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WHOSE TENDER SYMPATHY HAS HELPED TO 
^LIGHTEN THE DARKNESS OP THE*DAY^ 
DURING WHICH THESE PXoW * v 
WERE WRITTEN, 



ji the common operations and practices of chymists, almost 
letters of the alphabet, without whose knowledge 'tis very 
hard for a man to become a philosopher; and yet that knowledge is very far 
from being sufficient to make him one." 

ROBERT BOYLE. (The Sceptical Chymist.) 



PREFACE TO SECOND EDITION. 



"The theoretical side of physical chemistry is and will probably remain the 
dominant one; it is by this peculiarity that it has exerted such a great in; 
fluence upon the neighboring sciences, pure and applied, and on this ground 
physical chemistry may be regarded as an excellent school of exact reasoning 
for all students of natural sciences." Arrhenius. 

The demand for a second edition of this book has not only 
afforded the author an opportunity to thoroughly revise the 
original text, but also has made it possible to include such ne\v 
material as should properly find a place in an introductory test- 
book of theoretical chemistry. 

The arduousness of the task of revision and amplification has 
been appreciably lightened by the helpful criticisms and valuable 
suggestions which have been received from those who have used 
the first edition with their classes. 

So numerous were the additional topics suggested that the 
author found himself confronted with a veritable embarrassment 
of riches, and not the least difficult part of his task has been the 
attempt to weave in as many of these suggestions as seemed to be 
consistent with a well-balanced presentation of the entire subject. 

The features which distinguish this edition from the preceding 
edition may be briefly summarized as follows: 

1. The necessity of introducing a short chapter on the mod- 
ern conception of the atom and its structure involved the 
further necessity of including a preliminary chapter treat- 
ing of those radioactive phenomena upon which the greater 
part of our present atomic theory is based. 
The chapter on solids has been practically rewritten: 
the space formerly devoted to an outline of crystallog- 

vii 



viii PREFACE 

raphy being devoted in the present edition to a discussion 
of the absorption of heat by crystalline solids and the bear- 
ing of X-ray spectra on crystalline form. 

3. The increasing importance of colloidal phenomena, not 
only to the chemist but also to the biologist, to the physician, 
and to the technologist, has made it seem desirable to rewrite 
the entire chapter devoted to the chemistry of colloids. 

4. The Brownian movement and its bearing upon the exist- 
ence of molecules has been briefly presented in a separate 
chapter in order to emphasize the importance of th^ bril- 
liant experimental work of Perrin and others in confirming 
the kinetic theory. 

5. The chapter treating of electromotive force has been 
enlarged so as to include a discussion of some of the more 
valuable methods which have been proposed for deter- 
mining junction potentials and also to point out several 
useful applications of concentration cells. 

6. An entirely new chapter in which an attempt has been made 
to present the salient facts and more important theories 
of photochemistry in succinct form replaces the former 
chapter treating of the relations between radiant and 
chemical energy. 

Among the books to which the author has had frequent re- 
course in the preparation of this edition should be mentioned 
Rutherford's " Radioactive Substances and their Radiations/' W. 
H. and W. L. Bragg's "X-Rays and Crystal Structure/' Freund- 
lich's " Kapillarchemie," Perrin's "Les Atomes/' and Sheppard's 
" Photochemistry. " 

It is with a deep sense of gratitude that acknowledgment is 
made to all of those friends who have offered criticisms of the old 
edition and suggestions for the new. Special thanks are due 
to Dr. W. D. Harkins of the University of Chicago not only for 
suggestions but for permission to quote extensively from his 
papers, to Dr. J. Howard Matthews of the University of Wis- 
consin for numerous helpful suggestions, to Dr. Walter A. Patrick 
of Johns Hopkins University for criticisms of the former chapter 



PREFACE IX 

on colloids and to Mr. John McGavack for his conscientious work 
in checking the answers to all of the problems. In the preparation 
of the indices of names and subjects the author is indebted to his 
wife and to Miss Mary K. Pease who have given valuable assist- 
ance in that wearisome and exacting task, To the publishers, 
Messrs. John Wiley and Sons, Inc., acknowledgment is made 
of their kindness in permitting the use of Pig. 68 taken from 
Chamot's "Elementary Chemical Microscopy. " 

FREDERICK H. GETMAN. 
STAMFORD, CONN. 
Aug. 7, 1918. 



PREFACE. 



"The last thing that we find in making a book is to know what we* must 
put first/' PASCAL. 

THE present book is designed to meet the requirements of 
classes beginning the study of theoretical or physical chemistry. 
A working knowledge of elementary chemistry and physics has 
been presupposed in the presentation of the subject, the introduc- 
tory chapter being the only portion of the book in which space is 
devoted to a review of principles with which the student is assumed 
to be already fairly familiar. With the exception of a few para- 
graphs in which the application of the calculus is unavoidable, 
no use is made of the higher mathematics, so that the book should 
be intelligible to the student of very moderate mathematical 
attainments. Wherever the calculus has been employed, the 
student who is unfamiliar with this useful tool must accept the 
correctness of the results without attempting to follow the suc- 
cessive operations by which they are obtained. 

The contributions to our knowledge in the domain of physical 
chemistry have increased with such rapidity within recent years, 
that the prospective author of a general text book finds himself 
confronted \vith the vexing problem of what to omit rather than 
what to include. In selecting material for this book, the author 
has been guided in large measure by his own experience in teaching 
theoretical chemistry to beginners arid to advanced students. 
The attempt has been made to present the more difficult portions 
of the subject, such as the osmotic theory of solutions, the laws 
of equilibrium and chemical action, and the principles of electro- 
chemistry, in a clear and logical manner. While the treatment 
of each topic is necessarily brief yet the effort has been made to 
avoid the sacrifice of clearness to brevity, 

xi 



XII PREFACE 

The author is fully convinced from his own experience as 
as from that of his colleagues, that the complete mastery of the 
fundamental principles of the science is best attained through 
the solution of numerical examples. For this reason, typical 
problems have been appended to various chapters of the book. 

Numerous references to original papers have been given through- 
out, since the importance of literary research on the part of the 
student is conceded by all teachers to be of prime importance. 

While a brief account of radioactive phenomena might very 
properly be considered to lie within the scope of a general outline 
of theoretical chemistry, yet owing to the unparalleled growth 
of knowledge in this field during the last decade, the author has 
come to believe that a condensed statement of the main facts of 
radiochemistry would not be of sufficient value to justify the 
effort involved in its preparation. 

In the original preparation of his lectures, and in their evolution 
into book form, the author has had frequent occasion to consult 
Nernst's "Theoretische Chemie," Ostwald's "Lehrbuch der 
allgemeinen Chemie," and Van't Hoffs "Vorlesungen ueber 
theoretische und physikalische Chemie." Among other books 
to which the author is especially indebted are the following: 
Le Blanc's " Lehrbuch der Elektrochemie," Daneel'a "Elek- 
trochemie." Text books of Physical Chemistry edited by Sir 
William Rarnsay, Bigelow's "Theoretical and Physical Chem- 
istry." Jones' "Elements of Physical Chemistry," Reychler- 
Kuhn's " Physikalisch-chemische Theorieen," and Whetham's 
"Theory of Solution." 

In the preparation of the problems the author would record his 
indebtedness to Abegg and Sackur's "Physikalisch-chemische 
Rechenaufgaben," and to Morgan's "Elements of Physical 
Chemistry." 

It is a pleasure to acknowledge the valuable assistance rendered 
by Dr. Eleanor F. Bliss and Dr. Anna Jonas, who have read and 
revised the proof of the paragraphs treating of crystalline form. 
The index of titles and names has been prepared by the author's 
wife to whose untiring patience its completeness is due. The 
author would also record his thanks to those friends whose kindly 



PREFACE Xiii 

^iticism has helped to remove many blemishes. Finally, the 
author would express his appreciation of the kindness of Messrs, 
Adam Hilger of London, and Fritz Koehler of Leipzig who have 
rendered great assistance by permitting the reproduction of 
illustrations of apparatus from their catalogs. 

FREDERICK H. GETMAN. 

STOCKBRIDGE, MASS. 
Aug. 18, 1913. 



CONTENTS. 



CHAP. PAGE 

PREFACE TO SECOND EDITION vii 

PREFACE xi 

I.^Fundamental Principles 1 

^Classification of the Elements 20 

IirTThe Electron Theory 31 

IV. Radioactivity 42 

V.^tomic Structure 57 

VI. Gases 72 

VII. Liquids : 104 

VIII. Solids 153 

IX. Solutions 167 

X. Dilute Solutions and Osmotic Pressure . , 187 

XI. Association, Dissociation and Solvation 225 

XII. Colloids 237 

XIII. Molecular Reality , 279 

XIV. Thermochemistry >. 286 

XV. Homogeneous Equilibrium 312 

XVI. Heterogeneous Equilibrium X 328 

XVII. Chemical Kinetics < 359 

XVIII. Electrical Conductance 385 

XIX. Electrolytic Equilibrium and Hydrolysis 422 

XX. Electromotive Force 446 

XXI. Electrolysis and Polarization 492 

XXII. Photochemistry 504 

INDEX OF NAMES 529 

INDEX OF SUBJECTS * 533 



xv 



THEORETICAL CHEMISTRY. 



CHAPTER I. 
FUNDAMENTAL PRINCIPLES. 

Theoretical Chemistry. That portion of the science of chem- 
istry which has for its object the study of the laws controlling 
chemical phenomena is called theoretical or physical chemistry. 
The first attempt to summarize the more important facts and 
ideas underlying the science of chemistry was made by Dalton in 
1808 in his "New System of Chemical Philosophy." The birth 
of the science of theoretical chemistry may be considered to be 
coeval with the appearance of Dalton's epoch-making book. 

Theoretical chemistry is concerned with the great generaliza- 
tions of chemical science and bears the same relation to chemistry 
that philosophy bears to the whole body of scientific truth; it 
aims to systematize all of the established facts of chemistry 
and to discover the laws governing the various phenomena of 
chemical action. 

Law, Hypothesis and Theory. The science of chemistry is 
based upon experimentally established facts. When a number 
of facts have been collected and classified we may proceed to draw 
inferences as to the behavior of systems under conditions which 
have not been investigated. This process of reasoning by analogy 
we term generalization and the conclusion reached we call a law. 
It is apparent that a law is not an expression of an infallible truth, 
but it is rather a condensed statement of facts which have been 
discovered by experiment. It enables us to predict results with- 
out recourse to experiment. The fewer the number of cases in 
which a law has been found to be invalid, the greater becomes our 
confidence in it, until eventually it may come to be regarded as 
tantamount to a statement of fact. 

1 



2 THEORETICAL CHEMISTRY 

Natural laws may be discovered by the correlation of experi- 
mentally determined facts, as outlined above, or by means of a 
speculation as to the probable cause of the phenomena in question. 
Such a speculation in regard to the cause of a phenomenon is 
called an hypothesis. 

After an hypothesis has been subjected to the test of experiment 
and has been shown to apply to a large number of closely related 
phenomena it is termed a theory. 

In his address to the British Association (Dundee, 1912), Profes- 
sor Senier has this to say : " While the method of discovery in chem- 
istry may be described generally, as inductive, still ?11 the modes of 
inference which have come down to us from Aristotle, analogical, 
inductive, and deductive, are freely used. An hypothesis is framed 
which is then tested, directly or indirectly, by observation and 
experiment. All the skill, and all the resource the inquirer can 
command, are brought into his service; his work must be accurate; 
and with unqualified devotion to truth he abides by the result, 
and the hypothesis is established, and becomes a part of the 
theory of science, or is rejected or modified." 

Elements and Compounds. All definite chemical substances 
are divided into two classes, elements and compounds. 

Robert Boyle was the first to make this distinction. He de- 
fined an element as a substance which is incapable of resolution 
into anything simpler. The substances formed by the chemical 
combination of two or more elements he termed chemical com- 
pounds. This definition of an element as given by Boyle was 
later proposed by Lavoisier and, notwithstanding the vast accumu- 
lation of scientific knowledge since their time, the definition re- 
mains very satisfactory today. 

At the present time we have a group of about eighty substances 
which have resisted all efforts to decompose them into simpler 
substances. These are the so-called chemical elements. It 
should be borne in mind, however, that because we have failed to 
resolve these substances into simpler forms of matter, we are not 
warranted in maintaining that such resolution may not be effected 
in the future. 

Recent investigations of the radioactive elements have shown 



FUNDAMENTAL PRINCIPLES 3 

that they are continuously undergoing a series of transformations, 
one of the products of which is the inactive element helium. This 
behavior is contrary to the old view that transformation of one 
element into another is impossible. At first the attempt was 
made to explain it by assuming that the radioactive element was 
a compound of helium with another element, but since the radio- 
active elements possess all of the properties characteristic of 
elements as distinguished from compounds, and find appropriate 
places in the periodic table of Mendeteeff, the " compound theory " 
must be abandoned. Uranium and thorium, the heaviest ele- 
ments known, appear to be undergoing a process of spontaneous 
disintegration over which we have no control. The products 
of this disintegration have filled the gap in the periodic table 
between thorium -and lead with about thirty new elements, each 
of which is in turn undergoing transformations similar to those 
of the parent elements. Professor Soddy * says: "In spite cf 
the existence at one time of a vague belief (a belief which has no 
foundation), that all matter may be to a certain extent radioactive, 
just as all matter is believed to be to a certain extent magnetic, it 
is recognized today that radioactivity is an exceedingly rare prop- 
erty of matter." 

Notwithstanding these remarkable discoveries, we may still hold 
to the idea of an element as suggested by Boyle and Lavoisier. 
Professor Walker says: " The elements form a group of substances, 
singular not only with respect to the resistance which they offer 
to decomposition, but also with respect to certain regularities dis- 
played by them and not shared by substances which are designated 
as compounds." 

Law of the Conservation of Mass. In 1774, as the result of 
a series of experiments, Lavoisier established the law of the con- 
servation of mass which may be stated as follows: In a chemical 
reaction the total mass of the reacting substances is equal to the total 
mass of the products of the reaction. It is sometimes stated thus: 
the total mass of the universe is a constant; but this form of state- 
ment is open to the objection that we have no means of verification; 
it is a statement of a fact which transcends our experience. 

* Chemistry of the Radio-Elements, p. 2. 



THEORETICAL CHEMISTRY 



s 

A 



The law of the conservation of mass has been subjected to most 
rigid investigation by Landolt * in a series of experiments extend- 
ing over a period of fifteen years. 

The reacting substances, AB and CD, 
were placed in the two arms of the in- 
verted U-tube shown in Fig. 1 which was 
then sealed at S and the whole weighed 
upon an extremely sensitive balance. The 
vessel was then inverted when the following 
reaction took place: 

CD-+AD + CB. 



AB 



CD 



Fig. 1. 



When the reaction was complete and suf- 
ficient time had elapsed to allow the vessel 
to return to its original volume (this some- 
times required nearly three weeks), it was 
weighed again using every precaution to 
avoid errors and any gain or loss in weight 
noted. 



Landolt concluded from the thirty or more reactions which he 
studied that the gain or loss in weight was less than one ten- 
millionth of the total weight.f 

Law of Definite Proportions. The enunciation of the law of 
the conservation of mass and the introduction of the balance into 
the chemical laboratory marked the beginning of a new era in the 
history of chemistry, the era of quantitative chemistry. As 
the result of painstaking experimental work, Richter and Proust 
announced the law of definite proportions about the beginning 
of the nineteenth century. This law may be expressed thus: 
A definite chemical compound always contains the same elements 
united in the same proportion by weight. 

Shortly after the enunciation of this law its truth was questioned 
by the French chemist Berthollet.J From the results of a series 

* Zeit. phys. chem., 12, 1 (1893); 55, 589 (1906). 

t An excellent summary of this important investigation will be found io 
the Journal de Chimie physique, 6, 625 (1908). 
J Essai de statique chimique (1803) . * 



FUNDAMENTAL PRINCIPLES 5 

of brilliant experiments, he became convinced that chemical reac- 
tions are largely controlled by the relative amounts of the react- 
ing substances. As we shall see later, he really foreshadowed the 
work of Guldberg and Waage who were the first to correctly for- 
mulate the influence of mass on a chemical reaction. Berthollet 
argued that when two elements unite to form a compound, the 
proportion of one of the elements in the compound is conditioned 
solely by the amount of that element which is available. This 
led to the celebrated controversy between Berthollet and Proust 
which finally resulted in the establishment of the latter's original 
statement. Subsequent investigation has only strengthened our 
faith in the law of definite proportions. 

Law of Multiple Proportions. Elements are known to unite 
in more than one proportion by weight. Dalton analyzed the 
two compounds of carbon and hydrogen, methane and ethylene, 
and found that the ratio of the weights of carbon to hydrogen in 
the former was 6 : 2 while in the latter it was 6:1. That is, for the 
same weight of carbon, the weights of the hydrogen in the two 
compounds were in the ratio 2:1. 

A large number of compounds were examined and similar 
simple ratios between the masses of the constituent elements 
were found. As a result of these observations, Dalton * formu-, 
lated in 1808 the law of multiple proportions, as follows: When 
two elements unite in more than one proportion, for a fixed mass of 
one element the masses of the other element bear to each other a simple 
ratio. Notwithstanding the fact that Dalton was a careless 
experimenter the subsequent investigations of Marignac and 
others have established the validity of his law. 

Law of Combining Proportions. Dalton pointed out that it is 
possible to assign to every element a definite relative weight with 
which it enters into chemical combination. He observed that 
the weights or simple multiples of the weights of the different 
elements which unite with a given weight of a definite element, 
represent the weights of the different elements which combine 
with each other. The weights of the elements which combine 
with each other are termed their combining weights. This com- 

* A New System of Chemical Philosophy (1808). 



6 THEORETICAL CHEMISTRY 

prehensive law of chemical combination may be stated as follows: 
Elements combine in the ratio of their combining weights or in 
simple multiples of this ratio. It will be observed that this law 
really includes the law of definite and the law of multiple pro- 
portions. 

If we assume the combining weight of hydrogen to be unity, 
the combining weights of chlorine, oxygen and sulphur will be 
35.5, 8 and 16 respectively. These numbers represent the ratios 
in which the elements substitute each other in chemical com- 
pounds. Hydrochloric acid, for example, contains 35.5 parts by 
weight of chlorine to 1 part by weight of hydrogen and when 
oxygen is substituted for chlorine, forming water, the new com- 
pound contains 8 parts by weight of oxygen to 1 part by weight 
of hydrogen. Similarly, if the oxygen be substituted by sulphur, 
forming hydrogen sulphide, there will be found 16 parts by weight 
of sulphur to 1 part by weight of hydrogen. We may say, then, 
that 35.5 parts of chlorine, 8 parts of oxygen and 16 parts of sul- 
phur are equivalent. 

A chemical equivalent may be defined as the weight of an element 
which is necessary to combine with or displace 1 part by weight 
of hydrogen. 

The Atomic Theory. In very early times two different views 
were entertained by opposing schools of Greek philosophers as to 
the mechanical constitution of matter. According to the school 
of Plato and Aristotle, matter was thought to be continuous 
within the space it appears to fill and to be capable of indefinite 
subdivision. According to the other school, first taught by 
Leucippus, and afterwards by Democritus and Epicurus, matter 
was considered to be made up of primordial, extremely minute 
particles, distinct and separable from each other but in themselves 
incapable of division. These ultimate particles were called atoms 
(i^TOfios), signifying something indivisible. While the Aristote- 
lian doctrine held sway for many-centuries yet the notion of atoms 
was revived at intervals. Late in the seventeenth century, Boyle 
seems to have looked upon chemical combination as the result of 
atomic association. 

Guided by these early speculations as to the constitution of 



FUNDAMENTAL PRINCIPLES 7 

matter and influenced by his study of the writings of Sir Isaac 
Newton, Dalton seems to have formed a mental picture of the 
part played by atoms in the act of chemical combination. After 
a few carelessly performed experiments, the results of which ac- 
corded with his preconceived ideas, he formulated his atomic 
theory. 

1 According to this theory matter is composed of extremely mr- 
nute, indivisible particles or atoms. Atoms of the same element 
are all of equal weight, but atoms of different elements have 
weights proportional to their combining numbers. Chemical 
compounds are formed by the union of atoms of different kinds.* 
This theory offers a simple, rational explanation of the laws of 
chemical combination. 

Since a chemical compound results from the union of atoms, 
each of which has a definite weight, its composition must be in- 
variable, which is the law of definite proportions. Again, 
when atoms combine in more than one proportion, for a fixed 
weight of atoms of one kind, the weights of the other species of 
atoms must bear to each other a simple ratio, since the atoms are 
indivisible units. This is clearly the law of multiple proportions. 

Finally, the law of combining weights is seen to follow as a 
necessary consequence of the atomic theory, since the experimen- 
tally determined combining weights bear a simple relation to the 
relative weights of the atoms. 

At the time when Dalton. proposed his atomic theory, the 
number of facts to be explained was comparatively small, but 
with the enormous growth of the science of chemistry during the 
past century and with the vast accumulation of data, the theory 
has proved capable of affording adequate representation of all of 
the facts, and has opened the way to many important generaliza- 
tions. 

While the atomic theory has played a very important part in the 
development of modern chemistry, and while we recognize that it 
helps to clarify our thinking and enables us to construct a mental 
image of tiny spheres uniting to form a chemical compound, yet we 
must not forget the fact that these atoms are purely hypothetical. 

Faraday has said: "Whether matter be atomic or not, this 



8 THEORETICAL CHEMISTRY 

much is certain, that granting it to be atomic, it would appear 
as it now does." Ostwald believes that in the not distant future 
the atomic theory will be abandoned and chemists will free them- 
selves from the yoke of this hypothesis, relying solely upon the 
results of experiment. He says: "It seems as if the adaptabil- 
ity of the atomic hypothesis is near exhaustion, and it is well 
to realize that, according to the lesson repeatedly taught by the 
history of science, such an end is sooner or later inevitable/' 

Combining Weights and Atomic Weights. The problem of 
determining the relative atomic weights of the elements would at 
first sight appear to be a very simple matter. This might appar- 
ently be accomplished by selecting one element, say hydrogen, 
it being the lightest known element, as the standard; a compound 
of hydrogen and another element may then be analyzed and the 
amount of the other element in combination with one part by 
weight of hydrogen determined. This weight will be its atomic 
weight only when the compound contains but one atom of each 
element. To determine the relative atomic weight, therefore, we 
must know in addition to the chemical equivalent of the element, 
the number of atoms present in the compound. For example, the 
analysis of water shows it to contain 8 parts by weight of oxygen 
to 1 part by weight of hydrogen; the chemical equivalent of 
oxygen is, therefore, 8, and if water contained but one atom of 
hydrogen the atomic weight of oxygen would be 8. It can be 
shown, however,- that water contains two atoms of hydrogen 
and one atom of oxygen, therefore, the atomic weight of oxygen 
must be 16. It is evident, therefore, that neither the analysis nor 
the synthesis of a compound is sufficient to enable us to determine 
the number of atoms of an element combined with one atom of 
hydrogen. We shall proceed to the consideration of the methods 
by which this problem may be solved. 

Gay-Lussac's Law of Volumes. Gay-Lussac in 1808, while 
studying the densities of gases before and after reaction, announced 
the following law: When gases combine they do so in simple ratios 
by volume, and the volume of the gaseous product bears a simple 
ratio to the volumes of the reacting gases when measured under like 
conditions of temperature and pressure. Thus, one volume of hydro- 



FUNDAMENTAL PRINCIPLES 9 

gen combines with one volume of chlorine to form two volumes 
of hydrochloric acid; one volume of oxygen combines with two 
volumes of hydrogen to form two volumes of water (vapor) ; and 
one volume of nitrogen combines with three volumes of hydrogen 
to form two volumes of ammonia. 

In a previous investigation, Gay-Lussac had shown that all gases 
behave identically when subjected to changes of temperature and 
pressure. This fact, taken together with the simple volumetric 
relation just enunciated and the atomic theory, suggested a possible 
relation between the number of ultimate particles in equal vol- 
umes of different gases. 

Berzelius attempted to show that under corresponding condi- 
tions of temperature and pressure, equal volumes of different 
gases contain the same number of atoms, but he was compelled 
to abandon the assumption as untenable. 

Avogadro's Hypothesis. It remained for the Italian physicist, 
Avogadro,* in 1811, to point out the distinction between atoms 
and molecules, terms which had been used almost synonymously 
up to his time. He defined the atom as the smallest particle 
which can enter into chemical combination, whereas the molecule 
is the smallest portion of matter which can exist in a free state. 
He then formulated the following hypothesis :f Under the same 
conditions of temperature and pressure, equal volumes of all gases 
contain the same number of molecules. This hypothesis has been 
subjected to such rigid experimental and mathematical tests that 
its validity cannot be questioned. 

Avogadro's Hypothesis and Molecular Weights. According 
to Gay-Lussac when hydrogen and chlorine combine to form hydro- 
chloric acid, one volume of hydrogen unites with one volume of 
chlorine yielding two volumes of hydrochloric acid. 

According to the hypothesis of Avogadro, the number of mole- 
cules of hydrochloric acid is double the number of molecules of 
hydrogen or of chlorine, and, consequently, each molecule of the 
reacting gases must contain at least two atoms. If we take 
hydrogen as the unit of our system of atomic weights, its molec- 

* Jour, de Phys., 73, 58 (1811). 

f Ampere advanced nearly the same hypothesis in 1814. 



10 THEORETICAL CHEMISTRY 

ular weight must be 2. It is convenient to express molecular 
and atomic weights in terms of the same unit, for then the molec- 
ular weight of a substance will be simply the sum of the weights 
of the atoms contained in the molecule. The determination of 
the approximate molecular weight of a substance, therefore, re- 
solves itself into ascertaining the mass of its vapor in grams which, 
under the same conditions of temperature and pressure, will 
occupy the same volume as 2 grams of hydrogen. 

This weight is called the gram-mokcular weight or the molar 
weight of the substance, while the corresponding volume is known 
as the gram-mokcular or molar volume* It is nearly the same for 
all gases and at and 760 mm. it may be taken equal to 22.4 
liters. The molecular weights obtained from vapor density meas- 
urements are approximate only, because of the failure of most 
gases and vapors to obey the simple gas laws, a condition essen- 
tial to the strict applicability of Avogadro's hypothesis. 

Atomic Weights from Molecular Weights. While vapor 
density determinations as ordinarily carried out do not give exact 
molecular weights, it is an easy matter to arrive at the true values 
when we take into consideration the results of chemical analysis. 
It is apparent that the true molecular weight must be the sum of 
the weights of the constituent elements, these weights being exact 
multiples or submultiples of their combining proportions, which 
proportions have been determined by analysis alone. We select, 
as the true molecular weight, the value which is nearest to the 
approximate molecular weight calculated from the vapor density 
of the substance. For example, the molecular weight of ammonia, 
as computed from its vapor density, is 17.5 or, in other words, 
17.5 grams of ammonia occupy the same volume as 2 grams of 
hydrogen, measured under the same conditions of temperature 
and pressure. The analysis of ammonia shows us that for every 
gram of hydrogen, there are present 4.67 grams of nitrogen. 
Hence the true molecular weight must contain a multiple of 1 gram 
of hydrogen and the same multiple of 4.67 grams of nitrogen. 
The problem is, to find what integral value must be assigned to 
x In the expression, x (1 -f 4.67), in order that it may give the 
closest approximation to 17.5. Clearly if x = 3 the value of the 



FUNDAMENTAL PRINCIPLES 



11 



expression becomes 17, and this we take to be the true molecular 
weight. This gives 3 X 4.67 14 as the probable atomic weight 
of nitrogen. To decide whether the atomic weight of nitrogen is 
a multiple or a submultiple of 14, we must determine the molecu- 
lar weights of a large number of gaseous or vaporizable compounds 
of nitrogen and select as the atomic weight the smallest quantity 
of the element which is present in any one of them. 

The following table gives a list of seven gaseous compounds of 
nitrogen together with their gram-molecular weights, and the 
number of grams of the element in the gram-molecule. 



'Compound. 


Gram-mol. 

Wt. 


Grams Nitro- 
gen. 


Ammonia 


17 


14 


Nitric oxide . . . . 


30 


14 


Nitrogen peroxide 


46 


14 


Methyl nitrate , 


77 


14 


Cyanogen chloride 


61.5 


14 


Nitrous oxide. ... 


44 


28 


Cyanogen 


52 


28 


* 







- It will be observed that the least weight of nitrogen entering 
into a gram-molecular weight of any of these compounds is 14 
grams, and, therefore, we accept this value as the atomic weight 
of the element, although there is still a very slight chance that 
in some other compound of nitrogen a smaller weight of the ele- 
ment may be found. We shall proceed to point out that there 
are methods by which the probable values of the atomic weights 
may be checked. 

Specific Heat and Atomic Weight. In 1819 the French chem- 
ists, Dulong and Petit,* pointed out a very simple relation between 
the specific heats of the elements in the solid state and their 
atomic weights. This relation, known as the law of Dulong and 
Petit, is as follows: The product of the specific heat and the atomic 
weight of the solid elements is constant. The value of this constant, 
called the atomic heat, is approximately 6.4 A little reflection 
will show that an alternative statement of this law is that the 
* Ann. Chim. Phys., 10, 395 (1819). 



12 



THEORETICAL CHEMISTRY 



atoms of the elements in the solid state have the same thermal capac- 
ity. The specific heats, atomic weights and atomic heats of 
several elements are given in the subjoined table. 



Element. 


At. Wt. 


Sp. Ht. 


At. Ht. 


Lithium 


7 


940 


6 6 


Glucinum . . . 


9 


410 


3 7 


Boron (amorphous) 
Carbon (diamond) . . 


11 
12 


250 
140 


2 8 
1 7 


Sodium 


23 


290 


6 7 


Silicon (crystalline) ... 
Potassium 


28 
39 


160 
166 


4 5 
6 5 


Calcium 


40 


170 


6 8 


Iron 


56 


112 


6 3 


CoDDer 


63 


093 


5 9 


Zinc . . . . 


65 


093 


6 1 


Silver ... . 


108 


056 


6 


Tin 


119 


054 


6 5 


Gold 


197 


032 


6 3 


Mercury 


200 


032 


6 4 



It is truly remarkable that elements differing as greatly as 
lithium and mercury differ, not only in atomic weight but in 
other properties as well, should have identical atomic heats. It 
will be observed that the atomic heats of boron, silicon, carbon 
and glucinum are too low. This departure from the law of Dulong 
and Petit is more apparent than real, for in the statement of the 
law there is no specification as to the temperature at which the 
specific heat should be determined. The specific heats of all 
solids vary with the temperature, this variation being greater in 
the case of some elements than in that of others. It has been 
shown that the specific heats of the above four elements increase 
rapidly with rise of temperature and approach limiting values. 
As these values are approached the product of specific heat and 
atomic weight approximates more and more closely to the mean 
value of the constant, 6.4. 

The following table gives the values obtained by Weber * for 
carbon and silicon. 

* Pogg. Ann., 154, 367 (1875). 



FUNDAMENTAL PRINCIPLES 



13 



CARBON (DIAMOND). 



Temperature, 
degrees. 


Sp. Ht. 


At. Ht. 


-50 


0635 


76 


+10 


1128 


1 35 


85 


1765 


2 12 


206 


2733 


3 28 


607 


4408 


5.30 


806 


4489 


5 40 


985 


4589 


5 50 



CARBON (GRAPHITE). 



Temperature, 
degrees. 


Sp. Ht. 


At. Ht. 


-50 


0.1138 


1 37 


+ 10 


1604 


1 93 


61 


1990 


2 39 


202 


2966 


3 56 


642 


4454 


5.35 


822 


4539 


5 45 


978 


4670 


5 50 



SILICON. 



Temperature, 
degrees 


Sp. Ht. 


At. Ht 


-40 


136 


3 81 


+57 


183 


5 13 


129 


196 


5.50 


232 


203 


5 63 



It is evident that this empirical relation can be used to deter- 
mine the approximate atomic weight of an element when its 

specific heat is known, thus 

6.4 

atomic weight = ^~~r , 

specific heat 

The law of Dulong and Petit has been of great service in fixing 
and checking atomic weights. 

About twenty years after the law of Dulong and Petit was 
formulated, Neumann * showed that a similar relation holds for 
* Pogg. Ann., 23, 1 (1831). 



14 



THEORETICAL CHEMISTRY 



compounds of the same general chemical character. Neumann's 
law may be stated thus: Similarly constituted compounds in the 
solid state have the same mokcular heat. Subsequently Kopp * 
pointed out that the thermal capacity of the atoms is not appreciably 
altered when they enter into chemical combination, or in other words, 
the molecular heat of solid compounds is an additive property, 
being made up of the atomic heats of the constituent elements. 

For example, the specific heat of PbBr 2 is 0.054 and its molec- 
ular weight is 366.8, therefore, the molecular heat is 0.054 X 366.8 
= 19.9. Since there are three atoms in the molecule, 19.9 *- 3 
= 6,6 is their average atomic heat, a value in excellent agree- 
ment with the constant in the law of Dulong and Petit. Neu- 
mann's law may be used to estimate the atomic heats of elements 
which cannot be readily investigated in the solid state. The 
following table gives a list of atomic heats of elements in the solid 
state derived by means of Neumann's law. 



Element. 


At. lit. 


Element. 


At. Ht. 


Hydrogen . . . 


s 
2 3 


Carbon 


1.8 


Oxygen 


4 


Silicon 


4,0 


Fluorine 


5 


Phosphorus . 


5.4 


Nitrogen 


5.5 


Sulphur 


5.4 











Isomorphism. From a study of the corresponding salts of 
phosphoric and arsenic acids, Mitscherlich f observed that they 
crystallize with the same number of molecules of water and are 
nearly identical in crystalline form, it being possible to obtain 
mixed crystals from solutions containing both salts. This sug- 
gested to Mitscherlich a line of investigation which resulted, in 
1820, in the establishment of the law of isomorphism which bears 
his name. 

This law may be stated as follows: An equal number of atoms 
combined in the same manner yield the same crystal form 9 which is 
independent of the chemical nature of the atoms and dependent upon 
their number and position. Thus, when one element replaces 

* Lieb. Ann. (1864), Suppl., 3, 5. 

f Ann. Chim. Phys. (2), 14, 172 (1820). 



FUNDAMENTAL PRINCIPLES 



15 



another in a compound without changing its crystalline form, 
Mitscherlich assumed that one element has displaced the other, 
atom for atom. For example, having two isomorphous substances, 
such as BaCl2.2 H 2 and BaBr 2 .2 H 2 0, we assume that the brom- 
ine in the second compound has replaced the chlorine in the first 
and, if the atomic weights of all of the elements in the first com- 
pound are known, then it is evident that the atomic weight of the 
bromine in the second compound can be easily calculated. This 
method was largely used by Berzelius in fixing atomic weights and 
in checking [the values obtained by the volumetric method. It 
should be remembered that the converse of the law of isomorphism 
does not hold, since elements may replace each other, atom for 
atom, without preserving the same form of crystallization. Many 
exceptions to the law have been pointed out. For example, 
Mitscherlich himself showed that Na^SC^ and BaMn 2 8 are iso- 
morphous and yet the two molecules do not contain the same 
number of atoms. Furthermore, careful measurements of the 
interfacial angles of crystals have revealed the fact that sub- 
stances which have been regarded as isomorphous are only approx- 
imately so, thus the interfacial angles of the apparently isomorph- 
ous crystalline salts given in the following table differ appreciably. 



Salt. 


Interfacial Angle. 


MgSC 
ZnSO 
NiSO 


) 4 .7 H 2 O 

4.7 H 2 O 
4.7 H 2 


89 26' 
88 53' 
88 56' 



Ostwald has suggested that the term homeomorphous be applied to 
designate substances which have nearly identical form. At best 
the principle of isomorphism is only an approximation and should 
be employed with caution. 

-Valence. During the latter half of the nineteenth century 
the usefulness of the atomic theory was greatly enhanced by the 
introduction of certain assumptions concerning the combining 
power of the atoms. These assumptions, constituting the so- 
called doctrine of valence, were forced upon chemists in order 
that a satisfactory explanation might be offered of the phenomenon 



16 THEORETICAL CHEMISTRY 

of isomerism. A consideration of the following formulas, 
HC1, H 2 0, NH 3 , CH 4 , shows that the power to combine with 
hydrogen increases regularly from chlorine, which combines with 
hydrogen, atom for atom, to carbon, one atom of which is capable 
of combining with four atoms of hydrogen. Either hydrogen or 
chlorine, each of which is capable of combining with but one 
atom of the other, may be taken as an example of the simplest 
kind of atom. Any element like hydrogen or chlorine is called 
a univalent element, whereas elements similar to oxygen, nitrogen 
and carbon, which are capable of combining with two, three or 
four atoms of hydrogen, are called bivalent, trivalent and quadri- 
valent elements respectively. Most elements belong to one or 
the other of these four classes, although quinquivalent, sexivalent 
and septivalent elements are known. The familiar bonds or link- 
ages of structural formulas are graphic representations of the 
valence of the atoms constituting the molecule. This useful con- 
ception of valence has made possible the prediction of the prop- 
erties of many compounds before they have been discovered in 
nature or in the laboratory. 

Atomic Weights. Among the first to recognize the importance 
of Dalton's atomic theory was the Swedish chemist, Berzelius. 
He foresaw the importance for chemists of a table of exact atomic 
weights and in 1810 he undertook the task of determining the 
combining weights of most of the known elements. For nearly 
six years he was engaged in determining the exact composition of 
a large number of compounds and calculating the combining 
weights of their constituent elements, thus compiling the first 
table of atomic weights. 

Numerous investigators since Berzelius have been engaged in 
this important work, among whom should be mentioned Stas, 
Marignac, Morley and Richards. On two occasions special stimu- 
lus was given to such investigations. The first occasion was.in 
1815 when Prout suggested that the atomic weights of the elements 
are exact multiples of the atomic weight of hydrogen. The values 
obtained by Berzelius were incompatible with the hypothesis of 
Prout, although the atomic weights of several of the elements 
differed but little from integral values. To test the accuracy of 



FUNDAMENTAL PRINCIPLES 19 

atomic weight of oxygen is taken as 16, and the unit to which all 
atomic weights are referred is one-sixteenth of this weight. The 
atomic weight of hydrogen on this basis is 1.008. Aside from the 
fact that most of the elements form compounds with oxygen which 
are suitable for analysis, the atomic weights of more of the ele- 
ments approximate to integral values when oxygen instead of hy- 
drogen is used as the standard. 

The tab'e on page 18 gives the values of the atomic weights 
as published by the International Committee on Atomic 
Weights for 1917. 



CHAPTER II. 
CLASSIFICATION OF THE ELEMENTS. 

Early Attempts at Classification. Many attempts were made 
to classify the elements according to various properties, such as 
their acidic or basic characteristics or their valence. In all of 
these systems the same elements frequently found a place in more 
than one group, and elements bearing little resemblance to each 
other were classed together. The early attempts at classifica- 
tion based upon the atomic weights of the elements were not 
successful owing to the uncertainty as to the exact numerical 
values of these constants. 

Prout's Hypothesis. In 1815, W. Prout, an English physician, 
observed that the atomic weights of the elements, as then given, 
did not differ greatly from whole numbers when hydrogen was 
taken as the standard. Hence he advanced the hypothesis that 
the different elements are* polymers of hydrogen. As has already 
been pointed out this hypothesis led Stas to undertake his refined 
determinations of the atomic weights of silver, lithium, sodium, 
potassium, sulphur, lead, nitrogen and the halogens. As a result 
of his investigations he says: "I have arrived at the absolute 
conviction, the complete certainty, so far as is possible for a human 
being to attain to certainty in such matter, that the law of Prout 
is nothing but an illusion, a mere speculation definitely contra- 
dicted by experience/' Notwithstanding the fact that Prout's 
hypothesis as originally stated was thus disproved by Stas, it still 
survived in a modified form given to it by J. B. Dumas, who sug- 
gested that one-half of the atomic weight of hydrogen should be 
taken as the fundamental unit. When Stas showed that his 
experiments excluded this possibility, Dumas suggested that the 
fundamental unit be taken as one-quarter of the atomic weight 
of hydrogen. Having begun to divide and subdivide, there was no 
limit to the process, and the hypothesis fell into disfavor, although 

20 



CLASSIFICATION OF THE ELEMENTS 



21 



the belief in a primal element, something akin to the protyle 
(irpurrj vA.i;) of the ancient philosophers, has survived and in modern 
times has reappeared in the electron theory. 

Dobereiner's Triads. About 1817 J. W. Dobereiner * observed 
that groups of three elements could be selected from the list of the 
elements, all of which are chemically similar, and having atomic 
weights such that the atomic weight of the middle member is the 
arithmetical mean of the first and third members of the group. 
These groups of three elements he termed triads. In the follow- 
ing table a few of these triads are given. 



Element. 


At. Wt. 


Mean atomic 
weight of 
triads. 


Lithium . . . ... 


6 94 


. 


Sodium 


23 00 


I 23.02 


Potassium 


39 10 




Calcium. . . . 


40.07 


) 


Strontium . , . 


87 63 


> 88.72 


Barium 


137 37 




Chlorine 


35.46 


) 


Bromine . . 


79.92 


> 80.69 


Iodine 


126.92 




Sulphur 


32.07 


) 


Selenium . 


79 2 


> 78.78 


Tellurium 


127.5 




Phosphorus ... 


31.04 


) 


Arsenic 


74.96 


> 75 62 


Antimony 


120 2 


$ 









This simple relation, first pointed out by Dobereiner, is clearly 
a foreshadowing of the periodic law. 

The Helix of de Chancourtois. The idea of arranging the 
elements in the order of their atomic weights with a view to 
emphasizing the relationship of their chemical and physical prop- 
erties, seems to have first suggested itself to M. A. E. B. de Chan- 
courtois f in the year 1862, On a right-circular cylinder he traced 

* Pogg. Ann., 15, 301 (1825). 

f Vis Tellurique, Classement naturel des Corps Simples. 



22 



THEORETICAL CHEMISTRY 



what he termed a " telluric helix" at a constant angle of 45 to the 
axis. On this curve he laid off lengths corresponding to the 
atomic weights of the elements, taking as a unit of measure a 
length equal to one-sixteenth of a complete revolution of the 
cylinder. He then called attention to the fact that elements with 
analogous properties fall on vertical lines parallel to the generatrix. 
Being a mathematician and a geologist he did not express himself 
in such terms as would attract the attention of chemists and con- 
sequently his work remained unnoticed until recent times. 

The Law of Octaves. In 1864 J. A. R. Newlands* pointed 
out that if the elements are arranged in the order of their atomic 
weights, the eighth element has properties very similar to the 
first; the ninth to the second; the tenth to the third; and so on, 
or to employ Newlands' own words: "The eighth element starting 
from a given one is a kind of repetition of the first, like the eighth note of 
an octave in music." This peculiar relationship, termed by New- 
lands the law of octaves, is brought out in the following table. 



H 


Li 


Gl 


B 


C 


N 


O 


F 


Na 


Mg 


Ai 


Si 


P 


S 


Cl 


K 


Ca 


Cr 


Ti 


Mn 


Fe 



Notwithstanding the fact that its author was ridiculed and his 
paper returned to him as unworthy of publication in the proceed- 
ings of the Chemical Society, this generalization must be regarded 
as the immediate forerunner of the periodic law. 

The Periodic Law. Quite independently of each other and 
apparently in ignorance of the work of Newlands and de Chan- 
courtois, Mendel6eff f in Russia and Lothar Meyer in Germany, 
gained a far deeper insight into the relations existing between 
the properties of the elements and their atomic weights. In 1869 
Mendeteeff wrote: "When I arranged the elements according 
to the magnitude of their atomic weights, beginning with the 
smallest, it became evident that there exists a kind of periodicity 



* Chem. News, 10, 94 (1864), Ibid., 12, 83 (1865). 
t Lieb. Ann. Suppl., 8, 133 (1874). 



CLASSIFICATION OF THE ELEMENTS 23 

in their properties. I designate by the name 'periodic law' the 
mutual relations between the properties of the elements and their 
atomic weights; these relations are applicable to all the elements 
and have the nature of a periodic function/' This important 
generalization may be briefly stated thus: The properties of the 
elements are periodic functions of their atomic weights. 

The original table of Mendeleeff has been amended and modified 
as new data has accumulated and new elements have been dis- 
covered. The accompanying table, though containing several 
new elements and an entirely new group, is essentially the same as 
that of Mendeleeff. It consists of nine vertical columns, called 
groups, and twelve horizontal rows termed series or periods. The 
second and third periods contain eight elements each, and are 
known as short periods, while in the fourth series, starting with 
argon, it is necessary to pass over eighteen elements before another 
element, krypton, is encountered which bears a close resemblance 
to argon: such a series of nineteen elements is called a long period. 
The entire table is composed of two short and five long periods, 
the last one being incomplete. The positions of the elements are 
largely determined by their chemical similarity to those in the 
same group, the hyphens indicating the positions of undiscovered 
elements. The elements in Group VIII, presented difficulties 
when Mendel6eff attempted to place them according to their 
atomic weights and so he was obliged to group them by themselves. 
This group has wittily been designated as " the hospital for incur- 
ables. " An examination of the table shows that the valence of 
the elements toward oxygen progresses regularly from Group O, 
containing elements which exhibit no combining power, up to 
Group VIII, where it attains a maximum value of eight in the 
case of osmium. The valence toward hydrogen on the other hand 
increases regularly from Group VII to Group IV in which the 
elements are quadrivalent. 



24 



THEORETICAL CHEMISTRY 



o 



8 

o 



2ffl 
Off? 



1 S 






Stfl 

o5 



is 
M 



S 

r-< 

co 

On 



a 



a 





n 



00 

c59S 



o 

I 



lg 

O 



i-H ^1 * 



^ 



o 



ZH W 

O S 







8 



,-! 

Z! u2 v 

pn pn y^ ^^ 



? 

M 



M 



00 OS 



"8 "S "g 



OQ 



"8 "g " 
Jl |1 | 

M t-5 (-3 






CIASSIFICATION OF THE ELEMENTS 25 

The formulas of the typical oxides and hydrides of the elements 
in the several groups are indicated at the top of each vertical 
column in the table, where R denotes any element in the group. 
The valence of elements in the long periods arc apt to be variable. 
The elements in the second series are frequently called bridge 
elements, since they bear a closer relation to the elements in the 
next adjacent group than they do to any other members of the 
same group in succeeding series. The members of the third series 
are styled typical elements, because they exhibit the general prop- 
erties and characteristics of the group. Each group is divided into 
subgroups, the elements on the right and left sides of a column 
forming families, the members of which are more closely related 
than are all of the elements included within the group. In other 
words we detect a kind of periodicity within each group. 

In any given series the element with lowest atomic weight 
possesses the strongest basic character. Thus we find the strongly 
basic, alkali metals on the left side of the table, while on the right 
side are the acidic elements such as the halogens and sulphur. 
In fact, the strictly non-metallic elements are confined to the 
upper right-hand corner of the table. 

Similarly, as we pass from the top to the bottom of the table, 
we observe a progressive change in the base-forming tendency of 
the elements; i.e., as the atomic weight increases, the metallic 
character of the elements in each group becomes more pronounced. 

Periodicity of Physical Properties. Lothar Meyer, as has been 
pointed out, discovered the periodic relations of the elements at 
about the same time as Mendel^eff . His table differed but slightly 
from that already given. The most important part of Meyer's * 
work, however, was in pointing out that various physical proper- 
ties of the elements are periodic functions of their atomic weights 
We know today that such properties as specific gravity, atomic 
volume, melting point, hardness, ductility, compressibility, ther- 
mal conductivity, coefficient of expansion, specific refraction, and 
electrical conductivity are all periodic. When the numerical 
values of these properties are plotted as ordinates against their 
atomic weights as abscissae, we obtain wave-like curves similar to 
* Die Modernen Theorien der Chemie. 



26 



THEORETICAL CHEMISTRY 



those shown in Fig. 2. The specific heats of the elements are an 
exception to the general rule. According to the law of Dulong and 
Petit, the product of specific heat and atomic weight is a constant, 
and consequently the graphic representation of this relation must 
be an equilateral hyperbola. 

Applications of the Periodic Law. Mendeleff pointed out the 
four following ways in which the periodic law could be employed: 
(1) The classification of the elements; (2) The estimation of the 




A/I/ 



Fig. 2. 

atomic weights of elements; (3) The prediction of the properties 
of undiscovered elements; and (4) The correction of atomic 
weights. 

1. Classification of Elements. The use of the periodic law in 
this direction has already been indicated. It is without doubt 
the best system of classification known and is to be ranked among 
the great generalizations of the science of chemistry. 

2. Estimation of Atomic Weights. Because of experimental 
difficulties it is not always possible to fix the atomic weight of an 
element by determinations of the vapor densities of some of its 
compounds, or by a determination of its specific heat. In such 
cases the .periodic law has proved of great value. An historic 



CLASSIFICATION OF THE ELEMENTS 27 

example is that of indium, the equivalent weight of which was 
found by Winkler to be 37.8. The atomic weight of the element 
was thought to be twice the equivalent weight or 75.6. If this 
were the correct value it would find a place in the periodic table 
between arsenic and selenium. Clearly there is no vacancy in 
the table at this point and furthermore its properties are not 
allied to those of arsenic or selenium. Mendel<5eff proposed to 
assign to it an atomic weight three times its equivalent weight or 
113.4, when it would fall between cadmium and tin in the table. 
This would bring it in the same group with aluminium, the typical 
element of the group, to which it bears a close resemblance. This 
suggestion of Mendeteeff s was confirmed by a subsequent deter- 
mination of the specific heat of indium. 

3. Prediction of Properties of Undiscovered Elements. At the 
time when Mendeleeff published his first table there were many 
more vacant spaces than exist in the present periodic table. He 
ventured to predict the properties of many of these unknown 
elements by means of the average properties of the two neighbor- 
ing elements in the same series, and the two neighboring elements 
in the same subgroup. These four elements he termed atomic 
analogues. The undiscovered elements Mendel6eff designated by 
prefixing the Sanskrit numerals, eka (one), dwi (two), tri (three), 
and so on, to the names of the next lower elements of the sub- 
group. When the first periodic table was published there were 
two vacancies in Group III, the missing elements being called by 
Mendeleeff eka-alurninium and eka-boron, while in Group IV 
there was a vacancy below titanium, the missing element being 
called eka-silicon. The subsequent discovery of gallium, scandium 
and germanium, with properties nearly identical with those pre- 
dicted for the above hypothetical elements, served to strengthen 
the faith of chemists in the periodic law. The following table 
illustrates the accuracy of Mendel6efFs prognostications: in it is 
given a comparison of a few of the properties of the hypothetical 
element, eka-silicon, as predicted by Mendeleeff in 1871, and tbf* 
corresponding observed properties of germanium, discovered by 
Winkler fifteen years later. 



28 



THEORETICAL CHEMISTRY 



Eka-sihcon, Es. 



Germanium, Ge. 



Atomic weight, 72 

Specific gravity, 5.5. 

Atomic volume, 13. 

Metal dirty gray, and on ignition 

yields a white oxide, EsO 2 . 
Element decomposes steam with 

difficulty. 
Acids have slight action, alkalies no 

pronounced action. 



Action of Na on EsOz or on EsK 2 F 6 
gives metal. 

The oxide EsO 2 refractory. 

Specific gravity of EsO 2 , 4.7. 

Basic properties of EsO 2 less marked 
than Ti(>2 and SnO 2 , but greater 
than SiO 2 . 

Forms hydroxide soluble in acids, 
and the solutions readily decom- 
pose forming a metahydrate. 

EsCl 4 a liquid with a b.p. below 100 
and a sp. gr. of 1.9 at 0. 

EsF4 not gaseous. 

Es forms a compound Es(C2H 5 )4 boil- 
ing at 160, and with a sp. gr. 0.96. 



Atomic weight, 72.3. 
Specific gravity, 5.47. 
Atomic volume, 13.2. 
Metal grayish- white, and on igni- 
tion yields a white oxide, GeO 2 . 
Element does not decompose water. 

Metal not attacked by HC1, but 
acted upon by aqua regia. 

Solutions of KOH have no action. 
Oxidized by fused KOH. 

Ge obtained by reduction of 
with C, or of GeK 2 F 6 with Na. 

The oxide GeOa refractory. 

Specific gravity of GeO 2 , 4.703. 

Basic properties of GeO 2 feeble. 



Acids do not ppt. the hydroxide 
from dii. alkaline solutions," but 
from cone, solutions, acids ppt. 
GeO or a metahydrate. 

GeCl 4 a liquid with a b.p. of 86, 
and a sp. gr. at 18 of 1.887. 

GeF 4 .3 H 2 O a white solid. 
Ge forms a compound Ge(C 2 H 6 )4 
boiling at 160 and with a sp. gr. 
slightly less than water. 



4. Correction of Atomic Weights. When an element falls in a 
position in the periodic table where it clearly does not belong, 
suspicion as to the correctness of its atomic weight is immediately 
aroused. Frequently a redetermination of the atomic weight has 
revealed an error which, when corrected, has resulted in assigning 
the element to a place among its analogues. Formerly the 
accepted atomic weights of osmium, iridium, platinum and gold 
were in the order 

Os > Ir > Pt > Au. 

But from analogies existing between osmium, ruthenium and iron 
and the disposition of the preceding members of Group VIII, 
Mendel^eff predicted that the atomic weights were in error and 
that the order of the elements should be 

Os < Ir < Pt < Au. 



CLASSIFICATION OF THE ELEMENTS 29 

Subsequent atomic weight determinations by Seubert substantiated 
Mendel6eff's prediction. 

Defects in the Periodic Law. While the arrangement of the 
elements in the periodic table is on the whole very satisfactory, 
there are several serious defects in the system which should be 
pointed out. At the very outset there is difficulty in finding a 
place for hydrogen in the system. The element is univalent and 
falls either in Group I, with the alkali metals, or in Group VII with 
the halogens. While the element is electro-positive it cannot be 
considered to possess metallic properties. It forms hydrides with 
some of the metallic elements and can be displaced by the halo- 
gens from organic compounds. These facts make it extremely 
difficult to decide whether hydrogen should be placed in Group I 
or Group VII. The idea has been advanced that hydrogen is 
the only known member of the first series of the periodic table. 

These hypothetical elements have been styled proto-ekments, 
the successive members of the series being, proto-glucinum, proto- 
boron and so on to the last element in the series, proto-fluorine. To 
find a suitable location for the rare-earth elements in the periodic 
system is another difficulty which has not been satisfactorily met. 
Brauner considers that these elements should all be grouped 
together with cerium (at. wt. = 140.25), but owing to our limited 
knowledge of the properties of these elements it seems better to 
defer attempting to place them for the present. In the group of 
non-valent elements the atomic weight of argon is distinctly higher 
than that of potassium in the next group. There can be little 
doubt that the values of the atomic weights are correct and it 
is evidently impossible to interchange the positions of these two 
elements in the periodic table, since argon is as much the analogue 
of the rare gases as potassium is of the alkali metals. A similar 
discrepancy occurs with the elements, terrurium and iodine. The 
atomic weight of the former element is appreciably higher than 
that of the latter and, notwithstanding the attempts of numerous 
investigators to prove tellurium to be a complex of two or more 
elements, nothing but failure has attended their efforts. Still 
another anomaly is encountered in Group VII, where manganese 
is classed with the halogen family, to which it bears much less 



30 THEORETICAL CHEMISTRY 

resemblance than it does to chromium and iron, its two immediate 
neighbors. 

As has already been mentioned, Group VIII is made up of non- 
conformable elements. If the properties of the elements are 
dependent upon their atomic weights, it should be impossible for 
several elements having almost identical atomic weights and 
different properties to exist, and yet such is the case with the 
elements of Group VIII. The elements copper, silver and gold, 
while not closely resembling the other members of Group VIII, 
are much more closely allied to them than to the alkali metals 
with which they are also classed. Notwithstanding its imper- 
fections, the periodic law must be regarded as a truly wonderful 
generalization which future investigations will undoubtedly show 
to be but a fragment of a more comprehensive law. 



CHAPTER III. 
THE ELECTRON THEORY. 

Conduction of Electricity through Gases. Within recent 
years the discovery of new facts relative to the conduction of 
electricity through gases has led to the development of the so- 
called electron or corpuscular theory of matter. Under ordinary 
conditions gases are practically non-conductors of electricity, but 
when a sufficiently great difference of potential is established 
between two points within a gas it is no longer able to withstand 
the stress, and an electric discharge takes place between the points. 
The potential necessary to produce such a discharge is quite high, 




>pump 
Fig. 3. 

several thousand volts being required to produce a spark of one 
centimeter length in air at ordinary pressures. The pressure of 
the gas has a marked effect upon the character of the discharge 
and the potential required to produce it. If we make use of a 
glass vessel similar to that shown in Fig. 3, the effect of pressure 
on the nature of the discharge may be studied. 

This apparatus consists of a straight glass tube about 4 cm. in 
diameter and 40 cm. long, into the ends of which platinum elec- 
trodes are sealed. To the side of the vessel a small tube is sealed 
so that connection may be established with an air-pump and 
manometer. If the electrodes are connected with the terminals 
of an induction coil and the pressure within the tube be gradually 
diminished, the following changes in the character of the dis- 

31 



32 



THEORETICAL CHEMISTRY 



charge will be observed. At first the spark becomes more uni- 
form and then broadens out, assuming a bluish color. When a 
pressure of about 0.5 mm. is 
reached, the negative electrode 
or cathode will appear to be 
surrounded by a thin luminous 
layer; next to this will be a 
dark region, known as the 





Crookes' dark space; adjoining . 4 opump 

this will be a luminous portion, 

called the negative glow, and beyond this will be seen another dark 
region which is frequently referred to as the Faraday dark space. 
Between the Faraday dark space and the positive electrode or 
anode is a luminous portion, called the positive column. By a slight 
variation of the current and pressure the positive column can be 
caused to break up into alternate light and dark spaces or strice, 
the appearance of which is dependent upon various factors such as 




Fig. 5. 

the nature of the gas and the size of the tube. If we use a modi- 
fication of this tube, such as is shown in Fig. 4, and diminish the 
pressure to about 0.01 mm., a new phenomenon will be observed. 
The positive column will vanish and the walls of the tube opposite 
the cathode will become faintly phosphorescent. The color of 
the phosphorescence will depend upon the nature of the glass: 
if the tube is made of soda glass, the glow will be greenish yellow, 
while with lead glass the phosphorescence will be bluish. The 
phosphorescence is due to the bombardment of the walls of the 



THE ELECTRON THEORY 33 

tube by very minute particles projected normally from the cathode. 
These streams of particles are called the cathode rays. 

Some Properties of Cathode Rays. The following are among 
the most important properties of the cathode rays: 

1. The cathode rays travel in straight lines normal to the cathode: 
and they cast shadows of opaque objects placed in their path. This 
property may be demonstrated by means of the apparatus shown 
in Fig. 5, where a small metallic Maltese cross is interposed in the 
path of the rays, a distinct shadow being cast on the opposite wall 
of the tube. The cross may be hinged at the bottom so that 
it can be dropped out of the path of the rays, when the usual 
phosphorescence will be obtained. 

2, The cathode rays can produce mechanical motion. By means 






Fig. 6. 

of the apparatus due to Sir William Crookes, Fig. 6, this prop- 
erty of the cathode rays may be demonstrated. Within the 
vacuum tube is placed a small paddle wheel which rolls horizon- 
tally on a pair of glass rails. When the current is applied to the 
tube, the wheel will revolve, moving away from the cathode. By 
reversing the current, the wheel will stop and then rotate in the 
opposite direction owing to the reversal of the direction of the 
cathode stream. 

3. The cathode rays cause a rise of temperature in objects upon 
which they fall. In the tube shown in Fig. 7, the anode consists 
of a small piece of platinum: this is placed at the center of curva- 
ture of the spherical cathode. After pumping down to the 
proper pressure, if a strong discharge be sent through the tube, 



34 



THEORETICAL CHEMISTRY 



c 




Jr To pump 



the anode will begin to glow, and if the action of the current be 
continued long enough, the platinum plate may be rendered in- 
candescent, thus showing the 
marked heating effect of the 
cathode rays. 

4. Many substances become 
phosphorescent on exposure to 
the cathode rays. If the cathode 

rays be directed upon different p. 7 

substances, such as calc-spar, 

barium platino-cyanide, willemite, scheelite and various kinds 
of glass, beautiful phosphorescent effects may be observed. This 
phosphorescent property is useful in observing and experimenting 
with the cathode rays. 

5. The cathode rays can be deflected from their rectilinear path by 
a magnetic field. In studying the magnetic deviation of the 
cathode rays a tube similar to that shown in Fig. 8 has been 
found very satisfactory. An aluminium diaphragm, A, pierced 



C 



IJL 




Fig. 8. 

with a 1 mm. hole, is placed in front of the cathode while at the 
opposite end of tube is placed a phosphorescent screen, D. When 
the discharge takes place a circular phosphorescent spot will 
appear on D. If the tube be placed between the poles of an 
electromagnet, the phosphorescent spot will move at right angles 
to the direction of the magnetic field. On reversing the polarity 
of the magnet the spot will move in the opposite direction. 
Furthermore the direction of the deflection will be found to be 



THE ELECTRON THEORY 



35 



similar to that produced by a negative charge of electricity mov- 
ing in the same direction as the cathode ray. 

6. The cathode rays can be deflected from their rectilinear path 
by an electrostatic field. The same tube which was used in observ- 
ing the magnetic deflection may be employed in studying the 
effect of an electrostatic field. Two insulated metal plates, B 
and C, are placed on opposite sides of the tube and parallel to 
each other. When the tube is in action, if a difference of poten- 
tial of several hundred volts be applied to the plates, the phos- 
phorescent spot on D will be found to move, the direction of the 




Elect. 1 



Fig. 9. 

motion being the same as that of a negatively charged body under 
the influence of an electrostatic field. Reversal of the field causes 
the phosphorescent spot to move to the opposite side of the 
screen. 

7. The cathode rays carry a negative charge. Probably the most 
important charactew|tic of the cathode rays is their ability to 
carry a negative charge. While the magnetic and electrostatic 
deviation of the rays made this fact more than probable, it re- 
mained for Perrin to demonstrate that a negative electrification 
accompanies the cathode stream. A modification of Perrin's 
apparatus due to J. J. Thomson is shown in Fig. 9. It con- 



36 THEORETICAL CHEMISTRY 

sists of a spherical bulb to which is sealed a smaller bulb and a long 
side tube. The small bulb contains the cathode C and the anode 
A. The anode consists of a tight-fitting brass plug pierced by a 
central hole of small diameter. The side tube, which is out of 
the direct range of the cathode rays, contains two coaxial metallic 
cylinders insulated from each other, each being perforated with a 
narrow transverse slit: D is earth-connected and B is connected 
with an electrometer by means of the rod F. When the tube has 
been pumped down to the proper pressure for the production of 
cathode rays, a phosphorescent spot will appear at E directly 
opposite the cathode C. Upon testing B for possible electrifica- 
tion by means of the electrometer, it will be found to be uncharged. 
If the cathode stream be deflected by means of a magnet so that 
the rays fall upon J5, a sudden charging of the electrometer will be 
observed, proving that B is becoming electrified. Upon deflect- 
ing the rays still further so that they are no longer incident upon 
By the accumulation of charge will immediately cease. If the 
electrometer be tested for polarity, it will be found to be negatively 
charged, thus proving the charge carried by the cathode rays to 
be negative. 

8. The cathode rays can penetrate thin sheets of metal. In 1894 
Lenard constructed a vacuum tube fitted with an aluminium 
window opposite the cathode. He showed that the cathode rays 
passed through the aluminium and are absorbed by different sub- 
stances outside of the tube, the absorption varying directly with 
the density of the substance. 

9. The cathode rays when directed into moist air cause the forma- 
tion of fog. This phenomenon has been shown by C. T. R. Wilson 
to be due to the minute particles in the cathode stream acting as 
nuclei upon which the water vapor can condense. 

Velocity of the Cathode Particle. Since the cathode rays 
consist of minute, negatively-charged particles which can be 
deflected by a magnetic and an electrostatic field, it is possible 
to measure their speed and to compute the ratio of the mass of a 
particle to its charge. The special form of tube shown in Fig. 
10 was devised for the purpose by J. J. Thomson. It consists 
of a glass tube about 60 cm. in length, furnished with a flat cir- 



THE ELECTRON THEORY 



37 



cular cathode, C, and an anode, A, in the form of a cylindrical 
brass plug about 2.5 era. in length, pierced by a central hole 1 mm. 
in diameter. Another brass plug, B, is placed about 5 cm. away 
from A-, the two holes being in exactly the same straight line, so 
that a very narrow bundle of rays may pass along the axis of the 
tube and fall upon the phosphorescent screen at the opposite 
end of the tube. Upon this screen is a millimeter scale, SS'. 
Two parallel plates, D and E, are sealed into the tube for the pur- 
pose of establishing an electrostatic field. When the tube is 
connected with an induction coil or other source of high-potential, 
a phosphorescent spot will appear at F. If a strong magnetic 
field be applied, the lines of force being at right angles to the plane 




Fig. 10. 

of the diagram, the rays will be deflected vertically and the spot 
on the screen will move from F to G. 

Let H denote the strength of the magnetic field and let m, e 
and v represent respectively the mass, charge and velocity of a 
cathode particle. A magnetic field, H, acting at right angles to 
the line of flight of the cathode particle will exert a force, 
Hev, which will tend to deflect the particle from a rectilinear 
path. This force must be equal to the centrifugal force of the 
moving particle ^pttng outwards along its radius of curvature 
Therefore 



v 

Hev = > 

r 





Hr = 

e 



38 THEORETICAL CHEMISTRY 

Since H and r can both be measured, the ratio, , can be deter- 

mined. Now if a difference of potential be established between 
D and E, and the lines of force in the electrostatic field have the 
proper direction, it will be possible to alter the strength of the field 
so as to just counterbalance the effect of the magnetic field, and 
bring the phosphorescent spot back to F again. Under these 
conditions, if X denotes the strength of the electrostatic field, we 
have 

Xe = Hev, 
or 

*-f- (2) 

Since X and H can both be measured, v can be calculated, and by 
introducing the value so obtained into equation (1), the ratio e/m 
can be evaluated. By this method the average value of v has 
been found to be 2.8 X 10 9 cm. per second, while 1.7 X 10 7 is 
the mean value of a large number of determinations of the ratio 
e/m. 

Comparison of the Ratio of Charge to Mass for the Cathode 
Particle with that for the Ion in Electrolysis. The ratio of the 
charge carried by an ion in electrolysis to its mass can be easily 
computed. Thus it may be shown that the ratio of the charge E, 
of the hydrogen ion to its mass, M , in electrolysis is about 1 X 10 4 
C.G.S. units or 

E 
-z = I X 10 4 approximately. 



The mass of the hydrogen ion may be considered to be identical 
with that of the hydrogen atom, the lightest atom known. Com- 
paring the value of e/m for the cathode particle with the value of 
E/M for the hydrogen ion in electrolysis, it is evident that the 
former is about 1700 times greater than the latter. 

Charge Carried by the Cathode Particle. Until the value 
of the charge carried by the cathode particle has been determined, 
it is clearly impossible to compute its mass. Thus, if we consider 



THE ELECTRON THEORY 39 

the last statement of the preceding paragraph, which may be 
formulated as follows: 

e/m : E/M :: 1700 : 1, 

the proportion will remain unaltered whether m = Af/1700 and 
e = E, or e = 1700 and m = M. The method employed to deter- 
mine the charge carried by a cathode particle is too complicated 
for a detailed description in this place; merely the general outline 
will be given. Upon suddenly expanding a volume of saturated 
water vapor, its temperature is lowered, and a cloud forms, each 
particle of dust present serving as a nucleus for a fog particle. If 
sufficient time be allowed for the mist to settle and the vapor to 
become saturated again, a repetition of the preceding process" will 
result in the formation of less mist, owing to the presence of fewer 
dust particles. By repeating the operation enough times the 
space may be rendered dust free. As has already been pointed 
out, cathode particles serve as nuclei for the condensation of 
water vapor, their function being similar to that of dust particles. 
It has been shown by Sir George G. Stokes that if a drop of water 
of radius r, be allowed to fall through a gas of viscosity 17, then the 
velocity with which the drop falls is given by the equation 

.-?.* (3, 

where g is the acceleration due to gravity. The viscosity of air 
at any temperature being known, a cloud can be produced in an 
appropriate chamber by expansion of water vapor in the presence 
of cathode particles and the speed, y, with which the cloud falls 
can be measured, and hence r can be calculated by means of equa- 
tion (3). If m is the total mass of the cloud and n is the number of 
drops per cubic centimeter, then 

m = 4/3 mrr 3 (density of water = 1). 

From a simple application of thermodynamics m may be deter- 
mined. Knowing the values of m and r, the number of drops 
in the cloud, n, which is the same as the number of cathode par- 
ticles, can be calculated. It is a simple matter to measure the 
total charge in the expansion chamber, and dividing this by the 
total number of charged particles, gives the charge carried by a 



40 THEORETICAL CHEMISTRY 

single particle. The latest determinations of J. J. Thomson show 
this to be 3,4 X 10~ 10 electrostatic unit. This is practically 
identical with the calculated value of the charge on the hydro- 
gen ion in electrolysis, or e = E and therefore m = M/1700; the 
mass of the cathode particle is 1/1700 of the mass of the atom 
of hydrogen. The cathode particle has the smallest mass yet 
known and has been called the corpuscle or electron. 

An ingenious modification of the foregoing method devised by 
Millikan,* has made it possible to determine e with extreme 
accuracy. Millikan gives as the mean of a large number of deter- 
minations of e, the following value which he states is in error by 
less than 0.1 per cent: 

e = 4.4775 X 10~ 10 electrostatic units. 

For purposes of comparison, the following values of e obtained by 
other investigators using different experimental methods are here 
given: 

(a) Making use of available data on radiant energy, Planck 
calculated e = 4.69 X lO" 10 . 

(6) By counting the number of scintillations produced by a 
known weight of polonium and measuring the total charge, Regener 
found e = 4.79 X 1Q- 10 . 

(c) By counting the number of a-particles escaping from a given 
amount of radium bromide and measuring the total charge, 
Rutherford and Geiger calculated e = 4.65 X 10~ 10 . 

The Avogadro Constant. The actual number of molecules in 
one gram-molecule or the actual number of atoms in one gram- 
atom of a gas is called the Avogadro Constant. The most accurate 
method for the calculation of this constant involves the elementary 
charge of electricity, e. 

The quantity of electricity carried by one gram equivalent in 
electrolysis has been found to be 96,500 coulombs (see p. 390). 
This quantity, known as the faraday, and commonly designated 
by F, bears the same relation to e that the number expressing its 
atomic weight bears to the actual weight of an atom. The relation 

*Phil. Mag., 6, 19, 209 (1910); Phys. Rev., 39, 349 (1911); Trans. Am. 
Electrochem. Soc., 21, 186 (1912). 



THE ELECTRON THEORY 41 

between the Avogadro constant N and the ionic and electronic 
charges F and e, is given by the equation 

F 

N = L. 

e 

Substituting the above values in this equation and converting 
coulombs into electrostatic units, we have ' 

, r 96500 X 3 X HP _, - 1fm 
N = 4.4775 X 10-" = *** X **' 

Other Sources of Electrons. Electrons may be produced by 
other agents than cathode rays. Thus, electrons are emitted by 
radium and other radioactive substances, by metals and amalgams 
under the influence of ultra-violet light, and also by gas flames 
charged with the vapors of salts. It has been shown that from 
whatever source an electron is derived, the value of the ratio e/m 
remains constant. This interesting fact has led Thomson to sug- 
gest that the electron may be regarded as "one of the bricks of 
which the atom is built up." 

Before entering upon a discussion of modern views of atomic 
structure, however, it will be necessary to summarize very briefly 
some of the salient facts of radiochemistry. 



CHAPTER, IV. 
RADIOACTIVITY. 

Discovery of Radioactivity. The first radioactive substance 
was discovered by Henri Becquerel* in 1896. It had been shown 
by Roentgen in the previous year that the bombardment of the 
walls of a vacuum tube by the cathode stream, gives rise to a now 
type of rays, which, because of their puzzling characteristics, he 
called X-rays. The portion of the tube where these rays originate 
was observed to fluoresce brilliantly, and it was at once assumed that 
this fluorescence might be the cause of the new type of radiation. 

Many substances were known to fluoresce under the stimulus of 
the sun's rays, and it wa>s natural, in the light of Roentgen's dis- 
covery, that all substances which exhibit fluorescence should be 
subjected to careful examination. Among those who became inter- 
ested in these phenomena was BecquereL He studied the action of 
a number of fluorescent substances, among which was the double 
sulphate of potassium and uranium. This salt-, after exposure to 
sunlight, was found to emit a radiation capable of affecting a care- 
fully protected photographic plate. Further investigation proved 
that the fluorescence had nothing to do with the photographic 
action, since both uranous and uranic salts were found to exert 
similar photographic action, notwithstanding the fact that uranous 
salts are not fluorescent. The photographic activity of both 
uranous and uranic salts was found to be proportional to their 
content of uranium. Becquerel also showed that preliminary 
stimulation by sunlight was wholly unnecessary. Uranium salts 
which had been kept in the dark for years were found to be just as 
active as those which had been recently exposed to brilliant sun- 
light. The properties of the rays emitted by uranium salts differ 
in many respects from those of the X-rays. The rate of emission 
of the uranium rays remains unaltered at the highest or the lowest 

* Compt. rendus, 122, 420 (1896). 
42 



RADIOACTIVITY 43 

obtainable temperatures. The entire behavior of these salts justi- 
fies the conclusion that the continuous emission of penetrating 
rays is a specific property of the element uranium itself. This 
property of spontaneously emitting radiations capable of penetra- 
ting substances opaque to ordinary light is called radioactivity. 

Discovery of Radium. Shortly after the discovery of the 
radioactivity of uranium, the element thorium and its compounds 
were also found to be radioactive. As a result of a systematic 
examination by Mme. Curie * of minerals known to contain 
uranium or thorium, it was learned that many of these were much 
more radioactive than either uranium or thorium alone. Thus, 
pitchblende, one of the principal ores of uranium, was found to 
be four times more active than uranium alone, and chalcolite, a 
phosphate of copper and uranium, was found to be at least twice 
as active as uranium. On the other hand, when a specimen of 
artificial chalcolite, prepared in the laboratory from pure materials, 
was examined, its activity was found to be proportional to the 
content of uranium. Mme. Curie concluded from this result that 
natural chalcolite and pitchblende must contain a minute amount 
of some substance much more active than uranium. 

With the assistance of her husband, Mrne. Curie undertook the 
task of separating this unknown substance from pitchblende. 
Pitchblende is an extremely complex mineral and its systematic 
chemical analysis calls for skill and patience of a high order. With- 
out entering into details as to the analytical procedure, it must 
suffice here to state the results obtained. Associated with bismuth, 
a very active substance was discovered, to which Mme. Curie gave 
the name polonium in honor of her native land, Poland. In like 
manner, an extremely active substance was found associated with 
barium in the alkaline earth group. The substance was called 
radium because of its great radioactivity. 

While the isolation of pure polonium is extremely difficult and, 
while sufficient quantities have not been obtained to permit de- 
terminations of its physical properties, the isolation of radium 
in relatively large amounts is readily accomplished. The pure 
bromides of radium and barium are prepared together and the 

* Compt. rendus, 126, 1101 (1898). 



44 THEORETICAL CHEMISTRY 

two salts are then separated by a series of fractional crystalliza- 
tions. That the salts of barium and radium are very similar in 
chemical properties is shown by the fact that they separate to- 
gether from the same solution. The atomic weight of radium has 
been determined by several investigators, the accepted value being 
226. It is thus, with the exception of uranium, the heaviest known 
element. 

In 1910, Mme. Curie * succeeded in obtaining metallic radium. 
It is a metal possessing a silvery luster, dissolving in water with 
energetic evolution of hydrogen and tarnishing rapidly in air with 
the formation of the nitride. 

It is estimated that one ton of pitchblende contains approxi- 
mately 0.2 gram of radium. 

Other Radioactive Substances. Shortly after the discovery 
of radium and polonium by the Curies, Debierne f discovered 
another radioactive element in pitchblende. This element, which 
he named actinium, was found associated with the iron group in 
the course of the analysis of the mineral. 

In 1906, Boltwood f discovered in pitchblende, and in several 
other uranium^ minerals, the presence of still another radioelement 
which he named ionium. Ionium is much more active than 
thorium to which it bears such a close resemblance that the two 
elements cannot be separated from each other. 

The lead which is obtained from uranium and thorium minerals 
is found to be slightly radioactive, the activity being attributed 
to the presence of a small proportion of a constituent called radio- 
lead, the chemical properties of which resemble those of ordinary 
lead. It is interesting to note that recent determinations by 
Richards of the atomic weight of lead obtained from different 
sources, reveal differences greater than the possible experimental 
errors of the determinations. Thus, the values of the atomic 
weight of lead from pitchblende and from thorite were found to 
be 206.40 and 208.4 respectively, while the value of the atomic 
weight of ordinary lead is 207.15. 

* Compt. rendus, 151, 523 (1910). 

t Compt. rendus, 129, 593 (1899). 

i Am. Jour. ScL, 22, 537 (1906). 

Jour. Am. Chem. Soc., 36, 1329 (1914). 



RADIOACTIVITY 45 

About thirty other radioactive substances have been separated 
and many of their properties have been determined. AJ1 of these 
radio-elements have been shown to be the lineal descendants of one 
or the other of the two parent elements, uranium or thorium. 

lonization of Gases. The radiations emitted by radioactive 
substances have the power of rendering the air through which they 
pass conductors of electricity. To account for this action, Thom- 
son and Rutherford formulated the theory of gaseous ionization. 
According to this theory, which has since been experimentally 
confirmed, the radiations break up the components of the gas into 
positive and negative carriers of electricity called ions. If two 
parallel metal plates are connected to the terminals of a battery 
and a radioactive substance is placed between them, the air will be 
ionized and, owing to the movement of the positive and negative 
ions toward the plates of opposite sign, an electric current will 
pass between the plates. If the electric field is weak, the mutual 
attraction between the positive and negative ions will cause many 
of them to recombine before reaching the plates and the resulting 
current will be small. As the strength of the field is increased, the 
greater will be the speed of the ions toward the plates and the 
smaller will become the tendency to recombination. Ultimately, 
with increasing strength of field all of the ions will be swept to the 
plates as fast as they are formed and the ionization current will 
attain a maximum value. This limiting or saturation current 
affords the most accurate method for the measurement of radio- 
activity. 

The method is so sensitive that, by means of it alone, it is possible 
to detect amounts of radioactive products far beyond the reach of 
the balance or the spectroscope. The theory of gaseous ionization 
has been confirmed in several different ways, but one of the most 
striking verifications of the theory is that due to C. T. R. Wilson. 
Making use of the fact that the ions tend to condense water vapor 
around themselves as nuctei, Wilson succeeded in actually photo- 
graphing the path of an ionizing ray in air. 

Photographic Action of Radiations. It has already been 
pointed out that the radiations from radioactive substances are 
capable of affecting a photographic plate. The photographic 



46 THEORETICAL CHEMISTRY 

action of the radiations has been employed quite extensively in 
studying radioactive phenomena from a purely qualitative stand- 
point. The method employed, consists in exposing the photo- 
graphic plate, which has been previously wrapped in opaque black 
paper, to the action of the radiations. The time of exposure 
varies with the nature of the substance under examination, a few 
minutes being required for highly active preparations while sev- 
eral days or even weeks may be needed for preparations of low 
activity. 

Phosphorescence Induced by Radiations. A screen covered 
with crystals of phosphorescent zinc sulphide is rendered luminous 
when exposed to fairly intense radiation from a radioactive 
substance. This phenomenon has been shown to be due to the 
bombardment of the crystals of zinc sulphide by the so-called a- 
rays (see below). When the screen is examined with a lens the 
phosphorescence is seen to consist of a series of scintillations of 
very short duration. 

Nature of the Radiations. The ionizing, photographic, and 
luminescent properties of the radiations from radioactive sub- 
stances are not sufficient to differentiate them from cathode rays 
or X-rays, although each of these properties may be employed in 
determining their intensity. 

Evidence of the composite character of the radiations was fur- 
nished by a study of their penetrating power as well as by investi- 
gations of the behavior of the rays when subjected to the action of 
magnetic and electric fields. 

A thin sheet of aluminum or a few centimeters of air was found 
sufficient to cut off a large percentage of the rays. 

The unabsorbed portion of the radiation was found to consist of 
two distinct types, one of which was cut off by five or six millimeters 
of lead while the other possessed such great penetrating power 
that its presence could be readily detected after passing through a 
layer of lead fifteen centimeters thick. 

Rutherford named these three distinct types of radiation, the 
a-, 0-, and 7-rays, respectively. The penetrating powers of the 
-, 0-, and 7-rays may be approximately expressed by the propor- 
tion 1 to 100 to 10,000; i.e., the 0-rays are 100 times more pene- 



RADIOACTIVITY 47 

trating than the a-rays, while the 7-rays are 100 times more 
penetrating than the 0-rays. 

The general characteristics of the three kinds of rays may be 
briefly summarized as follows: 

(1) a-Rays. The a-rays consist of positively charged particles 
moving with speeds approximately one-tenth as great as that 
of light. These particles have been shown to be identical with 
helium atoms carrying two positive charges of electricity. They 
are appreciably deflected from a rectilinear path by magnetic and 
electric fields. They possess great ionizing power but relatively 
little penetrating power or photographic action. The depth to 
which an oc-particle penetrates a homogeneous absorbing medium 
before losing its ionizing power, is known as its " range." The 
range is proportional to the cube of the initial speed of the a- 
particle and is one of the characteristic properties of the radio- 
elements emitting a-rays. 

(2) 0-Rays. The 0-rays consist of negatively charged particles 
moving with speeds varying from two-fifths to nine-tenths of the 
speed of light. They are, in fact, electrons moving with much 
greater speeds than those shot out from the cathode in a vacuum 
tube. While the a-particles emitted by a particular radio-element 
have a definite velocity, the corresponding 0-ray emission consists 
of a flight of particles having widely different speeds. The pene- 
trating power of the 0-rays is conditioned by the speed of the 
particles, those which move most rapidly possessing the greatest 
penetrating power. 

The ionizing action of the 0-rays is much weaker than that of the 
a-rays, while exactly the reverse is true of the photographic action. 

(3) y-Rays. The 7-rays are identical with X-rays. They con- 
sist of extremely short waves of light, the wave-length varying 
from about 1 X 10~ 8 cm. for the rays of low penetrating power to 
about 1 X 10~ 9 cm. for the most penetrating rays. Obviously the 
7-rays cannot be deflected from a rectilinear path by either electric 
or magnetic fields. 

Uranium-X and Thorium-X. In 1900, Crookes * precipitated 
a solution of a uranium salt with ammonium carbonate; when an 

* Proc. Roy. Soc., 64, 409 (1900). 



48 THEORETICAL CHEMISTRY 

excess of the reagent had been added, all but a minute portion of 
the precipitate was found to have dissolved. This small insoluble 
residue, though chemically free from uranium, was found, when 
tested photographically, to be several hundred times more active, 
weight for weight, than the original salt. The solution, on the 
other hand, was found to have lost nearly all of its activity. At 
the end of a year, however, the solution had entirely regained its 
original activity, while the insoluble residue had become inactive. 
The active substance thus separated was called, on account of its 
unknown nature, uranium-X. 

Similarly, Rutherford and Soddy,* by precipitating a solution 
of a thorium salt with ammonium hydroxide, found that a large 
proportion of the activity remained behind in the thorium-free 
filtrate. On evaporating the filtrate to dryness and driving off 
the ammonium salts, a residue was obtained which was, weight 
for weight, several thousand times more active than the original 
solution. After standing for a month, this residue was found to 
have lost its Activity, while the precipitate had regained the ac- 
tivity of the original thorium compound. This active residue was 
called thorium-X from analogy to Crookes' uranium-X. 

The fact that uranium-X and thorium-X had each been obtained 
as the result of specific chemical processes, seemed to warrant the 
conclusion that they are new substances possessing well-defined 
properties. The manner in which these substances were obtained 
led to a variety of speculations as to the mechanism of the process 
involved in their production. In a subsequent paragraph it will 
be shown that the so-called disintegration theory offers a most 
satisfactory explanation of the foregoing experimental results. 

The Emanations. The element thorium was found by Ruther- 
ford to give off a radioactive gas or emanation which, when left tc 
itself, rapidly loses its activity in a similar manner to uranium-X 
and thorium-X. The thorium emanation was found to resemble 
the inactive gases in its chemical behavior. Thus, it can be sub- 
jected to the action of lead chromate, metallic magnesium and 
zinc dust at extremely high temperatures without undergoing 
change* The only other substances which, at the time of the dis- 
* Phil. Mag., VI, 4, 370 (1902). 



RADIOACTIVITY 49 

covery of the emanation, were known to resist the action of these 
reagents under the same conditions were the gaseous elements 
helium, neon, argon, krypton, and xenon. The most conclusive 
evidence of the gaseous character of the thorium emanation is the 
fact that it condenses to a liquid at very low temperatures. 

Rutherford and Soddy showed that the origin of the thorium 
emanation is thorium-X and not the element thorium itself. 
Freshly precipitated thorium hydroxide shows only a trace of 
emanating power, whereas the thorium-X separated from the 
filtrate possesses this power to a marked degree. As the thorium- 
X gradually loses its emanating power, the hydroxide shows a 
corresponding recovery. 

Radium was found to give off an emanation which behaves 
similarly to the thorium emanation except that it parts with its 
activity at a slower rate. The rate at which the radium emanation 
is produced, together with its longer life, enabled Ramsay and 
Soddy * not only to measure the volume of the emanation obtained 
from 60 mg. of radium bromide but also to establish the fact that 
it obeys Boyle's law. 

The Active Deposits. It was discovered^ by Rutherfordf for 
thorium, and by M. and Mine. Curie t for radium, that the emana- 
tions from these elements are ca-gable of imparting radioactivity 
to surrounding objects. On the other hand, uranium and polo- 
nium, which evolve no emanations, have no such influence on their 
environment. This fact is taken as a proof that induced radio- 
activity is due to actual contact with the emanations. If a nega- 
tively charged platinum wire be exposed to the thorium or radium 
emanation, the whole of its activity will be concentrated on the 
wire. This so-called active deposit may be transferred from the 
wii-e to a piece of sandpaper by rubbing. It may be sublimed 
from the wire to the walls of the tube in which it is heated, or it- 
may be dissolved from the wire by means of hydrochloric or sul- 
phuric acid. On evaporating the solution, the activity will be found 
to reside on the evaporating dish. Microscopic examination of 

* Proc. Roy. Soc., 73, 346 (1904). 
t Phil. Mag., VI, 23, 621 (1911). 
i Coirmt. rendus. 120. 714 (1899). 



50 THEORETICAL CHEMISTRY 

the wire fails to reveal any deposit, and the most sensitive balance 
is incapable of detecting any gain in weight after exposing the wire 
to the emanation. These experiments leave no room for doubt 
that the active deposits consist of infinitesimal amounts of solid 
substances possessing definite chemical properties. The active 
deposits undergo a gradual loss of activity similar to that observed 
with the emanations and with uranium-X arid thorium-X. 

The Disintegration Hypothesis. It was soon discovered that 
the active deposits undergo a series of additional radioactive 
changes. These subsequent transformations were found to be 
much more obscure and difficult to follow. In fact, it is only be- 
caiiife'of the ingenuity and mathematical acumen of those who 
undertooK^Jbhis difficult research that we are today in possession of 
such complete knowledge of the succeeding members of the radio- 
active serie& of elements. 

In*HK)3, Rutherford and Soddy * brought forward their theory 
of atoi^^fcijtegration which affords a perfectly satisfactory in- 
t&pjretatjg njp the complicated results already detailed, as well as 
ofthe-ewbafequent changes in the active deposits to which reference 
has just been made. According to this theory, radioactive change 
is assumed to be due to the spontaneous disintegration of the radio- 
elements with concomitant formation of new elements. These 
new elements, which are often less stable than the parent element, 
are assumed to undergo further disintegration with the production 
of still other elements, the end of the process ultimately being 
reached when a stable element is formed. 

The Radioactive Constant. The activity of a radio-element 
decays exponentially with time according to the equation 

It = Ioe~", (1) 

where Jo is the initial activity, It the activity at the end of an in- 
terval of time t, X a constant, and e is the base of the natural system 
of logarithms. The constant X, known as the radioactive constant, 
represents the fraction of the total amount of radioactive substance 
undergoing disintegration in unit of time, provided the latter is so 
small that the quantity at the end of time unit is only slightly 
* Phil. Mag., VI, 5, 576 (1903). 



RADIOACTIVITY 51 

different from the initial quantity. The reciprocal of the radio- 
active constant is called the average life of the element. Soddy 
defines the average life of a radio-element as "the sum of the sepa- 
rate periods of future existence of all the individual atoms divided 
by the number in existence at the starting point." 

If n t represents the number of atoms of a radio-element changing 
in unit time at the end of a time t, and no the corresponding value 
when t = 0, equation (1) may be written 

Tit == no6~~ 

In order to determine the initial rate of change, let N denote 
total number of atoms originally present, and Nt the nu 
maining unchanged at time t] we then have 



r" 

*-/ 



But when t = 0, N Q = N t . 

and No = * 



Hence N t 

On differentiating, we have 

f--xy. (2) 

Or, stated in words, the rate at which the atoms of a radio-element 
undergo disintegration at any given time is found to be propor- 
tional to the total number in existence at that time. 

This law of radioactive change is also peculiar to unimolecular 
chemical reactions (see p. 361). The velocity of a unimolecular 
reaction, however, is conditioned by the temperature, whereas 
the velocity of a radioactive change remains unaltered at the 
highest and the lowest attainable temperatures. 

The time required for one-half of a radio-element to undergo 
transformation is known as the period of half change T, and may be 
readily calculated from X as follows: 
log 0.5 = 0.4343 \T 

m 0.6932 

'i - 
* \ 



52 THEORETICAL CHEMISTRY 

It has been shown by Geiger and Nuttall * that the radioactive 
constant X and the range R of the a-particles shot out from a dis- 
integrating atom bear the following empirical relation to each 
other, 

X = aR\ 

where a and b are constants, the former constant being character- 
istic of the particular radioactive series to which the element be- 
longs. This formula has been found useful in calculating the 
values of X for the longest and shortest lived elements. 

Radioactive Equilibrium. It is evident that a state of equi- 
librium must ultimately be attained among the atoms of a radio- 
active substance. When the rate of production of a radio-element 
from its parent element is equal to its rate of disintegration into 
the next succeeding element of the series, the substance is said to 
be in radioactive equilibrium. The relative amounts of the suc- 
cessive members of a series of elements in radioactive equilibrium 
are inversely proportional to their radioactive constants. 

In order that measurements of the rate of radioactive change 
may be strictly comparable, it is necessary to make use of the 
amounts which are in equilibrium with a fixed amount of the 
parent element. Thus, the unit adopted for the measurement of 
the quantity of the radium emanation is the mass of emanation in 
equilibrium with one gram of radium. This unit is known as the 
curie. Its mass is Xi/X 2 gram, where Xi and X 2 are the radioactive 
constants of radium and its emanation respectively. One curie 
of radium emanation may be shown to occupy 0.63 cu. mm. under 
standard conditions of temperature and pressure. 

The Disintegration Series. It appears almost certain that the 
thirty or more radio-elements are disintegration products of one or 
the other of the two parent elements, uranium or thorium. One 
of the most convenient methods of classification of these elements 
is to arrange them in disintegration series, starting with the parent 
element and placing the succeeding elements in the order of their 
production. The accompanying table shows the three series of 
radio-elements as thus arranged by Soddy. The numbers within 
the circles are the atomic weights of the elements, while the small 
* Phil. Mag., VI, 22, 613 (1911). 



RADIOACTIVITY 
TABLE SHOWING DISINTEGRATION SERIES 



53 




54 THEORETICAL CHEMISTRY 

circles and dots at the right of the larger circles indicate the char- 
acter of the radiation given out at each stage of the disintegration 
process. The average life, 1/X, of each element in the series is 
given below the name of the element. While a detailed account 
of the properties of the different radio-elements included in this 
table cannot be given here, attention should be called to the com- 
plex transformations occurring in the active deposit. It is also of 
interest to observe that the end-products of the three series bear a 
striking resemblance to the element lead and that their atomic 
weights are approximately equal and nearly identical with that 
of ordinary lead. 

Counting the a-Particles. It has already been stated that 
when an a-particle strikes a screen coated with phosphorescent 
zinc sulphide, a distinct flash of light may be seen when the screen 
is viewed through a magnifying lens in a dark room. It is obvious 
that if one could count the number of these scintillations, it would 
be an easy matter to ascertain the total number of a-particles 
shot out from a radioactive substance in a given time. By using 
a phosphorescent screen and a microscope, Regener * has deter- 
mined in this manner the rate of emission of a-particles from 
polonium. He found from his different experiments an average 
emission of 3.94 X 10 5 a-particles per second. The total charge 
on the a-particles was then measured by collecting them in a suit- 
able measuring vessel. A charge of 37.7 X 10~~ 5 electrostatic 
units was found to be associated with the total number of a- 
particles emitted by polonium in one second. 

Rutherford and Geiger f devised an electrical method for count- 
ing the a-particles. In their experiment the source of the a- 
particles was a small disc which had been exposed to the radium 
emanation for some hours. This disc was placed in an evacuated 
tube at a measured distance from a small aperture of known cross- 
section. The aperture was closed with a thin plate of mica through 
which the a-particles could pass with Qase. After passing through 
th$ mica plate, the a-particles entered an ionization chamber 
filled with air at reduced pressure and fitted with two charged 

* Sitzunsbericht d. K. preuss. Akad., 38, 948 (1909). 
t Proc.Boy. Soc. A, 8x, 141 (1908), 



RADIOACTIVITY 55 

metal plates connected with appropriate apparatus for measuring 
ionization currents. Whenever an a-particle entered the ioniza- 
tion chamber, a momentary current passed, producing a sudden 
deflection of the needle of the electrometer. By counting the 
number of throws of the needle occurring in a definite interval of 
time, the total number of a-particles passing through the ionization 
chamber was determined. Knowing the distance of the source 
of the radiations from the aperture, together with the area of the 
aperture, the total number of a-particles emitted by the radio- 
active disc in a given time could be computed. Rutherford and 
Geiger thus found that one gram of radium emits very nearly 
107 X 10 16 a-particles per year. Having determined the total 
number of a-particles emitted, it only remained to measure the 
total charge, in order to calculate the charge carried by a single 
a-particle. From a series of very consistent measurements, the 
charge carried by a single a-particle was found to be 9.3 X 10~~ 10 
electrostatic units. Since the fundamental charge e has been 
shown to be 4.48 X 10~ 10 electrostatic units, it follows that the 
a-particle carries two ionic charges of electricity. 

Helium Atoms and a-Particles Identical. In 1909, Ruther- 
ford and Royds * performed a crucial experiment to determine the 
nature of the a-particle. A glass bulb was blown with walls thin 
enough to permit the passage of the a-particles but sufficiently 
strong to withstand atmospheric pressure. The bulb was filled 
with radium emanation and then enclosed in an outer glass tube 
to which a spectrum tube had been sealed. On exhausting the 
outer tube and examining the spectrum of the residual gas, no 
evidence of helium was obtained until after an interval of twenty- 
four hours. After four days the characteristic yellow and green 
lines were plainly visible and at the end of the sixth day, the com- 
plete spectrum of helium was obtained. The unavoidable 
elusion from this experiment is, that the presence of helium ir 
outer tube must have been due to the a-particles whiplr were 
projected through the thin walls of the inner tube. Jn anothfli 
experiment, the inner tube was filled with pure/^helium uifoer 
pressure while the exhausted outer tube was exaifuned fop^elium, 

* Phil. Mag., VI, 17, 281 



56 THEORETICAL CHEMISTRY 

No trace of helium could be detected spectroscopically even after 
an interval of several days, thus proving that the helium detected 
in the first experiment must have resulted from the a-particles 
which had been shot out from the radium emanation with sufficient 
energy to penetrate the thin walls of the inner tube. These experi- 
ments leave no room for doubt that an a-particle becomes a 
helium atom when its positive charge is neutralized. 

Rate of Production of Helium. The rate of production of 
helium from the series in equilibrium with one gram of radium 
has been determined experimentally by Rutherford and Bolt- 
wood * to be 156 cu. mm. per year. This result agrees closaly 
with the calculated rate of production, viz., 158 cu. mm. per gram 
of radium per year. 

Energy Evolved by Radium. Curie and Laborde * were the 
first to call attention to the interesting fact that the temperature 
of radium compounds was uniformly higher than that of their 
environment. Careful measurements have shown that one, gram 
of radium evolves heat at the rate of approximately 135 gram- 
calories per hour. That the greater part of this heat energy is due 
to the a-particles may be proven by a direct calculation of their 
mean kinetic energy. The magnitude of the store of energy con- 
tained in radium may be realized upon the statement/, that one 
gram of radium ; before it entirely disappears, evolves an amount 
of heat energy nearly one million times greater than that evolved 
in the formation of one gram of water from its elements. 

* Phil. Mag., VI, 22, 586 (1911). 
t Compt. rendus, 136, 673 (1904). 



CHAPTER V. 

ATOMIC STRUCTURE. 

The Modern Conception of Atomic Structure. As a result of 
the investigations of Thomson,* Rutherford,t Nicholson,} Bohr, 
and others, a theory of atomic structure has been developed which 
affords a satisfactory interpretation of many of the important 
relationships among the chemical elements. 

Briefly stated, this theory assumes that the atom consists of a 
central, positively charged nucleus, surrounded by a miniature 
solar system of electrons. The investigations of Rutherford and 
Geiger|| show that the character of the deflection of a-particles 
shot out from radioactive atoms at speeds approximating 20,000 
miles per second, and consequently completely penetrating other 
atoms, is sucBTSg~to indicate an extremely high concentration of 
positive electricity on the central nucleus. The central nucleus 
which is supposed to represent nearly the entire mass of the atom, 
is thought to be very small in comparison with the size of the atom 
as a whole. Recent investigations make it appear probable that 
the maximum diameter of the nucleus of the hydrogen atom is 
about one one-hundred-thousandth of the diameter commonly 
attributed to the atom. In commenting on this statement, Har- 
kins says: 1f "On this basis the atom would have a volume a 
million-billion times larger than that of its nucleus, and thus the 
nucleus of the atom is much smaller in comparison with the size 
of the atom than is the sun when compared with the dimensions, 
of its planetary system." It is highly probable that the central 
nucleusjg itself made up of a definite number of units of positive 
electricity together with a small number of attendant electrons. 

* Phil. Mag., 7, 237 (1904). 
t Popular Science Monthly, 87, 105 (1915) 
t Phil. Mag., 22, 864 (1911). 
Phil. Mag., 26, 476, 857 (1913). 
|| Phil. Mag., 21, 669 (1911). 
If Science, 66, 419 (1917). 
57 



58 THEORETICAL CHEMISTRY 

It is further assumed that the units of positive electricity are 
hydrogen atoms, each of which has been deprived of one electron. 
If the mass of an atom is largely due to the presence of hydrogen 
nuclei, then we should expect Prout's hypothesis to hold and the 
atomic weights of the elements to be exact multiples of the atomic 
weight of hydrogen. When we consider, however, that according 
to electromagnetic theory the total mass of a body composed of 
positive and negative units is dependent upon the relative posi- 
tions of these units when packed together, it is evident that the 
mass of the atom will not necessarily be an exact multiple of the 
mass of the hydrogen atom. 

It has already been pointed out that helium is a product of 
many radioactive transformations. This fact may be taken as an 
indication of the extraordinary stability of the helium atom. Be- 
cause of its stability, the nucleus of this atom has come to be con- 
sidered as a secondary unit of positive electricity. The nucleus 
of the helium atom, or the nucleus of an a-particle is assumed to 
consist of four hydrogen nuclei with two nuclear electrons. 

The Atomic Number. Since the algebraic sum of the positive 
and negative electrification on an atom must be zero, it follows 
that the charge resident upon the nucleus must be equal to the 
number of electrons outside the nucleus. This number, which has 
come to be recognized as more important and characteristic than 
the atomic weight, is known as the atomic number. 

X-Rays and Atomic Structure. The discovery by W. L. Bragg 
in 1912 that X-rays undergo reflection at crystal surfaces and the 
subsequent development by Mr. Bragg and his father, W. H. Bragg 
of the X-ray spectrometer, has led to a series of investigations of 
the utmost importance to both the chemist and the physicist. 

In order that the significance of these investigations may be 
understood it may be well to summarize very briefly some of the 
more important properties of the X-ray. 

The bombardment of metal plates, usually of platinum, by elec- 
trons give rises to X-rays. The radiation issuing from an X-ray 
tube is very far from homogeneous. When screens of different 
materials and varying thicknesses are interposed in the path of 
the rays, the degree of absorption is irregular. It has been found, 



ATOMIC STRUCTURE 59 

however, that every substance when properly stimulated is capable 
of emitting a homogeneous and characteristic X-radiation, the 
penetrating power of which is wholly determined by the nature of 
the elements of which the substance is composed. The pene- 
trating power of this typical X-radiation increases with the atomic 
weight of the radiating element. With elements whose atomic 
weights are less than 24, the radiation is too feeble to be measured. 
It is important to note that this property of the elements is not a 
periodic function of the atomic weight. This type of X-radiation 
is entirely independent of external conditions, indicating that it 
is closely connected with the internal structure of the atoms from 
which it emanates. The rays possess the power of affecting the 
photographic plate and also of rendering gases through which 
they pass conductors of electricity. 

It is estimated that the wave-length of an X-ray is about 1 X 10~ 8 
to 1 X 10~ 9 cm., or about one ten-thousandth of the wave-length 
of sodium light. It is obvious that the spacing of the lines of a 
grating capable of diffracting such short waves must be of the 
order of magnitude of interrnolecular distances. It is well known 
that a grating owes its power of analyzing a complex system of 
light waves into its component wave-trains, to the series of paral- 
lel lines which are ruled upon its surface at exactly equal intervals. 
When a train of waves is incident upon a grating, each line acts 
as a center from which a diffracted train of waves emerges. 

Such a grating is relatively simple in its action since it consists 
of a single series of centers of diffraction lying in one plane. The 
power of a crystal surface to reflect X-rays, however, is due to the 
fact that the crystal is in reality a three-dimensional diffraction 
grating, the atoms or molecules of which the crystal is built up, 
acting as the centers of diffraction. It must be borne in mind that 
the reflection of X-rays is in no way dependent upon the existence 
of a polished surface on the outside of the crystal, but rather upon 
the regularly spaced atoms or molecules within the crystal. To 
ordinary waves of light the atomic structure is so fine grained as 
to behave as a continuous medium, whereas to the short X-ray 
waves, the crystal acts as a discontinuous structure of regularly 
arranged particles, each of which functions as a diffraction center. 



60 



THEORETICAL CHEMISTRY 



X-Ray Spectra. By making use of the reflecting power of one 
of the cleavage planes of a crystal, and employing different metals 
as anti-cathodes in an X-ray tube, Moseley * succeeded in photo- 
graphing the X-ray spectra of the characteristic radiations of a 
number of the elements. He showed that the X-ray spectrum of 
an element is extremely simple and consists of two groups of lines 
known as the "K" and "L" radiations. As a result of careful 
study of the "K" radiations of thirty-nine elements from alumin- 
ium to gold, Moseley discovered that these radiations are char- 
acterized by two well-defined lines whose vibration frequency v 
is connected with the atomic number of the clement N, by the 
simple relation v = A (N IY 

where A is a constant. When the square roots of the frequencies 
of the elements are plotted as abscissae against their atomic num- 
bers as ordinates, 
the points are found 
to lie on a straight 
line as shown in Fig. 
11. On the other 
hand, if the square 
roots of the fi&quen- 
cies are plotted 
against the atomic 
weights of the ele- 
ments, the relation- 
ship is no longer rec- 
tilinear. When the 
elements are ar- 
ranged in the order 
of their atomic num- 
bers instead of in the 
order of their atomic weights, the irregularities f hitherto noted in 
connection with argon, cobalt, and tellurium entirely disappear. 
In reviewing Moseley's work on X-ray spectra, Soddy t says: 

* Phil. Mag., 26, 210 (1913); 27, 703 (1914). 

t See p. 29. 

j Ann. Reports on the Prog, of Chemistry, p. 278 (1914). 



fin 








4- 


+* 


100 










4- 

f 


/ 












>^ 




80 Jc 








/ 


x 




1 








^r 









90 




t 


*x^ 






60 5 






+ V^ 








-i; 






+*s 










90 




s 








in 


10 


X 














10 U 18 

Square Root of Frequency X 

Kg. 11. 



22 



ATOMIC STRUCTURE 61 

"A veritable roll-call of the elements has been made by this 
method. Thirty-nine elements, with atomic weights between 
those of aluminium and gold, have been examined in this way, and 
in every case the lines of the X-ray spectrum have been found to 
be simply connected with the integer that represents the place 
assigned to it by chemists in the periodic table." 

One of the most interesting results of this " roll-call" of the 
elements is the fixing of the number of possible rare-earth elements. 
Between barium and tantalum there are places for only fifteen 
rare-earth elements and fourteen of these places are filled. While 
future investigation may necessitate some rearrangement in the 
order of tabulation, the total number of these elements is limited 
to fifteen. 

Periodicity among the Radio-elements. The problem of plac- 
ing the newly discovered radio-elements in the periodic table re- 
mained unsolved until 1913, when Fajans * and Soddy,t working 
independently, discovered an important generalization concerning 
the changes in chemical properties resulting from the expulsion of 
a- and jft-particles during radioactive transformations. This impor- 
tant generalization may be stated as follows: The expulsion of 
an a-$article causes a radioactive element to shift its position in 
the periodic table two places in the direction of decreasing atomic 
weight, whereas the emission of a fi-particle causes a shift of one 
place in the opposite direction. This generalization not only agrees 
with our present theory of atomic structure, but may be shown 
to be a necessary consequence of this theory. 

The loss of an a-particle or helium atom involves a loss of 4 
units in atomic weight and of 2 units of positive electricity from 
the nucleus of the atom. In consequence of this loss, the atomic 
number is diminished by 2 units and the resulting new element 
will find a place in the periodic table two groups to the left of that 
occupied by the parent element. On the contrary, while the 
expulsion of a 0-particle, or electron, involves practically no change 
in mass, the nucleus of the parent atom suffers a loss of 1 unit of 
negative electricity. This loss is equivalent to a gain of 1 unit of 

* Physikal. Zeit., 14, 49 (1913). 
t Chem. News, 1-07, 97 (1913). 



62 



THEORETICAL CHEMISTRY 



positive electricity, or to an increase of 1 unit in the atomic number, 
and in consequence, the position of the new element in the periodic 
table will be shifted one group to the right of that occupied by the 
parent element. 

Soddy's arrangement of all of the radio-elements in accordance 
with this generalization is shown in Fig. 12. Thus, starting with 
the element uranium in Group VIA, we may follow the successive 



RADIO-ELEMENTS AND PERIODIC LAW 
ALL ELEMENTS IN THE SAME PLACE 

IN THE PERIODIC TABLE 
ARE CHEMICALLY NON-SEPARABLE 
^V AND (PROBABLY) 

>PECTROSCOPICALLY INDISTINGUISHABLE 




Relative No. of Negative Electrons 
5 4 82 1 



Fig. 12. 

steps in the radium disintegration series which was discussed in 
the preceding chapter. The element UXi resulting from U by 
the loss of an a-particle is placed in Group IVA. This element in 
turn undergoes a 0-ray change producing the element UX 2 which 
is accordingly placed in Group VA. The element UII in Group 
VIA is formed from UX 2 by #-ray disintegration, while the ele- 
ment lo results from the loss of an a-particle by UII with a con- 
sequent shifting of two places to the left in the table. A similar 
loss of an a-particle by lo brings us to the element Ra in Group 



ATOMIC STRUCTURE 63 

IIA. In the successive steps of this disintegration from U to Ra, 
three a-particles or 12 units of atomic mass are lost, and the atomic 
weight of Ra, as calculated from that of U, agrees with the atomic 
weight found by direct experiment. In like manner the remain- 
ing stages of the disintegration may be followed to the end-product 
in Group IVB. 

Isotopes. Perhaps the most striking feature in the table is 
the occurrence of several different elements in the same place, as 
for example in Group IVB, where in the place occupied by the 
element Pb, we also find RaB, RaD, ThB, and AcB, together 
with four other elements to which no names have been assigned, 
but which are none the less stable end-products. The individual 
members of such a group of elements occupying the same place in 
the periodic table, and being in consequence chemically identical, 
are known as isotopes. Isotopic elements have identical arc and 
spark spectra and, except for differences in atomic weight, are 
chemically indistinguishable. 

Making use of the fact that two gaseous elements having differ- 
ent atomic weights diffuse at different rates, Thomson and Aston 
have recently succeeded in separating neon into two isotopes 
having atomic weights 20 and 22 respectively. This is the only 
method which has thus far given promise of success in effecting 
isotopic separations. 

It is interesting to note in Soddy's table (Fig. 12), that "the ten 
occupied spaces (groups) contain nearly forty distinct elements, 
whereas if chemical analysis alone had been available for their 
recognition, only ten elements could have been distinguished." 

The Hydrogen-Helium System of Atomic Structure. A 
generalization similar to that just outlined for the radio-elements 
has been found by Harkins and Wilson * to hold true for the lighter 
elements which apparently do not undergo appreciable a-ray dis- 
integration. Beginning with helium and adding 4 units of atomic 
weight for each increase of 2 units in the atomic number, gives th^ 
atomic weights of the elements in the even-numbered groups of 
the periodic table, neglecting small changes in mass due to nuclear 
packing. This rule has been found to hold very closely for all of 
the elements having atomic weights below 60. 

* Proc. Nat. Acad., Vol. I, p. 276 (1915), 



64 THEORETICAL CHEMISTRY 

The atomic weights of the elements of the odd-numbered groups 
can be calculated by a similar rule, provided that the atom of 
lithium, the first member of the odd-numbered groups, be assumed 
to be made up of 1 hydrogen and 3 helium nuclei. The following 
table gives the results as calculated by Harkins and Wilson for the 
first three series of the periodic table. 

