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Full text of "An Introduction To Electrochemistry"

THE BOOK WAS 
DRENCHED 



[<OU_164128 



OUP 875 2-U-6&-- 6,00<). 

OSMANIA UNIVERSITY LIBRARY 

Call No. J&l '0*3 Accession No. ( 

Author 

Title 

This book should be "returned on or before th date latumarked below. 



AN INTRODUCTION 



TO 



ELECTROCHEMISTRY 



BY 

SAMUEL GLASSTONE, D.Sc., PH.D. 
Contultanl, Untied Stales Atomic Energy Commirsion 



TENTH PRINTING 




(AN EAST-WEST EDITION) 



AFFILIATED EAST-WEST PRESS PVT. LTD. 

NEW DELHI. 



Copyright 1942 by 
LITTON EDUCATIONAL PUBLISHING, INC. 



No reproduction in any form of this book, in whole 
or in part (except for brief quotation in critical 
articles or reviews), may be made without written 
authorization from the publishers. 



First Published May 1942 



AFFILIATED EAST-WEST PRESS PVT. LTD. 

East- West Student Edition - 1965 
Second East-West Reprint - 1968 
Third East-West Reprint 1971 
Fourth East-West Reprint - 1974 

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Reprinted in India with the special permission of the 
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Inc. New York, U.S.A. and the copyright holders. 

This book has been published with the assistance 
of the joint Indian-American Textbook Programme 



Published by K.S. Padmanabhan for AFFILIATED EAST- 
WEST PRESS PVT. LTD., 9 Nizamuddm East, New Delhi 13, 
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New Delhi. 



To V 



PREFACE 

The object of this book is to provide an introduction to electro- 
chemistry in its present state of development. An attempt has been 
made to explain the fundamentals of the subject as it stands today, de- 
voting little or no space to the consideration of theories and arguments 
that have been discarded or greatly modified. In this way it is hoped 
that the reader will acquire the modern point of view in electrochemistry 
without being burdened by much that is obsolete. In the opinion of the 
writer, there have been four developments in the past two decades that 
have had an important influence on electrochemistry. They are the ac- 
tivity concept, the interionic attraction theory, the proton-transfer theory 
of acids and bases, and the consideration of electrode reactions as rate 
processes. These ideas have been incorporated into the structure of the 
book, with consequent simplification and clarification in the treatment of 
many aspects of electrochemistry. 

This book differs from the au thor's earlier work, "The Electrochem- 
istry of Solutions/' in being less comprehensive and in giving less detail. 
While the latter is primarily a work of reference, the present book is 
more suited to the needs of students of physical chemistry, and to those 
of chemists, physicists and physiologists whose work brings them in con- 
tact with a variety of electrochemical problems. As the title implies, 
the book should also serve as an introductory text for those who in- 
tend to specialize in either the theoretical or practical applications of 
electrochemistry. 

In spite of some lack of detail, the main aspects of the subject have 
been covered, it is hoped impartially and adequately. There has been 
some tendency in recent electrochemical texts to pay scant attention to 
the phenomena at active electrodes, such as ovcrvoltage, passivity, cor- 
rosion, deposition of metals, and so on. These topics, vihich are of 
importance in applied electrochemistry, are treated here at Mich length 
as seems reasonable. In addition, in view of tho growing interest in 
electrophoresis, and its general acceptance as a branch of electrochem- 
istry, a chapter on clectrokinetic phenomena has boon included. 

No claim is made to anything approaching completeness in the matter 
of references to the scientific literature. Such reformers as arc given arc 
generally to the more recent publications, to review articles, and to 
papers that may, for one reason or another, have some special interest. 
References are also frequently included to indicate the sources from which 
data have been obtained for many of the diagrams and tables. Since 
no effort was made to be exhaustive in this connection, it was felt that 
an author index would be misleading. This has consequently been 



VI PREFACE 

omitted, but where certain theories, laws or equations are usually asso- 
ciated with the names of specific individuals, such names have been in- 
cluded in the general index. 

In conclusion, attention may be drawn to the problems which are to 
be found at the end of each chapter. These have been chosen with the 
object of illustrating particular points; very few are of the kind which 
involve mere substitution in a formula, and repetition of problems of the 
same type has been avoided as far as possible. Many of the problems 
are based on data taken directly from the literature, and their solution 
should provide both valuable exercise and instruction. The reference to 
the publication from which the material was taken has been given in the 
hope that when working the problem the student may become sufficiently 
interested to read the original paper and thus learn for himself something 
of the methods and procedures of electrochemical research. 

SAMUEL GLASSTONE 
NORMAN, OKLAHOMA 

March 1942 



CONTENTS 

CHAPTER PAGE 

PREFACE v 

I. INTRODUCTION 1 

II. ELECTROLYTIC CONDUCTANCE 29 

III. THE THEORY OF ELECTROLYTIC CONDUCTANCE 79 

t VI. THE MIGRATION OF IONS 107 

i.V. FREE ENERGY AND ACTIVITY 131 

REVERSIBLE CELLS 183 

ELECTRODE POTENTIALS 226 

VIII. OXIDATION-REDUCTION SYSTEMS 267 

IX. ACIDS AND BASES 306 

X. THE DETERMINATION OF HYDROGEN IONS 348 

XI. NEUTRALIZATION AND HYDROLYSIS 370 

XII. AMPHOTERIC ELECTROLYTES 418 

XIII. POLARIZATION AND OVERVOLTAGE 435 

XIV. THE DEPOSITION AND CORROSION OF METALS 482 

< XV. ELECTROLYTIC OXIDATION AND REDUCTION 504 

XVI. ELECTROKINETIC PHENOMENA 521 

INDEX 547 



CHAPTER I 
INTRODUCTION 

Properties of Electric Current. When plates of two dissimilar metals 
are placed in a conducting liquid, such as an aqueous solution of a salt 
or an acid, the resulting system becomes a source of electricity; this 
source is generally referred to as a voltaic cell or galvanic cell, in honor 
of Volta and Galvani, respectively, who made the classical discoveries in 
this field. If the plates of the cell are connected by a wire and a mag- 
netic needle placed near it, the needle will be deflected from its normal 
position ; it will be noted, at the same time, that the wire becomes warm. 
If the wire is cut and the two ends inserted in a conducting solution, 
chemical action will be observed where the wires come into contact with 
the liquid; this action may be in the form of gas evolution, or the libera- 
tion of a metal whose salt is present in the solution may be observed. 
These phenomena, viz., magnetic, heating and chemical effects, are said 
to be caused by the passage, or flow, of a current of electricity through 
the wire. Observation of the direction of the deflection of the magnetic 
needle and the nature of the chemical action, shows that it is possible 
to associate direction with the flow of electric current. The nature of 
this direction cannot be defined in absolute terms, and so it is desirable 
to adopt a convention and the one generally employed is the following: 
if a man were swimming with the electric current and watching a compass 
needle, the north-seeking pole of the needle would turn towards his left 
side. When electricity is passed through a solution, oxygen is generally 
liberated at the wire at which the positive current enters whereas hydro- 
gen or a metal is set free at the wire whereby the current leaves the 
solution. 

It is unfortunate that this particular convention was chosen, because 
when the electron was discovered it was observed that a flow of electrons 
produced a magnetic effect opposite in direction to that accompanying 
the flow of positive current in the same direction. It was necessary, 
therefore, to associate a negative charge with the electron, in order to be 
in harmony with the accepted convention concerning the direction of a 
current of electricity. Since current is carried through metals by means 
of electrons only, it means that the flow of electrons is opposite in direc- 
tion to that of the conventional current flow. It should be emphasized 
that there is nothing fundamental about this difference, for if the direc- 
tion of current flow had been defined in the opposite manner, the electron 
would have been defined as carrying a positive charge and the flow of 
electrons and of current would have been in the same direction. Al- 



2 INTRODUCTION 

though a considerable simplification would result from the change in 
convention, it is too late in the development of the subject for any such 
change to be made. 

E.M.F., Current and Resistance: Ohm's Law. If two voltaic cells 
are connected together so that one metal, e.g., zinc, of one cell is con- 
nected to the other metal, e.g., copper, of the second cell, in a manner 
analogous to that employed by Volta in his electric pile, the magnetic 
and chemical effects of the current are seen to be increased, provided 
the same external circuit is employed. The two cells have a greater 
electrical driving force or pressure than a single one, and this force or 
pressure * which is regarded as driving the electric current through the 
wire is called the electromotive force, or E.M.F. Between any two points 
in the circuit carrying the current there is said to be a potential difference, 
the total E.M.F. being the algebraic sum of all the potential differences. 

By increasing the length of the wire connecting the plates of a given 
voltaic cell the effect on the magnetic needle and the chemical action 
are seen to be decreased: the greater length of the wire thus opposes the 
flow of current. This property of hindering the flow of electricity is 
called electrical resistance, the longer wire having a greater electrical 
resistance than the shorter one. 

It is evident that the current strength in a given circuit, as measured 
by its magnetic or chemical effect, is dependent on the E.M.F. of the cell 
producing the current and the resistance of the circuit. The relationship 
between these quantities is given by Ohm's law (1827), which states that 
the current strength (/) is directly proportional to the applied E.M.F. (E) 
and inversely proportional to the resistance (R) ; thus 



is the mathematical expression of Ohm's law. The accuracy of this law 
has been confirmed by many experiments with conductors of various 
types: it fails, apparently, for certain solutions when alternating currents 
of very high frequency are employed, or with very high voltages. The 
reasons for this failure of Ohm's law are of importance in connection with 
tho theory of solutions (see Chap. III). It is seen from equation (1) that 
the E.M.F. is equal to the product of the current and the resistance: 
a consequence of this result is that the potential difference between any 
two points in a circuit is given by the product of the resistance between 
those points and the current strength, the latter being the same through- 
out the circuit. This rule finds a number of applications in electro- 
chemical measurements, as will be evident in due course. 

* Electrical force or pressure does not have the dimensions of mechanical force or 
pressure; the terms are used, however, by analogy with the force or pressure required to 
produce the flow of a fluid through a pipe. 



ELECTRICAL DIMENSIONS AND UNITS 3 

Electrical Dimensions and Units. The electrostatic force (F) be- 
tween two charges e and e' placed at a distance r apart is given by 



where K depends on the nature of the medium. Since force has the 
dimensions mlt~ 2 , where m represents a mass, I length and t time, it can 
be readily seen that the dimensions of electric charge are mWt* 1 **, the 
dimensions of K not being known. The strength of an electric current is 
defined by the rate at which an electric charge moves along a conductor, 
and so the dimensions of current are mWt-***. The electromagnetic 
force between two poles of strength p and p' separated by a distance r 
is pp'lnr*, where p, is a constant for the medium, and so the dimensions 
of pole strength must be mH*t~ l p*. It can be deduced theoretically that 
the work done in carrying a magnetic pole round a closed circuit is pro- 
portional to the product of the pole strength and the current, and since 
the dimensions of work are mPt~* 9 those of current must be m*W~V~~*- 
Since the dimensions of current should be the same, irrespective of the 
method used in deriving them, it follows that 



The dimensions l~ l t are those of a reciprocal velocity, and it has been 
shown, both experimentally and theoretically, that the velocity is that 
of light, i.e., 2.9977 X 10 10 cm. per sec., or, with sufficient accuracy for 
most purposes, 3 X 10 10 cm. per sec. 

In practice K and n are assumed to be unity in vacuum: they are then 
dimensionlcss and are called the dielectric constant and magnetic per- 
meability, respectively, of the medium. Since K and n cannot both be 
unity for the same medium, it is evident that the units based on the 
assumption that K is unity must be different from those obtained by 
taking /x as unity. The former are known as electrostatic (e.s.) and the 
latter as electromagnetic (e.m.) units, and according to the facts recorded 
above 

1 e.m. unit of current 

- ------ - = 3 X 10 l crn. per sec. 

1 e.s. unit of current 

It follows, therefore, that if length, mass and time are expressed in centi- 
meters, grams and seconds respectively, i.e., in the c.g.s. system, the 
e.m. unit of current is 3 X 10 10 times as great as the e.s. unit. The e.m. 
unit of current on this system is defined as that current which flowing 
through a wire in the form of an arc one cm. long and of one cm. radius 
exerts a force of one dyne on a unit magnetic pole at the center of the arc. 
The product of current strength and time is known as the quantity of 
electricity; it has the same dimensions as electric charge. The e.m. unit 
of charge or quantity of electricity is thus 3 X 10 10 larger than the corre- 



4 INTRODUCTION 

spending e.s. unit. The product of quantity of electricity and potential 
or E.M.F. is equal to work, and if the same unit of work, or energy, is 
adopted in each case, the e.m. unit of potential must be smaller than the 
e.s. unit in the ratio of 1 to 3 X 10 10 . When one e.m. unit of potential 
difference exists between two points, one erg of work must be expended 
to transfer one e.m. unit of charge, or quantity of electricity, from one 
point to the other; the e.s. unit of potential is defined in an exactly 
analogous manner in terms of one e.s. unit of charge. 

The e.m. and e.s. units described above are not all of a convenient 
magnitude for experimental purposes, and so a set of practical units have 
been defined. The practical unit of current, the ampere, often abbrevi- 
ated to " amp.," is one-tenth the e.m. (c.g.s.) unit, and the corresponding 
unit of charge or quantity of electricity is the coulomb ; the latter is the 
quantity of electricity passing when one ampere flows for one second. 
The practical unit of potential or E.M.F. is the volt, defined as 10 8 e.m. 
units. Corresponding to these practical units of current and E.M.F. there 
is a unit of electrical resistance; this is called the ohm, and it is the re- 
sistance of a conductor through which a current of one ampere passes 
when the potential difference between the ends is one volt. With these 
units of current, E.M.F. and resistance it is possible to write Ohm's law 
in the form 

volts 



By utilizing the results given above for the relationships between 
e.m., e.s. and practical units, it is possible to draw up a table relating 
the various units to each other. Since the practical units are most fre- 
quently employed in electrochemistry, the most useful method of ex- 
pressing the connection between the various units is to give the number 
of e.m. or e.s. units corresponding to one practical unit: the values are 
recorded in Table I. 

TABLE I. CONVERSION OF ELECTRICAL UNITS 

Practical Equivalent in 

Unit e.m.u. e.s.u. 

Current Ampere 10" 1 3 X 10 fl 

Quantity or Charge Coulomb 10~ l 3 X 10' 

Potential or E.M.F. Volt 10 8 (300)~ l 

International Units. The electrical units described in the previous 
section are defined in terms of quantities which cannot be easily estab- 
lished in the laboratory, and consequently an International Committee 
(1908) laid down alternative definitions of the practical units of elec- 
tricity. The international ampere is defined as the quantity of electricity 
which flowing for one second will cause the deposition of 1.11800 milli- 
grams of silver from a solution of a silver salt, while the international ohm 
is the resistance at c. of a column of mercury 106.3 cm. long, of uniform 
cross-section, weighing 14.4521 g. The international volt is then the 



ELECTRICAL ENERGY 5 

difference of electrical potential, or E.M.F., required to maintain a current 
of one international ampere through a system having a resistance of one 
international ohm. Since the international units were defined it has 
been found that they do not correspond exactly with those defined above 
in terms of the c.g.s. system; the latter are thus referred to as absolute 
units to distinguish them from the international units. The international 
ampere is 0.99986 times the absolute ampere, and the international ohm 
is 1.00048 times the absolute ohm, so that the international volt is 1.00034 
times the absolute practical unit.* 

Electrical Energy. As already seen, the passage of electricity through 
a conductor is accompanied by the liberation of heat; according to the 
first law of thermodynamics, or the principle of conservation of energy, 
the heat liberated must be exactly equivalent to the electrical energy 
expended in the conductor. Since the heat can be measured, the value 
of the electrical energy can be determined and it is found, in agreement 
with anticipation, that the heat liberated by the current in a given con- 
ductor is proportional to the quantity of electricity passing and to the 
difference of potential at the extremities of the conductor. The practical 
unit of electrical energy is, therefore, defined as the energy developed 
when one coulomb is passed through a circuit by an E.M.F. of one volt; 
this unit is called the volt-coulomb, and it is evident from Table I that 
the absolute volt-coulomb is equal to 10 7 ergs, or one joule. It follows, 
therefore, that if a current of / amperes is passed for t seconds through a 
conductor under the influence of a potential of E volts, the energy liber- 
ated (Q) will be given by 

Q = Elt X 10 7 ergs, (3) 

or, utilizing Ohm's law, if R is the resistance of the conductor, 

Q = PRt X 10 7 ergs. (4) 

These results are strictly true only if the ampere, volt and ohm are 
in absolute units; there is a slight difference if international units are 
employed, the absolute volt-coulomb or joule being different from the 
international value. The United States Bureau of Standards has recom- 
mended that the unit of heat, the calorie, should be defined as the equiva- 
lent of 4.1833 international joules, and hence 

Elt 

Q = j^gjj calories, (5) 

where E and / are now expressed in international volts and amperes, 
respectively. Alternatively, it may be stated that one international 
volt-coulomb is equivalent to 0.2390 standard calorie. 

* These figures are obtained from the set of consistent fundamental constants 
recommended by Birge (1941); slightly different values are given in the International 
Critical Tables. 



6 INTRODUCTION 

Classification of Conductors. All forms of matter appear to be able 
to conduct the electric current to some extent, but the conducting powers 
of different substances vary over a wide range; thus silver, one of the 
best conductors, is 10 24 times more effective than paraffin wax, which is 
one of the poorest conductors. It is not easy to distinguish sharply 
between good and bad conductors, but a rough division is possible; the 
systems studied in electrochemistry are generally good conductors. 
These may be divided into three main categories; they are: (a) gaseous, 
(6) metallic and (c) electrolytic. 

Gases conduct electricity with difficulty and only under the influence 
of high potentials or if exposed to the action of certain radiations. Metals 
are the best conductors, in general, and the passage of current is not 
accompanied by any movement of matter; it appears, therefore, that the 
electricity is carried exclusively by the electrons, the atomic nuclei re- 
maining stationary. This is in accordance with modern views which 
regard a metal as consisting of a relatively rigid lattice of ions together 
with a system of mobile electrons. Metallic conduction, or electronic 
conduction) as it is often called, is not restricted to pure metals, for it is 
a property possessed by most alloys, carbon and certain solid salts 
and oxides. 

Electrolytic conductors, or electrolytes, are distinguished by the fact 
that passage of an electric current through them results in an actual 
transfer of matter; this transfer is manifested by changes of concentration 
and frequently, in the case of electrolytic solutions, by the visible sepa- 
ration of material at the points where the current enters and leaves the 
solution. Electrolytic conductors are of two main types; there are, 
first, substances which conduct elcctrolytically in the pure state, such 
as fused salts and hydrides, the solid halides of silver, barium, lead and 
some other metals, and the a-form of silver sulfide. Water, alcohols, 
pure acids, and similar liquids are very poor conductors, but they must 
be placed in this category. The second class of electrolytic conductors 
consists of solutions of one or more substances; this is the type of con- 
ductor with which the study of electrochemistry is mainly concerned. 
The most common electrolytic solutions are made by dissolving a salt, 
acid or base in water; other solvents may be used, but the conducting 
power of the system depends markedly on their nature. Conducting sys- 
tems of a somewhat unusual type are lithium carbide and alkaline 
earth nitrides dissolved in the corresponding hydride, and organic acid 
amides and mtro-compounds in liquid ammonia or hydrazine. 

The distinction between electronic and electrolytic conductors is not 
sharp, for many substances behave as mixed conductors; that is, they 
conduct partly electronically and partly electrolytically. Solutions of 
the alkali and alkaline earth metals in liquid ammonia are apparently 
mixed conductors, and so also is the -form of silver sulfide. Fused 
cuprous sulfide conducts electronically, but a mixture with sodium or 
ferrous sulfide also exhibits electrolytic conduction; a mixture with nickel 



THE PHENOMENA AND MECHANISM OP ELECTROLYSIS 7 

sulfide is, however, a pure electronic conductor. Although pure metals 
conduct electronically, conduction in certain liquid alloys involves the 
transfer of matter and appears to be partly electrolytic in nature. Some 
materials conduct electronically at one temperature and electrolytically 
at another; thus cuprous bromide changes its method of conduction 
between 200 and 300. 

The Phenomena and Mechanism of Electrolysis. The materials, 
generally small sheets of metal, which are employed to pass an electric 
current through an electrolytic solution, are called electrodes; the one 
at which the positive current enters is referred to as the positive electrode 
or anode, whereas the electrode at which current leaves is called the 
negative electrode, or cathode. The passage of current through solu- 
tions of salts of such metals as zinc, iron, nickel, cadmium, lead, copper, 
silver and mercury results in the liberation of these metals at the cathode; 
from solutions of salts of the very base metals, e.g., the alkali and alka- 
line earth metals, and from solutions of acids the substance set free is 
hydrogen gas. If the anode consists of an attackable metal, such as one 
of those just enumerated, the flow of the current is accompanied by the 
passage of the metal into solution. When the anode is made of an inert 
metal, e.g., platinum, an element is generally set free at this electrode; 
from solutions of nitrates, sulfates, phosphates, etc., oxygen gas is liber- 
ated, whereas from halide solutions, other than fluorides, the free halogen 
is produced. The decomposition of solutions by the electric current, 
resulting in the liberation of gases or metals, as described above, is known 
as electrolysis. 

The first definite proposals concerning the mechanism of electrolytic 
conduction and electrolysis were made by Grotthuss (1800) ; he suggested 
that the dissolved substance consisted of particles with positive and 
negative ends, these particles being _ _L 

distributed in a random manner 
throughout the solution. When a 
potential was applied it was believed 



q f+q 



3- E3 E3 E3 E3 +E: 
ED ED E3 ED ED 



ii 



in 



IV 



that the particles (molecules) became 
oriented in the form of chains with 
the positive parts pointing in one di- In 
rection and the negative parts in the 
opposite direction (Fig. 1, I). It 

was supposed that the positive elec- T. * i. . * 

, , */. , ., '. , e FIQ. 1. Mechanism of 

trode attracts the negative part of Orotthuss conduction 

one end particle in the chain, resulting 

in the liberation of the corresponding material, e.g., oxygen in the elec- 
trolysis of water. Similarly, the negative electrode attracts the positive 
portion of the particle, e.g., the hydrogen of water, at the other end of 
the chain, and sets it free (Fig. 1, II). The residual parts of the end 
units were then imagined to exchange partners with adjacent molecules, 
this interchange being carried on until a complete series of new particles 



8 



INTRODUCTION 



is formed (Fig. 1, III). These are now rotated by the current to give 
the correct orientation (Fig. 1, IV), followed by their splitting up, and 
so on. The chief objection to the theory of Grotthuss is that it would 
require a relatively high B.M.F., sufficient to break up the molecules, 
before any appreciable current was able to flow, whereas many solutions 
can be electrolyzed by the application of quite small potentials. Al- 
though the proposed mechanism has been discarded, as far as most 
electrolytic conduction is concerned, it will be seen later (p. 66) that a 
type of Grotthuss conduction occurs in solutions of acids and bases. 

In order to account for the phenomena observed during the passage 
of an electric current through solutions, Faraday (1833) assumed that 
the flow of electricity was associated with the movement of particles of 
matter carrying either positive or negative charges. These charged 
particles were called ions ; the ions carrying positive charges and moving 

in the direction of the current, i.e., 
towards the cathode, were referred 
to as cations, and those carrying a 
negative charge and moving in the 
opposite direction, i.e., towards 
Cathode the anode, were called onions * (see 
Fig. 2). The function of the ap- 
plied E.M.F. is to direct the ions 
towards the appropriate electrodes 
where their charges are neutralized 
and they are set free as atoms or 
molecules. It may be noted that 
since hydrogen and metals are dis- 
charged at the cathode, the metal- 
lic part of a salt or base and the hydrogen of an acid form cations and 
carry positive charges. The acidic portion of a salt and the hydroxyl 
ion of a base consequently carry negative charges and constitute the 
anions. 

Although Faraday postulated the existence of charged material par- 
ticles, or ions, in solution, he offered no explanation of their origin: it was 
suggested, however, by Clausius (1857) that the positive and negative 
parts of the solute molecules were not firmly connected, but were each 
in a state of vibration that often became vigorous enough to cause the 
portions to separate. These separated charged parts, or ions, were be- 
lieved to have relatively short periods of free existence; while free they 
were supposed to carry the current. According to Clausius, a small 
fraction only of the total number of dissolved molecules was split into 

* The term "ion" is derived from a Greek word moaning "wanderer" or "traveler," 
the prefixes ana and cata meaning "up" and "down," respectively; the anion is thus 
the ion moving up, and the cation that moving down the potential gradient. These 
terms, as well as electrode, anode and cathode, were suggested to Faraday by Whewell 
(1834); see Oesper and Speter, Scientific Monthly, 45, 535 (1937). 




Direction of Election Flow 

FIG. 2. Illustration of 

electrochemical terms 



THE ELECTBOLYTIC DISSOCIATION THEORY 9 

ions at any instant, but sufficient ions were always available for carrying 
the current and hence for discharge at the electrodes. Since no electrical 
energy is required to break up the molecules, this theory is in agreement 
with the fact that small E.M.P/S are generally adequate to cause elec- 
trolysis to occur; the applied potential serves merely to guide the ions to 
the electrodes where their charges are neutralized. 

The Electrolytic Dissociation Theory. 1 From his studies of the con- 
ductances of aqueous solutions of acids and their chemical activity, 
Arrhenius (1883) concluded that an electrolytic solution contained two 
kinds of solute molecules; these were supposed to be " active" molecules, 
responsible for electrical conduction and chemical action, and inactive 
molecules, respectively. It was believed that when an acid, base or salt 
was dissolved in water a considerable portion, consisting of the so-called 
active molecules, was spontaneously split up, or dissociated, into positive 
and negative ions; it was suggested that these ions are free to move 
independently and are directed towards the appropriate electrodes under 
the influence of an electric field. The proportion of active, or dissoci- 
ated, molecules to the total number of molecules, later called the "degree 
of dissociation," was considered to vary with the concentration of the 
electrolyte, and to be equal to unity in dilute solutions. 

This theory of electrolytic dissociation, or the ionic theory, attracted 
little attention until 1887 when van't Hoff's classical paper on the theory 
of solutions was published. The latter author had shown that the ideal 
gas law equation, with osmotic pressure in place of gas pressure, was 
applicable to dilute solutions of non-electrolytes, but that electrolytic 
solutions showed considerable deviations. For example, the osmotic 
effect, as measured by depression of the freezing point or in other ways, 
of hydrochloric acid, alkali chlorides and hydroxides was nearly twice as 
great as the value to be expected from the gas law equation; in some 
cases, e.g., barium hydroxide, and potassium sulfate and oxalate, the 
discrepancy was even greater. No explanation of these facts was offered 
by van't Iloff, but he introduced an empirical factor i into the gas law 
equation for electrolytic solutions, thus 

n = iRTc, 

where II is the observed osmotic pressure of the solution of concentra- 
tion c; the temperature is T, and R is the gas constant. According to 
this equation, the van't Hoff factor i is equal to the ratio of the experi- 
mental osmotic effect to the theoretical osmotic effect, based on the ideal 
gas laws, for the given solution. Since the osmotic effect is, at least 
approximately, proportional to the number of individual molecular par- 
ticles, a value of two for the van't Hoff factor means that the solution 
contains about twice the number of particles to be expected. This result 

1 Arrhenius, J. Chem. Soc., 105, 1414 (1914); Walker, ibid., 1380 (1928). 



10 INTRODUCTION 

is clearly in agreement with the views of Arrhenius, if the ions are re- 
garded as having the same osmotic effect as uncharged particles. 

The concept of "active molecules/' which was part of the original 
theory, was later discarded by Arrhenius as being unnecessary; he sug- 
gested that whenever a substance capable of yielding a conducting 
solution was dissolved in water, it dissociated spontaneously into ions, 
the extent of the dissociation being very considerable with salts and with 
strong acids and bases, especially in dilute solution. Thus, a molecule 
of potassium chloride should, according to the theory of electrolytic 
dissociation, be split up into potassium and chloride ions in the following 
manner: 

KC1 = K+ + Cl-.' 

If dissociation is complete, then each " molecular particle " of solid potas- 
sium chloride should give two particles in solution; the osmotic effect 
will thus approach twice the expected value, as has actually been found. 
A bi-univalent salt, such as barium chloride, will dissociate spontaneously 
according to the equation 

BaCl 2 = Ba++ + 2C1~, 

and hence the van't Hoff factor should be approximately 3, in agreement 
with experiment. 

Suppose a solution is made up by dissolving m molecules in a gr/en 
volume and a is the fraction of these molecules dissociated into ions; if 
each molecule produces v ions on dissociation, there will be present in 
the solution m(l a) undissociated molecules and vma ions, making a 
total of m ma + vma particles. If the van't Hoff factor is equal to 
the ratio of the number of molecular particles actually present to the 
number that would have been in the solution if there had been no dis- 
sociation, then 



m ma + vma 
i = - = 1 a + va: 

m ' 



(6) 



Since the van't Hoff factor is obtainable from freezing-point, or analo- 
gous, measurements, the value of or, the so-called degree of dissociation, 
in the given solution can be calculated from equation (6). An alterna- 
tive method of evaluating a, using conductance measurements (see p. 51), 
was proposed by Arrhenius (1887), and he showed that the results ob- 
tained by the two methods were in excellent agreement: this agreement 
was accepted as strong evidence for the theory of electrolytic dissocia- 
tion, which has played such an important role in the development of 
electrochemistry. 

It is now known that the agreement referred to above, which con- 
vinced many scientists of the value of the Arrhenius theory, was to a 



EVIDENCE FOR THE IONIC THEORY 11 

great extent fortuitous; the conductance method for calculating the 
degree of dissociation is not applicable to salt solutions, and such solu- 
tions would, in any case, not be expected to obey the ideal gas law 
equation. Nevertheless, the theory of electrolytic dissociation, with 
certain modifications, is now universally accepted; it is believed that 
when a solute, capable of forming a conducting solution, is dissolved in 
a suitable solvent, it dissociates spontaneously into ions. If the solute 
is a salt or a strong acid or base the extent of dissociation is very con- 
siderable, it being almost complete in many cases provided the solution 
is not too concentrated; substances of this kind, which are highly dis- 
sociated and \\hich give good conducting solutions in water, are called 
strong electrolytes. Weak acids and weak bases, e.g., amines, phenols, 
most carboxylic acids and some inorganic acids and bases, such as hydro- 
cyanic acid and ammonia, and a few salts, e.g., mercuric chloride and 
cyanide, are dissociated only to a small extent at reasonable concentra- 
tions; these compounds constitute the weak electrolytes.* Salts of weak 
acids or bases, or of both, are generally strong electrolytes, in spite of 
the fact that one or both constituents are weak. These results are in 
harmony with modern developments of the ionic theory, as will be evident 
in later chapters. As is to be expected, it is impossible to classify all 
electrolytes as "strong" or "weak," although this forms a convenient 
rough division which is satisfactory for most purposes. Certain sub- 
stances, e.g., trichloroacetic acid, exhibit an intermediate behavior, but 
the number of intermediate electrolytes is not large, at least in aqueous 
solution. It may be noted, too, that the nature of the solvent is often 
important; a particular compound may be a strong electrolyte, being 
dissociated to a large extent, in one solvent, but may be only feebly 
dissociated, and hence is a weak electrolyte, in another medium (cf. 
p. 13). 

Evidence for the Ionic Theory. There is hardly any branch of elec- 
trochemistry, especially in its quantitative aspects, which does not pro- 
vide arguments in favor of the theory of electrolytic dissociation; without 
the ionic concept the remarkable systems tization of the experimental 
results which has been achieved during the past fifty years would cer- 
tainly not have been possible. It is of interest, however, to review 
briefly some of the lines of evidence which support the ionic theory. 

Although exception may be taken to the quantitative treatment given 
by Arrhenius, the fact of the abnormal osmotic properties of electrolytic 
solutions still remains; the simplest explanation of the high values can 
be given by postulating dissociation into ions. This, in conjunction with 
the ability of solutions to conduct the electric current, is one of the 
strongest arguments for the ionic theory. Another powerful argument is 

* Strictly speaking, the term "electrolyte" should refer to the conducting system 
as a whole, but it is also frequently applied to the solute; the word "ionogen," i.e., 
producer of ions, has been suggested for the latter [see, for example, Blum, Trans. 
Electrochem. Soc., 47, 125 (1925)], but this has not come into general use. 



12 INTRODUCTION 

based on the realization in recent years, as a result of X-ray diffraction 
studies, that the structural unit of solid salts is the ion rather than the 
molecule. That is to say, salts are actually ionized in the solid state, and 
it is only the restriction to movement in the crystal lattice that prevents 
solid salts from being good electrical conductors. When fused or dis- 
solved in a suitable solvent, the ions, which are already present, can move 
relatively easily under the influence of an applied E.M.F., and conductance 
is observed. The concept that salts consist of ions held together by 
forces of electrostatic attraction is also in harmony with modern views 
concerning the nature of valence. 

Many properties of electrolytic solutions are additive functions of the 
properties of the respective ions; this is at once evident from the fact 
that the chemical properties of a salt solution are those of its constituent 
ions. For example, potassium chloride in solution has no chemical reac- 
tions which are characteristic of the compound itself, but only those of 
potassium and chloride ions. These properties are possessed equally by 
almost all potassium salts and all chlorides, respectively. Similarly, the 
characteristic chemical properties of acids and alkalis, in aqueous solu- 
tion, are those of hydrogen and hydroxyl ions, respectively. Certain 
physical properties of electrolytes are also additive in nature; the most 
outstanding example is the electrical conductance at infinite dilution. 
It will be seen in Chap. II that conductance values can be ascribed 
to all ions, and the appropriate conductance of any electrolyte is equal 
to the sum of the values for the individual ions. The densities of elec- 
trolytic solutions have also been found to be additive functions of the 
properties of the constituent ions. The catalytic effects of various acids 
and bases, and of mixtures with their salts, can be accounted for by 
associating a definite catalytic coefficient with each type of ion; since 
undissociated molecules often have appreciable catalytic properties due 
allowance must be made for their contribution. 

Certain thermal properties of electrolytes are in harmony with the 
theory of ionic dissociation; for example, the heat of neutralization of a 
strong acid by an equivalent amount of a strong base in dilute solution 
Is about 13.7 kcal. at 20 irrespective of the exact nature of the acid or 
base. 2 If the acid is hydrochloric acid and the base is sodium hydroxide, 
then according to the ionic theory the neutralization reaction should be 
written 

(H+ + C1-) + (Na+ + OH-) = (Na+ + Cl~) + H 2 O, 

the acid, base and the resulting salt being highly dissociated, whereas 
the water is almost completely undissociated. Since Na+ and Cl~ ap- 
pear on both sideb of this equation, the essential reaction is 

H+ + OH- = H 2 O, 

8 Richards and Rowe, /. Am. Chem. Soc., *4, 684 (1922); see also, Lambert and 
Gillespie, ibid., 53, 2632 (1931); Rossini, /. Res. Nat. Bur. Standards, 6, 847 (1931); 
Pitzer, J. Am. Chem. Soc., 59, 2365 (1937). 



INFLUENCE OF THE SOLVENT ON DISSOCIATION 13 

and this is obviously independent of the particular acid or base em- 
ployed: the heat of neutralization would thus be expected to be constant. 
It is of interest to mention that the heat of the reaction between hydro- 
gen and hydroxyl ions in aqueous solution has been calculated by an 
entirely independent method (see p. 344) and found to be almost identical 
with the value obtained from neutralization experiments. The heat of 
neutralization of a weak acid or a weak base is generally different from 
13.7 kcal., since the acid or base must dissociate completely in order that 
it may be neutralized and the process of ionization is generally accom- 
panied by the absorption of heat. 

Influence of the Solvent on Dissociation. 8 The nature of the solvent 
often plays an important part in determining the degree of dissociation 
of a given substance, and hence in deciding whether the solution shall 
behave as a strong or as a weak electrolyte. Experiments have been 
made on solutions of tetraisoamylammonium nitrate in a series of mix- 
tures of water and dioxane (see p. 54). In the water-rich solvents the 
system behaves like a strong electrolyte, but in the solvents containing 
relatively large proportions of dioxane the properties are essentially those 
of a weak electrolyte. In this case, and in analogous cases where the 
solute consists of units which are held together by bonds that are almost 
exclusively electrovalent in character, it is probable that the dielectric 
constant is the particular property of the solvent that influences the 
dissociation (cf. Chaps. II and III). The higher the dielectric constant 
of the medium, the smaller is the electrostatic attraction between the 
ions and hence the greater is the probability of their existence in the free 
state. Since the dielectric constant of water at 25 is 78.6, compared 
with a value of about 2.2 for dioxane, the results described above can be 
readily understood. 

It should be noted, however, that there are many instances in which 
the dielectric constant of the solvent plays a secondary part: for example, 
hydrogen chloride dissolves in ethyl alcohol to form a solution which 
behaves as a strong electrolyte, but in nitrobenzene, having a dielectric 
constant differing little from that of alcohol, the solution is a weak elec- 
trolyte. As will be seen in Chap. IX the explanation of this difference 
lies in the ability of a molecule of ethyl alcohol to combine readily with 
a bare hydrogen ion, i.e., a proton, to form the ion C 2 H 6 OHt, and this 
represents the form in which the hydrogen ion exists in the alcohol 
solution. Nitrobenzene, however, does not form such a combination to 
any great extent; hence the degree of dissociation of the acid is small 
and the solution of hydrogen chloride behaves as a weak electrolyte. 
The ability of oxygen compounds, such as ethers, ketones and even 
sugars, to accept a proton from a strongly acidic substance, thus forming 
an ion, e.g., R 2 OH+ or R 2 COH+, accounts for the fact that solutions of 
such compounds in pure sulfuric acid or in liquid hydrogen fluoride are 
relatively strong electrolytes. 

See, Glasstone, "The Electrochemistry of Solutions/' 1937, p. 172. 



14 INTRODUCTION 

Another aspect of the formation of compounds and its influence on 
electrolytic dissociation is seen in connection with substituted ammonium 
salts of the type RaNHX; although they are strong electrolytes in hy- 
droxylic solvents, e.g., in water and alcohols, they are dissociated to only 
a small extent in nitrobenzene, nitromethane, acetone and acetonitrile. 
It appears that in the salts under consideration the hydrogen atom can 
act as a link between the nitrogen atom and the acid radical X, so that 
the molecule RsN-H-X exists in acid solution. If the solvent S is of 
such a nature, however, that its molecules tend to form strong hydrogen 
bonds, it can displace the X~ ions, thus 

R 3 N-H-X + S ^ K 3 N-H-S+ + X~ 

so that ionization of the salt is facilitated. Hydroxylic solvents, in virtue 
of the type of oxygen atom which they contain, form hydrogen bonds 
more readily than do nitro-compounds, nitriles, etc.; the difference in 
behavior of the two groups of solvents can thus be understood. 

Salts of the type R 4 NX function as strong electrolytes in both groups 
of solvents, since the dielectric constants are relatively high, and the 
question of compound formation with the solvent is of secondary impor- 
tance. The fact that salts of different types show relatively little differ- 
ence of behavior in hydroxylic solvents has led to these substances being 
called levelling solvents. On the other hand, solvents of the other group, 
e.g., nitro-compounds and nitriles, are referred to as differentiating 
solvents because they bring out the differences between salts of different 
types. The characteristic properties of the levelling solvents are due 
partly to their high dielectric constants and partly to their ability to act 
both as electron donors and acceptors, so that they are capable of forming 
compounds with either anions or cations. 

The formation of a combination of some kind between the ion and a 
molecule of solvent, known as solvation, is an important factor in en- 
hancing the dissociation of a given electrolyte. The solvatcd ions are 
relatively large and hence their distance of closest approach is very much 
greater than the bare unsolvated ions. It will be seen in Chap. V that 
when the distance between the centers of two oppositely charged ions is 
less than a certain limiting value the system behaves as if it consisted of 
undissociated molecules. The effective degree of dissociation thus in- 
creases as the distance of closest approach becomes larger; hence solvation 
may be of direct importance in increasing the extent of dissociation of a 
salt in a particular solvent. It may be noted that solvation does riot 
necessarily involve a covalent bond, e.g., as is the case in CuCNTIs)^ 
and Cu(H 2 0)t 4 "; there is reason for believing that solvation is frequently 
electrostatic in character and is due to the orientation of solvent molecule 
dipoles about the ion. A solvent with a large dipole moment will thus 
tend to facilitate solvation and it will consequently increase the degree of 
dissociation. 



FARADAY'S LAWS OP ELECTROLYSIS 15 

It was mentioned earlier in this chapter that acid amides and nitro- 
compounds form conducting solutions in liquid ammonia and hydrazine; 
the ionization in these cases is undoubtedly accompanied by, and is 
associated with, compound formation between solute and solvent. The 
same is true of triphenylmethyl chloride which is a fair electrolytic con- 
ductor when dissolved in liquid sulfur dioxide; it also conducts to some 
extent in nitromethane, nitrobenzene and acetone solutions. In chloro- 
form and benzene, however, there is no compound formation and no 
conductance. The electrolytic conduction of triphenylmethyl chloride in 
fused aluminum chloride, which is itself a poor conductor, appears to 
be due to the reaction 

Ph 3 CCl + A1C1 3 = Pb 3 C+ + AlClr; 

this process is not essentially different from that involved in the ioniza- 
tion of an acid, where the II f ion, instead of a Cl~ ion, is transferred 
from one molecule to another. 

Faraday's Laws of Electrolysis. During the years 1833 and 1834, 
Faraday published the results of an extended series of investigations on 
the relationship between the quantity of electricity passing through a 
solution and the amount of metal, or other substance, liberated at the 
electrodes: the conclusions may be expressed in the form of the two 
following laws. 

I. The amount of chemical decomposition produced by a current is 
proportional to the quantity of electricity passing through the electro- 
lytic solution. 

II. The amounts of different substances liberated by the same quan- 
tity of electricity are proportional to their chemical equivalent weights. 

The first law can be tested by passing a current of constant strength 
through a given electrolyte for various periods of time and determining 
the amounts of material deposited, on the cathode, for example; the 
weights should be proportional to the time in each case. Further, the 
time may be kept constant and the current varied; in these experiments 
the quantity of deposit should be proportional to the current strength. 
The second law of electrolysis may be confirmed by passing the same 
quantity of electricity through a number of different solutions, e.g., 
dilute sulfuric acid, silver nitrate and copper sulfate; if a current of one 
ampere flows for one hour the weights liberated at the respective cathodes 
should be 0.0379 gram of hydrogen, 4.0248 grams of silver and 1.186 
grams of copper. These quantities are in the ratio of 1.008 to 107.88 to 
31.78, which is the ratio of the equivalent weights. As the result of 
many experiments, in both aqueous and non-aqueous media, some of 
which will be described below, much evidence has been obtained for the 
accuracy of Faraday's laws of electrolysis within the limits of reasonable 
experimental error. Apart from small deviations, whic^gai^ J>e readily 
explained by the difficulty of obtaining pure deposit^ A^ WWi?* aiia ~ 



16 INTRODUCTION 

lytical problems, there are a number of instances of more serious apparent 
exceptions to the laws of electrolysis. The amount of sodium liberated 
in the electrolysis of a solution of the metal in liquid ammonia is less 
than would be expected. It must be remembered, however, that Fara- 
day's laws are applicable only when the whole of the conduction is 
electrolytic in character; in the sodium solutions in liquid ammonia some 
of the conduction is electronic in nature. The quantities of metal de- 
posited from solutions of lead or antimony in liquid ammonia containing 
sodium are in excess of those required by the laws of electrolysis; in these 
solutions the motals exist in the form of complexes and the ions are quite 
different from those present in aqueous solution. It is consequently not 
possible to calculate the weights of the deposits to be expected from 
Faraday's laws. 

The applicability of the laws has been confirmed under extreme con- 
ditions: for example, Richards and Stull (1902) found that a given quan- 
tity of electricity deposited the same weight of silver, within 0.005 per 
cent, from an aqueous solution of silver nitrate at 20 and from a solution 
of this salt in a fused mixture of sodium and potassium nitrates at 260. 
The experimental results are quoted in Table II. 

TABLE II. COMPARISON OF SILVER DEPOSITS AT 20 AND 260 

Deposit at 20 Deposit at 260 Difference 

1.14916 g. 1.14919 g. 0.003 per cent 

1.12185 1.12195 0.009 

1.10198 1.10200 0.002 

A solution of silver nitrate in pyridine at 55 also gives the same 
weight of silver on the cathode as does an aqueous solution of this salt 
at ordinary temperatures. Pressures up to 1500 atmospheres have no 
effect on the quantity of silver deposited from a solution of silver nitrate 
in water. 

Faraday's law holds for solid electrolytic conductors as well as for 
fused electrolytes and solutions; this is shown by the results of Tubandt 
and Eggert (1920) on the electrolysis of the cubic form of silver iodide 
quoted in Table III. The quantities of silver deposited in an ordinary 

TABLE III. APPLICATION OP FARADAY'S LAWS TO SOLID SILVER IODIDE 

Ag deposited Ag deposited Ag lost 

Temp. Current in coulometer on cathode from anode 

150 0.1 amp. 0.8071 g. 0.8072 g. 0.8077 g. 

150 0.1 0.9211 0.9210 0.9217 

400 0.1 0.3997 0.3991 0.4004 

400 0.4 0.4217 0.4218 0.4223 

silver coulometer in the various experiments are recorded, together with 
the amounts of silver gained by the cathode and lost by the anode, 
respectively, when solid silver iodide was used as the electrolyte. 

The Faraday and its Determination. The quantity of electricity 
required to liberate 1 equiv. of any substance should, according to the 



THE FARADAY AND ITS DETERMINATION 



17 



second of Faraday's laws, be independent of its nature; this quantity is 
called the faraday; it is given the symbol F and, as will be seen shortly, 
is equal to 96,500 coulombs, within the limits of experimental error. 
If e is the equivalent weight of any material set free at an electrode, then 
96,500 amperes flowing for one second liberate e grams of this substance; 
it follows, therefore, from the first of Faraday's laws, that 7 amperes 
flowing for t seconds will cause the deposition of w grams, where 



w = 



lie 
96,500* 



(7) 



If the product It is unity, i.e., the quantity of electricity passed is 1 
coulomb, the weight of substance deposited is e/96,500; the result is 
known as the electrochemical equivalent of the deposited element. If 
this quantity is given the symbol e, it follows that 



w = Ite. 



(7o) 



The electrochemical equivalents of some of the more common elements 
are recorded in Table IV; * since the value for any given element depends 



TABLE IV. ELECTROCHEMICAL EQUIVALENTS IN MILLIGRAMS PER COULOMB 



Element Valence 



Hydrogen 

Oxygen 

Chlorine 

Iron 

Cobalt 

Nickel 



e 

0.01045 
0.08290 
0.36743 
0.2893 
0.3054 
0.3041 



Element 
Copper 
Bromine 
Cadmium 
Silver 
Iodine 
Mercury 



Valence 
2 
1 
2 
1 
1 
2 



0.3294 
0.8281 
0.5824 
1.1180 
1.3152 
1.0394 



on the valence of the ions from which it is being deposited, the actual 
valence for which the results were calculated is given in each case. 

