact that a chemical equation showing the reaction
involved should always attend the formulation of an equilibrium

HETEROGENEOUS EQUILIBRIUM

401

constant is of such importance as to justify its repetition and
further illustration. For the reaction

FeO(s)

= Fe(s) + H2O(0)

KP — PH20/PH2; and the numerical value of this equilibrium
ratio is 0.332 at GOO°C. in the presence of FeO and Fe. For the
reaction

CoO(s) + H2(</) = Co(s) + H20(0)

Kp is again Pn2o/Pir2; but the equilibrium ratio at 600°C. is 46
when the solid phases are CoO and Co.

Supplementary Equilibriums in Vapor -solid Systems. — In the
illustrations of equilibrium between metals, metallic oxides,
hydrogen, and water vapor (or CO and C02), we have not men-
tioned the presence of oxygen in the vapor phase, and there is no
experimental evidence of its presence in these systems at these
temperatures. But it is well known that water vapor and CO2
dissociate at high temperatures and that the extent of dissocia-
tion changes with the temperature as required by van't Hoff's
equation. Hence there must be in these systems enough oxygen
to satisfy the dissociation equilibriums of H2O or CO2 and to main-
tain the dissociation pressures of the oxides, such as that of NiO
into Ni and oxygen. Moreover, if H2 and H2O in a certain ratio
are in equilibrium with Ni and NiO, they must be in equilibrium
also with a mixture of CO and CO2 that is in equilibrium with
Ni and NiO. The relations among the various constants are as
follows :

NiO(s) = Ni(«) + M02(g) (1) NiO(s) = Ni(s)

(2)

Adding (1) and (2),
NiO(s) + H,fo) = Ni(s)

(4)

(1)

V) = C02(flO (3)
Pco,

CO(sr) +

Adding (1) and (3),
NiO(s) + COfo) = Ni(s)

PCO

Subtracting equation (5) from equation (4), we have

(5)

(6)

402 PHYSICAL CHEMISTRY

for which the equilibrium constant is

The data on page 347 for the water-gas reaction were obtained
from the equilibrium ratios corresponding to K\ and K^ when the
solid phases are Fe and FeO, in the way indicated above.

It will be difficult for beginners to avoid making a simple
equilibrium system appear too complex or a complex system
appear too simple; for which purpose experience and a sense of
proportion are required. However, in most chemical systems
some possible equilibriums are not important, and in general only
one or two important reactions need be considered. In the
illustration above, p0z was a negligible part of the tota/ pressure,
and the sum of the pressures of H2 and H2O (or of CO and CO2)
in the systems shown by the chemical equations (2), (3), (4),
and (5) is equal to the total pressure. But po2 as a multiplying
factor in the equilibrium expressions for equations (2) and (3)
would obviously not be neglected when it is small. The follow-
ing routine procedure in solving problems should prove helpful:

(1) Write and balance the chemical equation for the important
chemical reaction involved. (2) Formulate the equilibrium
expression in the standard way. Substances present as pure
solid or liquid phase may have their partial pressures or concen-
trations included in the value of K, or they may appear in the
equilibrium expression as desired; but it is important to indicate
which procedure has been followed. "Mixed" constants, \vhich
contain the pressures of -some substances and the concentrations
of others, may be used to advantage in some problems. (3)
State the "working basis" for the problem, the initial state
of the system that reacts to equilibrium. (4) List all the molec-
ular species present, gases, liquids, solids, solutes, and solute
ions. (5) Cross out all the pressures or concentrations that are
negligible in condition equations; for example,

+ PCO, 4" PG^ = Ptotal

(6) Set up a "mole table" for the necessary pressures or concen-
trations in terms of a single unknown. (7) Consider all possible
equilibriums supplementary to the main equilibrium, and dis-

HETEROGENEOUS EQUILIBRIUM

403

card those which are not important. (8) Solve the problem,
and check the equilibrium pressures through a material balance
from the working basis.

Distribution between Two Liquid Phases. — The distribution
of a solute between two mutually insoluble solvents in which it
has the same molecular weight was considered on page 189.
When passage of the solute from one solvent to the other is
attended by partial ionization or dissociation or polymerization,
the distribution ratio Ci/C* is no longer a constant, if Ci and €2
denote total concentrations, for it will be remembered that the

TABLE 68. — DISTRIBUTION OFBENZOIC ACID BETWEEN WATER AND BENZENE1

AT 6°

rvoB

cwa*

Cw(l - «)/[CB(l - f)}*

0 00329

0 0156

0 210

0 0263

0245

0 00435

0 0275

0 158

0 0264

0246

0 00493

' 0 0355

0 139

0 0262

0245

0 00579

0 0495

0 117

0 0261

0244

0 00644

0 0616

0 105

0 0260

0244

0 00749

0 0835

0 089

0 0259

0244

0 00874

0 1144

0 076

0 0258

0243

0 00993

0 148

0 067

0 0258

0243

0 0114

0 195

0 058

0 0258

0244

distribution ratio is constant only with respect to a single molec-
ular species. Some slightly ionized organic acids exist almost
wholly as single molecules in water and almost wholly as double
molecules in some organic solvents. An illustration of this is the
distribution of benzoic acid, which is written HBz for C6H&COOH,
between water and benzene. The only relation given by the
distribution law is between the concentration of single molecules
of benzoic acid in water and the concentration of single molecules
of benzoic acid in benzene. We see from the equilibrium con-
stant for the reaction J^(HBz)2 = HBz in benzene, which we
call Ki, that (HBz)i = KI \/(HBz)2; and since nearly all the
acid in benzene is in the form of the dimer, (HBz)x in benzene is
nearly Jf£iCBH, in which CB is the total concentration in benzene.
Combining all the constants into a single one, we show, in the
fourth column of Table 68, CW/C*^, which is substantially con-

1 CREIGHTON, / Franklin Inst., 180, 63 (1915).

404 PHYSICAL CHEMISTRY

stant. The ratio C«,/CB, which takes no account of the different
molecular condition of the solute in the two layers, is not even
roughly constant, as may be seen in the third column. By
applying a correction for the small fraction of the acid ionized in
water and for the small part in the benzene that is not in the
form of dimer and by again grouping all the constants into a
single one, we obtain the figures in the last column of Table 68 1
A similar variation in the distribution ratio is shown by other
organic acids, though the explanation may not be the formation
of a dimer in the organic solvent. For example, the concentra-
tions of picric "acid" distributed between water and benzene
at 18° are as follows:

Cw 0 0334 '0 0199 0 0101 0 00327 0 00208

CB 0 1772 0 070 0 0199 0 00225 0 00093

From these data the simple distribution ratio CW/C^ varies
from 0 188 to 2 24, and the ratio Cw/\/C* is more nearly con-
stant but varies from 0 079 to 0.068. If complete ionization in
water is assumed, with no polymer in benzene, the equilibrium
relation is CW2/CB7 which is the square of the constant written
for the assumption of polymerization in benzene. Distribution
data alone do not allow us to chose between these possibilities
or to exclude the possibility of both effects to differing extents.

Freezing-point depressions for picric acid in water give A^/m
ratios that vary from 3.7 to 3.2, which is typical of the behavior
of strongly ionized solutes like HC1. If we suppose the impor-
tant equilibrium to be

HP (in benzene) = H+ + P~ (in water)

upon writing HP for HOC6H2(N02)3, the equilibrium expression
becomes

No activity coefficients are available for picric acid, but if we

1 WALL, /. Am. Chem. Soc., 64, 472 (1942). Equilibrium constants
K » (RCOOH)2/([RCOOH]2) f°r some other organic acids in benzene are
given by Wall and Banes, ibid., 67, 898 (1945): for example, this constant
for benzoic acid is 0.0023 at 32.5°C. and 0.00633 at 56.5°.

HETEROGENEOUS EQUILIBRIUM 405

use the ones for HC1, which are typical of uni-univalent elec-
trolytes in general, the calculated X2 is nearly constant, as follows:

Cw 0 0334 0 0199 0 0101 0 00327 0 00208

Act coeff 0 84 0 88 0 90 0 94 0 95

103#2 . 44 43 42 42 42

Thus assumptions of complete ionization in water, no
polymerization in benzene, and correction for activity coefficients
yield a satisfactory constant. It should be noted that without
supplementary data, such as freezing-point depressions in one
solvent or the other, polymerization in one layer and ionization in
the other are equally probable interpretations of the distribution
data alone.

There are other variations of the distribution ratio with chang-
ing concentration of the distributed solute for which neither
ionization in one phase nor polymerization in the other appears
a probable or acceptable explanation. Under such circum-
stances, a plot of the distribution ratio against the concentra-
tion in one layer or the other will be useful, even though the
explanation of the deviation is not known.

Solids and Dissolved Substances. — The simplest equilibrium
between a solid and a dissolved substance is that of a saturated
solution of a substance which does not ionize upon solution, the
concentration of this solution depending on the temperature
alone.1 For a given temperature the eauilibrium expression is

Kc = C (t const.)

When the nature of the crystalline phase is unchanged over a

1 Strictly speaking, it depends upon the pressure as well, but the small
changes in atmospheric pressure produce only a negligible effect that need
not be t'aken into consideration A suitably large increase in pressure will
cause considerable change in the solubility of a substance; for example, the
solubility of thallous sulfate in water at 25° changes with the pressure as
shown in the following table [Cohen and van den Bosch, Z. physik. Chem ,
114, 453 (1925)]:

Pressure . 1 500 1000 1500 atm

Solubility . 0 123 0 160 0 198 0 232m.

Increase of pressure usually increases the solubility in water for sulfates,
carbonates, sulfides, fluorides, and hydroxides of alkali, alkaline earth, and
heavy metals The solubilities of most other salts decrease with increase of
pressure. [GIBSON, Am. J. Sci., 35A, 49 (1938).]

406 PHYSICAL CHEMISTRY

temperature range, a plot of C against the temperature is a
smooth curve, and usually (but not always) it shows an increase
in solubility at higher temperatures.

An illustration of this simple equilibrium is the variation of
solubility of melamine in water with temperature.1 The solu-
bility, in moles per 1000 grams of water, is

T 293 .308 323 337 348 368 372

m ... 0 0257 0.0468 0 083 0 135 0 190 0 365 0 402

A plot of log m against 1/T, based on these solubilities, is a
straight line, of which the slope is — A///2.3/? and from which
A# = 8200 cal. per mole for the heat of solution.

When a saturated aqueous solution of a slightly ionized solute,
such as benzoic acid, and its crystalline phase are in contact,
the equilibriums may be represented as follows:

J_ A3 ^

HBz(s) ^± HBz (dissolved) ^± H+ + Bz~

These expressions represent (1) a constant equilibrium concen-
tration of un-ionized benzoic acid in all saturated aqueous solu-
tions containing other solutes or solute ions at low concentrations,
such as 0.01m. or less, (2) ionization equilibrium between the
dissolved acid molecules and its ions, whether derived from
benzoic acid or from small additions of other acids or of benzoates,
and (3) a constant solubility product for the H+ and Bz~ ions
in the presence of the crystalline phase, all for a constant tempera-
ture. According to this third equilibrium, addition of a little
hydrochloric acid or nitric acid to a saturated aqueous solution
of benzoic acid should decrease the Bz~ concentration materially.
In the absence of an added acid, benzoic acid in its saturated
solution (0.026m. at 25°) is about 4 per cent ionized; (H+) is
about 0.001, and Ks is about 10~6; hence, addition of 0.01 mole
of nitric acid to a liter of saturated benzoic acid solution would
reduce (Bz~) to 10~~4 and cause the precipitation of about 0.0009
mole of benzoic acid, but nitric acid in such a small concentration
would leave the concentration of un-ionized benzoic acid sub-
stantially unchanged at 0.025m.

1 CHAPMAN, AVEBILL, and HARRIS, Ind. Eng. Chem , 35, 137 (1943).

HETEROGENEOUS EQUILIBRIUM 407

When salts such as KC1 or KBr or BaCl2 are dissolved in satu-
rated aqueous solutions of benzoic acid in considerable quantities,
they materially reduce the solubility of the acid by changing
the activity coefficients for all the solutes present. If So is
the solubility of benzoic acid in pure water and S the solubility
in an electrolyte of molality m (both So and S being corrected
for ionization of benzoic acid), the decreased solubility is shown
by the relation log S/S0 — /cm, where k is a different constant
for each electrolyte. To illustrate the magnitude of this change
in solubility of benzoic acid in water by the addition of salts not
yielding benzoate ions, we may note that for 1m. solutions
of the added salts1 at 35° the ratio S/S0 is 0.8 for KBr, 0.7 for
LiCl, and 0.5 for BaCU. Salts cause similar changes in the
solubilities of other un-ionized solutes, but it would not be true
that S/So would be 0.8 for 1m. KBr with some other solute.

The recorded data on the change in solubility of benzoic acid
in water with changing temperature may be used to show the
necessity for a critical consideration of data and the use and the
limitations of calculations based on the van't Hoff equation.
The data are

m 0 0139 0 0172 0 0238 0 0282 0 0336 0 0458 0 0948 0 222 0.482
t ... 0° 10° 20° 25° 30° 40° 60° 80° 100°

If we calculate A// of solution from the solubilities at 0° and
10° we find 3240 cal., and from the solubilities at 30° and 40°
we calculate 5860. It is possible, but not probable, that the
heat of solution would change so much in this temperature range.
A plot of log m against l/T from these figures will show that all
of them except log m for 0° fall on a smooth curve, but not on a
straight line; and that from 10° to 40° the slope is substantially
constant, corresponding to A# = 5700 cal. The calculation of
the solubility at 0° from this A# shows that 0.012 is more rea-
sonable than 0.0139. The curve also shows that 5700 cal. is not
the proper heat of solution to use above 40°; for example, it
leads to a calculated solubility of 0.078 at 60° in place of 0.0948.

There is another fact which is not indicated by the data but
which is of the greatest importance, namely, that above 90° the

1 GOELLEB and OSOL, /. Am. Chem. Soc., 69, 2132 (1937). In these
experiments it is found that log -S is a linear function of m for these salts
and for KI and KC1 up to 2m.

408 PHYSICAL CHEMISTRY

equilibrium is between an aqueous solution and a liquid phase
containing mostly benzoic acid and some water and not between
an aqueous solution and crystalline benzoic acid. It is another
type of system. While the melting point of benzoic acid is
122°, water lowers the " freezing point" to about 90° when added
in sufficient quantity. Thus at 90° there may be three phases
present at equilibrium in a system of benzoic acid and water —
a liquid of 5.6 per cent acid, a liquid of about 80 per cent acid,
and crystalline benzoic acid. This is not the type of system
that we started out to discuss. We shall return to it in the
chapter on phase diagrams, but we must note here that the equi-
librium described on page 406 is not applicable above 90° in
this system.

Solubility Product for Ionized Solutes. — The current theory
of solutions assumes no appreciable concentration of nonionized
molecules in dilute solutions of "highly ionized" solutes; thus
equilibrium between a slightly soluble salt such as silver acetate
and its saturated solution is represented by an equation such as

AgAc(s) = Ag+ + Ac~

for which the equilibrium expression is

,*mA -TA,- congt>)

= I

If the solutions involved are " sufficiently" dilute, the change in
activity coefficient with slight additions of AgNO3 or KAc may
be ignored, and an approximation written

== Kc == mAK+^iAo- = solubility product

The data in Table 69 show that Kc is nearly constant in " dilute"
solutions of silver acetate to which potassium acetate has been
added but that in solutions over O.lm. in potassium acetate Kc
increases. On the assumption that the aativity coefficients for
silver nitrate at equivalent total molality apply to mixtures of
silver acetate and potassium acetate, the activity product Ka
remains practically constant, as is shown in the last column of
Table 69.

The use of a similar procedure for additions of silver nitrate
to silver acetate leads to a less satisfactory constant Ka and indi-

HETEROGENEOUS EQUILIBRIUM

409

cates that some further explanation is needed. The activity
coefficients for salts of the same ionic type at the same molality
are not quite the same, and this may be the explanation. It has
also been suggested that " complexes" are responsible for the
variation in Ka. But it will generally be true that the use of
activity coefficients for one salt in solutions of another salt is not
wholly justified, and variations in a quantity supposedly con-
stant will result to about the extent shown in solutions of silver
nitrate and silver acetate

TABLE 69 — SOLUBILITY OF SILVER ACETATE IN POTASSIUM ACETATE*

KAc

AgAc

Total Ac

(Ag+)(A<r)

Activity
coefficient

Activity
product

0 06674

44 5 X 10~4

0 76

25 4 X 10-

0 01144

0 06135

0 07279

44 6 X 10~4

0 75

25 0 X 10-

0 04956

0 04867

0 09821

47 7 X 10~4

0 72

25 0 X 10-

0 1028

0 03763

0 1404

52 8 X 10~4

0 68

24 4 X 10-

0 1965

0 02796

0 2245

63 8 X 10~4

0 63

25 4 X 10-

0 4828

0 01925

0 5021

96 7 X 10~4

0 51

25 0 X 10~

0 6751

0 01722

0 6923

119 0 X 10-*

0 45

24 8 X 10-

1 001

0 01575

1 0168

161 X 10~4

0 40

25 8 X 10"

SOLUBILITY OF SILVER ACETATE IN SILVER NITRATE*

AgNO,

AgAc

Total Ag

(Ag+)(Ac-)

Activity
coefficient

Activity
product

0 04920

0 05008

0 09928

49 7 X 10~4

0 72

26 X 10~4

0 07063

0 04555

0 11618

52 8 X 10-4

0 70

26 X 10~4

0 09491

0 04107

0.13598

55 9 X 10-

0 69

26 X 10~4

0 10590

0 03999

0 14589

58 4 X 10-

0 68

27 X 10~4

0 19900

0 03145

0 23045

72 6 X 10-

0 62

28 X 10~4

0 2009

0 03135

0 2322

72 6 X 10-

0 62

28 X 10-4

0 3104

0 02745

0 33785

92 7 X 10-

0 56

29 X 10~4

It should be understood that the variation of the activity
product in the last column of Table 69 arises from using esti-
mated activity coefficients, for the product a^+a^- is a constant
whenever equilibrium exists between solid silver acetate and its
saturated solution.

1 MAcDouGALL and ALLEN, / Phys Chem , 46, 730 (1942).
8 MACDOUGALL, ibid., 46, 738 (1942).

410 PHYSICAL CHEMISTRY

Application of the van't Hoff equation to changing "solubility
with changing temperature is rendered difficult by lack of data
on the heat effects attending the process of solution and by lack
of data on heat capacities of ions at temperatures other than
25°. The change in AjFf with temperature is somewhat compen-
sated by changing activity coefficients as the solubility changes ;
thus approximate agreement between experiment and solubilities
calculated from Kc, uncorrected for activity coefficients, is some-
times found. For example, the solubility of KC104 changes
with the temperature as follows:

t, °C 0 20 40 80 100

8 0 052 0 121 0 268 1 04 1 56

Taking Kc = $2, without correction for activity coefficients,
one calculates from the solubilities at 0° and 20° that A// is
13,700 cal., from which S is calculated to be 0.256 at 40°, com-
pared with 0.268 by experiment; and S at 100° is calculated
to be 1.51, compared with 1 .56 by experiment. Such close agree-
ment, which is somewhat due to compensations in the incorrect
assumptions, will not always be found, and in general the agree-
ment will be better for smaller solubilities.

Solubility products apply as well in systems containing ions
of valence other than unity, but the form of the expression is
different when some of the ions have unit valence and others
have not. For example, the solubility product for lead iodate,
for which the chemical equation is

Pb(IO«)2(«) = Pb++ + 2IOa-

is the product of the tead-ion activity and the square of the
iodate-ion activity. In the absence of added iodate the molality
of the iodate ion is twice that of the lead ion, and thus the equi-
librium expression is

in which SQ denotes the solubility in pure water, 3.6 X 10~6 m.
at 25°. The solubility of lead iodate in, say, 0.01m. KI08 would
then be shown by the equation

Ka - Sy[(2S + O.Olh]2

An equation of similar form would apply to the solubility product
for PbI2, Mg(OH)2, or Ag2S04. The exact application of solu-

HETEROGENEOUS EQUILIBRIUM 411

bility products such as these to calculations of solubility in the
presence of added salts with an ion in common is usually compli-
cated by a lack of accurate activity coefficients to use in them.
" Estimates" of activity coefficients from those of other salts
of the same ionic type are not very satisfactory. For example,
the activity coefficient for 0.01m. PbCl2 is 0.61, and that for
0.01m. Mg(NO3)2 is 0.71; and since these coefficients are raised
to the third power in the equilibrium expressions, the error is
too large for satisfaction.

Formation of "Complex" Ions. — Some slightly soluble salts
react with solutes to form " complexes " that result in an increased
solubility. For example, silver iodate (AgIO3) is soluble in water
only to the extent of 1.75 X 10~4 mole per liter at 25°. It reacts
with ammonia to form the familiar complex Ag(NH3)2+, thus
removing one of the ions of silver iodate and increasing the solu-
bility. The chemical reaction and its equilibrium constant Kc
are

Agl03(s) + 2NH3 = Ag(NH.),+ + IOr

[Ag(NH3)2+](T03-) _ ffl

_
Kc ~

(NH3)2

where m is the total ammonia. It will be noted that the activity
coefficient for the ions would appear in the numerator of this
expression as 72 and that the activity coefficient for nonionized
ammonia is substantially unity. In dilute ammonia another
correction is required for the formation of NH4+ and OH~ ions,
since these ions are not concerned in the reaction with AgIO3.
Thus the complete expression for the equilibrium constant in
terms of activities is

OS7)2

Ka =

[(m - 25) (1 - «)]«

in which a is the fractional ionization of the ammonia. The
data in Table 70 show that Kc is not constant and that Ka is
constant. Activity coefficients for the ions of Ag(NH8)2I08 are
not available, and those for AgNO3 at the same molality have
been used in the table.

The existence of another type of "complex" ion, which is
strictly an intermediate ion, is shown by the increased solubility
of Pb(IOa)2 in the presence of acetates. The increase is due to

412

PHYSICAL CHEMISTRY

the formation of PbAc+. Activity coefficients for mixtures such
as Pb (103)2 and NH4Ac are not available; but since the activity
coefficient is nearly constant in a mixture of constant ionic con-
centration, a simple expedient is available, namely, the solubility
of Pb(I03)2 is determined in a mixture of NH4C104 and NH4Ac
at a constant total molality that is high compared with the
molality of lead ion, with increasing proportions of acetate and
decreasing proportions of perchlorate. The perchlorate takes
no part in the formation of a complex, and there is no evidence

TABLE 70 — SOLUBILITY OF SILVER IODATE IN AMMONIA AT 2501

(NH.)
= m

(I0r)

= S

Kc =

7 = act.
coeff.

a — frac
ionized

Ka =

£V

(m - 2S)2

[(m-2S)(l~a)P

0.01241

0.003665

0 520

0 93

0 060

0 51

0.02481

0.007430

0 558

0.91

0.042

0.51

0.03085

0 009358

0.595

0.89

0 038

0.51

0 06180

0 01901

0.639

0.87

0 027

0 51

0 1028

0 03223

0 708

0.83

0 022

0 51

0 1847

0 05937

0 810

0.78

0 017

0 51

0.2487

0 08125

0 888

0 75

0 014

0 51

of the presence of ions such as PbClO4+ or PbIO3+. In Table 71
the third column gives the solubility (in moles per 1000 grams
of water) of Pb(IOs)2 at 25° in the mixtures of NH4Ac and
NH4C104 shown in the first two columns; the fourth column giv^es
the molality of lead ion calculated on the assumption that the
solubility product Kc of lead iodate is constant in this mixture;
the fifth column gives by difference the molality of PbAc+;andthe
last column gives Kc for the reaction PbAc+ = Pb++ + Ac"".
The fact that this Kc is substantially constant is evidence for the
formation of the ion PbAc+.

It should be noted that the solubility of lead iodate given in
the first line of Table 71 is not the solubility in pure water but a
much higher value because of the smaller activity coefficient in a
mixture of salts. The assumption is that since the total ionic
concentration is substantially constant in these mixtures the
activity coefficient will be constant, not that it will be nearly
unity, and thus that Kc will be constant throughout the series
of experiments.

1 DEBB, STOCKDALE, and VOSBUBGH, / Am Chem Soc., 63, 2670 (1941)

HETEROGENEOUS EQUILIBRIUM 413

TABLE 71. — SOLUBILITY or LEAD IODATE IN AMMONIUM ACETATE1

NH4Ac

NH4C1O4

Pb(I08)2
X 104

(Pb++)
X 10*

(PbAc+)
X 104

(Pb++)(Ac-)

(PbAc+)

1 0

1 950

0 05

0 95

3 557

0 586

2 97

9 86 X 10~3

0 10

0 90

4 370

0 388

3 98

9 75 X 10~3

0 20

0 80

5 584

0 238

5 35

8 89 X 10~3

0 50

7 265

0 141

7 12

9 85 X 10-3

1 00

0 0

9 11

0 089

9 02

9 92 X 10-'

Other lead salts would also react with acetate ions to form the
PbAc+ ion, as, for instance, in the procedure of qualitative analy-
sis in which lead sulfate is dissolved in ammonium acetate
solution.

The solubility of mercuric bromide (HgBr2) in potassium
bromide solutions is quantitatively explained by the formation
of a complex ion HgBr3~~. Since mercuric halides are substan-
tially un-ionized in aqueous solutions, it is the concentration of
HgBr2, and not the solubility product (Hg4"f)(Br~)2, that
remains constant in solutions in equilibrium with solid HgBr2.
The solubility of HgBr2 in KBr at 25° is as follows:2

KBr molahty 0 0 010 0 030 0 080 0 100 0 300

Total Hg dissolved. . 0 0170 0 0235 0 0365 0 0692 0 0825 0 213

Corresponding solubility data for HgI2 in KI are not so simply
interpreted and probably indicate two complexes HgI3~ and
Hgl4 — . The ratio of chloride ion to dissolved mercury in solu-
tions of KC1 saturated with HgCl2 varied fortyfold when the KC1
molality increased from 0.1 to 5.0.

There are numerous instances of increased solubility of salts
produced by adding comparatively large quantities of another
salt with one ion in common. Thus, silver chloride is much
more soluble in strong sodium chloride solution than in pure
water, and dilution causes the precipitation of silver chloride.
Similar behavior is shown by AgSCN dissolving in aqueous
solutions of KSCN as follows:3

KSCN, moles per liter 0 312 0 564 0 870 1 124

AgSGN, moles per liter .. 000202 00121 00458 00985

1 EDMONDS and BIRNBAUM, tbid , 62, 2367 (1940)

2 GARBETT, ibid., 61, 2745 (1939).

3 RANDALL and HALFORD, ibid., 62, 189 (1930).

414 PHYSICAL CHEMISTRY

The formation of " complex" salts such as NaAgCU or KAg-
(SCN)2 seems a logical assumption but does not account for the
facts, and no adequate quantitative explanation of either solubility
increase is known. But numerous increases in solubility are
quantitatively explained by the formation of similar compounds.

Solubility of Hydrolyzed Salts. — Salts of weak acids, such as
carbonates and sulfides, are hydrolyzed in aqueous solution to
an extent that increases as the concentration decreases. Hence
in the saturated solutions of such salts allowance must be made
for the hydrolytic reaction as well as for the equilibrium demanded
by the solubility product. In a saturated solution of CaC03,
for example, the equilibrium

K8P = (Ca++)(CO«— )

must of course be maintained, but the ion concentrations are not
equal because of the reaction

CO.— + H20 = HCOr + OH-
for which

(HCO3-)(OH~) Kw

(CO,—)

If S is the total calcium ion concentration, (C03 ) = S(l — h),
and (HCOs"") = (OH") = Sh. Upon substituting these quan-
tities and the numerical values of the appropriate constants for
25° into the above equilibrium equations, we have

iSf/?2

5 x 10-9 = S2(l - K) and ^~ = 1.8 X 10~4

1 — /i

These two equations contain only two unknowns, whence we
find h = 0.68 and S = 1.2 X 10~4. (The equations may, if pre-
ferred, be solved by successive approximations, on the basis of
a value of S greater than the square root of the solubility product,
h being solved in the hydrolytic equilibrium, and the process
repeated until a suitable value of S is found.) Thus we see
that the solubility, which is of course total calcium, is nearly
double the square root of the solubility product in this system.
A similar calculation could be made for any slightly soluble
carbonate, but one should not merely repeat these operations as
routine. For example, a similar calculation for the solution in

HETEROGENEOUS EQUILIBRIUM 415

equilibrium with MgCO3.3H2O(s) at 25°C. yields h = 0.2 and
S = 3.7 X 10~~3, but the concentrations of Mg+4" and OH~ corre-
sponding to these figures lead to a product (Mg++)(OH~)2 that
exceeds the recorded solubility product by a hundredfold. Thus
a new solid phase appears, which renders the calculation that
assumes no precipitation of Mg(OH)2 valueless. While a calcula-
tion of the concentrations of all the ions in a solution in equilibrium
with both hydroxide and carbonate as solid phases could readily be
carried out, one must first establish that these substances are the
solid phases present at equilibrium and that " basic carbonates"
are absent.

The recorded solubility product of PbS is 10~29, which is of
course (Pb++)(S ), but the square root of this solubility product
would have little relation to the solubility of lead sulfide in water.
A saturated solution of PbS in water would certainly contain
HS- and OH~ and probably H2S and PbOH+; therefore, the
equilibrium is a far more complex matter than merely the equi-
librium between a solid phase and the ions of which it consists.

Solubility of Carbonates in Carbonic Acid. — In the presence
of dissolved carbon dioxide at moderate concentration hydrolysis
of the carbonate ion is negligible, and the reaction that governs
the equilibrium is the formation of bicarbonate. Since most
bicarbonates are more soluble than the corresponding carbonates,
solubility increases are the result. For example, the equilibrium
corresponding to the reaction

FeCOs(s) + H2CO3 = Fe++ + 2HC03~

has been studied over a wide range of concentration,1 and the
equilibrium expression is

Let S denote the solubility of ferrous salt in the carbonic acid
solutions, i.e., its molal concentration in carbonic acid solution
in equilibrium with solid ferrous carbonate. If the notation
previously employed is followed, the ferrous-ion concentration
is S] the bicarbonate-ion concentration is 2$, since the solubility
of FeCOs as such is negligibly small. The ionization of carbonic

1 SMITH, H. J., ibid., 40, 879 (1918)

416

PHYSICAL CHEMISTRY

acid is slight and in the presence of dissolved ferrous bicarbonate,
which is highly ionized, may be neglected entirely in the calcula-
tion. Table 72 shows the results of experiments at 30°, where
the equilibrium constant is

cr/r»c*\9

(t const.)

"c (H2C03)
TABLE 72. — SOLUBILITY OF FERROUS CARBONATE IN CARBONIC ACID

Total concentrations at 30°

Equilibrium

H2CO3

Fc(HCO3)2

constant Kc

0 196

0 00256

34 2 X 10~8

0 230

0 00274

35 6 X 10-8

0 309

0 00304

36 5 X 10-8

0 326

0 00311

37 0 X 10-8

0 401

0 00332

36 5 X 10~8

0 655

0 00402

39 8 X 10~8

0 755

0 00434

43 3 X 10~8

This constant Kc is seen to be almost constant as long as the
molalities are low enough, which is to say that the activity
coefficients are almost constant (though not almost unity). At
higher molalities Kc increases, as is often true of such equilibrium
constants.

In this system we may equate Kc to KspKi/K^ as we have done
before and calculate the solubility product for ferrous carbonate.
This solubility product is 4.5 X 10~n, but the square root of
Ksp would have little relation to the solubility of ferrous carbonate
in water because of hydrolysis and the probable precipitation of
ferrous hydroxide.

Dissolved carbon dioxide also produces increased solubility for
other carbonates. Experimental studies for CaC03, MgCO3.-
3H20, and ZnC03 are given in the data for problems at the end
of the chapter, and other systems have also been studied.

Conversion of One Solid into Another. — A familiar example
of this type of reaction is the conversion of barium sulfate into
barium carbonate by boiling it with sodium carbonate solution
in excess. The chemical equation is

BaS04(s) + 2Na+ + C03— = BaCO3(s) + 2Na+ + S04—

HETEROGENEOUS EQUILIBRIUM 417

and the equilibrium expression in terms of activities is

Ka = ^££l°tl- (t const.)

Assuming that the ratio of the ion activities is equal to the ratio
of ion molalities and defining the activities of the solid phases
as unity, as we have done so often before, this reduces to

In the presence of the solid barium compounds there must be a
very small concentration of barium ion of such amount that the
solubility products KI = (Ba^+XSC^ ) for saturated barium
sulfate and K* = (Ba++)(COs ) for saturated barium carbonate
are both satisfied. These values are, respectively, 1 X 10~10 and
25 X 10~10 at 25°, and dividing KI by X2 we obtain a value of Kc.

(SO 4 )

I n n . 0_0

= = 0.04 at 25

In any solution in equilibrium with both barium sulfate and
barium carbonate, the carbonate-ion concentration must be 25
times the sulfate ion concentration. Therefore, for the complete
conversion of a mole of barium sulfate to barium carbonate a mole
of sodium carbonate will be required for the chemioal reaction,
and 25 moles of sodium carbonate will be required to maintain
the equilibrium ratio, or 26 moles in all. This calculation has
been made for 25°, but no smaller quantity of sodium carbonate
could be used in a boiling solution safely, since the solution is
cooled while filtering.

Suppose 2.33 grams (0.01 mole) of barium sulfate is shaken
a long time with 100 ml. of 1.0m. sodium carbonate solution,
which contains 0.1 mole of sodium carbonate. Let x be the moles
of sodium carbonate remaining in solution at equilibrium; then
(0.1 — x) moles of sodium sulfate are in solution, and

<ai - *> = 0.04

whence x = 0.0962 mole of sodium carbonate remaining. Then
0.0038 mole of sodium carbonate has reacted, forming 0.0038 mole

418 PHYSICAL CHEMISTRY

of barium carbonate, and leaving 0.0062 mole, or about two-
thirds of the original barium sulfate unchanged. It is clear
that too little carbonate solution has been used. As stated
above, the minimum quantity required is 26 times the moles
of barium sulfate to be converted to carbonate, or 260 ml. of
molal sodium carbonate solution. Any quantity of this solu-
tion greater than 2GO ml. will convert the sulfate completely
to carbonate. Experiments of this kind may be used to deter-
mine the solubility product of one salt when that of another
is known, since the equilibrium constant is the ratio of the two
solubility products.

Equilibrium between Metals and Ions. — Silver reacts with
ferric salts, forming ferrous salts and silver salts. The reaction
is

Ag(s) + Fe+++ + 3NO3~ = Ag+ + NO3~ + Fe++ + 2NO3-
and the equilibrium expression is1

(^0gp) = K< = 0.128 at 25°

From this value of Kc it may be seen that unless the silver-ion
concentration is very small, complete reduction of ferric nitrate
to ferrous nitrate will not take place. For example, suppose
0.2m. ferric nitrate to be shaken with an excess of silver until
equilibrium is reached. If x is the ferrous-ion concentration,
(0.2 — x) is the ferric-ion concentration, and x is the silver-
ion concentration, since the chemical equation shows that a silver
ion is formed for each ferrous ion. Substituting in the above
expression, we have x2/ (0.2 — x) = 0.128, whence x is 0.108,
the concentration of ferrous salt, and the ferric-salt concentration
is 0.092. This shows that about half the ferric salt has been
reduced by silver. If some salt is added that precipitates
silver ions as soon as formed and that does not react with the
iron salts, then in the presence of solid silver the ferric-salt
concentration must be very small compared with the ferrous-salt
concentration. Addition of a thiocyanate serves this purpose,2
and by this means ferric iron may be reduced for titration; the

1 NOTES and BRANN, ibid., 34, 1016 (1912).

2 EDGAR and KEMP, ibid., 40, 777 (1918).

HETEROGENEOUS EQUILIBRIUM

419

thiocyanate furnishes at the same time an indicator for com-
plete reduction. The excess thiocyanate is removed by adding
silver nitrate solution just before the titration.

TABLE 73 — EQUILIBRIUM BETWEEN TIN AND LEAD PERCHLORATE AT 25°C.

Molal concentration of solu-
tion at start of experiment

Equilibrium concentrations,
moles per liter

K — JiS, V i v

(Pb^"1")

Tin

Lead

perchlorate

0 094

0 0704

0 0233

3 02

0 050

0 0393

0 0123

3 19

0 050

0 0413

0 0132

3 14

0 096

0 0716

0 0237

3 04

0 060

0 0457

0 0148

3 08

0 050

0 0369

0 0119

3 11

0 038

0 019

0 0428

0 0145

2 96

0 051

0 037

0 0697

0 0239

2 92

0 066

0 027

0 0692

0 0235

2 95

0 086

0 024

0 0821

0 0275

2 98

Another reaction of this type is that between lead perchlorate
and metallic tin,1 according to the equation

2C104- = Pb(s)

2C1O4-

for which Kc = (Sn++)/(Pb++). The experimental results for
25° are shown in Table 73. In some of the experiments the
original solution contained lead perchforate alone or tin perchlo-
rate alone ; in other experiments both perchlorates were present in
solution; excess of both solid metals was always used, and the solu-
tions were shaken at 25° until they had reached equilibrium. As
shown by the value of Kc in the last column of the table, the ratio
of tin salt to lead salt in solution at equilibrium is about 3.0,
whether the reaction*proceeded in one direction or the other and
regardless of the relative quantities of tin perchlorate and lead
perchlorate in solution at the start.

The constancy of Kc in this system shows, not that the activity
coefficients are nearly unity, but only that they are nearly con-
stant. In a mixture of salts of the same ionic type, such as we
have in this system, the same activity coefficient would apply

1 NOTES and TOABE, ibid , 39, 1537 (1917).

420 PHYSICAL CHEMISTRY

to stannous ions and lead ions, whether or not their molalities
were the same. The exact equilibrium relation is

and the activity coefficients cancel to make Ka equal to Kc in
this particular system. For the solutions described in Table 73
the activity coefficients would be about 0.5.

This equilibrium ratio would also apply in the presence of
negative ions other than perchlorate, for instance, in dilute lead
chloride and stannous chloride. If 0.2m. SnCl2 and excess lead
react to equilibrium, the ratio (Sn++)/(Pb++) = 3.0 would pre-
vail; but under these conditions solid PbCl2 would form, and
thus the solubility relations of PbCU in the presence of excess
chloride ions would also prevail.

The need for considering activity coefficients is better illus-
trated by the reaction1

6C104- + 2Hg(Z) = 2Fe++ + Hg2++ + 6ClOr
for which the equilibrium constant Kc is

If this constant is calculated in the usual way, it varies with the
total iron as follows, in the presence of 0.01m. perchloric acid at
35°:

Total Fe 0004 0.002 0001

Kc . . * 0 0820 0.0862 0.0975

In these mixtures the activity coefficients are neither near
unity nor nearly constant. The correct equilibrium relation is

This equation shows how the equilibrium should be handled;
but it does not provide the data for carrying out the calculation,
since in a mixture of salts of three different ionic types the exact
calculation of activity coefficients is not within the powers of
the theory.

1 FLEHABTY, ibid., 56, 2647 (1933).

HETEROGENEOUS EQUILIBRIUM 421

It should be said again that, unless the solid phases concerned
in an equilibrium are correctly identified and are all present,
the equilibrium relations are not correctly given. As one more
illustration, consider the reaction

CaSO40) + 2Na+ + C03— = CaC03(s) + 2Na+ + S04—

If the sodium carbonate solution is dilute, we may evaluate Kc
from the ratio of the solubility products, as was done for the
reaction of BaS04 with Na2C03, as follows:

_ (B04-) _ (*ph _ 2.3 X 10-* ^
Ac ~ (CO,—) " (sp)2 ~ 5 X 10-9 - *'° X 1U

Thus in dilute solution the conversion is complete with a very
small excess of sodium carbonate. But in strong solutions a new
solid phase, CaCO3.Na2CO3.5H2O, appears, and the ratio of sul-
fate to carbonate at equilibrium is reduced from 46,000 to 191
in the presence of new solid phases.

Problems

1. For the reaction FeO(s; -j- H2(0) = Fe(s) + H2O(^), the equilibrium
constants are [EMMETT and SHULTZ, /. Am. Chem. Soc., 62, 4268 (1930)]

Ki = pn2o/pH2 ................ 0422 0499 0594 0669 078

T,°K ............ 973 1073 1173 1273 1400

and for the reaction FeO(s) -f CO(g) = Fe(«) -f CO2(0) the equilibrium
constants are [EASTMAN, /. Am. Chem. Soc , 44, 975 (1922)]

Kt « pco2/pco . 0 678 0 552 0.466 0 403 0.35

T, °K 973 1073 1173 1273 1400

(a) Calculate the equilibrium constant Kz for the reaction
C02(0) + H2(flf) = C0(g) + H20(gf)

at these temperatures from the above data. (6) Plot log K* against 1/T for
these constants, add those given on page 347, draw a "best straight line"
through the points, and determine AT/ for the reaction. (The result should
check that of Problem 21, page 328.) (c) How many moles of CO would be
required to reduce IFeO at 1273°K.? (d) From the data above and those
in Table 67, calculate the partial pressure of oxygen in equilibrium with Fe(s)
and FeO(s) at 1400°K. [For data on these systems at higher temperatures,
see Darken and Gurry, /. Am. Chem. Soc., 67, 1398 (1945).]

i HERTZ, Z. anorg. Chem., 71, 206 (1911).

422 PHYSICAL CHEMISTRY

2. (a) Calculate the equilibrium constant for the reaction HgBr2(s) -f-
Br" = HgBr8~ from the solubility data on page 413. (6) Calculate the
solubility of HgBr2 in 0 2m KBr.

3. Ten grams of Ag2S remains in contact with a liter of hydrogen at
873°K. and 1 atm. until equilibrium is established, (a) Calculate the equi-
librium constant for the reaction Ag2S + H2(0) = 2Ag -j- H2S(0) at 873°K ,
from the data on page 399. (6) Calculate the quantities of all four sub-
stances present at equilibrium, (c) What is the least quantity of hydrogen
required for reduction of all of the Ag2S at 873°K.?

4. Calculate A// for the dissociation of CaCOa into CaO and CO2 from
the data on page 395. (The result should check that of Problem 3, page
326.)

5. Hydrogen may be prepared by passing steam over hot iron and con-
densing out the unchanged water, (a) From the data in Problem 1 above,
calculate the moles of steam passing over iron per mole of hydrogen pro-
duced, if the reaction occurs at 1273°K. Calculate the composition of the
gas phase and the quantities of Fe(s) and FeO(s) present at equilibrium in
systems at 1273°K containing initially 1 mole of H2O(#) with (b) 0 3 atomic
weight of iron, (c) 0 5 atomic weight of iron, and (d) 0 8 atomic weight of
iron.

6. Experiments on the solubility of zinc carbonate in water containing
carbon dioxide in excess gave the following results at 29&°K , in moles per
liter of solution •

Total CO2 . 0 184 0 454 0 768

Total Zn 0 0021 0 0029 0 0034

(a) Calculate the equilibrium constant Kc for the reaction
ZriCO3« + H2COg = Zn++ + 2HCOr

at 298°K. (6) Calculate the solubility product for ZnCO8 (c) Calculate
the solubility of ZnCO3 in 0 25m. H2CCX, at 298°K

7. Ammonium carbamate dissociates completely in the vapor phase as
shown by the equation NH4CO2NH2(s) = 2NH8(0) + CO2(0), and at 25°
the dissociation pressure atr equilibrium is 0 117 atm. The dissociation
pressure at 25° for the equilibrium LiCl 3NH8(s) = LiCl.NH8(s) + 2NH3(0)
is 0.168 atm. (a) Neglecting the volume of the solid phases in comparison
with the volume of the vapor phase, calculate the final total pressure when
equilibrium is reached in a 24.4-liter vessel at 25° containing initially
0.050 CO2(0) and 0.20 LiC1.3NH8(s). The solid phases at equilibrium are
NH4C02NH2(s), LiCl.NH8(s), and LiCL3NH3«. (6) Calculate the moles
of each solid phase present at equilibrium, (c) Calculate the equilibrium
total pressure at 25° in a 24.4-liter vessel containing initially 0 050C02(gr)
and0.10LiC1.3NH8(s).

8. The equilibrium constant for the reaction

ZnO(s) + CO(0) - Zn(gf) -f CO2(0)

with partial pressures in atmospheres changes with the Kelvin temperature
as follows:

HETEROGENEOUS EQUILIBRIUM 423

T 1073 1173 1273 1373

KP 1.24 X 10-3 738X10-3 3.29 X 10~2 1.17 X KT1

(a) Determine AH for the reaction. (6) Calculate the ratio of COa to CO
at equilibrium with ZnO(s), Zn{7), and Zn(g) at 1173°K The boiling point
of zinc is 1 180°K , and its latent heat of evaporation is 29,170 cal. per atomic
weight near the boiling point. [TRUESDALE and WARING, J. Am. Chem.
Soc, 63, 1610 (1941).]

9. The equilibrium constant K r for the reaction

Ag(«) + Fe+++ = Ag+ -f Fe++

is 0 128 at 25°C (a) What fraction (x) of the ferric ion will be reduced when
0 Ira ferric nitrate is shaken with excess silver until equilibrium is estab-
lished? (b) What fraction of 0 1m ferric chloride will be reduced by excess
silver? (The solubility of AgCl m water at 25° is ] 3 X 10~B mole per liter )

10. The solubility product for CaCOs at 25°C is given in chemical liter-
ature as 5 X 10~9, and its solubility increases m the presence of dissolved
CO2 because of the chemical reaction

CaCO3(s) + H2CO8 = Ca++ -f 2HCOr

(a) Write the equilibrium expression for this reaction, and evaluate Ka at 25°,
assuming the solubility product is an activity product and using the lomza-
tion constants of carbonic acid (6) The measured solubility of CaC03 is
0 0039m when the equilibrium pressure of CO2 above the solution at 25° is
0 1 atm , and the solubility of C02 m water at 25° is 0 034m when the pres-
sure of CO2 is 1 0 atm Calculate the equilibrium constant Kt from these
facts (c) Calculate the value of the activity coefficient that would be
required to obtain the same* value of Ka from these measurements as from
part (a)

11. Calculate the equilibrium constant for the reaction

Pb(IO3)2(s) + Ac- = PbAc+ + 2I08~

from the solubility data in Table 71

12. The pressure in a 500-nil bulb containing 1.0 gram of NH4C1 changes
with temperature as follows .

p, atm 0 050 0 112 0 217 0 408 0 730 1 22

T, °K 520 540 560 580 600 620

The pressure in a 500-ml. bulb containing 0.091 gram of NEUCl changes
with temperature as follows:

p, atm 0 079 0 158 0 303 0 335 0 346 0.357

T, °K . . 530 550 570 590 610 630

(a) Plot both sets of data on the same paper, and draw lines that fit a
reasonable interpretation of the observed pressures. (6) The density of
saturated NEUCl vapor changes with the temperature as follows:

424 PHYSICAL CHEMISTRY

T, °K 555 585 593 608

p, atm 0 192 0 471 0 621 0 922

grams per liter 0 114 0 269 0 347 0 500

Determine the extent of dissociation of the vapor from these densities
(c) Calculate the equilibrium constant for the reaction

NH4C10) = NH3(0) -f HCKcr)

at several temperatures, plot log K against 1/T, and determine A// for the
reaction, (d) The dissociation pressure is 1 0 atm at 613°K How much
solid NH4C1 forms at equilibrium when 0 10 mole of NH3(0) and 0.15 mole
of HCl(p) are introduced into a 5-liter vessel at 613°K ?

[Data from Smits and deLange, / Chcm Soc (London), 1928, 2945 ]

13. The equilibrium pressure for the reaction 2NaH(s) = 2Na(0 -f H2(00
changes with the temperature as follows :

t, °C 300 320 340 360 380

p, mm 8 02 18 6 41 7 89 1 182

The boiling point of sodium is 878°C., and its latent heat of evaporation is
25,300 cal per atomic weight, (a) Determine whether the vapor pressure of
sodium is a negligible part of the dissociation pressures given above. (b)
Calculate A/7 for the reaction from a suitable plot (c) Calculate the dissoci-
ntion pressure at 400°C [Ans (b) 28,100 cal , (c) 355 mm ]

14. The density (in grams per liter) of the vapor in equilibrium with
NH4Br(s) and the total pressure (dissociation pressure) change with tem-
perature as follows:

•

T 631 645 653 7 668

d 0 346 0 474 0 590 0 820

p, atm 0 366 0 539 0 662 0 953

Problem basis 3 46 grams of saturated vapor at 631 °K. and 0.366 atm.
(a) Show whether the vapor consists of NH3 and HBr only or whether
NH4Br(0) is present in significant quantity (b) Calculate the equilibrium
constant from the dissociation pressure, (c) Calculate the total pressure
at equilibrium and the moles of NH4Br(s) present after 0.040 mole of HBr(#)
has been forced into the space [SMITS and PURCELL, J. Chem. Soc (Lon-
don), 1928, 2936 ]

15. For the reaction NiBr2 NH3(s) = NiBr2(s) -f NH3(gr), ACP = 0, A# =
20,600 cal , and the equilibrium pressure of NH3(0) is 0.50 atm at 609°K.
(a) Calculate the equilibrium pressure at 617°K. (b) At 617°K. the dissoci-
ation pressure for the reaction NH4Br(s) « NH3(0) -f HBr(0) is 0.243 atm.,
and the vapor is completely dissociated. Calculate the total pressure at
equilibrium and the moles of each solid phase present at 617°K in a space of
50.4 liters containing originally !NiBr,(s), 1NH3(0), and 0.25HBr(gr). The
only chemical reactions in the system are those given above, and all three
solids, NiBr2.NH8, NiBr2, and NH4Br, are present.

HETEROGENEOUS EQUILIBRIUM 425

16. The solubility product for MgCO3 3H2O in water at 25° is 1.1 X 10~6,
and the solubility of CO2 in water at 25° and 1 atm. is 0.034m. When
equilibrium is established between MgCO3 3H2O(s) and water over which a
partial pressure of CO2 of 0 05 atm is maintained, the molality of Mg(HCO3)2
is 0 049. (a) Calculate the equilibrium constant Kc from the solubility data
without allowance for activity coefficients (b) Calculate the solubility
when the equilibrium pressure of CO 2 is 0 01 atm. (The measured solubil-
ity at this pressure is 0.027 ) (c) Calculate the equilibrium constant Ka
in terms of activities, using the solubility product above as an activity
product and using KI and K% from Table 63 (d) What activity coefficient
is required to calculate the correct solubility from Ka when the equilibrium
pressure of CO 2 is 0.05 atm ? (c) The measured solubility is 0 217m when
the equilibrium pressure of C02 is 1 atm. Calculate this solubility from Kr
without allowance for activity coefficients. Calculate the solubility again
from Ka, taking 0 42 as the activity coefficient. [No measured activity
coefficients for Mg(HCO3)2 are available; 0.46 is the activity coefficient for
0 2m. Mg(N03)2 and 0 42 for 0 2m. Ca(NO3)2 ]

17. (a) Calculate A// for the dissociation of 2NaHCO3(s) from the dis-
sociation pressures given on page 398, and compare the result with that of
Problem 4, page 327 (6) Calculate the minimum quantity of CO 2(0) that
must be added to a 10-liter space at 100°C containing 0 10 mole of
Na2COa(«) and 0 20 mole of H2O(0) to convert the solid completely into
NaHCO3(s) (c) A cylinder with a movable piston is charged with 0 10 mole
Na2CO,(«), 0 20 mole of CO2(0), and 0 20 rnole of H2O(0) at 100°C. Cal-
culate the total pressure and the quantities of Na2CO3(s) and NaHCO3(s)
present when the volume is 15, 10, and 5 liters. (It is not assured that both
solids are present for every volume.) (d) In order to dry moist NaHCOa at
100°C. and 1 atm total pressure, a mixture of CO2(#) and H2O(0) containing
the minimum of water vapor necessary to prevent decomposition is passed
over the moist NaHCO3 How many moles of water vapor can be evapo-
rated into each mole of this mixture without decomposing any NaHCO3, the
total pressure being kept at 1 atm ?

18. Plot the dissociation pressures of Ag20(s), given on page 396 against
the absolute temperature, and determine A// at 463°K. for the dissociation of
2Ag2O (Reserve the plot for use in Problem 7, page 460 )

19. Determine the total pressure developed at equilibrium by the dis-
sociation of HgO into a space containing oxygen at 0.10 atm. and 400°C.
from the data on page 397.

20. When dilute HC1 is saturated with solid CuCl at 25°C the following
data are obtained [NOTES and CHOW, /. Am. Chem. Soc , 40, 739 (1918)]:

Total Cu 0 00596 0 0134 0 0198 atomic weights per liter

Total Cl . . . . 0 1038 0 2290 0 3364 atomic weights per liter

(a) Show that the formation of a complex, CuCl 2", explains this solubility
of CuCl in HC1, and calculate the equilibrium constant (CuCl2~)/(Cr) for
25°. (6) This equilibrium ratio is 0.0453 at 15°. Calculate A# for the
equilibrium reaction, and state explicitly the change in state to which this

426 PHYSICAL CHEMISTRY

A// applies, (c) At higher HC1 concentrations the data for 25°C are as
follows :

Total Cu . 0 047 0 15 0 29 atomic weights per liter

Total Cl ... 0 944 1 90 3 15 atomic weights per liter

These solutions precipitate cuprous chloride upon dilution with water
Would the reaction assumed in part (a) account for the precipitation?
Consider the possibility of a complex bivalent ion such as CuCl3 , and
state any conclusion to be drawn

21. When 0 2m SnCi2 is shaken to equilibrium at 25° with excess solid
lead, lead chloride precipitates and the concentration of stannous ion becomes
0.0465 mole per liter (a) Calculate the composition of the equilibrium solu-
tion from the data of Table 73 (6) Calculate the solubility product
Kc for lead chloride, and calculate from this the solubility of lead chloride
in water. (Ans.' About 004m) (c) Calculate the composition of the
equilibrium solution when 0 02m. SnCl2 is shaken to equilibrium with excess
solid lead.

22. For the reaction NH4HS(») = NH,(0) + H2S(0), A# - 22,400 cal.
and A(7P = 0. At 298°K the dissociation pressure of NH4HS is 0,592 atm ,
and the vapor contains only NH3(0) and H2S(0) Calculate the equilibrium
total pressure at 308°K. and the moles of solid NEUHS formed when 0 60
mole of H2S(0) and 0.70 mole of NH8(0) are added to a vessel of 25.25 liters
volume

23. The solubility product of PbI2 is 9 5 X 10"9 at 25°, and the solubility
product of PbS04 is 16 X 10~9 at 25° (a) What volume of 0 1m. K2S04 is
required for the complete conversion of 0.010 mole of PbI2(s) to PbS04(s)?
(b) What volume of O.lw. Kl is required for the complete conversion of
0.010 mole of PbS04(s) to PbI2(s)?

24. The solubility of AgI03 m water at 25° is 0.000175, and its solubility
in ammonia solutions is given in Table 70. Calculate Kc for the reaction
Ag(NH8)2+ = Ag+ + 2NH3

25. The equilibrium constant Kp (in atmospheres) for the reaction
C(.s) + C02(gr) = 2CO(0) changes with the Kelvin temperature as follows:

T * 1123 1173 1223 1273 1323

Kp. 14 1 43 1 73 8 167 268

(a) Calculate A/7 for this reaction from a plot of log K against l/T
(6) In the calculation of Problem 15, page 328, the partial pressure of C02
was neglected Estimate this pressure, assuming equilibrium was attained
in the reactor (c) Estimate the very small partial pressure of oxygen in
the mixture in this problem from the data in Table 67. [Data from "Inter-
national Critical Tables," Vol. VII, p. 243.]

CHAPTER XI
PHASE DIAGRAMS

In this chapter we are to consider another aspect of hetero-
geneous equilibrium, the change in the number and composition
of phases at equilibrium with changing temperature or pressure
or gross composition. The experimental facts are commonly
shown by "phase diagrams'7 that cover variations in composition
from 0 to 100 per cent of a given component. As a guiding
principle we have Gibbs's "phase rule/' which limits the number
of phases in terms of allowable variations of pressure or tempera-
ture. Before discussing these topics, it will be convenient to
define two or three new terms and to repeat the definitions of some
other terms previously used.

A system is any combination of matter on which we choose
to focus attention. For our own convenience we consider a
restricted system and study the effect of varying one or another
of the external conditions that govern its behavior; the con-
tainer and any other objects in contact with the system are con-
sidered as " surroundings."

The phases of a system are its homogeneous parts, separated
from one another by definite physical boundaries. A gas or a
gaseous mixture is a single phase, as is a liquid solution or solid
solution, but two mutually saturated liquid layers, such as ether
and water, constitute two phases. Each pure crystalline sub-
stance is a separate phase, and a mixture of rhombic and mono-
clinic sulfur, for example, is two phases.

The components of a system are the chemical substances
required to make each of its phases in whatever quantity they
may be present. Thus one substance, water, is capable of
forming all the phases of the water system; but if the system
under consideration is a solution, water and the solute are its
components. The number of components is defined as the smal-
lest number of chemical substances required to form all the parts
of the system in whatever proportion they may exist. For

427

428 PHYSICAL CHEMISTRY

example, one system composed of calcium oxide, calcium carbon-
ate, and carbon dioxide may be made from a single substance,
calcium carbonate. But it is possible for these three phases to
exist together when the amount of calcium oxide is not chemically
equivalent to the carbon dioxide present. Since all three sub-
stances may be formed in any desired quantity from calcium
oxide and carbon dioxide, these two substances may be called the
components of the system. It would serve equally well to
designate the components as calcium oxide and calcium carbon-
ate, for by adding or removing these two substances any desired
quantity of each phase could be brought into a system. The
three-phase system CaO(s), CaC03(s), C02(0) is thus a two-
component system.

The variance of a system, also called the degree of freedom,
is the number of intensive properties that can be altered inde-
pendently and arbitrarily (within certain limits) without causing
the disappearance of a phase or the appearance of a new phase.
For example, in a one-component liquid system both tempera-
ture and pressure may be varied within limits without causing
the appearance of solid or vapor, and hence the variance is
2. Since both these properties must be specified to define
completely the state of the system the variance is also the num-
ber of intensive properties that must be specified to define the
state of the system and to fix all its properties. In a two-phase
one-component system, such as a pure liquid in equilibrium with
its vapor, there is only one pressure for each temperature at which
the two phases exist in equilibrium or one temperature for a
specified pressure and thus the variance of the system is 1. The
Clapeyron equation has been used to describe such systems many
times in the preceding text. If three phases exist in a one-
component system, neither temperature nor pressure may be
varied without causing the disappearance of a phase, and the
variance of the system is zero. If there are two components
and only one phase, pressure, temperature, and composition may
be varied, and the variance of the system is 3.

It will be true of every statement in this chapter, as it was of
every statement in the two preceding chapters, that equilibrium
is a necessary condition. In spite of the repeated use of the
word equilibrium on almost every page, students sometimes fail
to realize that systems are not necessarily at equilibrium when

PHASE DIAGRAMS 429

no reaction or change is evident and that equilibrium considera-
tions do not apply to systems not yet at equilibrium. None of
the common metals is in equilibrium with air, and yet they exist
in contact with air for years without any evident change; the
calculated dissociation pressure of potassium chlorate exceeds
any attainable pressure of oxygen, and yet it does not dissociate
at an observable rate; sodium bicarbonate is not in equilibrium
with dry air, but it does not decompose under ordinary storage
for long periods of time. None of these systems is at equilibrium,
and accordingly none of the statements in this chapter would
apply until true equilibrium is established.

Gibbs's Phase Rule.1 — If the number of phases in a system is
denoted by P, the number of components by C, and the variance
by V, Gibbs's phase rule is expressed by the equation

p + v = C + 2

This is a law limiting the number of phases that may* exist
together at equilibrium in a system. It tells nothing as to what
phases exist, but only the maximum number that may exist
under specified conditions. Moreover, it is not concerned with
the relative proportions of the phases ; it relates only to intensive
properties of the phases. The three-phase two-component
system consisting of CaO(s), CO2(0), and CaCO3(s) would have
one degree of freedom, i.e., one may specify the pressure (say,
1 atm.) but not the temperature at which these three phases
exist under this pressure. If we specify 1 atm. pressure and
800°C., the phase rule says that two phases may exist, but it does
not say which phases. The data on page 395 show that these
phases may not be CaO(s) and C02(0), but the phase rule is
not capable of furnishing this information; it shows only that
some "two phases may exist. Actually CaCOs(s) and C0z(g)
or CaC03(s) and CaO(s) may exist together at 800° and 1 atm.,
but all three phases exist at 800° only when the pressure of C02
is 0.220 atm. It should be further noted that the phase rule
gives only the maximum number of phases permitted but does
not forbid a smaller number. For example, under 1 atm. pres-
sure at 800° the system might be CaCOsW alone. If all three

1 For a full discussion of this equation see Alexander Findlay, "The Phase
Rule and Its Applications," 1927; Marsh, "Principles of Phase Diagrams,"
McGraw-Hill Book Company, Inc., New York, 1935.

430 PHYSICAL CHEMISTRY

phases exist at equilibrium at 800° and 0.220 atm., the addition of
further quantities of solid CaO would change the composition of
the system as a whole but would not change the composition or
any intensive property of any phase; hence, this is still the same
system to the phase rule.

Phase Diagrams. — The quantitative relations in heterogene-
ous systems at equilibrium are frequently shown in the form of
phase diagrams in which (for plane diagrams) some two variables
which are of interest are plotted while the others are kept con-
stant. For systems of one component the common forms are
p-v isotherms (Fig. 10) and p-t diagrams (Figs 47, 48); for two-
component systems the usual variables are temperature-compo-
sition at constant pressure (Figs. 31 and 34, and most of those
in this chapter) or pressure-composition at constant temperature
(Fig. 28)

Solid models are, of course, required to show p-v-T relations in
a ong-component system, and they are also used to describe
temperature-composition equilibrium in systems of three compo-
nents. Perspective drawings of such models are difficult to draw
and to study except for the simplest systems. In this brief
treatment we shall not have space in which to consider either
the models or drawings of them, notwithstanding their great
practical importance. We turn first to pressure-temperature
diagrams for one-component systems and then to temperature-
composition diagrams for two-component systems at atmospheric
pressure.

SYSTEMS OF ONE COMPONENT

Pressure -temperature Diagrams. — Many pure substances have
two or more crystalline phases of different form (crystal habit),
solubility , and other physical properties. When these solids
have transition temperatures at which phase changes occur
reversibly among them or to liquid or vapor, the equilibrium
conditions may be shown on diagrams. Substances (such as
phosphorus) that do not have reversible transitions but do form
different solid phases are called monotropic ; those in which transi-
tions are reversible (tin and sulfur, for example) are called enan-
tiotropic. For substances of the latter class we may draw
diagrams showing the temperature and pressure corresponding

PHASE DIAGRAMS

431

to the stable existence of single phases, pairs of phases, and triple
points.