The so-called theoretical atomic weights are calculated on the 
basis H = 1, while the experimentally determined values are on 
the basis O = 16 or H = 1.0078. The remarkably close agree- 
ment between the two sets of values is taken as an indication that 
the packing effect, resulting from the formation of the elements 
from hydrogen nuclei and attendant electrons, is very small. 
This packing effect has been estimated to involve a decrease in 
atomic mass of about 0.77 per cent, and is believed to be due 
almost entirely to the formation of the helium atom. The hydro- 
gen-helium hypothesis of atomic structure offers a rational ex- 
planation of many interesting but hitherto obscure facts concern- 
ing the nature of the elements. 

Relation between Atomic Weights and Atomic Numbers. For 
all elements whose atomic weights are less than that of nickel, 
Harkins finds the following simple mathematical relation to hold, 



where W is the atomic weight and N is the atomic number. In 
other words, the atomic weights are a linear function of the atomic 
numbers. 

The Periodic Law. In the light of recent discoveries the 
periodic law acquires new significance; in fact to-day the periodic 
law may be regarded as the most comprehensive generalization in 
the whole science of chemistry. 

Attention has already been directed in an earlier chapter to the 
most apparent of the imperfections in Mendeteeff s system of 
classification of the elements. While the later tables are more 
complete than the original, owing in part to the discovery of new 
elements, it must be admitted nevertheless that relatively little 
real progress has been made until recently toward removing the 
seemingly inherent defects of the system. 



ATOMIC STRUCTURE 



65 



W 



ooO* 88 



MM 
00 



W 

CSI 



03 O) O O 






- 






tn 

,08+00 



ffi S QO 



W 



^00 



W 
+00 



o 

1 



tS 



o 

4J 

CO 



66 THEORETICAL CHEMISTRY 

A satisfactory periodic table should meet the following require- 
ments: 

(1) It should afford a place for isotopic elements such as lead. 

(2) The radio-elements together with their a- and j3-disinte- 
gration products should be shown. 

(3) It should contain no vacant spaces except those correspond- 
ing to the atomic numbers of undiscovered elements. 

(4) It should bring out the relation between the elements con- 
stituting a main group and those forming the corresponding sub- 
group. For example, the relation between the elements Be, Mg, 
Ca, Sr, Ba, and Ha on the one hand, and the elements Zn, Cd, and 
Hg on the other, should be emphasized. 

(5) The elements of Group O and Group VIII should fit natu- 
rally in the table. 

(6) All of the foregoing conditions should be shown by means of 
a continuous curve connecting the elements in the order of their 
atomic numbers, the latter having been shown to be more charac- 
teristic of an element than its atomic weight. 

A table which satisfactorily meets these requirements has re- 
cently been devised by Harkins and Hall. This table may be 
constructed in the form of a helix in space or as a spiral in a plane. 
The following description of the helical arrangement, shown in 
Fig. 13, is taken verbatim from the original paper of Harkins and 
Hall.* 

"The atomic weights are plotted from top down, one unit of 
atomic weight being represented by one centimeter, so the model 
is about two and one-half meters high. . . . 

' 4 The balls representing the elements are supposed to be strung 
on vertical rods. All of the elements on one vertical rod belong to 
one group, have on the whole the same maximum valence, and are 
represented by the same color. The group numbers are given at 
the bottom of the rods. On the outer cylinder the electro-nega- 
tive elements are represented by black circles at the back of the 
cylinder, and electro-positive elements by white circles on the front 
of the cylinder. The transition elements of the zero and fourth 
groups are represented by circles which are half black and half 

* Jour. Am. Chem. Soc., 38, 169 (1916). 



20 



40 



80 



100 



120 



140 



160 



180 



200 



220 



240 

II 

if 



Group 




Fig. 13. 



68 THEORETICAL CHEMISTRY 

white. The inner loop elements are intermediate in their proper- 
ties. Elements on the back of the inner loop are shown as heavily 
shaded circles, while those on the front are shaded only slightly. 
"In order to understand the table it may be well to take an 
imaginary journey down the helix, beginning at the top. Hydro- 
gen (atomic number and atomic weight = 1) stands by itself, and 
is followed by the first inert, zero group, and zero valent element 
helium. Here there comes the extremely sharp break in chemical 
properties with the change to the strongly positive, univalent 
element lithium, followed by the somewhat less positive bivalent 
element, beryllium, and the third group element boron, with a 
positive valence of three, and a weaker negative valence. At the 
extreme right of the outer cylinder is carbon, the fourth group 
transition element, with a positive valence of four, and an equal 
negative valence, both of approximately equal strength. The 
first element on the back of the cylinder is more negative than 
positive, and has a positive valence of five, and a negative valence 
of three. The negative properties increase until fluorine is reached 
'and then there is a sharp break of properties, with the change from 
the strongly negative, univalent element fluorine, through the zero 
valent transition element neon, to the strongly positive sodium. 
Thus in order around the outer loop the second series of elements 
are as follows: 

Group number 01234567 

Maximum valence. .. 01234567 

Element He Li Be B C N F 

Atomic number 2 3 4 5 6 7 89 

" After these comes neon, which is like helium, sodium which is 
like lithium, etc., to chlorine, the eighth element of the second 
period. For the third period the journey is continued, still on 
the outer loop, with argon, potassium, calcium, scandium, and- 
then begins with titanium, to turn for the first time into the inner 
loop. Vanadium, chromium, and manganese, which comes next, 
are on the inner loop, and thus belong, not to main but to sub- 
groups. This is the first appearance in the system of sub-group 
elements. Just beyond manganese a catastrophe of some sort 
seems to take place, for here three elements of one kind, and there- 



ATOMIC STRUCTURE 69 

fore belonging to one group, are deposited. The eighth group in 
this table takes the place on the inner loop which the rare gases of 
the atmosphere fill on the outer loop. The eighth group is thus a 
sub-group of the zero group. 

" After the eighth group elements, which have appeared for the 
first time, come copper, zinc, and gallium; and with germanium, a 
fourth group element, the helix returns to the outer loop. It then 
passes through arsenic, selenium, and bromine, thus completing 
the first long period of 18 elements. Following this there comes a 
second long period, exactly similar, and also containing 18 elements. 

"The relations which exist may be shown by the following 
natural classification of the elements. They may be divided into 
cycles and periods as follows: 

TABLE I. 

Cycle 1 = 4 2 elements. 

1st short period He F = 8 = 2X22 elements. 

2nd short period Ne Cl = 8 = 2X22 elements. 

Cycle 2 = 6 2 elements. 

1st long period A Br = 18 = 2 X 3 2 elements. 

2nd long period Kr I = 18 = 2 X 3 2 elements. 

Cycle 3 = 8 2 elements. 

1st very long period Xe Eka-I = 32 = 2 X 4 2 elements. 

2nd very long period Nt U. 

" The 'last very long period, and therefore the last cycle, is in- 
complete. It will be seen, however, that these remarkable relations 
are perfect in their regularity. These are the relations, too, which 
exist in the completed system,* and are not like many false nu- 
merical systems which have been proposed in the past where the 
supposed relations were due to the counting of blanks which do not 
correspond to atomic numbers. This peculiar relationship is un- 
doubtedly connected with the variations in structure of these com- 
plex elements, but their meaning will not be apparent until we 
know more in regard to atomic structure. 

* If elements of atomic weights two and three are ever discovered then the 
zero cycle would contain 2P elements, and period number one should then be 
said to begin with lithium. Such extrapolation, however, is an uncertain 
basis for the .prediction of such elements. 



70 THEORETICAL CHEMISTRY 

"The first cycle of two short periods is made up wholly of outer 
loop or main group elements. Each of the long periods of the 
second cycle is made up of main and of sub-group elements, and 
each period contains one-eighth group. The only complete very 
long period is made up of main and of sub-group elements, con- 
tains one-eighth group, and would be of the same length (18 ele- 
ments) as the long periods if it were not lengthened to 32 elements 
by the inclusion of the rare earths. 

"The first long period is introduced into the system by the in- 
sertion of iron, cobalt, and nickel, in its center, and these are three 
elements whose atomic numbers increase by steps of one while 
their valence remains constant. The first very long period is 
formed in a similar way by the insertion of the rare earths, another 
set of elements whose atomic numbers increase by one while the 
valence remains constant. 

"In this periodic table the maximum valence for a group of 
elements may be found by beginning with zero for the zero group 
and counting toward the front for positive valence, and toward 
the back for negative valence. 

"The negative valence runs along the spirals toward the back 
as follows: 

-1 -2 -3 -4 

Ne F O N C 

A Cl S P Si 

"Beginning with helium the relations of the maximum theoreti- 
cal valences run as follows: 

Case 1 . He-F 0, 1, 2, 3, 4, 5, 6, 7, but does not rise to 8. Drops by 7 to 0. 

Ne-Cl ... 0, 1, 2, 3, 4, 5, 6, 7, but does not rise to 8. Drops by 7 to 0. 
Case 2. A-Mn ... 0, 1, 2, 3, 4, 5, 6, 7, 8, 8. Drops by 7 to 1. 

Fe, Co, Ni. 
Case 1. Cu-Br. 
Case 2. Kr-Ru, Rh, Pd. 

"In the third increase, the group number and maximum valence 
of the group rise to 8, three elements are formed, and the drop is 
again by 7 to 1. 

"Thus in every case when the valence drops back the drop in 
maximum group valence is 7, either from 7 to 0, or from 8 to 1. 



ATOMIC STRUCTURE 71 

This is another illustration of the fact that the eighth group is a 
sub-group of the zero group. The valence of the zero group is 
zero. According to Abegg the contra-valence, seemingly not 
active in this case, is eight. 

"In Fig. 13 the table is divided into five divisions by four 
straight lines across the base. These divisions contain the fol- 
lowing groups: 

Division 1 2 3 4 

Groups 0,8 1,7 2,6 3,5 4,4 

"The two groups of any division are said to be complementary. 
It will be seen that the sum of the group numbers in any division 
is equal to 8, as is also the sum of the maximum valences. The 
algebraic sum of the characteristic valences of two complementary 
groups is always zero. In any division in which the group numbers 
are very different, the chemical properties of the elements of the com- 
plementary main groups are very different, but when the group 
numbers become the same, the chemical properties become very much 
alike. Thus the greatest difference in group numbers occurs in 
division 8, where the difference is 8, and in the two groups there is 
an extreme difference in chemical properties, as there is also in 
division 1 between Groups 1 and 7 

f( Whenever the two main groups of a division are very different in 
properties, each of the sub-groups is quite different from its related 
main group. Thus copper in Group IB is not very closely related 
to potassium Group IA in its properties, and manganese is not 
very similar to chlorine, but as the group numbers approach each 
other the main and sub-groups become much alike. Thus scandium 
is quite similar to gallium in its properties, and titanium and ger- 
manium are very closely allied to silicon. 

"One important relation is that on the outer cylinder the main 
groups I A, HA, III A, become less positive as the group number 
increases, while on the inner loop the positive character increases from 
Group IB to IIB, and at the bottom of the table the increase from 
IIB to IIIB is considerable. Thus thallium is much more posi- 
tive than mercury. It has already been noted that in the case of 
the rare earths also the usual rule is inverted, that is the basic 
properties decrease as the atomic weight increases.' 1 



CHAPTER VI. 
GASES. 

The Gas Laws. Matter in the gaseous state possesses the 
property of filling completely and to a uniform density any avail- 
able space. Among the most pronounced characteristics of 
gases are lack of definite shape or volume, low density and small 
viscosity. The laws expressing the behavior of gases under differ- 
ent conditions are relatively simple and to a large extent are 
independent of the nature of the gas. The temperature and 
pressure coefficients of aliases are very nearly the same. 
KTT662, Robert Boyle discovered the familiar law that at 
constant temperature, the volume of a gas is inversely proportional 
to the pressure upon it. This may be expressed mathematically 
as follows: 

v oc - (temperature constant) 

where v is the volume and p the pressure. 

In 1801, Gay-Lussac discovered the law of the variation of the 
volume of a gas with temperature. 

This law may be formulated thus: At constant pressure, the 
volume of a gas is directly proportional to its absolute temperature, 
or 

v <x T (pressure constant). 

There are three conditions which may be varied, viz., volume, 
temperature and pressure. The preceding laws have dealt with 
the relation between two pairs of the variables when the third 
is held constant. There remains to consider the relation between 
the third pair of variables, pressure and temperature, the volume 
being kept constant. Evidently a necessary corollary of the first 
two laws is that at constant volume, the pressure of a gas is directly 
proportional to its absolute temperature, or 

p oc T (volume constant), 
72 



GASES 73 

These three laws may be combined into a single mathematical 
expression as follows: 

v oc - (T const.) law of Boyle, 

v oo T (p const.) law of Gay-Lussac; 
combining these two variations we have, 

T 

00C-, 

P 
or introducing a proportionality factor ik, 

i T 

V = # 9 
P 

or 

vp = kT. (1) 

If the temperature of the gas be (273 absolute), and the corre- 
sponding volume and pressure VQ and p<> respectively, then (1) 
becomes 

z>o??o = 273 fc, 
and 



~ 273' 
eliminating the constant k between (1) and (2), we have 

- VQ P 
- 



For any one gas the term ~ is a constant. If V Q is the volume 
of 1 gram of gas at and 76 cm., we write 

vp = rT, (3) 

where v is the volume of 1 gram of gas at the temperature T and 
the pressure p, and r is a constant called the specific gas constant. 
On the other hand when v denotes the volume of one mol. of gas 
at and 76 cm. (22.4 liters), the equation becomes 

vp = RT, (4) 

where R is termed the molecular gas constant, which has the game 



74 THEORETICAL CHEMISTRY 

value for all gases. If M is the molecular weight of the gas, Mr 
= R. Equation (4) is the fundamental gas equation. 

Evaluation of the Molecular Gas Constant. Since the product 
of p and v represents work, and T is a pure number, R must be 
expressed in energy units. There are four different units in which 
the molecular gas constant is commonly expressed, viz., (1) gram- 
centimeters, (2) ergs, (3) calories, and (4) liter-atmospheres. 

1. R in gram-centimeters. The volume, v, of 1 mol. of gas at 
and 76 cm. is 22.4 liters or 22,400 cc. The pressure, p, is 76 cm. 
multiplied by 13.59, (the density of mercury), or 1033.3 grams per 
square centimeter. Substituting we obtain 

poV Q 1033.3 X 22,400 
# = -yT - 273 - = 84 > 760 ST- cm - 

2. R in ergs. To convert gram-centimeters into ergs we must 
multiply by the acceleration due to gravity, g = 980.6 cm. per 
sec. per sec., or 

R = 84,760 X 980.6 = 83, 150,000 ergs. 

3. R in calories. To express work in terms of heat, we must 
divide by the mechanical equivalent of heat, or since, 1 calorie is 
equivalent to 42,640 gr. cm. or 41,830,000 ergs, we have 

83 150 000 
R = / = *' ca *' (approximately 2 cal.) 



4. R in liter-atmospheres. A liter-atmosphere may be defined 
as the work done by 1 atmosphere on a square decimeter through 
a decimeter. If p is the pressure in atmospheres, and VQ is the 
volume in liters, we have 

R = -JT == -0 = 0.0821 liter-atmosphere. 



Deviations from the Gas Laws. Careful experiments by 
Amagat * and others on the behavior of gases over extended ranges 
of temperature and pressure have shown that the fundamental 
gas equation, pv = RT, is not strictly applicable to any one gas, 
the deviations depending upon the nature of the gas and the 
conditions under which it is observed. It has been shown that 
the gas laws are more nearly obeyed the lower the pressure, the 
* Ann. Chim. phys. (5) 19, 345 (1880). 



GASES 



75 



higher the temperature and the further the gas is removed from 
the critical state. A gas which would conform to the require- 
ments of the fundamental gas equation is called an ideal or per- 
fect gas. Almost all gases are far from ideal in their behavior. 
At constant temperature the product, pv, in the gas equation is 
constant, so that if we plot pressures as abscissae and the corre- 



40 



20 



10 




100 



P (atmospheres) 
Fig. 14. 



200 



800 



spending values of pv as ordinates, for an ideal gas we should 
obtain a straight line parallel to the axis of abscissae, as shown in 
Fig. 14. The results obtained by Amagat with three typical 
gases are also shown in the same diagram. It will be apparent 
that all of these gases depart widely from ideal behavior. In 
the case of hydrogen pv increases continuously with the pressure, 



76 THEORETICAL CHEMISTRY 

while with nitrogen and carbon dioxide it first decreases, attains 
to a minimum value and beyond that point increases with increas- 
ing pressure. With the exception of hydrogen, all gases show a 
minimum in the curve, thus indicating that at first the compress- 
ibility is greater than corresponds with the law of Boyle, but 
diminishes continuously until, for a short range of pressure, the law 
is followed strictly: beyond this point the compressibility is less 
than Boyle's law requires. 

Hydrogen is exceptional in that it is always less compressible 
than the law demands. This is true for all ordinary tempera- 
tures, but it is highly probable that at extremely low temperatures 
the curve would show a minimum. The two curves for carbon 
dioxide at 31.5 and 100 illustrate the fact that the deviations 
from the gas laws become less as the temperature increases. The 
deviations of gases from the laws of Boyle and Gay-Lussac, as well 
as their behavior in general, may be satisfactorily accounted for 
on the basis of the kinetic theory. 

Kinetic Theory of Gases. The first attempt to explain the 
properties of gases on a purely mechanical basis was made by 
Bernoulli in 1738. Subsequently, through the labors of Kroenig, 
Clausius, Maxwell, Boltzmann and others, his ideas were developed 
into what is known today as the kinetic theory of gases. Accord- 
ing to this theor^f gases are considered to be made up of minute, 
perfectly elastic particles which are ceaselessly moving about 
with high velocities, colliding with each other and with the walls 
of the containing vessel. These particles are identical with the 
molecules defined by Avogadro. The volume actually occupied 
by the gas molecules is supposed to be much smaller than the 
volume filled by them under ordinary conditions, thus allowing 
the molecules to move about free from one another's influence 
except when they collide. Tl^fi distance thrqugh which a molecule 
moves before ftqll^ing with another molecule is known as its meg/ 
freejQQtb. In terms of this theory, the pressure exerted by a gas 
is due to the combined effect of the impacts of the moving molecules 
upon the walls of the containing vessel, the magnitude of the 
pressure beinff dependent upon the kinetic energy of the mole* 
cules and their number^ 



GASES 



77 



Derivation of the Kinetic Equation. Starting with the assump- 
tions already made, it is possible to derive a formula by means of 
which the gas laws may be deduced. Imagine n molecules, each 
having a mass, w, confined within the cubical vessel shown in 
Fig. 15, the edge of which has a length, L While the different 
molecules are doubtless Mpdving with different velocities, there 
must be an average velocity for all of them. Let c denote this 
mean velocity of translation. The molecules will impinge upon 
tTRTwaJIiTin^ velocity of each may be resolved 

according to the well-known dy namdcalp^ com- 




Fig. 15. 



ponents, a, y and z, parallel .to-the 



and JL-JThe analytical expression for the velocity of a single 
molecule, M, is 

c 2 = a* + y* + z\ 

f jUJ 

In words, this means that the effect of the collision of the molecule 
upon the wall of the containing vessel, is equivalent to the com- 
bined effect of successive collisions of the molecule perpendicular 
to the three walls of the cubical vessel with the velocities x, y and 
z respectively! Fixing our attention upon the horizontal com- 
ponent, the molecule will collide with the wall with a velocity x, 
and owing to its perfect elasticity it will rebound with a velocity 



78 THEORETICAL CHEMISTRY 

X, having suffered no loss in kinetic energy;. The momentum 
before collision was mx and after collision it will be mx, the 
total change in momentum being 2 mx. The distance between 
the two walls being I, the number of collisions on a wall in unit 
time will be, x/l, and the total effect of a single molecule in one 
direction in unit time will be 2 mx^x/l = 2 mx 2 /l. The same 
reasoning is applicable to the other components, so that the com- 
bined action of a single molecule on the six sides of the vessel 
will be 

2 m , 9 . 9 . 9 . 2 me 2 



2 fYVYK? 

There being n molecules, the total effect will be -, . The 

entire inner surface of the cubical vessel being 6 P, the pressure p, 
on unit area, will be 



~ 6P ""3 I* ' 

but since P is the volume of the cube, which we will denote by v, 
we have 

1 mnc* 
P = 3 ' 
or 

pv = ^ wnc 2 . 

This is the fundamental equation of the kinetic theory of gases. 
While the equation has been derived for a cubical vessel, it is 
equally applicable to a vessel of any shape whatever, since the 
total volume may be considered to be made up of a large number 
of infinitesimally small cubes, for each of which the equation holds. 
Deductions from the Kinetic Equation. Law of Boyle. In 
the fundamental kinetic equation, pv = | mnc 2 , the right-hand 
side is composed of factors which are constant at constant temper- 
ature, and therefore the product, pv, must be constant also under 
similar conditions. This is clearly Boyle's law. 



GfASES 79 

Law of Gay-Lussac. The kinetic equation may be written in 
the form 

pv = 3 ' 2 mmj2 ' 

The kinetic energy of a single molecule being represented by 
1/2 me 2 , the total kinetic energy of the molecules of gas will 
be 1/2 mnc 2 . Therefore the product of the pressure and volume of 
the gas is equivalent to two-thirds of the kinetic energy of its molecules. 
A corollary to this proposition is that at constant pressure, the 
average kinetic energy of the molecules in equal volumes of different 
gases is the same. The law of Gay-Lussac teaches that at constant 
volume, the pressure of a gas is directly proportional to its abso- 
lute temperature. Taking this together with the fact that the 
pressure of a gas at constant volume is directly proportional to , 
the mean kinetic energy of its molecules, it follows that the mean 
kinetic energy of the molecules of a gas is directly proportional to its 
absolute temperature. Thus we see that the absolute temperature of 
a gas is a measure of the mean kinetic energy of its molecules. This 
deduction is partially based upon the experimentally-determined 
law of Gay-Lussac. Having obtained a definition of temperature 
in terms of kinetic energy, it is easy to derive Gay-Lussac's law 
from the fundamental kinetic equation. Writing the equation in 
the form 

2 1 



it is apparent that pv is directly proportional to the total kinetic 
energy of the gas molecules, or in other words, is directly propor- 
tional to its absolute temperature, which is the most general 
statement of Gay-Lussac's law. 

Hypothesis of Avogadro. If equal volumes of two different gases 
are measured under the^same pressure, we will have 



pv = 1/3 ftiWiCi 2 = 1/3 n 2 W2C2 2 , (1) 



where n\ and n 2 , mi and w 2 , and c\ and c 2 denote the number, mass 
and velocity of the molecules in the two gases. If the gases are 



80 THEORETICAL CHEMISTRY 

measured at the same temperature, the molecules of each possess 
the same mean kinetic energy, or 

1/2 mid 2 = 1/2W2C2 2 . (2) 

Dividing equation (1) by equation (2), we have 

HI = tt2, 

or under the same conditions of temperature and pressure equal 
volumes of the two gases contain the same number of molecules. 
This is the hypothesis of Avogadro. 

Law of Graham. If the fundamental kinetic equation be solved 
for c, we have 



mn 



but v/mn = 1/d, where d is the density of the gas, and therefore 
we may write 



If the pressure remains constant it is evident that the mean veloc- 
ities of the molecules of two gases are inversely proportional to 
the square roots of their densities, a law which was first enunciated 
by Graham in 1833 as the result of his experiments on gaseous 
diffusion. 

Mean Velocity of Translation of a Gaseous Molecule. By 
substituting appropriate values for the various magnitudes in the 
equation 



mn 

it is possible to calculate the mean velocity of the molecules of any 
gas. Thus, for the gram-molecule of hydrogen at and 76 cm. 
pressure, p = 76x13.59 = 1033.3 gr. per sq. cm. = 1033.3 X 980.6 
dynes per sq. cm., v = 22,400 cc., and mn = 2.016 gr. 
Substituting these values in the above equation we have, 



= /3 X 1033.3 X 980.6 X 22,400 _ mm cm 
V 2.016 



GASES 81 

Thus at the molecule of hydrogen moves with a speed slightly 
greater than one mile per second. This enormous speed is only 
attained along the mean free path, the frequent collisions with 
other molecules rendering the actual speed much less than that 
calculated. 

Equation of van der Waals. As has been pointed out in a 
previous paragraph, the gas laws are merely limiting laws and 
while they hold quite well up to pressures of about 2 atmospheres, 
above this pressure the differences between the observed and cal- 
culated values become steadily larger. In the case of hydrogen, 
Natterer was the first to show that the product of pressure and 
volume is invariably higher than it should be. A possible explana- 
tion of this departure from the gas laws was offered by Budde, 
who proposed that the volume, v, in the equation pv = RT, should 
be corrected for the volume occupied by the molecules them- 
selves. If this volume correction be denoted by 6, then the gas 
equation becomes 

p (v - 6) = RT, 

where b is a constant for each gas. Budde calculated the value 
of 6 for hydrogen and found it to remain constant for pressures 
varying from 1000 to 2800 meters of mercury. 

While Budde's modification of the gas equation is quite satis- 
factory in the case of hydrogen, it fails when applied to other gases. 
In general, the compressibility at low pressures is considerably 
greater than can be accounted for by Boyle's law. The compressi- 
bility reaches a minimum value, and then increases rapidly so that 
pv passes through the value required by the law. This suggests 
that there is some other correction to be applied in addition to 
the volume correction introduced into the gas equation by Budde. 
van der Waals pointed out in 1879, that in the deduction of Boyle's 
law by means of the fundamental kinetic equation, the tacit 
assumption is made that the molecules exert no mutual attraction. 
While this assumption is undoubtedly justifiable when the gas is 
subjected to a very low pressure, it no longer remains so when 
the gas is strongly compressed. A little consideration will make 
it apparent that wtien increased pressure is applied to a gas, the 



S2 THEORETICAL CHEMISTRY 

resulting volume will become less than that calculated, owing to 
molecular attraction. In other words the molecular attraction 
and the applied pressure act in the same direction and the gas 
behaves as if it were subjected to a pressure greater than that 
actually applied, van der Waals showed that this correction is 
inversely proportional to the square of the volume, and since it 
augments the applied pressure the expression p + a/v 2 is sub- 
stituted for p in the gas equation, a being the constant of molecular 
attraction. The corrected equation then becomes 

(p + A 2 ) (v - 6) = RT. 

This is known as the equation of van der Waals. It is applicable 
not only to strongly compressed gases, but also to liquids as well. 
While it will be given detailed consideration in a subsequent chapter, 
it may be of interest to point out at this time the satisfactory ex- 
planation which it offers of the experimental results of Amagat, 
to which we have already made reference, (page 72). When v is 
large, both 6 and a/v 2 become negligible, and van der Waals' 
equation reduces to the simple gas equation, pv = RT. We may 
predict, therefore, that any influence tending to, increase v will 
cause the gas to approach more nearly to the ideal condition. This 
is in accord with the results of Amagat's experiments, which show 
that an increase of temperature at constant pressure, or a diminu- 
tion of pressure at constant temperature, causes the gas to tend to 
follow the simple gas laws. The equation also offers a satisfactory 
explanation of the exceptional behavior of hydrogen when it is 
subjected to pressure. As we have seen, pv for all gases, except 
hydrogen, diminishes at first with increasing pressure, reaches a 
minimum value, and then increases regularly. Since the volume 
correction in van der Waals' equation acts in opposition to the 
attraction correction, it is apparent that at low pressures the effect 
of attraction preponderates, while at high pressures the volume 
correction is relatively of more importance. At some intermediate 
pressure the two corrections counterbalance each other, and it is 
at this point that the gas follows Boyle's law strictly. The 
exceptional behavior of hydrogen may be accounted for by making 
the very plausible assumption that the attraction correction is 



GASES 83 

negligible at all pressures in comparison with the volume correc- 
tion. 

Vapor Density and Molecular Weight. As has been pointed out 
in an earlier chapter, when a substance can be obtained in the gas- 
eous state, the determination of its molecular weight resolves itself 
into finding the mass of that volume of vapor which will occupy 
22.4 liters at and 76 cm. It is inconvenient to weigh a volume 
of gas or vapor under standard conditions of temperature and 
pressure, but by means of the gas laws the determination made 
at any temperature and under any pressure can be reduced to 
standard conditions. For example, suppose v cc. of gas are found 
to weigh w grams at t and p cm. pressure, then the weight in grams 
of 22.4 liters or 22,400 cc. at and 76 cm. will be given by the fol- 
lowing proportion, in which M denotes the molecular weight of the 
substance: 

pv _ , , 76 X 22,400 
Wl ~ M: 273 ' 



or 

^ X 76 X 22,400 X (t + 273) 

, , 273 pv 

1^ (x > i4 ^ 1+6C 

The ^termination of vapor density may be effected in either of 
two ways; (1) we may determine the mass of a known volume of 
vapor under definite conditions of temperature and pressure, or 
(2) we may determine the volume of a known mass under definite 
conditions of temperature and pressure. There are a variety of 
methods for the determination of vapor density; but for our pur- 
pose it will be necessary to describe but two of them. In the 
method of Regnault the mass of a definite volume of vapor is 
determined, while in the method due to Victor Meyer we measure 
the volume of a known mass. 

Method of Regnault. In this method which is especially adapted 
to permanent gases, use is made of two spherical glass bulbs 
(Fig. 16) of approximately the same capacity, each bulb being 
provided with a well-ground stop-cock. By means of an airpump 
one bulb is evacuated as completely as possible, and is then filled, 
at definite temperature and pressure, with the gas whose density 




84 THEORETICAL CHEMISTRY 

is~to be determined. The stop-cock is then closed and the bulb 
weighed, the second bulb being used as a counterpoise. The use 
of the second bulb is largely to avoid the 
troublesome corrections for air displacement 
and for moisture, each bulb being affected in 
the same way and to nearly the same extent. 
The volume of the bulb may be obtained by 
weighing it first evacuated, and then filled with 
distilled water at known temperature. From 
these results we may calculate the mass per unit 
of volume; or we may substitute the values of 
w, v, p and t in the above formula and calcu- 
late M, the molecular weight. This method 
was used by Morley * in his epoch-making re- 
search on the densities of hydrogen and oxygen. 
Method of Victor Meyer, In the method 
of Victor Meyer, a weighed amount of the lg * 

substance is vaporized, and the volume which it would have 
occupied at the temperature of the room and under existing 
barometric pressure is determined. The apparatus of Meyer, 
shown in Fig. 17, consists of an inner glass tube A, about 1 
cm. in diameter and 75 cm. in length. This tube is expanded 
into a bulb at the lower end, while at the top it is slightly en- 
larged and is furnished with two side tubes C and E. The tube 
A is suspended inside a heating jacket J3, containing some liquid 
the boiling point of which is about 20 higher than the vaporizing 
temperature of the substance whose vapor density is to be de- 
termined. The side tube E dips beneath the surface of water in a 
pneumatic trough G, and serves to convey the air displaced from 
A to the eudiometer F. By means of the side tube C, and the glass- 
rod Dy the small bulb containing the substance can be dropped to 
the bottom of A. To carry out a determination of vapor density 
with this apparatus, the liquid in B is heated to boiling and 
the sealed bulb V, containing a weighed amount of the substance, 
is placed in position on the rod D, the corks being tightly inserted. 

* Smithsonian Contributions to Knowledge, (1895). 



GASES 



85 



When bubbles of air cease to issue from E in the pneumatic trough, 
showing that the temperature within A is constant, the eudiometer 
F, full of water, is placed over the mouth of E, and the bulb V is 
allowed to drop by drawing aside the rod D. Air bubbles immedi- 
ately begin to issue from E and to collect in the eudiometer. When 
the air ceases to collect, the eudiometer is closed by the thumb and 




Fig. 17. 

is removed to a large cylinder of water where it is allowed to stand 
long enough to acquire the temperature of the room. It is then 
raised or lowered until the level of water inside and outside is 
the same, when the volume of air is carefully read off. In this 
method, the substance on vaporizing displaces an equal volume 
of air which is collected and measured, this observed volume being 



86 THEORETICAL CHEMISTRY 

that which the vapor would occupy after reduction to the condi- 
tions under which the air is measured. It is evident that in this 
method we do not require a knowledge of the temperature at 
which the substance vaporizes. Since the air is measured over 
water, the pressure to which it is subjected is that of the atmos- 
phere diminished by the vapor pressure of water at the temperature 
of the experiment. The method of calculating molecular weights 
from the observations recorded may be illustrated by the follow- 
ing example: 0.1 gram of benzene (C 6 H 6 ) was weighed out, and 
when vaporized, 32 cc. of air were collected over water at 17 and 
750 mm. pressure. The vapor pressure of water at 17 is 14.4 
mm., and the actual pressure exerted by the gas is 750 14.4 = 
735.6 mm. Substituting in the proportion 

pv .. 760X22,400 

" : + 273- M: 273 ' 

and solving for M we have 

M - 0-1 X 760 X 22,400 X (17 + 273) 

M ~ 273 X 735.6 X 32 "" ' b '*' 

The result agrees fairly well with the molecular weight of benzene 
(78.05) calculated from the formula. 

Unless a vapor follows the gas laws very closely, the value of the 
molecular weight obtained by the method of Victor Meyer will be 
only approximate, but this approximate value will be sufficiently 
near to the true molecular weight to enable us to choose between 
the simple formula weight, given by chemical analysis, and some 
multiple of it. 

Results of Vapor-Density Determinations. As the result of 
numerous vapor-density determinations extending over a wide 
range of temperatures, much important data has been collected 
concerning the number of atoms contained in the molecules of a 
large number of chemical compounds. The molecular weights 
of most of the elementary gases are double their atomic weights, 
showing that their molecules are diatomic. In like manner the 
molecular weights of mercury, zinc, cadmium and, in fact, all of 
the vaporizable metallic elements have been found to be identi- 
cal with their atomic weights. The molecules of sulphur, 



GASES 87 

arsenic, phosphorus and iodine are polyatomic, if they are not 
heated to too high a temperature. The investigations of Meyer 
and others have shown that the vapor densities of a large number 
of substances diminish as the temperature is increased. In other 
words as the temperature is raised the number of atoms contained 
in the molecules decreases. The molecular weight of sulphur, cal- 
culated from its vapor density at temperatures below 500, corre- 
sponds to the formula S 8 . If the vapor of sulphur is heated to 
1100, the molecular weight corresponds to the formula S 2 . In 
fact, sulphur in the form of vapor may be represented by the formu- 
las Sg, fi4, $2, or even S according to the temperature at which its 
vapor density is determined. Iodine behaves similarly, the mole- 
cules being diatomic between 200 and 600, while at temperatures 
above 1400 the vapor density has about one-half its value at the 
lower temperature, showing a complete breaking down of the dia- 
tomic molecules into single atoms. Heating to yet higher tem- 
peratures has failed to reveal any further decomposition. This 
phenomenon is not confined to the molecules of the elements alone, 
but is also met with in the case of the molecules of chemical com- 
pounds. The vapor density of arsenious oxide between 500 and 
700 corresponds to the formula As40 6 . As the temperature is 
raised, the vapor density becomes steadily smaller until, at 1800, the 
calculated molecular weight corresponds to the formula As 2 03. In 
like manner ferric and aluminium chlorides have been shown to 
have molecular weights at low temperatures corresponding to the 
formulas, Fe2Cl 6 and A1 2 C1 6 . The commonly-used formulas, FeCk 
and A1C1 3 , represent their molecular weights at high temperatures 
only. The experimental difficulties attending vapor density de- 
terminations increase as the temperature is raised, owing chiefly to 
the deformation of the apparatus when the material of which it is 
constructed approaches its melting point. Glass which can be used 
at relatively low temperatures only, has been replaced by specially 
resistant varieties of porcelain which may be used up to tempera- 
tures of 1500 or 1600. Platinum vessels retain their shape up to 
temperatures between 1700 and 1800. Measurements up to 
2000 have recently been effected by Nernst and his pupils.* In 
* Wartenberg. Zeit. anorg. Chem., 56, 320 (1907). 



88 THEORETICAL CHEMISTRY 

their experiments use was made of a vessel of iridium, the inside 
and outside of which was surrounded with a cement of magnesia 
and magnesium chloride, the entire apparatus being heated electri- 
cally. With this apparatus they showed that the molecular 
weight of sulphur between 1800 and 2000 is 48, indicating that 
the diatomic molecule is approximately 50 per cent broken down 
into single atoms. 

Abnormal Vapor Densities. In all of the cases cited above 
the molecular weight calculated from the vapor density corre- 
sponds either with the simple formula weight, as determined by 
chemical analysis, or with a multiple thereof. In no case is there 
any evidence of a breaking down of the simple molecule into its 
constituents. Substances are known, however, the molecular 
weights of which, calculated from their vapor densities, are less 
than the sum of the atomic weights of their constituents. For 
example, tne vapor density of ammonium chloride was found to 
be 0.89, while that corresponding to the formula NH 4 C1 should be 
1.89. Similar results have been obtained with phosphorus penta- 
chloride, nitrogen peroxide, chloral hydrate and numerous other 
substances. The phenomenon can be explained in either of the two 
following ways: (1) that the molecule has undergone a complete 
disruption, or (2) that the substance does not follow the law of 
Avogadro. Until the former explanation was shown to be correct, 
the latter was accepted and for a time the law of Avogadro fell into 
disrepute. In 1857, Deville showed that numerous chemical com- 
pounds are broken down or " dissociated " at high temperatures. 
Shortly afterward Kopp suggested that the abnormal vapor 
densities of such substances as ammonium chloride, phosphorus 
pentachloride, etc., might be due to thermal dissociation. If 
ammonium chloride underwent complete dissociation, one molecule 
of the salt would yield one molecule of ammonia and one molecule 
of hydrochloric acid gas, and the vapor density of the resulting 
mixture would be one-half of that of the undissociated substance, 
a deduction in complete agreement with the results of experiment. 
It remained to prove that the products of this supposed dissocia- 
tion were actually present. 

The first to offer an experimental demonstration of the simul- 



GASES 



89 



taneous formation of ammonia and hydrochloric acid, when ammon- 
ium chloride is heated, was PebaL* The apparatus which he de- 
vised for this purpose is shown in Fig. 18. It consisted of two tubes 
T and t, the latter being placed within the former as indicated in 
the sketch. Near the top of the inner tube, which was drawn down 
to a smaller diameter, was a porous plug of asbestos, C, upon which 
was placed a little ammonium chloride. A stream of dry hydro- 




Rydrogen 



Hydrogen 




Fig. 18 



gen was passed through the apparatus by means of the tubes A and 
B, the former entering the outer tube and the latter the inner 
tube. The entire apparatus was heated to a temperature above 
that necessary to vaporize the ammonium chloride. If the salt 
undergoes dissociation into ammonia and hydrochloric acid, the 
former being less dense than the latter, would diffuse more 
rapidly through the plug C and the vapor below the plug would 

* Lieb,, Ann., 123, 199 (1862). 



90 



THEORETICAL CHEMISTRY 



be relatively richer in ammonia than the vapor above it. The 
current of hydrogen through jB would therefore sweep out from 
the lower part of t an excess of ammonia, while the current through 
A would carry out from T an excess of hydrochloric acid. By 
holding strips of moistened litmus paper in the currents of gas 
issuing from E and F, it was possible for Pebal to test the correct- 
ness of Kopp's idea. He found that the gas issuing from E had 
an acid reaction while that escaping from F had an alkaline reac- 
tion. It would at first sight appear that Pebal had demonstrated 



Nitrogen 




Fig. 19. 



beyond question that ammonium chloride undergoes dissociation 
into ammonia and hydrochloric acid. 

It was pointed out, however,*that Pebal had heated the ammon- 
ium chloride in contact with a foreign substance, asbestos, and 
that this might have acted as a catalyst, promoting the decomposi- 
tion into ammonia and hydrochloric acid. This objection was 
removed by the ingenious experiment of Than.* He devised a 
modification of Pebal's apparatus, as shown in Fig. 19. In the 
horizontal tube, AB, the ammonium chloride was placed at F and a 

* Lieb. Ann., 131, 129 (1864). 



GASES 91 

porous plug of compressed ammonium chloride was introduced at 
G. The tube was heated and nitrogen passed in at C. The 
reactions of the currents of gas issuing at D and E were tested 
with litmus as in Pebal's experiment and it was found that the 
gas escaping from D was alkaline, while that issuing from E was 
acid. This experiment proved beyond question that the vapor 
of ammonium chloride is thermally dissociated into ammonia 
and hydrochloric acid. Experiments on other substances whose 
vapor densities are abnormally small show that a similar explan- 
ation is applicable, and thus furnish a confirmation of the law of 
Avogadro. 

Calculation of the Degree of Dissociation. Since the density 
of a dissociating vapor decreases with increase in temperature, 
it is important to be able to calculate the degree of dissociation at 
any one temperature. This is clearly equivalent to ascertaining 
the extent to which the reaction 



has proceeded from left to right. This can be determined easily 
from the relation of vapor density to dissociation. If we start 
with one molecule of gas and let a represent the percentage dis- 
sociation, then 1 a will denote the percentage remaining un- 
dissociated. If one molecule of gas yields n molecules of gaseous 
products, the total number of molecules present at any time will 
be 

(1 - a) + na = 1 + (n - 1) a. 

The ratio 1 : 1 + (n 1) a will be the same as the ratio of the 
density cfe of the dissociated gas to its density in the undissociated 
state do, or 

1 : 1 + (n - 1) a = 4 : A; 

solving this proportion for a, we have 



a = 



(n - 1) d* 

The vapor density of nitrogen peroxide has been measured by E. 

and L. Natanson,* and the degree of dissociation at the different 

* Wied. Ann., 24, 454 (1885); 27, 606 (1886). 



92 



THEORETICAL CHEMISTRY 



temperatures calculated by means of the preceding formula. The 
following table gives their results. 

The course of the dissociation is shown in the accompanying 
illustration, Fig. 20, in which the abscissae represent temperature 
and the ordinates, percentage dissociation. It will be observed 




80 100 

Temperature 

Fig. 20. 

that the dissociation of nitrogen peroxide is at first nearly pro- 
portional to the temperature. It then increases more rapidly 
until, when about four-fifths of the molecules of N 2 4 are broken 
down, the dissociation proceeds slowly to completion. 

Specific Heat. The addition of heat energy to a body causes 
its temperature to rise. The ratio of the amount of heat supplied 
to the resulting rise in temperature is called the heat capacity of 
the body; obviously its value is dependent upon the initial temper- 



GASES 



93 



DISSOCIATION OF NITROGEN PEROXIDE, N 2 O 4 . 

ATMOSPHERIC PRESSURE. 
(Density of N 2 O 4 =3.18; of NO 2 +NO 2 = 1.59; of air = 1.00.) 



Temperature, 
(degrees) 


Density of Gas. 


Percentage Dis- 
sociation. 


26.7 


2.65 


19 96 


35.4 


2 53 


25 65 


39 8 


2 46 


29 23 


49 6 


2 27 


40 04 


60.2 


2 08 


52 84 


70 


1 92 


65 57 


80 6 


1 80 


76 61 


90 


1 72 


84 83 


100 1 


1 68 


89 23 


111 3 


1 65 


92 67 


121 5 


1 62 


96 23 


135 


1.60 


98 69 


154 


1 58 


100 00 



ature of the body. The specific heat of a substance may be defined 
as the heat capacity of unit mass of the substance. If dt represents 
the increment of temperature due to the addition of dQ units of 
heat energy to m grams of any substance, then its specific heat, c, 
will be given by the equation 



.. 
m dt 

Specific Heat at Constant Pressure and Constant Volume. It 

is well known that the specific heat of a gas depends upon the 
conditions under which it is determined. If a definite mass of 
gas is heated under constant pressure, the value of the specific 
heat, c p , is different from the value of the specific heat, c v , ob- 
tained when the pressure varies and the volume remains con- 
stant. The value of c p is invariably greater than that of c v . 
When heat is supplied to a gas at constant pressure not only does 
its temperature rise, but it also expands, and thus does external 
work. On the other hand, if the gas be heated in such a way that 
its volume cannot change, none of the heat supplied will be used 
in doing external work, and consequently its heat capacity will 



94 THEORETICAL CHEMISTRY 

be less than when it is heated under constant pressure. The 
recognition by Mayer in 1841 of the cause of this difference between 
the two specific heats of a gas led him to his celebrated calculation 
of the mechanical equivalent of heat, and the enunciation of the 
first law of thermodynamics. Mayer observed that the differ- 
ence between the quantity of heat necessary to raise the temper- 
ature of 1 gram of air 1 C. at constant pressure, and at constant 
volume respectively, was 0.0692 calorie, or 
c p - c v = 0.0692 cal. 

That is to say, 0.0692 calorie is the amount of heat energy which 
is equivalent to the work Squired to expand 1 gram of air 1/273 
of its volume at 0. Imagine 1 gram of air at enclosed within 
a cylinder having a cross-section of one square centimeter, and 
furnished with a movable, frictionless piston. Since 1 gram of 
air under standard conditions of temperature and pressure occu- 
pies 773.3 cc., the distance between the piston and the bottom of 
the cylinder will be 773.3 cm. If the temperature be raised from 
to 1, the piston will rise 1/273 X 773.3 = 2.83 cm., and since 
the pressure of the atmosphere is 1033.3 grams per square centi- 
meter, the external work done by the expanding gas will be 

1033.3 X 2.83 = 2924.3 gr. cm. 

This is evidently equivalent to 0.0692 calorie and therefore, the 
equivalent of 1 calorie in mechanical units, J, will be 

0004. Q 



a value agreeing very well with the best recent determinations of 
the mechanical equivalent of heat. 

The difference between the two specific heats may be easily 
calculated in calories from the fundamental gas equation. Start- 
ing with 1 mol. of gas, and remembering that when a gas expands 
at constant pressure, the product of pressure and change in volume 
is a measure of the work done, we have, at temperature TV , 



where Vi is the molecular volume. Raising the temperature to 



GASES 95 

T 2 , the corresponding molecular volume being t; 2 , we have for 
the work done during expansion 



If r 2 TI = 1, then the equation reduces to 

p (^2 - vi) = R. 

Since the difference between the molecular heats * at constant 
pressure and constant volume is equivalent to the external work 
involved when the temperature of 1 mol. of gas is raised 1, we have 

M (c p - c v ) = p (v 2 - t>i), 

where M is the molecular weight of the gas; and therefore 
M (c p c v ) = R = 2 calories. 

In words, the difference of the molecular heats of any gas at 
constant pressure and at constant volume is 2 calories. The 
specific heat of a gas at constant pressure can be readily deter- 
mined, by passing a definite volume of the gas, heated under con- 
stant pressure to a known temperature, through the worm of a 
calorimeter at such a rate that a constant difference is maintained 
between the temperature of the entering and the temperature of 
the escaping gas. Thus the number of calories which causes a 
definite thermal change in a certain volume of the gas is deter- 
mined, and from this it is an easy matter to calculate the specific 
heat, c p . The molecular heat at constant pressure for all gases 
approaches the limiting value, 6.5, at the absolute zero. This 
relation, due to Le Chateiier, may be expressed thus, 

Mcp = 6.5 + aT, 

where a is a constant for each gas. The value of a tor hydrogen, 
oxygen, nitrogen and carbon monoxide is 0.001, for ammonia, 
0.0071 and for carbon dioxide, 0.0084. As the complexity of the 
gas increases the value of a becomes numerically greater. 

The experimental determination of the specific heat of a gas at 
constant volume is difficult and the results obtained are not 
trustworthy. The chief cause of the inaccuracy of the results 

* The molecular heat of a gas is equal to the product of its specific heat and its 
molecular weight. 



96 



THEORETICAL CHEMISTRY 



is that the vessel containing the gas absorbs so much more heat 
than the gas itself that the correction is many times larger than 
the quantity to be measured. The specific heat at constant vol- 
ume is almost always obtained by indirect methods, as for example 
by means of the preceding formula 

M (c p - c v ) = R = 2 cal., 

in which the values of M and c p are known. 

The molecular heats of some of the commoner gases and vapors 
are given in the subjoined table together with the ratio c p /c v . 

MOLECULAR SPECIFIC HEATS. 



Gas. 


Mc p 


Me. 


<>/<= 7 


Argon 






1 66 


Helium 






1 66 


Mercury 






1 66 


Hydrogen 


6 88 


4 88 


1 41 


Oxygen 


6 96 


4 96 


1 40 


Nitrogen 


6 93 


4 93 


1 41 


Chlorine 


8 58 


6,58 


1 30 


Bromine 


8 88 


6 88 


1 29 


Nitric oxide 


6 95 


4 95 


1 40 


Carbon monoxide 


6 86 


4 86 


1 41 


Hydrochloric acid 


6 68 


4 68 


1.43 


Carbon dioxide 


9 55 


7 55 


1 26 


Nitrous oxide 
Water 


9 94 
8 65 


7 94 
6 65 


1 25 

1 28 


Sulphur dioxide 


9 88 


7.88 


1 25 


Ozone 






1 29 


Ether ... 


35 51 


33.51 


1 06 











The Ratio of the Two Specific Heats. There are two methods 
by which the ratio c p /c v can be determined directly, one due to 
Clement and Desorrnes * and the other due to Kundt.f 

Method of Clement and Desormes. The apparatus devised by 
these investigators consists, as is shown in Fig. 21, of a glass 
balloon flask, A, of about 20 liters capacity, furnished with two 
stop-cocks, D and 13, and a manometer, C. The stop-cock D 
has an aperture nearly as large as the diameter of the neck of the 

* Jour, de phys., 89, 321, 428 (1819). 

t Pogg. Ann., 128, 497 (1866); 135, 337, 527 (1868). 



GASES 



97 



flask, B. To determine the ratio of the two specific heats, the 
flask is filled with the gas under a pressure slightly greater than 
barometric pressure. The manometer C serves to measure the 
pressure of the gas within A. After the value of the pressure 
has been read on the manometer, the stop-cock D is opened 
momentarily to the air, thus permitting the pressure of the gas 
to fall adiabatically to that of the atmosphere. The stop-cock 




Fig. 21. 

is then closed and the flask is allowed to stand for a few moments 
until its contents, which has cooled by adiabatic expansion, has 
regained the temperature of the room. The pressure on the 
manometer is then observed. Let the initial pressure of the gas 
be denoted by p Q , and atmospheric pressure by P. If the initial 
and final specific volumes are denoted by V Q and v\, then for an 
adiabatic process, we have 

P = /voV 

Po W ' 



98 THEORETICAL CHEMISTRY 

The value of the final specific volume is determined from the 
final pressure, pi, by an application of Boyle's law, the pressure pi 
being developed isothermally. 
Thus, 

VjO^Pl^ 
Vi po' 

and consequently, 



or 

= logP- logpo 
log pi- log po* 

Method of Kundt. According to the formula of Laplace for the 
velocity of transmission of a sound wave in a gas, we have 



in which p and d denote the pressure and density of the gas, and 
7 is the ratio of the two specific heats. If the wave velocities in 
two different gases, whose densities are d\ and cfe under the same 
conditions of temperature and pressure, be denoted by v\ and v z , 
we may write 





or replacing the densities of these gases by their respective molec- 
ular weights, MI and M 2 , we have 

TI/T 

(D 

The ratio of the velocities of the two waves can be measured by 
means of the apparatus shown in Fig. 22. A wide glass tube 
about \\ meters in length is furnished with two side tubes, E and 
F. Into one end of the tube is inserted the glass rod BD which 
is clamped at its middle point by a tightly fitting cork, C. The 
other end of the tube is closed by means of the plunger A. A 
small amount of lycopodium powder is placed upon the bottom of 



GASES 99 

the tube and is distributed uniformly by gently tapping the walls 
of the tube. The gas in which the velocity of the sound wave 
is to be determined is introduced into the tube through JB, and 
the displaced air escapes at F. When the tube is filled, E and F 
are closed by means of rubber caps, and a piece of moistened 
chamois leather is drawn along BD causing it to vibrate longitudi- 
nally and to emit a shrill note. The vibrations are taken up by the 




Fig. 22. 

gas in the tube and the powder arranges itself in a series of heaps 
corresponding to the nodes of vibration. If the nodes are not 
sharply defined, then A should be moved in or out until they 
become so. If Xi is the distance between two heaps or nodes, 
then 2 Xi will be the wave length of the note emitted by the rod 
BD, and if n represents the number of vibrations per second of 
the note emitted, we have for the velocity of sound in the gas 



Similarly if a second gas be introduced into the tube we shall 
have 



Therefore, 

? - r 

v t X 2 
Substituting in equation (1), we have 




or 



If the second gas is air, as is usually the case, 72 = 1.405 and M 2 = 
28.74, (mol. wt. of hydrogen -5- density of hydrogen referred to air, 
or 2 * 0.0696 = 28,74) or equation (3) becomes 



100 THEORETICAL CHEMISTRY 

1 A f\ K Xl -M- 1 

^ = L405 V > 2^7i- 

Thus, 7 for any gas can be determined by this method provided 
we know its value for another gas of known molecular weight. 

Specific Heat of Gases and the Kinetic Theory. In terms of 
the kinetic theory, the energy of a gas may be considered to be 
made up of three parts: (1) the translatioiial energy of the mole- 
cules, commonly termed their kinetic energy, (2) the intramolec- 
ular kinetic energy, and (3) the potential energy due to inter- 
atomic attraction within the molecules. When a gas is heated 
at constant volume all three of these factors of the total energy 
of the molecule may be affected. It is fair to assume, however, 
that when a monatomic gas, such as mercury vapor, is heated, 
all of the heat energy supplied is used to augment the translational 
kinetic energy of the molecules. As we have seen, the fundamental 

kinetic equation 

pv = 1/3 nmc 2 
may be written 

pv = 2/3 -1/2 nmc 2 , 

and since 1/2 nmc? represents the total kinetic energy of the gas, 

we have 

pv = 2/3 kinetic energy of 1 mol., 
or 

kinetic energy of 1 mol. = 3/2 pv. 

But pv = 2 T calories, therefore 

kinetic energy of 1 mol. = 3 T cal. 

The kinetic energy of a constant volume of any gas at the temper- 
tures TI and T 2 , is given by the following equations: 

3/2 2W = 3T 7 !, (1) 

and 

3/2 p 2 v = 3 !F 2 . (2) 

Subtracting (1) from (2) we obtain 

3/2 (ft - pi) t> = 3 (T 2 - TO, (3) 

and for an increase in temperature of 1, (3) becomes 
3/2 (p2 - Pi) v = 3 cal. 



GASES 101 

The molecular kinetic energy of one mole of a monatomic gas 
at constant volume is thus increased by 3 calories for each degree 
rise in temperature. As has already been shown, 

M (c p - c,) = 2 cal, 
therefore, since Mc v = 3 calories, Mc p = 3 + 2 = 5 calories, and 

/t/ r ^\ 
*, 1KiC P ~ 1 Afi 

7 ~ Mc v ~ 3 " Lbb * 

This value of 7 is in perfect agreement with the results of the 
experiments on mercury vapor which is known to be monatomic. 
The converse of this method has been employed by Ramsay to 
prove that the rare gases of the atmosphere are monatomic, the 
value of 7 for all of these gases being 1.66. In the case of poly- 
atomic molecules the heat energy supplied is not only used in 
increasing their translational kinetic energy, but also in the 
performance of work within the molecule. The value of the 
internal work is indeterminate, but it is without doubt constant 
for any one gas. If the internal work be represented by a, then 
the value of the ratio of the two specific heats will be 

7 = ^ = |+|< UK >l. 

Reference to the table on p. 96, giving the value of 7 for differ- 
ent gases, will show that this deduction from the kinetic theory 
is in perfect agreement with the experimental facts. With 
increasing complexity of the molecule, it is apparent that the 
amount of heat expended in doing internal work should increase, 
and therefore the specific heat should increase also. Inspection 
of the table confirms this deduction. The specific heat of mona- 
tomic gases is independent of the temperature while the specific 
heat of polyatomic gases increases slightly. These results may 
justly be regarded as among the greatest triumphs of the kinetic 
theory of gases. 



102 THEORETICAL CHEMISTRY 



PROBLEMS. 

1. The volume of a quantity of gas is measured when the barometer 
stands at 72 cm., and is found to be 646 cc.: what would its volume be 
at normal pressure? Ans. 612 cc. 

2. At what pressure would the gas in the preceding problem have a 
volume of 580 cc.? Ans. 80.19 cm. 

3. A certain quantity of oxygen occupies a volume of 300 cc. at 0: 
find its volume at 91. Ans. 400 cc. 

4. The weight of a liter of air under standard conditions is 1.293 grams: 
to what temperature must the air be heated so that it may weigh exactly 
1 gram per liter? Ans. 79.99. 

5. At what temperature will the volume of a given mass of gas be 
exactly double what it is at 17? Ans. 307. 

6. On heating a certain quantity of mercuric oxjde it is found to give 
off 380 cc. of oxygen, the temperature being^Fjand the barometric 
height 74 cm.; what would be the volume of tHe gas under standard 
conditions? Ans. 341.25 cc. 

7. A liter of air weighs 1.293 grains under standard conditions. At 
what temperature will a liter of air weigh 1 gram, the pressure being 72 cm.? 

Ans. 61.43. 

8. A quantity of air at atmospheric pressure and at a temperature of 
7 is compressed until its volume is reduced to one-seventh, the temper- 
ature rising 20 during the process: find the pressure at the end of the 
operation. Ans. 7.3 atmos. 

9. The weight of a liter of nitrogen under standard conditions is 1.2579 
grams. Calculate the specific gas constant, r. Ans. 3007 gr. cm. 

10. The time of outflow of a gas is 21.4 minutes, the corresponding 
time for hydrogen is 5.6 minutes. Find the molecular weight of the gas. 

Ans. 29.2. 

11. Calculate the molecular weight of chloroform from the following 
data: 

Weight of chloroform taken 0.220 gr. 

Volume of air collected over water 45.0 cc. 

Temperature of air 20 

Barometric pressure 755.0 mm. 

Pressure of aqueous vapor at 20 17.4 mm. 

Ans. 121.1. 



GASES 103 

12. The density of a gas is 0.23 referred to mercury vapor. What is 
its molecular weight? Ans. 46. 

13. Phosphorus pentachloride dissociates according to the equation 



The molecular weight PCl is 208.28. At 182 the density is 73.5 and 
at 203 it is 62. Find the degree of dissociation at the two temperatures. 

Ans. ai82 = 0.417, oW = 0.68. 

14. The specific heat at constant volume for argon is 0.075, and its 
molecular weight is 40. How many atoms are there in the molecule? 

Ans. 1. 

15. What is the specific heat of carbon dioxide at constant volume, its 
molecular weight being 44 and the temperature 50. Ans. 0.164. 

16. The specific heat for constant pressure of benzene is 0.295: what is 
the specific heat for constant volume? Ans. 0.27, 



CHAPTER VII. 
LIQUIDS. 

General Characteristics of Liquids. The most marked char- 
acteristic of the liquid state is that a given mass of liquid has a 
definite volume but no definite form. The volume of a liquid is 
dependent upon temperature and pressure but to a much smaller 
degree than is the volume of a gas. The formulas in which the vol- 
ume of a liquid is expressed as a function of temperature and pres- 
sure are largely empirical, and contain constants dependent upon 
the nature of the liquid. This is undoubtedly due to the fact that 
in the liquid state the molecules are much less mobile than in the 
gaseous state. The distance between contiguous molecules being 
much less in liquids than in gases, the mutual attraction is increased 
while the mobility is correspondingly diminished. That liquids 
represent a more condensed form of matter than gases is shown 
by the change in volume which results when a liquid is vaporized: 
thus, 1 cc. of water at the boiling point when vaporized at the same 
temperature occupies a volume of about 1700 cc. A liquid con- 
tains less energy than a gas, since energy is always required to 
transform it into the gaseous state. Since gases can be liquefied 
by increasing the pressure and lowering the temperature, and since 
liquids can be vaporized by lowering the pressure and increasing 
the temperature, it is apparent that there is no generic difference 
between the two states of matter. 

Connection Between the Gaseous and Liquid States. If a gas 
is compressed isothermally, its state may change in either of two 
ways depending upon the temperature: (1) The volume at first 
diminishes more rapidly than the pressure increases, then in the 
same ratio and lastly more slowly. When the pressure attains a 
very high value the volume is but slightly altered. This case 
has already been considered in the preceding chapter. (2) The 
volume changes more rapidly than the pressure until, when a cer- 

104 



LIQUIDS 105 

tain pressure is reached, the gas ceases to be homogeneous, partial 
liquefaction resulting. For a constant temperature, the pressure at 
which liquefaction occurs is invariable for a given gas, while the vol- 
ume steadily diminishes until liquefaction is complete. Only when 
the whole mass of gas has been liquefied is it possible to increase 
the pressure and then, owing to the small compressibility of liquids, 
a large increase in pressure is required to produce a slight dimin- 
ution in volume. If the temperature is above a certain point, 
dependent upon the nature of the gas, the phenomena of com- 
pression will follow (1) ; if below this point, the process will follow 
(2). That a gas may behave in either of the above ways was 
first clearly recognized by Andrews* in 1869, in connection with 
his experiments on the liquefaction of carbon dioxide. He found 
that if carbon dioxide was compressed, keeping the temperature 
at 0, the volume changes more rapidly than the pressure, lique- 
faction resulting when a pressure of 35.4 atmospheres was reached. 
As the temperature was raised, he found that a higher pressure 
was required to liquefy the gas, until at temperatures above 
30.92 it was no longer possible to condense the gas to the liquid 
state. The temperature above which it was no longer possible 
to liquefy the gas he termed the critical temperature. In like 
manner the pressure required to liquefy the gas at the critical 
temperature, he termed the critical pressure, and the volume 
occupied by the gas or the liquid under these conditions he called 
the critical volume. 

Isothermals of Carbon Dioxide. The results of Andrew's 
experiments f on the liquefaction of carbon dioxide are shown in 
Fig. 23, in which the ordinates represent pressures and the abscissae 
the corresponding volumes at constant temperature. The curves 
obtained by plotting volumes against pressures at constant 
temperatures are called isothermals. For a gas which follows 
Boyle's law, the isothermals will be a series of equilateral hy- 
perbolas. This condition is approximately fulfilled by air, for 
which three isothermals are given in the diagram. At 48. 1 the 
isothermal for carbon dioxide is nearly hyperbolic, but as the 

* Trans. Roy. Soc. 159, 583 (1869). 
t loc. cit. 



106 



THEORETICAL CHEMISTRY 



temperature becomes lower, the isothermals deviate more and more 
from those for an ideal gas. At the critical temperature, 30.92, 



Carbon Dioxide 



Air 




Volume 
Fig. 23. 



the curve is almost horizontal for a short distance, showing that for 
a very slight change in pressure there is an enormous shrinkage 
in volume. At still lower temperatures, 21.l and 13. 1, the 



LIQUIDS 107 

horizontal portions of the curves are much more pronounced, 
indicating that during liquefaction there is no change in pressure. 
When liquefaction is complete the curves rise abruptly, showing 
that the change in volume is extremely small for a large increase 
in pressure; in other words the liquefied gas possesses a small 
coefficient of compressibility. At any point within the parabolic 
area, indicated by the dotted line ABC, both vapor and liquid are 
coexistent; at any point outside, only one form of matter, either 
liquid or vapor, is present. Andrew's experiment show that 
there is no fundamental difference between a gas and a liquid. 
It is apparent from the diagram that when carbon dioxide is sub- 
jected to great pressures above its critical temperature it behaves 
more like a liquid than a gas, in fact it is difficult to determine 
whether a highly compressed gas above its critical temperature 
should be classified as a gas or as a liquid. 

Van der Waals' Equation and the Continuity of the Gaseous 
and Liquid States. In the preceding chapter we have learned 
that the fundamental gas equation 

pv = RT 

is only strictly applicable to an ideal gas, and that the behavior 
of actual gases is represented with considerable accuracy, even at 
high pressures, by the equation of van der Waals, 



If this equation be arranged in descending powers of v, we have 

3 .A.. RT \ . a ab n n\ 

z> 3 v 2 [ b H -- ) + v --- = 0. (1) 

\PIPP 

This being a cubic equation has three possible solutions, each 
value of p affording three corresponding values of v; a, 6, R and T 
being treated as constants. The three roots of this equation are 
either all real, or one is real and two are imaginary, depending upon 
the values of the constants. That is to say, at one temperature 
and pressure the values of a and b may be such, that v has three 
real values, while at another temperature and pressure, v may 
have one real and two imaginary values. In the accompanying 



108 



THEORETICAL CHEMISTRY 



diagram, Fig. 2$ I I#, series of graphs of the equation for different 
values of T is given. It will be observed that these curves bear 
a striking resemblance to the isotherms of carbon dioxide estab- 
lished by the experiments of Andrews. In the case of the theo- 




Volume 
Fig. 24. 

retical curves there are no sudden breaks such as appear in the 
actual discontinuous passage from the gaseous to the liquid state. 
Instead of passing from B to D along the wavelike path BaCbD, 
experiment has shown that the substance passes directly from the 
state B to the state D along the straight line BD. It is here 



LIQUIDS 109 

that van der Waals' equation fails to apply. As has been pointed 
out the substance between these two points is not homogeneous, 
being partly gaseous and partly liquid. Attempts have been 
made to realize the portion of the curve BaCbD experimentally. 
By studying supersaturated vapors and superheated liquids it 
has been found possible to follow the theoretical curve for short 
distances between B and D without discontinuity, but owing to 
the instability of the substance in this region, it is evident that 
the complete isothermal and continuous transformation of a gas 
into a liquid can never be effected, van der Waals has called 
attention to the fact that in the surface layer of a liquid, where 
unique conditions prevail, it is quite possible that such unstable 
states may exist, and that there the transition from liquid to gas 
may in reality be a continuous process. The diagram shows that 
as T increases, the wave-like portion of the isothermals becomes 
less pronounced and eventually disappears, when the points jB, 
C and D coalesce. At this point the three roots of the equa- 
tion become equal, the volume of the liquid becoming identical 
with the volume of the gas. The substance at this point is in the 
critical condition. Since under these conditions the three roots 
of the equation 

3 L i RT \ 2 i a ab A m 

v 3 { b H -- 1 v 2 + - v -- =0 (1) 

\ p / p p 

are equal, we may write #1 = # 2 = v 3 = v c , the subscript c indicat- 

ing the critical state. Then equation (1) must be equivalent to 

(v - Vc )3 = v * - 3 V jp + 3 V fy - Vc * = o. (2) 

Equating the corresponding coefficient of equations (1) and (2), 
we have 



8*- (4) 

and v c z = (5) 

PC 

Dividing equation (5) by equation (4), we have 

*> c = 36, (6) 



110 THEORETICAL CHEMISTRY 

and substituting this value in equation (4), we obtain 



Lastly, substituting the values of v c and p c , given in equations 
(6) and (7), in equation (3), we have 

T c = 8a (8) 

Therefore, 



"1 R - (9) 

Or expressing the constants a, b and R in terms of the critical 
values of pressure, temperature and volume, we have 

a = 3 p c v*, (10) 

6 = \ (11) 

and R = |^- (12) 

By means of equations (6), (7) and (8) it is possible to calculate 
the critical constants of a gas when the constants a and b of van 
der Waals' equation are known. If we take carbon dioxide as an 
example, for which a = 0.00874 and b = 0.0023 we obtain 
v c = 0.0069 (observed value = 0.0066), p c = 61 atmospheres, 
(observed value = 70 atmospheres), TV = 305.5 abs. (observed 
value 303.9 abs.) Conversely by means of equations (10) and 
(11) the value of a and 6 can be calculated when the critical data 
are given. 

Corresponding Conditions. If in the equation of van der 
Waals, the values of p, v and T be expressed as fractions of the 
corresponding critical values, we may write 

V = 0V C 

and 

T = yT c . 



LIQUIDS 111 

Substituting these values in the equation 



we have 

(p. + 7srH)G&-*>-yZ I . > 



and replacing p c , v c , and T c by their values given in equations (6), 
(7) and (8) of the preceding paragraph, we obtain 



which is van der Waals' reduced equation of condition. 

In this equation everything connected with the individual 
nature of the substance has vanished, thus making it applicable 
to all substances in the liquid or gaseous state in the same way 
that the fundamental gas equation is applicable to all gases irre- 
spective of their specific nature. It has been shown, however, 
that the equation is not entirely trustworthy and at best can be 
considered as little more than a rough approximation. The 
chief point to be observed in connection with this equation is 
that whereas for gases, the corresponding values of temperature, 
pressure and volume, measured in the ordinary units, may be 
compared, it is necessary in the case of liquids to make the com- 
parison under corresponding conditions. For example, the molec- 
ular volumes of two liquids are to be compared, not at room 
temperature but at temperatures which are equal fractions of 
their respective critical temperatures. Such temperatures van 
der Waals called corresponding temperatures. 

By way of illustration, suppose we wish to compare alcohol 
and ether with respect to some particular property, such as 
surface tension. If the surface tension of alcohol be measured 
at 60, at what temperature must a similar measurement be 
made with ether in order that the results may be comparable? 
The critical temperature of alcohol is 243 C. or 516 absolute; 
that of ether is 194 C, or 467 absolute. Then according to van 
der Waals' definition of corresponding conditions, the tempei^ture, 



112 



THEORETICAL CHEMISTRY 



t, at which measurements should be made with ether will be given 
by the proportion, 273 + t : 467 :: 273 + 60 : 516, or t = 28 C. 
By making comparisons of various properties at corresponding 
temperatures it has been found that greater regularities are 
observed than when comparisons are made at the same tempera- 
ture, thus justifying the claim of van der Waals. 

Liquefaction of Gases. The history of the liquefaction of 
gases has for a long time been regarded as one of the most 
interesting chapters of physical science. Among the first success* 




Kg. 25. 

ful workers in this field was Faraday.* He liquefied practically 
all of the gases which condense under moderate pressures and at 
not very low temperatures. A sketch of the apparatus used by 
Faraday is shown in Fig. 25. It consisted of an inverted 
V-shaped tube, in one end of which was placed some solid which 
would liberate the desired gas on heating, while the other end 
was sealed and immersed in a freezing mixture. When the sub- 
stance at A had been heated long enough to liberate considerable 
gas, the pressure within the tube became sufficiently high to cause 
the gas to liquefy at the temperature of the end B. Thus chlorine 

* PhU. Trans., 113, 189 (1823). 



LIQUIDS 113 

hydrate was heated in the tube and the liberated chlorine was 
condensed at B as a yellow liquid. IjnJ^^jJQiilQr^^ 
in liquefying carbon dioxide in quite large amounts by the use of 
a new form of apparatus. In connection with his experiments 
ori liquidTcarbon dioxide, he observed that when it was allowed to 
vaporize, enough heat was absorbed to lower the temperature 
below its freezing point, solid carbon dioxide being obtained. He 
discovered that a mixture of solid carbon dioxide and ether was a 
powerful refrigerant, and that under diminished pressure the 
mixture g&ve temperatures ranging from 100 C. to 110 C. 
This mixture is known today as Thilorier's mixture. Faraday f 
undertook the liquefaction of the so-called permanent gases in 1845. 
In this second series of experiments by Faraday he employed 
higher pressures than in his earlier experiments, and also made 
use of the newly discovered Thilorier mixture as a refrigerant. 
He was partially successful in his attempt to liquefy the hitherto 
noncondensible gases. He liquefied ethylene, phosphine and 
hydrobromic acid and also solidified ammonia, cyanogen, and 
nitrous oxide. He failed to liquefy hydrogen, oxygen, nitrogen, 
nitric oxide and carbon monoxide. No further advance in the 
liquefaction of gases was made until the year 1869 when Andrews 
pointed out the importance of cooling the gas below its critical 
temperature. This discovery explained why so many of the 
earlie^ experiments had failed, and opened the way to the brilliant 
successes of the latter part of the nineteenth century. In 1877, 
Cailletet J and Pictet, working independently, succeeded in 
liquefying oxygen. Cailletet subjected the gas to a pressure of 
about 300 atmospheres using boiling sulphur dioxide as a refriger- 
ant. The gas was further cooled by suddenly releasing the pres- 
sure and allowing it to expand, In addition to oxygen he also 
liquefied air, nitrogen and possibly hydrogen. Shortly afterward 
in 1883, the Polish scientists, Wroblewski and Olszewski, <[[ pub- 

* Lieb. Ann., 30, 122 (1839). 
t Phil. Trans., 135, 155 (1845). 
J Compt. rend., 85, 1217 (1877). 
Ibid., 85, 1214, 1220 (1877). 
| Wied. Ann., 20, 243 (18&3). 



114 THEORETICAL CHEMISTRY 

lished an account of their interesting and highly important work. 
In their experiments they subjected the gas to be liquefied to high 
pressure, and simultaneously cooled it to a very low temperature. 
Among the refrigerants used by them was liquid ethylene, which 
was allowed to boil off under diminished pressure, giving a temper- 
ature of 130 C. At this temperature, a pressure of only 
20 atmospheres was sufficient to condense oxygen to the liquid 
state. Having liquefied oxygen, nitrogen, air and carbon mon- 
oxide/ and having determined the boiling-points of these gases 
under atmospheric pressure, they proceeded to use the&e liquefied 
gases as refrigerants, allowing them to boil off undfcr diminished 
pressure, thus obtaining temperatures as low as 200 C. A 
very small amount of liquid hydrogen was obtained in this way. 
Subsequent attempts by these same experimenters to liquefy 
hydrogen, while not much more successful than their former 
attempts, enabled them to determine its boiling-point. Shortly 
after the publication of the first papers of Wroblewski and 
Ola&ewski, Dewar * devised a new form of apparatus for lique- 
fying air, oxygen and nitrogen on a comparatively large scale. 
He also introduced the well-known vacuum-jacketed flasks and 
tubes which greatly facilitated carrying out experiments with 
liquefied gases. In 1895, Linde in Germany and Hampson in 
England simultaneously and independently constructed machines 
for the liquefaction of air in large quantities. 

In the method devised by these experimenters the air is not 
subjected to a preliminary cooling, produced by the rapid evapora- 
tion of a liquefied gas under diminished pressure, as in the methods 
of Wroblewski and Olszewski. In the Linde liquefier, the air is 
compressed to about 200 atmospheres. It is then passed through 
a chamber containing anhydrous calcium chloride to remove the 
greater part of the moisture, after which it is cooled by allowing it 
to circulate through a coiled pipe immersed in a freezing mixture. 
Nearly all of the moisture remaining in the air is deposited on the 
walls of the pipe in the form of frost. The air then enters a long 
spiral tube jacketed with a non-conducting material, and is there 
allowed to expand to a pressure of about 15 atmospheres. 
* Proc. Roy. Inst., 1886, 560. 



LIQUIDS 115 

During this expansion the temperature of the air is appreciably 
lowered. When the air has traversed the spiral tube, it is still 
further cooled by allowing it to expand to a pressure equal to that 
of the atmosphere. The air which has been thus cooled is then 
passed backward through the annular space between the spiral tube 
and a concentric jacker, thus cooling the entering portion of air. 
Consequently this next portion of air expands from a lower initial 
temperature, and the cooling effect is increased. In like manner, 
when this cooler air passes backward it cools still further the next 
succeeding portion, and eventually the temperature is reduced 
sufficiently to cause a small amount of the air to liquefy as it 
issues from the end of the spiral tube. The remaining portion of 
the air which has not been liquefied, passes backward through the 
annular tube and cools the following portion to a still greater 
extent, causing a larger proportion to liquefy on expansion. With 
a 3-horse-power machine, a continuous supply of 0.9 liter per hour 
can be obtained. Further improvements in this apparatus have 
been made by Dewar and Hampson, and by means of it Dewar's 
brilliant successes in the liquefaction of gases have been achieved. 
The most efficient apparatus for the liquefaction of air and other 
gases is that developed by Claude.* The essential features of 
his liquefier are shown in Fig. 26. The air is first compressed to 
40 atmospheres pressure by means of an ordinary compression 
pump not shown in the diagram, the moisture and carbon dioxide 
being removed as in Linde's method. It then enters the tube A, 
which in reality is of a spiral form, and divides at B. A portion 
enters the cylinder D through a valve chest similar to that in a 
steam engine forces out the piston and causes the wheel, W, 
to revolve, thereby doing work and cooling the air. The cooled 
air escapes from the valve chest and circulates through the lique- 
fying chamber L, where it causes the portion of compressed air 
entering at B to liquefy. It then issues from the liquefier and 
traverses M, cooling the entering portion of air in A, and finally 
returns to the compressor. The pressure of the air when it issues 
from D is almost atmospheric, and its temperature is below 
140 C. About twenty-five per cent of the power consumed 

* Compt. rend., II, 500 (1900); I, 1568 (1902); II, 762, 823 (1905). 



116 



THEORETICAL CHEMISTRY 



in compression is regained by the motor. The apparatus pro- 
duces about 1 liter of liquid air per horse-power hour. By means 
of this improved apparatus, based upon the regenerative principle, 
all known gases have now been liquefied, the last to succumb being 




Liquid Air 
40atmoB..-140 



Fig. 26. 



helium which was liquefied in 1908, by Kammerlingh Onnes in the 
Ley den cryogenic laboratory. The subjoined table gives the 
critical data together with the boiling and freezing temperatures 
of some of the more common gases. 

CRITICAL DATA FOR GASES. 



Gas. 


Crit. Temp. 


Crit. Press. 

(Atmos.) 


Boiling Point 
(at 760 mm ) 


Freezing 
Point. 


Helium 


-267 








Hydrogen 


-242 


20 


-252 5 


-258 9 


Air 


-140 


39 


-191 




Nitrogen 


-146 


35 


-195 5 


-210 5 


Oxygen 


-118 


50 8 


-182 8 


-227 


Carbon monoxide .... 
Nitric oxide 
Carbon dioxide . . 


-141 
- 96 
-f 31 


36 
64 
73 


-190 
-153 6 

- 78 


-207 
-W7 
- 65 


Hydrochloric acid 
Ammonia 


+ 51. 3 
+130 


81 5 
115 


- 35 
- 33. 7 


-116 

- 77 













LIQUIDS 117 

Vapor Pressure of Liquids. According to the kinetic theory 
there is a continuous flight of particles of vapor from the surface 
of a liquid into the free space above it. At the same time the 
reverse process of condensation of vapor particles at the surface 
of the liquid is taking place. Eventually a condition of equilib- 
rium will be established between the liquid and its vapor, when 
the rate of escape will be exactly counterbalanced by the rate of 
condensation of vapor particles. The pressure exerted by the 
vapor of a liquid when equilibrium has been attained is known as 
its vapor pressure. The equilibrium between a liquid and its vapor 
is dependent upon the temperature. For every temperature 
below the critical temperature, there is a certain pressure at which 
vapor and liquid may exist in equilibrium in all proportions; and 
conversely for every pressure below the critical pressure, there is 
a certainjbemperature at which vapor and liquid may* exist in 
equiirBrium in all proportions. This latter temperature is termed 
the Boiling-point of the liquid. The vapor pressure of a liquid 
may be measured directly by placing a portion of it above the 
mercury in the vacuum of a barometer tube, heating to the desired 
temperature, and observing the depression of the mercury column. 
This is known as the static method. It is open to the objection 
that the presence of volatile impurities in the liquid causes too 
great depression of the mercury column, the vapor pressure of 
the impurity adding itself to that of the liquid whose vapor 
pressure is sought. A better method for the measurement of 
vapor pressure is that known as the dynamic method. In this 
method the pressure is maintained constant and the boiling 
temperature is determined with an accurate thermometer. The 
boiling temperatures corresponding to various pressures may be 
measured, provided we have a suitable device for changing and 
measuring the pressure. The results obtained by the static and 
dynamic methods agree closely if the liquid is pure, but if volatile 
impurities are present the results obtained by the dynamic method 
are more trustworthy. A method for the measurement of vapor 
pressure due to James Walker * is of considerable interest. In 
this method, a current of pure dry air is bubbled through a weighed 

* Zeit. phys. Chem., 2, 602 (1888). 



118 THEORETICAL CHEMISTRY 

amount of the liquid whose vapor pressure is to be determined. 
The liquid is maintained at constant temperature and its loss in 
weight is observed. In passing through the liquid the air will 
absorb an amount of vapor directly proportional to the vapor 
pressure of the liquid. If 1 mol. of liquid is absorbed by v liters 
of air, then we have 

pv = RT, 

where p is the vapor pressure of the liquid, and T its temperature. 
If Vi is the volume of air which absorbs g grams of the vapor of 
the liquid whose molecular weight is M , then 



or 



In this equation, Vi denotes the total volume containing g grams 
of the liquid in the form of vapor, or in other words it represents 
the air and vapor together. Since the volume of the air is in 
general so much greater than that of the vapor, v\ may be taken 
as that of the air alone. 

Heat of Vaporization. In order to transform a liquid into a 
vapor a large amount of heat is required. Thus, when a liquid 
is heated to the boiling-point, the volume must be increased against 
the pressure of the atmosphere, external work being done, and 
when the boiling temperature is reached the liquid must be vapor- 
ized; the heat expended in causing the change of physical state 
being much greater than that required to expand the liquid. An 
interesting relation between the heat of vaporization and the 
absolute boiling-point of a liquid was discovered by Trouton.* 
If T denotes the absolute boiling-point and w the heat of vapor- 
ization of 1 gram of liquid whose molecular weight is M, then 
according to Trouton 

Mw _ 

-y~ = ^1, 

or in words, the ratio of the molecular heat of vaporization to the 
absolute boiling temperature of a liquid is constant, the numerical 

* Phil. Mag. (5), 18, 54 (1884). 



LIQUIDS 



119 



value of the ratio being approximately 21. This is known as 
Trouton's law. While this relation holds quite well for many 
liquids, Nernst has pointed out that the constant varies with the 
temperature, and has proposed two other forms of the equation. 
Bingham has simplified the [equations of Nernst to the following 
form: 

^ = 17 + 0.011 T. 

While this modification of the Trouton equation has been found 
to hold for a large number of substances, there are other substances 
for which the left side of the equation has a value greater than 
that of the right side. Bingham infers that where this occurs, the 
substance in the liquid state has a greater molecular weight 
than it has in the gaseous state, or in other words, the liquid is 
associated. It is evident that an associated liquid will require 
an expenditure of energy over and above that required for vapori- 
zation, to break down the molecular complex. The difference 
between the values of the two sides of the equation may be 
taken as a rough measure of the degree of association. 

Boiling-Point and Critical Temperature. An interesting rela- 
tion has been pointed out by Guldberg * and Guye.f These 
two investigators have shown that the absolute boiling temper- 
ature of a liquid is about two-thirds of its critical temperature. 
That this empirical relation holds for a variety of different sub- 
stances is shown in the accompanying table. 

RELATION OF BOILING-POINT TO CRITICAL TEMPERATURE. 



Substance. 


T b 


T a 


JVTV, 


Oxygen 


90 


155 


0.58 


Chlorine 


240 


414 


58 


Sulphur dioxide 


263 


429 


0.61 


Ethyl ether 


308 


467 


0.66 


Ethyl alcohol . . 


351 


516 


0.68 


Benzene 


353 


562 


63 


Water 


373 


637 


0.59 


Phenol . 


454 


691 ' 


0.66 











* Zeit. phys. chem., 5, 376 (1890). 
f Bull. Soc. Chim., (3), 4, 262 (1890). 



120 THEORETICAL CHEMISTRY 

Molecular Volume. In dealing with the volume relations of 
liquids it is customary to employ the molecular volume, i.e., the 
volume occupied by the molecular weight of the liquid in grams. 
The justification of this procedure is that when we compare the 
gram-molecular weights of liquids, the comparison involves equal 
aumbers of molecules of the different substances. Since 

, mass 

volume = 



density ' 
we may write 

, molecular weight 

molecular volume = . r- a > 

density 

and similarly, 

atomic weight 



atomic volume = 



density 



Relations between the molecular volumes of liquids were first 
pointed out by Kopp.* On comparing the molecular volumes of 
different liquids at their boiling-points, he found that constant 
differences in composition correspond to constant differences in 
the molecular volumes. Thus the molecular volumes of the 
successive members of an homologous series differ by the same 
number of units, this difference corresponding, for example, to a 
CH 2 group. In like manner the molecular volumes of various 
groups have been determined, and from these in turn the atomic 
volumes of the constituent elements have been worked out. The 
atomic volumes assigned by Kopp to some of the elements com- 
monly entering into organic compounds are as follows : 

C = 11 Cl = 22 8 1=375 Hydroxyl = 7.8 

H = 5 .5 Br = 27 .8 S = 22 6 Carhonyl = 12 .2 

The value of the atomic volume is found to be dependent upon 
the manner of linkage; thus oxygen in the hydroxyl group has the 
atomic volume, 7.8, while oxygen in the doubly linked condi- 
tion, as in the carbonyl group, has the atomic volume, 12.2. By 
means of such a table of experimentally determined atomic 

* Lieb. Ann., 41, 79 (1842); 96, 153, 303 (1855); 96, 171 (1855). 



LIQUIDS 121 

volumes, Kopp showed that it is possible to calculate the molec- 
ular volume of a liquid with a fair degree of accuracy. For 
example, the molecular volume of acetic acid C 2 H 4 2 , may be 
calculated from the atomic volumes of its constitutent atoms as 
follows: 

2 C = 2 X 11 =22 
4H = 4 X 5.5 = 22 
IHydroxylO = 1 X 7.8 = 7.8 
1 Carbonyl = 1 X 12 2 = 12.2 
Molecular volume = 64 

The density of acetic acid at its boiling-point is 0.942, and its 
molecular weight is 00, therefore the observed value of the molec- 
ular volume is 60 -f- 0.942 = 63.7, a result which is in excellent 
agreement with that calculated from the atomic volumes of the 
constituents. The more recent investigations of Thorpe, Lossen, 
Schiff and Buff afford a confirmation of the conclusion reached by 
Kopp, that the molecular volumes of liquids are in general additive. 
While Kopp found that his results were most regular when the 
molecular volumes were determined at the boiling temperatures 
of the respective liquids, the reason for this did not appear until 
after van der Waals had developed his theory of corresponding 
states. As has been pointed out in the preceding paragraph the 
boiling-points of most liquids are approximately two-thirds of 
their respective critical temperatures, and therefore the boiling- 
points are corresponding temperatures. 

Co-volume. By studying various series of hydrocarbons, 
alcohols and ethers, Traube * has been led to suggest that the 
molecular volume of a liquid be looked upon as made up of the 
atomic volumes of its constituent elements and a magnitude 
which he terms the co-volume. This latter he defines as the space 
surrounding a molecule within which it is free to vibrate and from 
which other molecules are excluded. The co- volume appears to 
be nearly constant for a large number of substances, its mean 
value at a temperature of 15 C. being 25.9 cc. The values 

* Uber den Raum der Atome. J. Traube. Ahrens' Sammlung Chemischer 
und chemisch-technischer Vortraege, 4, 255 (1899). 



122 THEORETICAL CHEMISTRY 

assigned by Traube to the atomic volumes of some of the elements 
are as follows: 

C = 9 .9 0=5.5 Br = 17.7 N (trivalent) = 1 .5 
H = 3 . 1 Cl = 13 .2 I = 21 A N (pentavalent) = 10 .7 
Traube has worked out a series of constants which must be de- 
ducted to allow for ring formation and for double and treble link- 
ing. By means of these values, it is possible to calculate the 
molecular volume of a substance by adding together the respective 
atomic volumes of the constituents of the liquid and the co- 
volume, 25.9. It is of course necessary to know the molecular 
weight of the substance together with its constitution, so that 
due allowance may be made for unsaturation. For example, 
the molecular volume of ethyl ether, C 4 HioO, may be calculated 
by Traube's method as follows: 

4C = 4X9 9 = 39.6 
10H = 10X3.1 = 31 
10 = 1 X 5.5 = 5.5 
76.1 

Co- volume 25 . 9 
Molecular volume 102.0 

The molecular volume, as determined from the molecular weight 
and density at 15 C., is 74 -f- 0.7201 = 102.7. 

The method of Traube may be employed in roughly checking 
the accepted value of the molecular weight of a liquid provided 
its density at 15 C. is known, since in the equation 

M/d = atomic volumes + 25.9, expressing Traube's relation, 

M is the only unknown quantity. It is apparent that the liquid 
must be non-associated, since for an associated substance the 
normal co-volume must necessarily accompany the polymerized 
molecule. In this case the formula becomes 

M/d = S atomic volumes + 25.9/n, 

where n denotes the number of simple molecules in the polymer. 
Obviously when the molecular weight of a liquid is known, the 
experimental determination of the co-volume, (M/d S atomic 
vols.) may be used to estimate the degree of association. The 



LIQUIDS 

values thus obtained are not in satisfactory agreement with the 
factors of association derived by means of other methods. 

Refractive Power of Liquids. The velocity of transmission 
of light through any medium depends upon its nature, especially 




Fig. 27. 

upon its density. When a ray of light passes from one medium 
into another it is refracted, the degree of refraction being such 
that the ratio of the sines of the angles of incidence and refrac- 
tion is constant and characteristic for the two media. This 



124 



THEORETICAL CHEMISTRY 



fundamental law of refraction was discovered by Snell about 
1621. According to the wave theory of light, the ratio of the 
sines of the angles of incidence and refraction is identical with 
the ratio of the velocities of light in the two media. The ratio is 
termed the index of refraction and is usually denoted by the letter 
n. Representing by i and r, the angles of incidence and refraction, 
and by v\ and v%, the respective velocities of light in the two media, 

we have 

sin i V] 

n = -T = 
sin r v% 

Various forms of apparatus have been devised for the determina- 
tion of the refractive index of liquids. Of these the best known 
and most satisfactory is the refractometer of Pulfrich, an improved 
form of which is shown in Fig. 27. While the limits of this book 
prohibit a detailed description of the apparatus, the fundamental 
principles involved in its construction will be readily understood 
from the accompanying diagram, Fig. 28. The liquid or fused 
solid is placed in a small glass cell, C, which is cemented to a rec- 
tangular prism of dense optical 
glass, P, the refractive index of 
which is generally 1.61. A beam 
of monochromatic light, from a 
sodium flame of a spectrum-tube 
containing hydrogen, is allowed to 
enter the prism in a direction par- 
allel to the horizontal surface of 
separation between the glass and 
the liquid. After passing through 




Fig. 28. 



the liquid and the prism, the beam emerges making an angle i with 
its original direction. By means of a telescope, the emergent beam 
can be observed and its position noted, the angle of emergence 
being read on a divided circle attached to the telescope. From the 
angle of emergence thus determined, the index of refraction of the 
liquid can be calculated in the following manner. The value of 
the index of refraction, N, for air/glass being known, we have 



sin i 
sin r 



(i) 



LIQUIDS 125 

The angle of incidence of the last ray entering the prism from the 
liquid is 90, or sin ii = 1. The index of refraction, rai, for liquid/ 
glass may be calculated thus, 



sin 

n 




But __ 

(3) 

Transposing equation (1) and substituting in equation (3), we 
have 



sin 2 i 
smn = 



or 



_ _ __ __ . 

sin 7*1 = -T vW 2 sin 2 i. (4) 



Therefore, substituting equation (4) in equation (2), we have 

N 



= 

ni 



Remembering that n = JV/ni, we have for the index of refraction, 
n, for air/liquid, by substitution in equation (5), 



n = 
or if N = 1.61, ______ 

w = V2.592i-sin 2 i'. 



The values of ^/N* sin 2 ! are generally given, for different values 
of i, in tables supplied with the refractometer, thus saving the 
experimenter a somewhat laborious calculation. The value of n 
thus obtained is the index of refraction from air into the liquid; if 
the index from vacuum into the liquid, the so-called absolute index, 
is required, the value of n must be multiplied by 1.00029. The 
index of refraction is dependent upon temperature, pressure and 
in general upon all conditions which affect the density of the 
medium. Furthermore, it is dependent upon the wave-length of 
the light employed, the index for the red rays being greater than 
that for the violet rays. It is therefore necessary in making 



126 THEORETICAL CHEMISTRY 

measurements of refractive indices to use light of a definite wave- 
length, or what is termed monochromatic light. The sodium 
flame is most frequently used for this purpose, the wave-length 
being represented by the letter D. Measurements of the refrac- 
tive index referred to the D-line of sodium are commonly desig- 
nated by the symbol n/>. When incandescent hydrogen is employed 
as a source of light, the refractive index may be determined 
for the C-, F- and G-lines, the respective values being represented 
by n c , n F , and UQ. 

Specific and Molecular Refraction. Various attempts have 
been made to express the refractive power of a liquid by a formula 
which is independent of variations of temperature and pressure. 
Of the different formulas proposed but two need be mentioned. 
The first, due to Gladstone and Dale,* is as follows: 



d ' 

in which d denotes the density of the liquid and n is the so-called 
specific refraction. The other formula, proposed by Lorenz f an d 
Lorentz,{ has the following form: 

__ 1 n 2 - 1 
r2 ~d'n* + 2' 

This formula is superior to that of Gladstone and Dale which is 
purely empirical. It is based upon the electromagnetic theory 
of light and gives values of r 2 which are quite independent of the 
temperature. In order that we may compare the refractive 
powers of different liquids, the specific refractions are multiplied 
by their respective molecular weights, the resulting products 
being termed their molecular refractions. As the result of a large 
number of experiments, it has been shown that the molecular 
refraction of a compound is made up of the sum of the refractive 
constants of the constituent atoms, or in other words refractive 
power is an additive property. The values of the refractive con- 
stants of the elements and commonly-occurring groups have been 

* Phil. Trans. (1858). 

t Wied. Ann., n, 70 (1880). 

t Ibid., 9, 641 (1880). 



LIQUIDS 127 

determined with great care by Brtihl and others, the method 
employed being similar to that used by Kopp in connection with 
his investigations on molecular volumes. Thus, Brtihl found in 
the homologous series of aliphatic compounds that a difference 
of CH2 in composition corresponds to a constant difference of 
4.57 in molecular refraction. Then, having determined the 
molecular refraction of a ketone or an aldehyde of the composition, 
CnH 2 0, he subtracted n times the value of CH 2 and obtained the 
atomic refraction of carbonyl oxygen. By deducting the molecular 
refraction of the hydrocarbon, C n #2n+2, from that of the corre- 
sponding alcohol, Cn//2n+20, he obtained the atomic refraction of 
hydroxyl oxygen. By subtracting six times the value of CH 2 
from the molecular refraction of hexane, C 6 Hi 4 , he obtained the 
refractive constant for hydrogen or 2 H = 2.08. In like manner 
the refractions of other elements and groups of elements were 
determined. 

Just as in the case of molecular volumes so with molecular 
refractions, the arrangement of the atoms in the molecule must 
be taken into consideration. Briihl,* who has devoted much time 
to the investigation of the effect of constitution upon refraction, 
has pointed out that the molecular refraction of compounds con T 
taining double and triple bonds is greater than the calculated value, 
and he has assigned to these bonds definite constants of refrac- 
tion. The values of the atomic refractions for a few of the elements 
as given by Brtihl are as follows: 

C = 2.48 Hydroxyl O = 1.58 

H= 1.04 Carbonyl O -2.34 

CI = 6 .02 Double bond = 1 .78 

I = 13 . 99 Triple bond =2.18 

More recent investigations bring out the fact that, when double 
or triple bonds occupy adjacent positions in the molecule, the 
simple additive relations no longer obtain. The determination 
of the molecular refraction of a liquid affords a means of ascertain- 
ing or confirming its chemical constitution. For example, geraniol 
has the formula CioHisO, and its chemical behavior is such as to 

* Proc. Roy. Inst., 18, 122 (1906). 



128 THEORETICAL CHEMISTRY 

warrant the conclusion that it is a primary alcohol. The value 
of UD is 1.4745, from which the molecular refraction is calcu- 
lated to be 48.71. The molecular refraction calculated from the 
atomic refractions given in the preceding table is: 

IOC = 10 X 2.48 = 24.80 

18 H = 18 X 1.04 = 18 72 

1 Hydroxyl O = 1 X 1 58 = 1_58 

Molecular refraction 45 . 10 

The difference between the theoretical and experimental values 
of the molecular refraction is 48.71 45.10 = 3.61, which is 
approximately twice the value of a double bond, 1.78 X 2 = 3.56. 
From this we conclude that the molecule of geraniol contains two 
double bonds. Furthermore an alcohol of the formula, ( a Hi 8 (), 
containing two double bonds cannot possess a ring structure and 
therefore must be a member of the aliphatic group of compounds. 
This conclusion is supported by the chemical properties of the 
substance.* In a similar manner the Kekule formula for benzene 
has been confirmed, the difference between the theoretical and 
experimental -values of the molecular refraction indicating the 
presence of three double bonds in the molecule. 

Specific Refraction of Mixtures. The specific refraction of an 
homogeneous mixture or solution is the mean of the specific 
refractions of its constituents. Thus, if the specific refractions 
of the mixture and its two components are represented by r\, r 2 , 
and r 3 , then 

- P . 000 -p) 

~ ""~ 



where p denotes the percentage of the constituent whose specific 
refraction is r 2 . Hence it is possible to determine the specifiq 
refraction of a substance in solution by measuring the refractive 
indices and densities of the solution and solvent. If the refrac- 
tive indices of the solvent, solution and dissolved substance are 

* The accepted structural formula of geraniol is 

H - C - CH 2 OH 

II 
(CH 3 ) 2 C - CH-CH 2 -CH 2 - C - CH 8 . 



LIQUIDS 129 

represented by n\, ^ and n 3 respectively, and if di, da, and da 
denote the corresponding densities, then we have 

JZJ: _ 1QQ "" P V-- 1 
" 



where p is the percentage of the dissolved substance. As has 
already been mentioned, the formula of Lorenz-Lorentz is based 
upon the electromagnetic theory of light. According to this 
theory n* 1/n 2 + 2 expresses the fraction of the unit of volume 
of the substance which is actually occupied by it. From this it 

follows that the molecular refraction, -, is an expression 

d ti -]- & 

of the volume actually occupied by the atomic nuclei of the 
molecule. It is interesting to note that the ratio of the sum of 
the atomic volumes, calculated by the method of Traube, to the 
corrected molecular volume, as determined by the Lorenz-Lorentz 
formula, is approximately constant, or 

S atomic volumes . r . , , 

- rr run - ^ 3-45 approximately. 



This may be considered as the ratio of the volume within which 
the atoms execute their vibrations to their actual material volume. 
Rotation of the Plane of Polarized Light. Some liquids when 
placed in the path of a beam of polarized light possess the prop- 
erty of rotating the plane of polarization to the right or to the 
left. Such liquids are said to be optically active. Those substances 
which rotate the plane of polarization to the right are termed 
dextro-rotatory, while those which cause an opposite rotation are 
called levo-rotatory. The determination of the rotatory power of 
a liquid is made by means of an instrument known as a polarimeter, 
a convenient form of which is shown in Fig. 29. The essential 
parts of this instrument are two similar Nicol prisms placed one 
behind the other with their axes in the same straight line. The 
light after passing through the forward prism, P, known as the 
polarizer, has its vibrations reduced to a single plane; it is said 
to be plane polarized. On entering the rear Nicol prism, A, 



130 



THEORETICAL CHEMISTRY 



known as the analyzer, the light will either pass through or be 
completely stopped, depending upon the position of the prism. 
If the analyzer be slowly rotated, it will be observed that 
the position of maximum transmission and extinction occur at 
points 90 apart. If the analyzer be rotated, so that its axis is 
at right angles to the axis of the polarizer, the field observed will 
be dark, no light being transmitted. If now a tube similar to 
that shown in Fig. 30 be filled with an optically active liquid 
and placed between the polarizer and analyzer, the field will 




Fig. 29. 

become light again, due to the rotation of the plane of polariza-y 
tion by the optically-active substance. The extent to which the 
plane of polarization has been rotated can be determined by 
turning the analyzer until the field becomes dark again, and read- 
ing on the divided circle, K , the number of degrees through which 
it has been moved. When it is necessary to turn the analyzer 
to the right, the substance is dextro-rotatory, and when it is neces- 
sary to turn it to the left, the substance is levo-rotatory. Various 



LIQUIDS 131 

optical accessories have been added to the simple polarimeter 
described above to render the instrument more sensitive, but for 
these details the student must consult some special treatise.* 
The angle of rotation is dependent upon the nature of the liquid, 
the length of the column of substance through which the light 
passes, the wave-length of the light used, and the temperature at 
which the measurement is made. It is customary in polarimetric 





Fig. 30. 

work to employ sodium light and, unless otherwise specified, it 
may be assumed that a given rotation corresponds to the D-line. 

Specific and Molecular Rotation. The results of polarimetric 
measurements are expressed either as specific rotations or as 
molecular rotations, the latter being preferable since the optical 
activities of different substances may then be compared. 

The specific rotation is obtained by dividing the observed rota- 
tion by the product of the length of the column of liquid and its 
density, or 



where [a] t is the specific rotation at the temperature, t, a the 
observed angle, I the length of the column of liquid in decimeters, 
and d its density. If the specific rotation is multiplied by the 
molecular weight of the substance, the molecular rotation is ob- 
tained, but owing to the fact that the resulting numbers are too 
large, it is customary to express the molecular rotation as one 
one-hundredth of this value, thus 

Ma 



100H 

The specific and molecular rotations of solutions of optically 
active substances may also be determined, if we assume that the 

* See for example, 4C Thc Optical Rotatory Power of Organic Substances 
and its Practical Applications." H. Landolt, trans, by J. H. Long. 



132 



THEORETICAL CHEMISTRY 



solvent is without effect. While this assumption is justifiable 
with aqueous solutions, it is not so when non-aqueous solvents are 
used. If g grams of an optically active substance be dissolved in 
v cc. of solvent, then 

r , av j r i M av 
[],-, a*d [],= _._, 

or if the composition of the solution is expressed in terms of 
weight instead of volume, g grams of substance being dissolved 
in 100 grams of solution of density d, then 



100 a. 

gdl ' 



and 



Ma 



Optical Activity and Chemical Constitution. The fact that 
some substances have the power of rotating the plane of polarized 
light was first discovered by Biot, but the credit for recognizing 
the chemical significance of this fact belongs to Pasteur.* He 
discovered that ordinary racemic acid can be separated into two 
optically active modifications, one of which is dextro- and the 
other levo-rotatory, the numerical values of the two rotations 





Fig. 31. 

being identical. If a solution of sodium ammonium racemate 
be allowed to evaporate at a low temperature, crystals of the 
composition NaNH^EUOe . 4 H 2 O will separate. On close in- 
spection it will be found that the crystals are not all alike, but 
that they may be divided into two classes, one class showing 
some unsymmetrical crystal surfaces which are oppositely placed 
in the crystals of the other class. The crystals of one class may 

* Ann. Chim. Phys. (3), 24, 442 (1848); 28, 56 (1850); 31, 67 (1851). 



LIQUIDS 133 

be regarded as the mirror images of those of the other class: 
such crystals are said to be enantiomorphous. The forms usually 
assumed by the two enantiomorphous modifications of sodium 
ammonium racemate are shown in Fig. 31. After separating 
the two forms Pasteur dissolved each in water, making the solu- 
tions of the same strength. The solution of the crystals with the 
"right-handed faces" was found to be dextro-rotatory, while that 
of the crystals with the " left -handed faces" was found to be levo- 
rotatory. Pasteur then decomposed the two salts obtained from 
sodium ammonium racemate and obtained the corresponding 
acids, which he called dextro- and levo-racemic acids. It was 
subsequently shown that the two acids were identical with dextro- 
and levo-tartaric acids. Finally, when Pasteur mixed equiv- 
alent amounts of concentrated solutions of dextro- and levo- 
tartaric acids, an appreciable evolution of heat was observed, 
indicating that a chemical reaction had taken place. After 
allowing the solution to stand for some time, crystals of ordinary 
racemic acid were obtained. Thus it was clearly proven that an 
optically inactive substance may be separated into two opti- 
cally active modifications, possessing equal and opposite rotatory 
powers, and that by mixing equivalent quantities of the two 
optically active forms, the optically inactive substance may be 
recovered. 

Pasteur discovered and applied three other methods in addi- 
tion to the mechanical method already described, for the separation 
of a substance into its optically active modifications. These are 
as follows: (a) Method of Crystallization; (b) Method of Forma- 
tion of Derivatives; and (c) Methods of Ferments. 

Methods of Crystallization. To a supersaturated solution of the 
racemic modification a very small crystal of one of the active 
forms is added. This will induce the separation of crystals of 
the same form, inoculation with a dextro-crystal producing the 
dextro-form and inoculation with a levo-crystal producing the 
levo-form. 

Method of Formation of Derivatives. In this method an opti- 
cally active substance, generally an alkaloid, is added to the racemic 
modification, producing optically active derivatives having differ- 



134 THEORETICAL CHEMISTRY 

ent solubilities. Thus if cincbonine, an optically active alkaloid 
having the formula, CigEM^O, be added to the racemic modifica- 
tion of tartaric acid, the cinchonine salt of the levo-acid will 
crystallize first. The crystals of the cinchonine salt are then 
removed and after adding ammonia to displace the alkaloid, 
dilute sulphuric acid is added and the pure levo-tartaric acid is 
obtained. 

Methods of Ferments. Notwithstanding the fact that optical 
antipodes resemble each other so closely in most of their properties, 
Pasteur found that certain micro-organisms have the power of 
distinguishing sharply between these forms. For example, if 
penicillium glaucum be introduced into a solution of racemic 
tartaric acid, it thrives at the expense of the dextro-acid and 
eventually leaves the pure levo-form. In this method one of the 
active modifications is always lost. 

Pasteur was the first to point out that there must be some inti- 
mate connection between optical activity and the constitution of 
the molecule. It remained for Le Bel * and van't Hoff f to for- 
mulate independently and almost simultaneously an hypothesis 
to account for optical activity on the basis of molecular constitu- 
tion. Their important work laid the foundation of spatial chemis- 
try, commonly termed stereochemistry (derived from the Greek 
orrcpeos = a solid). Le Bel accepted Pasteur 's view that optical 
activity is dependent upon a condition of asymmetry, but whether 
'this asymmetry is a property of the crystal alone or whether it 
belongs to the molecule of the optically active substance, was the 
question he set himself to answer. He found, on dissolving certain 
optically active crystals in an inactive solvent, that the optical 
activity is imparted to the solution and therefore he concluded 
that the condition of asymmetry must exist in the chemical mole- 
cule. All of the optically active substances known to Le Bel were 
compounds of carbon. An examination of the formulas of these 
compounds led him to ascribe the cause of their optical activity 
to the presence of an asymmetric carbon atom, that is, a carbon 
atom combined with four different atoms or groups of atoms. 

* Bull. Soc. Chim, (2), 22, 337 (1874). 
t Ibid. (2), 23, 295 (1875). 



LIQUIDS 



135 



One of the simplest examples is afforded by lactic acid, the struc- 
tural formula of which is 

H 

CH 3 C COOH 

in 

In this formula the asymmetric carbon atom is placed at the 
center and is in combination with hydrogen, hydroxyl, methyl 
and carboxyl. In connection with his work on the relation 
between optical activity and asymmetry, Le Bel pointed out that 
active forms never result from laboratory syntheses, the racemic 
modification being invariably obtained. Van't Hoff reached con- 
clusions similar to those of Le Bel and proposed the additional 




OH 



HO 




COOH 



COOH 



Fig. 32. 



theory of the asymmetric tetrahedral carbon atom. Since the four 
valences of the carbon atom are equivalent, as the work of Henry 
on methane has shown them to be, van't Hoff pointed out that 
the only possible geometrical arrangement of the atoms in the 
molecule of methane must be that in which the carbon atom is 
placed at the center of a regular tetrahedron with the four 
hydrogen atoms at the four apices. He then pointed out that 
when the four valences of the tetrahedral carbon atom are satis- 
fied with different atoms or groups, no plane of symmetry can be 
passed through the figure, the carbon atom being asymmetric. 
This conception of Le Bel and van't Hoff forms the basis of all 
stereochemistry, and has proved of inestimable value to the 
organic chemist in enabling him to explain the existence of many 
isomeric compounds. Thus, ordinary lactic acid can be split into 



136 



THEORETICAL CHEMISTRY 



two optically active isomers. Aside from the fact that one acid 
is dextro- and the other is levo-rotatory, the properties of the 
two acids are practically identical. If the formulas are written 
spatially, the different groups can be arranged about the 
asymmetric carbon atom in such a way that the two tetrahedra 
shall be mirror images of each other, as shown in Fig. 32. It will 
be observed that these two tetrahedra can in no way be super- 
posed so that the same groups fall over each other, that is to 
say, they are enantiomorphous forms. In tartaric acid there are 
two asymmetric carbon atoms as is evident when its structural 
formula is written as follows: 

H H 

HOOC C C COOH 
OH OH 

If the stereochemical formulas of the dextro- and levo-acids be 
represented as in Fig. 33, (a) and (b), it will be apparent that 
the theory admits of the existence of another isomer with the 
atoms and groups arranged as in Fig. 33 (c). 

Bacemic Acid 
A 




GO OH 
d-TartaricAcid 

*> 




COOH 
2-Tartaric Acid 

ib) 
Fig. 33. 



COOH 
Meso-Tartaric Acid 



In this arrangement the asymmetry of the upper tetrahedron 
is the reverse of that of the lower, and consequently the optical 
activity of one-half of the molecule exactly compensates the optical 



LIQUIDS 137 

activity of the other half, and the molecule as a whole is inactive. 
It is evident that such a tartaric acid could not be split into two 
active forms. Actually there are four tartaric acids known, viz., 
(1) inactive racemic acid which is separable into (2) dextro-tartaric 
acid and (3) levo-tartaric acid; and (4) meso-tartaric acid, an in- 
active substance which has never been separated into two active 
forms, but which has the same formula, the same molecular 
weight and in general the same properties as the dextro- or levo- 
tartaric acids. Inactive forms, such as meso-tartaric acid, are said 
to be inactive by internal compensation. This constitutes one of 
many beautiful confirmations of the van't Hoff theory of the 
asymmetric tetrahedral carbon atom. 

Meso-tartaric acid furnishes an illustration of the fact that 
asymmetric carbon atoms may be present in the molecule with- 
out imparting optical activity to the substance. The converse 
of this proposition, however, that optical activity is dependent 
upon asymmetric carbon atoms, is generally true. Quite recently 
some substances apparently containing no asymmetric carbon 
atoms have been discovered which are optically active. An 
example of such a substance is 1-methyl cyclohexylidene-4 acetic 
acid, to which the following formula has been assigned: 

CH 2 CH 2 

CH 3 CH C : CH COOH 

CH 2 CH 2 

Other atoms aside from carbon may be asymmetric; thus certain 
compounds of nitrogen, sulphur and tin have been shown to be 
optically active. The theory also furnishes an explanation of 
the fact, pointed out by Le Bel, that optically active forms are 
never obtained by direct synthesis. Since the rotatory power is 
dependent upon the arrangement of the atoms and groups in the 
molecule, it follows from the doctrine of probability that as many 
dextro as levo configurations will be formed and consequently the 
racemic modification will be obtained. Up to the present time no 
satisfactory generalization has been discovered as to the factors 
determining the molecular rotation in any particular case. An 



138 



THEORETICAL CHEMISTRY 



attempt in this direction has been made by Guye,* in which he 
ascribes the magnitude of the observed rotation to the relative 
masses of the atoms or groups which are in combination with the 
tetrahedral carbon atom. But it cannot be mass alone which 
conditions optical activity, since substances are known which 
rotate the plane of polarization notwithstanding the fact that 
their molecules have two groups of equal mass in combination 
with the asymmetric carbon atom. The molecular rotations of 
the members of homologous series exhibit some regularities, but 
on the other hand many exceptions occur which cannot be satis- 
factorily explained. About all that can be said at the present 
time is, that optical activity is a constitutive property. 