The results given above, and equation (7) or (7a), are the quantita- 
tive expression of Faraday's laws of electrolysis; they can be employed 
either to calculate the weight of any substance deposited by a given 
quantity of electricity, or to find the quantity of electricity passing 
through a circuit by determining the weight of a given metal set free by 
electrolysis. The apparatus used for the latter purpose was at one time 
referred to as a "voltameter," but the name coulometer, i.e., coulomb 
measurer, proposed by Richards and Heimrod (1902), is now widely 
employed. 

The most accurate determinations of the faraday have been made 
by means of the silver coulometer in which the amount of pure silver 
deposited from an aqueous solution of silver nitrate is measured. The 
first reliable observations with the silver coulometer were those of 
Kohlrausch in 1886, but the most accurate measurements in recent years 
were made by Smith, Mather and Lowry (1908) at the National Physical 

* For a complete list of electrochemical equivalents and for other data relating to 
Faraday's laws, see Roush, Trans. Electrochem. Soc., 73, 285 (1938). 



18 



INTRODUCTION 



Laboratory in England, by Richards and Anderegg (1915-16) at Harvard 
University, and by Rosa and Vinal, 4 and others, at the National Bureau 
of Standards in Washington, D. C. (1914-16). The conditions for ob- 
taining precise results have been given particularly by Rosa and Vinal 
(1914) : these are based on the necessity of insuring purity of the silver 
nitrate, of preventing particles of silver from the anode, often known as 
the " anode slime," from falling on to the cathode, and of avoiding the 
inclusion of water and silver nitrate in the deposited silver. 

The silver nitrate is purified by repeated crystallization from acidified 
solutions, followed by fusion. The purity of the salt is proved by the 
absence of the so-called " volume effect," the weight of silver deposited 
by a given quantity of electricity being independent of the volume of 
liquid in the coulometer: this moans that no extraneous impurities are 
included in the deposit. The solution of silver nitrate employed for the 
actual measurements should contain between 10 and 20 g. of the salt in 
100 cc.; it should be neutral or slightly acid to methyl red indicator, after 
removal of the silver by neutral potassium chloride, both at the beginning 
and end of the electrolysis. The anode should be of pure silver with an 
area as large as the apparatus permits; the current density at the anode 
should not exceed 0.2 amp. per sq. cm. To prevent the anode slime 




II 



FIG. 3. Silver coulometers 



from reaching the cathode, the former electrode (A in Fig. 3), is inserted 
in a cup of porous porcelain, as shown at B in Fig. 3, 1 (Richards, 1900), 
or is surrounded by a glass vessel, B in Fig. 3, II (Smith, 1908). The 
cathode is a platinum dish or cup (C) and its area should be such as to 
make the cathodic current density less than 0.02 amp. per sq. cm. After 
electrolysis the solution is removed by a siphon, the deposited silver is 
washed thoroughly and then the platinum dish and deposit are dried at 
150 and weighed. The gain in weight gives the amount of silver de- 
posited by the current; if the conditions described are employed, the 
impurities should not be more than 0.004 per cent. 

4 Rosa and Vinal, Bur. Standards Bull, 13, 479 (1936); sec also, Vinal and Bovard, 
J. Am. Chem. Soc., 38, 496 (1916); Bovard arid Hulett, ibid., 39, 1077 (1917). 



THE FARADAY AND ITS DETERMINATION 



19 



If the observations are to be used for the determination of the faraday, 
it is necessary to know exactly the quantity of electricity passed or the 
current strength, provided it is kept constant during the experiment. 
In the work carried out at the National Physical Laboratory the absolute 
value of the current was determined by means of a magnetic balance, 
but at the Bureau of Standards the current strength was estimated from 
the known value of the applied E.M.F., based on the Weston standard 
cell as 1.01830 international volt at 20 (see p. 193), and the measured 
resistance of the circuit. According to the experiments of Smith, Mather 
and Lowry, one absolute coulomb deposits 1.11827 milligrams of silver, 
while Rosa and Vinal (1916) found that one international coulomb de- 
posits 1.1180 milligrams of silver. The latter figure is identical with 
the one used for the definition of the international coulomb (p. 4) and 
since it is based on the agreed value of the E.M.F. of the Weston cell it 
means that these definitions are consistent with one another within the 
limits of experimental accuracy. If the atomic weight of silver is taken 
as 107.88, it follows that 



107.88 
0.0011180 



= 96,494 international coulombs 



If allowance is 



8 5 






D' 



yr 



are required to liberate one gram equivalent of silver. 

made for the 0.004 per cent of impurity 

in the deposit, this result becomes 96,498 

coulombs. Since the atomic weight of 

silver is not known with an accuracy of 

more than about one part in 10,000, the 

figure is rounded off to 96,500 coulombs. 

It follows, therefore, that this quantity 

of electricity is required to liberate 1 

gram equivalent of any substance: hence 

1 faraday = 96,500 coulombs: 

The reliability of this value of the 
faraday has been confirmed by mea- 
surements with the iodine coulometer 
designed by Washburn and Bates, and 
employed by Bates and Vinal. 6 The ap- 
paratus is shown in Fig. 4; it consists of 
two vertical tubes, containing the anode 
(A) and cathode (C) of platinum-indium 
foil, joined by a V-shaped portion. 
A 10 per cent solution of potassium io- 
dide is first placed in the limbs and then 

6 Washburn and Bates, /. Am. Chem. Soc., 34, 1341, 1515 (1912); Bates and Vinal, 
ibid., 36, 916 (1914). 




FIG. 4. Iodine coulometer 
(Wushburn and Bates) 



20 INTRODUCTION 

by means of the filling tubes D and D' a concentrated solution of 
potassium iodide is introduced carefully beneath the dilute solution in 
the anode compartment, and a standardized solution of iodine in potas- 
sium iodide is similarly introduced into the cathode compartment. 
During the passage of current iodine is liberated at the anode while an 
equivalent amount is reduced to iodide ions at the cathode. After the 
completion of electrotysis the anode and cathode liquids are withdrawn, 
through D and D', and titrated with an accurately standardized solution 
of arsenious acid. In this way the amounts of iodine formed at one elec- 
trode and removed at the other can be determined; the agreement 
between the two results provides confirmation of the accuracy of the 
measurements. The results obtained by Bates and Vinal in a number 
of experiments, in which a silver and an iodine coulometer were in series, 
are given in Table V; the first column records the weight of silver de- 

TABLE V. DETERMINATION OF THE FARADAY BY THE IODINE COULOMETER 

Coulombs Passed 

From From Milligrams 

. Silver Iodine Silver E.M.F. and of Iodine 

mg. mg. deposited Resistance per Coulomb Faraday 

4099.03 482224 3666.39 3666.65 1.31526 96,498 

4397.11 5172.73 3933.01 .... 1.31521 96,502 

4105.23 4828.51 3671.94 3671.84 1.31498 96,518 
4123.10 4849.42 3687.92 .. . 1.31495 96,521 
4104.75 4828.60 3671.51 3671.61 1.31515 96,506 

4184.24 4921.30 3742.61 . 1.31494 96,521 
4100.27 4822.47 3667.50 3667.65 1.31492 96,523 
4105.16 4828.44 3671.88 3671.82 1.31498 96,519 

Mean 1.31502 96,514 

posited and the second the mean quantity of iodine liberated or removed; 
in the third column are the number of coulombs passed, calculated from 
the data in the first column assuming the faraday to be 96,494 coulombs, 
and in the fourth are the corresponding values derived from the E.M.F. 
of the cell employed, that of the Weston standard cell being 1.01830 
volt at 25, and the resistance of the circuit. The agreement between 
the figures in these two columns shows that the silver coulometer was 
functioning satisfactorily. The fifth column gives the electrochemical 
equivalent of iodine in milligrams per coulomb, and the last column is 
the value of the faraday, i.e., the number of coulombs required to deposit 
1 equiv. of iodine, the atomic weight being taken as 126.92. 

The faraday, calculated from the work on the iodine coulometer, is 
thus 96,514 coulombs compared with 96,494 coulombs from the silver 
coulometer; the agreement is within the limits of accuracy of the known 
atomic weights of silver and iodine. In view of the small difference 
between the two values of the faraday given above, the mean figure 
96,500 coulombs is probably best for general use. 

Measurement of Quantities of Electricity. Since the magnitude of 
the faraday is known, it is possible, by means of equation (7), to deter- 



MEASUREMENT OP QUANTITIES OF ELECTRICITY 21 

mine the quantity of electricity passing through any circuit by including 
in it a coulometer in which an element of known equivalent weight is 
deposited. Several coulometers, of varying degrees of accuracy and 
convenience of manipulation, have been described. Since the silver and 
iodine coulometers have been employed to determine the faraday, these 
are evidently capable of giving the most accurate results; the iodine 
coulometer is, however, rarely used in practice because of the difficulty 
of manipulation. One of the disadvantages of the ordinary form of the 
silver coulometer is that the deposits are coarse-grained and do not ad- 
here to the cathode; a method of overcoming this is to use an electrolyte 
made by dissolving silver oxide in a solution of hydrofluoric and boric 
acids. 6 

In a simplified form of the silver coulometer, which is claimed to give 
results accurate to within 0.1 per cent, the amount of silver dissolved 
from the anode into a potassium nitrate solution during the passage of 
current is determined volumetrically. 7 

For general laboratory purposes the copper coulometer is the one 
most frequently employed; 8 it contains a solution of copper sulfate, and 
the metallic copper deposited on the cathode is weighted. The chief 
sources of error are attack of the cathode in acid solution, especially in 
the presence of atmospheric oxygen, and formation of cuprous oxide in 
neutral solution. In practice slightly acid solutions are employed and 
the errors are minimized by using cathodes of small area and operating 
at relatively low temperatures; the danger of oxidation is obviated to a 
great extent by the presence of ethyl alcohol or of tartaric acid in the 
electrolyte. The cathode, which is a sheet of copper, is placed midway 
between two similar sheets which act as anodes; the current density at 
the cathode should be between 0.002 and 0.02 ampere per sq. cm. At 
the conclusion of the experiment the cathode is removed, washed with 
water and dried at 100. It can be calculated from equation (7) that one 
coulomb of electricity should deposit 0.3294 milligram of copper. 

In a careful study of the copper coulometer, in which electrolysis was 
carried out at about in an atmosphere of hydrogen, and allowance 
made for the. copper dissolved from the cathode by the acid solution, 
Richards, Collins arid Heimrod (1900) found the results to be within 0.03 
per cent of those obtained from a silver coulometer in the same circuit. 

The electrolytic gas coulometer is useful for the approximate meas- 
urement of small quantities of electricity; the total volume of hydrogen 
and oxygen liberated in the electrolysis of an aqueous solution of sulfuric 
acid or of sodium, potassium or barium hydroxide can be measured, and 
from this the quantity of electricity passed can be estimated. If the 
electrolyte is dilute acid it is necessary to employ platinum electrodes, 

6 von Wartenberg and Schutza, Z. Elektrochem., 36, 254 (1930). 

7 Kisti&kowsky, Z. Elektrochem., 12, 713 (1906). 

8 Datta and Dhar, J. Am. Chem. Soc., 38, 1156 (1916); Matthews and Wark, J. Phys. 
Chem., 35, 2345 (1931). 



22 



INTRODUCTION 




but with alkaline electrolytes nickel electrodes are frequently used. One 

faraday of electricity should liberate one gram equivalent of hydrogen 

at the cathode and an equivalent of oxygen at the anode, i.e., there 

should be produced 1 gram of hydrogen and 8 grams of oxygen. Allow- 
ing for the water vapor present in the liberated gases 
and for the decrease in volume of the solution as the 
water is electrolyzed, the passage of one coulomb of 
electricity should be accompanied by the formation 
of 0.174 cc. of mixed hydrogen and oxygen at S.T.P., 
assuming the gases to behave ideally. 

The mercury coulometer has been employed 
chiefly for the measurement of quantities of elec- 
tricity for commercial purposes, e.g., in electricity 
meters. 9 The form of apparatus used is shown in 
Fig. 5; the anode consists of an annular ring of mer- 
cury (A) surrounding the carbon cathode (C); the 
electrolyte is a solution of mercuric iodide in potas- 
sium iodide. The mercury liberated at the cathode 
falls off, under the influence of gravity, and is col- 
lected in the graduated tube D. From the height 
of the mercury in this tube the quantity of electricity 
passed may be read off directly. When the tube has 
become filled with mercury the apparatus is inverted 
and the mercury flows back to the reservoir J3. In 
actual practice a definite fraction only of the current 
to be measured is shunted through the meter, so that 
the life of the latter is prolonged. The accuracy of 
the mercury electricity meter is said to be within 
1 to 2 per cent. 

A form of mercury coulometer suitable for the 

measurement of small currents of long duration has also been described. 10 
An interesting form of coulometer, for 

which an accuracy of 0.01 per cent has 

been claimed, is the sodium coulometer; 

it involves the passage of sodium ions 

through glass. 11 The electrolyte is fused 

sodium nitrate at 340 and the electrodes 

are tubes of highly conducting glass, elec- 
trical contact being made by means of a 

platinum wire sealed through the glass and 

dipping into cadmium in the cathode, and 

cadmium containing some sodium in the 

anode (Fig. 6). When current is passed, sodium is deposited in the 



FIG. 5. Mercury 
coulometer electricity 
meter 





FIG. 6. 



Sodium roulometer 
(Stewart) 



Hatfield, Z. Elektrochem., 15, 728 (1909); Schulte, ibid., 27, 745 (1921). 

10 Lehfeldt, Phil. Mag., 3, 158 (1902). 

Burt, Phys. Rev., 27, 813 (1926); Stewart, /. Am. Chem. Soc., 53 % 3366 (1931). 



GENERAL APPLICABILITY OP FARADAY'S LAWS 



23 



glass of the cathode and an equal amount moves out of the anode tube. 
From the change in weight the quantity of electricity passing may be 
determined; the anode gives the most reliable results, for with the cathode 
there is a possibility of the loss of silicate ions from the glass. In spite 
of the great accuracy that has been reported, it is doubtful if the sodium 
coulometer as described here will find any considerable application be- 
cause of experimental difficulties; its chief interest lies in the fact that 
it shows Faraday's laws hold under extreme conditions. 

General Applicability of Faraday's Laws. The discussion so far has 
been concerned mainly with the application of Faraday's laws to the 
material deposited at a cathode, but the laws are applicable to all types 
of processes occurring at both anode and cathode. The experiments on 
the iodine coulometer proved that the amount of iodine liberated at the 
anode was equal to that converted into iodide ions at the cathode, both 
quantities being in close agreement with the requirements of Faraday's 
laws. Similarly, provided there are no secondary processes to interfere, 
the volume of oxygen evolved at an anode in the electrolysis of a solution 
of dilute acid or alkali is half the volume of hydrogen set free at the 
cathode. 

In the cases referred to above, the anode consists of a metal which is 
not attacked during the passage of current, but if an attackable metal, 
e.g., zinc, silver, copper or mercury, is used as the anode, the latter dis- 
solves in amounts exactly equal to that which would be deposited on the 
cathode by the same quantity of electricity. The results obtained by 
Bovard and Hulett 12 for the loss in weight of a silver anode and for the 
amount of silver deposited on the cathode by the same current are given 
in Table VI; the agreement between the values in the eight experiments 
shows that Faraday's laws are applicable to the anode as well as to the 
cathode. 



TABLE VI. COMPARISON OP ANODIC AND CATHODIC PROCESSES 



Anode loss 
4.18685 g. 
4.13422 
4.21204 
4.08371 



Cathode gain 

4.18703 g. 
4.13422 
4.21240 
4.08473 



Anode loss 
4.17651 g. 
4.14391 
4.08147 
4.09386 



Cathode gain 
4.17741 g. 
4.14320 
4.08097 
4.09478 



The results obtained at the cathode in the iodine coulometer show 
that Faraday's laws hold for the reduction of iodine to iodide ions; the 
laws apply, in fact, to all types of electrolytic reduction occurring at the 
cathode, e.g., reduction of ferric to ferrous ions, ferri cyanide to ferro- 
cyanide, quinone to hydroquinone, etc. The laws are applicable simi- 
larly to the reverse process of electrolytic oxidation at the anode. The 
equivalent weight in these cases is based, of course, on the nature of the 
oxidation-reduction process. 

* Bovard and Hulett, JT. Am. Chem. Soc., 39, 1077 (1917). 



24 INTRODUCTION 

In the discussion hitherto it has been supposed that only one process 
occurs at each electrode; there are numerous instances, however, of two 
or more reactions occurring simultaneously. For example, in the elec- 
trolysis of nickel salt solutions the deposition of the metal is almost 
invariably accompanied by the evolution of some hydrogen; when current 
is passed through a solution of a stannic salt there may be simultaneous 
reduction of the stannic ions to starinous ions, deposition of tin and 
liberation of hydrogen at the cathode. Similarly, the electrolysis of a 
dilute hydrochloric acid solution yields a mixture of oxygen and chlorine 
at the anode. The conditions which determine the possibility of two or 
more electrode processes occurring at the same time will be examined 
in later chapters; in the meantime, it must be pointed out that whenever 
simultaneous reactions occur, the total number of equivalents deposited 
or reduced at the cathode, or dissolved or oxidized at the anode, are equal 
to the amount required by Faraday's laws. The passage of one faraday 
of electricity through a solution of a nickel salt under certain conditions 
gave a deposit of 25.48 g. of the metal, instead of the theoretical amount 
29.34 g. ; the number of equivalents of nickel deposited is thus 25.48/29.34, 
i.e., 0.8684, instead of unity. It follows, therefore, that 0.1316 equiv., 
i.e., 0.1326 g., of hydrogen is evolved at the same time. The ratio of the 
actual amount of material deposited, or, in general, the ratio of the actual 
extent of any electrode reaction, to that expected theoretically is called 
the current efficiency of the particular reaction. In the case under con- 
sideration the current efficiency for the deposition of nickel under the 
given conditions is 0.8684 or 86.84 per cent. 

Ions in Two Valence Stages. A special case of simultaneous elec- 
trode processes arises when a given ion can exist in two valence stages, 
e.g., mercuric (Hg++) and mercurous (HgJ+) ;* the passage of one faraday 
then results in the discharge at the cathode or the formation at the anode 
of a total of one gram equivalent of the two ions. An equilibrium exists 
between a metal and the ions of lower and higher valence; thus, for 
example, 

Hg + Hg++ 



and if the law of mass action is applicable to the system, it follows that 

Concn. of mercurous ions 

7: - f - : -. - = constant, 

Concn. of mercuric ions 

the concentration of the metal being constant. By shaking a simple 
mercuric salt, e.g., the nitrate, with mercury until equilibrium was estab- 
lished and analyzing the solution, the constant was found to be 120, at 
room temperatures. When a mercury anode dissolves, the mercurous 
and mercuric ions are formed in amounts necessary to maintain the 

* There is much evidence in favor of the view that the mercurous ion has the for- 
mula H&*+ and not Hg+ (see p. 264). 



25 

equilibrium under consideration; that is, the proportion of mercurous 
ions is 120 to one part of mercuric ions. It would appear, therefore, that 
99.166 per cent of the mercury which dissolves anodieally should form 
mercurous ions: this is true provided no secondary reactions take place 
in the solution. If the electrolyte is a chloride, the mercurous ions are 
removed in the form of insoluble mercurous chloride, and in order to 
maintain the equilibrium between mercury, mercuric and mercurous 
ions, the anode dissolves almost exclusively in the mercurous form. On 
the other hand, in a cyanide or iodide solution the mercuric ions are 
removed by the formation of complex ions, and hence a mercury anode 
dissolves mainly in the mercuric form. In each case the electrode mate- 
rial passes into solution in such a manner as to establish the theoretical 
equilibrium, but the existence of subsidiary equilibria in the electrolyte 
often results in the anode dissolving in the two valence stages in a ratio 
different from that of the concentrations of the simple ions at equilibrium. 

With a copper electrode, the equilibrium is greatly in favor of the 
cupric ions and so a copper anode normally dissolves virtually completely 
in the higher valence (cupric) state, i.e., as a bivalent metal. In a 
cyanide solution, however, cuprous ions are removed as complex cupro- 
cyanide ions; a copper anode then dissolves as a univalent element. 
Anodes of iron, lead and tin almost invariably dissolve in the lower 
valence state. 

Similar arguments to those given above will apply to the deposition 
at the cathodo; the proportion in which the higher and lower valence ions 
are discharged is identical with that in which an anode would dissolve 
in the same electrolyte. Thus, from a solution containing simple mer- 
curous and mercuric ions only, e.g., from a solution of the perchlorates 
or nitrates, the two ions would be discharged in the ratio of 120 to unity. 
From a complex cyanide or iodide electrolyte, however, mercuric ions 
are discharged almost exclusive!}'. 

Significance of Faraday's Laws. Since the discharge at a cathode or 
the formation at an anode of one gram equivalent of any ion requires 
the passage of one faraday, it is reasonable to suppose that this represents 
the charge * carried by a gram equivalent of any ion. If the ion has a 
valence z, then a molo of those ions, which is equivalent to z equiv., carries 
a charge of z faradays, i.e., zF coulombs, where F is 96,500. The number 
of individual ions in a mole is equal to the Avogadro number N, and so 
the electric charge carried by a single ion is zF/N coulombs. Since z is 
an integer, viz., one for a univalent ion, two for a bivalent ion, three for 
a tervalent ion, and so on, it follows that the charge of electricity carried 
by any single ion is a multiple of a fundamental unit charge equal to 
FIN. This result implies that electricity, like matter, is atomic in nature 
and that F/N is the unit or " atom" of electric charge. There arc many 
reasons for identifying this unit charge with the charge of an electron 

* It was seen on page 3 that quantity of electricity and electric charge have the 
same dimensions. 



26 INTRODUCTION 

(*), so that 

F 
'-* W 

According to these arguments a univalcnt, i.e., singly charged, cation is 
formed when an atom loses a single electron, e.g., 

Na -> Na+ + . 

A bivalent cation results from the loss of two electrons, e.g., 

On -> ( 1 u++ + 26, 

and so on. Similarly, a univalent anion is formed when an atom gains 
an electron, e.g., 

Cl + -> C1-. 

In general an ion carries the number of charges equal to its valence, and 
it differs from the corresponding uncharged particle by a number of 
electrons equal in magnitude to the charge. 

Electrons in Electrolysis. The identification of the unit charge of a 
single ion with an electron permits a more complete picture to be given 
of the phenomena of electrolysis. It will be seen from Fig. 2 that the 
passage of current through a circuit is accompanied by a flow of electrons 
from anode to cathode, outside the electrolytic cell. If the current is to 
continue, some process must occur at the surface of the cathode in the 
electrolyte which removes electrons, while at the anode surface electrons 
must be supplied: these requirements are satisfied by the discharge and 
formation of positive ions, respectively, or in other ways. In general, 
a chemical reaction involving the formation or removal of electrons must 
always occur when current passes from an electronic to an electrolytic 
conductor. For example, at a cathode in a solution of silver nitrate, 
each silver ion takes an electron from the electrode, forming metallic 
silver; thus 

Ag+ + e -> Ag. 

At a silver anode it is necessary for electrons to be supplied, and this can 
be achieved by the atoms passing into solutions as ions; thus 

Ag - Ag+ + c. 

If the anode consisted of an unattackable metal, e.g., platinum, then the 
electrons must be supplied by the discharge of ariions, e.g., 

OH- -> OH + c, 
which is followed by 

2OH = H 2 O + 2 , 

resulting in the liberation of oxygen; or 

Cl- -> Cl + 6, 



PROBLEMS 27 

followed by 

2C1 = C1 2 , 

which gives chlorine gas by the discharge of chloride ions. Since the 
same number of electrons is required by the anode as must be removed 
from the cathode, it is evident that equivalent amounts of chemical 
reaction, proportional to the quantity of electricity passing, i.e., to the 
number of electrons transferred, must take place at both electrodes. The 
electronic concept, in fact, provides a very simple interpretation of 
Faraday's laws of electrolysis. It should be clearly understood that 
although the current is carried through the metallic part of the circuit by 
the flow of electrons, it is carried through the electrolyte by the ions; the 
positive ions move in one direction and the negative ions in the opposite 
direction, the total charge of the moving ions being equivalent to the 
flow of electrons. This aspect of the subject of electrolytic conduction 
will be considered more fully in Chap. IV. 

Equations involving electron transfer, such as those given above, are 
frequently employed in electrochemistry to represent processes occurring 
at electrodes, either during electrolysis or in a voltaic cell capable of 
producing current. It is opportune, therefore, to emphasize their sig- 
nificance at this point: an equation such as 



means not only that an atom of copper gives up two electrons and be- 
comes a copper ion; it also implies that, two faradays are required to cause 
one gram atom of copper to go into solution forming a mole, or gram-ion, 
of cupric ions. In general, an electrode process written as involving z 
electrons requires the passage of z faradays for it to occur completely in 
terms of moles. 

PKOHLKMS 

1. A constant current, which gave a reading of 25.0 inilliamp. on a milli- 
ainmeter, was passed through a solution of copper sulfate for exactly 1 hour; 
the deposit on the cathode weighed 0.0300 grain. What is the error of the 
meter at the 25 inilliamp. reading f 

2. An average cell, in which aluminum is produced by the electrolysis of 
a solution of alumina in fused cryolite, takes about 20,000 amps. How much 
aluminum is produced per day in each cell, assuming a current efficiency of 
92 per cent? 

3. A current of 0.050 amp. was passed through a silver titration coulometer, 
and at the conclusion 23 8 cc. of 0.1 N sodium chloride solution were required 
to titrate the silver dissolved from the anode. How long was the current 
flowing? 

4. What weights of sodium hydroxide and of sulfuric acid are produced 
at the cathode and anode, respectively, when 1,000 coulombs are passed through 
a solution of sodium sulfate? 

5. Calculate the amount of iodine that would be liberated by a quantity 
of electricity which sets free 34.0 cc. of gas, at S.T.P., in an electrolytic gas 
coulometer. 



28 INTRODUCTION 

6. In the electrolysis of a solution containing copper (cuprous), nickel and 
zinc complex cyanides, Faust and Montillon [Trans. Electrochem. Soc., 65, 361 
(1934)] obtained 0.175 g. of a deposit containing 72.8 per cent by weight of 
copper, 4.3 per cent of nickel and 22.9 per cent of zinc. Assuming no hydrogen 
was evolved, how many coulombs were passed through the solution? 

7. Anthracene can be oxidized anodically to anthraquinone with an effi- 
ciency of 100 per cent, according to the reaction CuIIio + 30 = Ci4lI 8 02 + H 2 O. 
What weight of anthraquinone is produced by the passage of a current of 
1 amp. for 1 hour? 

8. A current of 0.10 amp. was passed for two hours through a solution of 
cuprocyanide and 0.3745 g. of copper was deposited on the cathode. Calcu- 
late the current efficiency for copper deposition and the volume of hydrogen, 
measured at S.T.P., liberated simultaneously. 

9. The 140 liters of solution obtained from an alkali-chlorine cell, operating 
for 10 hours with a current of 1250 amps., contained on the average 116.5 g. 
of sodium hydroxide per liter. Determine the current efficiency with which 
the cells were operating. 

10. In an experiment on the electrolytic reduction of sodium nitrate 
solution, Muller and Weber [_Z. Elektrochem., 9, 955 (1903)] obtained 0.0495 g. 
of sodium nitrite, 0.0173 g. ammonia and 695 cc. of hydrogen at S.T.P., while 
2.27 g. of copper were deposited in a coulometer. Evaluate the current effi- 
ciency for each of the three products. 

11. Oxygen at 25 atrn. pressure is reduced cathodically to hydrogen 
peroxide: from the data of Fischer and Priess \_Bcr., 46, 698 (1913)] the follow- 
ing results were calculated for the combined volume of hydrogen and oxygon, 
measured at S.T.P., liberated in an electrolytic gas coulometer (I) compared 
with the amount of hydrogen peroxide (II) obtained from the same quantity 
of electricity. 

I. 35.5 200 413 583 1,670 cc. of ft as. 
II. 34.7 150 265 334 596 mg. of H 2 O 2 . 

Calculate the current efficiency for the formation of hydrogen peroxide in each 
case, and plot the variation of the current efficiency with the quantity of 
electricity passed. 

12 In the electrolysis of an alkaline sodium chloride solution at 52, 
Muller [Z. anory. Chcm., 22, 33 (1900)] obtained the following results: 

Active Oxygen as Copper in 

Hypochlonte Chlorate Coulometer 

0.001 5 g. 0.0095 g. O.U5g. 

0.0053 0.0258 0.450 

0.105 0.2269 3.110 

0.135 0.3185 4.SOO 

0.139 0.4123 7.030 

Plot curves showing the variation with the quantity of electricity passed of the 
current efficiencies for the formation of hypochlorite and of chlorate. 



CHAPTER II 
ELECTROLYTIC CONDUCTANCE 

Specific Resistance and Conductance. Consider a uniform bar of 
a conductor of length I cm. and cross-sectional area a sq. cm.; suppose, 
for simplicity, that the cross section is rectangular and that the whole is 
divided into cubes of one cm. side, as shown in Fig. 7, I. The resistance 



I II 

FIG. 7. Calculation of specific resistance 

of the bar is seen to be equivalent to that of / layers, such as the one 
depicted in Fig. 7, II, in series with one another; further, each layer is 
equivalent to a cubes, each of one cm. side, whoso resistances are in 
parallel. If p is the resistance, in ohms, of a centimeter cube, generally 
called the specific resistance of the substance constituting the conductor, 
the resistance r of the layer containing the a cubes is given by 



there being a terms on the right-hand side: it follows, therefore, that 

P 



r ~~~ 



If R is the resistance of the whole bar, which is equivalent to / layers 
each of resistance r in series, then 



R = Ir = p - ohms. 



(0 



This equation is applicable to all conductors, electronic or electrolytic, 
and for uniform conductors of any cross section, not necessarily rec- 
tangular. 

29 



30 ELECTROLYTIC CONDUCTANCE 

The specific conductance of any conducting material is defined as the 
reciprocal of the specific resistance; it is given the symbol K and is stated 
in reciprocal ohm units, sometimes called " mhos."* Since, by defini- 
tion, K is equal to 1/p, it follows from equation (1) that 

1 I 
R = ~ ohms. (2) 

K a ^ ' 

The conductance (C) is the reciprocal of resistance, i.e., C I//?, and 
hence 

C = K a ohms" 1 . (3) 

The physical meaning of the specific conductance may be understood by 
supposing an E.M.F. of one volt to be applied to a conductor; since E = 1, 
it follows, by Ohm's law, that the current 7 is equal to 1/R, and hence to 
the conductance (C). For a centimeter cube a and I are unity, and so 
C is equal to K. It is seen, therefore, that when a potential difference 
of one volt is applied to a centimeter cube of a conductor, the current 
in amperes flowing is equal in magnitude to the specific conductance in 
ohm" 1 cm.~ l units. 

Equivalent Conductance. For electrolytes it is convenient to define 
a quantity called the equivalent conductance (A), represent ing the con- 
ducting power of all the ions produced by 1 equiv. of electrolyte in a 
given solution. Imagine two large parallel electrodes set 1 cm. apart, 
and suppose the whole of the solution containing 1 equiv. is placed 
between these electrodes; the area of the electrodes covered will then be 
v sq. cm., where v cc. is the volume of solution containing the 1 equiv. 
of solute. The conductance of this system, which is the equivalent con- 
ductance A, may be derived from equation (3), where a is equal to v sq. cm. 
and I is 1 cm.; thus 

A = KV, (4) 

where v is the "dilution" of the solution in cc. per equiv. If c is the 
concentration of the solution, in equivalents per liter, then z> is equal to 
1000/c, so that equation (4) becomes 

A = 1000 - - (5) 

The equivalent conductance of any solution can thus be readily derived 
from its specific conductance and concentration. Since the units of K 
are ohm" 1 cm." 1 , those of A are seen from equation (4) or (5) to be 
ohm" 1 cm. 2 

* It will be apparent from equation (1) or (2) that if R is in ohms, and I and a are 
in cm. and sq. cm. respectively, the units of K are ohm" 1 cm" 1 . This exact notation will 
be used throughout the present book. 



DETERMINATION OF RESISTANCE 



31 



In some cases the molecular conductance (/x) is employed; it is the 
conductance of 1 mole of solute, instead of 1 equiv. If v m is the volume; 
in cc. containing a mole of solute, arid c is the corresponding concentra- 
tion in moles per liter* then 



= Kv m = 1000 



(6) 



For an electrolyte consisting of two 
univalent ions, e.g., alkali halides, 
the values of A and /u are, of course, 
identical. 

Determination of Resistance. 
The measurement of resistance is 
most frequently carried out with 
some form of Wheats to no bridge 
circuit, the principle of which may 
be explained with the aid of Fig. 8. 
The four arms of the bridge, viz., 
ab, ac, bd and cd, have resistances Ri, 
Rt, Rz and /2 4 , respectively; a source 
of current S is connected across the 
bridge between b and c, and a cur- 
rent detector D is connected between a and d. Lot E\, E 2 , E^ and 7 4 be 
the fall of potential across the four arms, corresponding to the resistances 
Ri, Ri, Rs and / 4 , respectively, arid suppose the currents in these arms 
are I\, 1 2, Is and 7 4 , then by Ohm's hiw: 




FIG. 8. Wheatstone bridge circuit 



#4 = /4#4. 

If the resistances are adjusted so that there is no flow of current through 
the detector D, that is to say, when the bridge is " balanced," the poten- 
tial at a must be the same as that at d. Since the arms ab arid bd are 
joined at b and tho potentials are the same at a and d, it follows that the 
fall of potential across ab, i.e., E ly must equal that across bd, i.e., E*. 
Similarly, the fall of potential across ac must be the same as that across 
cd, i.e., E* and A T 4 are equal. Introducing the values of the various 7's 
given above, it is seen that 

IiRi = 7 3 /e 3 and I 2 R 2 = 7 4 72 4 , 



Since no current passes through ad when the bridge is balanced, it is 

* In accordance with the practice adopted by a number of writers, the symbol c is 
used to represent concentrations in equivalents and c in moles, per liter. 



32 ELECTROLYTIC CONDUCTANCE 

evident that the current flowing in the arm ab must be the same as that 
in ac, i.e., /i = 7 2 , while that passing through bd must be identical with 
that in erf, i.e., /3 = /4. It follows, therefore, that at the balance point 

Kt Jt, 



and so if the resistances of three of the arms of the bridge are known, 
that of the fourth can be readily evaluated. In practice, R\ is generally 
the unknown resistance, and Ri is a resistance box which permits various 
known resistances to be used; the so-called " ratio arms" Its and R\ may 
be a uniform wire (bdc) on which the position of d is adjusted until the 
bridge is balanced, as shown by the absence of current in D. The ratio 
of the lengths of the two parts of the uniform wire, corresponding to bd 
and dc, gives the ratio Ra/R*. 

Resistance of Electrolytes: Introduction. In the earliest attempts 
to determine the resistance of electrolytic solutions the results were so 
erratic that it was considered possible that Ohm's law was not applicable 
to electrolytic conductors. The erratic behavior was shown to be due 
to the use of direct current in the measurement, and when the resulting 
errors were eliminated it became evident that Ohm's law held good for 
electrolytic as well as for metallic systems. The passage of direr-t cur- 
rent through an electrolyte is, as seen in Chap. I, accompanied by 
changes in composition of the solution and frequently by the liberation 
of gases at the electrodes. The former alter the conductance and the 
latter set. up an E.M.F. of" polarization" (see Chap. XIII) which tends to 
oppose the flow of current. The difficulties may be overcome by the use 
of nori-polarizable electrodes and the employment of such small currents 
that concentration changes are negligible; satisfactory conductance meas- 
urements have been made in this way with certain electrolytes by the 
use of direct current, as will be seen later (p. 47). 

The great majority of the work with solutions has, however, been 
carried out with a rapidly alternating current of low intensity, following 
the suggestion made by Kohlrausrh in 1868. The underlying principle 
of the use of an alternating current is that as a result of the reversal of 
the direction of the current about a thousand times per second, the 
polarization produced by each pulse of the current is completely neu- 
tralized by the next, provided the alternations are symmetrical. There 
is also exact compensation of any concentration changes which may 
occur. Kohlrausch used an induction coil as a source of alternating 
current (abbreviated to A.C.) and in his early work a bifllar galvanometer 
acted as detector; later (1880) he introduced the telephone earpiece, and 
this, with some improvements, is still the form of A.C. detector most 
frequently employed in electrolytic conductance measurements. The 
electrolyte was placed in a cell and its resistance measured by a Wheat- 
stone bridge arrangement shown schematically in Fig. 9. The cell C 



A.C. SOURCES AND DETECTORS 



33 




FIG. 9. Measurement of resistance of electrolyte 



is in the arm ab and a resistance box R constitutes the arm ac\ the source 
of A.C. is represented by S, and D is the telephone earpiece detector. 
In the simplest form of bridge, frequently employed for ordinary labora- 
tory purposes, the arms bd and dc are in the form of a uniform wire, 
preferably of platinum-iridium, stretched along a meter scale, i.e., the 
so-called " motor bridge," or suitably wound round a slate cylinder. The 
point d is a sliding contact 
which is moved back and forth 
until no sound can be heard 
in the detector; the bridge is 
then balanced. If the wire 
be is uniform, the ratio of the 
resistances of the two arms 
is equal to the ratio of the 
lengths, bd and dc, as seen 
above. If the resistance taken 
from the box R is adjusted so 
as to be approximately equal 
to that of the electrolyte in the 
cell C, the balance poiiiL d will 
be roughly midway between 6 
and c; a small error in the set- 
ting of d will then cause the 
least discrepancy in the final value for the resistance of C. If somewhat 
greater accuracy is desired, two variable resistance boxes may be used 
for bd and dc, i.e., /?s and #4 (cf. Fig. 8), the resistance taken from each 
being adjusted until the bridge is balanced. Alternatively, two resistance 
boxes or coils may be joined by a wire, whose resistance is known in 
terms of that of the boxes or coils, for the purpose of making the final 
adjustment. 

It will be seen shortly that for precision measurements of electrolytic 
conductance it is necessary to take special precautions to obviate errors 
due to inductance and capacity in the bridge circuit. One immediate 
effect of these factors is to make the minimum sound in the telephone 
earpiece difficult to detect; for most general purposes this source of error 
can be overcome by using a good resistance box, in which the coils are 
wound in such a manner as to eliminate self-induction, and to use a 
straight-wire bridge, if a special non-inductive bridge is not available. 
Further, a variable condenser K is connected across the resistance box 
and adjusted until the telephone earpiece gives a sharply defined sound 
minimum; in this way the unavoidable capacity of the conductance cell 
may be balanced to some extent. 

A.C. Sources and Detectors. Although the induction coil suffers 
from being noisy in operation and does not give a symmetrical alter- 
nating current, it is still often employed in conductance measurements 
where great accuracy is not required. A mechanical high-frequency 



34 



ELECTROLYTIC CONDUCTANCE 




To bridge 



Fia. 10. Vacuum-tube oscillator 



A.C. generator was employed by Washburn (1913), and Taylor and 
Acree (1916) recommended the use of the Vreeland oscillator, which 
consists of a double mercury-arc arrangement capable of giving a sym- 
metrical sine-wave alternating current of constant frequency variable at 

will from 160 to 4,200 cycles per sec- 
ond. These costly instruments have 
been displaced in recent years by some 
kind of vacuum-tube oscillator, first 
employed in conductance work by 
Hall and Adams. 1 Several types of 
suitable oscillators have been de- 
scribed and others are available com- 
mercially; the essential circuit of one 
form of oscillator is shown in Fig. 10. 
The grid circuit of the thermionic 
vacuum tube T contains a grid coil 
LI of suitable inductance which is 
connected to the oscillator coil L 2 in 
parallel with the variable condenser C. The output coil L 3 , which is 
coupled inductively with L 2 , serves to convey the oscillations to the con- 
ductance bridge. 

The chief advantages of the vacuum-tube oscillator are that it is 
relatively inexpensive, it is silent in operation and gives a symmetrical 
sinusoidal alternating current of constant frequency; by suitable adjust- 
ment of inductance and capacity the frequency of the oscillations may be 
varied over the whole audible range, but for conductance work frequencies 
of 1,000 to 3,000 cycles per sec. are generally employed. The disad- 
vantage of this type of oscillator is that it is liable to introduce stray 
capacities into the bridge circuit which can be a serious source of error 
in precision work. The difficulty may be overcome, however, by the 
use of special grounding devices (see p. 42). 

If properly tuned to the frequency of the A.C., the telephone earpiece 
can be used to detect currents as small as 10" 9 amp. ; it is still regarded 
as the most satisfactory instrument for conductance measurements. The 
sensitivity of the telephone can be greatly increased by the addition of 
a vacuum-tube (low frequency) amplifier; this is particularly valuable 
when working with very dilute solutions having a high resistance, for 
it is then possible to determine the balance point of the bridge with 
greater precision than without the amplifier. The basic circuit of a 
simple type of audio-frequency amplifier is shown in Fig. 11, in which 
the conductance bridge is connected to the primary coil of an iron-cored 
transformer (P); T is a suitable vacuum tube and C is a condenser. 
The use of a vacuum-tube amplifier introduces the possibility of errors 

1 Hall and Adams, J. Am. Chem. tfoc., 41, 1515 (1919); see also, Jones and Josephs, 
ibid., 50, 1049 (1928); Luder, ibid., 62, 89 (1940;; Jones, Mysels and Juda, ibid. t 62, 
2919 (1940). 



ELECTRODES FOR CONDUCTANCE MEASUREMENTS 



35 



due to capacity and interaction effects, but these can be largely elimi- 
nated by suitable grounding and shielding (see p. 42). 

If results of a low order of accuracy are sufficient, as, for example, 
in conductance measurements for analytical or industrial purposes, the 
A.C. supply mains, of frequency about 60 cycles per sec., can be em- 
ployed as a source of current; in this case an A.C. galvanometer is a 
satisfactory detector. A combination of a vacuum-tube, or other form 
of A.C. rectifier, and a direct current galvanometer has been employed, 




FIG. 11. Vacuum-tube amplifier 

and in some cases the thermal effect of the alternating current has been 
used, in conjunction with a thermocouple and a sensitive galvanometer, 
for detection purposes. 

Electrodes for Conductance Measurements. For the determination 
of electrolytic conductance it is the general practice to use two parallel 
sheets of stout platinum foil, that do not bend readily; their relative 
positions are fixed by sealing the connecting tubes into the sides of the 
measuring cell (of. Fig. 12). In order to aid the elimination of polariza- 
tion effects by the alternating current, Kohlrausch (1875) coated the 
electrodes with a layer of finely divided platinum black; these are called 
platinized platinum electrodes. The platinization is carried out by elec- 
trolysis of a solution containing about 3 per cent of chloroplatinic acid 
and 0.02 to 0.03 per cent of lead acetate; the lead salt apparently favors 
the formation of the platinum deposit in a finely-divided, adherent form. 
The large surface area of the finely divided platinum appears to catalyze 
the union of the hydrogen and oxygen which tend to be liberated by 
the successive pulses of the current; the polarization E.M.F. is thus 
eliminated. 

In some cases the very properties which make the platinized platinum 
electrodes satisfactory for the reduction of polarization are a disadvan- 
tage. The finely-divided platinum may catalyze the oxidation of organic 
compounds, or it may adsorb appreciable quantities of the solute present 



36 



ELECTROLYTIC CONDUCTANCE 



in the electrolyte and so alter its concentration. Some workers have 

overcome this disadvantage of platinized electrodes by heating them to 

redness and so obtaining a gray surface; the resulting electrode is prob- 

ably not so effective in reducing polarization, but it adsorbs much less 

solute than does the black deposit. Others have employed electrodes 

covered with very thin layers of platinum black, and sometimes smooth 

electrodes have been used. By making measurements with smooth 

platinum electrodes at various frequencies and extrapolating the results 

to infinite frequency, conductance values have been obtained which are 

in agreement with those given by platinized electrodes; this method is 

thus available when platinum black must not be used. For the great 

majority of solutions of simple salts and of inorganic acids and bases 

it is the practice to employ electrodes coated with a thin layer of plati- 

num black obtained by electrolysis as already described. 

Conductance Cells: The Cell Constant. The cells for electrolytic 
conductance measurements are made of highly insoluble glass, such as 

Pyrex, or of quartz; they should 
be very carefully washed and 
steamed before use. For general 
laboratory requirements the sim- 
ple cell designed by Ostwald (Fig. 
12, I)' is often employed, but for 
industrial purposes the "dipping 
cell" (Fig. 12, II) or the pipette- 
type of cell (Fig. 12, III) have been 
found convenient. By means of 
the two latter cells, samples ob- 
tained at various stages in a 
chemical process can be readily 
tested. 

The resistance (R) of the solu- 
tion in the cell can be measured, 

as already explained, and hence the specific conductance (K) is given by 
equation (2) as 

I 

K = = 

aR 

where I is the distance between the electrodes and a is the area of cross 
section of the electrolyte through which the current passes. For a given 
cell with fixed electrodes I/a is a constant, called the cell constant ; if this 
is given the symbol K cm.~ l , it follows that 




FIG. 12. Types of conductance cells 



K 



(8) 



It is neither convenient nor desirable, with the cells in general use, to 
measure / and a with any degree of accuracy, and so an indirect method 



DESIGN OP CELLS 37 

is employed for the evaluation of the cell constant. If a solution whose 
specific conductance is known accurately, from other measurements, is 
placed in the experimental cell and its resistance R is measured, it is 
possible to obtain K for the given cell directly, by means of equation (8). 
The electrolyte almost invariably used for this purpose is potassium 
chloride, its specific conductance having been determined with high pre- 
cision in cells calibrated by measurement with a concentrated solution 
of sulfuric acid, the resistance of which has been compared in another 
cell with that of mercury; the specific conductance of the latter is known 
accurately from the definition of the international ohm as 10629.63 
ohms" 1 cm.~ l at 0. 

The potassium chloride solutions employed in the most recent work 
contain 1.0, 0.1 or 0.01 mole in a cubic decimeter of solution at 0, i.e., 
0.999973 liter; these solutions, designated as 1.0 D, 0.1 D and 0.01 D, 
where D stands for " demal," contain 76.627, 7.4789 and 0.74625 grams of 
potassium chloride to 1000 grams of water, respectively. The specific 
conductances of these solutions at 0, 18 and 25 are quoted in Table 
VII; 2 the particular solution chosen for calibrating a given cell depends 

TABLE VII. SPECIFIC CONDUCTANCES OF POTASSIUM CHLORIDE SOLUTIONS 
IN OHM" 1 CM.~ l 

Temp. I.OD 0.1 D 0.01 D 

0.065176 0.0071379 0.00077364 

18 0.097838 0.0111667 0.00122052 

25 0.111342 0.0128560 0.00140877 

on the range of conductances for which it is to be employed. The values 
recorded in this table do not include the conductance of the water; when 
carrying out a determination of the constant of a given conductance cell 
allowance must be made for this quantity. 