Any single phase in a system composed of only one chemical
substance may exist throughout a certain temperature range and
under a variety of pressures; two phases coexist at a certain
definite pressure foK each temperature and cannot exist at any
other pressure at this temperature ; when three phases are present
at equilibrium in a system of one chemical substance, neither
the temperature nor the pressure can be varied. For example,
liquid water may exist under any pressure greater than its vapor

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pressure and at any temperature above the freezing point and
below the boiling point corresponding to the pressure imposed,
but liquid water and water vapor exist together at any chosen
temperature only under the vapor pressure. If at 100° the
external pressure is maintained at less than 1 atm., no liquid water
condenses; if the pressure is made greater than 1 atm., all the
vapor condenses. Only when the pressure is exactly 1 atm. can
both liquid water and water vapor exist at 100°. Under these
conditions, however, the two phases can exist at equilibrium in
any relative quantities whatever — a drop of liquid in contact with
a large volume of vapor, or a single bubble of vapor in equilibrium
with a large quantity of liquid.

Only at the triple point and under the vapor pressure of ice
can all three phases exist. Thus the presence of three phases
in a system of one component fixes both the temperature and

432 PHYSICAL CHEMISTRY

the pressure, and this is an invariant system. A two-phase
system of one component may exist at one particular pressure
for each temperature or at one particular temperature for each
chosen pressure. Since one condition (pressure or temperature)
of such a system may be arbitrarily varied, it is a univariant
system.

A simple diagram describing the phases of water is shown in
Fig. 47. The line BDE is a vapor-pressure line, i.e. , a line showing
the pressure at which liquid and vapor exist at equilibrium for
each temperature. It is a line on which a monovari^nt system
prevails, a line whose slope is shown by the equation

dp AH
dT T Av

which applies to any monovariant system. The diagram shows
that the vapor pressure of water at 60° is 0.196 atm.; accordingly
if water at 60° is acted upon by a greater pressure, all of it
remains as liquid; if the pressure is reduced below 0.196 atm ,
liquid vaporizes until the equilibrium pressure is reached or
until all the liquid is evaporated. At a lower pressure than
0.196 atm. the system composed of water at 60° consists of vapor
only. Hence the line BDE is a two-phase line, defining the pres-
sure at which two phases coexist for each temperature on the
diagram.

The temperature at which ice and water saturated with air
exist in equilibrium under a pressure of 1 atm. is defined as 0°
on the centigrade scale; but since the vapor pressure of ice at
0° is only 0.006 atm., this is not the temperature at which all
three phases exist. As calculated on page 148, the melting
point of ice is lowered 0.0075° for each atmosphere increase of
pressure; hence, at 0.006 atm. the equilibrium temperature is
raised +0.0075°, and after allowing for a further temperature rise
of 0.0023°, due to the removal of air as a solute +0.0098° is the
three-phase temperature or triple-point temperature. The pres-
sure at the triple point is 0.006 atm., which is the vapor pressure
of both ice and water at 0.0098°, since they are in equilibrium
with each other at this temperature. The slight effect of pressure
upon the melting point of ice is shown by the slope of the line
BC of Fig. 47 to the left. This effect becomes large for very high
pressures as may be seen from the data in Problem 23, page 462.

PHASE DIAGRAMS

433

A consideration of the phases of urethane
will further illustrate phase diagrams for a system of one compo-
nent. l It forms a vapor, a liquid, and three different solid phases,
which we may designate by I, II, and III. As urethane boils at
180°, the vapor field would occupy only a very small area at
the bottom of the diagram, corresponding to vapor pressures of
less than 1 atm. for the temperature range shown. The position
of this vapor field is indicated in Fig. 48, showing the pressures
and temperatures at which each of the other phases exists.

Between 52° and 70° equilibrium between liquid and solid I is
shown by the line ab. It will be noted that this line slopes in
the opposite direction to the liquid-solid line for water, indicating
that an increase of pressure raises the melting point. As increase
of pressure at constant temperature always results in the forma-
tion of a more dense substance, solid I is more dense than liquid
and will sink in it. At 70° and 2200 atm. (6) there is a change in
the character of the solid phase, and during transition from I
to II there are three phases present. This is an invariant point,
and neither temperature nor pressure can change until some

1 BRIDGMAN, Proc. Am. Acad. Arts Sri., 52, 57 (1916); Proc Nat. Acad.
Set,., 1, 513 (1915) The following diagrams show the phases for three other
systems of one component Recent work in this field is summarized in

0° 50° 100° 150°
Silver Iodide

0° 50° 100° 150°
Carbon Tetrachloride

0° 50° 100° 150°
Poha&&ium Nitrate

ibid., 23, 202 (1937) About 150 substances have been examined, of which
nearly half have shown unmistakable evidence of polymorphism at high
pressures. The distribution given by Dr. Bridgman is as follows:

1 2 345678
80 45 13 7 0 3 1 0

Number of solid phases ...
Number of examples

Experimental technique for pressures of 50,000 atm. is described in Phys.
Rev., 48, 893 (1935). Data for 35 new polymorphic solids and negative
results on about 60 others are given in Proc. Am. Acad. Arts Sci., 72, 45
(1937). Means of attaining pressures of 425,000 atm. are described by
Bridgman in J Applied Phys., 12, 461 (1941).

434

PHYSICAL CHEMISTRY

phase disappears. Which one will be exhausted first depends
on the conditions of experiment. If heat is added to the system,
and such a pressure is maintained that liquid is always present,
phase I disappears, and the equilibrium between II and liquid
is shown by the line be. The point c corresponds to another
triple point involving the liquid phase; point d is the triple point
of all three solids.

Suppose a quantity of urethane to be kept at 60° while (through
the steady motion of a piston in a cylinder) its volume is slowly
decreased. As the melting point is 52°, the system consists of
a liquid at the start- — a one-phase, one-component system that
may exist under various temperatures and pressures, but we have

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**^

LlCfL

Vac.

ori

/fffoK

° 20° 40° 60° 80° 10

FIG. 48. — Phase diagram for urethane.

fixed arbitrarily upon a temperature of 00°. The system
remains liquid as the volume decreases until a pressure of about
900 atm. is reached at a point on the line ab. Here phase 1
appears; and until all the liquid is changed to I, we have a
two-phase system at a fixed temperature. Hence the pressure
will remain constant while the volume decreases to that of the
solid alone. As heat is 'evolved during the solidification, it must
be removed from the system in order to keep it at 60°. Finally,
all the liquid changes to solid I, and a further movement by the
piston causes an increase of pressure in the system. When a
pressure of 2500 atm. is reached (line bd), I changes to II at a
constant temperature and pressure, with a further decrease in
volume. Then II is compressed until the pressure reaches
about 3800 atm. (line cfc), where it changes to III. Further

PHASE DIAGRAMS 435

decrease in volume does not cause the appearance of any new
phases.

Multiple solid phases at high pressures, as well as at 1 atm.
pressure, are formed by many substances. Problems 20, 23,
and 24 at the end of this chapter are illustrations, and many
others are known.

Heat Effects of Phase Changes. — The Clapeyron equation
may be used to calculate the heat absorbed during any of these
phase changes, since all of them are in monovariant systems,
when A?; and the change of transition pressure with temperature
are known. If Av is the increase in volume attending transition
of a gram of substance from one phase to another at the tem-
perature T and if dp/dT is the change in transition pressure in
atmospheres per degree, A// will be in milliliter-atmospheres
absorbed per gram. Calories may be converted to these units
by multiplying by 41.3.

SYSTEMS OF Two COMPONENTS1

Temperature-composition Diagrams. — Equilibrium in syn-
tems of two components is most commonly shown on diagrams
in which the temperature is plotted against the composition
of the whole system (gross composition) while the pressure is
kept constant (usually at 1 atm.). Although these are some-
times inaptly called " phase-rule diagrams," they furnish quan-
titative information as to how the compositions and quantities
of the phases in a system at equilibrium change with the tempera-
ture and composition of the system as a whole. The phase
rule cannot furnish such information. In the diagrams that
we now consider, the abscissas show the composition of the whole
system and the ordinates show temperature changes at a con-
stant pressure of 1 atm. A vertical line in such a diagram shows
the composition of a phase that is unchanging as the temperature
changes. Horizontal lines show a constant equilibrium tempera-
ture with changing gross composition, and since for the main-

1 Phase diagrams for metallic systems of two components are given by
M Hansen in " Aufbau der Zweistoffiegierungen," 1936, in which some hun-
dreds of systems are described See also " International Critical Tables,"
Vol. II, pp. 400-455. Silicate systems are described by Hall and Insley in
J. Am. Ceram. Soc., 16, 463 (1933), 21, 113-156 (1938); other inorganic
systems are given in "International Critical Tables," Vol. IV, pp. 77 'ff.

436

PHYSICAL CHEMISTRY

tenance of a constant temperature in a two-component system
at a fixed pressure the phase rule allows only three phases, these
lines show three-phase equilibriums. The compositions of two
of these phases are shown by the ends of the horizontal line, and
that of the third phase by an intermediate point where some
other line joins the horizontal line.

System: Cadmium and Bismuth. — The simplest systems of
two components are illustrated by two substances that mix
in all proportions in the liquid state and that do not form com-
pounds or crystals other than those of the two pure components.
Mixtures of cadmium and bismuth satisfy these conditions and
will be considered first. Cadmium melts at 323°, and a solution
containing increasing quantities of bismuth begins to deposit

20 40

Per cent Bismuto
FIG 49 — Phase diagram for bismuth and cadmium.

•

solid cadmium at lower and lower temperatures, as shown in
Fig. 49, in which equilibrium temperature is plotted against the
gross composition of the system. The left-hand portion of this
figure shows the depression of the freezing point of the metallic
solution, or the temperature at which solutions of increasing
bismuth content are in equilibrium with crystalline cadmium.
Bismuth melts at 273°, and equilibrium between solid bismuth
and a liquid mixture of bismuth and cadmium comes at lower
temperatures as the percentage of cadmium increases. Obvi-
ously any liquid mixture of these metals in any proportion
becomes solid at a sufficiently low temperature. The two
" freezing-point curves" intersect at this minimum temperature,
shown at c on the diagram. A liquid mixture containing 60 per
cent of bismuth deposits neither solid until 140°, but at this
temperature it deposits both solids at once. The field above

PHASE DIAGRAMS 437

abcdc is the " liquid field "; systems of any composition consist of
one liquid phase at all points above this line and below the
boiling points of the solutions for a pressure of 1 atm.

Let us study the behavior of a solution containing 25 per
cent of bismuth when it is cooled from 400° to 100° The
path of this process is indicated by the dot-and-dash line on
Fig. 49. The system under a pressure of 1 atm consists of
liquid until about 240° (point b on the figure) ; at this tempera-
ture solid cadmium begins to separate from the melt. The
composition of the system remains constant, but a new phase
appears whose composition is shown by the left-hand margin
(i.f., pure cadmium); and owing to the separation of cadmium
from the melt the percentage oi bismuth in the liquid increases.
At 200° considerable cadmium will have separated out, and the
liquid is about 40 per cent bismuth,

If heat has been withdrawn from the system at a uniform
rate of so many calories per minute, the fall of temperature will
take place more slowly after reaching &, owing to the "latent"
heat evolved when cadmium solidifies. As the cooling proceeds,
more solid cadmium separates, and the composition of the liquid
is shown for each temperature by the line be, until at 140° the
liquid is saturated with both metals Upon further cooling,
both metals solidify from the liquid, and the temperature remains
constant during the cooling until all the liquid phase disappears.
It should be noted that a system may evolve heat at a constant
temperature if a source of heat exists within it; for cooling con-
sists in taking away heat, and this may not cause a change in
temperature in all systems.

Let us return to a consideration of the system at 200°, which
is at the point n in the field ahc. The system contains 25 per
cent of bismuth, and 75 per cent of cadmium; but one phase of
the system is pure cadmium; hence the other phase must be
poorer than the system as a whole in this component. There is
anothei fact to be derived from the dimensions of the diagram,
namely, that the relative quantities of solid cadmium and of
solution at 200° are to each other as the lengths nr and nm.

The proof of this relation is as follows : Let w be the weight of
the system that at 200° consists of solid cadmium and 5 grams
of a solution of composition r. Note that hk corresponds to 100
per cent and that mn/hk is the fraction of bismuth in the whole

438 PHYSICAL CHEMISTRY

system. The weight of the bismuth in the system is w(mn/hk)
Since only solid cadmium has separated, all the bismuth is still in
the liquid, and s grams of the liquid contains s(mr/hk) grams of
bismuth. On equating these two expressions for the weight of
bismuth, we get

mn mr , mn

w — s whence w : s = mr: mn or 5 = — w

hk hk mr

The weight of solid cadmium that has separated from solution
must be w — s, and this is equivalent to w(nr/mr).

At b in Fig. 49 the length corresponding to nr is zero, which
means that no solid cadmium has yet separated; at 150° the
length nr is longer in proportion to that of nm, corresponding to a
further separation of solid cadmium at the lower temperature
For all temperatures and gross compositions shown by the field
ahc of Fig. 49, the phases at equilibrium are solid cadmium and
liquid solution. Horizontal lines drawn across a two-phase field
are called "tie lines," and the ends of a tie line show the composi-
tions of the phases in equilibrium at the temperature for which
it is drawn and for all gross compositions on the tie line. For
illustration, a tie line through the point t in Fig. 49 shows that at
180° solid bismuth is in equilibrium with a liquid containing 70
per cent bismuth. Since the qualitative significance of all the
tie lines in any one two-phase field is the same, we may mark each
field to show what phases are at equilibrium ID it. In Fig. 49
the area above abcde is the liquid field, ahc shows equilibrium
between cadmium and liquid, eke shows equilibrium between
bismuth and liquid, ancLthe area below hck that between the two
solid phases.

On the line hck three phases exist in equilibrium, solid cad-
mium, solid bismuth, and a liquid of composition c. When
heat is withdrawn from such a system, the temperature, the
composition of the system, and the composition of any phase
do not change; hence, neither a phase diagram nor the phase rule
can show the relative quantities of the three phases present at
equilibrium. It should be noted that this line hck is not a tie
line in a two-phase field but a three-phase line. The compo-
sitions of two of these three phases are shown by the ends of
the line, and that of the liquid is shown by the point c; but the
relative quantities of the phases present in systems of gross

PHASE DIAGRAMS

439

compositions shown on the line are not given by the lengths of
any lines on the diagram.

This diagram is typical of two-component systems if the sub-
stances mix in all proportions in the liquid state, and provided
that they do not form any crystalline phases other than the two
pure components ; it is the simplest type of such diagrams. Other
examples of mixtures with the same type of phase diagrams arc
shown in Fig. 50.

AVJ

I063Q

131°

Si
1410°

TL
302°

773'

JAu
Il063°

370°

KCL

LiCl

361

601°

FIG. 50. — Phase diagrams of simple two-component systems

The Eutectic Mixture. — A mixture of two solids such as
that which separates from the bismuth-cadmium system at
140° is usually referred to as "the eutectic," but it should be
clearly understood that it is a mixture of two separate phases.
There is no such thing as the "eutectic phase. " When a liquid is
cooled, the component that first separates usually appears in
larger crystals than those forming at the eutectic temperature.
Both solid phases separate at this temperature in an intimate mix-
ture that is of finer grains (smaller crystals) than the crystals of
the single component already separated, but this mixture may
be seen under a microscope to consist of separate crystals of
each substance. The phases of the eutectic are those indicated
by the two intersecting curves. The eutectic mixture is that
intimate mixture of two solid phases separating at the constant
temperature which marks the lower limit for the existence of
liquid.

Cooling Curves. — If the temperature of a system that is
evolving heat at a uniform rate is measured at suitable intervals
and a diagram is drawn showing these temperatures against time
as abscissas, abrupt changes in slope will indicate the processes
occurring during cooling. A group of such curves for a series
of mixtures of cadmium and bismuth is shown in Fig. 51,

440

PHYSICAL CHEMISTRY

In cooling a mixture containing 15 per cent of bismuth from
400° to 270° , there is no process occurring in the system that
evolves heat except loss of heat from the liquid phase. At 270°
solid cadmium begins to separate, giving rise to more heat, and
hence the rate at which the temperature falls is slower, though
heat is being withdrawn from the system at a constant rate.
There will, therefore, be a break in the slope of the curve at this
point (1 in Fig. 51). At 140°, where both solids are separating,
since this is a three-phase condition in a two-component system
under a specified pressure, the temperature remains constant
(2, 3), even though heat is being taken from the system. The
fourth curve of Fig. 51 is a cooling curve for a solution of 60 per
cent bismuth, which deposits both solids at 140°; the next one

15 25
500 - \-

100 t--\—

FIG 51.— Cooling curves for mixtures of c*adimum and bismuth.

is a cooling curve for a solution containing 75 per cent of bismuth.
At the point 12 (190°) solid bismuth begins to separate, and the
composition of the melt changes along the line edc of Fig. 49
as the temperature falls; at point 10 (140°) both solids separate,
as in cooling the other solutions. For pure bismuth all the
solid deposits at the melting point, 273°.

Thermal Analysis. — In a piece of apparatus in which a con-
stant rate of heat loss can be maintained, the time interval during
which eutectic mixture is separating is proportional to the weight
of the solids formed. In other words, the quantities of eutectic
mixtures are proportional to the lengths of the horizontal portions
of curves like those in Fig. 51. If these portions of curves for
the rate of cooling of a fixed quantity of the various mixtures —
such as (2, 3), (4, 5), and (6, 7) of the curves of Fig. 51 — are
plotted vertically against the composition of the system as a
whole, a triangle is obtained, as shown in Fig. 52, in which 8, 9 of

PHASE DIAGRAMS

441

Fig. 51 is the altitude. From this plot it is possible to make a
rough analysis of an unknown mixture of the two components by
determining the length of time required to solidify all its eutectic
in the standard apparatus. Knowing the weight of system
taken, we may estimate from the weight of eutectic the percent-
age of each component. It is necessary to determine in some
other way whether the given sample lies on the right-hand or
left-hand side of the eutectic, but this is usually known. The
chief uses of such diagrams are in locating the liquid composition
at the eutectic temperature and in showing the compositions of
the solid phases separating when they are not the pure crystalline
components.

0 20 40 60 80 100

Per cent Bismuth in the System

FIG. 52 — Eutectic pauses in cooling curves

Quenching Method. — The cooling-curve method has been
successfully applied to metal systems and to mixtures of crystal-
line salts It has not proved to be a useful means of studying
silicate systems, such as ceramic materials and other refractories r
partly because of the high viscosity of the liquids, with a resul-
tant undercooling and delayed crystallization, and partly because
the energy changes attending the chemical reactions involved
are not large in comparison with radiation losses from these
systems at high temperatures. Such systems are usually studied
by a "quenching method/' This method consists in heating a
finely ground charge or mixture of the appropriate solids until
equilibrium is reached at the desired temperature, after which
it is dropped from the furnace into cold mercury. In this
manner the system is "frozen" in the condition at which it
was in equilibrium in the furnace, and a microscopic examination
of the quenched material shows what phases were present at this
temperature. If it is found that the system contains more than
one crystalline phase, new charges of identical composition are

442

PHYSICAL CHEMISTRY

heated to higher temperatures, quenched, and examined until one
is found that contains only one crystalline phase, this being
the primary solid phase characteristic of that part of the system.
The process is continued until a temperature is found at which
the primary solid phase disappears from the quenched mixture
and leaves only liquid (glass at room temperature). This tem-
perature and the composition locate a point on the liquid-solid
curve of the system.

The quenching method is also applicable to metal systems
provided that quenching is carried out so rapidly as to " freeze"
the equilibrium. It involves the preparation of many charges
in order to determine the equilibrium points for one composition,
and is more laborious than the cooling-curve method, but it
furnishes much information not to be had from cooling curves.

Two -component Systems in Which a Compound Forms. —
If the two components of a system mix in all proportions in the
liquid state but form a solid compound, a diagram of somewhat
different character shows the phase equilibrium. Each compo-
nent dissolves in the compound to lower its freezing point, and
the compound dissolves in each pure component to lower its
freezing point. The phase diagram for such a system may be
constructed from breaks in the cooling curves, of the kind
described in Table 74. Magnesium melts at 651° and calcium
at 810°. It will be seen that a compound is formed which con-

TABLE 74 — SYSTEM MAGNESIUM AND CALCIUM

Weight percentage of
calcium in system

First break in cooling
curve

600°

525°

620°

700°

721°

650°

466°

720°

Horizontal portion of
cooling curve

514°

721°

466°

tains 55 per cent calcium by weight, or 55/40 = 1.38 atomic
weights of calcium to 45/24.3 = 1.85 of magnesium, that is,
Ca3Mg4, and that it melts at 721°. Further, the eutectic formed
of pure magnesium and compound solidifies at 514°, and that
composed of compound and calcium solidifies at 460°. TheTiew
diagram will consist of two portions, each similar to Fig. 49.
The composition of the second eutectic is shown by the 79 per

PHASE DIAGRAMS

443

cent solution solidifying all at one temperature; that of the other
eutectic is not given but may be obtained as shown below.
Inserting the known points on a diagram and connecting with
lines, we obtain the diagram of Fig. 53; and, by extending the
freezing curves smoothly, the first eutectic is seen to contain
about 19 per cent of calcium. The left-hand 55 per cent of this
diagram corresponds to the two-component system magnesium
+ Ca3Mg4; the right-hand 45 per cent, to a system Ca3Mg4 +
calcium; but it is convenient to cover the whole range of composi-
tion on a single sketch. Each portion of this diagram may be
treated exactly as was Fig. 49. The relative weights of com-

800

20 40 60 80
Per Cent Calcium
FIG. 53 — Phase diagram for calcium and magnesium.

pound and liquid melt, in a system consisting of 50 per cent by
weight of calcium at 600°, are to each other as the lengths nr
and mn on this diagram. When the cooling of a system con-
taining 70 per cent of calcium is carried out as indicated on
Fig. S3 by the dot-and-dash line, compound separates from the
melt at 590° and the melt changes composition during cooling
as shown by the line cd until 466° is reached, where both com-
pound and pure calcium separate at a constant temperature until
all the liquid phase is exhausted.

Systems which form a compound have, in general, this type
of diagram when the solid phases are those indicated, but before
applying these considerations to a given system it is first neces-
sary to ascertain that the solid phases are the pure components or
are compounds formed from them. As in the case of chemical

444

PHYSICAL CHEMISTRY

reactions involving solutions and solids, the equilibrium con-
ditions cannot be represented quantitatively on a diagram unless
the chemical composition of each phase is known.

Two other metallic systems in which stable compounds form
are shown in Fig. 54. Other systems in which this occurs are
Te + Bi, A1203 + Ti02, and T1C1 + BaCl2; H2O + S03 form
five compounds, as do CC14 and C12. It should be noted with
respect to Fig. 54 that because of the scale of the drawing no
triangular area for equilibrium between bismuth and liquid or
between sodium and liquid appears. Nevertheless, such areas
must exist. Note that the melting point of sodium is 97°, that
the eutectic temperature is given as 95°, and that the first hori-

710°

)52'

553°

0 14 35 100

Per cent1 Magnesium

632'

404'

65 8085 .
Per cent Antimony

FIG 54 — Compound foimatiori in metallic systems

zontal line on the Bi-Mg diagram is 2° below the melting point
of pure bismuth.

Peritectics ("Concealed Maxima").— Numerous examples are
known in which a compound does not melt upon being heated
but decomposes reversibly into a new solid phase and a liquid
phase saturated with respect to both solids. So far as phase
equilibrium is concerned, this condition is the same as that at a
eutectic point, but the term eutectic is restricted to the equi-
librium temperature below which no liquid phase exists; and
we shall see presently that liquid does exist below the decomposi-
tion temperature in certain ranges of gross composition.

For example, the compound Na2K, which contains 46 per cent
potassium, decomposes reversibly at 7° into solid sodium and a
liquid containing 56 per cent potassium. While this decomposi-
tion is going on there are three phases at equilibrium in a system
of two components at a specified pressure of 1 atm.7 which

PHASE DIAGRAMS

445

requires a constant temperature. Hence continued heating
causes all the compound to decompose at 7°, after which the
temperature rises and equilibrium prevails between solid sodium
and a solution of varying composition, as shown by the line ab
of Fig 55 The various areas below abed correspond to equi-
librium between different pairs of phases that may be identified
by drawing horizontal tie lines and by considering the gross com-
positions, as in the diagram for cadmium and bismuth.

If the lines be and cf of Fig. 55 are projected until they inter-
sect, an imaginary melting point for NasK is indicated, and thus

607oK

20 46 56 78

Per Cent Potassium
FJU 55 — Phase diagiarn and cooling curves for sodium and potassium.

what might have been a melting point is "concealed" by the
phase field ajb in which equilibrium prevails between solid
sodium and a liquid. For this reason such a decomposition is
sometimes called a " concealed maximum." No equilibrium rela-
tion is really concealed, and this " melting point" is not observed
when equilibrium prevails in the system. The term peritectic
is more suitable for this equilibrium; 7° is called the "peritectic
temperature," and the process observed when Na2K is heated at
7° is called peritectic decomposition. The line jeb indicates that
at 7° and for gross compositions up to 56 per cent potassium,
three phases may be at equilibrium. As usual, the compositions
of two of the phases are shown by the ends of the line and that
of the third phase by the point at which ef meets this line.

446

PHYSICAL CHEMIST&Y

Careful attention should be given the cooling curves at the
right of this figure. A system containing 40 per cent potassium
deposits pure sodium between 40° and 7°, and at 7° a liquid
of composition b reacts with solid sodium, producing the com-
pound Na2K until all the liquid is exhausted. Since this com-
pound contains 46 per cent of potassium and the system as a

sco

250

200

.
Q>

150

100

^iquid or
monoclm

id
ic Na2SO

Liquic

Liquic
rhom'

and
:>Ic Na25C

_ir^u\d an<
504.10H2(

1
3 Na2,

S04:10H5
nbic Nd2

0 and

S04

Ice and Na2S04 10H20 ' ""'

40 60 80

Per Cent Na2S04
FIG. 56 — Phase diagram for sodium sulfate and water.

100

whole has 40 per cent of this element, it is clear that excess solid
sodium remains. This describes the significance of the phase
area gjeh, in which solid sodium and solid compound Na2K exist
When any system of gross composition between 46 and 56 per
cent potassium is cooled, it deposits solid sodium until 7° is
reached, and at this peritectic temperature solid sodium reacts

PHASE DIAGRAMS 447

with liquid to form solid Na2K until the solid sodium is exhausted.
Since the system contains more potassium than does the com-
pound Na2K, some liquid remains, and this two-phase system
upon cooling deposits more Na2K, while the liquid composition
changes along the line be until — 12° is reached. At this eutectic
temperature the liquid deposits solid potassium and Na2K until
the liquid is exhausted. The cooling curve marked 50 per cent
in Fig. 55 applies to such a process.

Peritectics occur in many other systems, ferrous and nonferrous
alloys, inorganic salts, silicate systems, organic mixtures, and
salt hydrates. The phase diagram for Na2SQ4 and water is
shown in Fig. 56, in which the lines on the right-hand side of the
liquid field are the solubility curves for the various solid forms.1
The decahydrate Na2SO4.10H2O decomposes peritectically at
32.383° into rhombic anhydrous Na2SO4 and a liquid containing
about 32 per cent Na2S04. The solubility of this anhydrous form
decreases slightly with increasing temperature and goes through
a minimum of solubility at 120°, after which the solubility
increases slightly with temperature up to 241°C. At this tem-
perature Na2S04 undergoes another phase transition to mono-
clinic crystals of the same composition, and the slope of the
solubility curve changes abruptly. The solubility at this tem-
perature is 32 per cent by weight, and it decreases to 2.4 per
cent at 350°C. It will be understood, of course, that at these
temperatures the pressure is not 1 atm., but a sufficient pressure
to prevent boiling of the solution, namely, about 100 atm. at
310° and over 150 atm. at 350°. While solubilities change slightly
with pressure, no correction for these changes has been applied
to the data we are using.

The decomposition at 32.383° absorbs about 20,000 cal.; of
course, this quantity of heat is evolved by the reverse change.
Since this temperature has been well established,2 lies within
ordinary temperature range, occurs in a chemical system that is
readily available in a high state of purity, and is independent of

1 Solubilities below 150°C. are from ibid., Vol IV, p. 236, those above
150°C. are from Schroeder, Berk, and Partridge, /. Am Chem Soc., 59, 1790
(1937).

'2 RICHARDS and WELLS, Proc. Am Acad Arts Sri., 38, 431 (1903). For
the peritectic temperatures of other salt hydrate transitions see Richards
and Yngve, J, Am. Chem. Soc., 40, 89 (1918),

448 PHYSICAL CHEMISTRY

changes in atmospheric pressure, it is an accepted secondary
standard on the thermometric scale.

Many other salt hydrates show similar behavior. Those of
disodium hydrogen phosphate are Na2HPO4.12H2O (stable from
-2° to +3G°), Na2HPO4.7H20 (stable from 36° to 48°), and
Na2HP04.2H2O (stable from 48° to 95°). A plot of solubility
against temperature shows abrupt changes in slope at 36°, 48°,
and 95°, as would be true of any substance when there was a
change in the character or composition of the solid phase in
equilibrium with the solution. This system has three peritectic
transitions, and other systems also contain more than one. For
example, in the Au-Pb, Al-Co, and Ce-Fe systems two peritectic
transitions occur, as well as in many others. Two more illus-
trations are given in Problems 25 and 2(5 at the end of this
chapter.

Solid Solutions. — By analogy to liquid solutions, in which one
substance (a solute) is molecularly dispersed in another (a
solvent) to form a homogeneous liquid phase (a solution) of
variable composition, a crystalline phase of variable composition
in which molecules or atoms of one component are molecularly
dispersed in the other is called a solid solution or a crystalline
solution. Such a crystal is not a chemical compound, since a
substance is considered to be a compound only when it has a
constant composition. Solid solutions are not heterogeneous
mixtures of the crystals of two substances, and the term "mixed
crystals" that is sometimes used for solid solutions is an unfor-
tunate one in that it implies such a mixture. A solid solution
is a single crystalline phase in which the composition may vary
over a certain range when the substances have limited solubilities
or over the whole range from one pure substance to the other
when the solubilities are not limited. Intermetallic solid solu-
tions are somewhat better known than those involving inorganic
compounds or organic compounds, though the latter types of
solid solution are not uncommon.

For illustration, when a liquid mixture of 30 per cent copper
and 70 per cent nickel is cooled so slowly that equilibrium is
established, the composition of every crystal in the crystalline
phase is 30 per cent copper. If another liquid containing 29
per cent copper is cooled slowly, every crystal in the solid phase
contains 29 per cent copper. In the system Ni + Cu the atoms

PHASE DIAGRAMS 449

of nickel in the crystal space-lattice are replaceable by copper
to any extent , and a complete " series" of solutions ranging from
pure copper to pure nickel is formed.

In the formation of metallic solid solutions over any consider-
able range of composition the governing quantities appear to be
(1) the relative radii of the atoms, (2) the amount of distortion
that the crystal lattice can tolerate, and (3) the electronic struc-
ture of the atoms. Solid solubilities are usually very small
unless the radii of the atoms are within 14 or 15 per cent of one
another. Although this requirement seems to be of the greatest
importance, it must not be inferred that meeting it is alone
sufficient to produce unlimited solubility of one metal in the crys-
tals of another metal. For example, silver and copper both have
face-centered lattices, and their " atomic radii " are 1.44 X 10~8
and 1 28 X 10"~8 cm , respectively, which differ by 12 5 per cent
of the smaller radius. They do not form a continuous series of
solid solutions, as may be seen frc*m Fig. 58. Bismuth (1,82)
and antimony (1.61) have atomic radii that differ by 12 per cent
of the smaller quantity, and they form a complete series of solid
solutions as shown in Fig 57.

Some metals are able to enter to a limited extent the crystal
structure of others having a different structure. For example,
cobalt has a face-centered structure, which means a coordination
number of 12, with an atomic radius of 1 26 X 10~8; molybdenum
has a body-centered structure, which means a coordination
number of 8, with an atomic radius of 1.40 X 10~8; but these
metals form solid solutions of 0 to 29 per cent molybdenum. In
these crystals of varying composition molybdenum has 12 neigh-
borks when it replaces cobalt; it thus accepts a different coordina-
tion number in these solutions from the one in its own pure
crystals up to the limit of 29 per cent molybdenum. Beyond
this composition the system has other characteristics, to which we
uhall return a little later, and the series is " interrupted " at this
point.

We have considered here only " primary " solid solutions, those
in which substitution of one atom for another in the space-lattice
takes place. The more restricted interstitial solid solutions are
formed when the solute element is so small that it fits into the
spaces between those of the solvent. Only hydrogen, boron,
carbon, and nitrogen form important interstitial solid soliftions

450

PHYSICAL CHEMISTRY

in metallic solvents, and we shall not have space to discuss
them.1

Primary solid solutions do not form or at least are not likely
to form when there is a marked tendency to form stable com-
pounds. Thus elements which are strongly electropositive tend
to form compounds with those which are strongly electronega-
tive, even when the size factor is favorable for solid solutions.
Elements in columns of the periodic table that are far apart
usually tend to form compounds rather than solid solutions, but

800

200

FIG.

20 40 60 80 100 20 40 60 80

Per Cent Antimony
57 — Phase diagram and cooling curves for bismuth and antimony.

there are exceptions; and, of course, metallic elements in the
same column of the periodic table may form compounds rather
than solid solutions. This is true of sodium and potassium,
in which the size factor is unfavorable even though both elements
have body-centered lattices, and of calcium (face-centered) and
magnesium (body-centered). An illustration of near neighbors
forming a complete series of solid solutions is antimony and
bismuth, which do so over the whole range of composition.2

The equilibrium in bismuth-antimony systems is shown in
Fig. 57, of which the upper field shows liquid composition as

*See William Hume-Rothery, "The Structure of Metals and Alloys,"
Part IV, which is No. 1 of the Institute of Metals Monograph and Report
Series; 1936. He discusses the factors that determine solid solubilities of
both kinds.

2 Other substances forming complete series of solid solutions are Au -j- Pt,
A1208 + Cr208, ThO2 + Zr02, MgO -f NiO, C6H6 + C4H4S, SnBr4 + SnI4,
Cu + Mn, and Cu + Au. In the last two systems a minimum occurs
similafr to the minimum boiling system of Fig 34.

PHASE DIAGRAMS 451

usual and the lower field shows crystalline solutions of varying
composition from pure bismuth to pure antimony. This is a
one- phase area in which the gross composition is the composition
of every crystal. Within the area bounded by the two curved
lines, two solutions exist at equilibrium, one of which is crystal-
line. A system of 60 per cent antimony at 500° consists of a
liquid phase containing 43 per cent antimony and a crystalline
solution of 86 per cent antimony. When this system is cooled
to 400°, the phases at equilibrium contain 20 and 70 per cent
antimony, and the crystals deposited at higher temperature (and
therefore richer in antimony when deposited) have now changed
to crystals of 70 per cent antimony. This change in composition
probably takes place by diffusion in the crystalline phase, rather
than the re-solution into the liquid, and adequate time for this
adjustment must be allowed if equilibrium is to be attained.
When cooling is too rapid, the solid is not homogeneous and the
condition of equilibrium is not reached.

Liquids of other compositions show the same behavior. Some
typical cooling curves are shown in Fig. 57. It should be
noted that these curves have no horizontal portions, for this
would require three phases to maintain a constant temperature,
and the solid solution is a single phase. No more than two
phases exist at equilibrium in this system at any temperature or
gross composition.

Figures 31 and 34 in Chap. VI are also phase diagrams for
constant pressure, with two-phase equilibrium shown in the
area bounded by the liquid-composition and the vapor-composi-
tion lines and one phase in all other portions of the diagram.
Cooling curves for these systems would be similar tto those in Fig.
57. .Minimum melting solid solutions with phase diagrams simi-
lar in appearance to Fig. 34 are known; for example, chromium
and cobalt, nickel and manganese, arsenic and antimony, cop-
per and gold form systems in which a liquid phase exists below
the melting point of the lower melting component and in which
complete series of solid solutions form. Solid solutions over the
whole range of composition are also formed by metallic oxides,
by silicates, by other inorganic components, and by organic
compounds, sometimes with minimum melting, and occasionally
with maximum melting systems. The phase equilibrium has
the same general character in all of these systems.

452

PHYSICAL CHEMISTRY

Ag
960°

Fiu

9 285 92

Weight Per Cent Copper

58 — Phase diagram for copper
and silver

Solid Solutions of Incomplete Solubility. — Copper and silver
form solid solutions in one another to a limited extent only,
yielding the phase diagram of Fig. 58. At the eutectic tempera-
ture in this system the crystalline phases contain 9 and 92 per
cent copper by weight; and when a system of gross composition
between these limits is cooled, these mutually saturated solid
solutions separate from a liquid containing 28.5 per cent copper
at 779°. The areas at the right and left of the diagram, marked

with the Greek letters a and 0,
are one-phase areas in which sys-
tems of varying composition
consist of a single solid solution;
the area below the horizontal
line is marked a + ft to indicate
two saturated solid solutions.
Crystalline silver does not exist
in equilibrium with any liquid
phase other than pure liquid
silver, and at 960°. A liquid
containing 5 per cent copper
has a cooling curve of the same type as those in Fig. 57, with no
horizontal portion; and this system in equilibrium at 700° con-
sists of a single solid phase, with 5 per cent of copper and 95 per
cent of silver in every crystal.

The slanting lines separating the a field and the /3 field from
the a + 0 field indicate decreasing solid solubilities as the
temperature falls. In most phase diagrams where such lines
are vertical, the inference to be drawn is that the solubilities
have not been determined below the eutectic temperature,
rather than that they are constant. This is true of the vertical
lines bounding the solid-solution areas in Fig. 59, which indicate
only that two phases exist in the area below the horizontal line
at the eutectic. Their omission from the diagram would indicate
falsely a single phase in this area at lower temperatures, whereas
there are two phases at all points within it.

The cooling curves for systems containing between 9 and
92 per cent copper would be similar to those shown in Fig. 51.
Application of the method illustrated in Fig. 52, in which the
length of eutectic pause is plotted against the gross composition
of the system, yields a triangle whose base shows the com-

PHASE DIAGRAMS

453

positions of the saturated solid solutions at the eutectic
temperature.

10 20 30 40 50 60 70
Per Cent Molybdenum

1,000

900

800

700

600

3NaF-AlF3

500,

'0
NaF

Solid Solution*
3KlaF-AlF5

.Solid Solution
AIF5 in NaF

3NaF-AlF*+
5NaF3AlFx

AlFz+

20 30

Mole Per CentAlF3

FIG. 59. — Phase diagrams showing solid solutions of limited solubility.

Solid solutions of limited solubility also form between com-
pounds as components, sometimes only on one side of the dia-

454 PHYSICAL CHEMISTRY

gram, sometimes on both sides. A few illustrations are shown
in Fig. 39, a few more are described in the problems at the end
of the chapter, and hundreds of others are known.

Partially Soluble Liquids. — Many pairs of liquids, such as
ether and water, aniline and hexarie, aluminum and chromium,
lead and zinc, SO2 and TiBr4, have mutual solubilities that are
limited at certain temperatures and that increase as the tempera-
ture rises. These systems form two liquid layers when the com-
ponents are mixed in proportions lying between the mutual
solubilities. In these systems complete solubility is usually
attainable at sufficiently high temperatures, though this may not
occur below the boiling point for 1 aim. pressure. For example,
phenol and water at 25° form two layers containing 8 and 72 per
cent phenol by weight, respectively, when mixed in proportions
lying between these figures; as the temperature rises, each solu-
bility increases ; thus at 50° the layers contain 1 1 and 62 per cent
phenol, and solubility in all proportions prevails above 60.8°.
The layers in a mixture of aniline and water at 0° contain 3.3
and 95.6 per cent of aniline, respectively, at equilibrium; at
100° these solubilities are 7.2 and 89 7 per cent, and complete
solubility is attained at 167° with the application of sufficient
pressure to prevent the formation of vapor.

Bismuth and zinc arc completely soluble in one another above
825° and have limited solubilities below 825° as shown in Fig.
60, in which tie lines drawn in the dome-shaped area show these
solubilities. The horizontal line at 416° shows three phases at
equilibrium, solid zinc and two liquid layers containing 15 and
98 per cent zinc, respectiyely. The behavior of a system con-
taining 25 per cent zinc when cooled from 900° to 200° will serve
to describe the phase diagram. Such a system consists of a
single layer at temperatures above 600°; at this temperature a
second layer forms, containing at first 90 per cent zinc, and as
the temperature falls the equilibrium compositions of the two
liquid layers change along the right and left portions of the line
defining the two-liquid zone. At 416° crystalline zinc deposits;
and since the system then contains three phases at a fixed pres-
sure, the temperature remains constant while heat is withdrawn
from the system until the zinc-rich liquid is exhausted. As
zinc deposits, the bismuth in this liquid passes to the other liquid
with enough zinc to keep its composition 1 5 per cent zinc. When

PHASE DIAGRAMS

455

one liquid phase is exhausted, further removal of heat causes
the temperature to fall and zinc to deposit while the liquid
composition changes from 15 toward 2.7 per cent zinc. At
254°, which is the eutectic temperature, bismuth and zinc crystal-
lize from the liquid until it is exhausted, and cooling causes the
formation of no new phases. A liquid containing less than 15
per cent zinc does not separate into two liquids at any tempera-
ture; upon cooling, it deposits zinc first if its composition is
between 2.7 and 15 per cent, and it deposits bismuth first if it
contains less than 2 7 per cent zinc ; finally, it deposits both metals

800

600
§ Bi

•4-

400

"2730
200

254°

Zn
420°

100

20 40 60 60

Weight percent Zinc

FIG. 60---Phabe diagram for bismuth and zinc.

at 254°. The phase equilibrium diagrams for any of the other
systems described in the preceding paragraph are of the same
character.

SYSTEMS OF THREE COMPONENTS

Three substances may form three systems of two components
each; and if, for the simplest illustration, we choose three sub-
stances of complete solubility in one another, which form no
compounds and no solid solutions, the three two-component
systems may be represented by three diagrams similar to Fig. 49.
These are shown1 side by side at the top of Fig. 61 for biphenyl
(abbreviated BP), bibenzyl (BB), and naphthalene (N) and are
called edge sections.

(A much clearer understanding of the discussion that follows
may be obtained by making a copy of the upper part of Fig. 61

and WARNER, J Am Ckem. Soc., 67, 318 (1935).

456

PHYSICAL CHEMISTRY

on stiff paper 7*^ in. across, so that each of the two-component
diagrams is 2^ in. at the base, cutting it out along the upper
lines which show the liquid -solid equilibrium, folding it into a
triangular prism, and standing this up on the lower part of
Fig. 61.)

E
^
E

nit ,

£f\

J°

BB 20 40 60 80 BP 20 40 60 80 N 20 40 60 80 BB
Mole PerCent Mole PerCeni- Mole Per Cenf
BiphenyJ Naphthalene Bibenzyl

20 40 60 80

Mole Per Cent Naphthalene

FIG. 61. — Edge sections and composition triangle for the three-component
system . biphenyl-naphthalene-bibeiizyl

When all three substances are present in a single system, the
compositions of mixtures are shown in a triangular plot such as
that at the bottom of Fig. 61 and temperature is plotted ver-
tically. A ,solid figure results, of which the base is the composi-
tion plot and the three side elevations, or edge sections, are shown
at the top. The eutectic for BP and BB is marked a, that for

PHASE DIAGRAMS 457

BP and N is b, and that for N and BB is c. A system of three
components under 1 atm. pressure is univariant when a liquid
and two solid phases are present Thus a line ad begins at
point a (29.0°, 44 3 mole per cent BP), showing the changing com-
position of liquid in equilibrium with two solids, BP and BB, as
their mole fractions are decreased by the addition of N. It will
be seen that, while a is the eutectic point in a two-component
system, equilibrium along the ad line has not the properties of the
eutectic The addition of N introduces a new component, and
another degree of freedom. As the addition of N continues,
point d is approached. This point is the common intersection
of three 3-phase lines ad, bd, and cd, and it is the ternary eutectic
At d (1 7 4°, 33 8 mole per cent BP, 39 2 mole per cent BB, and 27
mole per cent N) three pure solid substances are in equilibrium
with a liquid under a fixed pressure, and the phase rule shows
that this is an invariant point As heat is withdrawn, all three
components solidify as pure crystalline phases at a constant
temperature.

In the triangular figure, all compositions within the area
UPbda will deposit BP as the first solid when cooled; those in the
area Ncdb will deposit N first; and those in the area BBcda will
deposit BB first. Which solid will deposit next will depend on
the composition For example, a mixture of 20 mole per cent N,
20 mole per cent BB, and 60 mole per cent BP (point h in Fig. 61)
will first deposit solid BP at about 57°; but as BP deposits, the
ratio of BB to N remains unity, and the liquid composition will
change along a straight line drawn from h toward k. Such a line
will intersect the bd line at about 40 mole per cent BP, 30 mole
per cent N; and, on further cooling, both these components will
separate as solid phases while the liquid composition changes
along bd to d, the ternary eutectic.

The student should draw a cooling curve for the process just
described, as a study exercise to clarify the phase equilibriums
involved.

In the system described, the components are chemically similar,
and the laws of ideal solutions apply closely. For example, the
calculated eutectic temperature a for BB and BP is 29.3°, and
the experimental temperature is 29.6°.

Similar phase diagrams describe the three-component mixtures
of metals lead, cadmium, and tin and mixtures of the salts

458

PHYSICAL CHEMISTRY

LiN03, NaNOs, and KNO3. But such ideal simplicity is rare.
Most mixtures of three components exhibit one or several of
the features for two components described earlier in this chapter.
They may form binary or ternary compounds; these compounds
may melt without decomposition or decompose at peritectic
temperatures; they may form two liquid phases of variable
solubility; the components or their compounds may form solid

2,100
1,900
1,700
1,500
1,300

1,100

2
^-C

^V-~To CaC
\ 21-

~"20~e>5±20
1900120

, 2572±

AT*

\
1

\
V

1475 t

10
cc2CaO-S

L /3Ca
\ / /a Cc

V *x 15z

1455i5r^'

ru i

C> O i

1
1

1 *

1" '

Liquid B , Liquids A + B
/ I698±5 1710!|-»N

- o
o

G
- O
CO

-*-

00
O

(XI

J''te\Uz + \\qu\d /ft 99

<0-5i 02 + liquid/
10±2 / Cnsiobalffe+Iiquid B

"^NV / 1470±!0

\^-S\^^f

•f
ru CJ
00

00
C O
00
CNfO

00
0 0
oo

'^Tridymitc t liquid B
Tridymiie+ot CaO'SiOa
r!200±2 "* 121015

1 Tndymite ^>OCo(0 SiOp
i

It 1 1 . t I.I -LI 1

0
aO

2Co.

40
OSt02

60 80 I0<
Weight Per Cent SiO^ SfC

3Ca05i02 3Ca025fOa

FIG 02 — Phase diagram foi calcium oxide and silica

solutions of partial solubility; they may do all these things in a
single system.

Thus the investigation of three-component systems may well
be a very complex problem. From the CaO-Si02 edge section of
the three-component system CaO-Si02-Al2O3 shown in Fig. 62
it will be evident this system is a complex one.1 Yet a full
understanding of this system is essential for much work in
ceramics, for example, the manufacture of portland cement,
and it has been completely worked out experimentally through

1 See "International Critical Tables," Vol. IV, p. 93, for this and similar
diagrams. Three-component metallic systems are given in Vol II, dia-
grams for ceramic materials are collected in /. Am, Ceram. Soc , 16, 463
.(1933).

PHASE DIAGRAMS 459

thousands of quenching experiments. Many of the three-com-
ponent metallic systems have also been studied experimentally,
but their consideration is beyond the scope of this text.1

Diagrams Involving Several Phases. — The various phenomena
that apply to phase equilibriums have now been described and
illustrated with data for systems involving each feature sepa-
rately. More complex systems involving several of these
features at the same time are frequently met in the study of
metallography or of ceramic materials, but the interpretation
of these more complicated diagrams does not involve any new
principles. Thus, mixtures of calcium oxide and silica are
described by the phase diagram of Fig. 62, with one peritectic,
two compounds that melt without decomposition, and two
liquid phases. As a study exercise the student should draw care-
fully to scale a set of idealized cooling curves that will describe
the whole system shown in Fig. 62, with intervals of 5 per cent
or less between curves. Note that any phase which separates
on cooling evolves heat as it separates and that the longest
interval of constant temperature for any three-phase equilib-
rium corresponds to the largest quantities of new phases being
formed at that temperature. (Refer if necessary to the preceding
pages describing the separate occurrences.)

Problems

For the systems described in Problems 1 to 18 draw phase diagrams rea-
sonably to scale, letter all the phase fields to show what phases are at equi-
librium within them, and draw a sufficient number of typical cooling curves
on the same temperature scale to correspond with all the important charac-
teristics of each system Choose reasonable points when data are lacking,
but do not include any features not required by the data. The centigrade
meltirig points of the elements involved in these problems are :

Aluminum

Antimony

Bismuth

Calcium

Cerium

Cesium

Chromium 1615° Molybdenum ... 2535°

Cobalt . .. 1480° Nickel . ... 1452°

^ee MARSH, op. cit.

658° Copper

1083° Palladium

1555C

630° Iron

1535° Silver

960'

273° Lead

327° Sodium

98<

810° Manganese

1260° Thallium .

303'

775° Magnesium

651° Tin.

. . 232C

26° Mercury

-39° Zinc

420C

460 PHYSICAL CHEMISTRY

1. Mercury and lead dissolve in all proportions in the liquid state, and
they form no compounds. A liquid phase is in equilibrium at —40° with
two crystalline phases, containing 35 and 100 per cent Hg, respectively

2. Iron and FesSb2 (m.p 1015°) form solid solutions in one another to a
limited extent, and FeSb2 decomposes at 728° into a liquid and the other
compound. The eutectics are at 1000° and 628°

3. NasBi rnelts at 775°, and NaBi decomposes at 446C into Na3Bi and a
liquid. The eutectio temperatures are 97° and 218°

4. A12O3 (m p 2050°) and Si<)2 (m p 1710°) form a compound 3A12O3-
2SiO2 known as mulhte, which decomposes at 1810° into Al2O(i and a liquid
phase containing about 40 per cent Si02 The eutectic is at 1545° and 93 per
cent SiO2

6. Lead and palladium form four compounds, PclPf)2 (m p 454°), PdPl>
(decomposes at 495° into a liquid and Pd-jPb), PdaPb (decomposes at 830°
into a liquid and Pd3Pb), and Pd3Pb (m p 1240°) Solid solutions from
77 to 100 per cent Pd are formed, but there is only one liqiud solution. The
eutectic temperatures are 260°, 450°, and 1185°

6. CoSb melts at 1190°, CoSb2 decomposes at 900° into OoSb and a
liquid containing 91 per cent Sb. There are eutectics at 1090° and 40 per
cent Sb and at 620° and 99 per cent Sb, and a solid-solution area exists up
to 12 per cent Sb

7. Silver oxide has the dissociation pressures given on page 396.

8. Aluminum and cobalt form three compounds, of which AlOo melts
at 1630°, Al &CO2 decomposes at 1 1 70°, and Al4Co decomposes at 945°. AICo
and Co form an incomplete series of solid solutions, with a of 84 per
cent Co, liquid of 89 per cent Co, and (3 of 92 per cent Co in equilibrium
at 1375°.

9. Magnesium and nickel form a compound MgNi2 that melts at 1145°
and a compound Mg2Ni that decomposes at 770° into a liquid containing
50 per cent Ni and the other compound The eutectics are at 23 per cent
Ni and 510° and at 89 per cent Ni and 1080°

10. Bismuth and lead form no compounds, solid solutions containing 1
and 63 per cent lead are in equilibrium with a liquid containing 43 per cent
lead at 125°

11. Calcium and sodium mix in all proportions in the liquid state above
1150°, the mutual solubilities are 33 per cent Na and 82 per cent Na at
1000°, liquids of 14 and 93 per cent Na are in equilibrium with solid calcium
at 710°, the eutectic temperature is 97.5°, and no compounds or solid solu-
tions form.

12. Al3Ca decomposes at 'TOO0 into Al2Ca and a liquid containing 14 per
cent Ca; Al2Ca melts at 1079°. The eutectics are at 616° and 7 per cent Ca
and at 545° and 73 per cent Ca.

13. Liquid and solid phases of the composition Hg&TU are in equilibrium
at 14°; the phases at 0° are a liquid of 40 per cent Tl and solid solutions of
32 and 84 per cent Tl; the phases at —59° are Hg, a liquid of 8 per cent Tl,
and a solution of 22 per cent Tl.

14. Cerium and iron form two compounds, of which CeFea decomposes
at 773° into Ce2Fe& and a liquid containing 91 per cent Ce and Ce2Fe6

PHASE DIAGRAMS 461

decomposes at 1094° into a liquid containing 65 per cent Ce and a solid
solution containing 15 per cent Ce

15. SbCr melts at 1110°, a compound Sb2Or decomposes at 675° into a
liquid and the other compound The eutectic temperatures are 620° and
1100°, and at the latter temperature the crystalline phases are solid solutions
containing 32 and 88 per cent Cr.

16. Nickel and molybdenum form one compound, MoNi, which decom-
poses at 1345° into molybdenum and a liquid containing 53 per cent Mo.
The phases at the only eutectic (1300°) are MoNi, a liquid of 49 per cent Mo,
and a solid solution of 32 per cent Mo.

17. Copper and cerium form four compounds, CeCus (m p 940°), OeOm
(which decomposes at 780° into CeCiu and a liquid containing 42 per cent
Ce), CeOu2 (m p 820°) and CeCu (which decomposes at 515° into CeCuz
and a liquid containing 79 per cent Ce). The eutectic points are at 16 per
cent Ce and 880°, 45 per cent Ce and 760°, and 85 per cent Ce and 415°.
There is but one liquid solution, and no solid solution.

18. MgZri2 melts at 590°, and there are four temperatures in the Mg-Zn
system at which three phases exist, with percentages of zinc as follows:

340°. . . a (8 per cent) -f liquid (53 per cent) + MgZn

354° . . . . liquid (55 per cent) + MgZn + MgZn2

380° liquid (96 per cent) -f MgZn2 -f MgZn6

364°. . . liquid (97 per cent) + MgZn* + Zn

19. (a) Describe in detail, by reference to Fig 48, what would happen if
urethane at 35° and 5000 atm were allowed to expand slowly while the
temperature remained constant (6) Do the same for a temperature of
100°C. (c) Draw diagrams showing the change of volume with change of
pressure for the compression of urethane at 20°, 30°, 60°, and 70°.

20. Potassium acid sulfate (KHSO4) forms four solid phases, and the triple
points are as follows*

I-II-IV 199° 1830 atm
II-III-IV 118° 2900 atm

Phase III is stable at room temperatures and pressures, and it changes to
IV at about 48° and 6000 atm. The transition points under 1 atm are
164° and 180°, and phases T and IV are in equilibrium at 220° and 2500 atm.
(a) On a diagram covering the range 40° to 350° and 0 to 6000 atm , draw
lines representing the equilibrium between the solid phases, and letter each
field to show what phase is stable within it. (6) The melting point is 210°,
ind the solid sinks in the liquid. Draw a short line (0 to 200 atm , say)
showing the equilibrium between liquid and solid, and show by the slope of
this line whether the melting point is raised or lowered with increase of
pressure, (c) Tell in detail all that would happen if KHSO4 were heated
very slowly from 40° to 260° under a pressure of 2500 atm., but do not draw
my conclusions that are not justified by the data given in the problem.

21. Phenol (m p 42°) and water dissolve in one another in all proportions
it temperatures above 67° but are only partly soluble below this temper-

462

PHYSICAL CHEMISTRY

ature. At 50° the liquid phases contain 11 and 62 weight per cent phenol,
at 5° they contain 7 and 75 weight per cent, and at 1.3° there are two liquid
phases containing 6.8 and 76 per cent phenol in equilibrium with solid
phenol. Ice and solid phenol are in equilibrium with a solution containing
5.8 per cent phenol at —1.3°. (a) Draw a temperature-composition dia-
gram for this system. (6) What would happen at 50° if successive small
portions of phenol were added to water until the system was 99 per cent
phenol? (c) Draw cooling curves for systems containing 6, 10, 60, and
80 per cent phenol, covering 70° to —10°. [CAMPBELL and CAMPBELL,
/. Am Chew. Soc , 69, 2481 (1937) ]

22. The cooling curves below are for mixtures of silver and tin containing
the indicated percentages of silver Construct the phase diagram, and letter
each field to show what phases exist within it

Per cent Silver

50 60 70 7E

23. The phase equilibrium for water involves a liquid and six solid phases
for pressures up to 45,000 atm. Denoting the liquid by L and the solids by
I, II, III, V, VI, and VII (no phase designated IV has been obtained), the
triple points in the system are at the following temperatures and pressures:

I-III-L

-22°

2,045 atm.

i-ii-m

-34 7°

2,100 atm.

III-V-L

-17°

3, 420 atm

n-m-v~

-24 3°

3,400 atm.

V-VI-L

+0 16°

6,175 atm.

VI-VII-L +81 6°

22,400 atm.

The pressures and temperatures of some two-phase equilibriums in this
system are as follows:

I-II -75°
II-V -32°
V-VI -20°
VI-VII -80°
VII-L +149°

1,800 atm.
4,000 atm
6,360 atm
20,000 atm.
32, 000 atm.

(a) Draw a phase diagram for this system in the range —80° to +160° and
1 to 45,000 atm., and letter the phase fields. (6) Which of the crystalline
forms will float in the liquid? [Data from Bridgman, Proc. Am, Acad., 47,

PHASE DIAGRAMS

463

440 (1912), and /. Chem. Phys., 5, 964 (1937); the diagram for deuterium
oxide (" heavy water") is given in ibid , 3, 597 (1935) ]

24. Carbon tetrabromide forms three solid phases. II changes to I at
50° and 1 atm.; 1 melts at 92° with an increase in volume; the liquid boils
at 190°. The triple point for I, II, and III is at 115° and 1000 atm , and
there are two phases at 2000 atm and 135° and at 2000 atm. and 200°. (a)
Draw the phase diagram, and letter its phase fields. (6) Draw a curve show-
ing how pressure changes with volume at 120° for a pressure increase from
1 atm. to 2000 atm.

25. Zinc nitrate forms hydrated crystals containing 9, 6, 4, 2, and 1H2O.
The solubility, in grams of Zn(NO3)2 per 100 grams of solution, and the
composition of the solid phase change with the temperature as follows:

% Zn-
(N08)2

Temp.

Solid
phase

%Zn-
(N03)2

Temp.

Solid
phase

%Zn-

(N0a)2

Temp.

Solid
phase

30 0

-16 0

Ice

66 2

34.6

VI-IV

81 6

50 6

39.6

-29 5

Icc-IX

67 9

40.0

84 0

55 4

40 1

-25 0

70 0

43.2

86 3

52 1

II-I

42 0

-20 0

72 5

44.7

87 6

59.2

44 6

-18.0

77.2

39 7

90 0

70.7

48 6

78.0

37.2

IV-II

63 4

36 1

79.7

43 6

Draw a phase diagram foi this system, and indicate the phases at equilib-
nurn in each phase area [The data are from Wiss. Abh Phys -Tech. Keich-
sanstalt, 3, 348 (1900), and /. Am. Chem Soc , 55, 4827 (1933) ]

26. Two substances, M (= MnSO4, mol wt 151) and W (= H2O), form
one liquid phase and three stable compounds MWi, MW&, and MW. The
equilibrium between liquid and solid phases is as follows:

Temperature

Per cent M by weight
in liquid

Solid phase or phases

-10 5°

32 2

MWi 4- W

0°

34 8

MWi

9°

37 0

MW7 + MWs

20°

38 5

MW,

27°

39 5

MW, + MW

40°

38 3

70°

33 3

100°

26 5

Mfr

Draw a phase diagram for this system, covering —20° to 100°, letter all
the phase fields, and draw cooling curves for systems containing 35, 38, 50,
60, and 70 per cent M

27. Draw diagrams similar to Fig. 52, which apply to the eutectic pauses
in the systems shown in Figs. 53, 56, and 58.

CHAPTER XII
KINETICS OF HOMOGENEOUS REACTIONS

This chapter presents the experimentally determined rates at
which some chemical reactions in gases or in solutions proceed
isothermally toward equilibrium, the effect of temperature upon
these rates, and some simple equations that are in approximate
agreement with the experiments. Although the rates of hun-
dreds of reactions have been studied, interpretation of the data
is often complicated by side reactions, by relictions proceeding
in steps of different velocities, by mechanisms other than those
indicated from the chemical equation expressing the initial and
final states, by the influence of the walls of the container upon
reactions involving only dilute gases as initial and final sub-
stances, and by many other factors that are not understood.
.Reactions among gases or in solutions sometimes proceed very
slowly, sometimes at measurable rates, sometimes so rapidly as
to make their measurement difficult or impossible. Reactions
involving only ions are usually too fast to be measured. Most
reactions increase in speed with increasing temperature, though
there are a few exceptional reactions that proceed more slowly
at higher temperatures.

The fact that the theory of reaction rates is still incomplete is
no indication of neglect of the field; it is an unavoidable conse-
quence of the complexity of the rate processes. Reacting mole-
cules must not only "collide"; they must collide with sufficient
energy or be sufficiently "activated"; they must be properly
" oriented " ; they must satisfy other conditions. The resources of
statistical mechanics, quantum mechanics, the kinetic theory,
and careful experimental research have been employed by many
capable investigators in an effort to develop an adequate theory.
Much progress has already been made, but much remains to be
done.1 In this brief chapter we must be content with some

1 See GLASSTONE, LAIDLEK, and EYKING, "The Theory of Rate Processes,"
McGraw-Hill Book Company, Inc., New York, 1941. The preface and
introduction of this excellent text present the nature of the problem, discuss
its difficulties, and outline current progress.

464

KINETICS OF HOMOGENEOUS REACTIONS 465

simple equations showing approximately the rates of reactions
involving one, two, or three molecules when these proceed iso-
thermally in one phase or when surface effects are relatively
unimportant.