Magnetic Rotation. That many substances acquire the power 
of rotating the plane of polarized light when placed in an intense 
magnetic field was first observed by Faraday f in 1846. 

The relation between chemical composition and magnetic 
rotatory power has since been investigated very exhaustively by 
W. H. Perkins,! his experiments in this field having been continued 
for more than fifteen years. In brief, Perkin's method of investi- 
gating magnetic rotatory power consisted in introducing the 
liquid to be examined into a polarimeter tube 1 decimeter in 
length and then placing the tube axially 
between the perforated poles of a 
powerful electromagnet, as shown in 
Fig. 34. Upon exciting the magnet it 
was found that the plane of polarization 
had been rotated, either to the right 
or the left, the direction of rotation 
depending upon the direction of the 
current, the intensity of the magnetic 




Fig. 34. 



field and the nature of the liquid. Perkin used the sodium flame 
as his source of light and carried out all of his experiments 
at 15 C. He expressed his results by jneans of the formula, 

* Compt. rend., no, 714 (1890). 

t Phil. Trans., 136, 1 (1846). 

j Jour, prakt. Chem. [2], 31, 481 (1885); Jour. Chem. Soc., 49, 777; 41, 
808; 53, 561, 695; 59, 981; 6x, 287, 800; 63, 57; 65, 402, 815; 67, 255; 69, 
1025 (1886-1896). 



LIQUIDS 139 

Ma/dj a being the observed angle of rotation, d the density 
of the liquid and M its molecular weight. All measurements 
were expressed in terms of water as a standard: thus if Ma/d 
is the rotation for any substance and M'a! '/d' is the corre- 
sponding rotation for water, then, according to Perkin, the 
molecular magnetic rotation will be given by the ratio, Ma/d: 
M'a'/d' or Mad'/M'a'd. 

The molecular magnetic rotation for a large number of organic 
compounds has been determined by Perkin, who has shown it to 
be an additive property. In any one homologous series the value 
of the molecular magnetic rotation is given by the formula 

mol. mag. rotation = a + rib, 

where a is a constant characteristic of the series, b is a constant 
corresponding to a difference of CH 2 in composition, its value 
being 1.023, and n is the number of carbon atoms contained in 
the molecule. This formula is applicable only to compounds 
which are strictly homologous, isomeric substances in two differ- 
ent series having quite different rotations. The constitution of 
the molecule exerts as great an influence on magnetic rotation 
as it does on refraction, a double bond causing an appreciable 
increase in the value of a. The results of experiments on mag- 
netic rotation show that nothing like the same regularities exist 
as have been discovered for molecular refraction and molecular 
volume. The rotatory powers of various inorganic substances 
have been determined, but the results are too irregular to admit 
of any satisfactory interpretation. 

Absorption Spectra. When a beam of white light is passed 
through a colored liquid or solution and the emergent beam is 
examined with a spectroscope, a continuous spectrum crossed by 
a number of dark bands is obtained. A portion of the light has 
been absorbed by the liquid. Such a spectrum is known as an 
absorption spectrum. If instead of passing the light through a 
liquid it is passed through an incandescent gas, a spectrum will 
be obtained which is crossed by numerous fine lines, termed 
Fraunhofer lines. Such lines occupy the same positions as the 
corresponding colored lines in the emission spectrum of the gas. 



140 



THEORETICAL CHEMISTRY 



It follows, therefore, that the absorption spectrum is quite as char- 
acteristic of a substance as its emission spectrum, and from a 
careful study of the absorption spectra of liquids we may expect 
to gain some insight into their molecular constitution. The 
pioneer workers in this field were Hartley and Baly * and it is 
largely to them that we owe our present experimental methods. 
The instrument employed for photographing spectra is called a 




Fig. 35. 

spectrograph, a very satisfactory form being shown in Fig. 35. 
It differs from an ordinary spectroscope in that the eye-piece is 
replaced by a photographic camera. This attachment is clearly 
shown in the illustration. The plateholder is so constructed that 
only a narrow horizontal strip of the plate is exposed at any one 
time, thus making it possible to take a series of photographs on 
the same plate by simply lowering the holder. By means of a 
millimeter scale, also shown in the illustration, the plateholder 
can be moved through the same distance each time before expos- 

* See numerous papers in the Jour. Chem. Soc., since 1880. 



LIQUIDS 141 

ing a fresh portion of the plate, thus insuring an equally-spaced 
series of spectrum photographs. In order that spectra in the ultra- 
violet region may be photographed, it is customary to equip the in- 
strument with quartz lenses and a quartz prism, ordinary glass not 
being transparent to the ultra-violet rays. Using a spectrograph 
furnished with a quartz optical system, it is possible to photograph 
on a single plate the entire spectrum from 2000 to 8000 Angstrom 
units. A scale of wave-lengths photographed on glass is provided 
with the instrument so that the wave-lengths of lines or bands 
can be read off directly by laying the scale over the photographs. 

The source of light to be used depends upon the character of 
the investigation. If a source rich in ultra-violet rays is desired, 
the light from the electric spark obtained between electrodes pre- 
pared from an alloy of cadmium, lead and tin is very satisfactory; 
or the light from an arc burning between iron electrodes may be 
used. For investigations in the visible region of the spectrum 
the Nernst lamp is unsurpassed. In using the spectrograph for 
the purpose of studying the constitution of a dissolved substance, 
it is necessary to determine not only the number and position of 
the absorption bands, but also the persistence of these bands as 
the solution is diluted. 

According to Beer's law the product of the thickness, t, of an 
absorbing layer of solution of molecular concentration, m, is con- 
stant, or mt = fc. If then the thickness of a given layer of solu- 
tion is diminished n times, its absorption will be the same as that 
of a solution whose concentration is only 1 /nth of that of the original 
solution. Thus, by varying the thickness of the absorbing layer 
we can produce the same effect as by changing the concentration. 
The convenient device of Baly for altering the length of the 
absorbing column of liquid is shown in Fig. 36 attached to the 
collimator of the spectrograph. It consists of two closely-fitting 
tubes, one end of each tube being closed by a plane, quartz disc. 
The outer tube is fitted with a small bulbed-funnel and is graduated 
in millimeters. The two tubes are joined by means of a piece of 
rubber tubing which prevents leakage of the contents, and at the 
same time admits of the adjustment of the column of liquid to 
the desired length by simply sliding the smaller tube in or out. 



142 



THEORETICAL CHEMISTRY 



Molecular Vibration and Chemical Constitution. There are 
two systems of graphic representation of the results of spectro- 
soopic investigations. In the first system, due to Hartley, the 
wave-lengths or their reciprocals, the frequencies, are plotted as 
abscissae and the thicknesses of the absorbing layers, in milli- 
meters, are plotted as ordinates. Such curves are known as 
curves of molecular vibration. The second system, due to Baly 
and Desch, is a modification of that developed by Hartley. 




Fig. 36. 

Baly and Desch suggested that for various reasons it would be 
more advantageous, if instead of plotting the thickness of the 
absorbing layers as ordinates, the logarithms of these thicknesses 
be plotted. Both methods have their advantages and both are 
used. As an illustration of the value of curves of molecular 
vibration in connection with questions of chemical constitution, 
we will take the case of o-hydroxy-carbanil. The constitution 
of this substance was known to be represented by one of the two 
following formulas : ' 



\NH 



or 



C,H 



-OH 



(a) 



(b) 



LIQUIDS 



143 



On comparing the curves of molecular vibration for the three sub- 
stances (Fig. 37), it is apparent that the curves for the lactam 
ether and o-hydroxy-carbanil bear a close resemblance to each 
other, while the curve for the lactim ether is very different from 
the curves for the other two substances. The constitution of 



4000 



' I I I I I I I I I" 

345978912 




Oscillation Frequencies 

4000 4000 

' ' ' I ' ' I I ' ' I I I ' I I I I I ' 
3 4567891 




345678y 123 




Ether 



Q-Hydroxy-CaiibaDil 
Fig. 37. 



Lactim Ether 



o-hydroxy-carbanil must then be very similar to that of the 
lactam ether. The formulas of the ethyl derivatives of the 
mother substance are known to be as follows: 




- C 2 H 6 
Lactam ether 



/\ 

/ \C 

N 



Lactim ether 



-0-C 2 H S 



144 THEORETICAL CHEMISTRY 

Hartley concluded, therefore, that formula (a), represents the 
structure of the molecule of o-hydroxy-carbanil. 

It is beyond the scope of this book to discuss at greater length 
the bearing of absorption spectra upon chemical constitution; but 
the student is earnestly advised to consult some book * treating of 
this important subject or to read some of the original papers. 

Surface Tension. The attraction between the molecules of a 
liquid manifests itself near the surface where the molecules are 
subject to an unbalanced internal force. The condition of a 
liquid near its surface is roughly depicted in Fig. 38, where the 
dots A, J3, and C represent molecules and the circles represent the 
spheres within which lie all of the other molecules which exert an 
appreciable attraction upon A , B, and C. The shaded portions rep- 









Fig. 38. 



resent those molecules whose attractions are unbalanced. These 
unbalanced forces will evidently tend to diminish the surface to 
a minimum value. That is, the contraction of the surface of a 
liquid involves the expenditure of energy by the liquid. The 
surface film of a liquid is consequently in a state of tension. 

Some liquids wet the walls of a glass capillary tube while others do 
not. When the liquid wets the tube, the surface is concave and the 
liquid rises in the tube; on the other hand, when the liquid does not 
adhere, the surface is convex and the liquid is depressed in the tube. 
The law governing the elevation or depression of a liquid in a 
capillary tube was discovered by Jurin and may be stated thus: 
The elevation or depression of a liquid in a capillary tube is inversely 
proportional to the diameter of the tube. Let Fig. 39 represent a 

* Relation between Chem. Constitution and Phys. Properties. Samuel 
Smiles. 



LIQUIDS 



145 



capillary tube of radius r, immersed in a vessel of liquid whose 
density is d, and let the elevation of the liquid in the capil- 
lary be denoted by h. Then the weight of the column of liquid 
in the capillary will be irr z h dg, where g is the acceleration due 
to gravity. The force sustaining this weight is 2 Trry cos 0, the 



Fig. 39. 

vertical component, of the force due to the tension of the liquid 
surface at the walls of the tube, 7 being the surface tension and 
the angle of contact of the liquid surface with the walls of 
the tube. 
Therefore 

7TT 2 /l dg = 2 7TT7 COS 0, 

or 

_ ^ dgr 
7 " 2cos0' 

In the case of water and many other liquids is so small that we 
may write = 0, the foregoing expression becoming 

7 = 1/2 h dgr. 

Thus the surface tension of a liquid can be calcuated provided 
its density and the height to which it rises in a previously calibrated 
tube is known. When h and r are expressed in centimeters, 7 
will be expressed in dynes per centimeter or &rgs per square centi- 
meter. A simple form of apparatus for the determination of 
surface tension used by the author is shown in Fig. 40. A capil- 
lary tube, A, of uniform bore is sealed to a glass rod, J5, which is 
held in position in the test tube, 5, by means of a cork stopper. 
A short right-angled tube, JD, and a thermometer, F, are also 



146 



THEORETICAL CHEMISTRY 



passed through the same cork stopper. The liquid whose surface 
tension is to be measured is introduced into the tube, -B, the cork 
inserted and the tube placed inside 
of the larger tube, C 9 containing a 
liquid of known boiling-point. 
When the thermometer, F, has 
become stationary, the capillary 
elevation of the liquid is measured 
with a cathetometer. The tube, 
D, permits the escape of vapor 
from the liquid in B and at the 
same time insures equality of pres- 
sure inside and outside of the ap- 
paratus. The spiral tube, <7, serves 
as an air condenser, preventing 
loss of vapor from the liquid in the 
outer tube. The surface tension 
of a liquid has been found to depend 
upon the nature of the liquid and 
also upon its temperature. 

Surface Tension and Molecular 
Weight. In 1886, Eotvos* showed 
that the surface tension multiplied 
by the two-thirds power of the mo- 
lecular weight and specific volume 
is a function of the absolute tem- 
perature, or 

7 (Mv)* = / (T), 

Fig. 40. 
where 7 is the surface tension, M 

the molecular weight, v the specific volume or reciprocal of the 
density, and T the absolute temperature. Ramsay and Shields f 
modified the equation of Eotvos as follows: 

7 (MvY =*k(t c -t 6), (1) 

t c being the critical temperature of the liquid, t the temperature 
of the experiment, and k a constant independent of the nature of 

* Wied. Ann., 27, 448 (1886). 

t Zeit. phys. Chem., 12, 431 (1893). 




-=3 c 



LIQUIDS 147 

the liquid. The physical significance of the two-thirds power of 
the molecular volume has been explained by Ostwald in the follow- 
ing manner: Assuming the molecules to be spherical, we shall 
have for two different liquids, the proportion 

Fi:F 2 ::n 3 :r 2 3 , 

where V\ and F 2 represent the volumes and r\ and r 2 the radii of 
their respective molecules. Similarly the ratio of the surfaces, 
Si and S%, of the molecules in terms of their respective radii, will 
be 

Si:&::ri:itf. 

From these two proportions it follows that the ratio of the molec- 
ular surfaces in terms of the molecular volumes, will be 

Si:&::yi*:Ft*. 

Making use of the value of M as determined in the gaseous state, 
Ramsay and Shields found the value of k for a large number of 
liquids to be equal to 2.12 ergs. Among the liquids for which 
this value of k was found were benzene, carbon tetrachloride, 
carbon disulphide and phosphorus trichloride. For certain other 
liquids such as water, methyl and ethyl alcohols and acetic acid, 
fc was found to have values much smaller than 2.12. Ramsay 
and Shields attributed these abnormalities to an increase in molec- 
ular weight due to association, and suggested that the degree of 
association might be calculated from the equation 

** = 2.12/fc', 
or 



x = (^rp (2) 

where x denotes the factor of association, and fc' is the value of 
the constant for the associated liquid in equation (1). It was 
further pointed out by Ramsay and Shields that equation (1) 
affords a means of calculating the molecular weight of a pure 
liquid, provided we assume that for a non-associated liquid the 
mean value of fc is 2.12. Since it is not an easy matter to deter- 
mine the critical temperature with accuracy, Ramsay and Shields 
made use of a differential method, and thus eliminated t c from 
equation (1). If the surface tension of a liquid be measured at 



148 THEORETICAL CHEMISTRY 

two temperatures t\ and fe, and the corresponding densities are d\ 
and ^2, we shall have 

7i (WO* = k (fc-fe-6), (3) 

and 

72(M/d 2 )|.==M*c-fe-6). (4) 

Subtracting equation (4) from equation (3), we obtain 

_ fc = 2.12, (5) 



or solving equation (5) for M , we have 



The method of Ramsay and Shields is the best known method for 
the determination of the molecular weight of a pure liquid. If 
M is known to be the same in the liquid and gaseous states, 
or in other words, if fc is independent of the temperature, even 
though its value is not exactly 2.12, the critical temperature of 
the liquid can be calculated by means of equation (1). In order 
that the correct value of the critical temperature may be obtained, 
Ramsay and Shields found it necessary to use the specific value 
of k for the liquid whose critical temperature is sought. As an 
illustration of the method of calculation, the following example is 
taken from the work of Ramsay and Shields. 
For carbon disulphide, 

7 at 19.4 = 33.58 7 at 46. 1 = 29.41 

d at 19.4 = 1.264 d at 46.l = 1.223. 

We have then for 7 (M/d)l, at the two temperatures, 

(76/1.264)* X 33,58 = 515.4, 
and 

(76/1.223)* X 29.41 = 461.4. 

Substituting in the equation 



= k ' 

we have, 

515.4-461.4 
46.1 - 19.4 ~ 2 - 022 ' 



LIQUIDS 149 

This value of k is so nearly equal to the mean value, 2.12, that we 
assume M to be the same in the liquid and gaseous states, and 
therefore we may substitute in equation (1) and calculate the 
critical temperature of carbon disulphide thus, 



or solving for t c , we have 






Substituting the data given above, in the preceding equation, we 

obtain 

t c = 515.4/2.022 + 6 + 19.4, 

or t e = 280.3 C. 

Surface Tension and Drop-Weight. Morgan and his co- 
workers,* from measurements of the volumes of a single drop 
falling from the carefully-ground tip of a capillary tube, have 
shown that the weight of the falling drop from such a tip can be 
used in place of the surface tension in the equation of Ramsay 
and Shields for the calculation of molecular weights and critical 
temperatures. The modified equation may be written thus: 



, 

where Wi and w z are the respective weights of the falling drop 
at the temperatures, t\ and 2 - The value of k obviously depends 
upon the tip employed. 

The results obtained by the drop-weight method have been 
shown to be more trustworthy than those obtained by the method 
of capillary elevation. Morgan has further pointed out that 
when the experimental data are substituted in the preceding 
formula, the magnification of the experimental errors is appreci- 
ably greater than when use is made of the original formula, 

w(M/d)l = fc( c --6). 

Morgan recommends therefore that this formula be used for 
the determination of molecular weights. After having calibrated 

* Jour. Am. Chem. Soc., 30, 360 (1908); 30, 1055 (1908). 



150 THEORETICAL CHEMISTRY 

a particular tip with pure benzene (a liquid which is known to be 
non-associated), and thus ascertaining the value of fc, the drop- 
weights at several different temperatures are determined. If 
we assume M to have the same value in the liquid and gaseous 
states, the value of l c can be computed by substituting the experi- 
mental data in the preceding equation. If at the different tem- 
peratures at which drop-weights are determined, the same value 
of t c is obtained, then we may infer that the liquid is non-asso- 
ciated and, therefore, that the assumption made as to the value of 
M is confirmed. It is a singular fact that the calculated value 
of t c for some liquids does not agree with the experimental value, 
although it remains constant throughout an extended range of 
temperatures. Morgan considers a constant value of t c to be an 
indication of non-association, even if the value is fictitious. In 
this method the constancy of the calculated value of the critical 
temperature becomes the criterion of molecular association, and 
thus affords a means of determining whether the molecular weight 
in the liquid state is identical with that in the gaseous state. The 
values of t c calculated from the drop-weights of an associated 
liquid become steadily smaller as the temperature increases. A 
large number of liquids have been studied by this method, and 
the results indicate that many of the substances which were con- 
sidered to be associated by Ramsay and Shields are in reality 
non-associated; in fact, it appears from the work of Morgan that 
association is much less common among liquids than has hitherto 
been supposed. 

Dielectric Constants. In 1837, Faraday discovered that the 
attraction or repulsion between two electric charges varies with 
the nature of the intervening medium or dielectric. If gi and q% 
represent two charges which are separated by a distance r, the 
force of attraction or repulsion, /, is given by the equation 



where D is a specific property of the medium known as the dielectric 
constant. The dielectric constant of air is taken as unity. Vari- 
ous methods have been devised for the experimental determination 



LIQUIDS 



151 



of the dielectric constant, but the scope of this book forbids 
even a brief description of the apparatus or an outline of the 
processes of measurement. For a description of these methods 
the student is referred to any one of the more complete physico- 
chemical laboratory manuals, or to the original communications 
of Nernst * and Drude.f 

The values of the dielectric constants for some of the more 
common solvents are given in the accompanying table. 

DIELECTRIC CONSTANTS AT 18 C. 



Substance. 


D 


Hydrogen dioxide 


92 8 


Water 
Formic acid 


77 
63.0 


Methyl alcohol 


33 7 


Ethyl alcohol 


25 9 


Ammonia, liquid 


22 


Chloroform 


5 


Ether . . 


4 4 


Carbon disulphide 


2.6 


Benzene 


2 3 







The importance of this property of liquids will become more 
apparent in subsequent chapters, especially in those devoted to, 
electrochemistry. 

PROBLEMS. 

1. It is desired to compare the molecular volumes of alcohol and ether. 
If the molecular volume of ether is determined at 20 C., at what temper- 
ature must the molecular volume of alcohol be determined? The boiling 
points of alcohol and ether are 78 and 35 respectively. Ans. 61 C. 

2. A volume of 50 liters of air in passing through a liquid at 22 C. 
causes the evaporation of 5 grams of substance, the molecular weight of 
which is 100. What is the vapor pressure of the liquid in grams per 
square centimeter? . Ans. 25. 



* Zeit. phys. Chem., 14, 622 (1894). 
t Ibid., 23, 267 (1897). 



152 THEORETICAL CHEMISTRY 

3. The boiling-point of ethyl propionate is 98.7 C. and its heat of 
vaporization is 77.1 calories. Calculate its molecular weight. 

4 The heat of vaporization of liquid ammonia at its boiling-point, 
under atmospheric pressure ( 33.5 C.) is 341 calories. Is liquid am- 
monia associated? 

5. Calculate the molecular volume of ethyl butyrate. The molecular 
volume determined by experiment is 149.1. 

6. For propionic acid, d = 1.0158 and nD = 1.3953. Calculate the 
molecular refraction by the formula of Lorenz-Lorentz and compare the 
value so obtained with that derived from the atomic refractions of the 
constituent elements. 

7. The density of ether is 0.7208, of ethyl alcohol, 0.7935 and of a 
mixture of ether and alcohol containing p per cent of the latter, 0.7389. 
At 20 C. the refractive indices for sodium light are, for ether, 1.3536, for 
alcohol, 1.3619, and for the mixture, 1.3572. Calculate the value of p, 
using the Gladstone and Dale formula. A us. 20.81. 

8. At 20 C. the density of chloroform is 1.4823 and the refractive 
index for the D-line is 1.4472. Given the atomic refractivities of carbon 
and hydrogen, calculate that of chlorine, using the Lorenz-Lorentz 
formula. Ans. 5.999. 

9. Calculate the surface tension of benzene in dynes per centimeter, 
the radius of the capillary tube being 0.01843 cm., the density of the 
liquid, 0.85, and the height to which it rises in the capillary, 3.213 cm. 

Ans. 24.71 dynes /cm. 

10. Find the molecular weight of benzene, the surface tension at 
46 C. being 24.71 dynes per centimeter, its critical temperature, 288.5 C,, 
its density, 0.85 and the value of k = 2.12. Ans. 77.7. 

11. At 14.8 C. acetyl chloride (density = 1.124) ascends to a height 
of 3.28 cm. in a capillary tube the radius of which is 0.01425 cm. At 
46.2 C. in the same tube the elevation is 2.85 cm. and the density = 
1.064. Calculate the critical temperature of acetyl chloride. 

Ans. 234.6C. 

12. From a certain tip the weights of a falling drop of benzene are 
35.329 milligrams (temp. = 11.4, density = 0.888) and 26.530 milli- 
grams (temp. = 68.5, density = 0.827). The molecular weight is the 
same in the liquid and gaseous states. Calculate the critical temper- 
ature of benzene. Ans. 286M C. 



CHAPTER VIII. 
SOLIDS. 

General Properties of Solids. Solids differ from gases and 
liquids in possessing definite, individual forms. Matter in the 
solid state is capable of resisting considerable shearing and tensile 
stresses. In terms of the kinetic theory of matter, the mutual 
attractive forces exerted by the molecules of solids must be re- 
garded as superior to the attractive forces between the molecules 
of gases and liquids. With one or two exceptions all solids ex- 
pand when heated, but there is no simple law expressing the relation 
between the increment of volume and the temperature. Rigidity is 
another characteristic property of solids, it being much more ap- 
parent in some than in others. Many solids are constantly under- 
going a process of transformation into the gaseous state at their 
free surfaces, such a change being known as sublimation. Just 
as when a gas is sufficiently cooled it passes into the liquid state, 
so on cooling a liquid below a certain temperature, it passes into 
the solid state. The reverse transformations are also possible, a 
solid being liquefied when sufficiently heated, and the resulting 
liquid completely vaporized if the heating be continued. Heat 
energy is required to effect transition from the solid to the liquid 
state, just as heat energy is required to effect transition from the 
liquid to the gaseous state. 

Obviously a substance in the solid state contains less energy 
than it does in the liquid state. The number of calories required 
to melt 1 gram of a solid substance is called its heat of fusion. 
It is often difficult to decide whether a substance should be classi- 
fied as a solid or as a liquid. For example the behavior of certain 
amorphous substances such as pitch, amber and glass, is similar 
to that of a very viscous, inelastic liquid. Solids are generally 
classified as crystalline or amorphous. In crystalline solids the 
molecules are supposed to be arranged in some definite order, this 

153 



154 THEORETICAL CHEMISTRY 

arrangement manifesting itself in the crystal form. An amor- 
phous solid on the other hand may be considered as a liquid 
possessing great viscosity and small elasticity. The physical 
properties of amorphous solids have the same values in all direc- 
tions, whereas in crystalline solids the values of these properties 
may be different in different directions. When an amorphous 
solid is heated it gradually softens and eventually acquires the 
properties characteristic of a liquid, but during the process of heat- 
ing there is no definite point of transition from the solid to the 
liquid state. On the other hand, when a crystalline solid is 
heated there is a sharp change from one state to the other at a 
definite temperature, this temperature being termed the melting- 
point. 

Crystallography. The study of the definite geometrical forms 
assumed by crystalline solids is termed crystallography. The 
number of crystalline forms known is exceedingly large, but it is 
possible to reduce the many varieties to a few classes or systems 
by referring their principal elements the planes to definite 
lines called axes. These axes are so drawn within the crystal that 
the crystal surfaces are symmetrically arranged about them. 
This system of classification was proposed by Weiss in 1809. 
He showed that notwithstanding the multiplicity of crystal forms 
encountered in nature, it is possible to consider them as belonging 
to one of six systems of crystallization. 

The six systems of Weiss are as follows: 

1. The Regular System. Three axes of equal length, inter- 
secting each other at right angles (Fig. 41). 

2. The Tetragonal System. Two axes of equal length and the 
third axis either longer or shorter, all three axes intersecting at 
right angles (Fig. 42). 

3. The Hexagonal System. Three axes of equal length, all in 
the same plane and intersecting at angles of 60, and a fourth axis, 
either longer or shorter and perpendicular to the plane of the 
other three (Fig. 43). 

4. The Rhombic System. Three axes of unequal length, all 
intersecting each other at right angles (Fig. 44). 



SOLIDS 



155 



5. The Monodinic System. Three axes of unequal length, two 
of which intersect at right angles, while the third axis is per- 
pendicular to one and not to the other (Fig. 45). 

6. The Tridinic System. Three axes of unequal length no two 
of which intersect at right angles (Fig. 46). 

The position of a plane in space is determined by three points 
in a system of coordinates, and consequently the position of the 



a 
Fig. 41. 



Q 

Fig. 42. 





Fig. 44. 



Fig. 45. 



Fig. 46. 



face of a crystal is likewise determined by its points of inter- 
section with the three axes, or by the distances from the origin 
of the system of coordinates at which the plane of the crystal 
face intersects the three axes. These distances are called the 
parameters of the plane. 
The fundamental law of crystallography discovered by Steno 



156 THEORETICAL CHEMISTRY 

in 1669 may be stated thus: The angle between two given crystal 
faces is always the same for the same substance. The fact that 
every crystalline substance is characterized by a constant inter- 
facial angle, affords a valuable means of identification which is 
used by both chemists and mineralogists. The instrument em- 
ployed for the measurement of the interfacial angles of crystals is 
called a goniometer. The crystal to be measured is mounted at 
the center of the graduated circular table of the goniometer, and 
the image of an illuminated slit, reflected from one surface of the 
crystal, is brought into coincidence with the cross-wires in the eye- 
'piece of the telescope. The table is then turned until the image 
of the slit, reflected from the adjacent face of the crystal, coincides 
with the cross-wires. The interfacial angle of the crystal is de- 
termined by the number of degrees through which the table has 
been turned. 

Properties of Crystals. The properties of all crystals, except 
those belonging to the regular system, exhibit differences, depend- 
ent upon the direction in which the particular moasurernents 
are made. Thus the elasticity, the thermal and electrical con- 
ductivities, and in fact all of the physical properties of crystals 
which do not belong to the regular system, have different values 
in different directions. Crystals whose physical properties have 
the same values in all directions are termed isotropic, while those 
in which the values are dependent upon the direction in which 
the measurements are made, are called anisotropic. Crystals 
belonging to the regular system, and amorphous substances are 
isotropic. Certain amorphous substances, such as glass, which 
are normally isotropic, may become anisotropic when subjected 
to tension or compression. The phenomenon of double refrac- 
tion observed in all crystals, except those belonging to the regular 
system, is due to their anisotropic character. Crystals belonging 
to the tetragonal and hexagonal systems resemble each other in 
one respect, viz. : that in all of them there is one direction, called 
the optic axis, or axis of double refraction (coincident with the prin- 
cipal crystallographic axis), along which a ray of light is singly 
refracted, while in all other directions it is doubly refracted. In 
crystals belonging to the rhombic, monoclinic, and triclinic systems, 



SOLIDS 157 

there are always two directions along which a ray of light is singly 
refracted. A crystal of Iceland spar (CaC0 3 ) affords a beauti- 
ful illustration of double refraction. On placing a rhomb of this 
substance over a piece of white paper on which there is an ink 
spot, two spots will be seen. On turning the crystal, one spot will 
remain stationary while the other spot will revolve about it. This 
property of Iceland spar is utilized in the construction of Nicol 
prisms for polariscopes. 

The examination of sections of anisotropic crystals in a polari- 
scope between crossed Nicol prisms, reveals something as to their 
crystal form. As has been stated, crystals of the tetragonal and 
hexagonal systems are uniaxiaL If a section is cut from such a 
crystal perpendicular to the optic axis, and this is placed between 
the crossed Nicol prisms of a polariscope, in a convergent beam 
of white light, a dark cross and concentric, spectral-colored circles 
will be observed, Fig. 47. Upon turning the analyzer through 
90 the colors of the circles will change to the respective comple- 
mentary colors and the dark cross will become light. Crystals 
of the rhombic, monoclinic, and triclinic systems are biaxial. If 
a section of a biaxial crystal, cut perpendicular to the line bisect- 
ing the angle between the two axes, be placed in. the polatiscope 






Fig. 46. Fig. 47. 

and examined as in the preceding case, a series of concentric 
spectral-colored lemniscates surrounding two dark centers and 
pierced by dark, hyperbolic brushes, will be observed, as shown in 
Fig 48. On rotating the analyzer, the colors will change to the 
corresponding complementary colors, as in the case of uniaxial 
crystals. The appearance of these figures is so varied and char- 
acteristic as to furnish, in many cases, a very satisfactory means 
of identifying anisotropic crystals. 



158 THEORETICAL CHEMISTRY 

Etch Figures. The solubility of crystals has been shown to 
be different in different directions. Thus, if the surface of a crys- 
talline substance be highly polished and then treated for a short 
time with a suitable solvent, faint patterns, known as etch figures, 
will appear as a result of the inequality of the rate of solution, in 
different directions. When these figures are examined under the 
microscope the crystal-form can generally be determined. The 
examination of etch figures has come to be of prime importance 
to the metallographer. Thus, when an appropriate solvent is 
applied to the polished surface of an alloy, not only is the crystal 
form revealed by the etch figures, but also the presence of various 
chemical compounds may be recognized. By a careful study 
of the etch figures developed on the surface of highly polished 
steel, the metallographer may gather important information as to 
its previous history, especially its heat treatment. 

Crystal Form and Chemical Composition. From the pre- 
ceding paragraphs it might be inferred that the same substance 
always assumes the same crystal form. While this is true 
in general, there are some substances which appear in several 
different crystal forms. This phenomenon is termed polymor- 
phism. 

Calcium carbonate is an example of a substance crystallizing in 
more than one form. As calcite, it crystallizes in the hexagonal 
system, while as aragonite, it crystallizes in the rhombic system. 
Such a substance is said to be dimorphous. Of the several factors 
controlling polymorphism, temperature is the most important. 
Thus sulphur crystallizes at temperatures above 95.6 in the mon- 
oclinic system, while at lower temperatures it assumes the 
rhombic form. The temperature at which it changes from one 
form into the other is termed its transition temperature. As has 
been mentioned in an earlier chapter (p. 14), some substances 
may crystallize in the same form, the characteristic interfacial 
angles being nearly identical. Such substances are said to be iso- 
morphous. This phenomenon, discovered by Mitscherlich, has 
been of great use in connection with the earlier investigations on 
atomic weights, as has already been pointed out. 

There can be little doubt as to the existence of an intimate 



SOLIDS 139 

* 

connection between crystalline form and chemical composition. 
Ever since the early part of the nineteenth century, when Hauy 
established the science of crystallography, various attempts have 
been made by chemists and crystallographers to connect crystal- 
line form with chemical constitution. In 1906, Barlow and Pope * 
made a most notable contribution to the theories concerning the 
relation between crystalline form and chemical constitution. 
Their ideas may be summarized as follows: If each atom be 
considered as appropriating a certain space, called its sphere of 
atomic influence, then (1) the spheres of atomic influence are so 
arranged as to occupy the smallest possible volume in every crystal; 
(2) the volumes of the spheres of atomic influences in any substance 
are proportional to the valences of the constituent atoms; (3) the 
volumes of the spheres of influence of the atoms of different elements 
of the same valence are nearly equal, any variation being in har- 
mony with their relations in the periodic system. Barlow and Pope 
have shown that the general agreement between theory and 
observation is most satisfactory, a particularly strong argument 
in favor of this theory being the very plausible explanation which 
it furnishes for a large number of crystallographic facts. It is 
without doubt the best working hypothesis which has yet been 
offered for the investigation of the dependence of crystalline form 
upon a definite chemical constitution. 

Compressibilities of the Solid Elements. A series of careful 
measurements of the compressibilities of the elements by T. W. 
Richards and his collaborators^ has revealed the fact that com- 
prefosibility is a periodic function of atomic weight. Richards 
has advanced some interesting suggestions as to the importance 
of compressibility in connection with intermodular cohesion 
and atomjjp volume. In fact, Richards' theory of compressible 
atoms may be regarded as a valuable supplement to the the- 
ory of Barlow and Pope and, taken together, these two theories 
constitute a rational basis for the science of chemical crystallog- 
raphy. 

X-Rays and Crystal Structure. In 191$fllaue pointed out 

* Jour. Chem. Soc., pi, 1150 (1907). 

t Zeit. phys. Chein'., 6x, 77, 100, 171, 183 (1008), 



160 THEORETICAL CHEMISTRY 

that the regularly arranged atoms or molecules of a crystal should 
act as a three-dimensional diffraction grating toward the X-rays. 
He showed mathematically that on traversing a thin section of a 
crystal, a pencil of X-rays should give rise to a diffraction pattern 
arranged symmetrically round the primary beam as a center. A 
photographic plate placed perpendicular to the path of the rays 
and behind the crystal should reveal, on development, a central 
spot due to the action of the primary rays, and a series of symmetri- 
cally grouped spots due to the diffracted rays. 

Laue's predictions were verified experimentally by Friedrich 
and Knipping, who obtained numerous plates showing a va- 
riety of geometrical patterns corresponding to the structural dif- 
ferences of the crystals examined. The analysis of the Laue 
diffraction patterns, while furnishing valuable information as to 
the internal structure of crystals, is nevertheless extremely 
complex. 

W. H. Bragg and his son W. L. Bragg * have devised an X-ray 
spectrometer in which use is made of the fact that the regularly 
spaced atoms of a crystal reflect the X-rays in much the same way 
that light is reflected (diffracted) by a plane grating. By observing 
the angles of reflection from the different faces of a crystal for an 
incident radiation of known wave-length, it is an easy matter to 
calculate the distances between the atoms of the crystal which 
function as diffraction centers. 

By means of the X-ray spectrometer the internal structure of a 
number of crystals has been determined. One of the most in- 
teresting results to the chemist is that obtained with the diamond. 
The X-ray spectra of the diamond reveal the fact that each carbon 
atom is situated at the center of a regular tetrahedron formed by 
four other carbon atoms. 

In commenting on this method of studying crystal structure, 
W, H. Bragg says: " Instead of guessing the internal arrange- 
ment of the atomtJrom the outward form assumed by the crystal, 
we find ourselvift^rae to measure the actual distances from atom 
to atom and to oMw a diagram as if we were making a plan of a 
building." 

* See "X-Rays and Crystal Structure," by W. H. Bragg and W. L. Bragg. 



SOLIDS 161 



Heat Capacity of Solids. Recent investigations of s 
heats of solids at extremely low temperatures have resulted in 
the formulation of several interesting relationships between heat 
capacity and temperature. 

At ordinary temperatures the molecules of a crystalline solid 
may be assumed to be in a state of violent, unordered motion. As 
the temperature is lowered, the amplitude of the molecular oscil- 
lations steadily diminishes until finally, at the absolute zero, there 
is in all probability a complete cessation of motion. In the neigh- 
borhood of the absolute zero, where the amplitude of the molecular 
oscillations is negligible, a crystalline solid may be assumed to 
possess the properties characteristic of a perfectly elastic body. 
In other wordtf, the crystalline forces holding the molecules to- 
gether would preponderate over the feeble thermal forces tending 
to initiate molecular oscillations within the solid. Under these 
conditions the solid as a whole would exhibit the same behavior as 
a single molecule, that is to say, the solid would function as a 
perfectly elastic body. 

pn this assumption Debye * has derived the following equation 
expressing the heat capacity of a solid, C v , in terms of its absolute 
temperature T, 

' 



In this equation is a constant characteristic of each solid and has 
the same dimensions as T. The value of 6 varies between the 
lllits 6 = 50 for calcium and 6 = 1840 for carbon. The agree- 
ment between the observed and calculated values of C v has been 
found to be excellent up to T = 6/12. 

Whea^us latter temperature is exceeded, the molecules of the 
solid begm to absorb more and more heat energy and to vibrate 
independently about their centers of oscillation. The failure of 
Debye's equation is to be expected under these conditions since 
the solid is no longer behaving as one large||ji&eule. Obviously 
the lighter the molecules and the greater "^P'crystalline forces 
. within the solid, the higher must the temperature become before 

* Ami. Physik., 39* 789 (1912). 



162 THEORETICAL CHEMISTRY 

the individual molecules can acquire appreciable kinetic energy. 
This is apparent from the familiar dynamical principle, that the 
kinetic energy of a vibrating particle is proportional to its mass 
and to the square of its vibration frequency. In the case of lead, 
which is a soft, malleable solid with a relatively low melting-point, 
it is reasonable to infer that the crystalline forces are feeble, and 
consequently we should expect that molecular and atomic vibra- 
tions would be set up at quite low temperatures. Furthermore, 
since the atoms of lead are extremely heavy, their kinetic energy 
must be great. The correctness of these conclusions is confirmed 
by the fact that the Debye equation when applied to lead has been 
found to hold only over a very short range _of temperature. On 
the other hand, the equation has. beeif found to hold for the 
diamond up to a temperature of about 'SOO 9 absolute. In this 
case we have a solid in which the crystalline forces are extremely 
powerful and in which the atoms are relatively light. A fairly 
high temperature must be attained before the energy absorbed 
by the individual atoms of the diamond acquires appreciable 
magnitude. 

The absorption of energy by the vibrating molecules continues 
to increase as the temperature is raised until ultimately, at the 
melting point of the solid, the crystalline forces become negligible. 
As this temperature is approached therefore, the intermolecu- 
lar restraint becomes less and less and the mean kinetic energy 
of the molecules approaches that of the molecules of the molten 
solid. 

As has already been stated in Chapter I (p. 11), Dulong and 
Petit, in 1819, discovered the interesting fact that the atomic heats 
of the solid elements have a constant value of 6.5. The importance* 
of this generalization in connection with the verification of atomic 
weights has already been pointed out. Quite recently, Lewis * 
has directed attention to the fact that it is much more rational to 
calculate the atomic heat of an element from the specific heat at 
constant volume rather than from the specific heat at constant 
pressure. While it is impossible to measure the specific heat at 
constant volume, its value may be derived from the specific heat 
* Jour. Am. Chem. Soc,, 29, 1165 (1907). 



SOLIDS 163 

at constant pressure by an application of the laws of thermo- 
dynamics. Thus, Lewis has obtained the formula 

c - c - ^!H 

" Uv ~ 41.78/3' 

where T denotes the absolute temperature, a the coefficient of 
expansion, the coefficient of compressibility, C p and C v the 
atomic specific heats at constant pressure and constant volume 
respectively and V the atomic volume. By means of this equa- 
tion, Lewis has established the following generalization: Within 
the limits of experimental error, the atomic heat at constant volume, 
at 20 C., is the same for all solid elements whose atomic weights are 
greater than that of potassium, and is equal to 5.9. In the case of a 
solid having a high melting-point, the violent agitation of its con- 
stituent molecules and atoms as the temperature is raised, will 
undoubtedly produce a corresponding increase in the amplitude of 
vibration of its electrons together with an increase in their trans- 
lational velocity among the molecules. Under these conditions, 
the specific heat at constant volume should be greater than 5.9 
calories. This conclusion cannot be verified experimentally until 
the values of a and /3 in the foregoing equation have been deter- 
mined at high temperatures. 

The complete heat capacity curves for three typical solid ele- 
ments, lead, aluminium, and carbon, are given in Fig. 49. It is 
apparent from these curves that the absorption of heat energy by 
a crystalline solid may be considered as taking place in three dis- 
tinct stages, as follows: (1) In the neighborhood of the abso- 
lute zero, the heat capacity remains practically zero; (2) the heat 
capacity increases rapidly with the temperature; and (3) the heat 
capacity increases slowly, approaching asymptotically the limiting 
value 5.9 for T = . For a malleable, low melting element of 
high atomic weight, such as lead, the first two stages are very 
short and the final stage commences at a low temperature. On 
the contrary, with a hard, high melting element of low atomic 
weight, such as carbon in the form of diamond, the final stage is 
not reached at any temperature within the range covered by the 
experiments. 



164 



THEORETICAL CHEMISTRY 



The Nernst-Lindemann Equation. Recently, several equa- 
tions have been derived expressing the heat capacity of a solid in 
terms of temperature and arbitrary constants. All of these 
equations are based upon the so-called "quantum theory" accord- 
ing to which the absorption of heat energy by matter is supposed 
to take place in a discontinuous manner, the discrete units of 
energy being termed quanta. While the discussion of the quantum 
theory and the equations connecting heat capacity and tempera- 
ture lies outside of the scope of this book, mention should never- 
theless be made of the empirical equation derived by Nernst and 
Lindemann.* This equation gives values of C v which are in re- 




500 



markably close agreement with the values determined by direct 
experiment. The equation may be written in the following form : 



c,= 



T 





pT 



'-I-' 



2T 



(e T I) 2 (e 2 T 1) 

In this expression, R is the molecular gas constant R = 2 calories, 
6 is the base of the natural system of logarithms and is a constant 
depending upon the nature of the solid. 

* Zeit. Elektrochem., 17, 817 (1911). 



SOLIDS 165 

The value of may be calculated with a fair degree of accuracy 
by means of the equation 



in which T/ denotes the absolute melting-point of the substance, 
A its atomic weight and d its density. 

Liquid Crystals. In addition to possessing well-defined geo- 
metrical forms, crystalline substances are characterized by their 
resistance to deformation when subjected to mechanical stress, 
and by the property of melting sharply at definite temperatures 
with the production of transparent liquids. 

In 1888, two substances, cholesteryl acetate and cholesteryl 
benzoate, were found by Reinitzer * to behave in an anomalous 
manner when heated. At definite temperatures these substances 
melted to turbid liquids which, in turn, became clear on further 
heating, the latter change also taking place at definite tempera- 
tures. On cooling the clear liquids, the reverse series of changes 
was found to occur. 

Examination of the turbid liquids revealed the fact that they 
resembled ordinary liquids in their general behavior, such as assum- 
ing the spherical shape when suspended in a medium of the same 
density, or of rising in a capillary tube under the influence of sur- 
face tension. But in addition to possessing the properties char- 
acteristic of the liquid state, Lehmann discovered that they 
possessed optical properties which had hitherto been observed 
only with solid, crystalline substances. Their behavior towards 
polarized light was such as to warrant the conclusion that these 
turbid liquids are anisotropic. In view of these facts, Lehmann 
proposed that liquids possessing these properties should be called 
liquid crystals, the term implying that under ordinary condi- 
tions, the crystalline forces in these substances are so feeble that 
the crystals readily undergo deformation and actually flow like 
liquids. That these turbid liquids are not emulsions, is proven 
by the fact that when they are examined under the micro- 
scope, the turbidity is found to be due to the aggregation of a 

* Monatshefte, 9, 435 (1888). 



166 THEORETICAL CHEMISTRY 

myriad of differently oriented transparent crystals. All subse- 
quent investigation of liquid crystals has failed to show any lack 
of homogeneity. 

The number of such substances known at the present time is 
fairly large. 



CHAPTER IX. 
SOLUTIONS. 

Classification of Solutions. Having dealt with the properties 
of pure substances in the gaseous, liquid and solid states we now 
proceed to the consideration of the properties of mixtures of two 
or more pure substances. When such a mixture is chemically 
and physically homogeneous, and no abrupt change in its prop- 
erties results from an alteration of the proportions of the com- 
ponents of the mixture, it is termed a solution. When one 
substance is dissolved in another, it is customary to designate as 
the solvent that component which is present in the larger proportion, 
the other component being termed the solute. When not more 
than one-tenth mol of solute is present in one liter of solution, the 
solution is said to be dilute. The detailed study of dilute solu- 
tions will be deferred until the next chapter. 

There are nine possible classes of solutions, as follows: 

(1) Solution of gas in gas; v^ 

(2) Solution of liquid in gas; v 

(3) Solution of solid in gas; > 

(4) Solution of gas in liquid; w^ 

(5) Solution of liquid in liquid; 

(6) Solution of solid in liquid; ^ 

(7) Solution of gas in solid; 

(8) Solution of liquid in solid; 

(9) Solution of solid in solid. 

While examples of all of these different types of solutions are 
known, only the more important classes will be considered here. 

Solutions of Gases in Gases. In solutions of this class the 
components may be present in any proportions, since gases are 
completely miscible. In a mixture of gases where no chemical 
action occurs, each gas behaves independently, the properties of 

167 



168 



THEORETICAL CHEMISTRY 



the gaseous mixture being the sum of the properties of the con- 
stituents. Thus, the total pressure of a mixture of several gases is 
equal to the sum of the pressures which each gas would exert were it 
alone present- in the volume occupied by the mixture. This law was 
discovered by Dalton * and is known as Dalton's law of partial 
pressures. If the partial pressures of the constituent gases be 
denoted by pi, p 2 , PS, etc., and P and V represent the total pres- 
sure and the total volume of the gaseous mixture, then 

PV = V (pi + p* + p 3 +) 

Dalton's law holds when the partial pressures are not too great, 

its order of validity being the same 
as that of the other gas laws. Dai- 
ton's law can be tested experimen- 
tally by comparing the total pressure 
of the gases with the sum of the pres- 
sures exerted by each gas before 
mixture. van't Hoff pointed out 
the possibility of measuring the par- 
tial pressure of one of the two com- 
ponents of a gas mixture, provided 
a diaphragm could be found which 
would be pervious to one of the 
gases but not to the other. It was 
shown shortly afterward by Ram- 
say,! that the walls of a vessel of 
palladium, when sufficiently heated, 
permit the free passage of hydrogen 
but not of nitrogen. The walls are 




Hydrogen 



Fig. 50. 



said to be semi-permeable. A sketch of the apparatus used by 
Ramsay in the verification of Dalton's law is shown in Fig. 50. 
A small vessel of palladium, P, containing nitrogen, is connected 
with a manometer AB, which serves to measure the pressure of the 
gas in P. The vessel P is enclosed within a larger vessel C, which 
can be filled with hydrogen at known pressure. On heating P and 

* Gilb. Ann., 12, 385 (1802). 
t Phil. Mag. (5), 38, 206 (1894). 



SOLUTIONS 169 

passing a current of hydrogen at a definite pressure through C, the 
hydrogen enters P until the pressures due to hydrogen inside and 
outside are equal. The total pressure in P, measured on the 
manometer, is greater than the pressure in C. The difference 
between the two pressures is very nearly equal to the partial pres- 
sure of the nitrogen. Conversely, if a mixture of the two gases 
be introduced into P, which is then heated and maintained at suf- 
ficiently high temperature to insure its permeability to hydrogen, 
the partial pressure of the nitrogen can be determined by passing 
a current of hydrogen at known pressure through C until equilib- 
rium is attained, as shown by the manometer. The difference 
between the external and internal pressures is the partial pressure 
of the nitrogen. This experiment has a very important bearing 
upon the modern theory of solution. 

Solutions of Gases in Liquids. The solubility of gases in 
liquids is limited, the extent to which they dissolve depending 
upon the pressure, the temperature, the nature of the gas, and 
the nature of the solvent. When a liquid cannot absorb any 
more of a gas at a definite temperature, it is said to be saturated, 
and the solution is called a saturated solution. The solubility of 
a gas in a liquid is defined by Ostwald as the ratio of the volume 
of the gas absorbed to the volume of the absorbing liquid at a 
specified temperature and pressure, or if the solubility of the gas 
be represented by S 9 we have 

S = vfV, 

where v is the volume of gas absorbed and V is the volume of the 
absorbing liquid. The " absorption coefficient" of Bunsen in 
terms of which he expressed the results of his measurements of 
the solubility of gases, may be defined as the volume of a gas, 
reduced to C. and 76 cm. pressure which is absorbed by unit 
volume of a liquid at a certain temperature and under a pressure 
of 76 cm. of mercury. In certain cases the volume of the gas 
absorbed is found to be independent of the pressure, so that if a 
is the coefficient of gaseous expansion, and ft Bunsen's coefficient 
of absorption, then 

S- 0(1 + 0. 



170 



THEORETICAL CHEMISTRY 



The solubilities of a few gases in water and alcohol as determined 
by Bunsen are given in the following table: 





Watei. 


Alcohol. 


Gas 











15 





15 


Hydrogen 


0215 


0190 


0693 


0673 


Oxygen . . 


0489 


0342 


2337 


2232 


Carbon dioxide. . . . 


1 797 


1002 


4 330 


3 199 



The solubility of gases in water is appreciably diminished by 
the presence of dissolved solids or liquids, especially electrolytes. 
Various theories have been proposed to account for the diminished 
solubility of gases in salt solutions but the most satisfactory is 
that due to Philip,* who suggests that the phenomenon is caused 
by the hydration of the dissolved salt. A portion of the water 
in the salt solution is supposed to be in combination with the salt, 
the water which is thus removed from the role of solvent, being 
no longer free to absorb gas. The solubility of a gas increases 
with increase in pressure. For gases which do not react chem- 
ically with the solvent, there exists a simple relation between 
pressure and solubility, discovered by Henry.f This relation, 
known as Henrys law may be stated as follows: When a gas is 
absorbed in a liquid, the weight dissolved is proportional to the pressure 
of the gas. Since pressure and volume, at constant temperature, 
are inversely proportional (Boyle's law), the law of Henry may be 
stated thus: The volume of a gas absorbed by a given volume of 
liquid is independent of the pressure. There is yet another form 
in which the law may be stated which is instructive in connection 
with the modern theory of solution. When a definite volume of 
liquid is saturated with a gas at constant temperature and pres- 
sure, a condition of equilibrium is established between the gas in 
solution and that in the free space over the solution, therefore, 
Henry's law may be stated as follows: The concentration of the 
dissolved gas is directly proportional to that in the free space above 

* Trans. Faraday Soc., 3, 140 (1907). 
t Gilb. Ann., 20, 147 (1805). 



SOLUTIONS 171 

the liquid. If Ci represents the concentration of the gas in the 
liquid and c% the concentration in the free space above the liquid, 
Henry 's law may be expressed thus: Ci/c% = k, 
where k is known as the solubility coefficient. 

Dalton showed that the solubility of the individual gases in a 
mixture of gases is directly proportional to their partial pressures, 
the solubility of each gas being nearly independent of the presence 
of the others. 

As will be seen from the foregoing table, the solubility of a gas 
in a liquid diminishes with increase in temperature.* Concerning 
the influence of the nature of the gas on its solubility, it may be 
said that those gases which exhibit acid or basic reactions are 
the most soluble, the solubilities of neutral gases being small. 
In the case of many of the very soluble ffasesJHenry's law does 
riot hold. For example, ammonia, a gas having marked basic 
properties and a large coefficient of solubility, does not obey 
Henry's law at ordinary temperatures, the mass of ammonia 
absorbed not being proportional to the pressure. The curve 
showing the variation in solubility with pressure at C. has 
two marked discontinuities. At temperatures above 100 C. 
the gas obeys Henry's law. Sulphur dioxide behaves similarly, 
the law holding only for temperatures exceeding 40 C. 

With regard to the connection between the solvent power of a 
liquid and its nature but little is known. About all that can be 
said is, that the order of aolubilitff_of gasejyn different J^uidsis^ 
the sam._ Thus in the preceding table the solubilities of hydro* 
gen, oxygen and carbon dioxide in water and in alcohol will be 
seen to be approximately proportional. A slight ' change in 
volume always results when a gas is dissolved in a liquid. In 
general it may be said that the less compressible a gas is, the 
greater is the increase in volume produced when it is absorbed by 
a liquid. It is of interest to note that the increase in volume 
caused by the solution of a gas is nearly equal to the value of b 
for the gas in the equation of van der Waals. This is shown m 
the following table: 

* Helium is an exception to the rule that the solubility of a gas in a liquid 
diminishes with increase in temperature. The absorption coefficient of He 
diminishes from to 25 and then increases again as the temperature is raised. 



172 THEORETICAL CHEMISTRY 



Gas. 


Increase in Vol. 


& 


Oxygen ... 


00115 


000890 


Nitrogen 


00145 


001359 


Hydrogen ... 


00106 


000887 


Carbon dioxide 


00125 


OQ0866 









solutions of Liquids in Liquids. Solutions of liquids in liquids 
can be divided into three classes as follows: (1) Liquids which 
are miscible in all proportions; (2) Liquids which are partially 
miscible; and (3) Liquids which are immiscible. Examples of 
these three classes in the order mentioned are, alcohol and water, 
ether and water, and benzene and water. As to the cause of 
miscibility and non-miscibility of liquids very little is known. 

Partial Miscibility. If a small amount of ether is added to 
a large volume of water in a separatory funnel and the mixture 
vigorously shaken, a perfectly homogeneous solution will be 
obtained. On gradually increasing the amount of ether, shaking 
after each addition, a concentration will eventually be reached 
at which a separation into two layers will take place. The upper 
layer is a saturated solution of water in ether and the lower layer 
is a saturated solution of ether in water. So long as the relative 
amounts of the two liquids is such that the mixture does not 
become homogeneous on standing, the composition of the two 
layers will be independent of the relative amounts of the two 
components. Measurements of the mutual solubility of liquids 
have been made by Alexieeff * by placing weighed amounts in 
sealed tubes and observing the temperature at which the mixture 
became homogeneous. In general the solubility of a pair of 
partially miscible liquids increases with the temperature, and 
therefore it may be inferred that at a sufficiently high temperature 
the mixture will become perfectly homogeneous. An example of 
this type of binary mixture is furnished by phenol and water, the 
solubility curve of which is shown in Fig. 51. In this diagram 
temperature is plotted on the axis of ordinates and percentage 
composition of the solution on the axis of abscissae. Starting 

* Jour, prakt. Chem., 133, 518 (1882); Bull. Soc. Chem., 38, 145 (1882). 



SOLUTIONS 



173 



with a small amount of phenol and adding it in increasing quan- 
tities to a large volume of water, a concentration will eventually 
be reached at which the solution will separate into two layers. 
This concentration is represented by the point A. On raising the 



100* 



Percentage Water in Phenol 




i Percentage Phenolln Water 

Fig. 51. 



100* 



temperature, the solubility of phenol in water increases, as shown 
by the curve AB. In like manner, starting with pure phenol 
and adding increasing amounts of water, separation into two layers 
will occur at a concentration represented by the point C. As the 
temperature is raised the solubility of water in phenol increases, 
as shown by the curve CB. When the temperature is raised 
above 68.4 C., corresponding to the point B, phenol and water 
become miscible in all proportions. 

If we start with a solution whose temperature and composition 
is represented by the point a, the addition of increasing amounts 
of phenol, at constant temperature will be represented by the 
dotted line afed. When the point / is reached, the solution will 
separate into two layers the composition of which will be inde- 
pendent of the relative amounts of phenol and water. At e the 
solution will again become homogeneous. If the solution repre- 



174 THEORETICAL CHEMISTRY 

sented by the point a be again chosen as the starting point, and its 
composition be kept unaltered while the temperature is raised to a 
value above 68. 4 C., the change will be represented by the dotted 
line ab. If now the temperature be maintained constant and the 
percentage of phenol increased, the alteration in composition will 
be effected without discontinuity, as represented by the dotted 
line be. On cooling the solution represented by the point c to the 
initial temperature of a, the point d will be reached. Thus it is 
possible to pass from a to d by the path abed without causing a 
separation of the components into two layers. There is an 
analogy between the solubility curve of a pair of partially mis- 
cible liquids and the dotted, parabolic curve in the diagram of 
the isothermals of carbon dioxide, shown in Fig. 23. In both 
cases there is but one phase outside of the curves, while two 
phases are coexistent within the area enclosed by the curves. The 
analogy may be traced further, since in each case only one phase 
can exist above a certain temperature. The temperature corre- 
sponding to the apex of the parabolic curve in Fig. 23, is termed 
the critical temperature of carbon dioxide and by analogy the 
temperature corresponding to the point, /?, in Fig. 51 is called the 
critical solution temperature. The mutual solubilities of some 
pairs of partially miscible liquids were found by Alexieeff to di- 
minish with increasing temperature. Thus a mixture of ether and 
water, which is perfectly homogeneous at ordinary temperatures, 
becomes turbid on warming. A specially interesting pair of 
liquids is nicotine and water. At ordinary temperatures these 
liquids are miscible in all proportions. If the temperature is 
raised above 60 C., the solution becomes turbid owing to incom- 
plete miscibility. On continuing to heat the mixture the 
mutual solubility of the liquids begins to increase, until at 
210 C. they become completely soluble again. The solubility 
relations of this binary mixture are shown in Fig. 52. The closed 
solubility curve defines the limits of the coexistence of two layers, 
all points outside of the curve representing homogeneous solu- 
tions. 

Complete Miscibility: The study of the vapor pressures of 
binary mixtures of completely miscible liquids is of great im- 



SOLUTIONS 



175 



portance in connection with the possibility of separating them 
by the process of distillation. The experimental investigations of 
Konowalow * on homogeneous binary mixtures of liquids have 
shown that such pairs of liquids may be divided into three classes 



100* 



Percentage Water in Nicotine 



810 



60' 



Percentage Nicotine in Water 

Fig, 52. 



100* 



as follows: (1) Mixtures having a maximum vapor pressure 
corresponding to a certain composition, e.g., propyl alcohol and 
water; (2) Mixtures having a minimum vapor pressure corre- 
sponding to a certain composition, e.g., formic acid and water, 
and (3) Mixtures having vapor pressures intermediate between 
the vapor pressures of the pure components, e.g., methyl alcohol 
and water. In considering the possibility of separating binary 
mixtures of liquids belonging to these three classes, it is essential 
to determine the composition of both solution and escaping vapor. 
When a pure liquid is boiled the composition of the escaping 
vapor is the same as that of the liquid itself, but this is, in general, 

* Wied. Ann., 14, 34 (1881). 



176 



THEORETICAL CHEMISTRY 



not the case when a binary mixture is distilled. The composition 
of the liquid mixture in the distilling flask generally alters contin- 
uously when such a mixture is distilled. 

(1) The relation between the vapor pressure and composition 
of all possible mixtures of propyl alcohol and water is represented 
graphically in Fig. 53. In this diagram the compositions of the 
mixtures are plotted as abscissae and vapor pressures as ordinates. 
The vapor pressures of the pure components, water and propyl 
alcohol, at a definite temperature are represented by A and C. 
The maximum in the yapor-pressurecurve corresponds to a mix- 



100* 



Water 




Propjl Alcohol 
Fig. 53. 



80* 



ture containing 80 per cent of propyl alcohol. The dotted curve 
represents the boiling-points of the various mixtures under normal 
atmospheric pressure. Konowalow has shown^jthat the vapor of 
abinary mixture)with JL minimum or maximum bqUing-pointfhas 
t^^m^jeomposition as_that of the liquid.) The vagor^of all 
mixtures containing less than 80 per cent of propyl alcohol will 
be relatively richer in alcohol than the liquid mixture, since the 
vapor of propyl alcohol is quite insoluble in water. If the amount 
of alcohol in the mixture exceeds 80 per cent, then the vapor will 
be relatively richer in water. Thus, whatever may be the com- 
position of the mixture in the distilling flask, the distillate will 
approximate to the composition of the mixture having the mini- 
mum boiling-point. The residue in the flask will gradually 



SOLUTIONS 



177 



change to pure water if the original concentration were below 
80 per cent, or to pure alcohol if the original concentration were 
above 80 per cent. 

(2) The second type of binary mixture of liquids is illustrated 
by formic acid and water, the vapor pressure and boiling-point 
curves for which are shown in Fig. 54. A mixture containing 
73 per cent of formic acid has a minimum vapor pressure and a 
maximum boiling-point. At this concentration the vapor and the 
liquid have the same composition. The vapor of mixtures con- 



100* 



Water 




formic Acid* 
Fig. 54. 

taining less than 73 per cent of acid is relatively richer in water 
than the liquid, while the vapor of mixtures containing more than 
73 per cent of acid contains relatively less water than the liquid. 
Any mixture of formic acid and water when distilled will thus 
leave a residue in the distilling flask containing 73 per cent of 
acid; this residue will distil at constant temperature like a homo- 
geneous liquid. It was thought for a long time that such constant 
boiling mixtures were definite chemical compounds of the two 
liquids. Thus a mixture of hydrochloric acid and water contain 
ing 20.2 per cent of acid boils at 110 C. under atmospheric pres- 
sure. The composition of such a mixture corresponds very nearly 
to the formula, HC1.8 H 2 O. Roscoe * showed that these mix- 

* Lieb. Ann., 116, 203 (1860). 



178 



THEORETICAL CHEMISTRY 



tures are not definite chemical compounds since the composition 
of the distillate changes when the distillation is carried out under 
different pressures. 

(3) The vapor-pressure and boiling-point curves for methyl 
alcohol and water, a mixture typical of the third class of com- 
pletely miscible liquids, are shown in Fig. 55, the heavy line 
representing vapor pressures at 65.2 C. and the dotted line the 
boiling points under normal, atmospheric pressure. In this case 



oo* 



Water 




OH Methyl Alcohol 100* 

Fig. 55. 

the composition of both vapor and liquid alter continuously on 
distillation. The distillate will contain a relatively larger amount 
of alcohol and the residue in the distilling flask, an excess of water. 
If this distillate be redistilled from a clean flask, a second dis- 
tillate still richer in alcohol will be obtained. By repeating this 
process a sufficient number of times, a more or less complete 
separation of the two components of the mixture can be effected. 
This process is termed fractional distillation. 

Immistibility. When two immiscible liquids are brought 
together, the total vapor pressure is equal to the sum of the vapor 
pressures of the components; hence when such a mixture is dis- 
tilled, the two liquids will pass over in the ratio of their respective 



SOLUTIONS 179 

vapor pressures, the boiling-point of the mixture being the temper- 
ature at which the sum of the vapor pressures of the two liquids 
is equal to the pressure of the atmosphere. The relation between 
vapor pressure and composition in this case will be represented 
by a horizontal line drawn at a distance above the axis of abscissae 
equal to the sum of the vapor pressures of the components. 

Nitrobenzene and water may be chosen as an example of a pair 
of liquids which are practically immiscible. Under a pressure 
of 760 mm. the mixture boils at 99 C. The vapor pressure of 
water at this temperature is 733 mm.; the vapor pressure of nitro- 
benzene muSt be 760 733 = 27 mm. Notwithstanding the 
relatively small vapor pressure of nitrobenzene in the mixture, 
considerable quantities of it distil over with the water. It is this 
fact that makes possible separations of liquids by the process of 
steam distillation so frequently employed by the organic chem- 
ist. The relative weights of water and nitrobenzene passing 
over in a steam distillation may be calculated as follows: The 
relative volumes of steam and vapor of nitrobenzene which distil 
over will be in the ratio of their respective vapor pressures at the 
temperature of the experiment, and consequently the relative 
weights of the two liquids which pass over will be in the ratio, 
p\di : ptfk, where pi and p 2 denote the respective vapor pressures 
of water and nitrobenzene, and d\ and d 2 the corresponding vapor 
densities. If w\ and w 2 denote the weights of the two liquids in 
the state of vapor, then 



or, since vapor density is proportional to molecular weight, we 
may write 



Substituting in this proportion the values given above for the vapor 
pressures of steam and nitrobenzene, we have 

MI : w 2 :: 733 X 18 : 27 X 123 
or, 

Wi : m :: 13,194 : 3321. 

Thus the weights of water and nitrobenzene in the distillate are 
approximately in the ratio of 4 to 1 notwithstanding the fact that 



180 



THEORETICAL CHEMISTRY 



the ratio of their vapor pressures at the boiling-point of the mix- 
ture is 27 to 1. If an organic substance is not decomposed by 
steam, it is possible to effect an appreciable purificationj>y steam 
distillation, even though its vapor pressure be relatively small. 
As will be seen from the above example, it is the 



weight^ of the nitrobenzene which compensates for its low vapor 
pressure. It is the small molecular weight ,of_ water which renders 
it so suitable for steam distillation. 

Finally, the vapor-pressure and boiling-point relations of binary 
mixtures of partially miscible liquids must be considered. In 



100* 



Water 






V 



700 mm.- 




Isobutyl Alcohol 
Fig. 56. 



100* 



general when two liquids are mixed, each lowers the vapor pressure 
of the other, so that the vapor pressure of the mixture is less than 
the sum of the vapor pressures of the components. As has al- 
ready been pointed out, the composition of the two layers in a 
binary mixture of partially miscible liquids is independent of 
the relative amounts of the components present; hence the vapor 
pressure remains constant so long as the solution remains hetero- 
geneous. The vapor-pressure and boiling-point curves for a 
binary mixture of partially miscible liquids (isobutyl alcohol and 
water) are shown in Fig. 56. The horizontal portion BC, repre- 
sents the vapor pressures, at 88.5 C., of mixtures of isobutyl 
alcohol and water where two layers are present. The vapor 



SOLUTIONS 181 

pressure of the homogeneous mixtures are represented by AB and 
CD, AB corresponding to solutions of isobutyl alcohol in water, 
and CD to solutions of water in isobutyl alcohol. The dotted 
line A'B'C'D' represents the boiling-points of all possible mix- 
tures of isobutyl alcohol and water, under normal atmospheric 
pressure. 

Solutions of Solids in Liquids. The solubility of a solid in a 
liquid is limited and is dependent upon the temperature, the 
nature of the solute and the nature of the solvent. When a 
solvent has taken up as much of a solute as it is capable of dis- 
solving at a definite temperature, the solution is said to be satu- 
rated. There are two general methods for the preparation of 
saturated solutions: (1) An excess of the finely-divided solute 
is agitated with a known amount of the solvent, at a definite 
temperature, until equilibrium is attained; (2) the solvent is 
heated with an excess of the solute to a temperature higher than 
that at which saturation is required, and then cooled in contact 
with the solid solute to the desired temperature. Both of these 
methods give equally satisfactory results provided sufficient 
time is allowed for the establishment of equilibrium, and provided 
the solid substance is always present in excess. The solubility 
of a solid in a liquid may be expressed as the number of grams of 
the solute in a given mass or volume of solvent or solution, but 
it is usually expressed as the number of grams of solute in 100 
grams of solution. The solubility of solids has recently been 
shown to be somewhat dependent upon their state of division. 
Thus, Hulett * has found that a saturated solution of gypsum at 
25 C. contains 2.080 grams of CaSO 4 per liter, whereas when 
very finely divided gypsum is shaken with this solution, it is 
possible to increase the content of dissolved CaSO 4 to 2.542 grams 
per liter. When a saturated solution is cooled, every _ trace of 
solid solMeJbSi^^ dissolved solid may not 

separate. jSuchLa solution is said" ^^bWsupef saturated. 

As ITgeneral r^^tTG^sduEniEy 15F's5li^ TiTTTq^ids increases 
with the temperature, as shown in Fig. 57. Several exceptions 
to this rule are known, among which may be mentioned calcium 

* Jour. Am. Chera. Soc., 27, 49 (1905). 



182 



THEORETICAL CHEMISTRY 



hydroxide, calcium sulphate above 40 C., and sodium sulphate 
between the temperatures of 33 C. and 100 C. 

Solubility curves are usually continuous, but exceptions to this 
rule are common: the solubility curve of sodium sulphate fur- 




40 00 80 100 

Temperature 

Fig. 57. 

nishes an illustration. The discontinuity in the solubility curve 
of sodium sulphate is due to the fact that we are not dealing with 
one solubility curve, but with two solubility curves. At temper- 
atures below 33 C., the dissolved salt is in equilibrium with the 
decahydrate, Na-jSO^lOH^O, whereas at temperatures above 
33 C. the dissolved salt is in equilibrium with the anhydrous 
salt, Na2S0 4 . The solubility of Na2S0 4 .10H 2 O increases with 
the temperature, while the solubility of Na^SC* diminishes. That 
we are actually dealing with two solubility curves, is proved by 
the fact that the solubility curves of the hydrated and anhydrous 
salts in supersaturated solutions are continuations of the corre- 
sponding curves for saturated solutions, as shown by the dotted 



SOLUTIONS 183 

curves in Fig. 57. If we select any point, such as p, lying between 
a dotted curve and a full curve, it is apparent that it represents a 
solution supersaturated with respect to Na^SO^lOH^O, but un- 
saturated with respect to Na2S0 4 . If pure anhydrous sodium 
sulphate be shaken with this solution it will slowly dissolve, where- 
as if a trace of the hydrated salt be added, the solution will deposit 
Na2SO 4 .10 H 2 0, until the amount remaining in solution corre- 
sponds to the solubility of the hydrate at that temperature. 
Supersaturated solutions of some substances can be preserved 
indefinitely, provided all traces of the solid phase are excluded. 
Such solutions are called metastable. On the other hand there 
are some supersaturated solutions which deposit the excess of 
solid solute even when all traces of it are excluded. These solu- 
tions are termed labile. The distinction between metastable and 
labile solutions is not sharp. If a metastable solution is suffi- 
ciently cooled, or if its concentration is sufficiently increased, it 
may be made to pass over into the labile condition. The concen- 
tration at which this tradition occurs is termed the metastable 
limit. The stability of supersaturated solutions has recently 
been shown by Young * to be greatly influenced by vibrations or 
sudden shocks within the solution. He has been able to control 
the amount of overcooling in a supersaturated solution, by alter- 
ing the intensity of the vibrations due to the friction between glass 
or metal surfaces within the solution. 

Very little is known concerning the relation between solubility 
and the specific properties of solute and solvent. 

Owing to the fact that the change in volume resulting .from the 
solution of a solid in a liquid is very small, the effect of pressure 
on the solution is almost negligible. The chief factors condition- 
ing the change in solubility due to increasing pressure, are the 
heat of solution of the solute in the nearly saturated solution, 
and the change in volume on solidification. Very few experiments 
have been made to determine the effect of pressure on solubility, 
van't Hoff states that the solubility of a solution of ammonium 
chloride, a salt which expands when dissolved, decreases by 1 per 
cent for 160 atmospheres, while the solubility of copper sulphate, 
* Jour. Am. Chem. Soc., 33, 148 (1911). 



184 



THEORETICAL CHEMISTRY 



a salt which contracts when dissolved, increases by 3.2 per cent 
for 60 atmospheres. 

Solid Solutions. In general, when a dilute solution is suffi- 
ciently cooled the solvent separates in the form of crystals which 
are almost entirely free from the solute. When, however, the 
temperature of a solution of iodine in benzene is reduced to the 
freezing-point, the crystals which separate are found to contain 
iodine^ Furthermore, the depression of the freezing-point of the 
solvent is found to be less than that calculated on the assumption 
that the solvent crystallizes uncontaminated with the solute. 
Such solutions were first studied by van't Hoff.* He found that 
when the concentration of such abnormal solutions is varied, the 
ratio of the amount of solute in the liquid solvent to the amount 
of solute in the solidified sol vent* remains constant. Thus, in solu- 
tions of iodine in benzene, the ratio of the concentration of iodine 
in the liquid to its concentration in the crystallized benzene is 
constant. In the following table Ci is the concentration of iodine 
in the liquid benzene, and fy is the concentration of iodine in the 
solid benzene. 









Cj/C 2 


3 39 
2 587 
0.945 


1 279 
0.925 
0.317 


377 
358 
336 



van't Hoff pointed out the analogy between the distribution of 
the solute between the solid and liquid solvent, and the distribution 
of a gas between a liquid and the free space above it. In other 
words, the distribution follows Henry's law for the solution of a 
gas in a liquid. Since the crystals containing both solute and 
solvent are perfectly homogeneous, van't Hoff suggested that 
they be regarded as solid solutions. The mixed crystals which 
separate from solutions of isomorphous substances being chem- 
ically and physically homogeneous, are to be considered as solid 
solutions. Many alloys possess the properties characteristic of 

* Zeit. phys. Chem., 5, 322 (1890). 



SOLUTIONS 185 

solid solutions; hardened steel, for example, being regarded as a 
homogeneous solid solution of carbon in iron. One of the char- 
acteristic properties of a dissolved substance is its tendency to 
diffuse into the pure solvent. Interesting experiments performed 
by Roberts- Austen * have shown that even solids have the prop- 
erty of mixing by diffusion. Thus, by keeping gold and lead in 
contact at constant temperature for four years, he was able to 
detect the presence of gold in the layer of lead at a distance of 
7 mm. from the surface of separation. Many other instances of 
diffusion in solids have been observed.f 

Instances of gases and liquids dissolving in solids are also 
known. Thus platinum, palladium, charcoal and other sub- 
stances have the property of taking up large volumes of hydrogen. 
This phenomenon, known as occlusion, is but little understood, 
van't Hoff has suggested that wllen~1iydrogen dissolves in palla- 
dium we are really dealing with two solid solutions: one a solu- 
tion of hydrogen in palladium and the other a solution of 
palladium in solid hydrogen, the system being analogous to 
that of two partially miscible liquids. 

Certain natural silicates, the so-called jeolites,_are transparent 
and homogeneous. Since they contain varying quantities of 
water they may be regarded as examples of solutions of liquids 
in solids. This classification is further justified by the fact that 
portions of the water may be removed and replaced by other 
substances, such as alcohol, with apparently no change in the 
transparency or homogeneity of the mineral. 

PROBLEMS. 

1. 2.3 liters of hydrogen under a pressure of 78 cm. of mercury, and 
5.4 liters of nitrogen at a pressure of 46 cm. were introduced into a ve&sel 
containing 3.8 liters of carbon dioxide under a pressure of 27 cm. What 
was the pressure of the mixture? Ans. 140 cm. of mercury. 

2. Air is composed of 20.9 volumes of oxygen and 79.1 volumes of 
nitrogen. At 15 C. water absorbs 0.0299 volumes of oxygen and 0.0148 

* Proc. Roy. Soc., 67, 101 (1900). 

t See Report on Diffusion in Solids, by C. H. Desch, Chem. News, 106, 
153 (1912). 



186 THEORETICAL CHEMISTRY 

volumes of nitrogen, the pressure of each being that of the atmosphere. 
Calculate the composition of the mixture of gases absorbed by the water. 
Ans. 34.8% by vol. of 0, and 65.2% by vol. of N. 

3. The vapor pressure of the immiscible liquid system, aniline-water, 
is 760 mm. at 98 C. The vapor pressure of water at that temperature is 
707 mm. What fraction of the total weight of the distillate is aniline. 

Ans. 0.28. 

4. The boiling-point of the immiscible liquid system, naphthalene-water, 
is 98 C. under a pressure of 733 mm. The vapor pressure of water at 
98 C. is 707 mm. Calculate the proportion of naphthalene in the dis- 
tillate. Ans. 0.207. 



CHAPTER X. 
DILUTE SOLUTIONS AND OSMOTIC PRESSURE. 

Osmotic Pressure. In the preceding chapter reference was 
made to the fact that diffusion is a characteristic property of solu- 
tions. If a few cubic centimeters of a concentrated solution of 
cane sugar are placed at the bottom of a tall cylinder, and water 
is added, care being taken to prevent mixture, the sugar immedi- 
ately begins to diffuse into the water, the process continuing until 
the concentration of sugar is the same throughout the liquid. 
The sugar molecules move from a region of high concentration 
to a region of low concentration, the rate of diffusion being rela- 
tively slow owing to the viscosity of the medium. A similar 
process is encountered in the study of gases, but the rate of gas- 
eous diffusion is extremely rapid. In terms of the kinetic theory, 
the movement of the molecules of a gas from regions of high 
concentration to regions of low concentration, is to be considered 
as due to the pressure of the gas. By analogy, we may regard the 
process of diffusion in solutions as a manifestation of a driving 
force, known as the osmotic pressure. 

Semi-permeable Membranes. The use of a semi-permeable 
membrane for the measurement of the partial pressure of nitrogen 
in a mixture of nitrogen and hydrogen, has already been explained. 
A similar method may be employed for the measurement of 
the osmotic pressure of a solution, provided a suitable semi-perme- 
able membrane can be found. Such a membrane must prevent 
the passage of the molecules of solute and must be readily perme- 
able to the molecules of solvent; it must exert a selective action 
on solute and solvent. If a solution is separated from the pure 
solvent by a semi-permeable membrane, diffusion of the solute 
is no longer possible. Since equilibrium of the system can only 
be attained when the concentrations on both sides of the mem- 
brane are equal, it follows that the solvent mult pass through 

187 



188 THEORETICAL CHEMISTRY 

the membrane and dilute the more concentrated solution. A 
number of semi-permeable membranes have been discovered 
which are readily permeable to water and nearly, if not entirely, 
impermeable to various solutes. About the middle of the eight- 
eenth century Abb6 Nollet discovered that certain animal mem- 
branes are permeable to water but not to alcohol. 

Artificial semi-permeable membranes were first prepared by 
M. Traube.* If a glass tube, provided with a rubber tube and 
pinch-cock, be partially filled, by suction, with a solution of 
and then immersed in a solution of potassium 



ferrocyanide, a thin film of copper ferrocyanide_ will be formed 
at the junction of the two solutions. When the film has once 
been formed, further precipitation of copper ferrocyanide will 
cease, the solutions on either side of the film remaining clear. 
Traube showed that this membrane is semi-permeable. He also 
showed that a number of other gelatinous precipitates possess 
the property of semi-permeability. A membrane formed in the 
above manner is easily ruptured and is wholly inadequate for 
quantitative or even qualitative experiments. Pfeffer f devised 
a method for strengthening the membrane. By depositing the 
precipitate in the walls of a porous clay cup, the area of unsup- 
ported membrane is greatly diminished and its resisting power 
correspondingly increased. Pfeffer directs that the cup to be 
used for this purpose must be thoroughly washed, and its walls 
allowed to become completely permeated with water. The cup 
is then filled to the top with a solution of copper sulphate, con- 
taining 2.5 grams per liter, and allowed to stand for several hours 
in a solution of potassium ferrocyanide, containing 2.1 grams 
per liter. The two solutions diffuse through the walls of the cup 
and on meeting, deposit a thin membrane of copper ferrocyanide. 
When precipitation is complete, the cup is thoroughly washed 
and soaked in water. The cup is then filled to the top with a 
solution of cane sugar, and a rubber stopper, fitted with a long 
glass tube of narrow bore, is inserted, care being taken to exclude 
air-bubbles. The stopper is then made fast with a suitable 

* Archiv. fur Anat. und Physiol., p. 87 (1867). 
t demotische Untersuchungen, Leipzig, 1877. 



DILUTE SOLUTIONS AND OSMOTIC PRESSURE 



189 




cement, and the cup completely immersed in a beaker of water. 
The completed apparatus is shown in Fig. 58. If the formation 
of the membrane has been successful, the level of the liquid in the 
vertical glass tube will slowly rise and will eventually attain a 
height of several meters. If the mem- 
brane is sufficiently strong and no leaks 
develop, the passage of water through the 
membrane will continue until the hydro- 
static pressure of the column of liquid in 
the tube is great enough to overcome the 
tendency of the water to force its way 
into the sugar solution. As a general 
rule, the membrane becomes ruptured be- 
fore equilibrium is attained. 

Measurement of Osmotic Pressure. 
The first direct measurements of osmotic 
pressure were made by Pfeffer. His ex- 
periments deserve brief consideration, 
since the results obtained furnish the 
basis of the modern theory of solution. 
The cell used was similar to that described 
above, but instead of employing a ver- 
tical glass tube as a manometer, the cup 
was connected, as shown in Fig. 59, with 
a closed mercury manometer. The sub- 
stitution of the closed for the open man- 
ometer is necessitated by the fact, that 
with an open manometer so much water entered the cell that 
the concentration of the solution became appreciably diminished, 
and the pressure actually measured corresponded to a solu- 
tion of smaller concentration than that introduced into the cell. 
With the closed manometer, when a trace of water has entered 
the cell, sufficient pressure is developed to prevent the further en- 
trance of more water. Pfeffer calculated that with a cell, the 
capacity of which was 16 cc., the volume of water entering before 
equilibrium was attained, did not exceed 0.14 cc. In his experi- 
ments, Pfeffer determined the density of the cell Contents before 



Fig. 58. 



190 



THEORETICAL CHEMISTRY 



and after measurement of the osmotic pressure, and corrected for 
any change in concentration. With this apparatus he made numer- 
ous measurements of the osmotic 
pressures of different solutions, the 
entire apparatus being immersed in 
a constant-temperature bath. With 
solutions of cane sugar he obtained 
the results given in the accompany- 
ing table, where C denotes the per- 
centage concentration of the solu- 
tion, and P the corresponding 
osmotic pressure, expressed in centi- 
meters of mercury. The tempera- 
ture varied from 13.5 C. to 14.7 C. 




C 


p 


P/C 


1 


53.5 


53.5 


2 


101.6 


50.8 


4 


208.2 


52.0 


6 


307.5 


51.2 



It is evident from these results, 
that the osmotic pressure is propor- 
tional to the concentration of the 
solution, since P/C is approximately 
constant. The deviations from con- 
stancy in the ratio of pressure to 
concentration may be ascribed to 
experimental errors, since the dif- 
ficulties involved in these measure- 

^ ments are very great. Pfeffer also 

Fig. 59. studied the influence of temperature 

on osmotic pressure, and showed 

that as the temperature is raised the pressure increases. The 
following table gives his results for a 1 per cent solution of cane 
sugar. 



DILUTE SOLUTIONS AND OSMOTIC PRESSURE 191 



Temperature. 


Osmotic Pressure. 


6 8 
13. 2 
14. 2 
22 
36. 


cin. 

50.5 
52 1 
53.1 
54 8 
56.7 



Osmotic Pressure and the Nature of the Membrane. Pfeffer 
also studied the effect of the nature of the membrane on osmotic 
pressure. In addition to copper ferrocyanide, he used membranes 
of calcium phosphate and Prussian blue. His results seemed to 
indicate that the magnitude of the osmotic pressure developed,/ 
was dependent upon the nature of the membrane used. 1 

The variations observed have since been shown to have been 
due to leakage of the calcium ^phosphate and Prussian blue mem- 
branes, the copper ferrocyanide membrane Being the only one 
which was capable of withstanding the pressure. Ogfagfrld *. has 
devisedjan ingepj^&Jheoretical demonstration of the Jact 
osm^c^pressurejnust^be independent of t&e natm^of_the 
brane employedjn^n^simng it r " Let A and B, in Fig. 60, repre- 



Fig. 60. 

sent two different semi-permeable membranes placed in a glass 
tube of wide bore. Let us imagine the space between the two 
membranes to be filled with a solution, and the tube immersed 
in a vessel of water. If the osmotic pressures developed at A 
and B are pi and p* respectively, and p 2 is less than pi, then 
water will pass through A until the pressure pi is reached. Since 
the pressure at B only reaches the value p 2 , however, the pres- 
sure pi can never be attained, and a steady stream of water from 
A to By under the pressure pi p 2 , will result. This, however, 
would be a perpetual motion, and since this is impossible, the 
osmotic pressures at the two membranes must be the same. 
* Lehrb. d. allg. Chem., I., p. 662. 



192 



THEORETICAL CHEMISTRY 



Theoretical Value of Osmotic Pressure. The physico-chem- 
ical significance of Pfeffer's results was first perceived by van't 
Hoff.* In a remarkably brilliant paper, he pointed out the 
existence of a striking parallelism between the properties of gases 
and the properties of dissolved substances. 

We have already called attention to the analogy between osmotic 
pressure and gas pressure: we now proceed to trace the connec- 
tion between osmotic pressure, volume and temperature, as first 
pointed out by van't Hoff. Pfeffer's experiments showed that at 
constant temperature, the ratio, P/(7, is constant for any one 
solute. Since the concentration varies inversely as the volume 
in which a definite amount of solute is dissolved, we obtain, by 
substituting l/V for C, the equation, PV = constant, which is 
plainly the analogue of the familiar equation of Boyle for gases. 
An examination of Pfeffer's data for osmotic pressures at differ- 
ent temperatures, convinced van't Hoff that the law of Gay- 
Lussac is also applicable to solutions. 

In the following table, the osmotic pressures in atmospheres for 
a 1 per cent solution of cane sugar at different temperatures are 
recorded, together with the pressures calculated on the assump- 
tion that the osmotic pressure is directly proportional to the 
absolute temperature. 



Temperature. 


/' (oba.) 


P (calc ). 


6 8 


664 


0.665 


13 7 


691 


681 


15. 5 


684 


686 


22 


721 


0.701 


32 


0.716 


725 


36 


0.746 


0.735 



Since the laws of Boyle and Gay-Lussac are both applicable, 
we may write an equation for dilute solutions corresponding to 
that already derived for gases, or 

PV = B'T. 

in which P is the osmotic pressure of a solution containing a defi- 
* Zeit. phys, Chem., i, 481 (1887). 



DILUTE SOLUTIONS AND OSMOTIC PRESSURE 193 

nite weight of solute in the volume, V, of solution, T being the 
absolute temperature of the solution and R l a constant corre- 
sponding to the molecular gas constant. 

The molecular gas constant R has already been evaluated and 
has been found to be equal to 0.0821 liter-atmosphere. 

Making use of Pfeffer's data, van't Hoff calculated the value 
of R f in the above equation, in the following manner: the osmotic 
pressure of a 1 per cent solution of cane sugar at C. is 0.649 
atmosphere, and since the concentration of the solution is 1 per 
cent, the volume of solution containing 1 mol of sugar, will be 
34,200 cc. or 34.2 liters. Substituting these values in the equa- 
tion, we have 

n , PV 0.649X34.2 AACM01 ., 

R' = -=- = - ;r-- - = 0.0813 hter-atmos., 

JL 



a value which is nearly the same as that of the molecular gas 
constant, R. The equality of R and R f leads to a conclusion of 
the greatest importance, as was pointed out by van't Hoff, viz., 
"the osmo&(L%ressure exerted by any substance in solution is the same 
as it u)ouldi exj^t if present as a gas in the same volume as that occupied 
by the solution, provideSTihat the solution is so dilute that the volume 

tSrojisMtaWMiMcffTOR'. v> w ' '"*** ' * 

occupied by the solute is negligible in comparison with that occupied 
by the solvent" It should be remembered that we are not justified 
in concluding from this proposition of van't Hoff, that osmotic 
pressure and gaseous pressure have a common origin. While 
the origin of osmotic pressure may be kinetic, it is also conceivable 
that it may result from the mutual attraction of solvent and 
solute, or that it may bear some relation to the surface ten- 
sion of the solution. Up to the present time no wholly satis- 
factory explanation of the cause of osmotic pressure has been 
advanced. 

Just as 1 mol of gas at C. and 760 mm. pressure occupies a 
volume of 22.4 liters, so when 1 mol of a substance is dissolved 
and the solution diluted to 22.4 liters at C., it will exert an 
osmotic pressure of 1 atmosphere. I^therj^ords, woZor weights, 
or quantities proportional to molar weights, of different substances, 
wKen diss^KTirTeqiu^ volumes of the same solvent exert the same 



194 THEORETICAL CHEMISTRY 

osmotic pressure. If we deal with n mols of solute instead of 1 mol 
the general equation becomes 

PV = nRT. 

But n = gf/M, where g is the number of grams of solute per liter, 
and M is its molecular weight. Substituting in the preceding 
equation, we have 



or, 

M - gRT - 
M ~~ 



Since P, V, g, R and T are all known, M can be calculated. The 
direct measurement of the osmotic pressure of a solution does not 
afford a practical method for the determination of the molecular 
weight of dissolved substances, because of the experimental 
difficulties involved and the time required for the establishment 
of equilibrium. There are other and simpler methods for deter- 
mining molecular weights in solution, based upon certain proper- 
ties of solutions which are proportional to their respective osmotic 
pressures. 

Recent Work on the Direct Measurement of Osmotic Pressure. 
It is only within the past decade that the investigations of Pfeffer 
have been confirmed and extended by elaborate and systematic 
experiments on the direct measurement of osmotic pressure. 
Morse and his co-workers,* while employing a method essentially 
the same as that of Pfeffer, have, as the result of much patient 
labor, brought the apparatus to such a high state of perfection, 
that the experimental errors are now estimated to affect only 
the second place of decimals in the numerical data expressing 
osmotic pressures in atmospheres. The most important of the 
improvements introduced by Morse are the following: (1) the 
improvement of the quality of the membrane; (2) the improve- 
ment of the connection between the cell and the manometer, 
and (3) the improvement of the means of accurately measuring 

* Am. Chem. Jour., 26, 80 (1901); 34, 1 (1905); 3$, 1, 39 (1906); 37, 324, 
425, 558 (1907); 381 175 (1907); 39, 667 (1908); 40, 1, 194 (1908); '41, 1, 257 
(1909). 



DILUTE SOLUTIONS AND OSMOTIC PRESSURE 195 



the pressure. The membrane of copper 
ferrocyaiiide is deposited electrolyti- 
cally. After thorough washing and 
soaking in water, the porous cup, made 
from specially prepared clay, is filled 
with a solution of potassium ferrocy- 
anide and immersed in a solution of cop- 
per sulphate. An electric current is 
then passed from a copper electrode 
in the solution of copper sulphate, to 
a platinum electrode immersed in the 
solution of potassium ferrocyanide. 
This drives the copper and ferrocy- 
anide ions toward each other, and the 
membrane of copper ferrocyanide is 
thus formed in the walls of the cup. 
The passage of the current is continued 
until the electrical resistance reaches a 
value of about 100,000 ohms. The 
cell is then rinsed, and soaked in water 
for several hours, and then the electro- 
lytic treatment is repeated until the 
electrical resistance attains a maximum 
value. A solution of cane sugar is now 
introduced into the cell, which is con- 
nected with the manometer and im- 
mersed in water. When the pressure 
has attained its maximum value, the 
apparatus is dismantled and the cell, 
after thorough washing and soaking in 
water, is again subjected to the electro- 
lytic process of membrane forming. In 
this way the weak places in the mem- 
brane which may have yielded to the 
high pressure, can be repaired, and by 
continued repetition of this treatment 
the membrane can ultimately be brought to its maximum power 




Fig. 61. 



196 



THEORETICAL CHEMISTRY 



of resistance. A sketch of the Morse apparatus is shown in Fig. 
61. A description of the details of this apparatus lies beyond the 
scope of this book. The results of the work of Morse and his 
students are of the highest importance. The osmotic pressures 
of solutions of cane sugar and dextrose have been shown to be 
proportional to the respective concentrations, provided the con- 
centration is referred to unit volume of solvent instead of unit 
volume of solution. Thus in their experiments, the solutions were 
made up containing from 0.1 to 1.0 mol of solute in 1000 grams of 
water. Morse calls such solutions weight-normal solutions in con- 
trast to volume-normal solutions, in which 1 rnol or a fraction of 
a mol of solute is dissolvpd in water and the solution diluted to 1 
liter. The following data taken from the work of Morse, shows 
that when concentration is expressed on the weight-normal basis, 
there is direct proportionality between osmotic pressure and con- 
centration. The figures refer to solutions of dextrose at 10 C. 



Molar Concentration. 


Osmotic Pressure. 


Per 1000 gm. 
Water. 


Per Liter of 
Solution. 


In Atinos. 


Relative to First 
as Unity. 


0.1 
2 
5 
1.0 


099 
196 
474 
901 


2 39 
4 76 
11 91 
23.80 


1.00 
1.99 
4.98 
9.96 



Morse and his co-workers also conclude from their experiments 
at temperatures ranging from C. to 25 C., that^hg^eigp^r- 
ature coefficients of osmotic pressure and gas pressure are prac- 
ticaUx <M Jdentical. In other words, their results confirm the 
conclusions of van't Hoff , that the law of Gay-Lussac is applicable 
to solutions. The results of the experiments of Morse are of 
special interest in connection with the proposition of van't Hoff, 
that the osmotic pressure of a dilute solution is the same as that 
which the solute would exert if it were gasified at the same temper- 
ature and occupied the same volume as the solution. The data 
in the following table is taken from the work of Morse on solutions 



DILUTE SOLUTIONS AND OSMOTIC PRESSURE 



197 



of cane sugar at 15 C. In addition to the observed osmotic 
pressures, the table contains the corresponding gas pressures, 
calculated (1) on the assumption that the solute when gasified 
occupies the same volume as the solution (proposition of van't 
Hoff), and (2) on the assumption that it occupies the same volume 
as the solvent alone. 



Molar Concentration. 


Osmotic Pressure in Atmos. 


Per 1000 gm. of 
Water. 


Per Liter of 
Solution. 


Obs. 


Calc. (a). 


Calc. Cb). 


0.1 


098 


2 48 


2 30 


2 35 


2 


192 


4 91 


4 51 


4 70 


4 


369 


9 78 


8 67 


9 40 


6 


533 


14 86 


12 51 


14 08 


8 


684 


20 07 


16 07 


18 79 


1 


825 


25 40 


19.38 


23.49 



The calculated pressures, recorded in the last column, are in 
much closer agreement with the observed osmotic pressures, than 
are the calculated pressures, recorded in the fourth column of the 
table. The proposition of van't Hoff should then be modified 
to read as follows: A dissolved substance in dilute solution^^x^rts 
an osmotic pressure equal to that which it wpuld exert if it were gas- 
ified at the same temperature, and the volume of the gas were reduced 
to that of the solvent in the pure state., The investigations of Morse 
and his co-workers may be summarized thus: (1) the law of 
Boyle is applicable to dilute solutions, provided the concentration 
is referred to 1000 grams of solvent and not to 1 liter of solution; 
(2) the law of Gay-Lussac is also applicable to dilute solutions, 
that is, the temperature coefficients of osmotic pressure and gas 
pressure are equal, and (3) the small departures from the theo- 
retical values of the osmotic pressures may be traced to hydration 
of the solutec 

Direct measurements of the osmotic pressure of concentrated 
solutions of cane sugar, dextrose and mannite have been made by 
the Earl of Berkeley and E. G. J. Hartley.* The method em- 

* Proc. Roy. Soc., 73, 436 (1904); Trans. Roy. Soc. A., 206, 481 (1906). 



198 THEORETICAL CHEMISTRY 

ployed by these investigators is slightly different from that of 
Pfeffer or Morse; the tendency of water to pass through the 
semi-permeable membrane is offset by the application of a counter 
pressure to the solution. A membrane of copper ferrocyanide 
is deposited electrolytically very near the outer surface of a tube 
of porous porcelain. This tube is placed co-axially within a large 
cylindrical vessel of gun metal, an absolutely tight joint between 
the two being secured by an ingenious system of dermatine rings 
and clamps. The open ends of the porcelain tube are closed bv 
rubber stoppers fitted with capillary tubes bent at right angles) 
one of the latter being provided with a glass stop-cock. When a 
determination of osmotic pressure is to be made, the apparatus is 
placed in a horizontal position and water is introduced into the 
porcelain tube, completely filling it and the connecting capillary 
tubes up to a certain level. The gun metal vessel is then filled 
with the solution, and connected with an auxiliary apparatus by 
means of which a gradually increasing hydrostatic pressure can 
be applied. If no pressure is applied to the solution, water will 
pass through the semi-permeable membrane into the solution, and 
the level of the water in the capillary tubes will fall. In carrying 
out a measurement, therefore, as soon as the solution is introduced 
into the gun metal vessel, hydrostatic pressure is applied, the mag- 
nitude of the pressure being so adjusted as to counterbalance the 
osmotic pressure of the solution. The level of the water in the 
capillary tubes serves to indicate the relative magnitudes of 
the osmotic and hydrostatic pressures. When the level of the 
water in the capillary tubes remains constant, the two pressures 
are in equilibrium. The following are the values of the equilibrium 
pressures of solutions of cane sugar, dextrose and mannite at C. 
It must be remembered that when the two pressures are in equi- 
librium, there is always a pressure of one atmosphere on the solvent. 
As will be seen the pressures developed in the more concen- 
trated solutions are enormous and it is a surprising fact, that even 
in cases where the highest pressures were measured, hardly a 
trace of sugar was found in the pure solvent, the membrane 
retaining its property of semi-permeability throughout the entire 
range of pressures* The figures in the third column are calculated 



DILUTE SOLUTIONS AND OSMOTIC PRESSURE 



CANE SUGAR. 





Osmotic Pressure in Atmospheres. 


Cone. gm. per 




Liter. 








Obs. 


Calc. 


180.1 


13 95 


13.95 


300.2 


26.77 


28 74 


420.3 


43.97 


32.55 


540.4 


67.51 


41 85 


660.5 


100 78 


51 16 


750.6 


133 74 


58.14 



DEXTROSE. 





* ^~* * 

Osmotic Pressure in Atmosphere. 


Cone. gm. per 




V- 


Liter. 








Obs. 


Calc. 


99 8 


13 21 


13 21 


199.5 


29.17 


26.41 


319 2 


53.19 


42.25 


448 6 


87.87 


59 28 


548 6 


121 . 18 


72 61 



MANNITE. 



Cone, gm, per 
Liter. 


Osmotic Pressure in Atmospheres. 


Obs. 


Calc. 


100 
110 
125 


13 1 
14 6 
16.7 


13.1 
14.4 
16.4 



on the assumption that there is direct proportionality between 
osmotic pressure and concentration. It is apparent that in every 
case the observed osmotic pressure is greater than the calculated. 
Even when the concentrations are expressed on the weight-normal 



200 



THEORETICAL CHEMISTRY 



basis, as recommended by Morse, the osmotic pressure increases 
more rapidly than the concentration. 

This is well shown in the accompanying diagram, Fig. 62, due 
to the Earl of Berkeley. In this diagram, the osmotic pressures 
of solutions of cane sugar are plotted against concentrations, curve 
A representing the actually observed osmotic pressures; curve C 



120 



100 



380 



E60 



40 




100 200 300 400 600 

Grams Cfetne Sugar per liter of Solution 

Fig. 62. 



600 



700 



being traced on tlie assumption that osmotic pressure may be 
calculated from the equation, PV = JK7 7 , where V denotes the 
volume of solvent containing 1 mol. of cane sugar; and curve B, 
a straight line, being drawn on the assumption that osmotic 
pressure may be calculated from the equation, PV = RT, where 
V represents the volume of solution containing 1 mol. 
While the theoretical and observed values of the osmotic pres- 



DILUTE SOLUTIONS AND OSMOTIC PRESSURE 201 

sure are approximately equal in the more dilute solutions, it is 
obvious that the observed values of the osmotic pressure of the 
concentrated solutions are always greater than the calculated 
values, even when the calculation is made on the assumption that 
V in the equation, PV = RT, is the volume of the solvent. The 
abnormally high osmotic pressures observed by the Earl of 
Berkeley have been discussed by Callendar * who suggests hydra- 
tion of the solute as a probable cause. 

He shows, that if 5 molecules of water are assumed to be asso- 
ciated with each molecule of cane sugar in the most concentrated 
solutions studied by the Earl of Berkeley, the discrepancy between 
the observed and calculated values of the osmotic pressure dis- 
appears. 

Comparison of Osmotic Pressures. Although the difficulties 
involved in the direct determination of osmotic pressure are 
many, these can be avoided by the employment of one of several 
indirect methods which have been devised for the comparison of 
osmotic pressures. All of these methods depend upon the exchange 
of water which occurs when two solutions are separated by a 
semi-permeable membrane. The movement of the water will 
always be in such a direction as to tend to equalize the osmotic 
pressures on opposite sides of the membrane, or, in other words, 
.the transfer of water will take place from the solution with the 
lesser osmotic pressure to the solution with the greater osmotic 
pressure. 

The Plasmolytic Method. In this method, solutions of various 
substances are prepared, the concentration of each being such 
that its osmotic pressure is the same as that of a particular plant 
cell. Obviously the osmotic pressures of all of these solutions 
must be equal: such solutions are said $P^i&li3ML* ? ' s s m ?*iJL 
The plasmolytic method for the comparison of osmotic ^pressures 
was developed by the Dutch botanist, De Vries.f This method 
depends upon the shrinking or swelling of the protoplasmic sac 
of plant cells when they are immersed in a solution whose osmotic 

* Proc. Roy. Soc. A., 80, 466 (1908). 

t Jahrb.wiss.Botanik., 14,427(1884); Zeit. phys. Chem., 2, 415 (1888); 3, 
103 (1889). 



202 THEORETICAL CHEMISTRY 

pressure differs from that of their own sap. De Vries found that 
the cells of Tradescantia discolor. Curcuma rubricaulis, and Begonia 
manicata fulfil the necessary conditions, viz.; the cell walls are 
strong and resist alteration when immersed in solutions, the cells 
are readily permeable to water, and the cell contents are colored, 
thus enabling the slightest contraction or expansion to be de- 
tected. The cell walls are lined on the inside with a thin, elastic, 
semi-permeable membrane which encloses the colored contents 
of the cell. The content of the cell consists of an aqueous solu- 
tion of several substances, among which may be mentioned 
glucose, potassium and calcium malate, together with coloring 
matter. The osmotic pressure of the cell contents ranges from 
four to six atmospheres. The semi-permeable membrane expands 
when the contents of the cell increases and contracts when the 
contents diminishes. In making a comparison of osmotic pres- 
sures by this method, tangential sections are cut from the under 
side of the mid-rib of the leaf of one of the above plants, e.g., 
Tradescantia discolor, and are placed in the solution whose osmotic 
pressure it is desired to compare with that of the cell contents. 
The cells are then observed under the microscope, any decrease 
in pressure below the normal resulting in a detachment of the 
semi-permeable membrane from one or more points of the cell 
wall. This contraction always occurs when the cells are im- 
mersed in a concentrated solution, the phenomenon being termed 
plasmolysis. When the solution in which the cells are placed 
has a lower osmotic pressure than the cell contents, no visible 
effect is produced, the increased pressure within the cell simply 
forcing the membrane closer to the rigid cell walls. By starting 
with a concentrated solution, the osmotic pressure of which is 
greater than that of the cell, and gradually diluting it, a concen- 
tration will ultimately be reached at which the elastic membrane 
will just completely fill the cell. This solution is isotonic with 
the cell contents. In this method the very reasonable assump- 
tion is made that all of the cells have the same osmotic pressure, 
any differences which might have existed having equalized them- 
selves in the living plant. The microscopic appearance of cells 
of Tradescantia discolor when immersed in different solutions is 



DILUTE SOLUTIONS AND OSMOTIC PRESSURE 203 

shown in Fig. 63. The appearance of the normal cell when im- 
mersed in water or in a solution whose osmotic pressure is less 
than that of the cell contents, is shown in A. When the cell is 
immersed in a 0.22 molar solution of cane sugar it appears as in 
JB, this solution having a greater osmotic pressure than the cell 
contents. When the cell is immersed in a molar solution of 




Fig. 63. 

potassium nitrate, there is marked plasmolysis, as shown in C. 
De Vries determined the concentrations of a large number of 
solutions which were isotonic with the cell contents. He ex- 
pressed his results in terms of the isotonic coefficient, which he 
defined as the reciprocal of the molar concentration. The iso- 
tonic coefficient of potassium nitrate was taken equal to 3. A 
few of De Vries' results are given in the following table. 



Substance. 


Formula. 


Isotonic 
Coefficient. 


Glycerol 


C 3 H 8 O 3 


1 78 


Glucose 


CeH^Oe 


1 81 


Cane sugar 


\*t 1 2-ti 22 *J 1 1 


1 88 


Malic acid 


C 4 HeO5 


1 98 


Tartaric acid 


CiHUOg 


2 02 


Citric acid 


CeHgO? 


2 02 


Potassium nitrate 


KNO 8 


3 00 


Magnesium chloride 


MgCl 2 


4 33 









204 THEORETICAL CHEMISTRY 

De Vries applied the plasmolytic method to the determination 
of the molecular weight of raffinose. At that time there was 
considerable uncertainty as to the correct formula of crystallized 
raffinose, three different formulas, all consistent with the results 
of analysis, having been proposed as follows: Ci8H 32 Oi 6 . 
5 H 2 0, Ci 2 H 22 Oii.3 H 2 0, and C 3 6H 6 40 32 . De Vries found that a 
3.42 per cent solution of cane sugar was isotonic with a 5.96 per- 
cent solution of raffinose. Letting the unknown molecular 
weight of raffinose be represented by M 9 then 

3.42 : 5.96 :: 342 : M. 

Solving the proportion we have, M - 596. This result has since 
been confirmed by purely chemical methods, and the formula, 
CisH.tfOie.5 H 2 O, the molecular weight of which is 594, is thus 
established. 

The Blood Corpuscle Method. The red blood corpuscle is a 
cell, the contents of which is enclosed by a thin elastic semi- 
permeable membrane. Unlike the plant cells, there is no resistant 
cell wall to give support to the membrane, so that when red blood 
corpuscles are immersed in water they at first swell, owing to the 
osmotic pressure developed, and finally burst. When the mem- 
brane is ruptured, the coloring matter of the cell, the haemoglobin, 
escapes and the water acquires a deep red color. 

Advantage of this behavior of red blood corpuscles was taken 
by Hamburger * for the comparison of osmotic pressures. He 
found that when a 1.04 per cent solution of potassium nitrate 
is added to the defibrinated blood of a bullock, the corpuscles 
will settle completely to the bottom, while the supernatant liquid 
will remain clear. On the other hand, if a 0.96 per cent solution 
of potassium nitrate is used, the corpuscles will not settle and the 
supernatant liquid becomes colored. If more dilute solutions of 
potassium nitrate are used, the solution acquires a still deeper 
color. By careful adjustment, a concentration of potassium 
nitrate can be found in which the red blood corpuscles will just 
settle. In like manner, the concentration of solutions of other 
substances can be so adjusted as to cause the precipitation of the 

* Zeit. phys. Chem., 6, 319 (1890). 



DILUTE SOLUTIONS AND OSMOTIC PRESSURE 205 

corpuscles. These solutions are isotonic. Without going into 
details, it may be said that the isotonic coefficients obtained by 
Hamburger, agree well with those obtained by the plasmolytic 
method, 

The Hffimatocrit Method* In this method developed by 
Hedin,* advantage is again taken of the properties of red blood 
corpuscles. As has already been stated, when red blood cor- 
puscles are immersed in solutions of gradually diminishing 
concentration of the same solute, they continue to swell and ulti- 
mately the semi-permeable envelope bursts. On the other hand, 
when the corpuscles are immersed in solutions of gradually increas- 
ing concentration, they shrink, owing to the transfer of water 
from the corpuscles. It is apparent that there must be a certain 
concentration for each solute which will cause no change in the 
volume of the corpuscles. To determine this concentration, use 
is made of an instrument known as an hcematocrit. This is 
simply a graduated thermometer-tube which may be attached to 
the spindle of a centrifugal machine. When the spindle is re- 
volved at high speed, the corpuscles collect in the bottom of the 
graduated tube. A measured volume of blood is centrifuged 
until no further shrinkage in volume of the corpuscles can be 
detected in the haematocrit. The same volume of blood is then 
added to each of a series of solutions whose concentration dimin- 
ishes progressively, and the volume of the corpuscles is determined 
as in pure blood. In this way the concentration of the solution 
is found, in which the volume of the corpuscles is the same as in 
the undiluted blood. By proceeding in a similar manner with 
solutions of different substances, a series of isotonic coefficients 
can be determined. The following table gives a comparison of 
the isotonic coefficients of various substances obtained by the 
plasmolytic, blood corpuscle and haematocrit methods. The iso- 
tonic coefficients are referred to that of cane sugar as unity. 

There are other methods which may be used for the comparison 
of osmotic pressures, among which may be mentioned that due to 
Wladimiroff,t involving the use of bacteria, and the interesting 

* Ibid., 17, 164 (1895). 

t Zeit. pltfs. Chem., 7, 529 (1891). 



206 



THEORETICAL CHEMISTRY 



Substance. 


Plasmolytic 
Method. 


Corpuscle 
Method. 


Haematocrit 
Method. 


C 12 H 2 2On . 


1.00 


1.00 


1 00 


MgSO 4 


1 09 


1 27 


1 10 


KNO 8 . 


1.67 


1 74 


1 84 


NaCl . . 


1 69 


1 75 


1 74 


CHs.COOK 


1 67 


1 66 


1 67 


CaCl 2 


2.40 


2,36 


2.33 











method develbped by Tammann,* in which artificially prepared 
membranes are employed. 

Osmotic Pressure and Diffusion. That there is a very close 
connection between osmotic pressure and diffusion, has already 
been pointed out. In fact the osmotic pressure of a solution may 
be regarded as the dnvmgTorce which causes the mdle^c^e^of a 
dissolved substance to distribute themselves uniformly through- 
out the solution. 

The process of diffusion was first systematically investigated 
by Graham f in 1850, but it was not until five years later that 
the general law of diffusion was enunciated by Fick.t He proved 
theoretically and experimentally that the quantity of solute, ds, 
which diffuses through an area A, in a time dt, when the concen- 
tration changes by an amount dc, in a distance dx, at right angles 
to the plane of A, is given by the equation 

ds= -DAdt, 

dx ' 

in which D is a constant, known as the coefficient of diffusion. 
Interpreting the equation of Fick in words, we see that the coeffi- 
cient of diffusion is the amount of solute which will cross 1 square 
centimeter in 1 second, if the change of concentration per centi- 
meter is unity. 
The phenomena of diffusion have also been investigated by 

* Wied. Ann., 34. 299 (1888). 

t Phil. Trans. (1850), p. 1, 805; (1861), p. 483. 