Design of Cells. In the design of conductance cells for precision 
measurements a number of factors must be taken into consideration. 
Kohlrausch showed theoretically that the error resulting from polariza- 
tion was determined by the quantity P 2 /o>/ 2 , where P is the polarization 
E.M.F., R is the resistance of the electrolyte in the cell and w is the fre- 
quency of the alternating current. It is evident that the error can be 
made small by adjusting the experimental conditions so that co/i! 2 is much 
greater than P 2 ; this can be done by making either or R, or both, as 
large as is reasonably possible. There is a limit to the increase in the 
frequency of the A.C. because the optimum range of audibility of the 
telephone earpiece is from 1,000 to 4,000 cycles per sec., and so it is 
desirable to make the resistance high. If this is too high, however, the 
current strength may fall below the limit of satisfactory audibility, and 
it is not possible to determine the balance point of the bridge. The 

Jones and Bradshaw, J. Am. CJiem. Soc., 55, 1780 (1933); see also, Jones and 
Prendergast, ibid., 59, 73> (1937); Bremner and Thompson, ibid., 59, 2371 (1937); 
Davies, J. Chem. Soc., 432, 1326 (1937). 



38 



ELECTROLYTIC CONDUCTANCE 



highest electrolytic resistances which can be measured with accuracy, 
taking advantage of the properties of the vacuum-tube audio-amplifier, 
are about 50,000 ohms. In order to measure low resistances the polari- 
zation P should be reduced by adequate platinization of the electrodes, 
but there is a limit to which this can be carried and experiments show 
that resistances below 1,000 ohms cannot be measured accurately. The 
resistances which can be determined in a given cell, therefore, cover a 
ratio of about 50 to unity. The observed specific conductances of electro- 
lytes in aqueous solution range from approximately 10" 1 to 10~ 7 ohms" 1 
cm." 1 , and so it is evident that at least three cells of different dimensions, 
that is with different cell constants, must be available. 

Another matter which must be borne in mind in the design of a con- 
ductance cell is the necessity of preventing a rise of temperature in the 
electrolyte due to the heat liberated by the current. This can be achieved 
either by using a relatively large volume of solution or by making the 
cell in the form of a long narrow tube which gives good thermal contact 
with the liquid in the thermostat. 

Two main types of cell have been devised for the accurate measure- 
ment of electrolytic conductance; there is the " pipette " type, used by 
Washburn (1916), and the flask type, introduced by Hartley and Barrett 
(1913). In the course of a careful study of cells of the pipette form, Parker 
(1923) found that with solutions of high resistance, for which the polari- 
zation error is negligible, there was an apparent decrease of the cell con- 
stant with increasing resistance. This phenomenon, which became 
known as the " Parker effect," was confirmed by other workers; it was at 

first attributed to adsorption of the 
electrolyte by the platinized electrode, 
but its true nature was elucidated by 
Jones and Bollinger. 3 The pipette type 
of cell (Fig. 13, I) is electrically equiv- 
alent to the circuit depicted in Fig. 13, 
II; the resistance R is that of the solu- 
tion contained between the electrodes 
in the cell, and this is in parallel with 
the resistance (R p ) of the electrolyte 
in the filling tube at the right and a 
capacity (C p ). The latter is equiv- 
alent to the distributed capacity be- 
tween the electrolyte in the body of 
the cell and the mercury in the con- 
tact tube, on the one hand, and the solution in the filling tube, on 
the other hand; the glass walls of the tubes and the thermostat liquid 
act as the dielectric medium. An analysis of the effect of shunting the 
resistance R Q by a capacity C p and a resistance R p shows that, provided 

Jones and Bollinger, /. Am. Chem. Soc., 53, 411 (1931); cf., Washburn, ibid., 38, 
2431 (1916). 






I H 

FIG. 13. Illustration of the 
"Parker effect" 



DESIGN OF CELLS 



39 



the cell is otherwise reasonably well designed, the error &R in the meas- 
ured resistance is given by 



- Aft 



(9) 



where o>, as before, is the frequency of the alternating current. Accord- 
ing to equation (9) the apparent cell constant will decrease with increasing 
resistance R , as found in the Parker 
effect. In order to reduce this source 
of error, it is necessary that RQ, w and 
C p should be small; as already seen, 
however, RQ and o> must be large to 
minimize the effect of polarization, 
and so the shunt capacity C p should be 
negligible if the Parker effect is to be 
eliminated. Since most of the shunt 
capacity lies between the filling tube 
and the portions of the cell of opposite 
polarity (cf. Fig. 13, I) it is desirable 
that these should be as far as possible 
from each other. This principle is em- 
bodied in the cells shown in Fig. 14, 
designed by Jones and Bollinger; the 
wider the tube and the closer the 
electrodes, the smaller the cell con- 
stant. These cells exhibit no appre- 
ciable Parker effect: the cell constants 
are virtually independent of the fre- 
quency of the A.C. and of the resist- 
ance of the electrolyte within reason- 
able limits. 

The Parker effect is absent from cells with dipping electrodes, such 
as in cells of the flask type; there are other sources, of error, however, as 
was pointed out by Shedlovsky. 4 In the cell represented diagram- 
matically in Fig. 15, I, the true resistance of the solution between the 
electrodes is R Qy and there is a capacity Ci between the contact tubes 
above the electrolyte, and a capacity C 2 in series with a resistance r 
between those parts immersed in the liquid; the equivalent electrical 
circuit is shown by Fig. 15, II. When the cell is placed in the arm of a 
Wheatstone bridge it is found necessary to insert a resistance R and a 
capacity C in parallel in the opposite arm in order to obtain a balance 
(cf . p. 33) ; it can be shown from the theory of alternating currents that 



FIG. 14. Cells for accurate con- 
ductance measurements (Jones and 
Bollinger) 






1 

R RQ \ a 
4 Shedlovsky, J. Am. Chem. Soc., 54, 1411 (1932). 



(10) 



40 



ELECTROLYTIC CONDUCTANCE 



where r is taken as proportional to /2o, the constant a being equal to 
r//? . It follows, therefore, that if the cell is balanced by a resistance and 
a capacity in parallel, no error results if part of the current through the 
cell is shunted by a pure capacity such as Ci, since the quantity Ci does 




FIG. 15. Equivalent resistance and capacity 
of flask cell 



FIG. 16. Shedlovsky flask cell 



not appear in equation (10). On the other hand, parasitic currents 
resulting from a series resistance-capacity path, i.e., involving r and C%, 
will introduce errors, since the apparent resistance R will be different 
from the true resistance R . In order to eliminate parasitic currents, 
yet retaining the advantages of the flask type of cell for work with a 
series of solutions of different concentrations, Shedlovsky designed the 
cell depicted in Fig. 16; the experimental solution contained in the flask A 
is forced by gas pressure through the side tube into the bulb containing 
the electrodes B and B' . These consist of perforated platinum cones 
fused to the walls of the bulb; the contact tubes C and C" are kept apart 
in order to diminish the capacity between them. The Shedlovsky cell 
has been used particularly for accurate determination of the conductances 
of a series of dilute solutions of strong electrolytes. 

Temperature Control. The temperature coefficient of conductance 
of electrolytes is relatively high, viz., about 2 per cent per degree; in 
order to obtain an accuracy of two parts in 10,000, which is desirable 
for accurate work, the temperature should be kept constant within 0.01. 
The use of water in the thermostat is not recommended; this liquid has 
an appreciable conductance and there is consequently a danger of current 
leakage leading to errors in the measurement, as explained below. The 
thermostatic liquid should, therefore, be a hydrocarbon oil which is a 
non-conductor. 

Design of the A.C. Bridge. Strictly speaking the condition of balance 
of the Wheatstone bridge given by equation (7) is applicable for alter- 
nating current only if R iy /2 2 , #3 arid R* are pure resistances. It is un 



DESIGN OF THE A.C. BRIDGE 41 

likely that the resistance coils will be entirely free from inductance and 
capacity and, in addition, the conductance cell and its connecting tubes 
are equivalent to a resistance shunted by a condenser. One consequence 
of this fact is that the alternating currents in the two arms (R\ and #2) 
of the bridge arc not in phase and it is found impossible to obtain any 
adjustment of the bridge which gives complete silence in the telephone 
earpiece. For most purposes, this difficulty may be overcome by the 
use of the condenser K in parallel with the resistance box / 2 , as suggested 
on page 33. 

For precision work it is necessary, however, to consider the problem 
in further detail. For alternating current, Ohm's law takes the form 
E = 7Z, where Z is the impedance of the circuit, i.e., Z 2 is equal to 
/j>2 _|_ x 2 , the quantities R and X being the resistance and reactance, 
respectively. The condition for balance of a Wheatstone bridge circuit 
with alternating current is, consequently, 



If there is no leakage of current from the bridge network to ground, or 
from one part of the bridge to any other part, and there is no mutual 
inductance between the arms, I\ is equal to /2, and /a to /4, so that 



at balance. It follows, therefore, that in a Wheatstone A.C. bridge, 
under the conditions specified, the impedances, rather than the resist- 
ances, are balanced. It can bo shown that if the resistances are also to 
be balanced, i.e., for RijRi to be equal to R^jR*, at the same time as 
Zi/Z 2 is equal to Z^/Z^ it is necessary that 

Xi X% A" 3 A 4 

-Rl = ltl and R- 3 = R~<- 

The fraction X/R for any portion of an A.C\ circuit is equal to tan 0, 
where 6 is the phase angle between the voltage and current in the given 
conductor. It is soon, therefore, that the conditions for the simple 
Whcatstorie bridge relationship between resistance*, i.e., for equation (7), 
to be applicable when alternating current is used, are (a) that there should 
be no leakage currents, and (6) that the phase angles should be the same 
in the tw r o pairs of adjacent arms of the bridge. 

These requirements have been satisfied in the A.(\ bridge designed 
for electrolytic conductance measurements by Jones and Josephs; 5 the 
second condition is met by making the two ratio arms (A* 3 and 7tJ 4 , Fig. 8) 
as nearly as possible identical in resistance and construction, so that any 

6 Jones and Josephs, J. Am. Chem. Sue., 50, 1049 (1928); see also, Luder, ilnd., 62, 
89 (1940). 



42 



ELECTROLYTIC CONDUCTANCE 




reactance, which is deliberately kept small, is the same in each case. 
In this way X 3 /Ra is made equal to Xt/R*. It may be noted that this 
condition is automatically obtained when a straight bridge wire is em- 
ployed. The reactance of the measuring cell, i.e., X\, should be made 
small, but as it cannot be eliminated it should be balanced by a variable 

condenser in parallel with the resistance 
box R z ; in this way Xi/Ri can be made 
equal to X Z /R2. 

It has often been the practice in con- 
ductance work to ground certain parts 
of the bridge network for the purpose 
of improving the sharpness of the sound 
minimum in the detector at the balance 
point; unless this is done with care it is 
liable to introduce errors because of the 
existence of leakage currents to earth. 
The telephone earpiece must, however, 
be at ground potential, otherwise the 
capacity between the telephone coils and 
the observer will result in a leakage of 
current. Other sources of leakage are 
introduced by the use of vacuum-tube 
oscillator and amplifier, and by various 
unbalanced capacities to earth, etc. 

The special method of grounding 
proposed by Jones and Josephs is illus- 
trated in Fig. 17. The bridge circuit 
consists essentially of the resistances 
7?i, # 2 , #3 and # 4 , as in Fig. 8; the re- 
sistances #6 and #6, with the movable 
contact g and the variable condenser C , constitute the earthing 
device, which is a modified form of the Wagner ground. By means 
of the switch Si the condenser C is connected either to A or to A', 
whichever is found to give better results. The bridge is first balanced 
by adjusting 7 2 in the usual manner;* the telephone detector D is then 
disconnected from R' and connected to ground by means of the switch /S 2 . 
The position of the contact g and the condenser C g are adjusted until 
there is silence in the telephone, thus bringing B to ground potential. 
The switch 82 is now returned to its original position, and 72 2 is again 
adjusted so as to balance the bridge. If the changes from the original 
positions arc appreciable, the process of adjusting (/.and C g should be 
repeated and the bridge again balanced. 

Shielding the A.C. Bridge. In order to eliminate the electrostatic 
influence between parts of the bridge on one another, and also that due 

* This adjustment includes that of a condenser (not shown) in parallel, as explained 
above; see also page 33 and Fig. 9. 



R 5 R 6 

vwwywwwsA/ 

4-> 



Fia. 17. Jones and Josephs bridge 



PREPARATION OF SOLVENT 43 

to outside disturbances, grounded metallic shields have sometimes been 
placed between the various parts of the bridge, or the latter has been 
completely surrounded by such shields. It has been stated that this 
form of shielding may introduce more error than it eliminates, on account 
of the capacity between the shield and the bridge; it has been recom- 
mended, therefore, that the external origin of the disturbance, rather 
than the bridge, should be shielded. According to Shcdlovsky 6 the 
objection to the use of electrostatic screening is based on unsymmetrical 
shielding which introduces unbalanced capacity effects s to earth; further, 
it is pointed out that it is not always possible to shield the disturbing 
source. A bridge has, therefore, been designed in which the separate 
arms of each pair are screened symmetrically; the shields surrounding the 
cell and the variable resistance (#2) are grounded, while those around the 
ratio arms (R 3 and 72 4 ) are not. The leads connecting the oscillator and 
detector to the bridge arc also screened and grounded. In this way 
mutual and external electrostatic influences on the bridge are eliminated. 
By means of a special type of twin variable condenser, connected across 
Ri and # 2 , the reactances in these arms can be compensated so as to give 
a sharp minimum in the telephone detector and also the correct con- 
ditions for Ri/Rz to be equal to Rs/R*. It is probable that the screened 
bridge has advantages over the unscreened bridge when external dis- 
turbing influences are considerable. 

Preparation of Solvent: Conductance Water. Distilled water is a 
poor conductor of electricity, but owing to the presence of impurities 
such as ammonia, carbon dioxide and traces of dissolved substances 
derived from containing vessels, air and dust, it has a conductance suffi- 
ciently large to have an appreciable effect on the results in accurate work. 
This source of error is of greatest importance with dilute solutions or 
weak electrolytes, because the conductance of the water is then of the 
same order as that of the electrolyte itself. If the conductance of the 
solvent were merely superimposed on that of the electrolyte the correc- 
tion would be a comparatively simple matter. The conductance of the 
electrolyte would then be obtained by subtracting that of the solvent 
from the total; this is possible, however, for a limited number of solutes. 
In most cases the impurities in the water can influence the ionization of 
the electrolyte, or vice versa, or chemical reaction may occur, and the 
observed conductance of the solution is not the sum of the values of the 
constituents. It is desirable, therefore, to use water which is as free as 
possible from impurities; such water is called conductance water, or 
conductivity water. 

The purest water hitherto obtained was prepared by Kohlrausch and 
Heydweiller (1894) who distilled it forty-two times under reduced pres- 
sure; this water had a specific conductance of 0.043 X 10~ 6 ohmr 1 cm.~ l 

Shedlovsky, J. Am. Chem. Soc., 52, 1793 (1930). 



44 



ELECTROLYTIC CONDUCTANCE 



at 18.* Water of such a degree of purity is extremely tedious to pre- 
pare, but the so-called " ultra-pure " water, with a specific conductance 
of 0.05 to 0.06 X 10~ 6 ohm- 1 cm.- 1 at 18, can be obtained without 
serious difficulty. 7 The chief problem is the removal of carbon dioxide 
and two principles have been adopted to achieve this end; either a rapid 
stream of pure air is passed through the condenser in which the steam 
is being condensed in the course of distillation, or a small proportion 
only of the vapor obtained by heating ordinary distilled water is con- 
densed, the gaseous impurities being carried off by the uncondensed 
steam. Ultra-pure water will maintain its low conductance only if air is 
rigidly excluded, but as such water is not necessary except in special 
c&ses, it is the practice to allow the water to come to equilibrium with 
the carbon dioxide of the atmosphere. The resulting "equilibrium 

water" has a specific conductance of 
0.8 X 10" 6 ohmr 1 cm." 1 and is quite 
satisfactory for most conductance 
measurements. 

The following brief outline will 
indicate the method 8 used for the 

J \_^\ ready preparation of water having a 

II __ specific conductance of 0.8 X 10~ 6 

' ' * " ohm~ l cm." 1 ; it utilizes both the air- 

stream and partial condensation meth- 
ods of purification. The 20-liter boiler 
A (Fig. 18) is of copper, while the 
remainder of the apparatus should be 
made of pure tin or of heavily tinned 
copper. Distilled water containing 
sodium hydroxide and potassium per- 
manganate is placed in the boiler and 
the steam passes first through the trap 
B t which collects spray, and then into 
the tube C. A current of purified 
air, drawn through the apparatus by 
connecting D and E to a water pump, 
enters at F; a suction of about 8 inches of water is employed. The tem- 
perature of the condenser G is so arranged (about 80) that approximately 
half as much water is condensed in // as in /; the best conductance 

* Calculations based on the known ionization product of water and the conductances 
of the hydrogen and hydroxyl ions at infinite dilution (see p. 340) show that the specific 
conductance of perfectly pure water should be 0.038 X 10~ ohm" 1 cm." 1 at 18. 

7 Kraus and Dexter, J. Am. Chem. Soc., 44, 2468 (1922); Bencowitz and Hotchkiss, 
J. Phys. Chem., 29, 705 (1925); Stuart and Worm well, J. Chem. Hoc., 85 (1930). 

8 Vogel and Jeffery, /. Chem. oc., 1201 (1931). 



I 



6 

fi^r 







Fia. 18. Apparatus for preparation 
of conductance water (Vogel and 
Jeffery) 



SOLVENT CORRECTIONS 45 

water collects in the Pyrcx flask ./, while a somewhat inferior quality is 
obtained in larger amount at K. 

For general laboratory measurements water of specific conductance 
of about 1 X 10~ 6 ohm" 1 cm." 1 at 18 is satisfactory; this can be obtained 
by distilling good distilled water, to which a small quantity of permanga- 
nate or Nessler's solution is added. A distilling flask of resistance glass 
is used and the vapor is condensed either in a block-tin condenser or in 
one of resistance glass. If corks are used they should be covered with 
tin foil to prevent direct contact with water or steam. 

Non-aqueous solvents should be purified by careful distillation, special 
care being taken to eliminate all traces of moisture. Not only are con- 
ductances in water appreciably different from those in non-aqueous 
media, but in certain cases, particularly if the electrolytic solution con- 
tains hydrogen, hydroxyl or alkoxyl ions, small quantities of water have 
a very considerable effect on the conductance. Precautions should thus 
be taken to prevent access of water, as well as of carbon dioxide and 
ammonia from the atmosphere. 

Solvent Corrections. The extent of the correction which must be 
applied for the conductance of the solvent depends on the nature of the 
electrolyte; 9 although not all workers are in complete agreement on the 
subject, the following conclusions are generally accepted. If the solute 
is a neutral salt, i.e., the salt of a strong acid and a strong base, the 
ionization and conductance of the carbonic acid, which is the main im- 
purity in water, arc no* affected to any great extent; the whole of the 
conductance of the solvent should then be subtracted from that of the 
solution. With such electrolytes the particular kind of conductance 
water employed is not critical. Strictly speaking the change in ionic 
concentration due to the presence of the salt does affect the conductance 
of the carbonic acid to some extent; when the solvent correction is a 
small proportion of the total, e.g., in solutions of neutral salts more con- 
centrated than about 10~ 3 N, the alteration is negligible. For more 
dilute solutions, however, it is advisable to employ ultra-pure water, 
precautions being taken to prevent the access of carbon dioxide. 

Salts of weak bases or weak acids are hydrolyzed in aqueous solution 
(see Chap. XI) and they behave as if they contained excess of strong 
acid and strong base, respectively. According to the law of mass action 
the presence of one acid represses the ionization of a weaker one, so that 
the effective conductance of the water, which is due mainly to carbonic 
acid, is diminished. The solvent correction in the case of a salt of a 
weak base and a strong acid should thus be somewhat less than the total 
conductance of the water. For solutions of salts of a weak acid and a 
strong base, which react alkaline, the correction is uncertain, but methods 
of calculating it have been described; they are based on the assumption 

'Kolthoff, Rec. trav. chim., 48, 664 (1929); Davies, Trans. Faraday Soc., 25, 129 
(1929); "The Conductivity of Solutions," 1933, Chap. IV. 



46 ELECTROLYTIC CONDUCTANCE 

that the impurity in the water is carbonic acid. 10 If ultra-pure water is 
used, the solvent correction can generally be ignored, provided the solu- 
tion is not too dilute. 

If the solution being studied is one of a strong acid of concentration 
greater than 10" 4 N, the ionization of the weak carbonic acid is depressed 
to such an extent that its contribution towards the total conductance is 
negligible. In these circumstances no water correction is necessary; at 
most, the value for pure water, i.e., about 0.04 X 10" 6 ohm" 1 cm." 1 at 
ordinary temperatures, may be subtracted from the total. If the con- 
centration of the strong acid is less than 10~ 4 N, a small correction is 
necessary and its magnitude may be calculated from the dissociation 
constant of carbonic acid. 

The specific conductance of a 10~ 4 N solution of a strong acid, which 
represents the lowest concentration for which the solvent correction may 
be ignored, is about 3.5 X 10~ 5 ohm" 1 cm." 1 Similarly, with weak acids 
the correction is unnecessary provided the specific conductance exceeds 
this value. For more dilute solutions the appropriate correction may be 
calculated, as mentioned above. 

The solvent correction to be applied to the results obtained with 
solutions of bases is very uncertain; the partial neutralization of the 
alkali by the carbonic acid of the conductance water results in a decrease 
of conductance, and so the solvent correction should be added, rather 
than subtracted. A method of calculating the value of the correction 
has been suggested, but it would appear to be best to employ ultra-pure 
water in conductance work with bases. 

With non-aqueous solvents of a hydroxylic type, such as alcohols, the 
corrections are probably similar to those for water; other solvents must 
be considered on their own merits. In general, the solvent should be 
as pure as possible, so that the correction is, in any case, small; as indi- 
cated above, access of atmospheric moisture, carbon dioxide and ammonia 
should be rigorously prevented. Since non-hydroxylic solvents such as 
acetone, acetonitrile, nitromethane, etc., have very small conductances 
when pure, the correction is generally negligible. 

Preparation of Solutions. When the conductances of a series of 
solutions of a given electrolyte are being measured, it is the custom to 
determine the conductance of the water first. Some investigators recom- 
mend that measurements should then commence with the most concen- 
trated solution of the series, in order to diminish the possibility of error 
resulting from the adsorption of solute from the more dilute solutions 
by the finely divided platinum on the electrodes. When working with 
cells of the flask type it is the general practice, however, to fill the cell 
with a known amount of pure solvent, and then to add successive small 
quantities of a concentrated solution of the electrolyte, of known cori- 

10 Davies, Trans. Faraday Soc., 28, 607 (1932); Maclnnes and Shedlovsky, /. Am. 
Chem. Soc., 54, 1429 (1932); Jeff cry, Vogel and Lowry, J. Chem. Soc., 1637 (1933); 166 
(1934); 21 (1935). 



DIRECT CURRENT METHODS 47 

centration, from a weight burette. When rolls of other types are used 
it is necessary to prepare a separate solution for each measurement; 
this procedure must be adopted in any case if the solute is relatively 
insoluble. 

Direct Current Methods. A few measurements of electrolytic con- 
ductance have been made with direct current and non-polarizable elec- 
trodes; the electrodes employed have been mercury-mercurous chloride 
in chloride solutions, mercury-mercurous sulfate in sulfate solutions, and 
hydrogen electrodes in acid electrolytes.* Two main principles have 
been applied: in the first, the direct current is passed between two elec- 
trodes whose nature is immaterial; the two non-polarizable electrodes 
are then inserted at definite points in the electrolyte and the fall of 
potential between them is measured. The current strength is calculated 
by determining the potential difference between two ends of a wire of 
accurately known resistance placed in the circuit. Knowing the poten- 
tial difference between the two non-polarizable electrodes and the current 
passing, the resistance of the column of solution separating these elec- 
trodes is obtained immediately by means of Ohm's law. The second 
principle which has been employed is to use the non-polarizable electrodes 
for leading direct current into and out of the solution in the normal 
manner arid to determine the resistance of the electrolyte by means of a 
Wheatstone bridge network. A sensitive mirror galvanometer is used 
as the null instrument and no special precautions need be taken to avoid 
inductance, capacity and leakage effects, since these do not arise with 
direct current. 11 

The cells used in the direct current measurements are quite different 
from those employed with alternating current; thore is nothing critical 
about their design, and they generally consist of horizontal tubes with 
the electrodes inserted either at the ends or at definite intermediate 
positions. The constants of the cells are determined either by direct 
measurement of the tubes, by means of mercury, or by using an electro- 
lyte whose specific conductance is known accurately from other sources. 
It is of interest to record that where data are available for both direct 
and alternating current methods, the agreement is very satisfactory, 
showing that the use of alternating current does not introduce any in- 
herent error. The direct current method has the disadvantage of being 
applicable only to those electrolytes for which non-polarizable electrodes 
can be found. 

The following simple method for measuring the resistance of solutions 
of very low specific conductance has been used. 12 A battery of storage 

* The nature of these electrodes will be understood better after Chap. VI has been 
studied. 

"Eastman, J. Am. Chem. Soc., 42, 1648 (1920); Br0nsted and Nielsen, Trans. 
Faraday Soc., 31, 1478 (1935); Andrews and Martin, J. Am. Chem. Soc., 60, 871 (1938). 

u LaMer and Downes, J. Am. Chem. Soc., 53, 888 (1931); Fuoss and Kraus, ibid.. 
55, 21 (1933); Bent and Dorfman, ibid., 57, 1924 (1935). 



48 



ELECTROLYTIC CONDUCTANCE 



cells, having an E.M.F. of about 150 volts, is applied to the solution whose 
resistance exceeds 100,000 ohms; the strength of the current which 
passes is then measured on a calibrated mirror galvanometer. From a 
knowledge of the applied voltage and the current strength, the resistance 
is calculated with the aid of Ohm's law. In view of the high E.M.F. 
employed, relative to the polarization E.M.F., the error due to polarization 
is very small; further, since only minute currents flow, the influence of 
electrolysis and heating is negligible. 

Conductance Determinations at High Voltage and High Frequency. 
The electrolytic conductances of solutions with alternating current of very 
high frequency or of high voltage have acquired special interest in con- 
nection with modern theories of electrolytic solutions. Under these 






Hf- 


<il VU, 
K 


7 


o 



o 




o 




1 Jl 

High | 
frequency 




4H 







FIG. 19. Barretter bridge 

extreme conditions the simple Wheatstone bridge method cannot be used, 
and other experimental procedures have been described. The chief diffi- 
culty lies in the determination of the balance point, and in this connection 
the "barretter bridge" has been found to be particularly valuable. 
A form of this bridge is shown in Fig. 19, II; it is virtually a Wheatstone 
bridge, one arm containing the choke inductances Si and $ 3 , and a small 
fine-wire filament "barretter" lamp (Zi), across which is shunted a coup- 
ling inductance MI and a condenser Ci; the corresponding arm of the 
bridge contains the chokes S 2 and S 4 , and the barretter tube Z 2 , which is 



RESULTS OF CONDUCTANCE MEASUREMENTS 49 

carefully matched with /i, shunted by the coupling inductance M^ and 
the condenser CY The ratio arms of the bridge consist of the variable 
resistances Jt 3 and R*. The actuating direct current voltage for the 
bridge is supplied by a direct current battery, and the detecting instru- 
ment is the galvanometer (7; an inductance in series with the latter 
prevents induced currents from passing through it. At the beginning of 
the experiment the resistances # 3 and R* are adjusted until the bridge 
is balanced. 

The actual resistance circuit is depicted in Fig. 19, I; K is the con- 
ductance cell and R is a variable resistance which are coupled to the 
barretter circuit by means of the inductances LI and L<>. The high fre- 
quency or high voltage is applied to the terminals of this circuit, and the 
currents induced in the bridge are restricted to the barretters li and Z 2 by 
the pairs of inductances NI *S 3 and *S f 2 -A>4, respectively. The heating 
effect of these currents causes a change of resistance of the barretters, and 
if the currents in L\ and L 2 are different, the bridge will be thrown out 
of balance. The resistance R is then adjusted until the bridge remains 
balanced when the current is applied to the cell circuit. The cell K is 
now replaced by a standard variable resistance and, keeping R constant, 
this is adjusted until the bridge is again balanced; the value of this re- 
sistance is then equal to that of the cell 7v. 13 

Results of Conductance Measurements. --The results recorded here 
refer to measurements made at A.C. frequencies and voltages that are 
not too high, i.e., ono to four thousand cycles per sec. and a few volts per 
cm., respectively. Under these conditions the electrolytic conductances 
are independent of the voltage, i.e., Ohm's law is obeyed, and of fre- 
quency, provided polarization is eliminated. Although the property of 
a solution that is actually measured is the specific conductance at a given 
concentration, this quantity is not so useful for comparison purposes as 
is the equivalent conductance; the latter gives a measure of the con- 
ducting power of the ions produced by one equivalent of the electrolyte 
at the given concentration and is invariably employed in electrochemical 
work. The equivalent conductance is calculated from the measured 
specific conductance; by means of equation (5). 

A largo number of conductance measurements of varying degrees of 
accuracy have been reported; the most reliable* results for some electro- 
lytes in aqueous solution at 25 are recorded in Table; VIII, the concen- 
trations being expressed in equivalents per liter. 14 

These data show that the equivalent conductance, and hence the con- 
ducting power o f the ions in one gram equivalent of any electrolyte, 
increases with decreasing concentration. The figures appear to approach 

1J Malsch and Wien, Ann. Physik, 83, 305 (1927); Neese, ibid., 8, 929 (1931); Wien, 
i/m/., 11, 429 (1931); Srhicle, Physik. Z. t 35, 632 (1934). 

14 For a critical compilation of recent accurate data, see Machines, "The Principles 
of Electrochemistry," 1939, p. 339; for other data International Critical Tables, Vol. 
VI, and the Ijtindolt-Boriistein Tabellen should be consulted. 



50 ELECTROLYTIC CONDUCTANCE 

TABLE VIII. EQUIVALENT CONDUCTANCES AT 25 IX OHMS" 1 CM. 8 



Concn. 


HCl 


KCl 


Nul 


NaOH 


AgNOa 


iBaClj 


iNiRO 


iLaCla 


iKFe(CN). 


0.0005 N 


422.74 


147.81 


125.36 


246 


131.36 


135.96 


118.7 


139.6 





0.001 


421.36 


140.95 


124.25 


245 


130.51 


134.34 


113.1 


137.0 


167.24 


0.005 


415.80 


143.55 


121.25 


240 


127.20 


128.02 


93.2 


127.5 


146.09 


0.01 


412.00 


141.27 


119.24 


237 


124.76 


123.94 


82.7 


121.8 


134.83 


0.02 


407.24 


138.34 


llti.70 


233 


121.41 


1 19.09 


72.3 


115.3 


122.82 


0.05 


399.09 


133.37 


112.79 


227 


115.24 


111.48 


59.2 


106.2 


107.70 


0.10 


391.32 


128.96 


108.78 


221 


109.14 


105.19 


50.8 


99.1 


97.87 



a limiting value in very dilute solutions; this quantity is known as the 
equivalent conductance at infinite dilution and is represented by the 
symbol A . 

An examination of the results of conductance measurements of many 
electrolytes of different kinds shows that the variation of the equivalent 
conductance with concentration depends to a great extent on the type of 
electrolyte, rather than on its actual nature. For strong uni-univalent 
electrolytes, i.e., with univalent cation and anion, such as hydrochloric 
acid, the alkali hydroxides and the alkali halides, the decrease of equiva- 
lent conductance with increasing concentration is not very large. As the 
valence of the ions increases, however, the falling off is more marked; 
this is shown by the curves in Fig. 20 in which the equivalent conduct- 



160 



120 



90 



i 



Potassium Chloride 




0.01 0.02 0.05 0.1 

Concentration in Equtv. per Liter 

Fio. 20. Conductances of electrolytes of different types 



THE CONDUCTANCE RATIO 51 

anccs of potassium chloride, a typical uni-univalont strong electrolyte, 
and of nickel sulfatc, a hi-bivalent electrolyte, are plotted as functions of 
the concentration. Electrolytes of an intermediate valence type, e.g., 
potassium sulfatc, a uni-bivalcnt electrolyte, and barium chloride, which 
is a bi-uriivalcnt salt, behave in an intermediate manner. 

The substances referred to in Table VIII are all strong, or relatively 
strong, electrolytes, but weak electrolytes, such as weak acids and bases, 
exhibit an apparently different behavior. The results for acetic acid, a 
typical weak electrolyte, at 25 are given in Table IX. 

TABLE IX. EQUIVALENT CONDUCTANCE OF ACETIC ACID AT 25 

Concn. 0.0001 0.001 0.005 0.01 0.02 0.05 0.10 N 

A 131.6 48.63 22.80 16.20 11.57 7.36 5.20 ohms- 1 cm. 2 

It is seen that at the higher concentrations the equivalent conductance 
is very low, which is the characteristic of a weak electrolyte, but in the 
more dilute solutions the values rise with great rapidity; the limiting 
equivalent conductance of acetic acid is known from other sources to be 
390.7 ohms" 1 cm. 2 at 25, and so there must be an increase from 131.6 
to this value as the solution is made more dilute than 10 4 equiv. per 
liter. The plot of the results for acetic acid, shown in Fig. 20, may be 
regarded as characteristic of a weak electrolyte. As mentioned in 
Chap. I, it is not possible to make a sharp distinction between electro- 
lytes of different classes, and the variation of the equivalent conductance 
of an intermediate electrolyte, such as trichloroacetic, cyanoacetic and 
mandelic acids, lies between that for a weak electrolyte, e.g., acetic acid, 
and a moderately strong electrolyte, e.g., nickel sulfate (cf. Fig. 20). 

The Conductance Ratio. The ratio of the equivalent conductance 
(A) at any concentration to that at infinite dilution (A )* has played an 
important part in the development of electrochemistry; it is called the 
conductance ratio, and is given the symbol a, thus 



In the calculations referred to on page 10, Arrhenius assumed the con- 
ductance ratio to be equal to the degree of dissociation of the electro- 
lyte; this appears to be approximately true for weak electrolytes, but 
not for salts and strong acids and bases. Quite apart from any theoreti- 
cal significance which the conductance ratio may have, it is a useful 
empirical quantity because it indicates the extent to which the equivalent 
conductance at any specified concentration differs from the limiting value. 
The change of conductance ratio with concentration gives a measure of 
the corresponding falling off of the equivalent conductance. In accord- 
ance with the remarks made previously concerning the connection be- 

* For the methods of extrapolation of conductance data to give the limiting value, 
see p. 54. 



52 ELECTROLYTIC CONDUCTANCE 

tween the variation of equivalent conductance with concentration and 
the valence type of the electrolyte, a similar relationship should hold for 
the conductance ratio. In dilute solutions of strong electrolytes, other 
than acids, the conductance ratio is in fact almost independent of the 
nature of the salt and is determined almost entirely by its valence type. 
Some mean values, derived from the study of a number of electrolytes 
at room temperatures, are given in Table X; the conductance ratio at any 

TABLE X. CONDUCTANCE RATIO AND VALENCE TYPE OP SALT 

Valence Type 0.001 0.01 0.1 N 

Uni-uni 0.98 0.93 O.S3 

21} 0.05 0.87 0.75 

Bi-bi 0.85 0.65 0.40 

given concentration is seen to be smaller the higher the valence type. 
For weak electrolytes the conductance ratios are obviously very much 
less, as is immediately evident from the data in Table IX. 

As a general rule increase of temperature increases the equivalent 
conductance both at infinite dilution and at a definite concentration ; the 
conductance ratio, however, usually decreases with increasing tempera- 
ture, the effect being greater the higher the concentration. These con- 
clusions are supported by the results for potassium chloride solutions in 
Table XI taken from the extensive measurements of Noyes and his 

TABLE XI. VARIATION OF CONDUCTANCE RATIO OP POTASSIUM CHLORIDE SOLUTIONS 

WITH TEMPERATURE 

18 100 150 21S 306 

0.01 N 0.94 0.91 0.90 0.90 0.8 i 

0.08 N 0.87 0.83 0.80 0.77 O.G4 

collaborators. 16 The falling off is more marked for electrolytes of higher 
valence type, and especially for weak electrolytes. A few cases are known 
in which the conductance ratio passes through a maximum as the tem- 
perature is increased; this effect is probably due to changes in the extent 
of dissociation of relatively weak electrolytes. 

Equivalent Conductance Minima. Provided the dielectric constant 
of the medium is greater than about 30, the conductance behavior in that 
medium is usually similar to that of electrolytes in water; the differences 
are not fundamental and are generally differences of degree only. With 
solvents of low dielectric constant, however, the equivalent conductances 
often exhibit distinct abnormalities. It is frequently found, for example, 
that with decreasing concentration, the equivalent conductance decreases 
instead of increasing; at a certain concentration, however, the value 
passes through a minimum and the subsequent variation is normal. In 
other cases, e.g., potassium iodide in liquid sulfur dioxide and tetra- 

14 Noyes et al., J. Am. Chem. Soc., 32, 159 (1910); sec also, Kraus, "Klectrioully 
Conducting Systems," 1922, Chap. VI. 



EQUIVALENT CONDUCTANCE MINIMA 



53 



propylammonium iodide in methylene chloride, the equivalent conduct- 
ances pass through a maximum and a minimum with decreasing concen- 
tration. The problem of the minimum equivalent conductance was 
investigated by Walden 16 who concluded that there was a definite rela- 
tionship between the concentration at which such a minimum could be 
observed and the dielectric constant of the solvent. If c m i n . is the con- 
centration for the minimum equivalent conductance, and D is the dielec- 
tric constant of the medium, then Walden's conclusion may be repre- 
sented as 

c mm . = kD\ (14) 

where k is a constant for the given electrolyte. It is evident from this 
equation that in solvents of high dielectric constant the minimum should 
be observed only at extremely high concentrations; even if such solutions 
could be prepared, it is probable that other factors would interfere under 
these conditions. It will be seen later that equation (14) has a theoreti- 
cal basis. 




-4.0 - 



-4.5 



-3.5 



-2.5 



-1.5 



logc 



Fia. 21. Influence of dielectric; constant on conductance (Kuoss and Kraus) 
M Walden, Z. physik. Chcm., 94, 263 (1920); 100, 512 (1922). 



54 ELECTROLYTIC CONDUCTANCE 

The influence of dielectric constant on the variation of equivalent 
conductance with concentration has been demonstrated in a striking 
manner by the measurements made by Fuoss and Kraus 17 on tetra- 
isoamylammonium nitrate at 25 in a series of mixtures of water and 
dioxane, with dielectric constant varying from 78.6 to 2.2. The results 
obtained are depicted graphically in Fig. 21, the dielectric constant of 
the medium being indicated in each case; in view of the large range of 
conductances and concentrations the figure has been made more compact 
by plotting log A against log c. It is scon that as the dielectric constant 
becomes smaller, the falling off of equivalent conductance with increasing 
concentration is more marked. At sufficiently low dielectric constants 
the conductance minimum becomes evident; the concentration at which 
this occurs decreases with decreasing dielectric constant, in accordance 
with the Walden equation. The theoretical implication of these results 
will be considered more fully in Chap. V. 

Equivalent Conductance at Infinite Dilution. A number of methods 
have been proposed at various times for the extrapolation of experi- 
mental equivalent conductances to give the values at infinite dilution. 
Most of the procedures described for strong electrolytes are based on the 
use of a formula of the type 

A = Ao ac n , (15) 

where A is the equivalent conductance measured at concentration c; the 
quantities a and n are constants, the latter being approximately 0.5, as 
required by the modern theoretical treatment of electrolytes. If data 
for sufficiently dilute solutions are available, a reasonably satisfactory 
value for A may be obtained by plotting the experimental equivalent 
conductances against the square-root of the concentration and performing 
a linear extrapolation to zero concentration. It appears doubtful, from 
recent accurate work, if an equation of the form of (15) can represent 
completely the variation of equivalent conductance over an appreciable 
range of concentrations; it follows, therefore, that no simple extrapola- 
tion procedure can be regarded as entirely satisfactory. An improved 
method 18 is based on the theoretical Onsagcr equation (p. 90), i.e., 

A' - A "*" ^ r 
A ~~ i _ /W 

where A and B are constants which may be evaluated from known 
properties of the solvent. The results for \' derived from this equation 
for solutions of appreciable concentration are not constant, and hence 
the prime has been added to the symbol for the equivalent conductance. 

17 Fuoss and Kraus, jr. Am. Chem. Soc., 55, 21 (1933). 
"Shedlovsky, J. Am. Chem. Soc., 54, 1405 (1932). 



THE INDEPENDENT MIGRATION OF IONS 



56 



For many strong electrolytes Aj is a linear function of the concentration, 
thus 

Ai = Ao + ac, 



so that if the values of AQ are plotted against the concentration c, the 
equivalent conductance at infinite dilution may be obtained by linear 
extrapolation. The data for 
sodium chloride and hydro- 
chloric acid at 25 are shown 
in Fig. 22; the limiting equiv- 
alent conductances at zero 
concentration are 126.45 and 
426.16ohm- l cm. 2 ,respectively. 
For weak electrolytes, no 
form of extrapolation is sat- 
isfactory, as will be evident 
from an examination of Fig. 
20. The equivalent conduct- 
ance at infinite dilution can 
then be obtained only from 
the values of the individ- 
ual ions, as will be described 
shortly. For electrolytes ex- 
hibiting intermediate behav- 
ior, e.g., solutions of salts in 
media of relatively low di- 
electric constant, an extrapo- 
lation method based on theo- 
retical considerations can bo 
employed (see p. 167). 

The Independent Migration of Ions. A survey of equivalent con- 
ductances at infinite dilution of a number of electrolytes having an ion 
in common will bring to light certain regularities; the data in Table XII, 




128 
0.02 0.04 0.06 0.08 

Concentration in Equiv. per Liter 
FIG. 22. Extrapolation to infinite dilution 



TABLE XII. 

Electrolyte 

KC1 
KNOj 



COMPARISON OF EQUIVALENT CONDUCTANCES AT INFINITE DILUTION 

A Electrolyte A Difference 

130.0 NaCl 108.9 21.1 

126.3 NaNO, 105.2 21.1 

133.0 NajS0 4 111.9 21.1 



for example, are for corresponding sodium and potassium salts at 18. 
The difference between the conductances of a potassium and a sodium 
salt of the same anion is seen to be independent of the nature of the latter. 
Similar results have been obtained for other pairs of salts with an anion 
or a cation in common, both in aqueous and non-aqueous solvents. 
Observations of this kind were first made by Kohlrausch (1879, 1885) by 



56 ELECTROLYTIC CONDUCTANCE 

comparing equivalent conductances at high dilutions; he ascribed them 
to the fact that under these conditions every ion makes a definite con- 
tribution towards the equivalent conductance of the electrolyte, irre- 
spective of the nature of the other ion with which it is associated in tho 
solution. The value of the equivalent conductance at infinite dilution 
may thus'be regarded as made up of the sum of two independent factors, 
one characteristic of each ion; this result is known as Kohlrausch's law of 
independent migration of ions. The law may be expressed in the form 

Ao = \+ + Ai, (16) 

where X+ and X?. are known as the ion conductances, of cation and anion, 
respectively, at infinite dilution. The ion conductance is a definite con- 
stant for each ion, in a given solvent, its value depending only on the 
temperature. 

It will be seen later that the ion conductances at infinite dilution are 
related to the speeds with which the ions move under the influence of an 
applied potential gradient. Although it is possible to derive their values 
from the equivalent conductances of a number of electrolytes by a method 
of trial and error, a much more satisfactory procedure is based on the use of 
accurate transference number data; these transference numbers are deter- 
mined by the relative speeds of the ions present in the electrolyte and 
hence are related to the relative ion conductances. The determination 
of transference numbers will be described in Chap. IV and the method of 
evaluating ion conductances will be given there; the results will, however, 
be anticipated and some of the best values for ion conductances in water 
at 25 are quoted in Table XIII. 19 It should be noted that since these are 

TABLE XIII. ION CONDUCTANCES AT INFINITE DILUTION AT 25 IN OHMS" 1 CM. 2 

Cation X^. a X 10 2 Anion Xi a X 10 2 

H+ 349.82 1.42 OH~ 198 1.60 

T1+ 74.7 1.87 Br~ 78.4 1.87 

K+ 73.52 1.89 I~ 76.8 1.86 

NH+ 73.4 1.92 Cl 76.34 1.88 

A- 61.92 1.97 NO 3 ~ 71.44 1.80 

Na + 50.11 2.09 ClOr 68.0 

Li+ 38.69 2.26 HCO 3 ~ 44.5 

JBa++ 63.64 2.06 }SO 4 79.8 1.96 

JCa++ 59.50 2.11 iFe(CN)jf 101.0 

JSr-n- 59.46 2.11 lFe(CN)<f 110.5 

53.06 2.18 



actually equivalent conductances, symbols such as JBa++and JFe(CN)if 
are employed. (The quantities recorded in the columns headed a are 
approximate temperature coefficients; their significance will be explained 
on page 61.) 

In the results recorded in Table XIII, there appears to be no con- 
nection between ionic size and conductance; for a number of ions be- 

19 See Maclnnes, J. Franklin Iwt., 225, 661 (1938); "The Principles of Electro- 
chemistry." 1939, p. 342. 



APPLICATION OF ION CONDUCTANCES 57 

longing to a homologous series, as for example the ions of normal fatty 
acids, a gradual decrease of conductance is observed and a limiting value 
appears to be approached with increasing chain length. The data for 
certain fatty acid anions are known accurately, but others are approxi- 
mate only; the values in Table XIV, nevertheless, show the definite trend 

TABLE XIV. ION CONDUCTANCES OP PATTY ACID IONS AT 25 

Anion Formula X_ 

Formate HCOr ~52 ohms" 1 cm. 1 

Acetate CH 8 COr 40.9 

Propionate CH 8 CH,CO^ 35.8 

Butyrate CH,(CH,)jCOf 32.6 

Valerianate CH 8 (CH a )aCOf ~29 

Caproate CH,(CH,)4COr ~28 



towards a constant ion conductance. A similar tendency has been ob- 
served in connection with the conductances of alkylammonium ions. 

A large number of ion conductances, of more or less accuracy, have 
been determined in non-aqueous solvents; reference to these will be made 
shortly in the section dealing with the relationship between the con- 
ductance of a given ion in various solvents and the viscosities of the latter. 