Although the rate of disappearance of reactants or the rate of
formation of products may usually be formulated in terms of the
concentrations or pressures of the reacting substances, as we shall
do bolow, there are many puzzling facts about these rates. For
example, the two chemical reactions

2NO + 02 = 2N02
2CO + O2 = 2C02

in the gaseous state each involve 2 moles of a lower oxide and
1 mole of oxygen, but it is not to be inferred from the similarity
in the equations expressing the over-all effects of the reactions
that the oxidations take place by molecular mechanisms which
are the same for both or at comparable rates. If molecular
collisions were the chief requirement for these reactions to
proceed, they should have comparable rates at the same tem-
perature. The experimental facts are that the oxidation of NO
at ordinary temperature is very rapid and the oxidation of
CO is immeasurably slow. Equilibrium requires substantially
complete oxidation in both systems. Although both reactions
probably require collisions among the molecules and in systems of
comparable compositions at the same temperature the number of
collisions would be approximately the same for both, the rates evi-
dently depend to a governing extent upon other factors. More-
over, the rate of the faster reaction is extremely slow compared
with that calculated tor a gaseous system in which every collision
causes a reaction. Thus, the number of collisions that are effec-
tive is very much smaller than, and must be clearly differentiated
from, the total collisions. Later in the chapter we shall attempt
an approximate estimate of this fraction in some simple systems
In the experiments discussed in this chapter it has been
possible to determine the change of concentration with time for
a reacting substance or a product of the reaction and then
through stoichiometry to express the concentrations of all the
reacting substances as functions of time. Interpretation of these
concentrations in terms of the chemical reaction expressing the
"over-all" change in state sometimes shows that the time

466 PHYSICAL CHEMISTRY

reaction is not the same as the reaction showing the change in
state but that some " intermediate" product forms slowly and
decomposes rapidly or forms rapidly and decomposes slowly.
We shall presume a mechanism for the time reaction that is in
harmony with the observed rate, but it must not be forgotten
that such a presumption may be wrong, even though probable in
the light of present knowledge. Additional experiments upon a
given system may require a revision of the interpretation placed
upon the data now available.

The "Order" of a Reaction. — Aside from complicating initial
conditions that are sometimes important and sometimes negli-
gible, all reactions proceed at rates that decrease with time if the
temperature is kept constant, and equations of different alge-
braic form apply to different types of reactions. The experiments
determine concentrations or pressures at suitable time intervals.
If the rate of a reaction is proportional to the fir^t power of the
concentration of some reacting substance, the reaction is said
to be of the first order with respect to that substance. When
the rate depends upon the first power of the concentration of
two substances or upon the square of the concentration of one
substance it is called a reaction of the second order. A reac-
tion whose rate depended upon CA and (V would be a third-
order reaction with respect to both substances but could be
considered a first-order reaction with respect to A alone or a
second-order reaction with respect to B alone For example, if
the initial concentration of B were very large compared with
that of A, the concentration of B would remain almost constant,
even though a large fraction of A had reacted, and the reaction
rate would be proportional to the momentary concentration of A

As has been said before, the "order " of a reaction as measured
by rate experiments may not be that expected from the chemical
reaction describing the over-all change in state. There are also
numerous observed reaction rates that do not conform to any
simple order, possibly because reactions of different order or of
different rates are proceeding consecutively, or for other reasons.
It has been possible to isolate consecutive reactions in enough
instances to show that this is one of the explanations. Other
reasons include influence of the walls of the reaction container,
self-catalysis by a reaction product, reverse reactions, simultane-
ous reactions, and factors not yet discovered.

KINETICS OF HOMOGENEOUS REACTIONS 467

Although it is not possible at present to predict the order of a
reaction from the over-all change in state, it is conversely true
that an experimental determination of the rate of a reaction often
furnishes an important clue as to the mechanism by which the
change in state occurs. Some examples will be given presently,
and many more are known.1

Reactions involving more than one phase, such as those between
gases or solutes reacting upon a solid surface and hence catalyzed
by the surface, are more complicated than the rates of homo-
geneous reactions, and they require special methods of treat-
ment 2 Many such reactions can be interpreted upon the
assumption that one particular step in the process is so slow
compared with the others that it governs the observed rate.
This step might be (1) the rate of adsorption of the reactants or
(2) the rate of desorption of a reaction product that covers the
surface or (3) the rate of reaction upon the surface by molecules
that adsorb and desorb rapidly. If (1) were the governing
process, the reaction might well appear to be homogeneous; if
(2) governed, the rate would be nearly independent of the con-
centrations or pressures of the reacting substances; if all three
processes had comparable rates, no simple equation could express
it. For example, the rate of reaction between CO and 02 on a
silver catalyst is independent of the pressure of COs, which indi-
cates that the desorption rate is rapid by comparison with the
rate-governing process; but the fact that the rate is independent
of the oxygen pressure also when the ratio of CO to 02 is high
is more difficult to interpret simply.3 Since glass may function
as a catalyzer, it is sometimes necessary to vary the ratio of
volume to surface exposed (for example, by " packing " the reac-
tion vessel with broken glass of the same composition) in a series
of experiments in order to demonstrate that the reaction is or
is not homogeneous.

In the discussion that follows we shall write the initial concen-
tration of a reacting substance as Co, meaning the concentration
for zero time, or its initial pressure as p0. When equal volumes

1 See especially HAMMETT, " Physical Organic Chemistry," Chap. IV,
McGraw-Hill Book Company, Inc., New York, 1940.

2 See HINSHELWOOD, "Kinetics of Chemical Change," Chap. VIII,
Oxford University Press, New York, 1940,

3 BENTON and BELL, J. Am. Chem. Soc , 66, 501 (1934).

408 PHYSICAL CHEMISTRY

of 0.10m. solutions of two reacting substances are mixed, Co
will thus be 0.050 for both. The concentration at a time t
will be written C, from which it will be evident that in any
given experiment C0 is a constant while C and t are variables.
The fraction reacted at a given time is (Co — C)/C0, which will
be denoted by x. We define the specific reaction rate as the rate
at an instant when the concentrations of all reacting substances
are unity, and we denote it by k. For a constant temperature k
will be constant; when the temperature changes, k will change,
but this change may not be calculated from the thermochemical
A// for the reaction.

Experimental Methods. — When there is a change in the num-
ber of molecules attending a homogeneous gaseous reaction, the
change of pressure with time at constant volume and constant
temperature may be used to follow the extent of a reaction. Simi-
larly, if the color, conductance, optical rotation, acidity, or any
quickly measurable property of the system changes as the reac-
tion proceeds, this property may be used to follow the reaction.
But it is not the pressure (or other property) that measures the
extent of the reaction — it is the change of pressure (or other
property) that does so. A few illustrations will make this
clearer. Suppose the reaction to be a gaseous one in which
one molecule yields three, A = 3B. If p0 is the initial pressure
of A, the final total pressure will be 3j>o> and the total increase
in pressure will be 2p0- At some time t the pressure is observed
to be p, and the increase in pressure for this time is Ap = p — p(),
whence the fraction reacted is x = Ap/2pQ. The partial pressure
of A is PQ times the fraction not reacted, p0(l — #), which is
;>o(2po — Ap)/2p0, or pQ — % Ap, and the partial pressure of B is
% Ap.

Let ao, ctt, and Oend represent the optical rotation of a reacting
system at the start, after the time £, and at the end of the reac-
tion. No one of these quantities measures the extent of the
reaction, but (a0 — QWd) measures the change in rotation for
the completed reaction, and («0 — ««) measures the change in
the time t, whence x — (a0 — «<)/(ao — ow) gives the fraction
changed at the time /.

If m is any measure of the concentration of a reacting substance,
this quantity will be ra0 at the start, mt at a later interval, and
when the reaction is completed, so that the fraction reacted

KINETICS OF HOMOGENEOUS REACTIONS 469

is (m0 — mt)/(niQ — mcnd). Whenever the progress of a reaction
is measured by the quantity of a reaction product formed, this
measure will be zero at the start of the reaction, and mt/mead
will give x, the fraction reacted.

Applying these relations to the decomposition A = 3B that
was our first illustration, the partial pressure of A is its measure,
namely, po at the start, (p0 — M &P) a^ ^ and zero at the end.
Then

In terms of the reaction product B, zero is its measure at the
start, % Ap is its measure at t, and 3po its measure at the end,
whence x — % Ap/3po = Ap/2p0 as before.

The choice of a suitable measure is not always easy, however;
for while the partial pressure of a gas above a solution measures
its concentration in solution at equilibrium, equilibrium is not
certainly attained quickly in a system in which a gas is increasing
its concentration with time. The measured pressure on a
gaseous system in which the pressure is changing must be
measured by a device in which there is no time lag if it is to be
an instantaneous pressure and therefore a definite quantity at a
fixed time. When the concentration of a substance is determined
by titration, the time consumed in the titration must not be long
enough for the reaction to proceed appreciably while titration is
in progress.

First -order Reactions. — A reaction whose rate is proportional
to the first power of the concentration of one substance is a
first-order reaction. A monomolecular reaction would be first
order, but there are reactions that conform to the first-order
equation in their rates and yet are not monomolecular. As the
reaction proceeds, the concentration of the reacting substance
decreases and the reaction proceeds more slowly, so that equi-
librium is approached at a decreasing rate. For such reactions
the rate at a. const ant temperature is given by the equation

- f = kC (1)

Upon integrating this equation between concentration limits
Co and C and time limits 0 and t we have

470 PHYSICAL CHEMISTRY

ln^° = fc (2)

Since (Co — C)/Co is the fraction reacted, the equation in terms
of this fraction is

In y-^ = tt or 2.3 log y-^ = kt (3)

It will be noted that equation (3) for the fraction reacting
in a time interval does not contain Co, which shows that in
first-order reactions the time required for a given fraction of the
substance to react is independent of the initial concentration.
This is not to say that the rate in moles per liter per minute is
independent of Co, for this is not true. Dilution with an equal
volume of solvent for a reaction in solution or reducing pQ to
half its value in a gaseous system reduces to half the rate in
moles per liter per minute and doubles the volume of the system,
so that the quantity per total system per minute is unchanged.

A common procedure for determining whether a reaction
is or is not of the first order is to determine the "half time/7
the time in which x = 0.50, for different initial pressures or
concentrations. If the half time is independent of Co or p0,
the reaction is shown to be of the first order. (We shall see
later that for reactions of the second order the half time is
inversely proportional to Co or p0.)

All the transformations of radioactive substances (to be
discussed later in Chap. XV) follow the first-order equation.
It is usual to describe their reaction rates in terms of "half life,"
or the time required for one-half the substance to be trans-
formed into its decomposition products, whether or not these
products undergo further decompositions at new character-
istic rates. By substituting x = 0.5 into equation (3) it will
be seen that the relation between t for half decomposition and
k is £0.6 = 0.693/fc.

Obviously these equations imply that the reaction velocity
at a given temperature depends only upon the concentration of
a single reacting substance. Otherwise, all the other factors
that influence the rate are collected into fc; and since it some-
times happens that not all these conditions are known and kept
constant, the "constant" derived from experimental data

KINETICS OF HOMOGENEOUS REACTIONS 471

proves to be a variable instead. For example, some reactions
involving only dilute gases as initial and final substances take
place upon (or at least under the influence of) the wall of the
reacting vessel, and thus their rates depend upon the ratio of sur-
face to volume of container. These reactions are not homogene-
ous reactions and are not to be described by equation (1) without
allowance for the "wall effect." Other reactions are accelerated
by solutes whose concentrations do not change as the time
reaction proceeds. Such solutes are called catalyzers and will
be discussed presently; we note here only that equation (1)
would apply to experimental data in a catalyzed homogeneous
first-order reaction only when the catalyzer concentration is kept
constant, and hence its effect is included in k.

The significance of fc, the reaction-rate constant, is that, when
C = 1, the reaction rate is equal to k. It is thus a specific reac-
tion ratCj which will have the dimensions of trl\ it will be (min.)"1
when time is expressed in minutes, or (sec.)"1 when time is
expressed in seconds. This rate will not be maintained when a
solution of unit concentration reacts, for C decreases with time,
and the rate —dC/dt = kC is no longer equal to k when C falls
below unity.

As has been said before, k includes the influence of every
factor other than the concentration of a reacting substance,
whether these factors are known or unknown. When variable
values of k are derived from a set of experimental data, this
shows some influence that has not been controlled in the experi-
ments and indicates the need of further experimentation.

Decomposition of Nitrogen Pentoxide. — This reaction has
been extensively studied,1 both in the gas phase and in solution.
The chemical reaction that describes the change is

2N,(M0) = 2N204(<7) + 02(<7)

4N02(<7)

but experiment shows that the rate is given by the first-order
equations

1 DANIELS and JOHNSTON*, ibid., 43, 53 (1921); RAMSPEEGEB and TOLMAN,
Proc. Nat. Acad. Sci., 16, 6 (1930); EYEING and DANIELS, J, Am. Chem,
Soc., 62, 1486 (1930),

472

PHYSICAL CHEMISTRY

One would expect from the chemical equation that a second-
order reaction is taking place, which is contrary to the experi-
mental evidence. If it is assumed that a first-order reaction
is a monomolecular one, the reaction governing the rate might "be

N2OB = NO2 + NO3 or N2O6 = N2O3 + O2

N2OB = any compounds of N and O

followed by secondary reactions of much higher velocities whose
final products are N204 and O2. The available experimental
facts do not indicate which reaction is more probable than the
others.

Some of the experimental data for 35°C. are given in Table 75.
It should be noted that in order to follow this reaction rate from
the pressure increase it was first necessary to show that the
equilibrium between N2O4 and NO2 is established instantly and
to determine the equilibrium constant for this reaction.1 The
values of k in the last column are obtained through equation
(3) in terms of the fractions decomposed at the designated times.
If the equation in terms of the partial pressure of N2O6 given
above is integrated between time limits t\ and £2, it is

In 21 = k(h - ti)
P*

and constants obtained from this equation by substituting

corresponding times and pressures are said to be calculated by

TABLE 75 2 — DECOMPOSITION OF NITROGEN PENTOXIDE AT 35°

Time, mm.

Total
pressure, mm.

Partial press.

N2Oc, mm.

Fraction
decomposed

(0)

308 2

(308 2)

368 1

254 4

0 175

0.0096

385 3

235 5

0 236

0.0089

400 2

218 2

0 292

0.0086

414 0

202 2

0 345

0.0084

426 5

186 8

0.394

0.0083

100

465.2

137 2

0 554

0.0080

140

492.3

101 4

0 672

0.0080

200

519 4

63 6

0 792

0.0078

1 The equilibrium constant pNo22/pN2o4 = 0 32 atm. or 243 mm. at 35°

2 DANIELS and JOHNSTON, ibid., 43, 53 (1921).

KINETICS OF HOMOGENEOUS REACTIONS 473

the " interval" method. They may magnify the errors of any
single experiment, but they are usually a more sensitive test
for "drift" in the constant. In the absence of experimental
errors this procedure obviously yields the same k as integration
from zero time.

Thermal Decomposition of Paraldehyde.1 — This reaction, for
which the chemistry may be abbreviated P = 3A, is also a
reaction of the first order which may be followed by observing
the total pressure. For constant volume and constant tempera-
ture the rate may be expressed, in terms of the partial pressure of
paraldehyde,

— -£p-a-r = kpvar (t const.)

but since the observed physical quantity is the total pressure p,
this equation may be expressed in terms of experimental data by
noting that at any moment the pressure of acetaldehyde is three
times the loss in ^pressure of paraldehyde. If p0 is the original
pressure of paraldehyde and pt its pressure at a time t, the
acetaldehyde pressure is 3(p0 — pt), whence 3(po — Pt) + Pt is
equal to py the total pressure, or pt = /4(&po — p) The fraction
decomposed is x = 1 — (p«/po), and this quantity may be sub-
stituted into equation (3) in order to calculate k. If preferred,
the expression for pt may be substituted directly into the rate
expression in terms of this quantity to attain the same result.
This equation then becomes

(3po - p)

Upon integration between the pressure limits p0 and p for total
pressure, and the time limits 0 and J, we have

2.3 log 5-22S- = kt .

3p0 - p

The value of k in this equation for time expressed in seconds
changes with the temperature as follows:

Absolute temp . , .501 9° 512 2° 519 3° - 526.8° 534 9° 542 8°
k X 104 . 0.634 1 61 3 05 5.44 10 2 19 3

1 COFFIN, Can J. Research, 7, 75 (1932).

474

PHYSICAL CHEMISTRY

First-order Reactions in Solutions. — When one molecule of a
dissolved substance changes into one or more new substances,
the rate of its reaction may also be expressed by equation (3)
The conversion of hydroxyvaleric acid into valerolactone is an
illustration of such a reaction, and it may readily be followed
by titrating samples with standard base from time to time. The
chemical change is shown by the equation

CH3CHOHCH2CH2COOH = CH8CHCH2CH2COO + H20

[The rate of decomposition of hydroxyvaleric acid, which is a
weak acid, is accelerated by the presence of hydrochloric acid
almost in direct proportion to the concentration of hydrogen
ion. In the presence of HC1 the ionization of hydroxyvaleric
acid is negligible, and therefore the rate at which the concentration
of the hydroxyvaleric acid changes with time is shown by the
equation

-dC
dt

= Jfc(H+)C

(4)

Substances that accelerate a reaction without changing their
concentrations as the reaction proceeds, as is true of HC1 in
these experiments, are called catalyzers and will be discussed in
the next section.]

As the reaction proceeds, hydroxyva-leric acid undergoes the
change shown in the chemical equation, and a sample of the
reacting mixture requires less standard base for its titration.
Complete reaction corresponds to titrating the hydrochloric acid
"catalyzer" only; hence the fraction of hydroxyvaleric acid

TABLE 76. — RATE OF CONVERSION OF HYDROXYVALERIC ACID TO VALERO-
LACTONE AT 25° (CATALYZED BY 0.025 N HYDROCHLORIC ACID)

Time,
min.

Fraction
changed

Time,
min.

Fraction
changed

0 173

0 158

0 166

0 157

0 257

0 156

125

0 388

0 157

124

0 389

0 158

174

0 498

0 158

204

0 556

0.159

221

0 583

0 158

238

0 613

0.159

262

0 643

0 157

289

0 681

0 158

307

0 703

0 158

KINETICS OF HOMOGENEOUS REACTIONS 475

decomposed at a time t is obtained by subtracting the volume of
base used by a sample at that time from the volume employed
in the initial titration of a portion of the same volume and by
dividing this difference by the difference between the first titra-
tion and that corresponding to complete reaction. Two series
of experiments are shown in Table 76. It will be seen from the
figures in the third and sixth columns of this table that a suffi-
ciently constant value for k is obtained by substitution in equa-
tion (4).

Catalysis. — Substances that accelerate chemical reactions with-
out being exhausted as the reaction proceeds are called catalyzers.
Gaseous substances that increase the speed of gaseous reactions
or solutes that accelerate reactions in solution are called " homo-
geneous " catalyzers, and in these systems the catalyzer concen-
tration remains constant as the reaction proceeds. A catalyzer
does not alter the nature of the reaction products or the equi-
librium relations of the final chemical system ; it must lead to the
formation of the same, and only the same, end products as the
slower reaction in its absence. There are also numerous " hetero-
geneous" catalyzed reactions, in which a solid serves as the
accelerator for reactions in the gas phase or in solution. The
mechanism whereby these effects are produced is unknown in
most systems; more or less plausible explanations are available
for a few systems.1

Nitrous oxide probably decomposes into its elements by a
primary process which is shown by the equation

N2O = N2 + O

which occurs as an aftermath of a collision in which the necessary
energy is given to the molecule and which is followed by the
reunion of oxygen atoms to form molecules through some suitable
mechanism. Since the energy requirement for this dissociation
is much higher than that of an average collision, only a small frac-
tion of the collisions is effective. Effective collisions may occur,

1 Attention should be called to the statement of Dr. C. N. Hinshelwood
in /. Chem. Soc. (London), 1939, 1203: "There is no theory of catalysis.
The only question is whether we understand catalytic phenomena well
enough to arrange them into a picture of which we like the pattern." A
survey of the field, with references to the literature, is given in the National
Research Council's "Twelfth Report of the Committee on Catalysis," 1940.

476 PHYSICAL CHEMISTRY

not only among N20 molecules themselves, but between them
and C02 or N2 or A; and the different substances are specific
in their action. The efficiency of such collisions must be con-
nected with their capacity for communicating energy directly
to the reacting molecules, but a full knowledge of the laws
governing these energy exchanges is lacking. The efficiency of
halogens in accelerating the decomposition of N2<3 is of a different
order of magnitude and probably through a different mechanism.
The activation energy of the reaction N2O = N2 + 0 is about
60,000 cal., and that of the reaction with a halogen atom X as
shown by the equation

N20 + X = N2 + XO

would be less by the energy of formation of XO. One may
assume a' minute dissociation of halogen gas molecules into
atoms, X2 = 2X, and that these free halogen atoms could seize
the oxygen of N20, giving halogen oxides which are more stable
than the free elementary atoms. Since these oxides are unstable
with respect to the molecules of halogen and oxygen, a supple-
mentary reaction such as 2XO = 02 + 2X, or 2X0 = X2 + 02,
takes place, and the series of reactions is then repeated.

Series of reactions whereby unstable compounds are formed
and then decomposed to regenerate the catalyzer are plausible
explanations of many reactions. Another illustration is the
oxidation of S02 by oxygen, which is accelerated by oxides of
nitrogen. A large amount of experimental work has been done
upon this important reaction, but a full explanation is still lack-
ing. The fact that a compound of the composition (N02)HOS02
(nitrosyl sulfuric acid) may be prepared from S02, N203, 02,
and H20 and decomposed by water into H2S04 and N203 is
often advanced as an explanation of this catalysis, and it is a
plausible one. It should be remembered in this connection that
in the actual operation of a sulfuric acid " chamber ," it is desirable
to prevent the formation of this compound. Other reactions of
equal plausibility may be written for the formation of sulfuric
acid which involve other mechanisms and the final results of
which are in conformity with the chemistry of the total change
in state.

In connection with the mechanism of any catalytic process
it should be borne in mind that the " intermediate " compounds

KINETICS OF HOMOGENEOUS REACTIONS 477

are not necessarily those which are stable with respect to the other
molecules in the system. In such a series as

N2O + Cl = N2 + CIO
2C10 = C12 + 02
C12 = 2C1

the progress of the primary reaction is accelerated if the tendency
of the first reaction to occur is greater than that of the reaction
N20 = N2 + 0, however unstable CIO may be with respect to
C12 and 02. If the second and third of these reactions are fast
enough to keep the concentration of Cl constant, the reaction
will appear to be accelerated by C12.

The numerous reactions in which water or the elements of
water enter into the change in state are often accelerated by H+
or OH~" almost in proportion to the strong acid or strong base
from which they come. The effect of the former is sometimes
ascribed to hydrated hydrogen ion, or hydronium ion H30+,
though it is often difficult to see how the assumption is helpful
in understanding the mechanism of water addition. As an
illustration of such a catalysis, the following scheme has been
used to explain the acid catalysis of ester hydrolysis:1

O O- O

Ri— C— 0— R2 -> Rr— C— 0+— R2 -> Ri— C + O— R2
H— OHH+ H+ OHH H+ OH H

In this scheme water yields only the OH~ to the hydrolysis, the
H+ comes from the catalyst, and a new H+, which is the remainder
of the water molecule, appears and is ready to catalyze again.
On the other hand, it is a permissible point of view that the
H+ which appears in the first stage is present in the second and
third and may thus be only a " bystander."

The rate of conversion of hydroly valeric acid to valerolac-
tone, which is accelerated by hydrogen ions, was shown by the
equations

- f = k^c or - f = k>c

1 HINSHELWOOD, /. Chem. Soc. (London), 1939, 1203.

478

PHYSICAL CHEMISTRY

In order to show that the rate is proportional to the hydrogen-
ion concentration, we quote the data of Table 77.

TABLE 77. — CATALYZER CONCENTRATION AND VELOCITY CONSTANT kr

Concentration of
catalyst

104/c'

/c7(H+) - k

0 WN

156

0 156

0 05

78 8

0 157

0 025

39 3

0 157

0 010

15 7

0 157

In this chapter we denote the specific reaction rate by k] and
whenever some other constant quantity such as a catalyzer con-
centration, or the logarithmic conversion factor, or an initial
concentration is combined with this k, we write it k'.

All the illustrations thus far mentioned are " homogeneous "
catalyzers, gases that accelerate gaseous reactions or solutes that
accelerate reactions in solution. Many other examples are
known, but " heterogeneous " catalyzers are much more common.
They are solids that accelerate reactions in gases or solutions,
and thousands of reactions catalyzed by solids are known.1 The
reactions include hydrogenation of double bonds, reduction of
benzene to cyclohexane, aromatics to aliphatics, and of unsatu-
rated acid to saturated acid or to unsaturated alcohol, reduction
of nitrobenzene to aniline, of acids to aldehyde, of aldehyde to
alcohol or acid to alcohol in one step, of heptane to toluene, of
methanol from CO and hydrogen, and of benzaldehyde from
C6H6 and CO, and countless organic syntheses, decompositions,
oxidations, and reductions. The catalyzers are metals, alloys,
metal oxides, charcoal, clay, silica gel, inorganic salts, and other
substances. Careful control of experimental conditions is essen-
tial. For example, hydrogen on a nickel catalyst may change
an unsaturated acid to a saturated acid or to an unsaturated
alcohol, depending on the temperature and hydrogen pressure.

For many of these reactions no explanation is known, though
plausible assumptions are sometimes offered, such as preferential
adsorption on the surface where reaction is favored, followed by

1 See, for example, Berkman, Morrell, and Egloff, "Catalysis/' Reinhold
Publishing Corporation, New York, 1940, for references.

KINETICS OF HOMOGENEOUS REACTIONS

479

desorption of the reaction product and adsorption of new quanti-
ties of reacting substances. Such an explanation is offered for
the decomposition of nitrous oxide by platinum. An estimate
of the activation energy for the reaction

N2O = N2 + O (on Pt)

is 30,000 cal., so that collisions with the solid surface capable of
supplying this smaller quantity of energy would be more numer-
ous than those from which the 60,000 cal for the direct decompo-
sition are available Oxygen atoms on platinum being unstable

Per Cent BorC
FIG. 63. — Catalytic effect of mixtures.

with respect to oxygen molecules, the latter form and clear
the platinum surface for fresh acceleration of the decomposition.

For many heterogeneous catalyzers, the effectiveness is propor-
tional to the exposed surface rather than to the weight of catalyst.
Some of them are " promoted" by the presence of small quantities
of substances that are not themselves catalyzers ; some catalysts
are " poisoned " by the presence of small amounts of other sub-
stances and regenerated when these " poisons" are removed;
some mixtures follow a simple mixture law. The general effects
are shown in Fig. 63, in which A is a moderately effective cata-
lyzer, C is a better one, and pure B has no effect.

Sugar Hydrolysis. — The hydrolysis of dilute solutions of
sucrose into dextrose and levulose as shown by the equation

CeHigOe 4~ CeH^Og

480

PHYSICAL CHEMISTRY

proceeds at a rate proportional to the sucrose concentration
The concentration (or activity) of the water is substantially
constant for this reaction, as it is in all reactions involving water
in dilute aqueous solutions , and thus its effect is commonly
included in k. This reaction is accelerated by hydrogen ions,
almost in proportion to the acid concentration for strong acids.
Thus, the rate at which the concentration of sucrose decreases is

-77

(5)

Upon integration of this equation between time limits 0 and t
and substitution of x for (Co — C)/C0, the fraction decomposed
in the time t, we have

log

- .r

2.3

t = k't

TABLE 78 — SUGAR HYDROLYSIS AT 30° IN 2 5w FORMIC Arm1

Initial sugar concentration 0 44m.

Initial sugar concentration 0 167m

Elapsed

Rotation

A-' =

Elapsed

Rotation

k' =

time,

of plane

l\nffan ~~ af

time,

of plane

1 i^^ <*0 — OLf

hr.

of light

log

I at ~ af

of light

t Oil ~ Oif

(57 90)

(22 10)

53 15

0 0146

20 30

0 0146

48 50

0 0149

17 85

0 0145

44 40

0 0147

14 15

0 0148

40 50

0 0147

11 10

0 0147

35 2d

0 0146

8 65

0 0145

28 90

0 0146

6 00

0 0146

20 70

0 0146

4 50

0 0147

13 50

0 0149

1 90

0 0146

6 75

0 0148

0 35

0 0149

3 40

0 0147

-1 80

0 0146

- 0 40

0 0149

-3 20

0 0148

- 2 95

0 0148

-4 30

0 0147

- 7 45

0 0146

133

-5 10

0 0147

-11 25

0 0146

Complete

-5 50

112

-13 80

0 0147

Complete

-15 45

lRosANQFF, CLARK, and SIBLEY, /. Am, Chem, Soc., 33, 1911 (1911).

KINETICS OF HOMOGENEOUS REACTIONS 481

It should be noted that in this equation the catalyzer concentra-
tion, the water " concentration/ ' and the logarithmic conversion
factor 2.3 are grouped with the specific reaction constant k and
denoted by the single constant k'.

The velocity of this reaction is generally followed by observing
the change in optical rotary power of the solution. Let O.Q
and a/ represent the initial rotation and final rotation, and let ctt
represent the rotation at any time t. Then x, the fraction of
sugar decomposed at £, is given by the equation

«o — a/

Values of x so derived may be substituted in equation (3), or
the expression may be rearranged to contain the observed rota-
tions. It then becomes

, , 1 , ao — «/
kf = - logio - -

t OLt — OLf

Table 78 shows the results of experiments at 30° on sugar
solutions in which the catalyzer is 2.50m. formic acid. It
will be observed that the values of kf are constant and inde-
pendent of the sugar concentration or the extent to which the
reaction has progressed.

Second-order Reactions. — We have already defined a reaction
as of the second order when its rate is proportional to the first
power of the concentrations of two reacting substances. For the
general reaction

A + B = products

the expression for its rate in terms of the momentary concen-
trations of A and B is

If COA and COB are the initial concentrations of A and B, the
integral of this equation between limits is

o Q r* c*

^r.O •. V-^flBv/A •» i sr\\

7i TT" l°g r n = ** (8)

^ OA v/ OB v> OA V--' B

482

PHYSICAL CHEMISTRY

In experimental work it is important that the initial concen-
trations be made distinctly different or exactly equal. For the
special condition of equal initial concentrations of A and B the
rate equation is

-
dt ~

and its integral between time limits 0 and t is

Co - C

CoC

= kt

(9)

This equation is readily transformed into one in terms of the
fraction reacted at a given time interval, whereas equation (8)
cannot be so treated since equal quantities of A and B are
unequal fractions of unequal initial concentrations. We note
that x = (Co — C)/Co, and equation (9) becomes

- x

= kCot

(10)

In treating any given set of data C0 may be combined with k
into a single constant k' if desired, but this has the disadvantage
of implying by the appearance of the equation that the fraction
decomposed in a given time interval is independent of Co, which
is not true.

Saponification of Esters. — Reactions between hydroxyl ions
and esters in aqueous solutions, such as

OH- + CH3COOC2H6 = CH3COO- + C2H6OH
TABLE 79. — SAPONIFICATION OF ESTERS AT 25°

Ethyl acetate

Methyl acetate

Time,

Fraction

kCQ =

Time,

Fraction

kCQ -

min.

saponified

1 x

mm.

saponified

1 x

i I — x

t 1 — x

0 245

0 0649

0 260

0 117

0 313

0 0651

0 366

0 115

0 367

0 0645

0 450

0 117

0 496

0 0650

0 536

0 115

0 566

0 0652

0 637

0 117

0 615

0 0642

0 712

0 118

0 680

0 0644

0 746

0.118

KINETICS OF HOMOGENEOUS REACTIONS

483

are second-order reactions. If the ester and base are mixed in
equivalent quantities, equation (10) is applicable; if unequal, we
use equation (8). For either condition the fraction of base
reacted may be determined from the conductance of the solution,
since esters and alcohols are not ionized. As the reaction pro-
ceeds, hydroxyl ion is replaced by acetate ion that has a much
slower mobility, and the conductance decreases as the reaction
proceeds. If L0, Lt, and L/ denote the initial, temporary, and
final conductances of the solution, the fraction x of the NaOH
that has reacted is

x =

Table 79 shows some data1 for methyl acetate and ethyl acetate
at 25°. The evident fact that the derived constants are sub-
stantially constant shows that these reactions are second order.
Formation of Carbonyl Chloride. — As an example of a second-
order reaction in the gas phase, we consider the formation of
carbonyl chloride (phosgene), as shown by the equation

CO + C12 = COC12
The rate of this reaction is shown by the equation

_ dCco = kCcoCc{
dt co c 2

Since there is a decrease in the number of moles when COCh
is formed, the progress of the reaction may be followed by

TABLE 80. — FORMATION OF PHOSGENE

Concentrations

Time, minutes

1 x

CO or C12

COC12

0.500

0.488

0.0115

0.00780

0.479

0 0205

0.00712

0.471

0.0286

0.00676

0.463

0.0371

0.00676

0.455

0.0452

0.00664

0.447

0.0528

0.00654

0.439

0.0606

0.00660

1 WALKER, Proc. Roy Soc. (London), (4)78, 157 (1906).

484 PHYSICAL CHEMISTRY

measuring the decrease in pressure with time at constant volume
and constant temperature Table 80 shows some data1 for
this system, from which it is seen that k is not constant. We
have assumed that the mechanism is direct union of 1 mole of
CO with 1 of chlorine and that the reaction takes place in the
gas phase uninfluenced by the walls of the vessel. The drift
in the supposed constant k indicates that one of these assump-
tions is not correct; or, at least, it shows that some important
factor is not controlled, though there is no indication from these
data alone as to what this factor may be.

Third-order Reactions. — A reaction whose mechanism is
shown by an equation such as A + B + C = products is of the
third order. Its rate is given by the equation

- ^ = kCAC*Cc (11)

For the special condition of equal initial concentrations of all three
substances, the fraction x changed at t is given by the equation

= k't (12)

As an illustration of a reaction that is third order, the change
NO + NO + O2 = 2NO2, or, as usually written,

2NO + 02 = 2N02

is a reaction whose rate is proportional to the oxygen concentra-
tion and the square of the NO concentration. Its rate is

If we start with an initial concentration Co for oxygen and 2C0
for NO, the fraction x decomposed at time t is given by equation
(12) above.

Application of this equation to the oxidation of NO by oxygen2
in an extended series of tests showed that the rate was correctly

1 ALYEA and LIND, /. Am. Chem. Soc., 52, 1853 (1930). The experiments
are at 27° and an initial pressure of 709 mm., under which conditions a molal
volume is 26.7 liters. In Table 80, Co is in moles per 26.7 liters.

2 WOUBTZEL, Compt. rend., 170, 229 (1930).

KINETICS OF HOMOGENEOUS REACTIONS 485

described by it and hence indicated the reaction to be a true
third-order reaction.

On the other hand, a reaction whose stoichiometry indicates
it to be of the third order is not always found to be third order
when studied. Collisions involving three molecules properly
oriented and of sufficient energy to react are very rare. .More
commonly these systems are found to react in steps of which
one is so much slower than the others that it determines the rate
of the whole series. Under these conditions the order of the
reaction is that corresponding to the mechanism of the slow
reaction.

Reactions of Higher Order. — The equation for a general
change in state which we have used before is

aA + 6B + • • • = dD + eE + • • •

In considering the kinetics of such a reaction, we must establish
that the mechanism of the process is that shown by the equa-
tion, or use an equation that fits an actual mechanism other than
this. It would be a very rare collision among a molecules of A
and b molecules of B that would bring so many molecules together
properly oriented and of sufficient energy to react, and thus an
expression such as

would have no practical value and might be definitely mislead-
ing. As was stated in the previous section, many reactions for
which the over-all change in state involves several molecules are
found to take place in steps of varying velocities. When the
rates of more than two steps are nearly equal, the experimental
difficulties involved are too great for reasonable solution. Most
of the available data are for systems in which one slow reaction
occurs, and the supplementary or preliminary ones are com-
paratively rapid. We consider now some examples.

Consecutive Reactions (Series Reactions). — If a chemical reac-
tion takes place in steps of widely different speeds, the measured
velocity will be that of the slowest step. For example, hydrogen
peroxide is decomposed catalytically by iodides, and the velocity
is proportional to the first power of the H202 concentration.
It is proportional to the iodide concentration as well, but this

486

PHYSICAL CHEMISTRY

remains constant during a reaction. It has been suggested1 that
the slow reaction is

H2O2 + I- = H20 + IO-

and that this is followed by the practically instantaneous reaction
10- + H2O2 - H20 + I- + 02

which regenerates the iodide ions. This suggestion is supported
by the experimental data. The rate of the slow reaction is

dC in r

— - = /CL/I-CH2O2

and since Cj- is constant, the integral in terms of the fraction
decomposed is

log

1 - X

2.3

t = k't

It will be seen from Table 81 that the value of /c', which includes
0.02m. KI, is constant and thus that the suggested mechanism
of the reaction is a probable one for this system.

The rate of decomposition of hydrogen peroxide is catalyzed
by HBr in proportion to the square of its concentration. A rea-
sonable interpretation is that the slow reaction is

H202 + H+ + Br~ = H20 + HBrO
TABLE 81. — DECOMPOSITION OF HYDROGEN PEKOXIDE*

Time

Fraction decomposed

k' ho. l

* - t ic)g ! __ x

0 130

0.0124

0.242

0.0122

0.339

0.0116

0.497

0.0119

0.620

0.0120

0.712

0.0120

0.782

0.0121

0.835

0.0120

0.885

0.0125

1 BREDIG and WALTON, Z. physik. Chem., 47, 185 (1904).

2 HABNED, /. Am. Chem. Soc., 40, 1467 (1918).

KINETICS OF HOMOGENEOUS REACTIONS 487

followed by the very rapid reaction

H2O2 + HBrO = H20 + 02 + H+ + Br-

which regenerates the catalyzing ions, and it is well known that
hypobromites rapidly decompose hydrogen peroxide. The rate,
in the presence of a constant concentration of HBr, may be shown
by any of the equations

CTO 7 ,-* .-Y 0 U\j in s>< s~i

~ ~dt = *^H«°^HBr" or "" "57 = *CW>,GH+CBr- or

_ _ = k'C^

of which the second form is preferable for clearness.

Another illustration is the reaction whereby chromic ion is
oxidized to dichromate by persulfate ion in the presence of silver
ion.1 The chemical change is shown by an equation not involving
the silver ion,

3S208— + 2Cr+++ 4. 7H2O = 6SO4— + Cr207— + 14H+

but the rate of the reaction is independent of the chromic ion
concentration and is shown by the equation

d(S208— ) _ 7 xQ r, .

dt ~ "^*

The interpretation of the experiments is that the rate-governing
reaction is

S208— + Ag+ = 2S04— + Ag+++

which is then followed by a rapid supplementary reaction that
oxidizes the chromic ion and regenerates the monovalent silver
ion, namely,

3Ag+++ + 2Cr+++ + 7H2O = Cr207— + 3Ag+ + 14H+

A similar rate equation applies to the oxidation of manganous
ion to permanganate by persulfate and to some other oxidations.
While trivalent silver ion will appear new and perhaps improbable
to students, there is ample evidence of its formation.

Of the many other instances of series reactions, we shall have
space for only two more, though many are known. The rate'

1 YOST, ibid., 48, 152 (1926).

488 PHYSICAL CHEMISTRY

of halogenation of acetone in alkaline solution, as shown by the
chemical equation

CH3COCH3 + Br2 + OH~ = CH3COCH2Br + Br~ + H20

is independent of the halogen concentration and the same for
bromine and iodine. The probable steps in the reaction are

CH3COCH3 + OH- = CHsCOCH2- + H2O (slow)
CH,COCHr + Br2 = CH3COCH2Br + Br~ (fast)

as indicated by the experimental fact that the observed rate is
proportional to the first power of the concentration of acetone
and th6 first power of the OH~ concentration.

The oxidation of arsenious acid by iodine, for which the over-all
chemical change is

H3As03 + I2 + H2O = H3AsO4 + 2H+ + 21-

has an observed rate that is shown by the equation

_ d(H,As(),) , , (H3As03)(I2)
dt K (H+)(I-)

The suggested explanation is a rapid approach to equilibrium in
the reaction

I2 + H20 = HIO + H+ + I-

for which the equilibrium constant is

followed by a slow reaction for which the chemistry and rate
equations are

HIO + HsAs08 = H3AsO4 + H+ + I-
d(H8As08)

= fc,(H«AfiO«)(HIO)

By solving the equilibrium equation for the concentration of HIO
and inserting this in the last equation for the rate, we obtain
the first equation, with k' = k^K.

KINETICS OF HOMOGENEOUS REACTIONS 489

This explanation requires that the rate of the reverse reaction

in order to agree with the equilibrium relation, and experiment
shows that this is the rate of the reverse reaction. Additional
confirmation of the correctness of the accepted explanation is
that the ratio fc'/fc8, the observed rates in opposite directions, is
0.15 and that the equilibrium constant K, which is k'/ka, is 0.16.

Other reactions are known in which more puzzling phenomena
may be observed. For example, the oxidation of acetylene by
oxygen in the gaseous phase occurs in stages that involve glyoxal,
formaldehyde, and formic acid.1 The rate of the reaction
is proportional to the square of the acetylene concentration and
independent of the oxygen concentration. This behavior is
incomprehensible in the light of the rate equations given above.

Reversible Reactions. — It has already been stated that a
chemical system at equilibrium is not one in which there is no
reaction proceeding, but one in which equal rates in opposite
directions produce a system of constant composition. Thus
when equilibrium in the system

A + B = D + E

is approached by mixing A and B, these substances react; when
it is approached by mixing D and E, these react. The rate from
left to right is

and the rate from right to left is

(15)

(16)

~
At equilibrium the opposing rates are equal, and hence

CT>C-E ki f „

^r-^r = T- = const. (17)

CACB #2

which is the expression we have used in preceding chapters for
chemical equilibrium. •

1 KISTIAKOWSKY and LKNHER, 7, Am. Chem. Soc., 52, 3785 (1930).

490

PHYSICAL CHEMISTRY

By the use of radioactive indicators,1 the rate of oxidation of
arsenious acid by iodine at equilibrium and the reverse reaction
rate at equilibrium have been measured. These rates are in agree-
ment with those observed for the oxidation of arsenious acid by
iodine in systems far from equilibrium and for the reverse reac-
tion far from equilibrium, by the usual kinetic methods.

Since experiments upon reaction rates are confined to systems
in which the mechanism of approach to equilibrium is known,
while equilibrium when reached is independent of mechanism,
the constants of equilibrium are seldom determined from the

150

£100

a 50

100

50 75

Time in Minutes
FIG. 64. — Pressui e-time curve for the decomposition of ethyl bromide.

rates of the opposing reactions involved. But it is important to
realize that at equilibrium the rates are not zero. It is still
more important to realize in connection with experimental work
that equilibrium js approached at a decreasing rate and that
adequate time must be allowed for its complete attainment.

Decomposition of Ethyl Bromide. — As an example of reaction
rate in a system that reaches equilibrium before decomposition
is complete, we may consider the thermal decomposition of
ethyl bromide, which has been studied2 near 400° by observing
the change in total pressure with time at constant volume and
constant temperature. The data for a typical experiment are
shown in Fig. 64, in which p* is the initial pressure and Pf the

1 WILSON and DICKINSON, ibid., 69, 1358 (1937).

2 VERNON and DANIELS, ibid., 66, 927 (193S); FUGASSI and DANIELS,
., 60, 771 (1938).

KINETICS OF HOMOGENEOUS REACTIONS 491

final pressure at equilibrium. From the reaction equation
C2H5Br = C2H4 + HBr

we see that for complete decomposition p/ should be 2p*, but the
observed final pressure is less than 2pt, which shows incomplete
decomposition. If the equation written for the process is cor-
rect, the equilibrium relations follow from Fig. 64, in which
pc2H5Br = 2pt — pf and pHBr = Pc2n4 is half of the difference
between pf and pc2HBBr, whence pHBr = Pf — PI. Then the
equilibrium constant, which is the ratio of the rates of decomposi-
tion (fci) and reunion (fc2), is

K = (Pf - ?*)2 = h

t
The rate of decomposition of ethyl bromide is given by the

niQ.l fiYTYrp«si rvn

usual expression

At any time t when the total pressure is pt, the pressure of
ethyl bromide is 2pl — pt, and the pressure of C2H* or HBr is
pt — pl. These pressures and the volume of the system serve
to calculate the concentrations of each substance in moles per
liter from the ideal gas law. Let Co be the initial concentration
of ethyl bromide, proportional to pt and therefore constant, and
let z be the concentration of HBr or C2H4, which is a variable.
The rate of increase of z, which is the rate of decomposition of
C2H6Br, is

The reverse reaction, whereby ethylene and HBr form C2HBBr,
is bimolecular, or of the second order. It proceeds at the rate

dz 7 2

~ Tt - **'

and the net rate is the sum of these two rates, or
= *!(C, - 2) - k*z* = MC* - z) -

492

PHYSICAL CHEMISTRY

The rather complex integral of this equation proved to be very
sensitive to slight errors in C0 and thus not satisfactory as a
means of determining ki from K and measured changes in total
pressure with time. Other methods1 were devised for treating
the data from which it was found that fci is 5.8 X 10~4 (sec."1).
Effect of the Solvent. — Reaction rates for very few chemical
systems have been studied in a variety of solvents. It would
appear that a first-order reaction in which the solvent takes no
chemical part should proceed at a rate independent of the nature
of the solvent, but the data for N2Os do not confirm this sup-
position. Experiments upon the rate of decomposition of N206
in several solvents at several temperatures have been used to
calculate 104fc (sec."1), and these values are shown in Table
82 The specific reaction rates are 30 to 100 per cent greater
in solution than in the gas phase for this reaction.

TABLE 82 — SPECIFIC DECOMPOSITION RATE OF NITROGEN PENTOXIDE IN
DIFFERENT SOLVENTS*

•

Solvent

Values of k X 104

15°

20°

25°

35°

40°

45°

Nitrogen tetroxide
Ethykdene chloride . .
Chloroform ...
Ethylene chloride . . .
Carbon tetrachloride . .
Pentachloroethane

0 159

0 114
0 079
0.0747

0.344
0.322
0.274
0.238
0.235
0.220
0.215
0.165

0.554
0.479
0.469
0.430

2 54

(4 22)
(3.78)
(3.70)
(3 62)
(3 26)

(7.26)
(7.05)
(621)
(6.29)
(6.02)

Bromine • .

Gas phase *.

2 52

(2.14)

4.73
4.33

Nitromethane

Effect of Temperature upon Reaction Rate. — The usual effect
of temperature increase is an increase in rate of reaction, but a
few reactions decrease in rate as the temperature is increased.
In general, the rate near room temperature increases 10 to
20 per cent for each degree rise in temperature; and for a few
reactions the increase is even greater. Since an increase of
1° at ordinary temperatures increases the frequency of collision

1 See the original paper, ibid., 66, 922 (1933) for these methods

2 EYRING and DANIELS, ibid., 62, 1472 (1930).

KINETICS OF HOMOGENEOUS REACTIONS 493

among the molecules only about 0.2 per cent, the increase in reac-
tion rate evidently arises from some cause other than increased
collision frequency. An empirical equation, first suggested by
Arrhenius,1 expresses the increase in the specific reaction rate k
with increasing temperature,

din k A

.
U '

In order to test the applicability of this equation, it may be
put in the form d In k = —A d(l/T\ when it will be seen that a
plot of the common logarithm of k against l/T will give a
straight line of slope — A/2,3 if the equation is valid. This
expectation of a linear plot is realized for most reactions whose
velocities have been studied over ranges of temperature. We
shall return to a discussion of the meaning of the equation a
little later; but since the plotting procedure above was also
applied to van't HofTs equation in Chaps. IX and X we may say
now that the quantity A is not the heat of the chemical reaction
or any quantity which may be calculated from thermal data.

After the quantity A has been shown independent of* the
temperature, the Arrhenius equation may be integrated between
limits, and it then becomes

Activated Molecules. — It is probable from the observed rates
of first-order reactions that the molecules which react are in
some exceptional state, perhaps one of high energy compared
with that of an average molecule. The collisions that cause
reactions between two or more molecules are exceptional ones;
they may be collisions between molecules of high energy. Mole-
cules that react are called " activated molecules/' and a collision

1 Z. physik. Chem., 4, 226 (1899). Equation (18) above is in the form
given by Arrhenius as his equation (1). Later in the paper he introduces
the form d In k - (E/RT*)dT, in which E is clearly stated not to be A# for
the reaction, and thus the equation is not derived from van't Hoff s equation.
If A; i is the specific rate for a reaction A + B * C + D and k2 the specific
rate for C + D = A + B, then din k^dT - Ei/RTz,dlnk2/dT - J0,/«3Pf,
whence din (ki/kz) — (Ei — E^/RT2. Since ki/k* is the equilibrium con-
stant, Ei — E« is A/7. But it will be evident that one may not calculate
either EI or E% from A/7 unless the other is known.

494 PHYSICAL CHEMISTRY

that causes reaction is called an energy-rich collision or an
" activated complex." The fraction of the collisions which pro-
duce reaction is approximately

Effective collisions __ _B/RT _ k

~~ ~~

Total collisions

in which e~E/RT is the fraction of the molecules having activation
energy E above the average, k is the specific reaction rate, and /c0
is the rate that would result if every collision were effective.
Taking logarithms, this equation becomes

In k - In k« = - (21)

The temperature coefficient of ko would be the rate of increase
of collisions with increasing temperature, which we have stated
to be about 0.2 per cent per degree near room temperature,
whereas the temperature coefficient of k at ordinary temperature
is of the order of 10 to 20 per cent or more per degree. As a
first^ approximation we neglect the change of kQ with T, and
upon differentiating (21) we obtain

T (22)

which is the equation found empirically by Arrhenius if we
substitute E/R for A in equation (18). The fact that plots of
In fc against the reciprocal of T for actual data are straight lines
shows that the temperature coefficient of fc0 is negligible, as we
have assumed it to be.

For reactions in which equation (20) is assumed to hold, it
has not been possible to calculate k theoretically, because, while
fco could be computed from the kinetic theory, there was no way
to calculate E. Even the principle of excess energy content as a
requirement for reaction is not valid for all reactions, for some
few of them proceed at decreasing rates with increasing tempera-
ture. The oxidation of NO to N(>2 by oxygen is an example,
for which k for the third-order reaction 2NO + 02 = 2N(>2 is
36 at 0° and 18 at 50°. Applying these constants to the integral
of equation (22), one obtains E = —2400, from which fc at 25°
is calculated to be 25; this agrees with experiments at 25°. But

KINETICS OF HOMOGENEOUS REACTIONS 495

substituting this value of E into equation (20) leads to the
absurdity of a collision efficiency greater than 1, which shown
that the interpretation of the equation is unjustified or incom-
plete in this instance, even though equation (22) correctly de-
scribes the changing rate with changing temperature.

An assumed but unproved explanation for this particular reac-
tion is a rapid polymerization to equilibrium with the evolution
of heat, as shown by the equation 2NO = N202, followed by a
slow reaction N202 + Oz = 2N02. Since the extent of poly-
merization would be less at higher temperature, the rate of
oxidation, wiiich depends upon the concentration of the hypo-
thetical N2O2, would also be less at higher temperatures.

This is, of course, merely a suggested explanation. Some
other mechanism, such as rapid approach to equilibrium by an
exothermic reaction NO + O2 = NOs, followed by a slow reac-
tion such as NO + NO3 = 2NO2, is equally plausible; and there
are other possibilities.

A common modification of equation (20) designed to allow for
circumstances such as negative temperature coefficients is

(23)
in which p is interpreted as a steric, or orientation, factor.1

More generally, p may be regarded as a term that includes all the
requirements that the activated complex must satisfy in order to decom-
pose into product molecules, other than the possession of the minimum
excess energy E necessary for its formation. The explanation of the
negative temperature coefficient in terms of this equation is simply that
the chance that the three molecules shall collide with the correct orienta-
tion decreases with rising temperature more rapidly than the factor
e-s/RT increases. The term e~E/RT is not increasing very rapidly with
temperature because E is very small, possibly zero.2

The fact that a straight line usually is obtained when log k is plotted
against \/T suggests that pko and E are comparatively insensitive to
temperature, or that both may be temperature functions in such a way
that their product is constant, or that E may vary with T in such a way
as to hide the temperature dependence on pko.

In order to show what an exceptional mplecule an activated
one is, note that the A of equation (18) is 22,000 for par aldehyde

1 Quoted from Sherman, Pub. Am. Assoc. Adv. Sci., No. 7, 126 (1939).

2 GEHSHINOWITZ and EYEING, J. Am. Chem. Soc., 57, 985 (1935).

496

PHYSICAL CHEMISTRY

decomposition, or E is 44,000, and e~E/RT at 520°K. is 4 X 10~19.
If we accept equation (20), only this fraction of the total mole-
cules is in a condition for reacting.

There is evidence that some of the activated molecules deacti-
vate without reacting, which is to say that, before a molecule

which has acquired sufficient
energy to be in a reactive condi-
tion has time to react, it may
divssipate enough of its energy to
bring it into a lower energy state
again. We do not imply that an
activated molecule is merely one
of exceptionally high velocity ; for
its extra energy may be in the
form of vibrational energy, and
its reaction may depend upon the

c.o

30
3.5

4.0

4.5
F>0

1,625 1.875 1.925 1.975 £025 2,075
(1/T)x 1,000

FIG. 65

accumulation of this energy at the chemical bond to be severed
in the reaction.

A plot of — In k against l/T for the decomposition of paral-
dehyde is given in Fig. 65, from which it may be shown that

In k = 34.83 -

44,160
RT

whence E is 44,160 cal. per mole, independent of T within this
range. For other reactions there is evidence of a variation of E
with temperature.1

If the energy of activation is taken as 24,700 cal. for the decom-
position of nitrogen pentoxide,2 the reaction constants calculated
at other temperatures from the value for 25° agree closely with
the measured constants, as may be seen from Table 83.

The fraction of the molecules " activated" to this additional
energy content above the average for 25° is exceedingly small;
it may be calculated to be e-(89o+24,7oo)/594) or j 6 x 10-i9 Thus?

the activated molecule is very exceptional indeed, and questions
arise as to its condition. How and in what form does it "con-
tain" so much energy? What can be the source of it? These

•

1 LAMER, /. Chem. Phys., 1, 289 (1933) ; HtteKEL, Ber., 67% (A) 129 (1934) ;
LAMER and MILLER, J. Am. Chem. Soc., 57, 2674 (1935).

2 DANIELS and JOHNSTON, ibid., 43, 53 (1921).

KINETICS OF HOMOGENEOUS REACTIONS 497

TABLE 83. — CHANGE OF VELOCITY OF REACTION WITH TEMPERATURE

Temperature

10*

Observed

Calculated

0°

0 047

0.0444

2 03

8 08

7 9

29 9

28 3

90 0

93 2

292

286

questions cannot be completely answered, though it seems prob-
able that some of the excess energy must come from collisions.
As shown in the distribution curve for velocities (Fig. 6 on page
75), there are a few molecules with very high velocities, and
the rare collision between two of them would certainly form at
least one that is highly energized. It has been suggested1 that,
even after a molecule has accumulated this most exceptional
amount of energy, it may be "deactivated" before it has time to
react. Atoms bound into a molecule by a valence bond cannot
fly apart in less time than the natural period of vibration of this
molecule, and before the energy of activation can be localized
in a given bond it may be dissipated to surrounding molecules.
Such statements sufficiently illustrate the lack of defmiteness
associated with the idea of activated molecules. The subject is
being investigated intensively by many workers at present; one
may expect further light upon it within a reasonable time.

References

BERKMAN, MORRELL, and EGLOFF, "Catalysis," 1940; HINSHELWOOD,
" Kinetics of Chemical Change," 1940, KASSEL, "Kinetics of Homogeneous
Gas Reactions", HAMMETT, "Physical Organic Chemistry," McGraw-Hill
Book Company, Inc., 1940. A symposium on kinetics m homogeneous
systems will be found in Chem. Rev., 10, February, 1932, and another in
ibid., 17, August, 1935.

Problems

1. A solution 0.167m. in sugar and 2.5m. in formic acid has at 30° a rota-
tion of 22.10 deg. Owing to the presence of acid in the solution, inversion
takes place at such a rate that the angle of rotation of polarized light is

1 EYRING and DANIELS, ibid., 52, 1472 (1930).

498 PHYSICAL CHEMISTRY

11.10 deg. after 15 hr. and 0.35 deg after 45 hr. (a) Calculate the angle of
rotation corresponding to complete inversion of the sugar, using the value
of kr from Table 78. (6) Calculate the time necessary for half the sugar to
be inverted, (c) The solution was 2.50m. in formic acid, whose loriization
constant is 1.7 X 10~4. Calculate the hydrogen-ion concentration in this
solution, and estimate the time required for inverting half the sugar when
the catalyzing acid is 0 Olm. hydrochloric acid

2. In a solution containing 0.1 mole of ethvl acetate and 0.1 mole of
sodium hydroxide per liter, 10 per cent of the ester is decomposed in 15 mm.
at 10° and 20 per cent at 25°. What fraction would be decomposed in 5 mm
at 55°?

3. The decomposition of paraldehyde vapor into acetaldehyde vapor,
for which the chemistry may be written P = 3^1, is a first-order reaction
At 262°C. the reaction-rate constant is 0 00102, when time is expressed hi
seconds. What will be the total pressure 1000 sec after paraldehyde is
introduced into a closed space at 262° and an ^n^tlal pressure of 0 10 atm ?

4. How long would it take to convert 40 per cent of hydroxyvalenc acid
into valerolactone at 25° in the presence of 0.075 N hydrochloric acid?

6. The oxidation of formaldehyde to formic acid by hydrogen peroxide is
a second-order reaction. When equal volumes of molal HCHO and m9lal
H2O2 are mixed at 60°, the concentration of formic acid is 0 215 after 2 hr.
(a) In what time would this reaction be 99 44 per cent completed? (6) If
equal volumes of O.lm. solutions are mixed at 60°, what time would be
required for the reaction to be 43 per cent complete? (c) In about what time
would the reaction be 43 per cent complete at 100°C if equal volumes of
molal solutions were mixed?

6. The decomposition of ethyhdene diacetate into acetaldehyde and
acetic anhydride is a first-order reaction occurring in the gas phase, in which
one molecule decomposes into two Equilibrium corresponds to complete
decomposition, and the progress of the reaction may be followed by observ-
ing the total pressure At 536°K the constant of reaction is 7.2 i X 10~4for
time in seconds, and this constant changes with the temperature as shown
by the equation d In k/dT = 16,450/T72. (a) Derive an expression for x,
the fraction decomposed at a time i, in terms of the initial pressure po and
the total pressure p ~ (6) What time would be required for 75 per cent
decomposition at 536°K if p0 were 0 10 atm.? (c) What time would be
required for 75 per cent decomposition at 573°K.?

7. A liter of a solution of "NzOz in CC14 at 40° decomposes with the evolu-
tion of oxygen at the following rate:

t, mm ... . . " 20 40 60 80 100 Complete

O2, ml 114 18.9 23.9 27.2 295 3475

Show whether the reaction is of the first or second order from a set of reaction
constants.

8. Nitrogen pentoxide decomposes slowly at 20°C. according to the equa-
tion (1) N2O6 = y2Oz + N2O4, and the reaction (2) N2O4 - 2NO2 reaches
equilibrium instantly. The equilibrium constant, Kp = 45, for the second

KINETICS OF HOMOGENEOUS REACTIONS 499

reaction is for pressures in millimeters. The rate at which the pressure of
N2O6 decreases is given by the equation — d In p/dt = 0,001 for time in
minutes. If the initial pressure is 100 mm. and the reaction is earned out
at constant volume at 20°, calculate the partial pressure of the gases N2OB,
N2O4, and NO2 at the end of 350 mm.

9. In a liter of solution at 65° containing 22.9 grams of ammonium cya»
nate, urea is formed as follows.

t, mm 0 20 50 65 150

Urea formed, grams 0 7 12 1 13 8 17.7

The equation for the reaction is NEUCNO = (NH2)2CO. (a) Determine
the order of the reaction by calculating a set of values of the specific reaction
constant. (6) Estimate the time that would be required to transform half
the ammonium cyanate to urea at 65° and at 25°.

10. The conversion of acetochloramlide into parachloroacetamlide m the
presence of HC1 (which is a catalyst only) proceeds at such a rate that the
fraction converted varies with time as follows*

t, min 77 15 8 32 2

x . 0 159 0 295 0 510

Determine whether the reaction is of the first or second order

11. The velocity constant for the (first-order) decomposition of NaOGl
in aqueous solution changes with the temperature as follows:

t 25° 30° 35° 40° 45° 50°

k 0 0093 0 0144 0 0222 0 0342 0 0530 0 0806

Show that this change takes place m accordance with the Arrhemus equa-
tion. [HOWELL, Proc Roy Soc. (London), (A) 104, 134 (1923).]

12. The second-order reaction between thiosulfate ion and bromoacetate
ion may be followed by titrating samples with iodine solution When equal
volumes of 0.1 m solutions are mixed at 25°, samples of the mixture required
the following quantities of iodine solution :

J, mm 0 20 35 End

Iodine, ml 2790 1616 12.27 0.0

(a) Calculate the specific reaction constant for this reaction at 25°. (b) The
energy of activation is 15,900 cal. for this reaction. What fraction of the
thiosulfate ion in the above system will have reacted in 20 mm. at 40°?

13. A solution of benzenediazomum chloride in isoamyl alcohol decom-
poses at 20° with the evolution of nitrogen gas at the following rate :

Time, min 0 100 200 300 410 End

Vol. N2; ml 0 15 76 28 17 37 76 45 88 69.84

(a) Determine whether the reaction is first order or second order, (b) The
rate at 40°C. is 18 2 times the rate at 20°C. Determine the energy of activa*
tion for the reaction. [WARING and ABRAMS, /. Am. Chem. Soc., 63, 2757
(1941).]

500 PHYSICAL CHEMISTRY

14. The specific reaction rate of ethyl acetate with NaOH is 6.5 moles per
liter per mm. at 25°. Calculate the specific conductance of the mixture
1 hr. after a liter of 0.03 N ethyl acetate is added to 500 ml of 0 06 N sodium
hydroxide. Ethyl acetate and alcohol do not ionize and do not appreciably
change the conductance. The limiting equivalent conductances at 25° are
Na+ = 50, OH~ = 197, and Ac~ = 41.

16. The decomposition of NO2 into NO and 02 has been found to be a
homogeneous reaction. When 0 105 gram of NO2 is introduced into a
liter bulb at 330°C., the initial rate of decomposition is 0 0196 mole per liter
per hr , and when the concentration of NO2 has become 0 00162 mole per
liter, the rate of decomposition has fallen to half the initial rate, (a) Show
whether the reaction is first order or second order. (6) Calculate the frac-
tion of the original N02 decomposed at the end of 30 mm. (c) If 70 per cent
of this sample of NO2 is decomposed at the end of 10 mm at 354°C., calcu-
late the temperature at which the same percentage decomposition would be
obtained in 15 min.

16. The reaction CH3CONH2 + H+C1~ + H2O = CH8COOH + NH4+-
Cl~ may be followed by observing the specific conductance of the mixtures,
which changes as follows when equal volumes of 2 AT solution are mixed at
63°:

t, mm 0 13 34 48

Specific conductance 0 409 0 374 0 333 0 313

Ao = 515 for H+, 133 for Cl~, and 137 for NH44 at 63°. (a) Determine the
order of the reaction. (6) How long would be required for 15 per cent to
react if equal volumes of 0 5 N solutions were mixed at 63°? (c) About how
long would be required to hydrolyze 0.005 mole of acetamide if 0.010 mole
were dissolved in a liter of normal HC1?