J Pogg. Ann., 94, 59 (1855). 



DILUTE SOLUTIONS AND OSMOTIC PRESSURE 207 

Nernst * and Planck.f If we have a tall cylindrical vessel con- 
taining a solution of a non-electrolyte in its lower part, and pure 
water at the top, the solute will slowly diffuse upward into the 
water. 

Assuming the osmotic pressure at a height x, to be P, and letting 
A denote the area of cross-section of the cylinder, the solute in 
the layer whose volume is A dx, will be subjected to a force equal 
to A dP y the negative sign indicating that the force acts in the 
direction of diminishing pressure. If c is the concentration in 
mols per cubic centimeter, the force acting on each molecule in 
this layer will be 

_ A.^= _!.*? 

cA dx c dx 

Let F denote the force necessary to drive a single molecule through 
the solution with the velocity of one centimeter per second. Since 
the velocity is constant, the resistance due to the viscosity of the 
medium must also be denoted by F. The velocity attained will 
be 

_J- <*? 

cF' dx 

If dN represents the number of molecules crossing each layer in 
a time dt, then, since the number crossing unit area per second 
is proportional to the concentration and to the mean velocity of 
the molecules, we shall have 

dN= -~-~ 
cF dx 

or > 

, Ar 1 A dP ,. 

dN = - nA-j-dt- 
F dx 

When the solution is dilute, we may apply the general equation, 
PV = RT, remembering that V = 1/c. Substituting in the pre- 
ceding equation, we have 

, RT dc,. 



* Zeit. phys. Chem., 2, 40, 615 (1888); 4, 129 (1889). 
t Wied. Ann., 40, 561 (1890). 



208 THEORETICAL CHEMISTRY 

Comparing this equation with that of Fick, we see that the 
coefficient of diffusion D, corresponds to the factor, RT/F. From 
the equation of Nernst it is possible to calculate the force required 
to drive a molecule of solute through the solution with unit veloc- 
ity. Thus, solving the above equation for F, we have 



By means of this equation, it has been calculated that the force 
necessary to drive one molecule of formic acid through water 
with a velocity of one centimeter per second at C. is equal to 
the weight of 4,340,000,000 kilograms. It is difficult at first to 
realize that such enormous forces are operative in solutions, but 
when one considers the minute size of the molecules and the great 
resistance offered by the medium, it becomes evident that a very 
large driving force must be applied to produce an appreciable 
movement of the solute through the solution. 

Principle of Soret. If a solution is maintained at a uniform 
temperature it will ultimately become homogeneous; if, on the 
other hand, two parts of a homogeneous solution are kept at 
different temperatures for some time, the solution will become 
more concentrated in the colder portion. This phenomenon was 
first investigated by Soret.* The experiments of Soret are of 
special interest, since they furnish a means of determining the 
influence of temperature on osmotic pressure. Thus, if the law 
of Gay-Lussac holds for osmotic pressure, the colder portion of a 
solution should increase in concentration by 1/273 for each degree 
of difference in temperature. The experimental results are in 
satisfactory agreement with the requirements of theory, and con- 
stitute another proof of the applicability of the gas laws to dilute 
solutions. 

Lowering of Vapor Pressure. It has long been known that 
the vapor pressure of a solution is less than that of the pure sol- 
vent, provided the solute is non-volatile. The investigations of 
von Babo and Wiillner f on the lowering of vapor pressure of 

* Ann. Chem. Phys. (5), 22, 293 (1881). 
t Pogg. Ann., 103, 529 (1858). 



DILUTE SOLUTIONS AND OSMOTIC PRESSURE 209 

, ' , ' ' A 

various liquids when non-vol^til'e substances are dissolved in 
them, resulted in the following generalizations: (1) The lower- 
ing of the vapor pressure of a solution is proportional to the amount 
of solute present; and (2) For the same solution, the lowering of the 
vapor pressure of the solvent by a non-volatile solute is at all tempera- 
tures a constant fraction of the vapor pressure of the pure solvent. 

In 1887, Raoult,* as the result of an exhaustive experimental 
investigation, enunciated the following laws: (1) When equi- 
mokcular quantities of different non-volatile solutes are dissolved in 
equal volumes of the same solvent^ the vapor pressure of the solvent is 
lowered by a constant amount; and (2) The ratio of the observed 
lowering of the vapor pressure to the vapor pressure of the pure sol- 
vent is equal to the ratio of the number of mols of solute to the total 
number of mols in the solution. The ratio of the observed lowering 
to the original vapor pressure is called the relative lowering of the 
vapor pressure. Letting pi and p 2 denote the vapor pressures of 
solvent and solution, Raoult's second law may be put in the 
form 

Pi Pz _ n 



PI 



n 



in which n and N represent the number of mols of solute and 
solvent respectively. Some of Raoult's results for ethereal solu- 
tions are given in the accompanying table. 









Pl-P2 




Mols of Solute 


Pi-Pt 


Pi 


Substance. 


per 100 mols 
of Solution. 


PI 

for Solution. 


for 1 molar 
per cent 
Solution. 


Turpentine 


8.95 


0.0885 


0099 


Methyl salicylic acid 


2.91 


0.026 


0.0089 


Methyl benzoic acid 


9.60 


0.091 


0.0095 


Benzoic acid 


7.175 


070 


0.0097 


Trichioracetic acid 


11.41 


0.120 


0.0105 


Aniline 


7.66 


0.081 


0.0106 











The results given in the fourth column of the table are nearly 

* Compt. rend., 104, 1430 (1887); Zeit. phys. Chem., 2, 372 (1888); Ann. 
Chem. Phys. (6), 15, 375 (1888). 



210 THEORETICAL CHEMISTRY 

constant, and are in close agreement with the theoretical value 
of the relative lowering of a 1 molar per cent solution calculated 
as follows: 

gL ~ ** 
pi 



When the solution is very dilute, the number of mols of solute 
is negligible in comparison with the number of mols of solvent, 
and the equation of Raoult may be written 

Pi Pz n_ 

Pi N' 

Since n = g/m, and N = W/M, where g and W are the weights of 
solute and solvent respectively, and m and M are the correspond- 
ing molecular weights, the above equation becomes 

PI ~ P2 = gM 
Pi Wm 

This equation enables us to calculate the molecular weight of a 
dissolved substance from the relative lowering of the vapor 
pressure produced by the solution of a known weight of solute in 
a known weight of solvent. Solving the equation for w, we 
have 

m - gM . Pl . 

'** fir 

W pi- pz 

As an illustration of the application of this equation, we may 
take the determination of the molecular weight of ethyl benzoate 
from the following experimental data : The vapor pressure at 
80 C. of a solution of 2.47 grams of ethyl benzoate in 100 grams 
of benzene is 742.6 rnm, : the vapor pressure of pure benzene at 
80 C. is 751.86 mm. Substituting in the equation, we have 

_ 2.47 X 78 751.86 



~ 751.86 - 742.6 

The molecular weight calculated from the formula, CcHs 
is 150. 

The difficulties which attend the accurate measurement of the 
vapor pressure of a solution by the static method have already 



DILUTE SOLUTIONS AND OSMOTIC PRESSURE 211 



been mentioned. While there are other methods which are pref- 
erable for the determination of the molecular weight of dissolved 
substances, the vapor pressure method has one marked advan- 
tage, it can be used for the same solution at widely divergent 
temperatures. The method devised by Walker and already 
described in connection with the determination of the vapor 
pressure of pure liquids (p. 92) is well adapted to the measure- 
ment of the vapor pressure of solutions. 

Connection between Lowering of Vapor Pressure and Osmotic 
Pressure. The relation between osmotic pressure and the 
lowering of vapor pressure has been derived 
in the following manner by Arrhenius.* 
Imagine a very dilute solution contained in 
the wide glass tube A> Fig. 64. The tube, 

A, is closed at its lower end with a semi- 
permeable membrane, and dips into a vessel, 

B, which contains the pure solvent. The 
entire apparatus is covered by a bell-jar C, 
and the enclosed space exhausted. Let 
h be the difference in level between the 
solvent and solution when equilibrium is 
established, that is, when the hydrostatic 
pressure of the column of liquid is equal to 
the osmotic pressure. When equilibrium is . 
attained, the vapor pressure of the solution 

at the height "h" will be equal to the pres- 
sure of the vapor of the solvent at this height. If the vapor pres- 
sure of the pure solvent in the vessel B is pi, and if p 2 is the vapor 
pressure of the solution at the height h, we shall have 

Pi p* = hd, 

where d denotes the density of the vapor. Let v be the volume of 
1 mol of solvent in the state of vapor, then 



Fig. 64. 



and 



piv = RT, 

RT 

v = 
Pi 

* Zeit. phys. Chem., 3, 115 (1889). 



212 THEORETICAL CHEMISTRY 

If the molecular weight of the solvent is M, we may replace v 
by M/d, when the preceding equation becomes 

M ^RT 

d pi ' 
or, 



The solution being very dilute the osmotic pressure may be cal- 
culated from the equation 

PV = nRT, 

where P is the osmotic pressure of the solution, V the volume 
of the solution containing 1 mol of solute, and n the number of 
mols of solute present. If s represents the density of the solvent 
and also of the solution, since it is very dilute, we may write 

P = hs, 
and 



where g is the number of grams of the solvent in which the n 
mols of solute are dissolved. Substituting these values of P and 
V in the general equation, we have 

PV = nRT = hg, 
and solving for A, 

h - nRT (to 

Q W 

if 

Substituting the values of d and h, given in equations (2) and 
(3), in equation (Hwe have 

nRT Mpi nMpi , A . 

*-- "Bf- , ' (4) 

Rearranging equation (4), and remembering that N = g/M, we 
obtain 

Pi P* __ n ( . 

~^ IT (5) 

This equation it will be seen, is identical with that derived experi- 
mentally by Raoult for very dilute solutions. 



DILUTE SOLUTIONS AND OSMOTIC PRESSURES 213 

van't Hoff showed, by an application of thermodynamics to 
dilute solutions, that the relation between osmotic pressure and 
the relative lowering of the vapor pressure is expressed by the 
equation 

Pi pz = MP 
pi sRT' 

in which the symbols have the same significance as above. This 
equation may be reconciled easily with the equation of Raoult. 
If n in equation (5) be replaced by its equal, PV/RT, the 
equation becomes 

pi-?>2 = PV 
pi NET* 

But V = NM/s, hence 

pi - p 2 = MP 

pi sRT' 

This equation shows that the relative lowering of the vapor pressure 
is directly proportional to the osmotic pressure. 

Elevation of the Boiling-Point. Just as the vapor pressure of 
a solution is less than that of the pure solvent, so the boiling-point 
of a solution is correspondingly higher than the boiling-point of 
the solvent. It follows that when equimolecular quantities of 
different substances are dissolved in equal volumes of the same 
solvent, the elevation of the boiling-point is constant. Thus, the 
molecular weight of any soluble substance may be determined by 
comparing its effect on the boiling-point of a particular solvent, 
with that of a solute of known molecular weight. The elevation 
in boiling-point produced by dissolving 1 mol of a solute in 100 
grams, or 100 cubic centimeters, of a solvent is termed the molec- 
ular elevation, or boiling-point constant of the solvent. In deter- 
mining the boiling-point constant of a solvent, a fairly dilute 
(solution is employed and the elevation in the boiling-point is 
observed; the value of the constant is then calculated on the 
assumption that the elevation in boiling-point is proportional to 
the concentration. 

If g grams of a substance of unknown molecular weight m, 
are dissolved in W grams of solvent, and the boiling-point is raised 



214 



THEORETICAL CHEMISTRY 



A degrees, then, since m grams of the substance when dissolved 
in 100 grams of solvent, produce an elevation of K degrees (the 
molecular elevation), it follows that 



1000. 



therefore, 



W 



: A :: m : K, 



The accompanying table gives the boiling-point constants for 
100 grams and 100 cubic centimeters of some of the more com- 
mon solvents. 



Solvent. 


Molecular Elevation. 


100 gr. 


100 cc. 


Water. 
Ethyl alcohol 
Ether 


5 2 
11 5 
21 
16 7 
26 7 
35 6 
30 1 


5 4 
15 6 
30 3 
22 2 
32 8 


Acetone ... . 


Benzene . . ... 
Chloroform . . 
Pyridine . . 



As an example of the calculation of the molecular weight of a 
dissolved substance by the above formula, we may take the 
calculation of the molecular weight of camphor in acetone from 
the following data: 

When 0.674 gram of camphor is dissolved in 6.81 grams oi 
acetone, the boiling-point of the solvent is raised 1.09. Substitut- 
ing in the formula, we have 

m = 100 X 16.7 X ' 674 



6.81 X 1.09 



= 151. 



The molecular weight of camphor according to the formula 
CioHieO, is 152. 

The molecular elevation of the boiling-point can be calculated 
by means of the formula, 

K = 0.02 T 2 
w 



DILUTE SOLUTIONS AND OSMOTIC PRESSURE 215 



T :: 



in which T is the absolute boiling-point of the solvent, and w is 
the heat of vaporization for 1 gram of the solvent at its boiling- 
point. This formula will be derived in a subsequent paragraph 
of this chapter. The calculated values 
of K are in close agreement with the 
values obtained experimentally by Raoult 
and others. As an example, the calcu- 
lated value of the molecular elevation 
for water, the heat of vaporization of 
which at 100 C. is 537 calories, is 

0.02 X (373) 2 = r 
537 "" ' 

a value in exact agreement with the exper- 
imental value given in the table. 

Experimental Determination of Mo- 
lecular Weight by the Boiling-Point 
Method. One of the simplest and most 
convenient of the various forms of appa- 
ratus which have been devised for the 
determination of the boiling-point of 
solutions, is that developed by Jones,* 
and shown in Fig. 65. The liquid whose 
boiling-point is to be determined is intro- 
duced into the vessel A, which already 
contains a platinum cylinder P, em- 
bedded in some glass beads. Sufficient 
liquid is introduced to insure the com- 
plete covering of the bulb of the ther- 
mometer, as shown in the sketch. 

The side tube of A is connected with a condenser, C. 




65. 



The 



vessel A, is surrounded by a thick jacket of asbestos J, and rests 
on a piece of asbestos board in which a circular hole is cut, and 
over which a piece of wire gauze is placed. The liquid is heated 
by means of a burner, B. The platinum cylinder is the feature 
which differentiates this apparatus from the various other forms 

* Am. Chem. Jour., 19, 581 (1897). 



216 THEORETICAL CHEMISTRY 

of boiling-point apparatus. It has the two-fold object of pre- 
venting the condensed solvent from coming in direct contact with 
the bulb of the thermometer, and of reducing the effect of radia- 
tion to a minimum. The liquid in A is boiled, using a very small 
flame, until the thermometer remains constant; this temperature 
is taken as the boiling-point of the liquid. The apparatus is now 
emptied and dried. A weighed amount of the liquid is then intro- 
duced into A, and to this is added a known weight of solute; the 
thermometer is replaced and the boiling-point of the solution is 
determined. The difference between the readings of the thermom- 
eter when immersed in the solution, and in the solvent alone, gives 
the boiling-point elevation. For further details concerning the 
boiling-point method as applied to the determination of molecular 
weights, the student is referred to any one of the standard labo- 
ratory manuals. 

Osmotic Pressure and Boiling-Point Elevation. Imagine a 
dilute solution containing n mols of solute in G grams of solvent, 
and let dT be the elevation in the boiling-point. Suppose a large 
quantity of the solution to be introduced into a cylinder, fitted 
with a frictionless piston, and closed at the bottom by a semi- 
permeable membrane. Let the cylinder and contents be raised 
to the absolute temperature T, the boiling-point of the solvent, 
and then let pressure be exerted on the piston just sufficient 
to overcome the osmotic pressure of the solution. In this way, let 
a quantity of solvent corresponding to 1 mol of solute be forced 
through the semi-permeable membrane. The volume V, thus 
expelled is the volume corresponding to G/n grams of solvent. 
If the osmotic pressure of the solution is P, then the work done in 
moving the piston and expelling the solvent is PV. Now let the 
portion of the solvent which has been forced through the semi- 
permeable membrane be vaporized. For this operation G/n . w 
calories will be required, w being the heat of vaporization for 1 
gram of solvent at its boiling-point. Then let the entire system 
be raised to the boiling-point of the solution (T + dT), the pre- 
viously expelled G/n grams of vapor being allowed to mix with 
the solution. The heat of vaporization lost at T is thus recov- 
ered at the slightly higher temperature, (T + dT). Finally, the 



DILUTE SOLUTIONS AND OSMOTIC PRESSURE 217 

entire system is cooled to T, and is thus restored to its original 
state. Applying the well-known thermodynamic relation, that 
the ratio of the work done to the heat absorbed, is the same as 
the ratio of the difference in temperature to the absolute initial 
temperature of the system, we have 



PV 




therefore, 



But, since PV = RT, equation (1) may be written 

n 



If n = I and G = 100 grams, then dT = K (the molecular eleva- 
tion of the boiling-point), or 



( 
(2) 



lOOll) 

Or putting R 2 calories, we have 

0.02 T* 



Equation (1) shows that the osmotic pressure of a solution is directly 
proportional to the elevation of the boiling-point. Equation (2) was 
originally derived by van't Hoff at about the time when Raoult 
determined the values of K experimentally. 

Lowering of the Freezing-Point. Of all the methods employed 
for the determination of molecular weights in solution, the freez- 
ing-point method is the most accurate and the most widely used. 
It was pointed out by Blagden * over a century ago, that the de- 
pression of the freezing-point of a solvent by a dissolved substance is 
directly proportional to the concentration of the solution. When 
equimolecular quantities of different substances are dissolved in 
equal volumes of the same solvent, the lowering of the freezing- 
point is constant. The molecular weight of any soluble sub- 

* Phil Trans., 78, 277 (1788). 



218 



THEORETICAL CHEMISTRY 



stance can be found, asrin the beilmg-point method, by comparing 
its effect on the freezing-point of a solvent with that of a solute 
of known molecular weight. The molecular lowering of the freezing- 
point, or the freezing-point constant, of a solvent is defined as the de- 
pression of the freezing-point produced by dissolving 1 mol of solute 
in 100 grams or 100 cubic centimeters of solvent. The freezing-point 
constants of a few common solvents are given in the following table. 



Solvent. 


Molecular Depression. 


100 gr. 


100 cc. 


Water 


18 5 
50 
74 
69 
39 


18 5 
56 

41 " 


Benzene. ... 


Phenol 
Naphthalene . . . . . ... 
Acetic acid. ..... . 



van't Hoff showed that the molecular lowering of the freezing- 
point of a solvent K, can be calculated from the absolute freez- 
ing-point T 7 , and the heat of fusion w, for 1 gram of solvent at 
the temperature T, by means of the formula 

= 0.02 T 2 
w 

This expression is analogous to that which applies to the molec- 
ular elevation of the boiling-point. The agreement between the 
observed and the calculated values of K is very satisfactory, as 
the following calculation for water shows: 

v 0.02 X (273) 2 



80 



18.6. 



It is of interest to note that the calculated value of K for water is 
lower than the experimental values originally obtained by Raoult 
and others. Subsequent experiments, carried out with greater 
care and better apparatus, by Raoult, Abegg and Loomis gave 
values in close agreement with that derived theoretically. A 
formula analogous to that employed for the calculation of the 
molecular weight of a dissolved substance from the elevation it 



DILUTE SOLUTIONS AND OSMOTIC PRESSURE 219 

produces in the boiling-point of a solvent, may be used for the 
calculation of molecular weight from freezing-point depression. 
Thus, if g grams of solute when dissolved in W grams of solvent 
produce a depression A of the freezing-point of the solvent, the 
molecular weight m, is given by the formula 



m= 



where K is the molecular lowering of the freezing-point. 

EXAMPLE. When 1.458 grams of acetone are dissolved in 
100 grams of benzene, the freezing-point of the solvent is de- 
pressed 1.22, therefore the molecular weight of acetone is 



m = 100 X 50 X j-oo - 59.8. 

The molecular weight of acetone, calculated from the formula 
C 3 H 6 0, is 58. 

In order to obtain trustworthy results with the freezing-point 
method, it is necessary that only the pure solvent separate out 
when the solution freezes, and that excessive overcooling be 
avoided. When too great overcooling occurs, the subsequent 
freezing of the solution results in the separation of so large an 
amount of solvent in the solid state, that the observed freezing- 
point corresponds to the equilibrium temperature of a more 
concentrated solution than that originally prepared. A formula 
for the correction of the concentration, due to excessive overcooling, 
has been derived by Jones.* If the overcooling of the solution 
in degrees be represented by u, the heat of fusion of 1 gram of sol- 
vent at the freezing-point by w, and the specific heat of the sol- 
vent by c, then the fraction of the solvent which will solidify, /, 
may be calculated by the formula, 



When water is used as the solvent, c = 1 and w = 80. There- 

fore, for every degree of overcooling, the fraction of the solvent 

separating as ice will be 1/80, and the concentration of the original 

* Zeit. phys. Chem., 12, 624 (1893). 



220 



THEORETICAL CHEMISTRY 



solution is increased by just so much. It is simpler, however, to 
apply the correction directly to the freezing-point depression in- 
stead of to the concentration. 

Experimental Determination of Molecular Weight by the 
Freezing-Point Method. The apparatus almost universally em- 
ployed for the determination of molecular weights by the freezing- 
point method is that devised by Beckmann,* 
and shown in Fig. 66. It consists of a thick- 
walled test tube A, provided with a side tube, 
and fitted into a wider tube Ai, thus surround- 
ing A with an air space. 

The whole is fitted into the metai cover of 
a large battery jar, which is filled with a freez- 
ing mixture whose temperature is several de- 
grees below the freezing-point of the solvent. 

The tube A is closed by a cork stopper, 
through which passes the thermometer and 
stirrer. The thermometer is generally of the 
Beckmann differential type. This instrument 
has a scale about 6 in length, each degree 
being divided into hundredths; the quantity 
of mercury in the bulb can be varied by means 
of a small reservoir at the top of the scale, so 
that the zero of the instrument can be adjusted 
for use with solvents of widely different freez- 
ing-points. In carrying out a determination 
lg * ' with the Beckmann apparatus, a weighed 

quantity of solvent is placed in A, and the temperature of the 
refrigerating mixture regulated so as to be not more than 5 below 
the freezing-point of the solvent. The tube A is removed from its 
jacket, and is immersed in the freezing mixture until the solvent 
begins to freeze. It is then replaced in the jacket AI, and the 
solvent is vigorously stirred. The thermometer rises during the 
stirring until the true freezing-point is reached, after which it 
remains constant. This temperature is taken as the "freezing 
temperature of the solvent. 

* Zeit. phys. Chem., 2, 683 (1888). 




DILUTE SOLUTIONS AND OSMOTIC PRESSURE 221 

The tube A is now removed from the freezing mixture, and a 
weighed amount of the substance whose molecular weight is to 
be determined is introduced. When the substance has dissolved, 
the tube is replaced in A\ and the solution cooled not more than 
a degree below its freezing-point. A small fragment of the solid 
solvent is dropped into the solution, which is then stirred 
vigorously until the thermometer remains constant. The maxi- 
mum temperature is taken as the freezing-point of the solution. 
The difference between the freezing-points of solution and 
solvent is the depression sought. For further details con- 
cerning the determination of molecular weights by the freezing- 
point method, the student is referred to a physico-chemical 
laboratory manual. 

Osmotic Pressure and Freezing-Point Depression. Let dT 
be the freezing-point depression produced by n mols of solute in 
G grams of solvent, the solution being dilute. Imagine a large 
quantity of this solution to be confined within a cylinder fitted 
with a frictionless piston, the bottom of the cylinder being closed 
by a semi-permeable membrane. Let the cylinder and contents 
be cooled to the freezing temperature of the solvent I 7 , and then 
let pressure be applied to the piston until a quantity of solvent 
corresponding to 1 mol of solute is forced through the semi-perme- 
able membrane. This requires an expenditure of energy equiva- 
lent to PV, where P is the osmotic pressure of the solution and V 
is the volume of solvent expelled. The volume V is clearly the 
volume of 0/n grams of solvent. Now let the expelled portion 

S*1 

of solvent be frozen and the system deprived of w calories of 

heat, where w is the heat of fusion of 1 gram of the solvent at the 
temperature T. 

The temperature of the solution is then lowered to its freezing- 
point (T dT)^ and the 0/n grams of solidified solvent dropped 
into it. The solidified solvent melts, thereby restoring to the 
system at the temperature (T dT), the heat of fusion formerly 
taken from it. Finally, the temperature of the system is raised 
to T f , the initial temperature of the cycle. Applying the familiar 
thermodynamic relation, that the ratio of the work done to the 



222 THEORETICAL CHEMISTRY 

heat absorbed, is the same as the ratio of the difference in temper- 
ature to the initial absolute temperature, we have 

PV _ dT. () 

G ~T~ W 

w 
n 

From which we obtain 



w G 
But PV = RT, hence equation (1) becomes 



UJ ~ w G 

If n = 1 and G = 100 grams, then dT = K, the molecular lower- 
ing of the freezing-point, and 

RT* 
100 W' 

Or putting R = 2 calories, we have 



A' = . (2) 

w ^ J 

An equation to which reference has already been made. 

It is evident from equation (1) that the osmotic pressure of a 
solution is directly proportional to the freezing-point depression. 

Molecular Weight in Solution. As has been pointed out, the 
molecular weight of a dissolved substance can be readily calcu- 
lated, provided that the osmotic pressure of a dilute solution of 
known concentration at known temperature is determined. But 
the experimental difficulties attending the direct measurement 
of osmotic pressure are so great, that it is customary to employ 
other methods based upon properties of dilute solutions which 
are proportional to osmotic pressure. We have shown that in 
dilute solutions osmotic pressure is directly proportional (1) to 
the relative lowering of the vapor pressure, (2) to the elevation 
of the boiling-point, and (3) to the depression of the freezing-point. 

From this it follows, that equimolecular quantities of different 
substances dissolved in equal volumes of the same solvent, exert the 



DILUTE SOLUTIONS AND OSMOTIC PRESSURE 223 

same osmotic pressure, and produce the same relative lowering of 
vapor pressure, the same elevation of boiling-point, and the same 
depression of freezing-point. Since equimolecular quantities of 
different substances contain the same number of molecules, it is 
evident that the magnitude of osmotic pressure, relative lowering of 
vapor pressure, elevation of boiling-point and depression of freezing- 
point, is dependent upon the number of particles present in the solu- 
tion and is independent of their nature. It has been pointed out 
by Nernst that any process which involves the separation of 
solvent from solute, may be employed for the determination of 
molecular weights. A little reflection will convince the reader 
that the four methods discussed in this chapter involve such separ- 
ation. Both van't Hoff and Raoult emphasized the fact that the 
formulas derived for the determination of molecular weights in so- 
lution depend upon assumptions which are valid only for dilute 
solutions. It follows, therefore, that we are not justified in apply- 
ing these formulas to concentrated solutions. Up to the present time 
we have no satisfactory theory of concentrated solutions, neither 
can we state up to what concentration the gas laws apply. 

PROBLEMS. 

1. At 10 C. the osmotic pressure of a solution of urea is 500 mm. 
of mercury. If the solution is diluted to ten times its original volume, 
what is the osmotic pressure at 15 C.? Ans. 50.89 mm. 

2. The osmotic pressure of a solution of 0.184 gram of urea in 100 cc. 
of water was 56 cm. of mercury at 30 C. Calculate the molecular weight 
of urea. Ans. 62.12. 

3. At 24 C. the osmotic pressure of a cane sugar solution is 2.51 atmos- 
pheres. What is the concentration of the solution in mols per liter? 

Ans. 0.103. 

4. At 25. 1 C. the osmotic pressure of solution of glucose containing 
18 grams per liter was 2.43 atmospheres. Calculate the numerical value 
of the constant R, when the unit of energy is the gram-centimeter. 

Ans. 84,231. 

Jb. The vapor pressure of ether at 20 C. is 442 mm. and that of a solu- 
tion of 6.1 grams of benzoic acid in 50 grams of ether is 410 mm. at the 
same temperature. Calculate the molecular weight of benzoic acid in 
ether. Ans. 124. 



224 THEORETICAL CHEMISTRY 

6. At 10 C. the vapor pressure of ether is 291.8 mm. and that of a 
solution containing 5.3 grams of benzaldehyde in 50 grams of ether is 
271.8 mm. What is the molecular weight of benzaldehyde? 

Ans. 106.6. 

7. A solution containing 0.5042 gram of a substance dissolved in 42.02 
grams of benzene boils at 80. 175 C. Find the molecular weight of the 
solute, having given that the boiling-point of benzene is 80.00 C.. and its 
heat of vaporization is 94 calories per gram. Ans. 181.9. 

8. A solution containing 0.7269 gram of camphor (mol. wt. = 152) 
in 32.08 grams of acetone (boiling-point = 56.30 C.) boiled at 56.55 C. 
What is the molecular elevation of the boiling-point of acetone? What is 
its heat of vaporization? 

Ans. K = 16.74; w = 129.5 cals. per gm. 

9. A solution of 9.472 grams of CdI 2 in 44.69 grams of water boiled at 
100.303 C. The heat of vaporization of water is 536 calories per gram. 
What is the molecular weight of CdI 2 in the solution? What conclusion 
as to the state of CdI 2 in solution may be drawn from the result? 

Ans. 363.2. 

10. The freezing-point of pure benzene is 5.440 C. and that of a solu- 
tion containing 2.093 grams of benzaldehyde in 100 grams of benzene is 
4.440 C. Calculate the molecular weight of benzaldehyde in the solu- 
tion. K for benzene is 50. Ans. 104.6. 

11. A solution of 0.502 gram of acetone in 100 grams of glacial acetic 
acid gave a depression of the freezing-point of 0.339 C. Calculate the 
molecular depression for glacial acetic acid. Ans. 39. 

12. By dissolving 0.0821 gram of m-hydroxybenzaldehyde (C 7 H 6 2 ) in 
20 grams of naphthalene (melting point 80. 1 C.) the freezing-point is 
lowered by 0.232 C. Assuming that the molecular weight of the solute 
is normal in the solution, calculate the molecular depression for naphtha- 
lene and the heat of fusion per gram. 

Ans. K = 68.96; w = 36.2 cals. per gm. 



CHAPTER XII. 
ASSOCIATION, DISSOCIATION AND SOLVATION. 

Abnormal Solutes. As has already been pointed out the 
acceptance of Avogadro's hypothesis was greatly retarded by the 
discovery of certain substances whose vapor densities were ab- 
normal. Thus, the vapor density of ammonium chloride is approx- 
imately one-half of that required by the formula NH 4 C1, while 
the vapor density of acetic acid corresponds to a formula whose 
molecular weight is greater than that calculated from the formula, 
C2H 4 O 2 . The anomalous behavior of ammonium chloride and 
kindred substances <has been shown to be due, not to a failure of 
Avogadro's law, but to a breaking down of the molecules a 
process known as dissociation. The abnormally large molecular 
weight of acetic acid on the other hand, has been ascribed to a 
process of aggregation of the normal molecules, known as asso- 
ciation. In extending the gas laws to dilute solutions similar 
phenomena have been encountered. 

Association in Solution. When the molecular weight of acetic 
acid in benzene is determined by the freezing-point method, the 
depression of the freezing-point is abnormally small and conse- 
quently, as the formula 



shows, the molecular weight will be greater than that correspond- 
ing to the formula, C 2 H 4 O 2 . Acetic acid in benzene solution is 
thus shown to be associated. Almost all compounds containing 
the hydroxyl and cyanogen groups when dissolved in benzene are 
found to be associated. Solvents, such as benzene and chloro- 
form, are frequently termed associating .solvents, although it is 
doubtful whether they exert any associating action. There is 
considerable experimental evidence to show that those substances 

225 



226 THEORETICAL CHEMISTRY 

whose molecules are associated in benzene and chloroform solu- 
tion, are also associated in the free condition. Just as the depres- 
sion of the freezing-point of a solution of an associated substance 
is abnormally small, so its osmotic pressure and other related 
properties will be less than the calculated values. 

Dissociation in Solution. Van't Hoff * pointed out that the 
osmotic pressure of solutions of most salts, of all strong acids, and 
of all strong bases is much greater for all concentrations than 
would be expected from the osmotic pressure of solutions of sub- 
stances, like cane sugar or urea, for corresponding concentrations. 
He was unable to account for this abnormal behavior, and in order 
to render the general gas equation applicable, he introduced a 
factor i t the modified equation being 

PV = iRT. 

If the osmotic pressure of some substance, like cane sugar, which 
behaves normally, be represented by P , the factor i is given by 
the expression 

. P 



Since the osmotic pressure of a solution is proportional to the 
relative lowering of its vapor pressure, to the elevation of its 
boiling-point, and to the lowering of its freezing-point, we may 
write 



. _ ^ Pl 

' 




where the symbols have their usual significance. The subscript 
refers in each case to a substance which behaves normally, and A 
denotes either boiling-point elevation or freezing-point depression. 
A. more definite conception of the abnormal behavior of salts 
will be gained by an inspection of the accompanying tables. In 
the first column is recorded the molar concentration of the solu- 
tion; the second column gives the observed depressions of the 
* Zeit. phys. Chem., i, 501 (1887). 



ASSOCIATION, DISSOCIATION AND SOLVATION 227 



freezing-point and the third column contains the values of the 
ratio of the observed depression to the normal depression, or i. 



POTASSIUM CHLORIDE. 



POTASSIUM SULPHATE. 



m 


A 





m 


A 


{ 


05 


1750 


1 88 


05 


2270 


2 33 


10 


3445 


1 85 


10 


4317 


2 32 


20 


6808 


1 83 


20 


8134 


2 18 


40 


1 3412 


1 80 


30 


1 1673 


2 09 



ALUMINIUM CHLORIDE. 



SODIUM CHROMATE. 



m 


A 


i 


m 


A 


i 


046 
0.076 
102 


276 
446 
578 


3 22 
3.15 - 
3.04 


0.1 
2 
5 
1 


450 
850 
1.960 
3 800 


2 42 
2 28 
2 11 
2 04 















It is apparent from the above data that the depression of the 
freezing-point of water caused by these salts is abnormally large, 
a fact which points to an increase in the number of dissolved units 
over that corresponding to the initial concentration. 

The Theory of Electrolytic Dissociation. In 1887, Arrhenius * 
advanced an hypothesis to account for the abnormal osmotic 
activity of solutions of acids, bases and salts. He pointed out 
that just as the exceptional behavior of certain gases has been 
completely reconciled with the law of Avogadro, by assuming a 
dissociation of the vaporized molecule into two or more simpler 
molecules, so the enhanced osmotic pressure and the abnormally 
great freezing-point depression of solutions of acids, bases and 
salts can be explained, if we assume a similar process of dissoci- 
ation. He proposed, therefore, that aqueous solutions of acids, 
bases and salts be considered as dissociated, to a greater or less 
extent, into positively- and negatively-charged particles or ions, 
and that the increase in the number of dissolved units due to this 



* Zeit. phys. Chem., i, 631 (1887). 



228 THEORETICAL CHEMISTRY 

dissociation is the cause of the enhanced osmotic activity. Accord- 
ing to this hypothesis, hydrochloric acid, potassium hydroxide and 
potassium chloride, when dissolved in water, dissociate in the 
following manner: 

HC1 -H'+C1' 



KC1 -K' + C1', 

where the dots indicate positively-charged ions and the dashes 
negatively-charged ions. 

In each of the above cases, one molecule yields two ions, so 
that, if dissociation is complete, the maximum osmotic effect 
should not be greater than twice that produced by an equimolecu- 
lar quantity of a substance which behaves normally. Reference 
to the preceding table shows that the value of i for potassium 
chloride approaches the limiting value of 2 as the solution is 
diluted. The other salts given in the table dissociate, according 
to Arrhenius, in the following way: 



AlCl 3 -Ar'+3Cl'. 

If these equations correctly represent the process of dissociation, 
then when dissociation is complete, the osmotic effect of infinitely 
dilute solutions of potassium sulphate and sodium chromate 
should be three times the effect produced by an equimolecular 
quantity of a normal solute, and in the case of aluminium chloride, 
the maximum effect should be four times the effect due to a normal 
substance. A glance at the table shows that the values of i for 
solutions of the three salts approach these respective limits. If 
this hypothesis of ionic dissociation be accepted, then it becomes 
possible to calculate the degree of ionization in any solution by 
comparing its freezing-point depression with the freezing-point 
lowering of an equimolecular solution of a normal substance. 

Let us suppose that the degree of dissociation of 1 molecule of 
a dissolved substance is a, each molecule yielding n ions. Then 



ASSOCIATION, DISSOCIATION AND SOLVATION 229 

1 a will be the undissociated portion of the molecule, and the 
total number of dissolved units will be 

1 a + na. 

If A is the depression of the freezing-point produced by the sub- 
stance, and AO the depression produced by an equimolecular 
quantity of an undissociated substance, then 

1 a + na __ A __ . 

i "~ "7T" l > 



or 



a 



i- 1 



n- I 

It will be observed that this formula is identical with that derived 
for the degree of dissociation of a gas (p. 91). If this formula 
be applied to the freezing-point data for solutions of potassium 
chloride given in the preceding table we find the following per- 
centages of dissociation corresponding to the different concentra- 
tions: 

POTASSIUM CHLORIDE. 



m 


A 


t 


a 


0.05 


1750 


1.88 


88 


0.10 


3445 


1.85 


85 


20 


6808 


1 83 


83 


40 


1 3412 


1.80 


80 



Thfc figures in the last column show that the degree of dissocia- 
tion increases as the concentration diminishes. It was further 
pointed out by Arrhenius that all of the substances which exhibit 
abnormal osmotic effects, when dissolved in water, yield solutions 
which conduct the electric current, whereas, aqueous solutions of 
such substances as cane sugar, urea and alcohol, exert normal 
osmotic pressures, but do not conduct electricity any better than 
the pure solvent. In other words, only electrolytes * are capable of 
undergoing ionic dissociation; hence Arrhenius termed his hypoth- 
esis the electrolytic dissociation theory. As we have seen, when 
potassium chloride is dissolved in water, it is supposed to dissoci- 

* The term electrolyte strictly refers to the solution of an ionized substance, 
although it is often applied to acids, bases and salts because, when dissolved, 
they produce electrolytes. To avoid confusion, the term "ionogen" (ion 
former) has been proposed for those substances which give conducting solu- 
tions. 



230 



THEORETICAL CHEMISTRY 



ate into positively-charged potassium ions and negatively-charged 
chlorine ions. Accordingly when two platinum electrodes, one 
charged positively and the other negatively, are introduced into 
the solution, the potassium ions move toward the negative elec- 
trode and the chlorine ions move toward the positive electrode, the 
passage of a current through the solution consisting in the ionic 
transfer of electric charges. Since the undissociated molecules, 
being electrically neutral, do not participate in the transfer of 
electric charges, it follows that the conductance of a solution of 
an electrolyte is dependent upon the degree of dissociation. The 
relation between electrical conductance and the degree of ioniza- 
tion will be discussed in a subsequent chapter. It may be stated 
at this point, however, that the values of a based upon measure- 
ments of electrical conductance, while showing some discrepancies 
in individual cases, are in general in good agreement with the 
values obtained by the freezing-point method. Furthermore, the 
values of a obtained from freezing-point measurements are in har- 
mony with those based upon De Vries' isotonic coefficients. 
It will be seen, on referring to the table of isotonic coefficients 
(p. 181), that solutions of electrolytes show enhanced osmotic 
activity. Thus, the osmotic pressures of equi-molecular solutions 
of cane sugar, potassium nitrate and calcium chloride are to each 
other as 1 : 167 : 2.40. 

The following table illustrates the general agreement between 
the values of i calculated, (a) from electrical conductance, (b) 
from freezing-point depression, and (c) from De Vries' isotonic 
coefficients. 



Substance. 


Molar Cone. 


(a) 


(b) 


fo) 


KC1 


14 


1 86 


1 82 


1 81 


LiCl 


13 


1 84 


1.94 


1.92 


Ca(NO 3 ) 2 


0.18 


2.46 


2 47 


2.48 


MgCU 


19 


2.48 


2.68 


2.79 


CaCl 2 


0.184 


2.42 


2.67 


2.78 













It must be remembered that the values of i derived from freez- 
ing-point measurements correspond to temperatures in the 



ASSOCIATION, DISSOCIATION AND SOLVATION 231 

neighborhood of C., while those derived from the other 
methods correspond to temperatures ranging from 17 C. to 
25 C. 

Chemical Properties of Completely Ionized Solutions. The 

chemical properties of an ion are very different from the properties 
of the atom or radical when deprived of its electrical charge. 
For example, the sodium ion is present in an aqueous solution of 
sodium chloride, but there is no evidence of chemical reaction with 
the solvent; whereas, the element in the electrically-neutral con- 
dition reacts violently with water, evolving hydrogen and form- 
ing a solution of potassium hydroxide. Again, take the element 
chlorine: when chlorine in the molecular condition, either as gas 
or in solution, is added to a solution of silver nitrate, no precipitate 
of silver chloride is formed. Further, chlorine in such compounds 
as dlVls, OClt, etc., is not precipitated by silver nitrate, since 
these compounds are not dissociated by water. Or, chlorine may 
bo present in a compound which is dissociated by water and yet 
not exhibit its characteristic reactions, because it is present in a 
complex ion. Thus, potassium chlorate dissociates in the follow- 
ing manner: - 



On adding silver nitrate, there is no precipitation, because the 
chlorine forms a complex ion with oxygen. 

Physical Properties of Completely Ionized Solutions. The 
physical properties of completely ionized solutions are, in general, 
additive. This is well illustrated by a series of solutions of 
colored salts, the color of which is due to the presence of a par- 
ticular ion. It is found, when the solutions are sirfficiently 
dilute to insure complete dissociation, that they all have the 
same color. The additive character of the colors of solutions of 
electrolytes is brought out in a striking manner by a comparison 
of their absorption spectra. Ostwald * photographed the absorp- 
tion spectra of solutions of the permanganates of lithium, cadmium, 
ammonium, zinc, potassium, nickel, magnesium, copper, hydrogen 
and aluminium, each solution containing 0.002 grain-equivalents 

* Zcit. phys. Chein., 9, 579 (1892). 



232 THEORETICAL CHEMISTRY 

of salt per liter. The absorption spectra, as shown in Pig. 67, 
will be seen to be practically identical, the bands occupying the 
same position in each spectrum. This affords a strong con- 
firmation of the theory of electrolytic dissociation, according to 

which a dilute solution is to be 
regarded as a mixture of elec- 
trically equivalent quantities of 
oppositely charged ions, each of 
which contributes its specific 
properties to the solution. The 
permanganate ion being colored, 
and common to all of the salts 
examined, and the positive ions 
of the various substances being 
colorless, it follows that when dis- 
sociation is complete, the absorp- 
tion spectra of all of the solutions 
must be identical. A number of 
other properties of completely 
dissociated solutions have been 
shown to be additive. Among 
these may be mentioned density, 
specific refraction, surface ten- 
sion, thermal expansion, and 
magnetic rotatory power. Addi- 
tional evidence in favor of the 
theory of electrolytic dissociation 
will be furnished in forthcoming 
.p. 67 chapters. Notwithstanding the 

large number of facts which can 
be satisfactorily interpreted by the theory, there are directions 
in which it requires amplification and modification. Of the 
various objections which have been urged against the theory of 
electrolytic dissociation, one is of sufficient weight to call for 
brief consideration here. When two elements, such as potassium 
and chlorine, Combine to form potassium chloride, the reaction 
is violent and a large amount of heat is developed. Nevertheless, 




ASSOCIATION, DISSOCIATION AND SOLVATION 233 

according to the ionization theory, the strong, mutual affinity of 
these two elements is overcome by the act of solution in water, 
the molecule being split into two oppositely-charged atoms. Obvi- 
ously such a separation calls for the expenditure of a large amount 
of energy, and the question naturally arises: What is the 
source of this energy? While this question cannot be fully 
answered here, it may be pointed out that we have abundant 
evidence to show that the ions are hydrated, each being surrounded 
by an "atmosphere" of solvent. In view of this fact, it has been 
suggested * that dissociation in aqueous solution is caused by the 
mutual attraction between the ions and the molecules of the sol- 
vent, the heat of ionic hydration furnishing the energy necessary 
for the separation of the ions. 

Freezing-Point Depressions Produced by Concentrated Solu- 
tions of Electrolytes. As has already been mentioned, the 
dissociation of electrolytes in aqueous solution increases with the 
dilution, becoming complete at a concentration of about 0.001 
molar. We should expect the dissociation to diminish with in- 
creasing concentration, until, if the electrolyte is sufficiently solu- 
ble, the depression of the freezing-point becomes normal. Recent 
investigations by Jones and his co-workers f have shown that the 
facts are contradictory to this expectation. They found that the 
value of the molecular depression of the freezing-point of water 
produced by a number of chlorides and bromides, diminished with 
increasing concentration up to a certain point, as would be ex- 
pected, and then increased again. The increase in the molecular 
depression became very marked at great concentrations; in fact, 
the molecular depression in a molar solution was frequently greater 
than the molecular depression corresponding to a completely 
dissociated salt. This phenomenon was systematically studied 
by Jones and the author J and the fact was established that it 
is quite general. 

* Trans. Faraday Soc., i, 197 (1905); 3i 123 (1907). 

t Am. Chem. Jour., 22, 5, 110 (1899); 23, 89 (1900). 

i Zeit. phys. Chem., 46, 244 (1903); Phys. Rev., 18, 146 (1904); Am. Chem. 
Jour., 31, 303 (1904); 32, 308 (1904); 33, 534 (1905); 34, 291 (1905); Zeit, 
phys. Chem., 49, 385 (1904); Monograph No. 60, Carnegie Institution of 
Washington. 



234 THEORETICAL CHEMISTRY 

This abnormal depression of the freezing-point may be accounted 
for by assuming that the dissolved substance has entered into com- 
bination with a portion of the water, thus removing it from the 
role of solvent. The formation of a loose molecular complex be- 
tween one molecule of the solute and a large number of molecules 
of water, acts as a single dissolved unit in depressing the freezing- 
point of the pure solvent. Evidently the total amount of water 
present, which functions as solvent, is diminished by the amount 
of water which has been appropriated by the solute. The abnor- 
malities observed in the depression of the freezing-point of con- 
centrated solutions of electrolytes can be explained by assuming 
that the molecules of solute, or the resulting ions, are in combina- 
tion with a number of molecules of solvent. This hypothesis is 
termed the solvate theory, and the loose molecular complexes are 
called solvates. Since the solvate theory was first proposed, con- 
siderable evidence has been accumulated to confirm its correct- 
ness. Reference has already been made to the work of Philip on 
the solubility of gases in saline solutions, from which he concludes 
that the dissolved salts enter into combination with a portion of 
the solvent. The experiments of Morse and the Earl of Berkeley 
on osmotic pressure, also seem to point to the solvation of the 
dissolved substance. 

PROBLEMS. 

1. At 18 C, a 0.5 molar solution of NaCl is 74.3 per cent dissociated. 
What would be the osmotic pressure of the solution in atmospheres at 
18 C.? Am. 20.79. 

2. A solution containing 3 mols of cane sugar per liter was found by 
the plasmolytic method to be isotonic with a solution of potassium nitrate 
containing 1.8 mols per liter. What is the degree of ionization of the 
potassium nitrate? Ans. 67 per cent. 

3. The vapor pressure of water at 20 C. is 17.406 mm. and that of a 
0.2 molar solution of potassium chloride is 17.296 mm. at the same tem- 
perature. Calculate the degree of dissociation of the salt. 

Ans. 75.38 per cent. 

4. The degree of dissociation of a 0.5 molar solution of sodium chloride 
at 25 is 74.3 per cent. Calculate the osmotic pressure of the solution at 
the same temperature. Ans. 21.32 atmos. 



ASSOCIATION, DISSOCIATION AND SOLVATION 235 

5. A solution containing 1.9 mols of calcium chloride per liter is isotonic 
with a solution of glucose containing 4.05 mols per liter. What is the 
degree of ionization of the calcium chloride? Ans. 56.6 per cent. 

6. At C. the vapor pressure of water is 4.620 mm. and of a solution 
of 8.49 grams of NaN0 3 in 100 grams of water 4.483 mm. Calculate the 
degree of ionization of NaNOs. Ans. 64.9 per cent. 

7. At C. the vapor pressure of water is 4.620 mm. and that of a 
solution of 2.21 grams of CaCl 2 in 100 grams of water is 4.583 mm. Cal- 
culate the apparent molecular weight and the degree of ionization of 
CaCl 2 . Ans. M = 49.66, a = 62 per cent. 

8. The boiling-point of a solution of 0.4388 gram of sodium chloride 
in 100 grams of water is 100.074 C. Calculate the apparent molecular 
weight of the sodium chloride and its degree of ionization. K = 5.2. 

Ans. M = 30.84, = 89.7 per cent. 

9. The boiling-point of a solution of 3.40 grams of BaCl 2 in 100 grams 
of water is 100.208 C. K = 5.2. What is the degree of ionization of 
the BaCl 2 ? Ans. 72.5 per cent. 

10. At 100 C. the vapor pressure of a solution of 6,48 grams of ammo- 
nium chloride in 100 grams of water is 731.4 mm. K = 5.2. What is the 
boiling-point of the solution? Ans. 101,086 C. ^ 

11. A solution of 1 gram of silver nitrate in 50 grams of water freezes 
at -0.348C. Calculate to what extent the salt is ionized in solution. 
K = 18.6. Ans. 59 per cent. 

12. A solution of NaCl containing 3.668 grams per 1000 grams of 
water freezes at -0.2207 C. Calculate the degree of ionization of the 
salt. K = 18.6. Ans. 89.2 per cent. 

13. The freezing-point of a solution of barium hydroxide containing 
1 mol in 64 liters is -0.0833 C. What is the concentration of hydroxyl 
ions in the solution? Take K = 18.9 for concentrations in mols per 
liter. Ans. 0.0284 gm.-ion per liter. 

14. The vapor*pressure of water at C. is 4.620 mm., and the lower- 
ing of the vapor pressure produced by dissolving 5.64 grams of sodium 
chloride in 100 grams of water is 0.142 mm. What is the freezing-point 
of the solution? K 18.6. Ans. -3.177 C. 

15. A solution containing 8.34 grams Na 2 S04 per 1000 grams of water 
freezes at 0.280 C. Assuming dissociation into 3 ions, calculate the 



236 THEORETICAL CHEMISTRY 

degree of ionization and the concentrations of the Na* and S(V' ions. 
K - 18.6. 

Am. a 78.2 per cent; cone. Na* = 0.0918 grrn.-ion per liter; cone. 
SO 4 " - 0.0459 gm.-ion per liter. 



CHAPTER XII. 
COLLOIDS. 

Crystalloids and Colloids. In the course of his investigations 
on diffusion in solutions, Thomas Graham * drew a distinction 
between two classes of solutes, which he termed crystalloids and 
colloids. Crystalloids, as the name implies, can be obtained in the 
crystalline form : to tliis class belong nearly all of the acids, bases 
and salts. Colloids, on the oilier hand, are generally amorphous, 
such substances as albumin, starch and caramel being typical of 
the class. Because of the gelatinous character of many of the sub- 
stances in this class, Graham termed them colloids (*o\Xa = glue, 
and erdos = form). The differences between the two classes are 
most apparent in the physical properties of their solutions. Thus, 
crystalloids diffuse much more rapidly than colloids; the velocity 
of diffusion of caramel being nearly 100 times slower than that of 
hydrochloric acid at the same temperature. While crystalloids 
exert osmotic, pressure, lower the vapor pressure and depress the 
freezing-point of the solvent, colloids have very little effect upon 
the properties of the solvent. The marked differences in the rates 
of diffusion of crystalloids and colloids render their separation 
comparatively easy. If a solution containing both crystalloids 
and colloids be placed in a vessel over the bottom of which is 
stretched a colloidal membrane, such as parchment, and the whole 
is immersed in pure water, the crystalloids will pass through the 
membrane, while the colloids will he left behind. This process 
was termed by Graham, dialysis, while the apparatus employed to 
effect such a separation was called a dialyzer. When a solution of 
sodium silicate is added to an excess of hydrochloric acid, the 
products of the reaction, silicic acid and sodium chloride, remain 
in solution. When the mixture is placed in a dialyzer, the sodium, 
chloride and the hydrochloric acid, being crystalloids, diffuse 

* Lieb. Ann., 121, 1 (1862). 
237 



238 THEORETICAL CHEMISTRY 

through the membrane of the dialyzer, leaving behind the colloidal 
silicic acid. 

The terms crystalloid and colloid, as used at the present time, 
have acquired different meanings from those assigned to them by 
Graham. The terms arc now considered to refer, not so much to 
different classes of substances, as to different states which almost 
all substances can assume under certain conditions. 

Colloidal Solutions. A colloidal solution is OIK* in which the 
solute is a colloid, although the latter may not he included among 
the substances classified as such by Graham. For example, 
arsenious sulphide, ferric hydroxide or finely-divided gold may 
form colloidal solutions. In bringing such substances into the 
colloidal state, mere agitation with water will riot suffice, but some 
indirect method must be employed. 

Nomenclature. Graham distinguished between two condi- 
tions in which colloids were obtainable, the term sol being applied 
to forms in which the system resembled a liquid, while the term 
gel was used to designate those forms which were solid arid jelly- 
like. When one of the components of the solution was water, the 
two forms were called a Jiydrosol and a Jn/drogcL In like manner, 
when alcohol was one of the components, the terms alcosol and 
alcogel were applied to the two forms. 

As the knowledge of colloids has developed it has become neces- 
sary to supplement Graham's nomenclature by the introduction 
of various other terms. It is known to-day that the essential 
difference between colloidal suspensions and solutions on the one 
hand, and true solutions on the other, is due to the difference in the 
degree of subdivision or degree of dixpersity of the dissolved sub- 
stance. In a true solution the dissolved substance is generally 
present either in the molecular or ionic condition, as may be shown 
by means of the familiar osmotic methods for molecular weight de- 
termination. In colloidal solutions, however, the degree of clis- 
persity is not so great and has been found to vary from above the 
limit of microscopic visibility (1 X 10~ 5 cm.) to that of molecular 
dimensions (1 X 10~~ 8 cm.). When the degree of dispersity varies 
from 1 X 10~ 3 cm. to 1 X 10~ 5 cm. the particles are termed 
microns. The properties of the disperse phase at this degree of 



COLLOIDS 239 

dispersity differ appreciably from the properties of the same sub- 
stance when present in large masses. When the degree of dis- 
persity lies between 1 X 10~ 5 cm. and 5 X 10~ 7 cm. the particles 
are known as submicrons. The existence of particles whose diam- 
eters are approximately 1 X 10~ 7 cm. has been demonstrated by 
Zsigmondy with the ultramicroscope; these minute particles are 
termed amicrons. When the degree of dispersity is increased 
beyond this limit all heterogeneity apparently vanishes and we 
enter the realm of true solutions. 

When the dispersion is not too great, colloidal solutions may be 
divided into suspensions and emulsions according to whether the 
disperse phase is a solid or a liquid. As the dispersion is increased 
we obtain suspension and emulsion colloids which may be con- 
veniently called suspensoids and emulsoids. Suspensoids and 
emulsoids are included under the general term dispersoids. In 
certain cases, although the disperse phase is unquestionably liquid, 
the systems resemble suspensoids in their behavior, while in other 
cases, where the disperse phase is solid, the systems exhibit proper- 
ties characteristic of emulsoids. For this reason the classification 
of sols as suspensoids and emulsoids is not entirely satisfactory. 
A better system is that in which the presence or the absence of 
affinity between the disperse phase and the dispersion medium is 
made the basis of classification. Where there is marked affinity 
between the two phases, the system is termed lyophile, and where 
such affinity is absent, the system is termed lyophobe. When the 
dispersion medium is water, the terms hydrophile and hydrophobe 
are employed. 

In the reversible transformation of a sol into a gel, we are 
not warranted in referring to the change from gel to sol as an 
act of solution, for if the gel really dissolved, a solution and 
not a sol would result. Various terms have been proposed for 
these reversible transformations but perhaps the most satisfac- 
tory are the terms gelation and solation, the former designating 
the formation of a gel from a sol and the latter the reverse 
process. 

Lyotrope Series. The differences between lyophile and lyo- 
phobe sols are frequently very marked, this being especially true 



240 THEORETICAL CHEMISTRY 

of their behavior toward chemical reagents. The action of chemi- 
cal reagents on lyophobe sols is almost wholly confined to the 
disperse phase, while the addition of reagents to lyophile sols fre- 
quently produces a more marked effect on the dispersion medium 
than on the disperse phase. It should be observed that the physi- 
cal properties of a lyophobe sol and of the pure dispersion medium 
are practically identical, while exactly the reverse is true of lyo- 
phile sols. It is well known that the addition of a foreign sub- 
stance to a reaction-mixture frequently exerts a marked influence 
on the speed of the reaction, notwithstanding the fact that the 
nature of the added substance may be such as to render its partici- 
pation in the reaction highly improbable. If a series of reagents 
are arranged in the order of their influence on a particular reaction, 
it has been found that the same sequence is preserved when the 
same reagents are added to other reactions of widely different 
character. In some reactions the reagents may produce effects 
directly opposite to those which they produce in others, but the 
sequence remains unchanged. For example, when the same re- 
action takes place either in an acid or in an alkaline medium, the 
substances which promote the reaction when the medium is acid, 
retard it when the medium is alkaline, but the sequence of the 
added substances remains the same under both conditions. These 
facts make it appear highly probable that the effects produced by 
the addition of foreign substances to a chemical reaction are to be 
ascribed to the changes which they produce in the pure solvent. 
It is not without significance that the same sequence of reagents is 
maintained whether we observe their influence on different chemi- 
cal reactions or on certain physical properties of the solvent, such 
as its density, viscosity, and surface tension. 

The following examples * afford a striking illustration of the 
persistence of the sequence of reagents, generally known as the 
lyotrope series. The ions which precede the formula (H^O) reduce 
the velocity of reaction or cause a diminution in the magnitude of 
the particular physical property tabulated. The ions which follow 
the formula (H 2 0) exert the opposite effect. 

* Freundlich, Kapillarchemie, p. 411. 



COLLOIDS 241 

1. The hydrolysis of esters by acids. 

Anions: S0 4 (H 2 O) Cl < Br. 
'Cations: (H 2 O) Li < Na < K < Rb < Cs. 

2. The hydrolysis of esters by bases. 

Anions: I > N0 3 > Br > Cl (H 2 0) S 2 3 < S0 4 . 
Cations: Cs > Rb > K > Li (H 2 O). 

3. The viscosity of aqueous solutions. 

Anions: NO 3 > Cl (H 2 0) S0 4 (potassium salts). 
Cations: Cs > Rb > K (H 2 0) Na < Li (chlorides). 

4. The surface tension of aqueous solutions. 

(H 2 0) I< N0 3 < Cl< S0 4 < C0 3 . 

It will be observed that although the ions which accelerate the 
acid hydrolysis retard the basic hydrolysis, the sequence of the 
ions nevertheless remains the same. 

The Ultramicroscope. When a narrow beam of sunlight is 
admitted into a darkened room, the dust particles in its path are 
rendered visible by the scattering of the light at the surface of the 
particles. If the air of the room is free from dust, no shining 
particles will be seen and the space is said to be " optically void." 
When the particles of dust are very minute, the beam of light 
acquires a bluish tint. The blue color of the sky is thus attrib- 
uted to the presence of extremely fine particles of dust in the air 
together with minute drops of condensed gases in the upper regions 
of the atmosphere. 

The visibility of a beam of light due to the scattering effect of 
minute particles, is known as the Tyndall phenomenon. Almost 
all colloidal solutions exhibit this phenomenon when a powerful 
beam of light is passed through them, thus proving the presence 
of discrete particles in the solutions. 

The ultramicroscope is an instrument devised by Siedentopf 
and Zsigmondy * for the detection of colloidal particles much too 
small to be seen by the naked eye. A powerful beam of light 
issuing from a horizontal slit is brought to a focus within the 
colloidal solution under examination by means of a microscope 
objective, and this image is viewed through a second micro- 

* "Colloids and the Ultramicroscope," by R. Zsigmondy. Trans, by 
Alexander, John Wiley & Sons, Inc. 



242 THEORETICAL CHEMISTRY 

scope, the axis of which is at right angles to the path of the 
beam. 

When examined in this way a colloidal solution appears to be 
swarming with brilliantly colored particles, moving rapidly in a 
dark field; whereas a true solution if properly prepared, appears 
optically void. With the ultramicroscope it is possible to count 
the number of particles present in a given volume of a col- 
loidal solution. By means of a chemical analysis, the mass of 
colloid per unit of volume can be determined and from this the 
average mass of each particle can be calculated. If the particles 
be assumed to be spherical in shape and to have the same density 
as larger masses of the same substance, we can calculate the 
volume of a single particle and from this its diameter. Thus, 
Burton * in his experiments on gold, silver and platinum sols, found 
the average diameter of the colloidal particles to range from 0.2 to 
0.6 micron. 

Zsigmondy's latest ultramicroscope, Fig. 68, consisting of two 
compound microscopes placed at right angles and having their 
objectives so cut away as to permit them to be brought together 
in focus, enables the observer to discern particles whose diameters 
range from 1 to 2 rnilli-microns. 

The ultramicroscopic character of emulsoids is by no means 
sharply defined, notwithstanding the fact that they exhibit the 
Tyndail phenomenon. It has been suggested by Zsigmondy that 
the lack of sharpness in definition observed with emulsoids is 
probably due to the relatively small difference between the re- 
fractive indices of the disperse phase and the dispersion medium. 
Where the difference between the refractive indices of the two 
phases is very great, as in the case of the metallic sols, excellent 
definition is obtained. It is of interest to note that although the 
basic hydroxides are apparently suspensoids, yet in their ultra- 
microscopic characteristics they closely resemble emulsoids. 

Ultrafiltration. Almost all sols can be filtered through ordinary 
filter paper without undergoing more than a slight change in con- 
centration due to initial adsorption. The rate of filtration varies 
widely, depending upon the viscosity of the sol. As a general rule, 

* Phil. Mag., ii, 425 (1906). 



COLLOIDS 



243 



emulsoids filter more slowly than suspensoids owing to the high 
viscosity of the former. 




Fig. 68. 

By filtering an arsenious sulphide sol through a porous earthen-- 
ware filter, Linder and Picton * succeeded in obtaining four differ- 

* Jour. Chem. Soc., 61, 148 (1892). 



244 THEORETICAL CHEMISTRY 

ent sizes of particles which they described as follows: (1) visible 
under the microscope, (2) exhibited the Tyndall phenomenon, (3) 
retained by porous plate, and (4) passed through porous plate un- 
changed. By employing plates of different degrees of porosity and 
determining the average size of the pores which just permit filtra- 
tion, it is possible to determine the size of the particles which con- 
stitute the disperse phase of a sol. If we make use of a series of 
graduated filters, prepared by impregnating filter paper with a 
solution of collodion in acetic acid, it is not only possible to sepa- 
rate suspensoids from their dispersion media, but also to effect the 
concentration of emulsoids. Furthermore such filters are useful 
in removing impurities from sols, the impurity passing through the 
filter in a manner similar to the passage of the solvent through the 
membrane in the process of dialysis. Ultrafiltration is an exceed- 
ingly complex process involving the phenomena of adsorption and 
dialysis in addition to the ordinary process of mechanical separation. 
The complexity of the process is well illustrated by the phenomena 
attendant upon the filtration of almost any positive hydrosol. 
Thus, if we attempt to filter a ferric hydroxide hydrosol through a 
porous plate, or even through an ordinary filter paper, we shall 
find that the colloid will be partially retained by the filter. This is 
due to the fact that the filter becomes negatively charged in con- 
tact with water and, on the entrance of the positively-charged sol 
into the pores of the filter, the colloid is immediately discharged 
and the disperse phase precipitated. After the pores of the filter 
become partially stopped with particles of the colloid, the sol will 
then pass through unchanged. 

^ Classification of Dispersoids. There is abundant evidence in 
favor of the view that colloidal solutions and simple suspensions 
are closely related. Suspensions of all grades exist, from those in 
which the suspended particles are coarse-grained and visible to 
the naked eye, down to those in which a high-power microscope 
is required to render the suspended particles visible. Colloidal 
solutions have also been shown to be non-homogeneous, the 
presence of discrete particles being revealed by means of the 
ultramicroscope. It follows, therefore, that the size of the particles 
in solution determines whether a substance is to be considered as 



COLLOIDS 



245 



a colloid or not. At one extreme we have true solutions in which 
no lack of homogeneity can be detected, even by the ultramicro- 
scope, and at the other extreme we have coarse-grained suspen- 
sions, in which the particles are visible to the naked eye. Between 
these two limits all possible degrees of subdivision are possible 
and it is a very difficult matter to draw sharp 1 nes of distinction 
between true solutions and colloidal solutions on the one hand, 
and between colloidal solutions and suspensions on the other. 
One of the most satisfactory schemes of classification is that of 
von Weimarn and Wo. Ostwald.* Because of the fact that sus- 
pensions, colloidal solutions, and true solutions represent varying 
degrees of dispersion of the solute, all three types of system are 
termed by these authors, dispersoids. The dispersoids are classi- 
fied as shown in the accompanying diagram. 

DISPERSOIDS. 



. 

Coarse disp 


versions (sus- 






pensions, 


emulsions, 






etc.). 








Magnitude 


of particles 






greater than 0.1 /*.* 







Colloidal solutions. Molecular 



Ionic 



Magnitude of particles dispersoids. dispersoids. 
between 0.1 /* and 1 /*/*. ' ' 



Magnitude of particles, 
about 1 nfjL and smaller. 



Decreasing degree of 

"colloidity." 

' - 

Increasing degree of dis- 
persion. 

* 1 n = 1 micron - 0.001 mm. 

Density of Colloidal Solutions. As we have seen, suspensoids 
are commonly regarded as sols in which the disperse phase is solid, 
while emulsoids are considered to be sols in which the disperse 
phase is liquid. While this distinction between the two classes of 
sols is generally well defined, it should be borne in mind that there 

* Koll. Zeitschrift, 3, 20 (1908). 



246 



THEORETICAL CHEMISTRY 



are colloidal solutions in which it is extremely difficult to determine 
the physical state of the disperse phase. 

The fundamental difference between suspensoids and emulsoids 
manifests itself most clearly in those properties which undergo 
appreciable change in consequence of solution. Among these may 
be mentioned density, viscosity and surface tension. 

It was shown by Linder and Picton * that the density of 
suspensoids follows the law of mixtures. This is clearly shown 
by the following table in which are given the observed and 
calculated values of the density of a series of arsenious sulphide 
sols. 

DENSITY OF ARSENIOUS SULPHIDE SOLS. 



As 2 S 3 (grams per 
liter). 


Density (obs.). 


Density (calc.). 


44 


1.033810 


1 033810 


22 


1 016880 


1 016905 


11 


1 008435 


1 008440 


2 45 


1 002110 


1 002100 


1719 


1.000137 


1.000134 



The density of emulsoids, on the other hand, cannot be calcu- 
lated from the composition of the sol. This fact may be taken as 
evidence in favor of the view that a closer relation exists between 
the disperse phase and the dispersion medium in emulsoids than in 
suspensoids. 

Viscosity of Colloidal Solutions. Owing to the fact that the 
concentrations of most suspensoids are relatively small, it follows 
that their viscosities differ but little from the viscosity of the pure 
dispersion medium. In general, it may be said that the viscosity 
of suspensoid sols is slightly greater than that of the dispersion 
medium. 

On the other hand, the viscosity of emulsoid sols is frequently 
much greater than that of the pure dispersion medium. The vis* 
cosity of emulsoids also increases with increasing concentration, 
as is shown by the data of the following tabLa. 

* Jour. Chem. Soc., 67, 71 



COLLOIDS 



247 



VISCOSITY OF EMULSOIDS. 



' So.. 


Temp. 20 
Concentration . 


Viscosity. 




Per cent 




Gelatine .... 


1 


021 


Gelatine .... 


2 


037 


Silicic acid. . . . 


81 


012 


Silicic acid .... 


99 


016 


Silicic acid. . . . 


1.96 


032 


Silicic acid. . . . 


3 67 


165 



Viscosity of water at 20 = 0.0120. 

Surface Tension of Colloidal Solutions. The surface tension 
of suspensoid sols has been shown by Linder and Picton to be 
practically identical with that of the dispersion medium. 

As a general rule, the surface tension of emulsoid sols is appre- 
ciably smaller than that of the dispersion medium. According to 
Quincke * the surface tensions of gelatine sols are appreciably less 
than the surface tension of the dispersion medium. The differ- 
ence between suspensoids and emulsoids in respect to surface 
tension undoubtedly accounts for the fact that, in general, the 
former are not adsorbed while the latter are. 

Osmotic Pressure of Colloidal Solutions. The osmotic pres- 
sure of colloidal solutions is very small. This is what we should 
expect with solutions of substances which exhibit a slow rate of 
diffusion. As has been pointed out, diffusion is closely connected 
with osmotic pressure; hence, if the rate of diffusion is slow, the 
osmotic pressure exerted by the solution should be small. In 
some cases the osmotic pressure is so small as to escape detection. 
The experimental determination of the osmotic pressure of col- 
loidal solutions is complicated by the difficulty of removing the 
last traces of electrolytes from the colloid. Owing to their great 
osmotic activity, the presence of the merest trace of electrolytes 
may mask the true osmotic effect of the colloid. It should be borne 
in mind, however, that semipermeable membranes are much less 
permeable by colloids than by electrolytes and, in consequence of 
this fact, the impurities in the colloid would be gradually removed 
by prolonged dialysis. If the total osmotic pressure were due to 
* WiecL Ann,, III, 35, 582 (1885). 



248 



THEORETICAL CHEMISTRY 



the presence of small amounts of impurities in the colloid, then as 
these are removed, the pressure should steadily diminish and ulti- 
mately become zero. As a matter of fact, the final value of the 
osmotic pressure of a colloidal solution, although generally very 
small is never zero. This final, positive value of the osmotic 
pressure has been shown to be wholly independent of the method 
of preparation of the sol. Although differences in the method of 
preparation may introduce different impurities which give rise to 
different initial values of the osmotic pressure, in each case the 
same final value is obtained. Of course it must be admitted that 
the possibility exists that a minute portion of electrolyte which 
cannot be removed by dialysis is retained by the colloid, but even 
then it is difficult to account for the constancy of the final value of 
the osmotic pressure irrespective of the method of preparation of 
the sol. The values of the osmotic pressure of suspensoids are 
invariably small and by no means concordant. 

The following table gives the results obtained by Duclaux * with 
colloidal solutions of ferric hydroxide. 

OSMOTIC PRESSURE OF COLLOIDAL FERRIC HYDROXIDE. 



Cone. Fe(OH) 3 
Per cent. 


Pressure in cm. 
of Water. 


1 08 


8 


2 04 


2 8 


3 05 


5 6 


5 35 


12 5 


8 86 


22 6 



Inspection of the table shows that, even in the most concen- 
trated solution, the osmotic pressure is very small. Furthermore, 
it is apparent that although the osmotic pressure increases with 
the concentration, the variables are not proportional. Observa- 
tions on the variation of the osmotic pressure of colloidal solu- 
tions with temperature, show that, in general, as the temperature 
is raised the pressure increases at a more rapid rate than that 
required by the law of Gay-Lussac. 

Employing membranes of collodion and parchment paper, 

* Compt. rend., 140, 1544 (1905); Jour. Chim. Phys., 7, 405 (1909). 



COLLOIDS 249 

Lillie * and others have demonstrated that the values of the os- 
motic pressure of emulsoids are, in general, considerably greater 
than the corresponding values obtained with suspensoids. The 
osmotic pressures of several typical emulsoids are given in the 
following table. 

OSMOTIC PRESSURES OF EMULSOIDS. 



Sol. 


^Concentration 
(grams per liter). 


Osmotic Pressure 
(mm. of mercury). 


Egg albumin 


12 5 


20 


Gelatine 


12 5 


6 


Starch iodide 


30 


15 


Dextrin . 


10 


165 



It will be observed that the values of the osmotic pressure in the 
preceding table are appreciably greater than those given for ferric 
hydroxide hydrosols. This is in agreement with the well-estab- 
lished fact that emulsoids diffuse more rapidly than suspensoids. 
It has been observed, that the value of the osmotic pressure of 
gelatine solutions at ordinary temperatures can be increased by 
maintaining the solutions at a higher temperature for a short time 
and then cooling to the initial temperature. After standing for 
several days at the original temperature, however, the osmotic 
pressure of the solution returns to its former value. This phenom- 
enon would seem to indicate that the osmotic pressure of colloidal 
solutions is not completely defined by the two variables, tempera- 
ture and concentration. It has been suggested that the degree of 
aggregation of the colloid is partially dependent upon the tempera- 
ture; the molecular aggregates tending to break up as the tempera- 
ture is raised, thus increasing the number of dissolved units and 
therefore causing a corresponding increase in the osmotic pressure. 

Molecular Weight of Colloids. We have already learned that 
the knowledge of the osmotic pressure of a solution enables us to 
calculate the molecular weight of the solute, provided the solu- 
tion is dilute and obeys the gas laws. As we have seen, other 

* Am. Jour. Physiol., 20, 127 (1907). 



250 THEORETICAL CHEMISTRY 

factors than concentration and temperature determine the osmotic 
pressure of colloidal solutions, so that we are not justified in 
attempting to calculate the molecular weight of a colloid from the 
observed value of the osmotic pressure of its solution. Values for 
the molecular weight of colloids calculated from their effect on the 
vapor pressure, the boiling-point, and the freezing-point of the 
solvent are also untrustworthy, since the same factors which 
influence the osmotic pressure necessarily affect these related 
properties. This becomes evident when we reflect that an osmotic 
pressure of 1 mm. of mercury corresponds to a depression of the 
freezing-point of about 0.0001. Owing to the difficulty of ob- 
taining absolutely pure emulsoid sols, all determinations of their 
freezing-point depressions must be affected with an experimental 
error appreciably larger than the observed depression. Hydrosols 
of albumin, gelatine, etc., prepared with extreme care by Bruni 
and Pappada * failed to produce any detectable depression of the 
freezing-point of water. 

Electroendosmosis. The movement of a liquid through a 
porous diaphragm, due to the passage of an electric current be- 
tween two electrodes placed on opposite sides of the diaphragm, is 
known as electroendosmosis. This phenomenon, which was first 
observed by Reuss in 1807, has since been made the subject of 
numerous investigations by Wiedemann,t Quincke t and Perrin, 
the latter having worked out a satisfactory theoretical interpreta- 
tion of the phenomenon. If a porous partition be placed in the 
horizontal portion of a U-tube and an electrode be inserted in each 
arm of the tube, it will be found, on filling the tube with a feebly 
conducting liquid and passing a current, that the liquid will com- 
mence to rise in one arm of the tube and will continue to rise until 
a definite equilibrium is established. For a given difference of 
potential between the two electrodes, there will be a definite differ- 
ence in the level of the liquid in the two arms of the tube. The 
majority of substances acquire a negative electric charge when 

* Rend. R. Accad. dei Lincei, (5), 9, 354 (1900). 

t Pogg. Ann., 87, 321 (1852). 

t Ibid., 113, 513 (1861). 

Jour. Chim. Phys., 2, 601 (1904). 



COLLOIDS 251 

immersed in water. The water, under these conditions, becomes 
positively charged and will, in consequence, migrate toward the 
cathode. On the other hand, certain substances acquire a positive 
charge on immersion in water arid in these cases the direction of 
migration will obviously be reversed. 

It has been found that acids cause negative diaphragms to be- 
come less negative and positive diaphragms to become more posi- 
tive. The action of alkalies is, as we should expect, the reverse of 
that of acids. There is an interesting connection between the 
valence of the ions resulting from the dissociation of dissolved 
salts, and the difference of potential existing between the liquid 
and the diaphragm. When the diaphragm is positively charged, 
the difference of potential is found to be conditioned by the 
valence of the anion, and when the diaphragm is negatively 
charged, the difference of potential is determined by the valence 
of the cation. 

^ Cataphoresis. When a difference of potential is established 
between two electrodes immersed in a suspension of finely-divided 
quartz or shellac, the suspended particles move toward the positive 
electrode. This phenomenon is called cataphoresis and was first 
observed by Linder and Picton.* They showed that when the 
terminals of an electric battery are connected to two platinum 
electrodes dipping into a colloidal solution of arsenious sulphide, 
there is a gradual migration of the colloid to the positive pole. A 
similar experiment with a solution of colloidal ferric hydroxide 
resulted in the transport of the dissolved colloid to the negative 
pole. It follows, therefore, that the particles of colloidal arsenious 
sulphide are negatively charged, while those of colloidal ferric 
hydroxide carry a positive charge. It has been found that most 
colloids ca^ry an electric charge. In the table on page 252, some 
typical colloids arc classified according to the character of their 
electrification in aqueous solution. 

The nature of the charge varies with the dispersion medium used, 
colloidal solutions in turpentine, for example, having charges oppo- 
site to those in water. 

Direct measurements of the velocity with which the particles 

* Jour. Chem. Soc., 61, 148 (1892). 



252 



THEORETICAL CHEMISTRY 



move in cataphoresis have been made by Cotton and Mouton. 
By observing with a microscope the distance over which a singk 

ELECTRICAL CHARGES OF HYDROSOLS. 



Electro-positive. 


Electro-negative- 


Metallic hydroxides 
Methyl violet 
Methylene blue 
Magdala red 
Bismarck brown 
Haemoglobin 


All the metals 
Metallic sulphides 
Aniline blue 
Indigo 
Eosine 
Starch 



particle traveled in a given interval of time under a definite po- 
tential gradient, they calculated the average velocity of migration 
of a number of suspensoids. The following table gives the velocity 
of migration of a few typical suspensoids. 

VELOCITY OF MIGRATION OF SUSPENSOIDS. f 



Suspensoid. 


Average Diameter 
of Particles. 


Velocity cm. /sec for 
Unit Potential 
Gradient. 


Arsenic trisulphide. . .... 
Quartz .... 
Gold (colloidal) 


50 M 
1 MM 

<100 MM 


22X10- 5 
30X10- 5 
40X10" 5 


Platinum (colloidal) 


< 100 MM 


30X10" 5 


Silver (colloidal) 
Bismuth (colloidal) 


<100MM 
< 100 MM 


23 6X10~ 8 
11.0X10 8 


Lead (colloidal)... . . 
Iron (colloidal) 
Ferric hydroxide (colloidal) . . 


<1(X)MM 
<100MM 
100 MM 


12.0X10- 6 
19.0X10- 5 
30.0X10- 5 



t Freundlich, Kapillarchemie, p. 234. 

It will be observed that not only are the velocities of migration 
nearly constant, but also that they are apparently independent of 
the size and nature of the particles. 

The presence of electrolytes, especially acids and bases, exercises 
a marked effect upon the electrical behavior of suspensoids. 
Owing to the comparative instability of suspensoids, the addition 
of electrolytes usually results in the complete precipitation of the 

colloid. 

* Jour. Chim. Phys., 4, 365 (1906). 



COLLOIDS 253 

Emulsoids also show the phenomenon of cataphoresis, but their 
velocities of migration are appreciably less than the corresponding 
velocities of suspensoids, and their behavior in an electric field is 
such as to make it appear quite probable that the character of their 
electric charge is entirely fortuitous. Furthermore, emulsoids are 
much more susceptible to the influence of electrolytes than are 
suspensoids. 

W. B. Hardy * has found that the direction of migration of 
albumin, modified by heating to 100 C., is dependent upon the 
reaction of the dispersion medium. A very small quantity of free 
base causes the particles of albumin to move toward the positive 
electrode, while the addition of an equally small amount of acid 
results in a reversal of the direction of migration. Similar reversals 
of charge have been observed by Burton f in colloidal solutions of 
gold and silver. When small amounts of aluminium sulphate are 
added to colloidal solutions of these metals, the charge is gradually 
neutralized and eventually the colloidal particles acquire a reversed 
charge. 

Electrical Conductance of Colloidal Solutions. The electrical 
conductance of suspensoids differs so slightly from that of the pure 
dispersion medium that it is difficult to decide whether the small 
increase in conductance may not be due to the presence of traces 
of adsorbed electrolytes. In order to ascertain to what extent the 
conductance of suspensoids is dependent upon the presence of 
adsorbed electrolytes, Whitney arid Blake t investigated the effect 
of successive electrolyses upon the conductance of a gold hydrosoL 
If the pure sol is incapable of enhancing the conductance of the 
dispersion medium, then as the sol is subjected to successive elec- 
trolyses, the conductance should steadily diminish and ultimately 
become identical with that of the dispersion medium. Whitney 
and Blake found that the conductance converged to a definite 
limiting value which was slightly greater than the conductance of 
the dispersion medium. From these experiments we seem to be 
warranted in concluding that suspensoids conduct the electric 
current very feebly. 

* Jour. Physiol., 24, 288 (1899). 

t Phil. Mag., 12, 472 (1906). 

J Jour. Am. Chem. Soc., 26, 1339 (1904). 



254 THEORETICAL CHEMISTRY 

Emulsoids appear to conduct rather better than suspensoids. 
Whitney and Blake measured the conductance of silicic acid and 
gelatine sols and found the specific conductance of the former to be 
100 X 10~ 6 reciprocal ohms and that of the latter to be 68 X 10~ 6 
reciprocal ohms. On the other hand, Pauli* found that an albumin 
sol which had been prepared with extreme care was virtually a 
non-conductor. It should be remembered, however, that the 
albumins are closely related to the simple ammo-acids which are 
known to be exceedingly poor conductors. 

Precipitation of Colloids by Electrolytes. One of the most 
important and interesting divisions of the chemistry of colloids is 
that which treats of the precipitation of suspensoids and emulsoids 
by electrolytes. In general, it may be said that the precipitation 
of colloids by electrolytes is an irreversible process. Colloidal 
solutions are more or less unstable systems irrespective of the 
methods employed in their preparation, and the addition of a small 
amount of an electrolyte is usually found to be sufficient to cause 
the sol immediately to become opalescent, and ultimately to pre- 
cipitate, leaving the dispersion medium perfectly clear and free 
from the disperse phase. Some exceptions to this general state- 
ment as to the behavior of colloids are known. For example, 
Whitney and Blake f found that precipitated gold could be caused 
to return to the colloidal state by treatment with ammonia, while 
Linder and Picton J discovered that a ferric hydroxide hydrosol, 
which had been precipitated with sodium chloride, could be re- 
stored to the colloidal condition by simply removing the electrolyte 
with water. The sedimentation of suspensions, such as kaolin in 
water, is also promoted by the addition of electrolytes. 

On the other hand, the addition of some non-electrolytes fre- 
quently causes an increase in the stability of a suspensoid. 

Precipitation of Suspensoids. The phenomenon of the pre- 
cipitation of suspensoids has been carefully investigated by 
Freundlich. He has found that an amount of electrolyte which 

* Beitrag, Chem. Phys. Path., 7, 531 (1906). 

t Jour. Am. Chem. Soc., 26, 1341 (1904). 

J Jour. Chem. Soc., 61, 114 (1892); 87, 1924 (1905). 

Zeit. phys, Chem., 44, 131 (1903). 



COLLOIDS 



255 



is incapable of bringing about an instantaneous precipitation, may 
become effective after an interval of time. He has also shown 
that the total quantity of electrolyte required to precipitate a 
suspensoid completely depends upon whether the electrolyte is 
added all at one time or in successive portions. In order to com- 
pare the precipitating action of various electrolytes, Freundlich 
proposed the following procedure, which prevents the possibility 
of irregularities due to the time factor: To 20 cc. of a solution 
of a suspensoid, 2 cc. of the solution of the electrolyte are added, 
the resulting solution being shaken vigorously ; the mixture is then 
set aside for two hours, after which a small portion is filtered off, 
and the filtrate is examined for the suspensoid. In the following 
table some of the results obtained by Freundlich with colloidal solu- 
tions of ferric hydroxide are given. The data represent the mini- 
mum concentration for each electrolyte which produced precipita- 
tion in two hours. 

It will be seen that very small amounts of the electrolytes are 
required to precipitate the suspensoid, and further, that the pre- 
cipitating power of an electrolyte is dependent upon the charge of 
the negative ion. The greater the charge, the smaller is the 
quantity of electrolyte required to produce precipitation. 

PRECIPITATING ACTION OF ELECTROLYTES ON FERRIC 

HYDROXIDE HYDROSOL. 
(16 milli-mols Fe(OH) 2 per liter.) 



Electrolyte. 


Concentration 
(milli-inols per 
hter). 


NaCl 


9 25 


KC1 


9 03 


BaCl 2 


9 64 


KNO 3 


11.9 


KBr 


12 5 


Ba(N0 3 ) 2 


14 


KI 


16 2 


HC1 


400 ca. 


Ba(OH) 2 


42 


H 3 (SO) 4 


204 


MgS0 4 


217 


K 2 Cr 2 O 7 


0.194 


H*SO 4 


0.5ca. 



256 



THEORETICAL CHEMISTRY 



The significance of the relation between ionic charge and pre- 
cipitating power was first pointed out by Hardy,* who formulated 
the following rule: The precipitation of a colloidal solution is de- 
termined by that ion of an added electrolyte which has an electric charge 
opposite in sign to that of the colloidal particles. 

It has already been pointed out that colloidal particles of arse- 
nious sulphide are negatively charged, hence, according to Hardy's 
rule, the positive ions of the added electrolyte will condition the 
precipitation of the suspensoid. The experiments of Freundlich 
confirm this prediction, as is shown by the following table: 

PRECIPITATING ACTION OF ELECTROLYTES ON ARSENIOUS 
SULPHIDE HYDROSOL. 

(7.54 milli-mols As 2 Ss per liter.) 



Electrolyte. 


Concentration 
(milh-mols per 
liter). 


KC1 


49 5 


KNO 3 


50 


KC 2 H 3 O 2 


110 


NaCl 


51 


LiCl 


58.4 


MgCl 2 


0.717 


MgS0 4 


810 


CaCl 2 


0.649 


SrCh 


635 


BaCi 2 


691 


Ba(N0 3 ) 2 


687 


ZnCl 2 


685 


AlClr 


093 


A1(N0 3 ) 3 


0095 



Precipitation and Valence* An examination of the preceding 
tables reveals the fact that although the ionic concentration neces- 
sary to bring about precipitation, in accordance with Hardy's rule, 
decreases with increasing valence, the diminution in concentration 
is not, as we might expect, inversely proportional to the valence of 
the precipitating ion. The absence of any simple quantitative 
relation between the valence of an ion and its precipitating con- 
* Zeit. phys. Chem., 33, 385 (1900), 



COLLOIDS 257 

centration is undoubtedly due to the influence of several potent 
factors, such as adsorption and the protective action of ions whose 
electric charge is of the same sign as that of the colloidal substance. 

Precipitation of Emulsoids. The action of electrolytes on 
emulsoids is much more obscure than the action of electrolytes on 
guspensoids. Nothing approaching a generalization similar to 
Hardy's rule for the precipitation of suspensoids has been found to 
apply to the precipitation phenomena manifested by emulsoids. 
Owing to the fact that emulsoids are liquids, and in consequence 
of their greater degree of dispersity, it has been suggested that 
emulsoids probably resemble true solutions more closely than sus- 
pensoids. In fact there is reason for assuming that a portion of 
the colloid is actually dissolved in the dispersion medium. This 
may account for the fact that the precipitation of emulsoids is 
sometimes reversible and sometimes irreversible. It is to be re- 
gretted that, up to the present time, so many of the investigations 
on emulsoids have been carried out with materials of questionable 
purity and of insufficient uniformity. 

The addition of salts to gelatine generally causes irreversible 
precipitation, provided the concentration of the salt is not too low. 

The precipitation of albumin by some salts is reversible while 
by others it is irreversible. Those transformations which are 
initially reversible gradually become irreversible on standing. 
Although relatively concentrated solutions of salts of the alkalies 
and alkaline earths are required to precipitate albumin, very dilute 
solutions of the salts of the heavy metals are found to be sufficient 
to bring about complete precipitation. 

Action of Heat on Emulsoids. When an albumin hydrosol is 
gradually heated, a temperature is ultimately reached at which 
coagulation occurs. The exact nature of this transformation is 
not understood, but it is believed to be largely chemical. This 
belief is based upon the fact that the reaction of the natural albu- 
mins toward litmus is altered by heating. Slight acidity of the sol 
is essential to complete coagulation, while an excess or a de- 
ficiency of acids causes a portion of the albumin to remain in the 
sol. The presence of various salts has been found to exert a 
marked influence on the temperature of coagulation of albumin. 



258 THEORETICAL CHEMISTRY 

The coagulation temperature is invariably raised at first, attaining 
a constant value in some cases, while in others it decreases after 
reaching a maximum temperature. It is especially interesting to 
note that the anions of the salts follow the usual lytropic sequence. 
The effect of heat on a gelatine sol is very different from its 
effect on an albumin sol. If a fairly concentrated gelatine sol is 
heated and then permitted to cool, it sets into a jelly which is not 
reconverted into a sol when the temperature is again raised. Fur- 
thermore, the change does not take place at a definite temperature. 
In studying the phenomenon of gelation, either the temperature 
or the time of gelation may be determined. The melting-point of 
pure gelatine ranges from 26 to 29 while the solidifying tempera- 
ture lies between 25 and 18. The melting and solidifying tem- 
perature of gelatine sols vary with the concentration ; a 5 per cent 
sol melts at 26. 1 and solidifies at 17.8, while a 15 per cent sol 
melts at 29.4 and solidifies at 25.5. The temperature of the gel- 
sol transformation is affected by the addition of salts, some tending 
to raise the temperature of gelation and others to lower it. The 
order of the anions arranged according to their influence on the 
gel-sol transformation is as follows: 

Raising temperature: S0 4 > Citrate > Tartrate > Acetate (H 2 0). 
Lowering temperature: (H 2 O) 01 < C10 3 < NO 3 < Br < I. 

The same lytropic order was found by Schroeder * in an investi- 
gation of the viscosity of gelatine sols. 

It is noteworthy that the influence of salts on the temperature 
of gelation of agar-agar and other similar substances is analogous 
to their effect on gelatine, the same lytropic sequence being main- 
tained. 

Protective Colloids. The precipitating action of electrolytes 
on suspensoids may be inhibited by adding to the solution of the 
suspensoid a reversible colloid. The protective action of a re- 
versible colloid is not due, as might be supposed, to the increased 
viscosity of the medium and the resultant resistance to sedimenta- 
tion, since amounts of a reversible colloid, too minute to produce 
any appreciable increase in the viscosity of the medium, can pre- 

* Zeit. phys. Chem., 45, 75 (1903). 



COLLOIDS 



259 



vent precipitation. Thus, Bechhold * has shown that while a 
mixture of 1 cc. of a suspension of mastic and 1 cc. of a 0.1 molar 
solution of MgSO 4 diluted to 3 cc. with water, is completely pre- 
cipitated in 15 minutes, no precipitation will occur within 24 hours, 
if two drops of a 1 per cent solution of gelatine be added before 
diluting to 3 cc. 

Gum arabic and ox-blood serum exert a similar protective action 
when added to a suspension of mastic. The protective power of 
reversible colloids differs widely and Zsigmondy f has attempted 
to make this the basis of a method of classification of colloidal sub- 
stances. A red solution of colloidal gold becomes blue on the 
addition of a small amount of sodium chloride, owing to the in- 
crease in the size of the colloidal particles. Various colloidal sub- 
stances when added to a red colored gold sol protect the colloidal 
particles from precipitation by a solution of sodium chloride, no 
change in color following the addition of the electrolyte. A definite 
amount of each colloidal substance is required to prevent the 
change from red to blue in the color of the gold sol. In employing 
this color change as a means of differentiating colloidal substances, 
Zsigmondy introduced the "gold number," which may be defined 
as the weight in milligrams of a colloidal substance which is just 
insufficient to prevent the change from red to blue in 10 cc. of a 
gold sol after the addition of 1 cc. of a 10 per cent solution of 
sodium chloride. The following table gives the gold numbers of a 

few colloids. 

GOLD NUMBERS OF COLLOIDS. 



Colloid. 


Gold Number. 




Gelatine 
Casein (in ammonia) .... 
Egg-albumin 


005-0 01 
01 
15-0 25 




Gum arabic 


15-0 25; 5-4 




Dextrin 


6-12; 10-20 




Starch, wheat 


4-6 (about) 




Starch, potato 


25 (about) 




Sodium stearate 


10 (at 60);0 01 (at 


100) 


Sodium oleate 


0.4-1 




Cane sugar 


8 




Urea 


8 











* Zeit. phys. Chem., 48, 408 (1904). 
t Zeit. analyt. Chem., 40, 697 (1901). 



260 



THEORETICAL CHEMISTRY 



The gold number has proven useful in differentiating the various 
kinds of albumin, as is shown in the following table. 

GOLD NUMBERS OF ALBUMINS. 



Albumin. 


Gold Number. 


Egg white (fresh) 


08 


Globulin . . 


02-0 05 


Ovomucoid . 


04-0 08 


Albumin (Merck) 
Albumin (cryst.) .... 
Albumin (alkaline) 


1-0 3 
2-8 
006-0 04 







The addition of alkali to any one of the first five albumins of the 
above table reduces the gold number to that of alkaline albumin. 
Sulphide sols may be protected as well as metallic sols, and further- 
more the ability to exert protective action is not confined to organic 
colloids alone. 

In general, it may be said that when a suspensoid sol is mixed 
with an emulsoid sol in the proper proportions, the suspensoid sol 
acquires most of the characteristic properties of the protecting 
colloid. The masking of the properties of a suspensoid sol by a 
protecting colloid is probably to be ascribed to the formation of a 
thin film of adsorbed emulsoid over the suspensoid. 

Reciprocal Precipitation. A further deduction of the elec- 
trical theory of precipitation is, that when two oppositely-charged 
colloids are mixed, they should precipitate each other, and the 
resulting precipitate should contain both colloids. Experiments 
carried out by Biltz * have confirmed these predictions. He 
showed that when a solution of a positively-charged colloid is 
added to a solution of a negatively-charged colloid, precipitation 
occurs, unless the quantity of the added colloid is either relatively 
very large or very small. He also showed that when two colloids 
of the same electrical sign are mixed no precipitation occurs. Just 
as the amount of precipitation caused by the addition of an elec- 
trolyte to a sol is conditioned by the rate at which the electrolyte 
is added, so also the precipitation of one colloid by another is de- 

* Berichte, 37, 1095 (1904). 



COLLOIDS 



261 



pendent upon the manner in which the two sols are mixed. The 
extent to which one sol is precipitated by another sol of opposite 
sign is largely determined by the amount of one that is added to a 
definite amount of the other. This is clearly shown by the data 
of the following table which give the results obtained by Biltz on 
adding ferric hydroxide sol to 2 cc. of an antimony trisulphide sol 
containing 2.8 mg. per cc. 

PRECIPITATION OF COLLOIDAL ANTIMONY TRISULPHIDE 
BY COLLOIDAL FERRIC HYDROXIDE. 



Fe 2 3 (m*.). 


Immediate Result. 


Result After One Hour. 


0.8 
3 2 
4.8 
6.4 
8.0 
12 8 
20.8 


Cloudy 
Small flakes 
Flakes 
Complete precipitation 
Slow precipitation 
Cloudy 
Cloudy 


Almost homogeneous 
Unchanged 
Yellow liquid 
Complete precipitation 
Complete precipitation 
Slight precipitation 
Homogeneous 



It will be seen that the addition of a small amount of ferric 
hydroxide produces hardly any precipitation. With the addition 
of larger amounts of ferric hydroxide, the amount of precipitation 
increases until finally complete precipitation is attained. The 
addition of larger quantities of ferric hydroxide produces either 
little or no precipitation. It has been found that at the concen- 
tration which just produces complete precipitation, the electrical 
charges of the two sols are equivalent. When the amount of ferric 
hydroxide exceeds that required for complete precipitation, it is 
more than probable that the particles of colloidal antimony tri- 
sulphide are completely enveloped by the particles of ferric hy- 
droxide and thereby rendered inactive. 

When we come to study the action of one emulsoid on another, 
we find, as might be expected from the general behavior of emul- 
soids toward electrolytes, that the phenomena are more complex 
and very much less well-defined than with suspensoids. Although 
mutual precipitation does take place with emulsoids, the close 
resemblance between emulsoids and true solutions renders the 
phenomenon more or less indistinct. 



262 THEORETICAL CHEMISTRY 

When a suspensoid sol is added to an emulsoid sol having an 
opposite electric charge, precipitation may or may not occur 
according to the relative amounts of the two colloids in the mix- 
ture. When the two colloids are present in electrically equivalent 
quantities precipitation occurs, otherwise one colloid exerts a 
protective action on the other. 

Characteristics of Gels. Gels are generally obtained by cooling 
or evaporating emulsoid sols and, since the latter are known to be 
two-phase liquid systems, it is natural to infer that gels may also 
be two-phase systems. According to this conception, the only 
difference between an emulsoid sol and a gel is, that in the latter 
the concentration of at least one of the phases is greatly increased 
and thereby imparts greater viscosity and rigidity to the system. 
It has been suggested that the more concentrated of the two phases 
forms the walls of an assemblage of cells within which the more 
dilute phase is enclosed. The view that gels possess a distinct 
cellular structure is fully confirmed by microscopic examination. 

The extreme sensitiveness of gels changes in temperature and 
to the presence of extraneous substances renders their investi- 
gation exceedingly difficult. Notwithstanding the experimental 
difficulties involved in the study of gels, sufficient knowledge has 
been gained of their properties to make a brief account of these 
necessary in any treatment of the subject of colloids. 

Physical Properties of Gels. The process of gel formation 
from a dry gelatinous colloid and water invariably involves con- 
traction. This statement should not be confused with the fact 
that a gel on immersion in water undergoes appreciable increase in 
volume. 

Gels have been shown by Barus * to be considerably more com- 
pressible than solids. The compressibility increases as the tem- 
perature is raised until, when the gel is transformed into the sol, 
the compressibility becomes equal to that of pure water. The 
temperature of some gels, such as rubber and gelatine, is lowered 
by compression and raised by tension. 

The thermal expansion of gels is nearly identical with that of 
the more fluid component of the gel. 

* Am. Jour. Sci., 6, 285 (1898). 



COLLOIDS 263 

The rate at which pure substances diffuse in gels differs only 
slightly from the rate of diffusion in pure water, provided the con- 
centration of the gel is not too great. The slight resistance offered 
by gels to diffusion of dissolved substances may be regarded as 
further evidence in favor of their cellular structure. 

The modulus of elasticity of a gel, cast in a cylindrical mold, is 
given by the formula 

E - Pl 

^" 



where P is the tension which produces the increase in length AH in 
a cylinder whose length is I and whose radius is r. It has been 
found that the modulus of elasticity in gelatine gels increases as the 
square of the concentration of the gel. The time of recovery, after 
releasing the tension, increases as the concentration of the gel 
increases. 

The shearing modulus for a gel is given by the formula 

* 



"-27T+7)' 

where /i denotes the ratio of the relative contraction of the diameter 
to the relative change in length. The viscosity of a gel may be 
calculated from the shearing modulus by means of the equation 

11 = E 8 T } 

where r denotes the time of recovery. Since both E 8 and r increase 
with the concentration of the gel it is apparent that the viscosity 
of the gel must also increase with the concentration. 

As is well known, when glass is subjected to pressure or is un- 
equally strained, it exhibits the phenomenon of double refraction. 
Since glass bears some resemblance to gels in being a highly viscous, 
supercooled liquid, it might reasonably be inferred that gels should 
also show double refraction. Experiments with collodion and 
gelatine have shown that these substances, when subjected to 
pressure, behave similarly to glass. 

Hydration and Dehydration of Gels. The complementary 
processes of hydration and dehydration of gels are extremely inter- 



264 THEORETICAL CHEMISTRY 

esting. Following Freundlich * we will consider the subject very 
briefly under the two following heads: (a) Non-elastic Gels and 
(b) Elastic Gels. 

(a) Non-elastic Gels. When freshly prepared aluminium or fer- 
ric hydroxide gels were placed in a desiccator, it was found by 
van Bemmelen f that the rate at which these substances lost water 
was continuous. Furthermore, on removing the dried gels from 
the desiccator it was found that the process of recovery of moisture 
was also continuous. It will be shown in a subsequent chapter 
(p. 333) that a definite hydrate in the presence of its products of 
dissociation, possesses a constant vapor pressure so long as any of 
that particular hydrate is present. The fact that the vapor 
pressure of the gels investigated by van Bemmelen did not remain 
constant but decreased continuously as the water was removed, 
proved conclusively that no chemical compounds were formed in 
the process of gel hydration. 

Another gel which has been made the subject of much careful 
investigation is silicic acid. The dehydration curve of silicic acid 
is continuous, with the exception of a short portion where an ap- 
preciable amount of water is lost without much of any change in 
the vapor pressure. This portion of the curve corresponds to a 
marked change in the appearance of the gel. The gel which had 
hitherto been clear and transparent became opalescent soon after 
the vapor pressure had attained a temporarily constant value. 
The opalescence gradually permeated the entire mass, until the 
gel acquired a yellow color by transmitted light, and a bluish color 
by reflected light. These colors suggest a marked increase in the 
degree of dispersity of the gel, an inference the correctness of 
which subsequent investigation has fully confirmed. 

The curve of hydration, while resembling the curve of dehydra- 
tion in many respects, departs quite widely from it in others. The 
change in dispersity, as indicated by the appearance in reverse 
order of the color and opalescent phenomena mentioned above, 
also was manifest. 

* Kapillarchemie, p. 486. 

t Zeit. anorg. Chemie, 5, 466 (1894); 13, 233 (1897); 18, 14, 98 (1898); 30, 

265 (1902). 



COLLOIDS 265 

(b) Elastic Gels. The absence in elastic gels of a horizontal 
portion in the dehydration curves, and the fact that although 
elastic gels may have become saturated with water vapor, they 
still retain the power of taking up large amounts of liquid water 
when immersed in that medium, constitute the chief differences 
between elastic and non-elastic gels. 

The amount of water which can be taken up by an elastic gel is 
exceedingly large. Thus, on exposing a plate of gelatine weighing 
0.904 gram for eight days in an atmosphere saturated with water 
vapor, Schroeder * found that it had taken up 0.37 gram of water. 
On exposing for a longer period of time under the same conditions, 
he found no further gain in weight, and on removing the gelatine 
plate from the moist atmosphere and placing it in a desiccator, the 
plate slowly gave up the absorbed moisture and regained its origi- 
nal weight. On the other hand, when the plate, after having 
absorbed the maximum weight of moisture from the air, was im- 
mersed in water, it was found to increase in weight very rap- 
idly. Thus, on immersing the above plate which weighed 1.274 
grams when saturated in moist air, and allowing it to remain 
in water for one hour, it was found to have taken up 5.63 
grams of water. After an immersion of twenty-four hours, the 
plate was found to have taken up the maximum weight of water 
it was capable of absorbing at that temperature. On remov- 
ing the plate, it was found to part with the absorbed water 
very readily, even in moist air, the greater part of the absorbed 
water being so loosely held that the vapor pressure of the gel 
remained the same as that of pure water at the temperature of 
the experiment. 

It is impossible to measure directly the pressure produced by gels 
when they take up water, but some idea of the magnitude of these 
pressures may be obtained by coating a glass plate with gelatine, 
which has imbibed the maximum amount of water, and observing 
the degree to which the glass plate is bent by the drying gelatine 
film. Frequently the elastic limit of the glass is exceeded and 
the plate breaks under the stress produced by the dehydration of 

the gel. 

* Zeit. phys. Chem., 45, 75 (1903). 



266 



THEORETICAL CHEMISTRY 



Velocity of Imbibition. The rate at which gels imbibe water 
has been studied by Hofmeister.* The velocity of imbibition of 
water by thin plates of gelatine and agar-agar was measured by 
removing the plates at definite intervals and determining their 
increase in weight. Owing to the time consumed in making the 
weighings, the time intervals are affected by an appreciable error. 
The following table gives the data of a single experiment with 
gelatine. 

VELOCITY OF IMBIBITION IN GELATINE. 

(Thickness of plate 0.5 mm.) 



Time (mm ) 


Water Imbibed 
(grains) 


k 


5 


3 08 


090 


10 


3.88 


084 


15 


4 26 


084 


20 


4 58 


064 


25 


4 67 


075 


00 


4 96 





If the weight of water imbibed in i minutes is Wt, and w x is the 
maximum weight of water which a gel can take up under the con- 
ditions, then the velocity of imbibition should be given by the 
equation 

dw , , , 

-fi = k(w-w t ), 

which on integration becomes 

"If __j[{-p* dO. t 

t (WQO Wt) 

The figures in the third column of the preceding table were cal- 
culated by means of this equation and, although the values are not 
strictly constant, the variation is no greater than might be expected 
where the experimental error is so large. 

Heat of Imbibition. The process of imbibition is accom- 
panied by an evolution of heat. Quantitative measurements 

* Arch. exp. Pathol. u. Pharmakol., 27, 395 (1890). 



COLLOIDS 



267 



of the heat evolved when gels take up water have been made 
by Wiedemann and Liideking * and also by Rodewald.f The 
following table gives Rodewald's data on the heat of imbibition 
of starch. 

HEAT OF IMBIBITION OF STARCH. 
(Weight of dry starch 100 grams.) 



Per Cenjl, Water. 


Heat in Calories per 
Gram of Starch. 


23 


28.11 


2 39 


22 60 


6 27 


15 17 


11 65 


8 43 


15 68 


5 21 


19 52 


2 91 



It will be observed that the greatest development of heat ac- 
companies the initial stages of imbibition where very small 
amounts of water are taken up. This is what we might expect 
when we remember that it is the last remaining portion of water 
which is most difficult to remove from a gel, and that it is only 
through the application of heat that its complete removal can be 
effected. 

Imbibition in Solutions. When a gel is immersed in a saline 
solution, the salt distributes itself between the solvent and the 
gel. The rate of imbibition is found to vary greatly according 
to the salt which is present in the solution. Thus, the velocity 
of imbibition has been found to be accelerated by the presence 
of the chlorides of ammonium, sodium and potassium and by 
the nitrate and bromide of sodium; on the other hand, the pres- 
ence of the nitrate, sulphate and tartrate of sodium retard im- 
bibition. 

The effect of acids and bases on imbibition appears to be similar 
to the influence of salts. 

Adsorption. The change in concentration which occurs at 
the boundary between two heterogeneous phases is termed adsorp- 

* Wied. Ann., 25, 145 (1885). 

t Zeit. phys. Chem., 24, 206 (1897). 



268 THEORETICAL CHEMISTRY 

twn. At the surface of a solid surrounded by a gas or vapor, the 
phenomenon is generally known as gaseous adsorption, since any 
difference which may occur in the concentration of the solid phase 
is much too small to be detected. At the boundary between 
liquid and gaseous phases, the concentration of each phase un- 
doubtedly undergoes alteration. In the case of the boundary be- 
tween solid and liquid phases, the only apparent inequality in 
concentration occurs on the liquid side of the boundary, notwith- 
standing the fact that the adsorbed substance is quite commonly 
regarded as being bound to the surface of the solid phase. The 
cause of this erroneous conception is, that the extremely thin layer 
of liquid in which the alteration in concentration actually occurs, 
is the layer which wets the surface of the solid and hence is the 
layer which adheres to the solid when it is removed from the 
liquid. 

The retention of gases by charcoal is a typical example of gaseous 
adsorption, while the removal of coloring matter by charcoal in the 
purification of various organic substances may be cited as an 
example of adsorption of a liquid by a solid. 

If the adsorbed substance increases in concentration in the 
vicinity of the boundary, the adsorption is said to be positive; if it 
decreases, the adsorption is said to be negative, 

Adsorption of Gases. In gases, adsorption-equilibrium is 
attained with remarkable rapidity. Thus, if a gas is admitted 
into a vessel containing some freshly prepared cocoanut charcoal, 
the pressure will fall immediately to a value which corresponds to 
the removal of the entire adsorbed volume of gas. 

The concentration of adsorbed gas on the surface of a solid, when 
equilibrium is attained, has been shown to be approximately 
1 X 10~ 7 gram per square centimeter. This value is of the same 
order of magnitude as the strength of the limiting capillary layer 
of a liquid and, therefore, lends support to the suggestion put for- 
ward some years ago by Faraday that an adsorbed film of gas may 
be present in the liquid state. 

The amount of gas adsorbed by a solid increases with the pres- 
sure and diminishes with increasing temperature. The following 
empirical equation, expressing the relation between the amount of 



COLLOIDS 269 

gas adsorbed by a solid and the pressure, has been proposed by 
Freundlich * 

= ap n , 

m ' 

where x is the total mass of gas adsorbed on a surface of m sq. 
cm. under a pressure p, and where a and n are constants. This 
equation, known as the adsorption isotherm, has been found to hold 
quite generally. 

Adsorption in Solutions. The adsorption phenomena occur- 
ring at the surface of contact of a solid with a solution are similar 
to the phenomena which have just been discussed. Because of the 
frequency of its occurrence in many of the more common operations 
of both laboratory and factory, the subject of adsorption in solu- 
tions deserves fuller treatment. 

The general characteristics of adsorption in solutions may be 
briefly summarized as follows: 

(1) Adsorption in solutions is generally positive, i.e., on shaking 
a solution with a finely-divided adsorbent, the volume concentra- 
tion of the solution will diminish. 

(2) The amount of positive adsorption may be sufficient to re- 
move almost all of the solute from a solution, especially if the solu- 
tion is dilute. On the other hand, negative adsorption is always 
very small, and frequently is immeasurable. 

(3) Adsorption is directly proportional to the so-called " specific 
surface," the latter term being defined as the ratio of the total sur- 
face of the adsorbent to its volume. 

(4) On shaking a definite weight of an adsorbent with a given 
volume of solution of known concentration, a definite equilibrium 
will be established. If the solution is then diluted with a known 
amount of solvent, the adsorption will decrease until it acquires the 
same value which it would have attained, had the same weight of 
adsorbent been introduced directly into the more dilute solution. 
For example, if 1 gram of charcoal is agitated with 100 cc. of a 
0.0688 molar solution of acetic acid for 20 hours, adsorption is 
found to reduce the original concentration of the acid to 0.0678 

molar. 

* Zeit. phys. Chem., 57, 385 (1906). 