Application of Ion Conductances. An important use of ion con- 
ductances is to determine the equivalent conductance at infinite dilution 
of certain electrolytes which cannot be, or have not been, evaluated from 
experimental data. For example, with a weak electrolyte the extrapo- 
lation to infinite dilution is very uncertain, and with sparingly soluble 
salts the number of measurements which can be made at appreciably 
different concentrations is very limited. The value of A can, however, 
bo obtained by adding the ion conductances. For example, the equiva- 
lent conductance of acetic acid at infinite dilution is the sum of the con- 
ductances of the hydrogen and acetate ions; the former is derived from 
a study of strong acids and the latter from measurements on acetates. 
It follows, therefore, that at 25 

Ao(CH s co t H) = XH+ + XCH,CO;, 

= 349.8 + 40.9 = 390.7 ohms" 1 cm. 2 

The same result can be derived in another manner which is often con- 
venient since it avoids the necessity of separating the conductance of an 
electrolyte into the contributions of its constituent ions. The equivalent 
conductance of any electrolyte MA at infinite dilution A (MA> is equal to 
XM* + X A -, where XM+ and X A - are the ion conductances of the ions M+ 
and A~ at infinite dilution; it follows, therefore, that 

Ao(MA) = Ao(MCl) + Ao(NaA) ~ Ao(NaCl), 

where A O <MCD, A (NaA) and AO(NCD are the equivalent conductances at 
infinite dilution of the chloride of the metal M, i.e., MCI, of the sodium 
salt of the anion A, i.e., NaA, and of sodium chloride, respectively. Any 



58 ELECTROLYTIC CONDUCTANCE 

convenient anion may be used instead of the chloride ion, and similarly 
the sodium ion may be replaced by another metallic cation or by the 
hydrogen ion. For example, if M+ is the hydrogen ion and A~ is the 
acetate ion, it follows that 



Ao(CH s CO,H) = Ao(HCl) + AocCHjCOjN*) ~ A (NC1) 

= 426.16 + 91.0 - 126.45 
= 390.71 ohms~ l cm. 2 at 25. 

In order to determine the equivalent conductance of a sparingly 
soluble salt it is the practice to add the conductances of the constituent 
ions; thus for silver chloride and barium sulfate the results are as follows: 

XA+ + Xcr 

61.92 + 76.34 = 138.3 ohms" 1 cm. 2 at 25, 



63.64 + 79.8 = 143.4 ohms~ l cm. 2 at 25. 

Absolute Ionic Velocities: Ionic Mobilities. The approach of the 
equivalent conductances of all electrolytes to a limiting value at very 
high dilutions may be ascribed to the fact that under these conditions 
all the ions that can be derived from one gram equivalent are taking part 
in conducting the current. At high dilutions, therefore, solutions con- 
taining one equivalent of various electrolytes will contain equivalent 
numbers of ions; the total charge carried by all the ions will thus be the 
same in every case. The ability of an electrolyte to transport current, 
and hence its conductance, is determined by the product of the number 
of ions and the charge carried by each, i.e., the total charge, and by the 
actual speeds of the ions. Since the total charge is constant for equiva- 
lent solutions at high dilution, the limiting equivalent conductance of an 
electrolyte must depend only on the ionic velocities: it is the difference 
in the speeds of the ions which is consequently responsible for the differ- 
ent values of ion conductances. The speed with which a charged particle 
moves is proportional to the potential gradient, i.e., the fall of potential 
per cm., directing the motion, and so the speeds of ions are specified 
under a potential gradient of unity, i.e., one volt per cm. These speeds 
arc known as the mobilities of the ions. 

If w+ and u*L are the actual velocities of positive and negative ions 
of a given electrolyte at infinite dilution under unit potential gradient, 
i.e., the respective mobilities, then the equivalent conductance at infinite 
dilution must be proportional to the sum of these quantities; thus 

Ao = k(u+ + u ) = fc< + ku-> (17) 

where k is the proportionality constant which must be the same for all 
electrolytes. The equivalent conductance, as seen above, is the sum of 
the ion conductances, i.e., 

Ao - 4 + X?., 



ABSOLUTE IONIC VELOCITIES 59 

and since Xl and u+ are determined only by the nature of the positive 
ion, while X_ and u!L are determined only by the negative ion, it follows 
that 

X3_ = ku\ and X?. = fci. (18) 

Imagine a very dilute solution of an electrolyte, at a concentration 
c equiv. per liter, to be placed in a cube of 1 cm. side with square elec- 
trodes of 1 sq. cm. area at opposite faces, and suppose an E.M.F. of 1 volt 
to be applied. By definition, the specific conductance (*) is the con- 
ductance of a centimeter cube, and the equivalent conductance of the 
given dilute solution, which is virtually that at infinite dilution, is 
1000 JC/G [see equation (5)], so that 

1000 - = Ao = \\ + X?., 
c(XJ + X ) 



" 1000 

It was shown on page 30 that when a potential difference of 1 volt is 
applied to a 1 cm. cube, the current in amperes is numerically equal to 
the specific conductance, i.e., 



1000 

and this represents the number of coulombs flowing through the cube 
per second. 

Since the mobilities u+ and u!L are the ionic velocities in cm. per sec. 
under a fall of potential of 1 volt per cm., all the cations within a length 
of u+ cm. will pass across a given plane in the direction of the current 
in 1 sec., while all the anions within a length of u*L cm. will pass in the 
opposite direction. If the plane has an area of 1 sq. cm., all the cations 
in a volume u+ cc. and all the anions in u*L cc. will move in opposite 
directions per sec.; since 1 cc. of the solution contains c/1000 equiv., it 
follows that a total of (u^ + w) c/1000 equiv. of cations and anions will 
be transported by the current in 1 sec. Each equivalent of any ion 
carries one faraday (F) of electricity; hence the total quantity carried 
per sec. will be F(u+ -f u?.) c/1000 coulombs. It has been seen that 
the quantity of electricity flowing per sec. through the 1 cm. cube is 
equal to 7 as given above; consequently, 

u!L)c c(X3. + X ) 



1000 1000 

/. F(4 + ui) - X$. + X?.. (19) 

It follows, therefore, that the constant k in equation (17) is equal to F, 
and hence by equation (18), 

X?. =* Fu+ and X?. = Fu. (20) 



60 ELECTROLYTIC CONDUCTANCE 

The absolute velocity ef any ion in cm. per sec. under a potential gradient 
of 1 volt per cm. can thus be obtained by dividing the ion conductance 
in ohms~ l cm. 2 by the value of the faraday in coulombs, i.e., 96,500. 
Since the velocity is proportional to the potential gradient, as a conse- 
quence of the applicability of Ohm's law to electrolytes, the speed of an 
ion can be evaluated for any desired fall of potential. It should be 
pointed out that equation (20) gives the ionic velocity at infinite dilution; 
the values decrease with increasing concentration, especially for strong 
electrolytes. 

The ion conductances in Table XIII have been used to calculate the 
mobilities of a number of ions at infinite dilution at 25; the results are 
recorded in Table XV. It will be observed that, apart from hydrogen 

TABLE XV. CALCULATED IONIC MOBILITIES AT 25 

Mobility Mobility 

Cation cm. /sec. Anion cm./sec. 

Hydrogen 36.2X10^ Hydroxyl 20.5X10^ 

Potassium 7.61 Sulfate 8.27 

Barium 6.60 Chloride 7.91 

Sodium 5.19 Nitrate 7.40 

Lithium 4.01 Bicarbonate 4.61 

and hydroxyl ions, most ions have velocities of about 5 X 10~ 4 cm. per 
sec. at 25 under a potential gradient of unity. The influence of tem- 
perature on ion conductance, and hence on ionic speeds, is discussed 
below. 

Experimental Determination of Ionic Velocities. An attempt to 
measure the speeds of ions directly was made by Lodge (1886) who made 
use of some characteristic property of the ion, e.g., production of color 
with an indicator or formation of a precipitate, to follow its movement 
under an applied field. In Lodge's apparatus the vessels containing the 
anode and cathode, respectively, were joined by a tube 40 cm. long filled 
with a conducting gelatin gel in which the indicating material was 
dissolved. For example, in determining the velocity of barium and 
chloride ions the gel contained acetic acid as conductor and a trace of 
silver sulfate as indicator; barium chloride was used in both anode and 
cathode vessels and the electrodes were of platinum. On passing current 
the barium and chloride ions moved into the gel, in opposite directions, 
producing visible precipitates of barium sulfate and silver chloride, re- 
spectively: the rates of forward movement of the precipitates gave the 
speeds of the respective ions under the particular potential gradient 
employed. 

Although the results obtained by Lodge in this manner were of the 
correct order of magnitude, they were generally two or three times less 
than those calculated from ion conductances by the method described 
above. The discrepancies were shown by Whetham (1893) to be due to 
a. non-uniform potential gradient and to lack of precautions to secure 



INFLUENCE OF TEMPERATURE ON ION CONDUCTANCES 61 

sharp boundaries. Taking these factors into consideration, Whetham 
devised an apparatus for observing the movement of the boundary be- 
tween a colorless and a colored ion, or between two colored ions, without 
the use of a gel. The values for the velocities of ions obtained in this 
manner were in satisfactory agreement with those calculated, especially 
when allowance was made for the fact that the latter refer to infinite 
dilution. The principle employed by Whetham is almost identical with 
that used in the modern "moving boundary" method for determining 
transference numbers and this is described in Chap. IV. 

Influence of Temperature on Ion Conductances. Increase of tem- 
perature invariably results in an increase of ion conductance at infinite 
dilution; the variation with temperature may be expressed with fair 
accuracy by means of the equation 

X? = XS 5 [1 + (* - 25) + 0(i - 25) 2 ], (21) 

where X? is the ion conductance at infinite dilution at the temperature t, 
and X 5 is the value at 25. The factors a and are constants for a given 
ion in the particular solvent; for a narrow temperature range, e.g., about 
10 on either side of 25, the constant ft may be neglected, and approxi- 
mate experimental values of a are recorded in Table XIII above. It 
vill be apparent that, except for hydrogen and hydroxyl ions, the tem- 
perature coefficients a. are all very close to 0.02 at 25. 

Since the conductance of an ion depends on its rate of movement, it 
seems reasonable to treat conductance in a manner analogous to that 
employed for other processes taking place at a definite rate which in- 
creases with temperature. If this is the case, it is possible to write 



X = Ae- E ' RT , (22) 

where A is a constant, which may be taken as being independent of 
temperature over a relatively small range; E is the activation energy of 
the process which determines the rate of movement of the ions, R is the 
gas constant and T is the absolute temperature. Differentiation of 
equation (22) with respect to temperature, assuming A to be constant, 
gives 

<HnX 1 d\ Q E 
dT ~ \' dT~ RT*' (23) 

Further, differentiation of equation (21) with respect to temperature, 
the factor being neglected, shows that for a narrow temperature range 



_ 
X ' dT " " 

and hence the activation energy is given by 

E - aRT*. 



62 



ELECTROLYTIC CONDUCTANCE 



Since a is approximately 0.02 for all ions, except hydrogen and hydroxyl 
ions, at 25, it is seen that for conductance in water the activation energy 
is about 3.60 kcal. in every case. 

Ion Conductance and Viscosity : Temperature and Pressure Effects. 
It is an interesting fact that the activation energy for electrolytic con- 
ductance is almost identical with that for the viscous flow of water, viz., 
3.8 kcal. at 25; hence, it is probable that ionic conductance is related to 
the viscosity of the medium. Quite apart from any question of mecha- 
nism, however, equality of the so-called activation energies means that 
the positive temperature coefficient of ion conductance is roughly equal 
to the negative temperature coefficient of viscosity. In other words, the 
product of the conductance of a given ion and the viscosity of water at 
a series of temperatures should be approximately constant. The results 
in Table XVI give the product of the conductance of the acetate ion at 



TABLE XVI. CONDUCTANCE-VISCOSITY PRODUCT OP THE ACETATE ION 

Temperature 18 25 59 75 100 128 156 

Xe*> 0.366 0.368 



0.366 0.368 0.369 0.368 0.369 0.369 



L05 



Ap 
AT 



0.95 



infinite dilution (Xo) and the viscosity of water (ijo), i.e., A i?o, at tempera- 
tures between and 156; the re- 
sults are seen to be remarkably con- 
stant. It is true that such constancy 
is not always obtained, but the con- 
ductance-viscosity product for infi- s 
nite dilution is, at least, approxi- 
mately independent of temperature 
for a number of ions in water. The 
data for non-aqueous media are less 
complete, but it appears that in gen- 
eral the product of the ionic conduc- 
tance and the viscosity in such media 
is also approximately constant over a 
range of temperatures.* 

Another fact which points to a 
relationship between ionic mobility 
and viscosity is the effect of pressure 
on electrolytic conductance. Data 
are not available for infinite dilution, 
but the results of measurements on 

a number of electrolytes at a concentration of 0.01 N in water at 20 are 
shown in Fig. 23; the ordinates give the ratio of the equivalent conduct- 
ance at a pressure p to that at unit pressure, i.e., A p /Ai, while the ab- 
scissae represent the pressures in kg. per sq. cm. 20 The dotted line 

* It should be emphasized that the conductance-viscosity product constancy is, on 
the whole, not applicable to solutions of appreciable concentration. 

M Data mainly from Kdrber, Z. physik. Chem., 67, 212 (1909); see also, Adams and 
Hall, J. Phy 9 . Chem., 35, 2145 (1931); Zisman, Phys. Rev., 30, 151 (1932). 




100 200 

Pressure 



k./cm. 2 



Fio. 23. 



Variation of conductance 
with pressure 



INFLUENCE OP SOLVENT ON ION CONDUCTANCE 63 

indicates the variation with pressure of the fluidity, i.e., the reciprocal 
of the viscosity, of water relative to that at unit pressure. The existence 
of a maximum in both the conductance and fluidity curves suggests that 
there is some parallelism between these quantities: exact agreement 
would be expected only at infinite dilution, for other factors which are 
influenced by pressure may be important in solutions of appreciable 
concentration. 

The relationship between viscosity and ion conductance has been 
interpreted in at least two ways; some writers have suggested that the 
constancy of the product Xoi?o proves the applicability of Stokes's law to 
ions in solution. According to this law 

/ - Gin,, (24) 

where u is the steady velocity with which a particle of radius r moves 
through a medium of viscosity 17 when a force / is applied. For a par- 
ticular ion, r may be regarded as constant, and since the conductance is 
proportional to the speed of the ion under the influence of a definite 
applied potential (see p. 58), it follows that according to Stokes's law 
X 7?o should be constant, as found experimentally. Another suggestion 
that has been made to explain this fact is that the ion in solution is so 
completely surrounded by solvent molecules which move with it, that is 
to say, it is so extensively "solvated," that its motion through the 
medium is virtually the same as the movement of solvent molecules past 
one another in viscous flow of the solvent. 

It is not certain, however, that either of these conclusions can be 
legitimately drawn from the results. Since the activation energies for 
ionic mobility and viscous flow are approximately equal, it is reasonable 
to suppose that the rate-determining stage in the movement of an ion 
under the influence of an applied electric field and that involved in the 
viscous flow of the medium are the same. It has been suggested that 
in the latter process the slow stage is the jump of a solvent molecule 
from one equilibrium position to another, and this must also be rate- 
determining for ionic conductance. It appears, therefore, that when an 
electric field is applied to a solution containing ions, the latter can move 
forward only if a solvent molecule standing in its path moves in the 
opposite direction. The actual rate of movement of an ion will depend 
to a great extent on its effective size in the given solvent, but the tem- 
perature coefficient should be determined almost entirely by the activa- 
tion energy for viscous flow. 

Influence of Solvent on Ion Conductance. In the course of his in- 
vestigation of the conductance of tetraethylammonium iodide in various 
solvents, Walden (1906) noted that the product of the equivalent con- 
ductance at infinite dilution and the viscosity of the solvent was approxi- 



64 



ELECTROLYTIC CONDUCTANCE 



mately constant and independent of the nature of the latter; 21 this 
conclusion, known as Walden's rule, may be expressed as 



constant, 



(25) 



for a given electrolyte in any solvent. The values of AOTJO for the afore- 
mentioned salt, obtained by Walden and others, in a variety of media are 
given in Table XVII; the viscosities are in poises, i.e., dynes per sq. cm. 



TABLE XVII. VALUES OF AQI?O FOR TETRAETHYLAMMONIUM IODIDE IN 
VARIOUS SOLVENTS 



Solvent 



CH 3 OH CHsCOCH, CH,CN 
0.63 0.66 0.64 



C 2 H 4 C1 2 
0.60 



CH 3 N0 2 
0.69 



C.H,N0 2 
0.67 



C 6 H 6 OH 
0.63 



The results were generally obtained at 25, but since X *?o is approximately 
independent of temperature, as seen above, it is evident that Aoijo will 
also not vary appreciably. 

If Walden's rule holds for other electrolytes, it follows, since A is the 
sum of the conductances of the constituent ions, that Xoijo should be 
approximately constant for a given ion in all solvents. The extent to 
which this is true may be seen from the conductance-viscosity products 
for a number of ions collected in Table XVIII; the data for hydrogen 

TABLE XVIII. ION CONDUCTANCE-VISCOSITY PRODUCTS IN VARIOUS SOLVENTS AT 25 



Solvent 


Na+ 


K+ 


Ag+ 


N(CiHi)/ 


I- 


cio 4 - 


Picrate 


H S 


0.460 


0.670 


0.563 


0.295 


0.685 


0.606 


0.276 


CH a OH 


0.250 


0.293 


0.274 


0.338 


0.334 


0.387 


0.255 


C,H 6 OH 


0.204 


0.235 


0.195 


0.310 


0.290 


0.340 


0.292 


CH,COCH 8 


0.253 


0.259 





0.284 


0.366 


0.366 


0.275 


CH 8 CN 


0.241 


0.296 





0.296 


0.347 


0.359 


0.268 


CH,N0 2 


0.364 


0.383 


0.326 


0.310 


0.403 





0.276 


C.H.NO, 








0.330 


0.322 





0.366 


0.277 


NH, (-33) 


0.333 


0.430 


0.297 





0.437 









and hydroxyl ions are deliberately excluded from Table XVIII, for 
reasons which will appear later. The results show that, for solvents 
other than water, the conductance-viscosity product of a given ion is 
approximately constant, thus confirming the approximate validity of 
Walden's rule. If Stokes's law were obeyed, the value of Xow would be 
constant only if the effective radius of the ion were the same in the 
different media; since there are reasons for believing that most ions are 
solvated in solution, the dimensions of the moving unit will undoubtedly 

a Walden et al. t Z. physik. Chem., 107, 219 (1923); 114, 297 (1925); 123, 429 (1926); 
"Salts, Acids and Bases," 1929; Ulich, Fortschritte der Chemie, Physik and phys. Chem., 
18, No. 10 (1926); Trans. Faraday Soc., 23, 388 (1927); Barak and Hartley, Z. phys. 
Chem., 165, 273 (1933); Coates and Taylor, /. Chem. Soc., 1245, 1495 (1936); see also 
Longsworth and Maclnnes, J. Phys. Chem., 43, 239 (1939). 



ABNORMAL ION CONDUCTANCES 65 

vary to some extent and exact constancy of the conductance-viscosity 
product is not to be expected. It should be pointed out, also, that the 
deduction of Stokes's law is based on the assumption of a spherical 
particle moving in a continuous medium, and this condition can be 
approximated only if the moving particle is large in comparison with the 
molecules of the medium. It is of interest to note in this connection that 
for large ions, such as the tetraethylammonium and picrate ions, the X O T;O 
values are much more nearly constant than is the case with other ions; 
further, the behavior of such ions in water is not exceptional. Stokes's 
law is presumably applicable to these large ions, and since they are 
probably solvated to a small extent only, they will have the same size 
in all solvents ; the constancy of the conductance-viscosity product is thus 
to be expected. For small ions the value of X O T?O will depend to some 
extent on the fundamental properties of the solvent, as well as on the 
effective size of the ion: for such ions, too, Stokes's law probably does not 
hold, and so exact constancy of the conductance-viscosity product is not 
to be expected. 

An interesting test of the validity of the Walden rule is provided by 
the conductance measurements, made by LaMer and his collaborators, of 
various salts in a series of mixtures of light water (H 2 0) and heavy water 
(D 2 0). The results indicate that, although the rule holds approximately, 
it is by no means exact. 22 

Although no actual tabulation has been made here of the ion con- 
ductances of various ions in different solvents, it may be pointed out that 
these values are implicit in Table XVIII; knowing the viscosity of the 
solvent, the ion conductance at infinite dilution can be calculated. 

Abnormal Ion Conductances. An inspection of the conductance- 
viscosity products for the hydrogen ion recorded in Table XIX imme- 

TABLB XIX. CONDUCTANCE-VISCOSITY PRODUCT OF THE HYDROGEN ION 

Solvent H 2 CH,OH C,HOH CH,COCH 3 CH 3 NO 2 CH 6 NO, NH, 
Xoi?o 3.14 0.774 0.641 0.277 0.395 0.401 0.359 

diately reveals the fact that the values in the hydroxylic solvents, and 
particularly in water, are abnormally high. It might appear, at first 
sight, that the high conductance-viscosity product of the hydrogen ion 
in water could be explained by its small size. In view of the high free 
energy of hydration of the proton (cf. p. 308), however, in aqueous solu- 
tion the reaction 

H+ + H 2 O * H 3 O+, 

where H+ represents a proton or "bare" hydrogen ion, must go to virtual 
completion. The hydrogen ion in water cannot, therefore, consist of a 

LaMer et a/., J. Chem. Phys., 3, 406 (1935); 9, 265 (1941); J. Am. Chcm. Soc., 
58, 1642 (1936); 59, 2425 (1937); see also, Longsworth and Maclnnes, ibid., 59, 1666 
(1937). 



66 ELECTROLYTIC CONDUCTANCE 

bare ion, but must be combined with at least one molecule of water. 
The hydrogen ion in water is thus probably to be represented by H 8 O+, 
and its effective size and conducting power should then be approximately 
the same as that of the sodium ion; it is, however, actually many times 
greater, as the figures in Table XIX show. It is of interest to note that 
in acetone, nitromethane, nitrobenzene, liquid ammonia, and probably 
in other non-hydroxylic solvents, the conductance-viscosity product, and 
hence the conductance, of the hydrogen ion, which is undoubtedly sol- 
vated, is almost the same as that of the sodium ion. It is doubtful, 
therefore, if the high conductance of the hydrogen ion in hydroxylic 
solvents can be explained merely by its size. 

The suggestion has been frequently made that the high conductance 
is due to a type of Grotthuss conduction (p. 7), and this view has been 
developed by a number of workers in recent years. 23 It is supposed, as 
already indicated, that the hydrogen ion in water is H 8 0+ with three 
hydrogen atoms attached to the central oxygen atom. When a potential 
gradient is applied to an aqueous solution containing hydrogen ions, the 
latter travel to some extent by the same mechanism as do other ions, but 
there is in addition another mechanism which permits of a more rapid 
ionic movement. This second process is believed to involve the transfer 
of a proton (H+) from a H 3 O+ ion to an adjacent water molecule; thus 

H H H H 

-> I + I 

H O H Q H. 



The resulting H 3 f ion can now transfer a proton to another water 
molecule, and in this way the positive charge will be transferred a con- 
siderable distance in a short time. It has been calculated from the known 
structure of water that the proton has to jump a distance of 0.86 X 10~ 8 
cm. from a HaO" 1 " ion to a water molecule, but as a result the positive 
charge is effectively transferred through 3.1 X 10~ 8 cm. The electrical 
conductance will thus be much greater than that due solely to the normal 
mechanism. It will be observed that after the proton has passed from 
the HaO" 1 " ion to the water molecule, the resulting water molecule, i.e., 
the one shown on the right-hand side, is oriented in a different manner 
from that to which the proton was transferred, i.e., the one on the left- 
hand side. If the process of proton jumping is to continue, each water 
molecule must rotate after the proton has passed on, so that it may be 
ready to receive another proton coming from the same direction. The 
combination of proton transfer and rotation of the water molecule, which 
has some features in common with the Grotthuss mechanism for conduc- 

Hiickel, Z. Ekktrochem., 34, 546 (1928); Bernal and Fowler, /. Chem. P%., 1, 
515 (1933); Wannier, Ann. Physik, 24, 545, 569 (1935); Steam and Eyring, J. Chem. 
Phys., 5, 113 (1937); see also, Glasstone, Laidler and Eyring, "The Theory of Rate 
Processes," 1941, Chap. X. 




Jit/ 

H H 



ABNORMAL CONDUCTANCES OF HYDROXYL AND OTHER IONS 67 

tion, is sufficient to account for the high conductance of the hydrogen 
ion in aqueous solution. 

The abnormal conductance of the hydrogen ion in methyl and ethyl 
alcohols, which is somewhat less than in water, can also be accounted 
for by a proton transfer analogous to that suggested for water; thus, if 
the hydrogen ion in an alcohol ROH is represented by ROHJ, the 
process is 

R R R R 

+ 1 - I + I 

H H H O H, 

e e 

followed by rotation of the alcohol molecule. To account for the de- 
pendence of abnormal conductance on the nature of R, it must be sup- 
posed that the transfer of a proton from one alcohol molecule to another 
involves the passage over an energy barrier whose height increases as R 
is changed from hydrogen to methyl to ethyl. The Grotthuss type of 
conduction, therefore, diminishes in this order. It is probable that the 
effect decreases steadily with increasing chain length of the alcohol. 

Abnormal Conductances of Hydroxyl and Other Ions. The con- 
ductance of the hydroxyl ion in water is less than that of the hydrogen 
ion; it is nevertheless three times as great as that of most other anions 
(cf. Table XIII). It is probable that the abnormal conductance is here 
also due to the transfer of a proton, in this case from a water molecule 
to a hydroxyl ion, thus 

H H H H 

I + I -> I +1 
O H O O H O, 



followed by rotation of the resulting water molecule. If this is the case, 
it might be expected that the anion RO~ should possess abnormal con- 
ductance in the corresponding alcohol ROH; such abnormalities, if they 
exist at all, are very small, for the conductances of the CPI 3 O~ and 
C 2 HsO~ ions in methyl and ethyl alcohol, respectively, are almost the 
same as that of the chloride ion which exhibits normal conductance only. 
The energy barriers involved in the abnormal mobility process must 
therefore be considerably higher than for water. 

These results emphasize the fact that ions produced by self-ionization 
of the solvent, e.g., H 3 0+ and OH~ in water, ROHt and R0~ in alcohols, 
and NHi" and NHJ" in liquid ammonia, do not of necessity possess ab- 
normal conductance, although they frequently do so. It is seen from 
Table XIX that the conductance of the hydrogen ion in liquid ammonia, 
i.e., NHi", is normal; the same is true for the NHjf ion. The anilinium 
and pyridinium ions also have normal conductances in the corresponding 
solvents. The conductance of the HSOr ion in sulfuric acid as solvent 
is, however, abnormally high; it is probable that a Grotthuss type of 



68 ELECTROLYTIC CONDUCTANCE 

conduction, involving proton transfer, viz., 

HSOr + H 2 S0 4 = H 2 SO 4 + HSOr, 

is responsible for the abnormal conductance. 24 

Influence of Traces of Water. The change in the equivalent con- 
ductance of a strong electrolyte, other than an acid, in a non-aqueous 
solvent resulting from the addition of small amounts of water, generally 
corresponds to the alteration in the viscosity. With strong acids, how- 
ever, there is an initial decrease of conductance in an alcoholic solvent 
which is much greater than is to be expected from the change in vis- 
cosity; this is subsequently followed by an increase towards the value in 
water. When acetone is the solvent, however, the conductance in the 
presence of water runs parallel with the viscosity of the medium. It 
should be noted that the abnormal behavior is observed in solvents in 
which the hydrogen ion manifests the Grotthuss type of conduction. 
The hydrogen ion in alcoholic solution is ROHt and the addition of water 
results in the occurrence of the reversible reaction 

ROUt + H 2 ^ ROH + H 3 0+. 

The equilibrium of this system lies well to the right, and so a large pro- 
portion of the ROH2" ions will be converted into H 3 0+ ions. Although 
the former possess abnormal conductance in the alcohol solution, the 
latter do not, since the proton must pass from H 3 O+ to ROH, and the 
position of the equilibrium referred to above shows that this process must 
be slow. The result of the addition of small quantities of water to an 
alcoholic solution of an acid is to replace an ion capable of abnormal 
conduction by one which is able to conduct in a normal manner only; 
the equivalent conductance of the system must consequently decrease 
markedly. As the amount of water present is increased it will become 
increasingly possible for the proton to pass from H 3 0+ to a molecule of 
water, and so there is some abnormal contribution to the conductance; 
the conductance thus eventually increases towards the usual value for 
the acid in pure water, which is higher than that in the alcohol. 

From the initial decrease in conductance accompanying the addition 
of small amounts of water to a solution of hydrochloric acid in ethyl 
alcohol, it is possible to evaluate the conductance of the H 3 O+ ion in the 
alcohol. The value has been found to be 16.8 ohms" 1 cm. 2 at 25, which 
may be compared with 18.7 ohms"" 1 cm. 2 for the sodium ion in the same 
solvent. It is evident, therefore, that the H 3 O+ ion poasesses only 
normal conductance in ethyl alcohol. 26 

Determination of Solubilities of Sparingly Soluble Electrolytes. 
If a slightly soluble electrolyte dissociates in a simple manner, it is 
possible to calculate the saturation solubility from conductance measure- 

* Hammett and Lowenheim, J. Am. Chem. Soc., 56, 2620 (1934). 
Goldschmidt, Z. phyrik. Chem., 89, 129 (1914). 



DETERMINATION OF SOLUBILITIES 69 

ments. If s is the solubility, in equivalents per liter, of a given substance 
and K is the specific conductance of the saturated solution, then the 
equivalent conductance of the solution is given by 

A = 1000 - (26) 

In general, the solution will be sufficiently dilute for the equivalent con- 
ductance to be little different from the value at infinite dilution: the 
latter can be obtained, as already seen, from the ion conductances of the 
constituent ions. It follows, therefore, since A is known and K for the 
saturated solution can be determined experimentally, that it is possible 
to evaluate the solubility s by means of equation (26). 

From Kohlrausch's measurements on the conductance of saturated 
solutions of pure silver chloride the specific conductance at 25 may be 
estimated as 3.41 X 10~ 6 ohm" 1 cm." 1 ; the specific conductance of the 
water used was 1.60 X 10~ 6 ohm" 1 cm." 1 , and so that due to the salt may 
be obtained by subtraction as 1.81 X 10~ 6 ohm" 1 cm." 1 This is the 
value of K to be employed in equation (26). From Table XIII the 
equivalent conductance of silver chloride at infinite dilution is 138.3 
ohms" 1 cm. 2 at 25, and so if this is assumed to be the equivalent con- 
ductance in the saturated solution of the salt, it follows from equation 
(26) that 

100 X L81 X 10" 6 



= 1.31 X 10~ 6 equiv. per liter at 25. 

By means of this first approximation for the concentration of the satu- 
rated solution of silver chloride, it is possible to make a more exact 
estimate of the actual equivalent conductance by means of the Onsager 
equation (p. 89); a more precise value of the solubility may then be 
determined. In the particular case of silver chloride, however, the differ- 
ence is probably within the limits of the experimental error. 

It should be realized that the method described actually gives the 
ionic concentration in the saturated solution, and it is only when dis- 
sociation is virtually complete that the result is identical with the solu- 
bility. This fact is brought out by the data for thallous chloride: the 
solubility at 18 calculated from Kohlrausch's conductance measurements 
is 1.28 X 10~ 2 equiv. per liter, but the value obtained by direct solubility 
measurement is 1.32 X 10~ 2 equiv. per liter. The discrepancy, which 
is not very large in this instance, is probably to be ascribed to incomplete 
dissociation of the salt in the saturated solution; the degree of dissocia- 
tion appears to be 128/132, i.e., 0.97. 

If the sparingly soluble salt does not undergo simple dissociation, the 
solubility obtained by the conductance method may be seriously in error. 
For example, the value found for lanthanum oxalate in water at 25 is 



70 ELECTROLYTIC CONDUCTANCE 

6.65 X 10~* equiv. per liter, but direct determination gives 2.22 X 10~ 6 
equiv. per liter. The difference is partly due to incomplete dissociation 
and partly to the formation of complex ions. In other words, the lantha- 
num oxalate does not ionize to yield simple La + ++ and C 2 O ions, as is 
assumed in the conductance method for determining the solubility; in 
addition complex ions, containing both lanthanum and oxalate, are 
present to an appreciable extent in the saturated solution. It is neces- 
sary, therefore, to exercise caution in the interpretation of the results 
obtained from conductance measurements with saturated solutions of 
sparingly soluble electrolytes. 

Determination of Basicity of Acids. From an examination of the 
conductances of the sodium salts of a number of acids, Ostwald (1887) 
discovered the empirical relation 

Aio24 - A 32 116, (27) 

where Aio24 and Aa 2 are the equivalent conductances of the salt at 25 at 
dilutions of 1024 and 32 liters per equivalent, respectively, and 6 is the 
basicity of the acid. The data in Table XX are taken from the work of 

TABLE XX. BASICITY OF ACID AND EQUIVALENT CONDUCTANCE OF SALT 

Sodium salt of: A 1024 A w Difference Basicity 

Nicotinic acid 85.0 73.8 11.2 1 

Quinolinic acid 104.9 83.4 21.5 2 

1:2: 4-Pyridine tricarboxylic acid 121.0 88.8 32.2 3 

1:2:3: 4-Pyridine tetracarboxylic acid 131.1 87.3 43.8 4 

Pyridine pentacarboxylic acid 138 1 83.9 54.2 5 

Ostwald, recalculated so as to give the equivalent conductances in ohmr 1 
cm. 2 units, instead of reciprocal Siemens units; they show that the equa- 
tion given above is approximately true, and hence it may be employed 
to determine the basicity of an acid. The method fails when applied 
to very weak acids whose salts are considerably hydrolyzed in solution. 
The results quoted in Table XX are perhaps exceptionally favorable, 
for the agreement with equation (27) is not always as good as these figures 
would imply. The Ostwald rule is, nevertheless, an expression of the facts 
already discussed, viz., that substances of the same valence type have 
approximately the same conductance ratios at equivalent concentrations 
and that the values diminish with increasing valence of one or both ions. 
The rule has been extended by Bredig (1894) to include electrolytes of 
various types. 

Mode of lonization of Salts. Most ions, with the exception of hydro- 
gen, hydroxyl and long-chain ions, have ion conductances of about 60 
ohms" 1 cm. 2 at 25, and this fact may be utilized to throw light on the 
mode of ionization of electrolytes. It has been found of particular value, 
in connection with the Werner co-ordination compounds, to determine 
whether a halogen atom, or other negative group, is attached in a co 
valent or an electrovalent manner. 



CONDUCTOMETRIC TITRATION 71 

Since the mode of ionization of the salt is not known, it is not possible 
to determine the equivalent weight and hence the equivalent conductance 
cannot be calculated ; it is necessary, therefore, to make use of the molar 
conductance, as defined on p. 31. In the simple case of a series of salts 
all of which have one univalent ion, either the cation or anion, whereas 
the other ion has a valence of z, the gram molecule contains z gram 
equivalents; the molar conductance is thus z-times the equivalent con- 
ductance. If the mean equivalent conductance of all ions is taken as 60, 
the equivalent conductance of any salt is 120 ohms~~ l cm. 2 , and the molar 
conductance is 120 z ohms" 1 cm. 2 The approximate results for a number 
of salts of different valence types with one univalent ion at 25 are given 
in Table XXI. The observed molar conductances of the platinosammine 

TABLE XXI. APPROXIMATE MOLAR CONDUCTANCES OF SALTS OF DIFFERENT 

VALENCE TYPES 

Type Molar Conductance 
Uni-uni 120 ohms" 1 cm.* 

Uni-bi or bi-uni 240 

Uni-ter or ter-uni 360 

Uni-tetra or tetra-uni 480 

series, at a concentration of 0.001 M, arc in general agreement with expec- 
tation, as the following data show: 

[Pt(NH 8 ) 4 ]++2Cl- [Pt(NH 3 ) 3 Cl]+Cl- 

260 116 

K+[Pt(NH 3 )Cl 3 ]- 2K+[PtCl 4 ] 

107 267 ohms~ l cm. 2 

The other member of this group, Pt(NH 3 ) 2 Cl 2 , is a non-clcctrolytc and 
so produces no ions in solution; the two chlorine atoms are thus held to 
the central platinum atom by covalent forces. 

Conductometric Titration: (a) Strong Acids. When a strong alkali, 
e.g., sodium hydroxide, is added to a solution of a strong acid, e.g., hydro- 
chloric acid, the reaction 



(H+ + Cl~) + (Na+ + OH-) = Na+ + Cl~ + H 2 O 

occurs, so that the highly conducting hydrogen ions initially present in 
the solution are replaced by sodium ions having a much lower con- 
ductance. In other words, the salt formed has a smaller conductance 
than the strong acid from which it was made. The addition of the alkali 
to the acid solution will thus be accompanied by a decrease of conduct- 
ance. When neutralization is complete the further addition of alkali 
results in an increase of conductance, since the hydroxyl ions are no 
longer used up in the chemical reaction. At the neutral point, therefore, 
the conductance of the system will have a minimum value, from which 
the equivalence-point of the reaction can be estimated. When the 



72 



ELECTROLYTIC CONDUCTANCE 




I \ Jl I 



specific conductance of the acid solution is plotted against the volume of 
alkali added, the result will be of the form of Fig. 24. If the initial 
solution is relatively dilute and there is no appreciable change in volume 
in the course of the titration, the specific conductance will be approxi- 
mately proportional to the concentration of unneutralized acid or free 

alkali present at any instant. The 
specific conductance during the course 
of the titration of an acid by an al- 
kali under these conditions will conse- 
quently be linear with the amount 
of alkali added. It is seen, therefore, 
that Fig. 24 will consist of two 
straight linos which intersect at the 
equivalence-point. 

If the strong acid is titrated 
with a weak base, e.g., an aqueous 
solution of ammonia, the first part 
of the conductance-titration curve, 
representing the neutralization of the 
acid and its replacement by a salt, 
will be very similar to the first part 
of Fig. 24, since both salts are strong 
electrolytes. When .the equivalence- 
point is passed, however, the con- 
ductance will remain almost con- 
stant since the free base is a weak 

electrolyte and consequently has d very small conductance compared with 
that of the acid or salt. 

The determination of the end-point of a titration by means of con- 
ductance measurements is known as conductometric titration. 26 For 
practical purposes it is not necessary to know the actual specific con- 
ductance of the solution; any quantity proportional to it, as explained 
below, is satisfactory. The conductance readings corresponding to vari- 
ous added amounts of titrant are plotted against the latter, as in Fig. 24. 
The titrant should be at least ten times as concentrated as the solution 
being titrated, in order to keep the volume change small; if necessary the 
titrated solution may be diluted in order to satisfy this condition, for 
the method can be applied to solutions of strong acids as dilute as 0.0001 N. 
Since the variation of conductance is linear, it is sufficient to obtain six 
or eight readings covering the range before and after the end-point, and 
to draw two straight lines through them, as seen in Fig. 24; the inter- 
section of the lines gives the required end-point. The method of con- 

* Kolthoff, Ind. Eng. Chem. (Anal. Ed.), 2, 225 (1930); Davies, "The Conductivity 
of Solutions," 1933, Chap. XIX; Glasstone, Ann. Rep. Chem. Soc., 30, 294 (1933); 
Britton, "Conductometric Analysis," 1934; Jander and Pfundt, Bottger's "Physi- 
kalische Methoden der analytischen Xtoemie," 1935, Part II. 



Alka'i Added 

FIG. 24. Conductance titration 
of strong acid and alkali 



CONDUCTOMETRIC TITEATION 



73 




ductometric titration is capable of considerable accuracy provided there 
is good temperature control and a correction is applied for the volume 
change during titration. It can be used with very dilute solutions, as 
mentioned above, but in that case it is essential that extraneous electro- 
lytes should be absent; in the presence of such electrolytes the change of 
conductance would be a very small part of the total conductance and 

would be difficult to measure with 

accuracy. 

(b) Weak Acids. If a moder- 
ately weak acid, such as acetic 
acid, is titrated with a strong base, 
e.g., sodium hydroxide, the form of 
the conductance-titration curve is 
as shown in Fig. 25, 1. The initial 
solution of the weak acid has a low 
conductance and the addition of 
alkali may at first result in a fur- 
ther decrease, in spite of the for- 
mation of a salt, e.g., sodium ace- 
tate, with a high 'conducting power. 
The reason for this is that the 
common anion, i.e., the acetate ion, 
represses the dissociation of the 
acetic acid. With further addition 
of alkali, however, the conductance 
of the highly ionized salt soon ex- 
ceeds that of the weak acid which it replaces, and so the specific conduc- 
tance of the solution increases. After the equivalence-point there is a 
further increase of conductance because of the excess free alkali; the 
curve is then parallel to the corresponding part of Fig. 24. 

When a weak acid is titrated with a weak base the initial portion of 
the conductance-titration curve is similar to that for a strong ba*se, since 
the salt is a strong electrolyte in spite of the weakness of the acid and 
base. Beyond the equivalence-point, however, there is no change in 
conductance because of the small contribution of the free weak base. 
The complete conductance-titration curve is shown in Fig. 25, II. It 
will be observed that the intersection is sharper than in Fig. 25, I, for 
titration with a strong base; it is thus possible to determine the end- 
point of the titration of a moderately weak acid by the conductometric 
method if a moderately weak, rather than a strong, base is employed. 
As long as there is present an excess of acid or base the extent of hy- 
drolysis of the salt is repressed, but in the vicinity of the equivalence- 
point the salt of the weak acid and weak base is extensively split up by 
the water; the conductance measurements do not then fall on the lines 
shown, but these readings can be ignored in the graphical estimation of 
the end-point. 



Base Added 

FIG. 25. Conductance titration of 
weak acid 



74 



ELECTROLYTIC CONDUCTANCE 



If the acid is very weak, e.g., phenol or boric acid, or a very dilute 
solution of a moderately weak acid is employed, the initial conductance 
is extremely small and the addition of alkali is not accompanied by any 
decrease of conductance, such as is shown in Fig. 25. The conductance 
of the solution increases from the commencement of the neutralization 
as the very weak acid is replaced by its salt which is a strong electrolyte. 
After the equivalence-point the conductance shows a further increase if 
a strong base is used, and so the end-point can be found in the usual 
manner. Owing to the extensive hydrolysis of the salt of a weak base 
and a very weak acid, even when excess of acid is still present, the titra- 
tion by a weak base cannot be employed to give a conductometric end- 

point. 

One of the valuable features of 
the conductance method of analysis 
is that it permits the analysis of a 
mixture of a strong and a weak acid 
in one titration. The type of con- 
ductance-titration curve using a 
weak base is shown in Fig. 26; the 
initial decrease is due to the neutral- 
ization of the strong acid, and this is 
followed by an increase as the weak 
acid is replaced by its salt. When 
the neutralization is complete there 
is little further change of conduct- 
ance due to the excess weak base. 
The first point of intersection gives 
the amount of strong acid in the 
mixture and the difference between 
the first and second is equivalent 
to the amount of weak acid. 
(c) Strong and Weak Bases. The results obtained in the titration 
of a base by an acid are very similar to those just described for the reverse 
process. When a strong base is neutralized the highly conducting hy- 
droxyl ion is replaced by an anion with a smaller conductance; the con- 
ductance of the solution then decreases as the acid is added. When the 
end-point is passed, however, there is an increase of conductance, just 
as in Fig. 24, if a strong acid is used for titration purposes, but the value 
remains almost constant if a weak or very weak acid is employed. With 
an acid of intermediate strength there will be a small increase of con- 
ductance beyond the equivalence-point. In any case the intersections are 
relatively sharp and, provided carbon dioxide from the air can be ex- 
cluded, the best method of titrating acids of any degree of weakness conduc- 
tometrically is to add the acid solution to that of a standard strong alkali. 
The conductometric titration of weak bases and those of intermediate 
strength is analogous to the titration of the corresponding acida. Simi- 




Baac Added 

FIG. 26. Conductance titration of mixture 
of strong and weak acid 



CONDUCTOMETRIC TITRATION 75 

larly, a mixture of a strong and a weak base can be titrated quantita- 
tively by means of a weak acid; the results are similar to those depicted 
in Fig. 26. 

(d) Displacement Reactions. The titration of the salt of a weak acid, 
e.g., sodium acetate, by a strong acid, e.g., hydrochloric acid, in which 
the weak acid is displaced by the strong acid, e.g., 

(CHaCOr + Na+) + (H+ + Cl~) = CH 3 C0 2 H + Na+ + C1-, 

can be followed conductometrically. In this reaction the highly ionized 
sodium acetate is replaced by highly ionized sodium chloride and almost 
un-ionized acetic acid. Since the chloride ion has a somewhat higher 
conductance than does the acetate ion, the conductance of the solution 
increases slowly at first, in this particular case, although in other in- 
stances the conductance may decrease somewhat or remain almost con- 
stant; in general, therefore, the change in conductance is small. After 
the end-point is passed, however, the free strong acid produces a marked 
increase, and its position can be determined by the intersection of the 
two straight lines. The salt of a weak base and a strong acid, e.g., 
ammonium chloride, may be titrated by a strong base, e.g., sodium 
hydroxide, in an analogous manner. It is also possible to carry out 
conductometrically the titration of a mixture of a salt of a weak acid, 
e.g., sodium acetate, and weak base, e.g., ammonia, by a strong acid; 
the first break corresponds to the neutralization of the base and the 
second to the completion of the displacement reaction. Similarly, it is 
possible to titrate a mixture of a weak acid and the salt of a weak base 
by means of a strong base. 

(e) Precipitation Reactions. In reactions of the type 



(K+ + C1-) + (Ag+ + NOr) = AgCl + K+ + NOr 
and 

(Mg++ + SO) + 2(Na+ + OH-) = Mg(OH) 2 + 2Na+ + S0i~, 

where a precipitate is formed, one salt is replaced by an equivalent 
amount of another, e.g., potassium chloride by potassium nitrate, and 
so the conductance remains almost constant in the early stages of the 
titration. After the equivalence-point is passed, however, the excess of 
the added salt causes a sharp rise in the conductance (Fig. 27, I) ; the 
end-point of the reaction can thus be determined. 

If both products of the reaction are sparingly soluble, as for example 
in the titration of sulfates by barium hydroxide, viz., 

(Mg++ + SO") + (Ba++ + 20H-) = Mg(OH) 2 + BaS0 4 , 

the conductance of the solution decreases right from the commencement, 
but increases after the end-point because of the free barium hydroxide 
(Fig. 27, II). 



76 



ELECTROLYTIC CONDUCTANCE 




Precipitant Added 

Fio. 27. Conductance titration of 
precipitation reactions 



Precipitation reactions cannot be carried out conductometrically with 
such accuracy as can the other reactions considered above; this is due to 

slow separation of the precipitate, 
with consequent supersaturation of 
the solution, to removal of titrated 
solute by adsorption on the precipi- 
tate, and to other causes. 27 The 
best results have been obtained by 
working with dilute solutions in the 
presence of a relatively large amount 
of alcohol; the latter causes a dimi- 
nution of the solubility of the precipi- 
tate and there is also less adsorption. 
Conductometric Titration: Ex- 
perimental Methods. The titration 
cell may take any convenient form, 
the electrodes being arranged verti- 
cally so as to permit mixing of the 
liquids being titrated (see Fig. 28). 
The conventional Wheatstone bridge, 
or other simple method of measur- 
ing conductance, may be employed. 
If the form of Fig. 9 is used and the resistance R is kept constant, the 
specific conductance of the solution in the measuring cell can be readily 
shown to be proportional to dc/bd. An alternative procedure is to make 
the ratio arms equal, i.e., Rz = R* in Fig. 8 
or bd = dc in Fig. 9; the resistance of the 
cell is then equal to that taken from the 
box # 2 in Fig. 8 or ft in Fig. 9 when the 
bridge is balanced. If two boxes, or other 
standard resistances, one for coarse and 
the other for fine adjustment, are used in 
series, it is possible to read off directly the 
resistance of the cell; the reciprocal of 
this reading is proportional to the specific 
conductance and is plotted in the titration- 
conductance curve. 