17. The hydrolysis of methyl bromide is a first-order reaction whose
progress may be followed by titrating samples of the reaction mixture with
AgNO3 The volumes required for 10-ml samples at 330°K in a typical
experiment are

*,mm 0 88 300 412 End

AgNO3, ml ~ 059173 22 1 49 5

Calculate a set of reaction-rate constants for this reaction

18. The decomposition of gaseous silicon tetramethyl may be followed by
the increase of pressure at constant volume and constant temperature In
an experiment at 679°C. the pressure was 330 mm. at the start, 620 mm. in
10 min., and 990 mm. at the end. (a) Calculate k for this first-order reac-
tion at 679°C. (b) Calculate the time required for 50 per cent decomposi-
tion at 700°C., taking 79,000 cal. as the energy of activation.

19. When COS is dissolved in water, the reaction COS + H2O = CO2 +
H2S occurs. If at 30° no gases are allowed to escape from this solution, the
concentration of H2S changes with time as follows:

t, min 0 80 280 525 End

Concentration H2S, moles per liter 0 0 119 0 342 0 496 0 696

KINETICS OF HOMOGENEOUS REACTIONS 501

(a) To what order does the reaction rate conform? (fe) For an initial con-
centration of 1 mole of COS per liter the initial rate of formation of H2S is
18 X 10~3 mole per liter per mm at 47° and 1 2 X 10" 3 mole per liter per
mm at 25°. Calculate from these data a value for the specific reaction
constant for 30°, and show that this value is in reasonable conformity with
that obtained in part (a).

20. The same reaction, COS + H2O = CO2 + H2S, occurs when dilute
solutions of water in alcohol and of COS in alcohol are mixed, and this reac-
tion in alcoholic solution is second order. When equal volumes of 0.20m.
alcoholic solutions of COR arid of water are mixed at 75°, the initial rate of
formation of H2S is 4 X 10~5 mole per liter per mm. (a) Calculate the
specific reaction rate at 75°. (6) What time would be required for the H2S
concentration to reach 0 020m. at 75°?

21. Tertiary butyl chloride decomposes thermally into HC1 and isobuty-
lene as shown by the equation (CH3)3CC1 = (CH3)2CCH2 + HC1.
The following data were obtained in a liter flask at 295°C,:

i, mm 30 50 60 80

p(CH8)jCCl, mm 28 20 18 10 14 20 9 13

(a) Show to what order the reaction rate conforms (b) Calculate the initial
pressure of i-butyl chloride in the flask [BREAKLEY, KISTIAKOWSKY, and
STAUFFER, J. Am. Chern. Soc., 58, 42 (1936). J

CHAPTER XIII
RADIATION AND CHEMICAL CHANGE

In addition to chemical reactions that take place whenever
the reacting substances are brought together, proceeding at a
rate governed by the concentration and approaching equilibrium
spontaneously, there are other reactions that depend upon the
absorption of light for their initiation and progress. When
the reactants are mixed as gases or in solution and no light is
supplied, no reaction takes place, even upon long standing.
But when the system is illuminated with light of the proper
wave length or "color," reaction occurs; and the extent of the
chemical reaction is governed by the quantity of radiant energy
absorbed into the reacting system, or the absorbed light may
increase the rate enormously from that of the "dark reaction,"
as in the formation of phosgene.

In general, the chemical reactions produced by the absorption
of light are of the same nature as reactions produced in other
ways. They include synthesis and decomposition, oxidation,
reduction, polymerization, rearrangement, and condensation.
Photochemical processes are sometimes more complex than one
would suppose from the chemical equation, arid the kinetics of
the reaction are often not obvious from the nature of the reacting
substances. However, photochemical research may assist in the
study of the mechanism of "dark reactions" as was found in the
formation of HBr. It will be necessary to distinguish clearly
between experimental fact and plausible explanation in this topic
as well as in others previously discussed, or perhaps to a greater
extent than usual in this particular case — for the study of photo-
chemical reactions is newly developed, and the experimental work
requires considerable skill. Some of the research reported in the
current literature of physical chemistry has been done with
inadequate apparatus, occasionally with insufficient skill as well,
and frequent discrepancies may be found in the reports of different
observers apparently studying the same reacting system. This

502

RADIATION AND CHEMICAL CHANGE 503

is not to question the integrity of any of them, but to emphasize
the difficulty of some of the measurements, the insufficient con-
trol over the experimental conditions, and the uncertainties
inherent in the exploration of a new field of research before
adequate methods of experimentation have been perfected,
Moreover, the theoretical interpretations have frequently
changed in the past few years, and there are indications that
further revision may be required.

The light energy absorbed by a molecule may be temporarily
stored as potential energy, which may redistribute itself in the
molecule, rupturing the molecule at its weakest link. Instead
of dissociation taking place, the absorbed energy may raise some
of the external electrons to a level such that the molecule is
temporarily more reactive.

It will be seen later that absorption of light by a system under-
going photochemical reaction is attended by a change in the
concentration of some reacting substance. Thus the kinetics
of a photochemical change are the same as for any other chemi-
cally reacting system; the absorption of more light produces more
active material and causes a more rapid reaction. The quantity
of reactive material is proportional to the quantity of light (of
the proper wave length) absorbed

The initial velocity of reaction between hydrogen and bromine
at 200° is proportionaKto the concentration of hydrogen and to
the square root of the bromine concentration. Since the dis-
sociation equilibrium J^Br2 = Bri is shown by the relation
(Bri) = K(~Brz)^, it is probable that the reaction whose velocity
controls the formation of HBr is between H2 and Bri and that
subsequent reactions (of much higher velocity) are necessary to
complete the over-all reaction. Bromine is dissociated into
atoms by the absorption of light, which thus changes the con-
centration of a reacting substance.

The discussion in this chapter will be limited to the simplest
aspects of a few chemical changes that are dependent upon the
absorption of light for their progress.1 It will be seen that
photochemical reactions are usually more sensitive to certain
frequencies or ranges of frequency of the absorbed light, and an

1 See Relief son and Burton, "Photochemistry and the Mechanism of
Chemical Reactions" (Prentice-Hall, New York, 1939), for an excellent
treatment of the theory and experimental data on many reactions.

504 PHYSICAL CHEMISTRY

explanation of this fact must be sought in the experimental
data. In making the experiments themselves, it is necessary to
work with monochromatic light or at least to limit the light
supplied to a rather narrow range of wave lengths, in order to
observe the changing photochemical effect that sometimes
accompanies change of color of the light.

The Grotthuss-Draper law states that only radiant energy
that is absorbed by a system can be used in producing chemical
changes in it; transmitted light can have no effect. This simple
fact makes it necessary to measure quite accurately the intensity
of the transmitted light as well as that of the entering light, in
order to determine the actual amount of energy absorbed by a
reacting system. In cloudy media, scattered light must not be
considered as absorbed One should not consider that light is
acting as a catalyst in photochemical reactions; for by definition
a catalyst accelerates a reaction without being exhausted as the
reaction proceeds, and it is required for a photochemical change
that light must be absorbed by the reacting system It may be
stated here and explained later that the absorption of hght by a
system is a necessary but not a sufficient condition for photo-
chemical change.

In connection with the absorption of light, Lambert's law
should be borne in mind. This states that equal fractions of
the incident light are absorbed by successive layers of a homo-
geneous material of equal thickness. Since the light transmitted
by the first layer is that incident upon the second layer, it will be
seen that, if half the entering light is absorbed by a first layer
of material, half the remainder will be absorbed by a second
layer of the same thickness, and so on. Thus, the intensity of
light transmitted through a medium is

7 = 70c-« (1)

where I is the length of path in which the intensity of the light
is reduced from 70 to I and k is the extinction coefficient. The
decrease of intensity for a given medium varies greatly with the
wave length of light considered. Values of k for various wave
lengths may be found in tables.1

1 See " International Critical Tables," Vol V, p. 268. There is difference
of usage in expressing absorption. For example, one may use 10 in place of
e and thus employ Briggs's logarithms, writing the absorption equation in

RADIATION AND CPIEMICAL CHANGE 505

Light absorption by a gas or by a dissolved substance usually
depends only upon the number of molecules in the absorbing
layer and is independent of the pressure or concentration of the
absorbing substance (Beer's law).

Energy Quanta. — The fundamental assumption of Planck's
quantum theory is that light consists, not of a continuous
"wave front," but of quanta or "particles" of energy.1 The
energy content of these quanta depends upon the frequency v.
Since the velocity of light (usually denoted by c) is 3 X 1010 cm.
per second, regardless of wave length (A), the frequency of any
radiation may be Calculated from the relation v — c/X. The
frequency of visible light includes only the fairly narrow range
of about 4 X 1014 per second (red) to 8 X 10U per second (violet),
corresponding to wave lengths of 7000 to 4000A, respectively
(or 700 mjji to 400 m^),2 but a very much wider range of frequency
must be considered in photochemistry. Ultraviolet light, which
is light shorter in wave length than 4000A, is frequently employed
in producing photochemical changes, for a reason that will be
evident from the calculations shortly to be presented.

In order to calculate the energy of a quantum, the frequency is
multiplied by a universal constant, Planck's constant A, whose

the form 1 = /olO Kl, in which A' is called the extinction coefficient and
/ is the length of path in centimeters, in which the intensity is reduced from
/o to 7. For dissolved absorbers a molal extinction coefficient € is also
recorded, and the intensity relation is 7 = 7010~€cZ In this equation c is
the molal concentration of the absorber and / the length of path as before.
The variation of e with wave length is strikingly shown by some of the
data for chlorine gas In this case (and in general when the molal extinction
coefficient is stated for a gas) c is in moles per liter of gas reduced to 0° and
1 atm.

X, i . . 2540 3030 3340 3360 4050 4080 5090 5790
e 0 239 35 2 65 5 27 17 3 99 0 234 0 0452 0 003

1 In the present state of development of physics, one may not say what
light consists of, but only that light has certain properties which resemble
those of a wave and certain properties of a particle or corpusle. The cor-
puscular properties of light are clearly presented in a form not too difficult
for beginners in Richtmyer, "Introduction to Modern Physics," 2d ed.,
p 173; see also A. H. Compton, Phya. Rev. SuppL, 1, 74 (1929).

2 The< symbol A denotes 1 angstrom unit, or 10~8 cm., but wave lengths
are sometimes expressed as millimicrons, for which the symbol is mju; since a
micron is 10~4 cm., a millimicron is 10~7 cm.

506 PHYSICAL CHEMISTRY

value is 6.542 X 10~27 erg-sec. A single quantum of frequency
4 X 1014 would thus be 4 X 1014 X 6.542 X 10~27 erg, or
2.62 X 10~12 erg. The results of photochemical experiments
are expressed as moles of substance decomposed per calorie of
absorbed radiation or more frequently as molecules decomposed
per quantum, of absorbed radiation. Since hv is the energy of
one quantum, Nhv ergs, or Nhv/(4t.l8 X 107) cal., of radiant
energy is required to supply one quantum to each molecule in a
mole, where N is Avogadro's number, 6 X 1023. Many reactions
take place upon the absorption of light over a range of wave
lengths. For example, light of all wave lengths between 3300 and
2070A decomposes hydrogen iodide, and the yield is 2.0 molecules
per quantum of energy absorbed. But since v = c/X, the energy
content of a quantum hv is greater for light of shorter wave
lengths. Hence more energy is absorbed, more calories per gram
molecule of hydrogen iodide decomposed, in the short wave-
length ultraviolet than in the longer ultraviolet region.

When a quantum is absorbed by a molecule or atom, the energy
of the system increases, as expressed by the relation

A# = hv

For a system of one gram molecule or one gram atom, the corre-
sponding expression is

A# = Nhv (2)

Einstein Photochemical Equivalence Law. — When a photo-
chemical reaction is produced by the absorption of radiant energy,
the yield is proportional to the number of quanta absorbed by
the system. Einstein postulated that the system absorbs a
quantum for each molecule that reacts, or Nhv for each gram
molecule. There is thus a definite relation between the energy
required in a photochemical change, such as the dissociation of a
molecule into atoms, and the frequency of radiation that will
be able to produce it.

Avogadro's number of quanta, Nhv, is sometimes called 1
"einstein," but it should be noted that this is not a constant
energy quantity. Since N and h are constants, the energy repre-
sented by Nhv increases as v increases, which is to say that it
increases as the wave length of the radiation decreases. The

RADIATION AND CHEMICAL CHANGE 507

energy in calories corresponding to N quanta changes with the
wave length as shown in the following table :

Wave Length, A Nhv, cal.

7000 (red) 40,500

6000 (orange) 47,500

5000 (green) 57,000

4000 (violet) 71,000

3000 (ultraviolet) 95,000

2500 (ultraviolet) 113,500

2000 (ultraviolet) 142,000

It does not follow simply from these figures that a given photo-
chemical reaction will be brought about if its thermochemical
requirements (translated into radiant energy) are met. For
example, the dissociation of iodine vapor into normal atoms
absorbs about 34,500 cal., and any wave length in the whole
visible range of light should be of sufficient energy to decompose
it. Radiation of 4300 to 7000A is absorbed by iodine vapor,
but orange light does not cause it to dissociate, even though the
quanta would seem to have sufficient energy. The union of
hydrogen with chlorine evolves energy and might be supposed to
proceed spontaneously, but radiant energy is required for the
initiation and progress of the reactiorf

The table above shows that red light corresponds to quanta
of the lowest energy in the visible region, but infrared radiation
would correspond to quanta of still lower energy, of course.
Many substances absorb in the red region, but as yet no photo-
chemical reaction has been found to occur under the influence of
light of wave length greater than 7000A. This illustrates the
statement above that the absorption of light by a system is not
a sufficient condition for photochemical reaction.

Instances in which the final result of photochemical process is
the decomposition (photolysis) of a single molecule for each quan-
tum absorbed are rare.1 The apparent deviations of experiment
from this expected yield are often so large that one might well
question whether the law has any value whatever. Some data

•

1 The Einstein, photochemical equivalence law has been found to apply
to the photolysis of malachite green leucocyanide. The yield is one mole-
cule decomposed per quantum within the limit of accuracy of the measure-
ments, which was about 2.4 per cent. HABBIS and KAMINSKY, J. Am.
Chem, Soc,, 67, 1154 (1935).

508

PHYSICAL CHEMISTRY

are given in Table 84. Actual yields in various processes experi-
mentally studied vary from less than 0.001 to 1,000,000 molecules
reacting per quantum, and the yields in a single process may
change greatly with experimental conditions. But constant

TABLE 84. — QUANTUM YIELDS IN PHOTOCHEMICAL REACTIONS

Wave

Reaction

length,

Absorber

Quantum

yield

H2 4 Br2 = 2HBr

Bromine

10-3

H2 4 C12 = 2HC1 .

43GO

Chlorine

Over 6 X 106

2HI =H2 +Ij....

3320-

III

2.0

2000

2HBr = H2 4 Br2

2530

HBr

2 0

2NOC1 = 2NO 4 Cl, .

6300-

NOC1

2 0

3650

Oxidation of benzaldehydc

3660

10,000

Oxidation of Na2SO3

3660

50,000

CO 4 Cl, = COC12

4360

1,000

Chlorination of benzene.

106

3O2 = 2O8

2070

2 3-3.1

Photolysis of uranyl oxalate

2540

U02C204

0 60

•

3660

U02C2O4

0 49

Maleic-fumaric transformation

Maleic

0 04-0 13

2Fe++ 4 L «= 2Fe+++ 4 21"

Is-

1 0

2O8— >3O2

4200

Clj

2 0

2O8-»3O2

4200

Br2

31 0

2N02 = 2NO 4 02

N02

2 0

2NH$ = N2 4- 3H2

^2000

NH3

2 5

yields have been obtained in a considerable number of reactions,
and plausible explanations of the deviations are available in
others. It is still the only theory available, and it is probably
a correct explanation of the photochemical " primary process."
Primary Processes. — ^The initial encounter between a quantum
of energy and an atom or molecule is usually called the primary
photochemical process, since through it energy is absorbed by
the reacting system. If the primary process is succeeded by
others that advance the chemical change under consideration
without absorbing more radiant energy, the equivalence of quanta
absorbed to molecules reacting would not apply to the complete
process initiated by the quantum. It is probable that something

RADIATION AND CHEMICAL CHANGE 509

of this kind is responsible for quantum yields greater than
unity. Low quantum yields may be due to deactivation of the
reacting substances before they have time to react or to side
reactions.

The actual yield in photochemical reactions often depends
on the thermal processes initiated by the absorption of the
quantum. Small quantum yields have been obtained in a suffi-
cient number of reactions to indicate that some process involving
one molecule per quantum takes place in all photochemical
processes This " primary " process may be the dissociation
of a molecule into atoms or free radicals; it may be the formation
of " excited7' molecules or atoms, sometimes called "activated"
molecules or atoms or molecules "in a higher quantum state.77
No distinction is yet implied by the use of three terms to describe
the unusual condition of a molecule that has "absorbed" a
quantum of energy

If the absorbed energy corresponding to an excited state
is not dissipated by collision of the excited molecule with other
molecules or if it is not reemitted before reaction can occur, the
molecule may decompose and thus give a quantum yield of 1.
However, the products of the primary decomposition may be so
reactive that they immediately take part in secondary reactions
and thus mask the applicability of the photochemical equivalence
law. For example, the decomposition of hydrogen iodide illus-
trates a quantum yield of exactly 2.0, and it probably involves
the decomposition of one molecule per quantum in the primary
process. A similar yield by a different mechanism is found for
the decomposition of NOC1.

It seems well established that a continuous absorption spec-
trum, without bands, indicates that the primary photochemical
process is the dissociation of a molecule and that a banded
absorption spectrum indicates the formation of an excited mole-
cule. When dissociation of a molecule into atoms occurs by the
absorption of a quantum of greater energy than the minimum
calculated from thermochemical data, an "ordinary77 atom and
an "excited77 atom are probably formed. The evidence for
this rests almost entirely on the interpretation of spectroscopic
data and cannot be discussed here,1 but the fact itself will be con-
sidered in connection with some photochemical decompositions.

1 See FRANCK, Trans, Faraday Soc., 21, 536 (1926).

510 PHYSICAL CHEMISTRY

Decomposition of Hydrogen Iodide. — The experiments of
many investigators have shown that the extent of this decom-
position is proportional to the amount of light absorbed and that
decomposition is complete for all wave lengths absorbed down to
2000A, over a considerable range of temperature and for moder-
ate variations in the intensity of illumination and in the partial
pressure of hydrogen iodide. The quantum yield is 2.0 over the
entire spectral range investigated. Tingey and Gerke1 have
shown that the absorption is continuous, that it begins at
about 3320A and extends down to 2000A, the limit set by their
apparatus.

It may be calculated thermochemically that the dissociation
HI = H + I absorbs about 68,000 cal., which would require a
value of Nhv equivalent to a wave length of about 4000A. This
lies in the violet end of the visible spectrum, but light of this
wave length is not absorbed by hydrogen iodide, which is color-
less. The continuous absorption does not begin until 3320A,
corresponding to 86,000 cal., which shows that more energy is
absorbed in the primary process than is required for simple
decomposition into atoms. It seems likely that the products of
decomposition are an iodine atom and a hydrogen atom. Sub-
sequent steps, not requiring more radiant energy, have been
suggested by Warburg2 as shown by the equations

HI + Nhv = H + I
II + HI = H2 + I
1 + 1 = 1,

The summation of these equations shows two molecules decom-
posed per quantum of energy absorbed, which is in agreement
with experiment. The energy evolved in the second and third
steps is probably dissipated as heat, which is to say that it may
be distributed among the molecules to increase their velocities.
Without introducing the quantum concept, the data may be
expressed as moles of HI decomposed per calorie of absorbed
energy of certain wave lengths as follows:

Wave length, A . . 2070 2530 2820

Moles HI >£ 10* per cal 1 44 1 85 2 09

1J. Am. Chem. Soc., 46, 1838 (1926).

2 Sibber, kgl preuss. Akad. Wiss., 1916, 300.

RADIATION AND CHEMICAL CHANGE 511

These figures show a smaller calorie efficiency in the shorter
wave regions, as was mentioned before, but they offer no clue
whatever as to the reason for this surprising fact, or for the
mechanism by which decomposition occurs. When translated
into quantum yields, however, the reason for the lower calorie
yield becomes evident, and the photochemical mechanism sug-
gested above appears reasonable.

The experimental fact should be emphasized that two molecules
of HI are decomposed for each quantum absorbed, of whatever
wave length. An interpretation has been given above that
seems the most probable, in view of our present information,
but that may require revision at a later time, when more facts
are available. It is improbable that the quantum yield will be
found to differ much from 2.0.

Hydrogen bromide shows a similar continuous absorption of
all wave lengths below 2640A, and the mechanism of its photo-
chemical decomposition is probably similar to that suggested
by Warburg above for hydrogen iodide.

The molecular mechanism by which nitrosyl chloride dis-
sociates is said to be different1 from that suggested for hydrogen
iodide above. A quantum yield of 2 has been obtained for wave
lengths from 6300 to 365oA. Since the absorption spectrum
of NOC1 is banded throughout the visible part of the spectrum,
this is an indication that activated molecules are formed in the
primary process. The process may be represented by the
equations

NOC1 + Nhv = NOC1*
NOC1* + NOC1 = 2ND + Cl«

where the activated molecule is denoted by NOC1*.

Ammonia, acetaldehyde, nitrogen dioxide, ozone, sulfur
dioxide, and other substances may be decomposed by light.
The quantum yield in these reactions, as in others where it is not
unity, depends on the thermal reactions that are subsequent
to the primary process.

Dissociation of Iodine Vapor. — The formation of normal
atoms from molecules of iodine vapor is attended by the absorp-
tion of 34,500 cal., as has been said above, and the Nhv value

1 KISTIAKOWSKY, /. Am. Chem. Soc., 52, 102 (1930).

512 PHYSICAL CHEMISTRY

calculated from this corresponds to a wave length in the red just
beyond visibility. Iodine vapor absorbs throughout the visible
range, but the longest wave length capable of decomposing iodine
vapor is about 5000A, corresponding to Nhv = 57,000 cal. The
suggested explanation is that the products of the dissociation
are a normal iodine atom and an "excited" atom of greater
energy content. If this excited atom is marked I*, the primary
process is

I2 + Nhv - I + I*

This "explanation" would not be very satisfactory if other
evidence were not available (from spectroscopic data) with
which to confirm it. The energy required for excitation of the
atom has been calculated1 at 21,600 cal., which is not far from
the difference between the two energy effects just given. Energy
equations for the separate effects will make this clearer.

I2 = I + I* Nhv = 56,800 cal absorbed

I* = I Nhv = 21,600 cal. emitted

1 2 = I + I A// = 35,200 cal. absorbed

The difference between this and 34,500 cal. is small enough to
indicate that the suggestion of excited atoms is near the truth,
for there is some uncertainty regarding the accuracy of the
thermal data.

"Chain" Reactions. — This term was first applied by Boden-
stein2 to interpret the fact that in many photochemical processes
the number of reacting molecules is much larger than the number
of absorbed quanta It is presumed that the quantum initiates
a series of reactions' which follow one another in such a way that
a very reactive intermediate substance is regenerated by a suc-
ceeding step. This reactive substance may be a free atom, a
free radical, or a highly energized molecule that is regenerated
again and again as the series of reactions proceeds. Conse-
quently, it is possible that the occurrence of one elementary reac-
tion will initiate a whole series of such reactions, proceeding until
the reactants are exhausted or until something breaks up the
chain of activations. This "something" may be the absorption

1 KUHN, Naturwtssenschaften, 14, 600 (1926).

2 Z. physik. Chem., 86, 329 (1913). For a review of the whole topic of
chain reactions, see Bodenstein, Chem. Rev., 7, 215 (1930).

RADIATION AND CHEMICAL CHANGE 513

of the activating energy by inhibitors, collisions with inert
molecules present or with the walls of the vessel, which dissipate
the energy among several molecules, or other causes. The com-
bination of hydrogen with chlorine is a well-known illustration.
Since energy is evolved in the synthesis, it is difficult to see why
the reaction series, once it has been started by a quantum, should
stop before the reactants are exhausted. But there is the experi-
mental fact that about a million molecules react per quantum,1
which indicates that the " chain" is brokeji after a certain length.
Two types of " chains" are described,2 which differ in the
mechanism of the series reactions. A " matter chain" consists
in the formation, again and again, of highly reactive intermediate
products, such as free hydrogen atoms or free chlorine atoms,
which perpetuate the reaction. For example, in the series

C12 + Nhv = 2C1
Cl + H2 = HC1 + H
H + C12 = HC1 + Cl

the second and third reactions may be repeated one after the other
until some disturbing factor "breaks the chain."

Or the series may result from the formation, reaction, and
regeneration of an excited intermediate product, which would
be called an "energy chain" as shown by the equations

C12 + Nhv = C12*

C12* + H2 - 2HC1* or HC1* + HC1
HC1* + C12 = HC1 + C12*

and the activated C12* molecule then repeats the cycle. Other
series of somewhat the same character have been suggested by
different investigators; but the mechanism has not yet been
definitely determined, and no very clear explanation is available
of how the chain is ended after a definite period. But there are
the experimental facts that the hydrogen chloride formed is
proportional to the amount of radiant energy absorbed by the
reacting system and that the reaction ceases while hydrogen
and chlorine remain uncombined unless energy is supplied to the
system.

1 HARRIS, Proc. Nat. Acad. Set., 14, 110 (1928); BODENSTEIN, Trans.
Faraday Soc., 27, 413 (1931).

2 BODENSTEIN, Chem. Rev., 7, 215 (1930).

514 PHYSICAL CHEMISTRY

This chain theory has been applied by Backstrom to explain
the oxidation of 10,000 molecules of benzaldehyde per quantum
when its reaction with oxygen is produced by light of 3660A.
Similarly, 50,000 molecules of sodium sulfite are oxidized per
quantum in the absence of inhibitors. In this latter reaction,
the effective inhibiting action of isopropyl and benzyl alcohols
has been shown1 to consist in breaking up the chain, with the
simultaneous oxidation of the inhibitor to acetaldehyde or
benzaldehyde, which are incapable of carrying on the chain.

Other photochemical reactions are open to the same inter-
pretation. Over 1000 molecules of phosgene are formed per
quantum absorbed by a mixture of carbon monoxide and chlorine,
1,000,000 molecules of benzene are chlorinated per quantum, and
high but less extreme yields result in other halogenations and in
the oxidation of oxalates by halogens.

This theory of chain reactions, originally developed for photo-
chemical processes, has also been applied to explosions and other
processes not dependent on the absorption of radiant energy.
The oxidation of acetylene2 involves the intermediate products
glyoxal, formaldehyde, and formic acid, and, in the presence of
reacting acetylene, formaldehyde reacts with oxygen many
times faster than when alone.

Sensitized Reactions. — It was stated earlier in the chapter
that a photochemical change did not necessarily take place
whenever radiant energy of a sufficiently high frequency was
supplied. The dissociation of hydrogen molecules requires
about 100,000 cal. per mole (as calculated from the data in Table
67 and the van't Hoff equation), and the Nhv equivalent of this
large heat absorption corresponds to a wave length of about
2600A. But hydrogen does not absorb until the extremely
short wave length 1200 A. Thus the absorption of radiant
energy, which is the primary requisite for this photochemical
process, does not take place in hydrogen alone between 2600
and 1200 A. An absorber capable of accepting the radiant
energy and delivering it to hydrogen molecules is required, and a
reaction produced by means of an absorber that is not con-
sumed is called a sensitized reaction.

1 ALYEA and BACKSTBOM, J. Am. Chem. Soc., 51, 90 (1929).

2 SPENCE and KISTIAKOWSKY. ibid.. 52. 4837 (1930}.

RADIATION AND CHEMICAL CHANGE 515

Mercury vapor absorbs radiation of 2536A, and Nhv equiva-
lent to this wave length is about 112,000 cal. When a vessel
containing both hydrogen and mercury vapor is illuminated
with light of 2536A, chemical effects are observed that indicate
the formation of atomic hydrogen, and in similar experiments
without the presence of mercury vapor no such chemical effects
occur. Thus, in the presence of tungstic oxide, this substance
is reduced, water is formed, and hydrogen disappears.

Hydrogen and oxygen form water and hydrogen peroxide
when illuminated with light of 2536A in the presence of mercury
vapor and do not form them in its absence. Similarly, ethane
forms photochemically from ethylene arid hydrogen, when
" sensitized" by mercury vapor. Other reactions are sensitized
by mercury vapor, and other instances of photosensitization are
known. For example, chlorine may act as a sensitizer in the
decomposition of ozone or of chlorine monoxide and in other
reactions; the photochemical decomposition of colorless N2O6
is sensitized1 by the brown NO2, etc.

The two most important photochemical reactions known occur
in heterogeneous media and are too complicated for a first discus-
sion; they are the change of silver halide on a photographic
plate and the reaction of water with carbon dioxide in plants.
Authorities by no means agree on the mechanism or quantum
yield involved in the reactions on a photographic plate, and a
study of the published work demonstrates the extreme difficulty
of interpreting the results of seemingly simple experiments.
Even the nature of the reaction products is still somewhat in
doubt. Of the complex processes that take place in living
plants, whereby sugars, cellulose, and the most varied substances
are built up from water and carbon dioxide with the absorption
of sunlight, even less can be said. Until much more is known
of the simpler reactions, it is hardly to be expected that a fair
understanding of plant photochemistry will be developed.

The examples of photochemical change already mentioned
form only a very small portion of the total already known, and
the investigation of light-sensitive chemical reactions has just
begun. These reactions, as has been said before, are not illustra-
tions of the catalytic effect of light; rather, they show the energiz-
ing of molecules by radiation. As our knowledge increases

1 BAXTER and DICKINSON, /. Am. Chem. Soc., 61, 109 (1929).

516 PHYSICAL CHEMISTRY

and as experimental skill develops through experience, it may be
expected that reactions so produced or controlled will be of
greater and greater importance.

References

The literature of photochemistry up to 1939 is summarized so completely
in "Photochemistry and the Mechanism of Chemical Reactions" by Rollef-
son and Burton that no other source is needed. This excellent book is
suggested for further reading on the topic.

CHAPTER XIV
PERIODIC LAW OF THE ELEMENTS

Mendelejeff's periodic law states that there is a periodic recur-
rence in properties of the elements when they are arranged in
the order of increasing atomic weights. In the few instances in
which recurrence came in the seventh element in place of the
eighth, MendelejefT rightly concluded that there was a missing
element yet to be discovered, and he predicted with reasonable
accuracy the properties that some of these elements were to
possess when discovered. By writing the elements in eight
columns, in the order of increasing atomic weight, and by leaving
blanks where the existence of a new element w^as indicated, he
obtained the periodic table, of which a common version is given
in Table 85. Two other versions are given in Tables 86 and 87.

The fact that the atomic weights of many of the elements are
"almost" whole multiples of that of hydrogen suggested to
Prout in 1815 that elements had structures and that all of them
might be built from hydrogen; but the fact that the atomic
weights of magnesium (24.32), chlorine (35.46), and some others
were definitely not " almost " whole multiples of hydrogen seemed
to discredit the assumption, and it was abandoned. The periodic
table also suggested that atoms were " built up" in some way;
the radioactive changes described in the next chapter furnished
another clue to the structure of atoms and a vast bulk of evidence
was soon to follow. After brief consideration of these two topics,
we shall return to the topic of atomic structure.

The " zero-group" elements were all unknown at the time the
periodic table appeared, and the column for these elements has
since been added to the table given on page 518.

In this arrangement, as in any form of the periodic table,
three " irregularities " appear in the atomic-weight order; argon
has a higher atomic weight than potassium, cobalt a higher one
than nickel, and tellurium a higher one than iodine ; errors in the
atomic weights large enough to bring these elements into a weight

517

518

PHYSICAL CHEMISTRY

KS
fcS

000

a-**

pqo>

• "»<^

So !-«

Q
O

ooo

U5T-I

s
a

PQOO
100

. T

JHO rH

ss
as

£\

,«

* t-

PERIODIC LAW OF THE ELEMENTS 519

order are quite out of the question. These "irregularities"
disappear when the elements are arranged in the order of increas-
ing net charge on the nucleus, which is the atomic number order,
as we shall see later in this chapter; but there is still no explana-
tion of why the weight order is valid in all but three instances or
why those "out o£ order " should all be out by only one
place.

In spite of certain peculiarities, due to our incomplete knowl-
edge of the fundamental law that the present arrangement
partly expresses, the periodic law is the most important generali-
zation in inorganic chemistry. Much study has been given the
elements in order to discover the full significance of this periodic-
ity, and some variations of the periodic table will be given later
in the chapter. In all these arrangements the periods contain,
respectively, 2, 8, 8, 18, 18, and 32 elements, with a final incom-
plete period of which only 5 elements are known.

The first period contains hydrogen and helium only, and there
is abundant evidence (some of which will be given presently and
more in Chap. XVI) which makes it very unlikely indeed that
there are missing elements between them. Two "short periods "
of eight elements follow helium — lithium to neon and sodium
to argon — with quite definite recurrence of physical and chemical
properties in each group or column.

The next two periods, beginning with potassium and rubidium,
contain 18 elements each and are usually called "long periods."
They include the groups Fe, Co, Ni and Ru, Rh, Pd, which are
placed together in a single column, the significance of which is
not well understood.

A certain artificiality appears in this pressing of 18 elements
into groups of 8 which is not wholly satisfactory, but other
arrangements are available in which this is avoided. Bohr's
table uses 2, 8, 8, 18, 18, 32, and 5 elements per "period," and
von Antropoff subdivides the periods into two portions. Other
expedients, among which it is difficult to choose, have also been
tried.

Two more "long periods," the second definitely incomplete,
include the remaining elements. The sixth period, beginning
with cesium, is broken by the intrusion of the rare earths, and
it contains no halogen heavier than iodine. A seventh period
contains only 5 elements in place of the expected 32 to match the

520

PHYSICAL CHEMISTRY

preceding period, but there is yet no evidence that 27 other
natural elements remain to be discovered.

Table 85 contains the rare gases ; several rare earths, the
radioactive isotopes, and the elements rhenium, masurium, gal-
lium, scandium, and germanium, which were unknown to the
discoverer of the periodic law, though he correctly predicted the
general properties that some of these elements would have when
discovered.

In the arrangement in Table 85, 14 rare earths occupy the
place between barium and hafnium. These elements are not
isotopes; they are elements of slightly but distinctly different
properties and different* atomic weights. They are as much
entitled to separate positions as chlorine and bromine, and in
some of the more complicated periodic arrangements they have
separate places. The same difficulty is encountered in \ on Antro-
pofFs arrangement, and in Bohr's arrangement a " period " of
32 elements results from giving them separate places. There
is good evidence from spectroscopy that this is not merely an
expedient for finding them places; it has to do with the energy
levels of electrons in the atom.

Atomic Numbers. — The order number in which elements

appear in the periodic table is called the atomic number; it is

^ also the net positive charge on

the atomic nucleus. The ex-
periments of Moseley,1 in which
elements or their compounds
were bombarded with electrons
of sufficiently high velocity,
showed definitely that the
atomic number is a fundamental
quantity. Under this bombard-
ment the elements emit X rays
of characteristic wave length in
addition to general X radiation. These X-ray spectrum lines
are as characteristic of the elements as are the flame colors
that identify some of them, such as yellow for sodium; and
the X-ray spectra are simpler than the visible spectra. Like
these colors, the X rays consist of more than one "series"
of lines. When the square root of the frequency in a given
1 MOSELEY, Phil. Mag., 26, 1024 (1913), 27, 703 (1914).

I > 0 4 8 12 16 20 24 28 32 36 40
l^ Z= Atomic Number

FIG. 66. — Linear relation of atomic
number to square root of character-
istic X-ray frequency.

PERIODIC LAW OF THE ELEMENTS

521

series is plotted against the atomic number, a straight line
is obtained, as shown in Fig. 66. Such a plot brings potassium,
cobalt, and iodine in the order in which they should appear in
the periodic table, as the weight order does not.

TABLE 86 — BOHR'S PERIODIC -TABLE OF THE ELEMENTS

Period Period
VI VII

55Cs 87—

56Ba 88Ra

89Ac
90Th
91Pa
92U

4J.OU

22Ti
23V
24Cr
25Mn
26Fe
27Co
28Ni

09 X

40Zr
4ICb
42Mo
43Ma
44Ru
45Rh
46Pd

— •—

47Ag

48Cd

49In

50Sn

51Sb

52Te

531

54Xe

58Ce

59Pr

60Nd

61U

62Sa

63Eu

64Gd

65Tb

66Dy

67Ho

68Er

69Tu

70Yb

71Lu

72Hf

73Ta

74W

75Ro

760a

77Ir

78Pt

79Au
80Hg

vxxxv 81TI
WA\82Pb

When the characteristic frequencies of the L series are used
in place of those of the K series, another straight line of different
slope is obtained, but the order number of the elements is the
same. There is other evidence that the atomic number is the
correct order to use in arranging the elements.

522 PHYSICAL CHEMISTRY

The relation of frequency to atomic number, which is known
as Moseley's law, is

v = a(Z - 6)2

in which a and b are constants for a given series of lines and Z is
the atomic number. For the~Ka series, for example,

v = 0.248 X 1016(Z - I)2

Bohr's Arrangement of the Elements. — In this scheme the
emphasis on eight columns is abandoned, and the periods con-
tain 2, 8, 8, 18, 18, 32, and 5 elements, as shown in Table 86.
Hydrogen and helium constitute the " first period/' and the
other periods begin and end with the same elements as in Table
85. A systematic increase would lead one to expect 32 elements
in the seventh period, but there is as yet no evidence that so
many unknown elements exist. The reasons for this arrange-
ment will be better understood after reading the chapter on
atomic structure, but its general relation to other periodic
tabulations will be evident from an examination of the table.
It does not explain the tellurium-iodine and similar irregulari-
ties in mass; it. groups the rare earths together, as does Table 85,
but it does show better than the other arrangements the relation
of atomic number to the arrangement of electrons in the atoms.

von Antropoff's Periodic Table. — Another interesting arrange-
ment of the periodic table has been devised by von Aptropoff,1
in which the left-hand and right-hand portions of each group
are listed separately after the third period. This arrangement
is shown in Table 87. The transitions, which are indicated by
arrows for the first-and fifth groups only, will be obvious in the
other cases from a study of the table. In common with the
other arrangements, it has nothing satisfactory with which to
replace the crowding of rare earths into a single position, but it
does eliminate the appearance of gaps when no elements are
missing. The periods contain 2, 8, 8, 18, 18, 32, and 5 elements
as before, and, of course, they begin and end with the same
elements.

Many other attempts to prepare periodic tables have been
made, by the use of plane diagrams, solid figures such as spirals,

* £, angew. Chem,} $9, 722 (1926),

ATOMIC STRUCTURE 539

Although it seems impossible at first thought that any knowledge
of the structure of a particle of this size could exist, yet the
technique of modern physics and its attending theory have led
to assumed structures which are in accord with practically all
the experimental data.

It was stated in an earlier chapter that light has certain
properties, such as interference, which are best explained by
assuming it to possess wavelike characteristics and has other
properties which seem to indicate that it is corpuscular. It is
even more difficult to understand how such " particles " as atoms
can show interference and have wavelike properties, as well as
kinetic energies; yet this appears to be true from experiments
on the interference of "rays" consisting of atomic "particles"
impinging on a grating.

We shall see below that an atom probably consists of a positive
nucleus which is not over 10~12 cm. in diameter, surrounded by an
"atmosphere" of electrons within a radius of 10~8 cm. of the
nucleus. Evidence on nuclear structure has been derived from
radioactivity or from experiments in which the nucleus is shat-
tered with explosive violence, and the disintegration products are
inferred from their penetration of air or other matter. Such
experiments cannot show how the constituents were arranged or
bound together before the shattering took place, any more than
the distribution and range of debris from the explosion of a larger
object could show its original structure. But this work does
show the units of which the nucleus was composed, insofar as
these survive the atomic explosion. One must be constantly on
guard not to mistake interpretation for experimentation; for
interpretation involves a hazardous completion of our understand-
ing that may change decidedly as experimentation proceeds
slowly but positively to establish unchanging facts.

Early Speculations. — The fact that so many of the atomic
weights are nearly whole numbers led Prout to suggest over a
hundred years ago that elements were made up from hydrogen
as a "fundamental" particle. As the atomic weights became
more precisely known, it was found that half of them were not
whole-number multiples of the atomic weight of hydrogen within
0.1 unit, and the hypothesis was abandoned. The periodic
table showed that with progressively increasing mass the chemical
properties of the elements were partly reproduced every eighth

540 PHYSICAL CHEMISTRY

element and that with each increase in mass a change in valence
took place. These facts also indicated that elements were
composed of some fundamental unit. When radioactivity was
shown to be an atomic disintegration and when the products
were shown to be electrons (beta rays) and charged helium atoms
(alpha particles), there could be no doubt that these radioactive
atoms had structures and that electrons and positively charged
masses were involved in them.

Since the atomic weights of many abundant elements are not
multiples of 4, 'the atomic weight of helium, some of the mass
must come from a lighter particle, and it was again suggested1
that the masses of light elements, such as nitrogen, are made up
of helium arid hydrogen, Prout's hypothesis being thus revived in
a modified .form. But atomic weights that were riot multiples
of the atomic weights of hydrogen and helium were an insur-
mountable difficulty for the general application of such a theory
unless one were prepared to discard the conservation of mass or
to accept the possibility that elements consisted of atoms which
were not of the same mass though identical in chemical properties.
The periodic table showed that increase in mass was attended
by a change in chemical properties, and loss of mass was so
improbable in the light of all evidence as to be unacceptable.
Here matters rested, awaiting new and fundamental discoveries,
one of which was shortly to be made and to the results of which
we now turn.2

Isotopes. — In the radioactive changes given in a previous
chapter, it was shown that the loss of one alpha particle and 2
electrons by successive reactions formed a new element of the
same atomic number and same chemical properties, occupying
the same place in the periodic table, but four units lighter than
the parent element. These elements were called isotopes of the
parent element, and their existence suggested the possibility
that other elements might consist of isotopes; but since all

1 RUTHERFORD, " Radioactive Substances and Their Transformations,"
p. 621, 1913. In 1919 Rutherford obtained traces of hydrogen by bom-
barding nitrogen (atomic weight 14) with alpha particles, and in similar
experiments upon elements whose atomic weights were multiples of 4 no
hydrogen was obtained. This is an early instance, probably the very first
instance, of atomic transmutation m a laboratory.

2 For this timely discovery F. W. Aston was awarded the Nobel Prize in
Chemistry in 1922.

ATOMIC STRUCTURE 541

attempts to resolve elements into different portions had failed,
it was evident that a method based upon some new principle
was urgently needed.1 In 1919, the Aston "mass spectrograph"
supplied such a method; it showed that some of the elements were
mixtures of atoms of different masses and the approximate (later
the exact) proportions in which these were present in the natural
elements. But the isolation of weighable quantities of these
isotopes was not accomplished by any method until 1934, and
not by the use of this method until 1936 2 The operation of the
mass spectrograph is shown diagrammatically in Fig. 67. Posi-
tive rays from a discharge tube (not shown in the figure) con-

PhofographJc
Si

FIG 67. — Diagram of Aston's positivo-ray spectrograph.

taining the vapor to be investigated are sorted into a thin
ribbon on passing through the parallel slits Si and S% and are
then spread into an electric " spectrum77 by means of the charged
plates PI and P2, of which the latter is negative. A portion of
this spectrum deflected through a given angle is selected by the
diaphragm D and passes between the circular poles of a powerful
electromagnet 0, the field of which is such as to bend the rays
back again through a greater angle than that of the first deflec-
tion. The result of this is that rays having a constant mass (or
more properly a constant ratio m/e of mass to charge) will con-
verge to a focus at F and indicate their position on a photo-
graphic plate placed as shown, giving a " spectrum" dependent
on mass alone. The instrument is called a positive-ray spectrom-
eter, and the spectrum produced is known as a mass spectrum.

1 Aston's first mass spectrograph is described in Phil Mag., 39, 454 (1920) ;
see also F. W. Aston, "Isotopes," Edward Arnold & Co., London, 1922. A
new instrument of high precision is described in Aston, "Mass Spectra and
Isotopes," 2d ed., 1942, which gives also the distribution of the isotopes of
various masses in all of the elements

2 Lithium was separated by Itumbaugh and Haf stead, Phys. Rev , 60, 681
(1936); potassium by Smythe and Hemmendinger, ^b^d., 51, 178 (1937);
rubidium by Hemmendinger and Smythe, ibid., 61, 1052 (1937).

542 PHYSICAL CHEMISTRY

Only relative masses are obtained by this method, but the
scale may be calibrated by introducing a small amount of some
substance of known mass. Oxygen is obviously the most suit-
able reference substance since it forms the basis of the atomic-
weight scale.

A sketch of the mass spectrum for chlorine is shown in Fig.
68. The spots at 28, 32, and 44 correspond to carbon monoxide,
oxygen, and carbon dioxide. It will be seen that the chlorine
mass spectrum consists of four strong lines at 35.0, 36.0, 37.0,
and 38.0; there is no line at 35 46, the accepted atomic weight of
chlorine. The lines at 35.0 and 37.0 are due to chlorine atoms;
the other lines one unit higher are their corresponding HC1
compounds. This is strong evidence that chlorine consists of
two isotopes whose atomic weights are whole numbers on the
oxygen scale. Of course, these two chlorines are chemically

OJ LO

ro ro

• i U M •

AtOtTIIC

FIG. 68 — Sketch of the mass spectrum of chlorine.

identical in every way and inseparable by chemical means, so
that the practical chemistry of chlorine is not disturbed in any
way. Since these atoms have different atomic weights, there
may be three kinds of chlorine molecules of molecular weight
70, 72, and 74. In the current notation, these molecules would be
written C1235, C135C137, and C1237.

In the discharge tube at such low pressures there will be
particles unknown to ordinary chemistry, such as C1+, HC1+,
C12+, Ne+, and the^charged products of dissociation of compounds.

Almost all the elements have now been examined in the
mass spectrograph, and a total of about 280 kinds of atoms com-
prise the 92 elements. Thus mercury has 9 isotopes, lead 4,
and tin 11. Table 92 shows the mass numbers of the atomic
nuclei occurring in nature in a stable state, but it omits radioac-
tive isotopes and the unstable synthetic nuclei that show induced
radioactivity. Brief mention of these synthetic isotopes will be
made later in the chapter.

The most surprising result of work with the high-precision
spectographs later developed is that the atomic masses are not
exactly whole numbers and do not differ by exactly whole num-

ATOMIC STRUCTURE

543

bers, when referred to O16, as might have been expected. Thus
the isotopes of chlorine have masses of 34.9803 and 36.9779 on
this scale. Some other isotopic masses are Shown in Table 93.
Studies with the mass spectrograph have shown that radiogenic
lead consists of isotopes mixed in varying proportion, thus
TABLE 92 — MASS NUMBERS AND ATOMIC NUMBERS OF THE IsoTOPES1

accounting for "the variable atomic weights given in Table 91.
The isotopic constitution of ordinary lead and of specimens of
radiogenic lead (atomic weight 207.85) from thorite and from
pitchblende (atomic weight 206.08) is as follows:

Mass number
Per cent in common lead2
Per cent in 207.85 "lead"
Per cent in 206.08 "lead11

204 206 207 208

13 27 3 20 0 51 4

0 46 1.3 94 1

0 89 9 79 23

1 Rev. Sri. Instruments, 6, 61 (1935).

2 This analysis is by Nier, /. Am. Chem. Soc., 60, 1571 (1938) A search
for isotopes of mass numbers 203, 205, 209, and 210 in lead showed that they
are very rare, if they exist at all. Others give slightly different proportions
of the isotopes; for example, Mattauch, Naturwissenschaften, 25, 763 (1937),

544

PHYSICAL CHEMISTRY

Such figures as these leave us completely in the dark as to the
way ordinary lead from all over the earth came to have the same
atomic weight. It could scarcely be by coincidence, and it
seems improbable now that radioactive end products could have
TABLE 93. — MASS NUMBER AND RELATIVE ABUNDANCE OF SOME ISOTOPES 1

Element

Mass

Relative
abundance

Element

Mass

Relative
abundance

On 1

1 00893

12 Mg 26

25 9898

11 1

1H 1

1 00813

99 98

13 Al 27

26 9899

100

1 H 2

2 01473

0 02

14 Si 28

27 9866

98 6

2 He 4

4 00389

100

14 Si 29

28 9866

6 2

3 Li 6

6 01682

7 5

14 Si 30

29 9832

4 2

3 Li 7

7 01814

92 1

15 P 31

30 9823

4 Be 9

9 01486

99 95

16 S 32

31.9823

97 0

5 B 10

10,01613

16 S 33

0 8

5B 11

11 01292

16 S 34

33 978

2 2

6 C 12

12 00398

99 3

17 Cl 34

33 981

6C 13

13 00761

0 7

17 Cl 35

34.9803

7N 14

14.00750

99.62

17 Cl 37

36.9779

7N 15

15 00489

0 38

17 Cl 38

37 981

8 O 16

16 00000

99.76

19 K 39

93 2

8O 17

17 00450

0 04

19 K 41

6 8

80 18

18 00369

0.20

24 Cr 52

51 948

83 8

9F 19

19 00452

100

28 Ni 58

57 942

10 Ne 20

19 99881

90.00

30 Zn 64

63 937

50.9

10 Ne 21

20 99968

0 27

33 As 75

74 934

100

10 Ne 22

21 99864

9 73

35 Br 79

78 929

50 7

11 Na 23

22 9961

100

35 Br 81

80 930

49 3

12 Mg 24

23 9924

77.4

53 I 127

126 993

100

12 Mg 25

24 9938

11 5

55 Cs 133

132 934

100

been so exactly mixed. Several radiogenic leads appear to
contain only isotopes of masses 206, 207, and 208, which is not
true of common lead.

Other elements have been similarly analyzed. Thus the per
cent of the isotopes of various mass numbers in molybdenum is2

Mass number 92 94 95 96 97 98 100

Per cent 15 5 7 7 16 3 16 8 8 7 25 4 8 6

gives 1.5, 24.55, 21.35, and 52.95 per cent in place of the above figures for
common lead.

1 A full table is given by the Committee on Atoms of the International
Union of Chemistry in /. Chem, Soc. (London), 1940, 1416,

» MATTAUCH, Z. physik, Chem,, 42B, 288 (1939),

ATOMIC STRUCTURE 545

Similar resolutions and " analyses" are available for most of
the elements, but it must be clearly understood that separation
of the element into its isotopes is not accomplished in this
resolution. The percentages are estimated from the intensities
of lines on photographic plates in the mass spectrum.

Some of the elements appear to contain no isotopes; for
example, F7 Na, Al, P, Mn, As, I, Cs, and Au have not yet
been shown to have stable atoms of different masses, though
experiments directed to their discovery have been made. Per-
haps all that can be said safely is that the experimental means
which have shown the existence of isotopes for other elements
have failed to show them for these elements.

When the weight order was not followed in arranging the
elements in the periodic table, it was stated that a reason would
be given for believing the atomic number to be more important.
This reason is evident from Table 92, in which elements of diffei-
ent properties have isotopes of the same mass. Single elements
may have isotopes of several masses, but all ol them have iden-
tical chemical properties and the same atomic number. Different
elements may have atoms of the same mass and different chemical
properties. These nuclei are called isobars, meaning elements
of the same mass and different atomic numbers. If we follow
the usual custom of indicating the atomic number by a subscript
preceding the symbol and the mass number by a superscript
following it, some examples of isobars are isA40, i^K40, 2oCa40;
26Fe67, 27Co67; 5iSb123, 52Te123; and some 60 other pairs besides
additional trios. Since all the isotopes of an element have
the same atomic number, this number is a more suitable quantity
to use in arranging them for chemical properties.

Atomic Weights from the Mass Spectrograph. — Results of
mass-spectrograph experiments of the kind shown in Table 93
should not be compared directly with atomic weights from
chemical analyses such as the entries in Table 4, for the mixture
known as " oxygen," which occurs in nature, is not wholly com-
posed of O16 but contains small quantities of the isotopes O17
and O18. The ratio of the atomic weight of O16 to ordinary
oxygen is 1 : 1.00027, and this correction should be applied before
making comparisons.

Atomic weights measured in the mass spectrograph may reveal
slight errors in the accepted weights based on chemical methods

546 PHYSICAL CHEMISTRY

as, for example, in the atomic weight of cesium, which was given
as 132.81 in the 1933 International Table of Atomic Weights.
Aston1 found no isotope of cesium and, after correcting his work
to the chemical scale by the factor 1.00027, as has been explained
above, suggested that the atomic weight of cesium should be
132.91 in place of 132.81. New experiments2 upon carefully
purified materials gave the ratio CsCl: Ag = 1 : 1.5607, correspond-
ing to an atomic weight of 132.91 for cesium, in confirmation of
the value obtained in the mass spectrograph.

Isotopes from Band Spectra. — It will be clear that the moment
of inertia of a molecule composed of HC135 would not be the same
as that of a molecule of HC137 Since band spectra are associated
with vibrations within the molecules and rotations of molecules,
the existence of isotopes may be shown from spectroscopic data,
and some indication of their relative abundance may also be
found in this way.3 Isotopes O18, O17, and N15 have been identi-
fied from band spectra. The fact is of interest as confirmation
of the existence of isotopes and as a means of finding new ones.
It will be noted that several kinds of nitric oxide may result from
these isotopes, of which N14016, N15016, N14018, and N14017 have
been indicated.

Separation of Isotopes.4 — From the first discovery of isotopes,
research has been directed toward means of separating an element
into its constituents of different mass, and fractionation into
portions of slightly different combining weight were early reported
for chlorine, mercury, and a few other elements The first com-
pletely successful preparation of a pure isotope was that of H2 or
deuterium,5 for which the symbol D is now in common use.

:Proc. Roy Soc (London), (A) 143, 573 (1932).

2 BAXTER and THOMAS, J.'Am. Chem. Soc., 56, 1108 (1934).

3 See JEVONS, "Report on Band Spectra of Diatomic Molecules."

4 See Aston, op. cit., for a full account of discoveries up to 1942 in this field,
for more recent work see Chaps. IX, X, and XI of " Atomic Energy for
Military Purposes" by H. D. Smyth (Princeton University Press, 1945)

6 The history of this discovery, for which the Nobel Prize was awarded to
Dr. H. C. Urey, is given by Urey and Teal in Rev. Modern Phys., 7, 34 (1935).
Concentration of the heavy isotope by fractional distillation of liquid hydro-
gen gave the first indication that successful isolation of it in a pure state
might be possible, but its isolation as nearly pure deuterium oxide (some-
what inaptly called "heavy water") was accomplished by Washburn, who
electrolyzed large quantities of water and obtained D2O from the last por-

ATOMIC STRUCTURE 547

These experiments showed that hydrogen contains about 99.98
per cent of atoms of weight 1.0081 and only about 0.02 per cent
of deuterium atoms of weight 2.0147. They do not show that
hydrogen contains 0.8 per cent of the heavier element and that
the lighter one is of mass 1.000, and thus they do not explain the
mass changes that must be assumed if the atoms of other ele-
ments are made up of hydrogen nuclei or protons. (This
"mass defect" will be discussed later in the chapter.) But
the experiments confirm the results of the mass spectrograph in
showing that natural elements are mixtures of particles of differ-
ent masses and identical chemical properties.

Complete separations have been accomplished for lithium/
neon/ rubidium, potassium, and chlorine;3 nearly complete
separation of some other elements has also been attained, and,
of course, the attempts are still being actively conducted. The
chief methods are electrolysis, centrifuging, the mass spectro-
graph, fractional distillation, and gaseous diffusion.

Two minor facts will illustrate the very slight chance of
separating isotopes except in experiments designed for the
purpose.4 (1) The residual brine in an electrolytic cell to which
KC1 and water had been added to produce KClOa for 30 yeais
without refilling showed an apparent separation of the isotopes
of chlorine about equal to the error of the experimental method,
which was 0.01 per cent. (2) The residual chlorine in a still
through which 2700 tons of liquid chlorine had been passed
showed possible increase of 0.1 per cent of Cl37 at the most.

Isotopes and the Law of Definite Proportions. — Experiments
quoted in the previous chapter have shown that the combining

tion of the residue. In the reference given above, the work bearing upon
deuterium through the end of 1934 is reviewed (279 papers). Later work
on iso to pic separation is given by Urey in Pub. Am. Assoc. Advancement Sci.,
No. 7, 73 (1939).

1 OLIPHANT, SHIRE, and CROWTHER, Proc. Roy. Soc. (London), (A) 146,
922 (1934).

2 HARMSEN, Z. Physik., 82, 589 (1933) (by using a high-intensity mass
spectrograph); HERTZ, ^b^d., 91, 810 (1934) (by diffusion against mercury
vapor).

* Hirschbold-Wittner, Z. anorg. allgem. Chem., 242, 222 (1939), using the
thermal-diffusion method of Clusius and Dickel, Natnrwissenschaften, 26,
546 (1938).

*Helv. Chim. Acta, 22, 805 (1939), through Chem. Abst., 33, 8064 (1939).

548 PHYSICAL CHEMISTRY

weight of lead from radioactive decomposition is not the same as
that of ordinary lead. Thus, lead bromide may contain a
variable proportion of "lead," depending on the source from
which it was derived. This constitutes a real exception to the
law of definite proportions, though one of no veiy great practical
importance in view ol the scarcity of radiogenic lead.

Since the atomic weight of deuterium is twice that of hydrogen,
the fraction of oxygen in "water" will vary from about l%& to
about ^GJ depending on the ratio of H1 to H2 (or of H to D) in
the specimen. So long as we leave hydrogen (or the other ele-
ments) in the state in which nature made them, the law of
definite proportions stands as a useful general law of chemistrtt
But it will be imperative to clarify our nomenclature with respect
to the products of isotopic separation as, for example, by reference
to the oxide Li6Li7016 rather than to "lithium oxide," which was
adequate until isotopic separations were accomplished.

Models of Atomic Structure. — All the isotopes carry positive
charges in the mass spectrograph, as do the mass-bearing products
of radioactive change when they are expelled. Since atoms as a
whole are not electrically charged, it follows that there must be
an equal number of positive and negative charges in the atom
structure. The experiments discussed so far do not show whether
the positive electricity is on the outside of the atom and the nega-
tive electricity within it or whether the positive electricity is
concentrated in the interior of the atom and the negative electric
charges are on the outside. The latter arrangement is now con-
sidered to be the correct one, and several "models" or proposals
for discussion have been suggested, of which those by'Rutherford
and Bohr are discussed briefly in this chapter.
j Thomson Atom Model. — This model, which was proposed as a
working hypothesis by Sir J. J. Thomson prior to 1907, assumed
that the atom was a sphere over which the positive charge was
uniformly distributed and within which the electrons were
symmetrically arranged. Experiments on the scattering of alpha
particles by thin metal foil could not be explained by a dis-
tribution of the positive charge over a sphere of radius 10~8
cm. as assumed in the Thomson model, and it was discarded.
It is of interest only as the first clearly described model to be
suggested.

ATOMIC STRUCTURE 549

N/Scattering of Alpha Particles by Matter.1 — When a beam of
swiftly moving alpha particles, or charged helium atoms, is made
to fall on thin gold foil, most of the particles pass through it,
showing that the greater part of the space within the gold is
" empty" or that the mass is concentrated in a very small por-
tion of the total volume. But while nearly all the particles pass
through or are slightly deflected, an occasional particle is deflected
through an angle greater than a right angle, presumably because
of having entered into the very core of an atom and there encoun-
tered an intense electric field. In order to account for the
intensity of this field it is necessary to suppose that the positive
electricity is concentrated within a region less than 10~12 cm.
in diameter. This led Sir Ernest Rutherford2 to propose the
model that forms the basis of the atomic structure now considered
most probable.

^Rutherford Atom Model. — It is now commonly accepted that
an atom consists of a small nucleus with which are associated the
mass of the atom and the positive charges; that this nucleus is at,
or very near, the center of the space available for the whole atom;
and that the exterior portion *of this space contains the negative
electrons. In Rutherford's model, it was assumed that the
electrons form the outer layer of the atom. The material in
the following pages relates (1) to the structure of this inner mass
nucleus; (2) to the number and arrangement or behavior of the
outer electrons, and the relation of this arrangement to chemical
behavior; or, alternatively, to the rotation of electrons about the
nucleus in orbits of different energy levels, and the relation of
this to atomic spectra.

J Nuclear Charge and Atomic Number. — If we define the atomic
number of an element as the number of positive charges on its
'niideuSj as determined in experiments on the scattering of alpha
particles, the same order is obtained as in the periodic system.
There is a simple relation between atomic number and the
frequency of characteristic X-ray spectra, as determined by
Moseley's experiments mentioned in the previous chapter. If

1 GEIGER and M ARSDEN, Proc. Roy. Soc. (London), (A) 82, 495 (1909) ,
Phil. Mag., 21, 669 (1911); 27, 488 (1914); see also RUTHERFORD, Proc.
Roy. Soc. (London), (A) 97, 378 (1920).

*PhiL Mag., 21, 669 (1911), 26, 702 (1913), 27, 488 (1914).

550 PHYSICAL CHEMISTRY

Z is the atomic number, which is the magnitude of the positive
charge on the nucleus of an atom, and v is the characteristic
X-ray frequency, this relation is

v = a(Z - fc)2

where a and b are constants. The elements when arranged
according to the atomic numbers fall inter their proper places
in the periodic table. Hence the atomic number of an element
is a more fundamental property than its atomic weight.
>^ Structure of Atomic Nuclei.1 — The early experiments of
Rutherford, in which hydrogen was obtained from nitrogen by
bombardment with alpha particles, as well as natural radioactive
decompositions that expel alpha particles, seemed to indicate
that hydrogen and helium nuclei were the constituents of atomic
nuclei responsible for the mass of these atoms. Among the
abundant elements, carbon, oxygen, silicon, and calcium have
atomic weights that are very close to multiples of 4, and alumi-
num and silicon have atomic weights that are nearly whole
numbers, not divisible by 4. It seemed reasonable to assign the
structure 3a to the carbon nucleus, 4a to oxygen, 7 a to silicon,
and lOo: to calcium. Nitrogen was assigned the structure
3a + 2H; and there was the possibility that helium itself might
be 4H, with the mass defects not explained Nuclear structures
such as the last two indicate more positive charges on the nucleus
than corresponded to the number of external electrons, and there-
fore " nuclear electrons" in sufficient number to make the atoms
neutral were also assumed. There were serious difficulties in
explaining the stability of a nucleus containing electrons to which
no one was blind but from which there was no evident escape at
the time. With the discovery of the neutron,2 a particle with
the mass of a hydrogen atom and no electric charge, these diffi-
culties vanished, and a more reasonable theory of nuclear struc-
ture became available.3

1 For an excellent discussion of the material presented so briefly in this
section, see Richtmyer and Kennard, "Introduction to Modern Physics,"
Chap. XI, McGraw-Hill Book Company, Inc., New York, 1942.

2 CHADWICK, CONSTABLE, and POLLAED, Proc. Roy Soc. (London), 130,
463, (1931); see also CHADWICK, ibid., 136, 692 (1932).