270 THEORETICAL CHEMISTRY 

In a second experiment, if 1 gram of charcoal is shaken for the 
same period of time with 50 cc. of 2 X 0.0688 = 0.1376 molar 
acetic acid, and then, after adding 50 cc. of water, the shaking is 
continued for an additional period of 3 hours, the final concentration 
of the acid will be found to be the same as in the first experiment. 

(5) It is impossible to determine the specific surface of an ad- 
sorbent directly owing to its porosity. However, according to (3) 
adsorption is directly proportional to the specific surface and there- 
fore the weights of different adsorbents which produce the same 
amount of adsorption may be assumed to possess equal specific 
surfaces. 

(6) Adsorption in solution is largely dependent upon the surface 
tension of the solvent. In solutions of the same substance in 
different solvents, the greatest adsorption occurs in that solution 
whose solvent possesses the highest surface tension. 

(7) The order of efficiency of adsorption is not only independent 
of the nature of the solvent but also of the nature of the adsorbed 
substance. 

The Adsorption Isotherm. The empirical equation of Freund- 
lich for gaseous adsorption has been found to apply equally well to 
adsorption equilibria in solutions. The equation may be written 
as follows: 

= ac n . 
m ' 

where x is the weight of substance adsorbed by a weight m of ad- 
sorbent from a solution whose volume-concentration at equilibrium 
is c, and where, as before, ex. and n are constants. The constant n 
varies in different cases from n = 2 to n = 10; within these limits 
the value of n is independent of the temperature and also of the 
natures of the adsorbed substances and the adsorbent. Although 
the value of the constant a varies over a wide range, the ratio of its 
values for two adsorbents in different solutions is practically con- 
stant. 

The following table contains the data given by Freundlich * on 
the adsorption of acetic acid by charcoal. 

* Kapillarchemie, p. 147. 



COLLOIDS 



271 



ADSORPTION OF AQUEOUS ACETIC ACID BY CHARCOAL. 
t = 25; = 2.606; 1/n = 0.425. 



Concentration 
(mols per liter). 


x/m (obs.). 


x/m (calc.). 


0181 


467 


474 


0309 


624 


596 


0616 


0.801 


798 


1259 


1 11 


1.08 


2677 


1 55 


1.49 


0.4711 


2.04 


1 89 


0.8817 


2 48 


2.47 


2.785 


3 76 


4 01 



The validity of the adsorption isotherm is best tested graphically, 
by plotting the logarithms of the experimentally determined values 
of x/m against the logarithms of the corresponding concentrations. 
If the equation holds, a straight line should be obtained. The 
curves shown in Fig. 69 represent x/m as a function of c, and 
log x/m as a function of log c: it will be observed that the loga- 
rithmic plot is practically rectilinear. 

log C 




c 
Fig. 69. 

Surface Energy of Colloids. In almost all colloidal solutions 
there exists a difference of potential between the particles of the col- 
loid and the surrounding medium. The importance of this factor 



272 THEORETICAL CHEMISTRY 

in interpreting the behavior of colloids has already been empha- 
sized. Another factor of equal importance in connection with 
colloidal phenomena, is that which depends upon the enormous 
surface of contact between the colloid and the surrounding medium 
There is an abundance of evidence showing that a colloidal solution 
is non-homogeneous, or in other words, that it is essentially a sus- 
pension of finely-divided particles in a fluid medium. An immense 
increase in superficial area results from the division and sub- 
division of matter. To bring about this comminution requires a 
large expenditure of energy. In a colloidal solution this energy is 
stored up in the colloidal particles in the form of surface energy, 
which may be defined as the product of surface area and surface 
tension. 

For example, suppose 1 cc. of a substance to be reduced to 
cubical particles measuring 0.1 M on each edge, and let the particle 
be suspended in water at 17 C, The total energy involved can 
be calculated as follows: The volume of a single particle is 
0.1 /i 3 or 1 X 10~ 15 cc.; hence the total number of particles is 
1 X 10 15 . The surface of a single particle is 6 X (0.1 /z) 2 , or 
6 X 10~ 10 sq. cm., and the total surface is 6 X 10 5 sq. cm. The 
surface tension of water at 17 C. is 71 dynes; hence the total 
surface energy is 71 X 6 X 10 5 = 4.32 X 10 7 ergs. This enormous 
figure shows that where the surface of the disperse phase is highly 
developed, as it is in colloidal solutions, the surface energy becomes 
a very important factor in determining the behavior of the system. 
This is especially the case when the degree of aggregation of the 
colloidal particles is changed, since a relatively small change in the 
amount of aggregation may involve a great change in the surface 
exposed and a corresponding change in the surface energy. A very 
close connection exists between the electrical and surface factors in 
a colloidal solution. 

i Surface Concentration. It has been pointed out in an earlier 
chapter (p. 144) that as the result of unbalanced molecular attrac- 
tion, the surface of a liquid behaves like a tightly stretched mem- 
brane. In consequence of this contractile force, or surface tension, 
the pressure at the surface of a liquid is greater than the pressure 
within the liquid. 



COLLOIDS 273 

The experiments of Soret (p. 208) and the theoretical deductions 
of van't Hofif have shown that when a dilute solution is unequally 
heated, the solute distributes itself in accordance with the gas laws, 
the solution becoming more concentrated in the cooler portion. 
Just as the homogeneity of a dilute solution has been shown to 
be disturbed by inequality of temperature, so also inequality of 
pressure may be assumed to cause differences in concentration in 
the solution. Although direct experimental verification is difficult, 
there is abundant evidence for the view that the concentration at 
the surface of solution differs from the volume-concentration of 
the solution in consequence of the greater pressure in the surface 
layer. 

The mathematical relation between surface concentration and 
surface tension was first deduced by J. Willard Gibbs * in 1876. 
The following simplified derivation of this important equation is 
due to Ostwald. Let s be the surface of a solution whose surface 
tension is 7, and let it be assumed that the surface contains 1 mol 
of the solute. If a very small portion of the solute enters the sur- 
face layer from the solution, thereby causing a diminution dy in 
the surface tension, the corresponding change in energy will be 
s dy. But this gain in energy must be equivalent to the osmotic 
work involved in effecting the removal of the same weight of solute 
from the solution. Let v be the volume of solution containing 
unit weight of solute, and let dp be the difference in the osmotic 
pressures of the solution before and after its removal; the osmotic 
work will be vdp. Since the gain in surface energy and the 
osmotic work are equal, we have 

sdy = vdp. 

The solutions being dilute, we may assume that the gas laws hold, 
and since v = RT/p, we may write 

RT , 
~dp, 



^ 
OT dp sp 



* Trans. Conn. Acad., Vol. Ill, 439 (1876). 



274 THEORETICAL CHEMISTRY 

Since pressure is directly proportional to concentration, the pre- 
ceding equation becomes 



_. 

dc sc 

But s has already been defined as the surface which contains 1 mol 
of solute in excess, from which it follows that the excess of solute 
in unit surface is I/a. Writing u = I/a, we have 

- -JL*L 

U ~~ RT dc 
which is the equation of Gibbs. 

From this equation it is evident that if the surface tension, 7, 
increases with the concentration, then u is negative and the surface 
concentration is less than the concentration of the bulk of the 
solution. This is clearly negative adsorption. On the other 
hand, if 7 decreases as the concentration increases, u is positive and 
the surface concentration is greater than the concentration of the 
bulk of the solution, or the adsorption is positive. Finally, if the 
surface tension is independent of the concentration, then the con- 
centration of the solute in both the surface layer and the bulk of 
the solution will be the same. 

Preparation of Colloidal Solutions. Since 1861, when Graham 
published his first paper on colloids, numerous investigators have 
devised methods for the preparation of colloidal solutions. Within 
recent years our knowledge of this class of solutions has been 
greatly increased, many crystalloidal substances having been 
obtained in the colloidal condition. As a result of these investi- 
gations, we no longer speak of crystalloidal and colloidal matter, 
but use the terms crystalloid and colloid to distinguish two dif- 
ferent states. In fact it is now recognized that it is simply a 
matter of overcoming certain experimental difficulties, before it 
will be possible to obtain all forms of matter in the colloidal 
state. The scope of this book forbids a detailed account of the 
various methods which have been devised for the preparation of 
colloidal solutions.* We must content ourselves with a general 
classification of these methods into two groups as follows: 

* See "Die Methoden zur Herstellung Kolloider Losungen anorganischer 
Stoffe," by Theodore Svedberg, Dresden, 1909. 



COLLOIDS 275 

(1) Crystallization Methods, and (2) Solution Methods. These 
two divisions are sufficiently comprehensive to include all of the 
known methods for the preparation of colloidal solutions, with the 
possible exception of the electrical methods which may be con- 
sidered as forming a separate group. 

Crystallization Methods. The crystallization methods include 
the following subdivisions: 

(1) Methods involving cooling of a liquid or solution. 
Example: On cooling an alcoholic solution of sulphur in liquid 

air, a transparent, highly dispersed, solid sol is obtained. 

(2) Methods involving change of medium. 

Example: On gradually adding a solution of mastic in alcohol 
to a large volume of water, the mastic is precipitated in a finely 
divided condition and a colloidal mastic hydrosol results. 

(3) Reduction methods. 

Example: On adding a cold, dilute solution of hydrazine 
hydrate to a dilute, neutral solution of auric chloride, a dark blue 
gold sol is obtained. 

In addition to hydrazine, numerous other reducing agents may 
be employed, such as phosphorus, carbon monoxide, hydrogen, 
acetylene, formaldehyde, acrolein, various carbohydrates, hy- 
droxylamine, phenylhydrazine, and metallic ions. 

(4) Oxidation methods. 

Example: On oxidizing a solution of hydrogen sulphide by air 
or sulphur dioxide, a colloidal solution of sulphur is obtained. 

(5) Hydrolysis methods. 

Example: When a solution of ferric chloride is slowly added to 
a large volume of boiling water, the salt undergoes hydrolysis, and 
on cooling the dilute solution, a reddish-brown ferric hydroxide 
sol is obtained. 

(6) Methods involving metathesis. 

Example: A colloidal solution of silver may be prepared by 
adding a few drops of a dilute solution of sodium chloride to a dilute 
solution of silver nitrate, provided the resulting solution of sodium 
nitrate is below the precipitating concentration. 



276 THEORETICAL CHEMISTRY 

Solution Methods. Under this heading are to be grouped the 
so-called "peptization" methods. The term, peptization, was intro- 
duced by Graham to express the transformation of a gel into a sol. 
To-day we understand a peptizer to be a substance which, if 
sufficiently concentrated, is capable of effecting the solution of a 
solid which is insoluble in its dispersion medium. A typical 
example of peptization is afforded by silver chloride which forms a 
sol on prolonged digestion with a solution which contains either 
Ag'or Cl'. It is apparent that the rate of peptization can be con- 
trolled by dilution of the peptizer, and that when the sol stage is 
attained, the peptizer may be readily removed by dialysis. 

Numerous reactions are known in which the conversion of an 
insoluble precipitate into a sol can only be effected through the 
removal of the excess of electrolyte by prolonged washing or dial- 
ysis. A familiar example of this type of peptization is furnished 
by the tendency of many precipitates to run through the filter 
after too prolonged washing with water. 

Electrical Methods. These methods of preparing colloidal solu- 
tions depend upon the dispersive action of a powerful electric 
discharge upon compact metals. In 1897 Bredig * discovered, 
while studying the action of the electric current on different 
liquids, that if an arc be established between two metallic wires 
immersed in a liquid, minute particles of metal are torn off from 
the negative terminal and remain suspended in the liquid indefi- 
nitely. In order to prepare a colloidal solution by the method of 
electrical dispersion, Bredig recommends that a direct current 
arc be established between wires of the metal of which a colloidal 
solution is desired, the ends of the wires being submerged in water 
in a well-cooled vessel, as shown in Fig. 70. The current employed 
ranges in strength from 5 to 10 amperes, and the voltage lies be- 
tween 30 and 110 volts. A rheostat and an ammeter are included 
in the circuit. 

The wires are brought in contact for an instant in order to 
establish the arc, after which they are separated about 2 nun. 
During the gentle hissing of the arc, clouds of colloidal metal are* 
projected out into the water from the negative wire, a portion of 

* Zeit. Elektrochem., 4, 514 (1897); Zeit. phys. Chem., 31, 258 (1899). 



COLLOIDS 277 

the metal torn off being distributed through the water as a coarse 
suspension. The size of the particles disrupted from the negative 
terminal is dependent upon the strength of the current, a current 



Ammeter 




Fig, 70. 

of 10 amperes producing a greater proportion of colloidal metal 
than a current of 5 amperes. The addition of a trace of potassium 
hydroxide to the water has been shown to facilitate the process of 
dispersion. When gold wires are used, deep red colloidal solu- 
tions are obtained, which after standing for several weeks, acquire 
a bluish-violet color. With extra precautions, the red colloidal 
gold solutions may be preserved for two years. These solutions 
have been shown by Bredig to contain about 14 mg. of gold per 
100 cc. In this manner Bredig prepared colloidal solutions of 
platinum, palladium, iridium, and silver. The method of Bredig 
has been improved and extended by Svedberg. 

A diagram of Svedberg's apparatus is shown in Fig. 71. The 
secondary terminals of an induction coil, capable of giving a spark 
ranging from 12 to 15 cm. in length, are connected in parallel with 
the electrodes and a glass plate-condenser having a surface of 
approximately 225 sq. cm. Minute fragments or grains of the 
metal of which a sol is desired are placed on the bottom of the 
vessel containing the dispersion medium. The electrodes, which 
need not necessarily be of the same metal, are immersed as shown 
in the diagram, and during the process of electrical dispersion, the 
contents of the vessel are gently stirred with one or the other of the 



278 



THEORETICAL CHEMISTRY 



electrodes. With this apparatus Svedberg has succeeded in pre- 
paring colloidal solutions of tin, gold, silver, copper, lead, zinc, 
cadmium, carbon, silicon, selenium, and tellurium. He has also 



Induction Coil 




Fig. 71. 

obtained all of the alkali metals in the colloidal state, ethyl ether 
being used as the dispersion medium. w An interesting observation 
made by Svedberg in the course of his experiments is that the color 
of a metal is the same in both the colloidal and gaseous states. 



CHAPTER XIII. 
MOLECULAR REALITY. 

The Brownian Movement. If a liquid in which fine particles 
of matter are suspended, such as an aqueous suspension of gam- 
boge, be examined under the microscope, the suspended particles 
will be seen to be in a state of ceaseless, erratic motion. This 
phenomenon was first observed in 1827 by the English botanist, 
Robert Brown, while examining a suspension of pollen grains, and 
has been called the Brownian Movement in honor of its discoverer. 

Ever since its discovery, the Brownian Movement has been the 
subject of numerous investigations. It was not until 1863, how- 
ever, that Wierner suggested that the cause of the phenomenon was 
the actual bombardment of the suspended particles by the mole- 
cules of the suspending medium. Twenty-five years later a similar 
conclusion was reached independently by Gouy, who showed that 
neither light nor convection currents within the liquid could 
possibly give rise to the motion. Furthermore, Gouy showed the 
movement to be independent of external vibration and only slightly 
influenced by the nature of the suspended particles. The smaller 
the particles and the less viscous the suspending medium, the more 
rapid the motion was found to be. By far the most striking feature 
of the phenomenon, however, is the fact that the motion is cease- 



Perrin's Experiments. The first quantitative investigation 
of the Brownian Movement was undertaken by Perrin in 1909. 
It has been shown (p. 100) that the mean kinetic energy Ek of one 
mol of a perfect gas is given by the expression 

#*=fpt>. (1) 

Since pv = RT, we may write 



279 



280 THEORETICAL CHEMISTRY 

where N denotes the Avogadro Constant, i.e., the number of 
molecules contained in one mol of any gas. It is evident that if 
Ek can be measured, equation (2) affords a means of calculating N, 
provided we are warranted in applying an equation which has been 
derived for the gaseous state to a suspension of fine particles in a 
liquid medium. It has already been shown that the simple gas 
laws hold for dilute solutions and therefore we may assume that, 
at the same temperature, the mean kinetic energy of the dissolved 
molecules is equal to that of the gaseous molecules. In other 
words, at the same temperature, the mean kinetic energy of all 
the molecules of all fluids is the same, and is directly proportional 
to the absolute temperature. Since the gas laws apply equally 
well to dilute solutions containing either large or small molecules, 
Perrin held that there was no a priori reason for assuming that the 
grains of a suspension should not conform to the same laws. If 
this assumption be correct, the grains of a uniform suspension 
should so distribute themselves under the influence of gravity 
that, when equilibrium is attained, the lower layers will have a 
higher concentration than the upper layers. In other words, the 
distribution should be strictly analogous to the distribution of the 
air over the surface of the earth, the density being greatest at the 
surface and diminishing as the altitude increases. 

Let us imagine a suspension to be confined within a tall 
vertical cylinder whose cross-sectional area is s sq. cm. As- 
suming that the suspension has come to equilibrium under the 
influence of gravitation, let n be the number of grains per unit of 
volume at a height h from the base of the cylinder. Since the 
concentration diminishes as the height increases, the number of 
grains at a height h + dh will be n dn. The osmotic pressure 
of the grains at the height h will be f nEk, where E k is the mean 
kinetic energy of each grain. In like manner, the osmotic pressure 
at the height d + dh will be f (n dn) E k . The difference in 
osmotic pressure between the two levels is f dnE k and since the 
pressure acts over a surface of s sq. cm., the difference of osmotic 
forces acting over the cross-sectional area of the cylinder is 
t dnE k . Since the system is in equilibrium, this difference in 
osmotic forces must be balanced by the difference in the attraction 



MOLECULAR, REALITY 281 

of gravitation at the two levels. Let < be the volume of a single 
grain, D its density, and 5 the density of the suspending medium. 
The resultant downward pull upon a single grain will be <j> (D 6)0, 
where g is the acceleration due to gravity. The volume of liquid 
between the two levels being s dh, it follows that the total down- 
ward pull upon all the grains included between the two levels must 
be nsh<t> (D 6) g. It is this force which opposes the tendency 
of the grains to distribute themselves uniformly throughout the 
entire volume of the suspending medium, or, in other words, it is 
the force which acts in opposition to the osmotic force f s dnE k . 
When equilibrium is established, these two forces must be equal, 
and we may then write 

- f s dnE k = ns dh<t> (D - 6) g. (3) 

If HQ and n denote the number of grains per unit of volume at each 
of two planes h units apart, we obtain, on integrating equation (3), 

| E k log e n /n = <t> (D - 6) gh. (4) 

On substituting in equation (4) the value of E k in equation (2), 
and transforming to Briggsian logarithms, we have 

2.303 RT/N log n,/n = J m*g (D - 6) h, (5) 

< being expressed in terms of the mean radius, r, of a single 
grain. It is evident that if we can measure n, no, D, and r in 
equation (5), the calculation of the Avogadro Constant, N, be- 
comes possible. 

The determination of the density of the grains, D, was carried 
out in two different ways with suspensions of gamboge and mastic 
which had been rendered uniform by a process of centrifuging. 
In the first method, the grains were dried to constant weight 
at 110, and then by heating to a higher temperature, a viscous 
liquid was obtained which, on cooling, formed a glassy solid. The 
density of this solid was determined by suspending it in a solution 
of potassium bromide of known density. 

In the second method for the determination of D, Perrin measured 
the masses mi and m^ of equal volumes of water and suspension 
respectively. On evaporating the suspension to dryness, the mass 
Ws of suspended solid contained in m^ grams of suspension was 



282 THEORETICAL CHEMISTRY 

obtained. If the density of water is d, the volume of the sus- 
pended grains will be 

_, m\_ wfa wiz 

V = ~d d ' 

and consequently the density of the grains will be m s /V. The 
values of D obtained by these two methods were found to be in 
excellent agreement. 

A microscope furnished with suitable micrometers was employed 
in the determination of n and n . With the high magnification 
employed, the depth of the field of view was limited: in fact, the 
measurements were carried out with a microscopic slide similar 
to those used for counting the corpuscles in the blood. By focus- 
sing the microscope at different depths, the average number of 
grains in the field of view at each level could be counted. Perrin 
was able to photograph the larger grains at different levels, whereas 
with the smaller grains it was necessary to reduce the field so that 
relatively few grains were visible. The average number of grains 
counted at any two different levels would of course give the de- 
sired ratio riQ/n. 

The only other quantity in equation (5) to be measured was the 
average radius of the grains r. To determine this quantity, 
Perrin made use of a method similar to that used by Thomson for 
counting the number of electrically charged particles in an ionized 
gas. Stokes has shown that the force required to impart a uniform 
velocity v, to a particle of radius r, moving through a liquid 
medium whose viscosity is T?, is given by the formula, 6 irqrv. If 
the motion be due to gravity, as in the case of suspensions of fine 
particles, obviously the foregoing expression must be equal to the 
right-hand side of equation (5), or 

Giyrv = $irr*(D-$)g. 

From this equation the value of r can be calculated. The rate 
at which the grains settled under the influence of gravity was 
determined by placing a portion of the uniform suspension in a 
capillary tube and observing the rate at which the suspension 
cleared, care being taken to keep the temperature constant. This 
method of determining r was open to the objection that Stokes' 



MOLECULAR REALITY 283 

law might not apply to particles as small as those of colloidal 
suspensions. 

In order to test the validity of Stokes' law under these conditions, 
the following modification of the method for the determination of 
r was introduced. It had been observed that when a suspension is 
rendered slightly acid, the grains, on coming in contact with the 
walls of the containing vessel, adhered, while the motion of the 
grains throughout the bulk of the liquid remained unaltered. In 
this way it was possible to gradually remove all of the grains from 
the suspension and count them and, knowing the total volume of 
suspension taken, the average number of grains per cubic centi- 
meter could be calculated. If the total mass of suspended matter 
is known it is an easy matter to calculate the volume of each grain, 
and from this to compute the radius, r. The value of r determined 
in this way was found to agree with that calculated by the first 
method, thus proving the validity of Stokes' law when applied to 
colloidal suspensions. 

Five series of experiments carried out by Perrin with gamboge 
suspensions in which several thousand individual grains were 
counted, gave as a mean value of N in equation (5), 69 X 10 22 . 
Similar experiments with mastic suspensions gave N = 70.0 X 10 22 . 
These values, it will be seen, are in close agreement with the value 
of Avogadro's Constant given on page 41. * 

The Law of Molecular Displacement. The actual movements 
of the individual grains of a suspension when observed under the 
microscope are seen to be exceedingly complex and erratic. The 
horizontal projections of the paths of three different grains in a 
suspension of mastic are shown in Fig. 72, the dots representing 
the successive positions occupied by the particles after intervals of 
30 seconds. The straight line joining the initial and final positions 
of a particle is called the horizontal displacement A, of the particle. 

If the time taken by the particle to move from its initial to its 
final position be t, Einstein * has shown that the mean value of the 
square of the horizontal displacement of a spherical particle of 
radius r ought to be 

"-TO" <6) 

* Zeit. Elektrochem., 14, 235 (1908). 



284 



THEORETICAL CHEMISTRY 




72. 



MOLECULAR REALITY 



285 



where 77 is the viscosity of the suspending medium and where the 
other symbols have their usual significance. 

This equation was tested by Perrin, using suspensions of gam- 
boge and mastic. Some of the results obtained are given in the 
following table: 

VALUES OF N CALCULATED BY EINSTEIN'S EQUATION. 



Suspension. 


. 

r in 
microns.* 


mXlO 16 . 


No of dis- 
placements 


A 7 XI 0-22. 


Gamboge in water . 


367 


246 


1500 


69 


Gamboge in 10% solution of 
glycerine 


385 


290 


100 


64 


Mastic in water .... 


52 


650 


1000 


73 


Mastic in 27% solution of urea. 


5.50 


750,000 


100 


78 



* The micron is one-millionth of a meter or one ten-thousandth of a centimeter. 

It will be seen that notwithstanding the large variations in the 
granular masses of the different suspensions recorded in the table, 
the values of JV, calculated by means of Einstein's equation, are 
quite concordant. Perrin gives as the mean value of all of his 
experiments, N = 68.5 X 10 22 . 

Recent Investigations of the Brownian Movement. Nord- 
lund * has recently repeated Perrin's experiments, employing a 
colloidal solution of mercury and an arrangement of apparatus 
whereby the movements of the particles could be recorded photo- 
graphically. The mean value of N derived from twelve carefully 
executed experiments was 59 X 10 22 , the average deviation of the 
results of the individual experiments from the mean being approxi- 
mately 10 per cent. 

The Brownian Movement in gases has been studied by Millikan f 
and by Fletcher J employing a minute drop of oil as the suspended 
particle. In the gaseous state, where the intermolecular distances 
are greater than in the liquid state, not only are the collisions less 
frequent but the mean free paths are appreciably longer. These 
conditions are favorable to the study of the Brownian Movement 
and offer an opportunity for the determination of the Avogadro 
Constant with a high degree of accuracy. As the mean of nearly 
six thousand measurements, Fletcher gives N = 60.3 X 10 22 , this 
value being accurate to within 1.2 per cent. 

* Zeit. phys. Chem., 87, 60 (1914). f Phys. Rev., i, 220 (1913). 
J Ibid., 4, 453 (1914). 



CHAPTER XIV 
THERMOCHEMISTRY. 

General Introduction. A chemical reaction is almost invari- 
ably accompanied by a thermal change. In the majority of 
cases heat is evolved; a violent reaction developing a large amount 
of heat, while a feeble reaction develops a comparatively small 
amount. Such reactions are said to be exothermic. A relatively 
small number of chemical reactions are known which take place 
with an absorption of heat. These are termed endothermic reac- 
tions. Instances of chemical reactions unaccompanied by any 
thermal change are very rare and are almost wholly confined to 
the reciprocal transformations of optical isomers. These facts, 
which were first observed by Boyle and Lavoisier, led to the view 
that the amount of heat evolved in a chemical reaction might be 
taken as a measure of the chemical affinity of the reacting sub- 
stances. However, with the advance of our theoretical knowledge, 
it is now known that this is not true, although a parallelism 
between heat evolution and chemical affinity frequently exists. 

Thermochemistry is concerned with the thermal changes which 
accompany chemical reactions. 

Thermal Units. Heat is a form of energy, and like other 
forms of energy it may be resolved into two factors; an intensity 
factor, the temperature, and a capacity factor, which may be 
measured in any one of several units. Among these units those 
defined below are the most frequently employed. 

The small calorie (cal.) is the quantity of heat required to raise 
the temperature of 1 gram of water from 15 C, to 16 C. The 
temperature interval is specified because the specific heat of water 
varies with the temperature. The large or kilogram calorie (Cal.) 
is the quantity of heat required to raise the temperature of 1000 
grams of water from 15 C. to 16 C. The Ostwald or average 

286 



THERMOCHEMISTRY 287 

calorie (K), is the quantity of heat required to raise the temper- 
ature of 1 gram of water from the melting-point of ice to the 
boiling-point of water under a pressure of 760 mm. of mercury. 
It is approximately equal to 100 cal. or to 0.1 Cal. The joule (j), 
a unit based on the C.G.S. system, is equal to 10 7 ergs. This 
being inconveniently small is generally multiplied by 1000, giving 
the kilojoule (J), which is therefore equal to 10 10 ergs. The last 
two units are open to the objection that their values are depend- 
ent upon the mechanical equivalent of heat, any change in the 
accepted value of which would involve a correction of the unit of 
heat. The different capacity factors of heat energy are related 
as follows: 
1 cal. = 0.001 Cal. = 0.01 K (approx.) = 4.183 j 0.004183 J. 

Thermochemical Equations. In order to represent the changes 
in energy which accompany chemical reactions, an additional 
meaning has been assigned to the chemical symbols. As ordina- 
rily used, these symbols represent only the molecular or formula 
weights of the reacting substances. In a thermochemical or 
energy equation the symbols represent not only the weight in 
grams expressed by the formula weights of the substances, but 
also the amount of heat energy contained in the formula weight 
in one state as compared with the energy contained in a standard 
state. For example, the energy equation, 

C + 2 = CO 2 + 94,300 cal., 

indicates that the energy contained in 12 grams of carbon and 
32 grams of oxgyen exceeds the energy contained in 44 grams of 
carbon dioxide, at the same temperature, by 94,300 calories. In 
writing energy equations it is very essential that we have some 
means of distinguishing between the different states of aggrega- 
tion of the reacting substances, since the energy content of a 
substance is not the same in the gaseous, liquid, and solid states. 
In the system proposed by Ostwald, ordinary type is used for 
liquids, heavy type for solids, and italics for gases. Another and 
more convenient system has been proposed, in which solids are 
designated by enclosing the symbol or formula within square 
brackets; liquids by the simple, unbracketed symbol or formula; 



288 



THEORETICAL CHEMISTRY 



B 



and gases by enclosing the symbol or formula within parentheses. 
The above equation should, therefore, be written in the following 

manner: 

|C] + (2 O) = (C0 2 ) + 94,300 cal. 

Thermochemical Measurements. In order to measure the 

number of calories evolved or ab- 
sorbed when substances react, it is 
necessary that the reaction should 
proceed rapidly to completion. This 
condition is fulfilled by two classes 
of processes. In the first class we 
may mention the processes of solu- 
tion, hydration, and neutralization; 
and in the second class, the process 
of combustion. 

The apparatus used for measur- 
ing the capacity factor of heat energy 
is a calorimeter. This instrument 
may be given a variety of forms, 
depending upon the particular use 
to which it is to be put. A simple 
form of calorimeter is shown in Fig. 
73. It consists of two concentric 
metal cylinders, A and 5, insulated 
from each other by an air jacket, 
the inner vessel being supported 
en vulcanite points. Through a 
vulcanite cover passes a thin walled 
test tube, in which the reaction is 
allowed to take place. An accurate thermometer and a ring-stirrer 
also pass through the cover of the calorimeter. In order to deter- 
mine the thermal capacity of the calorimeter, B is nearly filled with 
water, and a known mass of water, w, at a temperature t\ is intro- 
ducedintoC. Let the initial temperature of the water in B be 2 - The 
water in Bis stirred until the contents of both B and C have acquired 
the same temperature, fe. When thermal equilibrium has been estab- 
lished, it is evident that m (ti 3 ) calories are required to raise 



7\ 



Fig. 73. 



THERMOCHEMISTRY 289 

the temperature of the apparatus and the water in 5, (fe 3 ) 
degrees. From this data it is an easy matter to calculate the 
number of calories required to raise the temperature of the appar- 
atus and water in B 1 degree, this being the thermal capacity of 
the apparatus. The calorimeter may now be used to determine 
the heat evolved or absorbed in a reaction. Suppose, for exam- 
ple, that it is desired to measure the heat of neutralization of an 
acid by a base. Equivalent quantities of both acid and base are 
dissolved in equal volumes of water, care being taken to make the 
solutions dilute. A definite volume of one solution is introduced 
into C and an equal volume of the other solution is placed in a 
vessel from which it can be quickly and completely transferred to 
C. When both solutions have acquired the same temperature, 
the thermometer in B is read and then the two solutions are 
mixed. When the reaction is complete, the temperature of the 
water in B is again noted. If the thermal capacity of the calori- 
meter is Q, and the rise in temperature produced by the reaction 
is 0, then Qd is the amount of heat evolved by the reaction. To 
this quantity of heat must be added the number of calories re- 
quired to raise the temperature of the products of the reaction B 
degrees. The solutions of the products being dilute, their specific 
heats may be assumed to be equal to unity. From the total quan- 
tity of heat so obtained, the number of calories evolved when mo- 
lecular quantities react can be readily calculated. The chief source 
of error in calorimetric measurements is loss by radiation. This 
may be reduced to a minimum (1) by making the thermal capac- 
ity of the calorimeter large, and (2) by so arranging matters that 
the initial temperature of the water in the calorimeter is as much 
below the temperature of /the room as the final temperature is 
above it. 

The Combustion Calorimeter. The combustion of many sub- 
stances, such as organic compounds, proceeds very slowly in air 
under ordinary pressures. Such reactions can be accelerated, if 
they are caused to take place in an atmosphere of compressed 
oxygen. For this purpose the combustion calorimeter was de- 
vised by Berthelot.* In this apparatus the essential feature is 
* Ann. Chim. Phys., (5), 23, 160 (1881); (6), 10, 433 (1887). 



290 



THEORETICAL CHEMISTRY 



the so-called combustion bomb, shown in Fig. 74. This consists 
of a strong steel cylinder lined with platinum or gold, and fur- 
nished with a heavy threaded cover. The substance to be 
burned is placed in a platinum capsule fastened to the support 
R, and a short piece of fine iron wire of known mass is connected 
with the electric terminals Z, Z, the middle portion of the wire 
dipping into the substance. The cover is then screwed down 
tight, and the bomb is filled with oxygen under a pressure of from 
20 to 25 atmospheres. The bomb is then 
submerged in the calorimeter, as shown in 
Fig. 75. The mass of water in the calorim- 
eter being known and its temperature having 
been read, an electric current is passed through 
|S the iron wire in the bomb causing it to burn 
and thus ignite the substance. The rise in 
temperature due to the combustion is ob- 
served, and the quantity of heat evolved is 
calculated. Corrections must be applied for 
loss by radiation, for the heat evolved from 
the combustion of the iron, and for the heat 
evolved from the oxidation of the nitrogen of 
the residual air in the bomb. 

For the details of the method of determin- 
ing heats of combustion the student must 
consult a laboratory manual. 
Law of Lavoisier and Laplace. In 1780, Lavoisier and Laplace,* 
as a result of their thermochemical investigations, enunciated the 
following law: The quantity of heat which is required to decompose 
a chemical compound is precisely equal to that which was evolved in 
the formation of the compound from its elements. This first law of 
thermochemistry will be seen to be direct corollary of the law 
of the conservation of energy which was first clearly stated by 
Mayer in 1842. 

Law of Constant Heat Summation. A generalization of funda- 
mental importance to the science of thermochemistry was discov- 
ered in 1840 by Hess.f He pointed out that the heat evolved in a 

* Oeuvres de Lavoisier, Vol. II, p, 283. 
t Pnffff Ann.. n. 3ftfi HfttfN. 




Fig. 74. 



THERMOCHEMISTRY 



291 



chemical process is the same whether it takes place in one or in several 
steps. This is known as the law of constant heat summation. The 




Fig. 75. 

truth of the law may be illustrated by the equality of the heat of 
formation of ammonium chloride in aqueous solution, when pre- 
pared in two different ways. 



292 THEORETICAL CHEMISTRY 

Thus, 

(A) 



(NH 3 ) + (HC1) = [NH 4 C1] + 42,100 cal, 

[NH 4 C1] + aq. = NH 4 C1, aq. - 3,900 cal. 

38,200 cal. 

(B) 



(NH 3 ) + aq. = NH 3 , aq. + 8,400 cal. 

(HC1) + aq. = HC1, aq. + 17,300 cal. 

NH 3 , aq. + HC1, aq. = NH 4 C1, aq. + 12,300 cal. 

38,000 cal. 

It will be observed that the total amount of heat evolved in 
the formation and solution of ammonium chloride is the same 
within the limits of experimental error, whether gaseous ammonia 
and hydrochloric acid are allowed to react and the resulting prod- 
uct is dissolved in water, or whether the gases are each dissolved 
separately and then allowed to react. It should be noted that 
when a substance is dissolved in so much water that the addition 
of more water or the removal of a small portion of water produces 
no thermal effect, it is customary to denote it by the symbol aq. 
(Latin aqua = water). Thus, 

NH 4 C1, aq. + nH 2 = NH 4 C1, aq., 
NH 4 C1, aq. - nH 2 O = NH 4 C1, aq. 

By means of the law of constant heat summation it is possible to 
find indirectly the amount of heat developed or absorbed by any 
reaction, even though it is impossible to carry it out experimen- 
tally. For example, it is impossible to measure the heat evolved 
when carbon burns to carbon monoxide. But the heat evolved 
when carbon monoxide burns to carbon dioxide, and also the heat 
evolved when carbon burns to carbon dioxide, can be accurately 
determined. The energy equations are as follows: 

[C] + 2(0) = (CO,) + 94,300 cal. (1) 

(CO) + (0) = (C0 2 ) + 67,700 cal. (2) 



THERMOCHEMISTRY 293 

'Treating these equations algebraically, and subtracting equation 
(2) from equation (1), we have 

[C] + (0) (CO) + 26,600 cal., 

or, the heat of combustion of carbon to carbon monoxide is 26,600 
calories. Again, as a further illustration of the applicability of 
the law of Hess, we may take the calculation of the heat of forma- 
tion of hydriodic acid from its elements, making use of the follow- 
ing energy equations: 

2 KI, aq. + 2 (Cl) = 2 KC1, aq. + 2 [I] + 524 K (1) 

2 HI, aq. + 2 KOH, aq. = 2 KI, aq. + 2 H 2 O + 274 K (2) 

2 HC1, aq. + 2 KOH, aq. = 2 KC1, aq. + 2 H 2 + 274 K (3) 

2 (HI) + aq. = 2 HI, aq. + 384 K, (4) 

2 (HC1) + aq. = 2 HC1, aq. + 346 K, (5) 

2 (H) + 2 (Cl) = 2 (HC1) + 440 K, (6) 

adding equations (1) and (2), 

2 (Cl) + 2 HI, aq. + 2 KOH, aq. = 2 KC1, aq. + 2 [I] + 2 H 2 O 

+ 798 K. (7) 

Subtracting equation (3) from equation (7), 

2 (Cl) + 2 HI, aq. - 2 HC1, aq. = 2 [I] + 524 K, 
or 

2 (Cl) + 2 HI, aq. = 2 [I] + 2 HC1, aq. + 524 K, (8) 

adding equations (4) and (8), 

2 (HI) + aq. + 2 (Cl) - 2 [H + 2 HC1, aq. + 908 K, , (9) 
subtracting equation (5) from equation (9), 

2 (HI) + 2 (Cl) - 2 (HC1) = 2 [I] + 562 K, 
or 

2 (HI) + 2 (Cl) = 2 [I] + 2 (HC1) + 562 K, (10) 

subtracting equation (6) from equation (10), 

2 (HI) -2(H) = 2[I] + 122K, 
or 

2 (H) + 2 (I) = 2 (HI) - 122 K. 

In a similar manner, practically any heat of formation may be 
calculated, provided the proper energy equations are combined. 



294 THEORETICAL CHEMISTRY 

Heat of Formation. The intrinsic energy of the substances 
entering into chemical reaction is unknown, the amount of heat 
evolved or absorbed in the process being simply a measure of the 
difference between the energy of the reacting substances and the 
energy of the products of the reaction. Thus, in the equation 

[C] + 2 (O) = (C0 2 ) + 94,300 cal., 

the difference between the energy of a mixture of 12 grams of 
carbon and 32 grams of oxygen, and the energy of 44 grams of 
carbon dioxide is seen to be 94,300 calories. The equation is 
clearly incomplete since we have no means of determining the 
intrinsic energies of free carbon and oxygen. Furthermore, since 
the elements are not mutually convertible, we have no means of 
determining the difference in energy between them. It is cus- 
tomary, therefore, in view of this lack of knowledge, to put the 
intrinsic energies of the elements equal to zero. 

If the heats of formation of the substances present in a reac- 
tion are known, it is much simpler to substitute these in the 
energy equation and solve for the unknown term. This method 
avoids the laborious process of elimination from a large number of 
energy equations, as in the preceding pages. If all of the sub- 
stances involved in a reaction are considered as decomposed into 
their elements, it is evident that the final result of the reaction 
will be the difference in the sums of the heats of formation on the 
two sides of the equation. This leads to the following rule: 
To find the quantity of heat evolved or absorbed in a chemical reac- 
tion, substract the sum of the heats of formation of the substances 
initially present from the sum of the heats of formation of the products 
of the reaction j placing the heat of formation of all elements equal to 
zero. 

The energy equation for the formation of carbon dioxide from 
its elements may y .hen be written as follows: 

+ 0= (C0 2 )+ 94,300 cal., 
or 

(C0 2 ) = -94,300 cal. 

That is, the energy of 1 mol of carbon dioxide is 94,000 calories. 
Therefore in writing an energy equation we make use of the fol- 



THERMOCHEMISTRY 295 

lowing rule: Replace the formulas of each compound in the equa- 
tion representing the reaction by the negative values of their respective 
heats of formation and solve for the unknown term. This unknown 
term may be either the heat of a reaction or the heat of formation 
of one of the reacting substances. The following examples will 
serve to illustrate the application of the above rules: 

(1) Let it be required to find the heat of the following reaction 

[MgCl 2 ] + 2 [Na] - 2 [NaCl] + [Mg] + x, 

where x is the heat of the reaction. The heat of formation of 
MgCl 2 is 151 Cal., and that of NaCl is 97,9 Cal., therefore, 

- 151 + = - (2 X 97.9) + + x, 
or 

x = 44.8 Cal. 

(2) The heat of combustion of 1 mol of methane is 213.8 Cal., 
and the heats of formation of the products, carbon dioxide and 
water, are 94.3 Cal. and 68.3 Cal., respectively. Let it be required 
to find the heat of formation of methane. Representing the heat 
of formation of methane by x, we have 

(CH 4 ) + 2 (0 2 ) = (C0 2 ) + 2 (H 2 0) + 213.8 Cal., 

- x + o = - 94.3 - 2 X 69.3 + 213.8, 
or 

x = 17.1 Cal. 

(3) The heat of combustion of 1 mol of carbon disulphide is 
265.1 Cal., the thermochemical equation being 

' CS 2 + 3 (0 2 ) = (C0 2 ) + 2 (S0 2 ) + 265.1 Cal. 
The heats of formation of carbon dioxide and sulphur dioxide are 
94.3 Cal. and 71 Cal. respectively. The heat of formation of 
carbon disulphide x, may then be calculated as follows: 

_ x + o = - 94.3 - 2 X 71 + 265.1, 
or 

a; =*f28.8 Cal. 

Carbon disulphide is thus seen to be an endothermic compound. 

Heat of Solution. The thermal change accompanying the 

solution of 1 mol of a substance in so large a volume of solvent 

that subsequent dilution of the solution causes no further thermal 



296 



THEORETICAL CHEMISTRY 



change is termed the heat of solution. The solution of neutral 
salts is generally an endothermic process. This fact may be 
readily accounted for on the hypothesis that considerable heat 

HEATS OF FORMATION AND SOLUTION. UT\ 



Substance. 


Hoat of 
Formation, 


Heat of 
Solution. 


Water, vapor 


58.7 




Water, liquid 


68 4 




Hydrochloric acid 


22 


20.3 


Sulphuric acid 


193 1 


17 8 


Ammonia 


12 


8 4 


Nitric acid 


41 9 


7 2 


Phosphoric acid. . 


302 9 


2 7 


Potassium hydroxide 


103 2 


13.3 


Potassium chloride 


104 3 


-3.1 


Potassium bromide 


95 1 


-5.1 


Potassium iodide 


80 1 


-5 1 


Potassium nitrate 


119.5 


-8 5 


Sodium hydroxide 


101 9 


10 9 


Sodium chloride 


97.6 


1.2 


Sodium bromide 


85 6 


-0 2 


Sodium sulphate 


328 8 


2 


Sodium nitrate ... 


111 3 


-5 


Sodium carbonate 


272 6 


5 6 


Ammonium chloride 


75 8 


-4 


Ammonium nitrate 


88 


-6 2 


Calcium hydroxide 


215 


3.0 


Calcium chloride 


170 


17 4 


Magnesium sulphate 


502 


20.3 


Ferrous chloride 


82 


17 9 


Ferric chloride 


96.1 


63.3 


Zinc chloride 


97 


15.6 


Zinc sulphate 


30 


18.5 


Cadmium chloride 


293.2 


3 


Cupric chloride 


51 6 


11.1 


Cupric sulphate 


182 6 


15 8 


Mercuric chloride 


53.2 


-3.3 


Silver nitrate 


28.7 


-5.4 


Stannous chloride 


80 8 


0.3 


Stannic chloride . . 


127.3 


'29.9 


Lead chloride 


82.8 


-6.8 


Lead nitrate 


105 5 


-7.6 









must be absorbed as heat of fusion and heat of vaporization before 
the solid salt can assume a condition in solution which closely 
resembles that of a gas. The heat of solution of hydrated salts 
is less than the heat of solution of the corresponding anhydrous 



THERMOCHEMISTRY 



297 



salts. For example, the heat of solution of 1 mol of anhydrous 
calcium nitrate is 4000 calories, while the heat of solution of 1 mol 
of the tetrahydrate is 7600 calories. The difference between 
the heats of solution of the anhydrous and hydrated salts is termed 
the heat of hydration. The heats of formation and heats of solu- 
tion in water of some of the more common compounds are given 
in the preceding table, the values being expressed in large calories. 

Heat of Dilution* The heat of dilution of a solution is the 
quantity of heat per mol of solute which is evolved or absorbed 
when the solution is greatly diluted. Beyond a certain dilution, 
further addition of solvent produces no thermal change. While 
there is a definite heat of solution for a particular solute in a par- 
ticular solvent, the heat of dilution remains indefinite, since the 
latter is dependent upon the degree of dilution. Those gases 
which obey Henry's law are practically the only substances which 
have no appreciable heats of solution or dilution. 

The following tables give the heats of dilution of hydrochloric 
and nitric acids. 



HEAT OF DILUTION OF SOLUTIONS OF HYDROCHLORIC 

ACID. 

Heat of solution = 20.3 cal. 



HC1+H 2 O 5.37 

HC1+2H 2 O 11.36 

HC1+10H 2 16.16 

HC1+50H 2 17.1 

HC1+300H 2 17.3 



HEAT OF DILUTION OF SOLUTIONS OF NITRIC ACID. 
Heat of solution = 7.15 cal. 



HNO 3 4-H 2 O 


3 84 


HN0 3 -f-2H 2 O 


2 32 


HNO 3 +4H 2 O 


1 42 


HNO 8 -*-6H 2 


0.2 


HN0 3 -f8H 2 O 


-0 04 


HNO 3 -t-100H 2 O 


-0 03 







298 THEORETICAL CHEMISTRY 

Reactions at Constant Volume. When a chemical reaction 
takes place without__any change in volum^ or when the external 
work resulting from a change in volume is not included in the 
heat of the reaction, the process is said to take place at constant 
volume. That is to say, the condition of constant volume is a 
condition which involves no^ external jvprk^ either positive or 
negative. Under these conditions the total energy of the react- 
ing substances is equal to the total energy of the products of the 
reaction, plus the quantity of heat developed by the reaction. 

Reactions at Constant Pressure. When a chemical reaction 
is accompanied by a change in volume, the system suffers a loss 
of heat equivalent to the work done against the atmosphere, if 
the volume increases; or the system gains an amount of heat 
equivalent to the work done on the system by the atmosphere, 
if the volume decreases. Under these conditions the reaction is 
said to take place at constant pressure. The difference between 
constant volume and constant pressure conditions, then, is that 
under the former, the heat equivalent of the work corresponding 
to any change in volume which may occur is not considered as 
having any effect upon the energy of the system; whereas under 
the latter, due account is taken of the change in energy resulting 
from change in volume. Suppose that in a reaction, 1 mol of gas 
is formed. Under standard conditions of temperature and 
pressure the volume of the system will be increased by 22.4 liters. 
The formation of gas involves the performance of work against 
the atmosphere, this work being done at the expense of the heat 
energy of the system. To calculate the heat equivalent of the 
work done, let us imagine the gas enclosed in a cylinder fitted with 
a piston whose area is 1 square centimeter. The normal pressure 
of the atmosphere on the piston is 76 cm. of mercury or 1033.3 
grams per square centimeter. If the increase in the volume of 
the gas is 22.4 liters, the piston will be raised through 22,400 cm. 
and the work done will be 1033.3 X 22,400 gram-centimeters. 
The heat equivalent of this change in volume will be (1033.3 X 
22,400) -5- 42,600 = 542.3 calories or 0.5423 large calories. This 
amount of heat must be added to the heat of the reaction. It 
should be observed that this correction is independent of the actual 



THERMOCHEMISTRY 299 

value of the pressure upon the system. Thus, if the pressure is 
increased n times, the volume of the gas will be reduced to 1/n 
of its former value, and the work done will involve moving the 
piston through 1/n of the distance against an n-fold pressure, 
which is plainly equivalent to the former amount of work. While 
the correction is independent of the pressure it is not independent 
of the temperature. The familiar equation, pv ^RT shows us 
that the^work^done by a gas is directly proportional to its ab- 
solut^iaaiperatureT Thus, if a gas is evolved at 273, it will 
occupy double the volume it would occupy at 0, and the work 
done at 273 will involve moving the piston through twice the 
distance that it would have to be moved at 0. Theoretically, a 
gas evolved at the absolute zero would occupy no volume and 
hence no work would be done. Introducing the correction for 
temperature, we see that 

= 1.986 Teal., 

must be added to the heat of the reaction, where T is the absolute 
temperature at which the change in volume occurs. For all 
ordinary purposes it is sufficiently accurate to take 2 T calories 
as the correction. Thus, suppose n mols of gas to be formed in a 
reaction at 17 C., the amount of heat absorbed will be 

nx 2(273 + 17) = 580ncal. 

Under constant pressure conditions, the symbols, in addition to 
their usual significance, represent the energy plus or minus the 
term, 2 T per mol, the positive or negative sign being used 
according as the gas is absorbed or formed. Since the constant 
volume condition is a condition in which no account is taken of the 
external work, even if a change in volume does occur during the 
reaction, and the constant pressure condition is one in which 
the external work is taken into consideration, it is apparent that 
the relation of the heat energy of a reaction at constant volume, 
Q V) to the heat energy at constant pressure Q P) can be represented 
by the equation 

Q p = Q v -2n!TcaL, 



300 THEORETICAL CHEMISTRY 

where n denotes the number of mols of gas formed in excess of 
those initially present. This equation is of great importance in 
connection with the determination of heats of combustion in the 
bomb-calorimeter in which the reactions necessarily take place 
under constant volume conditions. Since it is customary to state 
heats of reaction under constant pressure conditions, the foregoing 
equation makes it possible to convert heats of combustion deter- 
mined under constant volume conditions into heats of combustion 
under constant pressure conditions. For example, the combustion 
of naphthalene takes place in accordance with the equation 

CioH 8 + 12 (0 2 ) = 10 (C0 2 ) + 4 (H 2 0) + 1242.95 Cal. 

It is apparent that the combustion is accompanied by the forma- 
tion of 2 mols of gas, and at 15 C. the correction will be 

Q p = 1242.95 - [2 X 0.002 (273 + 15)], 
or 

Q p = 1241.8 Cal. 

The volume occupied by solids or liquids is so small as to be 
negligible and does not enter into these calculations. 

Variation of Heat of Reaction with Temperature. If a chem- 
ical reaction be allowed to take place first at the temperature fo, 
and then at the temperature k, the amounts of heat developed in 
the two cases will be found to be quite different. Let Qi and Q 2 
represent the quantities of heat evolved at the temperatures ti 
and t% respectively. Let us imagine that the reaction takes place 
at the temperature ti, Qi units of heat being evolved; and then 
let the products of the reaction be heated to the temperature fe. 
If c' represents the total thermal capacity of the products of the 
reaction, then the quantity of heat necessary to produce this rise 
in temperature will be c' (h h). Now let us imagine the 
reacting substances, at the temperature h, to be heated to the 
temperature 2 , and then allowed to react with the evolution of 
Q 2 units of heat. The heat necessary to produce this rise in 
temperature in the reacting substances is c (fe 2i), where c 
is the total thermal capacity of the original substances. Having 
started with the same substances at the same initial temperature, 
and having obtained the same products at the same final temper- 



THERMOCHEMISTRY 301 

ature, we have, according to the law of the conservation of 
energy, 

Qi-c'(fe-fe) -&-c(fe-i), 
or 



or, where the change in temperature is very small, 

dQ 

, ~ / /' 

df " C ' 

If c' is greater than c then the sign of dQ/dt will be negative, or, in 
other words, an increase in temperature will cause a decrease in 
the heat of reaction. On the other hand, if c is greater than c', 
dQ/dt will be positive and the heat of reaction will increase with 
the temperature. 

EXAMPLE. The reaction between hydrogen and oxygen at 
18 C. is represented by the following equation: 

2 (H 2 ) + (0 2 ) = 2 (H 2 O) + 1367.1 K 

Suppose it is required to find how much heat will be evolved when 
equal masses of the two gases react at 110 C., the product of the 
reaction being maintained at this temperature, and the pressure 
remaining constant. The specific heats per gram of the different 
substances involved are as follows: 

Hydrogen = 3.409; Oxygen = 0.2175; Water (between 18 
and 100) = 1; (Water between 100 and 110) = 0.5. 
The heat of vaporization of water is 537 calories per gram. 

For liquid water per degree we have, 

dQ/dt = (4 X 3.409 + 32 X 0.2175) - (36 X 1) = - 15.404 cal. 

and for (100 -18) = 82, we have, 82 x(- 15.404) = -1263 cal. 
The heat of formation of liquid water at 100 is, therefore, 
1367.1- 12.63 = 1354.47 K. 

When the liquid water is vaporized at 100, (36 X 537) calories of 
this heat is absorbed, or the formation of steam at 100 from 
hydrogen and oxygen, evolves 

1354.47 - 193.32 = 1161.15 K. 



302 



THEORETICAL CHEMISTRY 



For steam per degree, we have, 

dQ/dt = (4 X 3.409 + 32 X 0.2175) - (36 X 0.5) = 2.596 cal;, 
and for the interval (110 - 100) = 10, 

10 X 2.596 = 25.96 cal. 
Or for the total heat evolved, we have 

1161.15 + 0.2596 = 1161.41 K. 

Heat of Combustion. The heat evolved during the complete 
oxidation of unit mass of a substance is termed its "heat of 
combustion. The unit of mass commonly chosen in all physico- 
chemical calculations is the mol. An enormous amount of 
experimental work has been done by Thomsen,* Berthelot,t and 
Langbein t on the determination of the heats of combustion of a 
large number of organic compounds. A few of their results are 
given in the accompanying tables. 

SATURATED HYDROCARBONS. 



Hydrocarbon. 


Heat, of 
Combustion. 


Difference. 


Methane CHU . 


Cal. 
211 9 


Cal. 


Ethane C2EU . ... 


370.4 


158 5 


Propane, C$Hs 


529 2 


158 8 


Butane, C4Hio . 


687 2 


158 


Petane, CsH^ 


847 1 


159.9 









UNSATURATED HYDROCARBONS. 



Hydrocarbon. 


Heat of 
Combustion 


Difference. 


Ethylene, C 2 H 4 


Cal. 

333 4 


Cal. 

1 CA O 


Propylene, CsHU 


492 7 


159 3 


Isobutylene, C^s 


650 6 


157.9 


Amylene, CsHio 


807 6 


157 


Acetylene, C2HU 


310 1 


1 C*7 C 


Allylene, C 3 H 4 . . ... 


467.6 


157 5 









* Thermochemische Untersuchungen, 4 Vols. 

t Essai de Mecanique Chimique, Thermochimie, Done*es et Lois Numeri- 
ques. 

f Jour, prakt, Chem., 1885 to 1895. 



THERMOCHEMISTRY 



303 



ALCOHOLS. 



Alcohol. 


Heat of 
Combustion. 


Difference. 


Methyl alcohol, CH*O 


Cal. 
182 2 


Cal. 


Ethyl alcohol, C2H 6 O 


340 5 


158.3 


Propyl alcohol, CsHsO 


498 6 


158.1 


Isobutyi alcohol, CiHioO 


658 5 


159.9 









It will be observed that a very nearly constant difference in 
the heat of combustion corresponds to a constant difference of a 
CH 2 group in composition. A number of interesting relations 
between heats of combustion of compounds and their differences 
in composition have been discovered, but these cannot be taken up 
at this time. It has also been pointed out that the heat of com- 
bustion of organic compounds is conditioned not only by their 
composition, but also by their molecular constitution. 

Some exceedingly interesting and important results have been 
obtained with the different allotropic forms of the elements. For 
example, when equal masses of the three common allotropic forms 
of carbon are burned in oxygen, the amounts of heat evolved are 
found to be quite different, as is shown by the following energy 
equations: 

[C] diamond + 2 (0) - (C0 2 ) + 94.3 Cal. 
[C] graphite + 2 (0) = (CO*) + 94.8 Cal. 
[C] amorphous + 2 (0) = (C0 2 ) + 97.65 Cal. 

It is apparent that amorphous carbon contains the greatest 
amount of energy of any one of the three allotropic modifications, 
and, therefore, when amorphous carbon is changed into diamond, 
the reaction must be accompanied by the evolution of (97.65 
94.3) = 3.35 Cal. In like manner, the allotropic forms of sulphur 
and phosphorus have different heats of combustion. The follow- 
ing equations show the heat equivalents of the differences in 
intrinsic energy between the allotropic forms: 

S (monoclinic) = S (rhombic) + 2.3 Cal. 
P (white) = P (red) + 3.71 Cal. 



304 THEORETICAL CHEMISTRY 

When the same substance is burned in oxygen and then in ozone, 
it is found that more heat is evolved in ozone than in oxygen. 
The energy equation expressing the change of ozone into oxygen 
may be written thus, 

(0 3 ) = lJ(Q) + 36.2Cal. 

All of the above facts illustrate the general principle that the 
larger amounts of intrinsic energy are associated with the more 
unstable forms. 

Thermoneutrality of Salt Solutions. In addition to the law 
of constant heat summation, Hess discovered two other important 
laws of thermochemistry, viz., the law of thermoneutrality of 
salt solutions, and the law governing the neutralization of acids 
by bases.* When two dilute salt solutions are mixed there is 
neither evolution nor absorption of heat. Thus when dilute solu- 
tions of sodium nitrate and potassium chloride are mixed, there is 
no thermal effect. The energy equation may be written as follows : 

NaN0 3 , aq. + KC1, aq. = NaCl, aq. + KNO d , aq. + Cal. 

According to this equation a double decomposition has taken 
place and we should naturally expect an evolution or an absorption 
of heat. While Hess could not account for the absence of any 
thermal effect, he recognized the fact as quite general and formu- 
lated the law of the thermoneutrality of salt solutions as fol- 
lows: The metathesis of neutral salts in dilute solutions takes place 
with neither evolution nor absorption of heat. 

The explanation of the phenomenon of thermoneutrality was 
furnished by the theory of electrolytic dissociation. When the 
above equation is written in the ionic form, it becomes 

Na* + N(Y + 1C + Cl' - Na- + Cl' + K' + N(Y. 

From this it is apparent that the same ions exist on both sides of 
the equation, and in reality no reaction takes place. 

There are numerous exceptions to the law of thermoneu- 
trality/ These can be satisfactorily accounted for by the theory of 
electrolytic dissociation. All of those salts, the behavior of which in 
dilute solution is contrary to the law, are found to be only partially 

* Pogg. Ann., 50, 385 (1840). 



THERMOCHEMISTRY 305 

ionized, and, therefore, when their solutions are mixed, a chem- 
ical reaction actually occurs. The exceptions must be considered 
as furnishing additional evidence in favor of the theory of elec- 
trolytic dissociation. 

Heat of Neutralization. Hess also discovered * that when 
dilute solutions of equivalent quantities of strong acids and 
strong bases are mixed, practically the same amount of heat is 
evolved. The following energy equations may be considered as 
typical examples of such neutralizations: 

HC1, aq. + NaOH, aq. = NaCl, aq. + H 2 + 13.75 CaL, 
HN0 3 , aq. + NaOH, aq. = NaN0 3 aq. + H 2 + 13.68 CaL, 
HC1, aq. + KOH, aq. = KC1, aq. + H 2 + 13.70 CaL, 
HN0 3 , aq. + KOH, aq. = KNO 3 , aq. + H 2 + 13.77 CaL, 
HC1, aq. + LiOH, aq. = LiCl, aq. + H 2 + 13.70 CaL 

Here again it would be difficult to explain the phenomenon with- 
out the theory of electrolytic dissociation. In terms of this 
theory, however, the explanation is perfectly plausible. If MOH 
and HA represent any strong base and any strong acid respectively, 
then when equivalent amounts of these are dissolved in water, 
each solution being largely diluted to the same volume, the reac-. 
tion may be written thus: 

M' + OH' + H* + A' = M* + A' + H 2 + 13.7 CaL 

Disregarding the ions which occur on both sides of the equality 
sign, we have 

H' = H 2 + 13.7Cal. 



It thus appears that the neutralization of a strong acid by a strong 
base in dilute solution consists solely in the combination of hydro- 
gen and hydroxyl ions to form undissociated water, the heat of 
this ionic reaction being 13.7 large calories. 

The heat of formation of water from its ions must not be con- 
fused with the heat of formation of water from its elements. 
When weak acids or weak bases are neutralized by strong bases 
or strong acids, or when weak acids are neutralized by weak bases, 

* Loc. cit. 



306 THEORETICAL CHEMISTRY 

the heat of neutralization may differ widely from 13.7 Cal. This 
is shown by the following thermochemical equations: 

H-COOH, aq. + NaOH, aq. = H-COONa, aq. + H 2 

+ 13.40 Cal., 
CHC1 2 -COOH, aq. + NaOH, aq. = CHCl 2 -COONa, aq. + H 2 O 

+ 14.83 Cal., 
H-COOH, aq. + NH 4 OH, aq. = H-COONH 4 , aq. + H 2 

+ 11.90 Cal., 
HCN, aq. + NaOH, aq. = NaCN, aq. + H 2 + 2.90 Cal. 

As will be seen, the heat of neutralization may be either greater 
or less than 13.7 Cal. The exceptions to the generalization of 
constant heat of neutralization are readily explained by the 
theory of electrolytic dissociation. Suppose a weak acid to be 
neutralized by a strong base. According to the dissociation 
theory, the acid is only slightly dissociated and, therefore, yields 
a comparatively small number of hydrogen ions to the solution. 
The base on the other hand is completely dissociated into hydroxyl 
and metallic ions. Therefore, as many hydroxyl ions disappear 
as there are free hydrogen ions with which they can combine to 
form water. When the equilibrium between the acid and the 
products of its dissociation has been thus disturbed, it undergoes 
further dissociation and the resulting hydrogen ions immediately 
combine with the free hydroxyl ions of the base. This process 
continues until all of the hydroxyl ions of the base have been 
neutralized. It is evident that the thermal effect in this case is 
the algebraic sum of the heat of dissociation of the weak acid, 
which may be positive or negative, and the heat of formation of 
water from its ions. A similar explanation holds for the neutrali- 
zation of a weak base by a strong acid, or for the neutralization 
of a weak acid by a weak base. This affords a method for estimat- 
ing the approximate value of the heat of dissociation of a weak 
acid or a weak base. For example, in the equation given above, 

HCN, aq. + NaOH, aq. = NaCN, aq. + H 2 + 2.90 Cal., 

the difference between 2.9 and 13.7 or -10.8 Cal. represents 
approximately the heat of dissociation of hydrocyanic acid. 



THERMOCHEMISTRY 



307 



Since the acid is initially slightly dissociated in dilute solution, 
it is apparent that in order to obtain the true heat of dissociation 
we must add to 10.8 Cal. the thermal value of the dissociation 
of that portion of the acid which has already become ionized. 
Heat of lonization. Since 13.7 Cal. is the heat of formation 
of water from its ions, this must also be the thermal equivalent of 
the energy required to dissociate one mol of water into its ions. It 
must be remembered that the dissociated molecule of water must 
be mixed with a very large volume of undissociated water, in order 

HEAT OF FORMATION OF IONS. 



Ion. 


Heat of 
Formation. 


Ion. 


Heat of 
Formation 


Hydrogen 





j Copper (ic) 


-15 8 


Potassium 
Sodium ... 
Lithium . ... 
Ammonium 
Magnesium 
Calcium .... 


61 9 
57 5 
62 9 
32 8 
109 
109 


i Copper (ous) . . 
Mercury (ous) 
Silver ... .... 
Lead . 
Tin (ous). 
Chlorine . . . . 


-16 

-19 8 
-25 3 
5 
3 3 
39 3 


Aluminium 


121 


Bromine 


28 2 


Manganese . . 
Iron (ous) 
Iron (ic) 
Cobalt 
Nickel 
Zinc 


50 2 
22 2 
-9 3 
17 
16 
35 1 


Iodine 
Sulphate 
Sulphite 
Nitrous 
Nitric 
Carbonate 


13 1 
214 4 
151 3 
27 
49 
161 1 


Cadmium 


18 4 


Hydroxyl . .... 


54 7 








* 



that the dissociation may be permanent. Reference to the table 
of heats of formation (p. 296), will show that 68.4 Cal. are required 
to form one mol of water from its elements. Hence, it follows that 
68.4 13.7 = 54.7 Cal., is the heat of formation of one equivalent 
of hydrogen and hydroxyl ions. It has been shown that an ex- 
tremely small amount of energy is necessary to ionize hydrogen 
when it is dissolved in water. It is evident, therefore, that 54.7 
Cal. is a close approximation to the heat of formation of one equiva- 
lent of hydroxyl ions. 

On the assumption that the heat of ionization of gaseous hydro- 
gen in solution is zero, the values of the other ionic heats of forma* 



308 THEORETICAL CHEMISTRY 

tion may be computed. For example, the heat of formation of 
KOH, aq. is 116.5 Cal. The ionic heat of formation of potassium 
ions must be 116.5 - 54.7 = 61.8 Cal. In like manner, the 
heat of formation of KC1, aq. is 101.2 Cal.; hence the ionic heat 
of formation of chlorine ions must be 101.2 61.8 = 39.4 Cal. 
The preceding table of ionic heats of formation has been calculated 
as in the above examples. 

The Principle of Maximum Work. A fundamental principle 
of the science of mechanics is, that a system is in stable equilib- 
rium when its potential energy is a minimum. In 1879, Ber- 
thelot * suggested that a similar principle applies to chemical 
systems. 

In terms of the kinetic theory, the temperature of a substance 
is to be regarded as a measure of the kinetic energy of its molecules. 
The development of heat by a chemical reaction would, therefore, 
be taken as an indication of a decrease in the potential energy of 
the system. Berthelot's theorem, known as the principle of 
maximum work, may be stated as follows: " Every chemical proc- 
ess accomplished without the intervention of any external energy 
tends to produce that substance or system of substances which evolves 
the maximum amount of heat. 11 The table of heats of formation 
(p. 296) illustrates the general truth of this principle, but as will 
be seen, the theorem precludes the possibility of spontaneous 
endotherrnic reactions. Thus, for example, the formation of 
acetylene* from its elements at the temperature of the electric 
arc is a well-known endothermic reaction, but according to the 
principle of maximum work, it would not take place spontaneously. 
Another serious objection to Berthelot's principle is, that accord- 
ing to it, all chemical reactions should proceed to completion, the 
reaction taking place in such a way as to evolve the greatest amount 
of heat. As is well known, many reactions, and theoretically all 
reactions, are never complete, but proceed until a condition of 
equilibrium is reached. The principle of maximum work, there- 
fore, denies the existence of equilibria in chemical reactions. 
Many attempts have been made to " explain away " these defects, 
but none of them have been successful. In referring to the generali- 

* Essai de Mecanique Chirmque. 



THERMOCHEMISTRY 309 

zation, Le Chatelier terms it "a very interesting approximation 
toward a strictly valid generalization. " 

^The Theorem of Le Chatelier. As a result of his attempts to 
modify the principle of maximum work and render it generally 
applicable, Le Chatelier was led to the discovery of a rigorous law 
of wide-reaching usefulness. His generalization may be stated 
as follows: Any alteration in the factors which determine an equi- 
librium, causes the equilibrium to become displaced in such a way 
as to oppose, as far as possible, the effect of the alteration. If the 
temperature of a system which is in equilibrium be raised or 
lowered, the resulting displacement of the equilibrium is accom- 
panied by such absorption or evolution of heat as will tend to 
maintain the temperature constant. An interesting illustration 
of the behavior of a system when one of the factors controlling 
the equilibrium is varied, is afforded by the system 

2N0 2 <=N 2 4 . 

The reaction proceeds in the direction indicated by the upper 
arrow with the evolution of 12.6 Cal. Increase of temperature 
favors the reaction which is accompanied by an absorption of 
heat, which in this case, is the reaction indicated by the lower 
arrow. Hence as the temperature rises, the percentage of N0 2 
increases at the expense of N 2 O 4 . This fact can be demonstrated 
by the following experiment: Some liquefied N 2 4 is placed in 
each of three long glass tubes, which are sealed at one end. When 
enough N 2 4 has vaporized to displace the air, the open ends of 
the tubes are sealed. Changes in the equilibrium caused by 
varying the temperature can be followed by noting the changes 
in the color of the mixture. N 2 4 is an almost colorless substance, 
while N0 2 is reddish brown. At ordinary temperatures the 
contents of the tubes will be brown in color. One tube is set 
aside as a standard of comparison, while the temperature of the 
second is lowered by surrounding it with a freezing mixture. As 
the temperature falls, the brown color of the contents of the tube 
becomes much lighter, showing an increased formation of N 2 4 . 
The third tube is heated by immersing it in a beaker of boiling 
water. As the temperature rises, the contents of the tube becomes 



SlO THEORETICAL CHEMISTRY 

much darker in color, indicating an increase in the amount of N(>2 
in the mixture. 

Another example is afforded by the equilibrium between ozone 
and oxygen, represented by the equation 



The reaction indicated by the upper arrow is exothermic. In- 
crease of temperature causes a displacement of the equilibrium 
in the direction of the lower arrow, since under these conditions 
heat is absorbed. Thus, as the temperature rises ozone becomes 
increasingly stable. Nernst has calculated that at 6000 C., 
the temperature of the photosphere of the sun, 10 per cent of the 
above equilibrium mixture would be ozone. Other applications 
of the theorem of Le Chatelier will be given in subsequent 
chapters. 

PROBLEMS. 

1. From the following data calculate the heat of formation of HNO2 
aq. 

[NH 4 N0 2 ] = (N 2 ) + 2 H 2 + 71.77 Gal, 

2 (H 2 ) + (O 2 ) = 2 H 2 + 136.72 Cal, 

(N 2 ) + 3 (H 2 ) + aq. == 2 NH, aq. + 40.64 Cal, 

NH 3 aq. + HN0 2 aq. = NH 4 N0 2 aq. + 9.110 Cal., 

[NH 4 N0 2 ] + aq. = NH 4 N0 2 aq. - 4.75 Cal. 

Ans. (H) + (N) + (0 2 ) + aq. = HN0 2 aq. + 30.77 Cal 

2. By the combustion at constant pressure of 2 grams of hydrogen 
with oxygen to form liquid water at 17 C., 68.36 Cal. are evolved. What 
is the heat evolution at constant volume? Ans. 67.49 Cal. 



3. The heats of solution of NajSO*, Na2S0 4 .H 2 0, and Na2S04.10 H 2 O 
are 0.46, -1.9 and -18.76 Cal. respectively. What are the heats of 
hydration of Na2S0 4 ; (a) to monohydrate, (b) to decahydrate? 

Ans. (a) 2.36 Cal., (b) 19.22 Cal. 

4. The heats of neutralization of NaOH and NH 4 OH by HC1 are 13.68 
and 12.27 Cal. respectively. What is the heat of ionization of NH 4 OH, 
if it is assumed to be practically undissociated? Ans. - 1.41 CaL 



THERMOCHEMISTRY 311 

5. From the following energy equations: 

[C] + (0 2 ) = (CO 2 ) + 96.96 Cal., 

2 (H 2 ) + (O 2 ) - 2 H 2 O + 136.72 Cal., 

2 C e H e + 15 (0 2 ) = 12 (C0 2 ) + 6 H 2 + 1598.7 Cal., 

2 (C 2 H 2 ) + 5 (O 2 ) - 4 (CO;) + 2 H 2 O + 620.1 Cal., 

all at 17 C. and constant pressure, calculate the heat evolved at 17 0. 
in the reaction 



(a) at constant pressure, and (b) at constant volume. 

Ans. (a) 130.8 Cal., (b) 129.07 Cal. 

6. Calculate the heat of formation of sulphur trioxide from the follow- 
ing energy equations: 

[PbO] + [S] + 3 (0) = [PbSOJ + 1655 K. 
[PbO] + H2S0 4 .5 H 2 O = [PbSOJ + 6 H 2 O + 233 K. 
[S] + 3 (O) + 6 H 2 H2S0 4 .5 H 2 + 1422 K. 
[S0 3 ] + 6 H 2 = H2S0 4 .5 H 2 + 411 K. 

Ans. [S] + 3 (0) = [S0 8 ] + 1011 K 

7. What is the heat of formation of a very dilute solution of calcium 
chloride? (See table on p. 307.) Ans. 187.6 Cal. 



CHAPTER XV. 
HOMOGENEOUS EQUILIBRIUM. 

Historical Introduction. In this and the two succeeding 
chapters, the conditions which affect the rate and the extent of 
chemical reactions will be considered. When two substances 
react chemically, it is customary to refer the phenomenon to the 
existence of an attractive force known as chemical affinity. 

Ever since the metaphysical speculations of the Greeks, who 
endowed the atoms with the instincts of love and hate, the nature 
of chemical affinity has been under discussion. So little has been 
learned as to the pause of chemical reactions, that in recent years 
this question has been dismissed and attention has been directed 
to the more promising question as to how they take place. New- 
ton's discovery of the law of gravitation led him to consider the 
attraction between atoms and the attraction between large masses 
of matter as manifestations of the same force. 

Although Newton found that chemical attraction does not 
follow the law of the inverse square, yet his suggestion exerted 
a profound influence upon the minds of his contemporaries. 

Geoffroy and Bergmann arranged chemical substances in the 
order of their displacing power. Thus, if we have three sub- 
stances, A y B, and C and the attraction between A and B is 
greater than that between A and C, then when B is added to AC 
it will completely displace C, as indicated by the following equa- 
tion: AC + B = AB + C. 

These investigators overlooked a factor of fundamental importance 
in conditioning chemical reactivity, viz., the influence of mass. 
The importance of the relative amounts of the reacting substances 
in determining the course of a reaction was first clearly recognized 
by Wenzel * in 1777. It remained for Berthollet,f however, to 

* Lehre von der chemischen Verwandtschaft der Korper. 
t Essai de Statique Chimique. 
312 



HOMOGENEOUS EQUILIBRIUM 313 

point out the significance of the views advanced by Wenzel. 
His first paper on this subject was published in 1799, while acting 
as a scientific adviser to Napoleon on his Egyptian expedition. 
Under ordinary conditions sodium carbonate and calcium chloride 
react according to the equation, 

NasCOs + CaClj = 2NaCl + CaCO 8 , 

the reaction proceeding nearly to completion. Berthollet observed 
the deposits of sodium carbonate on the shores of certain saline 
lakes in Egypt, and pointed out that this salt is produced by the 
reversal of the above reaction, the large excess of sodium chloride 
in solution in the water of the lakes conditioning the course of 
the reaction. 

The German chemist Rose* furnished much additional evidence 
in favor of the effect of mass on chemical reactions. He pointed 
out that in nature, the silicates, which are among the most stable 
compounds known, are undergoing a continual decomposition 
under the influence of such relatively weak agents as water and 
carbon dioxide. The relatively strong specific affinities of the 
atoms of the silicates are overcome by the preponderating masses 
of water and carbon dioxide in the atmosphere. In 1862 an 
important contribution to our knowledge of the effect of mass on 
the course of a chemical reaction was made by Berthelot and Pean 
de St. Gilles.f They investigated the formation of esters from 
alcohols and acids. The reaction between ethyl alcohol and 
acetic acid is represented by the equation 

2 H 5 OH + CHsCOOH <= CH 3 COOC 2 H5 + H 2 0. 

Starting with equivalent quantities of alcohol and acid, the reac- 
tion proceeds until about two-thirds of the reacting substances 
have been converted into ester and water. In like manner, if 
equivalent quantities of ethyl acetate and water are brought 
together, the reaction proceeds in the direction indicated by the 
lower arrow, until about one-third of the original substances have 
been converted into acid and alcohol. In other words the reac- 
tion is reversible, a condition of equilibrium resulting when the 

* Pogg. Ann., 94, 481 (1855); 95, 96, 284, 426 (1855). 

t Ann. Chim. Phys. [3], 65, 385; 66, 5; 68, 225 (1862-1863). 



314 



THEORETICAL CHEMISTRY 



speeds of the two reactions, indicated by the upper and lower 
arrows, become equal. If now a fixed amount of acid is taken, 
say 1 equivalent, and the quantity of alcohol is varied, a corre- 
sponding displacement of the equilibrium follows. 

The following table gives the results obtained by Berthelot and 
P6an de St. Gilles for ethyl alcohol and acetic acid. The first 
and third columns give the number of equivalents of alcohol to 
1 equivalent of acetic acid, and the second and fourth columns 
give the percentage of ester formed. 



Equivalents 
of Alcohol. 


Ester 
Formed. 


Equivalents 
of Alcohol. 


Ester 
Formed. 


0.2 


19 3 


20 


82.8 


0.5 


42 


4.0 


88.2 


1 


66 5 


12.0 


93 2 


1 5 


77.9 


50.0 


100 



The effect of increasing the mass of alcohol on the course of the 
reaction is very beautifully shown by the above results. 

The Law of Mass Action. While the influence of the relative 
masses of the reacting substances in conditioning chemical reac- 
tions was thus fully established, it was not until 1867 that the law 
governing the action of mass was accurately formulated. 

In that year Guldberg and Waage,* two Scandinavian investiga- 
tors, enunciated the law of mass action as follows: The rate, or 
speed, of a chemical reaction is proportional to the active masses of 
the reacting substances present at that time. Guldberg and Waage 
defined the term * 'active mass" as the molecular concentration 
of the reacting substances. It is to be carefully noted that the 
amount of chemical action is not proportional to the actual 
masses of the substances present, but rather to the amounts 
present in unit volume. The law is generally applicable to 
homogeneous systems; that is, to those systems in which ordi- 
nary observation fails to reveal the presence of essentially 
different parts. The amount of chemical action exerted by a 



' Etudes sur les Affinitfe Chimiques, Jour, prakt. Chem. [2], 19, 69 (1879). 



HOMOGENEOUS EQUILIBRIUM 315 

substance can be determined, either from its effect on the equili- 
brium, or from its influence on the speed of reaction. 

In order to apply the law of mass action practically, it must be 
formulated mathematically. Let a and b denote the molecular 
concentrations of the substances initially present in a reversible 
reaction. According to the law of mass action, the rate at which 
these substances combine is proportional to the active masses 
of each constituent, and therefore to their product, ab. The 
initial speed of the reaction at the time to is therefore, 

Speedy QO a6, or Speed/ = k 06, 

in which the proportionality factor fc, is known as the velocity 
constant. As the reaction proceeds, the molecular concentrations 
of the original substances steadily diminish, while the molecular 
concentrations of the products of the reaction steadily increase. 
Let us assume that after the interval of time t, x equivalents of 
the products of the reaction have been formed. The speed of 
the original reaction will now be 

Speed/ = k (a x) (b x). 

As the reaction proceeds, the tendency of the products to combine 
and reform the original substances increases. At the time t, 
when the concentration of the products is x, the speed of the 
reverse reaction will be 

Speed* = ki x 2 , 

where fci is the velocity constant of the reverse reaction. 

We thus have two reactions proceeding in opposite directions: 
the speed of the direct reaction continuously diminishes while 
that of the reverse reaction continually increases. It is evident 
that a point must ultimately be reached at which the speeds of 
the direct and reverse reactions become equal, and a condition 
of dynamic equilibrium will be established. Let Xi represent the 
value of x when equilibrium is attained; we then have 

Speed direct = k (a -Xi) (6 - Xi) = Speed reverse = 

or 

(a Xj) (b Xj) _ fci _ ~ 
xi 2 ~ k ~ A ' 



316 THEORETICAL CHEMISTRY 

in which K is known as the equilibrium constant. Since the veloc- 
ity constants k and k\, are independent of the concentration, it 
follows that the above equation holds for all concentrations. 
Therefore, if the value of the equilibrium constant of a reaction 
is known, the equilibrium conditions can be calculated for any 
concentrations of the reacting substances. When more than one 
mol of a substance is involved in a reaction, each mol must be 
considered separately in the mass action equation. 
Thus let 



represent any reversible reaction, in which n\ mols of A\ and n 2 
mols of At react to form n\ mols of A\ and n^ mols of A?,'. When 
equilibrium is attained, we shall have 



or 

Jii'tfr' I. 

C/ A 'V> t I ... /V 



C A 1 C A Z ' Al 

in which the symbol c is used to denote the active mass or molecular 
concentration of the substances involved in the reaction. This 
is a perfectly general form of the mass-action equation. Since at 
anyone temperature, concentration and pressure are proportional, 
we may write equation (1) in the following form: 



which, in the case of gaseous equilibria, is often a more convenient 
form of the equation. 

The relation between the two equilibrium constants, K c and K p , 

can be easily determined, as follows: Since c =~ ^ p^T 
have, on substituting this valye of c in equation (1), 

\RT) \RTt ' 

BT A 



\RT! ' ' ' 



HOMOGENEOUS EQUILIBRIUM 317 

or indicating the sum of the initial number of mols by Snand the 
sum of the final number of rnols by Sn', we have 
K f = 



It is evident, therefore, that in reactions where the same number 
of mols occur on both sides of the equality sign, K c = K p . Equa- 
tion (1) (or equation (2)) is sometimes known as the reaction 
isotherm. While the law of action may be proved thermody- 
namically, a much simpler kinetic derivation has been given by 
van't Hoff. If we assume that the rate of chemical change is pro- 
portional to the number of collisions per unit of time between the 
molecules of the reacting substances, then in the reaction 

niAi + nsA 2 + . . . = ni'Ai + n 2 'A 2 ' + . . . , 
the velocity of the direct change will be fcc^c^, and the 
velocity of the reverse reaction will be kic^,c^ n ,. . . . 
At equilibrium, the two velocities will be equal and, therefore, 



or 



As a consequence of the assumptions involved in both the thermo- 
dynamic and the kinetic proofs of the law of mass action, it fol- 
lows that the law is only strictly applicable to very dilute solutions. 
Notwithstanding this limitation, experimental results indicate 
that it frequently holds for moderately-concentrated solutions. 

Equilibrium in Homogeneous Gaseous Systems. 

(a) Decomposition of Hydriodic Acid. A typical example of 
equilibrium in a gaseous system is afforded by the decomposition 
of hydriodic acid, as represented by the equation 

Ha + !,<=* 2 HI. 

This reaction has been thoroughly investigated by Hautefeuille, 

Lemoine and Bodenstein.* The reaction is well adapted for 

investigation since it proceeds very slowly at ordinary temper- 

* Zeit. Phys. Chem., 22, 1 (1897). 



318 THEORETICAL CHEMISTRY 

atures, while at the temperature of boiling sulphur, 448 C., 
equilibrium is established quite rapidly. If the mixture of gases 
is maintained at 448 C. for some time and is then cooled quickly, 
the respective concentrations of the components of the mixture 
can be determined by the ordinary methods of chemical analysis. 
Various mixtures of the gases are sealed in glass tubes and heated 
for a definite time in the vapor of boiling sulphur. The tubes 
are then cooled rapidly to the temperature of the room and, after 
the iodine and hydriodic acid have been removed by absorption in 
potassium hydroxide, the amount of free hydrogen present in each 
tube is measured. 
Applying the law of mass action to the above equation, we have 

CH, c/ 2 _ ~ 
^72 -AC. 
^m 

Expressing the analytical results in mols, let a mols of iodine be 
mixed with b mols of hydrogen, and let 2 x mols of hydriodic acid 
be formed. Then when equilibrium is established, a x will be 
the amount of iodine vapor and b x will be the amount of hydro- 
gen present. The concentrations being directly proportional to 
the amounts present, we may substitute these values for c// 2 , c/ 2 , 
and CHI in the mass-action equation. The following expression 

is thus obtained: 

(b - x) (a - x) 
- - 



Solving the equation for x, we obtain 

a +b - Va? + b*-ab(2- 16 J 



= - __ -- 

Since, according to Avogadro's law, equal volumes of all gases 
contain the same number of molecules, volumes may be sub- 
stituted for a, b y and x. Bodenstein expressed his results in terms 
of volumes reduced to standard conditions of temperature and 
pressure. On analyzing equilibrium mixtures, Bodenstein found 
that at 448 C., K c 0.01984, and at 350 C., K e = 0.01494. 

Having determined the value of the equilibrium constant, he 
made use of this value in calculating the volume of hydriodic 



HOMOGENEOUS EQUILIBRIUM 



319 



acid which should be obtained from known volumes of hydrogen 
and iodine. A comparison of the calculated and observed values 
showed excellent agreement. The following table contains a few 
of the results obtained by Bodenstein at 448 C. 



Hydrogen, 
6. 


Iodine, 

0. 


HI calculated, 
2z. 


HI observed, 
2x. 


20.57 


5.22 


10 19 


10 22 


20 60 


14 45 


25 54 


25 72 


15 75 


11 90 


20 65 


20 70 


14.47 


38.93 


27 77 


27 64 


8.10 


2 94 


5 64 


5 66 


8 07 


9 27 


13 47 


13 34 



It is of interest to note that a change in pressure does not 
alter the equilibrium in this gaseous system. Making use of the 
partial pressures of the components of the gaseous system instead 
of the concentrations, we have 



PH, Pi 2 
PHI 



K p . 



Now let the total pressure on the system be increased to n times 
its original value; then the partial pressures are all increased in 
the same proportion, and we have 



K, 



which is equivalent to the original expression, since n cancels 
out. The equilibrium is thus seen to be independent of the pres- 
sure. This is only true for those systems in which a change in 
volume does not occur. 

(6) Dissociation of Phosphorus Pentachloride. When phos- 
phorus pentachloride is vaporized it dissociates according to the 
following equation 



Applying the law of mass action, we have 



320 THEORETICAL CHEMISTRY 

Starting with 1 mol of phosphorus pentachloride, which if undis- 
sociated would occupy the volume V, under atmospheric pressure, 
and letting a denote the degree of dissociation, the molecular con- 
centrations at equilibrium will be as follows: 






Letting (1 + a) V = V, and substituting in the above equation, 
we have 




(1 - a) F 



At 250 C. phosphorus pentachloride is dissociated to the extent 
of 80 per cent. Under atmospheric pressure 1 mol will be present 

273 I 250 
in 22.4 ^ liters = V. The final volume will, therefore, be 

V = (1 + 0.8) (22 A 



.8) (22. 



The value of the equilibrium constant usually designated in 
cases of dissociation, the dissociation constant is, therefore, 

_ (0.8)' 

"-- - - - - 



(1 - 0.8) (1 + 0.8) ( 22.4 273 + 3 250 ) 



Having obtained the value of K e , the direction and extent of the 
reaction at 250 C. can be determined, provided the initial molec- 
ular concentrations are known. The reaction is accompanied by 
a change in volume, and, therefore, the equilibrium is displaced by 
a change in pressure. Making use of the partial pressures of the 
components of the gaseous mixture, we have 



where p\ and p 2 are the partial pressures of phosphorus penta- 
chloride and the products of the dissociation, phosphorus tri- 



HOMOGENEOUS EQUILIBRIUM 321 

chloride and chlorine, respectively. Let the total pressure be 
increased n-times, then 



npi pi 

It is apparent from this equation, that the equilibrium is not 
independent of the pressure, an increase in pressure being accom- 
panied by a diminution of the dissociation. An important point 
in connection with dissociation, first observed by Deville,* is the 
effect on the equilibrium of the addition of an excess of one of the 
products of dissociation. For example, in the equilibrium 



an excess of chlorine or of phosphorus trichloride, drives back the 
dissociation. If p\ denotes the partial pressure of phosphorus 
pentaehloride, p z that of phosphorus trichloride, and p$ that of 
chlorine, then we have 



Now let an excess of chlorine be added; this will cause the value 
of PC to increase. Since the value of K p is constant, the value of 
p z must diminish and that of pi must increase. Hence, the addi- 
tion of an excess of either product of dissociation causes a diminu- 
tion of the amount of the dissociation. 

(c) Dissociation of Carbon Dioxide. Carbon dioxide dissociates 
according to the equation, 



This is a somewhat more complex gaseous system than either 01 
the foregoing systems. When equilibrium is established, let 
p L be the partial pressure of the carbon dioxide, p^ the partial pres- 
sure of carbon monoxide, and p 3 the partial pressure of oxygen, 
then we have 



_ 

- 5 Jt. 
Pi 2 

At 3000 C. and under atmospheric pressure, carbon dioxide is 
* Logons sur la dissociation, Paris (1866). 



322 THEORETICAL CHEMISTRY 

40 per cent dissociated. The partial pressures of each of the com- 
ponents may be readily calculated as follows : 

- 2(1-0.40) 

Pl 2 (1 - 0.40) + 3 X 0.40 ' 

_ _ 2 X 0.40 _ 

** 2 (1 - 0.40) + 3 X 0.40 ' 

' 40 -0.17. 



^ 2 (1 - 0.40) + 3 X 0.40 
Substituting these values in the above equation, we obtain 
_ (0.33)' X 0.17 



V -- (050)5 

The dissociation constant for carbon dioxide may have a different 
value if the equation is written in the form 



Applying the law of mass action, we have 



Substituting the above values of the partial pressures, we obtain 

K p = 0.272. 

Equilibrium in Liquid Systems. The reaction between an 
alcohol and an acid to form an ester and water may be taken as 
an example of equilibrium in a liquid system. In the reaction 

C 2 H 5 OH + CH 3 COOH ^CH 3 COOC 2 H 5 + H 2 0, 
let a, 6, and c represent the number of mols of alcohol, acid and 
water respectively, which are present in V liters of the mixture, 
and let x denote the number of mols of ester and water which 
have been formed when the system has reached equilibrium. 

The active masses of the components will then be, 

r - a ~~ x r - *>- x t r x . , r _ c + x 

^alc. -- y > v/acid -- y > ^ester ~ y > Jia U water y 

Applying the law of mass action, we obtain 

x (c + x) ~ 

(a -*)(&-*) "^ 



HOMOGENEOUS EQUILIBRIUM 



323 



In this case the value of the equilibrium constant is independent 
of the volume. This reaction has been studied, as already men- 
tioned, by Berthelot and Pan de St. Gilles.* They found that 
when equivalent amounts of alcohol and acid are mixed, the reac- 
tion proceeds until two-thirds of the mixture is changed into ester 
and water. Hence, we find 



Having determined the value of K c , it may now be used to cal- 
culate the equilibrium conditions for any initial concentrations 
of the substances involved in the reaction. As an illustration, 
we will take 1 mol of acetic acid and treat it with varying amounts 
of alcohol, the initial mixture containing neither of the products of 
the reaction. The equation takes the form 

2 

*' 



(a - s) (1 - x) 
Solving for x, we have 

x = | (1 + a - Va 2 - a + 1). 

A comparison of the observed and calculated values given in the 
accompanying table shows that the agreement is excellent, even 
in the more concentrated solutions, where we might reasonably 
expect that the mass law would cease to hold. 



Alcohol, 

0. 


Ester 
(observed), 
x. 


Ester 
(Calculated), 
x. 


0.05 


05 


049 


OS 


078 


078 


18 


171 


0.171 


28 


226 


232 


33 


293 


311 


0.50 


414 


523 


67 


519 


528 


1.0 


0.665 


0.667 


1 5 


819 


0.785 


2.0 


0.858 


0.845 


2 24 


0.876 


864 


8 


0.966 


0.945 



* Loc, cit. 



324 THEORETICAL CHEMISTR^ 

The Variation of the Equilibrium Constant with Temperature. 

van't Hoff showed that the displacement of equilibrium due to 
change in temperature is connected with the heat evolved or ab- 
sorbed in a chemical reaction by the equations 



and 

d(log.K p ) - 



tSL = 5?" d\ 

jrp T>Hnz ' \ A / 



(2) 



dT RT 2 ' 



where Q v and Q p are the heats of reaction at Constant volume and 
constant pressure respectively, and where R and T have their 
usual significance. Equation (1) is known as the reaction isochore. 
Both equations show that the rate of change of the natural log- 
arithm of the equilibrium constant with temperature is equal to 
the total heat of reaction divided by the molecular gas constant 
times the square of the absolute temperature at which the 
reaction takes place. Equations (1) arid (2) hold only for displace- 
ments of the equilibrium due to infinitely small changes in temper- 
ature. In order to render these equations applicable to concrete 
equilibria, it is necessary to integrate them. The integration of 
these expressions can only be performed if Q is constant. For 
small intervals of temperature, Q is practically independent of 
the temperature, and for larger intervals we may take the value 
of Q which corresponds to the mean of the two temperatures 
between which the integration is performed. Integrating equa- 
tions (1) and (2) on this assumption, we obtain 



log.1^ - log. X C| = jr ~ r, (3) 

and 

(4) 



Passing to Briggsian logarithms, and putting R = 1.99 calories, 
equations (3) and (4) become 

(5) 



HOMOGENEOUS EQUILIBRIUM 325 

and 

(6) 



We shall now proceed to show how these important equations 
may be applied to several typical, equilibria. 

(a) Vaporization of Water. The equilibrium between a liquid 
and its vapor is conditioned by the pressure of the vapor, 
this in turn being dependent upon the temperature. In this case 
of physical equilibrium, we have K pi = pi, and K P<1 = p 2 . The 
value of Q p for water can be calculated from the following data: 

Ti = 273, pi = 4.54 mm. of mercury, 

T* = 273 + 1T.54, p 2 = 10.02 mm. of mercury. 

Substituting in equation (6), we have 

4.581 (log 10.02 - log 4.54) 273 X 284.5 
y " 284.5 - 273 

or -Q P = -10,670 calories. 

The value of Q p obtained by experiment is 10,854 calories. 

(b) Dissociation of Nitrogen Tetroxide. In the reaction 

N 2 4 <& 2 NO 2 , 

the following values for the dissociation of N 2 4 have been ob- 

tained: 

r 1= =273+ 26.l, ! = 0.1986, 

T 2 = 273 + 111.3, 2 = 0.9267. 

If the dissociation takes place under a pressure of 1 atmosphere, 
then the partial pressures of the component gases will be 

Pw ' = l- 1 ~+2' and P^i-a + 2*- 

The values of K PI and K PI are, then, according to the law of 
mass action as follows: 



326 THEORETICAL CHEMISTRY 

and 






Substituting in equation (6) and solving for Q p , we obtain 

A C oi fi 4 X (0.9267) 2 , 4 X (0.1986) 2 ] OOQ , _ 
4.58l|log nrjp^-Wf^^ 

y * 384.3 - 299.1 

or 

Q P = 12,260 calories per mol of N 2 O 4 . 

In a reaction which is accompanied by no thermal change, 
Q = 0, and the right-hand side of equations (1) and (2) becomes 
equal to zero. In other words, in such a reaction a change in 
temperature does not cause a displacement of the equilibrium. 

The reaction, 

C 2 H 5 OH + CH 3 COOH * CH 3 COO.C 2 H 5 + H 2 0, 

is accompanied by such a small thermal change that it may be 
considered as zero, and according to the above reasoning there 
should be only a very slight displacement of the equilibrium when 
the temperature is varied. Berthelot found that at 10 C., 
65.2 per cent of the alcohol and acid are changed into ester, and 
at 220 C., 66.5 per cent of the mixture is transformed into ester. 
As will be seen, an increase of 210 produces hardly any displace- 
ment of the equilibrium. 

PROBLEMS. 

1. When 2.94 mols of iodine and 8.10 mols of hydrogen are heated at 
constant volume at 444 C. until equilibrium is established, 5.64 mols 
of hydriodic acid are formed. If we start with 5.30 mols of iodine and 
7.94 mols of hydrogen, how much hydriodic acid is present at equilibrium 
at the same temperature? Ans. 9.49 mols. 

2. At 2000 C., and under atmospheric pressure, carbon dioxide is 
1.80 per cent dissociated according to the equation 



Calculate the equilibrium constant for the above reaction using partial 
pressures. Ans. 3 X 10~ 6 . 



HOMOGENEOUS EQUILIBRIUM 327 

3. What is the equilibrium constant in the preceding problem, if the 
concentrations are expressed in mols per liter? Ans. 1.61 X 10~ 8 . 

4. When 6.63 mols of amylene and 1 mol of acetic acid are mixed, 
0.838 mol of ester is formed in the total volume of 894 liters. How much 
ester will be formed when we start with 4.48 rnols of amylene and 1 mol 
of acetic acid in the volume of 683 liters? Ans. 0.8111 mol. 

5. If 1 mol of acetic acid and 1 mol of ethyl alcohol are mixed, the 
reaction 

C 2 H 6 OH + CHaCOOH ? CH 3 COOC 2 H 5 + H 2 0, 

proceeds until equilibrium is reached, when -J- mol of ethyl alcohol, J mol 
of acetic acid, J mol of ethyl acetate, and f mol of water are present. If 
we start (a) with 1 mol of acid and 2 mols of alcohol; (b) with 1 mol of 
acid, 1 mol of alcohol, and 1 mol of water; (c) with 1 mol of ester and 3 
mols of water, how much ester will be present in each case at equilibrium? 
Ans. (a) 0.845 mol, (b) 0.543 mol, (c) 0.465 mol. 

6. In the reaction 



we find, since J O = I 2 , for 



__ 9 
PHCI ' V p 

the values 3.02 at 386 C. and 2.35 at 419 C. Calculate the heat evolved 
by the reaction under constant pressure. Ans. 6827 cal. 

7. Above 150 C. N0 2 begins to dissociate according to the equation 
N0 2 ^NO+|0 2 . 

At 390 C. the vapor density of N0 2 is 19.57 (H = 1), and at 490 C. 
it is 18.04. Calculate the degree of dissociation according to the above 
equation at each of these temperatures; the equilibrium constants 
expressing the concentrations in mols per liter; and the heat of dissocia- 

tion of N0 2 . 

Ans. on = 0,35, <* 2 = 0.55, Ki = 2.884 X 10~ 2 . 

K 2 = 7.173 X 10-*. Q -9407 cal. 



CHAPTER XVI. 
HETEROGENEOUS EQUILIBRIUM. 

Heterogeneous Systems. We have now to consider equilibria 
in systems made up of matter in different states of aggregation. 
Such systems are termed heterogeneous systems, as distinguished 
from those dealt with in the preceding chapter where the compo- 
sition is uniform throughout. The physically distinct portions of 
matter involved in a heterogeneous system are known as phases, 
each phase being homogeneous and separated from the other 
phases by definite bounding surfaces. Thus, ice, liquid water 
and vapor constitute a physically heterogeneous system. Another 
heterogeneous system is formed by calcium carbonate and its 
dissociation products, calcium oxide and carbon dioxide. The 
equilibrium between a solid, its saturated solution, and vapor 
affords an illustration of a still more complex heterogeneous 
system. 

Application of the Law of Mass Action to Heterogeneous 
Equilibria. It has been shown in the preceding chapter that 
the law of mass action may be applied to homogeneous equilibria 
provided the molecular condition of the reacting substances is 
known. 

When we attempt to apply the law of mass action to hetero- 
geneous equilibria, especially where solids are involved, the 
problem presents difficulties. In his investigation of the dis- 
sociation of calcium carbonate, according to the equation 



Debray * showed that just as every liquid has a definite vapor 
pressure corresponding to a certain temperature, so there is a 
definite pressure of carbon dioxide over calcium carbonate at a 
definite temperature. Furthermore, the pressure was found to 
be independent of the amount of calcium carbonate present. 

* Compt. rend., 64, 603 (1867). 
328 



HETEROGENEOUS EQUILIBRIUM 329 

Guldberg and Waage * showed that the law of mass action can 
be applied to such heterogeneous equilibria, provided that the 
active masses of the solids present are considered as constant. 

Nernst pointed out that this statement of Guldberg and Waage 
can be easily reconciled with experimental facts. In a hetero- 
geneous equilibrium involving solids, it is only necessary to con- 
sider the gaseous phase, the active mass of a solid being equivalent 
to its concentration in the gaseous phase. That is, every solid 
is to be looked upon as possessing, at a definite temperature, a defi- 
nite vapor pressure which is entirely independent of the amount of 
solid present. Such substances as arsenic, antimony, and cadmium 
are known to have appreciable vapor pressures at relatively low 
temperatures, and it is quite reasonable to suppose that every 
solid substance exerts a definite vapor pressure at a definite temper- 
ature, even though we have no method sufficiently refined to meas- 
ure such minute pressures. 

Since the active mass of a solid remains constant so long as 
any of it is present, the application of the law of mass action 
to certain heterogeneous equilibria is, in general, simpler than 
its application to homogeneous systems. The truth of this state- 
ment will be evident after a few typical heterogeneous systems 
have been considered. 

(a) Dissociation of Calcium Carbonate. In the reaction 



let TI and 7r 2 represent the pressures due to the vapor of calcium 
carbonate and calcium oxide respectively, and let p denote the 
pressure of the carbon dioxide. Applying the law of mass action, 
we obtain 



But since TI and x 2 are constant at any one temperature, the 
equation becomes 

P - K 9 ; 

or, the equilibrium constant at any one temperature is solely 
dependent upon the pressure of the carbon dioxide evolved. The 

* Loc. cit 



330 



THEORETICAL CHEMISTRY 



accompanying table gives the values of the pressure of carbon 
dioxide corresponding to various temperatures. 



Temperature, 
Degrees. 


Pressure in 
Millimeters of 
Mercury. 


547 


27 


610 


46 


625 


56 


740 


255 


745 


289 


810 


678 


812 


753 


865 


1333 



(b) Dissociation of Ammonium Hydrosulphide. When solid 
ammonium hydrosulphide is heated, it is almost completely dis- 
sociated into ammonia and hydrogen sulphide as shown by the 
following equation: 

[NH 4 HS]^(NH 3 ) 



This reaction was investigated by Isambert,* who found that the 
total gas pressure at 25. 1 C. is equal to 501 mm. of mercury. 
Since the partial pressures of the ammonia and hydrogen sulphide 
are necessarily the same, each must be approximately equal to 
250.5 mm., the relatively small pressure due to the undissociated 
vapor of the ammonium hydrosulphide being neglected. Let IT 
be the partial pressure of the vapor of ammonium hydrosulphide, 
and let pi and p^ be the partial pressures of the ammonia and 
hydrogen sulphide. Applying the law of mass action, we have 

Pi'p* _ v m 

- i^p. {i) 

Since ?r is constant at any one temperature, equation (1) becomes 

pi p 2 = K P '. 
According to Dalton's law of partial pressures, we have 

P = Pi + Pz + *, 
* Compt. rend., 93, 595, 730 (1881). 



HETEROGENEOUS EQUILIBRIUM 



331 



where P is the total pressure. Neglecting the relatively small 
pressure ?r, we may write 

P = Pi + ft. 
HencCj since pi = p%, 

p 

2 = Pi = Pa- 
Substituting these values in equation (1), we obtain 

P 2 _ , _ (501 
T -A P -- T 



^ = IT.' = Ml! = 62,750. 



The value of the equilibrium constant may be checked by observ- 
ing the effect on the system of the addition of an excess of either 
one of the products of the dissociation. The accompanying table 
gives the results of a few of Isambert's experiments. 



Pressure of 
Ammonia. 


Pressure of 
Hydrogen 
Sulphide. 


PNH 3 -PH 2 S = K p '. 


208 
138 
417 
453 


294 
458 
146 
143 


61,152 
63,204 
60,882 
64,779 


Mean 62,504 



As will be seen, the mean value of the equilibrium constant agrees 
well with the value found for equivalent amounts of the products 
of dissociation. 

(c) Dissociation of Ammonium Carbamate. The dissociation of 
ammonium carbamate takes place according to the equation 
ONH 4 



OC/ 
X 



NH 2 

This dissociation has been investigated by Horstmann.* Applying 
the law of mass action, we have 

' (2) 



* Lieb. Ann., 187, 48 (1877). 



332 THEORETICAL CHEMISTRY 

where pi and p 2 are the partial pressures of ammonia and carbon 
dioxide respactively, and where TT is the partial pressure of ammo- 
nium carbamate. Since T is constant, equation (2) becomes 



If P denotes the total gaseous pressure, and TT is neglected as in 
the preceding example, we have, since three mols of gas are 
formed 

9 4P 2 , P 

Pi ? = and p 2 = -3 

Substituting these values in equation (2), we have 



27 ~ j " 

This equation has also been tested by Isambert * by adding an 
excess of ammonia or carbon dioxide to the dissociating system. 
He found that the value of the equilibrium constant remains 
practically constant. The addition of a foreign gas was shown 
to be without effect on the dissociation. 

(d) Dissociation of the Hydrates of Copper Sulphate. Many inter- 
esting examples of heterogeneous equilibrium are furnished by hy- 
drated salts. Thus, if crystallized copper sulphate, CuS04.5 H 2 0, 
is placed in a desiccator, it gradually loses water of crystallization 
and ultimately only the anhydrous salt remains. If the desiccator 
be provided with a manometer and is so arranged that the tem- 
perature can be maintained constant, it is possible to observe the 
changes in vapor pressure accompanying the process of dehydra- 
tion. At the temperature of 50 C., the pressure over completely 
hydrated copper sulphate is found to remain constant at 47 mm. 
until the salt has been deprived of two molecules of water, when 
it drops abruptly to 30 mm. and remains constant until two more 
molecules of water have been lost. It then drops again to 4.4 mm. 
and remains constant until dehydration is complete. 

The successive stages of the dehydration are shown in the accom- 
panying diagram, Fig. 76. The constant pressures observed in 
the dehydration correspond to the successive equilibria involved. 

* Loc. cit. 



HETEROGENEOUS EQUILIBRIUM 



333 



At 50 C. the pentahydrate and the trihydrate are in equilibrium, 
a pressure of 47 mm. being maintained so long as any of the penta- 
hydrate is present. When all of the pentahydrate is used up, 
then the trihydrate begins to undergo dehydration into the 
monohydrate. This is a new equilibrium and the pressure of the 



47mm 



80 mm 



6H 2 O 



3H S O 
Composition 

Fig. 76. 



OH 2 O 



aqueous vapor necessarily changes, and remains constant so long 
as any trihydrate remains. The last stage corresponds to the 
equilibrium between the monohydrate and the anhydrous salt. 
The following equations represent the three successive equilibria: 

(1) CuS0 4 5 H 2 ^ CuS0 4 3 H 2 + 2 H 2 0, 

(2) CuS0 4 3 H 2 * CuS0 4 H 2 O + 2 H 2 O, 
(3) 



Applying the law of mass action to the first of the above equi- 
libria, we have 



tr\ 



in which ir\ and T 2 denote the partial pressures due to the hydrates 
CuS0 4 .5 H 2 O and CuSOi.3 H 2 O respectively, and p denotes the 



334 



THEORETICAL CHEMISTRY 



pressure of aqueous vapor. Since TTI and 7r 2 are constant, the 
above expression simplifies to the following 



In a similar manner it may be shown that the pressure of aqueous 
vapor in the other equilibria must be constant. It must be 
clearly understood that the observed pressure is only definite 
and fixed when two hydrates are present. If the dehydration 




Tee 



Temperature 

Fig. 77. 

were conducted at another temperature than 50 C. the equilibrium 
pressure would be different. The vapor pressure curves of the 
different hydrates are shown in the temperature-pressure diagram 
of Fig. 77. 

Heat of Dissociation of Solids. When the products of the 
dissociation of a solid are gaseous, it has been pointed out by De 
Forcrand * that the ratio of the heat of dissociation of 1 mol of 

* Ann. Chim. Phys. [7L 28, 545. 



HETEROGENEOUS EQUILIBRIUM 



335 



solid to the absolute temperature at which the dissociation pres- 
sure is equal to 1 atmosphere, is constant. Or, denoting the heat 
of dissociation by Q and the absolute temperature by T y De For- 
crand's relation may be expressed thus, 

~ = constant = 33. 

Nemst has shown that the value of the constant in this relation 
is not independent of the temperature. Thus, the value of the 
ratio at 100 C. is 29.7, while at 1000 C. it is 37.7. Up to the 
present time no expression has been derived in which the variation 
of the ratio with the temperature is included. 

Distribution of a Solute between Two Immiscible Solvents. 
When an aqueous solution of succinic acid is shaken with ether, 
the acid distributes itself between the ether and the water in such 
a way that the ratio between the two concentrations is always 
constant. It will be seen that the distribution of the succinic 
acid between the two solvents is analogous to that of a substance 
between the liquid and gaseous phases (see page 170), and there- 
fore the laws governing the latter equilibrium should apply equally 
to the former. Nernst * has shown that (a) // the molecular 
weight of the solute is the same in both solvents, the ratio in which it 
distributes itself between them is constant at constant temperature, 
or in other words, Henry's law is applicable; and (b) If there are 
several solutes in solution the distribution of each solute is the same as 
if it were present alone. This is clearly Dalton's law of partial 
pressures. The ratio in which the solute distributes itself between 
the two solvents is termed the coefficient of distribution or partition. 
The following table gives the results of three experiments on the 
distribution of succinic acid between ether and water. 



Concentration 
in Water. 


Concentration 
in Ether. 


Distribution 
Coefficient. 


43.4 

43. 8 
47.4 


7.1 
7.4 
7.9 


6.1 
5.9 
6.0 



Zeit. phys. Chem., 8, 110 (1891). 



336 



THEORETICAL CHEMISTRY 



As will be seen the distribution coefficient is constant, show- 
ing that Henry's law applies. When the molecular weight of a 
solute is not the same in both solvents the distribution coefficient 
is not constant, and conversely, if the distribution coefficient is 
not constant, we infer that the molecular weights of the solute 
in the two solvents are not identical. 

Let us assume that a solute whose normal molecular weight is 
A, when shaken with two immiscible solvents undergoes polymeri- 
zation in one of them, its molecular weight being A n . We then 
have the equilibrium 



applying the law of mass action, we have 

CA n rr 

= A c . 

C A 

If the molecular weight in one solvent is twice the molecular 
weight in the other, then n = 2, and 



Ci z 

or 



r= = constant. 



Thus Nernst found the following concentrations of benzoic acid 
when it was shaken with benzene and water. 



c, (Water). 


c 2 (Benzene). 


C 2 


C|. 


0150 
0.0195 
0.0289 


242 
0.412 
970 


0.062 
048 
0.030 


0305 
0304 
0293 



As will be seen, the values of the ratio c\/c^ steadily decrease, 
while on the other hand, the values of the ratio Ci/V^ remain 
constant. This shows, therefore, that benzoic acid has twice the 
normal molecular weight in benzene. 

The Solution of a Solid in a Non-dissociating Solvent When 
a solid is brought in contact with a non-dissociating solvent, it 
continues to dissolve until the solution becomes saturated. A 
condition of equilibrium then obtains, the rates of solution and 



HETEROGENEOUS EQUILIBRIUM 337 

precipitation being the same. This is plainly a case of hetero- 
geneous equilibrium. If c is the concentration of the dissolved 
substance, and TT is the concentration of the undissolved solid, then 
according to the law of mass action 

--K 

* 

or since T is constant, 

c = K c '. 