Since for most titration purposes it is unnecessary to have results of 
high precision, a certain amount of accuracy has been sacrificed to con- 
venience in various forms of conductometric apparatus. 28 In some cases 
the Wheatstone bridge arrangement is retained, but a form of visual 

27 van Suchtelen and Itano, J. Am. Chem. Soc., 36, 1793 (1914); Harned, ibid., 39, 
252 (1917); Freak, /. Chem. Soc., 115, 55 (1919); Lucasse and Abrahams, J. Chem. Ed., 
7, 341 (1930); Kolthoff and Kameda, Ind. Eng. Chem. (Anal Ed.), 3, 129 (1931). 

"Treadwell and Paoloni, Helu. Chim. Acta, 8, 89 (1925); Callan and Horrobin, 
J. Soc. Chem. Ind., 47, 329T (1928). 




FIG. 28. Vessel for con- 
ductometric titration 



PROBLEMS 77 

detector replaces the telephone earpiece (see p. 35). In other simplified 
conductance-titration procedures the alternating current is passed directly 
through the cell and its magnitude measured by a suitable instrument in 
series; if the applied voltage is constant, then, by Ohm's law, the current 
is proportional to the conductance of the circuit. For analytical pur- 
poses all that is required is the change of conductance during the course 
of the titration, and this is equivalent to knowing the change of current 
at constant voltage. The type of apparatus employed is shown in 
Fig. 29; the source of current is the alternating-current supply mains 



A.C. 




C 

FIG. 29. Conductometric titration using A.C. supply mains 

(A.C.), which is reduced to about 3 to 5 volts by means of the trans- 
former T. The secondary of this transformer forms part of the circuit 
containing the titration cell and also a direct current galvanometer G 
and a rectifier Z); the 400-ohm resistances A and B are used as shunts 
for the purpose of adjusting the current to a value suitable for the meas- 
uring instrument. The rectifier D may be a rectifying crystal, a copper- 
copper oxide rectifier or a suitable vacuum-tube circuit giving rectifica- 
tion and amplification; alternatively, D and G may be combined in the 
form of a commercial A.C. microammeter. The solution to be titrated 
is placed in the vessel C, the resistances A and B are adjusted and then 
the current on G is noted: the titration is now carried out and the gal- 
vanometer readings are plotted against the volume of titrant added. 
The end-point is determined, as already explained, from the point of 
intersection of the two parts of the titration curve. 

PROBLEMS 

1. A conductance cell has two parallel electrodes of 1.25 sq. cm. area placed 
10.50 cm. apart; when filled with a solution of an electrolyte the resistance was 
found to be 1995.6 ohms. Calculate the cell constant of the cell and the 
specific conductance of the solution. 

2. Jones and Bradshaw [J. Am. Chem. Soc. t 55, 1780 (1933)] found the 
resistance of a conductance cell (Z 4 ) when filled with mercury at to be 
0.999076 ohm when compared with a standard ohm. The cell Z 4 and another 
cell YI were filled with sulfuric acid, and the ratio of the resistances Fi/Z 4 was 
0.107812. The resistance of a third cell N* to that of Y lt i.e., Ni/Yi, was 
found to be 0.136564. Evaluate the cell constant of JV*, calculating the specific 
resistance of mercury at from the data on page 4. (It may be mentioned 
that the result is 0.014 per cent too high, because of a difference in the current 
lines in the cell Z 4 when filled with mercury and sulfuric acid, respectively.) 



78 ELECTROLYTIC CONDUCTANCE 

3. A conductance cell having a constant of 2.485 cm." 1 is filled with 0.01 N 
potassium chloride solution at 25; the value of A for this solution is 141.3 
ohms" 1 cm. 2 If the specific conductance of the water employed as solvent is 
1.0 X 10~* ohm" 1 cm." 1 , what is the measured resistance of the cell containing 
the solution? 

4. The measured resistance of a cell containing a 0.1 demal solution of 
potassium chloride at 25, in water having a specific conductance of 0.8 X 10~ 6 
ohm" 1 cm." 1 , was found to be 3468.86 ohms. A 0.1 N solution of another salt, 
dissolved in the same conductance water, had a resistance of 4573.42 ohms in 
the same cell. Calculate the specific conductance of the given solution at 25. 

5. A conductance cell containing 0.01 N potassium chloride was found to 
have a resistance of 2573 ohms at 25. The same cell when filled with a 
solution of 0.2 N acetic acid had a resistance of 5085 ohms. Calculate 
(a) the cell constant, (b) the specific resistances of the potassium chloride and 
acetic acid solutions, (c) the conductance ratio of 0.2 N acetic acid, utilizing 
data given in Chap. II. (The conductance of the water may be neglected.) 

6. Use the data in Tables X and XIII to estimate the equivalent conduct- 
ance of 0.1 N sodium chloride, 0.01 N barium nitrate and 0.001 N magnesium 
sulfate at 25. (Compare the results with the values in Table VIII.) 

7. The following values for the resistance were obtained when 100 cc. of a 
solution of hydrochloric acid were titrated with 1.045 N sodium hydroxide: 

1.0 2.0 3.0 4.0 5.0 cc. NaOH 

2564 3521 5650 8065 4831 3401 ohms 

Determine the concentration of the acid solution. 

8. A 0.01 N solution of hydrochloric acid (A = 412.0) was placed in a cell 
having a constant of 10.35 cm." 1 , and titrated with a more concentrated 
solution of sodium hydroxide. Assuming the equivalent conductance of each 
electrolyte to depend only on the total ionic concentration of the solution, 
plot the variation of the cell conductance resulting from the addition of 25, 50, 
75, 100, 125 and 150 per cent of the amount of sodium hydroxide required for 
complete neutralization. The equivalent conductance of the sodium chloride 
may be taken as 118.5 ohms" 1 cm. 2 ; the change in volume of the solution during 
titration may be neglected. 

9. The following values were obtained by Shedlovsky [V. Am. Chem. Soc., 
54, 1405 (1932)] for the equivalent conductance of potassium chloride at 
various concentrations at 25: 

0.1 0.05 0.02 0.01 0.005 0.001 N 

128.96 133.37 138.34 141.27 143.55 146.95 ohms- 1 cm. 2 

Evaluate the equivalent conductance of the salt at infinite dilution by the 
method described on page 54; the values of B and A may be taken as 0.229 
and 60.2, respectively. 

10. A potential of 5.6 volts is applied to two electrodes placed 9.8 cm. 
apart: how far would an ammonium ion be expected to move in 1 hour in a 
dilute solution of an ammonium salt at 25? 

11. A saturated solution of silver chloride when placed in a conductance 
cell whose constant is 0.1802 had a resistance of 67,953 ohms at 25. The 
resistance of the water used as solvent was found to be 212,180 ohms in the 
same cell. Calculate the solubility of the salt at 25, assuming it to be com- 
pletely dissociated in its saturated solution in water. 



CHAPTER III 
THE THEORY OF ELECTROLYTIC CONDUCTANCE 

Variation of Ionic Speeds. It has been seen (p. 58) that the equiva- 
lent conductance of an electrolyte depends on the number of ions, on the 
charge carried by each ionic species and^on their speeds. For a given 
solute the charge is, of course, constant, and so the variation of equiva- 
lent conductance with concentration means that there is either a change 
in the number of ions present or in their velocities, or in both. In the 
early development of the theory of electrolytic dissociation, Arrhenius 
made the tacit assumption that the ionic speeds were independent of the 
concentration of the solution; the change of equivalent conductance 
would then be due to the change in the number of ions produced from 
the one equivalent of electrolyte as a result of the change of concen- 
tration. In other words, the change in the equivalent conductance 
should then be attributed to the change in the degree of dissociation. 
All electrolytes are probably completely dissociated into ions at infinite 
dilution, and so, if the speeds of the ions do not vary with the concentra- 
tion of the solution, it is seen that the ratio of the equivalent conductance 
A at any concentration to that (A ) at infinite dilution, i.e., A/A , should 
be equal to the degree of dissociation of the electrolyte. For many years, 
therefore, following the original work of Arrhenius, this quantity, which 
is now given the non-committal name of " conductance ratio" (p. 51), 
was identified with the degree of dissociation. 

There are good reasons for believing that the speeds of the ions do 
vary as the concentration of the solution of electrolyte is changed, and 
so the departure of the conductance ratio (A/A ) from unity with in- 
creasing concentration cannot be due merely to a decrease in the degree 
of dissociation. For strong electrolytes, in which the ionic concentration 
is high, the mutual interaction of the oppositely charged ions results in 
a considerable decrease in the velocities of the ions as the concentration 
of the solution is increased ; the fraction A/A under these conditions bears 
no relation to the degree of dissociation. In solutions of weak electro- 
lytes the number of ions in unit volume is relatively small, and hence so 
also is the interionic action which reduces the ionic speeds. The latter, 
consequently, do not change greatly with concentration, and the con- 
ductance ratio gives a reasonably good value of the degree of dissociation ; 
some correction should, however, be made for the influence of interionic 
forces, as will be seen later. 

The Degree of Dissociation. An expression for the degree of dis- 
sociation which will be found useful at a later stage is based on a con- 

79 



80 THE THEORY OF ELECTROLYTIC CONDUCTANCE 

sidcration of the relationship between the equivalent conductance of a 
solution and the speeds of the ions. It was deduced on page 59 that 
the speed of an ion at infinite dilution under a potential gradient of 
1 volt per cm. is equal to X/^, the derivation being based on the 
assumption that the electrolyte is completely dissociated. A consid- 
eration of the arguments presented shows that they are of general appli- 
cability to solutions of any concentration; the only change is that if 
the electrolyte is not completely dissociated, an allowance must be made 
in calculating the actual ionic concentration. If a is the true degree of 
dissociation and c is the total (stoichiometric) concentration of the elec- 
trolyte, the ionic concentration ac equiv. per liter must be employed in 
evaluating the quantity of electricity carried by the ions; the total con- 
centration c is still used, however, for calculating the equivalent con- 
ductance. The result of making this change is that equation (19) on 
page 59 becomes 

aF(u+ + ii_) = X+ + X_ = A, (1) 

where A+ and X_ are the actual ion conductances and A the equivalent 
conductance of the solution; u+ and M_ are the mobilities of the ions in 
the same solution and a is the degree of dissociation at the given concen- 
tration. It follows, therefore, that 



-& <*> 

For a weak electrolyte the sum u*+ + ul, for infinite dilution, does not 
differ greatly from u+ + w_ in the actual solution, and so the degree of 
dissociation is approximately equal to the conductance ratio, as stated 
above. 

If equation (1) is divided into its constituent parts, for positive and 
negative ions, it is seen that 

aFu, = X, (3) 

for each ion ; hence 

X t 



where X< and Ui are the equivalent conductance and mobility of the ith 
ion in the actual solution. 

Interionic Attraction: The Ionic Atmosphere. The possibility that 
the attractive forces between ions might have some influence on electro- 
lytic conductance, especially with strong electrolytes, was considered by 
Noyes (1904), Sutherland (1906), Bjerrum (1909), and Milner (1912) 



INTERIONIC ATTRACTION 81 

among others, but the modern quantitative treatment o? this concept is 

due mainly to the work of Debye and Hiickel and its extension chiefly 

by Onsager and by Falkenhagen. 1 The essential postulate of the Debye- 

Hiickel theory is that every ion may be considered as being surrounded 

by an ionic atmosphere of opposite sign: this atmosphere can be regarded 

as arising in the following manner. Imagine a positive ion situated at 

the point A in Fig. 30, and consider a small volume element dv at the 

end of a radius vector r; the distance r is supposed to be of the order of 

less than about one hundred times the diameter of an ion. As a result 

of thermal movements of the ions, there will sometimes be an excess of 

positive and sometimes an excess of negative ions in the volume element 

dv; if a time-average is taken, however, it will be found to have, as a 

consequence of electrostatic attraction by 

the positive charge at A, a negative charge 

density. In other words, the probability of 

finding ions of opposite sign in the space sur- 

rounding a given ion is greater than the prob- 

ability of finding ions of the same sign; every 

ion may thus be regarded as being associated 

with an ionic atmosphere of opposite sign. 

The net charge of the atmosphere is, of course, 

equal in magnitude but opposite in sign to 

that of the central ion: the charge density will FlG 30 Tlie ionic 

obviously be greater in the immediate vicinity atmosphere 

of the latter and will fall off with increasing 

distance. It is possible, nevertheless, to define an effective thickness of 

the ionic atmosphere, as will be explained shortly. 

Suppose the time-average of the electrical potential in the center of 
the volume element dv in Fig. 30 is $] the work required to bring a posi- 
tive ion from infinity up to this point is then z+ef/ and to bring up a 
negative ion it is z_c^, where z+ and z- are the numerical values of the 
valences of the positive and negative ions, respectively, and c is the unit 
charge, i.e., the electronic charge. If the Boltzmann law of the distri- 
bution of particles in a field of varying potential energy is applicable to 
ions, the time-average numbers of positive ions (dn+) and of negative 
ions (dnJ) present in the volume element dv are given by 




dn+ = 

and 

dn, = n-.e-<-'-'+ /kT >dv, 

where n+ and n_ are the total numbers of positive and negative ions, 

1 Debye and Hiickel, Physik. Z., 24, 185, 305 (1923); 25, 145 (1924); for reviews, 
see Falkenhagen and Williams, Chem. Revs., 6, 317 (1929); Williams, ibid., 8, 303 (1931); 
Hartley et al., Ann. Rep. Chem. Soc., 27, 326 (1930); Falkenhagen, Rev. Modern Phys., 
3, 412 (1931); "Electrolytes" (Translated by Bell), 1934; Maclnnes et al, Che.m. Revs., 
13, 29 (1933); Trans. Electrochem. Soc., 66, 237 (1934); J. Franklin Inst., 225, 661 (1938). 



82 THE THEORY OP ELECTROLYTIC CONDUCTANCE 

respectively, in unit volume of the solution; k is the Boltzmann constant, 
i.e., the gas constant per single molecule, and T is the absolute tempera- 
ture. The electrical density p, i.e., the net charge per unit volume, in 
the given volume element is therefore given by 

c(z+dn+ 



dv 

itikT __ ri-Z-e*-*+ lkT ). (5) 

For a uni-univalent electrolyte z+ and z_ are unity, and n+ and n_ must 
be equal, because of electrical neutrality; hence equation (5) becomes 

p = n<(e-'+' kT - e ikT ), (6) 

where n is the number of either kind of ion in unit volume. Expanding 
the two exponential series, and writing x in place of aff/kT, equation (6) 
becomes 

P == i. rp ^ 

and if it is assumed that x, i.e., c\fs/kT, is small in comparison with unity, 
all terms beyond the first in the parentheses may be neglected, so that 

In the general case, when z+ and 2_ are not necessarily unity, if the 
assumption is made that zel/jkT is much less than unity in each case, the 
corresponding expression for the electrical density is 

p = Sn2?, (8) 

where n,- and 2,- represent the number (per unit volume) and valence of 
the ions of the ith kind. The summation is taken over all the types of 
ions present in the solution, and equation (8) is applicable irrespective 
of the number of different kinds of ions. 

In order to solve for ^ it is necessary to have another relationship 
between p and ^, and this may be obtained by introducing Poisson's 
equation, which is equivalent to assuming that Coulomb's law of force 
between electrostatic charges also holds good for ions. This equation in 
rectangular coordinates is 



__ 

dx* "" dy* "" dz* ~ D ' 

x, y and z are the coordinates of the point in the given volume element, 
and D is the dielectric constant of the medium. Converting to polar 
coordinates, and making use of the fact that the terms containing ty/dO 
and d^/d^ will be zero, since the distribution of potential about any point 



INTERIONIC ATTRACTION 83 

in the electrolyte must be spherically symmetrical, and consequently 
independent of the angles 8 and 0, equation (9) becomes 



If the value of p given by equation (8) is inserted, this becomes 
1 d 



= *V, (11) 

where the quantity K (not to be confused with specific conductance) is 
defined by 

/ A-~2 \ \ 

(12) 

The differential equation (11) can be solved, and the solution has the 
general form 

Ae~* r A'e"' 



where A and A' are constants which can be evaluated in the following 
manner. Since ^ must approach zero as r increases, because the poten- 
tial at an infinite distance from a given point in the solution must be 
zero, it follows that the constant A' must be zero; equation (13) conse- 
quently becomes 

Ap~ itr 
#-^ (14) 

For a very dilute solution 2n t z? is almost zero, and hence so also is K, 
as may be seen from equation (12) ; the value of the potential at the point 
under consideration will then be A/r, according to equation (14). In 
such a dilute solution the potential in the neighborhood of any ion will 
be due to that ion alone, since other ions are too far away to have any 
influence: further, if the ion is regarded as being a point charge, the 
potential at small distances will be z l /Dr. It follows, therefore, that 

A Z 

7 = Wr' 

Zt * 



and insertion of this result in equation (14) gives 



84 THE THEORY OF ELECTROLYTIC CONDUCTANCE 

This equation may be written in the form 

.*!!_*!(!_ e -.r) 

* Dr Dr^ e )j 

and if the solution is dilute, so that * is small and 1 tr* is practically 
equal to *r, this becomes 

*-- < 

The first term on the right of equation (16) is the potential at a 
distance r due to a given point ion when there are no surrounding ions; 
the second term must, therefore, represent the potential arising from the 
ionic atmosphere. It is seen, therefore, that ^, the potential due to the 
ionic atmosphere, is given by 



for a dilute solution. Since this expression is independent of r, it may 
be assumed to hold when r is zero, so that the potential on the ion itself, 
due to its surrounding atmosphere, is given by equation (17). If the 
whole of the charge of the ionic atmosphere which is e t c, since it is 
equal in magnitude and opposite in sign to that of the central ion itself, 
were placed at a distance I/K from the ion the potential produced at 
it would be z t K/D, which is identical with the value given by equation 
(17). It is seen, therefore, that the effect of the ion atmosphere is equiva- 
lent to that of a single charge, of the same magnitude, placed at a distance 
I/K from the ion; the quantity I/K can thus be regarded as a measure 
of the thickness of the ion atmosphere in a given solution. 

According to the definition of K, i.e., equation (12), the thickness of 
the ionic atmosphere will depend on the number of ions of each kind 
present in unit volume and on their valence. If c t is the concentration 
of the ions of the ith kind expressed in moles (gram-ions) per liter, then 

N 



where N is the Avogadro number; hence, from equation (12), after making 
a slight rearrangement, 

l_(DT 1000* \ 

-' 



The values of the universal constants are as follows: k is 1.38 X 10~ 18 
erg per degree, e is 4.802 X 10~ 10 e.s. unit, and N is 6.025 X 10 23 ; hence 



- = 2.81 X 10- 10 | 
x 



( DT V 
I ^ 2 I cm. 
\ ZcA 2 ) 



TIME OF RELAXATION OF IONIC ATMOSPHERE 85 

For water as solvent at 25, D is 78.6 and T is 298, so that 

1 4.31 X 10-* , 1ft . 

- = 2Ti cm. (19) 

K (2c t Z|)* 

The thickness of the ionic atmosphere is thus seen to be of the order of 
iO~~ 8 cm. ; it decreases with increasing concentration and increasing va- 
lence of the ions present in the electrolyte, and increases with increasing 
dielectric constant of the solvent and with increasing temperature. The 
value of I/K in Angstrom units for solutions of various types of electro- 
lytes at concentrations of 0.1, 0.01 and 0.001 moles per liter in water at 
25 are given in Table XXII. 

TABLE XXII. THICKNESS OF THE IONIC ATMOSPHERE IN WATER AT 25 

Concentration of Solution 

Valence Type 0.10 M 0.01 M 0.001 M 

Uni-uni 9.64A 30.5A 96.4A 

Uni-bi and bi-uni 5.58 19.3 55.8 

Bi-bi 4.82 15.3 48.2 

Uni-ter and ter-uni 3.94 13.6 39.4 

Time of Relaxation of Ionic Atmosphere. As long as the ionic at- 
mosphere is "stationary," that is to say, it is not exposed to an applied 
electrical field or to a shearing force tending to cause movement of the 
ion with respect to the solvent, it has spherical symmetry. When the 
ion is made to move under the influence of an external force, however, 
e.g., by the application of an electrical field, the symmetry of the ionic 
atmosphere is disturbed. If a particular kind of ion moves to the right, 
for example, each ion will constantly have to build up its ionic atmos- 
phere to the right, while the charge density to the left gradually decays. 
The rate at which the atmosphere to the right forms and that to the left 
dies away is expressed in terms of a quantity called the time of relaxation 
of the ionic atmosphere. The decay of the ionic atmosphere occurs 
exponentially, and so the return to random distribution is asymptotic in 
natuie; it follows, therefore, that the time required for the ionic atmos- 
phere to fall actually to zero is, theoretically, infinite. It has been shown, 
however, that, after the removal of the central ion, the surrounding 
atmosphere falls virtually to zero in the time 4q6, where 9 is the time of 
relaxation of the ionic atmosphere and q is defined by 

g-r^-.Mh ; (20) 



z is the valence, excluding the sign, and X is the ion conductance, of the 
respective ions. For a binary electrolyte, i.e., one yielding only two ions, 
Zf and Z- are equal and q is 0.5; the time for the ionic atmosphere to 
decay virtually to zero is then 26. 

When an ion of valence z is moving with a steady velocity through 
a solution, under the influence of an electrical force tzV y where V is the 



86 THE THEORY OF ELECTROLYTIC CONDUCTANCE 

applied potential gradient, this force must balance the force due to re- 
sistance represented by Ku\ K is the resultant coefficient of frictional 
resistance and u is the steady velocity of the ion. It follows, therefore, 
that 

zV = Ku, 



If the potential gradient is 1 volt per cm., then V is 1/300 e.s. unit; 
further the velocity u is then given, according to equation (20), Chap. II, 
by X/F, where F is 96,500, and hence 



since c is 4.802 X 10~ l e.s. unit. It has been shown by Debye and 
Falkenhagen 2 that the relaxation time is related to the frictional coeffi- 
cients K+ and /_ of the two ions constituting a binary electrolyte by the 
expression 

6 - scc - ^ 



where K has the same significance as before. Utilizing equation (21) 
and remembering that z+ is equal to z_ for a binary electrolyte and that 
A+ + A_ is equal to A, the equivalent conductance of the electrolyte, 
equation (22) becomes 

= 30.8 X 10-' - - 2 scc. (23) 



Introducing the value of I/K for aqueous solutions at 25, given by equa- 
tion (19), into equation (23), the result is 

A 71.3 X 10- 10 

6 = - : --- sec., (24) 

cz\ 

where c is the concentration of the solution in moles per liter. For most 
solutions other than acids and bases, A is about 120 ohms" 1 cm. 2 at 25, 
so that 

0.6 X 10- 10 

* - sec. 

cz 

The time of relaxation of the ionic atmosphere for a binary electrolyte 
is thus seen to be inversely proportional to the concentration of the 
solution and to the valence of the ions. The approximate relaxation 
times for 0.1, 0.01 and 0.001 N solutions of a uni-univalent electrolyte 
are 0.6 X 10"*, 0.6 X 10~ 8 and 0.6 X 10~ 7 sec., respectively. 

* Debye and Falkenhagen, Physik. Z., 29, 121, 401 (1928); Falkenhagen and Wil- 
liams, Z. phyrik. Chem., 137, 399 (1928); J. Phys. Chem., 33, 1121 (1929). 



MECHANISM OF ELECTROLYTIC CONDUCTANCE 87 

Mechanism of Electrolytic Conductance. The existence of a finite 
time of relaxation means that the ionic atmosphere surrounding a moving 
ion is not symmetrical, the charge density being greater behind than in 
front; since the net charge of the atmosphere is opposite to that of the 
central ion, there will be an excess charge of the opposite sign behind the 
moving ion. The asymmetry of the ionic atmosphere, due to the time 
of relaxation, will thus result in a retardation of the ion moving under 
the influence of an applied field. This influence on the speed of an ion is 
called the relaxation effect or asymmetry effect. 

Another factor which tends to retard th,e motion of an ion in solution 
is the tendency of the applied potential to move the ionic atmosphere, 
with its associated solvent molecules, in a direction opposite to that in 
which the central ion, with its solvent molecules (cf. p. 114), is moving. 
An additional retarding influence, equivalent to an increase in the viscous 
resistance of the solvent, is thus exerted on the moving ion; this is known 
as the electrophoretic effect, since it is analogous to the resistance acting 
against the movement of a colloidal particle in an electrical field (cf. 
p. 530). 

An attempt to calculate the magnitude of the forces opposing the 
motion of an ion through a solution was made by Debye and Htickel: 
they assumed the applicability of Stokes's law and derived the following 
expression for the electrophoretic force on an ion of the ith kind: 

Electrophoretic Force = K t V, (25) 

where , z l and K have their usual significance, the latter being taken as 
equal to the reciprocal of the thickness of the ionic atmosphere; 77 is the 
viscosity of the medium, /< is the coefficient of frictional resistance of 
the solvent opposing the motion of the ion of the ith kind, and V is the 
applied potential gradient.* The same result was derived in an alter- 
native manner by Onsager, 3 who showed that it is not necessary for 
Stokes's law to be strictly applicable in the immediate vicinity of an ion. 
In the first derivation of the relaxation force Debye and Huckel did 
not take into account the natural Brownian movement of the ions; allow- 
ance for this was made by Onsager who deduced the equation: 

f^Z If 

Relaxation Force = n * wV, (26) 



* The coefficient Ki given here differs somewhat from that (K) employed on page 
86; the latter is defined as the resultant frictional coefficient, based on the tacit assump- 
tion that all the forces opposing the motion of the ion in a solution of appreciable con- 
centration are frictional in nature. An attempt is made here to divide these forces into 
the true frictional force due to the solvent, for which the coefficient Ki is employed, and 
the electrophoretic and relaxation forces due to the presence of other ions. At infinite 
dilution, K and Ki are, of course, identical. 

'Onsager, Physik. Z., 27, 388 (1926); 28, 277 (1927); Trans. Faraday Soc., 23, 341 
(1927). 



88 THE THEORY OF ELECTROLYTIC CONDUCTANCE 

where D is the dielectric constant of the medium and w is defined by 



the value of q being given by equation (20). 

It is now possible to equate the forces acting on an ion of the ith 
kind when it is moving through a solution with a steady velocity w,; 
the driving force due to the applied electrical field is zF, and this is 
opposed by the frictional force of the solvent, equal to X t w,-, together 
with the electrophoretic and relaxation forces; hence 

~ wV. (28) 

^ 



On dividing through by K V V and rearranging, this becomes 

U v Z t 2 t K 3 Z,K W 

V " K~> " 6^ ~ QDkT ' JT V ' 

If the field strength, or potential gradient, is taken as 1 volt per cm., 
i.e., V is 1/300, then 



J E !?i_ J^.\ 

KDkT ' KJ 



1 300 A t 300 V GTnf ^ GDkT 

At infinite dilution K Ls zero, and so under these conditions this equa- 
tion becomes 



"* 300/v, 
and since Fifi Ls equal to X?, it follows that 

&->' w 

Further, according to equation (3), u l is equal to \ l /aF, where a is the 
degree of dissociation; and if this result and that of equation (30) are 
introduced into (29) the latter becomes 

X t X? ex / z t , tz l 



For simplicity, Me assumption is now made that the electrolyte is com- 
pletely dissociated, that is to say, a is assumed to be unity; this, as will be 
evident shortly, is true for solutions of strong electrolytes at quite appre- 
ciable concentrations. Equation (31) can then be put in the form 



MECHANISM OF ELECTROLYTIC CONDUCTANCE 89 

making use of equation (30) to replace Z i /K l by 300\t/F. Introducing 
the expression for K given by equation (12), and utilizing the standard 
values of 6, k and N (p. 84), equation (32) becomes 



, [29.15z< 9.90X10* 

= x ~ ~ "- 



1 

^ w J 



The quantities c f and c_ represent the concentrations of the ions in moles 
per liter; these may be replaced by the corresponding concentrations c 
in equivalents per liter, where c, which is the same for both ions, is equal 
to c t 2 t ; hence 



X, = X? - + - X " w V^TPT). (34) 



The equivalent conductance of an electrolyte is equal to the sum of 
the conductances of the constituent ions, and so it follows from equation 
(34) that 



20.15(2, + O 9.90 X 
----- - --- 



. . 

A = Ao - ----- j- --- + , Aow> Vc(z f + z_). (35) 



X 10* 1 
j, } Aow> J 



In the simple case of a uni-univalent electrolyte, z+ and z_ are unity, 
and w is 2 A/2; equation (35) then reduces to 

I" 82.4 , 8.20 X 10 5 "l r 
A = Ao - [ pfft + -^r)T- A J VC ' (36) 

the concentration c, in equivalents, being replaced by c, in moles, since 
both are now identical. This equation and equations (33), (34) and (35) 
represent forms of the Debye-Hiickel-Onsager conductance equation; 
these relationships, based on the assumption that dissociation of the 
electrolyte is complete, attempt to account for the falling off of the 
equivalent conductance at appreciable concentrations in terms of a de- 
crease in ionic velocity resulting from interionic forces. The decrease of 
conductance due to those forces is represented by the quantities in the 
square brackets; the first term in the brackets gives the effect due to the 
olectrophoretic force and the second term represents the influence of 
the relaxation, or asymmetry, force. It will be apparent from equation 
(35) that, for a givon solvent at a definite temperature, the magnitude 
of the interionic forces increases, as is to be anticipated, with increasing 
valence of the ions and with increasing concentration of the electrolyte. 
Before proceeding with a description of the experiments that have 
been made to test the validity of the Onsager equation, attention may 
be called to the concentration term c (or c) which appears in the equa- 
tions (33) to (36). This quantity arises from the expression for K 
[equation (12)], and in the latter it represents strictly the actual ionic 
concentration. As long as dissociation is complete, as has been assumed 
above, this is equal to the stoichiometric concentration, but when cases 



90 THE THEORY OF ELECTROLYTIC CONDUCTANCE 

of incomplete dissociation are considered it must be remembered that 
the actual ionic concentration is c, and this should be employed in the 
Onsager equation. 

Validity of the Debye-Huckel-Onsager Equation. For a uni-univa- 
lent electrolyte, the Onsager equation (36), assuming complete dissocia- 
tion, may be written in the form 

A = Ao - (A +Ao)Vc, (37) 

where A and B are constants dependent only on the nature of the solvent 
and the temperature; thus 

82.4 

A 



and 

8.20 X 10 s 



B = 



(D2 1 ) 1 



The values of A and B for a number of common solvents at 25 are given 
in Table XXIII. 

TABLE XXIII. VALUES OF THE ONSAGER CONSTANTS FOR UNI-UNIVALENT 
ELECTROLYTES AT 25 

Solvent D 17 X 10* A B 

Water 78.5 8.95 60.20 0.229 

Methyl alcohol 31.5 5.45 156.1 0.923 

Ethyl alcohol 24.3 10.8 89.7 1.33 

Acetone 21.2 3.16 32.8 1.63 

Acetonitrile 36.7 3.44 22.9 0.716 

Nitromethane 37.0 6.27 125.1 0.708 

Nitrobenzene 34.8 18.3 44.2 0.776 

(a) Aqueous Solutions. In testing the validity of equation (37), it 
is not sufficient to show that the equivalent conductance is a linear func- 
tion of the square-root of the concentration, as is generally found to be 
the case (cf. p. 54); the important point is that the slope of the line 
must be numerically equal to A + #A , where A and B have the values 
given in Table XXIII. It must be realized, further, that the Onsager 
equation is to be regarded as a limiting expression applicable to very 
dilute solutions only; the reason for this is that the identification of the 
ionic atmosphere with !/*, where K is defined by equation (12), involves 
simplifications resulting from the assumption of point charges and dilute 
solutions. It is necessary, therefore, to have reliable data of conduct- 
ances for solutions of low concentration in order that the accuracy of the 
Onsager equation may be tested. Such data have become available in 
recent years, particularly for aqueous solutions of a few uni-univalent 
electrolytes, e.g., hydrochloric acid, sodium and potassium chlorides and 
silver nitrate. The experimental results for these solutions at 25 are 
indicated by the points in Fig. 31, in which the observed equivalent 



VALIDITY OF THE DEBYE-HUCKEL-ONSAGER EQUATION 



91 



conductances are plotted against the square-roots of the corresponding 
concentrations. 4 The theoretical slopes of the straight lines to be ex- 
pected from the Onsager equation, calculated from the values of A and B 
in Table XXIII in conjunction with an estimated equivalent conductance 




NaCl 



0.02 



0.04 



0.06 



^Concentration 
Fia. 31. Test of the Onsager equation 

at infinite dilution, are shown by the dotted lines. It is evident from 
Fig. 31 that for aqueous solutions of the uni-univalent electrolytes for 
which data are available, the Onsager equation is very closely obeyed at 
concentrations up to about 2 X 10~ 3 equiv. per liter. 

For electrolytes of unsymmetrical valence types, i.e., z. and z_ are 
different, the verification of the Debye-Hiickel-Onsager equation is more 
difficult since the evaluation of the factor w in equation (35) requires a 
knowledge of the mobilities of the individual ions at infinite dilution; 
for this purpose it is necessary to know the transference numbers of the 

4 Shedlovsky, J. Am. Chem. Soc*, 54, 1411 (1032); Shedlovsky, Brown and Maclnncs, 
Trans. Ekttrockem. Soc., 66, 165 (1934); Krieger and Kilpatrick, J. Am. Chem. Soc., 
59, 1878 (1937). 



92 THE THEORY OP ELECTROLYTIC CONDUCTANCE 

Ions constituting the electrolyte (see Chap. IV). The requisite data for 
dilute aqueous solutions at 25 are available for calcium and lanthanum 
chlorides, i.e., CaCl 2 and LaCl 3 , and in both instances the results are in 
close agreement with the requirements of the theoretical equation at 
concentrations up to 4 X 10~ 6 equiv. per liter. 5 It is apparent that the 
higher the valence type of the electrolyte the lower is the limit of con- 
centration at which the Onsager equation is applicable. 

Less accurate measurements of the conductances of aqueous solutions 
of various electrolytes have been made, and in general the results bear 
out the validity of the Onsager equation. 6 A number of values of the 
experimental slopes arc compared in Table XXIV with those calculated 

TABLE XXIV. COMPARISON OF OBSERVED AND CALCULATED ONSAGER SLOPES IN 
AQUEOUS SOLUTIONS AT 25 

Electrolyte Observed Slope Calculated Slope 
LiCl 81.1 72.7 

NnNO 3 82.4 74.3 

KBr 87.9 S0.2 

KCNS 76.5 77.8 

CsCl 76.0 80.5 

MgCl 2 144.1 145.6 

Ba(NO 3 ) 2 160.7 150.5 

K 2 SO 4 140.3 159.5 

theoretically; the agreement is seen to be fairly good, but it may be even 
better than would at first appear, owing to the lack of data in sufficiently 
dilute solutions. It is of interest to record in this connection that the 
experimental slope of the A versus Vc curve for silver nitrate was given 
at one time as 88.2, compared with the calculated value 76.5 at 18; more 
recent v, T ork on very dilute solutions has shown much better agreement 
than these results would imply (see Fig. 31). 

Further support for the Onsager theory is provided by conductance; 
measurements of a number of electrolytes made at and 100. At both 
temperatures the observed slope of the plot of A against Vc agrees with 
the calculated result within the limits of experimental error. The slope 
of the curve for potassium chloride changes from 47.3 to 313.4 within 
the temperature range studied. 

The data recorded above indicate that the Onsager equation repre- 
sents in a satisfactory manner the dependence on the concentration of the 
equivalent conductances of uni-univalent and uni-bi- (or bi-urii-) valent 
electrolytes. With bi-bivalent solutes, however, very marked discrep- 
ancies are observed; in the first place the plot of the equivalent con- 

1 Jones and Bickford, /. Am. Chem. Soc., 56, 602 (1934); Shedlovsky and Brown, 
ibid., 56, 1066 (1934). 

See, Davies, "The Conductivity of Solutions," 1933, Chap. V; Hartley et a/., 
Ann. Rep. Chem. *S'oc., 27, 341 (1930); /. Chem. Soc., 1207 (1933); Z. physik. Chem., 
165A, 272 (1933). 



VALIDITY OF THE DEBYK-HUCKEL-ONSAGKR EQUATION 



93 



ductance against the square-root of the concentration is not a straight 
lino, but is concave to the axis of the latter parameter (Fig. 32). Further, 
the slopes at appreciable concentrations are much greater than those 
calculated theoretically. It 



144 r 



128 



112 



IS 

probable that these results are 
to be explained by incomplete 
dissociation at the experimental 
concentrations : the shapes of 
the curves do in fact indicate 
that in sufficiently dilute solu- 
tions the slopes would probably 
be very close to the theoretical 
Onsager values. 

(b) N on- Aqueous Solutions. 
A number of cases of satisfactory 
agreement with theoretical re- 
quirements have been found in 
methyl alcohol solutions; this is 
particularly the case for the 
chlorides and thiocyanates of the 
alkali metals. 7 Other electro- 
lytes, such as nitrates, tetralkyl- 
ammoniurn salts and salts of 
higher valence types, however, 
exhibit appreciable deviations. These discrepancies become more marked 
the lower the dielectric constant of the medium, especially if the latter is 
non-hydroxylic in character. The conductance of potassium iodide has 
been determined in a number of solvents at 25 and the experimental and 
calculated slopes of the plots of A against Vc are quoted in Table XXV, 

TABLE XXV. OBSERVED AND CALCULATED OXSAGER SLOPES FOR POTASSIUM 




0.02 



0.04 



0.06 



Fm. 32. Deviation from Onsager equation 



Solvent 
Water 

Methyl alcohol 
Kthyl cyanoacetate 
Ethyl alcohol 
Benzomtrile 
Acetone 



IODIDE AT 25 

D 

7S.6 
31.5 
27.7 
25.2 
25.2 
20.9 



Onsager Slope 
Observed Calculated 



73 
260 
115 
209 
263 
1000 



SO 
268 

63 
153 
142 
638 



together with the dielectric constant of the medium in each case. At 
still lower dielectric constants, and for other electrolytes, even greater 
discrepancies have been recorded : in many cases substances which are 
strong electrolytes, and hence almost completely dissociated in water, 
behave as weak, incompletely dissociated electrolytes in solvents of low 

7 Hartley et al, Proc. Roy. Soc., 127A, 228 (1930); 132A, 427 (1931); J. Chun. Soc., 
2488 (1930). 



94 THE THEORY OF ELECTROLYTIC CONDUCTANCE 

dielectric constant. It is not surprising, therefore, to find departures 
from the theoretical Onsager behavior. 

Deviations from the Onsager Equation. Two main types of devia- 
tion from the Onsager equation have been observed: the first type is 
exhibited by a number of salts in aqueous solution which give conduct- 
ances that are too large at relatively high concentrations, although the 
values are in excellent agreement with theory in the more dilute solutions. 
This effect can be seen from the results plotted in Fig. 31; it is probably 
to be ascribed to the approximations made in the derivation of the 
Onsager equation which, as already explained, can only be expected to 
hold for point ions in dilute solution. An empirical correction, involving 
c'and logc, has been applied to allow for these approximations in the 
following manner. Solving equation (37), for a uni-univalent electro- 
lyte, for AO it is found that 

*- 



according to the simple Onsager theory, and after applying the correc- 
tions proposed by Shedlovsky, 8 this becomes 

Ao = - p - Cc - DC log c + Ec\ (39) 

1 J5Vc 

where C, D and E are empirical constants. In some cases D and E are 
very small and equation (39) reduces to the form 

* 



which was employed on page 55 to calculate equivalent conductances 
at infinite dilution. Its validity is confirmed by the results depicted in 
Fig. 22. In general, the Shedlovsky equation (39) adequately represents 
the behavior of a number of electrolytes in relatively concentrated solu- 
tions; it reduces to the simple Onsager equation at high dilutions when 
c is small. It is of interest to call attention to the fact that if the term 
in equation (39) involving log c is small, as it often is, and can be neg- 
lected, this equation can be written in the form of the power series 

A = Ao - 4'c* + B'c - C'c* + DV - #V, (41) 

where A' t B', etc., are constants for the given solute and solvent. 

For many electrolytes the plot of the equivalent conductance against 
the square-root of the concentration is linear, or slightly concave to the 
concentration axis, but the experimental slopes are numerically greater 

'Shedlovsky, J. Am. Chem. Soc., 54, 1405 (1932); Shedlovsky and Brown, ibid., 
56, 1066 (1934); cf., Onsager and Fuoss, J. Phys. Chem., 36, 2689 (1932). See, how- 
ever, Jones and Bickford, J. Am. Chem. Soc., 56, 602 (1934). 



DEVIATIONS FROM THE ON8AGER EQUATION 95 

than those expected theoretically; this constitutes the second type of 
deviation from the Onsager equation, instances of which are given in 
Table XXV. In these cases the conductance is less than required by 
the theory and the explanation offered for the discrepant behavior, as 
indicated above, is that dissociation of the electrolyte is incomplete : the 
number of ions available for carrying the current is thus less than would 
be expected from the stoichiometric concentration. It will be seen from 
the treatment on page 89 that, strictly speaking, the left-hand side of 
equation (32), and hence of all other forms of the Onsager equation, 
should include a factor I/a, where a. is the degree of dissociation of the 
electrolyte; further, it was noted on page 90 that the concentration 
term should really be ac. It follows, therefore, that for a uni-univalent 
electrolyte the correct form of equation (37), which makes allowance for 
incomplete dissociation, is 

A = a[A - (A + A )Vac]. (42) 

This equation is sometimes written as 

A = aA', (43) 

where A', defined by 

A' s Ao - (A + #A )V^c, (44) 

is the equivalent conductance of 1 equiv. of free ions at the concentration 
ac equiv. per liter, i.e., at the actual ionic concentration in the solution^ 

It is not evident from equation (42) that the plot of A against Vc 
will be a straight line, since a varies with the concentration; but as a is 
less than unity, it is clear that the observed values of the equivalent 
conductance will be appreciably less than is to be expected from the 
simple Onsager equation. The second type of deviation, which occurs 
particularly with salts of high valence types and in media of low dielectric 
constant, can thus be accounted for by incomplete dissociation of the 
solute. It is seen from equation (43) that the degree of dissociation a. 
is numerically equal to A/A', instead of to A/A as proposed by Arrhenius. 
It is apparent from equation (44) that for all electrolytes, and especially 
those which are relatively strong, A' is considerably smaller than AoJ the 
true degree of dissociation (A/A') is thus appreciably closer to unity than 
is the value assumed to be equal to the conductance ratio (A/A ). For a 
weak electrolyte, the degree of dissociation is in any case small, and 
ac will also be small; the difference between A' and A is thus not large 
and the degree of dissociation will be approximately equal to the con- 
ductance ratio. The values for the degree of dissociation obtained in 
this way are, however, in all circumstances too small, the difference being 
greater the more highly ionized the electrolyte. 

The fact that the type of deviation from Onsager's equation under 
discussion is not observed, at least up to relatively high concentrations, 
with many simple electrolytes, e.g., the alkali halides in both aqueous 



96 THE THEORY OF ELECTROLYTIC CONDUCTANCE 

and methyl alcohol solutions, shows that these substances are completely 
or almost completely dissociated under these conditions. At appreciable 
concentrations the degree of dissociation probably falls off from unity, 
but the value of a is undoubtedly much greater than the conductance 
ratio at the same concentration. 

Significance of the Degree of Dissociation. The quantity a, referred 
to as the degree of dissociation, represents the fraction of the solute 
which is free to carry current at a given concentration. The departure 
of the value of a from unity may be due to two causes which are, how- 
ever, indistinguishable as far as conductance is concerned. Although 
many salts probably exist in the ionic form even in the solid state, so 
that they are probably to be regarded as completely or almost completely 
ionized at all reasonable concentrations, the ions are not necessarily free 
to move independently. As a result of electrostatic attraction, ions of 
opposite sign may form a certain proportion of ion-pairs; although any 
particular ion-pair has a temporary existence only, for there is a con- 
tinual interchange between the various ions in the solution, nevertheless, 
at any instant a number of ions are made unavailable in this way for the 
transport of current. In cases of this kind the electrolyte may be com- 
pletely ionized, but riot necessarily completely dissociated. At high dilu- 
tions, when the simple Onsager equation is obeyed, the solute is both 
ionized and dissociated completely. 

In addition to the reason for incomplete dissociation just considered, 
there are some cases, e.g., weak acids and many salts of the transition 
arid other metals, in which the electrolyte is not wholly ionized. These 
substances exist to some extent in the form of un-ionizcd molecules; 
a weak acid, such as acetic acid, provides an excellent illustration of this 
type of behavior. The solution contains un-ionized, covalent molecules, 
quite apart from the possibility of ion-pairs. With sodium chloride, and 
similar electrolytes, on the other hand, there are probably no actual 
covalent molecules of sodium chloride in solution, although there may 
be ion-pairs in which the ions are held together by forces of electrostatic 
attraction. 

The quantity which has been called the " degree of dissociation 7 ' rep- 
resents the fraction of the electrolyte present as free ions capable of 
carrying the current, the remainder including both un-ionized and un- 
dissociated portions. Neither of the latter is able to transport current 
under normal conditions, and so the ordinary conductance treatment is 
unable to differentiate between them. 

The experimental data show that the deviations from the Onsager 
equation which may be attributed to incomplete dissociation occur more 
readily the smaller the ions, the higher their valence and the lower the 
dielectric constant of the medium. This generalization, as far as ionic 
size is concerned, appears at first sight not to hold for the salts of the 
alkali metals, for the deviations from the Onsager equation become more 
marked as the atomic weight of the metal increases; owing to the effect 



DETERMINATION OF THE DEGREE OF DISSOCIATION 97 

of hydration, however, the effective size of the ion in solution decreases 
with increasing atomic weight. It is consequently the radius of the ion 
as it exists in solution, i.e. together with its associated solvent molecules, 
and not the size of the bare ion, that determines the extent of dissociation 
of the salt. 

According to the concept of ion association, developed by Bjerrum 
(see p. 155), small size and high valence of the ions and a medium of low 
dielectric constant are just the factors that would facilitate the formation 
of ion-pairs. The observed results are thus in general agreement with 
the theory of incomplete dissociation due to the association of ions in 
pairs held together by electrostatic forces. The theory of Bjerrum leads 
to the expectation that the extent of association of an electrolyte con- 
sisting of small or high-valence ions in a solvent of low dielectric constant 
would only become inappreciable, and hence the degree of dissociation 
becomes equal to unity, at very high dilutions. It follows, therefore, 
that the simple Onsager equation could only be expected to hold at very 
low concentrations; under these conditions, however, the experimental 
results would not be sufficiently accurate to provide an adequate test of 
the equation. 

Determination of the Degree of Dissociation. The determination of 
the degree of dissociation involves the evaluation of the quantity A' at 
the given concentration, as defined by equation (44) ; as seen previously, 
A' is the equivalent conductance the electrolyte would have if the solute 
were completely dissociated at the same ionic concentration as in the 
experimental solution. Since the definition of A' involves a, whereas 
A' is required in order to calculate a, it is evident that the former quantity 
can be obtained only as the result of a series of approximations. Two 
of the methods that have been used will be described here. 