3 No more striking illustration could be found of the changing interpre-
tation required by additional experimentation than the radical revision of
ideas of nuclear structure that followed the discovery of the neutron.

ATOMIC STRUCTURE 551

If we denote a hydrogen nucleus or proton by p, a neutron by
w, and an electron by e, the nuclear structures already given
become 2p + 2n for He, 6p + 6n for C, 7p + 7n for N, 8p + 8n
for 0, etc., and the electron is not required in any nucleus. The
atomic structures are 2p + 2n + 2e for He, 6p + 6n + Ge for C,
etc. In general, an atom of atomic number Z has Z protons in
the nucleus arid enough neutrons to supply the remainder
of the mass, with Z electrons outside the nucleus Thus C13&
is supposed to be 17 p + ISn + 17?, and Cl37 is I7p + 20n + I7e.

Of course, the carbon nucleus may contain 3a rather than
Op + 6n ; but, since one of the nuclear reactions to be given in
a later section synthesizes alpha particles by a reaction that we
shall write Li7 + H1 = 2He4, it is unnecessary and possibly mis-
leading to make this assumption. On the other hand, there is the
fact that the decompositions of naturally radioactive elements
expel alpha particles and never protons; and this seems to indicate
that the alpha particle is a constituent of these elements stable
enough to survive the violent atomic explosion which expels it
from the nucleus. If an alpha particle consists of two protons
and two neutrons, the decrease in mass attending its formation
(0.03 gram per mole) indicates that the energy necessary for
decomposing an alpha particle is 28 X 106 electron volts

The "Packing Effect" or Mass Defect. — There are some
mass discrepancies in these assumed constitutions that are of
the greatest importance and some others that are only apparent
mass discrepancies. If we consider the helium nucleus first,
its formation may be indicated by 2n + 2p = a, but the mass
2n + 2p exceeds the atomic weight of helium by about 0.034,
which is at least a hundred times the error of the atomic-weight
^determinations. According to an important equation of the
theory of relativity, mass is convertible into energy, and the ergs
obtained by the conversion of m grams of mass into energy is me2,
where c is the velocity of light in centimeters per second. Hence
0.034 (3 X 1010)2 c.g.s. units of energy, or 7 X 1011 cal., should
be evolved by this synthesis, and this quantity of energy would
be absorbed in decomposing 4 grams of helium into neutrons and
protons. If these statements are accepted, it is easy to under-
stand that helium nuclei are very stable indeed and that it will
be difficult to decompose them. While this has never been
accomplished in the laboratory, the synthesis of helium nuclei

552 PHYSICAL CHEMISTRY

from lithium and hydrogen leaves no doubt that the helium
nucleus is not a "fundamental" particle but only one of excep-
tional stability Moreover, the decrease in mass attending this
synthesis explains quantitatively the energy of the new products
formed and thus confirms the belief that mass is converted into
energy in these atomic reactions

The decrease in mass that attends the formation of helium
from two protons and two neutrons is called the mass defect, or
the binding energy. A similar mass defect could be computed
for the nitrogen nucleus or for any nucleus; but, since the stand-
ard reference mass is the O1G isotope of oxygen, a slightly different
procedure is usually followed in computing the mass changes
It is assumed in calculating mass defects that the O16 nucleus
contains 8 neutrons and 8 protons, each of ] ^ G the mass of 01(),
and the fractional decrease in mass that results from the union
of these fictitious particles is recorded as the " packing fraction "
This has the advantage of retaining the same mass standard
that is used in the mass spectrograph and for atomic weights,
but it gives the largest packing fraction to II1, which contains
only one proton or hydrogen nucleus, and leads to negative pack-
ing fractions for certain elements. There is no known particle
of the exact mass used as the basis of the packing fraction ; the
closest approach to it is the hydrogen nucleus of mass 1.0081.

Nuclear Reactions. — In addition to the formation of hydrogen
from nitrogen by bombardment with alpha particles,1 in which
the projectiles came from a natural source, there are many reac-
tions in which nuclei are synthesized or shattered by particles
accelerated to suitable velocities in the laboratory. A cyclotron is
one of the instruments for providing high-velocity particles for
this purpose. The nature of the particles formed in these reac-
tions is usually inferred from their penetration of air or other
matter, since the quantities produced are usually too small for
chemical identification.

Some elements from which protons hav been derived by atomic
shattering through their use as targets for alpha particles are B10,
N14, F,19 Na23, Al27, P31, and Mg26, with Ne, S, Cl, A, and K
doubtful. Neutrons have been derived from the alpha-particle
bombardment of Li7, Be9, B10, B11, N14, F19, Na23, Mg24, Al27,

1 RUTHERFORD, Phil. Mag., 37, 571 (1919) Science, 60, 467 (1919); Proc.
Moy. Soc. (London), (A) 97, 374 (1920).

ATOMIC STRUCTURE 553

P31, and others. When neutrons are produced by the bombard-
ment of atoms with high-velocity alpha particles from the dis-
integration of polonium, the processes are atomic transmutations
that may be shown by equations such as1

Li7 + He4 = B10 + n1
Be1' + He4 = C12 + n1

in which the small decreases in mass (the masses of isotopes are
not quite whole numbers) account for extremely high energies
of the neutrons formed Neutrons have also been produced
by the impact of deuterons (H2 nuclei of unit charge) upon metal
targets and from other reactions.

One transmutation that seems to prove beyond doubt that
the helium nucleus contains protons is the nuclear reaction2

Li7 + H1 = 2He4

The mass decrease in this reaction is about 0.018, which should
(and did) give the alpha particles energy corresponding to over
8,000,000 electron volts, whereas the energy oi the bombarding
particles was less than 1,000,000 electron volts.3 Since the
bombarding protons here concerned were energized in the
laboratory, this reaction constitutes atomic transmutation wholly
by laboratory means.

Other nuclear reactions consist in adding neutrons to existing
nuclei with no change in atomic number, ejecting neutrons from
stable nuclei by gamma rays, adding protons to nuclei with an
increase in atomic number, and proton emission by neutron
bombardment. One typical example of each reaction is given
for illustration, but many other examples are well known. In
each equation the subscript preceding the symbol is the atomic
number, and the superscript is the mass number of the nucleus.

nNa23 4- on1 = uNa24 (1)

801G + 7 = 8016 + on1 (2)

6C12 + xH1 = 7N» (3)

i2Mg24 + on1 = nNa" + iH* (4)

1 CHADWICK, ibid, (A) 142, 1 (1933).

2 COCKCROFT and WALTON, ibid f (A] 136, 619 (1932).

8 The energy that a particle of unit charge would acquire by falling
through a field of 1,000,000 volts is equivalent to the disappearance of
0,001074 mass unit, or about 1,500,000 ergs for 6 X 1023 particles.

554 PHYSICAL CHEMISTRY

These reactions produce unstable nuclei that decompose at
characteristic rates. They are artificial radioactive elements,
but their important feature for this discussion is that they prove
the presence of neutrons and protons in atomic nuclei. The
complexity of the field is made evident by the fact that over 350
artificial nuclei, not known to exist in nature, have been added
to the natural atomic nuclei, of which there are about 280.

The nuclear "chain reactions/' which have recently attracted
so much attention, form more than one neutron in a nuclear
reaction initiated by one neutron, and some of them have com-
paratively large conversions of mass into energy An illustra-
tion of such a reaction is

92U235 + o^1 = r,6Ba140 + 36Kr93 + Son1

The actual reaction is much more complicated than this in
that fission products other than barium and krypton may form
and the number of neutrons may not be 3, but the simplified
illustration will show the principle of self-perpetuating nuclear
reactions. If the reaction can be so arranged that the efficiency
of the generated neutrons in continuing the initial reaction
(assuming 3 to be the number formed) is higher than one-third,
the reaction "builds up," and an explosive reaction may result
If the efficiency is less than one-third, the reaction stops when
the supply of initiating electrons stops.

Since uranium has the highest atomic number (92) of any
natural element now known, an artificial nucleus of higher
atomic number is called a " trans-uranium " element. The
unstable artificial element 92U239, which is formed by the reaction

U238 + on1 - 92U2

gives off electrons in its decomposition and leads to trans-
uranium elements as shown by the reaction?

92U239 = 93Np239 + e-
93Np239 = 94Pu239 + e-

These reactions are similar to that given above for the formation
of nNa24.

The possibility of a nuclear reaction arises whenever protons,
neutrons, electrons, or alpha particles strike a nucleus with
sufficient velocity to overcome the repulsive forces. Either

ATOMIC STRUCTURE 555

shattering of a nucleus or synthesis may result. Some of the
nuclei so produced are identical with natural nuclei, but about
350 nuclei not known to exist in nature have also been synthesized.
Artificial nuclei may be "stable " or radioactive, with the emission
of electrons or positrons Element 85, which is not known to
exist in nature, has been synthesized, and it is the only " artificial "
nucleus yet made that decomposes with the expulsion of an
alpha particle. Polonium has been synthesized from bismuth
and neutrons; of course, it gives off alpha particles, as does
"natural" polonium.

Artificially radioactive elements may be mixed with their
naturally occurring isotopes and made into compounds in which
they still retain their radioactivity. Radioactive iron, iodine,
carbon, sulfur, and other elements have long been used as tracers
in studying animal metabolism; sodium, phosphorus, bromine,
and other tracer elements have been used in plant metabolism;
still others have been used in radiotherapy; and the possibility
of other uses is fascinating.

vNumber and "Arrangement" of Electrons in Atoms. — The
atomic number of an element is the net positive charge on its
nucleus and hence also the number of electrons in the space sur-
rounding the nucleus Since the radius of the nucleus is of the
order 10~12 cm. and the minimum distance between atomic cen-
ters is about 10~8 cm , the volume available for the electrons is
large relative to the volume of the nucleus. Interpretations
based on spectroscopy seem to require orbits of different energy
levels in which the electrons revolve about the nucleus. Such a
picture is not well adapted to chemical interpretations, and for
this purpose the electrons are treated as if they were in shells
tor layers of different quantum levels with "positions," which
means average densities higher in some parts of the orbits than
in others, for a reason that will presently appear. We consider
the spectroscopic model first.

^/Bohr's Atom Model. — In order to explain the spectra of the
elements, Bohr assumes orbits for the electrons, with radii
restricted to certain discrete values, and that while revolving
in these orbits the electrons do not radiate. An electron revolv-
ing in any one of these orbits is in a "stationary state"; i.e.,
it possesses an integral number of quanta of energy. This is
contrary to the classical electrodynamics, and there is experi-

556

PHYSICAL CHEMISTRY

mental evidence that these laws are not applicable to atomic
systems in these circumstances. The picture of this atom model
that we shall present here is oversimplified in that only the
" principal" quantum numbers are considered, but it is probably
sufficient for a first consideration of the spectra. We discuss
first the hydrogen atom, in which a single electron revolves about
a nucleus of unit positive charge.

Bohr assumes that the rotating electron in the hydrogen atom
is restricted to definite, stable orbits whose radii are proportional
to I2, 22, 32, . . . within any one of which the electron rotates
continuously without loss of energy (see Fig 69). These integers
1, 2, 3, ... are called the principal quantum numbers of the

FIG 69 — Orbits of the election around a hydrogen nucleus.

orbits. It is further assumed that the electron may pass from
one orbit to another and that the energy of the atomic system
is greater for orbits of greater radius. Little is known as to
how the electron passes or what causes it to pass to a "higher"
orbit, but it is assumed that energy is absorbed in the transfer
and radiated when the electron passes to a "lower" orbit. It
may be calculated that the diameter of the innermost orbit for
the hydrogen atom is very close to 1 X 10~8 cm., while the diam-
eter of the nucleus is of the order 10~12 cm. or less. Thus the
estimates of molecular diameter based on the kinetic theory
correspond to those required for the orbits of the electrons.
While the electron rotates in a given orbit, it radiates no energy
and the atom is in a "stable state"; but wrhen the atom passes
from one stable state to another of lower energy, or when the

ATOMIC STRUCTURE 557

electron falls to a lower energy level, energy must be lost by the
atomic system. It may be shown that for the transition from
the orbit of quantum number n2 to that of number n\ the energy
lost is

- El = k f — 2 -
\ni2

and the numerical value of k may be calculated from physical
constants.1 The numbers n\ and n2 may be any integers, and so
long as nz is greater than HI energy will be radiated by the atomic
system.

If it be further assumed that the quantum theory is applicable,
this lost energy appears as a quantum of frequency p, and

The spectrum of hydrogen has been carefully studied, and lines
corresponding to many frequencies (v — c/\) are known. Spec-
troscopic data are more commonly give'n in terms of the wave
number v rather than the frequency, where v is the number of
waves per centimeter and vc = v. The equation above may
therefore be written in the form

The value of R, calculated from physical constants not involving
spectroscopic data, is 110,500, and that derived from spectro-
scopy is 109,737. This is usually called the Rydberg constant.
J3y choosing the proper whole numbers for n\ and n2 it should be
possible to calculate wave numbers for hydrogen spectrum lines
from this equation that are in agreement with spectroscopically
measured wave numbers, if the Bohr theory is correct.

A series of lines in the visible spectrum of hydrogen, discovered
by B aimer, may be described quantitatively by the equation
above, if HI = 2 and n2 is successively 3, 4, 5, 6, .... Simi-
larly, by taking HI = 1 and n2 = 2, 3, 4, 5 . . . the Lyman

1 See RICHTMYEB and KENNARD, op. cit. The quantity is
k = 2ir*me*/ch* = 1 105 X 106 cm.'1

558 PHYSICAL CHEMISTRY

series (discovered later in the ultraviolet region) is accurately
described; and the lines of the Paschen infrared series are in
agreement with wave numbers calculated from HI = 3 and
n2 = 4, 5, 6, 7, . . . , the same value for R being used in all three
series. Brackett's series (or Bergman's series) follows similarly
if n, = 4

It is evident that a fundamental truth is partly revealed
by Bohr's model, which deserves serious attention. The theory
has been applied successfully to ionized helium and somewhat
loss satisfactorily (owing to the complexity of the phenomena) to
heavier elements as well. It has been necessary to assume
elliptical orbits as well as circular orbits and to use more than
one set of quantum numbers, in addition to other complications,
to explain even the simplest spectra. Elements with several
electrons, revolving in orbits that require four or more quantum
numbers, necessarily present complications and are best excluded
from a first consi delation of atoms.

J Electron "Shells." — The Bohr atom model is less useful to
chemists than another concept, in which the energy levels, or
" shells," of electrons are considered. These shells are considered
to be complete for the rare gases and incomplete in the outer
layers for all other elements to an extent that offers a partial
explanation of their chemical properties. The maximum number
of electrons in the shells increases as twice the squares of natural
numbers, 2(1 2, 22, 32, 42) = 2, 8, 18, 32; and it will be recalled
that in Bohr's arrangement of the elements in Table 86 the
numbers of elements in the periods were 2, 8, 8, 18, 18, 32 (and
5 for the incomplete seventh period). Thus in each period after
the first the maximum number of electrons in the main group is
repeated once before going on to the next highest number in the
series 2(12, 22, 32, 42). Through the first three periods the lowest
shell is completed before any electrons are added at a higher
level. In the language of spectroscopy these levels are desig-
nated K} L, M , N, 0, P, and Q, with subdivisions in all the levels
except K.

The elements that end the various periods are all rare gases
of the zero group, elements numbered 2, 10, 18, 36, 54, and 86;
and hence the elements containing these numbers of electrons are,
respectively, He, Ne, A, Kr, Xe, and Rn. The simplified discus-
sion of the distribution of electrons at the various levels that is

ATOMIC STRUCTURE 559

now to be given is not, of course, a complete explanation of
chemical properties, or even a close approach to completeness,
and there are difficulties in applying the concept, even to simple
systems. An attempt to present the experimental evidence on
which the picture is based or to consider bonds that are neither
polar nor covalent would be quite out of place in a first discussion
such as we are attempting here. Nevertheless, the simplified
concept is worthy of careful study, and we now turn to a dis-
cussion of it.

First Period. — The hydrogen atom consists of a single proton
and a single K electron, or electron at the first quantum level,
or one electron in the first shell, or one Is electron. A helium
atom consists of a nucleus of net charge +2 and two electrons
at the first level. Thus a hydrogen atom must acquire an elec-
tron to complete the first shell, but the fact that H~ is not a
familiar chemical substance indicates that a hydrogen atom has
little tendency to acquire the electron.

The fact that H+ is a familiar chemical substance shows that it
has a greater tendency to lose its electron under favorable cir-
cumstances, and we shall soon come to a consideration of what
these circumstances are First, we may consider the hydrogen
molecule, in which the electron density between the two nuclei
is at a maximum, causing a "covalent" or "nonpolar" bond.
In the common terminology these atoms "share" a pair of elec-
trons, and the bond is written H:H, in which the two dots indi-
cate the pair of shared electrons. (This notation should not be
confused with a double bond such as exists in ethylene and which
is two pairs of shared electrons; such a bond would be written
C::C.)

The helium atom consists of two protons and two neutrons, with
two Is electrons, or K electrons, to complete its electrical neu-
trality. Since two is the number of electrons for the completed
first shell, this is a very stable system. Helium has practically
no tendency in the ordinary chemical sense to lose or acquire or
share electrons, and thus there are no stable compounds of
helium. The ionization potential of helium for removal of the
first electron is 24.46 volts, which is the highest of any element
and which indicates again that helium has very little tendency
to lose an electron and become He+. Helium has no tendency to
share electrons, even with another helium atom, and it forms no

500

PHYSICAL CHEMISTRY

chemical compounds. Thus it is indicated that an atom with a
complete electron shell is an inert, stable atom. We shall soon
see that neon and argon also have complete outer shells, though
not shells of two electrons, and they are likewise chemically
inert.

Second Period. — The elements of the second period in their
normal states, considering only the isotope of mass number
nearest the atomic weight, have the compositions shown in the
following table, in which p is a proton and n a neutron •

Atomic
number

Element

Mass
number

Nuclear
composition

Elections

3p + 4n

2 + 1

4p + 5n

2+2

5p + 6w

2+3

6p + 6n

2 +4

7p + 7n

2+5

8p + 8n

2+6

9p + lOn

2 +7

lOjo + lOw

2+8

In a first consideration of atomic structure it seems advisable
to omit the distinction between the electrons at the second level
usually designated 2s and 2p and to list them all merely as of the
second level, or in the second shell. The table above shows 2
electrons in the first shell and those of the second shell increasing
from 1 to 8. Since neon, with eight in the second shell, has an
ionization potential (21.47 volts) higher than any element except
helium, it will be evident that 2 electrons in the first shell and
eight in the second constitute also a very stable system, as was
helium with two in the first shell and no others. Neither helium
nor neon forms stable compounds or molecules.

A lithium atom could acquire the stable electron structure of
helium by losing the electron in the second shell and becoming
Li+, and* a fluorine atom could acquire the complete second shell
that is possessed by neon if it accepted the electron lost by lithium
and became F~. We have already seen in Chap. V that NaCl
crystals consist of ions and not of molecules of NaCl, and LiF
has the same crystal structure. We customarily write Li+ and
F~~ as the solutes in an aqueous solution of LiF, just as we write

ATOMIC STRUCTURE 561

Na+ and Cl~~ for an aqueous solution of NaCl. Thus the assumed
electron structures of Li atoms and F atoms are in harmony with
the known chemistry of these elements. Chemical union of Li
and F is assumed to be attended by the transfer of an electron
from one atom to the other; such a "bond" is called "polar/'
in contrast to the nonpolar bond in H2.

Beryllium atoms have 2 external electrons, and become Be++
when these electrons are transferred, say, to 2 fluorine atoms. An
aqueous solution of BeF2 probably contains Be++ and F~, but
the salt is largely hydrolyzed. (The Be=F bond is not wholly
polar, nor are the Be — Cl and Be — 0 bonds, but the distinction
is best ignored at first.1 The same statement applies to bonds
between boron and the halogens and to a lesser extent to some
of the other bonds.)

Boron, with 3 external electrons, has little tendency to lose
them and become B+++ and a strong tendency to share them in
forming compounds such as BF3, even though no shell is com-
pleted by doing so The probable structure of BFs is indicated
by the arrangement

:F:B:F:

The halides of boron hydrolyze completely in aqueous solution,
BF3 forming boric acid and fluoboric acid, with no ~B+++ ions,
and the other halides forming boric acid and the ions of hydrogen
halides.

Carbon, with 4 external electrons, likewise has no tendency
to lose them and form C~H""H", but it readily completes its shell
by sharing 4 pairs of electrons with any of several elements having
incomplete shells. There are four covalent bonds in all the
compounds CH4, CH2C12, CFC13, CO2, and CS2. No single
electron structure is satisfactory for carbon monoxide, and there
may be several arrangements with resonance between them.
The remarkable resemblance of CO to N2 in many physical
properties has often been cited as evidence that they have the
same electron structure.

The next two elements, nitrogen and oxygen, commonly form
compounds in which the bonds are covalent or nonpolar. The

1 See PAULING, "Nature of the Chemical Bond," 2d ed., Chap. II, Cornell
University Press, Ithaca, N. Y.T 1940.

562

PHYSICAL CHEMISTRY

arrangement of electrons in nitrogen molecules is not known,
and there may be two arrangements as with CO; oxygen mole-
cules are probably formed, not merely by sharing two pairs of
electrons, but by some other arrangement that is uncertain.

Fluorine readily accepts an electron to complete its shell and
become F"", and it shares a pair of electrons in F2.

Third Period. — Sodium begins this period, and argon ends the
period. If we consider only the isotope with mass number
nearest the atomic weight and ignore the distinction between
p and s electrons, as was done for the second period, the elements
in their normal states have the following compositions :

Atomic
number

Element

Mass
number

Nuclear
composition

Electrons

lip + 12n

2 +8 -f 1

12p + 12»

2+8+2

13p + 14n

2+8 + 3

Up + 14n

2+8 + 4

15p + 16n

2+8 + 5

16p -f 16n

2 + 8 + 6

I7p + 18n

2+8 + 7

18p + 22n

2 + 8 + 8

In general, the discussion for each element in this period is the
same as that for the element above it in each column of Table
85. Sodium tends to lose its single outer electron, assume the
electron arrangement of neon, and become Na+ in solution or in
a crystal. Chlorine tends to acquire an electron, assume the
electron arrangement of argon, and become Cl~ in solution or in
a crystal. Thus NaCl has a polar bond. The ions Mg++ and
A1+++ similarly result from the loss of electrons and reversion to
the stable electron arrangement of neon, and these ions are found
in solution and in most of the crystals

Silicon, like carbon, does not lose 4 electrons and form positive
ions, but it shares electrons to form compounds such as SiH4,
SiCl4, and SiHCl3 with covalent or nonpolar bonds. " Phosphorus
forms PH8 and completes its shell just as nitrogen forms NH3
Sulfur may complete its shell, as in H2S with two covalent bonds,
or become S . In HS~ it probably shares one pair of electrons
and loses one electron. Polysulfides up to 85 are also known,

ATOMIC STRUCTURE 563

and the electron arrangement for sulfur in all of them is probably
close to that for covalent bonds.

Fourth Period. — Eighteen elements fall in this period, two
each in groups I to VII (thus beginning the subgroups), three
elements in group VIII, and one element in the zero group as
shown in Table 85. In discussing the fourth period we cannot
ignore the separation of electrons into s and p groups, and thus
we must now refer briefly to the distinction in the second and
third periods of the elements.

In the second period lithium has one 2s electron, beryllium
and all the other elements in this period have two 2s electrons,
boron has in addition one 2p electron, and the following elements
have, successively, two, three, four, five, and six 2p electrons.
Thus in neon, which has a complete shell, there are two Is, two
2s, and six 2p electrons to form the "neon core/'

The succession in the third period is the same for electrons
outside the "neon core"; sodium has one 3s electron, magnesium
has two 3s electrons, as do all the other elements in the third
period, and the elements from aluminum to argon add one to six
3p electrons. Throughout the third period the "neon core"
persists, throughout the fourth period the "argon core" persists,
while outside of each core the electrons of the next shell are suc-
cessively added.

Electron structures for the first four periods are shown in
Table 94, which the student should study before reading the
next section and to which he should refer while reading it.

Potassium (atomic number 19) begins the fourth period by
adding a single 4s electron to the argon core of element 18, and
calcium (atomic number 20) has two 4s electrons; but with
scandium (21), titanium (22), and vanadium (23) a new cir-
cumstance is met. These three elements have two 4s electrons,
and, respectively, one, two, and three electrons at the third
level, designated 3d. Chromium (24) has not one additional 3d
electron, but two more, or five altogether, and only one 4s elec-
tron. Manganese (25) adds one 4s electron to restore the number
to two, and iron (26), cobalt (27), and nickel (28) retain two at
the 4s level while increasing, respectively, to six, seven, and eight
at the Zd level.

All the elements in the second line of the fourth period of the
periodic table as shown in Table 85, the elements copper (29)

564

PHYSICAL CHEMISTRY

TABLE 94. — SOME ELECTRON STRUCTURES FOR ATOMS IN THEIR NORMAL

STATES1

2s 2p

3s 3p 3d

4s 4p 4d 4/

H 1

He 2

Li 3

Be 4

B 5

2 I

0 6

2 2

N 7

2 3

O 8

2 4

F 9

2 5

Ne 10

2 6

Na 11

Mg 12

2 26

Al 13

neon

2 1

Si 14

core

2 2

P 15

2 3

S 16

2 4

01 17

2 5

Ai c

2 R

K 19

Ca 20

2 26 26

Sc 21

argon 1

Ti 22

core 2

V 23

Cr 24

Mn 25

Fe 26

Co 27

Ni 28

Pn 9Q

v/U. £i\J

Zn 30

2 26 2 6 10

Ga 31

copper

2 1

Ge 32

core

2 2

As 33

2 3

Se 34

2 4

Br 35

2 5

Kr 36

2 6

1 For a full table see Richtmyer and Kennard, op. cit , Appendix III.

ATOMIC STRUCTURE 565

to krypton (36), have the same " copper core" of electrons, while
the additional electrons increase as in the second and third
periods. Copper has one 4s electron, zinc (30) has two 4s, gal-
lium and all of the remainder have two 4s and successively one
4p for gallium, two 4p for germanium (32), up to six 4p for
krypton (36) to complete the period, and a new stable " krypton
core" that persists through the next 10 elements.

This detailed discussion of the elements of the fourth period is
given to point out the fact that the addition of electrons at a
fourth level does not exclude further additions at the third level.
In the next period additions at the fifth level do not exclude
further additions at the fourth level. It should also be noted that
a number of electrons once reached at a given level is not always
maintained. In the period beginning with cesium the first addi-
tion of an electron is at the sixth level, and subsequent additions
at both the fourth and fifth levels are found.

The rare earths have the same number of electrons in all levels
up through 6s and 6p except the 4/, with different numbers of
4f electrons, which is a partial explanation of their chemical
similarity, since chemical behavior is largely determined by
electrons in the outer levels.

As was said at the beginning of the chapter, the purpose of the
discussion has been to give support to the belief that atoms consist
of neutrons, protons, and electrons, to obtain a general picture
of their structures, to point out that the electrons largely govern
the properties of the elements while the protons and neutrons
supply substantially all the mass, and to indicate the sources of
the information on which these beliefs are based. It may be
worth repeating that the concept of electron shells is not wholly
free from objections, that, since the evidence regarding electrons is
almost all spectroscopic, the conclusions apply to gaseous atoms,
and that there is some danger of error in the literal acceptance
of a simplified picture of a complicated situation.

References

For a discussion of nuclear reactions see Ilichtmyer and Kennard, " Intro-
duction to Modern Physics," McGraw-Hill Book Company, Inc., New York,
1942; for the separation of isotopes see Aston, "Mass Spectra and Isotopes,"
Edward Arnold & Co , London, 1942; for a discussion of the chemical bonds
see Pauling, "Nature of the Chemical Bond," Cornell University Press,
Ithaca, N.Y., 1940.

CHAPTER XVII
COLLOIDS. SURFACE CHEMISTRY

In this chapter we discuss very briefly some " heterogeneous "
systems of a special type, systems in which a substance is so
finely dispersed in a liquid that surface effects become of first
importance. Any attempt to discuss such a vast field of chem-
istry in a few pages must necessarily be only a consideration of a
few principles and their application to a few simple systems, with
an almost total neglect of the complex experimental technique
and the many important industrial applications. It must be
remembered, however, that there are experimental techniques
of preparation and study which are of the greatest importance
and applications of the widest variety in plastics, adhesives,
pharmacy, textiles, ceramics, and many other fields.

Colloidal systems are intermediate between true solutions,
homogeneous dispersions of ionic or molecular solutes, and mix-
tures in which phase boundaries are evident and to which the
principles of heterogeneous equilibrium apply. Since many col-
loidal systems are not at equilibrium, their study is complicated
by a change of properties with time. There is no sharp dividing
line between solutions, colloids, and gross suspensions except
by arbitrary definition that would serve no useful purpose. As
polymerization (for example) proceeds from single to double or
triple or multiple molecules, to "low" polymers, to "high"
polymers, to visible droplets or crystals, the change is attended
by a gradual change of properties. One of the important prob-
lems in industry is control of such a process and its restraint
in order to produce a polymer of the desired properties. Since
the mechanism of the process is commonly not known, it is
difficult to apply rate considerations such as were discussed in
Chap. XII to them; and since the composition or structure of
the colloid changes with time, it is also difficult to consider
adsorption isotherms, intermolecular forces, oriented monolayers
at interfaces, and other apparently applicable principles.

566

COLLOIDS. SURFACE CHEMISTRY 567

Of the eight possible types of disperse systems [(1) liquid in
gas, (2) solid in gas, (3) gas in liquid, (4) liquid in liquid, (5)
solid in liquid, (6) gas in solid, (7) liquid in solid, (8) solid in
solid] only the fourth and fifth are of such general importance as
to be considered in this brief discussion. While the word col-
loid, which is commonly applied to these systems, is derived
from the Greek word for glue, it is now used to classify almost
any system in which particles significantly larger than molecules
but small enough to be invisible in a microscope are dispersed
in a nearly stable form. Almost any liquid may be the dis-
persion medium, or "solvent," but we shall consider mostly
aqueous dispersions; and almost any insoluble substance may
be the " dispersed part" of a colloidal system.

As a rough classification, particles that are of greater diame-
ter than 10~4 cm. or IJJL are considered coarse suspensions, and
particles 10~~6 to 10~7 cm. (100 to 1 mju) are called colloidal sus-
pensions. Since molecular dimensions are about 10~8 cm., it
will be clear that a particle of 10~7 cm. diameter might contain
only a few molecules of a substance of high molecular weight.
We have seen in an earlier chapter that the thickness of some
of the monolayers exceeded 10A or 1/x, and hence a particle of I/*8
volume might be a single molecule. Thus there is no sharp
dividing line between colloids and true molecular dispersion of
large molecules. We shall see presently that colloidal suspen-
sions have some of the properties of dilute solutions of very large
molecules.

Aqueous suspensions, or hydrosols, may be divided into two
classes called hydrophobic, or electrocratic (when the attraction
between water and the colloid is slight), and hydrophilic (when
there is a strong attraction between water and the colloid) ; but
since slight and strong are not precisely definable, there are
colloidal systems whose classification in this way might be arbi-
trary or misleading. Typical examples of hydrophobic colloids
are gold, platinum, ferric hydroxide, arsenious sulfide, sulfur,
bentonite, silver iodide, and ferric ferrocyanide. Stable aqueous
suspensions of these substances, when the particles are 10~5 cm.
or less in diameter, appear transparent when viewed in trans-
mitted light, but they may be opalescent or opaque when viewed
at a right angle to the transmitted light. The viscosity or
vapor pressure or surface tension of any of them would be

568 PHYSICAL CHEMISTRY

almost the same as for pure water, but methods to which we shall
come presently show that they are not true solutions. Through
X-ray diffraction it has been shown that the particles of many
colloids are small crystals presumably held in suspension because
of the extreme fineness of subdivision and probably stabilized by
selective adsorption on the large surface exposed Even such
typically crystalline substances as sodium chloride have been
prepared in colloidal form in nonaqueous dispersion media.

When the particle edge for a centimeter cube of material is
reduced to 10~5 cm., the surface exposed is multiplied by 106,
HO that about 10 sq yd. of surface become exposed for each
original square centimeter, and surface effects become of the
first importance in determining the properties of these systems

Typical hydrophyllic colloids are aqueous gelatin, agar,
starch, proteins, and soap. These systems at moderate dilutions
would have almost the same vapor pressure as pure water,
since the mole fraction of the colloid is very small, but the surface
tension is usually much less, and the viscosity much greater,
than for pure water.

Degree of Dispersion. — The diameter of particles concerned
in suspension formation depends on the method of preparation;
thus a gold suspension may be red, purple, violet, or black,
according to the average size of particle produced, though the
color also depends on the concentration of colloid and its method
of preparation. The average diameter may be determined by
counting the particles in a known volume of solution (by a
method to be described presently),^ then evaporating a portion
of sol, and weighing the resulting deposit. From the number of
particles- per cubic centimeter, their weight, and the density
of the dispersed substance, the average diameter is readily
calculated. It should be borne in mind that the diameters of
individual particles in a sol may be very much larger and very
much smaller than an average thus determined unless special
precautions are taken to ensure a nearly uniform size This is
accomplished by fractional settling, usually with the aid of a
powerful centrifuge, or by the use of selective filters called
ultrafilters. Diameters of particles may also be obtained from
the density distribution of a sol under the influence of gravity
and in another way that will be described in connection with
Brownian movement in a later paragraph.

COLLOIDS. SURFACE CHEMISTRY 569

In this discussion the word diameter should not be taken too
literally, for while some dispersed solids behave as if they were
of approximately the same size in sfll directions others do not.
Some "high polymers " made by condensation of molecules may
be 100 or 1000 times as long as the dimensions of molecules and
of approximately molecular cross section, and the cube root of
the volume of such a particle would have little meaning as a
diameter.

Surface Phenomena. — According to Langmuir's theory of the
structure of a solid surface, outlined in Chap. V, an atom or
molecule in the surface of a crystal has an attractive force
reaching into space for a distance comparable to the diameter
of a molecule and capable of holding molecules in adsorbed
monolayers upon the surface. A dispersed solid, such as a
hydrosol, exposes a very large surface per unit quantity of dis-
persed solid and is thus able to adsorb solvent molecules or
whatever solute may be in the suspending medium to a much
greater extent than the same quantity of gross matter. The
adsorptive forces are selective in character; they may attract
one kind of ion to the nearly total exclusion of others present in
equivalent or greater concentration; they may hold only solvent
molecules and ignore moderate concentrations of solutes. In
the latter circumstance that part of the liquid in immediate
contact with the dispersed part of the colloidal system may be
wholly free of solute, and the effect of adding a solute may not
be appreciable. If the adsorption is confined to a given ion,
addition of very small quantities may alter the stability of the
sol and cause coagulation, while the addition of a larger quantity
of some other solute may produce almost no effect.
^ Dialysis. — The existence of colloidal materials was first shown
by their failure to diffuse through membranes of parchment
paper, collodion films, and animal membranes, while salts,
alcohol, sugar, and most "true" solutes passed through such
membranes when they were used to separate a solution from
pure water. Such a separation of solutes from colloids by allow-
ing the former to pass through the membranes is called dialysis,
and the process itself is still of common application in colloid
chemistry whenever it is desired to free a hydrosol from dissolved
salts or other solutes.

Dialysis is a slow process, requiring many days when a sol is to

570 PHYSICAL CHEMISTRY

be freed from dissolved substances completely. It cannot, in
general, be accelerated by immersing a sol in hot water, since this
is likely to precipitate the *sol. As dialysis depends on diffusion
of a dissolved substance through a membrane into a region where
its concentration is lower, the rate of dialysis depends on the
area of membrane used and on the difference in concentration
between the inside and outside liquid. Hence a vessel composed
entirely of membrane is used to enclose the sol, and a stream of
distilled water is sometimes passed into the outer vessel. Toward
the end of such a dialyzing process the difference in concentration
of diffusing substance becomes very small, and the rate very slow
- Methods of Preparing Sols.1 — Since the dispersed part of a
colloidal system consists of particles that are smaller than
ordinary crystals and larger than single molecules, the obvious
methods of preparation are dispersion of larger particles and
condensation of molecules. Dispersion by mechanical grinding
in " colloid mills" usually fails to reach true colloidal dimensions
These mills afe shearing mills rather than grinders; they find
application in decreasing the particle size of emulsions and thus
increasing their stability. Electrical dispersion is accomplished
by striking an arc between metal poles immersed in the suspend-
ing medium. Gold, platinum, silver, and other metals have
been made into hydrosols by this procedure; electrodes of
oxidizable metals form hydroxide or oxide hydrosols. Low-
frequency alternating current or direct current gives similar
sols, but high-frequency alternating current is said to produce
smaller particles.

Condensation methods include precipitation by chemical
reactions", as in the formation of As2S3 by passing H2S into
arsenious acid, of colloidal sulfur by pouring an alcoholic solution
of sulfur into a large quantity of water, of ferric hydroxide by
the hydrolysis and dialysis of ferric chloride, and of other sub-
stances by the ordinary reactions such as oxidation, reduction,
and -metathesis. The insoluble substances familiar in analytical
chemistry are usually precipitated under conditions designed to

1 Stable colloids must be prepared with care by special methods, with
attention to many details. Only a bare outline of the general methods can
be included here, but there are several books readily available.. Hauser
and Lynn, "Experiments in Colloid Chemistry," McGraw-Hill Book Com-
pany, Inc., New York, 1940, gives many of these methods, with ample refer-
ences to the literature.

COLLOIDS. SURFACE CHEMISTRY 571

avoid the formation of colloids that are difficult to 'filter, but
most of them may be prepared in colloidal form under other
conditions of precipitation. Removal of electrolytes by dialysis
usually increases the stability of these colloids up to a certain
point, but complete removal may cause flocculation. The ferric
hydroxide hydrosol formed by the hydrolysis of ferric chloride
is more stable in the presence of some ferric chloride than after
its nearly complete removal. Other colloids, such as platinum,
silica, and some sulfides, are unstable if electrolytes are removed.
Certain solutes act on precipitates in such a way as to con-
vert them into nearly stable hydrosols; the most common
examples are inorganic salts of which the solute has an ion in
common with the precipitate. Thus silver halides are converted
into sols by dilute silver ilitrate or the corresponding potassium
halide; sulfides, such as cadmium sulfide, zinc sulfide, mercuric
sulfide, and lead sulfide, are rendered colloidal by hydrogen
sulfide; metallic oxides, by strong alkali hydroxides. In some
sols this action, which is called peptizing, is reversible, as in that
of metallic sulfides, which may be made into colloidal suspensions
by hydrogen sulfide, thrown down by boiling it out, and taken
up again by passing hydrogen sulfide into a suspension of the
precipitate, and this process may be repeated over and over.
[Another type of "colloidal" particle may be built up through
the usual methods of organic chemistry, a chain such as

— Si— O— Si— O— Si— O—

being formed with organic radicals on the silicon atoms. The
first step is shown by the equation

R
SiCU + 2RMgBr - Cl— Si— Cl + 2MgClBr

R
followed by partial hydrolysis

R R

Cl— Si— Cl + H20 = Cl— Si— OH + HC1

''T

572 PHYSICAL CHEMISTRY

Two of tKese molecules then split out HC1, uniting the silicon
atoms and leaving a terminal — OH on which further condensa-
tion takes place, and this may be continued as long as desired
The third step is

R R R R

I I i

Cl— Si— OH + Cl— Si -OH - ( 1— Si— 0— Si— OH + HC1

I I i

R R R R

These "silicones" may contain only one organic radical or
several, and "branched" chains may be formed by using RSiCl3
as the starting material

Solutions oi isobutylene in volatile solvents yield polyiso-
butylene when treated with BF3, the number of molecules in the
polymer depending on experimental conditions. These polymers
are also probably chain molecules Other materials may likewise
be polymerized under suitable conditions.!
/Determination of Molecular Weights.-^ols do not appreciably
lower the vapor pressures or freezing points of the solvents in
which they are dispersed; their osmotic pressures are very small,
and their molecular weights are very high.

The diameters of colloidal particles in the finest suspensions are
ten times those of molecules and much larger in the ordinary
colloid; the "molecular weights" would be thousands of times
those of molecules of ordinary solutes, and thus the mole fractions
corresponding to small weight percentages would be vei;y small
Since the ordinary molecular-weight methods measure the mole
fraction of the solute, it is uncertain whether the osmotic pressure
or freezing-point measurements carried out on these substances
really represent the osmotic pressure of the colloidal substance
itself, and not that of some contaminating solute, in spite of
great care used in purifying the sols. Molecular weights so
determined are often many thousands and far from concordant.
It will be clear from considerations to be given presently that
colloidal particles are far larger than ordinary molecules, and
that ordinary molecular-weight methods are quite unsuited to
studying them. Molecular weights of certain colloidal sub-
stances may be determined from osmotic-pressure measurements
in which the ratio of osmotic pressure to concentration is plotted
against the concentration, as was explained in Chap. VI. This

COLLOIDS. SURFACE CHEMISTRY 573

procedure has been particularly successful in studying some of the
"high polymers/'

K Viscosity and Density. — Densities of colloidal suspensions,
calculated on the assumption that the sol is a mixture of solid
particles in suspension in a liquid, and without any effect upon it,
agree with those based on experiment. This is not surprising in
view of the small concentrations of suspended material usually
encountered, as these are usually less than a tenth of 1 per
cent. The viscosity of dilute suspensions is usually only slightly
greater than that of the suspending medium, and the increase
in viscosity depends on what fraction of the total volume is
solid, rather than on its fineness of dispersion. A relation due
to Einstein, T? = -70(1 + %<£), where </> is the volume of suspended
material, is approximately true under certain restricted condi-
tions, but greatly in error if the particles carry electric charges.
An approximate relation of some usefulness in following the
extent of polymerization of long-chain molecules is1

*? - ^0 I

- - - = ken

in which (77 — 770) /T?O is the fractional change in the viscosity
of the solution produced by the solute, i?0 is the viscosity of
the solvent and 17 is that of the solution, k is a constant, c is
the concentration of the solution expressed as moles of single
molecules, and n is the number of molecules in the chain. Other
factors also influence the change in viscosity, so that the relation
is only a rough guide. For instance, the viscosity change is not
the same when " branched chains" are formed as when straight
chains are formed, and thus the molecular weight of the con-
densation product is not n times that of the single molecule
when n is determined from the viscosity change, unless proper
allowance is made for the structure of the condensed molecule.

These polymers have " molecular " dimensions in two direc-
tions and "colloidal" dimensions in their length. Their
solutions have some of the properties of "true" solutions and
some of the properties of colloids, as is true of other organic
compounds of high molecular weight.

Rate of Settling of Suspensions. — If it be assumed that a
particle is a sphere of radius a and density d and that it is settling

1 STA-UDINQER, Kolloid Z., 82, 129 (1938).

574 PHYSICAL CHEMISTRY

through a gaseous or liquid dispersing medium of density d'
and viscosity TJ under the influence of gravity gr, its rate of settling
is given by Stokes 's law,

- d')g

9rj

Thus the rate at which a particle settles becomes slower as
the density of the particle approaches that of the suspending
medium Under the influence of a force greater than gravity
(for example, in a centrifuge) the rate of settling can be cor-
respondingly increased.

Experiments have shown that this equation describes the rate
of settling of some dilute suspensions and that the radius of the
particles as determined from the rate of settling agrees with tnat
from other methods. "Very small" particles settle faster than
Stokes's law requires, but particles 10~B cm. in diameter or smaller
remain permanently in suspension, probably because of their
Brownian movement. In very concentrated suspensions the
particles settle with a uniform velocity more slowly as the con-
centration increases. For example, in an aqueous suspension
containing 25 per cent silica by volume, the rate of settling is
about half that calculated from Stokes 's law. The law also
applies to fog or dust particles settling in air, provided that the
particles are large compared with the mean free path of the gas
molecules.

'Electrical Properties. — Sols exert a slight effect on the elec-
trical conductance, and part of this small increase is probably
due to traces of electrolyte adsorbed by the particles. Either
because of adsorbed ions, or from frictional electricity, suspen-
sions bear charges that cause them to migrate in an electric
field. Most colloidal metals, As2Ss, and Agl are examples of
colloids that move toward and precipitate upon the anode;
most hydroxide sols move toward the cathode. The phenomenon
is called cataphoresis, and it should not be confused with the
movement of ions as in transference. There is no relation such
as Faraday's law between the weight of colloid precipitated and
the quantity of electricity. In other words, the charge upon
a colloidal particle depends not on its weight, but on the amount
and charge of adsorbed ions, which vary with the conditions
under which the colloid is prepared. I

COLLOIDS. SURFACE CHEMISTRY 575

The motion of water toward the cathode through a porous
clay separator when a potential is applied to electrodes on
opposite sides of it is called electroendosmosis. When a fine
suspension of clay in water is placed between electrodes, the
clay moves toward the anode. Thus the displacement of clay
relative to water by the electric field is the same, whether the
clay or the water moves. A similar effect is observed when any
other suspension, such as arsenious sulfide, is held stationary
in an electric field; water is displaced in the opposite direction
This movement in an electric field is applied industrially in
purifying china clay, in tanning, in medicine, for separating
water from peat, and in several other wrays. The mechanism of
the process is probably similar to that in the Cottrell precipitator
for smoke and dust, in which fine particles suspended in air are
caused to precipitate on a charged netting or set of chains.

Electrical Double Layer. — This expression is commonly used
to describe the condition around a colloid particle that has
adsorbed ions of one charge, leaving the corresponding ions of
opposite charge in the solution free to migrate as much as the
electrical attractions permit. If to a suspension that has
adsorbed positive ions one adds a small amount of an electrolyte
whose negative ions are adsorbed, equal amounts of positive and
negative ions may be acquired by the particles at a characteristic
(small) concentration, and, when they have no net charge,
ftocculation usually results.

To account for the existence and formation of the electrical
double layer at colloid surfaces, two theories have been proposed.
The " adsorption theory" postulates that the ionic layer which
confers the fundamental charge is firmly held at the surface by
means of the preferential adsorption of ions from the dispersion
medium, whereupon the ions of opposite charge form a diffuse
system about the particle, owing to electrostatic attraction.
The solubility, or uionogenic complex/ ' theory attributes the
formation of the diffuse layer to ions dissociated from the col-
loidal particles, which themselves are considered as complex
colloidal salts. The charge on the particles exists because of
the free valence ions on the surface of the complex salt. The
experimental evidence seems to favor the adsorption theory.1

1 See Hauser and Hirshon, J. Phys. Chem., 43, 1015 (1939), for a discussion
of these theories and the Intel-attraction of colloidal micelles.

576

PHYSICAL CHEMISTRY

In an attempt to go one step further in explaining the stability
of colloidal suspensions, the interaction of "long-range" van der
Waals7 forces and electrostatic repulsions has been brought
into the discussion; but the evidence so far accumulated is not
very convincing, and direct proof is wholly lacking.
^ The Ultramicro scope. — This instrument does not render
particles visible that are invisible in an ordinary high-power
microscope, but it shows that such particles are present by a
bright spot of light radiated from each particle. Nothing
whatever as to the size or color or shape of a particle is learned
from its effect upon the eye when viewed through an ultra-
microscope; yet the apparatus is justly entitled to its name,

^Control sl/fs^

M/croscope

Beam

of light =-^'

Colloidal
suspension

FIG. 70 — Diagram of an ultramicroscope.

since it shows the presence of a particle that cannot be seen at
all in an ordinary microscope A rough illustration of the prin-*
ciple on which it is based is afforded by the beam of light from
a projection lantern in a darkened room, the so-called Tyndall
effect. This shows particles of dust or smoke suspended in the
air that are quite invisible when the room is thoroughly lighted
but does not show the color of the particles. The ultramicro-
scope merely magnifies highly a small portion of such an illumi-
nated volume of suspension in a liquid medium, which is made so
dilute that light radiated from each particle reaches the eye
without interference from some other particle, as shown in Fig.
70. l Under similar conditions a concentrated suspension gives
only a uniformly bright field in which no individual particles
are rendered visible.

1 Special methods have been developed for accurate control of the slit,
which governs the depth of liquid illuminated, and for intense illumination
These are described in any of the larger reference books on colloids men-
tioned at the end of this chapter.

COLLOIDS. SURFACE CHEMISTRY 577

A particle 10~5 cm. in diameter is invisible under the highest
power of a microscope, but the effect of such n particle is clearly
>seen under an ultramicroscope. Particles far smaller in diam-
eter than a wave length of visible light are able to show their
presence by radiating light in the ultramicroscope, and the
number of such particles in an illuminated volume may be
counted. When the area of field under the microscope is known
and the depth of illuminated, area is measured and regulated by
a micrometer slit between the arc light and vessel containing a
sol, a count of the spots of light in such a field gives the number of
particles in a known volume Even with very dilute sols it is
often necessary to dilute them with large quantities of pure
water before a count is possible. For this dilution ordinary
distilled water is quite unsuited, as it contains thousands of
visible particles in a drop. Specially prepared " optically
empty" water is required, and its preparation involves special
methods.

The size of particle detected by an ultramicroscope depends
chiefly on the intensity of illumination; the lower limit is not
far from 10~7 cm., which is about 0.2 per cent of the wave length
of visible light.

^ Brownian Movement. — The molecules in a liquid are in rapid
though tumultuous motion of the kind outlined in connection
with the kinetic theory of gases. A colloidal particle is very
large compared with the diameter of a single molecule, and it is
continuously bombarded on all sides by great numbers of mole-
cules. Occasionally, the pressure due to this bombardment is
for the moment greater on one side of the particle than on the
other, and the particle is urged forward until a new distribution
of impacts hurls it in another direction. The excursions due to
these movements depend mainly on the size of the particles, and
the movement corresponds exactly with that predicted by the
molecular theory.

Here we have reproduced in a way visible to our eyes the
random unordered continuous motion of molecules postulated in
connection with the kinetic theory of gases. This motion takes
place as a result of impacts with real molecules, but it makes a
colloidal particle behave as if it were a single molecule. The
metion was first observed by the botanist Brown on plant cells
that were visible in an ordinary microscope; the movement was

578 PHYSICAL CHEMISTRY

little more than an irregular oscillation, whose real cause remained
long unsuspected From equations based on the kinetic theory
it may be shown that the amplitude of this vibration is directly
related to the diameter of the particle and the viscosity of the
suspending medium. Thus what in an ordinary high-power
microscope is a slow-oscillating effect produced on a plant cell
or small bacillus becomes, for a much smaller colloidal particle,
a lively zigzag motion, as shown by the cone of light radiated
from it in an ultramicroscope.

A reliable method of determining the size of suspended particles
is based on their Brownian movement, the equation for which
is used in another way in the next paragraph. In this equation
the radius of a particle may be determined if we assume a value
for Avogadro's number of molecules in a gram-molecular weight
of gas; or from counting particles and an analysis of the sol we
may determine the radius, perform the reverse calculation, and
compute a value of Avogadro's number. The latter procedure
is more interesting.

^Brownian Movement and Avogadro's Number. — A relation
may be derived between the intensity of Brownian movement,
the radius of the particle, the viscosity of the dispersing fluid,
and the number of molecules of gas in a gram-molecular weight.
Since colloidal particles are bombarded by molecules in a wholly
random way, they will have the random motions of a large gas
particle and will behave as such. Upon this assumption, the
equation, in terms of the mean displacement d in a unit of time
tj is

rf2 RT
t

where r is the radius of a particle, N is Avogadro's number,
and 77 is the viscosity of the liquid suspending medium. Experi-
ments based on observation of displacements in small time
intervals lead to values jof Avogadro's number between 6.2 X 1023
and 6.9 X 102S, in fair agreement with other methods.
^ Distribution of Particles under the Influence of Gravity. —
A suspension of colloidal particles tends to separate out the solid
under the influence of gravitational attraction and is partly
prevented from doing so by the Brownian movement, in much
the way that molecules of the atmosphere are attracted to

COLLOIDS SURFACE CHEMISTRY 579

the earth by gravity and prevented from settling upon it by the
intensity of their molecular motion. The equation expressing
the variation in density of the atmosphere with the altitude
contains JV, the number of molecules in a gram-molecular weight.
A colloidal suspension of particles of uniform size that has
reached settling equilibrium distributes itself in the way that
the atmosphere is distributed under the action of gravity, thus
reproducing within reasonable space the effect for which the
atmosphere requires several miles of altitude. From determina-
tions of the number of particles per milliliter at equilibrium,
the variation of density with altitude may be established and
used to calculate a value of Avogadro's number N. If n\ is
the number of particles per unit volume at a level that we may
call zero height, and n2 is the number at another level h cm.
above the first one, the equation for change of concentration
with h is

where m is the difference in mass between a colloidal particle and
the volume of solvent it displaces, g is the acceleration of gravity,
and N is Avogadro's number. Investigations based on this
equation lead to a value for N of 6.8 X 1023.

The derivation is based on the assumption that the colloid particles
exhibit the same behavior as molecules of an ideal gas. Let p be the density
of colloidal particles in the mixture; then p dh is the mass of an element of
thickness The change of pressure with h is then shown by the equation
on page 69, namely,

dp = -pgdh

For p substitute Nm/v, where N is Avogadro's number, v is the molecular
volume, and m is the apparent mass of a particle, i.e., the difference between
its mass and that of the solvent it displaces. It should be noted that Nm
is the molecular weight. If, now, we divide the above equation by pv = RT,
we have

On integrating between limits and noting that the ratio of pressures is equal
to the ratio of the number of particles per milliliter, we have

where (hz — hi) is the h of the equation in the above text.

580 PHYSICAL CHEMISTRY

Precipitation of Colloids. — As has been mentioned before,
most suspensions are electrically charged, probably as a result
of adsorbed ions on the surface of the particles. Ionic adsorption
is a selective process, some ions being more strongly adsorbed
than others When a solution containing readily adsorbed
negative ions at low concentration is added to a positively
charged sol, these ions are adsorbed and neutralize the electric
charge of the particles, so that they no longer repel each other.
Coagulation or precipitation takes place, and it has long been
recognized that this ionic adsorption is highly specific, in regard
to both the colloid and the ions. The significant ion in the
precipitation of colloids by electrolytes is the one having a
charge opposite in sign to that of the particle A general rule,
to which there ftre occasional exceptions, is that ions of higher
valence are more strongly adsorbed (and therefore more effective
in producing precipitation) than ions of lower valence. Thus, for
most negatively charged suspensions, ferric salts, aluminum salts,
and trivalent cations in general are most effective as precipi-
tants, i.e., produce coagulation when added in the smallest
concentrations; lead and barium salts come next, and then heavy
monovalent ions, such as silver; finally, the alkali ions are least
effective. The negative ions play only a minor part in these
precipitations. Similarly, positively charged sols are more
readily precipitated by sulfates or phosphates than by mono-
valent anions at equivalent concentrations, and the positive
ion exerts a secondary effect or one that is negligible. Among
anions the order of decreasing precipitating effect is sulfocyanate,
iodide, chlorate, nitrate, chloride, acetate, phosphate, and sulfate
for albumin and certain other colloids; but the precipitating
power of these ions is in the reverse order for some colloids.

The term precipitation is not used in the same sense as in
analytical chemistry, for there is no stoichiometric relation
between the weight of " precipitate " and the quantity of salt
producing it. There is ra/ther an aggregation of the particles,
which depends on the concentration of reagent to a greater
degree than on its quantity. .Ordinarily the salt, such as MgSO4,
that is used as the precipitant largely remains in solution after
the suspension has settled out.

Precipitation also takes place when a positively charged sol is
added in proper quantity to a negatively charged sol, each

COLLOIDS. SURFACE CHEMISTRY 581

neutralizing the charge carried by the other. It does not follow
that a chemical compound is formed, though the coagulated
material may seem to be a compound. For example, ferric
hydroxide sol precipitates arsenious sulfide sol but probably
does not form ferric thioarsenite. It seems probable that
precipitation is due to a reaction of the adsorbed stabilizing
electrolyte. In general, suspensions are much more sensitive to
electrolytes at very small concentrations than are emulsions.
v Protective Colloids. — Certain substances have a conspicuous
property of stabilizing colloidal suspensions. Thus a dispersion
of silver chloride is maintained in a stable state by gelatin in a
photographic film, and the success of a film is largely dependent
on its retaining a uniform dispersion of this silver chloride.
Lyophilic colloids such as gelatin, gum arabic, protein, starch,
casein, and soap are among the common protective colloids;
tannic acid stabilizes the aqueous suspensions of graphite used as
commercial lubricants, though other substances are also effective.
Electrolytes that stabilize colloids probably do not form a
protective film but owe their effectiveness to the adsorption of a
common ion, by which repulsive forces are set up between the
particles that increase dispersion and thus increase stability.
There is no reason to doubt that adsorption is also active in the
mechanism of protective colloids, though a simple and quite
plausible explanation is that the protective substance coats
the suspended particles with a very thin layer of protecting
colloid. Substances that are effective in this respect are them-
selves able to form stable gels.

Soap Solutions. — The extensive researches of McBain and his
associates have brought to light another colloidal condition
that seems to be characteristic of soaps in aqueous solution.
These solutions conduct electricity to about the same extent
as other salts at the same equivalent concentration but they
produce a depression of the vapor pressure of solvent that would
be expected of a nonionized solute.1 His studies have shown
that soaps are not hydrolyzed to the large extent formerly
assumed but that a colloidal aggregate of the negative ions
forms, which he calls an "ionic micelle. " If we take sodium

1 McBAiN and others, /. Chem. Soc. (London}, 101, 106, 113, 115, 117,
119, 121. See especially pp. 1-31 of the "Report on Colloid Chemistry,"
Brit Assoc. Advancement Sci. (1920), and pp. 244-263 of the 1922 Report.

582 PHYSICAL CHEMISTRY

palmitate as an example and let P~ denote the palmitate ion,
Ci&HsiCOO", an important part of the effect produced when soap
dissolves in water may be represented by the equation

The chief difference between this condition and that of an ordi-
nary ionized solute is the aggregation of negative ions into a
large (colloidal) group possessing about the same equivalent
conductance as a negative ion. There are also present in a soap
solution simple sodium palmitate molecules, colloidal soap
(NaP)y, and simple palmitate ions The proportion of these
various solutes present in solution varies greatly with the con-
centration. In dilute solutions NaP and P~ predominate, and
in a normal solution 50 per cent exists as (NaP)y at 90° and about
30 per cent as the ionic micelle. Aqueous solutions of soaps
when functioning as detergents are seldom at concentrations
greater than O.Olw. or less, so that neither the colloidal soap
nor the ionic micelle contributes very largely to the useful proper-
ties commonly associated with soap Probably the effect of
simple sodium palmitate molecules upon the surface tension is
chiefly responsible for the cleansing action of soap.

Experiments upon sodium laurate, CnH^COONa, which is
abbreviated NaL, show the molecular species HL2-, NaHL2,
HL.SNaL, and Le6"" are present1, and it is probable that similar
solutes exist in other soap solutions.

In a study2 of the potassium salts of the long-chain acids con-
taining 6 to 12 carbon atoms, the conductances and freezing
points are said to show that only simple ions and simple mole-
cules are present; and solutions up to 0.5m. contain very small
amounts of micelles if any. The soaplike properties of salts are
not important for chains much shorter than that of lauric acid,
which is CuH23COOH, so that these statements are not applicable
to the true soaps in common use.

Donnan Equilibrium. 8— We may consider here the equilibrium
that prevails on the two sides of a dialyzing membrane that

1 EKWALL, and LINDBLAD, Kolloid. Z., 94, 42 (1941).

* McBAiN, /. Phys. Chem., 43, 671 (1939).

3 Z. Mektrochem., 17, 572 (1911). For a detailed discussion of this equilib-
rium and its bearing on colloid chemistry, see Bolam, "The Donnan Equilib-
rium" (1932).

COLLOIDS. SURFACE CHEMISTRY 583

is permeable to ordinary ions but not to a colloid or its ion. If
congo red is taken as an illustration, we may write its formula
NaR to indicate that it is a sodium salt of a radical of colloidal
character. We shall assume that a solution containing this salt
and sodium chloride is separated from pure water by a membrane
permeable to sodium chloride and its ions but not to NaR or
to the colloidal ion R~ that is formed when congo red ionizes.
It may be that this ion forms a micelle (R~)*, as in the case of
soaps. Dialysis will proceed, and at equilibrium some of the
sodium chloride and all of the congo red and its negative ion
will be on the original side of the membrane and sodium chloride
alone will have diffused through the membrane. The equilibrium
condition may be shown as follows, if the dotted line represents
the membrane:

Na+R-

Na+Cl-

(1)

Na+Cl-

(2)

Since a positive ion may not diffuse through the membrane with-
out a negative ion except by overcoming very large electrostatic
forces, the ions of sodium chloride must diffuse through together.
If (Na+)i and (Cl~)i represent the concentrations of sodium ions
and chloride ions on the left-hand side at equilibrium, the rate
of diffusion through the membrane into the right-hand side is
proportional to the product of these concentrations, (Na+)i(Cl~)i.
But since equilibrium prevails, diffusion in the reverse direction
takes place at the same rate, this rate must be proportional
to the product of the concentrations on the right, and the same
proportionality constant applies. That is, at equilibrium

(Na+MCl-)! = (Na+)2(Cl-)2

The concentrations (Na+)2 and (Cl~)2 are necessarily equal,
since only sodium chloride has diffused through the membrane,
but (Na+)i = (Cl~)i + (R~). Thus the concentration of sodium
chloride on the side of the membrane where it alone is present
is greater than its concentration on the side with the colloid, but
the total solute concentration is greater on the side containing
the colloid.

This equilibrium may be applied to the swelling of gelatin
immersed in an acid solution, for the proteins are amino acids

584 PHYSICAL CHEMISTJiT

that are combined with hydrogen ions (above a certain con-
centration) to form salts. If P denotes the protein molecule,
P + H+C1- = PH+C1-. When the gelatin has swelled to
equilibrium, the product (H+)(C1~) in the solution within the
gelatin must be the same as in the external liquid. Denoting the
concentration in the presence of the colloid by the subscript 1,
in the inside solution we should have

(CT-)i = (H+), + (PH+)
whence at equilibrium

(H+MC1-)! = (H+)2(Cl-)2
or

i = (H+)22

It has been shown by Loeb that, when the hydrogen-ion
concentration is greater than 2 X 10~6 (i e., pll = 4.7, to use the
original notation), gelatin combines with hydrogen ions and
forms gelatin chloride; at a lower hydrogen-ion concentration
metal proteinates form, and at pH = 47 protein combines
equally with hydrogen ions and hydroxyl ions This is called
the isoelectric point for gelatin. Thus, whether the gelatin
combines to form a complex positive ion or a complex negative
ion, the total solute concentration within it is greater than in the
outer solution with which it is in equilibrium. In other words,
the activity of water, as measured by its vapor pressure, is less
within the gelatin, and water tends to pass into the gelatin.
This is probably the explanation of the swelling of gelatin in
water.