Variation of the Constant of Heterogeneous Equilibrium with 
Temperature. The reaction isochore equation of van't Hoff 



dT 

which has been shown to connect the displacement of a homo- 
geneous equilibrium with change in temperature, applies equally 
well to heterogeneous equilibria. The following examples will 
serve to illustrate its application in such cases. 

(a) Dissociation of Ammonium Hydrosulphide. In the reaction 
representing the dissociation of ammonium hydrosulphide, 



let pi and p 2 be the partial pressures of ammonia and hydrogen 
sulphide, and let w be the partial pressure of ammonium hydro- 
sulphide. Then as has been shown (see page 331), 

Jf/- T , 

where P is the total gaseous pressure. From the following data : 

Ti = 273 + 9.5, Pi = 175 mm. of mercury, 
and 

T z = 273 + 25. 1, P 2 = 501 mm. of mercury, 

we have, on applying the reaction isochore equation, and solving 
forQ p , 

298.1 



" 298.1 - 282.5 

or Q p = - 22,740 calories. 



338 THEORETICAL CHEMISTRY 

This result agrees well with the value, -22,800 calories, found 
by direct experiment. 

(b) Solution of Sucdnic Add. The concentration of succinic 
acid (in a saturated solution) and the temperature, are the factors 
which determine the equilibrium in this case. In the equation 

d(\Og e K c ) = -Q v 

dT RT 2 ' 

K c = c, where c is the concentration of succinic acid in a saturated 
solution. The following experimental data, due to van't Hoff, 
enables us to calculate the heat of solution of the acid. 

T! = 273 c = 2.88 grams per 100 grams of water, 

and 

T 2 = 273 + 8.5, c = 4.22 grams per 100 grams of water. 

Substituting in the reaction isochore equation and solving for Q c , 
we have 

_ 4.581 (log 4.22 - log 2.88) 273^X 281.5 

y " " 281.5 - 273 

or 

Q v = - 6871 calories. 

The value of the heat of solution for 1 mol of succinic acid as 
found by direct experiment is 6700 calories. 

The Phase Rule. While it is possible to apply the law of 
mass action to certain heterogeneous equilibria there are numerous 
cases where its application is either difficult or impossible. To 
deal with such heterogeneous systems we make use of a general- 
ization discovered by J. Willard Gibbs,* late professor of mathe- 
matical physics in Yale University. This generalization was first 
stated by Gibbs in 1874, and is commonly known as the phase rtile. 
Before entering upon a discussion of the phase rule, it will be 
necessary to define a few of the terms employed. 

The composition of a system is determined by the number of 
independent variables or components involved. Thus in the 
system ice, water, and vapor there is but a single com- 
ponent. In the system 



Trans. Connecticut Academy, Vols. II and III, 1875-8. 



HETEROGENEOUS EQUILIBRIUM 339 

while there are three constituents of the equilibrium, only two of 
these need, be considered as components, for the amount of any 
one constituent is not independent of the amounts of the other 
two, as the following equations show: 

CaO + CO 2 = CaCO 3 , 
CaC0 3 - CaO = C0 2 , 
CaC0 3 - CO 2 = CaO. 

In general, the components are chosen from the smallest number 
of independently-variable constituents required to express the 
composition of each phase entering into the equilibrium, even 
negative quantities of the components being permissible. 

The number of variable factors, temperature, pressure, and 
concentration, of the components which must be arbitrarily fixed 
in order to define the condition of the system, is known as the 
degree of freedom of the system. For example, a gas has two 
degrees of freedom since two of the variables, temperature, pres- 
sure or volume, must be fixed in order, to define it; a liquid and its 
vapor has only one degree of freedom, since for equilibrium at a 
certain temperature, there can be but a single pressure; and in a 
system consisting of a substance in the three states of aggregation 
equilibrium can only exist at a single temperature and pressure. 

Derivation of the Phase Rule. The following derivation of 
the phase rule is due to Nernst. Let us assume a complete hetero- 
geneous equilibrium made up of y phases of n components, and let 
us fix our attention upon one single phase. This phase will con- 
tain a certain amount of each one of the n components, the con- 
centrations of which may be designated by Ci, C2, c 3 , . . . c n . 
Since we have assumed complete equilibrium to exist, the slightest 
change in concentration, temperature or pressure will alter the 
composition of this phase. 

This may be expressed by the equation 

/ (ci, c 2 , CB, . . . c n , p, T) = 0, 

where / is any function of the variables. Since any change in 
one phase implies a corresponding change in the remaining y 1 
phases, it follows that the composition of all the phases is a certain 
determined function of the same variables. 



340 THEORETICAL CHEMISTRY 

The above equation is, then, of the form ascribed to each sepa- 
rate phase, and since there are y phases we have y separate equa- 
tions. There are, however, n + 2 variables in each equation, so 
that if y = n + 2, that is if we have two more phases than com- 
ponents, each unknown quantity has a definite known value. 
In this case there is only one value for ci, C2, c 3 , c 4 , . . . c n , p and 
T at which the system can be in equilibrium. Hence when n 
components are present in n + 2 phases, we have equilibrium only 
for a certain temperature, a certain pressure, and a certain ratio 
of concentrations of the single phases. That is, n + 2 phases of n 
substances can only exist at a certain point in a coordinate 
system. This point is termed the transition point. If one value 
be altered then one phase vanishes, and there remain n + 1 phases 
of n components, and the problem becomes indeterminate. Thus 
it is proved that n components are necessary in order that a system 
containing n + 1 phases may exist in complete equilibrium. 

The phase rule may be stated as follows: A system made up 
of n components in n + 2 phases can only exist when pressure, 
temperature and concentration have definite fixed values; a system 
of n components in n + 1 phases can exist only so long as one of the 
factors varies; and a system of n components in n phases can exist 
only so long as two of the factors vary. If P denotes the number of 
phases, C the number of components, and F the number of degrees 
of freedom, then the phase rule may be conveniently summarized 
by the expression, 



Equilibrium in the System, Water, Ice, and Vapor. In this 
system we may have one, two, or three phases present, according 
to the conditions. Under ordinary circumstances of temper- 
ature and pressure, water and water vapor are in equilibrium. 
The vapor pressure curve of water is represented by the line OA 
in the pressure-temperature diagram (Fig. 78). It is only at 
points on this curve that water and its vapor are in equilibrium 
Thus, if the pressure be reduced below that corresponding to any 
point on OA, all of the water will be vaporized; if, on the other 
hand, the pressure be raised above the curve, all of the vapor will 



HETEROGENEOUS EQUILIBRIUM 



341 



ultimately condense to the liquid state. When the temperature 
is reduced below C., only ice and vapor are present, the curve 
OC representing the equilibrium between these two phases. It is 
to be observed that the curve OC is not continuous with OA. At 




0:0075 
Temperature 
Fig. 78. 

the point 0, where the two curves intersect, ice, water, and water 
vapor are in equilibrium. At this point ice and water must have 
the same vapor pressure, otherwise distillation of vapor from the 
phase having the higher vapor pressure to that with the lower 
vapor pressure would occur, and eventually the phase having the 
higher vapor pressure would disappear. This result would be 
in contradiction to the experimentally-determined fact that 
both solid and liquid phases are in equilibrium at the point 0. 
The temperature at which ice and water are in equilibrium with 
their vapor under atmospheric pressure is C. Since increase 
of pressure lowers the freezing-point of water, the point 0, repre- 
senting the equilibrium between ice and water under the pressure 
of their own vapor, viz., 4.57 mm., must be a little above C. 
The exact temperature corresponding to the point has been 
found to be O.0075 C. 



342 THEORETICAL CHEMISTRY 

The change in the melting-point of ice due to increasing pressure 
is represented by the line OB. This line is inclined toward the 
vertical axis because the melting-point of ice is lowered by in- 
creased pressure. The point is called a triple point because 
there, and there only, three phases are in equilibrium. As is well 
known, water does not always freeze exactly at C. If the 
containing vessel is perfectly clean, and care is taken to exclude 
dust, it is possible to supercool water several degrees below its 
freezing-point and measure its vapor pressure. 

The dotted curve OA', which is a continuation of OA, represents 
the vapor pressure of supercooled water. It will be noticed that 
(1) there is no break in the vapor-pressure curve so long as the 
solid phase does not separate, and (2) the vapor pressure of super- 
cooled water, which is an unstable phase, is greater than that of 
ice, the stable phase, at that temperature. 

We now proceed to apply the phase rule to this system. In the 
formula, C - P + 2 = F, C = 1. It is evident that if P = 3, 
then F = 0; or the system has no degree of freedom. We have 
seen that the triple point represents such a condition. At this 
point ice, water, and water vapor are co-existent, and if either 
one of the variables, temperature or pressure, is altered, one of 
the phases disappears; in other words, the system has no degree 
of freedom. Such a system is said to be non-variant. If in the 
above formula, P = 2, then F = 1, and the system has one degree 
of freedom, or is univariant. Any point on any one of the curves 
OAj OBj or OC represents a univariant system. Take, for exam- 
ple, a point on the curve OA . In this case the temperature may 
be altered without altering the number of phases in equilibrium. 
If the temperature is raised, a corresponding increase in vapor 
pressure follows and the system will adjust itself to some other 
point on the curve OA. In like manner, the pressure may be 
altered without causing the disappearance of one of the phases. 
If, however, the temperature is maintained constant, then a change 
in the pressure will cause either condensation of water vapor or 
vaporization of liquid water. Under these conditions the system 
has only one degree of freedom. Again, if P = 1, then F =; 2, 
and the system is bivariant, or has two degrees of freedom. The 



HETEROGENEOUS EQUILIBRIUM 343 

areas included between the curves in the diagram are examples 
of bi variant systems. Consider the vapor phase; the temperature 
may be fixed at any desired value within the vapor area AOC. 
and the pressure may be altered along a line parallel to the vertical 
axis without causing a change in the number of phases, provided 
the curves OA and OC are not intersected. 

The System, Sulphur (Rhombic, Monoclinic), Liquid and 
Vapor. This system is more complicated than the preceding 
one-component system, since there are two solid phases in addition 
to the liquid and vapor phases. At ordinary temperatures, rhom- 
bic sulphur is the stable modification. When this is heated 
rapidly it melts at 115 C., but if it is maintained in the neighbor- 
hood of 100 C. it gradually changes into monoclinic sulphur 
which melts at 120 C. Monoclinic sulphur can be kept indefi- 
nitely at 100 C. without undergoing* change into the rhombic 
modification, or in other words it is the stable phase at this temper- 
ature. 

It is evident, therefore, that there must be a temperature above 
which monoclinic sulphur is the stable form and below which 
rhombic sulphur is the stable modification. This temperature 
at which both rhombic and monoclinic modifications are in equi- 
librium with each other and with their vapor, is termed the 
transition point. Its value has been determined to be 95. 6 C. 
The change from one form into the other is relatively slow, so 
that it is possible to measure the vapor pressure of rhombic sul- 
phur up to its melting-point, and that of monoclinic sulphur 
below its transition point. The vapor pressure of solid sulphur, 
although very small, has been measured as low as 50 C. 

The complete pressure-temperature diagram for sulphur is 
shown in Fig. 79. At the point 0, rhombic and monoclinic sul- 
phur are in equilibrium with sulphur vapor, this being a triple 
point analogous to the point in Fig. 78. The vapor pressure 
curves of rhombic and monoclinic sulphur are represented by OB 
and OA respectively. The dotted curve OA' which is a continu- 
ation of OA is the vapor-pressure curve of monoclinic sulphur in 
a metastable region. In like manner OB' represents the vapor- 
pressure curve of rhombic sulphur in the metastable condition, B' 



344 



THEORETICAL CHEMISTRY 



being a metastable melting-point. As in the pressure-temper- 
ature diagram for water, the metastable phases have the higher 
vapor pressures. The effect of increasing pressure on the transi- 
tion point 0, is represented by the line OC. This is termed a 



Bhombic Sulphur 




Vapot 



Temperature 
Fig. 79. 

transition curve, and, since increase in pressure raises the transi- 
tion point, the line slopes away from the vertical axis. The 
effect of increased pressure on the melting-point of monoclinic 
sulphur is shown by the curve AC. 

This also slopes away from the vertical axis, but the change in 
the melting-point of monoclinic sulphur produced by a given 
change in pressure being less than the corresponding change in the 



HETEROGENEOUS EQUILIBRIUM 345 

transition point, the two curves, OC and AC, intersect at the 
point C. The point C corresponds to a temperature of 131 C. 
and a pressure of 400 atmospheres. The vapor-pressure curve 
of stable liquid sulphur is represented by the curve AD. The 
vapor-pressure curve of the metastable liquid phase is represented 
by the curve AB' which is continuous with AD. The diagram is 
completed by the curve J5'C which represents the effect of pressure 
on the metastable melting-point of rhombic sulphur. Mono- 
clinic sulphur does not exist above the point C; hence when 
liquid sulphur is allowed to solidify at pressures exceeding 400 at- 
mospheres, the rhombic modification is formed, whereas under 
ordinary pressures the monoclinic modification appears first. 

The phase rule enables us to state the exact conditions required 
for equilibrium in this system and to check the results of exper- 
iment. Thus, according to the formula, C P + 2 = JF, since 
C = 1, the system will be non-variant when P = 3. Since there 
are four phases involved, theoretically any three of these may 
be co-existent and four triple points are possible. The theoreti- 
cally-possible triple points are as follows: 

(2) Rhombic sulphur, monoclinic sulphur, and vapor (0); 

(2) Rhombic sulphur, monoclinic sulphur, and liquid (C7); 

(3) Rhombic sulphur, liquid, and vapor (J30; 

(4) Monclinic sulphur, liquid and vapor (A). 

In this particular system all of the four possible triple points can 
be realized experimentally. That this is the case is due to the 
comparative slowness of the change from rhombic to monoclinic 
sulphur above the triple point. If this change were rapid it is 
evident that all of the theoretically-possible non-variant systems 
could not be realized experimentally. 

As in the case of water, the curves in the diagram represent 
univariant systems and the areas bivariant systems. The student 
is advised to tabulate the univariant and bivariant systems repre- 
sented in the pressure-temperature diagram for sulphur. 

Two-component Systems. Turning now to two-component 
systems we are confronted with a more difficult problem, and one 
which includes many special cases. Thus, we may have cases of 



346 



THEORETICAL CHEMISTRY 



anhydrous salts and water, hydrated salts and water, volatile 
solutes, two liquid phases, consolute liquids, and solid solutions. 
To enter upon a discussion of these would not be profitable, since 
they only serve to give greater emphasis to the general truth of 
the phase rule. We shall select a few typical two-component 
systems for consideration here. 

(a) Anhydrous Salt and Water. In the equilibrium diagram 
of water (here represented by dotted lines, Fig. 80), we desig- 




T Temperature 

Fig. 80. 

nate the triple point by 0. At this point ice and water have the 
same vapor pressure. Similarly, a solution at its freezing-point 
has the same vapor pressure as the ice which separates. The 
intersection of the vapor-pressure curve for ice, OB, and the vapor- 
pressure curve of the solution of the anhydrous salt, 0"A", deter- 
mines a new triple point 0". Since the presence of the dissolved 
salt tends to diminish the vapor pressure of water, the curve 0" A " 
is situated below the curve OA, and for the same reason the triple 
point 0" is found to the left of 0. If now we keep an excess of 
dissolved substance continually present, all of the liquid phases 



HETEROGENEOUS EQUILIBRIUM 347 

which are formed will of necessity be saturated solutions. When 
these solutions finally freeze they will furnish, not pure ice, but a 
mixture of ice and solid salt, known as a cryohydrat'e. By a 
partial freezing we can therefore obtain the system: Solid salt, 
ice, saturated solution and vapor, or in other words, a system of 
n + 2 phases of which the existence is only possible at the freez- 
ing temperature T f of the saturated solution, and under the pres- 
sure p f corresponding to the vapor pressure of ice and the saturated 
solution. These conditions are represented in the diagram by 
the quadruple point 0'. If now we pass from the point 0', increas- 
ing the temperature and pressure as prescribed by the curve 
O'A', the ice disappears, while the salt, the saturated solution, 
and the vapor furnish a series of 3-phase systems. Again start- 
ing from the point 0' and lowering the temperature and the pres- 
sure as indicated by the curve O'B, the liquid phase disappears, 
while the solid salt, ice, and vapor constitute another series of 
3-phase systems. This, of course, is on the supposition that the 
vapor pressure of the solid salt is negligible. Finally, a consider- 
able increase in pressure causes a slight lowering of the temper- 
ature corresponding to the quadruple point, the conditions being 
represented by the curve O'C'. 

All possible non-saturated solutions of the salt will be repre- 
sented by points within the area, AOO'A'. Thus, let 0" A 11 repre- 
sent the vapor-pressure curve of a dilute solution of the salt in 
water. The freezing-point of this solution is represented by the 
point 0", while 0"C" represents the variation of the freezing-point 
of the solution with pressure. 

The following table summarizes the possibilities indicated by 
the phase rule: 

4 phases; salt, ice, saturated solution, vapor (point, 0'); 

3 phases; salt, saturated solution, vapor (curve, O'A'); 

3 phases; salt, ice, vapor (curve, 0'J3); 

3 phases; salt, ice, saturated solution (curve, O'C'); 

2 phases; salt, saturated solution (area, C'O r A f )\ 

2 phases; salt, water vapor (area, BO'A'); 

2 phases; salt, ice (area, 50'C'); 



348 



THEORETICAL CHEMISTRY 



3 phases; ice, non-saturated solution, vapor (curve, 00'); 
2 phases; non-saturated solution, vapor ) . . nn'A '\ 

1 phase; non-saturated solution > ' '* 

2 phases; non-saturated solution, ice > , cnn f r f \ 
1 phase; non-saturated solution S ' 

As will be seen, there is only one non-variant point in the entire 
diagram, viz., the point 0'. In this system there are three degrees 
of freedom, since in addition to temperature and pressure the con- 
centration of the solution may also be varied. 

The pressure-temperature diagram (Fig. 80) having been dis- 
cussed, we now turn to the concentration-temperature diagram 
for the same system, Fig. 81. In this diagram the abscissae 




Temperature 
Fig. 81. 

represent temperatures and the ordinates, concentrations. For 
convenience, corresponding points in Figs. 80 and 81 will be desig- 
nated by the same letters. The equilibrium between ice, water 
and water vapor is represented by the point 0. If now a small 



HETEROGENEOUS EQUILIBRIUM 349 

amount of anhydrous salt be added to the water, the freezing-point 
will be lowered to 0". As the proportion of salt is increased the 
temperature of equilibrium is lowered along the curve 00"0'. A 
point is ultimately reached at which the solution becomes saturated, 
and on further addition of salt it is not dissolved, but remains in 
contact with the ice and saturated solution. This is the cryo- 
hydric point, and represents the lowest temperature which can be 
obtained in this particular system. The diagram is completed 
by the solubility curve of the salt, O'A'. Each point on this 
curve represents the concentration of the saturated solution at 
all temperatures, from the critical temperature of the solution to 
the cryohydric temperature. The meaning of the concentration- 
temperature diagram may be made clearer by a consideration of 
the behavior of a solution when gradually cooled. Let a repre- 
sent a dilute solution of the anhydrous salt. On lowering the 
temperature along ab, no change will occur until the curve 00' 
is reached; then ice will begin to separate and as the cooling is 
continued, the composition of the solution will change along 00' 
until it reaches the cryohydric point 0'. Here both salt and ice 
will separate, and the solution will solidify completely at the 
temperature corresponding to the point 0'. In like manner, if 
we start with a concentrated solution represented by the point c 
and cool along cd no change will take place until the curve O'A ' 
is reached; then solid salt will separate and the composition of 
the solution will alter along O'A' until the temperature is reduced 
to that corresponding to the cryohydric point, when the whole 
solution will solidify as in the previous case. This phenomenon 
was first systematically investigated by Guthrie * who concluded 
that such mixtures of constant composition and definite melting- 
point are chemical compounds, and, therefore, he proposed to call 
them cryohydrates. It has since been shown that cryohydrates 
are not definite chemical compounds. Among the various reasons 
which have been advanced to prove the incorrectness of Guthrie's 
views, the following are the most cogent: (1) the physical 
properties of a cryohydrate are the mean of the corresponding 
properties of the constituents, this being rarely true of chemical 

* Phil. Mag. [4J, 49, 1 (1875); [5], i, 49 and 2, 211 (1876). 



350 



THEORETICAL CHEMISTRY 



compounds; (2) the lack of homogeneity of a cryohydrate can 
be detected under the microscope; and (3) the constituents are 
seldom present in simple molecular proportions. 

Applying the phase rule to the above two-component system, 
it is evident that there is but one non- variant system: this is 
represented by the point 0'. When three phases are co-existent 
the system is univariant, when only two phases are present the 
system is bivariant, and finally, when only one phase is present 
the system acquires three degrees of freedom or is trivariant. 
It is evident that a system having three degrees of freedom cannot 
be completely represented by a diagram in a single plane. It is 
possible, however, to construct a three-dimensional model which 
will represent the equilibrium very satisfactorily. Such a model is 




Fig. 82. 

I. Unsaturated Solution. 

II. Salt and Saturated Solution. 

III. Ice and Unsaturated Solution, 

IV. Ice and Cryohydrate. 
V. Salt and Cryohydrate. 

shown in Fig. 82, the lettering being made to correspond with 
that of the two diagrams, Figs, 80 and 81, from which it is derived. 



HETEROGENEOUS EQUILIBRIUM 



351 



(b) Hydrated Salt and Water. An interesting example is fur- 
nished by the system ferric chloride and water. This system 
has been very carefully investigated by Roozeboom.* The con- 
centration-temperature diagram, plotted from Roozeboom's data, 




Temperature - 

Fig. 83. 

is given in Fig. 83. The freezing-point of pure water is repre- 
sented by Ay and the lowering of the freezing-point produced by 
the addition of ferric chloride is indicated by the curve AB. At 
the cryohydric temperature, 55 C., ice, Fe 2 Cl 6 12 H 2 0, sat- 
urated solution, and vapor are in equilibrium, and the system is 
non-variant. On adding more ferric chloride, the ice phase dis- 
appears, and the univariant system, Fe 2 Cl 6 12 H 2 0, saturated 
solution, and vapor results. The equilibrium is represented by the 
curve BC which may be regarded as the solubility curve of the 
dodecahydrate. On continuing the addition of ferric chloride, 
the temperature continues to rise until the point C is reached. 
Here the composition of the solution is identical with that of the 
dodecahydrate, and, therefore, the temperature corresponding to 
this point, 37 C., may be looked upon as the melting-point of 
Fe 2 Cl 6 12 H 2 0. Further addition of ferric chloride will naturally 

* Zeit. phys. Chem., 4, 31 (1889); 10, 477 (1892). 



352 THEORETICAL CHEMISTRY 

lower the melting-point and the equilibrium will alter along the 
curve CD. It is thus possible to have two saturated solutions, 
one of which contains more water and the other less, than the 
hydrate which is in equilibrium with the solution. These solu- 
tions are both stable throughout and are nowhere supersaturated. 
Roozeboom was the first investigator to discover a saturated 
solution containing less water than the solid hydrate with which 
it is in equilibrium. This discovery led him to define supersatu- 
ration as follows: "A solution is supersaturated with respect 
to a solid phase at a given temperature if its composition is between 
that of the solid phase and the saturated solution." At the point 
D the curve reaches another minimum which is analogous to the 
point J5, except that the heptahydrate, Fe 2 Cle 7 H 2 0, takes the 
place of ice. Here we have equilibrium between the dodccahydrate, 
the heptahydrate, saturated solution, and vapor, and the system 
is non-variant. On further addition of ferric chloride another 
maximum is reached at E, corresponding to the melting-point of 
the heptahydrate. In a similar manner, two other maxima at 
greater concentrations of ferric chloride reveal the existence of the 
hydrates, Fe 2 Cl 6 5 H 2 0, and Fe 2 Cl 6 4 H 2 0. 

At the three remaining quadruple points the following phases 
are in equilibrium : At F, Fe 2 Cl 6 7 H 2 0, Fe 2 Cl 6 5 H 2 0, saturated 
solution and vapor; at H, Fe2Cl 6 5 H 2 0, Fe 2 Cl 6 4 H 2 0, saturated 
solution and vapor; and at K, Fe 2 Cl 6 4 H 2 0, Fe 2 Cl 6 , saturated 
solution and vapor. The solubility of the anhydrous salt is 
represented by the curve KL. Metastable solubility and melting- 
point curves are represented by dotted lines. 

The student should apply the phase rule to this system. If a 
fairly dilute solution of ferric chloride is evaporated at 31 C., the 
water gradually disappears and a residue of the dodecahydrate 
remains. This residue then liquefies and again dries down, the 
composition of the residue corresponding to the heptahydrate: 
on further standing the phenomenon is repeated, the final and 
permanent residue having a composition corresponding to the 
pentahydrate. The dotted line ab shows the isothermal along 
which the composition varies. It would have been a difficult 
matter to explain the alternations of moisture and dryness ob- 



HETEROGENEOUS EQUILIBRIUM 353 

served in this experiment without the concentration-temperature 
diagram. 

Alloys. Among the most interesting two-component systems 
known are those involving mixtures of metals, or alloys. These 
have been made the subject of systematic investigations by num- 
erous experimenters among whom may be mentioned Roberts- 
Austen, Charpy, Roozeboom, and Heycock and Neville. We have 
space to consider only two comparatively-simple cases. 

(a) Alloys of Silver and Copper. The conditions of equilibrium 
in this binary system have been studied by Heycock and Neville.* 
The two components, silver and copper, are not miscible in the 
solid state and do not combine chemically. To determine the 
curves of equilibrium, mixtures of the two metals in varying pro- 
portions were fused and then allowed to cool slowly, the rate of 
cooling being observed with a thermocouple, one junction of which 
was maintained at constant temperature, while the other junction 
was placed in the mixture of molten metals. The terminals of 
the thermocouple were connected to a sensitive galvanometer 
graduated to read directly in degrees, and the rate of cooling 
was followed by the movement of the needle of the galvanometer. 
As the mixture cooled, two "breaks" were observed; the first of 
these varied with the composition of the mixture, while the second 
remained practically constant at 777 C. When the temperatures 
corresponding to the first break are plotted as ordinates against 
the composition of the mixture as abscissae, the diagram shown in 
Fig. 84 is obtained. 

The point A represents the freezing-point of pure silver, B that 
of pure copper, the curve AO represents the effect of the gradual 
addition of copper upon the freezing-point of silver, and BO the 
effect of silver on the freezing-point of copper. The intersection 
of the two curves at corresponds to an alloy containing 40 atomic 
per cent of copper. This lowest melting mixture is known as 
the eutectic (c? = well, and -n/icciv = melt) mixture. At the 
system is non-variant, silver, copper, solution and vapor being in 
equilibrium. The solid which separates at 0, having a more 
uniform texture than that of all other mixtures of the two com- 

* Phil. Trans., 189, 25 (1897). 



354 



THEORETICAL CHEMISTRY 



ponents, is known as the eutectic alloy. When the composition 
of a mixture of two metals corresponds to that of the eutectic 
alloy, the two rnetals crystallize simultaneously in minute separate 




Cu 



40 at. per cent 
Concentration 
Fig. 84. 

crystals. When examined under the microscope the solid eutectic 
alloy will be seen to be a conglomerate of very small crystals, 
whereas all of the other alloys of the same metals will be found to 
contain large crystals of either one or the other component em- 
bedded in the conglomerate. While the composition of the eutec- 
tic alloy in the above system is found to correspond very closely 
to the formula Ag 3 Cu 2 , yet the nature of the equilibrium curves 
proves it to be nothing more than a mechanical mixture of the 
two metals. The meaning of the diagram will be clearer from a 
careful consideration of the phenomena accompanying the cooling 
of a mixture of the molten metals. 

Take for example, a fused mixture relatively rich in silver. As 
the temperature falls, a point will ultimately be reached at which 



HETEROGENEOUS EQUILIBRIUM 



355 



pure silver begins to separate, and since the temperature remains 
constant during the solidification, a break occurs in the cooling 
curve. This first break corresponds to a point on the curve AO. 
As silver continues to separate, the composition of the mixtures 
changes along AO, until when is reached, the mixture is satu- 
rated with respect to copper, and both metals separate as a con- 
glomerate having the same composition as the fused mixture. 
The separation of the eutectic alloy causes the second break in 
the cooling curve, the temperature remaining constant until the en- 
tire mass has solidified. It will be noticed that this system is the 
exact analogue of the system anhydrous salt and water; the eutec- 
tic point and the cryohydric point representing identical conditions. 
(b) Alloys of Gold and Aluminium. This system has been 
studied by Roberts- Austen.* The equilibrium curves in the 
concentration temperature diagram, Fig. 85, reveal the existence 




B 



Au Composition 

Fig. 85. 

* Phil. Trans. A., 194, 201 (1900). 



356 THEORETICAL CHEMISTRY 



of definite compounds, AusAU, Au 2 Al, and AuAl 2 , corresponding 
to the points D, E and H respectively. The discontinuities at B 
and G suggest the possibility of two other compounds, viz., Au4Al 
and AuAl. The diagram shows that the following substances 
will crystallize in succession from the molten alloy, these being 
the different solids with which the liquid mixture is saturated in 
its successive stages of equilibrium: 

Curve AB, pure gold at A ; 

Curve BC, Au4Al, nearly pure at J5; 

Curve CD, Au5Al 2 or AusAl 3 , nearly pure at D; 

Curve DEF, Au 2 Al, pure at E\ 

Curve FG, AuAl, maximum undetermined, 

Curve GHI, AuAl 2 , pure at H] 

Curve IJ, Al, pure at J. 

The points C, F, and / represent non-variant systems, the melt- 
ing-points of the respective eutectic alloys being 527, 569, and 
647. This system in many respects resembles the system ferric 
chloride and water. 

Three-component Systems. When three components are pres- 
ent, the equilibria become much more complicated. Applying 
the formula, C P + 2 = F, we find that it is necessary to 
have five phases co-existent for a non-variant system, four for a 
univariant, three for a bivariant, and two for a trivariant. The 
most satisfactory method of representing equilibria in three-com- 
ponent systems is that in which use is made of the triangular 
diagram. The three corners of an equilateral triangle are taken 
to represent the pure components, and the composition of any 
mixture, expressed in atomic percentages, is represented by the 
position of the center of mass of the three components within 
the triangle. 

For example, in the system, potassium nitrate, sodium nitrate, 
and lead nitrate, carefully investigated by Guthrie,* the three 
components are placed at the corners of the triangle shown in 

* Phil. Mag., 5, 17, 472 (1884). 



HETEROGENEOUS EQUILIBRIUM 357 

Fig. 86. The melting-point of pure potassium nitrate is 340 
and that of pure sodium nitrate is 305. The melting-point of 




Fig. 86. 

pure lead nitrate cannot be determined since the salt decomposes 
before its melting-point is reached. The eutectic mixtures of 
the three pairs of salts are represented by the points D, E, and F 
respectively. In like manner represents the melting-point of 
the non- variant system, potassium nitrate, sodium nitrate, lead 
nitrate, fused mixture of the three salts, and vapor. In order 
to represent temperature, use is frequently made of a triangular 
prism in which the altitude is taken as the temperature axis, the 
resulting surface within the prism representing the variation of 
the equilibrium with temperature.* 

PROBLEMS. 

1. The vapor pressure of solid NH^HS at 25. 1 is 50.1 cm. Assuming 
that the vapor is practically completely dissociated into NH 3 arid H 2 S, 
calculate the total pressure at equilibrium when solid NH 4 HS is allowed 

* For a complete treatment of three-component systems as well as for a 
clear presentation of the phase rule, the student should consult "The Phase 
Rule and Its Applications," by Alexander Findlay. 



358 THEORETICAL CHEMISTRY 

to dissociate at 25. 1 in a vessel containing ammonia at a pressure of 
32 cm. Am. 59.5 cm. 

2. In the partition of acetic acid between CCU and water, the con- 
centration of the acetic acid in the CCU layer was c gram-molecules per 
liter and in the corresponding water layer w gram-molecules per liter. 

c 0.292 0363 0.725 1.07 1.41 

w 4.87 5.42 7.98 9.69 10.7 

Acetic acid has its normal molecular weight in aqueous solutions. From 
these figures show that, at these concentrations, the acetic acid in the 
carbon tetrachloride solution exists as double molecules. 

3. Acetic acid distributes itself between water and benzene in such a 
manner that in a definite volume of water there are 0.245 and 0.314 gram 
of the acid, while in an equal volume of benzene there are 0.043 and 
0.071 gram. What is the molecular weight of acetic acid in benzene, 
assuming it to be normal in water? Ans. 121.3. 

4. The salt Na2HPOi.l2 H 2 has a vapor pressure of 15 of 8.84 mm., 
and at 17.3 of 10.53 mm. Calculate the heat of vaporization, i.e., the 
thermal change during the loss of 1 mol of water of crystallization by 
evaporation. Ans. 12,651 cal. 

5. The solubility of boric acid in water is 38.45 grams per liter at 13, 
and 49.09 grams per liter at 20. Calculate the heat of solution of boric 
acid per mol. Ans. 5822 cal. 

6. Plot the pressure-temperature diagram for calcium carbonate from 
the table given on p. 280, and apply the phase rule. 

7. Is it possible to decide by the phase rule whether the eutectic alloy 
is a mixture or a compound? 



CHAPTER XVII. 
CHEMICAL KINETICS. 

Velocity of Reaction. In the two preceding chapters we have 
considered the equilibrium which is established when the speeds 
of the direct and reverse reactions have become equal. We now 
proceed to consider the velocity of individual reactions. By far 
the greater number of the reactions between inorganic substances 
proceed with such rapidity that it is impossible to measure their 
velocities. Thus, when an acid is neutralized by a base, the indi- 
cator changes color almost instantly. There are a few well- 
known reactions which are exceptions to this rule; among these 
may be mentioned the oxidation of sulphur dioxide and the de- 
composition of hydrogen peroxide. Both of these reactions are 
well adapted to kinetic experiments. In organic chemistry, on 
the other hand, slow reactions are the rule rather than the excep- 
tion. Thus the reaction between an alcohol and an acid forming 
an ester and water, proceeds very slowly under ordinary condi- 
tions and the progress of the reaction may be easily followed. 
By means of the law of mass action it is possible to derive equations 
expressing the velocity of a reaction at any moment in terms of 
the concentrations of the reacting substances present at that time. 

Let the equation 



represent a reversible reaction and let a, &, c, and d be the respec- 
tive initial concentrations of the reacting substances Ai, A 2, Ai, 
and A 2 '. The velocity of the direct reaction will then be 

^j-jb(a-*) (6-x), (1) 

where k is the velocity constant, and dx is the infinitely small 
increase in the amount of x during the infinitely small interval 



360 THEORETICAL CHEMISTRY 

of time dt. Similarly the velocity of the reverse reaction will 
be 

^ = k i (c+x) (d + x). (2) 

It is evident that the substances on the right-hand side of the equa- 
tion will exert an ever-increasing influence upon the velocity of 
the direct reaction, which must accordingly decrease. When, 
however, the velocities of the direct and reverse reactions become 
equal, equilibrium will be established, and the ratio of the amounts 
of the reacting substances on the two sides of the equation will 
remain constant. The total velocity due to these opposing reac- 
tions will be 

H = ~ = k (a ~~ x} (b " x} ~ kl " (c + x} (d+ X} (3) 
and at equilibrium, when ~-^- = 0, 

k (a - x) (b - x) = fci (c +x) (d + x), 
or 

(c + x) (d + x) k ^ , . 

(a-x)(b~x) k, c * w 

This equation has been thoroughly tested in the two preceding 
chapters. Thus, in the reaction 

C 2 H 6 OH + CH 3 COOH ? CH 3 COOC 2 H 6 + H 2 U, 

K c has been shown to have the value, 2.84, at ordinary temper- 
atures. The velocity constants of the direct and reverse reactions 
have also been determined, the values being, k = 0.000238 and 
fci = 0.000815. When these values are substituted in the equa- 

k 

tion, T- = K c , we obtain K c = 2.92, a value which agrees well 
KI 

with that found by direct experiment. The application of equa- 
tion (3) is much simplified by the fact that most reactions proceed 
nearly to completion in one direction, so that the term k\ (c + x) 
(d + x) will be so small that it may be neglected. We then have 

), (5) 



CHEMICAL KINETICS 



361 



an equation expressing the velocity of the direct reaction in terms 
of the concentrations of the reacting substances. 

Unimolecular Reactions. The simplest type of chemical 
reaction is that in which only one substance undergoes change 
and in which the velocity of the reverse reaction is negligible. The 
decomposition of hydrogen peroxide is an example of such a reac- 
tion. In the presence of a catalyst, such as certain unorganized 
ferments or colloidal platinum, hydrogen peroxide decomposes 
as represented by the equation, 



This reaction is usually allowed to take place in dilute aqueous 
solution so that there is no appreciable alteration in the amount 
of solvent throughout the entire course of the reaction. Further- 
more, the activity of the catalyst remains constant so that the 
course of the reaction is wholly determined by the concentration 
of the hydrogen peroxide. A very satisfactory catalyst is catalase, 
an enzyme derived from blood. The concentration of hydrogen 
peroxide present at any time during the reaction can be deter- 
mined very simply by removing a definite portion of the reaction 
mixture, adding an excess of sulphuric acid to destroy the activity 
of the hsemase, and then titrating with a standard solution of 
potassium permanganate. 
The following table gives the results of such an experiment: 



t (minutes). 


a x, 
cc. KMn0 4 . 


z 
cc. KMnO<. 


k 


o 


46 1 


o 




5 
10 
20 
30 
50 


37.1 
29.8 
19 6 
12 3 
5.0 


9 
16 3 
26.5 
33.8 
41.1 


0435 
0.0438 
0.0429 
0440 
0444 








Mean 0.0437 



The second column of the table gives the number of cubic 
centimeters of the potassium permanganate solution required to 
oxidize 25 cc. of the reaction mixture when the time intervals 



362 THEORETICAL CHEMISTRY 

recorded in the first column have elapsed after the introduction 
of the catalyst. Since the numbers in the second column repre- 
sent the actual concentration of hydrogen peroxide present at 
the end of the successive intervals of time, it is evident that the 
difference between these numbers and 46.1 cc. the initial 
concentration of hydrogen peroxide will give the amounts of 
peroxide decomposed in those intervals. These numbers are 
recorded in the third column of the table. It will be seen that 
as the concentration of the hydrogen peroxide decreases the rate 
of the reaction diminishes. Thus, in the first interval of 10 min- 
utes, an amount of hydrogen peroxide corresponding to 46.1 
29.8 = 16.3 cc. of potassium permanganate is decomposed, while 
in the second interval of 10 minutes, the amount of hydrogen 
peroxide decomposed is equivalent to 29.8 19.6 = 10.2 cc. of 
potassium permanganate. Since only a single substance is under- 
going change, equation (5) simplifies to the following form: 

dx j ( , 
# = *(-*> 

It is impossible to apply the equation in this form, since in order 
to obtain accurate titrations, dt must be taken fairly large and 
during this interval of time ax would have diminished. Approx- 
imate values of k may be obtained by taking the average value 
of a x during the interval dt within which an amount dx of hydro- 
gen peroxide is being decomposed. For example, let us take the 
interval between 5 and 10 minutes; dx = 16.3 9.0 = 7.3 cc., 
dt = 5 min., and the average value of a x is 

37.1 + 29.8 



Substituting in the equation 

we have 
and 



33.45 cc. 



dx j , ^ 
*= fc (a -a), 

- k X 33.45, 



k - 0.0436. 



CHEMICAL KINETICS 363 

Similarly taking the next interval between 10 and 20 minutes; 
dx = 26.5 16.3 = 10.2 cc., dt = 10 minutes, and the average 

29 g 4- 19 6 
value of a x is : ^ "~ ~ 24.7 cc. Substituting in the 

equation as before, we obtain 

and 

k = 0.0413. 

As will be seen these two values of k are not in good agreement, 
although the first value of k agrees closely with the mean value 
of k given in the fourth column of the table. 
In order to apply the equation 

dx_ , 
dt -b(P> x) 

it must be integrated.* 

The integration of this equation may be performed as follows: 

^ = k (a - x), 

therefore 

dx , , 

= k dt. 

& x 

Integrating, we have 

/j n> 

- / k dt = constant = C. 
a x J 

therefore 

- log, (a - x) - kt = C. 

In order to determine C, the constant of integration, we make 
use of the experimental fact that when t = 0, x = 0. Substitut- 
ing these values, we have 

loga = C. 
Consequently 

log, a log, (a x) = kt, 
or 

1 , a , 
7 log, ^j- = k. 

* The student who is unfamiliar with the Calculus must take the result 
of this calculation for granted. 



364 THEORETICAL CHEMISTRY 

Passing to Briggsian logarithms, we obtain 



-log = 0.4343 k. 
t & a x 



By substituting in this equation the corresponding values of a, 
a x, and t from the preceding table, the values of k given in the 
fourth column of the table are obtained. 
The equation 

dx j , ^. 
_ = fc(-*), 

may also be thrown into an exponential form, as follows: 
Since 

1 , _ a _ , 
7 10 S<^~T ~ x - *> 

we may write, 

7 . a x 

Kt = lOge 



) 



or 

a x = ae~~ kt , 
and 



In this equation k may be regarded as the fraction of the total 
amount of substance decomposing in the unit of time, provided 
this unit is so small that the quantity at the end of the time unit 
is only slightly different from that at the beginning. The time 
required for one-half of the substance to change, is known as the 
period of half-change, T, and may be calculated from k by means 
of the equation 

log 2 = 0.4343 fcT, 
therefore 



0.6943 J, 
k 



or 

1 



~ - 1.443 T. 



Reactions in which only one mol of a single substance undergoes 
change are known as unimolecular reactions, or reactions of the 



CHEMICAL KINETICS 365 

first order. In a unimolecular reaction, the velocity constant k 
is independent of the units in which concentration is expressed. 
If, in the integrated equation . 

kt = log , 

a x 

t becomes infinite, then x = a. In other words, for finite values 
of t y x must always remain less than a and the reaction will never 
proceed to completion. 

Another unimolecular reaction which has been thoroughly 
investigated, is the hydrolysis of cane sugar. When cane sugar 
is dissolved in water containing a small amount of free acid it is 
slowly transformed into d-glucose and d-fructose. The velocity 
of the reaction is very small and is dependent upon the strength 
of the acid added. The progress of the reaction may be very easily 
followed by means of the polarimeter. Cane sugar itself is dex- 
tro-rotatory, while d-fructose rotates the plane of polarization 
more strongly to the left than d-glucose rotates it to the right. 
Therefore, as the hydrolysis proceeds, the angle of rotation to the 
right steadily diminishes until, when the reaction is complete, the 
plane of polarization will be found to be rotated to the left. On 
this account the hydrolysis of cane sugar is commonly termed 
inversion and the molecular mixture of d-fructose and d-glucose 
constituting the product of the reaction is called invert sugar. 
Let a denote the initial angle of rotation, at the time t = 0, 
due to a mols of cane sugar, let a ; denote the angle of rotation 
when inversion is complete and let a be the angle of rotation at 
any time t] then since rotation of the plane of polarization is pro- 
portional to the concentration x, the amount of cane sugar in- 
verted, will be 

Ofo "- Oi 

x = a ; 7- 

OiQ + OQ 

In the equation 

CaHiAi + H 2 <=> CcHuOe + C 6 H, 2 6 , 

representing the inversion of cane sugar, the velocity of the 
reaction will be, according to the law of mass action, proportional 
to the molecular concentrations of the cane sugar and the water. 



366 



THEORETICAL CHEMISTRY 



Since the reaction takes place in the presence of such a large excess 
of water, its effect may be considered to be constant. The 
velocity of the reaction is then proportional to the active mass 
of the sugar alone, or in other words the reaction is umimolecular. 
In the differential equation expressing the velocity of a unimolec- 
ular reaction, 

dx 



we have 



dt 



= k (a x) y 



k = -. 



and since a and x are measured in terms of angles of rotation of 
the plane of polarization, we have 



The following table gives the results obtained with a 20 per cent 
solution of cane sugar in the presence of 0.5 molar solution of lactic 
acid at 25 C. 



t (minutes). 


a 


k 


o 


34 5 




1,435 


31 1 


2348 


4,315 


25. 


0.2359 


7,070 


20. 16 


2343 


11,360 


13 98 


2310 


14,170 


10 01 


0.2301 


16,935 


7. 57 


0.2316 


19,815 


5. 08 


0.2991 


29,925 


- 1.65 


0.2330 


Inf. 


-10. 77 





Bimolecular Reactions. When two substances react and the 
concentration of each changes, the reaction is bimolecular or of 
the second order. Let a and b represent the initial molar con- 
centrations of the two reacting substances and let x denote the 
amount transformed in the interval of time t\ then the velocity 
of the reaction will be expressed by the equation 

- - k (a - x) (b - x). 



CHEMICAL KINETICS 367 

The simplest case is that in which the two substances are present 
in equivalent amounts. Under these conditions the velocity 
equation becomes 



This equation may be integrated as follows: * 

7 j* dx 
kdt = 



therefore 



7 ----- -\i> 

(a - z) 2 

A/ I (tt == I / xo > 

Jti Jx, (a-z) 2 



a -a 
or 



(k ~~ ^i) ( a "~ &i) (# ~~~#2) 

If time be reckoned from the beginning of the reaction, then x\ = 
and t = 0, and we have 

k - -7 7-^ r 
2 a (a x) 

If the reacting substances are not present in equivalent amounts 
then the velocity equation becomes 

-- = k (a - x) (b - a). 

Assuming that time is measured from the beginning of the reaction, 
the integration of this equation may be performed as follows: 

rt rx d x 

Jo Jo (CL x) (b x) 

Decomposing into partial fractions, 

&* = -- 1 i . I I > 

ct o L/O o x t/o & xj 

* The student who is unfamiliar with the Calculus must take the results 
of these calculations for granted. 



368 THEORETICAL CHEMISTRY 

therefore 

7 If. a - xf 

fc < = ^L log '^d' 

or 

, . 1 , b (a x) 

- _ -^ ' 



= - _ - 

a b & a (b 
Or passing to Briggsian logarithms, 

0.4343 k = 4( l , , log 
* 



4( , , - 

t (a b) *a(b x) 

The value of fc in a bimolecular reaction is not independent of 
the units in which the concentration is expressed, as is the case 
with a unimolecular reaction. Suppose that a unit I/nth of 
that originally selected is used to express concentration, then the 
value of k in the equation 

z ! x 

fv = - -- -, - r ) 

t a (a x) 
becomes 

nx x 



t na - n(a x) t na(a x) 

Thus, the value of k varies inversely as the numbers expressing 
the concentrations. 

As an illustration of a bimolecular reaction we may take the 
hydrolysis of an ester by an alkali. The reaction 

CH 3 COOC 2 H 5 + NaOH ? CH 3 COONa + C 2 H B OH, 

has been studied by Warder,* Reicher,f Arrhenius,t Ostwald 
and others. Arrhenius employed in his experiments 0.02 molar 
solutions of ester and alkali. These solutions were placed in 
separate flasks and warmed to 25 C. in a thermostat maintained 
at that temperature; equal volumes were then mixed, and at 
frequent intervals a portion of the reaction mixture was removed 

* Berichte, 14, 1361 "(1881). 

t Lieb. Ann., 228, 257 (1885). 

} Zeit. phys. Chem., i, 110 (1887). 

Jour, prakt. Chem., 35, 112 (1887). 



CHEMICAL KINETICS 



369 



and titrated rapidly with standard acid. The accompanying 
table contains some of the results obtained: 



/ (minutes). 


a x 


k 





8 04 




4 


5 30 


o oieo 


6 


4 58 


0156 


8 


3 91 


0164 


10 


3 51 


0160 


12 


3 12 


0162 






Mean 0.0160 



The numbers in the second column of the table represent the 
concentrations of sodium hydroxide and of ethyl acetate, expressed 
in terms of the number of cubic centimeters of standard acid 
required to neutralize 10 cc. of the reaction mixture. Owing to 
the high velocity of the reaction it is difficult to avoid large experi- 
mental errors, nevertheless the values of fc given in the third column 
of the table will be observed to differ very slightly from the mean 
value. 

Reicher investigated the same reaction when the reacting sub- 
stances were not present in equivalent proportions. In this case, 
the progress of the reaction was followed by titrating definite 
portions of the reaction mixture from time to time, the excess of 
sodium hydroxide being determined by titrating a portion of 
the mixture at the expiration of twenty-four hours, when the 
ester was completely hydrolyzed. His results are given in the 
following table: 



t (minutes). 


fl X 

(alkali). 


b-x 
(ester). 


it 


o 


61 95 


47 03 




4.89 
11.36 
29.18 
Inf 


50.59 
42.40 
29.35 
14 92 


35.67 
27.48 
14.43 

o 


00093 
00094 
00092 











370 THEORETICAL CHEMISTRY 

Reicher also studied the effect of different bases upon the ve- 
locity of the reaction. He found for strong bases approximately 
equal values of fc, but for weak bases the values were irregular 
and smaller than those obtained with the more completely ionized 
bases. Arrhenius pointed out that the hydrolyzing power of a 
base is proportional to the number of hydroxyl ions which it 
yields. Writing the equation for the above hydrolysis in terms of 
ions, we have 

CH 3 COOC 2 H 5 + Na' + OH 7 * CH 3 COO' + Na' + C 2 H 5 OH. 

It is evident from this equation that all bases furnishing the same 
number of hydroxyl ions should give identical values of k. We 
may, therefore, modify the fundamental differential equation as 
follows: 

~ = k'a (a -x) (b - a;), 

where a is the degree of ionization of the base. 

Trimolecular Reactions. When equivalent quantities of three 
substances react, the reaction is trimolecular or of the third order. 
If the initial molar concentrations of the reacting substances are 
denoted by a, b, and c, and if x denotes the proportion of each 
which is transformed in the interval of time t, the velocity of the 
reaction will be represented by the differential equation 

= k (a - x) (b - x) (c - re). 

If the substances are present in equivalent amounts, the equation 
becomes 

fa = k (a _ x y 

an expression which is much less difficult to integrate. 
The integration of this equation may be performed as follows.* 



(a - xr 

* The student who is unfamiliar with the Calculus must take the results 
of these calculations for granted. 



CHEMICAL KINETICS 371 

therefore, 





2 >- 



hence 



i 1 1] 

2L(a-z) 2 a 2 J 



t 2 a 2 (a -a;) 2 

When the reacting substances are not taken in equivalent amounts, 
the integration of the velocity equation may be performed as 
follows: 

- 1c - 

therefore 

j~ 
kdt = 7 



Decomposing into partial fractions, 

(a x) (b x) (c x) a x b x c x' 
Multiplying through by (a x), we obtain 

* >*!/ _\ 1 ** I ^ 



(6-s)(c-s) " T ^ -'^ft-x^^T^ 
Let x = a, then 



(a - 6) (c - a) 

Similarly, multiplying by (b - x) and (c - x), and then placing 
x = b, and x = c, we have 



~ (a-b)(b-c)' 
and 



(6-c)(c-a) 
Then we obtain by substitution 

f _ dx __ _ 1 f dx 

Jo (a x)(b x) (c x) (a 6) (c a) Jo a x 
__ 1 C* dx ___ 1 /* dx 

(a - b) (b - c) Jo b - x (b - c) (c - a) Jo c - x' 



372 

Therefore, 



THEORETICAL CHEMISTRY 



or 



(c ~ a) 



I* (a -6) (b-c) (c-a) 

In a trimolecular reaction, k is inversely proportional to the square 
of the original concentration. 

A typical trimolecular reaction is that between ferric and 
stannous chlorides. This reaction, represented by the following 
equation 

2 Fed* + SnCl 2 <=* 2 FeCL 2 + SnCU, 

has been investigated by A. A. Noyes.* Dilute solutions of the 
reacting substances were mixed at constant temperature, and 
definite portions of the reaction mixture were removed at meas- 
ured intervals of time and titrated for ferrous iron. Before 
titrating with a standard solution of potassium permanganate it 
was necessary to decompose the* stannous chloride present with 
mercuric chloride. The following table gives the results obtained 
with 0.025 molar solutions of ferric chloride and stannous chloride. 



/ (minutes). 


a x 


X 


k 


2 5 


'0 02149 


00351 


113 


3 


02112 


00388 


107 


6 


01837 


00663 


114 


11 


01554 


00946 


116 


15 


01394 


01106 


118 


18 


01313 


01187 


117 


30 


01060 


01440 


122 


60 


0.00784 


0.01716 


122 








Mean 116 



Noyes also found that the velocity of the reaction is accelerated 
more by an excess of ferric chloride than by an equal excess of 
stannous chloride. 

* Zeit. phys. Chem., 16, 546 (1895). 



CHEMICAL KINETICS 373 

Reactions of Higher Orders. Reactions of the fourth, fifth 
and eight orders have recently been investigated, but examples 
of reactions of orders higher than the third are extremely rare. 
This fact is at first sight surprising since the equations of many 
chemical reactions involve a large number of molecules, and we 
would naturally expect the order of such reactions to be corre- 
spondingly high. For example, the reaction represented by the 

equation, 2 PH 3 + 4 O 2 = P 2 O 6 + 3 H 2 0, 

involves six molecules of the substances initially present and, 
therefore, we should infer it to be a reaction of the sixth order. 

Kinetic experiments by van der Stadt have shown it to be a 
bimolecular reaction, the velocity of reaction being proportional 
to the concentration of the phosphine and the oxygen. On 
allowing the gases to mix slowly by diffusion, it was discovered 
that the reaction actually takes place in several successive stages, 
the first stage being represented by the equation of the bimolec- 
ular reaction 

PH 3 + 2 = HP0 2 + H 2 . 

The subsequent changes, involving the oxidation of the products 
of this reaction, take place with great rapidity. It is highly 
probable that the equations which are ordinarily employed to 
represent chemical reactions really represent only the initial and 
final stages of a series of relatively simple reactions. Larmor * 
has shown that when chemical reactions are considered from the 
molecular standpoint, the bimolecular reaction is the most prob- 
able. He says, " Imagine a substance, say gaseous for simplicity, 
formed by the immediate spontaneous combination of three gas- 
eous components A, 5, and C. When these gases are mixed, the 
chances are very remote of the occurrence of the simultaneous 
triple encounter of an A, a 5, and a C, which would be necessary 
to the immediate formation of an ABC] whereas if ever formed, 
it would be liable to the normal chance of dissociating by collisions; 
it would thus be practically non-existent in the statistical sense. 
But if an intermediate combination AB could exist, very tran- 
siently, though long enough to cover a considerable fraction of the 
* Proc. Manchester Phil. Soc., 1908. 



374 THEORETICAL CHEMISTRY 

mean free path of the molecules, this will readily be formed by 
ordinary binary encounters of A and B, and another binary 
encounter of AB with C will now form the triple compound ABC 
in quantity/' 

Determination of the Order of a Reaction. It has been 
shown in the foregoing pages that the time required to complete 
a certain fraction of a reaction is dependent upon the order of 
the reaction in the following manner: 

(1) In a unimolecular reaction the value of k is independent of 
the initial concentration; 

(2) In a birnolecular reaction the value of k is inversely pro- 
portional to the initial concentration; 

(3) In a trimolecular reaction the value of k is inversely pro- 
portional to the square of the initial concentration. 

Hence, in general, in a reaction of the nth order, the value of 
k is inversely proportional to the (n 1) power of the initial con- 
centration. If the value of k is determined with definite concen- 
trations of the reacting substances, and then with multiples of 
those concentrations, the order of the reaction can be determined 
according to the above rules by observing the manner in which k 
varies with the concentration. 

The order of a reaction may also be readily determined by means 
of a graphic method. Thus, to determine the order of a reaction 
we ascertain by actual trial which one of the following expressions, 
in which C denotes concentration, will give a straight line when 
plotted against times as abscissae: 

(1) log C reaction unimolecular; 

(2) 1/C reaction bimolecular; 

(3) 1/C 2 reaction trimolecular; 

(4) I/O reaction n + 1 molecular. 

Complex Reaction Velocities. Thus far we have considered 
the velocity of reactions which are practically complete. There 
are numerous cases, however, in which the course of the reaction 
is complicated by such disturbing factors as (1) counter reactions, 
(2) side reactions, and (3) consecutive reactions. These disturb- 
ing causes will now be considered. 



CHEMICAL KINETICS 375 

(1) Counter Reactions. In the chemical change represented by 
the equation 



CH 3 COOH + C 2 H & OH *=> CH3COOC 2 H 5 + H 2 0, 

the speed of the direct reaction steadily diminishes owing to the 
ever-increasing effect of the reverse or counter reaction. Ulti- 
mately, when two-thirds of the acid and alcohol are decomposed, 
the velocities of the two reactions become equal and a condition 
of equilibrium results. Starting with 1 mol of acid and 1 mol 
of alcohol, and letting x represent the amount of ester formed, 
we have 

~~j] == \J- 3s) K> X * 

When equilibrium is attained, 



By observing the change for any time t, we have 

39 T 
i *i J/ 

Having the values of k/k' and k fc', the velocity constant fc 
of the direct reaction can be determined. The value of k so 
obtained has been shown by Knoblauch * to vary in those reac- 
tions where the concentration of the hydrogen ion changes. 

(2) Side Reactions. When the same substances are capable of 
reacting in more than one way with the formation of different 
products, the several reactions proceeds side by side. Thus, 
benzene and chlorine may react in two ways as shown by the 
equations, 

(1) C 6 H 6 + C1 2 = C 6 H 6 C1 + HC1, 
and 

(2) 



It is generally possible to regulate the conditions under which 
the substances react so as to promote one reaction and retard the 
other. 

* Zeit. phys. Chem., 22, 268 (1897). 



376 THEORETICAL CHEMISTRY 

(3) Consecutive Reactions. By consecutive reactions we under- 
stand those reactions in which the products of a certain initial 
chemical change react, either with each other or with the original 
substances to form new substances. Attention has already been 
called to the fact that many of our common chemical equations 
really represent the summation of a number of consecutive reac- 
tions. If the system A is transformed into the system C through 
an intermediate system J3, then we shall have the two reactions 

(1) A-+B 
and 

(2) B-*C. 

If reaction (1) should have a very much greater velocity than 
reaction (2), then the measured velocity of the change from A to 
C will be practically the same as that of the slower reaction. 
This fact has been illustrated by means of the following analogy, 
due to James Walker: * " The time occupied by the transmission 
of a telegraphic message depends both on the rate of transmission 
along the conducting wire, and on the rate of progress of the 
messenger who delivers the telegram; but it is obviously this 
last, slower rate that is of really practical importance in determin- 
ing the time of transmission." The saponification of ethyl 
succinate may be taken as an illustration of consecutive reactions. 
This reaction proceeds in two stages as follows : 



COOC2H5 
(1) C 2 H 4 <; + NaOH^C 2 H 4 +C 2 H 5 OH, 



/ /COONa 

(2) C 2 H 4 / + NaOH^C 2 H/ + C 2 H 5 OH. 

X COONa N^OONa 

In this case the product of the first reaction reacts with one of the 
original substances. 

Velocity of Heterogeneous Reactions. It has been shown that 
when a solid, such as calcium carbonate, is dissolved in an acid, 

* Proc. Roy. Soc., Edinburgh, 22 (1898.) 



CHEMICAL KINETICS 377 

the rate of solution is dependent upon the surface of contact 
between the solid and liquid phases, and also upon the strength 
of the acid. If the surface is large so that it undergoes relatively 
little change during the reaction, it may be considered as constant. 
If S represents the area of the surface exposed and x denotes the 
amount of solid dissolved in the time t, the velocity of the reaction 
will be represented by the differential equation 

dx j a , N 
-r:=kS(a x). 

Integrating this equation, we have 

7 cr 1 i a 

M.-.log.. 

This formula has been tested by Boguski * for the reaction 
CaC0 3 + 2 HC1 = CaCl 2 + CO 2 + H 2 O, 

and is found to give constant values of k. Furthermore, Noyes and 
Whitney f have shown that the rate of solution of a solid in a liquid 
at any instant, is proportional to the difference between the con- 
centration of the saturated solution and the concentration of the 
solution at the time of the experiment. 

Velocity of Reaction and Temperature. It is a well-estab- 
lished fact that the velocity of a chemical reaction is accelerated 
by rise of temperature. Thus, the rate of inversion of cane sugar 
is increased about five times for a rise in temperature of 30. It 
has been shown as the result of a large number of observations on 
a variety of chemical reactions, that in general the velocity of a 
reaction is doubled or trebled for an increase in temperature of 
10. It is of interest to note that the rate of development of 
various organisms, such as yeast cells, the rate of growth of the 
eggs of certain fishes, and the rate of germination of certain 
varieties of seeds is either doubled or trebled for a rise in temper- 
ature of 10. Up to the present time no wholly satisfactory form- 
ula, connecting the rate of reaction with the temperature, has been 
derived, although several purely-empirical expressions have been 

* Berichte, 9, 1646 (1876). 

t Zeit. phys. Chem., 23, 689 (1897). 



378 



THEORETICAL CHEMISTRY 



suggested. Of these formulas the most widely applicable is that 
proposed by van't Hoff and verified by Arrhenius. If fc and k\ 
represent the velocity constants at the respective temperatures 
To and Ti, then 



where e is the base of the Naperian system of logarithms and A is 
a constant. The following table gives the calculated and observed 
values of k at various temperatures for the reaction 



/NH 2 



NH 4 CNO^OC< 



when T = 273 + 25, k = 0.000227 and A = 11,700. 



T t 
Degrees. 


K (observed). 


Jk (calculated). 


273 + 39 


00141 


00133 


273 + 50 1 


00520 


0.00480 


273 + 64 5 


0.0228 


0.0227 


273 + 74 7 


062 


0623 


273 + 80 


100 


0.105 



In this case the agreement between the observed and calculated 
values is all that could be desired. 

Influence of the Solvent on the Velocity of Reaction. The 
velocity of a chemical reaction varies greatly with the nature of 
the medium in which it takes place. This subject has been 
studied by Menschutkin * who has collected much valuable data, 
as the result of a large number of experiments, on the velocity of 
the reaction between ethyl iodide and triethylamine, as represented 
by the equation 

C 2 H 6 I + (C 2 H 6 ) 3 N = (C 2 H 5 ) 4 NL 

This reaction was allowed to take place in a large number of 
different solvents and the velocity at 100 was measured. A few 



* Zeit. phys. Chem,, 6, 41 (1890). 



CHEMICAL KINETICS 



379 



of Menschutkin's results are given in the accompanying table, in 
which k denotes the velocity constant: 



Medium. 


k 


Medium. 


ft 


Hexane 


00018 


Ethyl alcohol 


0366 


Ethyl ether . . 


000757 


Methyl alcohol 


0516 


Benzene 


00584 


Acetone 


0608 











These figures show that the velocity of the reaction is greatly 
modified by the nature of the medium in which it takes place, the 
velocity in hexane being less than one three-hundreth of that in 
acetone. It is of interest to note that there is an approximate 
parallelism between the values of fc, and the values of the dielec- 
tric constant of the different media. 

Catalysis. It is a familiar fact that the velocity of reaction 
is frequently greatly accelerated by the presence of a foreign sub- 
stance which apparently does not participate in the reaction, and 
which remains unchanged when the reaction is complete. For 
example, cane sugar is inverted very slowly by pure water alone, 
but when a trace of acid is added the reaction is greatly acceler- 
ated. A substance which is capable of exerting such an acceler- 
ating action is termed a catalyst, and the process is known as 
catalysis. In addition to the fact that a relatively-small amount 
of a catalyst is capable of effecting the transformation of large 
amounts of material, there are two other important character- 
istics of catalytic action which should be mentioned: viz., (a) a 
catalyst does not initiate a reaction but simply promotes it; and 
(6) the equilibrium is not disturbed by the presence of a catalyst, 
since the velocities of the direct and reverse reactions are each 
altered to the same extent. As the result of a series of experi- 
ments, Ostwald concludes that the catalytic effect of acids in 
hastening the inversion of cane sugar is directly proportional to 
the concentration of the hydrogen ion, and, in general, is inde- 
pendent of the nature of the anion. Similarly, the catalytic action 
of bases may be attributed to the hydroxyl ion, the effect being 
proportional to the concentration of this ion. In fact we may 



380 THEORETICAL CHEMISTRY 

formulate the following fundamental law of catalysis: The 
degree of catalytic action is directly proportional to the concentration 
of the catalytic agent. Almost every chemical reaction can be 
accelerated by the addition of an appropriate catalyst. A few 
typical reactions which are accelerated catalytically are here 
given, together with the catalyst employed : 

Catalyst hydrogen ion, 

CH 3 COOC 2 H 5 + H 2 O = CH 3 COOH + C 2 H 5 OH, 

Catalyst hydroxyl ion, 

2 CH 3 -CO-CH 3 = CH 3 CO-CH 2 C(CH 3 ) 2 OH, 

Catalyst finely divided platinum, 

2 SO 2 + 2 = 2 SO 3 , 
2 CH.OH + O 2 = 2 H-COH + 2 H 2 O, 
2 H 2 + O 2 = 2 H 2 O, 

Catalyst water vapor, 

2 CO + O 2 = 2 CO 2 , 
NH 4 C1 = NH 3 + HC1, 

Catalyst copper sulphate, 

4 HC1 + 2 = 2 H 2 + 2 C1 2 , 

(Deacon Process) 

Catalyst mercury salts, 

2 C 10 H 8 + 9 2 = 2 C 6 H 4 / + 2 H 2 + 4 CO 2 

\COOH 

(First step in the synthesis of indigo; 

Catalyst colloidal platinum, 

2 H 2 2 = 2 H 2 + O 2 , 

Catalyst enzymes, 

C 6 H 12 6 = 2 C 2 H 5 OH + 2 CO 2 , 

(zymase) 

C 2 H 8 OH = C 3 H 7 COOC 2 H 6 + H 2 O. 

(lipase) 



CHEMICAL KINETICS 381 

It will be seen that catalysis is of great importance in connection 
with many industrial processes as well as in the field of pure 
chemistry. The majority of the reactions occurring within 
living organisms are accelerated catalytically by unorganized 
ferments or enzymes. Thus, before the process of digestion can 
proceed, starch must be changed into sugar. This transformation 
is accelerated by an enzyme called ptyalin occurring in the saliva, 
and by other enzymes found in the pancreatic juice. The digestion 
of albumen is hastened by the enzymes, pepsin and trypsin. As 
a rule each enzyme acts catalytically on just one reaction, or in 
other words the catalytic action of enzymes is specific. Enzymes 
are very sensitive to traces of certain toxic substances such as 
hydrocyanic acid, iodine, and mercuric chloride. 

An interesting series of experiments by Bredig * on the catalytic 
action of colloidal metals, established the fact that these sub- 
stances resemble the enzymes very closely in their behavior. 
Thus, they are " poisoned " by the same substances which inhibit 
the activity of the enzymes, and they show the same tendency to 
recover when the amount of the poison does not exceed a certain 
limiting value. Because of this close similarity, Bredig called the 
colloidal metals inorganic ferments. 

It sometimes happens that one of the products of a chemical 
reaction functions as a catalyst to the reaction. Thus, when 
metallic copper is dissolved in nitric acid, the reaction proceeds 
slowly at first and then, after a short interval, the speed of the 
reaction is greatly augmented. The acceleration is due to the 
catalytic action of the nitric oxide evolved. This phenomenon 
is known as autocatalysis. In reactions where autocatalysis 
occurs,the velocity increases with the time until a certain maximum 
value is reached, after which the velocity steadily diminishes. In 
ordinary reactions the initial velocity is the greatest. 

It sometimes happens that the speed of a reaction is retarded 
by the presence of a trace of some foreign substance. Thus, 
Bigelow f has shown that the rate of oxidation of sodium sulphite 
is retarded by the presence in the solution of only one one-hundred- 

* Zeit. phys, Chem., 31, 258 (1899). 
t Zeit. Dhvs. Chem.. 26. 493 (1898). 



382 THEORETICAL CHEMISTRY 

and-sixty-thousandth of a formula weight of mannite per liter. 
Such a substance is termed a negative catalyst. 

Mechanism of Catalysis. As to the cause of catalytic action 
very little is known. In fact it is more reasonable to suppose that 
the mechansim of catalysis varies with the nature of the reaction 
and the nature of the catalyst, than to conceive all catalytic effects 
to be traceable to a common origin. One of the earliest hypotheses 
as to the mechanism of catalysis was put forward by Liebig. He 
suggested that the catalyst sets up intramolecular vibrations which 
assist chemical reaction. The vibration theory was gradually 
abandoned as its inadequacy came to be recognized. Of the many 
explanations which have been offered to account for catalytic 
acceleration, that involving the formation of hypothetical inter- 
mediate compounds with the catalyst has been accepted with the 
greatest favor. Thus, if a reaction represented by the equation 

A + B = AB, 

takes place very slowly under ordinary conditions, it is possible 
to accelerate its velocity by the addition of an appropriate cat- 
alyst (7. According to the theory of intermediate compounds, 
the catalyst is supposed to act in the following manner: 

(1) A + C = AC, 
(2) 



As will be seen, the catalyst is regenerated in the second stage 
of the reaction. In 1806 Clement and Desormes suggested that 
the action of nitric oxide in promoting the oxidation of sulphur 
dioxide in the manufacture of sulphuric acid was purely catalytic. 
As is well known, the rate of the reaction represented by the 
equation 

2 SO 2 + O 2 = 2 S0 3 , 

is very slow. The accelerating action of nitric oxide on the 
reaction may be represented in the following manner: 



(1) 2NO + O 2 = 2N0 2 , 
and 

(2) S0 2 + N0 2 - S0 3 + NO. 



CHEMICAL KINETICS 383 

This explanation, first offered by Clement and Desormes, is still 
regarded as the most plausible explanation of the part played by 
the oxides of nitrogen in the synthesis of sulphuric acid. It is 
apparent that this so-called explanation is far from complete. In 
fact, it must be admitted that we have no adequate explanation 
of the phenomenon of catalysis. When we are able to answer 
the question "Why does a chemical reaction take place?" 
then we may be able to explain the accelerating and retarding 
influences of certain foreign substances on the speed of reactions. 
Ostwald likens the action of a catalyst to that of a lubricant on a 
machine it helps to overcome the resistance of the reaction. 
If the velocity of a reaction is represented by an equation similar 
to that expressing Ohm's law, we have 

, ., f , . driving force 

velocity of reaction = ~ 

resistance 

The driving force is the same thing as the free energy or chemical 
affinity of the reacting substances; of the resistance we know 
practically nothing. The velocity, according to the above expres- 
sion, can be increased in either of two ways, viz., (1) by increas- 
ing the driving force, or (2) by diminishing the resistance. It is 
inconceivable that a catalyst can exert any effect upon the chem- 
ical affinity of the reacting substances, so that we are forced to 
conclude that its action must be confined to lessening the 
resistance.* 

PROBLEMS. 

1. When a solution of dibromsuccinic acid is heated, the acid decom- 
poses into brom-maleic acid and hydrobromic acid according to the 
equation 

CHBr-COOH CH-COOH 



| =|| + HBr. 

CHBr-COOH CBr-COOH 



* For an excellent review of the subject of catalysis the student is advised 
to consult "Die Lehre von der Reaktionsbeschleunigung durch Fremdstoffe," 
by W. Herz. Ahrens* "Sammlung chemischer and chemisch-technischer 
Vortraege." 



384 



THEORETICAL CHEMISTRY 



At 50 the initial titre of a definite volume of the solution was jT = 
10.095 cc. of standard alkali. After t minutes the titre of the same 
volume of solution was Tt cc. of standard alkali. 

t 214 380 

T t 10.095 10.37 10.57 

(a) Calculate the velocity-constant of the reaction. 

(b) After what time is one-third of the dibromsuccinic acid decom- 
posed? An*, (a) 0.000260; (b) 1559 minutes. 

2. From the following data show that the decomposition of H 2 2 in 
aqueous solution is a unimolecular reaction: 

Time in minutes 10 20 

n 22 8 13.8 8.25cc. 

n is the number of cubic centimeters of potassium permanganate required 
to decompose a definite volume of the hydrogen peroxide solution. 

3. In the saponification of ethyl acetate by sodium hydroxide at 10, 
y cc. of 0.043 molar hydrochloric acid were required to neutralize 100 cc. 
of the reaction mixture t minutes after the commencement of the reaction. 

t 4 89 10 37 28 18 infinity 

y 61.95 50.59 42.40 2935 14.92 

Calculate the velocity-constant when the concentrations are expressed 
in mob per liter. Ans. Mean value of k = 2.38. 

4. The velocity-constant of formation of hydriodic acid from its ele- 
ments is 0.00023; the equilibrium constant at the same temperature is 
0.0157. What is the velocity-constant of the reverse reaction? 

Ans. 0.0146. 

5. Determine the order of the following reaction: 

6 FeCl 2 + KC10 3 + 6 HC1 = 6 FeCl s + KC1 + 3 H 2 0. 
When the initial concentration of the reacting substances is 0.1, the 
changes in concentration at successive times are as follows : 



Time (minutes). 


Change in Ccn 
centration. 


5 


0048 


15 


0122 


35 


0238 


60 


0329 


110 


0452 


170 


0525 



Ans. Third order. 



CHAPTER XVIII. 
ELECTRICAL CONDUCTANCE. 

Historical Introduction. In a book of this character it is 
impossible to give anything like a complete historical sketch of 
electrochemistry. Before entering upon an outline of this inter- 
esting division of theoretical chemistry, however, it is desirable 
to consider very briefly a few of the theories which have played a 
prominent part in the development of our modern views concern- 
ing electrochemical phenomena. While the early observations of 
Beccaria and others pointed to the probability of the existence 
of some relation between chemical and electrical phenomena, it 
was not until the beginning of the nineteenth century that the 
science of electrochemistry had its birth. The epoch-making 
discovery by Volta of a means of obtaining electrical energy from 
chemical energy, gave the initial impulse to all the brilliant dis- 
coveries and investigations upon which the modern science of 
electrochemistry is based. The apparatus devised by Volta, 
known as the voltaic pile, consisted of disks of zinc and silver 
placed alternately over one another, the silver disk of one pair 
being separated from the zinc disk of the next by a piece of 
blotting paper moistened with brine. Such a pile, if composed 
of a sufficient number of pairs of disks, will produce electricity 
enough to give a shock, if the top and bottom disks, or wires 
connected with them, be touched with the moist fingers. This 
discovery placed in the hands of the investigator a source of 
electricity by means of which experiments could be performed 
which had hitherto been impossible. Shortly after the discovery 
of the voltaic pile, Nicholson and Carlisle * effected the decom- 
position of water, and Davyf isolated the alkali metals As 
a result of these experiments, Davy was led to formulate his 

* Nich. Jour., 4, 179 (1800). 

t Ibid., 4, 275, 326 (1800); Gilb, Ann., 7, 114 (1801). 
385 



386 THEORETICAL CHEMISTRY 

electrochemical theory. According to this theory, the atoms of 
different substances acquire opposite electrical charges by con- 
tact, and thus mutually attract each other. If the differences 
between the charges are small, the attraction will be insufficient 
to cause the atoms to leave their former positions; if it is great, 
a rearrangement of the atoms will occur and a chemical com- 
pound will be formed. In terms of this theory, electrolysis con- 
sists in a neutralization of the charges upon the atoms. 

The theory of Davy was soon superseded by that of Berzelius.* 
According to the latter theory, every atom is charged with both 
kinds of electricity which exist upon the atoms in a polar arrange- 
ment, the electrical behavior of the atom being determined by the 
kind of electricity which is in excess. Chemical attraction is merely 
the electrical attraction of oppositely-charged atoms. Since each 
atom is endowed with both positive and negative electrification, 
one charge being in excess, it follows that the compound formed 
by the union of two or more atoms will be positively or negatively 
charged according to whichever charge remains unneutralized after 
the atoms have combined. Two compounds, the one charged pos- 
itively and the other negatively, may thus in turn combine, a 
more complex compound being formed. Shortly after Berzelius 
formulated his theory, it became the subject of much discussion 
and was severely critized. Thus, it was pointed out that if 
chemical combination results from the neutralization of oppo- 
sitely-charged atoms, then as soon as the charges have become 
equalized, there no longer exists any attractive force and the com- 
pound must again decompose. This objection was easily overcome 
by assuming that as soon as the union between the atoms is 
broken, they again acquire their original charges and, in conse- 
quence, recombine. In other words, a chemical compound is to 
be regarded as existing in a state of unstable equilibrium. An- 
other, and apparently insurmountable, objection to the -theory 
resulted from the exceptions presented by acetic acid and some of 
its substitution products. 

According to the theory of Berzelius, chemical combination is 
entirely dependent upon the nature of the electrical charges resid- 
* Gilb. Ann., 27, 270 (1807). 



ELECTRICAL CONDUCTANCE 387 

ing on the atoms. From this statement it follows that the prop- 
erties of a chemical compound must be a function of the electrical 
charges upon the atoms of its constituents. It was shown that 
when the three hydrogen atoms of the methyl group in acetic 
acid are successively replaced by chlorine, the chemical properties 
of the original substance are not materially altered. According 
to Berzelius, the three hydrogen atoms are positively charged 
while the three chlorine atoms are negatively charged. That 
three negative charges could be substituted for three positive 
charges in acetic acid without producing a more marked change 
in its properties, could not be satisfactorily accounted for by the 
theory. This criticism was for a long time considered as an 
insuperable barrier to the acceptance of the theory. Shortly 
before the close of the nineteenth century, J. J. Thomson * showed 
that this objection has little or no weight. When hydrogen gas 
is electrolyzed in a vacuum-tube and the spectra at the two elec- 
trodes are compared, Thomson found them to differ widely. 
From this he concluded that the molecule of hydrogen gas is in 
all probability made up of positively- and negatively-charged parts 
or ions. He then extended his experiments to the vapors of cer- 
tain organic compounds. In discussing these experiments he 
says: "In many organic compounds, atoms of an electro- 
positive element, hydrogen, are replaced by atoms of an elec- 
tronegative element, chlorine, without altering the type of the 
compound. Thus, for example, we can replace the four hydrogen 
atoms in CH 4 by chlorine atoms, getting, successively, the com- 
pounds CHsCl, CH 2 C1 2 , CHC1 3 , and CC1 4 . It seemed of interest 
to investigate what was the nature of the charge of electricity on 
the chlorine atoms in these compounds. The point is of some 
historical interest, as the possibility of substituting an electro- 
negative element in a compound for an electropositive one was 
one of the chief objections against the electrochemical theory of 
Berzelius." 

"When the vapor of chloroform was placed in the tube, it was 
found that both the hydrogen and chlorine lines were bright on 
the negative side of the plate, while they were absent from the 

* Nature, 52, 451 (1895). 



388 THEORETICAL CHEMISTRY 

positive side, and that any increase in brightness of the hydrogen 
lines was accompanied by an increase in the brightness of those 
due to chlorine. The appearance of the hydrogen and chlorine 
spectra on the same side of the plate was also observed in methyl- 
ene chloride and in ethylene chloride. Even when all the 
hydrogen in methane was replaced by chlorine, as in carbon tetra- 
chloride, the chlorine spectra still clung to the negative side of 
the plate. The same point was tested with silicon tetrachloride 
and the chlorine spectrum was brightest on the negative side of 
the plate. From these experiments it would appear, that the 
chlorine atoms in the chlorine derivatives of methane are charged 
with electricity of the same sign as the hydrogen atoms they 
displace. " 

Electrical Units. In 1827, Dr. G. S. Ohm enunciated his well- 
known law of electrical conductance, viz.: The strength of the 
electric current flowing in a conductor is directly proportional to the 
difference of potential between the ends of the conductor, a?id inversely 
proportional to its resistance. If C represents the strength of the 
current, E the difference of potential, and R the resistance, then 
Ohm's law may be formulated thus: 

E 

C = - 
C R 

The unit of resistance is the ohm, that of difference of potential 
or electromotive force, the volt, and that of current, the ampere. 
The ohm is defined as the resistance of a column of mercury 
106.3 cm. long and 1 sq. mm. in cross section at C. s The 
ampere is defined as the current which will cause the deposition 
of 0.001118 gram of silver from a solution of silver nitrate in 1 
second. The volt may be defined as the electromotive force 
necessary to drive a current of 1 ampere through a resistance of 
1 ohm. The unit quantity of electricity is the coulomb. This 
amount of electricity passes when a current of a strength of one 
ampere flows for one second. One gram equivalent of any ion 
carries 96,500 coulombs, a quantity of electricity known as the 
faraday = F . As has already been pointed out, any form of 
energy may be considered as the product of two factors, a capac- 
ity factor and an intensity factor. 



ELECTRICAL CONDUCTANCE 389 

The capacity factor of electrical energy is the coulomb while 
the intensity factor is the volt, i.e., 

electrical energy = coulombs X volts. 

The unit of electrical energy, therefore, is the volt-ampere-second 
commonly called the watt-second. One watt-second is the elec- 
trical work done by a current of 1 ampere flowing under an elec- 
tromotive force of 1 volt for 1 second, and is equivalent to 1 X 10 7 
C.G.S. units. The thermal equivalent of electrical energy may be 
calculated from the relation 

electrical energy in absolute units , , . - , , 
,& . = h ea t eq uiv . of elect, energy, 

mechanical equiv. of heat 
or 

42^T = 0-2394 cal. = l watt-second. 

Faraday's Laws. When two platinum plates or electrodes, one 
connected to the positive and the other to the negative terminal 
of a battery, are immersed in a solution of sodium chloride, it 
will be found that hydrogen is immediately evolved at the nega- 
tive electrode and oxygen at the positive electrode. If the salt 
solution is previously colored with a few drops of a solution of 
litmus it will be observed that the portion of the solution in the 
neighborhood of the positive electrode will turn red, indicating 
the formation of an acid, while that in the neighborhood of the 
negative electrode will turn blue, showing the formation of a 
base. The same changes will take place whether the electrodes 
are placed near together or far apart, and furthermore, the evolu- 
tion of gas and the change in color at the electrodes commences 
as soon as the circuit is closed. The study of these phenomena 
led Faraday * to the conclusion, that when an electric current 
traverses a solution, there occurs an actual transfer of matter, 
one portion travelling with the current and the other portion 
moving in the opposite direction. At the suggestion of the philol- 
ogist Whewell, Faraday termed these carriers of the current, ions 
= to wander). He also called the electrode connected to 
* Experimental Researches (1834). 



390 



THEORETICAL CHEMISTRY 



the positive terminal of the battery, the anode (dm = up and 
809 = way), and the electrode connected to the negative terminal 
the cathode (/cara = down and 6805 = way). The ions which 
move toward the anode he called anions, while those which 
migrate toward the cathode he called cations. The whole process he 
termed electrolysis. The question of the relationship between the 
amount of electrolysis and the quantity of electricity passing 
through a solution was investigated by Faraday. As a result of 
his experiments he enunicated the following laws which are com- 
monly known as the laws of Faraday: 

(1) For the same electrolyte, the amount of electroysis is propor- 
tional to the quantity of electricity which passes. 

(2) The amounts of substances liberated at the electrodes when the 
same quantity of electricity passes through solutions of different 
electrolytes, are proportional to their chemical equivalents. The 
chemical equivalent of any ion is equal to the atomic weight divided 
by its valence. If the same quantity of electricity is passed 
through solutions of hydrochloric acid, silver nitrate, cuprous 
chloride, cupric chloride, and auric chloride, the relative amounts 
of the different cations liberated will be as follows: 



Electrolyte. 


Chem. Equiv. of 
Cation. 


HC1 


H' = 1 


AgNOs 


Ag' 108 


Cu 2 Cl 2 


Cu' = 63.4 


CuCl 


Cu" =63.4-r2 


AuCl 3 


Aif" = 197-^3 



The electrochemical equivalent of an element or group of elements 
is the weight in grams which is liberated by the passage of one 
coulomb of electricity. The electrochemical equivalents are, 
according to Faraday's second law, proportional to the chemical 
equivalents. The quantity of electricity necessary to liberate one 
chemical equivalent in grams is called a faraday. This is a very 
important unit in electrochemical calculations. Since one coulomb 
liberates 0.00001036 gram of hydrogen, 1 -5- 0.00001036 = 96,500 



ELECTRICAL CONDUCTANCE 391 

coulombs of electricity will be required to liberate one gram equiv- 
alent of hydrogen. The same quantity of electricity will liberate 
35.45 X 0.00001036 = 0.000368 gram of chlorine, and 108 X 
0.00001036 = 0.001118 gram of silver. Or, in general, since one 
coulomb of electricity liberates 0.00001036 gram of hydrogen, it 
will cause the liberation of 0.00001036 w grams of any other ele- 
ment whose equivalent weight is w. 

The Existence of Free Ions. When an electrolyte is de- 
composed by the electric current, the products of decomposition 
appear at the electrodes. The fact that the liberation of the prod- 
ucts of decomposition is independent of the distance between 
the electrodes caused considerable difficulty in the early history 
of electrolysis. It was evident that the two products could 
hardly be derived from the same molecule, but must come from 
two different molecules. Several theories were advanced to 
account for the experimental results. Thus, in the electrolysis of 
water it was suggested that the two gases, hydrogen and oxygen, 
were not derived from the water but that electricity itself pos- 
sessed an acid character. Grotthuss * was the first to propose a 
rational hypothesis as to the mechanism of electrolysis. He 
assumed that when the electrodes in an electrolytic cell are con- 
nected with a source of electricity, the molecules of the electrolyte 
arrange themselves in straight lines between the electrodes, the 
positive poles being directed toward the negative electrode and 
the negative poles toward the positive electrode. When elec- 
trolysis begins, the cation of the molecule nearest the cathode 
is liberated at the cathode and the anion of the molecule nearest 
the anode is liberated at the anode. The anion which is left 
free near the cathode then combines with the cation of the next 
adjoining molecule, the anion thus left uncombined uniting with 
the cation of its nearest neighbor, a similar exchange of partners 
continuing throughout the entire molecular chain. Under the 
directive influence of the two electrodes, the newly-grouped mole- 
cules then rotate so that the positive poles all face the negative 
electrode and the negative poles all face the positive electrode. 
The process is then repeated, another molecule being electrolysed. 

* Ann. de Chim. [1], 58, 54 (1806). 



392 THEORETICAL CHEMISTRY 

This theory of electrolysis appears to have been accepted by 
Faraday. Its inherent defect was first pointed out by Grove.* 
From his experiments with the oxy-hydrogen cell, which derives 
its energy from the union of hydrogen and oxygen, he pointed out 
that a decomposition of the molecules of water is not essential 
for the evolution of these two gases, but that the molecules must 
be already in a state of partial decomposition. This suggestion 
was followed up by Clausius. f He argued that if an expenditure 
of energy is necessary to decompose the molecules, electrolysis 
should be impossible at very low voltages. Experiment showed 
that when silver nitrate is electrolyzed between silver electrodes, 
decomposition takes place at voltages which are much below the 
voltage corresponding to the energy of formation of silver nitrate. 
In other words, it requires very little energy to decompose a salt 
which is formed with the evolution of a large amount of energy, 
a result which is in contradiction to the principle of the conserva- 
tion of energy. Clausius was thus forced to conclude "that the 
supposition that the constituents of the molecule of an electrolyte 
are firmly united and exist in a fixed and orderly arrangement is 
wholly erroneous." 

As a result of his investigation of the synthesis of ethyl ether 
from alcohol and sulphuric acid, Williamson f concluded "that in 
an aggregate of the molecules of every compound, a constant inter- 
change between the elements contained in them is taking place." 
In the same paper he writes, "each atom of hydrogen does not 
remain quietly attached all the time to the same atom of chlorine, 
but they are continually exchanging places with one another." 
This view was accepted by Clausius, although he had no means of 
determining the extent to which the electrolyte was broken down 
or dissociated into free ions. 

In 1887, Arrhenius developed the views of Clausius by showing 
how the degree of dissociation of the molecules of an electrolyte 
can be deduced from measurements of the electrical conductance 

* Phil. Mag., 27, 348 (1845). 
t Pogg. Ann., 101, 338 (1857). 
t Lieb. Ann., 77, 37 (1851). 
Zeit. phys. Chem., i, 631 (1887). 



ELECTRICAL CONDUCTANCE 



393 



of its solutions, as well as from measurements of osmotic pressure 
and freezing-ptfint lowering. The important generalization sum- 
marizing these conceptions is known as the theory of electrolytic 
dissociation, to. which reference has already been made in earlier 
chapters (see page 227). 

The Migration of the Ions. Since the passage of a current of 
electricity through a solution of an electrolyte causes the dis- 
charge of equivalent amounts of positive and negative ions at the 
electrodes, it might be inferred that the ions all move with the 
same speed. That this inference is incorrect, was first shown by 
Hittorf * as the result of his observations on the changes in con- 
centration of the solution in the neighborhood of the electrodes 







00 l 



O 010 O 0j0 O O 
I n ! 

| |0 

0|0 00 



III J 

I 
0J0 0J0 

I (5) i 
I j 

00000 010 0000 0[0 



Fig. 87. 

during electrolysis. He showed that different ions migrate with 
different speeds, and that the faster moving ions carry a greater 
proportion of the current than the slower moving ions. The 
effect of unequal ionic velocities on the concentrations of the 
solutions around the electrodes is clearly shown by the ac- 
companying diagram (Fig. 87) due to Ostwald. The anode and 

* Pogg. Ann., 89, 177; 98, 1- 103, 1; 106, 337, 513 (1853-1859). 



394 THEORETICAL CHEMISTRY 

cathode in an electrolytic cell are represented by the vertical lines 
A and C respectively. The cell is divided into three compart- 
ments by means of porous diaphragms,* represented by the ver- 
tical dotted lines. The cations are represented by dots (*) and 
the anions by dashes ('). Before the current passes through the 
cell, the concentration of the solution is uniform throughout, the 
conditions being represented by I. Now let us imagine that only 
the anions move when the current is established. The conditions 
when the chain of anions has moved two steps toward the anode 
are shown in II. Each ion which has been deprived of a partner 
is supposed to be discharged. It will be observed that although 
the cations have not migrated toward the cathode, yet an equal 
number of positive and negative ions are discharged, and that 
while the concentration in the anode compartment has not changed, 
the concentration in the cathode compartment has diminished to 
one-half its original value. 

Let us now suppose that both anions and cations move with the 
same speed, and as before, let each chain of ions move two steps 
toward their respective electrodes, as indicated in III. It will be 
seen that four positive and four negative ions have been dis- 
charged, and that the concentration of the electrolyte in the anode 
and cathode compartments has diminished to the same extent. 
Finally, let us assume that the ratio of the speeds of the cations 
to that of the anions is as 3 : 2. When the cations have moved 
three steps toward the cathode and the anions have moved two 
steps toward the anode, the conditions will be as shown in IV. 
It is evident that five positive and five negative ions have been 
discharged, and that the concentration in the cathode compart- 
ment has diminished by two molecules while the concentration 
in the anode compartment has diminished by three molecules. 
It will b*e observed that the change in concentration in either of 
the electrode compartments is proportional to the speed of the 
ion leaving it. Thus, in II, the concentration in the cathode 
compartment diminishes while that in the anode compartment 
remains unchanged, since only the anion moves. In like manner, 
the change in concentration about the electrodes in III corre- 
sponds with the fact that both ions migrate .t the Same rate. 



ELECTRICAL CONDUCTANCE 395 

In IV the ratio of the change in concentration in the cathode 
compartment to that in the anode compartment is as 2 : 3. It 
will be apparent from tfyese examples, that the relation between 
the speeds of the ions and the corresponding changes in concen- 
tration at the electrodes may be expressed by the following pro- 
portion: 

Change in concentration at anode __ speed of cation 
Change in concentration at cathode speed of anion 

If the relative speed of the cations is represented by u, and that 
of the anions by v, then the total quantity of electricity trans- 
ported will be proportional to u + v: of this total, the fractions 

carried by the anion and cation respectively, will be n = : , 



and 1 w = -p The values of these ratios, n and 1 n, 
u ~i v 

are called the transport numbers of the anion and cation respec- 
tively. It is apparent from the diagram, that if the electrolysis 
is not carried too far, the concentration of the solution in the inter- 
mediate compartment will undergo no change. In order to deter- 
mine transport numbers, therefore, it is simply necessary to 
remove portions of the solutions in the immediate vicinity of the 
two electrodes and determine the concentration of the electrolyte 
analytically. The success of the experiment depends upon keep- 
ing the concentration of the intermediate compartment unaltered. 
Experimental Determination of Transport Numbers. Various 
forms of apparatus have been constructed for the determination 
of transport numbers, among which one of the most satisfactory 
is that devised by Jones and Bassett,* and shown in Fig. 88. It 
consists of two vertical tubes of wide bore connected by a U-tube 
fitted with a stop-cock. Into each of two electrodes, made of 
a suitable metal, is riveted a short piece of stout platinum wire, 
which is then sealed into heavy-walled glass tubes. The exposed 
end of the platinum wire on the under side of each electrode is 
covered with a drop of fusion glass. The tubes carrying the 
electrodes are fitted into holes bored through the ground glass 
* Am. Chem. Jour., 32, 409 (1904). 



396 



THEORETICAL CHEMISTRY 



stoppers which close the right and left arms of the apparatus. 
Two small graduated tubes are sealed to the two vertical tubes 
just below the stoppers. These tubes allow for any slight dis- 
placement of the solution due to expansion or the formation of 






Fig. 88. 

gas, and at the same time make it possible to level the apparatus 
accurately. When electrolysis has proceeded far enough, the 
circuit is broken and the stop-cock closed, thus preventing the 
mixing of the solutions in the anode and cathode compartments. 
The solutions in the two halves of the apparatus are then rinsed out 
into separate beakers and the concentration of each is determined 
analytically. Knowing the initial concentration of the solution 
and the final concentrations at the two electrodes, together with 
the total quantity of electricity which has passed through the 
apparatus during the experiment, we have all of the data neces- 
sary for the calculation of the transport numbers of the two ions. 



ELECTRICAL CONDUCTANCE 397 

The following example will serve to make the method of calcu- 
lation clear: In an experiment to determine the transport 
numbers of the ions of silver nitrate, a solution containing 0.00739 
gram of that salt per gram of water was prepared. The solution 
was introduced into the migration apparatus and, after inserting 
silver electrodes, a small current was passed through the appa- 
ratus for two hours. A silver coulometer was included in the cir- 
cuit, and 0.0780 gram of silver was deposited by the current. 

This mass of silver is equivalent to 0.000723 gram-equivalent. 
After the circuit was broken, the anode solution was rinsed out and 
its concentration determined analytically. It was found to con- 
tain 0.2361 gram of silver nitrate to 23.14 grains of water. This 
amount of solution contained originally 23.14 X 0.00739 = 
0.1710 gram of silver nitrate. Thus, the amount of silver nitrate 
in the anode compartment had increased by 0.2361 0.1710 = 
0.0651 gram of silver nitrate, or 0.000383 gram-equivalent of 
silver. Obviously the increase in the concentration of the nitrate 
ion must have been the same. The amount of silver dissolved 
from the anode must have been equal to that deposited in the 
coulometer, or since 0.000723 gram-equivalent of silver was 
deposited and the actual increase found was 0.000383 gram- 
equivalent, the difference, 0.000723 - 0.000383 = 0.000340 gram- 
equivalent, is the amount of silver which migrated away from the 
anode. At the same time 0.000383 gram-equivalent of nitrate 
ions migrated into the anode compartment. The ratio of the 
speed of migration of the silver ions to that of the nitrate ions is 
as 0.000340 : 0.000383. - Since 0.000723 gram-equivalent of silver 
ions measures the total quantity of electricity transported, the 
transport numbers of the two ions will be as follows: 



Transport number of Ag- = 1 - n = Q = 0.470, 

Transport number of NO 8 '= n = S| - 0.530. 



These numbers can be checked by a similar calculation based on 
the change in concentration in the cathode compartment. 



398 



THEORETICAL CHEMISTRY 



The following table gives the transport numbers of the anions 
of various electrolytes at different dilutions, V being the number 
of liters of solution containing one gram-equivalent of solute. 
The transport numbers of the corresponding cations can be found 
by subtracting the transport numbers of the anions from unity. 

TRANSPORT NUMBERS OF ANIONS. 



y= 


100 


50 


20 


10 


5 


2 


i 


0.5 


KC1 I 
KBr 


0.506 


507 


507 


508 


509 


0.513 


0.514 


515 


KI [ 
NH 4 C1 J 
NaCl 






614 


617 


0.620 


626 


637 




KNO, 








497 


0.496 


492 


0.487 


0.479 


AgNOa 


528 


528 


0.528 


0.528 


527 


0.519 


501 


0.476 


KOH . . 








0.735 


736 








HC1 






0.172 


172 


0.172 


O.i73 


176 




iBaCU .... 














0.640 


0.657 


K 2 CO 3 . . 












435 


434 


413 


CuSO 4 . 




620 


626 


0.632 


6.643 


668 


696 


0.720 


H 2 SO 4 












182 


174 























It is apparent from the table that the transport numbers are 
not entirely independent of the concentration. They also vary 
slightly with the temperature and approach the limiting value, 
0.5, at high temperatures. 

Specific, Molar and Equivalent Conductance. As is well 
known, the resistance of a metallic conductor is directly propor- 
tional to its length and inversely proportional to its area of cross- 
section. Similarly, the resistance of an electrolyte is proportional 
to the length and inversely proportional to the cross-section of 
the column of solution between the two electrodes. The specific 
resistance of an electrolyte may be defined as the resistance in 
ohms of a column of solution one centimeter long and one square 
centimeter in cross-section. Specific conductance is the reciprocal 
of specific resistance. Since the conductance of a solution is 
almost wholly dependent upon the amount of solute present, it 
is more convenient to express conductance in terms of the molar 
or equivalent concentration. The molar conductance n, is the 



ELECTRICAL CONDUCTANCE 



399 



conductance in reciprocal ohms, of a solution containing one mol 
of solute when placed between electrodes which are exactly one 
centimeter apart. The equivalent conductance A is the conduc- 
tance in reciprocal ohms of a solution containing one gram- 
equivalent of solute when placed betweep electrodes which are 
one centimeter apart. If K denotes the specific conductance of a 
solution and F m , the volume in cubic centimeters which contains 
one mol of solute, then 



and in like manner 



A = 



where V e is the volume of solution in cubic centimeters which 
contains one gram-equivalent of solute. The following table 
gives the specific and molar conductance of solutions of sodium 
chloride at 18 C.: 



Concentration. 


Dilution. 


Sp. Cond. 


Molar Cond. 


1 


1,000 


0.0744 


74 4 


1 


10,000 


0.00925 


92 5 


0.01 


100,000 


001028 


102 8 


001 


1,000,000 


0.0001078 


107.8 


0001 


10,000,000 


0.00001097 


109 7 



It will be observed that the molar conductance increases with the 
dilution up to a certain point beyond which it remains nearly 
constant. That the molar conductance should change but little 
will become apparent from the following considerations: 
Imagine a rectangular cell of indefinite height and having a cross- 
sectional area of one square centimeter, and further assume that 
two opposite walls can function as electrodes. Let 1000 cc. of a 
solution containing one mol of solute be introduced into the cell, 
and let its conductance be determined. Now let the solution be 
diluted to 2000 cc. and the conductance of the diluted solution be 
measured. While the specific conductance of the diluted solution 
is reduced to one-half of its original value, yet since the electrode 



400 



THEORETICAL CHEMISTRY 



surface in contact with the solution is doubled, owing to the fact 
that the solution stands at twice the original height in the 
cell, the total conductance due to one mol of solute remains un- 
changed. This, of course, is only the case with completely ionized 
solutes. 

Determination of Electrical Conductance. The determination 
of the electrical conductance of a solution resolves itself into 
the determination of its resistance by a simple modification of 
the familar Wheatstone-bridge method. The arrangement of the 
apparatus for this method devised by Kohlrausch * is represented 
diagrammatically in Fig. 89, where ab is the bridge wire, B is a 





Fig. 89. 

resistance box, and C is a cell containing the solution whose 
resistance is to be measured. The points d and c are connected 
to a small induction coil / which gives an alternating current. 
This is necessary in order to prevent polarization which would 
occur if a direct current were used. The use of the alternating 
current necessitates the substitution of a telephone, T, for the 
galvanometer usually employed in measuring resistance. The 
positions of the induction coil and telephone are sometimes inter- 
changed, but the arrangement shown in the diagram is to be pre- 
ferred, since it insures a high electromotive force where the sliding 

* Wied. Ann., 6, 145 (1879); n, 653 (1880); 26, 161 (1885). 



ELECTRICAL CONDUCTANCE 



401 



contact c touches the wire, this being the most uncertain connec- 
tion in the entire arrangement. A small accumulator A, serves 
to operate the induction coil. In making a measurement, the 
coil is connected with the accumulator and the vibrator adjusted 
so that a high mosquito-like tone is emitted; then the sliding 
contact c is moved along the wire ab until the sound in the tele- 
phone reaches a minimum, the position of the point of contact 
with the bridge-wire being read on the millimeter scale placed 
below. According to the principle of the Wheatstone bridge, it 
follows that 

C^fa 

B~ ac' 

Since the resistance B a*id the lengths be and ac are known, the 
resistance C can be calculated. Various types of conductance 
cells are in use, depending upon whether 
the solution has a high or a low resistance. 
The form shown in Fig. 90 is widely used. 
The two electrodes are made of platinum 
foil, connection with the mercury in the two 
glass tubes it being established by means of 
two pieces of stout platinum wire sealed 
through the ends of these tubes. The tubes it 
are fastened into a tight-fitting vulcanite 
cover so that the electrodes may be re- 
moved, rinsed and dried without altering 
their relative positions. Before the cell is 
used, the electrodes must be coated electro- 
lytically with platinum black. It is not 
necessary to know the area of the elec- 
trodes or the distance between them, since 
it is possible to determine a factor, termed 
the resistance capacity, by means of which 
the results obtained with the cell can be ^ig. 90. 

transformed into reciprocal ohms. To this 
end the specific conductances of a number of standard solu- 
tions have been carefully determined by Kohlrausch; thus, for 




402 THEORETICAL CHEMISTRY 

a 0.02 molar solution of potassium chloride he found th* Allowing 
values: 

K 18 o = 0.002397 and * = 0.002768, 
or 

A l8 o = 119.85 and A 2B o = 138.4. 

Let the resistance of the cell when filled with 0.02 molar potassium 
chloride be C, then according to the principle of the Wheatstone 
bridge we have 

C = B , 
ac 

or denoting the conductance of the solution by L, we obtain 

T 1 ac 

JU = 77 = 



C B be 

f 

Since the specific conductance K must be proportional to the 
observed conductance, we have 



~-^B^bc' 

where K is the resistance capacity of the cell. If the measure- 
ment is made at 18 C., then we have 

= 0.002397 B be 
ac 

Having determined the resistance capacity of the cell we may 
then proceed to determine the conductance of any solution. For 
example, suppose that when the resistance in the box is B', the 
point of balance on the bridge-wire is at c', then the specific con- 
ductance of the solution will be 



* B'bc' 

If k' is multiplied by the volume of the solution, we obtain the 
equivalent conductance, or 

A = 



Relative Conductances of Different Substances. The study 
of the electrical conductance of various solutes in aqueous solu- 
tion, reveals the fact that electrolytes differ greatly in their con- 
ducting power. They may be roughly divided into two classes: 



ELECTRICAL CONDUCTANCE 



403 



those with high conducting power, such as strong acids, strong 
bases, and salts; and those with low conducting power, such as 
ammonia and most of the organic acids and bases. Further- 
more, the equivalent or molar conductance increases with the dilu- 
tion until a dilution of about 10,000 liters is reached, beyond 
which it remains constant. The following table gives the equiv- 
alent conductances of three typical electrolytes, V representing 
the volume of the solution in liters, and A the equivalent con- 
ductance: 

HYDROCHLORIC ACID. 



V. 


A (18) 


333 


201 Q 


1 


278 


10 


324 4 


100 


341 6 


1000 


345 5 



SODIUM HYDROXIDE. 



V. 


A (18). 


333 


100 7 


1 


149 


10 


170 


100 


187 


500.0 


186.0 



POTASSIUM CHLORIDE. 



V. 


A 08). 


333 


82 7 


1 


91.9 


10 


104.7 


100 


114.7 


1,000 


119.3 


10,0000 


120.9 



404 THEORETICAL CHEMISTRY 

The curves shown in Fig. 91 are plotted from the data of the 
foregoing table, and bring out very clearly the differences in con- 
ducting power possessed by the three electrolytes. 

In general the conductance of pure liquids is small. Thus, the 
specific conductance of pure water at 18 is approximately 1 X 10~ 6 




Dilution, V 

Fig. 91. 

reciprocal ohms and, as Walden * has shown, the specific conduct- 
ance of a number of other solvents is of the same order as that 
for water. Mixtures of two liquids, each of which is practically 
non-conducting, may have a conductance differing but little from 
that of the two components; or the mixture may have a very 
high conductance. For example, the conductance of a mixture 

* Zeit. phys. Chem., 46, 103 (1903). 



ELECTRICAL CONDUCTANCE 405 

of water and ethyl alcohol is of the same order of magnitude as 
that of the two components, while on the other hand, a mixture 
of water and sulphuric acid, each of which in the pure state is 
practically a non-conductor, has great conducting power. The 
variation of the specific conductance of mixtures of water and 
sulphuric acid is represented in Fig. 92, the concentrations of sul- 
phuric acid being plotted on the axis of abscissae and the specific 
conductances on the axis of ordinates. It appears that as the 




10 20 30 40 60 60 70 80 90 100 110 
Per Cent Sulphuric Acdcl 

Fig. 92. 

Concentration of the sulphuric acid increases, the specific conduct- 
ance of the mixture increases until 30 per cent of acid is present, 
Beyond which point it gradually diminishes. When pure sul- 
Dhuric acid is present the value of the specific conductance is 
practically zero. On dissolving sulphur trioxide in the pure acid, 
the specific conductance increases slightly to a maximum and then 
falls rapidly to zero. There is a minimum in the curve corre- 
sponding to about 85 per cent of acid, a concentration which 



406 



THEORETICAL CHEMISTRY 



corresponds almost exactly with the hydrate H 2 S0 4 H 2 0. Why 
some liquid mixtures should have marked conducting power and 
others hardly any, it is difficult to explain. Many fused salts, 
such as silver nitrate and lithium chloride, are excellent conductors 
and are thus exceptions to the general rule, that pure substances 
belonging to the second class of conductors possess little conduct- 
ing power. 

The Law of Kohlrausch. The electrical conductance of solu- 
tions was systematically investigated by Kohlrausch who showed 
that the limiting value of the equivalent conductance, which may 
be represented by A*, is different for different electrolytes and 
may be considered as the sum of two independent factors, one of 
which refers to the cation and the other to the anion. This experi- 
mental result is commonly known as the law of Kohlrausch. 

The limiting value of the equivalent conductance is reached 
when the molecules are completely broken down into ions, and 
under these conditions the whole of the electrolyte participates 
in conducting the current. The accompanying table, giving the 
equivalent conductances at infinite dilution of several binary 
electrolytes, illustrates the truth of the law of Kohlrausch. 



EQUIVALENT CONDUCTANCES AT INFINITE DILUTION. 





K 


Na 


Li 


NH 4 


H 


Ag 


Cl . 


1231 


103 


95 


122 


353 




NO 8 . . 
OH 


ll&i 

228 


.98' 
201 






350 


109 


C10 8 


115 










103 


C 2 H 3 2 


94 


73 








83 

















The differences between two corresponding sets of numbers in 
the same vertical column- and of any two corresponding sets of 
numbers in the same horizontal, row. will be found to be nearly 
equal. This could only occur when the limiting conductance is 
the sum of two entirely independent quantities. Each ion 
invariably carries the same charge of electricity and moves with 



ELECTRICAL CONDUCTANCE 407 

its own velocity quite independent of the nature of its compan- 
ion ion. Therefore^ at infinite dilution, we have 



in which l c and l a are the equivalent conductances of the ions of 
the electrolyte at infinite dilution. From this it follows that 

ypu k - A 
n ^l c + l a ~ A.' 
and 



or 

l a = 
and ~ 

l c = (1 - n) Aoo. 

Thus, the equivalent conductance of silver nitrate at infinite dilu- 
tion at 18 is 115.5, while n = 0.518 and 1 - n = 0.482; there- 
fore 

Z = 0.518 X 115. 5 = 59.8, 
and 

l c = 0.482 X 115.5 = 55.7; 

or one gram-equivalent of N(V ions possesses a conductance of 
59.8 when placed between electrodes one centimeter apart and 
large enough to contain between them the entire volume of solu- 
tion in which the NO 3 ' ions exist; and one gram^equivalent of 
Ag" ions under the same conditions have a conductance equal 
to 55.7. 

The values of the ionic conductances at infinite dilution remain 
constant in all solutions in the same solvent at the same temper- 
ature, so that it is possible to calculate the equivalent conductance 
for any substance at infinite dilution. 

In the subjoined table are given the ionic conductances of 
various ions at 18 and infinite dilution, together with their temper- 
ature coefficients. 



408 THEORETICAL CHEMISTRY 

IONIC CONDUCTANCES AT, INFINITE DILUTION. 



Ion. 


i a 


Temp. Coeff . 


Li* 


33 44 


0265 


Na* 


43 55 


0244 


K* 


64 67 


0217 


Rb* 


67 6 


0214 


Cs' 


68 2 


0212 


NH 4 * 


64 4 


0222 


Tl* 


- 66 


0215 


Ag* .*... 


54.02 


0229 


F 


46 64 


0.0238 


Cl' 


65.44 


0216 


Br' s. 


67.63 


0215 


I'. 


66.40 


0213 


SCN' 


56 63 


0211 


C1O 8 ' 


55 03 


0215 


IO 3 ' 


33 87 


0234 


NO 8 ' 


61 78 


0.0205 


H' 


318 




OH' 


174.0 




| Zn" 


45 6 


0251 


| Mg" 


46.0 


0256 


|Ba" 


56 3 


0238 


J Pb" 


61.5 


0243 


JSO 4 " 


68.7 


0227 


\ CO 3 " 


70 


0270 









In the case of weak electrolytes the value of A* cannot be 
determined directly from conductance measurements, since before 
the limiting value is reached, the solution has become so dilute as 
to render accurate measurements of the specific conductance 
impossible. The law of Kohlrausch enables us to get around 
this difficulty. Thus, the value of Aoo for acetic acid must be 
equal to the sum of the conductances of the H* and CHsCOO' 
ions. The conductance of the H* ion at 18 is, according to 
the preceding table, 318. The value of the conductance of the 
CHsCOO' ion can be determined from the conductance of sodium 
acetate at infinite dilution, Aoo for this salt being 78.1 at 18. 
Since the ionic conductance of the Na* ion is 43.55 at 18, it 
follows that the conductance of the CH 3 COO' ion must be 78.1 
43.55 = 34.55. Therefore, for acetic acid we have 

Aoo - lc + L - 318 + 34.55 = 352.55 at 18. 