If Kohlrausch's law of independent ionic migration is applicable to 
solutions of appreciable concentration, as well as to infinite dilution, as 
actually appears to be the case, the equivalent conductance of an electro- 
lyte MA may be represented by an equation similar to the one on page 
57, viz., 

AM A = AMCI + ANHA ANUCI, (45) 



where the various equivalent conductances refer to solutions at the same 
ionic concentration. If MCI, NaA and NaCl are strong electrolytes, they 
may be regarded as completely dissociated, provided the solutions are 
not too concentrated; the equivalent conductances in equation (45) con- 
sequently refer to the same stoichiometric concentration in each case. 
If MA is a weak or intermediate uni-univalent electrolyte, however, the 
value of AMA derived from equation (45) will be equivalent to AMA, the 
corresponding ionic concentration being ac, where a is the degree of dis- 
sociation of MA at the total concentration c moles per liter. 9 

f Maclnnes and Shedlovsky, J. Am. Chem. Soc., 54, 1429 (1932). 



98 THE THEORY OF ELECTROLYTIC CONDUCTANCE 

The equivalent conductances of the three strong electrolytes may be 
written in the form of the power series [cf. equation (41)], 

A = Ao - A'c* + B'c - C'c* + -, (41a) 

where c is the actual ionic concentration, which in these instances is 
identical with the stoichiometric concentration. Combining the values 
of AMCI, A Na A and A Na ci expressed in this form, it is possible by adding 
AMCI and A Nft A and subtracting A Na ci to derive an equation for AM A; thus 

A MA = AO ( MA) + A"(ac) + JB"(ac) - C"(ac) + , (46) 

the c terms being replaced by ac to give the actual ionic concentration 
of the electrolyte MA. Since AO<MA) is known, and A", B", C", etc., are 
derived from the A', B', C', etc. values for MCI, NaA and NaCl, it follows 
that AMA could be calculated if a. were available. An approximate esti- 
mate is first made by taking a as equal to A/A for MA, and in this way a 
preliminary value for AMA is derived from equation (46) ; a. can now be 
obtained more accurately as AMA/AMA, and the calculations are repeated 
until there is no change in AMA. The method may be illustrated with 
special reference to the determination of the dissociation of acetic acid. 
The conductances of hydrochloric acid (MCI), sodium acetate (NaA) and 
sodium chloride (NaCl) can be expressed in the form of equation (4 la) : 
thus, at 25, 

ACHCD = 426.16 - 156.62A^ + 169.0c (1 - 0.2273>Tc), 

A (C H 3 co 2 Na) = 91.00 - 80.46V^ + 90.0c (1 - 0.2273 Vc), 

A(Naci) = 126.45 - 88.52Vc + 95.8c (1 - 0.2273Vc), 

.'. A' (C H 3 co,H) = 390.7 - 148.56V^ + 163.2c (1 - 0.2273\^). 

At a concentration of 1.0283 X 10~ 3 equiv. per liter, for example, the 
observed equivalent conductance of acetic acid is 48.15 ohms~ l cm. 2 and 
since A is 390.7 ohms" 1 cm. 2 , the value of a, as a first approximation, 
is 48.15/390.7, i.e., 0.1232; inserting this result in the expression for 
A(CH 3 co,H)> the latter is found to be 389.05. As a second approximation, 
a. is now taken as 48.15/389.05, i.e., 0.1238; repetition of the calculation 
produces no appreciable change in the value of A', and so 0.1238 may 
be taken as being the correct degree of dissociation of acetic acid at the 
given concentration. The difference between this result and the con- 
ductance ratio, 0.1232, is seen to be relatively small in this instance; for 
stronger electrolytes, however, the discrepancy is much greater. 

If there are insufficient data for the equivalent conductances to be 
expressed analytically in the form of equation (4 la), the calculations 
described above can be carried out in the following manner. 10 As a 
first approximation the value of a is taken as equal to the conduct- 

"Sherrffl and Noyes, J. Am. Chem. Soc., 48, 1861 (1926); Maclnnes, ibid., 48, 
2068 (1926). 



CONDUCTANCE RATIO AND THE ONBAGER EQUATION 99 

ance ratio and from this the ionic concentration ac is estimated. By 
graphical interpolation from the conductance data the equivalent con- 
ductances of MCI, NaA and NaCl are found at this stoichiometric con- 
centration, which in these cases is the same as the ionic concentration, 
and from them a preliminary result for A'MA) is obtained. With this a 
more accurate value of a is derived and the calculation of A' (M A) is re- 
peated; this procedure is continued u.itil the latter quantity remains 
unchanged. The final result is utilized to derive the correct degree of 
dissociation. This method of calculation is, of course, identical in prin- 
ciple with that described previously; the only difference lies in the fact 
that in the one case the interpolation to give the value of A' at the ionic 
concentration is carried out graphically while in the other it is achieved 
analytically. 

In the above procedure for determining the degree of dissociation, 
the correction for the change in ionic speeds due to interionic forces is 
made empirically by utilizing the experimental conductance data: the 
necessary correction can, however, also be applied with the aid of the 
Onsager equation. 11 Since A/A' is equal to a, equation (44) can be 
written as 

A' = Ao - A; VAc/A', (47) 

where k represents A + #A and is a constant for the given solute in a 
particular solvent at a definite temperature. The value of AO for the 
electrolyte under consideration can, in general, be obtained from the ion 
conductances at infinite dilution or from other conductance data (see 
p. 54); it may, therefore, be regarded as known. As a first approxi- 
mation, A' in the term VAc/A' is taken as equal to A , which is equivalent 
to identifying the degree of dissociation with the conductance ratio, and 
a preliminary value for A' can be derived from equation (47) by utilizing 
the experimental equivalent conductance A at the concentration c. This 
result for A' is inserted under the square-root sign, thus introducing a 
better value for a, and A' is again computed by means of equation (47). 
The procedure is continued until there is no further change in A' and this 
may be taken as the correct result from which the final value of a is 
calculated. 

Conductance Ratio and the Onsager Equation. Equation (42) can 
be written in the form 

(48) 

which is an expression for the conductance ratio, A/A ; the values, clearly, 
decrease steadily with increasing concentration. For weak electrolytes, 
the degree of dissociation decreases with increasing temperature, since 

"Davies, Trans. Faraday Soc., 23, 351 (1927); "The Conductivity of Solutions/' 
1933, p. 101; see also, Banks, J. Chem. Soc., 3341 (1931). 



100 THE THEORY OP ELECTROLYTIC CONDUCTANCE 

these substances generally possess a positive heat of ionization. It is 
apparent, therefore, from equation (48), that the conductance ratio will 
also decrease as the temperature is raised. For strong electrolytes, 
a being virtually unity, equation (48) becomes 



the influence of temperature on the conductance ratio is consequently 
determined by the quantity in the parentheses, viz., (.4/A ) + B. In 
general, this quantity increases with increasing temperature. That this 
is the case, at least with water as the solvent, is shown by the data in 
Table XXVI, for potassium chloride and tetraothylammonium picrate in 

TABLE XXVI. INFLUENCE OF TEMPERATURE ON CONDUCTANCE RATIO 

Temp. Potassium Chloride Tetraethylammonium Picratc 

Ao r-+* A T + B 

AO AO 

81.8 0.54 31.2 1.16 

18 129.8 0.61 53.2 1.17 

100 406.0 0.77 196.5 1.30 

aqueous solution. It follows, therefore, that the conductance ratio for 
strong electrolytes should decrease with increasing temperature, as found 
experimentally (p. 52). It will be evident from equation (49) that the 
decrease should be greater the more concentrated the solution, and this 
also is in agreement with observation. It may be noted that the quan- 
tity (-A/Ac) + B is equal to (A + J5Ao)/A , in which the numerator is a 
measure of the decrease in equivalent conductance due to the diminution 
of ionic speeds by interionic forces (p. 89) : it follows, therefore, that as 
a general rule the interionic forces increase with increasing temperature. 
Introducing the expressions for A and B given on page 90, it is 
seen that 

A . P _ 82 ' 4 4. 8 ' 20 * 1Q 6 
A + 



and since ryAo is approximately constant for a given electrolyte in different 
solvents (cf. p. 64), this result may be written in the form 



(50) 

where a and 6 are numerical constants. It is at once evident, therefore, 
that the smaller the dielectric constant of the solvent, at constant tem- 
perature, the greater will be the value of (A/Ao) + B, and hence the 
smaller the conductance ratio. The increase of ion association which 
accompanies the decrease of dielectric constant will also result in a de- 
crease of the conductance ratio. 



DISPERSION OF CONDUCTANCE AT HIGH FREQUENCIES 101 

The discussion so far has referred particularly to uni-univalent elec- 
trolytes; it is evident from equation (35) that the valences of the ions 
are important in determining the decrease of conductance due to inter- 
ionic forces and hcnco they must also affect the conductance ratio. The 
general arguments concerning the effect of concentration, temperature 
and dielectric constant apply to electrolytes of all valence types; in order 
to investigate the effect of valence, equation (35) for a strong electrolyte 
may be written in the general form 



+ z_), (51) 

Q 

where A' and B' are constants for the solvent at a definite temperature. 
It is clear that for a given concentration the conductance ratio decreases 
with increasing valence of the ions, since the factors z+ + z_ and w both 
increase. It was seen in Chap. II that the equivalent conductances of 
most electrolytes, other than acids or bases, at infinite dilution are approx- 
imately the same; in this event it is apparent from equation (51) that for 
electrolytes of a given valence type the conductance ratio will depend 
only on the concentration of the solution (cf. p. 52). 

In the foregoing discussion the Onsager equation has been used for 
the purpose of drawing a number of qualitative conclusions which are 
in agreement with experiment. The equation could also be used for 
quantitative purposes, but the results would be expected to be correct 
only in very dilute solutions. At appreciable concentrations additional 
terms must be included, as in the Shedlovsky equation, to represent more 
exactly the variation of conductance with concentration; the general 
arguments presented above would, however, remain unchanged. 

Dispersion of Conductance at High Frequencies. An important con- 
sequence of the existence of the ionic atmosphere, with a finite time of 
relaxation, is the variation of conductance with frequency at high fre- 
quencies, generally referred to as the dispersion of conductance or the 
Debye-Falkenhagen effect. If an alternating potential of high fre- 
quency is applied to an electrolyte, so that the time of oscillation is small 
in comparison with the relaxation time of the ionic atmosphere, the un- 
symmetrical charge distribution generally formed around an ion in 
motion will not have time to form completely. In fact, if the oscillation 
frequency is high enough, the ion will be virtually stationary and its 
ionic atmosphere will be symmetrical. It follows, therefore, that the 
retarding force due to the relaxation or assymmetry effect will thus dis- 
appear partially or entirely as the frequency of the oscillations of the 
current is increased. At sufficiently high frequencies, therefore, the con- 
ductance of a solution should be greater than that observed with low- 
frequency alternating or with direct current. The frequency at which 
the increase of conductance might be expected will be approximately 
1/0, where 6 is the relaxation time; according to equation (24) the relaxa- 



102 



THE THEORY OF ELECTROLYTIC CONDUCTANCE 



tion time for a binary electrolyte is 71.3 X 10~ lo /czA sec., and so the 
limiting frequency v above which abnormal conductance is to be expected 
is given by 



_ 

/ l.o 



X 10 10 oscillations per second. 



The corresponding wave length in centimeters is obtained by dividing 
the velocity of light, i.e., 3 X 10 l cm. per sec. by this frequency; the 
result may be divided by 100 to give the value in meters, thus 



2.14 

r- 
czA 



meters. 



For most electrolytes, other than acids and bases, in aqueous solutions 
A is about 120 at 25, and hence 



2 X 



cz 



meters. 



If the electrolyte is of the uni-univalent type and has a concentration 
of 0.001 molar, the Debye-Falkenhagen effect should become evident 
with high-frequency oscillations of wave length of about 20 meters or 
less. The higher the valence of the ions and the more concentrated the 
solution the smaller the wave length, and hence the higher the frequency, 
of the oscillations required for the effect to become apparent. 




1 10 100 1.000 meters 

Wave Length 
FIG. 33. High frequency conductance dispersion of potassium chloride 

The dispersion of conductance at high frequencies was predicted by 
Debye and Falkenhagen, 12 who developed the theory of the subject; the 
phenomena were subsequently observed by Sack and others. 18 The 

"Debye and Falkenhagen, Phyaik. Z., 29, 121, 401 (1928); Falkenhagen and 
Williams, Z. phyrik. Chem., 137, 399 (1928); J. Phys. Chem., 33, 1121 (1929); Falken- 
hagen, Physik. Z., 39, 807 (1938). 

" Sack et al., Physik. Z., 29, 627 (1928); 30, 576 (1929); 31, 345, 811 (1930); Brendel, 
ibid., 32, 327 (1931); Debye and Sack, Z. Ekktrochem., 39, 512 (1933); Arnold and 
Williams, /. Am. Chem. Soc., 58, 2613, 2616 (1936). 



CONDUCTANCE WITH HIGH POTENTIAL GRADIENTS 



103 



nature of the results to be expected will be evident from an examination 
of Figs. 33 and 34, in which the calculated ratio of the decrease of con- 
ductance due to the relaxation effect * at a short wave length X, i.e., 
), to that at long wave length A#, i.e., at low frequency, is plotted as 




10 100 LOGO meters 

Wave Length 
FIG. 34. High frequency conductance dispersion of salts at 10~ 4 mole per liter 

ordinate against the wave length as abscissa. The values for potassium 
chloride at concentrations of 10~ 2 , 10~ 3 , and IQr 4 mole per liter are 
plotted in Fig. 33, and those for potassium chloride, magnesium sulfate, 
lanthanum chloride and potassium ferrocyanide at 10~ 4 mole per liter 
in water at 18 are shown in Fig. 34. It is seen that, in general, the 
decrease of conductance caused by the relaxation or asymmetry effect 
decreases with decreasing wave length or increasing frequency; the actual 
conductance of the solution thus increases correspondingly. The effect 
is not noticeable, however, until a certain low wave length is reached, 
which, as explained above, is smaller the higher the concentration. 
The influence of the valence of the ions is represented by the curves in 
Fig. 34; the higher the valence the smaller the relative conductance change 
at a given high frequency. 

The measurements of the Debye-Falkenhagen effect are generally 
made with reference to potassium chloride; the results for a number of 
electrolytes of different valence types have been found to be in satis- 
factory agreement with the theoretical requirements. Increase of tem- 
perature and decrease of the dielectric constant of the solvent necessitates 
the use of shorter wave lengths for the dispersion of conductance to be 
observed; these results are also in accordance with expectation from 
theory. 

Conductance with High Potential Gradients. When the applied 
potential is of the order of 20,000 volts per cm., an ion will move at a 
speed of about 1 meter per sec., and so it will travel several times the 
thickness of the effective ionic atmosphere in the time of relaxation. 

* At low frequencies this quantity is equal to the second term in the brackets in 
equation (35), multiplied by Vc(z+ -f z_). 



104 



THE THEORY OF ELECTROLYTIC CONDUCTANCE 



As a result, the moving ion is virtually free from an oppositely charged 
ion atmosphere, since there is never time for it to be built up to any 
extent. In these circumstances both asymmetry and electrophoretic 
effects will be greatly diminished and at sufficiently high voltages should 



0.04 



0.03 

AA 
A 

0.02 



0.01 




100.000 200,000 

Volte per cm. 

FIG. 35. Wien effect for potassium ferncyariide 

disappear. Under the latter conditions the equivalent conductance at 
any appreciable concentration should be greater than the value at low 
voltages. The increase in conductance of an electrolyte at high potential 
gradients was observed by Wien 14 before any theoretical interpretation 
had been given, and it is consequently known as the Wien effect. 

It is to be expected that the Wien effect will be most marked under 
such conditions that the influence of the intcrionic forces resulting from 
the existence of an ionic atmosphere is abnormally large; this would be the 
case for concentrated solutions of high-valence ions. The experimental 
results shown in Figs. 35 and 36 confirm these expectations; those in 
Fig. 35 are for solutions of containing potassium ferricyanide at concen- 
trations of 7.5, 3.7 and 1.9 X 10~ 4 mole per liter, respectively, and the 
curves in Fig. 36 are for electrolytes of various valence types in solutions 
having equal low voltage conductances. The quantity AA is the increase 
* of equivalent conductance resulting from the application of a potential 
gradient represented by the abscissa. 

"Wien, Ann. Physik, 83, 327 (1927); 85, 795 (1928); 1, 400 (1929); Physik. Z., 
32, 545 (1931); Falkenhagen, ibid., 32, 353 (1931); Schiele, Ann. Physik, 13, 811 (1932); 
Debye, Z. Elektrochem., 39, 478 (1933); Mead and Fuoss, J. Am. Chem. Soc., 61, 2047, 
3257, 3589 (1939); 62, 1720 (1940); for review, nee Eckstrom and Schmelzer, Chem. 
Revs., 24, 367 (1939). 



PROBLEMS 



105 



o.io 



0.05 




100.000 200.000 

Volte per cm. 

FIG. 36. Wien effect for salts of 
different valence types 



It will be observed that the values of AA tend towards a limit 
at very high potentials; the relaxation and electrophoretic effects are 
then virtually entirely eliminated. For an incompletely dissociated elec- 
trolyte the measured equivalent conductance under these conditions 
should be A , where a is the true 
degree of dissociation; since AO is 
known, determinations of con- 
ductance at high voltages would 
seem to provide a method of ob- 
taining the degree of dissociation 
at any concentration. It has 
been found, however, that the 
Wien effect for weak acids and 
bases, which are known to be 
dissociated to a relatively small 
extent, is several times greater 
than is to be expected; the dis- 
crepancy increases as the voltage 
is raised. It is very probable 
that in these cases the powerful 
electrical fields produce a temporary dissociation into ions of the 
molecules of weak acid or base; this phenomenon, referred to as the 
dissociation field effect, invalidates the proposed method for calculating 
the degree of dissociation. With strong electrolytes, which are believed 
to be completely dissociated, the conductances observed at very high 
potential gradients are close to the values for infinite dilution, in agree- 
ment with anticipation. 

It may be pointed out in conclusion that the conductance phenomena 
with very high frequency currents and at high potential gradients pro- 
vide striking evidence for the theory of electrolytic conductance, based 
on the existence of an ionic atmosphere surrounding every ion, proposed 
by Debye and Hiickel and described in this chapter. Not only does the 
theory account qualitatively for conductance results of all types, but 
it is also able to predict them quantitatively provided the solutions are 
not too concentrated. 

PROBLEMS 

1. Calculate the thickness of the ionic atmosphere in 0.1 N solutions of a 
uni-univalent electrolyte in the following solvents: nitrobenzene (D = 34.8); 
ethyl alcohol (D = 24.3); and ethylene dichloride (D = 10.4). 

2. Utilize the results obtained in the preceding problem to calculate the 
relaxation times of the ionic atmospheres and the approximate minimum fre- 
quencies at which the Debye-Falkenhagen effect is to be expected. It may 
be assumed that A O T?O has a constant value of 0.6. The viscosities of the sol- 
vents are as follows: nitrobenzene (0.0183 poise); ethyl alcohol (0.0109); and 
ethylene dichloride (0.00785). 



106 THE THEORY OF ELECTROLYTIC CONDUCTANCE 

3. The viscosity of water at is 0.01793 poise and at 100 it is 0.00284; 
the corresponding dielectric constants are 87.8 and 56. Calculate the values 
of the Onsager constants A and B for a uni-univalent electrolyte at these 
temperatures. Make an approximate comparison of the slopes of the plots 
of A against Vc at the two temperatures for an electrolyte for which A is 
100 ohms" 1 cm. 2 at 0, assuming Walden's rule to be applicable. 

4. Make an approximate comparison, by means of the Onsager equation, 
of the conductance ratios at 25 of 0.01 N solutions of a strong uni-univalent 
electrolyte in water and in ethyl alcohol; it may be assumed that A i?o has the 
constant value of 0.6 in each case. 

5. The following values were obtained by Martin and Tartar [J. Am. 
Chem. Soc., 59, 2672 (1937)] for the equivalent conductance of sodium lactate 
at various concentrations at 25: 

c X 10 J 0.1539 0.3472 0.6302 1.622 2.829 4.762 
A 87.89 87.44 86.91 85.80 84.87 83.78 

Plot the values of A against Vc and determine the slope of the line; estimate 
AO and compare the experimental slope with that required by the Onsager 
equation. 

6. Calculate the limiting theoretical slope for the plot of A against Vc for 
lanthanum chloride (Lads) in water at 25 ; A for this salt is 145.9 ohms" 1 cm. 9 
and X- for the chloride ion is 76.3 ohms" 1 cm. 2 

7. Saxton and Waters [V. Am. Chem. Soc., 59, 1048 (1937)] gave the 
ensuing expressions for the equivalent conductances in water at 25 of hydro- 
chloric acid, sodium chloride and sodium a-crotonate (Naa-C.) : 

AHCI - 426.28 - 156.84^ + 169.7c (1 - 0.2276^) 

ANECI = 126.47 - 88.65Vc + 94.8c (1 - 0.2276Vc) 

A N a-c. - 83.30 - 78.84Vc + 97.27c (1 - 0.2276Vc). 

The equivalent conductances of a-crotonic acid at various concentrations were 
as follows: 

c X 10 1 A c X 10* A 

0.95825 51.632 7.1422 19.861 

1.7050 39.473 14.511 14.053 

3.2327 29.083 22.512 11.318 

4.9736 23.677 33.246 9.317 

Calculate the degree of dissociation of the crotonic acid at each concentration, 
making due allowance for interionic attraction. Compare the values obtained 
with the corresponding conductance ratios. 

(The results of this problem are required for Problem 8 of Chap. V.) 

8. Employ the data of the preceding problem to calculate the degree of 
dissociation of a-crotonic acid at the various concentrations using the method 
of Davies described on page 99. 



CHAPTER IV 
THE MIGRATION OF IONS 

Transference Numbers. The quantity of electricity g carried 
through a certain volume of an electrolytic solution by ions of the ith 
kind is proportional to the number in unit volume, i.e., to the concen- 
tration d in gram-ions or moles per liter, to the charge z carried by each 
ion, and to the mobility w, i.e., the velocity under unit potential gradient 
(cf.'p. 58); thus 

, (1) 



where k is the proportionality constant, which includes the time. The 
total quantity of electricity Q carried by all the ions present in the elec- 
trolyte is thus the sum of the q % terms for each species; that is 



Q = kciziui + kc&tu* + fccjZsWs + - (2) 

(2a) 



the proportionality constant being the same for all the ions. It follows, 
therefore, that the fraction of the total current carried by an ion of the 
ith kind is given by 



This fraction is called the transference number, or transport number, 
of the given ion in the particular solution and is designated by the symbol 
ti', the sum of the transference numbers of all the ions present in the 
solution is clearly equal to unity. In the simplest case of a single electro- 
lyte yielding two ions, designated by the suffixes + and _, the corre- 
sponding transference numbers are given, according to equation (3), by 

C+Z+U+ , C-Z-U- 

t. = and <_ = ; 

C+2+U+ + C-Z-U- C+Z+U+ + C-Z-U- 

The quantities c+z+ and c~Z- 9 which represent the equivalent concentra- 
tions of the ions, are equal, and hence for this type of electrolyte, which 
has been most frequently studied, 

* +== u + + u- and ^"ut + uJ (4) 

and 

t+ + t. = 1. 

107 



108 



THE MIGRATION OP IONS 



The speed of an ion in a solution at any concentration is proportional to 
the conductance of the ion at that concentration (p. 80), and so the 
transference number may be alternatively expressed in the form 



t - + 

t+ ~ A 



and 



(5) 



where the values of the ion conductances X+ and X_, and the equivalent 
conductance A of the solution, are those at the particular concentration 
to which the transference numbers are applicable. These values are, of 
course, different from those at infinite dilution, and so it is not surprising 
to find, as will be seen shortly, that transference numbers vary with the 
1 concentration of the solution ; they approach a limiting value, however, 
at infinite dilution. 

Three methods have been generally employed for the experimental 
determination of transference numbers : the first, based on the procedure 
originally proposed by Hittorf (1853), involves measurement of changes 
of concentration in the vicinity of the electrodes; in the second, known 
as the "moving boundary " method, the rate of motion of the boundary 
between two solutions under the influence of current is studied (cf. p. 
116); the third method, which will be considered in Chap. VI, is based 
on electromotive force measurements of suitable cells. 

Faraday's Laws and Ionic Velocities. It may appear surprising, at 
first sight, that equivalent quantities of different ions are liberated at 
the two electrodes in a given solution, as required by Faraday's Jaws, 



Anode 



Cathode 



II 



III 






+ 4- + + 



+ + + + + 



4- + + V 






FIG. 37. Migration of ions 

in spite of the possible difference in the speeds of the ions moving towards 
the respective electrodes. The situation can, however, be understood 
by reference to the diagram in Fig. 37; this represents an electrolytic cell 
in which there are an equivalent number of positive and negative ions, 



THE HITTORP METHOD 109 

indicated by plus and minus signs. The condition of the system at the 
commencement of electrolysis is shown in Fig. 37, I. Suppose that the 
cations only are able to move under the influence of an applied potential, 
and that two of these ions move from left to right; the condition attained 
will then be as at Fig. 37, II. At each electrode there are two ions un- 
paired and these must be considered to be discharged; the two electrons 
given up by the negative ions at the anode may be imagined to travel 
through the external circuit and discharge the two positive ions at the 
cathode. It is seen, therefore, that although only the positive ions are 
able to move, equivalent amounts of the two ions are discharged at the 
respective electrodes. A condition of this kind actually arises in cer- 
tain solid and fused electrolytes, where all the current is carried by the 
cations. 

If while the two cations are moving in one direction, three anions are 
carrying electricity in the opposite direction, so that the ionic velocities 
are in the ratio of 2 to 3, the result will be as in Fig. 37, III. Five ions 
are seen to be discharged at each electrode, in spite of the difference in 
speeds of the two ions. There is thus no difficulty in correlating Fara- 
day's laws with the fact that the oppositely charged ions in a solution 
may have different velocities. Incidentally it will be noted that the con- 
clusions to be drawn from Fig. 37 are in harmony with the results derived 
above, e.g., equation (4); the fraction of the total current carried by 
each ion, i.e., its transference number, is proportional to its speed. In 
the condition of Fig. 37, III, the total quantity of electricity passing 
may be taken as five faradays, since five ions are discharged; of these 
five faradays, two are carried by the cations in one direction and three 
by the anions in the opposite direction. 

Attention may be called here to a matter which will receive further 
discussion in Chap. XIII; the ions that carry the current through the 
solution are not necessarily those to be discharged at the electrodes. 
This is assumed to be the case here, however, for the sake of simplicity. 

The Hittorf Method. Suppose an electric current is passed through 
a solution of an electrolyte which yields the ions M+ and A~; these ions 
are not necessarily univalent, although a single + or sign is used for 
the sake of simplicity of representation. The fraction of the total cur- 
rent carried by the cations is t+ and that carried by the anions is L.; 
hence when one faraday of electricity is passed through the solution, 
t+ faradays are carried in one direction by t+ equivalents of M+ ions and 
J_ faradays are carried in the other direction by J_ equivalents of A" ions. 
At the same time one equivalent of each ion is discharged at the appro- 
priate electrode. The migration of the ions and their discharge under 
the influence of the current bring about changes of concentration in the 
vicinity of the electrodes, and from these changes it is possible to calcu- 
late the transference numbers. 

Imagine the cell containing the electrolyte to be divided into three 
compartments by means of two hypothetical partitions; one compart- 



110 THE MIGRATION OP IONS 

ment surrounds the cathode, another the anode, and the third is a middle 
compartment in which there is no resultant change of concentration. 
The effect of passing one faraday of electricity through the solution of 
the electrolyte MA can then be represented in the following manner. 

Cathode Compartment (I) Middle Compartment Anode Compartment (II) 

1 equiv. of M f is discharged f+ equiv. of M + migrate to I 1 equiv. of A" is discharged 

<+ equiv. of M+ migrate in *_ equiv. of A~ migrate from I J_ equiv. of A~ migrate in 

I- equiv. of A~ migrate out < h equiv. of M+ migrate from 1 1 t+ equiv. of M+ migrate out 

/_ equiv. of A" migrate to II 
Net Result: 

Loss of 1 1+ J.equiv. of M" 1 " No change of concentration LOBS of 1 <_ =/+equiv. of A" 
Loss of *_ equiv. of A~ Loss of t+ equiv. of M + 

.*. Net loss is /-equiv. of MA .'. Net loss is t+ equiv. of MA 

It follows, therefore, if the discharged ions may be regarded as being 
completely removed from the system and the electrodes are not attacked, 
as is tacitly assumed in the above tabulation, that 

Equiv. of electrolyte lost from anode compartment t+ 
Equiv. of electrolyte lost from cathode compartment ~~ t~ 

The total decrease in amount of the electrolyte MA in both compart- 
ments of the experimental cell is equal to the number of equivalents 
deposited on each electrode; if a coulometer (p. 17) is included in the 
circuit, then by Faraday's laws the same number of equivalents of ma- 
terial, no matter what its nature, will be deposited. It follows, therefore, 
that 

Equiv. of electrolyte lost from anode compartment 

Equiv. deposited on each electrode of cell or in coulometer + ' 

and 

Equiv. of electrolyte lost from cathode compartment __ 
Equiv. deposited on each electrode of cell or in coulometer ~~ 

By measuring the fall in concentration of electrolyte in the vicinity of 
anode and cathode of an electrolytic cell, and at the same time deter- 
mining the amount of material deposited on the cathode of the cell or of 
a coulometer in the circuit, it is possible to evaluate the transference 
numbers of the ions present in solution. Since the sum of t+ and _ must 
be unity, it is not necessary to measure the concentration changes in both 
anode and cathode compartments, except for confirmatory purposes; 
similarly, if the changes in both compartments are determined it is not 
strictly necessary to employ a coulometer in the circuit. It is, however, 
more accurate to evaluate the total amount of material deposited by the 
current by means of a coulometer than from the concentration changes. 
Chemical Changes at the Electrodes. Although the discharge of a 
cation generally leads to the deposition of metal on the cathode and its 
consequent removal from the system, this is not true for anions. If the 
anode consists of an attackable metal which does not form an insoluble 



HITTORF METHOD 111 

compound with the anions present in the solution, these ions are not 
removed on discharge but an equivalent amount of the anode material 
passes into solution. In these circumstances the concentration of the 
anode solution actually increases instead of decreasing, but allowance 
can be readily made for the amount of dissolved material. In the sim- 
plest case the anode metal is the same as that of the cations in the electro- 
lyte, e.g., a silver anode in silver nitrate solution; the changes in the 
anode compartment resulting from the passage of one faraday of elec- 
tricity are as follows : 

1 equiv. of M + dissolves from' the electrode 
t- equiv. of A~ migrate in 
t+ equiv. of M+ migrate out 
Net gain is t- equiv. of MA. 

It is thus possible to determine the transference number of the cation 
from the increase in concentration of the anode compartment. An alter- 
native way of treating the results is to subtract from the observed gain 
in amount of electrolyte the number of equivalents of M 4 " dissolved from 
the anode; the net result is a loss of 1 t-, i.e., t+, equiv. of MA per 
faraday, as would have been the case if the anions had been completely 
removed on discharge and the anode had not dissolved. It should be 
noted that the general results derived are applicable even if the anode 
material consists of a metal M' which differs from M; the increase or 
decrease of concentration now refers to the total number of equivalents 
of MA and M'A, but the presence of the extraneous ions will affect the 
transference numbers of the M+ and A~ ions. 

When working with a solution of an alkali or alkaline-earth halide, 
the anode is generally made of silver coated with the same metal in a 
finely-divided state, and the cathode is of silver covered with silver halide. 
In this case the discharged halogen at the anode combines with the silver 
to form the insoluble silver halide, and so is effectively removed from 
the anode compartment. At the cathode, however, the silver halide is 
reduced to metallic silver and halide ions pass into solution; there is con- 
sequently a gain in the concentration of the cathode compartment for 
which allowance must be made. 

Hittorf Method: Experimental Procedure. In Hittorfs original de- 
termination of transference numbers short, wide electrolysis tubes were 
used in order to reduce the electrical resistance, and porous partitions 
were inserted to prevent mixing by diffusion and convection. These 
partitions are liable to affect the results and so their use has been avoided 
in recent work, and other precautions have been taken to minimize 
errors due to mixing. Many types of apparatus have been devised for 
the determination of transference numbers by the Hittorf method. One 
form, which was favored by earlier investigators and is still widely used 
for ordinary laboratory purposes, consists of an H-shaped tube, as shown 



112 



THE MIGRATION OF IONS 



in Fig. 38, or a tube of this form in which the limbs are separated by a 
U-tube. The vertical tubes, about 1.5 to 2 cm. in width and 20 to 25 
cm. approximately in length, contain the anode and cathode, respectively. 
If the electrolyte being studied is the salt of a metal, such as silver or 
copper, which is capable of being deposited on the 
II || cathode with 100 per cent efficiency, the metal itself 

"^ r "^ may be used as anode and cathode. Transference 
numbers can be calculated from the concentration 
changes in one electrode compartment only; if this 
procedure is adopted the nature of the electrolyte and 
of the electrode in the other compartment is imma- 
terial. With certain solutions, e.g., acids, alkali 
hydroxides and alkali halides, there is a possibility 
that gases may be liberated at one or both electrodes; 
the mixing thus caused and the acid or alkali set free 
will vitiate the experiment. Cadmium electrodes have 
been employed to avoid the liberation of chlorine at 
the anode, and cathodes of mercury covered with con- 
centrated solutions of zinc chloride or copper nitrate 
have been used to prevent the evolution of hydrogen. 
In the latter cases the change in the concentration of 
the experimental electrolyte in the anode compartment 
only can be utilized for the calculation of the trans- 
ference numbers, as indicated above. For alkali 
halides the best electrodes are finely divided silver as 
anode and silver coated with silver halide by electro- 
lysis (p. 234) as cathode; the behavior of these electrodes has been ex- 
plained previously. 

The apparatus is filled with the experimental solution whose weight 
concentration is known, and the electrodes are connected in series with a 
copper or silver voltameter; a current of 0.01 to 0.02 ampere is then 
passed for two to three hours. Too long a time must not be used, other- 
wise the results will be vitiated by diffusion, etc., and too large a current 
will produce mixing by convection due to heating. If both the time and 
current are too small, however, the concentration changes will not be 
appreciable. At the conclusion of the experiment a sufficient quantity 
of solution, believed to contain all that has changed in concentration 
during the electrolysis, is run off slowly from each limb, so as to avoid 
mixing, and analyzed. A further portion of liquid is removed from each 
limb; these represent the "middle compartment" and should have the 
same concentration as the original solution. The amount of metal de- 
posited in the coulometer during the electrolysis is determined and 
sufficient data are now available for the calculation of the transference 
numbers. 

Since the gain or loss of electrolyte near the electrode is accompanied 
by changes of density and hence in the volume of the solution, the con- 



FIG. 38. Simple 
apparatus for trans- 
ference numbers. 



IMPROVED APPARATUS FOR THE HITTORF METHOD 



113 



ccntration changes resulting from the passage of current must be deter- 
mined with reference to a definite weight of solvent present at the con- 
clusion of the electrolysis. Thus, if analysis of x grams of the anode 
solution showed it to contain y grams of the electrolyte at the end of the 
experiment, then the latter was associated with x y grams of water. 
The amount of electrolyte, say z grams, associated with this same amount 
of water at the beginning, is calculated from the known weight composi- 
tion of the original solution. The decrease of electrolyte in the anode 
compartment, assuming due allowance has been made for the amount, 
if any, of anode material that has dissolved, is thus z y grams or 
(z ~ y)/e equivalents, where e is the equivalent weight of the experi- 
mental substance. If c is the number of equivalents of material de- 
posited in the coulomcter during the electrolysis, it follows from equation 
(6) that the transference number of the cation (t+) is given by 



- y 

ec 



(8) 




The transference number of the anion (t.) is of course equal to 1 t+. 

Improved Apparatus for the Hittorf Method. Recent work on trans- 
ference number determinations of alkali and alkaline-earth chlorides by 
the Hittorf method has been made with a form of apparatus of which 
the principle is illustrated by Fig. 39. l It 
consists of two parts, each of which contains 
a stopcock of the same bore as the main 
tubes; the anode is inserted at A and the 
cathode at C, the parts of the apparatus being 
connected by the ground joint at B. The 
possibility of mixing between tho anode aiid 
cathode solutions is obviated by introducing 
right-angle bends below the anode, above the 
cathode and in the vertical tube between the 
two portions of the apparatus. For the study 
of alkali and alkaline-earth chlorides the anode 
is a coiled silver wire and the cathode is 
covered with silver chloride. In these cases 
the anode solution becomes more dilute and 
tends to rise, while the cathode solution in- 
creases in concentration during the course of 
the electrolysis and has a tendency to sink; 
the consequent danger of mixing is avoided 
by placing the anode at a higher level than the cathode, as shown in 
Fig. 39. 

1 Jones and Dole, J. Am. Chem. A'oc., 51, 1073 (1929); Maclnnes and Dole, ibid., 
53, 1357 (1931); Jones and Bradshaw, ibid., 54, 138 (1932). 



(fciS, 



u 

6 


8 




[l 


= 





M 


) 


i 


C 



FIG. 39. Apparatus for 
application of Hittorf method 



114 THE MIGRATION OF IONS 

When carrying out a measurement the two parts of the apparatus, 
with the electrodes in position and the stopcocks open, are fitted to- 
gether, placed in a thermostat, and filled with the experimental solution. 
A silver coulometer is connected in series with each electrode to insure 
fche absence of leakage currents. A quantity of electricity, depending in 
amount on the concentration of the solution, is passed through the circuit, 
and the stopcocks are then closed. The liquid isolated above Si is the 
anode solution and that below 82 is the cathode solution; these are 
removed and analyzed. Quantities of liquid are withdrawn from the 
intermediate portion between Si and S z by inserting pipettes through the 
openings shown; these should have the same concentration as the original 
electrolyte. 

Although the Hittorf method is simple in principle, accurate results 
are difficult to obtain; it is almost impossible to avoid a certain amount 
of mixing as the result of diffusion, convection and vibration. Further, 
the concentration changes are relatively small and any attempt to increase 
them, by prolonged electrolysis or large currents, results in an enhance- 
ment of the sources of error just mentioned. In recent years, therefore, 
the Hittorf method for the determination of transference numbers has 
been largely displaced by the moving boundary method, to be described 
later. 

True and Apparent Transference Numbers. The fundamental as- 
sumption of the Hittorf method for evaluating transference numbers 
from concentration changes is that the water remains stationary. There 
is ample evidence, however, that ions are solvated in solution and hence 
they carry water molecules with them in their migration through the 
electrolyte; this will result in concentration changes which affect the 
measured or " apparent " transference number. Suppose that each cation 
and anion has associated with it w+ and w- molecules of water, respec- 
tively; let T+ arid T, be the "true" transference numbers, i.e., the actual 
fraction of current carried by cations and anions, respectively. For the 
passage of one faraday of electricity the cations will carry w+T+ moles 
of water in one direction and the anions will transport w-T- moles in the 
opposite direction; there will consequently be a resultant transfer of 

w+T+ - w-T- = x (9) 

moles of water from the anode to the cathode compartment. The trans- 
ference number t+ is equal to the apparent number of equivalents of 
electrolyte leaving the anode compartment, for the passage of one fara- 
day, whereas T+ is the true number of equivalents; the difference between 
these two quantities is equal to the change of concentration resulting 
from the transfer to the cathode compartment of x moles of water. If 
the original solution contained N 9 equiv. of salt associated with N w moles 
of water, then the removal of x moles of water from the anode compart- 
ment, for the passage of one faraday, will increase the amount of salt by 



TRUE AND APPARENT TRANSFERENCE NUMBERS 115 

(N t /N w )x equiv. The apparent transference number of the cation will 
thus be smaller than the true value by this amount; that is, 

T, = t+ + jx. (10) 

In exactly the same way it may be shown that the water transported by 
the ions will cause a decrease of concentration in the cathode compart- 
ment; hence the transference number will be larger * than the true 
value, viz., 



If the net amount of water (x) transported were known, it would thus 
be possible to evaluate the true and apparent transference numbers from 
the results obtained by the Hittorf method. 

The suggestion was made by Nernst (1900) that the value of x could 
be determined by adding to the electrolyte solution an indifferent 
"reference substance," e.g., a sugar, which did not move with the current; 
if there were no resultant transfer of water by the ions, the concentration 
of the reference substance would remain unchanged, but if there were 
such a transfer, there would be a change in the concentration. From 
this change the amount of water transported could be calculated. The 
earliest attempts to apply this principle did not yield definite results, but 
later investigators, particularly Washburn, 2 were more successful. At 
one time the sugar raffinose was considered to be the best reference sub- 
stance, since its concentration could be readily determined from the 
optical rotation of the solution; more recently urea has been employed 
as the reference material, its amount being determined by chemical 
methods. 3 

The mean values of x obtained for approximately 1.3 N solutions of a 
number of halidcs at 25 are quoted in Table XXVII, together with the 

TABLE XXVII. TRUE AND APPARENT TRANSFERENCE NUMBERS IN 1.3 N 
SOLUTIONS AT 25 

Electrolyte x t+ T+ 

HC1 0.24 0.820 0.844 

LiCl 1.5 0.278 0.304 

NaCl 0.76 0.366 0.383 

KC1 0.60 0.482 0.495 

CsCl 0.33 0.485 0.491 

apparent transference numbers (t+) of the cations and the corrected 
values (r+) derived from equation (10). The difference between the 

* The terms "smaller" and "larger" are used here in the algebraic sense; they also 
refer to the numerical values if x is positive. 

* Washburn, J. Am. Chem. Soc., 31, 322 (1909); Washburn and Millard, ibid., 37, 
694 (1915). 

Taylor et al, J. Chem. Soc., 2095 (1929); 2497 (1932); 902 (1937). 



116 



THE MIGRATION OP IONS 





Fia. 40. Determination of transport of water 



transference numbers t+ and T+ in the relatively concentrated solutions 
employed is quite appreciable; it will be apparent from equation (10) 
that, provided x does not change greatly with concentration, the differ- 
ence between true and apparent transference numbers will be much less 
in the more dilute solutions, that is when N 9 is small. 

Another procedure for determining the net amount of water trans- 
ported during electrolysis is to separate the anode and cathode compart- 
ments by means of a parchment membrane and to measure the change 

in volume accompanying the 
passage of current. This is 
achieved by using closed ves- 
sels as anode and cathode 
compartments and observing 
the movement of the liquid 
in a capillary tube connected 
with each vessel (Fig. 40). 
After making corrections for 
the volume changes at the 
electrodes due to chemical 
reactions, the net change is 

attributed to the transport of water by the ions. 4 The results may bo 
affected to some extent by electro-osmosis (see p. 521) through the mem- 
brane separating the compartments, especially in the more concentrated 
solutions, but on the whole they are in fair agreement with those given 
in Table XXVII. 

The Moving Boundary Method. The moving boundary method for 
measuring transference numbers involves a modification and improve- 
ment of the idea employed by Lodge and by Whetham (cf. p. 60) for the 
study of the speeds of ions. On account of its relative simplicity and the 
accuracy of which it is capable, the method has been used in recent years 
for precision measurements. 5 

If it is required to determine the transference numbers of the ions 
constituting the electrolyte MA, e.g., potassium chloride, by the moving 
boundary method, it may be supposed that two other electrolytes, desig- 
nated by M'A and MA', e.g., lithium chloride arid potassium acetate, 
each having an ion in common with the experimental solute MA, are 
available to act as "indicators." Imagine the solution of MA to be 
placed between the indicator solutions so as to form sharp boundaries 
at a and 6, as shown in Fig. 41; tho anode is inserted in the solution of 
M'A and the cathode in that of MA'. In order that the boundaries 

4 Remy, Z. physik. Chem., 89, 529 (1915); 118, 161 (1925); 124, 394 (1926); Trans. 
Faraday Soc., 33, 381 (1927); BaborovskJ et al., Kec. trav. chim., 42, 229, 553 (1923); 
Z. physik. Chem., 120, 129 (1927); 131, 129 (1927); 163A, 122 (1933); Trans. Electrochem. 
Soc., 75, 283 (1939); Hepburn, Phil. Mag., 25, 1074 (193S). 

6 Maclnnes and Longsworth, Chem. Revs., 11, 171 (1932); Longsworth, J. Am. 
Chem. Sor., 54, 2741 (1932); 57, 1185 (1935). 



THE MOVING BOUNDARY METHOD 



117 



between the solutions may remain distinct during the passage of the 
current, the first requirement is that the speed of the indicator ion M' 
shall be less than that of M, and that the speed of A' 
shall be less than that of the A ions. If these condi- 
tions hold, as well as another to be considered shortly, 
the M' ions do not overtake the M ions at a, and 
neither do the A' ions overtake the A ions at 6; the 
boundaries consequently do not become blurred. In 
view of the -slower speeds of the indicator ions, they v M A 

are sometimes referred to as "following ions." Under 
the influence of an electric field the boundary a moves 
to a', while at the same time 6 moves to 6'; the dis- 
tances aa f and 66' depend on the speeds of the ions 
M and A, and since there is a uniform potential 
gradient through the central solution MA, these will 
be proportional to the ionic velocities u+ and w_. It MA 

follows, therefore, from equation (4) that 



aa 



and 



aa' + 66' u+ + u_ "*" 
66' u. 



aa' + 66' u f + w_ 



= <-, 



a' 



6' 



MA' 



T 



FIG. 41. Prin- 
ciple of the moving 
boundary method 



so that the transference numbers can be determined 
from observations on the movements of the bounda- 
ries a and 6. 

In the practical application of the moving boundary 
method one boundary only is observed, and so the 
necessity of finding two indicator solutions is obvi- 
ated ; the method of calculation is as follows. If one 
faraday of electricity passes through the system, t+ equiv. of the cation 
must pass any given point in one direction; if c equiv. per unit volume 
is the concentration of the solution in the vicinity of the boundary formed 
by the M ions, this boundary must sweep through a volume t+fc while 
one faraday is passing. The volume <f> swept out by the cations for the 
passage of Q coulombs is thus 

* = ?,-' (12) 

r C 

where F is one faraday, i.e., 96,500 coulombs. If the cross section of the 
tube in which the boundary moves is a sq. cm., and the distance through 
which it moves during the passage of Q coulombs is I cm., then <t> is equal 
to /a, and hence from equation (12) 



laFc 
Q 



(13) 



118 THE MIGRATION OF IONS 

Since the number of coulombs passing can be determined, the trans- 
ference number of the ion may be calculated from the rate of movement 
of one boundary. 

In accurate work a correction must be applied for the change in 
volume occurring as a result of chemical reactions at the electrodes and 
because of ionic migration. If At; is the consequent increase of volume 
of the cathode compartment for the passage of one faraday, equation (12) 
becomes 

. , Q Q'corr. 

- 



> 



v r F "" Fc 

.' korr. = Jobs. + CAtf, (14) 

where t corTm is the corrected transference number and / O b a . is the value 
given by equation (13); the difference is clearly only of importance in 
concentrated solutions. 