Isoelectric Point. — Gelatin and other proteins probably con-
sist of complicated "molecules" having the character of amino
acids that may be represented by (NH2RCOOH)a;. In the
presence of acids the protein particles become " neutralized "
and function as cations such as (NHsRCOOH)/*. Of course,
the electrical balance is maintained by xd~. These ions
are positively charged and migrate toward the cathode. Simi-
larly, in the presence of bases, proteins form negative ions
such as (NH^RCOO)**"", and the opposite movement in an electric
field is observed. The extensive researches of Loeb1 and others

1 This work is described in detail in Loeb, "Proteins and the Theory of
Colloidal Behavior," McGraw-Hill Book Company, Inc., 1927.

COLLOIDS. SURFACE CHEMISTRY 585

have shown that for proteins there is a certain characteristic
acidity of the suspending medium, called an uisoelectric point, "
at which no migration takes place in either direction. It is
probable that at this hydrogen-ion concentration the acidic and
basic dissociations of the amino acids which make up the protein
" molecules7' are equal. This effect is observed when the hydro-
gen-ion concentration is 2 X 10~5, or at pH = 4.7. Other
substances also have characteristic isoelectric points.

When wool in a finely divided condition is suspended in a buffer
solution of pH 2 or 3, it moves toward the cathode1 but much
more slowly in the pH 3 solution. When pH is increased to 3.4,
no motion is perceptible. As pH changes from 3.6 to 5.5, the
suspended wool particles move toward the anode at increasing
velocities, indicating that the " isoelectric point" wras passed at
pH 3.4. Other experiments2 indicate that the isoelectric pH
may be nearer 4.8, and further work seems required before any
more definite statement may be made. But, regardless of the
numerical significance, it is evident that wool has amphoteric
properties similar to simpler amino acids. Analogous behavior
has been observed with silk.

Emulsions. — It is commonly, though not necessarily, true of
emulsions that both parts are liquid, and the ratio of dispersed
part to dispersing medium is much greater than in suspensions.
In the hydrosols that we have been discussing, the dispersed
part is usually not more than one-thousandth of the whole, but
emulsions may be prepared in which as much as 99 per cent is
the dispersed part and 1 per cent or less is dispersing medium
or continuous part. But suspensions of liquid oil in water in
which the dispersed part is only a small portion of the whole are
properly considered suspensions and not emulsions. Stable
emulsions usually require low surface tension between the parts
of the system, which is commonly brought about by dissolving
soap or some other "emulsifying agent" in the dispersing medium.

Such systems have also very large interfacial areas, and it is
probable that the orientation which was found in the monolayers
on liquids is established in emulsions. If soap is taken as a
typical stabilizer, it is to be expected that the hydrocarbon por-
tion of the soap will be toward the oil layer in the emulsion and

1 M. HARRIS, Am, Dyestuff Reptr., 21, 399 (1932).

2 SPEARMAN, Trans. Faraday Soc., 30, 539 (1934).

586 PHYSICAL CHEMISTRY

that the carboxyl group attached to sodium will be toward the
aqueous layer. The experimental evidence seems to show also
that the " concentration " of soap in the interface is very much
greater than that in the bulk of either liquid part of the emulsion,
probably forming a surface layer which is nearly saturated long
before saturation is attained in the liquid as a whole.

Concentration in Surfaces. — It is a general law that substances
which lower the surface tension of a solution accumulate in the
surface, producing there a higher concentration of solute than is
present in the bulk of liquid. Any substance that will lower
the surface tension may act as an emulsifying agent. The rela-
tion between u, the excess of solute per unit of surface or inter-
face; c, the concentration; and the rate at which surface tension
changes with concentration, dy/dc, is

_ _ c dy
u ~ ~ ~RTTc

which is called the "Gibbs adsorption equation." From this
equation it will be seen that, if dy/dc is positive, surface tension is
increased by the solute, u is negative, and there is no accumula-
tion of solute in the surface, but a deficiency of it. When the
solute lowers the surface tension, dy/dc is negative and u is posi-
tive; i.e., solute accumulates in the surface in excess. If a
froth is formed on such a liquid in which the surface tension has
been lowered, excess solute will be found in the froth.

In moderately strong solutions of substances that depress
the surface tension, the surface probably consists of a layer of the
dissolved substance one molecule deep,1 and there is no transition
layer ir^ which the concentration varies progressively at points
farther from the surface into the solution. The amount of
solute required to form this layer may be calculated from the
Gibbs equation, and from this quantity of solute in the surface
layer may be calculated the diameter and cross section of the
molecules forming the layer. The data so found are in agreement
with those obtained from other methods of measuring molecular
diameters. For example, Langmuir found that the molecular
cross section in the surface was 24 X 10~16 sq. cm. per molecule
for palmitic acid. In Chap. IV, 21 X 10""16 was found from the
spreading of a film of palmitic acid on the surface of water.

1 LANGMUIR, Proc. Nat, Awd, 8d., 3, 251 (1917).

COLLOIDS. SURFACE CHEMISTRY 587

Measurements of the. surface tension of soap solutions against
a benzene interface1 have given 40 to 47 X 10"" 16 for the molecular
cross section of sodium oleate adsorbed into the interface. With
solutions of inorganic salts, in which the surface tension is greater
than that of water and increases linearly with the concentration,
the "concentration" of solute in the surface layer is less than in
the solution as a whole. This does not explain the increase of
surface tension; for if no solute at all were in the surface layer
the interface tension would be that of pure water, and the surface
tension of some salt solutions is greater than that of pure water.

Surface Tension and Emulsion Formation. — It has long been
known2 that a decrease of surface tension is produced by those
substances which aid emulsification and that a lowering of
surface tension is essential to the formation of most stable
emulsions. The first sodium salt of the series of fatty acids to
produce appreciable lowering of surface tension when it is added
to water is sodium laurate, and this is the first salt in such a series
to aid appreciably in forming emulsions of oil in water. It is the
first to form a soap with marked froth formation and having
cleansing properties. There is at least a rough proportionality
between surface-tension decrease and emulsifying power so far as
emulsions of oil in water are concerned.

When sodium oleate is dissolved in water, a very rapid decrease
in surface tension takos place at the interface between solution
and vapor with increasing concentration of the soap, until at
0.002 N the surface tension reaches its minimum value3 of about
25 dynes per cm. Further additions of sodium oleate produce
no significant change, from which it may be concluded that the
surface is saturated. In other words, the interface contains all
the soap it is capable of holding when the bulk of the solution is
very far from saturated. The arrangement is probably similar
to that assumed by oleic acid films spreading upon water; a
monomolecular layer of soap exists at the interface, which is
saturated with sodium oleate at any concentration over 0.002 N.
Water and oil, when shaken together, do not form a stable
emulsion; i.e., the layers separate soon after shaking is discon-
tinued. If a little sodium oleate is dissolved in the water layer

1 HABKINS and ZOLLMAN, J. Am. Chem. Soc., 48, 58 (1926).

2 DONNAN and POTTS, Kolloid-Z., 9, 159 (1911).

* HABKINS, DAVIES, and CLABK, /. Am. Chem. Soc., 39, 541 (1917).

588 PHYSICAL CHEMISTRY

and this is then shaken with oil, a more stable emulsion forms in
which droplets of oil are suspended in a continuous solution of
dilute aqueous soap

If a solution of sodium oleate is 0.005m., the interfacial ten-
sion between benzene and the solution is about 5 dynes per
cm.; the interfacial tension between benzene and pure water is
35 dynes. Since the oleic group is highly soluble in benzene
and the sodium or NaCOO — group is not, the molecules of soap
in the interface probably arrange themselves with the latter
group toward the aqueous layer and the oleic groups toward the
benzene. Inversion of the emulsion takes place, and benzene
becomes the continuous part in which droplets of water are
suspended, when magnesium oleate (which is insoluble as a whole
in water and soluble in benzene) is substituted for sodium oleate
as the emulsifying agent.

Structure of Emulsions. — Emulsions of one liquid in another
probably consist of microscopic droplets of the dispersed liquid
in the continuous liquid. It is not necessary that the continuous
part be present in greater quantity than the dispersed part, so
long as there is enough of the continuous part to fill the voids
between the droplets. Stiff, nonflowing emulsions have been
prepared in which 99 per cent of mineral oil is dispersed in 1
per cent of dilute soap solution.1 Probably in such a system a
magnified cross section would look something like a section
through a comb of honey, with thin films of soap solution repre-
sented by the wax and the oil droplets by the honey. The
viscosity of such a system of droplets in a continuous liquid
would probably be much higher than that of either liquid part
in gross "form; it is sometimes so high that a "jelly" is formed.
But while it has been suggested that "gels" in general are emul-
sions of submicroscopic droplets, this has not been proved and
there is evidence that it cannot be true of all gels. Another
theory is that rodlike particles in a sol make contact with one
another when the sol gels, an effect perhaps roughly analogous
to that of a pile of matches strewn at random. The gelation of a
bentonite suspension has been thus explained,2 but another study3

1 PICKERINQ, /. Chem. Soc. (London), 92, 2001 (1907).

2 GOODEVE, Trans. Faraday Soc., 35, 3421 (1939).

SHAUSER and LEBEAU, /. Phys. Chem., 42, 961 (1938); HAUSER and
HIRSHON, ibid., 43, 1015 (1939).

COLLOIDS. SURFACE CHEMISTRY 589

of bentonite suspensions has shown that after gelation the
individual particles are separated from one another, which is
incompatible with a mechanical theory of gelation assuming a
continuous " scaffolding " structure.

Either theory might apply to the structure of the agar " jelly"
used in bacterial culture. The usual solution is 1.5 per cent by
weight ; solution in water does not occur at a reasonable rate much
below 100°C , but the solution so formed remains fluid until
cooled to about 35°. After " solidification" is produced by
cooling to room temperature, the culture medium does not
again become fluid when incubated at 37° or even higher It
has not been established that this system is either an emulsion
of a more fluid liquid in a less fluid one or a scaffolding of com-
paratively rigid rods supporting a fluid portion by something like
capillarity. Other rigid systems of high water content, such as
silica gel or gelatin or table jellies, have also been studied, but
no agreement as to a general theory of structure has yet been
reached. l

Gels in the Ultramicroscope. — Gelatin and other gels show
under the ultramicroscope a slight Tyndall effect that increases
with concentration, but these gels do not show individual par-
ticles as in the case of sols. Such light as is seen in an ultra-
microscope is probably due to a difference in index of refraction
of the liquid parts forming a gel There is no Brownian move-
ment of the droplets of disperse part. In very dilute disper-
sions of oil in water Brownian movement is observed, but
these are not properly considered gels, since the quantity of
disperse part is very small, and these emulsions have the proper-
ties of suspensions to a far greater extent than they resemble gels.

Viscosity of Emulsions. — Emulsions have viscosities which,
even for very dilute gels, are much higher than that of the
"solvent" and which seem to depend on the rate of shear within
the fluid dispersing medium. No satisfactory theory relating
to the viscosity of colloids has been developed; but it is known
that very slight changes in a gel produce a marked effect upon
its viscosity, and hence viscosity measurements are a delicate

'BoouE, " Colloidal Behavior," Vol. I, p. 378. Chapter XV of this
volume (by H. B Weiser) discusses at length the rather inconclusive evi-
dence in support of the various views regarding gel structure and gives
references to the voluminous literature devoted to it.

590 PHYSICAL CHEMISTRY

means of tracing such changes. About all that can be deduced,
however, is that a change has taken place, the nature of which
is matter for speculation or empirical interpretation. Use of
such methods is extensive in the rubber and nitrocellulose labora-
tories, where the age of an emulsion is a very important factor
in determining its properties. Lack of a satisfactory theory
does not interfere with the use of these measurements as control
methods.

References

A roviow of the field is given in "Surface Chemistry/' edited by F E
Moulton, Am Assoc Advancement Sci , Pub 21 (1943), containing papers
by 15 leading investigators in the field Some oi the many important texts
in the field are as follows.

ADAM: "The Physics and Chemistry of Surfaces," Oxford Universitv Press,

New York, 1932.
HAUSER: "Colloidal Phenomena," McGraw-Hill Book Company, Inc , New

York, 1939.
HOLMES: "Introduction to Colloid Chemistry," John Wiley & Sons, Ine ,

New York, 1934.
KRUYT: "Colloids," translated by van Klooster, 2d ed , John Wiley & Sons,

Inc., New York, 1930

RIDKAL: "An Introduction to Surface Chemistry,'1 2d ed., Cambridge Uni-
versity Press, London, 1930
THOMAS: "Colloid Chemistry," McGraw-Hill Book Company, Inc., New

York, 1934.
WEISER: "Inorganic Colloid Chemistry," John Wiley & Sons, Inc , New

York, Vol. I, "The Colloidal Elements," 1933, Vol. II, "The Hydrous

Oxides and Hydroxides," 1936.

CHAPTER XV 111
FREE ENERGY OF CHEMICAL CHANGES

In this chapter we consider some simple applications of thermo-
dynamics to changes in state involving chemical reactions.
Isothermal changes in state will be considered first and then the
effect of changing temperature on the values of the thermody-
namic properties. It should be remembered that changes in all
the thermodynamic properties p, v, T, E, Hy S, A, and F depend
only on the change in state, that AH and AE may be evaluated
along paths which are not thermodynamically reversible when
convenient, and that AS, A^4, and AF must be evaluated along
reversible paths. The definitions and most of the equations
that are to be used have been developed in Chap. II, but it will
be profitable to give some further discussion of them before
entering upon the calculations.

Maximum Work of Isothermal Changes in State.1 — The ideal
reversible process is one in which the pressure (or temperature
or potential or other property) of the working system differs
only by an infinitesimal amount from the pressure (or tempera-
ture or potential or other property) of the system on which the
work is done. Such a change may be reversed by an infinitesimal
change in the pressure, and in a change in state taking place
reversibly the work done is the maximum obtainable. Expendi-
ture of this work upon the system will restore it to its original
state. Although no actual process is reversible, yet by eliminat-
ing friction, electrical resistance, and other factors involved in
inefficiency this ideal type of change may be closely approached.

1 This section and the following one are quoted from Lewis, /. Am. Chem.
Soc., 35, 1 (913), with only minor changes. Readers of this chapter will not
need to be reminded that this brief treatment makes no pretense of being
complete. Its purpose is to illustrate a few of the simple operations that
may be carried out with free-energy data and to stimulate students who find
these calculations attractive to read further in the field. Five excellent
books in which to do further reading are given on page 49.

591

592 PHYSICAL CHEMISTRY

The maximum work that can be obtained from a system on
passing reversibly from state 1 to state 2 at the same temperature
is of great importance, for it is independent of the particular
reversible process employed. If this were not true, then by
proceeding from 1 to 2 by one isothermal reversible process
and returning from 2 to 1 by another isothermal reversible
process requiring less work a certain net amount of work would
be gained. This work could come only from heat absorbed
from the surroundings according to the first law of thermo-
dynamics, since the whole process is an isothermal cycle for
which fdE = 0 and dq must be equal to dw. But the second,
law of thermodynamics asserts the impossibility of converting
heat into work by an isothermal cycle of changes. Note that the
second law does not say that $dw = 0 for an isothermal cycle,
nor does it forbid the conversion of work into heat by an iso1
thermal cycle. It says that the work done by the system in an
isothermal cycle is zero or negative.

Since no work is obtainable from an isothermal reversible
cycle, it follows that the reversible work done by a system in
passing isothermally from state 1 to state 2 is the same by all
paths. We may then consider the maximum work as the differ-
ence between two quantities that are properties of the system
in the specified states. One of these, A], is completely deter-
mined by the initial state of the system, and the other, A2, is
determined by the final state of the system. These quantities
AI and A 2 may be designated the isothermal work contents of
the system before and after the change took place. Neither of
the values is determinable for the system; we are to consider
changes in A, just as we considered changes in H or E in earlier
chapters. The maximum work to be derived from an isothermal
change in state is

wm« = — AA = AI — A 2 (t const.)

It will not necessarily be true that — AA is the actual work per-
formed in an isothermal change in state, for many changes take
place while performing less work than the maximum that could
be obtained in an ideally reversible process. Even if the work
done were zero, — AA for the change would be equal to w^,
and at least this amount of work would be required to reverse
the change in state. The ratio of the actual to the maximum

FREE ENERGY OF CHEMICAL CHANGES 593

work is the efficiency of the process, but — AA is the decrease in
the capacity of the system to do work at constant temperature
and is independent of the work efficiency of the process, for there
is no law of conservation of work. In conformity with the
custom already followed for E and //, we write equations in terms
of &A rather than — A.4, so that this equation is

A.4 = Az - A i = -ww (t const.) (It)1
The definition of the quantity A given on page 45 was

A = E - TS

For a reversible process at constant temperature
dA - dE - T dS

and the last term is equal to the heat absorbed, dqnv. Hence
by substituting dqTOV — dwm&x for dE above, we have

dA = dgrev — c?u>mB* ~ T dS = — dwm^ (t const.)

which upon integration gives equation (It) above.

Free-energy Increase in Isothermal Changes in State. — For
many calculations in chemistry there is another quantity that
is more convenient to use than the isothermal work-content
increase, especially since the tabulated data are in terms of this
quantity. The quantity is related to A in the same way as H
is related to E ; but before giving a mathematical expression for it,
its significance may be illustrated by a concrete example. Sup-
pose that an electric cell operates isothermally and reversibly
under atmospheric pressure, producing the electrical work we and
at the same time undergoing a change in volume. The quantity
of work we represents all the work reasonably available from
the cell, for example, that which could be obtained by operating
an electric motor. But it is not we that we have denned as
— AA, for a certain amount of mechanical work is also involved
in the change in state, owing to the volume change against the
atmospheric pressure. If At; represents the increase in volume
when chemical substances react isothermally and at constant

1 The letter t included with the number of an equation indicates the restric-
tion of the equation to changes at constant temperature,

594 PHYSICAL CHEMISTRY

pressure through the operation of an electrical cell, the mechanical
work done by the system is p AT, whence

AA — — wc — p Av
or

-we = &A + p At; (20

This important quantity, which in general represents the work
actually available from an isothermal change, is itself dependent
only on the initial and final states of the system and is thus a
property of the system in a specified state, for — Ayl, p, and Av
depend only on the initial and final states. It is commonly called
the free-energy change We shall write as our formal definition
of the free energy

F = A + pv (3)

and for isothermal changes

AF - A4 + AO) (40

This definition of F is in no sense a retraction or a revision of the
definition F = H — TS given in Chap. II, where the definition
A = E — TS was also given. Since H and E differ by pvt it
will be seen that F and A must differ by the same quantity.

By substituting AA = — wmai from equation (10, we have
another equation for isothermal free-energy increase,

AF = -uw + A(p») (50

The quantity F will be called the free energy and AF the free-
energy increase accompanying a change in state.1

Electrical work is the product of potential and quantity of
electricity, we = ENF, in which E is the potential, N is the num-
ber of faradays of electricity required to produce the change in
state, and F is Faraday's constant. The maximum work obtain-
able from the isothermal operation of a cell at constant pressure
has already been given as A A = — we — p Av, and on substituting

1 This is the definition of free energy given by Lewis and followed in
"International Critical Tables*' and in the publications of the American
Chemical Society; it is the Gibbs £ and is written G in some recent books
Some European chemists call our A the free energy, following Helmholtz,
but most American chemists calf our F the free energy. Our E is Gibbs's €,
our H is his x, and our A is his ^.

FREE ENERGY OF CHEMICAL CHANGES 595

this in equation (40 we have another means of evaluating an
isothermal free-energy increase,1

AF = -ENF (t const.) (60

It will be recalled from Chap. VIII that the increase in enthalpy,
Ajy, accompanying an isothermal change in state is the negative
of the heat evolved. Similarly, the increase in free energy, AF,
of an isothermal change in state is the negative of the available
maximum work derived from it, other than that due to changes of
p or v, and AJ. is the negative of reversible work of all kinds
available from the change in state. When work (electrical
work, for example) is done upon a system at constant tempera-
ture, its free-energy content increases, and it is capable of per-
forming this work again when it is desired.

The condition of reversibility should be kept in mind con-
stantly in connection with changes in A or F. A system decreases
its work content and its free-energy content during a spontaneous
change in state by the maximum amount, whether it does the
maximum amount of work or a smaller quantity. Thus w^
depends only on the change in state that takes place, but
the actual work done may be any amount smaller than wmAT;
it may even be zero. The least work that will reverse the
change in state is wm&3i, regardless of the work efficiency of the
first change.

We have already seen that, because of the way free energy
is defined, the work of reversible isothermal expansion at con-
stant pressure is added to the work-content increase in evaluating
AF, so that AF = A^4 + A(py) becomes

AF = — wmax + p(vz — t>i) (t const.)

when the pressure remains constant, as required by equation
(40. If we apply this equation to the reversible isothermal
evaporation of a liquid, p is the vapor pressure at the tempera-
ture of the evaporation, (t>2 — t>i) is the difference between the
volume of the saturated vapor and the liquid from* which it forms,
so that p(v<L — Vi) is equal to wm. Hence for such a change in
state AF = 0, and the molal free-energy content of a liquid is

1 In these equations and throughout the book, the italic letter F denotes
Faraday's constant, 96,500 amp.-sec.; and the bold-faced letter F denotes
the free energy.

596 PHYSICAL CHEMISTRY

equal to that of its saturated vapor. For example, in the change
in state,

H2O(/, 100°, 1 atm ) = H2O(0, 100°, 1 atm.) AF = 0

This is not to say that the free-energy contents of liquid water
and water vapor at 1 atm. pressure are equal at any other tem-
perature than 100° or that liquid water at 100° arid water vapor
at 100° and some pressure other than 1 atm. have the same free-
energy contents, for these statements would be untrue. (Some
illustrations are given in the next section.) But at 25° the free-
energy contents of liquid water and water vapor at 0.0313 atm
would be equal; for this is the vapor pressure of water at 25°, and
evaporation at this temperature would be a reversible isothermal
process for which wmax — p(v2 — Vi). Hence, for the change in
state,

H2O(Z, 25°, 0.0313 atm.) = H2O(0, 25°, 0.0313 atm.)
AF = -it>ma* + 0.03 13 (t>, - vi) = 0

It will be true, in general, that AF is positive when an isothermal
change in state requires the expenditure of work from an outside
source in order to produce it; that AF = 0 for any equilibrium
change in state; and that AF is negative for spontaneous changes,
i.e., for changes that are capable of doing work in approaching
equilibrium. Thus, a solute at a greater pressure than its equilib-
rium pressure above a solution may be expanded roversibly with
the production of work and a decrease in its free-energy content
and then pass into solution reversibly under its equilibrium pres-
sure. But to remove a solute from a solution to a vapor phase
in which its pressure is higher than its equilibrium pressure
requires work from an outside source. The mechanism would
consist in removing the solute at its equilibrium pressure, for
which

AF = — tew + p At> = 0

followed by isothermal reversible compression, which would
require work and increase the free-energy content of the substance.
Isothermal Change of Free Energy with Pressure. — Con-
sider the change in free energy in an isothermal process, of which
the net result is the expansion of n moles of a pure substance
from the pressure p\ to the pressure p2. This may be done

FREE ENERGY OF CHEMICAL CHANGES 597

reversibly by allowing the substance to expand or contract under
an external pressure that is always kept equal within an infin-
itesimal amount to the pressure of the substance. Then

AA — — jp dv
and, from equation (4tf),

AF = -Jp dv + J d(pv) = Jv dp (70

This same relation follows from equation (31), page 47,
dF = -SdT +v dp

for in an isothermal reversible process the first term on the
right-hand side is zero, and thus

« / = v or dF = v dp (t const.)

OP/ T

Over moderate pressure ranges the isothermal change in free-
energy content of liquids and solids is very small. For .example,
in the change in state

H20(J, 25°, 5 atm.) = H20(Z, 25°, 1 atm.)
the volume is substantially constant at 18 ml. per mole;
JV dp = v(pz — pi) and AF = — 72 ml.-atm. or —1.7 cal.

Hence in chemical changes, in which AF is commonly several
thousand calories, the change in free-energy content of a liquid
phase or solid phase with changing pressure is usually negligible.
But AF would not be negligible for large pressure changes, and
for such changes v must be expressed as a function of p before
integrating equation (70-

When an ideal gas undergoes isothermal reversible expansion,
its volume is given as a function of the pressure by the relation
v = nRT/p, and for this change equation (7t) becomes

AF = nRTln («)

For the isothermal expansion of a mole of nearly ideal gas as
shown by a change in state such as

1O2(0, 25°, 5 atm.) = 102(0, 25°, 1 atm.)

598 PHYSICAL CHEMIST RY

AF is —954 cal , and since in this change A(jn>) is nearly zero,
AA is — 954 cal. and wmax is 954 cal. At high pressures equation
(8/5) would be inaccurate, and some adequate means of expressing
v as a function of the pressure must be found before performing
the integration of equation (70

Free Energy and Activity. — It will be recalled from previous
chapters that the activity a of an ideal solute is equal to its
molality and that for one which is not ideal a = my, in which 7 is
the activity coefficient, a number by which the molality must be
multiplied to correct it for deviation from the behavior of an ideal
solute For an ideal solute that has a vapor pressure, the
activity is proportional to the vapor pressure. Since AF for
any change must be the same by all paths, we may transfer a
solute from an activity «i to an activity a2 by the following
isothermal reversible steps:

1 Evaporate n moles of solute from a large quantity of solu-
tion in which its activity is ai and over which its vapor pressure
is pi The quantity of solution is assumed to be so large that
the molality is substantially constant duiing removal of n moles
of solute For this process the maximum work is PI(VI — raoiute),
in which Vi is the volume of n moles of vapor at pi and vno\^ is
the change in volume of the solution caused by the removal of
the solute; and for this change A(pv) is also pi(v} — f.oim»)
Hence

AF = -uu« + A(pf) = 0

This calculation shows that the molal free-energy content of a
solute is the same in a solution and in the vapor in equilibrium
with the solution, as was shown in an earlier paragraph to be true
of a pure liquid and its vapor. It is a general truth that the
molal free-energy content of any substance is the same in two
phases which are in equilibrium, and hence AF = 0 for the
transfer of it from one phase to another phase with which it is
in equilibrium.

2. Expand or compress the vapor from pi to p^ for which AF is
nRT In (PZ/PI) by equation (8/).

3. Condense the solute into so large a quantity of solution
of the solute at a2 that the molality is substantially unchanged
by the addition of n moles of solute. For this reversible process

— t>2), and this is also the value of A(^); hence

FREE ENERGY OF CHEMICAL CHANGES 599

AF = 0. The summation of free-energy changes for the entire
isothermal change in state,

n moles of solute at ai — > n moles of solute at a^

is AF = nRT In (p2/pi); and since the ratio pi/p\ is equal to
a^/di (i.e., since Henry's law applies to ideal solutes), we may
write

AF = nRT In- (90

Though we have chosen a volatile solute for this illustration,
we might have transferred a nonvolatile solute by an electro-
chemical reaction, as we shall do in the next chapter, to obtain
the same equation. This equation (9/) is in fact applicable to
the transfer of a solute by any reversible means, and regardless
of whether it has a vapor pressure or not, since AF has the same
value for any change in state by all paths

An ideal solute is one for which a = m. This relation is
almost satisfied by nonionized solutes in water at moderate con-
centration, so that m2/mi or C%/Ci may be used for nonionized
solutes in place of a2/ai in equation (9£), with little error. For
ionized solutes the ratios m2/mi and az/ai are not equal until
extreme dilutions are reached, and thus m2yz/miyi is required
for a2/ai in exact calculations In solutions containing a single
solute, activity coefficients may be estimated by means of the
equations .given at the end of Chap. VII, and we are to take up
other means oi obtaining them in the next chapter. While
the exact calculation of activity coefficients in mixtures of elec-
trolytes is too difficult for beginners, a suitable estimate may
usually be made. For example, it will be better practice to use
0.8 for the activity coefficient in a mixture of uni-univalent
solutes at O.lm. and 0.9 for such a mixture at O.Ol^n. than to
omit the correction entirely, though it will be still better prac-
tice to use the measured activity coefficients 0.796 for 0.10m.
HC1, 0.778 for 0.10m. NaCl, 0.765 for 0.10m. KBr, etc. One
must remember also that these estimated activity coefficients
do not apply in solutions of other ionic types such as O.lm.
H2SO4, in which the activity coefficient is 0.27, O.lm. ZnCl2, in
which it is 0.50, or O.lm. ZnS04, in which it is 0.15. Students
should refer to Table 98 on page 641 for data of this type.

600 PHYSICAL CHEMISTRY

Both of the equations (80 and (90 must be applied to a single
molecular species. For example, if a mole of nitrous acid is to
be transferred isothermally from a solution in which its molality
is mi and the fractional ionization is a\ to a solution in which
its molality is ra2 and the fractional ionization is «2, this may be
accomplished by transferring a mole of HNO2 from mi(l — «i) to
7712(1 — #2) or by transferring a mole of H+ and a mole of NO 2"
from mini to m^o^. The corresponding free-energy increases are

AF = ]flrin^|^ and AF = 2RT In

mi(l — oil) .

Since the same change in state is accomplished by either pro-
cedure, the free-energy increases must be equal, and on equating
them we have

m\(\ — oil) m2(l — a 2)

which is required by the ionization equilibrium. Since nitrous
acid conforms rather closely to the requirement of the equation
Kc = (H+)(N02~)/(HNO2), either procedure is satisfactory.
If hydrogen chloride, or H+ and Cl~~, is to be transferred,
the activity coefficients of the ions may not be canceled from
equation (90, as was done for nitrous acid above. Let the change
in state at 25° be

1HC1 (4m.) -* 1HC1 (6m.)

The vapor pressure of HC1(0) is 0.24 X 10~4 atm. above a 4m.
solution of HC1, and so HC1(00 at this pressure has the same
molal free-energy content as H+ + Cl~" at 4m. But the activi-
ties of H+ and Cl~~ in 4/n. HC1 are not 4 — they are about 7.0.
Similarly, in 6m. HC1 the activities of the ions are about 20.1,
and HCI(0). at 1.84 X 10~4 atm. (the vapor pressure) is in
equilibrium with HC1 (6m.) or with H+ and Cl~ at activities
of 20.1. If the transfer is brought about isothermally and
reversibly through the vapor, we see from equation (80 that

L84X10-4

and if it is brought about isothermally and reversibly by the
transfer of the ions (for example, through the operation of two

FREE ENERGY OF CHEMICAL CHANGES 601

opposed electrolytic cells), equation (90 applies, and

These free-energy increases are equal, of course, but they do
not lead to an ionization constant for HC1 when equated, for
this substance has no ionization constant. Activities in con-
centrated hydrochloric acid are obtained from the potentials
of cells in a way explained in the next chapter.

Free-energy Increase and Chemical Equilibrium. — The free-
energy increase for an isothermal change in state that involves
a chemical reaction is related to the equilibrium constant of the
reaction by an important equation that is now to be derived.
It will be recalled that AF for a specified change in state is
independent of the path or process by which the change occurs
but that in order to evaluate this change we must proceed by
some path which is reversible in the thermodynarnic sense
For our convenience we may choose any reversible path for which
the calculation is readily performed. Let the chemical reaction
be

aA + MB = dD + eE

and assume that the substances involved are ideal gases to
which we may apply equation (80 The equilibrium constant
for this chemical reaction is

A chemical equation does not adequately specify a change in
state, for the partial pressures of the substances and the tem-
perature must also be given. The isothermal change in state
at the temperature T is

aA(at PA') + &B(at PB') = dD(at pD') + eE(at p*')

For the ideal process by which this change in state is conceived
to occur reversibly, we may assume an " equilibrium box" con-
taining an equilibrium mixture of the substances and fitted with
four cylinders. Each cylinder connects to the box through a
membrane permeable to one substance only; each has an arrange-
ment for closing the membrane and a movable piston for altering

602 PHYSICAL CHEMISTRY

the pressure. At the start one cylinder contains a moles of sub-
stance A at a pressure p& ', a second contains b moles of substance
B at a pressure p&', and the membranes between these cylinders
and the equilibrium mixture are closed. The pistons of the
third and fourth cylinders are in contact with the membranes
permeable to C and D, so that these cylinders are empty. The
primed pressures pA' and pB' are the ones arbitrarily specified in
the change in state, and they do not satisfy the equilibrium rela-
tion; the pressures without primes, pA, etc., do satisfy this relation
As the first step of the reversible process, let a moles of A
expand (or be compressed) isothermally and reversibly from pA'
to pA, and let b moles of B expand (or be compressed) isother-
mally and reversibly from PB' to PB while the membranes remain
closed. The free-energy increases for these processes are

AFi = aRT In 2± and AF2 = bRT In -^
PA PB

Now open the membranes of these cylinders, and force A at
PA and B at PB into the equilibrium mixture through their
respective membranes; as they react, withdraw d moles of D
through its membrane at the pressure pD and e moles of E
through its membrane at the pressure pE. At the A cylinder, the
maximum work performed by the system is — PA^A, and A(pv) is
also — PA^A, whence AF = — wmax + A(py) = 0. It is also true
of each of the other cylinders that the work performed is only
that of a change of volume under constant pressure, so that
ww = p Av and AF = 0 for the entire second step.

The change in state is completed by closing the membranes
of the D and E cylinders, compressing (or expanding) d moles
of D isothermally and reversibly from pD to pD' and e moles of
E isothermally and reversibly from PE to pE'. For these steps

AF3 = dRT In & and AF4 = eRT In ^

PD PE

Upon adding the free-energy increases for all the steps and
rearranging so that all the initial or final pressures specified*
in the change in state appear in one term and all the equilibrium
pressures appear in another term, the summation becomes

RT in _ RT ln

PA °PB*

FREE ENERGY OF CHEMICAL CHANGES 603

This equation allows us to calculate AF for any gaseous isothermal
change in state for which the equilibrium constant is known.
We shall consider in the next section the use of tabulated data
that allow the calculation of equilibrium constants at a single
standard temperature in much the way that enthalpy changes at a
standard temperature were calculated from molal enthalpy
tables in Chap. VIII. We shall also have later in this chapter
an equation for calculating AF at any temperature from its value
at the standard temperature. Hence equation (lOt) is an impor-
tant one.

In order to save labor when equation (100 ig to be written
often, it has become fairly common practice to write it

AF - RT In Q - RT In K

in which Q indicates a fraction containing the pressures appear-
ing in the formulation of the change in state, and arranged
according to the same conventions as in the equilibrium constants,
and K is the equilibrium constant.

For changes in state involving solutes, an equation of similar
form involving the activities of solutes may be derived. For
the general change in state at T,

dD(at aD') + rE(at aE') =» 0G(at aG') + AH (at an')
the increase in free energy of the isothermal change is

AF _ RT In SS^Hi; _ RT ln 22^ (Ut)

CD d0E ' aDdaEe

where the activities not primed satisfy the equation for equilib-
rium

Since the activities of nonionized solutes at moderate con-
centrations are nearly equal to their concentrations, equation
(lit) may be altered by substituting concentrations or molalities
for the activities. In some approximate calculations involving
ions this may also be done. For example, it will matter little
whether the equilibrium concentration of a substance is 10~6w.
or 10~~7w. if the object of a process is to precipitate it completely.

604 PHYSICAL CHEMISTRY

But there are also many equilibriums involving ionized solutes
in which a rough approximation is inadequate, and for such cal-
culations activities must be used in equation (lit).

Free Energy and the Third Law of Thermodynamics. — In
Chap. II we defined the free energy as

F = H - TS
and, for isothermal changes, this becomes

AF = AH - T AS (I2t)

Thus we may calculate AF attending any isothermal change in
state for which A/7 and AS are known.

It will be recalled that, according to the third law of thermo-
dynamics, the entropy of any pure crystal is zero at the absolute
zero of temperature. It will also be evident that at a standard
temperature and pressure, such as 298°K. and 1 atm. pressure,
entropies are not zero; they are fCpd In T between 0° and 298°K.
Both free energies and entropies at a given temperature change
with pressure, and for liquids and solids the changes in entropy
or free energy are small for moderate changes in pressure. For
ideal gases the change of entropy with pressure at constant tem-
perature is given by the equation

•

AS = -nRln^
Pi

This equation follows from equations (80 and (I2t), since AH
is zero for the isothermal expansion of an ideal gas. It also
follows from the fourth "Maxwell relation" as shown in the
footnote on page* 607.

Entropies for elements or compounds are usually given in
tables in calories per mole per degree at 298°K. and 1 atm. pres-
sure for the state of aggregation stable under these conditions
and are designated $°298. A few are given in Table 96, and many
more are known. Since AS = $a — Si for any change in state,
an entropy table and an enthalpy table provide data for cal-
culating AF. *

Standard Isothermal Changes in State. — The changes in state
with which we are to be concerned in this section and in the
next three sections are called "standard changes in state. " In
such changes each substance, element or compound, appearing

FREE ENERGY OF CHEMICAL CHANGES 605

in the description of a change in state, is in its stable state of
aggregation at 1 atm. pressure for the temperature concerned.
Following the common custom, we take as our standard tempera-
ture 25°C. or 298°K., since this is the temperature for which
tabulated data are available. Solutes in a standard change in
state are used or formed at unit activity. Some illustrations of
standard changes in state are

MH2(1 atm.) + HC12(1 atm.) = HC1(1 atm.)

2Ag(s) + l/202(l atm.) = Ag20(s)
Ag2O(s) + 2H+Cl-(w.a.) = 2AgCl(» + H2O(Z)

H2O2(w.a. = 1m.) = H2O(/) + MO2(1 atm.)

It has become common practice in physical chemistry to desig-
nate the changes in enthalpy, free energy, entropy, etc., for
standard changes by a superscript zero attached to the symbol
for the quantity, followed by specification of the temperature
with a subscript, A//°298, AF°298, AS°298, etc. Standard changes
in state may of course be subtracted or added, with addition
or subtraction of the AF°s, as is true of any other changes.
They may be added to changes that are not standard; but the
sum of a AF and a AF° is a new AF and not a new AF°,

Standard Free-energy Contents of Elements. — In Chap. VIII
we defined the enthalpy of an elementary substance at 1 atm.
pressure and the standard temperature as zero and we compiled
a table of molal enthalpies of compounds relative to this stand-
ard. For moderate changes in pressure the variation of H with
pressure was negligible for liquids and solids, and for ideal gases
(dH/dp)T = 0, so that the enthalpies of the elements were sub-
stantially zero at any moderate pressure, and the enthalpies of
compounds were substantially the same at any moderate pressure
as at 1 atm. pressure. Enthalpies so calculated were relative
and not absolute, since they were based on a standard arbitrarily
defined as zero for the elements at the standard temperature.

For the purpose of preparing a table of standard molal free
energies of compounds we shall also define' the free energy as zero
for an elementary substance in its stable state of aggregation at
1 atm. pressure and the standard temperature - as zero. The
molal free energies of compounds at 1 atm. and the standard
temperature will thus be the free-energy increases attending their

606 PHYSICAL CHEMISTRY

formation at 1 atm. pressure from the elements at 1 atm. pressure.
Variations in pressure of a few atmospheres will cause negligible
changes in the free energies of liquids and sohds, as was shown on
page 597. This will not be true of gaseous compounds, nor will
the molal free energy of gaseous elements be zero at the standard
temperature and any moderate pressure, since (d¥/dp)T = t>,
from page 597. It may be seen from equation (St) that, if the
free energy of a mole of oxygen (for example) is zero at 1 atm.
arid 298°K, its free energy will be 1365 cal. at 10 atm., -410 ca).
$t 0.5 atm., -1365 cal. at 0.1 atm., and -2730 cal. at 0.01 atm.,
all for 298°K.

The molal free-energy content of Br2(0) at 25° and 1 atm
is given in Table 95 as 755 cal. Since this is a positive free-
energy content, bromine vapor does not assume this condition
spontaneously, and it is a familiar fact that the vapor pressure
of bromine is less than 1 atm. at 25°. The experimental fact
recorded by this free-energy content is the vapor pressure of
bromine at 25°. We shall use this molal free-energy content to
calculate the vapor pressure, though it will be understood that
this is the reverse of the actual procedure by which the free-
energy content of bromine in the. imaginary state of vapor at 1
atm. pressure at 25° was calculated from the measured vapor
pressure.

Let the changes in state at 25° be

Br2(7) — >Br2(g, satd vapor, p atm.) — * Br2(0r, 1 atm.)

Since bromine at 25° and 1 atm. is a liquid, the free-energy
content of the system in its original state is zero by the conven-
tion we have adopted. When it evaporates isothermally to form
saturated vapor, the only work done is p At;, so that

AF = -wmax + p At; = 0

and the free-energy content of saturated vapor is also zero. For
the second step AF is RT In (1/p) from equation (8Z), which is
755 cal., whence log p = —0.553 and p = 0.280 atm.

The molal free-energy content of I2(gr, 1 atm.) is given as
4630 cal. in Table 95, and this is another example of a free-energy
content ascribed to a substance in an imaginary state. It
records the experimental fact that the sublimation pressure of
iodine at 25° is 0.309 mm., and the entry itself is useful in making

FREE ENERGY OF CHEMICAL CHANGES 607

calculations which involve iodine vapor. There is no implication
that iodine vapor has been observed in this condition.

Standard Entropies of Elements. — The standard entropy of
oxygen gas at 298°K. and 1 atm. pressure is /S°298 = 49.03 cal.
per mole per deg. Its molal entropy at 298°K. and some other
pressure, such as 0.1 atm., will differ from 49.03 by an amount
shown by the equation1

AS = -#ln£-2
Pi

which is 4.57 e.u. lor the change in state

O2(0r, 298°K., 1 atm.) = O2(flf, 298°K, 0.1 atm.)

whence the entropy of oxygen at 298°K. and 0.1 atm. is 53.6 e.u.
The same result is obtained, of course, from the equation

AF = AH - T AS (120

From equation (St) we calculate AF = — 1365 cal. for the expan-
sion of a mole of gas from 1 atm. to 0.1 atm. at 298°K., and since
AH = 0 for the expansion,

- 1365 cal. =0-298 AS AS = 4.57 e.u.

As another illustration, we may calculate /S°298 for I2(g) in the
imaginary state of vapor at 298°K. and 1 atm. fromj^$° for the
standard change in state

120) = l*(g, 1 atm.)

1 For the isothermal expansion of an ideal gas, AE = 0, and if the expan-
sion takes place reversibly as well, <?rev = ^rev Since grev = T A/S at con-
stant temperature and wnv — nRT In (v2/Vi) = —nRT In (PZ/PI} = TAS,

AS = -nR In ^
Pi

This equation also follows froftn the fourth "Maxwell relation" given on
P 48,

dpT
For an ideal gas pv = nRT and —(dv/dT)p = —nR/p, whence

dS ^ -—dp and AS * -nR In 2?
fl Pi

for isothermal changes in pressure.

608 PHYSICAL CHEMISTRY

for which AF°298 = 4630 cal. was calculated on page 606. The
heat of sublimation at 298°K. is A/f = 14,877 cal,, and when
these quantities are substituted in the equation

AF° = A#° - T
4630 = 14,877 - 298 AS0

A$° is 34.4 e.u. Since the standard entropy of the solid is
S°298 = 27.9, /S°298 = 62.3 for I2(0).

At the risk of some repetition, it must be pointed out that AF
and A/S for the sublimation to yield saturated vapor at 25°, i.e.,
for the change in state

!»(«) = I*(0, 4.07 X 10~4 atm.)

are not the same as AF° and A£° for the change in state which
forms the vapor at 1 atm. pressure. For the formation of
saturated vapor, at 298°K. AF = 0, A# = 14,877 cal., AS = 49.9,
and the entropy of the saturated vapor is 77.8 e.u. For the
compression of the vapor to 1 atm. from the saturation pressure,

I2(0, 4.07 X 10~4 atm.) = I2(gr, 1 atm.)

AF = 0, AF = 4630 cal., AS = - 15.5, and S\w is 62.3 as before.
Since these last two changes in state are not standard ones, no
values of AF° and A/S° may be assigned to them.

StandarcOYee Energies of Compounds. — The standard free
energy of a compound is defined as the free energy 01 its forma-
tion from the elements by a standard change in state. The
fundamental equations for these calculations have all been given,
and we have already seen that for the evaluation of free energy we
must proceed along reversible paths. The standard free energy
of an ion in aqueous solution is its free energy of formation from
the elements in a standard change as well, and the standard for
ions is unit activity.

For the special condition in a gaseous reaction that the pressure
of each reacting substance is 1 atm. and the pressure of each reac-
tion product is 1 atm., i.e., for standard changes in state, equa-
tion (100 reduces to

AF° = -firing (130

It must be understood that this equation applies, not if the total
pressure of a mixture is 1 atm., but only when the pressure of each

FREE ENERGY OF CHEMICAL CHANGES 609

substance is 1 atm. The standard temperature for which free
energies are recorded is 298°K., but equation (130 may be used
for any constant temperature, provided that the initial and
final pressures of each substance involved are 1 atm. at this
temperature.

A corresponding equation may be written for changes in state
in which solutes are used or formed at unit activity. The general
change in state is

dD(a»f = 1) + eE(aE' = 1) = gG(aGf = 1) + /iH(aH' = 1)

and for this change the first logarithmic term in equation (110
becomes zero, so that

AF° = ~RTInKa (130

For many approximate calculations molalities or concentra-
tions may be used, and for standard changes in state in terms of
these quantities the free-energy equation is

AF° = -RTlnKc (130

We designate by (130 the equation in any terms. The super-
script zero on the AF° is intended to indicate that the first term in
equations (100 or (110 nas been made zero by the way in which
the change in state has been formulated, namely, by making it
a standard one. This superscript should always be written for
standard changes in state and omitted when the change in state
is not standard, as is the usual custom in physical chemistry.

For standard changes in state taking place in an electrolytic
cell, equation (60 becomes

AF° = -EQNF (uo

and the equation applies only to cells in which standard changes
in state take place reversibly with the development of a maxi-
mum or reversible potential E®.

For standard changes in state equation (120 becomes

AF° = AH ° - T AS0 (150

and the equation likewise applies only when standard entropies
are used. As has been pointed out so often before, the distinc-
tion between AH ° and Afl" is usually not required, since enthalpy
changes are small for moderate changes in pressure. Since we

610 PHYSICAL CHEMISTRY

have used AHQ) written with a subscript of zero, as an integration
constant in expressing AH as a function of the temperature, it
must be observed that A//° with the superscript of zero is not
this integration constant but A// for a standard change in state.
When it is necessary to indicate the integration constant in a
standard change in state, this is written with zero as both sub-
script and superscript, A//°0.

It is seldom possible to determine the free energy of formation
of a given compound directly by all three of the equations (130,
(140, and (150, though free energies determined by two of them
may usually be checked for the difference between them by the
third method. Before making any calculations, we summarize
the standard conventions for elements

H = 0 at any temperature and 1 atm. pressure for elements in
the state of aggregation stable at that temperature (Changes
in H with moderate changes in pressure may be neglected in all
but the most precise calculations.)

F = 0 at any temperature and 1 atm. pressure for elements in
the state of aggregation stable at that temperature. [Changes
in F with moderate changes in pressure are negligible for liquids
and solids; they are given for gases by equation (8t).]

S = 0 only at absolute zero.

Some calculations of standard free energies of compounds at
298°K. will now be given to illustrate the methods.

Silver Oxide. — 1. By plotting the logarithm of the dissociation
pressures for silver oxide given on page 396 against 1/T, we find
that A// = —7250 cal. and ACP is zero or very small for the
reaction

2Ag(s) + ^0,(0) = Ag20(s)

Through the van't Hoff equation we calculate the equilibrium
pressure of oxygen at 298°K. to be 1.66 X 10~4 atm Since
the equilibrium constant for the change in state as written is
the reciprocal of the square root of this pressure, equation (130
gives

AF°298 = -/erin— = = -2580 cal.
Vpo2

We define the free standard energies of the elements as zero, and

FREE ENERGY OF CHEMICAL CHANGES 611

thus the free energy of Ag20(s) is —2580 cal. per mole at 298°K.
from this calculation.

2. A cell of which the anode is silver and silver oxide, the
electrolyte dilute sodium hydroxide, and the cathode oxygen
gas bubbling over platinum would appear to be a means of
determining the free energy of silver oxide, since the cell reac-
tion is the formation of a mole of Ag20 for 2 faradays. But it
is a requirement in free-energy calculations that a reversible
process be used, and neither of the electrode reactions is reversi-
ble in the thermodynamic sense. Operation of the cell forms
silver oxide but does not form it reversibly, and thus the measured
potential (which is erratic) is not the maximum potential.
Accepting the free energy as determined by the other two
methods, one may calculate that the reversible potential should
be 0.055 volt, and such a potential is sometimes recorded for
this cell in tables of oxidation potentials. No harm is done by
such an entry if one understands that the potential has been
calculated and is not a measured reversible potential.

3. The standard entropies at 298°K. are 10.2 for silver, 49.03
for oxygen, and 29 1 for silver oxide, from which we may calcu-
late an entropy balance for the formation of silver oxide as
follows :

2Ag(s) + M02(<7) = Ag20(s)
2(10.2) + ^(49.03) = 29.1 - AS0
AS0 = - 15.81 cal. per deg.

Taking AH = —7250 cal. for the reaction, as before, we have
AF°298 = A# - T AS° = -7250 - 298(-15.81) = -2530 cal.

If A// is taken from Table 58, where it is given as —7300 cal.,
AF°298 becomes —2580, which is substantially the value given in
Table 95. *

Silver Chloride. — 1. Direct equilibrium measurements are not
available for calculating the free energy of formation of silver
chloride, since the pressure of chlorine at equilibrium is too small
for measurement. The theoretical equilibrium pressure for the
reaction

Ag(«) + MChfo) = AgCl(s)
PITZER and SMITH, J. Am. Chem. Soc., 69, 2633 (1937).

612 PHYSICAL CHEMISTRY

may be calculated from the free energy derived from other
methods through equation (130?

AF° = -RTlnK = -RTln— J= = -26,200 cal.

to be 10~88-4 atm., but such a quantity has no meaning as a
pressure. In a table of equilibrium constants this pressure might
be given as a record of the molal free energy derived from other
methods, and no harm IB done in recording it so long as it under-
stood that no pressure measurement is implied.

2. The potential of a cell in which silver chloride forms reversi-
bly is 1.136 volts at 298°K. For 1 faraday the change in state
and the free-energy increase are shown by the equations

Ag(s) + MC12(1 atm.) = AgCl(«)
AF° = -E°F = -109,600 joules = -26,220 cal.

Details of the method will be given in the next chapter.

3. The standard entropies of all the substances involved are
well known; therefore, through an entropy balance and the
enthalpy of formation, which is —30.300 cal., we obtain

Ag(«) + 1AC\2(1 atm.) = AgCl(s)
10.2 + 26.65 = 23.0 - AS0

and, on substituting AS0 = —13.85 in the equation (150,
AF°298 = Atf ° - r AS0

= -30,300 - 298(- 13.85)
= -26,270 cal.

which agrees with the value derived from cell potential.

Chloride Ion. — The molal free energies of ions are mostly
derived from the potentials of cells in which the ions are formed
reversibly from the elements or from equilibrium reactions in
which ions are involved. Since the procedures and conventions
used in this type of work require some explanation and since
some of the derived quantities are difficult to understand with-
out this explanation, we shall postpone our consideration of cell
potentials until the next chapter and be content to use the free
energies of ions before studying the methods by which they are

FREE ENERGY OF CHEMICAL CHANGES 613

obtained. It will suffice to point out here that, when suitable
conditions prevail, the potential of the cell

H2(0, 1 atm.), H+Cl-(unit activity), Cl2(g, 1 atm.)'

is 1.358 volts at 298°K. and that when 1 faraday passes through
this cell the change in state and the free-energy increase are

1 atm.) + MC12(1 atm.) = H+Cl-(w.a.)
AF° = -E°F = -131,000 joules = -31,350 cal.

The free energy of hydrogen ion at unit activity is defined as
zero by the convention that the hydrogen electrode H2 (1 atm.),
H+(tfc.a.), has zero potential, and the free energy of chlorine is
zero for 1 atm. pressure by definition, so that the free energy
of chloride ion at unit activity is given as —31,350 cal. by this
cell potential.

Water. — Since the free energies of water and water vapor
appear in many chemical calculations, they have been determined
with care by several methods. The calculation for the vapor,
based on high-temperature measurements of the dissociation,
is complicated by the fact that two reactions take place simul-
taneously, namely,

H20(?) = H,(ff) + K02(!7)
and

H20(0) = HH,(0) + OH(<7)

The older calculations, which did not take account of the second
reaction, were almost correct through a curious compensation of
errors. Since the oxygen electrode is not reversible, calculations
based on the potential of an oxygen-hydrogen cell and the equa-
tion AF° = —EQNF are not available.

An entropy balance and A# for the reaction

H2(l atm.) + KO2(1 atm.) = H20(/)

31.23 + 24.51 = 16.75 - AS0

gives AS0 = -38.99 e.u., ^AS0 = -11,625 cal.; and since A# =
-68,318 cal., AF° = -56,693 cal. at 298°K. for the formation
of liquid water.

Confirmation of this value is obtained by adding four standard
reactions and their free-energy changes.1

1 PITZER and SMITH, ibid., 69, 2633 (1937).

614 PHYSICAL CHEMISTRY

H2(l atm.) + 2AgCl(s) = 2Ag(«) + 2R+C\~(u.a.)

AF°298 = -10,259 cal.
Ag200) + H20(Z) + 2Cl-(w.a.) = 2AgCl(s) + 20R~(u.a )

AF°298 = -5596 cal.
2H+(w.a.) + 2OH~(u.a ) = 2H2O(/)

AF°298 = -38,186 cal.
2Ag(s) + KO2(1 atm ) = Ag2O(s)

AF°298 = -2585 cal

H2(l atm ) + MO2(1 atm ) = H2O(/)

AF°298 = -56,626 cal

The first of these reactions takes place in an electrolytic cell
that will be described on page 633, the second is from a measured
chemical equilibrium quoted in Problem 7 on page 626, the third
comes from Kw, which has been determined by several methods,
and the fourth from the calculation given on page 611. Other
means of confirming it are given in the next chapter.

The standard free energy of water vapor in the imaginary
state of a gas at 1 atm. and 298°K. is obtained by the method
used in calculating the standard free energy of bromine vapor.
The "changes in state and their free-energy increases are

H20(/) = H20(0, 0.0313 atm ) AF = 0
H2O(g, 0.0313 atm.) = H2O(g, 1 atm ) AF = RTln — ^

U.Uol-'4

= 2057 cal.
and, upon addition,

H2O(0 = H2O(0, 1 atm.) AF° - 2057 cal.
and the standafd free energy of water vapor is
-56,693 + 2057 = -54,636 cal.

These calculations will suffice to show the methods used in
measuring the standard free energies of substances. A short list
to be used in problems is given in Table 95, and many others
are known.1 A short list of standard entropies is given in Table
96, and many others are likewise known.2

1 See, for example, LATIMEB, "Oxidation Potentials," pp. 302-308, Pren-
tice-Hall, Inc., New York, 1938.

2 The best compilation of standard entropies is by Kelley, U.S. Bur. Mines,
Bull., 434, (1941). All the entropies in Table 96 are from this publication.

FREE ENERGY OF CHEMICAL CHANGES 615

TABLE 95 — SOME STANDARD FREE-ENERGY CONTENTS AT 298°K.1

Substance

AF°298

Substance

AF°298

Substance

AF°298

H20(0)

- 54,636

HNO2(w.o )

- 13,020

Br-(w.o.)

- 24,568

H20(0

- 56,690

HCN(0)

27,730

I-(w.a.)

- 12,340

H202(Z)

- 28,230

HCN(u.o )

26,340

Is~(u.a.)

- 12,295

H2O2(w.a )

- 31,470

C'Ofo)

- 32,787

HS-(w a.)

2,985

Oato)

39,400

C02(0)

- 94,239

HSOr(w a )

-125,870

C12(/)

1,146

CO2(u a.)

- 92,229

S04-~(M a.)

-176,100

Cl2(wa)

1,630

C0012(flf)

- 48,960

NH4+(w a )

- 18,830

Hcifo)

- 22,770

CH4(0)

- 12,085

NO8-(w a )

- 8,450

HClO(w.a )

- 19,110

OJIoto)

- 7,790

NOr(w a.)

- 26,345

Br2(flf)

755

C2H4(<7)

16,280

CN-(w a )

39,140

Br2(w a.)

977

C2H2((7)

50,030

HCOr(w a )

-140,270

HBr(0)

- 12,540

NaCl(s)

- 91,770

CO3— (M.a )

-126,170

IIBrO(w.a.)

- 19,680

KCl(s)

- 97,555

Li+(w.a.)

- 70,700

J2(0)

4,630

KC103(s)

- 67,960

Na+(w.a.)

- 62,590

I2(t*.a.)

3,926

AgCl(s)

- 26,200

K+(w.a.)

- 67,430

HI to)

315

Ag2O(s)

- 2,585

Cu+(a.a )

12,040

H2S(0)

- 7,865

Ou2O(s)

- 35,150

Cu++(w.o )

* 15,910

H2S(u.a.)

- 6,515

CaCO3(«)

-269,940

Ag4- (w.a )

18,441

S02fo)

- 71,750

Hg2Cl,(«)

- 50,310

Ca++(w.a.)

-132,430

SO2(z* a )

- 71,870

TlCl(s)

- 44,190

Zn++(w.a )

- 35,110

H2SO8(w.a.)

-128,563

PbCl2(«)

- 75,050

Cd++(i^ o.)

- 18,550

NH3(0)

- 3,864

CuC1l(«)

- 28,490

Hgs4-+(w « )

36,850

NH8(0

- 2,574

HgO(s)

- 13,940

Tl+(7/ a )

- 7,760

NH,(w.a.)

- 6,257

H+(M a )

Sn++(tt a.)

- 6,490

NOG;)

20,650

OH-(w.a.)

- 37,585

Pb + +(w.a.)

- 5,840

N02(0)

12,275

Cl-(u a )

- 31,340

Fe++(*/ a.)

- 20,310

N204(0)

23,440

ao-o/ a )

- 9,200

Fe+++(uo.)

- 2,530

Calculation of Chemical Equilibrium. — Free-energy changes and
entropy changes for isothermal changes in state, whether stand-
ard or not, may be evaluated by the procedure that was used in
Chap. VIII for enthalpy changes, namely, AF = F2 — FI and
AS = Sz — Si, and chemical equations may be added as was
done there, with addition of AF or AS. For standard changes,
a free-energy balance gives the equilibrium constant at 298°K.
for the reaction through equation (13i), and an entropy balance
gives the equilibrium constant at 298°K. through equation (150
when A// is known or can be calculated from tables. Thus a

1 In calories per mole, s — sohd; I = liquid, g = gas, u.a. = aqueous
solution at unit activity. For additional free energies, see Latimer, op. cit.,
Appendix II.

616 PHYSICAL CHEMISTRY

TABLE 96 — SOME STANDARD ENTROPIES AT 298°K.1

Substance

£°298

Substance

S°«8

Substance

s°298

H2fa)

31 23

H20(<7)

45 13

KCl(s)

19 76

Oifo)

49 03

H,0(Z)

16 75

KCIO.W

34 2

N,fo)

45 79

HClto)

44 66

KClO4(s)

36 1

C12(0)

53 31

HBr(0)

47 48

Ag20(s)

29 1

Br2(<7)

58 63

H2S(0)

49 1

AgCl(s)

23 0

Br,(0

36 7

NH.to)

46 03

AgBr«

26 1

I,(s)

27 9

C0(0)

47 32

HgO«

17 6

C (diamond)

0 585

C02(<7)

51 08

Hg2Cl2(s)

47 0

G (graphite)

1 36

S02(<7)

59 2

Pb012(«)

32 6

K«

15 2

CH4(flr)

44 5

MgO(s)

6 66

Na(«)

12 2

OHsOHfo)

56 66

Mg(OH),W

15 09

S(«)

7 62

C2H4(0)

52 3

MgCOa(«)

15 7

Mg(«)

7 77

02H6OHto;

67 3

CaO(«)

9 5

Ag«

10 20

NOfo)

50 34

CaC^O8(s)

22 2

Hg«)

18 5

N02(0)

57 47

ZnO(s)

10 4

Pb(«)

15 49

Zn(«) *

10 0

table of free energies, or of entropies and enthalpies, provides a
convenient means of recording a vast number of equilibrium
constants through a reasonable number of entries. The equilib-
rium constants at 298°K. for the hundreds of chemical reactions
involving the substances in Table 95 are all available from a
simple calculation involving this table, and the addition of one
more free-energy content to this list makes available the equi-
librium constants for all possible reactions of that substance with
all those in the table. A direct tabulation of all these equilibrium
constants would fill many pages, and the constants for a single
additional substance would fill more pages still.

The usefulness of these tables will be greatly extended by some
simple equations to be given presently, which allow the calcula-
tion of AF or AF° at any temperature from their values at a given
temperature by means of enthalpies and heat capacities. We
have already had one way of doing this through the van't Hoff
equation; the new equations are only more convenient means
for accomplishing the same end with a smaller number of inter-
mediate calculations through the use of data tabulated in other
forms. A few illustrations for constant temperature will be

1 In calories per mole per degree, a =* solid, I = liquid, g = gas.

FREE ENERGY OF CHEMICAL CHANGES 617

given before deriving the equations applicable to changing
temperature.

The standard free energies of HC1(0) and of H+ and Cl" at
unit activity enable us to calculate the activity my and the
activity coefficient y in solutions of HC1 for which vapor pressures
have been measured. For example, the activity of the ions in
6m. HC1 was given as 20.1 on page 600, which means an activity
coefficient of 3.35. This coefficient is calculated through
the following reversible path for the transfer of the gas to the
solution :

HC%, 1 atm.) = HC%, 1.84 X 10~4 atm.)

F! = -22,692 (1) F2 = -27,792

= H+Cl-(6m.) = H+Cl-(u.a.)

(2) F3 = -27,792 (3) F4 = -31,340

AFi for the first change in state is RT In 1.84 X 10~4 = -5100
cal., AF2 for the passage of HC1 into solution under, the equilib-
rium pressure is zero, and AF3 is —3553 cal., the difference
between the calculated F3 and F4, the free energies of the ions
from Table 95. From equation (90, -3553 = 2RT In I/ (my),
we find my = 20.1 and y = 20.1/6.0 = 3.35.

The standard free energy of lead ion at unit activity is —5840
cal. as calculated from its standard electrode potential. From
Table 73 we see that the equlibrium constant for the reaction

Sn(s) + Pb++ = Pb(s) + Sn++

is 3.0 at 298°K, and thus AF° for this reaction is -6&0 cal.,
which is the difference between the standard free energies of these
ions. This gives —6490 cal., or —27,200 joules, as the standard
free energy of stannous ion, which in turn gives EQ for the elec-
trode reaction Sn(a) = Sn++ + 2e~ as 27,200/2 X 96;500 = 0.140
volt from equation (140- Direct measurement of the standard
electrode potential for tin is excluded by the hydrolysis of
stannous ion in the absence of excess acid and by direct dis-
placement of hydrogen ion by tin in the presence of acid. Since
there are some calculations in which it is desirable to have this
standard potential available, this calculated potential is an
important one.