ELECTRICAL CONDUCTANCE 



409 



Bredig * has shown that the ionic conductance of elementary 
ions is a periodic function of the atomic weight. When the ionic 



I 



80- 
60- 
40 
80- 




B* 



Cd 



40 



80 120 

Atomic Weight 

Fig. 93. 



100 



800 



conductances are plotted as ordinates against the atomic weights 
as abscissae, the curve shown in Fig. 93 is obtained. A glance at 
the curve shows the periodic nature of the relation. 

Absolute Velocity of the Ions. Thus far we have considered 
only the relative velocities of the ions and their conductances; 
we now proceed to the consideration of their absolute velocities 
in centimeters per second. 

Let a current of electricity pass through a centimeter cube of a 
solution of a binary electrolyte. If the solution contains m mols 
of solute per liter, then w/1000 will be the number of mols in the 
centimeter cube. The charge on either the cation or the anion 

m : , where F = 96,540 coulombs. If C represents the total 



F 1000 
current, we have 



, m 
1000 



V), 



Zeit. phys. Chem., 13, 242 (1894). 



410 THEORETICAL CHEMISTRY 

since the current is the charge which passes through one face of 
the cube in one second. In a centimeter cube, the current is 
equal to the product of the specific conductance and the difference 
of potential E, the latter being numerically equal to the potential 
gradient, the distance between the electrodes being one centi- 
meter. Hence, we have 

1000 K E = Fm (U + V). 

If E is expressed in volts and K in reciprocal ohms, U and V will be 
expressed in centimeters per second, for on passing to absolute 
electromagnetic units, we have 

1000 (K X 10- 9 ) (E X 10 8 ) ,_ 7 . . 

- (F3ooH - "- m(u + y >> 

or 



where U and V are the ionic velocities for unit potential gradient 
1 volt per centimeter. 
From this it follows that 



The equivalent conductance of a 0.0001 molar solution of potas- 
sium chloride at 18 is 128.9; the total velocity of the two ions 
is then, 

0.001345 cm. per sec. 



This total velocity is made up of the two individual ionic velocities. 
The transport numbers of the two ions, K* and Cl ; , are respectively 
0.493 and 0.507. Hence the absolute velocities of the ions, ex- 
pressed in centimeters per second, in a 0.0001 molar solution of 
potassium chloride at 18 are as follows: 

U = 0.001345 X 0.493 = 0.00066 cm. per sec., 
and 

V = 0.001345 X 0.507 = 0.00068 cm. per sec. 



ELECTRICAL CONDUCTANCE 



411 



The absolute velocities of some of the more common ions at 18 C 
are given in the following table: 

ABSOLUTE IONIC VELOCITIES. 



Ion. 


Velocity. 


Ion. 


Velocity. 


K* 


cm per sec. 
00066 


H" 


cm. per sec. 
0.00320 


NH 4 * 


00066 


cr 


00069 


Na* 


00045 


NO 3 ' 


00064 


Li* 


00036 


CUV 


00057 


Ag- 
Cr 2 O 7 " 


00057 
000473 


OH' 
Cu" 


0.00181 
00031 











The velocities of certain ions have been determined directly. 
Thus, the velocity of the hydrogen ion was measured by Lodge * 
in the following manner: The tube 5, Fig. 94, 40 cm. long and 
8 cm. in diameter, was graduated and bent at right angles at the 

B 





Fig. 94. 

ends. This was filled with an aqueous solution of sodium chloride 
irf gelatine, colored red by the addition of an alkaline solution of 
phenolphthalein. When the contents of the tube had gelatinized, 
the latter was placed horizontally, connecting two beakers filled 
with dilute sulphuric acid as shown in the diagram. A current 
of electricity was passed from one electrode A to the other elec- 
trode C. 

The hydrogen ions from the anode vessel were thus carried along 
the tube, and discharged the red color of the phenolphthalein as 
they migrated toward the cathode. In this manner the velocity 

* Brit. Assoc. Report, p. 393 (1886). 



412 THEORETICAL CHEMISTRY 

of the hydrogen could be observed under a known potential gra- 
dient. The observed and calculated values agree excellently. It 
was shown that the velocity of the hydrogen ions suffered almost 
no retardation from the high viscosity of the gelatine solution. 
Whetham,* in his experiments on ionic velocity, employed two 
solutions one of which possessed a colored ion, the progress of the 
latter being observed and its velocity determined under unit 
potential gradient. For example, consider the boundary line 
between two equally dense solutions of the electrolytes AC and 
BC, C being a colorless and A a colored ion. When a current 
passes through the boundary between the two electrolytes, the 
anion C will migrate toward the positive electrode while the two 
cations, A and J3, will migrate toward the negative electrode 
and the color boundary will move with the current, its speed being 
equal to that of the colored ion A . In this way Whetham measured 
the absolute velocities of the ions, Cu", C^CV', and Cl'. Ionic 
velocities have also been determined by Stcele f who observed 
the change in the index of refraction of the solution as the ions 
migrated. The accompanying table gives a comparison of the 
calculated and observed velocities of some of the ions. 



Ion, 


Velocity (obs.). 


Velocity (calc.). 


H* 


cm. per sec. 
0026 


cm. per sec. 
0032 


Cu" 


00029 


00031 


Cl' 


00058 


00069 


Cr 2 7 " 


00047 


000473 









Conductance and lonization. We have already seen that 
solutions of strong acids, strong bases and salts exert abnormally- 
great osmotic pressures. According to the molecular theory, this 
abnormal osmotic activity has been ascribed to the presence in 
the solutions of a greater number of dissolved particles than would 
be anticipated from the simple molecular formulas of the solutes. 
The ratio of the observed to the theoretical osmotic pressure was 
represented, according to van't Hoff, by the factor "i." 

* Phil. Trans. A., 184, 337 (1893); 196, 507 (1895). 
t Phil. Trans. A., 198, 105 (1902). 



ELECTRICAL CONDUCTANCE 413 

In 1887, Arrhenius showed that there is an intimate connection 
between electrical conductance and abnormal osmotic activity, 
only those solutions conducting the electric current which exert 
abnormally-high osmotic pressures. It had already been pointed 
out by Kohlrausch, that the equivalent conductance of a solution 
increases at first with the dilution and then ultimately becomes 
constant. Arrhenius explained this behavior by assuming that 
the molecules of the solute are dissociated into ions, the con- 
ductance of the solution being solely dependent upon the number 
of ions present. The dissociation increases with the dilution until 
finally, when the equivalent conductance has reached its maximum 
value, it is complete, the molecules of solute being entirely broken 
down into ions. This theory of Arrhenius, known as the theory 
of electrolytic dissociation, is based, as has been pointed out, 
upon the views advanced by Clausius. Arrhenius showed how 
the degree of dissociation of an electrolyte can be calculated 
from the electrical conductance of its solutions. According to the 
theory of electrolytic dissociation, the conductance of a solution 
is dependent upon the number of ions present in the solution, 
upon their charges, and upon their velocities. Since the electric 
charges carried by equivalent amounts of the ions of different 
electrolytes are equal, and since the velocities of the ions for the 
same electrolyte are practically independent of the dilution of the 
solution, it follows that the increase in equivalent conductance 
with dilution must depend almost wholly upon the increase in the 
number of ions present. 

The equivalent conductance at infinite dilution has been shown 
by the law of Kohlrausch to be 

AQO = l c + l a , 

and, therefore, the equivalent conductance at any dilution v, must 
be 

At, = a (l c + la), 

where a is the degree of dissociation of the electrolyte. Dividing 
the second equation by the first, we obtain 



A oo 



414 



THEORETICAL CHEMISTRY 



This equation enables us to calculate the degree of ionization of 
an electrolyte at any dilution, provided the conductance of the 
solution at the particular dilution is known, together with its 
conductance, at infinite dilution. For example, A, at 18 for a 
molar solution of sodium chloride is 74.3, and A^ is 110.3; there- 
fore, a = 74.3 -5- 110.3 = 0.673, or in a molar solution, the mole- 
cules of sodium chloride are dissociated to the extent of 67.3 per 
cent. A comparison of the values of i based upon conductance 
and osmotic data has already been given in the table on page 230. 
Since A v = a (l c + i a ), we may also write 



The Dissociation of Water. Water behaves as a very weak 
binary electrolyte, dissociating according to the equation, 

H 2 <=> H* + OH'. 

The specific conductance of water, purified with the utmost care, 
has been determined by Kohlrausch and Heydweiller.* Their 
results are given in the following table: 



Temperature, 
degrees. 


Specific Conduct- 
ance xio-. 





014 


18 


040 


25 


055 


34 


084 


50 


170 



The conductance of pure water at is so small that one milli- 
meter of it has a resistance equal to that of a copper wire of the 
same cross-section and 40,000,000 kilometers in length, or in 
other words, long enough to encircle the earth one thousand times. 
Knowing the specific conductance of water, its degree of dissoci- 
ation can be easily calculated. The ionic conductances of the 
two ions of water at 18 are as follows: H* = 318, and OH' = 

* Zeit. phys. Chem., 14, 317 (1894). 



ELECTRICAL CONDUCTANCE 415 

174. Therefore, the maximum equivalent conductance of water 

should be 

A^ = 318 + 174 = 492. 

The equivalent conductance at 18, of a liter of water between 
electrodes 1 cm. apart is, according to the data of Kohlrausch, 

0.04 X 10~ 6 X 10 3 - 0.04 X 10~ 3 ; 
therefore 

04 X 10~ 3 

~ Zoo ---- = 0-8 x 1Q~ 7 ^ c > the concentration of the ions, 

H* and OH', in mols per liter at 18. 

Conductance of Difficultly-Soluble Salts. In a saturated 
solution of a difficultly-soluble salt, the solution is so dilute that in 
general we may assume complete ionization, or A B = A^. 

When this is the case, we have 



^solution 

and 

A v = A^ = 1000 KV. 
Hence 

V = AQO 
1000 jc ' 

or if m denotes the concentration in gram-equivalents per liter, 
we have 

1 1000 K 



Thus, Bottger found for a saturated solution of silver chloride at 
20, K' = 1.374 X 10~ 6 . Deducting the specific conductance of 
the water at this temperature, we have 

K = 1.374 X 10- 6 - 0.044 X 1Q- 6 = 1.33 X 1Q- 6 . 

Since the value of A^, at 20, for silver chloride, determined from 
the table of ionic conductances, is 125.5, we have 

1000 X 1.33 X 10~ 6 , AA w ln , . A ni r , 

m = - T255 " = gr.-equiv. AgCl per liter. 



416 



THEORETICAL CHEMISTRY 



Temperature Coefficient of Conductance. When the temper- 
ature of a solution of an electrolyte is raised, the equivalent con- 
ductance usually increases. The increase in conductance is due, 
not to an increase in the ionization, but to the greater velocity 
of the ions caused by the diminution of the viscosity of the solution. 
According to Kohlrausch, the relation between conductance and 
temperature may be approximately expressed by the following 
equation, 

A, = Ai8'{l+|S(*- 18)1, 

where ft is the temperature coefficient, or change in conductance 
for 1 C. Solving the equation for 0, we have 



The temperature coefficients of several of the more common elec- 
trolytes are given in the accompanying table. 



TEMPERATURE COEFFICIENTS OF CONDUCTANCE. 



Electrolyte 



Nitric acid 
Sulphuric acid 

Hydrochloric acid 

Potassium hydroxide 

Potassium nitrate 

Potassium iodide 

Potassium bromide 

Potassium chlorate 

Silver nitrate . 

Potassium chloride 

Ammonium chloride .... 
Potassium sulphate 
Copper sulphate 
Sodium chloride 
Sodium sulphate 
Zinc sulphate 



Temperature 
Coefficient. 



0163 
0164 
0.0165 
0190 
0211 
0212 
0216 
0216 
0216 
0217 
0219 
0223 
0.0225 
0.0226 
0.0234 
0250 



The temperature coefficient of conductance is not, however, a 
simple linear function of the temperature. The following empiri- 



ELECTRICAL CONDUCTANCE 417 

cal equations, expressing equivalent conductance at infinite dilu- 
tion at any temperature t in terms of the conductance at 18, have 
been derived by Kohlrausch: 

Aoo* = Aoo 18 o { 1 + a (t - 18) + ft (t - 18) 2 }, 
and 

P = 0,0163 (a - 0.0174). 

When the values of A 18 o, a, and /3, as determined for a large 
number of electrolytes, are substituted in the above equation, he 
showed that Aoo* becomes equal to zero at a temperature approx- 
imating to 40. Kohlrausch suggested that each ion moving 
through the solution carries with it an " atmosphere" of solvent, 
and that the resistance offered to the motion of the ion is simply 
the frictional resistance between masses of pure water. This 
view is in harmony with the solvate theory discussed in an earlier 
chapter. Washburn * has calculated the degree of ionic hydration 
for several ions. He finds, for example, that the hydrogen ion 
carries with it 0.3 molecule of water, while the lithium ion is 
hydrated to the extent of 4.7 molecules of water. 

Conductance at High Temperatures and Pressures. The 
conductance of several typical electrolytes, at temperatures rang- 
ing from that of the room up to 306, have been measured by A. A. 
Noyes and his co-workers.f These determinations were made in 
a conductance cell especially constructed to withstand high 
pressures. 

The results show that the values of A> for binary electrolytes 
become more nearly equal with rise of temperature. This may 
be taken as an indication of the fact that the ionic velocities tend 
to become more nearly equal as the temperature rises. The 
conductance of ternary electrolytes increases uniformly with the 
temperature, and attains values which are considerably greater 
than those reached by binary electrolytes. This is what might 
be expected, since if an ion is bivalent, as in a ternary electrolyte, 
the driving force is greater, and the ion must move faster, and, 
consequently, the conductance must be greater. 

* Jour. Am. Chem. Soc., 30, 322 (1909). 

t Publication of Carnegie Institution, No. 63. 



418 THEORETICAL CHEMISTRY 

The temperature coefficient of conductance for binary elec- 
trolytes is greater between 100 and 156, than below or above 
these temperatures. The temperature coefficients of ternary 
electrolytes increases uniformly with rising temperature. In the 
case of acids and bases, the rate of increase in conductance steadily 
diminishes as the temperatures rises. The ionization decreases 
regularly with rise in temperature, the temperature coefficient of 
ionization being small between 18 and 100. The effect of pres- 
sure on conductance was studied by Fanjung.* He found that 
the conductance increases slightly with increasing pressure. 
This result he interprets as being due to increased ionic velocity 
rather than to an increase in the number of ions present in the 
solution. 

Conductance of Non-aqueous Solutions. A large amount of 
interesting and important work has been done in recent years 
upon the electrical conductance of solutions in non-aqueous sol- 
vents. 

It is impossible to give even a brief survey of the results of these 
investigations, and we must limit ourselves to the statement of 
the following general conclusions : f 

(1) The conditions in non-aqueous solutions are much more 
complex than in aqueous solutions. 

(2) In general, the laws which have been found to apply to aque- 
ous solutions also apply to non-aqueous solutions. 

(3) Different solvents appear to have different dissociating 
powers. 

(4) The dissociating power appears to run parallel with the 
dielectric constant of the solvent. 

Many interesting phenomena present themselves in connec- 
tion with the conductance of electrolytes in mixed solvents, but 
for an account of this work the student must consult the original 
papers of Jones and his students. J 

* Zeit. phys. Chem., 14, 673 (1894). 

t "Elektrochemie der nichtwassrigen Losungen," by G. Carrara, Ahren's 
'Sammlung Chemischer und chemisch-technischer Vortraege," Vol. XII. 
t Publication of the Carnegie Institution, No. 80. 



ELECTRICAL CONDUCTANCE 



419 



Ionizing Power of Solvents. Thomson * and Nernst f pointed 
out that if the forces which hold the atoms in the molecule are of 
electrical origin, then those liquids which possess large dielectric 
constants should have correspondingly great ionizing power. 
This is a direct consequence of Coulomb's law of electrostatic 
attraction, which may be expressed by the equation, 



in which q\ and # 2 denote two electric charges, d the distance 
between them, / the force of attraction and K the dielectric con- 
stant. Obviously the larger K becomes, the smaller will be the 
value of/; i.e., the more likely the molecule will be to break down 
into ions. That the above relation is approximately true may be 
seen from the following table: 

DIELECTRIC CONSTANTS. 



Solvent. 


K 


Ionizing Power. 


Benzene 


2 3 


Extremely weak 


Ethyl ether 


4.1 


Weak 


Ethyl alcohol 


25 


Fairly strong 


Formic acid .... .... 


62 


Strong 


Water . . 


80 


Very strong 


Hydrocyanic acid 


96 


Very strong 









Dutoit and Aston J have suggested that there is a connection 
between the ionizing power of a solvent and its degree of associa- 
tion, and Dutoit and Friderich conclude that the values of AGO, 
for a given electrolyte dissolved in different solvents, are a direct 
function of the degree of association and an inverse function of 
the viscosity of the solvents. Water and the alcohols furnish 
good illustrations of the truth of this generalization. 

* Phil. Mag., 36, 320 (1893). 

t Zeit. phys. Chem., 13, 531 (1894). 

j Compt. rend., 125, 240 (1897). 

Bull. Soc. Chim. [3], 19, 321 (1898). 



420 



THEORETICAL CHEMISTRY 



Conductance of Fused Salts. While solid salts are exceed- 
ingly poor conductors of electricity, yet as the temperature is 
raised their conductance increases until at their melting-point 
they may be grouped with good conductors. There is no sudden 
increase in conductance at the melting-point. The specific 
conductance of a fused salt may exceed the specific conductance 
of the most concentrated aqueous solutions, but owing to the high 
concentration the equivalent conductance is much less. The 
following table gives the specific and equivalent conductance of 
fused silver nitrate: 



Temperature, 
degrees 


Sp. Corid. 


Equiv Cond. 


218 (inelt.-pt.) 


681 


29 2 


250 


834 


36 1 


300 


1 049 


46 2 


350 


1.245 


55 4 



The specific conductance of a 60 per cent aqueous solution of 
silver nitrate at 18 is 0.208 reciprocal ohm. 

If the salts are impure the conductance is raised, the effect of 
impurities being apparent even before the salts have reached their 
melting-points. This is analogous to the behavior of solutions, 
and suggests that the impurity functions in the salt mixture as a 
dissolved solute.* 

PROBLEMS. 

1. An aqueous solution of copper sulphate is electrolyzed between 
copper electrodes until 0.2294 gram of copper is deposited. Before elec- 
trolysis the solution at the anode contained 1.1950 grams of copper, after 
electrolysis 1.3600 grams. Calculate the transport numbers of the two 
ions, Cu" and S0 4 ". Ans. n = 0.28, 1 - n = 0.72. 

2. A solution containing 0.1605 per cent of NaOH was electrolyzed 
between platinum electrodes. After electrolysis 55.25 grams of the 
cathode solution contained 0.09473 gram of NaOH, whilst the concen- 
tration of the middle portion of the electrolyte was unchanged. In a 

* For a complete treatment of fused electrolytes the student is advised to 
consult, "Die Elektrolyse geschmolzener Salze," by Richard Lorenz. 



ELECTRICAL CONDUCTANCE 421 

silver coulometer the equivalent of 0.0290 gram of NaOH was deposited 
during electrolysis. Calculate the transport numbers of the Na" and OH' 
ions. Ans. n = 0.791, 1 n = 0.209. 

3. In a 0.01 molar solution of potassium nitrate, the transport num- 
bers of the cation and anion are, respectively, 0.503 and 0.497. Find the 
equivalent conductances of the two ions in this solution having given that 
its specific conductance is 0.001044. Ans. l c = 52.5, l a = 51.9. 

4. The absolute velocity of the Ag* ion is 0.00057 cm. per sec., and that 
of the Cl' ion is 0.00069 cm. per sec. Calculate the equivalent con- 
ductance of an infinitely dilute solution of silver chloride. 

5. The equivalent conductance of an infinitely dilute solution of am- 
monium chloride is 130; the ionic conductances of the ions OH' and Cl' 
are 174 and 65.44 respectively. Calculate the equivalent conductance 
of ammonium hydroxide at infinite dilution. Ans. Aoo = 238.56. 

6. The equivalent conductance of a molar solution of sodium nitrate 
at 18 is 66; its conductance at infinite dilution is 105.3. What is the 
degree of ionization in the molar solution? Ans, a = 62.6 per cent. 

7. The specific conductance of a saturated solution of AgCN at 20 
is 1.79 X 10~ 6 and the specific conductance of water at the same temper- 
ature is 0.044 X 10~ 6 reciprocal ohms. The equivalent conductance at 
infinite dilution is 115.5. Calculate the solubility of AgCN in grams per 
liter. Ans. 2.02 X 10~ 3 gram/liter. 

8. The equivalent conductance at 18 of a solution of sodium sulphate 
containing 0.1 gram-equivalent of salt per liter is 78.4, the conductance 
at infinite dilution is 113 reciprocal ohms. What is the value of i for 
the solution? What is its osmotic pressure? 

Ans. i = 2.388; osmotic pressure = 2.85 atmos. 

9. The freezing-point of a 0.1 molar solution of CaCl 2 is 0.482. 
(a) Calculate the degree of ionization (freezing-point constant = 1.89 
for one mol per liter), (b) Calculate the degree of ionization from the 
equivalent conductance at 18, which is 82.79 reciprocal ohms, whilst the 
equivalent conductance of CaCl 2 at infinite dilution is 115.8 reciprocal 
ohms. Ans. (a) a = 0.774; (b) = 0.715. 



CHAPTER XIX. 
ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS. 

Ostwald's Dilution Law. It has been shown in preceding 
chapters that the law of mass action is applicable to chemical 
equilibria in both gaseous and liquid systems. We now proceed 
to show that it applies equally to electrolytic equilibria. When 
acetic acid is dissolved in Water it dissociates according to the 
equation 

CH 3 COOH <F CH 3 COO' + H\ 

Let one mol of acetic acid be dissolved in water and the solution 
diluted to v liters, and let a denote the degree of dissociation. 

Then, the concentration of the undissociated acid is - and 

v 

the concentration of the ions is . Applying the law of mass 
action, we have 



or 



where K is the equilibrium or ionization constant. 

This equation expressing the relation between the degree of 
ionization and dilution, was derived by Ostwald * and is known 

as the Ostwald dilution law. Since a = - , we may substitute 

Aoo 

this value of a in equation (1) and obtain the expression 

A, 2 



-A.) " 

* Zeit. phys. Chem., 2, 36 (1888); 3, 170 (1889). 
422 



(2) 



ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 423 

The dilution law may be tested by substituting the value of a, 
corresponding to any dilution v, in the equation and calculating 
the value of the ionization constant, K; the value of a at any 
other dilution may then be calculated and compared with the 
value determined by direct experiment. The following' table gives 
the results obtained with acetic acid at 14. 1, K being equal to 
0.0000178: 



v (in liters) 


X102 (calc). 


<*Xl02 (obs.). 


0.994 


42 


40 


2 02 


60 


614 


15 9 


1 67 


1 66 


18 1 


1 78 


1.78 


1,500 


15 


14.7 


3,010 


20.2 


20 5 


7,480 


30 5 


30 1 


15,000 


40 1 


40 8 



As will be seen, the agreement between the observed and cal- 
culated values is very close. The table also shows to how small 
an extent the molecules of acetic acid are broken down into ions, 
a molar solution being dissociated less than 0.5 per cent. The 
dilution law holds for nearly all organic acids and bases, but fails 
to apply to salts, strong acids, and strong bases. When a is 
small, the term (1 a) does not differ appreciably from unity, 
and equation (1) becomes 



or 



a = VvK. 



(3) 



On the other hand, when a cannot be neglected, we have, on 
Bolving equation (1) for a, 



(4) 



The method of derivation indicates that the dilution law is 
only strictly applicable to binary electrolytes, and therefore, 



424 THEORETICAL CHEMISTRY 

it is improbable that it will hold for electrolytes yielding more 
than two ions. It has been found, however, that organic acids 
whether they are mono-, di-, or polybasic always ionize as 
a monobasic acid up to the dilution at which a. = 50 per 
cent. This means that the dilution law is applicable to poly- 
basic acids up to that dilution at which the acid is 50 per cent 
ionized. 

Strength of Acids and Bases. There are several methods by 
which the relative strengths of acids can be estimated. A method 
which has proved of great value is that in which two different 
acids are allowed to compete for a certain base, the amount of 
which is insufficient to saturate both of them. Suppose equiva- 
lent weights of nitric and dichloracetic acids together with sufficient 
potassium hydroxide to saturate one acid completely are taken: 
we then determine the position of the equilibrium represented by 
the equation 

HN0 3 + CHC1 2 COOK * CHC1 2 COOH + KN0 3 . 

In order to determine the conditions of equilibrium we may make 
use of any method which does not disturb this equilibrium. Since 
ordinary chemical methods are excluded on this account, we 
employ any physical property which is capable of exact measure- 
ment and differs sufficiently in the two systems, as for example, 
the change in volume, or the thermal change, accompanying 
neutralization. Thus, Ostwald * found that when one mol of 
potassium hydroxide is neutralized by nitric acid in dilute solu- 
tion, the volume increases approximately 20 cc. When one mol 
of potassium hydroxide is neutralized by dichloracetic acid, how- 
ever, the increase in volume is 13 cc. It is evident, therefore, that 
if nitric acid completely displaces dichloracetic acid as represented 
by the above equation, the increase in volume will be 20 13 
= 7 cc.; if no displacement occurs, then the volume will remain 
constant. He found that the volume actually increased 5.67 cc. 
Therefore, the reaction represented by the upper arrow has pro- 
ceeded to the extent of 5.67 *- 7 = 80 per cent. That is to say, 

* Jour, prakt. Chem. [2], 18, 328 (1878). 



ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 425 

in the competition of the two acids for the base, the nitric acid 
has taken 80 per cent and the dichloracetic acid has taken 20 per 
cent, or the relative strengths of the two acids are in the ratio of 
80 : 20, or 4 : 1. 

The relative strengths of acids can also be determined from their 
catalytic effect on the rates of certain reactions, such as the 
hydrolysis of esters or the inversion of cane sugar. 

The order of the activity of acids is the same whether measured 
by equilibrium or kinetic methods. Arrhenius pointed out that 
the relative strengths of acids can be readily determined from 
their electrical conductance. The order of the strengths of acids 
as determined by equilibrium and kinetic methods is the same as 
that of their electrical conductances in equivalent solutions. 
This is well illustrated by the following table in which the three 
methods are compared, hydrochloric acid being taken as the 
standard of comparison: 



Acid. 


Method Employed. 


Equilibrium. 


Kinetic. 


Conductance. 


HC1 


100 
100 
49 
9 


100 
100 
53.6 
4.8 
0.4 


100 
99.6 
65.1 
4.8 
1.4 


HNOs 


H 2 SO 4 


CH 2 C1COOH 


CHsCOOH 







The results of these and other experiments warrant the con- 
clusion that the strength of an acid is determined by the number 
of hydrogen ions which it yields. It is important to note that the 
electrical conductance of an acid is not directly proportional to 
its hydrogen ion concentration; the relatively high velocity of 
the H ion is the cause of the approximate proportionality between 
these two variables. In the case of a weak acid, the value of the 
ionization constant may be taken as a measure of the strength of 
the acid. The following table gives the values of the ionization 
constants at 25 for several different acids. 



426 



THEORETICAL CHEMISTRY 
IONIZATION CONSTANTS OF ACIDS. 



Acid. 


lonization 
Constant. 


Acetic acid 


0000180 


Monochloracetic acid .... 
Trichloracetic acid 
Cyanacetic acid 
Formic acid 


0.00155 
1.21 
0037 
000214 


Carbonic acid 
Hydrocyanic acid . 


3040X10- 10 
570X10~ 10 


Hydrogen sulphide ... 
Phenol . 


13X10- 10 
1 3X10~ 10 







Since for a weak acid, a = vWf, it follows that for two weak 
acids at the same dilution, we may write 



or the ratio of the degrees of ionization of the two acids is equal to 
the square root of the ratio of their ionization constants. Thus, 
from the data given in the foregoing table for acetic and mono- 
chloracetic acids, we have 



ai =1 /0. 
2 V 0. 



0.000018 



_ 
00155 9.3' 

or the effect of replacing one atom of hydrogen in the methyl 
group of acetic acid increases the strength of the acid about nine 
times. 

Just as the hydrogen ion concentration of acids determines their 
strength, so the strength of bases is determined by the concen- 
tration of hydroxyl ions. The strength of bases may be estimated 
by methods similar to those employed in determining the strength 
of acids. Thus, two different bases may be allowed to compete 
for an amount of acid sufficient to saturate only one of them; or 
a catalytic method developed by Koelichen * may be used. This 
method is based upon the effect of hydroxyl ions on the rate of 
condensation of acetone to diacetonyl alcohol, as represented by 
the equation 

2CH 3 COCH 3 - CH 3 COCH 2 C (CH 3 ) 2 OH. 
* Zeit. phys. Chem., 33, 129 (1900). 



ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 427 



In addition to these two methods, the method of electrical con- 
ductance is also applicable. The agreement between the results 
obtained by the three methods is quite satisfactory. The alkali 
and alkaline earth hydroxides are very strong bases and are dis- 
sociated to about the same extent as equivalent solutions of 
hydrochloric and nitric acids, while on the other hand, ammonia 
and many of the organic bases are very weak. The following 
table gives the ionization constants of several typical bases: 

IONIZATION CONSTANTS OF BASES. 



Base. 


Ionization 
Constant. 


Ammonia 


000023 


Methyl amine . 


00050 


Trimcthylamine 


000074 


Pyridine 


2 5X10~ JO 


Aniline 


1.1X10" 10 







Mixtures of Two Electrolytes with a Common Ion. Just as 
the dissociation of a gaseous substance is diminished by the addi- 
tion of an excess of one of the products of dissociation, so the 
ionization of weak acids and bases is depressed by the addition of 
a salt with an ion common to the acid or the base. If the degree 
of ionization of a salt with an ion in common with an acid or a 
base is represented by a', and n denotes the number of molecules 
of salt present, then the equation of equilibrium of the acid or 
base will be 

(not + a} a = Kv (1 - a), 

where a is the degree of ionization of the acid or base. For very 
weak acids and bases, a is so small that 1 a does not differ 
appreciably from unity, and since a! is practically independent of 
the dilution, we obtain 

na = Kv 
or 

Kv 
a = 

n 



428 THEORETICAL CHEMISTRY 

That is, the ionization of a weak acid or base, in the presence of 
one of its salts, is approximately inversely proportional to the 
amount of salt present. 

In many of the processes of analytical chemistry, advantage is 
taken of the action of neutral salts on the ionization of weak acids 
and bases. Thus, while the concentration of hydroxyl ions in 
ammonium hydroxide is sufficient to precipitate magnesium hy- 
droxide from solutions of magnesium salts, the presence of a small 
amount of ammonium chloride depresses the ionization of the 
ammonium hydroxide to such an extent that precipitation no 
longer takes place. 

Isohydric Solutions. Arrhenius * was the first to point out 
what relation must exist between solutions of two electrolytes 
with a common ion, in order that, when mixed in any proportions, 
they may not exert any mutual influence. He showed that when 
the concentration of the common ion in each of the two solutions 
is the same before mixing, no alteration in the degree of ionization 
will occur after mixing. Such solutions are said to be isohydric. 
Thus, an aqueous solution containing one mol of acetic acid in 
8 liters, is isohydric with an aqueous solution containing one mol 
of hydrochloric acid in 667 liters. On mixing these two solutions 
the hydrogen ion concentration remains unchanged, and if the 
mixture is treated with a small amount of sodium hydroxide, 
equal amounts of sodium acetate and sodium chloride will be 
formed. 

That isohydric solutions may be mixed without altering their 
respective ionizations may be shown in the following manner: 
Let C and c denote the concentrations of the undissociated por- 
tions, and CA, C 2 , CA, and C2 denote the concentrations of the dis- 
sociated portions of two electrolytes, and let C 2 and 02 correspond 
to two different ions. 

Then, we have 

kc = CA<%, (1) 

and 

KC = CUC,. (2) 

* Wied. Ann., 30, 51 (1887). 



ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 429 

If v liters of the first solution be mixed with V liters of the second 
solution, the concentrations of the undissociated portions, and of 
the dissimilar ions, will be 

Cv cv C^V , c^v 

jf+V v~+^> T+v' T+v* 

while the concentration of the common ion A, becomes 



Applying the law of mass action, we have 

JL-- 
fee- 

and 



But equations (3) and (4) only become identical with equations (1) 
and (2) when C^ = c A j or, in other words, no change in the degree 
of dissociation takes place after the two solutions are mixed. 

lonization of Strong Electrolytes. It has already been men- 
tioned that the Ostwafd dilution law, which is a direct conse- 
quence of the law of mass action, applies only to weak electrolytes. 
Just why the law of mass action should fail to apply to strong 
electrolytes is not known, but several possible causes have been 
suggested to account for its failure. One of the most plausible 
explanations is that advanced by Biltz,* who attributes the 
failure of the law of mass action when applied to strong electrolytes, 
to hydration of the solute. If the ions become associated with 
a large proportion of the solvent, the effective ionic concentration 
would then be the ratio of the amount of the ion present to that 
of the free solvent, instead of to the total solvent, as ordinarily 
calculated. This view is in harmony with certain facts which 
have been adduced in favor of the theory of solvation. While 
the Ostwald dilution law does not apply to strongly ionized elec- 
trolytes, certain empirical expressions have been derived which 

* Zeit. phys. Chem., 40, 218 (1902). 



430 



THEORETICAL CHEMISTRY 



hold fairly well over a wide range of dilution, 
showed that the equation 

a 2 

_ rrt 

(1 - a) Vt) ' 



Thus, Rudolphi 



gives approximately constant values for K ' for strong electrolytes 
The following table gives the results obtained with solutions o 
silver nitrate at 25; the numbers in the third column being cal 
culated by means of the Ostwald dilution law, while those in th 
fourth column are calculated by means of Rudolphi's dilution law 



V 


a 


K 


K' 


16 


8283 


253 


1.11 


32 


8748 


191 


1 16 


64 


8993 


127 


1.06 


128 


9262 


122 


1 07 


256 


9467 


124 


1 08 


512 


9619 


125 


1 09 



The Rudolphi equation was modified by van't Hofff to th< 
form 



This equation holds even more closely than that of Rudolphi 
Of the more recent empirical equations which have been derivec 
to express the change of conductance of an electrolyte with dilu- 
tion, the equations of Kraus * and Bates t deserve special men- 
tion. 
The equation of Kraus has the following form: 

/ Aw \ 2 C 7 , , , 



\AoW A _ A?;^ 



AOW 

In this equation A is the conductance of the electrolyte whose con- 
centration C is expressed in mols per liter, TJ/TJO is the ratio of the 
viscosity of the solution to that of the solvent, and k, k f , h, and AQ 

* Jour. Am. Chem. Soc., 35, 1412 (1913). 
t Ibid., 37, 1421 (1915). 



ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 431 



are empirical constants the values of which are so chosen as to 
insure close agreement between the observed and calculated values 
of the conductance. 

The equation of Bates is similar to that of the preceding equa- 
tion, except that the logarithm of the left-hand side of the equation 
is substituted for the original expression of Kraus. The equation 
of Bates takes the form : 



l&o TI- 



i _ f A?? ^ 

\AoW 

The constants fc, fc', and h are purely empirical as in the equation 
of Kraus, but A denotes the equivalent conductance at infinite 
dilution. 

A comparison between the two equations is afforded by the fol- 
lowing table in which is recorded the observed and calculated 
values of the " corrected " equivalent conductance, Aiy/r/o, for 
solutions of potassium chloride at 18. 

COMPARISON OF THE EQUATIONS OF KRAUS AND BATES. 



c 


n/'-no 


A (obs.) 


Af/170 


AWlo (K.) 


A*/ (B.) 


3 


9954 


88.3 


87.89 


87.4 


89.3 


2 


9805 


92 53 


90 73 


90 9 


91 9 


1 


982 


98 22 


96 5 


96 4 


96 53 


5 


9898 


102 36 


101 32 


101.1 


101 29 


2 


9959 


107 90 


107 46 


107.6 


107 43 


1 


9982 


111 97 


111 77 


111 9 


111 73 


05 


9991 


115 69 


115 59 


115 5 


115 58 


02 


9996 


119 90 


119 85 


119.8 


119 83 


01 


9998 


122 37 


122 35 


122 4 


122 32 


005 


9999 


124 34 


124 33 


124 4 


124 38 


0.002 


1.0000 


126 24 


126 24 


126 3 


126 31 


001 




127 27 


127.27 


127 2 


127 32 


0005 




128 04 


128 04 


127.6 


128 05 


0002 




128 70 


128 70 


127 9 


128.68 


0001 




129 00 


129 00 


128.1 


128.96 







129 50 


129 50 


128 3 


129.50 















It will be observed that for dilute solutions, the ratio 17/170 is 
practically unity and, furthermore, that the value of CA/Ao is so 
small that the second term on the right-hand side of both equations 
is negligible in comparison with the value of k. 



432 



THEORETICAL CHEMISTRY 



Therefore, under these conditions, both equations reduce to the 
form 

2 . . 

= constant, 



which will be recognized as identical with Ostwald's dilution law 
as given on page 422. 

Heat of Ionization. The heat of ionization of an electrolyte 
can be calculated by means of the reaction isochore equation of 
van't Hoff (see p. 324), provided the degree of ionization at two 
different temperatures is known. 

Since I 

and 



(l-ajv' 



it follows that the heat of ionization may be calculated by means 
of the equation 



2.3026 R \ log 



' (1 - 



-log 



(1 - ) V 



Arrhenius * has shown that this equation also applies to those 
electrolytes which do not obey the Ostwald dilution law. Some 
of the results obtained by Arrhenius are given in the accompany- 
ing table: 



Electrolyte 




Temperature. 


Calories. 


Acetic acid 


\ 


35 


386 


Propi'onic acid 


\ 


21. 5 
35 


-28 
557 


Butyric acid 


i 
J 


21. 5 
35 


183 
935 


Phosphoric acid 


I 
\ 


21. 5 
35 


427 
2458 


Hydrochloric acid 


\ 


21. 5 
35 


2103 
1080 


Potassium chloride 




35 


362 


Potassium bromide 




35 


425 


Potassium iodide 




35 


916 


Sodium chloride . . . ." 




35 


454 


Sodium hydroxide 




35 


1292 


Sodium acetate 




35 


391 











* Zeit. phys. Chem., 4, 96 (1889). 



ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 433 

It will be found that the values of the heats of ionization given 
in this table do not agree with the values calculated for these 
same substances from the data given in the table on page 307. 
The reason for this lack of agreement is, that the data of the earlier 
table refer to the heat of formation of the ions from the dissolved 
substance, whereas the data of the table just given represent the 
combined thermal effects of solution and ionization. 

The Solubility Product. While the law of mass action does 
not in general apply to the equilibrium between the dissociated 
and undissociated portions of an electrolyte, except in the case 
of organic acids and bases, it does apply with a fair degree of 
accuracy to saturated solutions of electrolytes. 

A saturated solution of silver chloride affords an example of 
such an equilibrium. This salt is practically completely ionized 
in a saturated solution, as represented by the equation 



Applying the law of mass action to this equilibrium, we obtain 

CAg- X CGI TT 

- = A. 

CAgCl 

Since the solution is saturated, the value of CA K CI must remain 
constant at constant temperature, and therefore 

Ag* X ccr = constant = s, 

where the product of the ionic concentrations s is called the solu- 
bility or ionic product. 

The equilibria in the above heterogeneous system may be repre- 
sented thus: 



(in solution) (solid) 

The solubility product for silver chloride at 25 is 1.56 X 10~ 10 , 
the ionic concentrations being expressed in mols per liter. Hence, 
since the two ions are present in equivalent amounts, a saturated 
solution of silver chloride at 25 must contain Vl.56 X 10~ 10 
= 1.25 X 10~ 5 mols per liter of Ag* and Cl' ions. In general, if 



434 THEORETICAL CHEMISTRY 

represents the equilibrium between an electrolyte and its products 
of dissociation in saturated solution, we have 



The solubility product may be defined as the maximum product of 
the ionic concentrations of an electrolyte which can exist at any one 
temperature. 

Just as the dissociation of a gaseous substance or of an organic 
acid is depressed by the addition of one of the products of dis- 
sociation, so when a substance with a common ion is added to the 
saturated solution of an electrolyte, the dissociation is depressed 
and the undissociated substance is precipitated. 

The following example will serve to illustrate how the solu- 
bility product of a substance can be determined, and how the 
change in solubility due to the addition of a substance containing 
a common ion may be calculated. The solubility of silver bromate 
at 25 is 0.0081 mol per liter. If we assume complete ionization, 
the concentration of the ions, Ag* and Br(V will be the same 
and equal to 0.0081 mol per liter, or 

(0.0081) (0.0081) = s. 

The solubility in a solution of silver nitrate containing 0.1 mol 
of Ag' ions can be calculated from the equation 

(0.0081) 2 = (0.0081 + 0.1 - x) (0.0081 - x), 

where x represents the amount of silver bromate thrown out of so- 
lution by the addition of 0.1 mol of Ag* ion. Since (0.0081 x) 
represents the concentration of BrO 3 ' ions after the addition of 
the silver nitrate, it also represents the solubility of silver bromate 
under similar conditions. The effect of adding a solution of a 
soluble bromate containing 0.1 rnol of Br0 3 ' ion will be the same 
as that produced by 0.1 mol of Ag' ion. 

The Basicity of Organic Acids. The Ostwald dilution law 
holds strictly for all monobasic organic acids, and also for poly- 
basic organic acids which are less than 50 per cent ionized. The 
neutral salts of these acids, however, are much more highly ionized, 
and the difference in conductance between two dilutions of a neu- 
tral salt of a polybasic acid is greater than the difference in 



ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 435 



conductance between the same dilutions of a neutral salt of 
a monobasic acid. Ostwald * has shown that it is possible 
to estimate the basicity of an organic acid from the difference 
in the equivalent conductance of its sodium salt at two different 
dilutions. 

As the result of a long series of experiments, he found that the 
difference between the equivalent conductance of the sodium salt 
of a monobasic organic acid at v = 32 liters and at v = 1024 liters 
is approximately 10 units. Similarly, the difference for a dibasic 
acid between the same dilutions is 20 units, and for an w-basic 
acid the difference is 10 n. Hence, to estimate the basicity of an 
organic acid, the equivalent conductance of its sodium salt at v = 
32 liters and at v = 1024 liters is determined; then, if A is the dif- 
ference between the values of the conductance at the two dilutions, 

the basicity will be n = -^ - 

The following table gives the values of A and n for the sodium 
salts of several typical organic acids: 



Acid. 


A 


n 


Formic 


10 3 


1 


Acetic 


9 5 


1 


Propionic 


10 2 


1 


Benzole 


8 3 


1 


Quininic 


19 8 


2 


Pyridine-tricar boxy lie (1, 2, 3) 


31 


3 


Pyridine-tricarboxylic (1, 2, 4) . ... 


29.4 


3 


Pyridine-tetracarboxylic . . 


41 8 


4 


Pyridine-pentacarboxylic 


50 1 


5 









Influence of Substitution on lonization. Attention has already 
been called to the marked difference in the strength of acetic 
acid produced by the replacement of the hydrogen atoms of the 
methyl group by chlorine. In the accompanying table the ioniza- 
tion constants for various substitution products of acetic acid 
are given: 

* Zeit. phys. Chem., i, 105 (1887); 2, 902 (1888). 



436 



THEORETICAL CHEMISTRY 



Acid. 


lonization 
Constant (25). 


Acetic. CHsCOOH 


000018 


Propionic, CH 8 CH 2 COOH 


000013 


Chloracetic, CH 2 C1COOH 


00155 


Bromacetic, CH 2 BrCOOH.. . 


00138 


Cyanacetic, CH 2 CNCOOH 


00370 


Glycollic, CH 2 OHCOOH 


0.000152 


Phenylacetic, C 6 H 6 CH 2 COOH 


0.000056 


Amidoacetic, CH 2 NH 2 COOH 


3 4 X 10~ 10 







This table affords an interesting illustration of the influence of 
different substituents on the strength of acetic acid. Thus, the 
activity of the acid is increased by the replacement of alkyl hydro- 
gen atoms by Cl, Br, CN, OH, or C 6 H 5 , while the substitution of 
the CH 3 or NH 2 groups diminishes its activity. If we assume 
that the substituents retain their ion-forming capacity on enter- 
ing into the molecule of acetic acid, these differences in activity 
can be readily explained. Thus, Cl, Br, CN, and OH tend to 
form negative ions, and hence increase the negative character of 
the group into which they enter. On the other hand, basic groups, 
such as NH 2 , diminish the tendency of the group into which they 
enter to yield negative ions. 

The influence of an alkyl residue on the strength of an organic 
acid is conditioned by its distance from the carboxyl group. This 
is well illustrated by the ionization constants of propionic acid 
and some of its derivatives. 



Acid. 


Ionization 
Constant (26). 


Propionic acid, CH 8 CH 2 COOH 


0.0000134 . 


Lactic acid, CHaCHOHCOOH 


0.000138 


0-oxypropionic acid, CH 2 OHCH 2 COOH 


0.0000311 







The effect of the OH group in the a-position is seen to be much 
more marked than when it occupies the 0-position. 

The position of a substituent in the benzene nucleus exerts a 
marked influence on the strength of the derivatives of benzoic 



ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 437 

acid. The ionization constants of benzoic acid and the three 
chlorbenzoic acids are given in the following table: 



Acid. 


Ionization 
Constant (25). 


Benzoic acid, C 6 H 6 COOH 


000073 


o-Chlorbenzoic acid, C 6 H 4 C1COOH 


0.00132 


m-Chlorbenzoic acid, CeH^ClCOOH 


000155 


p-Chlorbenzoic acid CeH^ClCOOH 


000093 







When the halogen enters the ortho-position, the strength of the 
acid is greatly augmented, while in the meta- and para-positions 
the effect is much smaller, meta-chlorbenzoic acid being stronger 
than para-chlorbenzoic acid. It is a general rule that the influence 
of substituents is always greatest in the ortho-position, and least 
in the meta- and para- positions, the order in the two latter being 
uncertain. 

Hydrolysis. When a salt formed by a weak acid and a strong 
base, such as sodium carbonate, is dissolved in water, the solution 
shows an alkaline reaction, while on the other hand, when a salt 
formed by a strong acid and a weak base, such as ferric chloride, 
is dissolved in water, the solution shows an acid reaction. 

The process which takes place in the aqueous solution of a salt, 
causing it to react alkaline or acid, is termed hydrolysis or hydro- 
lytic dissociation. If MA represents a salt, in which M is the basic 
and A is the acidic portion, then the hydrolytic equilibrium may 
be represented by the equation 

MA + H 2 < MOH + HA. 

If the base formed is insoluble or undissociated and the acid is 
dissociated, the solution will react acid. If the acid formed is 
insoluble or undissociated and the base is dissociated, the solution 
will react alkaline. Finally, if both base and acid are insoluble 
or undissociated, the salt will be completely transformed into base 
and acid, and, as there will be no excess of either H* or OH' ions, 
the solution will remain neutral. 

It is evident, then, that hydrolysis is due to the removal of 
either one or both of the ions of water by the ions of the salt to 



438 THEORETICAL CHEMISTRY 

form undissociated or insoluble substances. As fast as the ions 
of water are removed, the loss is made good by the dissociation of 
more water, until eventually a condition of equilibrium is estab- 
lished. The conditions governing hydrolytic equilibrium may be 
determined from a knowledge of the solubility or ionic constant 
of the substances involved. Thus, if the product of the concen- 
trations of the ions M" and OH' exceeds that which can exist in 
pure water, then some undissociated or insoluble substance will 
be formed. This will disturb the equilibrium of H" and OH' 
ions, and a further dissociation of water must occur until the 
ionic product of water is just reached. 

If now the ions H* and A 7 do not unite to form undissociated 
acid, the presence of an excess of H* ions will disturb the equi- 
librium between pure water and its products of dissociation; or, 
since 

CH- X COH' = H 2 o, 

the concentration of OH' ions present, when CH- represents the 



total concentration of H" ions, will be - 

CH 

A similar readjustment will take place when an undissociated 
or insoluble acid and a dissociated base are formed. 

We may now proceed to consider three different cases of hydroly- 
sis, viz., when the reaction is caused (1) by the base, (2) by the 
acid, and (3) by both base and acid. 

CASE I. The formation of an undissociated or insoluble base is pri- 
marily the cause of the hydrolysis, the acid formed being dissociated. 

Let the hydrolytic equilibrium be represented by the equation 

MA + H 2 0<:MOH + HA. 

The reaction will proceed in the direction of the upper arrow until 
the product, CM* X COH', exceeds that which can exist in the ab- 
sence of an undissociated base. When equilibrium is established, 
we have 

final CM- X final COH' = KUOH X CMOH formed, (1) 

or if the base formed is practically insoluble, the equilibrium equa- 
tion simplifies to the form 

final CM- X final COH' = MOH, (2) 



ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 439 

where SMOH is the solubility product of the base. The condition 
of equilibrium represented by the equation 



CH* X COH' = $H 2 o 

must* be fulfilled. It follows that the final concentration of the 
OH' ions will be the quotient obtained by dividing the ionic product 
for water, at the temperature of the experiment, by the final con- 
centration of the H* ion, this latter being wholly dependent upon 
the extent of the reaction and the degree of ionization of the acid 
formed. If the degree of hydrolysis of the salt be represented by 
x, and the degree of dissociation of the unhydrolyzed portion of 
the salt be denoted by then, if one mol of salt be dissolved 
in V liters of solution, the final concentration of M" ions will be 

y - and the final concentration of the undissociated base 

X /> 

will be y . The total acid formed will be y , and if a a denotes 

the degree of dissociation of the acid, the concentration of the H* 

y 
ions will be <x a y~ Substituting these values in equations (1) and 

(2), we obtain 

, (3) 



and 



Simplifying equations (3) and (4), we have 

(T^xyv ' ^7 = K^ = K * (5) 

and 



(1 - x) a, 

From equations (5) and (6) it appears that the constant of hydrol- 
ysis can be found either from the ionic product for water and the 



440 THEORETICAL CHEMISTRY 

ionization constant of the base, or from the ionic product for 
water and the solubility product of the base. Furthermore, if the 
base formed is insoluble, equation (6) shows that the degree of 
hydrolysis, z, is independent of the dilution of the salt, V. 

CASE II. The formation of an undissociated or insoluble add 
is primarily the cause of the hydrolysis, the base formed being dis- 
sociated. In this case hydrolysis takes place until the product 
CH' X CA' exceeds that which can exist in the absence of undis- 
sociated acid. When equilibrium is established, we have 

final CH X final CA' = ^HAX CHA formed, (7) 

or if the acid formed is practically insoluble, the equilibrium equa- 
tion simplifies to the form 

final CH- X final CA' = SHA (8) 

Since the final CH = SH Z O -*- final COH', we have, final CA' = y , 

oc 
final COH' = <*b -y , where a h is the degree of dissociation of the base 

formed, and the final CHA = y Substituting these values in equa- 
tions (7) and (8), we obtain 



and 

-"-*) -*_... (10) 



Simplifying equations (9) and (10), we have 



-x a, 
and 



(1 x) a, SHA 

It is evident from equations (11) and (12), that the constant of 
hydrolysis can be found either from the ionic product for water and 



ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 441 

the ionization constant of the acid, or from the ionic product for 
water and the solubility product of the acid. 

CASE III. The formation of an acid and a base, both being 
slightly dissociated, is the cause of the hydrolysis. 

In this case let us assume that K HA is smaller than KMOH. 

Since the final c O ir = K * XCMQH , and since both HA and MOH 

CM* 

are slightly dissociated, we may write CHA = CMOH = |F, and 

a 9 (1 - x) 
^- - 



Substituting these values in equation (7), we obtain 

<** ~ x) s H2 o x , Q , 

- y --- = X HA X y- (13) 

V j-r X V 

A MOH y 



V 
Simplifying equation (13), we obtain 



_ _ 

(1 - xY ~a* KJUL X KMOH 

From equation (14) we see that the constant of hydrolysis can be 
found from the ionic product for water and the ionization constants 
of the acid and the base. If both acid and base are practically 
insoluble, the reaction will be complete at all dilutions. 

As an illustration of the application of the foregoing equations, 
we may take the calculation of the degree of hydrolytic dissociation 
of potassium cyanide in 0.1 molar solution at 25. Potassium 
cyanide being a salt of a weak acid, the degree of hydrolysis can 
be calculated by means of the equation 



(l-x)V' a .~K^~ k " 

Since at 25 C, KHA = 7.2 X 1Q- 10 and sn,o = 1.05 X 10~ 7 ) 2 , we 
have 

(1.05 X 10-') 2 



K - 
* 



7.2 X 10 



- 10 ' 



442 THEORETICAL CHEMISTRY 

and since in dilute solution a a = o& = 1, we have 

s 2 (LOS X 10~ y ) 2 

(1 - x) 10 7.2 X 10- 10 ' 
or 

x = 0.0123 

Experimental Determination of Hydrolysis. The degree of 
hydrolysis can be determined experimentally in several different 
ways. A very convenient method is that based upon measure- 
ments of electrical conductance. When a salt reacts hydrolyti- 
cally with one mol of water, the limiting value of its equivalent 
conductance will be A^ + A#, where A^ and A# denote the equiv- 
alent conductances of the acid and base formed. If A is the equiv- 
alent conductance of the unhydrolyzed salt, and A h is the actual 
conductance of the salt at the same dilution, then the increase in 
conductance corresponding to a degree of hydrolysis x will be 
Afc A, The value of A may be found by determining the 
conductance of the salt in the presence of an excess of one of the 
products of hydrolysis and deducting from it the conductance of 
the substance added. Since if the hydrolysis were complete, the 
equivalent conductance would be A^ + A# A, we have 



A A +A B - A 

all conductances being measured at the same dilution and the 
same temperature. The following example will illustrate the 
use of this equation: At 25, the equivalent conductance of an 
aqueous solution of aniline hydrochloride is 118.6, the dilution 
being 99.2 liters. The equivalent conductance in the presence of 
an excess of aniline is 103.6, while the equivalent conductance of 
hydrochloric acid at the same dilution is 411. The conductance 
of pure aniline is so small as to be negligible. Substituting these 
values in the equation, we find 

118.6-103.6 ArvlQQ 
* = 411 - 103.6 - - 0488 - 

Lunden * has shown how this method may be extended to cases 
where both acid and base are slightly dissociated. 
* Jour. chim. phys., 5, 145, 574 (1907). 



ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 443 

The lonization Constant of Water. One of the most accurate 
methods known for the determination of the ionization constant 
of water is based upon measurements of the degree of hydrolytic 
dissociation of different salts. Thus Shields* found that a 0.1 
molar solution of sodium acetate is 0.008 per cent hydrolyzed at 
25. We may consider the salt, as well as the sodium hydroxide 
formed from its hydrolysis, to be completely dissociated at this 
dilution. The ionization constant of the acetic acid formed is 
0.000018 at 25. Solving equation (11) (on page 440) for sn 2 o, 
and remembering that , = & = 1, we have 



Substituting the above values in this expression, we obtain 



SH ,o - 0.000018 . . 10 - L16 X 



and since the ions, H* and OH', are present in equivalent amounts 
we have _ 

CH* = COIF = Vl.16 X 10" 14 = 1.1 X 10- 7 mol per liter. 

Kohlrausch obtained from his measurements of the conductance 
of pure water at 25, CH = COH' = 1.05 X 10~ 7 mol per liter (see 
p. 415). 

PROBLEMS. 

1. At 25 the specific conductance of butyric acid at a dilution of 
64 liters is 1.812 X 10~ 4 reciprocal ohms. The equivalent conductance 
at infinite dilution is 380 reciprocal ohms. What is the degree of ioniza- 
tion and the concentration of H" ions in the solution? What is the ioni- 
zation constant of the acid? 

Ans. a = 0.0305, CH- = 4.765 X 10~ 4 mol per liter, K = 1.5 X 10-*. 

2. The heat of neutralization of nitric acid by sodium hydroxide is 
13,680 calories, and of dichloracetic acid, 14,830 calories. When one 
equivalent of sodium hydroxide is added to a dilute solution containing 
one equivalent of nitric acid and one equivalent of dichloracetic acid, 
13,960 calories are liberated. What is the ratio of the strengths of the 
two acids? Ans. HN0 8 : CHC1 2 COOH :: 3.1 : 1. 

* Zeit. phys. Chem., 12, 167 (1893). 



444 THEORETICAL CHEMISTRY 

3. For potassium acetate we have the following data: 



V 


At)(l8) 


2 


67 1 


10 


78 4 


100 


87 9 


10000 


91 9 



and Zoo = 64.67, and 7oo = 35. Compare the constants obtained by the 

K' CH 3 COO' 

Ostwald, Rudolphi, and van't Hoff dilution laws. 

4. The ioriization constant of a 0.05 molar solution of acetic acid is 
0.0000175 at 18, and 0.00001624 at 52. Calculate the heat of ionization 
of the acid. To what temperature does this value correspond? 

Ans. 416 calorics at 35. 

5. At 20 the specific conductance of a saturated solution of silver 
bromide was 1.576 X 10~ 6 reciprocal ohms, and that of the water used 
was 1.519 X 10~ 6 reciprocal ohms. Assuming that silver bromide is 
completely ionized, calculate the solubility and the solubility product of 
silver bromide, having given that the equivalent conductances of potas- 
sium bromide, potassium nitrate, and silver nitrate at infinite dilution 
are 137.4, 131.3, and 121 reciprocal ohms respectively. 

Ans. CAgBr = 4.49 X 10~ 7 mol per liter, SAgBr = 2.02 X 10~ 13 . 

6. The solubility of silver cyanate at 100 is 0.008 mol per liter. Cal- 
culate the solubility in solution of potassium cyanate containing 0.1 mol 
of K* ions. Ans. 6.4 X 10" 4 mol per liter. 

7. Calculate the degree of hydrolytic dissociation of a 0.1 molar solu- 
tion of ammonium chloiide, having given the following data: a a = 0.86, 
Q = 0.87,#NH 4 OH = 0.000023, and H 2 o = (0.91 X 10- 7 ) 2 at25. 

Ans. x = 0.006 per cent. 

8. In the reaction represented by the equation 

MA 3 + 3 H 2 = M (OH), + 3 HA, 

the base formed is insoluble. Derive an expression for the constant of 
hydrolysis. 

Ans. K p n 



SM(OH) 8 (1 

9. The equivalent conductance of aniline hydrochloride at a dilution 
of 197.6 liters is 126.7 reciprocal ohms, at 25. The equivalent con- 



ELECTROLYTIC EQUILIBRIUM AND HYDROLYSIS 445 

ductance of aniline hydrochloride in the presence of an excess of aniline 
is 106.6; and the equivalent conductance of hydrochloric acid at the 
same dilution is 415. If the conductance of pure aniline is negligible, 
calculate the degree of hydrolytic dissociation and the constant of hydrol- 
ysis, assuming , = = 1. 

Ans. x = 6.52 per cent, K h = 2.30 X 10~ 5 . 

10. The hydrolysis constant of aniline is 2.25 X 10~ 6 , and the ioniza- 
tion constant is 5.3 X 10~ 10 . Calculate the concentration of the H* 
and OH' ions in water. Ans. CCH* = CCH' = 1.09 X 10~ 7 . 



CHAPTER XX. 
ELECTROMOTIVE FORCE. 

Galvanic Cells. Since the year 1800, when Volta invented 
his electric pile, many different forms of galvanic cell have been 
introduced. 

It is not our purpose to give a detailed account of these cells, 
but rather to give a brief outline of the theories which have been 
advanced in explanation of the electromotive force developed in 
such cells. When two metallic electrodes are immersed in a solu- 
tion of an electrolyte, a current will flow through a wire connect- 
ing the electrodes, provided the two metals are dissimilar, or that 
a difference exists between the solutions surrounding the electrodes. 
An electric current can be obtained from a combination of two 
different metals in the same electrolyte, from two different metals 
in two different electrolytes, from the same metal in different elec- 
trolytes, or from the same metal in two different concentrations of 
the same electrolyte. 

In order that the electromotive force of the combination shall 
remain constant, it is necessary that the chemical changes involved 
in the production of the current shall neither destroy the difference 
between the electrodes, nor deposit upon either of them a non- 
conducting substance. A galvanic combination which fulfils 
these conditions very satisfactorily is the Daniell cell. This cell 
consists of zinc and copper electrodes immersed in solutions of 
their salts, as represented by the scheme 

Zn - Sol. of ZnS0 4 1| Sol. of CuS0 4 - Cu, 

in which the two vertical lines indicate a porous partition separat- 
ing the two solutions. When the zinc and copper electrodes are 
connected by a wire, a current of positive electricity passes from 
the copper to the zinc along the wire. Zinc dissolves from'the zinc 
electrode, an equivalent amount of copper being displaced from 

446 



ELECTROMOTIVE FORCE 447 

the solution and deposited simultaneously on the copper electrode. 
As long as only a moderate current flows through the cell, the 
original nature of the electrodes is not modified, the only change 
which occurs being the gradual dilution of the copper sulphate, 
owing to the separation of copper and its replacement by zinc. 
If the loss of copper sulphate is replaced, the electromotive force 
of the cell will remain constant. If, after the cell is assembled no 
current be allowed to flow, the copper sulphate will slowly diffuse 
into the solution of zinc sulphate, and metallic copper will ulti- 
mately be deposited on the zinc electrode. In this way miniature, 
local galvanic cells will be formed on the surface of the zinc, and 
the metal dissolves as though the main circuit were closed. Until 
this deposition takes place, the cell may be left on open circuit 
without danger of deterioration. Unless chemically pure zinc is 
used, local action is likely to occur, owing to the formation of local 
galvanic couples between the impurities in the electrode, chiefly 
iron, and the zinc. This action may be prevented by amalga- 
mating the zinc electrode. In this process the mercury dissolves 
the zinc and not the iron, a uniform surface of the former metal 
being produced. 

An interesting experiment due to Ostwald * illustrates the con- 
ditions essential to the continuous production of an electric current. 
Two electrodes, one of amalgamated zinc and the other of platinum, 
are each immersed in a solution of potassium sulphate, the two 
solutions being separated by a porous cup. When the two elec- 
trodes are connected by means of a wire, no permanent current 
passes. An inappreciable quantity of zinc goes into solution, 
since any current must necessarily first liberate potassium at the 
platinum electrode, the potassium thus set free reacting with the 
water. This process requires the expenditure of more energy 
than the solution of the zinc supplies. If sulphuric acid is added 
to the compartment containing the zinc, the condition of the 
system will be unchanged, the zinc remaining undissolved. If, 
on the other hand, a few drops of sulphuric acid are added to the 
compartment containing the platinum electrode, bubbles of 
hydrogen will appear and the zinc will dissolve with the simulta- 

* Phil. Mag. [5], 32, 145 (1891). 



448 THEORETICAL CHEMISTRY 

neous development of an electric current. This experiment shows 
that in order that positively charged ions may enter a solution, 
an equivalent amount of negatively charged ions must be intro- 
duced, or an equivalent amount of positively charged ions must 
be removed. 

Reversible Cells. Galvanic cells are either reversible or non- 
reversible, according as the processes taking place within them can 
be reversed or not. If we disregard the slow processes of diffusion, 
the Daniell cell may be taken as an example of an almost perfect 
reversible element. If an electromotive force slightly less than 
that of the cell be applied to it in the reverse direction, the current 
within the cell will flow from the zinc to the copper electrode as 
usual. On the other hand, if the external electromotive force 
slightly exceeds that of the cell, the current within the cell will 
flow in the reverse direction, zinc being deposited and copper 
dissolved. 

Any cell from which gas is evolved is non-reversible, since the 
passage of a current in the reverse direction cannot restore the 
cell to its original condition. 

Relation between Chemical Energy and Electrical Energy. 
Helmholtz and Thomson were the first to propose a theory of the 
action of the reversible cell. According to this theory the energy 
of the chemical process taking place within the cell was considered 
to be completely transformed into electrical energy. It was soon 
shown that this theory is inadequate, since, with the exception of 
the Daniell cell, the chemical energy is not equivalent to the elec- 
trical energy produced. Subsequently, Gibbs * and Helmholtz f 
showed independently that only in those cells in which the elec- 
tromotive force does not vary with the temperature, is the chem- 
ical energy completely transformed into electrical energy. They 
also derived an equation expressing the relation between the 
chemical and electrical energies in any reversible cell. Let us 
imagine a reversible element in which an amount of heat g, is either 
liberated or absorbed, when one faraday of electricity has passed 
through the cell. Let the cell be immersed in a bath, which is so 

* Proc. Conn. Acad., 3, 501 (1878). 

t Sitzungsbericht., Ber. Akad., 22, 825 (1882). 



ELECTROMOTIVE FORCE 449 

arranged that the temperature of the cell can be maintained con- 
stant under any working conditions. If the chemical process 
within the cell is accompanied by an evolution or an absorption 
of heat, then of necessity, heat energy must be removed or 
supplied in order to maintain the temperature of the system 
constant. It is evident that this will involve a corresponding 
decrease or increase in the electrical energy produced by the cell. 
The effect of the evolution or absorption of heat upon the 
electrical energy of the cell may be derived in the following 
manner: Let the cell be heated from its initial temperature T 
to the temperature (T + dT), and let the corresponding change 
in the electromotive force of the cell be dE. If now the circuit be 
closed and one faraday of electricity be allowed to pass through 
the cell, F (E + dE) units of electrical work will be done. In 
order that the temperature of the cell may not change, (q + dq) 
units of heat must be absorbed. The cell is now cooled to the 
temperature !T, at which the electromotive force of the cell is E, 
and F units of electricity are sent through the cell in the reverse 
direction, thus increasing the energy of the cell by FE. In order 
to maintain the temperature of the cell unchanged, q units of 
heat must be removed. If the cell is completely reversible, when 
this cycle of operations is completed, it will be restored to its 
original condition. The total work done during the cycle is 
F (E + dE) FEj and the amount of heat transformed into work 
is (q + dq) g; therefore, applying the second law of thermo- 
dynamics, we have 

dq_FdE _dT 
q q - T> 
or 



Since the electrical energy is equal to FE, the relation between 
this and Q, the chemical energy of the cell, expressed in calories, 
becomes 

Q + q. (2) 



450 THEORETICAL CHEMISTRY 

Substituting in equation (2) the value of q given in equation (1), 
we obtain 



F- Q +T dE M\ 

E ~F +T dT' (3) 

When -j?fi SB 0, E becomes equal to ~, or, when the temperature 
al V 

coefficient of the cell is zero, the electrical energy is equal to the 
chemical energy. This is true of the Daniell cell, which has an 
extremely small temperature coefficient. 

For cells in which the electromotive force varies appreciably 
with the temperature, it is possible to calculate the value of the 
electromotive force at any temperature by means of the Gibbs- 
Helmholtz equation, provided the temperature coefficient is known. 
In the Grove gas cell, E = 1.062 and Q = 34,200 calories, hence 



- 418 
U.413. 



The value determined by direct experiment is 0.416 volt. The 
Gibbs-Helmholtz equation shows that the amount of heat accom- 
panying a chemical process does not alone furnish a measure of 
the electrical energy which may be obtained from it, since the 
heat which is absorbed from the surrounding medium may also 
be transformed into electrical energy, or the output of electrical 
energy may be less than the heat evolved by the chemical reaction 
within the cell. 

Solution Pressure. It is a familiar fact that water has a 
tendency to assume the form of vapor, and if the vapor be contin- 
ually removed from its surface, a definite mass of water will grad- 
ually be completely transformed into the state of vapor. The 
pressure of the vapor at any one temperature is a measure of the 
tendency of water to undergo this transformation. This tendency 
of water, to assume another form than that in which its actually 
exists, is typical of all substances. Attention has already been 
directed to this fact in connection with the application of the law 



ELECTROMOTIVE FORCE 451 

of mass action to heterogeneous equilibria. It was then pointed 
out that all solids have a definite vapor pressure at a definite 
temperature, which is independent of the amount of solid present. 
When a solid, such as cane sugar, is brought in contact with water, 
it tends to pass into solution. This tendency is constant at 
constant temperature, since the active mass of the solid is constant. 
From the close analogy between the vapor state and the dissolved 
state, the tendency of a solid to pass into solution is termed the 
solution pressure. A dissolved solid, on the other hand, also shows 
a tendency to separate from the solution as the concentration 
is increased. When the solution becomes supersaturated, the 
tendency of the solute to separate in the solid form is greater than 
the tendency of the solid to dissolve. It is evident from these 
considerations that the pressure exerted by the dissolved solid 
is its osmotic pressure, and whether the solid will dissolve or 
separate from the solution depends upon whether the solution 
pressure is greater or less than the osmotic pressure. 

This conception of solution pressure was introduced by Nernst,* 
and in conjunction with the theory of electrolytic dissociation it 
has proved of great value in affording a much deeper insight into 
the mechanism of the development of differences in potential 
within a galvanic cell. Thus, when a metal is dipped into water 
it tends to dissolve owing to its solution pressure P and, in con- 
sequence of this tendency, it sends a certain number of positive 
ions into solution. The solution thus becomes positively charged, 
and the metal, which was initially neutral, acquires a negative 
charge due to the loss of a certain amount of positive electricity. 
This process will cease when the solution becomes so strongly 
charged with positive electricity that it prevents the separation 
of any more positive ions from the metal. Relatively few ions 
leave the metal before equilibrium is established, since the charge 
on each ion is so great; in fact, the concentration of metal ions in 
the solution is much too small to be detected analytically. When 
a metal is dipped into a solution of one of its salts, the conditions 
are altered. In this case, the positive ions of the metal already 
present in the solution oppose the entrance of more positive ions, 

* Zeit. phys. Chem., 4, 150 (1889). 



452 



THEORETICAL CHEMISTRY 



and the equilibrium between these two opposing tendencies will 
be conditioned by the relative values of the solution pressure P, 
of the metal, and the osmotic pressure p, of the ions of the dissolved 
salt. 

It is evident that the three following conditions are possible: 

(1) If P > p, the metal will continue to send ions into the 
solution until the accumulated charges in the solution oppose 
further action. The solution acquires a positive charge and the 
metal a negative charge. 

(2) If P < p, the positive ions of the dissolved salt will sepa- 
rate on the metal until the accumulated charges oppose further 
action. The metal acquires a positive charge and the solution a 
negative charge. 

(3) If P == p, no action will take place and no difference of 
potential will be established between the metal and the solution. 
These three cases are represented diagrammatically in Fig. 95. 



-f 

4- 



Fig. 95. 

When equilibrium is established and the metal is negative against 
the solution, the metal is surrounded by a layer of positively 
charged ions. This constitutes what is known as a Helmholtz 
electrical double layer. If positive electricity be communicated to 
the metal, the double layer will be broken and more ions will 
pass from the metal into the solution, but as soon as the supply 



ELECTROMOTIVE FORCE 



453 



of positive electricity is cut off, the double layer will again be 
formed. Similarly, when the metal is positive against the solu- 
tion, an electrical double layer will be formed, the metal being 
surrounded by a layer of negatively charged ions. 

The actual existence of a Helmholtz double layer has been 
demonstrated by Palmaer.* In his experiments, Palmaer allowed 
exceedingly minute globules of mercury to fall into a dilute solu- 
tion of mercurous nitrate contained in a tall vessel, the bottom 
of which was covered with a layer of pure mercury, as shown in 
Fig. 96. Since the solution pressure of mercury is less than the 





Fig. 96. 

osmotic pressure of the Hg* ions, each drop of mercury as it 
enters the solution will acquire a positive charge, and if the theory 
of the electrical double layer is correct, this positively charged 
globule should attract negatively charged ions and drag them down 
through the solution. When the globule reaches the mercury at 

* Zeit. phys. Chem., 25, 265 (1898); 28, 257 (1899); 36, 664 (1901), 



454 THEORETICAL CHEMISTRY 

the bottom of the vessel, it will give up its positive charge and as 
many Hg* ions will pass into solution as there are N(V ions in the 
double layer. The solution will thus become more concentrated 
just above the layer of mercury on the bottom of the vessel. Pal- 
maer's experiments showed that this difference in concentration 
is actually produced, in some cases the concentration in the upper 
part of the solution being reduced as much as 50 per cent. 

The metals sodium, potassium, . . . zinc, cadmium, cobalt, 
nickel, and iron are negative against solutions of their salts, or 
P > p. The noble metals are generally positive against solutions 
of their salts, or P < p. The anions are, so far as is known, posi- 
tive to solutions of their salts. Electrolytic solution pressure 
varies with the temperature, with the nature of the solvent, and 
also with the concentration of the active substance in the elec- 
trode. 

The Difference of Potential between a Metal and a Solution. 
From the foregoing considerations, it is possible to derive an 
equation expressing the difference of potential between a metallic 
electrode and a solution of one of its salts. 

Let us imagine one gram-ion of a metal to be transferred from 
the electrolytic solution pressure P, to the osmotic pressure p. 
The osmotic work done will be 



Integrating this expression, we have 

P 

Osmotic work = RTloge 

P 

The corresponding electrical energy gained is nFir, where w is the 
difference of potential between the metal and the solution, F = 1 
faraday = 96,540 coulombs, and n is the valence of the metal. 
Since the osmotic work done is equivalent to the electrical energy 
gained, we may equate these two expressions, as follows: 

nFir = RT \oge~ > 
P 
or 

RT P 



ELECTEOMOTIVE FORCE 455 

Expressing both sides of equation (1) in electrical units, and trans- 
forming to Briggsian logarithms, we obtain 

2 P 

* = 96,540 X n X 0.4343 X 0.2394 ri g p' 
or 



For univalent ions at 17, we have 

* = 0.0575 log-. (3) 

In a galvanic cell composed of two metals, each immersed in a 
solution of one of its salts, a difference of potential may be estab- 
lished (1) at the junction of two metals, (2) at the junction of 
the two solutions, and (3) at the points of contact of the metals 
with their respective solutions. If the temperature remains con- 
stant, (1) is negligible, and in general, (2) is exceedingly small; 
therefore, the electromotive force of the cell may be considered as 
due to the differences of potential arising at the two electrodes. 
Assuming the temperature to be 17, the electromotive force of 
the cell will be 

-, 0.0575, Pi 0.0575, P 2 

E = TTi ~ 7T 2 = - log --- lOg 

n & pi n & p 2 

The Measurement of Electromotive Force. The value of the 
electromotive force of a cell may vary with the conditions of meas- 
urement. Since, according to Ohm's law, E = C (R + r), where 
R is the resistance of the external circuit and r is the internal 
resistance of the cell, it follows that the fall of potential Cfi, in 
the external circuit, will only be equal to E when r is negligible 
in comparison with R. Furthermore, when the circuit is closed, 
the electrodes of the cell frequently become polarized, owing to 
the deposition of the products of electrolysis, and an opposing 
electromotive force is set up. 

To avoid these difficulties, the electromotive force is usually 
measured on open circuit by the Poggendorfif compensation method. 
In this method the electromotive force to be measured is just 



456 



THEORETICAL CHEMISTRY 



balanced by an equal and opposite electromotive force, so that no 
current passes. The arrangement of the apparatus for such 
measurements is shown in Fig. 97. If the two ends of the wire AB 




Fig. 97. 

of a Wheatstone bridge are connected to a lead accumulator (7, 
there will be a uniform fall of potential along its length. The 
amount of fall along any portion AD will be proportional to the 

AD 
length AD, and equal to the fraction -j~^-oi the total fall of poten- 



tial along the entire length of the wire. Now let one terminal of 
a cell whose electromotive force is less than that of C be con- 
nected to A, and the other terminal be connected through a 
galvanometer G, with a sliding contact D, the two cells E and 
C working in opposition. A current will flow through the cir- 
cuit AEGD, and will be indicated by the galvanometer at all 
positions, except that at which the fall of potential along the wire 
from A to D is equal to the electromotive force of E. Hence we 
have 

e.m.f. of C : e.m.f. of E :: AB : AD, 

from which the value of the electromotive force of the cell E y can 
be calculated. Since the electromotive force of a lead accumu- 
lator is not quite constant, it is customary, after having deter- 
mined the point D, to substitute a standard cell for E, and balance 



ELECTROMOTIVE FORCE 



457 



this against the accumulator, finding a new point of balance D'. 
We now have the proportion 

e.m.f. of C : e.m.f. of standard :: AB : AD'. 
Combining these two proportions, we obtain 

e.m.f. of E : e.m.f. of standard :: AD : AD'. 
Instead of using a galvanometer as a "null" instrument for indi- 
cating when the point of balance has been reached, a capillary 
electrometer may be employed. 

Standard Cells. It is apparent that the accuracy of all 
measurements of electromotive force is dependent upon the cell 
employed as a standard. Much time has been devoted to the 
study of various reversible elements with a view to establishing a 
standard of electromotive force. As a result, we have the com- 
plete specifications for two standard cells, either of which may be 
readily reproduced. 

(a) The Western, or Cadmium Standard Cell. The most widely 
used standard of electromotive force is the so-called Weston cell, 
made up according to the scheme 

Hg - Solution Hg 2 S0 4 || Solution CdSO 4 - Cd. 
A diagram of the usual form of the Weston cell is given in Fig, 98. 




Fig. 98. 



458 THEORETICAL CHEMISTRY 

A short platinum wire is sealed through the bottom of each limb 
of the H-shaped vessel. In one limb is placed a small amount of 
a 10 to 15 per cent cadmium amalgam, A ; B is a layer of small 

o 

crystals of CdS0 4 H 2 O. In the other limb is placed a small 
o 

amount of pure mercury, over which is a layer, D, of a paste 
composed of solid mercurous sulphate and a saturated solution 
of cadmium sulphate. The cell is then filled with crystals of 
cadmium sulphate and a saturated solution of cadmium sulphate^ 
The two limbs of the cell are closed with a thin layer of paraffin 
E, cork F, and sealing wax G. If carefully prepared, this cell will 
remain unaltered for years and will have an electromotive force 
at 20 of 1.0183 volts. In addition to the fact that it can be so 
easily reproduced, the temperature coefficient of the cell is almost 
negligible. 

The electromotive force of a Weston standard cell at any temper- 
ature t } is given by the formula 

e.m.f.aU = 1.0183 - 0.000038 (t - 20). 

(b) The Clark, or Zinc Standard Cell. Until about ten years 
ago, the Clark cell was considered to be the most trustworthy 
standard of electromotive force. This cell is made up according 
to the scheme 

Hg - Solution Hg 2 S0 4 1| Solution ZnS0 4 - Zn. 

The construction of the cell is similar to that of the Weston cell. 
It may be reproduced with great accuracy and with no more 
trouble than the Weston cell, but its relatively large temperature 
coefficient renders it less satisfactory. The electromotive force 
of the Clark standard cell at any temperature t f may be calculated 
by means of the formula 

e.m.f. at f = 1.4328 - 0.00119 (t - 15) - 0.00007 (* - 15) 2 . 

The Capillary Electrometer. When pure mercury is covered 
with sulphuric acid, its surface tension is diminished. This may 
be shown by the following experiment: In a small evaporating 
dish place about 5 cc. of pure mercury, and cover it with a 10 per 



ELECTROMOTIVE FORCE 



459 



cent solution of sulphuric acid to which has been added enough 
potassium dichromate to impart a light yellow color to the so- 
lution. The globule of mercury will immediately flatten out, 
indicating that its surface tension has diminished. If now the mer- 
cury be touched with a piece of iron wire, it will instantly contract 
until the contact with the wire is broken; it will then flatten out, 
until it again comes in contact with the wire, when the globule 
of mercury will once more contract. In this way a regular pul- 
sation of the mercury may be obtained. This interesting phenom- 
enon was observed early in the nineteenth century by Henry, but 
was first satisfactorily explained by Lippmann * in 1873. Lipp- 
mann showed that when the globule of mercury is negatively 
electrified, its surface tension increases and the drop shrinks. 
If sufficient negative electricity is imparted to the mercury it 
is possible to restore the globule to its original form. On ap- 
plying more negative electricity, the globule of mercury again 
expands. When the iron wire touches the globule it charges 
it negatively, because when the iron dis- 
solves, it furnishes positively charged ions 
to the solution and thus acquires a negative 
charge which it imparts to the mercury. At 
the same time, the chromic acid in the so- 
lution undergoes reduction to chromium 
sulphate. Lippmann concluded from his 
experiments that the difference of poten- 
tial arises at the surface of contact between 
the mercury and the solution of the electro- 
lyte, and that the surface tension of the 
mercury is a function of the difference of 
potential. Making use of this principle he 
constructed the capillary electrometer, a con- 
venient form of which is shown in Fig. 99. 
The bulb A, through the bottom of which 
is sealed a platinum wire, contains pure 
mercury and dilute sulphuric acid (1 : 6). 




Fig. 99. 



Pure mercury is poured into the other limb of the electrometer 
* Pogg. Ann., 149, 546 (1873). 



460 



THEORETICAL CHEMISTRY 



until it stands at B in that tube, and at C in the capillary tube. 
Owing to the capillary depression of the mercury, C lies below B. 
Electrical connection with the mercury at B is established by 
means of a platinum wire. 

The position of the mercury in the capillary is determined by 
its surface tension; if the surface tension is increased, the mercury 
will descend; if it is diminished, the mercury will ascend. If a 
negative charge is communicated to the mercury at JS, the surface 
tension will be increased and the meniscus will descend; if a posi- 
tive charge is imparted to the mercury at B, the surface tension 
will be diminished and the meniscus will ascend. 

The amplitude of the movement of the meniscus is an inverse 
function of the diameter of the capillary tube. If the meniscus 
be observed through a microscope 
provided with an eye-piece microm- 
eter, Fig. 100, it is possible to detect 
very slight movements, and to meas- 
ure differences of potential less than 
0.0001 volt. 

The capillary electrometer is an ex- 
cellent "null" instrument. In using 
the electrometer no large electromo- 
tive force should be applied, since 
the meniscus surface becomes polar- 
ized very easily. If this should occur, 
a new surface may be secured by 
blowing gently at B and forcing a 
drop of mercury out of the capillary 
into the bulb. Lippmann studied the 
effect of steadily increasing potentials 
on the movement of the meniscus. 
Plotting movements of the meniscus 
on the axis of ordinates, and potentials 

on the axis of abscissae, he found that there is a maximum in the 
curve corresponding to about 0.8 volt. This is the electromotive 
force which must be applied in order to counterbalance the differ- 
ence of potential produced by the contact of dilute sulphuric acid 




Fig. 100. 



ELECTROMOTIVE FORCE 



461 




Sulphuric Acid- 



Mercury- 



Fig. 101. 



with the surface of the mercury. At the meniscus surface an 
electrical double layer is formed. The mercury is positively 

charged, and above it there must be a 
layer of negatively charged ions, as 
shown in Fig. 101. Just how this double 
layer is f onned is not known with cer- 
tainty, but it has been suggested that 
the slight film of oxide which is prob- 
ably present on the surface of the pur- 
est mercury, dissolves in the sulphuric 
acid forming a solution of mercurous 
sulphate, and from this solution the 
positively charged Hg" ions deposit on 
the mercury, giving it a positive charge. 
Whether this explanation is correct or 
not, the fact remains that the mercury 
is positive against the solution. 

Normal Electrodes. The method commonly employed for 
the measurement of the difference of potential between a metal 
and a solution, is based upon the use of an electrode in which the 
difference of potential between the electrode and a certain solution 
of one of its salts is known. Such an electrode is called a normal 
electrode. If a cell is made up by combining the normal electrode 
with the electrode whose potential is to be determined, it is possible, 
from measurements of the resulting electromotive force, to cal- 
culate the value of the unknown difference of potential. The most 
convenient electrode to prepare is the normal calomel electrode, a 
satisfactory form of which is shown in Fig. 102. The bottom of 
the electrode vessel is covered with a layer of pure mercury, upon 
which is poured a paste, prepared by rubbing together in a mortar 
mercury and calomel, moistened with a molar solution of potassium 
chloride. The vessel is then filled with a molar solution of the 
same salt which has been saturated with calomel by prolonged 
shaking with the latter. Connection with the mercury is estab- 
lished by means of a platinum wire sealed into a glass tube A, the 
latter being passed through the rubber stopper which closes the 
vessel. In using the calomel electrode, the bent side tube C is 



462 



THEORETICAL CHEMISTRY 



filled with molar potassium chloride by applying suction at the 
side tube 5, which is then closed by means of a pinch-cock. 

The difference of potential, at any temperature t, of the calo- 
mel electrode prepared as described, and represented by the 
scheme 

Hg - Solution HgCl in molar KC1, 
is 

TT = + 0.56 {1 + 0.0006 (t - 18)} volt. 

The positive sign before the value 0.56 indicates that the electrode 
is positive to the solution. In order to measure the potential of 




W 



Fig. 102. 



103. 



another electrode by means of the calomel electrode, the arrange- 
ment shown in Fig. 103 is commonly used. Here A represents the 
"half-element" of which the potential is to be determined, B repre- 
sents the side tube of the calomel electrode, and C represents an 
intermediate, connecting vessel containing a molar solution of 
potassium chloride. In cases where potassium chloride forms a 
precipitate with the electrolyte in A, the solution in C may be re- 



ELECTROMOTIVE FORCE 463 

placed by a molar solution of potassium nitrate without altering the 
value of the electromotive force of the cell. The original measure- 
ment of the potential of the calomel electrode was made by forming 
a cell with this and another electrode whose potential against its 
solution is zero. Such an electrode is known as a null electrode. 
Thus, if a copper electrode is immersed in a solution of copper 
sulphate, the Cu" ions will leave the solution and charge the 
electrode positively. If now a solution of potassium cyanide is 
added, the nearly undissociated salt, K 2 Cu 2 (CN) 4 , will be formed, 
and by adding a sufficient amount of the solution, the concentration 
of the Cu" ions may be reduced until the metal and the solution 
have the same potential. The addition of more potassium cyanide 
will still further diminish the osmotic pressure of the Cu" ions, 
and the electrode will acquire a negative charge. Similarly, mer- 
cury in a solution of a double cyanide may be used as a null elec- 
trode. 

Another form of null electrode is the so-called dropping elec- 
.trode of Helmholtz.* The principle involved in this electrode 
has already been discussed in connection with Palmaer's experi- 
ment (p. 453). An extremely fine stream of mercury is allowed 
to flow from a funnel having a minute capillary orifice: the stem 
of the funnel dips below the surface of a molar solution of potas- 
sium chloride containing mercurous ions. As each little globule 
enters the solution, it acquires a positive charge and attracts the 
negatively charged ions of the electrolyte, dragging them down 
with itself. When the globule reaches the layer of mercury at 
the bottom of the vessel, its surface and capacity are diminished, 
and as many Hg* ions leave the layer of mercury and enter the 
solution as there were negatively charged ions carried down by 
the globule. This process continues until the osmotic pressure 
of the remaining ions is equal to the solution pressure of the metal: 
the mercury, both in the stream and at the bottom of the vessel, 
has the same potential as the solution. If now the difference of 
potential between the mercury in the funnel and the mercury in 
the vessel be measured, we shall obtain the potential of mercury 
against a molar solution of potassium chloride. The dropping 
* Ann, der Phys., 44, 42 (1890). 



464 THEORETICAL CHEMISTRY 

electrode was for a long time regarded as an ideal standard of 
potential, but Nernst has quite recently pointed out a number 
of serious objections to it. Until a wholly satisfactory standard 
of potential is obtained, he proposes that the potential of the 
hydrogen electrode be adopted as the standard. This consists 
of a strip of platinized platinum, half in pure hydrogen gas and 
half in a solution of sulphuric acid of such concentration that 
it shall contain 1 gram of hydrogen ions per liter. The use of 
the hydrogen electrode as a standard is, of course, purely arbi- 
trary, but there are many advantages in referring differences of 
potential to this standard. Owing to certain experimental diffi- 
culties attending the use of this electrode, it is customary to 
make the actual measurements with the calomel electrode, and 
then refer them to the hydrogen standard, taking the potential 
of the calomel electrode to be + 0.283 volt when referred to 
the hydrogen electrode as zero. The positive sign indicates that 
the electrode is positive toward the solution. 

Measurement of the Difference of Potential between a Metal 
and a Solution. The difference of potential between a metal 
and a solution of one of its salts is easily determined by means of 
the calomel electrode. For example, in order to determine the 
potential of zinc against a molar solution of zinc sulphate, the 
electromotive force E, of the combination 

Zn - m ZnS0 4 1| m KC1, HgCl - Hg, 

(cal. electrode) 

is measured and found to be 1.08 volts, the mercury being the 
positive terminal of the cell. Applying the Nernst equation, 



RT , /2 RT , PI 



in which PI and pi denote the solution pressure and the osmotic 
pressure of the zinc ions, and P% and p% denote the solution pressure 
and the osmotic pressure of the mercury ions, we have 



T> 

1.08 -0.66- log. , 



ELECTROMOTIVE FORCE 465 

or 



T> 

loge = 0.56 - 1.08 = - 0.52 volt. 
2/ y pi 

That is, the zinc electrode is negative against a molar solution of 
zinc sulphate, the difference of potential being 0.52 volt. As an 
example of a cell in which the mercury of the calomel electrode 
is the negative terminal of the cell, we may take the following 
combination: 

Cu - m CuSO || m KC1, HgCl - Hg. 

The electromotive force of this cell is 0.025 volt. Since the 
current flows from the copper to the mercury, we have 



E = 0.025 -log.- 0.56, 
or 

T>fp f> 

~~Uog<, - 1 =0.025 + 0.56 = 0.585 volt. 
&r Pi 

That is, the copper electrode is positive against a molar solution 
of copper sulphate. From the above results it is possible to cal- 
culate the electromotive force of the combination 

Zn - in ZnSO 4 1| m CuS0 4 - Cu. 
Since, according to Nernst's equation 

w RT } P, RT, P 2 

E - 1T1 - 7T 2 - 2f loge- ~ 2l?loge-, 

where PI and pi refer to the copper, and P 2 and p* refer to the zinc, 
we have 

E = 0.585 - ( - 0.52) = 1.105 volts. 

Concentration Elements. We now proceed to consider cells in 
which the electromotive force depends primarily on differences 
in concentration, the so-called " concentration elements." 

Concentration elements may be conveniently divided into two 
classes: (a) elements in which the electrodes are of different concen- 
trations, and (b) elements in which the solutions are of different 
concentrations. 



466 



THEORETICAL CHEMISTRY 



(a) Elements in which the Electrodes are of Different Concentra- 
tions. (Amalgams and Alloys.) If in the equation 
RT 1 PI RT , P 2 



Pi = P2, as is the case when the ionic concentrations of the two 
solutions are identical, then we have 

a.^iog.g, 

where PI and P 2 are the respective solution pressures of the metal 
dissolved in the electrodes. If the amalgams are dilute, the 
osmotic pressure of the dissolved metal will be proportional to the 
solution pressure of the electrode, and since osmotic pressure 
is proportional to concentration, we may replace PI and P 2 in the 
above formula by the proportional terms, Ci and c 2 , the respec- 
tive concentrations of the metal in the two electrodes. Hence, 

we have 

RT , Ci 



The accuracy of this equation has been fully established by the 
experiments of Meyer,* and Richards and Forbes.f 

Meyer's results for zinc amalgams in solutions of zinc sulphate 
are given in the accompanying table. 



T, 
degrees. 


1 


Cz 


E (oba.). 


E (calc.). 


284 6 


0.003366 


0.00011305 


0419 


0.0416 


291 


003366 


00011305 


0.0433 


0.0425 


285 4 


002280 


0000608 


0474 


0.0445 


333 


0.002280 


0000608 


0.0520 


0.0519 



The agreement between the observed and calculated values of E 
is all that can be desired. That the above formula holds for 
zinc amalgams may be considered as a proof of the fact that the 
zinc dissolves in the mercury as monatomic molecules. Thus, 
suppose the zinc to be present in the mercury in the form of dia- 

* Zeit. phys. Chem., 7, 477 (1891). 

t Publication of the Carnegie Institution, No. 56. 



ELECTROMOTIVE FORCE 467 

tomic molecules; then while the electrical energy would be equal 
to 2 FE, the osmotic work required to develop this energy would 

be = R T log c - 1 , hence we should have 

L C% 

IRT 



or the calculated value of the electromotive force would be just 
one-half of the observed value. The mercury in the amalgam 
has been shown to exert no effect upon the electromotive force of 
the cell so long as the dissolved metal has the greater potential. 

(b) Elements in which the Solutions are of Different Concentra- 
tions. In this type of cell we have two electrodes of the same 
metal immersed in solutions of different ionic concentrations of 
the metal. Hence, we may put PI = F 2 in the equation 

,, RT, P l RT 



which then takes the form 

tf 

11 - 



Since osmotic pressure is proportional to concentration, pi and 
p 2 may be replaced by the proportional terms Ci and eg, and the 
foregoing equation becomes 

RT 
E ~nF 
or 

T? RT 

/& = 

nF 

where mi and m 2 are the molar concentrations of the two solutions 
and ai and 2 are the corresponding degrees of ionization. As an 
example of a concentration element of this class we may take the 
following: 

Ag - 0,01 m AgN0 3 1| 0.1 m AgN0 8 - Ag. 



468 THEORETICAL CHEMISTRY 

The degrees of ionization of the two solutions at 18 are as follows: 
for 0.01 m AgNO 3 , a = 0.93, and for 0.1 m AgNO 2 , a = 0.81. 
Substituting in the equation 

E = 0.058 log 
we have 

= 0.058 iog = 0.0545 volt. 



The value of E found by direct experiment is 0.055 volt. 

In the example just given, the electrodes are reversible with 
respect to the positive ion of the electrolyte. Such electrodes are 
known as electrodes of the first type. It is also possible to construct 
cells with electrodes which are reversible with respect to the nega- 
tive ion of the electrolyte. These are termed electrodes of the second 
type. The calomel electrode is an example of an electrode of this 
latter type. If positive electricity passes from the metal to the 
solution, the mercury combines with the Cl' ions forming mercu- 
rous chloride, and if positive electricity passes in the reverse 
direction, chlorine dissolves and mercurous chloride is formed. 
In other words the electrode behaves like a chlorine electrode, 
giving up or absorbing the element according to the direction of 
the current. A typical combination involving an electrode of 
the second type is the following: 

Ag - 0.1 m AgN0 3 - KNO 3 - 0.1 m KC1, AgCl - Ag. 

This particular combination was studied by Goodwin * with a 
view to determining the solubility of silver chloride. If we assume 
a saturated solution of silver chloride to be completely ionized, 
then the solubility product will be 

X Cor = s. 



Since the concentrations of the two ions, Ag* and Cl', are equal, 
it follows that Vs will be equal to the solubility of the silver 
chloride. The electromotive force of a concentration cell at 25 
is given by the equation 

0.0595, WK*! 

Jb = ---- log 7 



n 
* Zeit. phya. Chem., 13, 577 (1894). 



ELECTROMOTIVE FORCE 469 

or 

1 
g 



00595 

The value of E at 25 for the above cell was found to he 0.450 volt. 
The degrees of ionization of the two electrolytes are as follows: 
for 0.1 molar AgNO 3 , a = 0.82, and 0.1 molar KC1, a = 0.85. 
Substituting in the preceding equation, we obtain 

, 0.1 X 0.82 0.450 * 
tog _ ~ 0^595' 

therefore, 

C2 = 2.24 X 10~ 9 . 

Or, 2.24 X 10~~ 9 is the concentration of the Ag* ion in mols per 
liter in a 0.1 molar potassium chloride solution of silver chloride. 
Hence the solubility product s will be 

s = 2.24 X 10-* X 0.085 = 1.91 X 10" 10 , 
and 

V^ - 1.38 X 10-*; 

that is, the solubility of silver chloride in a saturated aqueous 
solution is 1.38 X 10~ 5 mol per liter at 25. 

The Difference of Potential at the Junction of the Solutions 
of Two Electrolytes. Thus far we have not taken into consider- 
ation the potential differences which may be established at the 
junction of two solutions. Nernst * has shown that in many cases 
it is possible to calculate these differences of potential by means 
of his osmotic theory of the origin of electromotive force, and the 
values obtained are in close agreement with the results of experi- 
ment. Let us imagine that two solutions of hydrochloric acid of 
different concentrations are brought together so as to avoid mix- 
ing, the acid in each solution being highly ionized. The hydro- 
gen and chlorine ions will diffuse independently, and since the 
former move with the greater velocity, the more dilute solution 
will soon contain an excess of H* ions and the more concentrated 
solution an excess of Cl' ions. The more dilute solution will be- 
come positively charged owing to the presence of an excess of H* 

* Ztit, phys, Chem,, 4, 129 (1889). 



470 THEORETICAL CHEMISTRY 

ions, while the more concentrated solution will acquire a negative 
charge due to the presence of an excess of Cl' ions. The accumu- 
lation of positive electricity, however, will retard the velocity of 
the H' ions and accelerate the velocity of the Cl' ions, so that 
ultimately the two ions will move with the same velocity. The 
difference of potential produced in this way will cease to exist 
when the two solutions have acquired the same concentration. 

In general, it may.be stated that the difference of potential set 
up at the junction of the two solutions is caused by the differ- 
ence in the rates of migration of the two ions, the more dilute 
solution acquiring a charge corresponding to that of the faster 
moving ion. 

Electromotive Force of Concentration Cells with Transference. 
Let u and v be the migration velocities of the cation and the 
anion respectively, and let pi be the osmotic pressure of the ions 
in the concentrated solution and p% the osmotic pressure of the 
ions in the dilute solution. Now let one faraday of electricity pass 
through the two solutions, the current entering on the concen- 

trated side; then -p- gram-equivalents of positive ions will 

U ~| V 

migrate from the concentrated to the dilute solution, while 

gram-equivalents of negative ions will migrate from the dilute to 
the concentrated solution. The maximum work done by the 
process will be 



. 

u + v & p 2 u 

The corresponding electrical energy developed is irF, hence we 
have 

u-v RT, pi 
n = --- i g *L. 

u + v F 6 p 2 

or, since osmotic pressure is proportional to concentration, we 
may substitute Ci and 02 for pi and p z in the preceding equation, 
and obtain the following expression for the electromotive force 
at the junction of the two solutions: 

u-v RT, 



ELECTROMOTIVE FORCE 471 

As an example of the use of the above formula, we may take the 
calculation of the electromotive force of the following combina- 

tion: 

Ag - 0.1 m AgN0 3 - 0.01 m AgNO 3 - Ag. 

(a) (b) (c) 

Taking the junctions (a), (b), and (c) in order, we obtain 

RT . C u - v RT . d RT, C 
. log.-- log.- 



.- -^^ .- . 

2v RT 



-^TV' F log 't 

2v 

But = 2 n a , the transport number of the anion, and there- 

fore we may write the preceding equation in the form 

a) 



Assuming the temperature to be 17 and passing to Briggsian 
logarithms, we have 

E = 0.116 n a log-- 

Ci 

The transport number for the anion of silver nitrate is 0.522, 
while ci = miai = 0.1 X 0.82 = 0.082, and c 2 = m 2 a 2 = 0.01 X 
0.94 = 0.0094. Hence, 

E = - 2 X 0.522 X 0.058 X l 

or 

E = - 0.057 volt. 

The value found by direct experiment is 0.055 volt. 

It is to be noted that if the electromotive force at the junction 
of the two solutions is negative, equation (1) takes the form 



where n e is the transport number of the cation. 

Electromotive Force of Concentration Cells without Trans- 
ference. A less familiar type of concentration cell is that which 



472 THEORETICAL CHEMISTRY 

does not involve transference. The following combination may 
be taken as an example of such a cell: 

Ag | AgCl, KC1 1 K (Hg) 4 1] K (Hg) 4 1 KC1, AgCl | Ag. 



It will be observed that this cell is in reality made up of two 
independent cells and involves no liquid junction. While the 
elimination of diffusion appreciably simplifies the theoretical 
treatment of cells of this type, practically it is difficult to find 
electrodes which are reversible toward both ions of the electrolyte. 
The passage of one faraday of electricity through the cell results 
in the formation on the dilute side of one equivalent of potassium 
chloride from the silver chloride and the amalgam, while on the 
concentrated side a corresponding amount of potassium chloride 
is decomposed. If we assume the solutions to be so dilute as to 
be completely ionized, one equivalent of potassium chloride will 
function as two mols of gas, and the electromotive force of the cell 
will be 



1- !*. (2) 



Formulas for the Difference of Potential at Liquid Junctions. 

As has been pointed out, the total electromotive force of a con- 
centration cell with transference is made up of the algebraic sum 
of the potentials at the two electrodes and the potential at the 
junction of the two solutions. Since the value of the potential at 
the electrodes alone is often desired, numerous formulas have been 
derived for the calculation of the potential at the liquid junction. 
In addition to the formula of Nernst, already mentioned, for- 
mulas have been proposed by Planck,* Henderson,f Gumming, { 
Lewis and Sargent, and Maclnnes.|| Of these different formulas 
that of Maclnnes possesses distinct advantages. 

* Wied. Ann., 40, 561 (1890). 

t Zeit. phys. Chem., 59, 118 (1906); 63, 325 (1908). 

t Trans. Faraday Soc., 8, 86 (1912); 9, 174 (1913). 

Jour. Am. Chem. Soc., 31, 363 (1909). 

|| Ibid., 37, 2301 (1915). 



ELECTROMOTIVE FORCE 473 

If one faraday of electricity be passed through the cell 
Ag|AgCl,KCl|KCl,AgCl|Ag, 

Cl Cz 

one equivalent of chloride ions will enter the dilute solution while 
a corresponding amount will be electrolyzed out of the more con- 
centrated solution. The current will be carried across the liquid 
junction by the migration of n c equivalents of potassium ions in 
the direction of the current and by the migration of (1 n c ) 
equivalents of chloride ions in the opposite direction. The total 
effect of the passage of one faraday of electricity will be the trans- 
ference of n c equivalents of salt from the more concentrated to the 
more dilute solution. The osmotic work at the junction of the 
two solutions will correspond to the algebraic sum of the number 
of ion equivalents which have migrated from the concentrated to 
the dilute solution, or n c (1 n c ) = 2n c 1. 

In order to obtain the electrical energy necessary to perform 
this amount of osmotic work, let us consider the following cell 
involving no transference: 

Ag | AgCl, KC1 1 K (Hg), || K (Hg) 4 1 KC1, AgCl | Ag. 

Ci C2 

The passage of one faraday of electricity through this cell in- 
volves, as we have seen, the formation of one equivalent each of 
potassium and chloride ions in the dilute solution, and the re- 
moval of one equivalent of each ion from the more concentrated 
solution. The electrical energy accompanying the transfer of two 
ion equivalents from one solution to the other will be equal to 
the product of the electromotive force of the cell and one faraday, 
orEF. 

The electromotive force ?r, at the liquid junction, can now be 
obtained by the simple proportion 

EF :*F = 2 :2n c - 1, 






Since the ratio of E t , the electromotive force of the cell with trans- 
ference, to E, the electromotive force of the cell without trans- 



474 THEORETICAL CHEMISTRY 

ference, is equal to n c , it follows that equation (1) may be written 
in the form 

E t (2n c 1) _ w , 
K _ - _ - -. & t {i 

It is to be observed that equations (1) and (2) involve no assump- 
tions concerning the concentrations of the ions in the solutions, one 
ot the characteristics which distinguishes these formulas from any 
of the others hitherto proposed. 

Lewis and Sargent * have proposed a modification of a formula 
derived by Planck for the difference of potential at the junction of 
two equally concentrated electrolytes having a common ion. If 
Ai and A 2 are the equivalent conductances of the two electrolytes, 
the value of the liquid junction potential TT is given by the formula 

RT , AI /0 , 



The junction between two different concentrations of two elec- 
trolytes with a common ion may be replaced by two junctions, 
one of which connects two different concentrations of the same 
electrolyte, and the other connects two different electrolytes of the 
same concentration. For example, the junction 

0.1NNaCl|0.05NHCl 
may be replaced by 

0.1 N NaCl 1 0.05 N NaCl 1 0.05 N HC1 

(A) (B) 

in which the potential of the junction (A) may be calculated by 
equations (1) or (2) and the potential of junction (B) by equa- 
tion (3). 

The electromotive force at the junction of solutions of differently 
concentrated uni-univalent ionogens which do not contain a com- 
mon ion may, in like manner, be calculated by employing three 
junctions. Thus, the junction 

0.1 NKNOsl 0.05 N NaCl 
* loc. cit. 



ELECTROMOTIVE FORCE 475 

may be replaced by 

0.05 N NaCl 1 0.1 N NaCl 1 0.1 N KC1 1 0.1 N KNO 3 . 
(A) (B) (B') 

The potential at (A) may be calculated by means of equations (1) 
or (2) and the potentials at (B) and (B') by means of equation (3). 

By the use of equations (1), (2), and (3) in the manner indi- 
cated, it is possible to calculate the junction-potential between the 
solutions of any two uni-univalent ionogens to within a few tenths 
of a millivolt. 

Normal Electrode Potential. The difference of potential at 
a reversible electrode, when the concentration of the ions of the 
electrolyte is normal, is known as the normal electrode potential. 

According to Ncrnst, the expression for the difference of potential 

at a single electrode is 

RT P 



RT , C /1N 

= ^ lo ^c' (1) 

where P is the solution pressure of the electrode, p is the osmotic 
pressure of the ions of the electrolyte, and where C and c are con- 
centrations proportional to P and p respectively. If the concen- 
tration of the ions is unity, c 1, and the expression for the 
normal electrode potential E Q becomes 

geC. (2) 

On subtracting (1) from (2), we obtain 

J?T 



~, (3) 

where c is the concentration of the ions in a solution whose elec- 
trode potential is TT. In the actual determination of normal elec- 
trode potentials, the value of c in equation (3) is usually very much 
less than unity. 

The following table gives the most important electrode poten- 
tials, the data in the second column being referred to the normal 



476 



THEORETICAL CHEMISTRY 



calomel electrode as zero, and the data of the third column being 
referred to the normal hydrogen electrode as zero. 

NORMAL ELECTRODE POTENTIALS. 



Electrode. 


N. E. Potential, 
N. Calomel Electrode 
= 0. 


N. E. Potential, 
N. Hydrogen Electrode 
= 0. 



Cations. 



Copper . . . . 


+ 046 


+0 329 


Hydrogen 


-0 283 


_l_o ooo 


Lead 


-0 412 


129 


Mercury 


4- 0.467 


-fO 750 


Platinum . . . 
Potassium 
Silver 


+0 580 ca. 
-3 48 
4-0 515 


+0 863 ca. 
-3 20 
+0 718 


Sodium 


+2 998 


-f-3 281 


Tin 


0.475 


192 


Zinc .... . .... 


-1 053 


-0 770 



Anions. 



Bromine 


-f 797 


4-1 08 


Chlorine 


4-1 067 


+ 1 35 


Iodine 


-f 345 


+0 628 


Oxygen 


+0 110 


+0 393 









It should be observed that the value of the normal electrode 
potential cannot be calculated directly by means of equation (2) 
owing to the uncertainty attaching to the quantity C. In fact the 
best interpretation that we have of the physical significance of C 
is, that it is a quantity whose logarithm is proportional to the 
normal electrode potential. 

Electrometric Determination of Valence. The equation of 
Nernst for the electromotive force of a concentration cell may be 
employed to determine the valence of the ions. 

For example, the chemical behavior of mercurous salts is such 
as to justify the use of single or double formulas involving either 
Hg" or Hg 2 ". To determine which of these two formulas is correct, 
Ogg * set up the following cell : 

Hg 1 0.5 N Mercurous Nitrate || 0.05 N Mercurous Nitrate | Hg 
0.1NHNO 3 0.1NHN0 3 

* Zeit. phys. Chem., 27, 285 (1898). 



ELECTROMOTIVE FORCE 477 

and found its electromotive force at 17 to be 0.029 volt. If we 
neglect the difference of potential at the junction of the two solu- 
tions, the electromotive force of this cell may be calculated by the 
familiar formula 



Assuming that c^/Ci is equal to 10, and passing to Briggsian loga- 
rithms, we have 

0.029 = 0.058/n, 

or n = 2. 

Hence the valence of the mercurous ion is 2 and it must, in con- 
sequence, be represented by Hg 2 ". It follows that the correct for- 
mula of mercurous nitrate is Hg 2 (NO 3 ) 2 . 

Electrometric Determination of Transport Numbers. The 
formula for the electromotive force of a concentration cell with 
transference has been shown to be 

^ 2n c RT, C2 /1X 

*'--aH*ft- 

If the electromotive force of such a combination is measured and 
the values of Ci and GZ are accurately known, obviously the value 
of the transport number of the cation n c can be calculated. If, 
after having measured E t , a similar cell without transference be 
set up and its electromotive force measured, it is possible to obtain 
an expression for n c which does not involve either Ci or (%. As has 
been shown, the electromotive force of a cell without transference, 
E, is given by the formula 



Dividing equation (1) by equation (2), we obtain 

n c - , (3) 

which expression gives the transport number in terms of the two 
measured electromotive forces. This method for the determina- 
tion of transport numbers was first suggested by Helmholtz.* 
The transport number of the lithium ion has recently been de- 

* Ges. Abhl. I, 840: II, 979. 



478 



THEORETICAL CHEMISTRY 



termined in this manner by Pearce and Mortimer * using cells made 
up according to the following schemes: 

Ag | AgCl, LiCl 1| LiCl, AgCl | Ag (with transference), 

Cl C2 

Ag | AgCl, LiCl | Li (Hg), || Li (Hg), | LiCl, AgCl | Ag / without \ 
Ci C2 ^transference/ 

A comparison between their results and the values given by 
Kohlrausch and Holborn f is afforded by the accompanying table. 
Concentration Ratio . . . 10-01 0.5-0 05 1-0.01 0.05-0 005 
Mean n c (K. and H.) . ... 285 300 340 360 

Mean n c (P. and M.) . . .0.279 0.322 0343 0365 

It will be observed that the agreement between the two series of 
results is satisfactory. 

Concentration and Activity. If in the equation for the elec- 
tromotive force of a concentration cell without transference, viz., 

c 2 



we substitute the observed value of E and solve the equation for 
the ratio, C2/ci, we find that the value thus obtained does not agree 
with the value derived from conductance measurements. In other 
words, if Wi and m^ are the actual molar concentrations of the two 
solutions forming the cell, and i and 2 are the corresponding de- 
grees of ionization, we find that the ratio m^a^/miOLi is not equal 
to the ratio c^/c\ as calculated from the electromotive force of 
the cell. This is shown by the following table which gives the 
ratios of the ionic concentrations of solutions of potassium chloride 
corresponding to various ten-to-one salt concentrations. 