The Kohlrausch Regulating Function. An essential requirement for 
a sharp boundary is that the cations M and M', present on the two 
sides of the boundary, should move with exactly the same speed 
under the conditions of the experiment. It can be deduced that the 
essential requirement for this equality of speed is given by the Kohlrausch 
regulating function, viz., 

^ = !/' (15) 

where t+ and c are the transference number and equivalent concentra- 
tion, respectively, of the ion M in the solution of MA, and t+ and c' 
are the corresponding quantities for the ion M' in the solution of M'A; 
the solutions are those constituting the two sides of the boundary. The 
equivalent concentration of each electrolyte at the boundary, i.e., of 
MA and M'A should be proportional to the transference number of its 
cation. Similarly, at the boundary between the salts MA and MA', the 
concentrations should be proportional to the transference numbers of 
the respective anions. The reason for this condition may be seen in an 
approximate way from equation (3) : the transference number divided 
by the equivalent concentration of the ion, which is equal to cz, is pro- 
portional to the speed of the ion; hence, when //c is the same for both 
ions the speeds will be equal. 

The indicator concentration at the boundary should, theoretically, 
adjust itself automatically during the passage of current so as to satisfy 
the requirement of the Kohlrausch regulating function. Suppose the 
indicator were more concentrated than is necessary according to equa- 
tion (15) ; the potential gradient in this solution would then be lower than 
is required to make the ion M' travel at the same speed as M. The 
M' ions would thus lag behind and their concentration at the boundary 
would fall; the potential gradient in this region would thus increase until 
the velocity of the M' ions was equal to that of the leading ion. Similar 



EXPERIMENTAL METHODS 



119 




automatic adjustment would be expected if the bulk of the indicator 
solution were more dilute than necessary to satisfy equation (15). 

It would appear, therefore, that the actual concentration of the indi- 
cator solution employed in transference measurements is immaterial: 
experiments show, however, that automatic attainment of the Kohl- 
rausch regulating condition is not quite complete, for the transference 
numbers have been found to be 
dependent to some extent on the 
concentration of the bulk of the 
indicator solution. This is shown 
by the results in Fig. 42 for the 
observed transference number of 
the potassium ion in 0.1 N potas- 
sium chloride, with lithium chlo- 
ride of various concentrations as 
indicator solution. The concen- 
tration of the latter required to 
satisfy equation (15) is 0.064 N, 
and hence it appears, from the 
constancy of the transference 
number over the range of 0.055 
to 0.075 N lithium chloride, that 
automatic adjustment occurs only 
when the actual concentration of 
the indicator solution is not 
greatly different from the Kohl- 

rausch value. The failure of the adjustment to take place is probably 
due to the disturbing effects of convection resulting from temperature 
and density gradients in the electrolyte. 6 

When carrying out a transference number measurement by the moving 
boundary method the bulk concentration of the indicator solution is 
chosen so as to comply with equation (15), as far as possible, using ap- 
proximate transference numbers for the purpose of evaluating c'. The 
experiment is then repeated with a somewhat different concentration of 
indicator solution until a constant value for the transference number is 
obtained; this value is found to be independent of the applied potential 
and hence of the current strength. 

Experimental Methods. One of the difficulties experienced in per- 
forming transference number measurements by the moving boundary 
method was the establishment of sharp boundaries; recent work, chiefly 
by Maclnnes and his collaborators, has resulted in such improvements 
of technique as to make this the most accurate method for the deter- 
mination of transference numbers. Since the earlier types of apparatus 

6 Maclnnes and Smith, /. Am. Chem. Soc., 45, 2246 (1923); Maclnnes and Longs- 
worth, Chem. Revs., 11, 171 (1032); Hartley and Moilliet, Proc. Roy. Soc., 140A, 141 
(1833). 



0.607 



0.604 



! 0.601 



I 0.498 



0.495 



0.492 



0.489 

0.46 0.66 0.66 0.76 0.85 0.95 

Concentration of Lithium Chloride 

FIG. 42. Variation of transference number 
with concentration of indicator solution 



120 



THE MIGRATION OF IONS 




FIG. 43. Sheared boundary apparatus 
(Maclnnes and Brighton) 



have been largely superseded, these will not be described here; reference 
will be made to the more modern forms only. 

The apparatus used in tho 
sheared boundary method is 
shown diagrammatically in Fig. 
43. 7 The electrode vessel A is 
fitted into the upper of a pair of 
accurately ground discs, B and C, 
which can be rotated with respect 
to each other. Into the lower disc 
is fixed the graduated tube D in 
which the boundary is to move, 
and this is attached by a similar 
pair of discs, $and F, to the other 
electrode vessel G. The vessel A 
is filled with the indicator solution 
and a drop is allowed to protrude 
below the disc B, while the exper- 
imental solution is placed in the 
vessel G and the tube D so that a drop protrudes above the top of C Y ; the 
discs are so arranged that tho protruding drops d and d' are accommodated 
in the small holes, as shown in the enlarged diagram at the right of 
Fig. 43. The disc B is now rotated, with the result that the electrode 
vessel A fits exactly over />, as shown by the dotted lines at A'] in the 
process the protruding drops of liquid are sheared off and a sharp bound- 
ary is formed. The above procedure is employed for a falling boundary, 
moving down the tube D under the influence of current, i.e., when the 
indicator solution has a lower density than the experimental solution. 
If the reverse is the case, a rising boundary must be used, arid thin is 
formed in a similar manner between the two lower discs E and F\ the 
indicator solution is now placed in G and the experimental solution in 
A and D. 

If the ions of a metal, such as cadmium or silver, which forms an 
attackable anode, are suitable as indicator cations, it is possible to use 
the device of the autogenic boundary. 8 No special indicator solution is 
required, but a block of the metal serves as the anode and the experi- 
mental solution is placed in a vertical tube above it. For example, with 
nitrate solutions a silver anode can be used, and with chloride solutions 
one of cadmium can be employed; the silver nitrate or cadmium chloride, 
respectively, that is formed as the anode dissolves acts as indicator 
solution. It is claimed that there is automatic adjustment of the con- 
centration in accordance with the Kohlrausch regulating function, and 
a sharp boundary is formed and maintained throughout the experiment. 

7 Maclnnes and Brighton, J. Am. Chem. Soc., 47, 994 (1925). 
Cady and Longsworth, J. Am. Chem. Soc., 51, 1656 (1929); Longsworth, ibid., 57, 
1698 (1935); J. Chem. Ed., 11, 420 (1934). 



EXPERIMENTAL METHODS 



121 



The method is capable of giving results of considerable accuracy, although 
its application is limited to those cases for which a suitable anode mate- 
rial can be found. 

An alternative, somewhat simple- but less accurate, procedure for 
measuring transference numbers by the moving boundary principle, 
utilizes the air-lock method of estab- 
lishing the boundary. 9 The appara- 
tus for a rising boundary is shown in 
Fig. 44; the graduated measuring tube 
A has a bore of about 7 mm., whereas 
E and F are fine capillaries; the top of 
the latter is closed by rubber tubing 
with two pinchcocks. The electrodes 
are placed in the vessels B and C. 
With electrode B in position, and the 
upper pinchcock at the top -of F 
closed, the apparatus is filled with the 
experimental solution. By closing the 
lower pinchcock a small column of air 
G is forced into the tube where F 
joins A 9 thus separating the solutions 

A and D. The solution in CDE 




FIG. 44. Air-lock method for estab- 
lishing boundary (Hartley and Donald- 
son) 



in 

is then emptied by suction through 

C and E, care being taken not to 

disrupt the air column G. The tube 

CDE is now filled with the indicator solution, the electrode is inserted 

in C, and the lower pinchcock at the top of F is adjusted so that the 

air column G is withdrawn sufficiently to permit a boundary to form 

between the indicator solution in CDE and the experimental solution 

in A. Even if the boundary is not initially sharp, it is soon sharpened 

by the current. 

In following the movement of the boundary, no matter how it is 
formed, use is made of the difference in the refractive indices of the 
indicator and experimental solutions; if the boundary is to be clearly 
visible, this difference should be appreciable. If the distance (I) moved 
in a given time and the area of cross section (a) of the tube are measured, 
and the equivalent concentration (c) of the experimental solution is 
known, it is only necessary to determine the number of coulombs (Q) 
passed for the transference number to be calculated by equation (13). 
The quantity of electricity passing during the course of a moving bound- 
ary experiment is generally too small to be measured accurately in a 
coulometer. It is the practice, therefore, to employ a current of known 
strength for a measured period of time; the constancy of the current can 
be ensured by means of automatic devices which make use of the proper- 
ties of vacuum tubes. 

Hartley and Donaldson, Trans. Faraday Soc., 33, 457 (1937). 



122 



THE MIGRATION OP IONS 



Results of Transference Number Measurements. Provided the 
measurements are made with great precision, the results obtained by 
the Hittorf and moving boundary methods agree within the limits of 
experimental error; this is shown by the most accurate values for various 
solutions of potassium chloride at 25 as recorded in Table XXVIII. 

TABLE XXVIII. TRANSFERENCE NUMBERS OF POTASSIUM CHLORIDE SOLUTIONS AT 25 

Concentration 0.02 0.05 0.10 0.50 l.ON 

Hittorf method 0.489 0.489 0.490 0.490 0.487 

Moving boundary method 0.490 0.490 0.490 0.490 0.488 

It is probable that, on the whole, transference numbers derived from 
moving boundary measurements are the more reliable. 

It may be noted that the values obtained by the moving boundary 
method, like those given by the Hittorf method, are the so-called "appar- 
ent" transference numbers (p. 114), because the transport of water by 
the ions will affect the volume through which the boundary moves. It is 
the practice, however, to record observed transference numbers without 
applying any correction, since much uncertainty is attached to the deter- 
mination of the transport of water during the passage of current. Fur- 
ther, in connection with the study of certain types of voltaic cell, it is 
the "apparent" rather than the "true" transference number that is 
involved (cf. p. 202). 

Some of the most recent data of the transference numbers of the 
cations of various salts at a number of concentrations at 25, mainly 
obtained by the moving boundary method, are given in Table XXIX; 10 

TABLE XXIX. TRANSFERENCE NUMBERS OF CATIONS IN AQUEOUS SOLUTIONS AT 25 



Concn. 


HC1 


LiCl 


NaCl 


KCl 


KNOa 


AgNOs 


BaCU 


K 2 SO 4 


LaCb 


0.01 N 


0.8251 


0.3289 


0.3918 


0.4902 


0.5084 


0.4648 


0.440 


0.4829 


0.4625 


0.02 


0.8266 


0.3261 


0.3902 


0.4901 


0.5087 


0.4652 


0.4375 


0.4848 


0.4576 


0.05 


0.8292 


0.3211 


0.3876 


0.4899 


0.5093 


0.4664 


0.4317 


0.4870 


0.4482 


0.1 


0.8314 


0.3168 


0.3854 


0.4898 


0.5103 


4682 


0.4253 


0.4890 


0.4375 


0.2 


0.8337 


0.3112 


0.3821 


0.4894 


0.5120 





0.4162 


0.4910 


0.4233 


0.5 





0.300 





0.4888 








0.3986 


0.4909 


0.3958 


1.0 





0.287 





0.4882 








0.3792 









the corresponding anion transference numbers may be obtained in each 
case by subtracting the cation transference number from unity. 

Influence of Temperature on Transference Numbers. The extent of 
the variation of transference numbers with temperature will be evident 
from the data for the cations of a number of chlorides at a concentration 
of 0.01 N recorded in Table XXX; these figures were obtained by the 
Hittorf method and, although they may be less accurate than those in 
Table XXIX, they are consistent among themselves. The transference 

Longsworth, J. Am. Chem. Soc., 57, 1185 (1935); 60, 3070 (1938). 



TRANSFERENCE NUMBER AND CONCENTRATION 123 

TABLE XXX. INFLUENCE OF TEMPERATURE ON CATION TRANSFERENCE NUMBERS 

IN 0.01 N SOLUTIONS 

Temperature HC1 NaCl KC1 BaClj 

0.846 0.387 0.493 0.437 

18 0.833 0.397 0.496 

30 0.822 0.404 0.498 0.444 

50 0.801 0.475 

numbers of the ions of potassium chloride vary little with temperature, 
but in sodium chloride solution, and particularly in hydrochloric acid, 
the change is appreciable. It has been observed, at least for uni-univa- 
lent electrolytes, that if the transference number of an ion is greater than 
0.5, e.g., the hydrogen ion, there is a decrease as the temperature is 
raised. It appears, therefore, in general that transference numbers 
measured at appreciable concentrations tend to approach 0.5 as the 
temperature is raised; in other words, the ions tend towards equal speeds 
at high temperatures. 

Transference Number and Concentration : The Onsager Equation. 
It will be observed from the results in Table XXIX that transference 
numbers generally vary with the concentration of the electrolyte, and 
the following relationship was proposed to represent this variation, viz., 

t = to - AVc, (16) 

where t and / are the transference numbers of a given ion in a solution 
of concentration c and that extrapolated to infinite dilution, respectively, 
and A is a constant. Although this equation is applicable to dilute 
solutions, it does not represent the behavior of barium chloride and other 
electrolytes at appreciable concentrations. A better expression, which 
holds up to relatively high concentrations, is 



_ _ 

'l + B+c ' 

where B is a constant for the given electrolyte. 11 This equation may be 
written in the form 



- 1 + Bc, (18) 



so that the plot of l/(t + 1) against Vc should be a straight line, as has 
been found to be true in a number of instances. Equation (17) can also 
be expressed as a power series, thus 

t = to ~ (to + l)#c + (<o + l)# 2 c + (fc + l)#'c + , 

and when c is small, i.e., for dilute solutions, so that all terms beyond 
that involving c* can be neglected, this reduces to equation (16) since 
(t Q + 1)5 is a constant. 

"Jones and Dole, J. Am. Chem. Soc., 51, 1073 (1929); Jones et al, ibid., 54, 138 
(1932); 58, 1476 (1936); Dole, J. Phys. Chem., 35, 3647 (1931). 



124 THE MIGRATION OF IONS 

The Onsager equation for the equivalent conductance X< of an ion 
may be written in the form [cf. equation (34), p. 89] 

X. = X? - A&, (19) 

where X? is the ion conductance at infinite dilution and A* is a constant. 
Introducing the expression for the transference number given by equa- 
tion (5), that is ti = X t /A, where A is the equivalent conductance of the 
electrolyte at the experimental concentration, it follows that 



The value of A can be expressed in terras of A by an equation similar to 
(19), and then equation (20) can be written in the form 

+ D, (21) 



1 XJ\C 

where A, B and D are constants. This equation derived from the 
Debye-Hiickel-Onsager theory of conductance is of the same form as the 
empirical equation (17), and hence is in general agreement with the facts; 
the constants A, B and D, however, which are required to satisfy the 
experimental results differ from those required by theory. This dis- 
crepancy is largely due to the fact that the transference measurements 
were made in solutions which are too concentrated for the simple Onsager 
equation to be applicable. 

Since the Onsager equation is, strictly speaking, a limiting equation, 
it is more justifiable to see if the variation of transference number with 
concentration approaches the theoretical behavior with increasing dilu- 
tion. The equivalent conductance of a univalent ion can be expressed 
in the form of equation (37), page 90, viz., 



A, = X?-QA+X?)c, (22) 

where .A and B as used here are the familiar Onsager values (Table 
XXIII, p. 90); the transference number (t+) of the cation in a uni- 
umvalent electrolyte can then be represented by 

X+ X + 



where A and B are the same for both ions. Differentiating equation 
(23) with respect to Vc, and introducing the condition that c approaches 
aero, it is found that 



.~ 2A, - 
It follows, therefore, that the slope of the plot of the transference number 



EQUIVALENT CONDUCTANCES OF IONS 



125 



of an ion against the square-root of the concentration should attain a 
limiting value, equal to (2J+ 1)4/2A as infinite dilution is approached; 
the results in Fig. 45, in which the full lines are drawn through the experi- 
mental cation transference numbers in aqueous solution at 25 and the 




0.316 



0.10 



0.20 



0.30 



0.40 



\J Concentration 
FIG. 45. Transference numbers and the Onsager equation (Longsworth) 

dotted lines represent the limiting slopes, are seen to be in good agree- 
ment with the requirements of the inter-ionic attraction theory, 1 - 

Equivalent Conductances of Ions. Since transference numbers and 
equivalent conductances at various concentrations are known, it should 
be possible, by utilizing the expression X = $,A, to extrapolate the re- 

"Longsworth, J. Am. Cham. Soc., 57, 1185 (1935); see also, Hartley and Donald- 
son, Trans. Faraday Soc., 33, 457 (1937); Samis, ibid., 33, 469 (1937). 



126 THE MIGRATION OF IJNS 

suits to give ion conductances at infinite dilution. Two methods of 
extrapolating the data are possible. In the first place, the equivalent 
conductances and the transference numbers may be extrapolated sepa- 
rately to give the respective values at infinite dilution; the product of 
thase quantities would then be equal to the ionic conductance at infinite 
dilution. The data from which the conductance of the chloride ion can 
be evaluated are given in Table XXXI; the mean value of the conduct- 

TABLE XXXI. CALCULATION OP CHLORIDE ION CONDUCTANCE AT 25 

Electrolyte &r A Afcr 

HC1 0.1790 426.16 76.28 

LiCl 0.6633 115.03 76.30 

NaCl 0.6035 126.45 76.31 

KC1 0.5097 149.86 76.40 

ance of the chloride ion at infinite dilution at 25, derived from measure- 
ments on solutions of the four chlorides, is thus found to be 76.32 ohms" 1 
cm. 2 The results in the last column are seen to be virtually independent 
of the nature of the chloride, in agreement with Kohlrausch's law of the 
independent migration of ions. 

The second method of extrapolation is to obtain the values of X t at 
various concentrations and to extrapolate the results to infinite dilution. 
The equivalent conductances of the chloride ion at several concentrations 
obtained from transference and conductance measurements, on the four 
chlorides to which the data in Table XXXI refer, are given in Table 
XXXII. These results can be plotted against the square-root of the 

TABLE XXXII. EQUIVALENT CONDUCTANCES OP CHLORIDE ION AT 25 

Electrolyte 0.01 0.02 0.05 0.10 N 

HC1 72.06 70.62 68.16 65.98 

LiCl 72.02 70.52 67.96 65.49 

NaCl 72.05 70.54 67.92 65.58 

KC1 72.07 70.56 68.03 65.79 

concentration and extrapolated to infinite dilution, thus giving 76.3 
ohms~ l cm. 2 for the ion conductance, but a more precise method is similar 
to that described on page 54, based on the use of the Onsager equation. 
The conductance of a single univalent ion, assuming complete dissocia- 
tion of the electrolyte, is given by equation (22), the values of A and B 
being known; if the experimental data for \ v at various concentrations, 
as given in Table XXXII, are inserted in this equation, the corresponding 
results for X< can be obtained. If the solutions were sufficiently dilute 
for the Onsager equation to be strictly applicable, the values of X? would 
all be the same; on account of the incomplete nature of this equation in 
its simple form, however, they actually increase with increasing concen- 
tration (cf. p. 55). By plotting the results against the concentration 
and extrapolating to infinite dilution, the equivalent conductance of the 
chloride ion in aqueous solution has been found to be 76.34 ohms~ l cm. 2 



TRANSFERENCE NUMBERS IN MIXTURES 127 

at 25; this is the best available datum for the conductance of the chlo- 
ride ion. 13 

Since the ion conductance of the chloride ion is now known accu- 
rately, that of the hydrogen, lithium, sodium, potassium and other 
cations can be derived by subtraction from the equivalent conductances 
at infinite dilution of the corresponding chloride solutions; from these 
results the values for other anions, and hence for further cations, can be 
obtained. The data recorded in Table XIII, page 56, were calculated 
in this manner. 

It is of interest to note from Table XXXII that the equivalent con- 
ductance of the chloride ion is almost the same in all four chloride solu- 
tions at equal concentrations, especially in the more dilute solutions. 
This fact supports the view expressed previously that Kohlrausch's law 
of the independent migration of ions is applicable to dilute solutions of 
strong electrolytes at equivalent concentrations, as well as at infinite 
dilution. 

Transference Numbers in Mixtures. Relatively little work has been 
done on the transference numbers of ions in mixtures, although both 
Hittorf and moving boundary methods have been employed. In the 
former case, it follows from equation (3) that the transference number 
of any ion in a mixture is equal to the number of equivalents of that ion 
migrating from the appropriate compartment divided by the total num- 
ber of equivalents deposited in a coulometer. It is possible, therefore, 
to derive the required transference numbers by analysis of the anode and 
cathode compartments before and after electrolysis. 

The moving boundary method has been used to study mixtures of 
alkali chlorides and hydrochloric acid, a cadmium anode being employed 
to form an "autogenic" boundary. After electrolysis has proceeded for 
some time two boundaries are observed; the leading boundary is due to 
the high mobility of the hydrogen ion and is formed between the mixture 
of hydrochloric acid and the alkali chloride on the one side, and a solution 
of the alkali chloride from which the hydrogen ion has completely mi- 
grated out on the other side. The rate of movement of this boundary 
gives the transference number of the hydrogen ion in the mixture of 
electrolytes. The slower boundary is formed between the pure alkali 
chloride solution and the cadmium chloride indicator solution, a,nd gives 
no information concerning transference numbers in the mixture. The 
transference number of the alkali metal ion cannot be determined directly 
from the movement of the boundaries, and so the transference number 
of the chloride ion in the mixed solution is obtained from a separate 
experiment with an anion boundary using a mixture of potassium iodate 
and iodic acid as indicator. Since the transference numbers of the three 

13 Longsworth, J. Am. Chem. Soc., 54, 2741 (1932); Maclnnes, /. Franklin InsL, 
225, 661 (1938); see also, Owen, J. Am. Chem. Soc., 57, 2441 (1935). 



128 THE MIGRATION OF IONS 

ions must add up to unity, the value for the alkali metal can now be 
derived. 14 

Abnormal Transference Numbers. In certain cases, particularly 
with solutions of cadmium iodide, the transference number varies mark- 
edly with concentration, and the values may become zero or even appar- 
ently negative; the results for aqueous solutions of cadmium iodide at 
18 are quoted in Table XXXIII. At concentrations greater than 0.5 N, 

TABLE XXXIII. CATION TRANSFERENCE NUMBERS IN CADMIUM IODIDE AT 18 

Concn. 0.0005 0.01 0.02 0.05 0.1 0.2 0.5 N 

t+ 0.445 0.444 0.442 0.396 0.296 0.127 0.003 

the transference number of cadmium apparently becomes negative : this 
means that in relatively concentrated solutions of cadmium iodide, the 
cadmium is being carried by the current in a direction opposite to that 
in which positive electricity moves through the solution. In other words, 
cadmium must form part of the negative ion present in the electrolyte. 
A reasonable explanation of the results is that in dilute solution cadmium 
iodide ionizes to yield simple ions; thus 

CdI 2 ^ Cd++ + 2I-, 

and so the transference number of the cadmium ion, in solutions con- 
taining less than 0.02 equiv. per liter, is normal. As the concentration 
is increased, however, the iodide ions combine with unionized molecules 
of cadmium iodide to form complex Cdl ions, thus, 

CdI 2 + 21- ^ Cdli~, 

with the result that appreciable amounts of cadmium are present in the 
anions and hence are transferred in the direction opposite to that of the 
flow of positive current. The apparent transference number of the 
cadmium ion is thus observed to decrease; if equal quantities of elec- 
tricity are carried in opposite directions by Cd+ + and Cdlr~ ions the 
transference number will appear to be zero. The proportion of Cdl" 
ions increases with increasing concentration and eventually almost the 
whole of the iodine will be present as Cdl" ions; the current is then 
carried almost exclusively by Cd +4 ~ and Cdlr~ ions. If the speed of the 
latter is greater than that of the former, as appears actually to be the 
case, the apparent cation transference number will be negative. A simi- 
lar variation of the cation transference number with concentration has 
been observed in solutions of cadmium bromide and this may be attrib- 
uted to the existence of the analogous CdBr^" ion. Less marked changes 
of transference number have been observed with other electrolytes; these 
are also probably to be ascribed to the presence of complex ions in con- 
centrated solutions. 

" Longsworth, J. Am. Chem. Soc., 52, 1897 (1930). 



PROBLEMS 129 

PROBLEMS 

1. Maclnnes and Dole [/. Am. Chem. Soc., 53, 1357 (1931)] electrolyzed 
a 0.5 N solution of potassium chloride, containing 3.6540 g. of salt per 100 g. 
solution, at 25 using an anode of silver and a cathode of silver coated with 
silver chloride. After the passage of a current of about 0.018 amp. for ap- 
proximately 26 hours, 1.9768 g. of silver were deposited in a coulometer in the 
circuit and on analysis the 119.48 g. of anode solution were found to contain 
3.1151 g. potassium chloride per 100 g. solution, while the 122.93 g. of cathode 
solution contained 4.1786 g. of salt per 100 g. Calculate the values of the 
transference number of the potassium ion obtained from the anode and cathode 
solutions, respectively. 

2. Jones and Bradshaw [J. Am. Chem. /Soc., 54, 138 (1932)] passed a 
current of approximately 0.025 amp. for 8 hours through a solution of lithium 
chloride, using a silver anode and a silver chloride cathode; 0.73936 g. of silver 
was deposited in a coulometer. The original electrolyte contained 0.43124 g. 
of lithium chloride per 100 g. of water, and after electrolysis the anode portion, 
weighing 128.615 g., contained 0.35941 g. of salt per 100 g. water, while the 
cathode portion, weighing 123.074 g., contained 0.50797 g. of salt per 100 g. 
of water. Calculate the transference number of the chloride ion from the 
separate data for anode and cathode solutions. 

3. In a moving boundary experiment with 0.1 N potassium chloride, using 
0.065 N lithium chloride as indicator solution, Maclnnes and Smith [_J. Am. 
Chem. Soc., 45, 2246 (1923)] passed a constant current of 0.005893 amp. 
through a tube of 0.1142 sq. cm. uniform cross section and observed the 
boundary to pass the various scale readings at the following times: 

Scale reading 0.5 5.50 5.80 6.10 6.40 6.70 7.00cm. 
Time 1900 2016 2130 2243 2357 2472 sec. 

Calculate the mean transference number of the potassium ion. The potential 
gradient was 4 volts per cm.; evaluate the mobility of the potassium ion for 
unit potential gradient. 

4. The following results were recorded by Jahn and his collaborators 
[Z. phyaik. Chem. t 37, 673 (1901)] in experiments on the transference number 
of cadmium in cadmium iodide solutions using a cadmium anode: 

Original Anode solution Silver 

solution after electrolysis deposited in 

Cd per cent * Weight Cd per cent coulometer 

2.5974 138.073 2.8576 0.7521 g. 

1.3565 395.023 1.4863 0.9538 

0.8820 300.798 1.0096 0.9963 

0.4500 289.687 0.5654 0.9978 

0.2311 305.750 0.3264 0.9604 

0.1390 301.700 0.1868 0.5061 

* The expression "Cd per cent" refers to the number of grams of Cd per 100 g. of 
solution. 

Evaluate the apparent transference number of the cadmium ion at the different 
concentrations, and plot the results as a function of concentration. 

5. A0.2 N solution of sodium chloride was found to have a specific con- 
ductance of 1.75 X 10~* ohm~ l cm." 1 at 18; the transference number of the 



130 THE MIGRATION OF IONS 

cation in this solution is 0.385. Calculate the equivalent conductance of the 
sodium and chloride ions. 

6. A solution contains 0.04 N sodium chloride, 0.02 N hydrochloric acid 
and 0.04 N potassium sulfate; calculate, approximately, the fraction of the 
current carried by each of the ionic species, Na+, K+, H+, Cl~ and S0r~, in 
this solution. Utilize the data in Tables X and XIII, and assume that the 
conductance of each ion is the same as in a solution of concentration equal to 
the total equivalent concentration of the given solution. 

7. The equivalent conductances and cation transference numbers of ammo- 
nium chloride at several concentrations at 25 are as follows [Longsworth, 
/. Am. Chem. Soc., 57, 1185 (1935)]: 

c 0.01 0.02 0.05 0.10 N 

A 141.28 138.33 133.29 128.75 ohms' 1 cm. 2 

t+ 0.4907 0.4906 0.4905 0.4907 

Utilize the results to evaluate the equivalent conductance of the ammonium 
and chloride ions at infinite dilution by the method described on page 126. 

8. Use the results of the preceding problem to calculate the limiting slope, 
according to the Onsager equation, of the plot of the transference number of 
the ammonium ion in ammonium chloride against the square-root of the 
concentration. 

9. Hammett and Lowenheim [J. Am. Chem. Soc., 56, 2620 (1934)] electro- 
lyzed, with inert electrodes, a solution of Ba(HS0 4 )2 in sulfuric acid as solvent; 
1 g. of this solution contained 0.02503 g. BaS0 4 before electrolysis. After the 
passage of 4956 coulombs, 41 cc. of the anode solution and 39 cc. of the cathode 
solution, each having a density of 1.9, were run off; they were found on analysis 
to contain 0.02411 and 0.02621 g. of BaS0 4 per gram of solution, respectively. 
Calculate the transference number of the cation. 

10. A solution, 100 g. of which contained 2.9359 g. of sodium chloride and 
0.58599 g. urea, was electrolyzed with a silver anode and a silver chloride 
cathode; after the passage of current which resulted in the deposition of 
4.5025 g. of silver in a coulometer, Taylor and Sawyer [/. Chem. Soc., 2095 
(1929)] found 141.984 g. of anode solution to contain 3.2871 g. sodium chloride 
and 0.84277 g. urea, whereas 57.712 g. of cathode solution contained 2.5775 g. 
sodium chloride and 0.32872 g. urea. Calculate the "true" and "apparent" 
transference numbers of the ions of sodium chloride in the experimental solution. 



CHAPTER V 
FREE ENERGY AND ACTIVITY 

Partial Molar Quantities. 1 The thermodynamic functions, such as 
heat content, free energy, etc., encountered in electrochemistry have the 
property of depending on the temperature; pressure and volume, i.e., 
the state of the system, and on the amounts of the various constituents 
present. For a given mass, the temperature, pressure and volume are 
not independent variables, and so it is, in general, sufficient to express 
the function in terms of two of these factors, e.g., temperature and 
pressure. If X represents any such extensive property, i.e., one whose 
magnitude is determined by the state of the system and the amounts, 
e.g., number of moles, of the constituents, then the partial molar value 
of that property, for any constituent i of the system, is defined by 






> 



and is indicated by writing a bar over the symbol for the property. The 
partial molar quantity is consequently the increase in the particular 
property X resulting from the addition, at constant temperature and 
pressure, of one mole of the constituent i to such a large quantity of the 
system that there is no appreciable change in its composition. 

If a small change is made in the system at constant temperature and 
pressure, such that the number of moles of the constituent 1 is increased 
by dn\, of 2 by dra 2 , or, in general, of the constituent i by dn, the total 
change dX in the value of the property X is given by 



(dX) T , P = Xidni + 2 dn 2 + - rfZn< + (2) 

In estimating dX from equation (2) it is, of course, necessary to insert a 
minus sign before the Xdn term for any constituent whose amount is 
decreased as a result of the change in the system. 

Partial Molar Free Energy: Chemical Potential. The partial molal 
free energy is an important thermodynamic property in connection with 
the study of electrolytes; it can be represented either as G, where G is 
employed for the Gibbs, or Lewis, free energy,* or by the symbol /i, 
when it is referred*to as the chemical potential; thus the appropriate form 

1 Lewis and Randall, "Thermodynamics and the Free Energy of Substances," 
1923, Chap. IV; Glasstone, "Text-book of Physical Chemistry," 1940, Chap. III. 

* Electrochemical processes are almost invariably carried out at constant tempera- 
ture and pressure; under these conditions G is the appropriate thermodynamic function. 
The symbol F has been generally used to represent the free energy, but in order to 
avoid confusion with the symbol for the faraday, many writers now adopt G instead. 

131 



132 FREE ENERGY AND ACTIVITY 

of equation (2), for the increase of free energy accompanying a change in 
a given system at constant temperature and pressure, is then 



(dG) T . p = mdni + p*dn* + /i<dn< + (3) 

One of the thermodynamic conditions of equilibrium is that (dG)r.p is 
zero; it follows, therefore, that for a system in equilibrium at constant 
temperature and pressure 

tJLidni + ndn 2 + /*dn< + - = S/i*dn< = 0. (4) 



The partial molal volume of the constituent i in a mixture of ideal 
gases, which do not react, is equal to its molar volume ;,- in the system, 
since there is no volume change on mixing; if p t is the partial pressure of 
the constituent, then t\ is equal to RT/pi, where R is the gas constant 
per mole and T is the absolute temperature. It can be shown by means 
of thermodynamics that the partial molal volume (D) is related to the 
chemical potential by the equation 



,. ......... 

and so it follows that, for an ideal gas mixture, 

RT 



Integration of equation (6) then gives the chemical potential of the gas i 
in the mixture, thus 

/u-Mf + Brinp,, (7) 

where /i? is a constant depending only on the nature of the gas and on 
the temperature of the system. It is evident that M? is equal to the chem- 
ical potential of the ideal gas at unit partial pressure. 

Activity and Activity Coefficient. 2 When a pure liquid or a mixture is 
in equilibrium with its vapor, the chemical potential of any constituent 
in the liquid must be equal to that in the vapor; this is a consequence of 
the thermodynamic requirement that for a system at equilibrium a small 
change at constant temperature and pressure shall not be accompanied 
by any change of free energy, i.e., (dG)T. P is zero. It follows, therefore, 
that if the vapor can be regarded as behaving ideally, the chemical po- 
tential of the constituent i of a solution can be written in the same form 
as equation (7), where pt is now the partial pressure of the component 
in the vapor in equilibrium with the solution. If the vapor is not ideal, 
the partial pressure should be replaced by an ideal pressure, or " fugacity," 
but this correction need not be considered further. According to Raoult's 

1 Lewis and Randall, J. Am. Chem. Soc., 43, 1112 (1921); "Thermodynamics and 
the Free Energy of Substances," 1923, Chaps. XXII to XXVIII; Glasstone, "Text- 
book of Physical Chemistry," 1940, Chap. IX. 



ACTIVITY AND ACTIVITY COEFFICIENT 133 

law the partial vapor pressure of any constituent of an ideal solution is 
proportional to its mole fraction (z<) in the solution, and hence it follows 
that the chemical potential in the liquid is given by 



Xi. (8) 

The constant /z? for the particular constituent of the solution is inde- 
pendent of the composition, but depends on the temperature and pres- 
sure, for the relationship between the mole fraction and the vapor pressure 
is dependent on the total pressure of the system. 

If the solution under consideration is not ideal, as is generally the 
case, especially for solutions of electrolytes, equation (8) is not applicable, 
and it is modified arbitrarily by writing 

M i= d + RT Inzi/i, (9) 

where / is a correction factor known as the activity coefficient of the 
constituent i in the given solution. The product xf is called the activity 
of the particular component and is represented by the symbol a, so that 

n* - vt + RTlnat. (10) 

As may be seen from equations (8) and (10), the activity in this particular 
case may thus be regarded as an idealized mole fraction of the given 
constituent. A comparison of equations (8) and (9) shows that for an 
ideal solution the activity coefficient/ is unity; in general, the difference 
between unity and the actual value of the activity coefficient in a given 
solution is a measure of the departure from ideal behavior in that solution. 
For a system consisting of a solvent, designated by the suffix 1, and 
a solute, indicated by the suffix 2, the respective chemical potentials are 



lZi/i (11) 

and 

M2 = A) + RTlnxtf*. (12) 

It is known that a solution tends towards ideal behavior more closely 
the greater the dilution ; hence, it follows that / 2 approaches unity as x 2 
approaches zero, and /i approaches unity as x\ attains unity. It is 
convenient, therefore, to adopt the definitions 

/i 1 as x\ 1 and /z > 1 as x z 0. 

Since /i and Xi become unity at infinite dilution, i.e., for the pure solvent, 
it follows from equation (11) that the chemical potential of a pure liquid 
becomes equal to /i2(n> and hence is a constant at a given temperature 
and pressure. By considering the equilibrium between a solid and its 
vapor, it can be readily shown that the same rule is applicable to a pure 
solid. 



x>^ 

\ n <K<. - 



134 FREE ENERGY AND ACTIVITY 

Forms of the Activity Coefficient. The equations given above are 
satisfactory for representing the behavior of liquid solutes, but for solid 
solutes, especially electrolytes, a modified form is more convenient. In a 
very dilute solution the mole fraction of solute is proportional both to its 
concentration (c), i.e., moles per liter of solution, and to its molality (m), 
i.e., moles per 1000 g. of solvent; hence for such solutions, which are 
known to approach ideal behavior, it is possible to write either 

M = + RTlnx, (13a) 

or 

M = Mc + flZMnc, (136) 

or 

M = f& + RTlnm, (13c) 

where v&, M? and & are constants whose relationship to each other 
depends on the factors connecting x, c and m in dilute solutions. Since 
solutions of appreciable concentration do not behave ideally, it is neces- 
sary to include the appropriate activity coefficients ; thus 

M = M 2 + RTlnxf x = + RTlna I9 

/* = /z? + RT In cf c = + RT In a e , ^ c ^(146) 
and v N v 

where the a terms are the respective activities. / x ^ 

It is evident from the equations (14) that the activity of a constituent 
of a solution can be expressed only in terms of a ratio * of two chemical 
potentials, viz., ju and /i, and so it is the practice to choose a reference 
state, or standard state, for each constituent in which the activity is 
arbitrarily taken as unity. It can be readily seen from the equations given 
above that in the standard state the chemical potential n is equal to the 
corresponding value of ff. The activity of a component in any solution 
is thus invariably expressed as the ratio of its value to that in the arbi- 
trary standard state. The actual standard state chosen differs, of course, 
according to which form of equation (14) is employed to define the 
activity. 

At infinite dilution, when a solution behaves ideally, the three activity 
coefficients of the solute, viz.,/ x ,/ c and/ m , are all unity, but at appreciable 
concentrations the values diverge from this figure and they are no 
longer equal. It is possible, however, to derive a relationship between 
them in the following manner. The mole fraction x, concentration c, and 
molality m of a solute can be readily shown to be related thus 

0.001 cM ! 0.001 mM l 

X ~ P - 0.001 cM 2 + 0.001 cMi ~~ 1 + 0.001 mM l ' ( ' 

* It is a ratio, rather than a difference, because in equations (14) the activity appears 
in a loganthmic term. 



FORMS OF THE ACTIVITY COEFFICIENT 135 

where A is the density of the solution, and MI and M 2 are the molecular 
weights of solvent and solute, respectively. In very dilute solutions the 
three related quantities are x , c and mo, and the density is po, which is 
virtually that of the pure solvent ; since the quantities 0.001 cM i, 0.001 
cM 2 and 0.001 mM i are then negligibly small, it follows from equation 
(15) that 



Po 

Incidentally this relationship proves the statement made above that in 
very dilute solutions the mole fraction, concentration and molality are 
proportional to each other. 

If /no is the chemical potential of a given solute in a very dilute solu- 
tion, to which the terms Xo f CQ and mo apply, the three activity coefficients 
are all unity ; further, if ju is the chemical potential in some other solution, 
whose concentration is represented by x, c or m, it follows from the three 
forms of equation (14) that A* Mo may be written in three ways, thus 



XQ Co 



x Co mo 



Combination of equations (15), (16) and (17) then gives the relationship 
between the three activity coefficients for the solute in the given solution : 



_ ,- 0.00. cM, + 0.001 _ m 

Po 

It is evident from this expression that f c and f m must be almost identical 
in dilute solutions, and that f x cannot differ appreciably from the other 
coefficients for solutions more dilute than about 0.1 N. 

The arguments given above are applicable to a single molecular 
species as solute, but for electrolytes it is the common practice to em- 
ploy a mean activity coefficient (see p. 138) ; in this event it is necessary 
to introduce into the terms 0.001 cM\ and 0.001 mM\ the factor v which 
is equal to the number of ions produced by one molecule of electrolyte 
when it ionizes. The result is then 



. . p - 0.001 cM 2 + 0.001 , ,, , n Ani lf , /im 

f x = fc --------- = /(! + 0.001 vmMi). (19) 

Po 

The activity coefficient f x is sometimes called the rational activity 
coefficient, since it gives the most direct indication of the deviation from 
the ideal behavior required by Raoult's law. It is, however, not often 
used in connection with measurements on solutions of electrolytes, and 
so the coefficients f c and/*,, which are commonly employed, are described 
as the practical activity coefficients. The coefficient / c , from which the 



136 FREE ENERGY AND ACTIVITY 

suffix is dropped, is generally used in the study of electrolytic equilibria 
to represent the activity of a particular ionic species; thus, the activity 
of ions of the ith kind is equal to c</,, where d is the actual ionic concen- 
tration, due allowance being made for incomplete dissociation if necessary. 
On the other hand / m , which is given the symbol 7, is almost invariably 
used in connection with the thermodynamics of voltaic cells; the activity 
of an ion is expressed as my , where m is the total molality of the ionic 
constituent of the electrolyte with no correction for incomplete dissocia- 
tion. For this reason 7 is sometimes called the stoichiometric activity 
coefficient. 

Equilibrium Constant and Free Energy Changes. If a system in- 
volving the reversible chemical process 

aA + 6B + ^ IL + mM + 

is in a state of equilibrium, it can be readily shown, by means of equations 
(4) and (14), that 



where K is the equilibrium constant for the system under consideration. 
Equation (20) is the exact form of the law of mass action applicable to 
any system, ideal or not. Writing fc or ym in place of the activity a, 
the following equations for the equilibrium constant, which are frequently 
employed in electrochemistry, are obtained, viz., 

v C LM ' * ' /L/M 

AC ~~ ~*J 

CA C B ' ' ' 

and 



/01 |v 
1 (216) 

' ' 

If the components of the system under consideration are at their 
equilibrium concentrations, or activities, the free energy change resulting 
from the transfer from reactants to resultants is zero. If, however, the 
various substances are present in arbitrary concentrations, or activities, 
the transfer process is accompanied by a definite change of free energy; 
thus, if a moles of A, b moles of B, etc., at arbitrary activities are trans- 
ferred to I moles of L, m moles of M, etc., under such conditions that the 
concentrations are not appreciably altered, the increase of free energy 
(AC?) at constant temperature is given by the expression 

- AG= RTln K- RTln*?? '" ; (22) 

a A a B 

this equation is a form of the familiar reaction isotherm. If the arbi- 
trary activities of reactants and resultants are chosen as the respective 



ACTIVITIES OF ELECTROLYTES 137 

standard states, i.e., the a's in equation (22) are all unity, it follows that 

- A<? = RTln K, (23) 

where A(J is the standard free energy change of the process. 

Activities of Electrolytes. When the solute is an electrolyte, the 
standard states for the ions are chosen, in the manner previously indi- 
cated, as a hypothetical ideal solution of unit activity; in this solution 
the thermodynamic properties of the solute, e.g., the partial molal heat 
content, heat capacity, volume, etc., will be those of a real solution at 
infinite dilution, i.e., when it behaves ideally. With this definition of the 
standard state the activity of an ion becomes equal to its concentration 
at infinite dilution. 

For the undissociated part of the electrolyte it is convenient to define 
the standard state in such a way as to make its chemical potential equal 
to the sum of the values for the ions in their standard states. Consider, 
for example, the electrolyte M^A,. which ionizes thus 



to yield the number v+ of M + ions and v-. of A~ ions. The chemical 
potentials of these ions are given by the general equations 

MM+ = & + RT\na+ (24a) 

and 

MA- = + RT\na-, (246) 

where a+ and a_ are the activities of the ions M+ and A~ respectively. 
If /i2 is the chemical potential of the undissociated portion of the elec- 
trolyte in a given solution and /z is the value in the standard state, then 
by the definition given above, 

M? = *+/4 + "-M-. (25) 

When the system of undissociated molecules and free ions in solution 
is in equilibrium, a small change at constant temperature and pressure 
produces no change in the free energy of the system; since one molecule 
of electrolyte produces v+ positive and v- negative ions, it is seen that 
[cf. equation (4)] 

v+(\ + RT In a+) + _( M - + RT In a.) = + RT In a 2 . (26) 

Introducing equation (25) it follows, on the basis of the particular stand- 
ard states chosen, that 

v+RT In a+ + v-RT In a_ = RT In a 2 , 

/. afa- = a 2 . (27) 

If the total number of ions produced by a molecule of electrolyte, 
i.e., v+ + v~, is represented by v, then the mean activity a of the elec- 



138 FREE ENERGY AND ACTIVITY 

trolyte is defined by 

a m (<#!-)'/', (28) 

and hence, according to equation (27), 

a = (a,) 1 " or a 2 = a v . (29) 

The activity of each ion may be written as the product of its activity 
coefficient and concentration, so that 

a+ = y+m+ and a_ = 7_w_, 

a+ , a- 

. . 7+ = and 7- = -- 

m+ m- 

The mean activity coefficient y of the electrolyte, defined by 

y s (7?7-) 1 ", (30) 

can consequently be represented by 

r 

If m is the molality of the electrolyte, m+ is equal to WP+ and m_ is equal 
to wy_, so that equation (31) may be written as 



(32) 



The mean molality m of the electrolyte is defined, in an analogous 
manner, by 

m* s (m^-ml-) 1 /" = m(v>!r) 1/ '', 

so that it is possible to write equation (32) as 

"*- (33) 

Relationships similar to those given above may, of course, be derived 
for the other activity coefficients. 

Values of Activity Coefficients. Without entering into details, it 
is evident from the foregoing discussion that activities and activity 
coefficients are related to chemical potentials or free energies; several 
methods, both direct and indirect, are available for determining the 
requisite differences of free energy so that activities, relative to the 
specified standard states, can be evaluated. In the study of the activity 
coefficients of electrolytes the procedures generally employed are based 
on measurements of either vapor pressure, freezing point, solubility or 
electromotive force. 3 The results obtained by the various methods are 

1 See references on page 132, also pages 200 and 203. For a valuable summary of 
data and other information on activity coefficients, see Robinson and Harned, Chem. 
Revs., 28, 419 (1941). 