From the free energies of a solid salt and of its ions at unit
activity one may calculate the activity product in a saturated

618 PHYSICAL CHEMISTRY

solution, and for slightly soluble salts that do not hydrolyze Ka
will be almost equal to Kc. When the molality in the saturated
solution is high enough so that allowance for activity coefficients
is required, the activity product and a solubility product in terms
of molalities will not be the same. Calculations for silver chloride
and for lead chloride will illustrate these two situations. For
the former the free-energy balance is

AgCl(s) = Ag+ (u.a ) + Cl~ (u.a.)
-26,200 = 18,441 - 31,340 - AF°

whence AF° = 13,301 cal = -RT In (aAe+)(aCi-), the activity
product is 1.75 X 10~10, and the square root of this product is
1.32 X 10~5, which is the solubility of silver chloride in water at
298°K.

For lead chloride the free-energy balance is

PbClaGO = Pb++ (u.a.) + 2C1- (u.a.)
'-75,050 = -5840 - 62,680 - AF°

whence

AF° = 6530 cal = -RT In (mPb^yPb

and my = 0.0158. We are unable to calculate the molality with-
out an activity coefficient or the activity coefficient without an
experimental solubility; since the measured solubility at 298°K. is
0.039, we calculate the activity coefficient as 0.0158/0.039 = 0 41
for the ions in a saturated solution of lead chlorfde. Without
allowance for the activity coefficients, the " calculated" solu-
bility would be-more than double the actual one.

The solubility of CO 2 in water as a function of the pressure is
recorded by the entries for C0%(g) and CQz(u.a.), as may be
seen by calculating AF° for the standard change in state at 25°,

C0,fo) = C02(w.a.)
-94,239 = -92,229 - AF°

for which K = mcojpco, and AF° = 2010 cal. = -RT In K,
whence K = 0.034, in agreement with the solubility used in earlier
chapters. A word of caution regarding such tabulated free ener-
gies as C02(w.a.) and H2CO8(w.a.) will not be out of place at this
point, and it will also apply to the difference between

FREE ENERGY OF CHEMICAL CHANGES 619

and NH4OH(w.a.) or between S02(u.a.) and H2S03(^.a.)- There
is no information on the fraction of the dissolved gas that is
hydrated for any of these systems, and the notations CO2(w.a.)
and H2C03(w.a.) both mean unit activity of the dissolved non-
ionized gas in the two forms together. Hence for all three of
the hydrates the free-energy content is merely that of the unhy-
drated solute plus —56,690 for a mole of liquid water. Thus for
the two forms of equation expressing the ionization of carbonic
acid,

CO2(w.a.) + H20(Z)' = H+(t*.a.) + HCO8-(u.a.)
and

H2C08(w.a.) = H+(w.a.)

AF° will be the same, and the ion activities calculated from
AF° = — RT In K will be the same, as they should be. But this
does not mean that we may use these free energies for such a
calculation as

C02(w.a.) + H2O(Z) = H2CO3(w.a.) AF° = 0

from which K = 1 = «H2co3/«co2 is justified; for this calculation
leads to the fiction that half the CO 2 is in the hydrated form, and
we have no information on this fraction.

A corresponding calculation for the solubility of chlorine in
water gives the equilibrium concentration of C12 molecules in
water when the pressure of chlorine gas is 1 atm., but it does not
give the total solubility of chlorine in water, for almost a third of
the total dissolved chlorine is hydrolyzed. The concentrations
of H+, Cl~, and HC10 in equilibrium with chlorine gas at 1 atm.
can of course be calculated from the free-energy tables ; and since
one mole of chlorine gives one mole of each of these solutes upon
hydrolysis, the total dissolved chlorine is the sum of the hydro-
lyzed and unhydrolyzed quantities, or (C12) + (H+).

A solubility product for CaCO3 may be calculated from the
standard free energies

CaC03(s) = Ca++(w.a.) + C03— (u.a.)
-269,940 = -132,430 - 126,170 - AF°

from which AF° = 11,340 caL at 25° and the solubility product
is 5 X 10~9 = (Ca++)(CO8— ). The product of these molalities
in a saturated solution of CaC03 is thus correctly given by the

620 PHYSICAL CHEMISTRY

calculation, but the square root of the solubility product will not
give the solubility of calcium carbonate in water, since more than
half the dissolved material is in the form of hydrolysis products,
as was explained on page 414.

Entropies and free energies may sometimes be used to deter-
mine enthalpies to advantage. For example, AH for the forma-
tion of PbS(s) is given by one source as —24,800 cal. and by
another source as —20,600 cal., with little indication as to which
is the better value. From the equilibrium

PbS + H2(<7) = Pb(«) + H2S(0)

and the well-known free energy of H2S one calculates the standard
free energy of PbS(s) as - 21,735 cal. at 298°K., and from entropy
data one calculates Pb(«) + S(«) = PbS(s), AS°298 = -1.3 e.u.,
and T AS0 = —390 cal., whence from equation (152)

A#° = AF° + T AS0 = -21,735 - 390 = -22,125 cal.

There are other reactions for which AH so determined will be a
better value than the direct calorimetric determination for one
reason or another. Precise calorimetry is difficult at tempera-
tures much above room temperature, and there are many reac-
tions that proceed too slowly for direct measurement of their
heat effects until high temperatures are reached. The experi-
mental difficulties of high-temperature equilibrium measurements
and low-temperature heat capacities have been so completely
solved as to open up a new means of determining enthalpies of
reactions through the relation AF = AH — T AS.

Change of Free Energy with Temperature. — This important
relation will be derived in two 'ways, first from a. reversible cycle
of changes in which a reacting system performs a Carnot cycle
with the absorption of heat at one temperature, the conversion
of part of the heat into work, and the rejection of the remainder
of the heat at a lower temperature, and then from the defined
relation F = H - TS.

In the first derivation, the maximum work of the reversible
cycle will be expressed in terms of the free-energy change, which
will then be related to the heat absorbed at the higher tempera-
ture through the second law of thermodynamics.

1. We begin the cycle with a system in state 1 at the tempera-
ture T, where its volume is vi and its pressure p\. The system

FREE ENERGY OF CHEMICAL CHANGES 621

changes to state 2 at T for the first step in the cycle, by which its
pressure becomes pz and its volume v^} and for this change in
state the heat absorbed is #, the enthalpy increase is A//, and the
free-energy increase is AF. The maximum work done by the
system in this step is wi = — AF + £2^2 — p\v\.

2. We cool the system under the constant pressure p2 to
T — dT, by which the volume becomes v% — dvz and for which
the work done by the system is Wz = — pz dv%.

3. We change the system back to state 1 at T — dT, where its
volume is t>i — dvi and its pressure pi; for this change in state
the free-energy increase is — (AF — dAF), since AF — d AF is
the smaller increase in AF upon going from state 1 to state 2 at
T — dT , and the free energy for the change from state 2 to state 1
has the opposite sign. The maximum work of this change is
w3 = (AF — dAF) + PI(VI — dvi) — p^(v2 — dv2).

4. Finally, we return the system to its original condition by
heating it at the constant pressure pi to T, for which w4 = pi dvi.

The summation of work quantities for the cycle is — d AF,
which by the second law of thermodynamics in equal to q dT /T.
This quantity q is equal to AH + [w — A (pi;)] in view of the
definition A// = &E + A(pv) = q — w + A(pu). But since the
change in state at T took place reverszbly, the quantity in square
brackets is — AF and hence q = A// — AF. Thus, the desired
relation is

(16)

U, AM.' fjj

d AF AF —

dT ~~ T

Upon rearranging and dividing through by !T2, this equation
becomes

Td AF - AF dT . /AF\ A//

. /AF\

= \r) ~ ~~

The formal definition of free energy, F = II — TS, given on
page 45, may also be used to derive equation (16). Upon differ-
entiating with respect to T at constant pressure, we have

**\ =(»Ji\ -T(^\ -s

622 PHYSICAL CHEMISTRY

But at constant pressure dH — T dS for a reversible process, and
thus this equation becomes

(18)

Upon summation of free-energy changes for the following paths,
State 1 at T + dT AF + d AF Stat* 2 at 71 + d?1

i .-h

State 1 at T __ AF __ State 2 at T

Fi ""* F2

we see that, for the change at T followed by heating to T + dT, the
free-energy change is AF + dF2 and, for heating first to T + dT
and then undergoing change, the free energy is d¥i + AF + d AF;
upon equating these,

d AF = c?F2 - dFj = -S2 dT7 + Si dT
whence

Before integrating equation (17), &H must be expressed as a
function of the temperature by the method given on page 321.
The equation for A# will usually have the form

AF = A#0 + aT + bT* + cT*

in which A/fo is the integration constant that appears when
d(A#) = ACp c?!T is integrated. Upon substituting this in equa-
tion (17), integrating, and multiplying through by T, we have

AFr = A#0 - aT In T - 6712 - Y%cT* + IT (19)

If A# is independent of temperature or sufficiently constant
over the temperature range involved, the simpler integral of
equation (17) is

AFr = A// + IT (20)

FREE ENERGY OF CHEMICAL CHANGES 623

When AF at a single temperature is to be calculated from AF at
the standard temperature and provided that AH is constant, one
may, of course, integrate equation (17) between limits and obtain

— — - T^

From the equation d(AH)/dT = ACP we see that, when A// is
constant, ACP is zero, and that the heat capacities of the system in
its initial and final states are the same. And since the entropy
increase on heating any system reversibly is JCP d In T between
the temperatures involved in the heating, it follows that the
entropies of the system in its initial and final states increase by
the same amount when heated through the same temperature
range, if Cp is the same for both, and thus that AS for the iso-
thermal change in state is the same at all temperatures. This
fact shows that the integration constant 7 in equation (20) is —AS
when AH is independent of temperature, since AF = AH — T AS
for an isothermal change.

Thus, for reactions in which ACP is zero or negligible, equation
(20) has the convenient forms

AFr = A// - T AS (22)

AFV = A7/° - T AS0 (23)

These equations are, respectively, (12£) and (15£) for isothermal
changes in state, but when AH is constant they are also the equa-
tions for changing AF with changing temperature. When AH
is not constant, these equations may not be used and equation
(19) must be used.

The van't Hoff Equation. — In order to show the relation of
these equations to the van't Hoff equation for the change of
equilibrium constant with temperature, equation (102) on page
602 may be put in the form

AF p in Pp'W* r, ln K

T = p^p7b ~

All the pressures in the first term on the right are the initial
or final pressures appearing in the change in state; and since they
are kept constant when the system changes temperature, the
derivative of this term with respect to T is zero. By differen-

624 PHYSICAL CHEMISTRY

tiating (24) and combining with (17), we have

which rearranges to give the van't Hoff equation

d]nK=jjjrtdT (26)

Thus the equations derived in this section are only more con-
venient ones for calculating change of equilibrium with tempera-
ture from free-energy tables or entropy and enthalpy tables.

A few illustrations will not be out of place. The dissociation
pressure of silver oxide is 1 atm. at 463°K., and AH is constant
for the reaction

2Ag(s) + J£08(0) = Ag20(s) Aff = -7300 cal.
At 463°K., AF° = 0, and thus from the substitution

AF° = A//° - T AS0
0 = -7300 - 463 A5°

we find AS0 = — 15.8 for all temperatures. From this we calcu-
late the standard free energy at 298°K.,

AF°298 = -7300 - 298(-15.8) = -2580 cal.

which is the same as the result obtained on page 611 from equa-
tion (13<) and the van't Hoff equation.

From a standard entropy balance at 298°K. for the reaction

HgO(a) = Hgfo)
17.6 = 41.8 + 24.5 - AS0

AS0 is 48.7 e.u. The heat absorbed is A// = 36,200 caL at 298°K.
It seems unlikely that ACP is zero for this reaction, but there are
no reliable data for the heat capacity of HgO as a temperature
function. We may make an approximate calculation of AF° at
713°K., at yrhich the measured dissociation pressure is 0.845 atm.,
K = HP V^ip = 0.30, and AF° = -RT In K = 1710 cal.
From equation (23) we calculate

AF°713 = 36,200 - 713(48.7) = 1500 cal.

FREE ENERGY OF CHEMICAL CHANGES 625

from which .£713 = 0.35 and the calculated dissociation pressure
is 0.94 atm. Such a calculation is not very satisfactory, but it
should be noted that AF°7i3 is the small difference between two
larger quantities, AH and T AS0, and small errors in either of
them have a large effect upon the difference. The assumption
that A// is constant is probably not the chief source of the error
in the calculated dissociation pressure; for changing AH to
36,400 cal , which is a change of less than 1 per cent, changes
AF°7i3 to 1700 cal. and gives perfect agreement between the
calculated and measured dissociation pressures. It is probable
that the actual error in AH is as great as 200 cal., but this "is
not to say that an approximate calculation such as we have
made above shows that this error exists.

It may be profitable to close this discussion of free-energy
data with a word of caution based upon ,the calculation just
given and other similar ones throughout the text, a word that
is applicable to the data in any field. Tables often include
entries of high accuracy with others of questionable accuracy
but give no indication of their comparative reliability. Entries
are sometimes admittedly uncertain but the only ones available.
Actual errors are sometimes increased by the necessity of taking
the small difference between two large quantities Under these
circumstances one must do the best he can with the data he
has, he must realize that the final result is no better than the
data on which it is based and discard digits that are not truly
significant, and above all he must maintain a sense of proportion
tempered with patience. The quantity of good data is increas-
ing rapidly; many of the older measurements are being repeated
with better instruments and higher skill; and many new quanti-
ties are being measured. We have attempted to show how the
data we have may be used; the appearance of new data will not
change the method of use.

Problems

Numerical data should be sought in the tables in the text.

1. (a) Calculate AH for the evaporation of a mole of bromine at 298°K.
from the data in Tables 95 and 96. (b) Calculate the entropy of saturated
bromine vapor at 298°K. (c) The density of liquid bromine is 2.93 grams
per ml. at 298°K. Estimate AF for the change in state Br2(Z, 1 atm.) =»
Bn(i, 10 atm.) at 298°K.

626 . PHYSICAL CHEMISTRY

2. (a) Calculate AF at 298°K for the change in state C12(0, 1 atm.) =
C12(0, 7.0 atm.), neglecting the deviation of chlorine from ideal gas behavior.
(6) The vapor pressure of chlorine at 298°K. is 7.0 atm. Calculate the
molal free-energy content of C12(/)

3. The partial pressure of HBr(0) above an aqueous solution of HBr at
298 °K changes with the molahty as follows:

m 6 8 10

106p, atm 1 99 117 77 6

(a) Calculate the activity coefficients in these solutions from the data in
Table 95. (In these solutions the activity coefficient will be greater than
unity ) (6) Calculate the pressure of HBr above a solution 1.0m. in HBr at
298°K.7 taking 0.80 as the activity coefficient for the ions

4. Calculate the free-energy increase at 298°K for the reaction H2O(/)
+ %®2(ff, 1 atm.) = HjjOsCw a ) and the pressure of oxygen in equilibrium
with H2O and H2O2(w a )

6. (a) Calculate the lemzation constant for water at 298°K from free-
energy data. (6) From A// for the lomzation of water given on page 320,
calculate Kw at 323°K

6. The ratio of CO2(#) to CO(0) in equilibrium with Zn(s) and ZnO(s) at
693°K.is5.5 X 10~5, A// for the reaction ZnO(s) + CO(0) = Zn(«) + CO2(0)
is 15,500 cal , and ACP = 0 (a) Calculate the standard free energy of
ZnO(s) at 298°K. (b) Calculate another value of the free energy of ZnO(s)
from the data in Tables 58 and 96.

7. Calculate the equilibrium constant at 298°K. for the reaction AgCl(s)
+ NaOH = KAgjO(fi) + NaCl + MH2O. [The measured ratio (Cl~)/
(OH-) is 0.00893. /. Am. Chem. Soc., 60, 3528 (1928).]

8. Using the free-energy data, calculate the pressure of oxygen required
to make the reaction KCl(s) + %O2(0) = KClO3(s) proceed, (b) Calcu-
late the free energies of KCl(s) and KClO3(s) from the entropies and enthalpy
data, and recalculate the pressure of oxygen required for the first reaction

9. Calculate the standard free energy and standard entropy of SO3(#)
at 298°K. from^he following data: The equilibrium constants for the reaction
SO2fo) + JiOjfo) = SOifo) are 31.3 at 800°K. and 6.56 at 900°K., and
ACP for the reaction is zero.

10. From the solubility data in Problem 20, page 425, calculate the molal
free-energy content for the complex ion CuCl2~~(w.a.) at 298°K.

11. (a) Show by free-energy calculations whether a catalyst could cause
the "fixation" of nitrogen as ammonia at 298°K (b) Show whether a
catalyst could form NO or NO2 in appreciable quantities from air at 298°K.

12. The chemical reaction N2(0) + C2H2(0) »= 2HCN(gr) is a possible one
for the fixation of nitrogen, (a) Given A//298 = 7700 cal., A£°298 =32,
and ACP = 0, calculate the equilibrium constant for this reaction and
the fraction of nitrogen reacting in a mixture of 1 mole of N2 and 1 mole
of C2H2 at 700°K. and at 1100°K. (b) Recalculate AF°70o, assuming
ACP » 2.6 - 0.00277 for the reaction.

IS. (a) Calculate the solubility of H2S in water at 1 atm. pressure and
298°K. (b) Calculate the solubility of bromine in water at 298°K., neglect-

FREE ENERGY OF CHEMICAL CHANGES 627

ing the small hydrolysis, (c) Calculate the fraction of the dissolved bromine
that is hydrolyzed.

14. (a) Calculate AF° as a function of the temperature for the reaction
COfo) + 2H2(0) = CHaOH(g), A#298 - -21,660 cal., taking Cp - 2.0 +
0.03077 for CH3OH(gr) and Cp = 6.5 + 0.00171 for the other gases. (6)
Calculate the equilibrium constant for the reaction at 473°K.

16. The solubility product of Mg(OH)2 is 5.5 X 10~12 at 298°K. (a)
Calculate the free energy of Mg++(w.a.). (&) Calculate the solubility
product for MgCO3 at 298°K. (Note that this should not agree with the
solubility product for MgCO3.3H2O given in Problem 16 on page 425.)

16. Calculate the hydrolysis constants for the ions CN~ and HCOs" from
the free-energy data.

17. (a) Calculate the dissociation pressure of MgCOs at 612°K. and at
681 °K., taking AH from Table 58 and assuming ACP = 0. (6) The recorded
dissociation pressure at 681 °K. is 1.00 atm. On the assumption that the
entropy data are correct, what value of AH would be required to show a
calculated dissociation pressure of 1 atm. at 681°K.? (The recorded AH
for the dissociation is given as 28,300 ± 850 cal.)

18. Calculate the dissociation pressure of Mg(OH)2(s) at 485°K., assum-
ing AH constant. (The measured dissociation pressure at 485°K. is 0.0717
atm )

19. Show that KClOs is thermodynamically unstable with respect to its
decomposition into KC1O4 and KC1 at 298°K.

20. (a) Calculate the entropy of H2O(0) at 298°K. and 0.0313 atm. (which
is the vapor pressure of water at this temperature) from the standard
entropy. (6) Calculate A// for the evaporation of water at 298°K.

21. (a) Calculate the quantities A/7, AZ?, AA, AF, and AS for the change in
state H2Oft 423°K , 4.7 atm.) = H2O(gr, 423°K , 4.7 atm.) from the experi-
mental data on page 108. (6) Estimate these quantities for the change in
state H2O(Z, 423°K., 4.7 atm.) = H2O(0, 423°K., 1 atm.) by devising a
reversible path for the change and assuming the vapor an ideal gas.

22. Calculate the standard free energies of I2(00 and I2(0 at 114.15°C.,
taking !,(«) = 0 at 114.14°C. (See page 146 for data.)

23. For the change in state N2(0, 1 atm.) = N2(gr, 0.1 atm,) at 25°C.,
calculate AT/, AE, AA, A/S, and AF, assuming nitrogen to be an ideal gas.
What are the upper and lower limits of q and w for the isothermal process?

24. (a) Calculate the equilibrium constant at 25° for the reaction CuCl(s)
+ ^H2O = HCuzO(s) + H+C1-. (b) Calculate the solubility product
for cuprous chloride in aqueous solution at 25°. (c) The solubility of
cuprous oxide in water is negligibly small. Calculate the concentration of
cuprous ion in a solution made by saturating water with cuprous chloride,
allowing for the hydrolysis shown in part (a). The activity coefficients may
be assumed unity in these dilute solutions.

25. For the chemical reaction 2NaH(s) - 2Na(Z) + H2(y), AH « 30;500
cal., and ACP == 0. The equilibrium pressure (in atmospheres) changes with
the absolute temperature as follows:

T 573 593 613 633 653 673

p 00105 00245 00549 0117 0.240 0.467

628 PHYSICAL CHEMISTRY

The vapor pressure of sodium is negligible in this temperature range, (a)
Calculate AF° at 371°K. for the reaction. (6) The latent heat of fusion of
sodium at 371°K. is 630 cal. per atomic weight Calculate AH for the
reaction 2NaH(s) = 2Na(s) + H2(0) at 371°K. and AF° for the reaction at
298°K., again assuming A(7P = 0.

26. From the data on page 64 and in Tables 58 and 95, calculate an
approximate value of the molal enthalpy of COC12(0) at 25°, neglecting ACP,
for which there are no data

27. (a) Calculate the equilibrium concentration of chlorine molecules in a
solution at 298°K. when the partial pressure of chlorine above the solution
is 1 atm (b) The measured solubility of chlorine is 0 094m. at 298°K. for

1 atm. pressure. Calculate the fraction of chlorine hydrolyzed in the solu-
tion, (c) Calculate the standard free energy of HC1O, taking 0.85 as the
activity coefficient for the ions and neglecting the very small lomzation of
HC1O in the solution, (d) Calculate the lomzation constant of HC1O.

28. The enthalpy of combustion of graphite is —94,030 cal. at 298°K ,
and that of diamond is —94,484 cal. (a) Calculate AF°298 for the transition
Cgraph — Cdiam. (6) Calculate roughly the pressure that would be required
to give AF a negative sign at 298°K. for this transition, taking the density
of diamond as 3.51 and that of graphite as 2 26 and neglecting the compressi-
bilities. (Note that AF must be negative for a spontaneous process )

29. (a) Derive an expression for AF° as a function of the temperature for
the reaction C2H4(0) + H2O(0) = C2H6OH(0), A// = -11,000 cal., assum-
ing ACP = 0. (b) Calculate the equilibrium constant for the reaction at
500°K. (c) Calculate the fraction of ethylene hydrated when a mixture of

2 moles of C2H4, 2 moles of H2O(g), and 6 moles of inert gas reach equilibrium
at a total pressure of 10 atm. at 500°K. [Parks, Ind. Eng. Chem., 29, 845
(1937), estimates ACP = -643 + 0 0133T, AF° « -9674 + 6.43T7 In
T - 0.00665 7'2 - 9 01 T7, and finds AF060o = 4139 cal.]

30. (a) Calculate the equilibrium constant at 25° for the reaction NOBr(^)
— NO(0) + KBr2(gr), taking 19,260 cal. as the standard free energy of
NOBrfo). (6) What fraction of NO will be converted to NOBr at 25° in
contact with liquid bromine?

31. Calculate the equilibrium constant for the reaction 2H2S(g) + SO2(g)
= 2H2O(0) + 3S(«) at 25°.

32. The molahty of a solution in equilibrium with C$SO4(s) at 298°K. is
0.0202, and the activity coefficient in this solution is 0.32; the molality of a
solution in equilibrium with CaSO4.2H2O(s) at 298°K. is 0 0153, and the
activity coefficient in this solution is 0.35. (a) Calculate AF°298 for each of
the reactions

CaSO4(s) - Ca++(t*.a.) + SO4— (u a.)
• CaSO4.2H2O(s) = Ca++(w a ) + SO4" (u.a.) + 2H,O(2)
CaSO4(s) + 2HiO(i) = CaSO4.2H2O(s)

taking the activity of water in the saturated solution equal to that of pure
water, (b) For the reaction CaSO4(s) + 2H2O(0 = CaSO4.2H2O(s), AH -
—4040 cal., and ACP =» 0. Calculate the transition temperature of the
dihydrate to anhydrous salt, this being the temperature at which AF° = 0.

FREE ENERGY OF CHEMICAL CHANGES 629

(c) The standard entropies at 298°K. are 25.5 for CaSO4, 46.4 for CaSO4.~
2H2O, and 16.75 for H2O(Z). Calculate A£°298 for the reaction CaSO4(s)
+ 2H2O(Z) = CaSO4.2H2O(s), and calculate another value of AF0298.

33. From the data in Tables 58 and 96 calculate the standard free
energies of HClfo), KCl(s), NH,(0), and KClO4(s)

34. (a) Calculate AF°298 for the reaction Na2SO4(s) + 10H2O(0 * Na-r
SO4.10H2O(s), for which A//29g = -19,400 cal. and A/S°298 = -61.4. (6)
The vapor pressure of water at 298°K. is 0.0313 atm. Calculate AF°29S for
the reaction Na2SO4(s) -f 10H2O(0) = Na2SO4(s). (c) Calculate the dis-
sociation pressure of the hydrate at 298°K.

36. (a) The vapor of NH4C1 is completely dissociated into NH8 and HC1,
and at 610°K. the equilibrium pressure is 1 0 atm. for the reaction NH4Cl(s)
= NH8(00 + HCl(flr); AH - 40,000 cal., ACP « 0. Calculate the standard
free energy of NH4Cl(s) at 298°K. (6) The standard entropy of NH4Cl(s)
at 298°K. is 22.6. Calculate the standard free energy of NH4Cl(s), using
the same free energies for NHa(gr) and HCl(^) as in part (a).

36. The enthalpy of PbS(s) is -22,160 cal. at 298°K., and the standard
entropy is 21.8. (a) Assume ACP = 0, and calculate the equilibrium con-
stant at 600°K for the reaction PbS(s) -f H2(0r) = Pb(s) + H2S(g). (6)
The heat of fusion of lead at 600°K is 1200 cal per atomic weight. Calcu-
late A//, AF°, and AS0 at 600°K for the reaction PbS(s) + H2(0)

37. (a) Calculate the standard free energy of S (u a.) from the data in
Tables 63 and 95. (6) Calculate the solubility product for PbS(s).

CHAPTER XIX
POTENTIALS OF ELECTROLYTIC CELLS

The purpose of this chapter is to consider the potentials of
electrolytic cells in which chemical changes take place isother-
mally and reversibly and through these potentials to evaluate
the free-energy changes of the chemical reactions. The equa-
tions and measurements will confirm some of the standard free
energies of substances obtained in the previous chapter and
furnish activity coefficients for ions, transference numbers, solu-
bilities, ionization constants, equilibrium constants, and other
important quantities. Cell potentials are one of the most impor-
tant sources of precise data for the calculations of physical
chemistry, and thus it is important to understand the underlying
theory and the limitations of the theory, in order to make full
use of the measurements.1 Since cell potentials change with
the nature of the electrodes, with the nature and molality of the
solutes, and with the temperature, it is evident that a record
of all cells at all molalities and all temperatures is not to be
compiled in a limited space. The expedient used is one that
has been used before, namely, a record of standard potentials
at a standard temperature and some simple equations through
which to calculate the change of potential with molality or
temperature.

Electrode reactions such as were considered in Chap. VII will
apply in this chapter also, as will Faraday's law and the law of
transference. The passage of electricity through a cell will
require chemical reactions in which one equivalent of chemical
change is produced at each electrode for each faraday passing;
and the ions in the solution will carry fractions of the total elec-

1 The precision of modern data is not to be judged from the fact that most
of the potentials in this chapter are given to a millivolt or 0 1 mv. ; for most
of them are known with higher precision. In a first meeting with the subject
it will not be important to know that the potential of a given cell is 0.46419
volt, and we have been content to state it as 0.4642 volt or as 0.464 volt
The additional figures will be found in the original sources of data quoted for
the cells.

630

POTENTIALS OF ELECTROLYTIC CELLS 631

tricity that are equal to their transference numbers in the solu-
tion. These transference numbers will be in the same ratio as
the mobilities of the ions, so that

+ - N+ + N- ~ A+ + A_

The potential of an electric cell depends on the rate at which
current is drawn from it, and experiment shows that the poten-
tial produced approaches a maximum value when the current
drawn from the cell becomes very small. When electricity is
passed through a cell to reverse the chemical change, the poten-
tial required decreases as the current decreases; and the maximum
potential produced by the cell approaches the minimum poten-
tial required to reverse it when the smallest measurable currents
are employed. This maximum potential is the one with which
we shall be concerned in this chapter.

Such potentials are measured on a potentiometer, a device for
opposing the potential of a cell with another which is slightly
greater or slightly less and in which the difference decreases until
the current flowing through the cell is no longer measurable.
The opposing potential is regulated by a sliding contact along
a wire of uniform resistance or through dial resistances, and the
actual potential is derived from a similar procedure with a cell
of standard potential. The common standard is the "Weston
cell/' whose potential is 1.0181 volts at 25°.

The electrical energy derived from a cell is the product of
potential, current, and time For most of the cells we are to
consider the quantity of electricity will be 96,500 amp.-sec., or
1 faraday, so that 96,500^ joules will be produced. When N
faradays passes through a cell, the electric energy is ENF joules.

The relation of the free-energy change to the electric energy
produced in a cell in which a chemical change takes place iso-
thermally and reversibly was given in the previous chapter,
but it is repeated here, since we are to use it extensively.1

AF = -ENF (It)

1 The italic letter F denotes Faraday's constant of 96,500 amp.-sec. per
equivalent of reacting substance, the bold-faced letter F the free-energy
content of a system, and AF the free-energy increase attending a change in
state, as was done in the previous chapter. The letter t with the number of
an equation indicates its restriction to a process taking place at constant
temperature.

632 PHYSICAL CHEMISTRY

A spontaneous process is one for which AF is negative; and
since AF = —ENF, a positive cell potential means a free-energy
decrease and a spontaneously operating cell. A negative poten-
tial means a free-energy increase attending operation of the cell
or that an opposing potential must be applied to the cell to
cause it to operate.

Formulation of Cells. — It has become common practice to
describe a cell in terms of the substances involved in the change
in state, with those portions at which a potential difference
exists separated by commas or vertical bars and with the anode
written at the left. For example,

Ag + AgCl, HCl(0.1m.), C12(1 atm.); E298 = 1.136 volts

Ag + AgCI | HCl(0.1m.) | C12(1 atm.); #298 = 1.136 volts

This notation means that an anode of silver coated with silver
chloride is dipping into O.lm. hydrochloric acid and that the
cathode is a platinum or other inert metal plate dipping into
O.lm. hydrochloric acid saturated with chlorine at 1 atm. pres-
sure and over which chlorine gas is bubbling. The subscript
attached to E defines the temperature at whixsh the potential
was measured, and the positive sign of the potential means that
the cell will operate and produce this maximum potential.
If the cell above were written

C12(1 atm.), HCl(0.1m.), AgCl + Ag; #298 = -1.136 volts

with the chlorine electrode as the anode, the negative sign of the
potential means that at least this potential must be applied to
the cell to make it operate with the silver chloride electrode as a
cathode.

In another type of cell the electrode materials are the same
for anode and cathode, and two solutions are involved. An
example is the cell

Ag + AgCl, HCl(0.10m.), HCl(0.020m.), AgCl + Ag;

£298 = 0.0645 volt

to which we shall come later in the chapter. The notation
means that a silver chloride electrode is in contact with O.lm.
HC1 at the anode of the cell, another silver chloride electrode is

POTENTIALS OF ELECTROLYTIC CELLS 633

in contact with 0.020m. HC1 at the cathode, and the two solu-
tions meet in a "liquid junction."

Throughout the chapter we shall follow the custom, which is
now standard, of considering the left-hand electrode as the
anode, which is to say that oxidation takes place at this electrode,
or that negative charges are given to the metal electrode at this
point, or that the electrode reaction is written with the symbol
e~ for a f araday of electricity on the right-hand side of the chem-
ical equation expressing the change in state. The cathode
reaction is that one which occurs at the right-hand electrode, and
the symbol e~ is written on the left side of the chemical equation
to indicate the acceptance of negative charges from the metal at
this electrode.

Cell Reactions. — Electrode reactions will be written in the
same manner as in Chap. VII, with the additional specification
of the molality at which ions are formed or used. So long as
we were concerned only with Faraday's law, this specification
was unnecessary, for the quantity of solute formed by an elec-
trode reaction is independent of molaiity. But the potential
of an electrode or a cell is the chief topic of the present discussion;
and since potentials depend on the molality in some of the cells
to be considered, we must always specify the molality of the
solution in a cell. For example, in the cell

Ag + AgCl, HCl(0.10m.), C12(1 atm.); E298 = 1.136 volts

the chemical change attending the passage of 1 faraday through
the cell is the sum of two electrode reactions, as follows:

Ag + Cl-(0.10m.) = AgCl(s) + cr
1 atm.) = Cl-(0.10m.)

Ag(«) + MC1«(1 atm.) = AgCl(«)

In this cell the potential is independent of molality, but this
will not be true of cells in general. As examples, the cells

H2(l atm.), HCl(0.10m.), AgCl + Ag; #298 = 0.3524 volt
H2(l atm.), HCl(0.01m.), AgCl + Ag; £298 = 0.4642 volt

differ only in the molality of the acid. The cell reaction for the
first one is again the sum of two electrode reactions:

634 PHYSICAL CHEMISTRY

l atm.) = H+(0.10m.) + <r
AgCl(g) + er = Cl-(0.1Qm.) + Agp)
atm.) + AgCl(a) = H+Cl-(0.10m.) + Ag(«)

By writing the corresponding reactions for the second cell, it
will be seen that it forms 0.010m. HC1 from hydrogen gas and
silver chloride. The free-energy changes in the two cells differ
by the free energy of transferring a mole of H+ and a mole of
Cl"~ from one molality to the other.

One further requirement in the operation of cells is illustrated
by the difference between the potentials above, namely, that
the molality of the acid must remain constant as the cell oper-
ates. In laboratory practice this is accomplished by passing
such a small quantity of electricity through the cell that the
change in acid molaiity is negligible. But it is convenient to
write the cell reactions in terms of a faraday of electricity and
to assume such large cells that the formation of a mole of solute
produces no change in the molality. If this condition is not met,
passage of electricity will cause a change in the molality, the
measured potential will not apply to any particular molality of
acid, and it will thus have no clear meaning. During the passage
of 1 faraday through the cell

H2, HCl(0.01m.), AgCl + Ag

if it contained a liter of 0.010m. HC1, the molality would increase
to 1.01 and the cell potential would fall gradually from 0.4642
volt to about 0.23 volt; and it should be understood that while
this is a possible occurrence in a laboratory it is not the procedure
that is being discussed here.

A " standard" cell reaction is one that conforms to the defini-
tion of a "standard" change in state given in the previous chap-
ter, namely, one in which all the reacting substances and all the
reaction products are gases at 1 atm. pressure, pure liquids or
pure solids, or solutes at unit activity. We shall write E® for
the potential of a cell when the cell reaction is a, standard one
in the sense of the definition, in conformity to the use of AF°
and A/S° for standard changes in state. For such changes

AF° = -E°NF

POTENTIALS OF ELECTROLYTIC CELLS 635

Whether or not the cell reaction is a standard one, it should
be written down fully and completely after the cell is completely
described and before any discussion or calculation is attempted.
Students are advised to make this a matter of rigid routine, both
to promote their understanding of the cell reaction and to save
needless labor in the solution of problems.

When two cells are connected in series, the potential of the
pair is the sum of the individual potentials, the change in state
is the sum of the individual changes in state, and the total AF
is the sum of those for the two cells. For example, if the silver
chloride electrodes of the cells

H2(l atm.), HCl(0.10m.), AgCl + Ag; E298 = 0.3524 volt
AgCl + Ag, HCl(0.10m.), C12(1 atm.); E™ = 1 130 volts

are connected together, the total potential is 1.488 volts at 298°K.
and the total change in state in the pair of cells for 1 f araday is

HH2(1 atm.) + KC12(1 atm.) = H+Cl-(0.10m.)

The same change in state would be produced by the passage
of 1 faraday through the cell

H2(l atm.), HCl(0.1ra.), C12(1 atm.)

and thus the potential of this cell is also 1.488 volts at 25°C.
For this particular cell the potential derived from the other two
is a better value than could be obtained by direct measurement,
since the first cell is accurately known and the second one is
independent of the molality of the acid so that no correction
for the hydrolysis of chlorine is required. There are serious
experimental difficulties in working with a chlorine electrode,
most of which are avoided by using silver chloride or mercurous
chloride electrodes when chloride ions are involved. Since these
electrodes are the ones commonly used in cells reversible to
chloride ions, their potentials have been determined with particu-
lar care.

The cells that we have been discussing also change potential
when the partial pressure of the gas at the electrode changes;
for the potential is a measure of the free-energy increase attend-
ing the operation of the cell, and the free-energy content of a
gas changes with pressure at constant temperature. A cell in

636 PHYSICAL CHEMISTRY

which the potential is due only to a difference in the pressure of
a gas is

H2(l atm.), HCl(0.1m.), H2(0.1 atm.)
for which the electrode reactions on a basis of 1 faraday are

(Anode) ^H2(l atm.) = H+(0.1m.) + <r

(Cathode) H+(0.1w.) + <r = ^H2(0.1 atm.)

Change in state: %H*(1 atm.) = ^H2(0.1 atm.)

The sum of the electrode reactions, which gives the net change in
state, shows that when 1 faraday passes through the cell % mole
of hydrogen gas is expanded by a reversible process from 1 atm.
pressure to 0.1 atm., and this is a change in state to which the
equation

AF298 = nRTln = -ENF (2t)

applies. In this equation R must be expressed in joules. The
free-energy increase for this change in state is thus —2855 joules,
and E = 2855/96,500 = 0.0296 volt. We shall have occasion
to use this equation later in making corrections for the partial
pressures of gas electrodes when the total pressure is given.
The potential of a hydrogen electrode, for example, depends on
the partial pressure of the hydrogen, and when the atmospheric
pressure is 1 atm. the partial pressure of hydrogen will be less
than 1 atm. by the pressure of water vapor from the solution.
At an electrode such as H2, HCl(14ra.), the partial pressure of
HCl(g) would also have to be subtracted from the barometric
pressure to give the partial pressure of hydrogen gas.

Standard Free Energies from Cell Potentials. — The isothermal
reversible operation of a cell in which a single substance forms
from its elements gives through the measured potential the free
energy of formation of the compound. As an illustration, the
cell1

Ag + AgCl, HCl(1.0m.), C12(1 atm.); #2*8 « 1.1362 volts

1 All the cells quoted in this section were measured by Gerke, /. Am.
Chem. Soc., 44, 1684 (1922).

POTENTIALS OF ELECTROLYTIC CELLS 637

forms silver chloride by its reversible operation, as shown by the
sum of the electrode reactions.

Ag(s) + Cl-(1.0m.) == AgCl(s) + er
1 atm.) + <r = Cl-(1.0m.)

Ag(«) + MCli(l atm.) = AgCl(s)

Since this is a " standard7' change in state, E is also EQ and
AF° = ~EWF = -109,640 joules, or -26,220 cal. This is the
free-energy content of silver chloride calculated on page 612
Another cell, from which we obtain the free energy of forma-
tion of mercurous chloride,1 is

Hg(Z) + Hg2Cl2(», HCl(0.1m.), C12(1 atm.); E298 = 1.0904 volts

The chemical reaction that attends the passage of 2 faradays
through this cell is

2Hg(0 + C12(1 atm.) = Hg2Cl2(s)

for which E is again E°, since this is a standard change in state.
Then AF° = -2E°F = -210,460 joules = -50,290 cal. This
free energy is readily confirmed by the enthalpy data in Table
58 and the entropy data in Table 96 through the equation

AF° = A# ° - T AS0 (30

Writing the chemical equation, appending A/I, and making an
entropy balance below the equation we have

2Hg(Z) + C12(1 atm.) = Hg2Cl2(s); A# = -63,150 cal.
37.0 + 53.3 = 47.0 - A/S°

from which AS0 = -43.3, T AS* = -12,900 cal., and
AF°298 = -63,150 - (-12,900) = -50,250 cal.

Later in the chapter we shall come to yet another means of
obtaining AS0 for this and other reactions through the tempera-
ture coefficient of the cell potential. Since the calomel and

1 The evidence that mercurous chloride is HgzCU rather than HgCl comes
from the X-ray diffraction pattern, which shows a linear molecule CIHgHgCl,
with the distance between mercury atoms smaller (by about 35 per cent)
than other atomic distances in the crystal. Mercurous ion is shown to be
Hga"1"*" and not Hg+ from the potentials of concentration cells [LINHAET,
Md., 38, 2356 (1916)] in which Hg2++ is transferred from one molality to
another. These cells are described on p. 653.

638 PHYSICAL CHEMISTRY

silver chloride electrodes are so extensively used, a further check
on their potentials has been obtained from the cell

Ag + AgCI(s), KCl(lm.), Hg2Cl2(s) + Hg; Ew = 0.0455 volt

for which E is equal to £r°, since the change in state is the stand-
ard one

Ag(«) + KHg2Cl2(s) = AgCl(«) + Hg(Z)

for which AF° = -W/4.18 = -1050 cal. If -26,220 cal. is
accepted as the free-energy content of AgCl(s), that of Hg2Cl2(s)
is 2(- 26,220 + 1050) = -50,340 cal.

One more illustration of cells of this particular type will suffice,
though there are of course many more available. The standard
free energy of lead chloride is measured in the cell

Pb + PbCl2(s), HCl(1.0m ), AgCl + Ag; E\^ = 0.4900 volt
in which the cell reaction and the free-energy balance are

Pb(s) + 2AgCl(s) = PbCl2(s) + 2Ag(s); AF<U = -2ff°F
0 + 2(- 26,220) = AF°Pbci2 + 0 - AF°oell

and, since AF°0<jn = -2 X 0.4900 X 96,500/4.18 = -22,600 cal.,
the standard free energy of PbCl2(s) is —75,040 cal. as calculated
from the potential of this cell.

In order to calculate the standard free energies of ions we
shall first need a means of determining their activity coefficients
at different molalities, and we now turn to a means of determining
them from the potentials of cells.

Activity Coefficients from Concentration Cells. — One of the
most direct means of determining activity coefficients for ions
is through the free-energy changes attending the operation of
cells that transfer the solute isothermally and reversibly from
one molality to another. Such a cell is

H,(l atm.), HCl(m«), AgCl + Ag— Ag + AgCl,

Ha(l atm.)

When 1 faraday passes through the whole combination considered
as a single cell, the net effect is the transfer of two moles of solute
from m\ to m^ one mole of hydrogen ion and one mole of chloride

POTENTIALS OF ELECTROLYTIC CELLS 639

ion. The free-energy relation that applies is

AF = -ENF = nRT In ^ = nRT In ^-2 (40

In this equation n is the number of moles of solute transferred
from mi to m2 when N faradays passes through the cell.

It should be remembered that an activity has the dimensions
of a molality and that, since a — my, the activity coefficient 7
is a number. For a given solute this coefficient varies with the
molality, the temperature, and the molality of any other solutes
present in the solution with it.

A " concent ration cell without transference/' such as the one
we are now considering, is really two opposed cells with identical
electrodes and a solute at different molalities, of which the net
effect is the transfer of a solute from one molality to another.
The name is a somewhat unfortunate choice but the one com-
monly applied to these cells. (A cell "with transference" is
one in which the transference numbers of the ions are involved in
the formulation of the change in state. We shall come to them
later in this chapter.) Data are commonly reported as in
Table 97. Thus the potential for the cell above is the difference
between E for m2 and E for mi in this table. The change in
state for the passage of 1 faraday through the whole cell is
obtained by adding the four electrode reactions, as follows:

atm.) - H+(m2) + er
AgCl(s) + er - Ag(«) + Cl-(ma)
Ag(«) + Cl-Cm^ = AgCl(s) + e-

l atm.)

Net change in state: H+Cl-(mi) = H+Cl-(m2)

We shall use the data of Table 97 and the potentials at 25°
first to show how the activity coefficient at any molality may be
calculated if a "reference" value of 7 is assumed for one molality,
such as 7 = 0.796 for O.lm. at 25° and then to show how this
standard is itself obtained. In the above cell let mx = 0.10 and
m2 = 0.4897, and assume 7 = 0.796 in O.lm. HC1 at 25°. The
potential of the u concentration cell " is the difference between the
tenth and seventh cells in the fourth column of Table 97,
0.27342 - 0.35240 = -0.07898 volt, and the net change in state

640 PHYSICAL CHEMISTRY

for 1 f araday is

H+Cl-(0.1ro.) - H+Cl-(0.4897m.)
for which the free-energy change is

AF = -EF = 2RT\n = 7622 joules

TABLE 97. — CHANGE OF POTENTIAL WITH MOLALITY IN THE CELL
H2, (1 atm ), HCl(rw), AgCl + Ag1

Electromotive force at

0°

15°

25°

35°

0 0050

0 48916

0 49521

0 49844

0 50109

0 0070

0 47390

0 47910

0 48178

0 48389

0.0100

0 45780

0 46207

0 46419

0 46565

0.020

0 42669

0 42925

0 43022

0 43058

0 050

0.38586

0 38631

0.38589

0 38484

0 070

0 37093

0 37061

0 36965

0 36808

0 100

0 35507

0 35394

0.35240

0 35031

0.2030

0 32330

0 32057

0 31803

0 3189

0 30239

0 29862

0 29545

0.4897

0 28193

0 27727

0.27342

0 6702

0 26616

0.26076

0 25644

0 9699

0 24623

0 23998

0 23513

1 2045

0 23362

0 22691

0 22174 '

1.4407

0 22253

0 21536

0 20992

2.3802

0.18684

0 17858

0 17245

4.0875

0.13594

0 12648

0 11968

Upon solving for 7, we find 0.756 for 0.4897m. HC1 at 25°, and
similar treatment of the1 other cells yields a table of activity
coefficients for the several molalities. The activity coefficients
thus obtained are given for HC1 in Table 53 and repeated with
coefficients for other solutes in Table 98, which will be useful
in solving problems.2

and EHLERS, ibid., 54, 1350 (1932), 65, 2179 (1933); for data
at higher molalities see Akerlof and Teare, ibid., 59, 1855 (1937); for other
cells see "International Critical Tables," Vol. VI, p. 321, and the current
chemical literature.

2 For an extensive table of activity coefficients, see Latimer, "Oxidation
Potentials," pp. 323#, Prentice-Hall, Inc., New York, 1938.

POTENTIALS OF ELECTROLYTIC CELLS

641

Of the many determinations of this type, we quote one more
for sodium hydroxide concentration cells " without transference"
at 25°, in which the potential of the whole concentration cell
was directly measured1 and 7 was taken as 0.920 in 0.010m.
NaOH. The cell measured was

H,(l atm.), NaOH(wti), NaHgar— NaHg,, NaOH (0.010m.),

H2(l atm.)

and some of the data are as follows :

.5/298.

mi. .

7298-

-0 0315 +0 0338 0 0796 0 1116 0 1416 0 1672 0 2103 0.2221
0 0053 0 0202 0.0527 0 1047 0 1934 0 3975 0 807 1 020
0 951 0 880 0.822 0 768 0 748 0 714 0 678 0 680

In both illustrations, all the activity coefficients depend on a
single one assigned to a " reference" solution. This is not an
arbitrary choice, but a quantity derived from the experimental
data, which are so treated as to provide a means of determining
E° for a cell in which the activity of the ions is unity (though the
molality is not unity), as shown in the following section.

TABLE 98 — SOME ACTIVITY COEFFICIENTS AT 25°

0 001

0.01

0 05

0 10

0 50

1.0

2 0

3 0*

HBr

0 966

0 906

0 838

0 804

0.79

0 87

1 17

1 7

HC1

0 966

0 904

0 823

0 796

0.758

0 809

1 01

1 32

NaOH

0 92

0.82

0 77

0 69

0 68

0.74

0 84

KOH

0.90

0 82

0 80

0 73

0 76

0 89

1 08

NaCl

0 966

0 904

0 82

0 78

0.68

0 66

0 67

0 71

KC1

0 964

0.90

0 81

0 77

0.65

0 61

0 58

0 57

H2S04

0 83

0 54

0 34

0 27

0 15

0 13

0 12

0 14

Mg(N03)2

0 88

0 71

0 55

0 51

0 44

0 50

0 69

0 93

PbCl2

0 86

0 71

ZnCl2

0 88

0 71

0.56

0 50

0 38

0 33

CuS04

0 74

0.41

0.21

0 16

0 068

0 047

ZnSOi

0 70

0.39

0 15

0 065

0 045

0.036

"Standard" Cell Potentials. — We have already considered some
cell potentials in which the cell reaction was a "standard" one
not involving solutes. But in many cells the reaction forms or
uses ions, and for these cells a " standard " reaction requires the

1 HARNED, /, Am. Chem. Soc., 47, 676 (1925).

642 PHYSICAL CHEMISTRY

formation or use of ions at unit activity. In order to evaluate
EQ for a cell of this type, such as

PI2(1 atm.), H+Cl-(M.a.), AgCl + Ag

in which (u.a.) indicates unit activity of the ions, and at the
same time determine the chemical composition of the solution
in which the activities are unity, we consider a concentration cell
in which this cell is opposed to another in which the molality
of the acid is m and the activity of the ions is 7717, as follows:

H2(l atm.), II+Cl-(a.a.), AgOl + Ag— Ag + AgCl,

m), II2(1 atm

The potential of this concentration cell is obviously E° — Em,
the difference between the potential of the cell containing ions
at unit activity and that of the cell

H2(l atm.), H+Cl-(m molal), AgCl + Ag

The change in state and the free-energy increase for 1 faraday
passing through the whole concentration cell are

lH+Cl-(m molal) - 1 H+Cl-fc a.)

AF = -(E° - Em)F = 2/2 r In — (5J)

7717

This equation may be rearranged with the experimentally deter-
mined quantities Em and m on the left-hand side as follows:

T, . 2RT , T,n , ,,,

- Em + -j- In m = E° - - -j- In 7 ((St)

By plotting the left-hand side of this equation against some
function of the molality suitable for extrapolation and by extend-
ing the curve to zero molality the potential E° is evaluated, since
7 becomes unity and In 7 becomes zero at zero molality by defini-
tion. There are theoretical as well as practical advantages (see
page 286) in plotting Em + 0.1183 log m against the square
root of the molality, of which the important one for our purpose
is that the plot for dilute solutions is almost a straight line.
Figure 71 shows such a plot for the potentials at 25° given in
Table 97, from which we find E°m = 0.2224 volt, in close agree-

POTENTIALS OF ELECTROLYTIC CELLS

643

ment with the value obtained by others.1 Upon substituting
this value in equation (5t) and rearranging, we have

Em = 0.2224 -

(70

Application of this equation to the cells at 25° yields the activity
coefficients given in Table 98, including the coefficient 0.796 used
in the previous section for O.lm. HC1.

UZ3*

(X232
0.230

^ 0.228
_o

i 0.226

LL.I 0.224
0222
0.220

) 004 Q08 012 016 Q20 02

FIG 71

Square Root of Molahty
-Determination of #°2»8 for H2, HCl(u.o.),-AgCl + Ag.

It will be noted that no cell whose potential is 0.2224 volt at
25° appears in Table 97, and it will seldom be required to prepare
a solution in which the activities of the ions are unity. By plot-
ting E against ra, one may determine that m = 1.19 when
E = 0.2224 and a = 1.00 for HC1 at 25°C., but it must be noted
that because of the definition of activity there is no assurance
that 1.19m. HC1 will have ions of unit activity at any temperature

1 For example, PBENTISS and SCATCHABD, Chem. Rev., 13, 139 (1933);
SHEDLOVSKY and MAC!NNES, J. Am. Chem. Soc., 58, 1970 (1936); CARMODY,
ibid., 54, 188 (1932).

644 PHYSICAL CHEMISTRY

other than 25° or that any other solute will be at unit activity
for this molality at 25°.

The same procedure may be applied to other cells. When ions
of valence higher than 1 are involved, the expression derived
for plotting has a slightly different form, but the method is
otherwise the same An illustration is the cell

Pb(«), PbCla(m ), AgCl + Ag

for which the electrode reactions for 2 faradays and the cell reac-
tion obtained by adding them are

Pb(» = Pb++(m ) + 2e~

2AgClQ) + 2c- = 2Ag(s) + 2Cl-(2m.) _
Pb(«) + 2AgCl(,s/ = 2Ag(s) + Pb++(m.) + 2Cl'(2m )

In such solutions the molality of the chloride ion is twice the
molality of lead ion, and thus the equivalent of equation (7t)
for this cell is

Em = E° - ~ In (my)(2myY = #° - In

Rearrangement with the experimental quantities on the left as
before and the substitution of numerical values of the constants
for 25° give s as the equivalent of equation (60

Em + 0.0887 log m + 0.0178 = EQ - 0.0887 log 7 (90

The measured potential at 298 °K. changes with the molality of
lead chloride as follows:1

m * 0 0390 0 0296 0.0205 0.0104 0.00516 0.00262

£298 . . . 0 490 0.496 0.507 0.526 0 548 0.570

Extrapolation to zero molality gives E° for the cell as 0.348 volt.
This method may be applied to any cell in which the potential
varies with the molality of the solution, such as

Pb, PbCl,(nt.), Hg2Cl2 + Hg
Pb + PbSO4(s), H2SO4(m.), H2
Zn, ZnCl2(m.), AgCl(a) + Ag
Tl + TlCl(s), HCl(m.), H2

1 CABMODY, ibid , 51, 2905 (1929).

POTENTIALS OF ELECTROLYTIC CELLS 645

to determine EQ for the cell; and it is unnecessary for cells such as

Pb + PbCl,(«), HCl(m.), Hg2Cl2(s) + Hg
Tl + TlCl(s), KCl(m.), AgCl(s) + Ag

in which the potential is independent of the molality of the
solution. However, the custom in physical chemistry is to
record not the standard potentials of cells but the standard
potentials of single electrodes, all of them being written as anodes.
Then in a given cell E° = E°i — E\ the difference between the
anodic potentials of the first and second electrodes. Since it is
impossible to determine the potential of a single electrode, the
expedient is to define one arbitrarily and to express all the others
in terms of this denned potential, as explained in the next section.
Standard Electrode Potentials. — In conformity to the usual
custom in physical chemistry, the potential of the single electrode
H2(l atm.), H+(w.a.) is taken as zero. This definition was con-
tained in the specification that the standard free energy of
hydrogen ion at unit activity is zero. The potential of a whole
cell of which this standard is a part is thus the potential of the
other electrode; but since all electrode potentials are listed as
anodes, the standard potential of the cell

H2(l atm.), H+Cl-(w.a.), AgCl + Ag; #0298 = 0.2224 volt

of which the hydrogen electrode is the anode, is given by the
relation E^ = E\ - E\ for which E\ is zero, and thus #°2
is -0.2224 volt.

The standard potential for chlorine1 may be calculated from
the potential of the cell

Ag(s) + AgCl(s), HCl(0.1m.), C12(1 atm.); #298 = 1.136 volts

The change in state for 1 faraday is the sum of the anode and
cathode reactions.

Ag(s) + Cl-(0.1m.) = AgCl(s) + «r
1 atm.) + er = Cl-(0.1m.)

Ag(8) + MCli(l atm.) = AgCl(s)

1 The electrode C12(1 atm.), Cl~(w.a.) might also be described as Clr-
(0.062m.), Cl~(u.a.), since this is the equilibrium molality for a chlorine
pressure of 1 atm. But the electrode potential CU(U.CL), Cl~(w.a.) is not
— 1.358 volts at 25°, for chlorine gas at 1 atm. is not in equilibrium with
chlorine as a solute at unit activity.

646 PHYSICAL CHEMISTRY

Since this is a standard change in state, in which the molality
of the acid solution cancels when the total change in state is
written, EQ for the cell is E\ — E°2, and E\ is -0.2224, whence
E°ci is -1.136 - 0.222 = -1.358 volts.

The standard potential of the lead electrode may be deter-
mined from the cell

Pb(«) + PbCl2(m.), AgCl(s) + Ag

for which EQw was found to be 0.348 volt on page 644. Since
E^ = #°Pb - (-0.222), EQn = 0.348 - 0 222 = 0.126 volt at
298°K.

The standard potential of Hg + HgsCU, G\~~(u.a ) is obtainable
from the cell

Ag + AgCl(s), HCl(0.1m.), Hg2Cl2 + Hg; #298 = 0.0455 volt
for which the change in state for 1 faraday is

Ag + Cl-(0.1m ) = AgCl(s) + <r

l2 + e~ = Hg(J) + Cl-(0.1w.)

Ag + MHg2Cl2« = AgCl(«)

These equations show that the potential of the cell is independent
of the molality1 and that E° would be 0.0455 volt for the cell

Ag(s) + AgCl(s), H+C1-(M a ), Hg2Cl2(s) + Hg(Z)
whence 0.0455 = E\^ - E\mf*» or

= -0.2224 - 0 0455 = -0.2679 volt

A brief list of standard potentials for 25*0. is given in Table 99,
and many others are available. Not all of them are derived
from cells *in which standard changes take place, as will be seen
in the next section. They are useful for calculating the poten-
tials of cells in which cell reactions are not standard, through a
relation that we now derive.

1 Experiment likewise shows that the potential of this cell is independent
of the molality of the acid. The data of Randall and Young, ibid., 50, 989
(1928), at 25° are

m. . 0.0974 0 1233 4.095

E 0.0456 0 0455 0.0455

POTENTIALS OF ELECTROLYTIC CELLS 647

TABLE 99. — SOME STANDARD ELECTRODE POTENTIALS AT 2501

Electrode reaction

#%8

Electrode reaction

^°298

Li = Li+ 4- e~

3 024

I- - ^I2« + e-

-0 535

K = K+ 4- e~

2 924

Br- = JiBrj(J) + e~ .

-1 065

Na = Na+ H- e~

2 715

ci- = Hcisto) + f-

-1 358

Zn = Zn++ -f 2e~

0 762

Ag + I- = AgT -f e~

0 151

Fe = Fe++ -f- 2e~

0 440

Ag + Br- = AgBr + e~

-0 073

Cd = Cd++ -f 2e~

0 402

Ag -f (;i- = AgCl H- f-

-0 222

Sn = Sn++ -f 2e~

0 140

Cu + 01- = CuCl -f e-

-0 124

Pb = Pb+4 -f 2e

0 126

Hg + Cl- = ^Hg2Cl2 + e~

-0 268

HH2 = H+ + e-

0 000

Normal calomel electrode

-0 280

Cu = Cu M 4- 2e~

-0 347

Cu+ = (^u++ + e~

-0 167

Cu = Cuf + e~

-0 522

ye++ = Fe+++ 4- c"

-0 771

Ag = Ag+ 4- e~

-0 799

Sn++ = Sn++++ + 2c~

-1 256

Hg = ^Hg2^+ +c-

-0 799

OH- 4- MHg -

KHgO 4- 12H20 +^~

-0 098

Change of Potential with Molality. — Since the free energy of
hydrogen gas is not zero at a pressure other than 1 atm. and since
the free energy of hydrogen ion is not zero at a molality other than
unity, it will be evident that the potential of the electrode

atm.),

molal)

is not zero In order to show the relation of E for this electrode
to EQ for the electrode at which the reaction is

atm.) =

we note that AF = -EF for the first electrode and AF = —E*F
for the standard electrode. Since the free-energy increases for
a series of changes add to that of a single step causing the same
net change in state, we may calculate them for the three following
steps and their sum as follows:

1 Compiled from various sources; for example, Li, Li4" is from Maclnnes,
" Principles of Electrochemistry," p. 256, Remhold Publishing Corporation,
1939, New York, where values for several other potentials will be found, Pb,
Pb++ is by Lingane, J. Am. Chem. Soc., 60, 724 (1938); Ag 4- Agl, I" is from
Cann and Taylor, ibid., 59, 1484 (1937); Cu, Cu++ is from Adams and
Brown, ibid., 59, 1387 (1937). A compilation of about 400 potentials is
given in Latimer, op. cit.

648 PHYSICAL CHEMISTRY

p atm.) = ^H2(l atm.) AFX = ]4RT In -

I atm.) = H+(w.a.) + e~ AF2 = -EQF

H+(u.a.) = H+(m molal) AF3 =' RT In my

i^H2(p atm.) = H+(ra molal) + e~ AF = -EF

Upon equating AF to the sum of the other three and rearranging,
we obtain the relation of E to EQ,

A corresponding relation is readily derived for the potential
of any electrode at which the reaction is not a standard one, for
example,

AgCl(s) + e~ = Ag(s) + Cl~(m molal) AFX = -EF

This reaction may be treated as the sum of two, of which one
involves the standard potential of the silver chloride electrode
and the other a transfer of ions from unit activity to an activity
my as follows:

AgCl(s) + e- = Ag(«) + Cl-(w.o.) AF2 = -E»F
* Cl-(u.a.) = Cl~(a = my) AF3 = RT In (my)

Since AFi = AF2 + AF3, we add them and solve for #, which
gives

^ABCI = ^°AKCI - =Y In (my) (lit)

The potential of any single electrode is related to the standard
potential by a similar equation. The routine procedure is to
write the reaction for the electrode as an anode and to obtain E
by subtracting from E* a term that is RT/NF times the natural
logarithm of a fraction in which the activities (or pressures) of the
reaction products appear in the numerator raised to the powers
that are the coefficients in the electrode reaction and the reacting
substances appear in the denominator under the same restric-
tion. This term is thus similar to the Q used in the previous
chapter for the relation of AF to AF°. The potential will of
course be independent of the quantity of electricity passing,
and the calculated potential will be the same whether the elec-

POTENTIALS OF ELECTROLYTIC CELLS 649

trode reaction is written for 1 faraday or 2 faradays. As an
illustration, we may calculate the potential of a zinc electrode,
writing the reaction first for 1 faraday and then for 2 faradays.
The electrode is

Zn(«), Zn++(w molal)
and the anode reactions and anode potentials are

EZn = #°zn - - In
Zn(s) = Zn++(m.) + 2e~

7~»/TT

It is common practice to write the reaction for the number of
faradays that corresponds to the valence of the ion involved,
but in cells such as

Zn(s), ZnCl2(m.), AgCl + Ag

it will make no difference whether the cell reactions are written
for 1 faraday or 2 faradays so far as the potential is concerned.
It will usually be more convenient to calculate the cell potential
in a single step, rather than to use equations such as (100 and
(110 for the individual potentials and then obtain that of the
cell from E^ = E\ — E2. Thus the potential of the cell

H2(p atm.), HCl(m molal), AgCl + Ag

follows directly from the difference between Jf£H calculated in
equation (10£) and Ej^a. in equation (110? namely,

#011 = (#°H - #°A,CI) - In - (130

It will be seen by writing the cell reaction for the whole cell,
which is the difference between the two anode reactions or the
sum of an anode reaction and a cathode reaction,

J^H2(p atm.) = H+(m molal) + e~
_ AgCl + <r = Cl-(m molal) + Ag
p atm.) + AgCl = H+Cl-(m molal) + Ag

650 PHYSICAL CHEMISTRY

that the logarithmic term contains the reaction products in the
numerator, each raised to the power that is the coefficient in the
cell reaction (that is, 7717 for H+ and my for Cl~, since each is a
separate solute), and the reacting substances in the denominator,
similarly treated. The solids are given unit activity as usual,
and thus equation (130 is only a special form of the general
equation

In Q (140

which applies to any cell reaction. A few illustrations of the
use of this important equation will not be out of place, for it
may be used to calculate cell potentials when the quantities in Q
are known or may be estimated or to obtain E° values from the
measured potentials of cells.