RATIOS OF ION CONCENTRATIONS IN SOLUTIONS OF 
POTASSIUM CHLORIDE. 



Concentration 
Ratio. 


C 2 /Ci 

(Conductance). 


c 2 M 
(Electromotive 
Force). 


5 :0.05 
01 : 01 
05:0.005 
01:0 001 


8.85 
9 16 
9 30 
9 62 


8 09 

8 33 
8 64 
9 04 



* Jour. Am. Chem. Soc., 40, 518 (1918). 
t Leitvermogen der Elektrolyte, p. 201. 



ELECTROMOTIVE FORCE 



479 



If the values of c*/ci in the table are plotted as ordinates against 
the higher concentration of each pair of solutions as abscissae, we 
obtain the curves shown in Fig. 104. It will be seen that notwith- 
standing the fact that the ratios differ widely when calculated by 
the two methods, both sets of figures approach the limiting value, 
10, at infinite dilution. From this it is evident that the equation 



10.0 - 



9.5 



9.0 



8.5 



8.0 




0.1 0.2 0.3 

Concentration 

Fig. 104. 



of Nernst is only applicable to cells involving solutions which are 
practically infinitely dilute. 

The figures in the last column of the table are known as activity 
ratios, and the individual values of Ci and c% are termed activi- 
ties. If it be assumed that in 0.001 KC1 the activity of the ions 
is identical with their concentration, as obtained by multiplying 
the total concentration of the salt by the ionization as determined 
from the conductance ratio, then the activities in 01 N KC1 and 
0.1 N KC1 can be calculated. The values so obtained are given in 
the following table together with the corresponding values of the 
ionization calculated, (1) from the conductance ratio, and (2) from 
the so-called "thermodynamically effective" ionization, this latter 
being the quotient obtained by dividing the activity by the corre- 
sponding total concentration. 

That the degree of ionization determined from conductance 
measurements does not agree with the value obtained from meas- 
urements of electromotive force has been known for some time. 



480 



THEORETICAL CHEMISTRY 



ACTIVITY AND IONIZATION OF POTASSIUM CHLORIDE 
SOLUTIONS. 



Salt Concentration. 


Activity. 


A/Ac- 


" Thermodynamically 
Effective " lonization. 


001 N 
01 N 

1 N 


000979 
00885 
0738 


97 9 
94 1 
86.1 


(97.9) 
88 5 
73.8 



It is thought that the decrease in activity of the ions with increas- 
ing concentration is due to a change in the nature of the solvent 
brought about by the electric charges on the ions. 

Electrometric Determination of Hydrolysis. One of the most 
satisfactory methods which we possess for the determination of the 
degree of hydrolysis of salts depends upon the measurement of the 
electromotive force of cells made up as follows: 

Pt m | salt solution | sat. NH 4 N0 3 * || N KC1, Hg 2 Cl 2 1 Hg. 

From the measured electromotive force of the cell, the concentra- 
tion of the hydrogen ion in the solution can be determined, and 
from this the degree of hydrolytic dissociation of the salt can be 
calculated. The method is especially valuable in cases where the 
concentration of the hydrogen ion is very small. Unfortunately 
the application of the method is restricted to the salts of metals less 
noble than hydrogen. In other words, it cannot be used to deter- 
mine the hydrolysis of the salts of metals which would be precipi- 
tated upon the platinum electrode. The applicability of the 
method is further limited by the fact that certain ions, such as Fe"", 
N(V, and C1CV, are either wholly or partially reduced by the 
hydrogen of the hydrogen electrode. 

The method has been successfully applied by Denhamf to 
the determination of the hydrolysis of various salts, among which 
may be mentioned, aluminium chloride, aluminium sulphate, nickel 
chloride, nickel sulphate, cobalt sulphate, and ammonium chloride. 
As an illustration of the method we may take the case of am- 

* A saturated solution of ammonium nitrate is interposed between the 
solutions to eliminate any difference of potential at the junction of the solu- 
tions. 

t Jour. Chem. Soc., 93, 41 (1908); Zeit. anorg. Chem., 57, 361 (1908). 



ELECTROMOTIVE FORCE 481 

monium chloride which dissociates hydrolytically according to 

the equation 

NH 4 C1 + H-OH <F NH 4 OH + HC1. 

Here we have a weak base and a very strong acid as the products 
of hydrolytic dissociation. While the ammonium hydroxide may 
be regarded as practically non-ionized, the hydrochloric acid is to 
be considered as having undergone complete ionization. If one 
mol of ammonium chloride is dissolved in v liters of water and the 
degree of hydrolytic dissociation is x, then the concentrations of 
the products of the reaction, ammonium hydroxide and hydro- 
chloric acid, will be x/v. Since the acid is completely ionized, x/v 
will also represent the concentrations of the hydrogen and chloride 
ions. Assuming that the active mass of the water remains con- 
stant, we have, according to the law of mass action, 

K - x * 
Kk ~v(l-x)' 

where Kh is the hydrolytic constant. Since v is known, it only re- 
mains to determine x/v, or the concentration of the hydrogen ions, 
in order to be able to calculate Kh- The potential at the hydrogen 
electrode is given by the familiar formula 

CH ' 
"C ' 

JD/TT 

or denoting the normal electrode, ~ log e C, by TT O , we may write 

. RT , x 
T = 7ro + _ Tlogc .. 

According to Denham, the electromotive force of the cell, 



Ptn, | N/32 NH 4 C1 1 sat. NH 4 N0 3 1| N KC1, Hg 2 Cl 2 1 Hg, 

at 25 is 0.6056 volt, the current flowing outside the cell in the 
direction of the arrow. The potential of the normal calomel 
electrode being + 0.56 volt, it follows that the potential of the 
hydrogen electrode, TT = 0.56 - 0.6056 = - 0.0456 volt. There- 
fore, 



- 0.0456 =7T + ~ log, 



482 THEORETICAL CHEMISTRY 

The absolute value of the potential of the normal hydrogen elec- 
trode TO, referred to the normal calomel electrode as + 0.56 volt 
and not as zero, is + 0.277 volt. Substituting this value in the 
foregoing equation and passing to Briggsian logarithms, we have 

0.059 log x/v = - 0.0456 - 0.277 = - 0.3226. 

Solving this equation we find x/v = 0.3406 X 10~ 6 gram-ions of 
hydrogen per liter. Since the value of x/v would be $% if the salt 
were completely hydrolyzed, the percentage of hydrolysis under 
the conditions of the experiment, i.e., when v = 32, is 

0.3406 X 10 ~ 6 X 100 nmnft 

Too " ^ 0-0109 percent. 



The value of the hydrolytic constant is given by the equation 
_ ** - (0.3406 X 10-) _ 



32 (1 0.3406 X 10~ 6 



X 1U . 



The degree of hydrolytic dissociation of ammonium chloride has 
been determined by Noyes from measurements of electrical con- 
ductance. The value of x for 0.01 N NH 4 C1 was found by him to 
be 0.02 at 18, while Denham found x = 0.018 at 25 by extrapola- 
tion of his electrometric data. 

Oxidation and Reduction Elements. When a dissolved sub- 
stance passes from a lower to a higher state of oxidation, the 
change in the positive ion may be considered as due to an increase 
in the number of electrical charges on the km; thus, when a ferrous 
salt is oxidized to the ferric state, the change may be represented 
by the equation 



Similarly, the reduction of a ferric salt to the corresponding ferrous 
salt may be represented by the reverse equation, or 



The formation of zinc ions from metallic zinc may be considered 
as an oxidation, and may be represented by the equation 



ELECTROMOTIVE FORCE 483 

The formation of negative ions from a non-metallic element may 
be considered as a reduction, as for example, the change of potas- 
sium ferricyanide to potassium ferrocyanide, which may be rep- 
resented by the equation 

Fe(CN)."' +(-)- Fe(CN) 6 "". 

The foregoing considerations lead to the following definition of 
the terms oxidation and reduction: Oxidation is the process in 
which a substance takes up positive or parts with negative charges, 
and reduction is the process in which a substance takes up negative 
or parts with positive charges. 

The potential difference between a metal and a solution of one 
of its salts is a measure of the tendency of the metal to form ions, 
or in other words is the criterion of its tendency towards oxidation. 
The tendency of an ion in a lower state of oxidation to pass over 
into a higher state of oxidation may be determined by means of 
electromotive force measurements. 

A typical oxidation cell is the following: 

Pt-Fe"||Fe'"-Pt. 

If the platinum electrodes are connected to a lead accumulator, 
and a current is passed in the direction of the arrow 

Pt-Fe"||Fe--Pt, 



the ferrous ions on one side of the element will all be oxidized to 
the ferric state, while the ferric ions on the other side will all be 
reduced to the ferrous state, thus: 

Ft-Fe*"||Fe"-Pt. 

If now the connection with the accumulator is broken and the two 
platinum electrodes are connected with a wire, the following proc- 
ess will take place: ferric ions will give up positive charges to 
one electrode, thereby being reduced to ferrous ions, while the 
charges given up pass along the wire to the other electrode, there 
converting some ferrous ions to ferric ions. This process will 
continue until an equilibrium is established, in which the ratio 



484 THEORETICAL CHEMISTRY 

Fe*" : Fe"* will be the same in both solutions. The electromotive 
force corresponding to this change gives a measure of the tendency 
of the ions to undergo oxidation, and is directly proportional to 
the ionic concentrations. 

The following modification of the Nernst equation gives the 
relationship between ionic concentrations and the resulting elec- 
tromotive force: 



where J5?( Co -c t ) is the electromotive force produced by the passage 
from the lower, or "ous" state of oxidation to the higher, or "ic" 
state of oxidation; P is the potential when the concentrations of 
the "ous" and "ic" ions are equal, CQ and d are the ionic con- 
centrations of the lower and higher states of oxidation respectively, 
and n is the difference in valence of the two kinds of ions. A 
number of oxidation and reduction elements have been studied 
by Bancroft.* 

Heat of lonization. If the difference of potential between a 
metal and the solution of one of its salts is known, together with 
its temperature coefficient, it is possible to calculate the heat of 
ionization of the metal by means of the Gibbs-Helmholtz equation, 

Q d* 

v ~nF + ^ar 

Solving the equation for Q, the heat evolved when one mol of ions 
is formed at the electrode, we have 



For example, the potential of zinc against a molar solution of zinc 
chloride is 0.497 volt at 25, and the temperature coefficient of 
electromotive force is 0.000664 volt per degree. Substituting in 
the equation, we have 

Q = [_ 0.497 - (273 + 25) X 0.000664] (2 X 96,540 X 0.2394) 
or Q = 32,120 calories. 

* Zeit. phys. Chem., 10, 387 (1892). 



ELECTROMOTIVE FORCE 
That is, the heat of the reaction 



485 



is 32,120 calories per mol of zinc. 

Gas Cells. It is interesting to note that gases may function 
as electrodes in much the same way as metals or amalgams. Gas 
electrodes are usually prepared by partially 
immersing strips of platinized platinum in 
a solution of a suitable electrolyte, and 
bubbling the gas through the solution until 
a constant difference of potential is estab- 
lished between it and the electrode. A very 
satisfactory form of gas electrode is shown in 
Fig. 105. Reference has already been made 
to the hydrogen electrode in connection 
with the measurement of single electrode 
potentials. 

This electrode is completely reversible 
and behaves like a plate of metallic hydro- 
gen, the reaction at the electrode being 
represented by the equation 




The amount of energy developed by the 
passage of a certain quantity of gas into 
the ionic state is precisely the quantity nec- 
essary and sufficient to produce the reverse Fig. 105. 
action. 

This being true, the metal of the electrode can exert no influence 
upon the electromotive force. A hydrogen concentration ceil 
can be formed by connecting two hydrogen electrodes, containing 
the gas at different pressures, through an intermediate electrolyte. 
The direction of the current is such that the pressures on the two 
sides of the cell tend to become equal, molecular hydrogen being 
ionized on the high pressure side, and ionized hydrogen being dis- 
charged on the low pressure side. 

The electromotive force of such a cell can be calculated by means 



486 THEORETICAL CHEMISTRY 

of the Nernst equation. Let us consider a cell composed of two 
hydrogen electrodes, each at atmospheric pressure, the H* ion 
concentration in each being Ci, then 

_, RT, Ci RT, Ci n 
E = 2F lo& ft ~ 2F l ^ = ' 

where Ci is the molecular concentration of the hydrogen dissolved 
in the platinum at gaseous pressure pi. Since the hydrogen 
is present in the form of diatomic molecules, n = 2. 

If now the pressure of the gas at one electrode be increased to 
p 2 , and the corresponding molecular concentration of the hydro- 
gen in the electrode be C 2 , then we shall have 

**?:,< ft 

* 2F 10g ' Cl 
or since 



Equation (1) applies equally well to cells in which two different 
gases are employed. If solution pressures be used in the calcu- 
lation of the electromotive force of a gas cell, equation (1) becomes 

= ^log e ;pi, (2) 

where PI and P 2 are the solution pressures of the two gases. Since 
the values of E obtained by equations (1) and (2) must be equal, 
we may write 



and 
therefore 



That is, the ratio of the actual gas pressures is equal to the ratio 
of the squares of the corresponding solution pressures. 



ELECTROMOTIVE FORCE 487 

lonization of Water. An important application of the gas 
cell is its use in determining the degree of ionization of water. 
If we measure the electromotive force of the cell 

Ptn 2 - 0.1 m NaOH - 0.1 m HC1 - Ptn 2 , 

and determine the concentration of the H" ions on one side and the 
concentration of the OH' ions on the other side, we can calculate 
the concentration of the H* ions in the sodium hydroxide solution. 
The reaction which produces the current is represented by the 
equation 

NaOH + HC1 = NaCl + H 2 0, 

or more correctly 

H' + OH' = H 2 O. 

At 25 the electromotive force of the above cell is 0.646 volt. 
At the junction of the two solutions an electromotive force of 
0.0468 volt is set up; hence the true electromotive force of the 
cell is 0.646 + 0.0468 = 0.6928 volt. The degree of ionization 
of a 0.1 molar solution of hydrochloric acid is i = 0.924, and 
the degree of ionization of a 0.1 molar solution of sodium 
hydroxide is 2 = 0.847. Introducing these values into the 
Nernst equation 



we have 

0.6928 = 0.0595 log ' 1 X ' 924 . 

C 2 

Solving this equation for c^ the concentration of the H* ions 
in the sodium hydroxide solution, we find c 2 to be equal to 
1.66 X 10~ 13 . Therefore 

* = CH- X coir = 1.66 X 10~ 13 X 0.1 X 0.847 = 1.406 X 10~ 14 , 
and 

CH- = COH' = Vs = Vl.406 X 10~ 14 = 1.187 X 10~ 7 . 



This value is in excellent agreement with the values obtained from 
measurements of the electrical conductance of water and the speed 



488 THEORETICAL CHEMISTRY 

of hydrolysis of esters. The values of the degree of ionization of 
water- at 25, as determined by these three methods, are as fol- 
lows: 

Electrical conductance of pure water ......... 1.05 X 10~ 7 

Velocity of hydrolysis of methyl acetate ...... 1.2 X 10~ 7 

E.M.F. of hydrogen-oxygen cell ............. 1.18 X 10~ 7 

When we consider the exceedingly small extent to which water 
is ionized, the close agreement between these results is most satis- 
factory. The correctness of these figures can be further checked 
by taking the values of the degree of ionization of water at two 
temperatures, and calculating the heat of the reaction 

H' + OH' = H 2 0, 

by means of van't Kofi's isochore equation. 

Thus according to Kohlrausch, the degree of ionization of water 
is 0.35 X 10- 7 at 0, and 2.48 X 10~ 7 at 50. Introducing these 
values into the equation 

_ - 2.3026 fi (logj 2 - 



and solving for Q, we obtain 13,810 calories, a value agreeing 
well with that found by the direct measurement of the heat of 
neutralization of completely ionized acids and bases, viz., 13,700 
calories. 

Storage Cells or Accumulators. Storage cells or accumulators, 
as the name implies, are devices for the storage of electrical energy 
in the form of chemical energy. Any reversible cell may be 
employed as an accumulator. Thus, the oxygen-hydrogen cell 

Pto 2 Solution of Sulphuric Acid Ptn 2 

may be used as an accumulator, if the gases resulting from the 
electrolysis of water are collected at the electrodes and then used 
to produce a current. Practically, the lead accumulator is used 
almost exclusively. If two lead plates are immersed in a 20 per 
cent solution of sulphuric acid, a minute amount of lead sulphate 
will be formed on the surface of each plate. If now a current of 
electricity is passed through the solution, the lead sulphate on the 



ELECTROMOTIVE FORCE 489 

cathode will undergo reduction to metallic lead, and the lead sul- 
phate on the anode will be oxidized to lead peroxide. In this way 
we form the cell 

Pb - 20 per cent Sol. H 2 SO 4 - PbO 2 , 

the electromotive force of which is about 2 volts. The amounts 
of lead and lead peroxide produced in this way are so small that 
the cell can only furnish a very small amount of electrical energy. 
In order to increase its capacity, the electrodes should be given 
as large an amount of surface as possible. This may be brought 
about by the method of Plant6, in which the solution is elec- 
trolyzed first in one direction and then in the other, thus causing 
the plates to become spongy; or by the method of Faure, in which 
a lead "grid" is charged with a paste of lead oxide and red lead, 
and is then introduced into the solution and the current passed 
until we obtain spongy lead at the cathode and lead peroxide at the 
anode. If now the charging circuit is broken and the two elec- 
trodes are connected by a wire, a current will flow from the peroxide 
plate to the lead plate, lead sulphate being slowly formed at each. 
In charging the accumulator, the lead sulphate on the negative 
electrode is reduced to metallic lead, the reaction being represented 
by the following equation: 

PbS0 4 + 2 (-) = Pb- + SO/'. 

At the positive electrode S0 4 " ions are liberated and they react 
with the lead sulphate and the water of the electrolyte in the 
following manner: 

. PbS0 4 + 2 H 2 + S0 4 " + 2 (+) = Pb0 2 + 4 H' + 2 S0 4 ". 

When the cell is discharged SO 4 " ions are liberated at the lead 
plate forming lead sulphate, as shown by the equation 

Pb + S0 4 " + 2 (+) = PbS0 4 . 

At the peroxide plate H* ions are discharged and, in the presence 
of the electrolyte, -convert the lead peroxide into lead sulphate, 
according to the equation 

"' Pt>0 2 + 2 H' + H 2 S0 4 + 2 (-) = PbS0 4 + 2 H 2 0. 



490 THEORETICAL CHEMISTRY 

Combining the foregoing equations, we obtain the following single 
equation summarizing the chemical changes involved in the pro- 
duction of the current: 

PbO 2 + Pb + 2 H 2 S0 4 <=* 2 PbS0 4 + 2 H 2 0. 

The upper arrow represents the reaction on discharging, while 
the lower arrow represents the reaction on charging. 

The electromotive force of the storage cell is approximately 
2 volts. It is not completely reversible, but under favorable 
conditions its efficiency is about 90 per cent; that is, 90 per cent 
of the electrical energy supplied to it in charging can be recov- 
ered on discharging.* 

PROBLEMS. 

1. Calculate the heat of amalgamation of cadmium at from the 
following data: the electromotive force of a cell made up of a 1-per cent 
cadmium amalgam in a solution of cadmium sulphate is 0.06836 volt at 
0, and 0.0735 volt at 24.45. Ans. 510 calories per mol of Cd. 

2. The electromotive force of the cell 

Pb - 0.01 m Pb(N0 3 ) 2 - sat. NH 4 N0 3 - m KC1, HgCl - Hg 

is - 0.469 volt at 25. The lead nitrate is 62 per cent ionized. What 
is the potential of lead against a solution containing 1 mol of Pb ions 
per liter, referred to the calomel electrodes as zero? 

Ans. -0.405 volt. 

3. Calculate the electromotive force of the cell 

Cu amalgam (a) Solution CuS0 4 Cu amalgam (b), 
at 20.8, having given that the concentrations of the amalgams (a) and 
(b) are 0.0004472 and 0.00016645 respectively. Ans. 0.0125 volt. 

4. Calculate the electromotive force of the cell 

Cu - m CuS0 4 - 0.01 m CuS0 4 - Cu, 

at 25, having given the following values for the degree of ionization of 
the two solutions: for m copper sulphate, = 0.21, and for 0.01 m 

* For a thorough treatment of the theory of the lead accumulator the 
student is recommended to consult "Die Theorie des Bleiaccumulators," 
by F. Dolezalek. 

For a detailed account of the primary cells in common use the reader will 
find Carhart's "Primary Batteries" most satisfactory. 



ELECTROMOTIVE FORCE 491 

copper sulphate, a = 0.61. The electromotive force at the junction of 
the two solutions may be neglected. Ans. 0.0458 volt. 

5. At 25 the electromotive force of the cell 

Zn - 0.5 m ZnS0 4 - 0.05 m ZnS0 4 - Zn 

is 0.018 volt. Neglecting the potential developed at the junction of the 
solutions, and assuming the dilute solution of zinc sulphate to be ionized 
to the extent of 35 per cent, find the degree of ionization of the concen- 
trated solution. Ans. 0.142. 

6. What is the electromotive force of the cell 

Zn - 0.1 m ZnS0 4 - 0.01 m ZnS0 4 - Zn, 

at 18? For ZnS0 4 , ^ = 0.601, for 0.1 molar ZnS0 4 , * = 0.39, and 
for 0.01 molar ZnS0 4 , = 0.63. Ans. 0.078 volt. 

7. The electromotive force of the cell 

Ag - 0.001 m AgN0 3 - m KN0 3 - m KI, Agl - Ag 
is 0.22 volt at 18. A molar solution of KI is 78 per cent ionized, and a 
0.001 molar solution of AgN0 3 is 98 per cent ionized. Calculate the solu- 
bility of Agl. Ans. 3.53 X 10~ 3 mols per liter. 

8. The potential of zinc against a solution containing one mol of 
Zn ions is 0.493 volt, at 18. Assuming complete dissociation, calculate 
the solution pressure of zinc in atmospheres. 

9. The electromotive force of the Daniell cell 

Cu - CuS0 4 - ZnS0 4 - Zn 

is 1.0960 volt at 0, and 1.0961 volt at 3. Calculate the heat of the reac- 
tion taking place in the cell. Ans. 51,070 calories. 

10. Calculate the electromotive force of the cell 

Ptn 2 - 0.1 m KOH - m HC1 - Ptn 2 , 

at 25 having given that 0.1 molar KOH is 85 per cent ionized and molar 
HC1 is 70 per cent ionized. Ans. 0.757 volt. 



CHAPTER XXI. 
ELECTROLYSIS AND POLARIZATION. 

Polarization. If a difference of potential of about 1 volt is 
applied to two platinum electrodes immersed in a concentrated 
solution of hydrochloric acid, it will be found that the current 
which passes at first, steadily diminishes and ultimately becomes 
zero. The cessation of the current has been shown to be due to 
the accumulation of hydrogen on the cathode and chlorine on the 
anode, these two gases setting up an opposing electromotive force 
called the electromotive 'force of polarization. 

If in the above case the applied electromotive force is increased 
to 1.5 volts, the counter electromotive force is no longer sufficient 
to reduce the current to zero. In fact, at any voltage above 1.35 
volts a continuous current passes; this is termed the decomposition 
potential of hydrochloric acid. 

At all voltages above the decomposition potential, the current C 
may be calculated by means of the formula 

E - e = CR, 

where jEJ'is the applied electromotive force, e the counter electro- 
motive force, and R the resistance of the electrolyte. 

As the applied electromotive force and the current increase, the 
polarization increases, since the gases are liberated under a pres- 
sure greater than that of the atmosphere; but since the gases 
escape from the solution the value of e can never become equal 
to E. The decomposition potential of an electrolyte can be 
determined in two different ways, viz.: (1) by gradually raising 
the applied electromotive force E until it exceeds e, when the 
current will suddenly increase; or (2) by charging the electrodes 
up to atmospheric pressure by means of an electromotive force 
greater than e, and then breaking the external circuit and measur- 
ing the counter electromotive force. 

492 



ELECTROLYSIS AND POLARIZATION 



493 



The arrangement of apparatus for the measurement of the 
electromotive force of polarization, as suggested by Le Blanc,* 
is indicated in Fig. 1Q6. A is the cell in which polarization 
occurs, B is the source of external electromotive force, C is a capil- 
lary electrometer, D is a source of variable potential, and E is one 
prong of an electrically-driven tuning fork which serves to make 
and break contact with the points F and G in rapid alternation. 
When the tuning fork makes contact at F the polarizing current 




B 



Fig. 107. 



flows through A, polarizing the electrodes; when contact is made 
at G, the counter electromotive force due to this polarization 
causes a current to flow through D and C. The counter electro- 
motive force is balanced by varying D, until C indicates zero 
current: the potential of D is then equal to the electromotive force 
of polarization. Just as the electromotive force of a galvanic cell 
is due to the combined action of several differences of potential, 
so also the electromotive force of polarization is due to the individ- 
ual differences of potential located at the electrodes. The method 
employed for the measurement of polarization at a single elec- 
trode was devised by Fuchs, and is illustrated diagrammatically 
in Fig, 107. Into the vessel containing the electrolyte, dip the 
* Zeit. phys, Chem., 5, 469 (1890). 



494 THEORETICAL CHEMISTRY 

two electrodes A and B, and the side tube of the calomel normal 
electrode C. D is the source of external electromotive force, E is 
a capillary electrometer, and F is a source of variable potential. 
Before closing the external circuit DAJ9, the potential of the 
electrode B against the solution is first measured. Then the cir- 
cuit DAB is closed, thereby polarizing the electrode 5. The 
external circuit is again broken and the* potential of B against the 
solution remeasured. The difference between the final and initial 
values of the electrode potential gives the polarization at B. In 
like manner, the polarization at the electrode A can be measured. 
The small amount of electricity which is necessary to polarize 
an electrode is termed the polarization capacity of the electrode. 
This factor is dependent upon the extent of surface of the electrode, 
and also upon the nature of the metal of which the electrode is 
made. For electrodes of equal surface, the polarization capacity 
of palladium is greater than that of platinum, when hydrogen 
is liberated on each. The solubility of hydrogen is greater in 
palladium than in platinum, and consequently, because a larger 
amount of hydrogen is dissolved, a greater quantity of electricity 
will be required to bring the pressure of the hydrogen up to that 
of the hydrogen dissolved in the platinum. If, through the 
processes of solution or diffusion, or through chemical action, the 
substance which causes the polarization is removed, the electrode 
is said to be depolarized. Thus, when a reducing agent, such as 
ferrous chloride, is electrolyzed, the oxygen liberated at the anode 
immediately combines with the electrolyte, forming ferric chloride 
and preventing polarization of the electrode. 

If water be electrolyzed between platinum electrodes, the cathode 
becomes saturated with hydrogen and the anode with oxygen, 
until, when the electromotive force of polarization becomes equal 
to that of the external circuit, the current ceases. The two 
gases, hydrogen and oxygen, are soluble, however, and conse- 
quently diffuse away from the electrodes, either escaping from 
the solution or recombining to form water. In order to compen- 
sate for this continuous loss of gas at the electrodes, a small current 
continues to flow, thus maintaining the initial electromotive force 
constant. This small current is termed the residwl current. 



ELECTROLYSIS AND POLARIZATION 



495 



If oxygen is bubbled over the surf acer of the cathode during elec- 
trolysis, the hydrogen is removed as rapidly as it is liberated. 
Such an electrode on which no new substance is formed during 
electrolysis is called an unpolarizable electrode. 

Decomposition Potentials. The decomposition potential of an 
electrolyte can be determined, as has already been pointed out, 
by immersing two platinum electrodes in the solution and connect- 
ing with a source of electricity, the electromotive force of which 
can be varied at will. The voltage is gradually increased and the 
corresponding current is observed. The current increases at first 




Electromotive Force 
Fig. 108. 



and then drops almost to zero every time the voltage is raised, 
until the decomposition potential is reached. Beyond this point 
the current is directly proportional to the electromotive force. 
If the applied electromotive forces are plotted as abscissae and 
the corresponding currents as ordinates, we obtain curves of the 
form shown in Fig. 108. Some of the decomposition potentials 
of molar solutions determined by Le Blanc * are given in the accom- 
panying tables. 

* Zeit. phys. Chem., 8, 299 (1891). 



496 



THEORETICAL CHEMISTRY 
SALTS. 



Salt. 


Decornp. 
Potential. 


Salt. 


Decomp. 
Potential. 


ZnSO 4 


Volts. 
2 35 


Cd(N0 3 ) 2 . . 


Volts. 
1 98 


ZnBr2 


1 80 


CdSO 4 


2 03 


NiSO 4 


2 09 


CdCl 2 


1 88 


NiCl 2 


1 85 


CoSQ 4 . . . 


1 92 


Pb(NO 3 ) 2 


1 52 


CoCl 2 


1 78 


AgN0 3 


0.70 







ACIDS. 



Acid. 


Decomp. 
Potential. 


H 2 S0 4 


Volts. 

1 67 


HNO 3 


1 69 


H 3 P0 4 
CHaCl.COOH 


1.70 
1 72 


CHC1 2 .COOH 
CH 2 (COOH) 2 


1 66 
1 69 


HC1O 4 


1 65 


HC1 


1 31 


(COOH) 2 


95 


HBr 


94 


HI 


0.52 







BASES. 



Base. 


Decomp. 
Potential. 


NaOH 


Volts. 
1 69 


KOH 


1 67 


NH 4 OH 


1 74 







It will be observed that while there is considerable variation in 
the decomposition potentials of salts, there is very little variation 
in the decomposition potentials of acids and bases. There is a 
maximum value of about 1.70 volts to which many acids and 
bases closely approximate. It is found that all acids and bases 
which decompose at 1.70 volts give off hydrogen and oxygen at 



ELECTROLYSIS AND POLARIZATION 



497 



the electrodes. Those acids and bases which decompose at 
potentials less than the maximum, do not liberate hydrogen and 
oxygen. When their solutions are sufficiently diluted, however, 
hydrogen and oxygen are evolved and the decomposition potential 
rises to the maximum value. Thus, Le Blanc found the following 
values for the decomposition potential of different dilutions of 
hydrochloric acid. 



Concentration . 


Decomp. 
Potential. 




Volts 


2mHCl 


1 26 


J m HC1 


1 34 


fcmHCl 


1 41 


A m HC1 


I 62 


& in HC1 


1 69 



When 2 m hydrochloric acid is electrolyzed, hydrogen and 
chlorine are given off at the electrodes, whereas when the concen- 
tration of the acid is reduced to 1/32 m, hydrogen and oxygen 
are the products of electrolysis, and the decomposition potential 
increases to 1.70 volts. It is found that the values of the decom- 
position potentials vary slightly with the nature of the electrodes 
used. The above values were determined with platinum elec- 
trodes. 

The Theory of Polarization. Our knowledge of the process 
taking place at the electrodes during electrolysis is largely due 
to the investigations of Le Blanc. He determined the electro- 
motive force of polarization at each electrode, varying the external 
electromotive force from zero up to the decomposition "potential 
of the solution. When the decomposition value was reached, he 
found the potential of the electrode against the solution to be the 
same as the difference of potential between the solution and the 
element liberated at the electrode. Thus, the decomposition 
potential of a molar solution of zinc sulphate is 2.35 volts; the 
corresponding difference of potential between the electrode and 
the solution is found to be 0.493 volt. If a piece of pure zinc 
is immersed in a molar solution of zinc sulphate, the difference 



498 THEORETICAL CHEMISTRY 

of potential is found to be 0.493 volt, the metal being negative 
to the solution. It frequently happens that the electrode exhibits 
the potential due to the deposited metal before the decomposition 
point of the solution is reached. For example, in a molar solution 
of silver nitrate the electrode acquires the potential of pure silver 
in molar silver nitrate before the decomposition value, 0.70 volt, 
is reached. This is due to the fact that the osmotic pressure of the 
silver ions exceeds the solution pressure of the metal, resulting in 
the deposition of the ions of the metal without the application of 
any external electromotive force. When an indifferent electrode, 
such as platinum, is immersed in a solution of a salt, a very small 
amount of ionic deposition must occur, otherwise, according to the 
Nernst equation, an infinite electromotive force must be established. 
Thus, in the equation 

RT, P 
^^P 10 ^' 

if the solution pressure P = 0, it is evident that TT = oo and a 
perpetual motion must result. We are thus forced to the con- 
clusion that when an indifferent electrode is immersed in a salt 
solution, ions will continue to separate upon it until the tendency 
for the deposited metal to go back into solution in the ionic state 
exactly counterbalances the tendency to separation. Hence, the 
electrode will become positive toward the solution. The magni- 
tude of this difference of potential will be dependent upon the 
amount of metal deposited, It is to be noted that this difference 
of potential need not be equal to that between the massive metal 
and the solution. If the electrodes be connected with an external 
source of electromotive force, the value of which can be varied at 
will, and a small electromotive force be applied, more metal will 
separate on the cathode. This will cause an increase in the solu- 
tion pressure P, tending to offset further deposition. A still 
further increase in the external electromotive force will cause the 
deposition of more metal, and as a result of the corresponding 
increase in P, further deposition at that voltage will be pre- 
vented. Ultimately, when the applied electromotive force is 
such that P acquires its maximum value, equivalent to that of 
the massive metal, continuous deposition will occur. An exactly 



ELECTROLYSIS AND POLARIZATION 499 

analogous process takes place at the anode. If a gas is liberated, 
its concentration steadily increases until the maximum pressure 
is reached, when it will escape from the solution. When strong 
currents are employed, P does not remain constant, as has been 
assumed above, but gradually diminishes causing the difference of 
potential at the electrode to increase. 

From the above considerations it becomes clear why a definite 
electromotive force is necessary to bring about a continuous 
decomposition of an electrolyte: this will only take place when 
the concentrations of the substances separating at the electrodes 
have attained their maximum values. When the decomposition 
point is reached, the electrode exhibits the potential characteristic 
of the massive metal. It is evident from the behavior of silver 
nitrate and the salts of other metals, for which the osmotic pres- 
sure of the metal ions is greater than the solution pressure of the 
metal, that the maximum values of concentration at the electrodes 
need not necessarily be attained simultaneously. 

When the products of electrolysis are gaseous, the value of the 
decomposition potential depends upon the nature of the electrodes. 
Thus, the cell 

Pt Ha - m H 2 S0 4 - Pto 2 

gives an electromotive force of 1.07 volts if platinized platinum 
electrodes are used. If an external electromotive force slightly 
greater than 1.07 volts be applied to this cell in the reverse direc- 
tion, water will be steadily decomposed, hydrogen and oxygen 
being evolved at the electrodes. If on the other hand, the plati- 
nized electrodes are replaced by electrodes of polished platinum, 
the decomposition potential rises to 1.68 volts. The reverse 
electromotive force of polarization, however, is only 1.07 volts. 
That is, the liberation of gas at a polished platinum electrode is 
an irreversible process. The difference in the behavior of the 
two electrodes is explicable when it is remembered that platinum 
is capable of occluding large amounts of gas. A platinized elec- 
trode absorbs the liberated gas very slowly, and when thoroughly 
saturated, if it is not entirely immersed in the solution, it gradually 
gives up the gas by diffusion, no bubbles being formed. Thus, if 
the external electromotive force be raised to 1.07 volts, the system 



500 THEORETICAL CHEMISTRY 

will be in equilibrium, while if the applied electromotive force be 
greater or less than the equilibrium value, a current will flow in 
one direction or the other, gas being either liberated or dissolved. 
In other words, the cell is completely reversible. 

Where polished platinum or gold electrodes are used, however, 
the decomposition potential is, as has been stated, 1.68 volts. 
Polished electrodes have relatively small absorbing power. Hence, 
if an electromotive force between 1.07 and 1.68 volts be applied, 
the gases cannot diffuse away from the electrode rapidly enough, 
and, when the solution in the vicinity of the electrodes becomes 
saturated with gas, the current ceases to flow. 

A very slow process of diffusion from the solution into the air 
is constantly taking place however, and this permits the continu- 
ous evolution of an exceedingly small amount of gas, while a corre- 
spondingly small current traverses the solution, 

In order to produce a steady electrolysis, it is necessary to raise 
the external electromotive force to such an intensity that it is 
able to bring about the formation of bubbles at the surface of the 
electrodes. This calls for the expenditure of an amount of work 
depending upon the condition of the electrode surfaces, the sur- 
face tension of the solution, and various other factors. In cases 
where bubbles are formed, a portion of the available energy of 
the chemical process is not expended in effecting electrical separa- 
tion; consequently the reverse electromotive force is less than the 
applied, and the system is irreversible. 

The reactions at the electrodes are catalytically accelerated by 
the metal of which the electrodes are made. Thus, platinized 
platinum is the most effective catalyst for the reaction represented 
by the equation 



Hydrogen is liberated on platinized platinum at the potential 
volt, on polished platinum at 0.09 volt, and on zinc at 0.70 volt. 
The electromotive force necessary to overcome the resistance of 
the chemical reaction at an electrode is termed the overvoltage. 
Thus, we say that hydrogen is liberated on polished platinum with 
an overvoltage of 0.09 volt, and on zinc with an overvoltage of 



ELECTROLYSIS AND POLARIZATION 



501 



0.70 volt. The following table gives the overvoltage necessary 
for the liberation of hydrogen and oxygen on electrodes of differ- 
ent metals. 

ELECTRODE OVERVOLTAGES. 



Hydrogen Liberation. 


Oxygen Liberation. 


Metal. 


Overvoltage. 


Metal. 


Overvoltage 


Pt (platinized) 
Au 


00 
01 

0.08 
0.09 
15 
0.21 
23 
0.46 
53 
64 
0.70 
0.78 


Au 


1 75 
1 67 
1.65 
1 65 
1.63 
1 53 
1 48 
1.47 
1 47 
1 36 
1 35 
1.28 


Pt (polished) 
Pd . 


Fe (inNaOH)... 
Pt (polished) ... 
Ag 


Cd 


Ag 
Pb 
Cu 


Ni 
Cu 
Pd 
Sn 
Pb 


Fe 
Pt (platinized). . . . 
Co. . . ... 
Ni (polished) 
Ni (spongy) ... 


Zn 


Hg.. .. 





Primary Decomposition of Water in Electrolysis, The decom- 
position potential of an electrolyte giving off hydrogen and oxygen 
at the electrodes, is dependent upon the concentrations of the two 
ions, H* and OH', and is independent of the nature of the electro- 
lyte. As has already been stated, the decomposition potential of all 
acids and bases giving off hydrogen and oxygen approximates to 
1.70 volts. According to the law of mass action, the product of 
the concentrations of the H* and OH' ions is constant and inde- 
pendent of the other substances which may be present; hence, 
although the potentials of the individual electrodes may differ 
considerably, their sum remains practically constant. 

Excluding solutions of salts which undergo reduction by hydro- 
gen, and solutions of chlorides, bromides, and iodides reducible 
by oxygen, the ions H* and OH', according to Le Blanc, are to 
be regarded as the sole factors in the electrolysis of solutions, and 
not the ions of the dissolved electrolyte. In other words, elec- 
trolysis involves a primary decomposition of water. 

The electrical conductance of the solution is due to the ions of 
the electrolyte together with the ions of water, but at the electrode 



502 THEORETICAL CHEMISTRY 

that process takes place which involves the expenditure of the 
minimum amount of energy, and this is, under ordinary conditions 
the separation of the H* and OH' ions. 

Thus, when a solution of potassium sulphate is electrolyzed, 
only a moderately strong current being used, it is not rational to 
assume the discharge of the K* and SO 4 " ions at the electrodes, 
and then subsequent reaction between these discharged ions and 
water. This may be made clear by considering the process taking 
place at the cathode. According to the explanation based upon 
so-called " secondary action/ 7 the K* ions give up their positive 
charges to the electrode and then react with water as indicated 
by the equation 

K + H' + OH'-+K' + OH' + H. 

This explanation involves the transfer to the potassium atom of 
the positive charge of the H* ion of water; this can only take place 
if the H" ion holds its charge less tenaciously than the K* ion. 
Hence, if the H* ion parts with its charge more readily than the 
K* ion, the former will be discharged primarily at the cathode. 
Similar reasoning may be employed to explain the action at the 
anode. Therefore, in electrolysis all of the ions participate in 
conducting the current and collect around the electrodes, but 
since the H* and OH' ions separate more easily, these are dis- 
charged. With stronger currents it is possible to cause the separ- 
ation of the K* and SO 4 " ions also, since the number of H" and OH' 
ions present is too small to carry all of the current, and the energy 
required to discharge the ions of the electrolyte is less than that 
necessary to remove the small number of residual H* and OH' 
ions. The formation and decomposition of water are reversible 
processes, so that no loss of energy is involved, as would be the 
case if secondary actions occurred. 

Electrolytic Separation of the Metals. Freudenberg * was the 
first to recognize the possibility of effecting the quantitative 
separation of different metals by means of graded electromotive 
forces. He showed that it was only necessary to select a salt of 
each metal the decomposition potentials of which differ as widely 
as possible, and ejectrolyze at an electromotive force intermediate 

* Zeit. phys. Chem., 12, 97 (1893). 



ELECTROLYSIS AND POLARIZATION 



503 



7T = 



between these potentials. The salt having the lower decomposi- 
tion potential will decompose first, and when the deposition of 
the metal is complete, the current will practically cease; then if 
the applied electromotive force be raised above the decomposition 
potential of the second salt, the second metal will be deposited. 
In practice it is found necessary to increase the applied electro- 
motive force slightly because of the gradual decrease in the num- 
ber of ions of the salt having the lower decomposition potential. 
The amount of this increase may be readily calculated from the 
familiar equation 

RT P 

nF ge p 

Suppose a mixture of the nitrates of cadmium, lead and silver is 
subjected to electrolysis, the decomposition potentials of the salts 
being as follows: Cd(N0 3 ) 2 = 1.98 volts, Pb(N0 8 ) a = 1.52, 
volts, and AgN0 3 = 0.70 volt. The applied electromotive force 
is made a little less than 1 volt and all of the silver is deposited; 
then the electromotive force is raised to about 1.6 volts, thus 
depositing all of the lead; and finally, with an electromotive force 
of about 2 volts the cadmium is deposited. 

In the subjoined table are given the separation potentials of 
some of the ions, the separation potential of the H* ion being 
assumed to be equal to zero. 

SEPARATION VALUES OF IONS FOR MOLAR CONCENTRA- 
TION. 



Ion. 


Separation 
Potential. 


Ion. 


Separation 
Potential. 


Ae* 


-0 78 


I'. 


52 


Cu" 


-0 34 


Br' 


94 


H* 


0.0 


O" 


1 08 (in acid) 


Fb" 


17 


cr... 


1 31 


Cd" 


0.38 


OH' 


1 68 (in acid) 


Zn" 


0.74 


OH' 


88 (in base) 






SO/'.. . 


1 9 



According to this table the decomposition potential of water is 
equal to the sum of the separation potentials of its ions, or 1.68 
volts. 



CHAPTER XXII. 
PHOTOCHEMISTRY. 

Radiant Energy. The visible portion of the spectrum is com- 
prised between the extreme red at one end and the extreme violet 
at the other; the wave-length corresponding to the former is 
approximately 0.7 micron, while that corresponding to the latter 
is about 0.4 micron. The visible portion of the spectrum, how- 
ever, is but a small fraction of the entire spectrum. Beyond the 
red of the visible spectrum lies the region of the so-called infra- 
red, comprising all wave-lengths from 0.76 micron up to 300 mi- 
crons. Beyond the infra-red, between 300 and 2000 microns, is 
an unmeasured region, which is succeeded by the region of elec- 
trical waves, extending from 2000 microns to an undetermined 
maximum. On the other hand, extending beyond the violet of 
the visible spectrum is the so-called ultra-violet or actinic region, 
comprising all wave-lengths, between 0.4 micron and 0.1 micron. 
It thus appears that heat, light, and electricity are all forms of 
radiant energy, the only distinction between them being a differ- 
ence in wave-length. Very little is known concerning radiant 
energy, and up to the present time all attempts to resolve it into 
a capacity and an intensity factor have failed. Whatever may be 
the nature of this form of energy, we know that the effects produced 
by it are dependent upon the wave-length of the radiation. 

We have already devoted several chapters to the consideration 
of thermochemistry and electrochemistry, and it now remains to 
study very briefly the connection between chemical energy and 
that subdivision of radiant energy called light. This branch of 
theoretical chemistry is termed photochemistry. The ultra-violet 
or actinic rays are the most active chemically, although light of 
every wave-length, including the invisible infra-red, is capable of 
producing chemical action. When light falls upon a substance, a 
portion of the incident radiation is reflected, a portion is absorbed, 
and a portion is transmitted. It has been shown that only that 

504 



PHOTOCHEMISTRY 505 

portion of the incident radiation which is absorbed is effective in 
producing chemical change. 

Radiant energy has been shown by Lebedew,* and also by 
Nichols and Hull,t to exert a definite, though extremely small 
pressure. Thus, the pressure of solar radiations on the earth is 
equivalent to that of a column of mercury 1.4 x 10~~ 9 mm. high. 

Source of Radiant Energy. According to the electromagnetic 
theory of light, the emission of waves of light from a material 
source is due to the vibrations of minute charged particles called 
radiators. These radiators, which may be either atoms or elec- 
trons, give rise to electromagnetic waves of the same period as 
their own, that is, to light waves of definite length. The energy 
required to produce these electromagnetic waves is derived from 
the vibrating system itself, and unless an equivalent amount of 
energy is constantly supplied to the system, the amplitude of the 
vibrations will steadily diminish and ultimately cease. 

There are two different ways in which this supply of energy can 
be maintained. First, the temperature of the vibrating system 
as a whole may be kept high. This type of radiation, which is 
maintained by purely physical means, is called pure temperature 
radiation. Every substance whose temperature is above the abso- 
lute zero ( 273) gives rise to pure temperature radiation. The 
higher the temperature, the more rapid and the more energetic the 
atomic and electronic vibrations become. With increase in rapid- 
ity of vibration, there results a corresponding diminution in wave- 
length, so that, as the temperature is raised, the longer heat waves 
are succeeded by the shorter waves of the visible region of the 
spectrum. When a sufficiently high temperature is reached, the 
period of vibration becomes so rapid as to cause the radiation of 
waves corresponding to the entire range of the visible spectrum. 
At higher temperatures, the rate of vibration is such that the 
vibrating particles must of necessity possess extremely small mass: 
it is commonly believed that under these conditions the vibration 
is wholly electronic. 

The second way in which energy may be supplied to the vibrating 

* Rapp. pres au Congres de Physique, 2, 133 (1900). 
t Phys. Rev., 13, 293 (1901). 



506 THEORETICAL CHEMISTRY 

atoms or electrons is by chemical or electrical means. The general 
term luminescence has been proposed by Wiedemann for all cases 
where luminous energy is derived from other sources than high 
temperature. It is to be observed that luminescence is frequently 
exhibited by systems whose temperatures are comparatively low. 
For example, notwithstanding the fact that the flame resulting 
from the combustion of carbon disulphide has a temperature of 
only 150, it has been found to be capable of affecting the photo- 
graphic plate. Pure temperature radiation alone at 150 would 
correspond to long waves in the infra-red region of the spectrum 
and, as is well known, such waves are incapable of exerting appre- 
ciable photographic action. 

Emission and Absorption. The relation between the emissive 
and absorptive powers of different bodies was first clearly enun- 
ciated by Kirchhoff * in 1859. This law may be stated as follows: 
Light of any given wave-length emitted by a body can also be 
absorbed by the same body at a lower temperature. This law, it will 
be seen, offers a satisfactory explanation of the Fraunhofer lines in 
the solar spectrum. The sun is surrounded by a gaseous atmos- 
phere resulting from the vaporization of the elements present in the 
body of the sun. Each element in the cooler gaseous envelope, 
according to KirchhofFs law, absorbs those wave-lengths which it 
emits at the higher temperature of the solar nucleus. The result- 
ing dark lines of the solar spectrum have enabled the astronomer to 
determine the elementary composition of the sun. 

If the emissive and absorptive powers of a body be denoted by 
E and A respectively, then according to KirchhofFs law, 

E/A = /?, 

where S is a constant. When absorption is complete, A is unity 
and S = E. Under these conditions the constant, S, may be de- 
fined as the emissivity of a body which absorbs all of the incident 
radiation and reflects none. Such a body was called by Kirch- 
hoff a perfectly black body. The emissivity of a perfectly black 
body is equal to the ratio of the emissive to the absorptive power of 
any body at the same temperature. A familiar qualitative illus- 

* Ostwald's Klassiker, No. 100 (1898). 



PHOTOCHEMISTRY 507 

tration of KirchhofPs law is that afforded by the appearance of 
a fragment of white chinaware possessing a dark pattern when 
heated to a high temperature. The dark parts of the design absorb 
light, while the white parts reflect it. On heating the fragment to 
redness, the pattern will be reversed, the dark portions of the 
design appearing bright and the white portions dark. 

It has been shown that the law of Kirchhoff is a necessary con- 
sequence of the application of the second law of thermodynamics 
to the thermal equilibrium within an enclosure whose walls are 
impervious to heat. 

The Stefan-Boltzmann Law. From a study of the experi- 
ments of Dulong and Petit on the rate of cooling of different bodies, 
Stefan * discovered an empirical relation between the total radia- 
tion of a body and its temperature. Later, Boltzmann f derived 
the same relation thermodynamically and showed that instead of 
being general, as Stefan supposed, it is only strictly applicable 
to a perfectly black body. The Stefan-Boltzmann law may be 
stated as follows: The total radiation from a perfectly black body 
is directly proportional to the fourth power of the absolute tempera- 
ture. If the total radiation be denoted by S, we may write 

S = C7 74 , 

where C is a constant. If the radiation from the sun be con- 
sidered solely as a temperature effect, its temperature may be 
calculated by the above equation expressing the Stefan-Boltzmann 
law. Employing available bolometric data, the temperature of 
the sun may thus be shown to be 6200 absolute. 

The Displacement Law of Wien. Having considered the total 
energy radiated by a given source, we now come to the considera- 
tion of the distribution of energy throughout the entire spectrum 
in its relation to temperature. The results of the experiments 
of Lummer and Pringsheim J on the distribution of energy in the 
normal spectrum of a black body are shown by the curves of Fig. 
109. The values of the energy radiated by the source are plotted 

* Sitz. Ber. Wiener" Akad., 79 (II), 391 (1879). 

t Wied. Ann., 22, 291 (1884). 

j Verb, deutsch. phys. Ges., i, 230 (1889); 3, 36 (1901). 



508 



THEORETICAL CHEMISTRY 



as ordinates against the corresponding values of the wave-length 
as abscissae. It will be observed that the energy corresponding to 
a definite wave-length increases with the temperature, and that 
each curve, or isothermal, exhibits a distinct maximum. The 



1646T 




Wave-lengthT 
Fig. 109. 

position of this maximum is displaced in the direction of decreasing 
wave-length as the temperature is raised. 

In 1893, Wien * discovered the law governing this displacement 
of the energy maximum with temperature. If X max . denotes the 

* Wied. Ann., 58, 662 (1896). 



PHOTOCHEMISTRY 509 

wave-length corresponding to the energy maximum, and T is the 
absolute temperature of the radiating black body, Wien showed 
that X m ax. X T = constant. 

Furthermore, Wien found that 



where S max . is the maximum emissivity corresponding to X max .. 
In general, the emissivity 8 is the amount of energy radiated per 
second from a narrow strip of the spectrum corresponding to the 
mean wave-length X. It should be mentioned that while the 
experimental realization of a perfectly black body is impossible, 
a very close approximation can be obtained by the employment of 
a hollow blackened sphere perforated by a small hole to permit the 
passage of the radiation. When this sphere is heated, practically 
all of the radiation is absorbed by multiple reflections at the inner 
surface. It has been found that where such a black body is not 
available, or where extreme accuracy is not required, a thin strip 
of platinum foil coated with ferric oxide and heated electrically 
proves a satisfactory substitute. It is hardly necessary to call 
attention to the fact that the term black body, as here used, does 
not imply a total absence of color. At high temperatures, a 
" black" source of radiation may be red or even white. To avoid 
confusion, it has been proposed to substitute the term full radia- 
tor for the older term, black body. 

Distribution of Energy throughout the Spectrum. The experi- 
mental determination of the radiant energy corresponding to a 
given wave-length, really resolves itself into the measurement of 
the energy emitted between two contiguous wave-lengths, X and 
X + dX. In other words, we actually measure the energy of a 
very small portion of the spectrum included between two wave- 
lengths which lie very close together. 

Several different formulae have been proposed for the calcula- 
tion of the distribution of energy throughout the spectrum. Of 
these, the formulae of Rayleigh and Wien have been found to 
reproduce experimental values with considerable accuracy. The 
formula of Rayleigh has the following form: 

CT -- 
S^~-e , 



510 THEORETICAL CHEMISTRY 

while that of Wien may be written thus: 

r 

O _ Si -XT 1 . 
X- X5 ^ 

In these two formulas, C and c' are constants, while the other 
symbols have their usual significance. The formula of Rayleigh 
has been found to hold better in the region of the longer wave- 
lengths while the reverse is true of the formula of Wien. It should 
be remembered that both of these formulas apply only to bodies 
emitting continuous spectra. 

High Temperature Thermometry.* Various optical methods 
for the measurement of high temperatures have been developed, 
but a detailed treatment of these methods is obviously out of place 
in a book of this character. Mention should be made, however, 
of two instruments which have proven of great value in high tem- 
perature measurements. 

The optical pyrometer of Fe*ry is based upon Stefan's law of 
total radiation. It consists of a telescope fitted with an objective 
of fluorite, at the focus of which is placed a sensitive thermo- 
couple. In order to determine the temperature of a source of 
radiant energy, such as a crucible of molten metal, the telescope 
is directed toward the contents of the crucible and the image is 
focussed on the thenno- junction by means of an adjustable eye- 
piece. The resulting electric current is then measured by means 
of a galvanometer. 

In the optical pyrometer of Holborn and Kurlbaum, use is made 
of the luminous radiations only. In this instrument the current 
through a small incandescent lamp is varied until its light is just 
eclipsed by that from the hot body. When this point of balance 
has been reached, the incandescent filament and the hot body 
have the same temperature. The pyrometer is calibrated by de- 
termining the current necessary to raise the filament of the lamp 
to the temperature of the standard black body, the temperature 
of the latter being determined by means of a thermocouple. Of 

* For a detailed account of optical pyrometers, the student is referred to 
"High Temperature Measurements" by Le Chatelier and Boudouard, trans- 
lated by Burgess (John Wiley and Sons, Inc.). 



PHOTOCHEMISTRY 511 

course, when the instrument is used to determine the temperature 
of sources of radiant energy which differ widely in character from 
that of a perfectly black body, the accuracy of the measure- 
ments is lessened, but even in an extreme case, such as that pre- 
sented by polished platinum at 950, the error does not exceed 
74. The importance of optical pyrometers in photochemical 
investigations lies chiefly in the determination of energy curves 
of light sources and in the absolute measurement of radiant 
energy. 

Luminescence. As has already been pointed out, it is cus- 
tomary to distinguish between pure temperature radiation and 
luminescence. The latter term is applied to all cases where chemi- 
cal or electrical energy is transformed directly into radiant energy. 
The various types of luminescence may be conveniently classified 
in the following manner: % 

Type of Luminescence. Origin of Radiation. 

(1) Photo-luminescence. Preliminary exposure of the lumi- 

(a) Fluorescence, nous substance to some external 

(6) Phosphorescence. source of radiant energy. 

(2) Thermo-luminescence. Stimulation by heat, but at a tem- 

perature considerably lower than that 
required for pure temperature radia- 
tion. 

(3) Chemi-luminescence. Chemical reaction. 

(4) Tribe-luminescence. Fracture or cleavage of crystals. 

(5) Cathode-luminescence. Electric discharge. 

(6) Radio-luminescence. Radioactivity. 

By the term fluorescence is meant the phenomenon ot the emission, 
by an illuminated medium, of light of a different wave-length from 
that of the incident radiation. In general, the wave-length of the 
transformed radiation is greater than that of the incident radiation. 
This law, to which several exceptions have been discovered, was 
first enunciated by Stokes. When the incident radiation is cut 
off, fluorescence ceases. 

On the other hand, there are many substances which continue 
to emit light for some time after the external light-stimulus is 
removed. This phenomenon is termed phosphorescence and 
appears to be governed by Stokes law for fluorescence. The 



512 THEORETICAL CHEMISTRY 

property of phosphorescence appears to be limited to anhydrous 
substances. 

Among the numerous substances which are known to exhibit the 
phenomenon of fluorescence may be mentioned fluorite (from 
which the phenomenon derived its name), uranium glass, petro- 
leum, solutions of organic dyestuffs, and quinine sulphate. The 
vapors of sodium, mercury, and iodine have recently been found 
by Wood to fluoresce brilliantly. 

The sulphides of the alkaline earths may be mentioned as ex- 
amples of phosphorescent substances. The investigations of 
Lenard and Urbain have revealed the interesting fact that the 
presence of a trace of one of the heavy metals greatly intensifies 
the light emitted by a phosphorescent substance. 

The phenomenon of thermo-luminescence calls for little com- 
mefit. There seems to be an intimate connection between thermo- 
luminescence and phosphorescence, since the substances exhibit- 
ing the former phenomenon must be exposed initially to light, 
otherwise they do not emit any visible radiation on gentle heating. 

Chemi-luminescence is a phenomenon accompanying many 
chemical reactions. Thus, the precipitation of sodium chloride 
from its saturated solution by hydrochloric acid gas is accompanied 
by an emission of light which may readily be seen if the reaction is 
carried out in a dark room. 

When certain crystals, such as those of cane sugar, are either 
crushed or simply rubbed together, flashes of light are emitted. 
This phenomenon is known as tribo-luminescence. 

Cathodo- and radio-luminescence may be considered as sub- 
divisions of the more inclusive term, electro-luminescence. At- 
tention has already been called to the fact that the residual gas in 
a vacuum tube is rendered luminous by the passage of the electric 
discharge, and also that certain minerals become phosphorescent 
when placed in the path of the cathode rays. These may be taken 
as examples of cathodo-luminescence. The luminosity of a screen 
coated with crystals of zinc sulphide, when subjected to the action 
of the a-particles shot out from a radio-active substance, has also 
been mentioned in a previous chapter. This is clearly an instance 
of radio-luminescence. 



u i o 



Having briefly reviewed the different processes involved in the 
production of light we now turn to a consideration of the chemical 
phenomena resulting from exposure to light. 

Photochemical Action. The development of the green color 
of plants under the influence of the rays of the sun, and the reverse 
process of bleaching in darkness, were probably the first photo- 
chemical reactions to be observed. To-day it is known that light 
has the power of initiating or accelerating every variety of chemical 
change. This statement may be illustrated by the following typi- 
cal photochemical reactions: The polymerization of anthracene, 
the depolymerization of ozone, the transformation of male'inoid 
into fumaroid forms, the hydrolysis of acetone, the oxidation of lead 
sulphide, and the reduction of silver salts. That such a variety of 
photochemical reactions should result from exposure to mixed, or 
heterogeneous, light is due to the selective absorption of each par- 
ticular chemical system. 

Photochemical action is not limited to the waves of the visible 
spectrum alone, but extends from the red end of the spectrum 
(wave-length 800 ML) * into the ultra-violet region (wave-length 
300 fjLfj,). In fact, the shorter wave-lengths of the ultra-violet 
region of the spectrum have been found to be the most active 
photochemically. 

Laws of Grotthuss. The two fundamental generalizations 
of photochemistry were first enunciated by Grotthuss in 1818. 
These generalizations may be stated as follows: 

(1) Only those rays of light which are absorbed produce chemical 
action. 

(2) The action of a ray of light is analogous to that of a voltaic cell 
It has recently been shown by Bancroft f that the second of 

these two laws is inadequate to account for all of the known facts 
and, therefore, he proposes the following modification: All of 
the radiations which are absorbed by a substance tend to eliminate 
that sitbstance. It is merely a question of chemistry whether any re- 
action occurs and what the products of the reaction will be. 



io- 7 cm. 

t Jour. Phys. Cfaem., 12, 209, 318, 417 (1908); 13, 1, 181, 269, 149, 538 
(1909); 14,292(1910). 



514 THEORETICAL CHEMISTRY 

Quantitative Relations Concerning the Absorption of Light. 

When a ray of light enters an absorbing medium only a certain 
proportion of the incident radiation is absorbed. The intensity of 
the light entering an absorbing medium is not equal to that which 
is incident on the surface of the medium, owing to the fact that a 
portion of the incident beam is reflected. It has been found that 
if the thickness of the medium be increased in arithmetical pro- 
gression, the intensity of the transmitted light decreases in geomet- 
rical progression. If the intensity of the light traversing a layer 
dl be denoted by 7, then 

- All dl = kl, (1) 

where k is a constant depending upon the nature of the absorbing 
medium and the wave-length of the light. This constant k is 
known as the absorption index. If the initial intensity of the light 
is Jo and the total thickness of the medium is d, equation (1) be- 

comes, on integrating, 

/ = Jo - r-w (2) 

Or equation (2) may be written in the form, 

*-3i*- - (3) 

If we replace er k by a, then equation (2) becomes 

///o = a*. (4) 

The constant a is called the transparency or the transmission co- 
efficient. Bunsen and Roscoe introduced the term extinction co- 
effitie-nt. This quantity may be defined as the reciprocal of that 
thickness of the medium which reduces the intensity of the trans- 
mitted light to one-tenth of its initial value. If the extinction 
coefficient be denoted by e, its value may be calculated from equa- 
tion (2) in the following manner: 

I = Jo 10-*, (5) 

or e=-3log7-- (6) 



It is evident that e~ k is identical with 10 " e or c = mk, where m 
represents the modulus of the Naperian system of logarithms. 



PHOTOCHEMISTRY 515 

In 1852 Beer * enunciated an important law concerning the in- 
fluence of concentration on absorption. Beer's law may be stated 
as follows: The absorption of light by different concentrations of 
the same solute dissolved in the same solvent is an exponential function 
of the concentration y provided the thickness of the absorbing medium 
be maintained constant. It follows from this law that 

/ = Ia>, (7) 

where c is the concentration of the solution. 

If Beer's law is valid, the ratio, c/c = A, known as the absorption 
ratio, should be constant. Beer's law has been tested with a large 
number of solutions and has been found to hold quite generally 
where no change in the solute occurs when the concentration of the 
solution is altered. 

Photochemical Extinction. According to the first law of 
Grotthuss, only the absorbed light is chemically active. The con- 
verse of this law, viz., that every substance which absorbs light 
undergoes chemical change, apparently does not hold. Further- 
more, only a portion of the rays absorbed by a light-sensitive sub- 
stance are directly involved in effecting chemical change. For 
example, while an alkaline copper tartrate solution shows marked 
absorption in the infra-red, red, yellow, and ultra-violet, it has 
been proven that the photochemical reduction of the copper salt 
to cuprous oxide is due to the action of the ultra-violet rays alone. 

The first quantitative measurements of the absorption of light 
by a reacting system were made by Bunsen and Roscoe.f These 
investigators found that the absorption of light by hydrogen and 
chlorine, taken separately, was less than that of the reacting 
mixture of the two gases. From this result they concluded that 
in a photochemical reaction, the absorption of light by the re- 
acting system is greater than the sum of the individual absorptions 
of the reacting substances. The absorption of light by the re- 
acting system, over and above the ordinary or "optical" absorp- 
tion of the reacting substances, they called photochemical extinc- 
tion. While the phenomenon of photochemical extinction is re- 

* Pogg. Ann., 86, 78 (1852). 
t Ostwald's Klassiker, No. 38. 



516 



THEORETICAL CHEMISTRY 



garded by many physical chemists as having doubtful significance, 
nevertheless the distinction between purely optical absorption 
on the one hand, and chemical absorption on the other, is quite 
generally accepted.* 

If the distinction drawn by Bunsen and Roscoe between optical 
and chemical absorption be accepted, then all photochemical re- 
actions may be regarded as belonging to one or the other of two 
classes, as follows: (1) reactions in which light does work 
against chemical affinity, the work being equivalent to the photo- 
chemical extinction; or (2) reactions in which the light functions 
merely as a catalyst. In reactions belonging to the first class, the 
light is considered to be the agent which actually initiates chemical 
change, whereas in reactions of the second class, the light is 
assumed to accelerate reactions which would otherwise proceed at 
a slower rate. 

Kinetics of Photochemical Reactions. Let A and B repre- 
sent two chemically distinct substances, and let us assume that the 
reaction 

A-+B 

takes place under the influence of light. The course of the re- 
action can be followed in the usual manner by determining the 

amount of either constituent 
which is present at any definite 
time. There are certain factors, 
however, which render such kin- 
etic measurements more difficult 
in the case of a photochemical 
reaction than in that of an ordi- 
nary chemical reaction. Let us 
assume that the above reaction 



d 



Fig. 110. 



is homogeneous and of the first order and also that it takes place 
in homogeneous solution. It is apparant that if the system is 
illuminated from one side, as shown in Fig. 110, the rate at which 
A undergoes transformation into B will depend upon the thick- 

* For a condensed summary of the different theories of light-absorption, 
the student should consult Sheppard's " Photochemistry " t (Longmans, Green 
& Co.). 



PHOTOCHEMISTRY 517 

ness, d, of the absorbing layer. Hence, if the velocity of the 
transformation be denoted by dB/dt, we may write 

dB/dt = k[A], 

in which the value of the constant k will not only be a function of 
the intensity and wave-length of the light but also of position. 
Furthermore, the variation in the velocity of the reaction in suc- 
cessive layers will cause differences in concentration which will tend 
to become equalized by the process of diffusion. From this ex- 
ample it is obvious that the extent to which light is absorbed is 
dependent upon the dimensions of the absorbing system. 

If light be regarded as a material substance, then we may con- 
veniently consider its absorption as analogous to the diffusion of a 
gas into a liquid and the light intensity at any point as the analogue 
of concentration or active mass. According to the law of mass 
action, the velocity of a chemical reaction is expressed by the 
equation, 

dx/dt = /cci n * . C2 wa - fc'ci" 11 ' c^' 

where ci, c 2 , etc., are the concentrations of the substances entering 
into the reaction, and ni, n%, etc., are the coefficients derived from 
the chemical equation and indicating the order of the reaction. In 
applying this equation to photochemical reactions, the following 
possibilities must be borne in mind: 

(1) The reaction may take place in successive stages. Under 
these conditions the experimentally measured velocity will corre- 
spond to that of the slowest reaction. 

(2) Side reactions may occur with the formation of products 
quite different from those resulting from the main reaction. 

(3) There may be catalysis. In fact, this phenomenon is fre- 
quently met with in the study of photochemical reactions. 

Nernst has pointed out that the velocity constants, k and fc', in 
the foregoing equation, may be conveniently considered as being 
directly proportional to the intensity of the light, for light of the 
same kind. Owing to absorption, this intensity will be a function 
of position in the absorbing medium. 

In applying the law of mass action to photochemical reactions, 
it is important to note that, as a general rule, the exponents n\* n% 



518 THEORETICAL CHEMISTRY 

etc., in the equation of the so-called dark-reaction are not identical 
with the exponents v\, v^ etc., of the light-reaction. In other 
words, the order <^f a chemical reaction is usually different in the 
light from what it is in the dark. The photochemical exponents 
i/i, j> 2 , etc., are never greater than, and are seldom equal to, the 
corresponding exponents ni, n 2 , etc., of the dark-reaction. 

Thus, Bodenstein * showed that the dissociation of hydriodic 
acid, in the dark, is a reaction of the second order and may be rep- 
resented by the equation 

2 HI ^ H 2 + I 2 , 

whereas when hydriodic acid dissociates in the light, the reaction 
is of the first order, as shown by the equation 

HI <= H + I. 

In this case, the light acts as a catalyst, merely accelerating the 
velocity of the dark-reaction. 

The action of light on the reverse reaction may be such as to 
oppose its usual course in the dark. In this case the resulting 
photochemical equilibrium, or so-called photo-stationary state, will 
not be identical with the corresponding chemical equilibrium. It 
should be observed that the photo-stationary state differs from 
ordinary chemical equilibrium, in that its permanency is wholly 
dependent upon the constancy of the source of illumination; i.e., 
when the light is cut off, the photo-stationary state shifts to the 
ordinary chemical equilibrium, provided the reaction is reversible. 
It has also been found that the temperature coefficients of most 
photochemical reactions are negligible. 

Becquerel f discovered that silver chloride, which had been 
precipitated in the dark, was only sensitive to short wave-lengths 
of light, whereas silver chloride, which had been exposed for a few 
moments to sunlight, became sensitive to all wave-lengths in the 
visible spectrum and to the shorter wave-lengths of the infra-red. 
This phenomenon, which has been observed with various sub- 
stances, was first studied systematically by Bunsen and Roscoe J 
who termed it photochemical induction. 

* Zeit. phys. Chem., 22, 23 (1897). 
t Ann. Chim. Phys. [3], 9, 257 (1843). 
t Pogg. Ann., 100, 481 (1857). 



PHOTOCHEMISTRY 519 

Employing a mixture of hydrogen and chlorine gases they found 
that under constant illumination, the velocity of formation of 
hydrochloric acid, which was hardly appreciable at first, increased 
rapidly to a maximum and then 'emained constant. The interval 
of time required for the reaction to attain its maximum velocity 
is known as the period of induction. 

It has been found that in almost every photochemical reaction 
there is a similar period of initial perturbation. The phenomenon 
has been thoroughly investigated by Burgess and Chapman * who 
arrived at the conclusion that induction effects are to be ascribed 
to the presence of minute traces of various impurities, such as 
gases or water vapor, adsorbed by the walls of the reaction-vessel. 
We may therefore conclude that induction effects are not char- 
acteristic of photochemical reactions. 

Classification of Photochemical Reactions. According to 
Sheppard ("Photochemistry") photochemical reactions may be 
conveniently classified in the following manner : 

(1) Reversible reactions, i.e., reactions in which the products 
formed under the influence of light react to reproduce the original 
system when the light is removed. 

(2) Irreversible reactions, i.e., reactions in which the light pro- 
motes transformation to a more stable system. Irreversible re- 
actions are subdivided into 

(a) Complete reactions; and 

(b) Pseudo-reversible reactions. 

The polymerization of anthracene may be taken as an example 
of a reversible photochemical reaction. This reaction may be 
represented as taking place according to the equation 

hKht 

2 CuHio *=^ C2sH20, 
dark 

the upper arrow indicating the direction of the light-reaction 
(polymerization) and the lower arrow that of the dark-reaction 
(de-polymerization) . 

An illustration of a completely irreversible photochemical re- 
action is afforded by a mixture of hydrogen and chlorine gases 
* Jour. Chem. Soc., 89, 1402 (1906). 



520 THEORETICAL CHEMISTRY 

which combine, on exposure to light, to form hydrochloric acid gas, 
according to the equation 

light 

H 2 + C1 2 -* 2 HC1. 

If the hydrochloric acid is removed by solution in water as 
fast as it is formed, the velocity of the reaction will be directly 
proportional to the intensity of the light. Bunsen and Roscoe * 
made use of the foregoing facts in the construction of their 
actinometer. 

The reduction of ferric oxalate may be taken as an example of 
a pseudo-reversible photochemical reaction. This substance is 
reduced to ferrous oxalate on exposure to light as shown by the 
equation 

light 

3 -> 2 Fe (C 2 4 ) + C0 2 . 



In the dark, ferrous oxalate is re-oxidized to ferric oxalate by the 
oxygen of the air. While the initial substance is reproduced, it is 
evident that the reaction is not strictly reversible. 

Actinometers. A number of different forms of apparatus have 
been devised for measuring the chemical action of light: such in- 
struments are known as actinometer $. 

The hydrogen-chlorine actinometer is based upon the well- 
known fact that the speed of the reaction between hydrogen and 
chlorine varies greatly with the intensity of illumination. Bun- 
sen and Roscoe,f guided by the experiments of Draper, con- 
structed an actinometer in which the rate of combination of hydro- 
gen and chlorine could be measured by allowing the hydrochloric 
acid formed to dissolve in water, and noting the resulting diminu- 
tion of volume. A diagram of this apparatus is given in Fig. 111. 
The apparatus is filled with a mixture of equal parts of hydrogen 
and chlorine, obtained by the electrolysis of a solution of hydro- 
chloric acid. The bulb A, containing water, is connected at one 
end with a tube fitted with a stop-cock 5, and at the other end with 
a horizontal tube terminating in a reservoir D, which also contains 
water. When the water has become saturated with the constitu- 

* " Photochemische Untersuchungen," Ostwald's Klassiker, No. 34. 
t Pogg. Ann., xoo, 43 (1857); 101, 235 (1857). 



PHOTOCHEMISTRY 521 

ents of the gaseous mixture, B is closed and the entire apparatus is 
protected from light. When it is desired to measure the photo- 
chemical action of a source of light, the bulb A is uncovered and 
the light is allowed to fall upon it. Some of the hydrogen and 
chlorine will combine, and the hydrochloric acid formed will be 
absorbed by the water in A ; the column of water in the horizontal 





Fig. 111. 

tube will move to the right, the magnitude of the movement being 
measured on the scale C. In this way the amount of photochemi- 
cal action can be detennined. An objection to the use of hydrogen 
and chlorine in the actinometer is the danger of violent explosions 
when the illumination is too intense. To remove this objection, 
Burnett replaced the hydrogen of the mixture by carbon monoxide. 
Action of Light on the Silver Halides. Probably the most 
familiar, and undoubtedly one of the most important, photo- 
chemical reactions is that which takes place on the exposure of a 
photographic plate.* Luther f has shown that when a pure silver 
halide is exposed to light, it undergoes reduction according to the 
equation 

light 

2AgX-Ag 2 X + X, 

where X may be chlorine, bromine, or iodine. On removing the 
light, the sub-halide recombines with the free halogen as shown 
by the equation 

Ag 2 X + X^ 2 AgX. 

In other words, the reaction is strictly reversible and a well- 
defined photo-stationary state results from a given intensity of 

* For an excellent treatment of the chemistry of photography the student 
is recommended to consult " Photography for Students of Physics and Chem- 
istry/' by Louis Derr (Macmillan); or " Photochemie und Beschreibung der 
photographischen Chemikalien," by H. W. Vogel. 

t Zeit. phys. Chem., 30, 628 (1899). 



522 THEORETICAL CHEMISTRY 

illumination. The investigations of Baker * make it appear quite 
probable that the photochemical reduction of the silver halides is 
dependent upon the presence of a minute trace of water vapor as a 
catalyst. 

In the photographic plate, the silver halidc is embedded in gela- 
tine, which not only accelerates the rate of reduction of the silver 
halide but also causes the reaction to become irreversible. The 
alteration in the behavior of the silver halide, brought about by 
the presence of gelatine, is due to the fact that the latter reacts with 
the free halogen according to the equation 

light 

2 AgBr + gelatine > Ag 2 Br + brominated gelatine. 

The continuous removal of the liberated bromine by the gelatine 
is an example of what is known as photochemical sensitization. 
Another example of photochemical sensitization is afforded by a 
mixture of benzene and silver chloride. The normal darkening 
of the silver salt is markedly increased by the presence of the 
benzene, which combines with the chlorine as rapidly as it is set 
free by the action of the light on the silver halide. 

While the silver bromide of the photographic plate is extremely 
sensitive to the shorter wave-lengths corresponding to the violet and 
ultra-violet regions of the spectrum, it is only reduced by the longer 
wave-lengths after prolonged exposure. It has been found, how- 
ever, that the addition of certain dyestuffs, such as eosine and 
Congo-red, renders the silver halide sensitive to the longer wave- 
lengths of the spectrum. This phenomenon, which is known as 
optical sensitization , must be carefully distinguished from chemical 
sensitization to which reference has already been made. It is to 
be noted that an optical sensitizer does not absorb chemically any 
of the products of the reaction. Photographic plates which have 
been sensitized in this way are commonly known as ortho-chro- 
matic plates. 

All of the dyestuffs which can function as optical sensitizers 
have been found to exhibit anomalous refraction; i.e., for wave- 
lengths slightly longer than those absorbed, these substances 
possess an abnormally large refractive index, in consequence of 
* Jour. Chem. Soc., 6x, 782 (1892). 



PHOTOCHEMISTRY 523 

which the refracted waves exert the same effect upon the silver 
halide as the shorter wave-lengths of the spectrum. Although it 
is not an essential property, it is generally found that optical sen- 
sitizers are fluorescent. 

As an example of optical sensitization, where the sensitizer is 
non-fluorescent, we may take the photochemical reduction of 
mercuric chloride in the presence of ammonium oxalate. This re- 
action takes place according to the equation 

light 

2 HgCl 2 + (NH 4 ) 2 C 2 4 -* Hg 2 Cl 2 + NH 4 C1 + 2 C0 2 . 

The presence of the non-fluorescent ferric ion, Fe*" ? has been 
shown by Winther * to be an effective optical sensitizer in the re- 
action. The ferric ion is reduced to the ferrous state while the 
oxalic acid undergoes oxidation by the mercuric chloride. It 
was pointed out by Eder f that this reaction is well adapted for 
actinometric measurements, since the amount of mercurous chlo- 
ride precipitated is directly proportional to the intensity of the 
light. 

Photochemical After-Effect. Certain photochemical reac- 
tions have been discovered in which the reaction proceeds even 
after the light stimulus is removed. This phenomenon is known 
as the photochemical after-effect. The velocity of a reaction of 
this kind is different from that of either the light- or the dark- 
reaction. It has also been found that if a portion of the reac- 
tion-mixture, which has already been exposed to the light, be 
added to a fresh unexposed portion, the latter immediately com- 
mences to decompose. Thus, a solution of iodoform in chloro- 
form becomes brown, on exposure to light, due to liberation of 
iodine. 

If the solution is removed from the light, the decomposition will 
continue for several days, and if a small portion of the partially de- 
composed solution be added to a freshly prepared solution, the 
latter will commence to decompose. The photochemical after- 
effect can be readily explained if we assume that the action of light 
gives rise to heterogeneous nuclei which persist for a sufficient time 

* Zeit. wiss. Phot., 7, 409 (1909). 
t Sitz. her. Wien. Akad. (1879). 



524 THEORETICAL CHEMISTRY 

after the light is removed to act as centers around which the 
reaction can proceed throughout the unexposed portion of the 
mixture. 

Assimilation of Carbon Dioxide. A photochemical reaction 
of special interest to the biologist is that in which atmospheric 
carbon dioxide is taken up by plants under the influence of solar 
radiation. While the reaction is exceedingly complex, it may be 
regarded as taking place according to the hypothetical equation 

light 

6 C0 2 + 5 H 2 - C 6 H 10 5 + 6 2 . 

(starch) 

This reaction represents a gain in energy amounting to approxi- 
mately 685 calories per formula-weight of starch. It is quite 
probable, however, that the initial product of the reaction is for- 
maldehyde and that subsequently the latter substance undergoes 
polymerization with the formation of starch and other carbo- 
hydrates. The green coloring matter of the leaves of plants, 
known as chlorophyll, also plays an important part in the assimi- 
lation of carbon dioxide, but beyond the fact that its action is not 
catalytic, little can be stated as to the manner in which it functions 
in the reaction. The velocity of the reaction has been found to 
attain its maximum value in yellow and green light, a result 
which is in complete agreement with the fundamental law of Crott- 
huss that only those rays which are absorbed arc active chemi- 
cally." As has already been stated, the temperature coefficients 
of photochemical reactions are generally very small. This is 
not true, however, of the reaction under consideration. It has 
been found that the rate at which carbon dioxide is assimilated 
by a plant is nearly doubled for a rise in temperature of 10 
(see p. 377). 

Photochemical Synthesis. While it has long been known 
that light is capable of effecting the synthesis of complex organic 
compounds in the living plant from carbon dioxide and water, it 
is only recently that its efficiency in bringing about a great variety 
of organic reactions has been fully recognized. Owing to the re- 
searches of Ciamician and others in this field, numerous photo- 
chemical syntheses of considerable practical value to the organic 



PHOTOCHEMISTRY 525 

chemist have been discovered. It must suffice here to mention a 
few typical photochemical syntheses. Alcohols may be oxidized 
in successive steps, the action of the light closely resembling the 
action of ferments. Methyl alcohol and acetone react to form iso- 
butylene glycol according to the equation 

GH 3 light CH 3 

CH 3 - OH + CO - COH - CH 2 OH 
CH 3 CH 3 

The action of light has been found to be especially favorable to such 
processes as the foregoing, involving reciprocal oxidation and re- 
duction. 

Ortho-nitro-benzaldehyde in the presence of ethyl alcohol reacts 
to form, the ethyl ester of o-nitroso-benzoic acid as shown by the 
equation 

NO 2 ^0 hght NO 

/ C CH 3 
H I 

+ CH 2 OH 

Photoelectric Cells. The absorption of light by any one of 
the three states of matter is invariably accompanied by a change 
in electrical condition. The different electric effects accompany- 
ing the absorption of radiant energy have been classified by Shep- 
pard in the following manner: 

(1) lonization with a corresponding increase of conductance in 
gases, liquids, and solids caused by transmission of light. 

(2) Indirect ionization of a gas due to reflection or emission of 
electrons from the surface of a contiguous denser phase. 

(3) Development of electromotive forces in cells of the following 
type: 




Electrode A 

(light) 



Conductor, 
Di-electric 



Electrode A 

(dark) 



where the two electrodes are separated by a medium which is par- 
tially a conductor and partially a di-electric. 



526 THEORETICAL CHEMISTRY 

The treatment of the first and second of these three classes of 
photoelectric phenomena properly lies within the domain of pure 
physics. The third class, however, includes a number of photo- 
galvanic combinations of considerable interest and importance to 
the physical chemist. 

The first investigation of photoelectric combinations was under- 
taken by E. Becquerel * in 1839. He prepared a cell consisting of 
identical plates of pure silver, coated with a silver halide, and 
immersed in dilute sulphuric acid as an electrolyte. On exposing 
one electrode to the light, while the other was kept in the dark, an 
appreciable electromotive force was developed, the current flowing 
in the solution from the darkened to the illuminated electrode. If 
a galvanometer be included in the circuit, the deflection of the 
needle may be taken as a measure of the intensity of the light. 

The action of the Becquerel actinometer has been explained by 
Ostwald as follows: The incident light lessens the stability of 
the silver iodide, which undergoes ionization according to the 
equation 

AgI-*Ag +1': 

the Ag* ions give up their charges to the electrode while the I' 
ions enter the solution. For every Ag" ion which is discharged at 
the illuminated electrode, an equal number of Ag* ions enter the 
solution at the darkened electrode, thus charging the latter nega- 
tively. Hence the current flows in the solution from the dark- 
ened to the illuminated electrode. Rigollot t has constructed an 
electrical actinometer in which the silver plates used by Becque- 
rel are replaced by two oxidized copper electrodes, while instead 
of dilute sulphuric acid a dilute solution of sodium chloride is used. 
A number of photoelectric combinations have been investigated 
by Wildermann, { among which he recommends the following as 
being especially suitable for actinometers: 

* Ann. Chim. Phys. [3], 9, 257 (1843). 

t Jour, de Phys. [3], 6, 520 (1897). 

j For a thorough treatment of theoretical photochemistry, the student is 
referred to Sheppard's " Photochemistry," while Plotnikow's " Photochemische 
Versuchstechnik" is recommended as an excellent guide to practical laboratory 
methods. 



PHOTOCHEMISTRY 527 

(a) Ag I AgBr, 0.1 N KBr | AgBr | Ag 

(light) (dark) 

Reaction, 2 AgBr Ag2Br, 

(b) Cu | CuO | NaOH | CuO | Cu 

(light) (dark) 

Reaction, 2 CuO -> Cu 2 O. 

Of these two cells, the latter is perhaps the better, since it gives an 
appreciable electromotive force for light of the same intensity and 
varying wave-length. 

While all of the theoretical interpretations of the action of photo- 
electric cells are more or less inadequate, the investigations of 
Scholl and others make it appear probable that negative electrons 
enter the electrolyte from the electrode, while positive ions are 
discharged on the electrode, thereby causing anodic polarization. 
In general the direction of the photoelectric current inside the cell 
is from the non-exposed to the exposed electrode. 



INDEX OF NAMES 



Abegg, 218. 

Alexieeff, 172, 174. 

Amagat, 74, 75. 

Andrews, 105. 

Angstrom, 141. 

Arrhenius, 211, 227, 368, 377, 392, 

413, 425, 428, 432. 
Aston, 63, 419. 
Avogadro, 9, 76. 

Babo, 208. 

Baker, 521. 

Baly, 140, 141, 142. 

Barlow, 159. 

Barnes, 262. 

Bassett, 395. 

Bates, 430. 

Beccaria, 385. 

Beohhold, 259. 

Beckmann, 220. 

Becquerel, 42, 518, 526. 

Beer, 515. 

Bergrnann, 312. 

Berkeley, Earl of, 197, 200, 201. 

Bernoulli, 76. 

Berthelot, 289, 30S, 313, 323, 326. 

Berthollet, 4, 312. 

Berzelius, 16, 386. 

Bigelow, 381. 

Biltz, 260, 429. 

Bingham, 119. 

Biot, 132. 

Blagden, 217. 

Blake, 253, 254. 

Bodenstein, 317, 518. 

Boettger, 415. 



Boguski, 377. 

Bolt-wood, 44, 56. 

Boltzmann, 76, 507. 

Boyle, 2, 49, 72, 76, 170, 192, 197, 

286. 

Bragg, W. H, 160. 
Bragg, W. L., 160. 
Bredig, 276, 381, 409. 
Brown, 279. 
Bruhl, 127. 
Bruni, 250. 
Budde, 81. 
Buff, 121. 

Bunsen, 169, 170, 514, 515, 518, 520. 
Burgess, 519. 
Burton, 253. 

Cailletet, 113. 
Callendar, 201. 
Carlisle, 385. 
Chapman, 519. 
Charpy, 353. 
Ciamician, 524. 
Claude, 115. 
Clausius, 76, 392, 413. 
Clement, 96, 382, 383. 
Cotton, 252. 
Crookes, 33, 47, 48. 
Cumming, 472. 
Curie, 43, 44, 49, 56. 

Dale, 126, 152. 
Dalton, 1, 7, 16, 168. 
Davy, 385, 386. 
Debierne, 44. 
Debray, 328. 



529 



530 



INDEX OF NAMES 



Debye, 161. 

De Chancourtois, 21. 

De Forcrand, 334. 

Denham, 480. 

Desch, 142. 

Desormes, 96, 382, 383. 

Deville, 88, 321 . 

De Vries, 201, 202, 203, 204, 230. 

Dewar, 114. 

DSbereiner, 21. 

Draper, 520. 

Drude, 151. 

Duclaux, 248. 

Dulong, 11, 162. 

Dumas, 20. 

Dutoit, 419. 

Eder, 523. 
Einstein, 283, 285. 
Eotvos, 146. 

Fanjung, 418. 

Faraday, 7, 138, 150, 268, 389. 

Faure, 499. 

F6ry, 510. 

Fick, 206, 208. 

Forbes, 466. 

Fletcher, 285. 

Freudenberg, 502. 

Freundlich, 254, 264, 269, 270. 

Friderich, 419 

Fuchs, 493. 

Gay-Lussac, 8, 76, 192, 196,197,208. 

Geiger, 40, 52, 54, 55, 57. 

Geoffrey, 312. 

Gibbs, 273, 274, 338, 448. 

Gladstone, 126, 152. 

Goodwin, 468. 

Gouy, 279. 

Graham, 206, 237, 274, 276. 

Grotthuss, 391, 513, 524. 

Grove, 392. 

Guldberg, 314, 329. 

Guthrie. 349. 



Hall, 66. 

Hamburger, 204, 205. 
Hampson, 114. 
Hardy, 253, 256. 
Harkins, 63, 66. 
Hartley, 140, 142, 144, 197. 
Hautefeuille, 317. 
Hatty, 159. 
Hedin, 205. 
Helmholtz, 448, 477. 
Henry, 170, 184. 
Henderson, 472. 
Hess, 290, 304, 305. 
Heycock, 353. 
Hcydweiller, 414. 
Hittorf, 393. 
Holborn, 478, 510. 
Horstmann, 331. 
Hulett, 181. 
Hull, 505. 

Isambcrt, 330. 

Jones, 219, 233, 395, 418. 
Jurin, 144. 

Kirchhoff, 506. 

Knoblauch, 375. 

Koelichen, 426. 

Kohlrausch, 400, 413, 414, 415, 416, 

417, 478, 488. 
Konowalow, 175, 176. 
Kopp, 14, 88, 120, 121, 127. 
Kraus, 430. 
Kroenig, 76. 
Kundt, 98. 
Kurlbaum, 510. 

Laborde, 56. 
Landolt, 4. 
Langbein, 302. 
Laplace, 98, 290. 
Larmor, 373. 
Lavoisier, 2, 286, 290. 
Lebedew, 505. 



INDEX OF NAMES 



531 



Le Bel, 134, 135, 137. 

Le Blanc, 493, 495, 497, 501. 

Le Chatelier, 95, 309. 

Lehmann, 165. 

Lemoine, 317. 

Lenard, 36. 

Lewis, 162, 472, 474. 

Liebig, 382. 

Lillie, 249. 

Linde, 114. 

Lindemann, 164. 

Linder, 243, 251, 254. 

Lippmann, 459 

Lodge, 411. 

Loomis, 218. 

Lorentz, 126, 152. 

Lorenz, 126, 152. 

Lessen, 121. 

Liideking, 267. 

Lummer, 507. 

Lunde"n, 442. 

Luther, 521. 

Mac Innes, 472. 

Marignac, 5, 16. 

Maxwell, 76. 

Mayer, 94, 290. 

Mendeleeff, 22. 

Menschutkin, 378. 

Meyer, 466. 

Meyer, Lothar, 22, 28, 

Meyer, Victor, 83. 

Millikan, 40, 285. 

Mitscherlich, 14. 

Morgan, 149, 150. 

Morley, 16. 

Morse, 194, 196, 197, 198, 200. 

Moseley, 60. 

Mouton, 252. 

Natanson, 91. 

Nerast, 119, 141, 151, 164, 297, 335, 

336, 419, 451, 472, 475, 476, 486, 

517. 
Neumann! 13. 



Neville, 353. 
Newlands, 22. 
Newton, 312. 
Nicholson, 385. 
Nollet, Abbe, 188. 
Nordlund, 285. 
Noyes, 372, 377. 
Nuttall, 52. 

Ohm, 388. 

Olszewski, 113. 

Onnes, 116. 

Ostwald, W., 8, 147, 169, 191, 231, 

273, 287, 368, 379, 383, 393, 422, 

424, 435, 447, 526. 
Ostwald, Wo., 245. 

Palmaer, 453. 

Pappada, 250. 

Pasteur, 132, 133, 134. 

Pauli, 254. 

Pebal, 90. 

Perkin, 138. 

Perrin, 35, 250, 279, 280, 281, 282, 

283,285. 
Petit, 11, 162. 
Pfeffer, 188, 189, 190, 191, 192, 193, 

194, 198. 
Philip, 170. 
Pictet, 113. 
Picton, 243, 251, 254. 
Planck, 40, 207, 472, 474. 
Plants, 499. 
Pope, 159. 
Pringsheim, 507. 
Proust, 5. 
Prout, 20. 
Pulfrich, 124. 

Quincke, 250. 

Ramsay, 49, 146, 147, 148, 149, 150, 

168. 
Raoult, 209, 210, 212, 215, 217, 218, 

223. 



532 INDEX OF NAMES 

Rayleigh, 509. Thorpe, 121. 

Regener, 40, 54. Traube, 122, 129, 188. 

Regnault, 83. Trouton, 118. 

Reicher, 368. 

Reinitzer, 165. Van der gta( ^ 3?3 

Reuss, 250. v d Waals 31 81 82 107 121, 

Richards, 16, 159, 466. m >>> 

Rigollot, 526. Van , t Hofl ^ ^ ^ lg ^ Ig5 ^ 1Q2j 

Roberts-Austen, 185, 353, 355. ^ ^ ^ 213> 217? ^ ^ 

Rodewald, 267. 226> 324? 337> 378> 412> 43Q 

Roentgen, 42. Volta, 385, 446. 
Roozeboom, 351, 353. 
Roscoe, 177, 514, 515, 518, 520. 

Rose, 313. Waage, 314, 329. 

Royds, 55. Walden, 404. 

Rudolphi, 430. Walker, 117, 211, 376. 
Rutherford, 40, 45, 46, 48, 49, 50, 54, Warder, 368. 

55 5 57. Washburn, 417. 

Weber, 12, 13. 

Sargent, 472, 474. Weimarn, 245. 

Schiff, 121. Wenzel, 312. 

Scholl, 527. Whetham, 412. 

Schroeder, 265. Whitney, 253, 254, 377. 

Shields, 146, 147, 148, 149, 150. Wiedemann, 250, 267- 

Siedentopf, 241. Wien, 508, 509. 

Snell, 124. Wiener, 279. 

Soddy, 48, 49, 50, 51, 52, 60. Wildermann, 526. 

Soret, 208. Williamson, 392. 

St. Gilles, Pean de, 313, 323. Wilson, C. T. R., 45. 

Stas, 16, 20. Wilson, E. D., 63. 

Stefan, 507. Winther, 523. 

Steno, 155. Wladimiroff, 205, 

Stokes, 39, 283. w <> d > 512. 

Svedberg, 277. Wroblewski, 113. 

Wullner, 208. 
Tammann, 206. 

Thilorier, 113. Young, 183. 
Thomsen, 302. 

Thomson, J. J., 35, 45, 63, 282, 387, . , rt ^ 

*! Zsigmondy, 241, 259. 



INDEX OF SUBJECTS 



Abnormal solutes, 225. 
Absolute index of refraction, 125. 
Absorption and emission, 506, 
Absorption index, 514. 
Absorption of light, quantitative re- 
lation concerning, 514. 
Accumulators, 488. 
Actinometcis, 520. 
Active deposits, 49. 
Activity and concentration, 478. 
Adsorption, 267. 

of gases, 268. 

in solution, 269. 

isotherm, 269, 270. 
Alloys, 353. 
Amicrons, 239. 
Association in solution, 225. 
Atomic theory, 6, 7. 

number, 58. 

structure, 57. 

structure, hydrogen-helium system 
of, 63. 

weights, 16, 18. 

weights and atomic numbers, re- 
lation between, 64. 
Autocatalysis, 381. 
Avogadro constant, 40. 

hypothesis of, 9, 79. 

Basicity of organic acids, 434. 
Bimolecular reactions, 366. 
Boiling point constant, 213. 

and critical temperature, 119. 

elevation of, 213. 

elevation and osmotic pressure, 
216. 



Blood corpuscle method, 204. 
Biownian movement, 279. 
recent investigations of, 285. 

Calorimeter, combustion, 289. 
Capillary electrometer, 458. 
Carbon dioxide, assimilation of, 523. 
Catalysis, 379. 

mechanism of, 382. 

negative, 382. 
Cataphoresis, 251. 

Cathode particle, charge carried by, 
38. 

ratio of charge to mass of, 38. 

velocity of, 36. 
Cathode rays, 33. 

properties of, 33. 
Cathodo-luminescence, 512. 
Cells, electromotive force of con- 
centration, 411. 

galvanic, 446. 

gas, 485. 

photo-electric, 525. 

reversible, 448. 

standard, 457. 

storage, 488. 

Chemical constitution, 132, 142. 
Chemi-luminescence, 512. 
Colloidal solutions, 238. 

density of, 245. 

electrical conductance of, 253. 

osmotic pressure of, 247. 

preparation of, 274. 

surface tension of, 247. 

viscosity of, 246. 
Colloids, 237. 



533 



534 



INDEX OF SUBJECTS 



Colloids, molecular weight of, 249. 

precipitation by electrolytes, 254. 

surface energy of, 271. 
Components, 338. 
Compounds, 2, 7. 

Compressibilities of solidelements,159. 
Concentration and activity, 478. 

elements, 465. 
Conductance, and ionization, 412. 

at high temperatures and pressures, 
417. 

determination of, 400. 

equivalent, 398. 

molar, 398. 

specific, 398. 

Conductance of difficultly soluble 
salts, 415. 

of different substances, 402. 

of fused salts, 420. 

of non-aqueous solutions, 418. 

temperature coefficient of, 416. 
Conduction of electricity through 

gases, 31. 
Connection between gaseous and 

liquid states, 104. 
Consecutive reactions, 376. 
Corresponding conditions, 110. 
Co- volume, 121. 
Critical pressure, 105. 

temperature, 105. 

volume, 105. 
Crookes' dark space, 32. 
Cryohydrate, 347. 

Crystal form and chemical composi- 
tion, 158. 

Crystallography, 154. 
Crystalloids, 237. 
Crystals, liquid, 165. 

mixed, 14. 

properties of, 156. 
Crystallization methods, 275. 

Decomposition potential, 495. 
Degree of dissociation, calculation of, 
91. 



Degree of dissociation, freedom, 339. 
Density of colloidal solutions, 245. 
Dialysis, 237. 
Dielectric constants, 150. 
Diffusion, coefficient of, 206. 
Dilution law, 422. 
Disintegration hypothesis, 50. 

series, 52. 
Dispersity, 238. 
Dispersoids, 239, 245. 

classification of, 244. 
Dissociation constant, 320. 

hypothesis, 437. 

in solution, 226. 
Distillation fractional, 178. 

steam, 179. 
Distribution of a solute between two 

immiscible solvents, 335, 
Drop weight, 149. 

Electrical conductance of colloidal 
solutions, 253. 

double layer, 452. 
Electrode, dropping, 463. 

hydrogen, 464. 

normal, 461. 

null, 463. 
Electrodes, of first type, 468. 

of second type, 468. 
Electroendosmosis, 250. 
Electro-luminescence, 512. 
Electrolytes, ionization of strong, 
429. 

mixtures with a common ion, 427. 
Electrolytic dissociation theory, 227, 
229. 

separation of metals, 502. 
Electrometer capillary, 458. 
Electromotive force, measurement of, 

455. 

Electrons, sources of, 41. 
Electron theory, 31. 
Elements, 2. 

classification of, 20. 

oxidation and reduction, 482. 



INDEX OF SUBJECTS 



535 



Elements, periodic series of, 24. 

Elevation of boiling point, 213. 

Emanations, 48. 

Emission and absorption, 506. 

Emulsions, 239. 

Emulsoids, 239. 

^action of heat on, 257. 

precipitation of, 257. 
Energy evolved by radium, 56. 

distribution in spectrum, 509. 
Equation of condition, reduced, 111. 
Equation of Nernst-Lindemann, 164. 

van der Waals, 81, 107. 
Equilibrium constant, 316. 

variation with temperature, 324. 
Equilibrium, homogeneous, 312. 

in homogeneous gaseous systems, 
317. 

in liquid systems, 322. 
Equivalent, chemical, 6. 

electrochemical, 390. 
Etch figures, 158. 
Eutectic, 353. 

Faraday dark space, 32. 
Ferments, inorganic, 381. 
Fluorescence, 511. 
Formation, heat of, 294. 
Freezing point, lowering of, 217. 
of concentrated solutions, 233. 
Fusion, heat of, 153. 

Galvanic cells, 440. 
Gas, ideal, 75. 

perfect, 75, 
Gases, adsorption of, 268. 

specific heat of, 100. 
Gas constant, molecular, 73. 

specific, 73. 

Gas laws, deviations from, 74. 
Gaseous and liquid states, connection 

between, 104. 
Gaseous molecule, mean velocity of 

translation of, 80. 
Gelation, 239. 



Gels, elastic, 265. 

hydration and dehydration of, 263. 

non-elastic, 264. 

physical properties of, 262. 
Gold-number, 259. 

Hajmatocrit method, 205. 
Heat, of combustion, 302. 

of dilution, 297. 

of dissociation of solids, 334. 

of formation, 294. 

of fusion, 153. 

of imbibition, 266. 

of ionization, 397, 432. 

of neutralization, 305. 

of reaction, variation with tempera-* 
ture, 300. 

of solution, 295. 

of vaporization, 118. 
Helium atoms and a-particles iden- 
tical, 55. 

Helium, rate of production of, 56. 
Helix of de Chancourtois, 21. 
Heterogeneous systems, 328. 
Homogeneous gaseous systems, equi- 
librium in, 317. 
Hydrogel, 328. 
Hydrolysis, 437. 

determination of, 442, 480. 
Hydrosol, 238. 

Imbibition, 266. 

in solutions, 267. 

velocity of, 266. 
Immiscibility, 178. 
Inorganic ferments, 381. 
Ionic product, 433. 
Ionization constant, 422. 

of water, 443. 
Ionization of gases, 45. 
Ionization, heat of, 432, 484. 

influence of substitution on, 435. 

of water, 487. 
lonogen, 229. 
Ions, 45. 



536 



INDEX OF SUBJECTS 



Ions, absolute velocity of, 409. 

existence of free, 391. 

migration of, 393. 
Isohydric solutions, 428. 
Isomorphism, 14. 
Isothermals, 105. 
Isotopes, 63. 
Isotonic coefficient, 203. 

Kinetic equation, deductions from, 78. 

derivation of, 77. 

theory of gases, 100. 
Kenetics, chemical, 359. 

Labile solutions, 183. 
Law of Beer, 141, 515. 

of Boyle, 73, 78. 

of combining proportions, 5. 

of conservation of mass, 3. 

of constant heat summation, 290. 

of definite proportions, 4. 

of Dulong and Petit, 11. 

of Faraday, 389. 

of Gay-Lussac, 73, 79. 

of Graham, 80. 

of Grotthuss, 513. 

of Guldberg and Waage, 314. 

of Henry, 170, 171. 

of Hess, 290, 305. 

of Jurin, 144. 

of Kirchhoff, 506. 

of Kohlrausch, 406. 

of Lavoisier and Laplace, 290. 

of mass action, 314. 

of Mitscherlich, 14, 15. 

of molecular displacement, 283. 

of multiple proportions, 5. 

of Neumann, 14. 

of octaves, 22. 

of Ohm, 388. 

of Raoult, 209. 

of Richter, 4. 

of Stefan-Boltzmann, 507. 

of volumes, 8. 

of Wien, 507. 



Light, action on silver halides, 521. 
Liquefaction of gases, 112. 
Liquids, characteristics of, 104. 

refractive power of, 123. 

vapor pressure of, 117. 
Liquid systems, equilibrium in, 322. 
Luminescence, 511. 
Lyotrope series, 239. 

Magnetic rotation, 138. 
Mass, conservation of, 3. 
Mass action, law of, 314. 
Maximum work, principle of, 308. 
Mechanism of catalysis, 382. 
Membranes, semi-permeable, 171. 
Metastable solutions, 183. 
Microns, 238. 
Miscibility, complete, 174. 

partial, 172. 

Mixtures, specific refraction of, 128. 
Molar volume, 10. 

Molecular gas constant, evaluation 
of, 74. 

magnetic rotation, 139. 

refraction, 126. 

rotation, 131. 

vibration, 142. 

volume, 120. 

weight, 83, 146. 

weight by boiling point method, 
215. 

weight by freezing point method, 
220. 

weight in solution, 222. 

weight of colloids, 249. 

weight in solution, 222. 

vibration, 142. 

volume, 120. 

Neutralization, heat of, 305. 
Nitrogen p, xide, dissociation of, 

93. 
Non-dissociatiug solvent, solution of 

solid in, 336. 
Normal electrodes, 461. 



INDEX OF SUBJECTS 



537 



Octaves, law of, 22. 
Optical activity, 132. 
Osmotic pressure, 187, 191. 

and boiling point elevation, 216. 

and diffusion, 206. 

and freezing point depression, 221. 
Osmotic pressure and lowering of 
vapor pressure, 211. 

comparative values of, 201. 

measurement of, 189. 

of colloidal solutions, 247. 

recent work on direct measure- 
ment of, 194. 

theoretical value of, 192. 
Oxidation and reduction elements, 
482. 

Particles, -, 55. 
Peptization, 276. 
Periodic law, 22, 64. 

applications of, 26. 

defects in, 29. 
Periodicity of physical properties, 25. 

of radio-elements, 61. 
Phase rule, 338. 

derivation of, 339. 
Phosphorescence, 511. 
Photochemical action, 513. 

after effect, 523. 

extinction, 515. 

induction, 518. 

reaction, kinetics of, 516. 

sensitization, 522. 

synthesis, 524. 
Photo-electric cells, 525. 
Photo-stationary state, 518. 
Physical properties of ionized solu- 
tions, 231. 

Plasmolytic methods, 201. 
Polarization, 492. *. 

capacity of electrode? i4. 

electromotive force of, 492. 

theory of, 497. <r 
Polarized light, rotation of plane of, 
129. 



Polymorphism, 158. 
Potential, between metal and solu- 
tion, 464. 
- decomposition, 495. 

difference at junction of two solu- 
tions, 469. 

difference at liquid junctions, 472. 

normal electrode, 475. 
Precipitation and valence, 256. 
Precipitation of colloids by electro- 
lytes, 254. 

of emulsoids, 257. 

of suspensoids, 254. 
Preparation of colloidal solutions, 

274. 
Principle of maximum work, 308. 

of Soret, 208. 
Properties of cathode rays, 33. 

of gels, 262. 
Proportions, law of combining, 5. 

law of definite, 4. 

law of multiple, 5. 
Protective colloids, 258. 
Prout's hypothesis, 20. 

Quantum Theory, 164. 

Radiant energy, 504. 

source of, 505. 
Radiations, nature of, 46. 

phosphorescence induced by, 46. 

photographic action of, 45. 
Radioactive constant, 50. 

equilibrium, 52. 

substances, 44. 
Rodioactivity, 42. 

discovery of, 42. 

Radio-elements, periodicity of, 61. 
Radio-luminescence, 512. 
Radium, 43. 

discovery of, 43. 

energy evolved by, 56. 
Rays, CB-, 0-, and 7-, 47. 
Ratio of charge to mass, 38. 

of specific heats, 96. 



538 



INDEX OF SUBJECTS 



Ratio of cnarge to mass, of specific 

heats, determination of, 97, 98. 
Reaction, determination of order of, 
374. 

influence of solvent on velocity of, 
378. 

velocity, 359, 374, 377. 
Reactions, at constant pressure, 298. 

at constant volume, 298. 

bimolecular, 366. 

counter, 374, 

heterogeneous, velocity of, 376. 

of first order, 364. 

of higher orders, 373, 

of second order, 366. 

of third order, 370. 

photochemical, classification of, 
519. 

side, 374. 

trimolecular, 370. 
Reciprocal precipitation, 260, 
Refraction, index of, 124. 

molecular, 126. 

specific, 126. 
Residual current, 495. 
Resistance capacity, 401. 
Reversible cells, 448. 
Rotation, magnetic, 139. 

molecular, 131. 

of plane polarized light, 129. 

specific, 131. 

Salt solutions, thermoneutrality of, 

305. 

Saturated solution, 181. 
Semi-permeable membranes, 187. 
Side reactions, 374. 
Solation, 239. 
Solids, general properties of, 153. 

heat capacity of, 161. 
Solubility coefficient;, 171. 

product, 433. 
Solutes, abnormal, 225. 
Solution, association in, 225. 

dissociation in, 226. 



Solution, heat of, 295. 

methods, 276. 

pressure, 450. 
Solutions, 167. 

absorption in, 269. 

classification of, 167. 

colloidal, 238. 

conductance of non-aqueous, 418. 

imbibition in, 267. 

of gases in gases, 167. 

of gases in liquids, 167. 

of liquids in liquids, 172. 

of solids in liquids, 181. 

isohydric, 428. 

labile, 183. 

metastable, 183. 

properties of ionized, 231. 

saturated, 169. 

solid, 184. 

supersaturated, 169. 

volume-normal, 196. 

weight-normal, 196. 
Solvate theory, 234. 
Solvent, ionizing power of, 419. 
Specific heat, 92, 93. 

and atomic weight, 11. 

at constant pressure and volume, 
93. 

of gases, 100. 

of solid elements, 11, 161. 

ratio of, 96. 
Specific refraction, 126. 

rotation, 131. 
Spectra, absorption, 139. 
Standard cells, 457. 
Steam distillation, 179. 
Strength of acids and bases, 424. 
Sublimation, 153. 
Submicrons, 239. 
Sulphur, equilibrium between the 

phases of, 343. 
Surface concentration, 272. 
Surface energy of colloids, 271. 
Surface tension, 144, 146, 149. 

of colloidal solutions, 247. 



INDEX OF SUBJECTS 



539 



Suspensions, 239. 
Suspensoids, 239. 

precipitation of, 254. 
Systems, three-component, 356. 

two-component, 345. 

Theorem of Le Chatelier, 309. 
Theory, electron, 31. 

kinetic, 100. 

of electrolytic dissociation, 227, 
229. 

of solvation, 234. 
Thermal units, 286. 
Thermochemical equations, 287. 

measurements, 288. 
Thermo-luminescence, 512. 
Thermoneutrality of salt solutions, 

304. 

Transition point, 340. 
Transmission coefficient, 514. 
Transport numbers, 395. 

determination of, 395. 

electrometric determination of, 477. 
Triads, of Doberciner, 21. 
Tribo-luminescence, 512. 
Trimolecular reactions, 370. 
Triple point, 342. 
Tyndall phenomenon, 241. 

Ultrafiltration, 242. 
Ultramicroscope, 241. 
Unimolecular reactions, 361, 364. 
Units, electrical, 388. 

Valence, 15. 
electrometric determination of, 476. 



Vapor density, 83. 

abnormal, 88. 

determinations of, 86. 
Vaporization, heat of, 118. 
Vapor pressure, 117. 

and osmotic pressure, connection 
between, 211. 

lowering of, 209. 

of liquids, 117. 

Variation of equilibrium constant 
with temperature, 337. 

of heat of reaction with tempera- 
ture, 300. 
Velocity constant, 315. 

of gaseous molecule, 80. 

of imbibition, 266. 
Vibration, curves of molecular, 142. 
Viscosity of colloidal solutions, 246. 
Voltaic pile, 385. 
Volume, molar, 10. 
Volumes, Gay-Lussac's law of, 8. 

Water, dissociation of, 414. 

equilibrium between phases of, 
340. 

ionization constant of, 443. 

primary decomposition of, 501. 
Weight, atomic, 8, 10, 17, 18. 

combining, 5, 8. 

gram-molecular, 10, 17. 

molar, 10, 17. 

X-rays and atomic structure, 58. 
and crystal structure, 159. 
spectra, 60.