In any cell involving gases at the electrodes the partial pres-
sure of the gas will be lower than the barometric pressure by the
pressure of water vapor and that of any volatile solute. Thus if
a hydrogen electrode and a chlorine electrode in O.lm. HC1 form
a cell and if the barometric pressure is 1 atm., the cell at 25° will
be

H2(0.967 atm.), HCl(0.1m.), C12(0.9G7 atm.)

in which each gas pressure is 1 atm. minus the vapor pressure
of water at 25°. The cell reaction and the potential of the cell
as calculated from equation (140 are

atm.) + ^C12(0.967 atm.) =

RT (0.0796)2

£oeii = (£°H - £uci) — Y ln "TO 957)

If the gas at the cathode were a mixture of 1 mole of chlorine and
9 moles of nitrogen and the remainder of the cell were the same,
the potential would then be

(0.0796)2

•

~w F (0.967)^(0.0967)^

The reduction of silver chloride to silver and chloride ion by
zinc takes place in the cell

, ZnCl2(0.01m.), AgCl(s) + Ag

POTENTIALS OF ELECTROLYTIC CELLS 651

for which the change in state for 2 faradays is the sum of the
electrode reactions

Zn(s) = Zn++(0.01m.) + 2<r

2AgCl(s) -f 2<r = 2Cl-(0.02m.) + 2Ag __ *
Zn(«) + 2AgCl(«) = Zn++(0.01m.) + 2Cl-(0.02m.) + 2Ag

The potential of this cell is

E = (#V - E'w) - £jjrln (0.01T)(0.027)2 (160

It should be noted that in 0.01m. ZnCl2 the molality of chloride
ion is 0.02 and not 0.01.

Standard Potential and Standard Free Energy. — The standard
potential of silver against silver ion may be obtained from the
potential of the cell

H2(l aim.), HCl(0.1m.), AgCl + Ag; #298 = 0.3524 volt

through the solubility product of silver chloride. This product1
is 1.77 X 10~10 at 25°, and in O.lm. HC1 the equilibrium

aAe(0.079G) = 1.77 X lO"10

requires that a^ = 2.22 X 10~9. We may then describe the
same cell in terms of this activity as follows:

H,(l atm.), ffixiO-' **'' B = °'3524 volt

in which the cell reaction is
>£H2(1 atm.) + Ag+(a = 2.22 X 10~9) = Ag(«) + H+(a = 0.0796)

and upon substituting the measured cell potential and these
quantities into equation (140 and taking E° == 0 for the hydrogen
electrode,

°-°796

fi ^9J. — (ft — FQ \ — . 1

0.3524 - (0 E A,) -jr 1- 2.22 xT(P»

we obtain an expression from which to calculate 2?°Ag = —0.799
volt. The standard free energy of silver ion is given as 18,441
cal. in Table 95, which is merely another way of recording this

1 PITZEB and SMITH, /. Am. Chem. Soc.t 69, 2633 (1937).

652 PHYSICAL CHEMISTRY

standard potential, as may be seen by calculating the free-energy
change for the standard reaction

Ag(fi) = A.g+(u.a.) + e~
AF° = -£°F = 77,190 joules = 18,448 cal.

This statement also applies to the other standard free energies
of the ions, for they are mostly from cell -potential measurements l
The standard potential £r°298 = —1.358 volts for the chlorine
electrode, which was obtained on page 646, corresponds to the
electrode reaction

and since the free energy of chlorine gas at 1 atm. is zero by
definition, the standard free energy of chloride ion results from
AF° = -E*F = +1.358 X 96,500/4.18 = 31,310 cal., which
requires —31,310 cal. for the standard free-energy content of
chloride ion. (The entry —31,340 cal. in Table 95 corresponds
to a derived E° = —1.3583 volts, but we have not attempted to
carry so many significant figures in the calculations in this text.
Similar slight differences between other calculated potentials or
free energies in other parts of the text arise from the same
source.)

Substantially the same standard free-energy content of chloride
ion may be derived from E° for Ag + AgCl, Cl~~(w.a.), for which
— EQF/4.1S gives the difference in calories between the standard
free energies of AgCl(s) and Cl~~(u.a.). If we accept —26,200
cal. for AgCl(s), the standard free energy of chloride ion is

Cl-(w.a) = -26,200 - (0.222 X 96,500/4.18 « 5130)

= -31,330 cal.

For further illustration, the formation of cupric ion from copper
as shown by the reaction

Cu = Cu++ + 2e—} E° = -0.345 volt

gives AF° = r2E*F = 66,600 joules, or 15,900 cal., for a change
in state in which the free energy of the initial system is zero,
and thus 15,900 cal. is the standard free energy of cupric ion.

1 They may also be calculated through the third law of thermodynamics
from solubility measurements and activity coefficients based on vapor pres-
sures or freezing points,

POTENTIALS OF ELECTROLYTIC CELLS 653

It will be seen from these examples that the potentials in
Table 99 are only another record of the free energies in Table
95 and that many of the entries in one table could have been
derived from the other.

The Composition of Mercurous Ion. — Concentration cells
without transference supply the best reason for writing mer-
curous ion as Hg2++, rather than the apparently simpler Hg+.
Consider, for example, two cells containing perchloric acid of
uniform concentration throughout but small concentrations of
mercurous perchl orate in the ratio 2:1. Two such cells are1

TJ n , N TinirwAnci? ^ ( HC104 (0.0817m )
H,(l atm),HC104(0.0817tn.), (Hg2(ClO4)2(0.00275m.

Em = o 7777 volt
and

TT n + >> TrnirwnAQi7 >> /HC104(0.0817m.) ) m

H2(l atm.), HC104(0.0817m.), (Hg2(C104)2(0.001375m.)j> Hg(I);

E298 = 0.7688 volt

Let these cells be opposed to one another by connecting the two
mercury electrodes, and consider the change in state resulting
when 1 faraday passes through the opposed cells. The four
separate electrode reactions are

atm.) = H+(0.0817m.) + er

.) = Hg©

e~ + H+(0.0817m.) = JiH2(l atm )

It will be seen upon addition of these equations that the net
change in state per faraday is

upon the assumption that the mercurous ion is Hg2++. The
potential of the whole concentration cell, calculated from the
free-energy increase attending this change in state,

--" "*>

1 Linhart, ibid., 38, 2356 (1916), records these cells with others in which
the molahties of HC104 and Hg2(C104)2 are varied over wide ranges, while
for each pair of cells the ratio of Hg 2(0104)2 remains 2:1. All these data
support the formula Hg2++ for mercurous ion,

654 PHYSICAL CHEMISTRY

is 0.0089 volt, which is the difference between the measured
potentials of the cells. Since the mercurous-ion concentration
is a small part of the total ion concentration on which the activity
coefficient depends, we may assume 72 = TI without appreciable
error in calculating the potential of the concentration cell. When
this is done, the calculated and observed potential differences
agree.

If the mercury electrode reactions are written upon the assump-
tion that the mercurous ion is Hg+, the second and third equa-
tions above become

e~ + Hg+ (0.0055m.) = Hg(/)
Hg(Z) = Hg+(0.00275w ) + <r

and 1 mole of Hg+ is transferred per iaraday Upon this
assumption,

and E = 0.0178 volt, which is twice the measured difference in
the cells. Thus it is indicated that Hg2+"H represents the composi-
tion and charge of the mercurous ion.

Relation of Electrode Potential to Electrolysis. — The standard
potentials in Table 99 are arranged in the order of decreasing
anode potential, which is the same order as in the " electromotive
series.'7 The maximum potential of the cell at 25°

H2(l atm.), H+Cl-(u.a.), AgCl + Ag

is 0.2224 volt, and the application of a higher opposing potential
would cause the silver chloride electrode to function as an anode,
with the electrode reaction Ag + Cl~ = AgCl + e~ taking place,
and with the evolution of hydrogen gas at the cathode. But
the evolution of chlorine gas at the silver chloride electrode does
not take place and could not take place until the opposing poten-
tial exceeded that of the cell

Hs(l atm.), H+Cr-(tt.o.), C12(1 atm.); E° = 1.358 volts

It is true of this cell, as it is true of cells in general, that the
electrode reactions requiring the lowest opposing potentials take
place first during electrolysis. Since the formation of silver

POTENTIALS OF ELECTROLYTIC CELLS 655

chloride from silver and chloride ions requires a potential about
1.14 volts less than that required for the evolution of chlorine,
no chlorine is evolved. If the silver chloride electrode were
replaced by platinum or any other inert metal, the evolution of
chlorine in this cell would require an opposing potential exceeding
1 358 volts.

As another illustration, consider a cell composed of sodium
chloride at unit activity (1.53m ) with two platinum electrodes.
When electricity is passed between these electrodes, hydrogen
gas is evolved at the cathode and chlorine gas at the anode.
We may calculate the minimum opposing potential required to
start this electrolysis, which is that of a cell

fH+(10~7m.) from water) „. ,

for which El is 0.414 volt, E2 is - 1.358 volts, and E is 1.762 volt§.
Hence, if the opposing potential is greater than 1.762 volts,
electrolysis will begin, chlorine gas will be evolved at the anode,
hydrogen gas will be evolved at the cathode, and hydroxide ion
forms in solution around the cathode. The use of an opposing
potential of 2,2 volts (to overcome the extra hydrogen potential
required as the solution around the cathode becomes more
alkaline) would continue the electrolysis until the sodium
hydroxide becomes about 2m. But the deposition of sodium
upon the cathode would require a potential of at least

2.713 + 1.358 = 4.071 volts

for this is the back potential of the cell Na, Na+Cl™(^.a.),
CUCI atm.). Since the actual potential required to electrolyse
aqueous sodium chloride with the evolution of hydrogen gas at a
platinum electrode is less than half of this potential, it will be
clear that there is no call for the " explanation " that sodium
deposits and then reacts with water to form sodium hydroxide
and hydrogen when salt brine is electrolysed.

In certain commercial cells, sodium amalgam, which is a dilute
solution of sodium in mercury, is formed when sodium chloride
is electrolysed with a mercury cathode, but there are several
circumstances that prevent direct comparison of this process
with the one discussed in the previous paragraph. In the first

656 PHYSICAL CHEMISTRY

place, the potential of the cell

Na(s), Nal in ethylamine, NaHgx (0.2 per cent Na in Hg)

is about 0.85 volt,1 and the amalgams are usually kept below this
sodium content; in the second place, the potential required for
the evolution of hydrogen gas upon mercury is 0.8 volt or more
above that for hydrogen upon platinum, depending on the
current density; and, in the third place, the actual potential
applied to the commercial cells exceeds 4 volts. The potentials
of these cells are no reflections on the statement in the previous
paragraph that sodium metal does not deposit during the elec-
trolysis of aqueous sodium chloride with inert electrodes; they
make this statement more probable.

Concentration Cells with Transference. — Cells that consist
of two identical electrodes dipping into solutions of the same
electrolyte at different molalities, and with the two solutions
in contact, are called cells "with transference." An example is
the cell

Ag + AgCl, HCl(0.10m.), HCl(0.020m ), AgCl + Ag

for which the measured potential is EW = 0.0645 volt. The
potential of this cell is not Ei — Ez, when the potentials for the
separate electrodes are •computed in the way already explained,
for the liquid junction is also a source of potential. We shall see
the calculation of liquid junction potentials in the next section,
but the cell will first be used for another purpose. The transfer-
ence number 77H being assumed constant over the concentration
range 0.10 to 0.020, the change in state for 1 faraday through the
cell is the sum of the effects at the anode, the liquid junction, and
the cathode, as shown by the equations

(Anode) Ag(«) + Cl~(0.10m.) = AgCl(s) + <r

/T. ., . .. , / rHH+(0.10m.) = !rHH+(0.02m.)
(Liquid junction) ) T^-(*SX*m.) = TCICI~ (0.10m.)
(Cathode) AgCl(s) + e~ = Ag(s) + Cl"(0.02m.)

Net change in state: rHH+Cl-(0.10m.) = 77HH+Cl-(0.02m.)

For this change, AF = -EtF = 2THRT In (0.0272)/(0.107i),and
this gives for the potential of the cell with transference

1 LEWIS and KBAUS, ibid., 32, 1459 (1910).

POTENTIALS OF ELECTROLYTIC CELLS 657

In the corresponding cell "without transference,"

Ag + AgCl, HCl(0.10m.), H2— H2, HCl(0.02m.), AgCl + Ag;

#298 = 0.0778 volt

(of which the potential is the difference between the fourth and
seventh cells in Table 97), the net change in state is

H+Cl-(0.10m.) = H+Cl-(0.02m.)

The free-energy change is AF = 2RT In (0.0272)/(0.107i), from
which the potential of the cell is seen to be

" F 0.0272

Upon dividing the expression for Et by that for E of the cell with-
out transference and inserting the measured potentials we have

Et _ 0.0645

E - 00778 ~ TH - °'829 (21°

The transference number for hydrogen ion was given on page 266
as 0.827 at 0.02m and 0.831 at 0.10m., based upon the moving-
boundary method, and it will be seen that the value derived
from the cells is in agreement with these figures.

This is the third method of determining transference numbers
that we have had, the others being a gravimetric method in
which the actual quantities of ions gained or lost near the elec-
trodes are determined, and the moving-boundary method in
which the relative velocities of the ions in a solution are measured.

If in the cell with transference the left-hand solution is kept
O.lm. HC1 and m2 is the molality of HC1 on the right, the poten-
tial of the concentration cell at 25° changes with m2 as follows:1

m2 0 00526 0 0100 0.02004 0 0598 0.0781

#298 . 0 118 0.0925 0 06446 0.0206 0.00995

1 SHEDLOVSKY and MAC!NNES, ibid., 68, 1970 (1936).

658 PHYSICAL CHEMISTRY

Corresponding data for cells with the same electrodes, with
O.lm. KC1 on the left and w2 molal KC1 on the right are1

w2 0 0100 0 0200 0 0500 0 200 0 500

tfm . 0 0540 0 0375 0 01591 -0 01576 -0.03645

Liquid -junction Potentials. — The potential at a liquid junction
depends upon the nature and concentration of the ions on the
two sides of the boundary and upon the temperature. In
order to show the relation of the sources of potential in cells with
transference to that of the whole cell, consider another cell
similar to the one in the previous section,

Ag+AgOl, HCl(0.1()m.), HCl(0.01m.), AgCl-f Ag; E298 = 0.0925 v
EI -f- EI, — EZ = E

in which the separate sources of potential are indicated. The
values of EI and Ez are compirted in the usual way, and that of
EL is, of course, E — EI + Ez. At the liquid junction, as
through all parts of the cell, electricity is carried by the ions in
proportion to their transference numbers. The transference
number being assumed constant over the concentration range
involved, the change in state at the liquid junction is shown by
the two equations

= rHH+(0.01m.)
raCl-(0.01m ) = TaCl-(

The free-energy increase is

AF = TiRT in ^ + TaRT In -i = -Rf (22*)

Upon rearranging and solving for E^

RT 1n .

£L = UH - Joi) -r In

Substitution of the numerical quantities Tn = 0.83, TCi — 0.17,
7i = 0.796, 72 = 0.904 yields EL = 0.0367. From the values

1 SHEDLOVSKY and MAC!NNKS, ibid., 60, 503 (1937) Data for NaCl cells
with transference are given by Brown and Maclnnes, ibid., 67, 1356 (1935),
and by Janz and Gordon, ibid., 66, 218 (1943).

POTENTIALS OF ELECTROLYTIC CELLS 659

Ei = -0.2874 and E2 = -0.3432, computed in the standard
way, we confirm the computed value of the liquid junction, since

#oeii = -0.2874 + 0.0367 - (-0.3432) = 0.0925

which is the measured potential of the cell.

The recorded potentials of cells with liquid junctions are some-
times " corrected for liquid potential" by subtracting the cal-
culated liquid potential from the measured potential. When
this has been done, the common notation is to insert a double
bar between the solutions written in the cell, as follows:

Ag + AgCl, HCl(0.1m.)||HCl(001m), AgCl + Ag; E = 0.0558

This indicates that the recorded potential is EI — E^ and the
notation

Ag + AgCl, HCl(0.1m.), HCl(0.01m.), AgCl + Ag; E = 0.0925

with a comma separating the two solutions, indicates that
the recorded potential has not been so corrected and that it is
Ei + £L — EI.

Liquid junctions are also found in cells that are not merely
concentration cells with transference, of course; and thus their
calculation is desirable. It will usually be true that such junc-
tions are avoided when possible, but in some types of work
this is difficult or impossible. When the junction is between
two solutions of the same solute at different concentrations,
the liquid potential is independent of the way in which the
junction is made or whether the boundary is sharp or not.
If the transference number is sufficiently constant over the
concentration range involved, the general expression for this
type of junction is

(240

where mi is the molality on the left-hand side of the junction.

There is another type of junction in which the solutes on the
two sides are not the same and for which the junction potential
depends on the way in which the junction is made. When
both the molality and the solute are different, the calculation is
uncertain at best, and such junctions are usually avoided in

6GO PHYSICAL CHEMISTRY

experimental work by the expedient of two junctions. For exam-
ple, HCl(0.1m.), KCl(O.lra), KCl(1.0m.) shows the way in
which the first and third solutions would be connected. For the
junction in which all the ions are univalent, the concentrations
are the same on both sides, and one of the ions is common to
both sides while the other is not, the liquid potential is given by
equations such as

7? _ RT , AHCI * scyr-^

&L = -™- In - — (25$)

r AKCI

which applies to the first junction listed above.1 When the ions
have valences other than unity, when both ions are different on
the sides of the junction, or when one ion is different and the
molalities are not the same, the relations for calculating junction
potentials are complicated and best not considered by beginners.
lonization Constant of Water. — The potential of a hydrogen
electrode is determined by the partial pressure of the hydrogen
gas and by the activity of hydrogen ion in the solution, even if
the solution is alkaline. Hence a properly designed cell may be
used to determine the ion product for water, and a suitable one
for the purpose is2

H2(l atm.), AgC1 + Ag; Ez9B = °'9916 volt

For this cell El = -0.0592 log (OH* in 0.01m. KOH) and E2 =
—0.222 — 0.0592 log (l/oci-). In this solution the activity
coefficient is determined by the total molahty and is very close
to 0.80. Upon equating the measured potential of the cell to
Ei — Et, we. find log aH+ = — 11.90; and since log a0n- is —2.097,
log Kw is the sum of these quantities, or —13.99, and Kw is
1.03 X 10~14. In the paper from which this cell is quoted, the
measurements were upon a series of cells in which bath molalities
varied over considerable ranges,

TT /^ x N
H,(latmO,

1 LEWIS and SARGENT, ibid., 31, 363 (1909). For the general equations
for liquid junctions and their integration, see Maclnnes, op. cit., Chap. 13.

1 HABNED and HAMER, /. Am. Chem. Soc., 55, 2194 (1933). This paper
gives data for a series of KC1 molalities, for temperatures from 0 to 60°, and
data for other cells from which K* may be determined.

POTENTIALS OF ELECTROLYTIC CELLS

661

In place of assuming activity coefficients in anjr solution, the
potentials were expressed as

E = E0-^ln

Eliminating mH and rearranging,

Won

«H20

RT , „
~^lnA- JVF

The left-hand side of this equation being plotted against the ionic
strength ju, its intercept at ju = 0 is (-RT/NF) In Kw, since,
at /z = 0, aHjo = 1? and by reason of the definition of activity
coefficients, the two other members on the right-hand side of the
equation vanish

From such a procedure for the data at several temperatures,
the value of Kw was determined over the range 0 to 60°. Finally,
from the data and equation (30), which will be given on page
666, the value of A// for the ionization of water as a function
of the temperature was calculated. The results given in Table
100 agree with those determined directly in a calorimeter. For
example, the figures on page 318 are 13,610 for 20° and 13,360 for
25°, an agreement as close as is ordinarily found.

TABLE 100. — IONIZATION CONSTANT OF WATER

«,°c.

Kv X 1014

A//t, cal.

0 115

14,513

0.293

14,109

0 681

13,692

1 008

13,481

1.471

13,267

2 916

12,833

5 476

12,390

9 614

11,936

Ionization Constants of Weak Acids. — Cells without liquid
junction, containing mixtures of a weak acid and its salt, may
also be used to determine the ionization constants of weak acids.
As an illustration, the ionization constant of acetic acid has been
measured1 through the potential of a cell in which the solution

1 BATES, SIEGAL, and AGREE, J Research Nail. Bur. Standard*, 30, 347
(1943).

662

PHYSICAL CHEMISTRY

is 0.049m. in sodium acetate and 0.050m. in sodium chloride,
with hydrogen and Ag + AgCl for electrodes. Addition of stand-
ard nitric acid (containing 0.05m. NaCl and 0.049m. NaNO3)
in known quantity displaces acetic acid from its salt and keeps
the ionic strength and chloride molality constant; and the activity
of hydrogen ion is determined from the potential of the cell.
In effect, the cell is

H2(l atm.),

H4" (variable a)
Cl~(0.050m.)

AgCl + Ag

With a constant chloride-ion molality and constant total-ion
molality the activity coefficient is constant, E2 for the silver
chloride electrode is constant, and the cell potential is

= (E °H ~ E 0A

- ~ In 0.057ci- - ~r In

By assuming 0.78 for the activity coefficient, substituting the
required numerical quantities, and rearranging, we have for 25°

log aH+ =

0.305 - E
0.0592

(26/)

The activity of acetate ion is my, and that of acetic acid is sub-
stantially equal to its molality; thus all the quantities necessary
for computing Ka for the weak acid are at hand. Some of the
measured potentials of this cell and the derived ionization con-
stant Ka are shown in Table 101.

TABLE 101. — IONIZATION CONSTANT OF ACETIC ACID

Ez9t

log OH+

WHAc

™Ae-

log Ka

Ka X 10&

0.6026

-5.014

0.00149

0 00305

-4 75

1.8

0.5803

-4 638

0 00250

0 00248

-4 75

1.8

0.5439

-4 021

0 00402

0 00096

-4 75

1 8

0.5231

-3.670

0 00450

0 00049

-1 75

1.8

These figures confirm the ionization constant of acetic acid
used in Chap. IX. Application of the same method to other
acids also yields ionization constants that are satisfactory.

POTENTIALS OF ELECTROLYTIC CELLS 663

Solubility Product of Lead Sulfate. — The potential at 25° for
the cell

Pb(s) + PbSO4(s), H2S04(m molal), H2(l atm.)
in which the cell reaction for 2 faradays is

Pb(«) + H2S04(m molal) = PbS04(s) + H2(l atm.)
changes with the molality of sulfuric acid as follows:1

m 0 001 0.002 0 005 0 010 0.020

EM . 0 1017 0 1248 0 1533 0 1732 0 1922

Extrapolation oi these potentials in the way already explained
gives E° = 0.356 volt for this cell; and since #°H = 0, E° = 0.356
volt for the electrode Pb(s) + PbSO4(s), SO4— (u.a.). , This
potential may be considered as that of an electrode at which the
reaction is Pb(s) = Pb++ (in SO4 — u.a.) + 2e~, and to which the
equation E = EQPb — (RT/2F) In aPb++ applies. Upon equating
these potentials and substituting EQFb = 0.126, we find

a™** = 1.6 X 10~8

in a solution containing S04 (u.a.)r and therefore this is the
activity product for lead sulfate. The square root of this
activity product is not the solubility in pure water, for even in
solutions 10~4m. of this ionic type the activity coefficient is about
0.9, and thus it is (0.9S)2 which is equal to 1.6 X 10~8, or
S = 1.5 X 10-4.

Electrometric Titration and pH Measurement. — It will be
recalled from Chap. IX that there is some confusion in the use of
the term "pH, " which is sometimes defined as pH = — log mH+
and sometimes as pH = — log aH+, which is — log mH+yn+.
These definitions obviously differ by — log 7, which is usually
0.05 or less for solutions in which the total ion molality is 0.1 or

1 Shrawder and Cowperthwaite, J. Am. Chem. Soc., 56, 2340 (1934),
measured this cell with a two-phase lead-mercury amalgam. Their poten-
tials have been increased by 0.0058 volt, which is the potential of the cell
Pb(s), PbCl2, PbHgx (two-phase) to give the potentials above. Since the
phases in. the amalgam are a liquid solution and a solid solution, the poten-
tial of Pb(s) and PbHgx cannot be the same. The phases in an electrode
Zn-Hg (two-phase) are solid zinc and a liquid solution, and thus the poten-
tials of Zii (s) and ZnHg* (two-phase) are the same. The potential 0.0058 is
given by Carmody, ibid., 61, 2905 (1929).

664 PHYSICAL CHEMISTRY

less, and in which the ions have unit valence; but — log 7 may
be a much larger quantity in strong salt solutions or in the pres-
ence of ions of higher valence. For some purposes the distinc-
tion between — log mH+ and — log a^ is not important; for some
it is less than other errors inherent in the measurements and
calculation, and for some it requires attention.1

When a titration of acid with base is being made through the
equivalent of a hydrogen electrode and another reference elec-
trode dipping into the solution, the change of cell potential as
base is added will often be a sufficient indication 0f the end point.
In a cell such as

f N (H+ (variable a) ) „. , .

H2(l atm.), { ,11 / , 4. \\> AgCl + Ag

^ ' [ Cr~ (constant a) j b &

•

the relation of E to pH on an activity scale is

E — EQ

and the cell potential changes as base is added in the way shown
in Fig. 72. As the end point is approached, small additions of
base produce large changes in the hydrogen-ion activity. For
example, in the titration of HC1, pH changes from 4 to 7 and
the cell potential changes about 0.18 volt when the fraction of
acid titrated changes from 0.998 to 1.000, and a similar change is
produced by the addition of 0.2 per cent more base, so that the
end point is determined without considering an activity coeffi-
cient. But in precise determinations of iomzation constants,
like the one described on page 662, it was desirable to keep the
activity coefficient constant or to allow for its influence on the
cell potential.

It should be observed that the above definition of pH in terms
of a cell potential is valid only when the electrode reaction is
3^H2 = H+ + e~, and hence any other substance that oxidizes
or reduces at the electrodes will "interfere" if it is present in the
solution. This important qualification is sometimes overlooked,
and it may lead to serious errors in pH evaluations. Among the
substances that must not be present at a hydrogen electrode are
dissolved air, H2S, organic substances, ions of metals below

1 See MAC!NNBS, BELCHER, and SHEDLOVSELY, ibid., 60, 1094 (1938).

POTENTIALS OF ELECTROLYTIC CELLS

665

hydrogen in the potential scale, and oxidizing or reducing sub-
stances in general. Exclusion of air is especially troublesome
from an experimental point of view, but it is quite necessary if
precision is desired. The use of KC1 "salt bridges " or other
means of separating the reference electrode from the unknown
solution is a further complication that is sometimes difficult to
avoid and always difficult to interpret.

0591

<L>
T3

_£ 0532

C 0473

g4 0414
u

x 0355
o
•5 0296

I 0237
0178

8
pH

100.8

996 1000 1004

Per Cent of /Acid Titrated

FIG. 72 — Change of hydrogen electrode potential and pH with progress of a

titration of acid.

Some equivalents of a hydrogen electrode will be briefly dis-
cussed, and it may be said that all of them have their own peculiar
virtues and restrictions For details, the student is referred to
the special works devoted to the topic.1 The " antimony elec-
trode " is a metal + oxide electrode for which the reaction is
2Sb + 3H20 = Sb2O2 + 6H+ + 6e~. Statements as to its use
are a little conflicting, but it is commonly said to be of moderate
but not high precision in the presence of air over the pH range
2 to 7 if oxidizing and reducing substances are absent. The
"quinhydrone electrode" consists of a gold plate in contact with
a molecular compound of 1 mole of quinone and 1 mole of hydrp-

1 See KOLTHOFF and LAITINEN, "Electrometric Titrations," John Wiley &
Sons, New York, 1941; DOLE, "The Glass Electrode," John Wiley & Sons,
New York, 1941; MAC!NNES, op. ctt., Chap. XV. A review of pH methods
with bibliography (630 references) is given by Furman in Ind. Eng. Chem.,
Anal. Ed., 14, 367 (1942).

666 PHYSICAL CHEMISTRY

quinone in the unknown solution, for which the electrode reac-
tion is C6H4O2H2(s) = C6H4O2(s) + 2H+ + 2e~. The unknown
solution is connected to the reference electrode through a "salt
bridge " of KC1, so that it is better suited to measuring changes
in pH than to their precise determination unless a standard
buffer is used for calibration. Interfering substances include
amines, oxidizing and reducing agents, phenols, and other sub-
stances, and the pH must be below 7 when this electrode is used.
Another common device is the so-called "glass electrode/ ' in
which a silver chloride electrode in O.lm. HC1 is separated from
the unknown solution by a glass barrier about 0.001 mm. thick,
and with a calomel or other reference electrode in the unknown
solution. The assembly functions as a concentration cell without
transference, and the equation

E ~ °'352

applies.1 The chief virtue of the glass electrode is that it per-
mits pH determinations in the presence of air, organic matter,
oxidizing or reducing agents, and metals below hydrogen in the
potential series over a pH range of 1 to 9, with the widest general
applicability of any method. Like any other method, it has its
restrictions, and there are some experimental difficulties that
require attention. In alkaline solutions it requires large cor-
rections for sodium ions and less important corrections for other
substances. It is probably the best means of determining
hydrogen-ion activities available at the present time.

Change of Potential with Temperature. — The equations
derived in the previous chapter for the change of free energy
with temperature become the equations for the change of poten-
tial with temperature when the relation AF = —ENF is com-
bined with them. By making this change in equations (16) and
(17) on page 621, we have the necessary relations

T dT

^ = AS (31)

U,I

1 MAC!NNES and LONGSWORTH, Trans. Electrochem. Soc., 19S7, 73.

POTENTIALS OF ELECTROLYTIC CELLS 667

As a direct check upon equation (31), we return to the cells
quoted on page 637, one of which was

Hg + Hg2Cl2, HCl(1.0m.), C12(1 atm.); E\n = 1.0904 volts

for which dE/dT = -0.000945 volt per deg.1 By substituting
the numerical quantities into equation (31), we find AS°298 =
— 43.6 cal. per mole per deg. for the cell reaction

2Hg(0 + C12(1 atm.) = Hg2Cl2(s)

and from the entropy data we calculated AS°298 = —43.3 for this
reaction on page 637.
Another confirmation of equation (31) is obtained from the cell

Ag + AgCl, HCl(1.0m.), CI2(1 atm.); #°298 = 1.1362 volts

for which dE/dT = -0.000595 volt per deg. These measured
quantities give AS°298 = — 13.7 for the cell reaction

Ag(«) + MC12(1 atm.) = AgCl(s)

in confirmation of AS0 = —13.8 calculated on page 612.

Application of equation (29) to these same cells leads to the values
A// = —63,200 cal. for the enthalpy of mercurous chloride and
AH = — 30;300 for silver chloride, and these are very close to
the entries in Table 58

When A// is sufficiently constant, equation (30) may be inte-
grated between limits to yield

T* Ti NF\ 7\1\

and equation (31) may be integrated to yield

ENF = T AS + const. (33)

By comparison with equation (22) on page 623, which is
AF = A// - T AS

it is at once evident that the integration constant in equation
(33) is — AH, and thus the equation may be written

ENF = -A# + TAS (34)

1 GKRKE, /. Am Chem Soc., 44, 1684 (1922).

668 PHYSICAL CHEMISTRY

We may illustrate the use of equation (32) by applying it to
the seventh cell in Table 97, taking the potentials at 15° and 35°.
The cell reaction for 2 faradays is

H2(l atm ) + 2AgCl(«) = 2HCl(in O.lm. HC1) + 2Ag(s)

for which we obtain AH by substituting the cell potentials in
equation (32), finding A// = —18,730 cal. The cell reaction
consists in forming two moles of HC1 from hydrogen and AgO
and introducing them into O.lm. HC1. If A// is calculated from
the data in Table 58 and the partial molal heat of solution based
on the data on page 314, the result is —18,820 cal., which is a
satisfactory check. For dilute solutions, such as the O.lm. HC1
in this cell, the difference between the partial molal heat of solu-
tion and A// for the change in state HCIQ?) + 555H20 = O.lm.
HC1 is small (about 100 cal. in this case) and is perhaps best
ignored by beginners. But if the temperature coefficient of
potential for a cell such as

H2(l atm ), HCl(10m.), AgCl + Ag

is used in the calculation of A#, the difference between a partial
molal heat of solution and the " integral" heat of solution is an
important one. It should be understood that Aff calculated
from the cell potentials involves this partial molal heat of solu-
tion, and not the integral heat of solution. Partial molal heat
quantities and the partial molal entropies of ions derived from
them are better reserved for more advanced courses.

Problems

Numerical data should be obtained from tables in the text.

1. The potential of the cell Zn(«), ZnCl«(ro molal), AgCl(s) + Ag(s)
changes with the molajity as follows:

#298... 1 1650 1 1495 1 1310 1 1090 1 0844 1 0556 1 0327 0.9978
m .... 0 00781 0 01236 0 02144 0 04242 0 0905 0 2211 0 4500 1.4802

(a) Calculate the mean activity coefficients for the ions in the first, third,
and last of these cells from the standard potentials in Table 99. (6) Show
that the relation of Em to E° for the cell is E° - (RT/2F) In 7 3 = Em -f 0.0886
log m •+• 0.0178. (c) Plot the right-hand side of this equation against *\/m
for the first four cells, extrapolate the curve to zero molahty, and obtain a
confirmation of the value of J£°zn — J£°A*CI used in the first part of the
problem, [SCATCHABD and TEFFT, J. Am Chem. Soc., 52, 2272(1930).]

POTENTIALS OF ELECTROLYTIC CELLS 669

2. For the cell H2(l atm.), HBr(0.100m.), Hg2Br2 + Hg, E™ - 0.2684
volt. Calculate j£°29&for the electrode Hg -f Hg2Br2, Br-(w.a.). [CROWELL,
MERTES, and BURKE, /. Am. Chem. Soc., 64, 3021 (1942).]

3. Thfe potential of the cell H2(l atm.), HCl(m molal), AgCl + Ag at
298°K. changes with the molahty of HC1 as follows:

m . 4 6 8 10 12 14

#298 0 1299 0.0704 0 0241 -0 0166 ~0 0525 -0 0839

(a) Calculate the standard free energy of HC1 (g) from some of these poten-
tials and the vapor-pressure data on page 188, taking —26,200 cal. for the
standard free energy of AgCl (s) . (b) Calculate the partial pressure of HC1 (g)
above the 12m. solution, (c) Calculate the activity coefficient for the ions in
the 10m solution.

4. Calculate the potential of the cell H2(0 1 atm ), HCl(0.001ro ),
C12(0 2 atm ) at 25°

6. Calculate the equilibrium constant for the chemical reaction ZnSC>4
-f Cd = CdSO4 -f- Zn at 25° from the electrode potentials.

6. Compute the potential of the concentration cell H2(l atm ), HC1-
(0 1m ), Hg2Cl2 + Hg— Hg -f HgaCla, HCl(0.001m ), H2(l atm ) at 25°.

7. (a) Compute the potential of the concentration cell H2(l atm.),
HCI(0 1m.), HCl(0.001m ), H2(l atm ) at 25° (b) Compute the potential
of the cell Hg + Hg2Cl2, HC1(0 1m ), HC1 (0.001m ), Hg2Cl2 + Hg at 25°.

8. The potential of the cell Zn(«), ZnSO4(0 010m ), PbS04(s) -f Pb(«)
is 0.5477 volt at 25°, and the activity coefficient in 0.010m ZnSO4 is 0 38.
(a] Calculate the standard electrode potential for Pb(«) -f- S04 (UM ) =
PbSO4(s) -f- e- (b) The activity product (apb++)(aso<--) = 1.58 X 10~8 at
25° in saturated PbSO4 solution. Calculate the standard for potential
Pb(s) = Pb++(w.a) +2e~ (c) The potential of the cell becomes 0.5230 volt
when the ZnS04 is 0 050w. Calculate the activity coefficient for this solu-
tion. [Data from Cowperthwaite and LaMer, J Am Chem. Soc., 63, 4333
(1931) ]

9. The cell Cu(s) -f CuCl(s), K+C1-(0 1m ), C13(1 atm ) has a potential
of 1.234 volts at 25°. (a) Calculate the solubility product for CuCl m water
at 25°. (b) What is the concentration of cuprous ion at the anode of this
cell?

10. Calculate the standard electrode potential Br2ff)» Br~(w.a.) from the
cell H»(l atm.), HBr(0.02m.), Br2(0; ^293 * 1.287 volts

11. Given #298 » 0.1116 for the cell

H,(l atm.), NaOH(0.105w.), NaHgx— NaHg,, NaOH (0.010m.), H2(l atm.)
calculate Ew% for the cells

(a) Hg-fHgO,NaOH(0.105m.),NaHgx-NaHgI,NaOH(0.010m.),Hg+HgO
(6) H2(l atm.), NaOH (0.105m.), NaOH(0.010m ), H2(l atm.)
(c) NaHgz, NaOH(O.lOSm), NaOH (0.010m.), NaHg,
In all these cells the activity of water may be assumed equal in the two solu-
tions, and the transference number of sodium ion may be assumed constant
at 0.20.

12. The cell Ag + AgCl(s), NaCl(0.050w ), NaCl (0.010m.), AgCl« 4- Ag
has a potential of 0.0304 volt at 25°C., and m this range of molality the, trans-

070 PHYSICAL CHEMISTRY

ference number of the sodium ion is 0.390. (a) Calculate the potential at
25° of the cell Ag-hAgClM,Na01(0.050m.),NaHg*-- NaHga;,NaCl(0.010w.),
AgCl(s) -f Ag. (b) Calculate another value of the potential from the data
in Table 98.

13. The potential at 298°K. of the cell Ag + AgBr(s), KBr(0.050m.),
KBr(0 010m ), AgBr(s) -f Ag is 0.0375 volt. Write the change in state per
faraday for the cell, and calculate the transference number of potassium ion,
assuming it constant in this concentration range, and assuming the activity
coefficients for KBr the same as those for KC1. [MAC WILLIAM and GORDON,
J Am. Chem. Soc., 65, 984 (1943) ]

14. Calculate the standard potential ChClw ), Cl~(w.a.) discussed in
footnote 1 on page 645 from the entries in Tables 95 and 99.

16. The potential at 298°K of the cell Hg -f HgO, NaOH(m.), H2(l atm.)
for some molahties of NaOH is as follows:

#2*8 ..... -0 9255 -0 9255 -0 9255 -0 9255

m 0 0487 0 2000 0.2737 0 5000

Calculate Kw, the ion product for water, from this potential and the
standard potentials in Table 99.

16. Calculate the potential at 25° of the cell ZnHg(amalg , 0.001m.),
ZnCl2(0 1m ), ZnHg(amalg., 0003m.). In the amalgams containing 0.001
and 0.003 mole of zinc per 1000 grams of mercury, zinc is an ideal monatomic
solute,

17. (a) The activity coefficient for all the ions in the cell

H,(l atm ), ' > AgCl + Ag; EMt = 0.992 volt

is 0.80. Calculate EQ for this cell with both negative ions at unit activity.
(b) Calculate the potential of this cell with unit activities in series with the
cell Ag 4- AgCl, HCl(w.a ), H2(l atm.), write the change in state for the
two cells, and calculate Kw

18. The potential of the cell H2(l atm ), NaOH(0.02m.), ZnO(a) -f Zn(«)
is —0.420 volt at 298°K. Calculate the standard free energy of ZnO(s),
taking —56,690 cal as the standard free energy of H20(Z). [The answer
should check that of Problem 6, page 626.]

19. Calculate the potential at 25° of the cell Ag + AgCl(s), NaCl (0.10m ),
NaCl(0.010m.), AgCl -f Ag. (The measured potential is 0.0430 volt.)

20. The potential of the cell Tl -f TlCl(s), KC1, (0.02m ), Clafo, 1 atm.)
is 1.91 volts at 25°. (a) Calculate AF for the change in state occurring in
the cell and the standard free energy of TlCl(s) at 25°. (b) Calculate the
solubility of T1C1 in water at 25°. (c) Calculate E for the electrode TlCl(s)
-f Tl(«)* HCl-(0.10m ), taking <y as 0.80.

21. For the cell Pb, PbCl2(s), HCl(lm.), AgCl + Ag; Em = 0.4900 and
dE/dT - -0.000186 volt per deg (a) Calculate A/7 and &S for the cell
reaction, (b) Calculate A/f and A£ for the cell reaction from the data in
Tables 58 and 96.

POTENTIALS OF ELECTROLYTIC CELLS 671

22. Calculate the temperature coefficient of potential for the cell Ag
+ AgCl, KCl(lm.), HgsCl2 -f- Hg, J0m - 0.0455 volt from the data in
Table 96

23. At 25° the cell Ag + AgCl, NaCl(m»), NaHg*— NaHg*, NaCl (0.10m ),
AgCl + Ag changes with mz as follows:

mz 0200 0500 1000 2000 3000 4,000

j&298 0.03252 0 07584 0 10955 0 14627 0 17070 0.19036

(a) Given the activity coefficient 0 773 in O.lm NaCl, calculate the
activity coefficients for 0 2 and 3 Ow NaCl (6) Calculate the potential of
the cell, at 25°, NaHg,, NaCl (0.20m ), NaCl (0 10m.), NaHg,, using the
transference numbers in Table 48. [HARKED and NIMB, J. Am. Ghent. Soc.,
54, 423 (1932) ]

24. Calculate the potential of the cell Hg -f Hg2Cl2, HC1(0 Olm ),
H£(! atm ) at 298°K., first from the free-energy table and again from the
standard electrode potentials.

25. Calculate the potential of the cell H2, HC1{0 1m ), AgCl -f Ag if it
operates under a barometric pressure of 700 mm., taking 23 mm as the
vapor pressure of water above the solution

26. Write the change in state for 1 faraday passing through the sodium
hydroxide concentration cell described on page 641 when m\ is 0 1934, and
confirm the calculated activity coefficient for this solution

27. Write the cell reactions for the six cells described on pages 644-645.

28. The potential of the cell

is 0 699 volt at 25 °C. (a) Calculate AF and AF° for the cell reaction. (6)
Note that the free 'energy of NH4C1 in its saturated solution is the same as
that of NH4Cl(s), refer to Table 95 for additional data, and calculate the
standard free energy of NH4Cl(s) at 298°K. (Compare the result with
that of Problem 35 on page 629.

29. Calculate the standard free energy of ferrous hydroxide from the cell
Fe(«) + Fe(OH)«(*), Ba (OH) 2 (0.05m.), HgO(s) -f Hg; EW - 0.973 volt
and such other data as are required.

30. Calculate E° for the cell H2(l atm.), HCl(w.« ), AgCl(s) -f- Ag at
273°K. from the data in Table 97.

31. Confirm the potential of the electrode Pb(s) -f PbSO4(s), SO4~~(?^.a.)
given on'page 646 from a suitable plot of the data for the cell Pb -f- PbSO4j
H2SO4, H2 given on page 663.

AUTHOR INDEX

Abrams, 499
Acree, 372, 661
Adam, 135
Adams, 395, 647
ikerlof, 640
Allen, 409
Allgood, 266
Alter, 533, 534
Alyea, 514
Anderson, 136
Archbold, 393
Arrhemus, 493
Aston, 541
Avenll, 406

Backstrom, 514

Bacon, 205

Baker, 208, 210

Banes, 404

Barnes, 303

Barry, 299

Bartlett, 55

Bates, 188, 246, 372, 661

Batson, 219

Batuecas, 27

Baxter, 8, 13, 14, 16, 25, 27, 53, 246,

515, 533, 546
Bearden, 71, 159

Beattie, 55, 57, 95, 97, 120, 344, 383
Beebe, 208

Belcher, 356, 372, 664
Bell, 467
Benton, 396, 467
Berg, 212
Berk, 447

Berkley, 184, 221, 224
Berkman, 478, 497
Bichowsky, 311, 315

Bickford, 251

Bingham, 132

Bird, 25

Birge, 70

Birnbaum, 413

Bjerrum, 361

Blair, 349

Blaisse, 115

Bliss, 533, 534

Blodgett, 136

Bobalek, 188

Bodenstein, 33, 512, 513

Bogart, 203

Bogue, 589

Bohr, 522

Bongart, 8

Born, 176

Bounon, 270

Bradshaw, 254

Bragg, 159, 160

Brann, 418

Brass, 349

Brearley, 501

Bredig, 486

Bndgeman, 95, 97

Bridgman, 131, 433, 462

Bntton, 376

Brown, 62, 125, 126, 141, 252, 647,

658

Brunjes, 203

Bryant, 299, 303, 346, 347
Burgess, 59
Burke, 669
Burnstall, 537
Burt, 53
Burton, 224, 503, 516

Campbell, 462
Cann, 647

673

674

PHYSICAL CHEMISTRY

Carmody, 643, 644- 663
Caven, 398
Chadwell, 220
Chadwick, 537, 550, 553
Chapman, 406
Chase, 374
Chaudhari, 176
Chow, 425
Clark, 376, 480, 587
Clarke, 14
Classen, 8
Clusius, 547
Cockroft, 553
Coe, 130
Coffin, 473
Cohen, 405
Colemaii, 272
Collins, 78

Cornpton, 153, 159, 505
Constable, 550
Cook, 282

* Coohdge, 170, 175, 190
Cornell, 210
Cottrell, 200
Coulter, 208

Cowperthwaite, 663, 669
Cox, 126
Creighton, 403
Crowell, 669
Crowther, 547

Dale, 126

Daniels, 101, 3^4, 471, 490, 492, 496

Darken, 421

Davies, 252, 287, 587

Debye, 284

deLange, 424

Derr, 412

Deschner, 62

Dickel, 547

Dickinson, 185, 490, 515

Dietrichson, 15

Doan, 159

Dodge, 49, 342

Dole, 252, 665

Donnan, 587

Dorsey, 112
Drake, 396
Dnesbach, 195
Dunphy, 185
Dushman, 70, 76

Eastman, 43, 153, 303

Edgar, 418

Edgerton, 126

Edmonds, 413

Edwards, 78

Egan, 146

Egloff, 478, 497

Egner, 132

Ehlers, 640

Ekwall, 582

Ellis, 537

Embree, 385

Evans, 166

Ewcll, 212

Eyring, 464, 471, 492, 495, 49

Falk, 251

Felsing, 399

Ferguson, 123

Findlay, 429

Flannagan, 303, 304, 327

Fleharty, 420

Flock, il5

Flory, 222

Flugel, 218

Forbes, 190

Fornwalt, 170, 175

Forsythe, 50

Foulk, 209

Franck, 509

Fraser, 146, 224

Frazer, 183

Friedrich, 160

Fugassi, 490

Furman, 665

Gaddy, 187
Gamow, 537

AUTHOR INDEX

675

Garrett, 413

Geary, 361

Geiger, 549

Gerke, 510, 636, 667

Gerry, 108, 116

Gershwinowitz, 495

Giauque, 50, 113, 139, 143, 146, 378,

393

Gibson, 405
Giddings, 105
Gilbert, 140

Gillespie, 67, 68, 146, 344, 383
Oilman, 184, 199
Glasstone, 88, 98, 464
Godfrey, 130
Goeller, 407
Goodeve, 588

Gordon, 256, 266, 303, 378, 658, 670
Gottlmg, 159
Gotz, 271
Graham, 78
Gramkee, 305
Gross, 184, 199
Grover, 13, 533
Gunning, 256
Gunther-Schulze, 234
Gurry, 421
Guye, 15

Hafstead, 541

Hahn, 346

Hale, 25

Halford, 413

Hall, 236, 435, 534

Hamer, 282, 372, 660

Hammett, 467, 497

Hansen, 435

Harkins, 125, 126, 136, 141, 236, 587

Harmsen, 547

Harned, 246, 252, 282, 361, 385, 486,

640, 660, 671
Harrington, 14
Harris, 406, 507, 513, 585
Harrison, 212
Hartley, 184, 221, 224
Harvey, 139

Hatschek, 129

Hauser, 125, 126, 570, 575, 588

Hemmindinger, 541

Hertz, 421, 547

Herzfeld, 103

Hess, 305

Heuse, 82

Hildebrand, 117, 128, 355

Hinshelwood, 467, 475, 477

Hirschbold-Wittner, 7, 547

Hirshon, 575

Hoenshel, 50

Hoff, 223

Holhngsworth, 209

Holt, 126

Homgschmid, 26

Hosking, 130

Hovorka, 195

Howell, 499

Rowland, 188

Hubbard, 210

Huckcl, 284, 496

Huguet, 210

Hulett, 397

Humo-Rothery, 450

Hutchmson, 20

Insley, 435

Jacques, 364

Janz, 658

Jevons, 546

Johnston, 20, 472, 496

Johnstone, 188

Jones, 123, 236, 251, 254

Joule, 87

Kahlenberg, 241
Kaminsky, 507
Kassel, 346, 378, 383, 497
Kegels, 318

Kelley, 43, 44, 108, 116, 149, 151,
152, 303, 304, 306, 614

676

PHYSICAL CHEMISTRY

Kemp, 78, 139, 418

Kendall, 132

Kennard, 98, 550, 557, 564

Kenny, 365

Keyes, 95, 108, 116, 272, 399

Kharasch, 307

Kilpatrick, 347

Kirschman, 188

Kistiakowsky, 489, 501, 511, 514

Knipping, 160

Kohler, 346

Kohlrausch, 251

Kolthoff, 665

Kovarik, 537

Kraus, 219, 246, 252, 274, 656

Kuhn, 78, 512

Laidler, 464
Laitmen, 665
Lamb, 364
LaMer, 496, 669
Landolt, 5

Langmuir, 86, 134, 170, 173, 378, 586
Lannung, 235
Laplace, 82
Larson, 342
Lassettre, 185

Latimer, 50, 311, 355, 614, 640, 647
LeBeau, 588
Lee, 229, 455
Lehmann, 176
Lembert, 534
Lenher, 489
Leppla, 188

Lewis, 29, 45, 153, 277, 281, 283, 311,
334, 336, 357, 591, 594, 656, 660
Lind, 484
Lindblad, 582
Lindsay, 208
Lingane, 647
Linhart, 637, 653
Loeb, 584
Lohnstein, 125
Longsworth, 265, 266, 666
Lovelace, 183
Lowenstein, 68

Lowry, 245
Lundstedt, 14, 246
Lurie, 68
Lynn, 570

Maass, 28, 127, 187

McAlpme, 25

McBain, 272, 581

McDonald, 203

MacDougall, 29, 49, 99, 409

Maclnnes, 250, 252, 265, 267, 280,

284, 356, 372, 643, 647, 657, 660,

664, 666
McKeehan, 537
McMillan, 203
McMorns, 390
Mac William, 670
Maier, 151
Manley, 5
Mannweiler, 361
Manov, 372
Marble, 534
Marcelm, 134
Marsden, 549
Marsh, 429
Martin, 353, 385
Mather, 245
Matheson, 114
Mattauch, 543, 544
Maxwell, 75 ,
Meads, 50
Menn, 26
Mertes, 669
Michalowski, 210
Michels, 115
Miller, 15, 89, 188, 496
Millikan, 71
Mochel, 108, 194
Moles, 12, 26, 27
Monroe, 132
Montonna, 210
Morgan, 127, 537
Morrell, 478, 497
Morse, 221
Moseley, 520

AUTHOR INDEX

677

Murrell, 89
Myrick, 224

Neumann, 346

Nier, 543

Nims, 282, 671

Noyes, 49, 251, 418, 425

O'Brien, 188, 365
Ohphant, 547
Onsager, 251, 284
Osborne, 115
Osol, 407
Owen, 246, 252, 256

Ritchie, 26
Hitter, 386
Roebuck, 89
Rogers, 183

Rollefson, 391, 503, 516
Rosanoff, 185, 205, 480
Ross, 28

Rossini, 305, 308, 311, 315
Rotarski, 177
Roth, 218
Rothrock, 219
Rouyer, 270
Royster, 395
Ruark, 160
Rumbaugh, 541

Rutherford, 71, 527, 535, 537, 540,
549, 552

Pamfil, 271

Partridge, 447

Pauling, 561, 565

Perlman, 391

Pickering, 588

Pitzer, 318, 319, 324, 611, 613, 651

Pohl, 338

Pohti, 220

Pollard, 550

Porter, 136

Potts, 587

Prentiss, 236, 643 *

Purcell, 424

Rabinowitch, 364

Ramsperger, 471

Randall, 29, 49, 281, 311, 413, 646

Ray, 123

Raymond, 195

Read, 199

Richards, 8, 18, 241, 245, 299, 447,

534
Richtmyer, 159, 160, 505, 550, 557,

564
Rideal, 122

Sameshima, 195

Sancho, 27

Sand, 398

Sargent, 660

Scatchard, 108, 194, 236, 282, 284,

643, 668
Schaefer, 136
Scholes, 384
Schroeder, 447
Schiibel, 153
Schulze, 194
Seldham, 115
Shedlovsky, 250, 252, 356, 372, 643,

657, 658, 664
Sherrill, 49
Shire, 547
Shrawder, 663
Sibley, 480
Siegal, 661
Simard, 120
Simons, 386
Slater, 153
Smith, 108, 114, 116, 245, 415, 611,

613, 651
Smits, 424
Smyth, 395, 546
Smythe, 541
Soddy, 532, 537

678

PHYSICAL CHEMISTRY

Southard, 395

Speakman, 585

Spence, 514

Spencer, 303, 304, 327

Starkweather, 15, 16, 27, 53

Staudmger, 133, 573

Stauffer, 501

Sterner, 29, 44

Stephenson, 103, 143

Stern, 74

Stillwell, 166

Stimson, 115

Stockdale, 412

Stockmayer, 364

Stookey, 132

Stall, 241

Sturtevant, 314

Su, 120

Swartout, 126

Sweeton, 256

Swietoslawski, 308

Tartar, 353, 385
Taylor, 299, 397, 647
Teal, 7, 546
Teare, 640
Tefft, 668
Thomas, 546
Thomsen, 87
Thomssen, 127
Thornton, 308
Tingey, 510
Titus, 14
Toabe, 419
Tolman, 471
Toral, 12, 26

U
Urey, 7, 160, 546

Vernon, 490
Vinal, 246
Virgo, 70
Voigt, 176
von Antropoff, 522
von Laue, 160
Vosburgh, 412

Walker, 483

Wall, 404

Walton, 486, 553

Warburg, 510

Waring, 499

Warner, 229, 445

Warren, 138

Washburn, 130, 185, 199, 262, 270,

299

Weber, 29, 49
Weibe, 187
Wells, 447
Whitcher, 15
White, 205
Willard, 18, 188
Williams, 130, 140
Wilson, 490
Winkler, 127
Wmninghoff, 272
Wood, 108, 194, 282
Wourtzel, 484
Wouters, 115
Wright, 132, 187
Wyckoff, 159, 160, 169

Yngve, 447

Yost, 349, 390, 487

Young, 646

van den Bosch, 405
Verhoek, 101, 344

Zawidski, 195
Zollman, 587

SUBJECT INDEX

Activated molecules, 493
Activity, 281

of ions, 284, 638

of solid phases, 393
Activity coefficient, 282 ^

from cell potentials, 638

table of, 284, 641

from vapor pressures, 282
Adsorption, 171
Alpha particles, 525
Atomic nuclei, 550
Atomic numbers, 520
Atomic structure, 538
Atomic weights, 12, 14, 246, 545

table of, 21 and inside front cover
Avogadro's law, 10, 70
Avogadro's number, 71, 169, 242, 578
Azeotropes, 208

Beattie-Bridgeman equation, 95
Beta particles, 526
Bohr atom model, 555
Boiling point, 113

and pressure, 114

of solutions, 197
Boiling-point constants, 202
Boyle's law, 53
Bragg's law, 160
Brownian movement, 577
Buffer solutions, 369

Calorimeter, 299
Carnot cycle, 38
Catalysis, 475

Cell potentials, 630-671

and temperature, 666
Cell reactions, 633
Chain reactions, 512
Change in state, 30, 31, 293
Charles's law, 56
Clapeyron equation, 109
Colloids, 566
Complete lonization, 277
Complex ions, 411
Composition of matter, 9
Compounds, 6
Concentration, 24, 180
Concentration cells, 638
Conductance, 247

equivalent, 248

limiting, 266

of liquids, 273

measurement, 253

of mixtures, 268

of nonaqueous solutions, 272

ratio, 276

of separate ions, 266

standards, 254

and temperature, 255

of water, 255

Conductimetric titration, 270
Consecutive reactions, 485
Conservation of matter, 5
Cooling curves, 439
Coordination number, 163
Corresponding states, 121
Coulometer, 245
Critical density, 117
Critical pressure, 117
Critical temperature, 117
Crystal structure, 154

of compounds, 166

of elements, 165
Crystals, properties, 144-178

679

680

PHYSICAL CHEMISTRY

Dalton's law, 66

Debye-Htickel theory, 285

Dialysis, 569

Dissociation of gases, 64, 210, 337,

511

Dissociation pressures, 395
Distillation, 196

fractional, 202

steam, 212
Distribution between solvents, 189,

403
Donnan equilibrium, 582

Effusion of gases, 78
Einstein's law, 506
Electrical conductance, 247
Electrical double layer, 575
Electrode reactions, 242, 633
Electrometric titration, 663
Electron shells, 558
Elements, 6
Emulsions, 585
Enthalpy, 35

of combustion, 307

of compounds, 310
table, 315

of dilution, 316

of formation, 308

of ionization, 319

of neutralization, 318

of solution, 314

table of standard, 315

and temperature, 320
Entropy, 41
Equilibrium, 333

heterogeneous, 392

homogeneous, 332

for ions, 358

between metals and ions, 418

phase, 427

solids and gases, 399

in solutions, 351

and temperature, 378
Equivalent conductance, 248
Eutectic, 439

Faraday's law, 239

First law of thermodynamics, 32, 294

Forces between atoms, 153

Fractional distillation, 202

Free energy, 46, 591

and activity, 598

and chemical equilibrium, 601

of isothermal changes, 593

and maximum work, 591

and temperature, 620

and third law, 604
Freezing points, 214

constants for, 217

of electrolytes, 235

of solutions, 214

Gamma rays, 526

Gas constant, 61

Gas dissociation, 64, 210, 337, 511

Gas thermometer, 58

Gases, 51-101

Gay-Lussac's law, 56

Heat, of combustion, 307

of evaporation, 115

of formation, 308

of fusion, 148

of neutralization, 318

of reaction, 307-320
and temperature, 320

of solution, 314
Heat capacity, 36, 79, 81, 300

of crystals, 149

of gases, 303

of solutions, 305
Henry's law, 185

Heterogeneous equilibrium, 392-426
Homogeneous equilibrium, 332-391
Hydration of ions, 270
Hydrolysis, 362

Ice point, 57
Ideal gas, 60

SUBJECT INDEX

681

Ideal solutions, 181
Indicators, 373
Intenonic attraction, 284
Ionic conductances, 267
Ionic strength, 24
Ionic theory, 274
lonization, of salts, 357

of water, 360, 660

of weak acids, 353
lonization constants, 355, 661
Ionized solutes, 231
Isoelectnc point, 584
Isotopes, 12, 532, 540

in periodic table, 523

Joule effect, 86
Joule-Thomson effect, 87

Kelvin scale, 40, 59

Keyes' equation, 95

Kinetic theory, 74

Kinetics, 464

first order, 469, 527
second order, 481
third order, 484

Latent heat, of evaporation, 115

of fusion, 148
Limiting conductance, 249, 266

table, 267

Limiting densities, 15
Liquefaction of gases, 89
Liquid crystals, 175
Liquid junctions, 658
Liquid solubilities, 103, 454
Liquids, 102-143

Mass defect, 551
Mass numbers, 544
Mass spectrograph, 541

Maxwell distribution law, 75

Maxwell equations, 48

Melting point, 147

Mercurous ion, 653

Mole, 23X 234

Mole fraction, 63

Mole numbers, 234, 237, 276

Molecular attraction, 127, 133

Molecular cross' section, 135

Molecular theory, 8

Molecular weights, 14, 72, 197, 214,

231, 572
Monolayers, 134

Nuclear reactions, 552
Nuclear structure, 550

Ohm's law, 247

Orientation in interfaces, 134

Osmotic pressure, 220

Partial pressure, 66

Periodic law, 517

Pentectics, 444

pH scale, 371

Phase diagrams, 430-455

Phase rule, 429

Phases, 52

Poiseuille's law, 129

Potentials' of cells, 630-671

Process, 508

Quanta, 505
Quenching method, 441

Radiation and chemical change, 502
Radioactive changes, 525
Radioactive series, 529

682

PHYSICAL CHEMISTRY

Rankine scale, 61
Raoult's law, 182
Reaction rate, 464

Second law of thermodynamics, 37,

110, 593, 621
Sensitized reactions, 514
Soap solutions, 581
Solubility, 179, 405

of carbonates, 415

of hydrolyzed salts, 414
Solubility product, 408
Solutions, 179

ionized, 231

solid, 448
Standard cell potentials, 641

and free energy, 651
Standard changes in state, 604
Standard electrode potentials, 645
Standard entropies, 607

table of, 616
Standard free energy, 636

and cell potential, 636

of compounds, 608

of elements, 605

table of, 615
Standards, 22
Steam distillation, 212
Stokes' law, 131
Structure of surfaces, 170
Surface tension* 122

and drop weight, 125

and temperature, 126

Temperature measurement, 58
Theories, 4

Thermochemistry, 292-331
Thermodynamic equations, 46
Thermodynamic properties, 45

Thermodynamic temperature, 40
Thermodynamics, 29

first law of, 32, 294

second law of, 37, 110, 593, 621

third law of, 43, 604
Titrations, 376

by conductance, 270

by potentials, 663

Transference numbers, 256, 264, 656
Trouton's law, 116
Types of electrolytes, 233

Ultramicroscope, 576
Unit cells in crystals, 161
Units, 22

van der Waals' equation, 91 ~

constants for, 93 •

reduced form, 120 •/
van't Hoff equation, 379, 392, 623
Vapor pressure, 104

of binary mixtures, 192 •

of crystals, 145

of electrolytic solutions, 235

measurement of, 106

and pressure, 107

of solute, 185

of solvent, 183

table, 108

and temperature, 109
Victor Meyer method, 73
Viscosity, 128

of emulsions, 589

of mixtures, 132

of suspensions, 573

X-ray diffraction, of crystals, 159
of liquids, 138

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