VALUES OP ACTIVITY COEFFICIENTS 



130 



TABLE XXXIV. MEAN ACTIVITY COEFFICIENTS OF ELECTROLYTES IN AQUEOUS 

SOLUTION AT 25 



Molality 


HC1 


NaCl 


KC1 


HBr 


NaOH 


CaCli 


ZnCU 


HS04 


ZnSO* 


LaCli 


Int(80)i 


0.001 


0.966 


0.966 


0.966 








0.888 


0.881 





0.734 


0.853 





0.005 


0.930 


0.928 


0.927 


0.930 





0.789 


0.767 


0.643 


0.477 


0.716 


0.16 


0.01 


0.906 


0.903 


0.902 


0.906 


0.899 


0.732 


0.708 


0.545 


0.387 


0.637 


0.11 


0.02 


0.878 


0.872 


0.869 


0.879 


0.860 


0.669 


0.642 


0.455 


0.298 


0.552 


0.08 


0.05 


0.833 


0.821 


0.816 


0.838 


0.805 


0.584 


0.556 


0.341 


0.202 


0.417 


0.035 


0.10 


0.798 


0.778 


0.770 


0.805 


0.759 


0.524 


0.502 


0.266 


0.148 


0.356 


0.025 


0.20 


0.768 


0.732 


0.719 


0.782 


0.719 


0.491 


0.448 


0.210 


0.104 


0.298 


0.021 


0.50 


0.769 


0.679 


0.652 


0.790 


0.681 


0.510 


0.376 


0.155 


0.063 


0.303 


0.014 


1.00 


0.811 


0.656 


0.607 


0.871 


0.667 


0.725 


0.325 


0.131 


0.044 


0.387 





1.50 


0.898 


0.655 


0.586 





0.671 





0.290 





0.037 


0.583 





2.00 


1.011 


0.670 


0.577 





0.685 


1.554 





0.125 


0.035 


0.954 





3.00 


1.31 


0.719 


0.572 








3.384 





0.142 


0.041 









in good agreement with each other and hence they may be regarded aa 
reliable. Although the description of the principles on which the deter- 
minations of activity coefficients are 
based will be considered later, it 
will be convenient to summarize in 
Table XXXIV some actual values 
of the mean activity coefficients at 
25 obtained for a number of electro- 
lytes of several valence types in aque- 
ous solution at various molalities. 
Some of the results are also depicted 
by the curves in Fig. 46; it will be 
observed that the activity coeffi- 
cients may deviate appreciably from 
unity. The values always decrease 
at first as the concentration is in- 
creased, but they generally pass 
through a minimum and then increase 
again. At high concentrations the 
activity coefficients often exceed 
unity, so that the mean activity of 
the electrolyte is actually greater than 
the concentration; the deviations 
from ideal behavior are now in the 
opposite direction to those which 
occur at low concentrations. An ex- FIQ. 46. Activity coefficients of electro- 
amination of Table XXXIV brings lytes of different valence types 

to light other important facts: it is 

seen, in the first place, that electrolytes of the same valence type, e.g , 
sodium and potassium chlorides, etc., or calcium and zinc chlorides, etc.. 




0.60 



1.60 2.0 
Molality 



140 FREE ENERGY AND ACTIVITY 

have almost identical activity coefficients in dilute solutions. Secondly, 
the deviation from ideal behavior at a given concentration is greater the 
higher the product of the valences of the ions constituting the electrolyte. 
The Ionic Strength. In order to represent the variation of activity 
coefficient with concentration, especially in the presence of added elec- 
trolytes, Lewis and Randall introduced the quantity called the ionic 
strength^ which is a measure of the intensity of the electrical field due 
to the ions in a solution. 4 It is given the symbol y and is defined as half 
the sum of the terms obtained by multiplying the molality, or concen- 
tration, of each ion present in the solution by the square of its valence; 
that is 

(34) 



In calculating the ionic strength it is necessary to use the actual ionic 
concentration or molality; for a weak electrolyte this would be obtained 
by multiplying its concentration by the degree of dissociation. 

Although the importance of the ionic strength was first realized from 
empirical considerations, it is now known to play an important part in 
the theory of electrolytes. It will be observed that equation (12) on 
page 83, which gives the reciprocal of the thickness of the ionic atmos- 
phere according to the theory of Debye and Hiickel, contains the quan- 
tity ^riiZi, where n,- is the number of ions of the zth kind in unit volume 
and hence is proportional to the concentration. This quantity is clearly 
related to the ionic strength of the solution as defined above; it will be 
seen shortly that it plays a part in the theoretical treatment of activity 
coefficients. 

It was pointed out by Lewis and Randall that, in dilute solutions, 
the activity coefficient of a given strong electrolyte is approximately 
the same in all solutions of a given ionic strength. The particular ionic 
strength may be due to the presence of other salts, but their nature does 
not affect the activity coefficient of the electrolyte under consideration. 
This generalization, to which further reference will be made later, holds 
only for solutions of relatively low ionic strength; as the concentration is 
increased the specific influence of the added electrolyte becomes manifest. 

The Debye-Hiickel Theory. The first successful attempt to account 
for the departure of electrolytes from ideal behavior was made by Milner 
(1912), but his treatment was very complicated; the ideas were essen- 
tially the same as those which were developed in a more elegant manner 
by Debye and Hiickel. The fundamental ideas have already been given 
on page 81 in connection with the theory of electrolytic conductance, 
and the application of the Dcbye-Hiickel theory to the problem of ac- 
tivity coefficients will be considered here. 6 

4 Lewis and Randall, J. Am. Chem. Soc., 43, 1112 (1921). 

6 Debye and Huckel, Physik. Z., 24, 185, 334 (1923); 25, 97 (1924); for reviews, see 
LaMer, T*ans. Electrochem. Soc., 51, 507 (1927); Falkenhagen, "Electrolytes" (trans- 
lated bj Bell), 1934; Williams, Chem. Revs., B, 303 (1931); Schingnitz, Z. Ekktrochem., 
36, 861 (1930). 



THE DEBYE-HtfCKEL THEORY 141 

According to equation (16), page 84, the potential $ due to ions of 
the ith kind may be represented by 



where the first term is the potential at a distance r from the central ion 
when there are no surrounding ions, and the second term is the contribu- 
tion of the ionic atmosphere; K is defined by equation (12), page 83. 
Suppose that all the ions are discharged and that successive small charges 
are brought up to the ions from infinity in such a way that at any instant 
all the ions have the same fraction X of their final charge z t . It follows, 
therefore, from equation (35), that at any stage during the charging process 
the potential fa due to ions of the ith kind is given by 



where K\ is the value of the quantity K at this stage. It can be seen, 
from the definition of *, that since the charge on the ion is then Xz, 
the value of KX will be a fraction X of the final value; the term KX in equa- 
tion (36) may thus be replaced by X*c. Making this substitution, equa- 
tion (36) becomes 

*x = gx- 2 fx> (37) 

If z#d\ is the magnitude of the small charge brought up to each ion 
of the ith kind, the corresponding work done is z t edX X ^x, and hence 
the total electrical work (Wi) done in charging completely, i.e., from 
X = to X = 1, an ion of the ith kind is 



p-i 

Wi = I z % Gl/\d\ 
Jx-o 



2Dr 3D 



D 

(38) 



If Ni is the total number of ions of the ith kind,* the total electrical 
work (W) done in charging completely all the ions of the solution is 
obtained by multiplying equation (38) by Ni and summing over all the 
ions, thus 



* This should not be confused with n,, the number of these ions in unit volume. 



142 FREE ENERGY AND ACTIVITY 

At infinite dilution there is no ionic atmosphere, and so K is zero and 
the second term on the right-hand side of equation (39) disappears; since 
the dielectric constant is that of the pure solvent, i.e., D , the electrical 
work (Wo) done in charging the ions at infinite dilution is 



Provided the solution is not too concentrated, D and Do are approxi- 
mately equal, and hence the difference in the electrical work of charging 
the same ions at a definite concentration and at infinite dilution is 
given by 



The volume change accompanying the charging process at constant 
pressure is negligible, and so W Wo may be identified with the differ- 
ence between the electrical free energy of an ionic solution at a definite 
concentration and at infinite dilution. 

The free energy (G) of a solution containing ions may be regarded as 
being made up of two parts: first, that corresponding to the value for an 
ideal solution at the same concentration as the ionic solution (G ), and 
second, an amount due to the electrical interaction of the ions (G e i.) ; thus 

G = Go + Gei., (42) 

where G e i. may be taken as being equal to W T7o, as given by equation 
(41). Differentiating with respect to N if the number of ions of the ith 
kind, at constant temperature and pressure, the result is 

dG dGo dGei. 
= ~~ ' 



or 

M = Mt(0) + Mt(el.). (43) 

According to the definition of the chemical potential /*, which now applies 
to a single ion y instead of to a g.-ion, 

M< = MI + kT In a t 

= d + kTlnx t + kTl*f>, (44) 

where k is the Boltzmann constant, i.e., the gas constant per single 
molecule. Further, since Go refers to an ideal solution, it follows that 

M.(O) = rf + kT\nx it (45) 

and hence from equations (43), (44) and (45), 

(46) 



Introducing the value of G e i. as given by equation (41), it is found on 



THE DEBYE-HtJCKEL LIMITING LAW 143 

differentiating with respect to N { , remembering that K involves ^Ui and 
hence V#i, that 






(47) 



N being the Avogadro number and R, equal to kN, the gas constant 
per mole.* 

The Debye-Hiickel Limiting Law. The value of K as given on page 
83 is 

(48) 

and if n,- is replaced by cJV/1000, where c< is the ionic concentration in 
moles per liter, and R/N is written for k, equation (48) becomes 

(49) 

The quantity 2c2j is seen to be analogous to twice the ionic strength as 
defined by Lewis and Randall [equation (34)]; the only difference is that 
the former involves volume concentrations whereas in the latter molalities 
are employed. For dilute aqueous solutions, such as were used in the 
work from which Lewis and Randall made the generalization given on 
page 140, the two values of the ionic strength are almost identical. It 
has been stated that if the Debye-Hiickel arguments are applied in a 
rigid manner the expression for K will actually involve molalities; never- 
theless, it is the practice in connection with the application of the equa- 
tions derived by the method of Debye and Hiickel to use an ionic strength 
defined in terms of molar concentrations, viz., 

(50) 



so that equation (49) can be written as 

/ arjw V 
* = \IOOODRT* ) m (51) 

Introducing this value for K into equation (47) and at the same time 
dividing the right-hand side by 2.303 to convert natural to common 
logarithms, the result is 

i * JVV / 2* V z\ r 

log/ ' = " 303tf>V 1000/ (DT)* * ( } 



* It should be noted that in the differentiation the summation in equation (41) 
has been reduced to a single term. This is because the numbers of all the ions except 
of the ith kind remain constant, and so all the terms other than the one involving n 
will be zero. 



144 PEBB ENERGY AND ACTIVITY 

The universal constants AT, c, v and R, as well as the numerical quantities, 
may be extracted from equation (52), and if the accepted values are 
employed, this equation becomes 

log/, = - 1.823 X 10' ~ V. (53) 



For a given solvent and temperature D and T have definite values which 
may be inserted; equation (53) then takes the general form 

log/. = - Az 2 ^, (54) 

where A is a constant for the solvent at the specified temperature. 

This equation, which represents what has been called the Debye- 
Hiickel limiting law, expresses the variation of the activity coefficient 
of an ion with the ionic strength of the medium. It is called the limiting 
law because, as seen previously, the approximations made in the deriva- 
tion of the potential at an ion due to its ionic atmosphere, can be expected 
to be justifiable only as infinite dilution is approached. The general 
conclusion may be drawn from equation (53) or (54) that the activity 
coefficient of an ion should decrease with increasing ionic strength of the 
solution: the decrease is greater the higher the valence of the ion and the 
lower the dielectric constant of the solvent. 

It will be seen later (p. 230) that there does not appear to be any 
experimental method of evaluating the activity coefficient of a single 
ionic species, so that the Debye-Hiickel equations cannot be tested in 
the forms given above. It is possible, however, to derive very readily 
an expression for the mean activity coefficient, this being the quantity 
that is obtained experimentally. The mean activity coefficient / of an 
electrolyte M+A~ is defined by an equation analogous to (30), and 
upon taking logarithms this becomes 



The values of log/+, which is equal to Ass+Jy, and of log/_, i.e., 
AzlVy, as given by equation (54) can now be inserted in (55); making 
use of the fact that z+v+ must be equal to Z-v-, it is found that 

log/ = ~ Az^.^, (56) 

which is the statement of the Debye-Huckel limiting law for the mean 
activity coefficient of an electrolyte whose ions have valences of z+ and z_, 
respectively. The values of the constant A for water at a number of 
temperatures are given in Table XXXV below. 

Attention should be drawn to the fact that the activity coefficients 
given by the Debye-Htickel treatment are the so-called rational coeffi- 
cients (p. 135) ; to express the values in the form of the practical activity 
coefficients, it is necessary to make use of equation (26). If the solvent 



DEBYE-HUCKEL EQUATION FOR APPRECIABLE CONCENTRATIONS 145 

is water, so that M i is 18, it is seen that 

log 7 = log/ - log (1 + 0.018m), 

where y is the activity coefficient in terms of molalities, / is the value 
given by the Debye-Huckel equations, and v is the number of ions pro- 
duced by one molecule of electrolyte on dissociation. As already seen, 
however, the difference between the various coefficients is negligible in 
dilute solutions, and it is in such solutions that the most satisfactory tests 
of the Debye-Hiickel theory can be made. 

Debye-Hiickel Equation for Appreciable Concentrations. In the 
derivation of equation (12), page 83, the approximation was made of 
regarding the ion as being equivalent to a point charge; this will result 
in no serious error provided the radius of the ionic atmosphere is large in 
comparison with that of the ion. An examination of Table XXII, page 
85, shows that this condition is satisfied in 
dilute solutions, but when the concentration 
approaches a value of about 0.1 molar the 
radius of the ionic atmosphere is about the 
same order as that of an ion, i.e., 2 X 10"" 8 cm. 
It follows, therefore, that in such solutions the 
approximation of a point charge is liable to 
lead to serious errors. A possible method of 
making the necessary correction has been pro- FlG 47 Mean distance of 
posed by Debye and Hiickel ; 6 it has been found approach of ions 

that if a is the mean distance of approach of 

other ions, e.g., B to the central ion A 9 as shown in Fig. 47, the potential 
due to ions of the zth kind is given by the expression 

2 2,K 1 

+ -*--D'TTZ' (57) 

instead of by equation (35). The mean distance of approach a is often 
referred to as the "average effective diameter" of the ions, although its 
exact physical significance probably cannot be expressed precisely. It is 
seen that the correction term is (1 + KCL)~ I , which approaches unity in 
dilute solutions when K is small. 

By following through the derivation on page 141, using equation (57) 
instead of (35), the final result is 

I ,_ 

' (58) 




in place of equation (47). It is apparent from equation (51) that, for a 
given solvent and a definite temperature, K is equivalent to #V|i, where 
B is a constant; hence 1 + KCI may be replaced by 1 + aB'fy. Making 
Debye and Huckel, Physik. Z. t 24, 185 (1923). 



146 FREE ENERGY AND ACTIVITY 

this substitution in equation (58), 



1 /KQ . 

' (59) 



and hence the Debye-Hiickel limiting law, corresponding to equation 
(54), now becomes 

' (60) 



where A has the same significance as before. The expression for the 
mean activity coefficient of an electrolyte is then 

' (61) 

a\v 

Both the constants A and B depend on the nature of the solvent and 
the temperature; the values for water at several temperatures arc given 
in Table XXXV; the corresponding dielectric constants arc also recorded. 

TABLE XXXV. DEBYE-HUCKEL CONSTANTS AND DIELECTRIC CONSTANT OF WATER 

Temp. DAB 

88.15 0.488 0.325 X 10 8 

15 82.23 0.500 0.328 

25 78.54 0.509 0.330 

30 76.76 0.514 0.331 

40 73.35 0.524 0.333 

50 70.10 0.535 0.335 

It will be observed from Table XXXV that at ordinary temperatures 
the value of B with water as solvent is approximately 0.33 X 10 8 ; for 
most electrolytes the mean ionic diameter a is about 3 to 4 X 10~ 8 cm. 
(see Table XXXVI), and hence aB does not differ greatly from unity. 
A reasonably satisfactory and simple approximation of equation (61) is 
therefore 



The Hiickel and Breasted Equations. A further correction to the 
Debye-Hiickel equation has been proposed in order to allow for the polari- 
zation of the solvent molecules by the central ion; since these molecules 
are, in general, more polarizable than the ions themselves, there will be 
a tendency for the solvent molecules to displace the other ions from the 
vicinity of a particular ion. The dipolar nature of the solvent molecules 
will also facilitate the tendency for these molecules to orient themselves 
about the central ion. It has been suggested that the result of this 
orientation is equivalent to an increase in the dielectric constant in the 
immediate vicinity of the ion above that in the bulk of the solvent. By 



QUALITATIVE VERIFICATION OF THE DEBYE-HUCKEL EQUATIONS 147 

assuming the increase to be proportional to the ionic concentration of the 
solution, it has been deduced that an additional term CV> where C" is 
an empirical constant, should be added to the right-hand side of equa- 
tions (60) and (61) ; hence, the latter now becomes 

+ c '*- (62) 



This result has sometimes been called the Hiickel equation. 7 

It is not certain that the theoretical arguments, which led to the 
introduction of the term C't*> are completely satisfactory, but it seems to 
be established that the experimental data require a term of this type. 
The aggregation of solvent molecules in the vicinity of an ion is the factor 
responsible for the so-called "salting-out effect," namely, the decrease in 
solubility of neutral substances frequently observed in the presence of 
salts; the constant C 1 is consequently called the salting-out constant. 
The activity coefficient of a non-electrolyte, as measured by its solubility 
in the presence of electrolytes, is often given by an expression of the form 
log/ = C"|i; this is the result to which equation (62) would reduce for the 
activity of a non-electrolyte, i.e., when z+ and z_ are zero, in a salt solu- 
tion of ionic strength y. 

By dividing through tfce numerator of the fraction on the right-hand 
side of equation (62) by the denominator, and neglecting all terms in the 
power series beyond that involving p, the result is 

+ (aABz+Z- + C")u 

+ CV, (63) 

where C is a constant for the given electrolyte, equal to aABz+z_ + C'. 
This relationship is of the same form as an empirical equation proposed by 
Br0nsted, 8 and hence is in general agreement with experiment; it has 
been called the Debye-Hiickel-Br^nsted equation. In dilute solution, 
when y is small, the term Cy can be neglected, and so this expression then 
reduces to the Debye-Hiickel limiting law. 

Qualitative Verification of the Debye-Hiickel Equations. The gen- 
eral agreement of the limiting law equation (54) with experiment is shown 
by the empirical conclusion of Lewis and Randall (p. 140) that the 
activity coefficient of an electrolyte is the same in all solutions of a given 
ionic strength. Apart from the valence of the ions constituting the 
particular electrolyte under consideration, the Debye-Hiickel limiting 
equation contains no reference to the specific properties of the salts 
that may be present in the solution. It is of interest to record that the 

7 Hiickel, Physik. Z., 26, 93 (1925); see also, Butler, /. Phys. Chem., 33, 1015 (1929); 
Scatchard, Physik. Z., 33, 22 (1932). 

8 Br0nsted, J. Am. Chem. Soc., 44, 938 (1922); Br0nsted and LaMer, ibid., 46, 555 
(1924). 



148 



FREE ENERGY AND ACTIVITY 



empirical equation proposed by Lewis and Linhart 9 to account for their 
results on the freezing points of dilute solutions of various electrolytes 
is of the form log/ = ftc a , where a was found to be about 0.4 to 0.5 
for several salts and ft depended on their valence type. Further, as 
already mentioned, Br0nsted's empirical equation for more concentrated 
solutions is in agreement with the extended equation (62). It can be 
seen from the Debye-Hiickel limiting law equation that at a definite ionic 
strength the departure of the activity coefficient of a given electrolyte 
from unity should be greater the higher the valences of the ions con- 
stituting the electrolyte; this conclusion is in harmony with the results 
given in Table XXXIV which have been already discussed. 

It was noted on page 139 that although activity coefficients generally 
decrease with increasing concentration in dilute solutions, in accordance 
with the requirement of equation (58), the values frequently pass through 
a minimum at higher concentrations. It is of interest, therefore, to see 
how far this fact can be explained, at least qualitatively, by means of the 
Debye-Huckel theory. According to the limiting law equation, the plot 
of log/ against Vy should be a straight line of slope AZ+Z-] for a uni- 

univalent electrolyte in water at 
25 this is equal to approximately 
0.51, as shown in Fig. 48, 1. If 
the ionic size factor is introduced, 
as in equation (61), the plot of 
log/ against Vy becomes of the 
form of Fig. 48, II, representing 
a type of curve which is often 
obtained experimentally. Finally, 
the addition of the salting-out fac- 
tor, as in equation (62), results in 
a further increase of the activity 
coefficient by an amount propor- 

0.2 o.4 0.6 0.8 1.0 1.2 tional to the ionic strength; the 

result is that the log/ against Vjt 
curve becomes similar to Fig. 48, 
III. It may be mentioned that 
the latter curve duplicates closely 

the variation of the activity coefficient of sodium chloride with concen- 
tration up to relatively high values of the latter. 

Quantitative Tests of the Debye-Hiickel Limiting Equation. Al- 
though the Debye-Huckel equations are generally considered as applying 
to solutions of strong electrolytes, it is important to emphasize that they 
are by no means restricted to such solutions; they are of general applica- 
bility and the only point that must be noted is that in the calculation of 
the ionic strength the actual ionic concentrations must be employed. 
9 Lewis and Linhart, J. Am. Chem. Soc., 41, 1951 (1919). 



-0.2 



-0.4 



-0.6 




Fia. 48. Simple (I) and extended (II and 
III) Debye-Huckel equations 



QUANTITATIVE TESTS OF THE DEBYE-HtfCKEL LIMITING EQUATIONS 149 



For incompletely dissociated electrolytes this involves a knowledge of the 
degree of dissociation, which may not always be available with sufficient 
accuracy. It is for this reason that the Debye-Htickel equations are 
generally tested by means of data obtained with strong electrolytes, since 
they can be assumed to be completely dissociated. It is probable that 
some of the discrepancies observed with certain electrolytes of high 
valence types are due to incomplete dissociation for which adequate allow- 
ance has not been made. 



1.0 
0.9 
0.8 
0.7 
0.6 
0.5 
0.4 




J_ 



0.10 



0.40 



0.20 0.30 

vr 

FIG. 49. Test of the limiting Debye-Huckel equation 

The experimentally determined activity coefficients, based on vapor 
pressure, freezing-point and electromotive force measurements, for a 
number of typical electrolytes of different valence types in aqueous 
solution at 25, are represented in Fig. 49, in which the values of log / 
are plotted against the square-root of the ionic strength; in these cases 
the solutions contained no other electrolyte than the one under considera- 
tion. Since the Debye-Hiickel constant A for water at 25 is seen from 
Table XXXV to be 0.509, the limiting slopes of the plots in Fig. 49 should 
be equal to -0.509 z+z_; the results to be expected theoretically, cal- 
culated in this manner, are shown by the dotted lines. It is evident that 
the experimental results approach the values required by the Debye- 
Hiickel limiting law as infinite dilution is attained. The influence of 
valence on the dependence of the activity coefficient on concentration is 
evidently in agreement with theoretical expectation. Another verifica- 
tion of the valence factor in the Debye-Hiickel equation will be given 
later (p. 177). 

A comparison of equations (52) and (53) shows that, for electrolytes 
of the same valence type, the limiting slope of the plot of log/ against Vy 
at constant temperature should be inversely proportional to Z> f , where D 



160 



FREE ENERGY AND ACTIVITY 



is the dielectric constant of the medium. A stringent test of the Debye- 
Hiickel equation is, therefore, to determine the activity coefficients of a 
given electrolyte in a number of different media of varying dielectric 
constant; the results are available for hydrochloric acid in methyl and 
ethyl alcohols, in a number of dioxane-water mixtures, as well as in pure 
water at 25. Some of the data are plotted in Fig. 50; the limiting slopes, 




Fia. 50. Limiting Debye-Hiickel equation at different dielectric constants 
(Earned, et al) 

marked with the appropriate value of the dielectric constant, are indi- 
cated by the dotted lines in each case. The agreement with expectation, 
over a range of dielectric constant from about 10 to 78.6, is very striking. 10 
The influence of one other variable, namely, the temperature, re- 
mains to be considered. It is not an easy matter to vary the temperature 
without changing the dielectric constant, and so these factors may be 
considered together. From equations (55) and (56) it is evident that 
the limiting slope of the plot of log/ against Vy should vary as 1/(DT)*, 
where T is the absolute temperature at which the activity coefficients 
are measured. The experimental results obtained under a wide variety 
of conditions, e.g., in liquid ammonia at 75 and in water at the boiling 

w Earned et al., J. Am. Chem. Soc., 61, 49 (1939). 



THE OSMOTIC COEFFICIENT 



151 



point, are generally in satisfactory agreement with theoretical require- 
ments. 11 Where discrepancies are observed they can probably be ex- 
plained by incomplete dissociation in media of low dielectric constant. 

The Osmotic Coefficient. Instead of calculating activity coefficients 
from freezing-point and other so-called osmotic measurements, the data 
may be used directly to test the validity of the Debye-Hiickel treatment. 
If is the depression of the freezing point of a solution of molality m of an 
electrolyte which dissociates into v ions, and X is the molal freezing-point 
depression, viz., 1.858 for water, a quantity <, called the osmotic 
coefficient, may be defined by the expression 



e 



(64) 



This coefficient is equivalent to the van't Hoff factor i (see p. 9) di- 
vided by v. It can be shown by means of thermodynamics that if 
log / is proportional to the square-root of the ionic strength, as it 
undoubtedly is in dilute solutions, then 



1 - * = - iln/. (65) 

Introducing the Debye-Hiickel limiting law for log /, it is seen that 



2.303 



(66) 



where A has the same significance as before. Since <t> can be determined 
directly from freezing-point measurements, by means of equation (64), 



0.5 



0.4 



0.3 



0.2 



0.1 




x Lithium Chloride 

o Lithium Bromide 

+ Lithium Perchlorate 

o Guanidine Nitrate 



0.05 



0.10 



0.15 



FIG. 51. Test of Debye-Huckel equation by freezing-point measurements in 
cyclohexanol (Schreiner and Frivold) 

"Saxton and Smith, /. Am. Chem. Soc., 54, 2626 (1932); Webb, /. Am. Chem. 
Soc., 48, 2263 (1926). 



152 



FREE ENERGY AND ACTIVITY 



it is possible to test the Debye-Huckel theory in the form of equation 
(66); the plot of 1 against Vy should approach a limiting value of 
0.768 A z+z_. The experimental results for electrolytes of different 
valence types in aqueous solutions are in agreement with expectation; 
since the data are in principle similar to many that were used in the 
compilation of Fig. 49, they need not be considered further. It is of 
interest, however, to examine the values derived from freezing-point 
measurements in a solvent of low dielectric constant, viz., cyclohexanol, 
whose dielectric constant is 15.0 and freezing point 23.6; the full curve 
in Fig. 51 is drawn through the results for a number of uni-univalent 
electrolytes, while the dotted curve shows the limiting slope required by 
equation (66). 12 

Activities at Appreciable Concentrations. A comparison of the ex- 
perimental curves in Figs. 49 and 50 with the general form of curve II 
in Fig. 48 suggests that equation (61) might represent the variation of 
activity coefficient with concentration in solutions of electrolytes that 



1.4 



1.3 




1.1 



I 



I 



I 



I 



0.05 



0.20 



0.10 0.15 

vr 

FIG. 52. Determination of mean ionic diameter 

were not too concentrated; by a slight rearrangement this equation can 
be put in the form 

A 9 . 9 A/M . 

(67) 



log/ ^ 

so that if the left-hand side of equation (67) is plotted against Vy the 
result should be a straight line of slope aB. Since the value of B is 

"Schreiner and Frivold, Z. physik. Chem., 124, 1 (1926). 



ACTIVITIES IN CONCENTRATED SOLUTIONS 153 

known (cf. Table XXXV), the magnitude of the mean ionic diameter 
required to satisfy the experimental results can be obtained. The data 
for aqueous solutions of hydrochloric acid at 25 are shown in Fig. 52; 
the points are seen to fall approximately on a straight line so that an 
equation of the form of (61) and (67) is obeyed. The slope of this line 
is about 1.75 and since B is 0.33 X 10 8 , it follows that for hydrochloric 
acid a is equal to 5.3 X 10~~ 8 . It has been found in a number of cases 
that by using values of a that appear to be of a reasonable magnitude it 
is possible to represent quantitatively the activity coefficients of a num- 
ber of electrolytes up to ionic strengths of ^about 0.1. Some of the mean 
values, collected from those reported in the literature, are given in Table 
XXXVI. It must be pointed out, however, that such satisfactory results 

TABLE XXXVI. MEAN EFFECTIVE IONIC DIAMETERS 

Electrolyte a Electrolyte a 

HC1 5.3 X 10- cm. CaCl 2 5.2 X lO" 8 cm. 

NaCl 4.4 MgS0 4 3.4 

KC1 4.1 K 2 SO 4 3.0 

CsNO, 3.0 La 2 (S0 4 ) 3.0 

are not always obtained; in order to satisfy the experimental data in the 
case of silver nitrate, for example, a should be 2.3 X 10~ 8 cm., and for 
potassium nitrate 0.43 X 10~ 8 cm., both of which values are lower than 
would be expected. It is nevertheless of interest that the figures are at 
least of the correct order of magnitude for an ionic radius, namely about 
10~~ 8 cm. In some instances, particularly with salts of high valence types, 
it is found necessary to employ variable or even negative values of a; 
this may be attributed either to incomplete dissociation or to the ap- 
proximations made in the Debye-Huckel derivation. 

Activities in Concentrated Solutions. For relatively concentrated 
solutions it is necessary to use the complete Hlickel equation (62); by 
choosing suitable values for the two adjustable parameters a and C", 
it has been found possible to represent the variation of activity coeffi- 
cients with concentration of several electrolytes from 0.001 to 1 molal, 
and sometimes up to 3 molal. The values of C' seem to lie approximately 
between 0.05 and 0.15 in aqueous solution. At the higher concentrations 
it is necessary to make allowance for the difference between the rational 
and stoichiometric activity coefficients; the latter, which is the experi- 
mentally determined quantity, is represented by an extension of equa- 
tion (62); thus (cf. p. 135), 

log 7 = - - + C"Y - 1<* (1 + - 001 



where v is the number of ions produced by one molecule of electrolyte on 
dissociation, m is the molality of the solution and M i is the molecular 



154 FREE ENERGY AND ACTIVITY 

weight of the solvent. This equation has been employed for the purpose 
of extrapolating activity coefficient data to dilute solutions from ac- 
curate measurements made at relatively high concentrations. It is not 
certain that this procedure is altogether justifiable, for the value of a 
obtained from activity data at high concentrations is often different 
from that derived from measurements on the same electrolyte in dilute 
solutions. 

Extension of the Debye-Hiickel Theory. In the calculation of the 
electrical density in the vicinity of an ion (p. 82), it was assumed that 
ZiGp/kT was negligible in comparison with unity, so that all terms beyond 
the first in the exponential series could be neglected. According to 
calculations made by Miiller (1927), the neglect of the additional terms 
is justifiable provided that 



a > 

that is, if the mean ionic diameter a is greater than about 1.4 X 10- 8 z 2 /D 
at 25. It follows, therefore, that the additional terms are negligible in 
aqueous solution if a/2 2 exceeds 1.6 X 10~ 8 ; for a uni-univalent salt, 
therefore, a should exceed 1.6 X 10~~ 8 cm., but for a bi-bivalent electro- 
lyte a must exceed 6.4 X 10~ 8 cm. if the Debye-Hiickel approximation is 
to be valid. Since ionic diameters are rarely as high as the latter figure, 
it is seen that salts of high valence type might be expected to exhibit dis- 
crepancies from the simple Debye-Hiickel behavior. Since the limiting 
values of a are larger the smaller the dielectric constant D of the medium, 
the deviations become more marked and will occur with electrolytes of 
lower valence type in media of low dielectric constant. 

The potential ^ is given approximately by equation (15) on page 83, 
and hence the assumption, made by Debye and Hiickel, that 2c^//cT is 
small compared with unity, is equivalent to stating that 



- 

D r 

and this is less likely to be true the higher the valence of the ion and the 
smaller its radius, and the smaller the dielectric constant of the medium. 
In order to avoid the approximation involved in neglecting the higher 
terms in the exponential series, Gronwall, LaMer and Sandved w used 
the complete expansion for the electrical density, and solved the differen- 
tial equation, following the introduction of the Poisson equation, in the 
form of a power series. The result obtained for a symmetrical valence 
type electrolyte, that is one with both ions of the same valence, is given 
by the following expression, which should be compared with equation 

"Gronwall, LaMer and Sandved, Physik. Z., 29, 358 (1928); see also, LaMer, 
Gronwall and Greiff, /. Phys. Chem., 35, 2345 (1031). 



ION-ASSOCIATION 155 

(58), viz., 

Nft* 1 



2DRT I + *a 

00 / Nz 2 * 2 \* m+1 



- 2m y 2m+1 (Ka)], (69) 

where X(KCL) and 7(ica) are known, but complicated, functions of *a. 
The summation in equation (69) should be carried over all integral values 
of m from unity to infinity, but it is found that successive terms in the 
series decrease rapidly and it is sufficient, in general, to include only two 
terms. 

In the application of equation (69) an arbitrary value of a is chosen 
so as to give calculated activity coefficients which agree with those de- 
rived by direct experiment; the proper choice of a is made by a process 
of trial and error until a value is found that is satisfactory over a range of 
concentrations. There is no doubt that the Gronwall-LaMer-Sandved 
extension represents an important advance over the simple Debye- 
Hlickel treatment, for it frequently leads to more reasonable values of the 
mean ionic diameter. 14 The validity of equation (69) has been tested 
by a variety of activity measurements and the results have been found 
satisfactory; were it not for the tedious nature of the calculations it 
would probably be more widely used. 

It is necessary to call attention to the fact that equation (69) was 
deduced for symmetrical valence electrolytes; for unsymmetrical types 
the corresponding equation is of a still more complicated nature. 

Ion-Association. A device, proposed by Bjerrum, 15 for avoiding the 
difficulty of integrating the Poisson equation when it is not justifiable to 
assume that z^lkT is much smaller than unity, involves the concept of 
the association of ions to form ion-pairs (cf. p. 96). It may be remarked 
that, in a sense, a solution, such as that of Gronwall, Sandved and LaMer, 
of the differential equation resulting from the use of the complete expres- 
sion for the electrical density, makes the Bjerrum treatment unnecessary. 
The results obtained are, nevertheless, of interest, especially in connection 
with their application to media of low dielectric constant. 

According to the Boltzmann distribution law, the number drii of 
ions of the iih kind in a spherical shell of radius r and thickness dr, sur- 
rounding a specified ion, is given by 

dm = n % 4an*c~ w i kT dr, (70) 

" LaMer et al., /. Phys. Chem., 35, 1953 (1931); 40, 287 (1936); /. Am. Chem. Soc. t 
53, 2040, 4333 (1931); 54, 2763 (1932); 56, 544 (1934); Partington et al., Trans. Faraday 
Soc., 30, 1134 (1934); Phil Mag., 22, 857 (1936). 

Bjerrum, K. Danske Vidensk. Selsk. Mat.-fys. Medd., 7, No. 9 (1926); Fuoss and 
Kraus, J. Am. Chem. Soc., 55, 1019 (1933); Fuoss, Trans. Faraday Soc., 30, 967 (1934); 
Chem. Revs., 17, 227 (1935). 



156 



FREE ENEROT AND ACTIVITY 



where n t is the number of ions of the z'th kind in unit volume and W 
is the work required to separate one of these ions from the central ion; 
k is the Boltzmann constant and T is the absolute temperature. The 
central ion, supposed to be positive, carries a charge z+e and that of the 
ith ion, which is of opposite sign, is 2_c; if Coulomb's law is assumed to 
hold at small interionic distances and the ions are regarded as point 
charges separated by a medium with an effective dielectric constant (D) 
equal to that of the solvent, then the work required to separate the ions 
from a distance r to infinity, and hence the value of W, is given by 



W 



(71) 



The influence of ions other than the pair under consideration is neglected 
in this derivation. Substituting this result for W in equation (70), it 
follows that 

dni = n % 4wr*e-+'-'*' DrkT dr. (72) 

The fraction dn^dr is a measure of the probability P(r) of finding an ion 
of charge opposite to that of the central ion at a distance r from the 

latter; thus 



P(r) 




P(r) 



. (73) 



If the right-hand side of this equa- 
tion, for various values of r, is plotted 
against r, the result is a curve of the 
type shown in Fig. 53, the actual 
form depending on the valences z+ 
and Z-. of the oppositely charged 
ions, and also on the dielectric con- 
stant of the medium. It will be 
observed that at small distances of 
approach there is a very high prob- 
ability of finding the two ions to- 
gether, but this probability falls 
rapidly, passes through a minimum 
and then increases somewhat for in- 
creasing distances between the ions. 
The interionic distance r m i n ., for 
which the probability of finding two oppositely charged ions together is a 
minimum, can be obtained by differentiating equation (73) with respect 
to r and setting the result equal to zero ; in this way it is found that 

(74) 



FIG. 53. Distribution of oppositely 
charged ions about a central ion (Bjerrum) 



min * 2DkT 

The suggestion was made by Bjerrum that all ions lying within a 
sphere of radius r m i n . should be regarded as associated to form ion-pairs, 



THE FRACTION OP ASSOCIATION 157 

whereas those outside this sphere may be considered to be free. The 
higher the value of r m [ n . the greater the volume round a given ion in 
which the oppositely charged ions can be found, and hence the greater 
the probability of the occurrence of the ion-pairs. It is evident, there- 
fore, from equation (74) that ion association will take place more readily 
the higher the valences, z+ and z_, of the ions of the electrolyte and the 
smaller the dielectric constant of the medium. This conclusion is in 
general agreement with experiment concerning the deviations from the 
behavior to be expected from the Debye-Hiickel treatment based on the 
assumption of complete dissociation. Attention may be called to the 
fact, the exact significance of which is not altogether clear, that the value 
of r min . given by equation (74) is about twice the mean ionic diameter a 
which must be exceeded if the additional terms in the Debye-Hiickel 
expansion may be neglected (see p. 154). 

The Fraction of Association. If equation (72) is integrated between 
r = a, where a is the effective mean diameter of the ions, or their dis- 
tance of closest approach, and r = r m i n ., the result should give the num- 
ber, which will be less than unity, of oppositely charged ions that may be 
regarded as associated with a given ion. In other words, this quantity is 
equal to the fraction of association (6) of the strong electrolyte into ion- 
pairs; thus 






rmin. 
rV-'+'-^^r. (75) 



If JVc/1000, where c is the concentration in moles per liter, is written in 
place of n,, and if both ions are assumed to be univalent, equation (75) 
may be expressed in the form 



where 

and 



The values of Q(b) as defined above have been tabulated for various 
values of b from 1 to 80, and so by means of equation (76) it is possible 
to estimate the extent of association of a uni-univalent electrolyte con- 
sisting of ions of any required mean diameter a, at a concentration c 
in a medium of dielectric constant D. It will be seen from equation (76) 
that in general B increases as b increases, i.e., 6 increases as the mean 
diameter a of the ions and the dielectric constant of the solvent decrease. 
The values for the fraction of association of a uni-univalent electrolyte 
in water at 18 have been calculated by Bjerrum for various concentra- 
tions for four assumed ionic diameters ; the results are recorded in Table 
XXXVII. The extent of association is seen to increase markedly with 
decreasing ionic diameter and increasing concentration. The values are 



158 FREE ENERGY AND ACTIVITY 

appreciably greater in solutions of low dielectric constant, as is apparent 
from the factor 1/D 8 in equation (76). 

TABLE XXXVII. FRACTION OP ASSOCIATION (0) OF UNI-UNIVALBNT ELECTROLYTE 

IN WATER AT 18 



Concentration 


a 


0.001 


0.005 


0.01 


0.05 


0.1 


0.5 


1.0 H 


2.82A 





0.002 


0.005 


0.017 


0.029 


0.090 


0.138 


2.35 


0.001 


0.004 


0.008 


0.028 


0.048 


0.140 


0.206 


1.76 


0.001 


0.007 


0.012 


0.046 


0.072 


0.204 


0.286 



The Association Constant. Suppose that a salt MA is completely 
ionized in solution and that a certain fraction of the ions are associated as 
ion-pairs; an equilibrium may be supposed to exist between the free 
M+ and A~ ions, on the one hand, and ion-pairs on the other hand. If 
the law of mass action [cf. equation (20)] is applied to this equilibrium, 
the result is 

_ Activity of M+ X Activity of A~ 
Activity of ion-pairs 

where K is the dissociation constant (cf. p. 163). If c is the concentra- 
tion of the salt MA, the concentration of associated ions is 6c while that 
of each of the free ions is (1 0)c; further, if fi represents the mean 
activity coefficient of the ions and /2 is that of the ion-pairs, then 

(1 - 8)c X (1 - fl)c ft (1 - OYc /! 

*~ fe u = ~~e -- 7T (77) 

For very dilute solutions, i.e., when c is small, the activity coefficients 
are almost unity, while 8 is negligible in comparison with unity (see 
Table XXXVII); equation (77) then reduces to 

*-! 

.'. K- - j. (78) 

where Jf" 1 , the reciprocal of the dissociation constant, is called the 
association constant of the completely ionized electrolyte. Introducing 
the value of given by equation (76), the result is 



and so the dissociation constant K can be calculated for any assumed 
value of the distance of closest approach of the ions in a medium of known 
dielectric constant. 



TRIPLE IONS 



159 



15 



10 



$ 
I . 




I 



0.5 



L5 



A test of equation (79), based on the theory of ion association, is 
provided by the measurements of Fuoss and Kraus 16 of the conductance 
of tetraisoamylammonium nitrate in a series of dioxane-water mixtures 
of dielectric constant ranging from 2.2 to 78.6 (cf. Fig. 21) at 25. From 
the results in dilute solution the dissociation constants were calculated 
by the method described on page 158. 
The values of log K, plotted against 
log D of the medium, are indicated by 
the points in Fig. 54, whereas the full 
curve is that to be expected from equa- 
tion (79) if a is taken as 6.4A. The agree- 
ment between the experimental and theor- 
etical results is very striking. It will be 
observed that as the dielectric constant 
increases the curve turns sharply down- 
wards and crosses the log D axis at a value 
of the dielectric constant of approximately 
41. The significance of this result is that 
for ions of mean effective diameter equal 
to 6.4A, the dissociation constant of the 
electrolyte is very large, and hence the 
extent of association becomes negligible 
when the dielectric constant of the solvent 
exceeds a value of about 41.* For smaller 
ions or for ions of higher valence, the di- 
electric constant would have to attain a larger value before an almost 
completely ionized electrolyte would be also completely dissociated. 

Triple Ions. The concept of ion-pairs has been extended to include 
the possibility of the presence in solution of groups of three ions, viz., 

H h or | , i.e., triple ions, held together by electrostatic forces. 17 

Such triplets might be expected to form most readily in solvents of low 
dielectric constant, for it is in such media that the forces of electrostatic 
attraction would be greatest. Consider an electrolyte MA in a medium 
of low dielectric constant ; there will be an equilibrium between the ions 
M+ and A~ and the ion-pairs, as described above. In this case, however, 
the ion-pair formation will be considerable and will approach unity. 
If 1 6 is replaced by , the fraction of the electrolyte present as free 
ions, and if both 6 and the activity coefficient factor are assumed to be 

18 Fuoss and Kraus, J. Am. Chem. Soc., 55, 1019 (1933). 

* According to the simple calculations on page 156 the dielectric constant necessary 
for the solvent in which a uni-univalent electrolyte whose mean ionic diameter is 
6.4 X 10~* cm. should be dissolved in order that there may be no appreciable association 
is 2.79 X 10-/6.4 X Ifr* i.e., 42. 

"Fuoss and Kraus, J. Am. Chem. Soc., 55, 2387 (1933); Fuoss, ibid., 57, 2604 
(1935); Chem. Revs., 17, 227 (1935); for reviews, see Kraus, /. Franklin Inst., 225, 687 
(1938); Science, 90, 281 (1939). 



1.0 
lo? D 

FIG. 54. Association constant 
and dielectric constant (Fuoss 
and Kraus) 



160 FREE ENERGY AND ACTIVITY 

unity, equation (77) can be written as 

k 2 c, (80) 

where k is the approximate dissociation constant and c is the total elec- 
trolyte concentration. If in addition to ion-pairs (dual ions) there are 
present triple ions, viz., MAM+ and AMA", the following equilibria. 

MAM+ ^ MA + M+, 
AMA- ^ MA + A-, 

also exist. If the formation of MAM+ and AMA~ is due to electrical 
effects only, there will be an equal tendency for both these ions to form ; 
the mass action constant & 8 of the two equilibria may thus be expected 
to be the same. Hence, neglecting activity coefficients, 

, CMACM+ 



CMAM+ CAMA" 

.-.-^--J-. (82) 

CMAM+ CAMA- 

The triple ions should consequently be formed in the same ratio as that 
in which the simple ions are present in the solution. If ot 3 is the fraction 
of the total electrolyte existing as either of the triple ions, e.g., MAM+, 
then CMAM+ is equal to <x 3 c. Since the amount of these ions will be small, 
CMA may be taken as approximately equal to the total concentration c, 
and CM* can be assumed to remain as ac. Substituting these results in 
equation (81), it follows that 

k, = -> (83) 

s 

and since k, by equation (80), is equal to 2 c, i.e., a is Vfc/c, it is found that 

Vfo 
3 = -r-- (84) 

A/3 

Although dual ions have no conducting power, since they are elec- 
trically neutral, triple ions are able to carry current and contribute to 
the conductance of the solution. If A is the sum of the equivalent con- 
ductances of the simple ions at infinite dilution, and Xo is the sum of the 
values for the two kinds of triple ions, then since the latter are formed 
in the same ratio as the simple ions, it follows that the observed equiva- 
lent conductance is given by 

A = AO + 0(3X0, 

interionic effects being neglected. Substituting Vfc/c for a, and Vfcc/fc* 



TRIPLE IONS AND CONDUCTANCE MINIMA 

for as, it is seen that 



A A /*.* ^ 

A = A \h + A -r~ 

* C A/3 



.'. AVc = 



, 
+ 



c. 



161 



(85) 



(86) 



If AVc is plotted against c for media of low dielectric constant, in which 
triple ions can form to an appreciable extent, the result should be a 
straight line; this expectation has been cpnfirmed by experiment, as 



1.66 



1.55 



0.25 



0.50 
C X 10* 



0.75 



Fia. 55. Test of triple-ion theory (Fuoss and Kraus) 

shown by the points in Fig. 55 which are for tetrabutylammonium 
picrate in anisole. The deviation from the straight line becomes evident 
only at high concentrations. 

Triple Ions and Conductance Minima. Since equation (85) is of the 
form 



A-A + 

Vc 



(87) 



where A and B are constants, it is evident that the first term on the right- 
hand side decreases and the second term increases as the concentration 
is increased; it is possible, therefore, for a minimum in the equivalent 
conductance to occur, as has been found experimentally (p. 52). The 
physical significance of this result is that with increasing concentration 
the single ions are replaced by electrically neutral ion-pairs, and so the 
conductance falls; at still higher concentrations, however, the ion-pairs 
are replaced by triple ions having a relatively high conducting power, 
and so the equivalent conductance of the solution tends to increase. 



162 FREE ENERGY AND ACTIVITY 

The condition for the conductance minimum is found by differentiating 
equation (87) with respect to c and setting the result equal to zero; this 
procedure gives 

_ A 

min. n 

= * (88) 

AO 

By substituting this value in equation (85), and utilizing the relation- 
ships given above for a and 3 , it is found that 

A min . = 2(A a) min . = 2(Xoa 3 ) mi n.. (89) 

It is seen from equation (88) that the concentration for the minimum 
conductance is proportional to fc 3 , and so is inversely proportional to the 
stability of the triple ions. The minimum occurs when the conductance 
due to these ions, i.e., Xoa, is equal to that due to the single ions, 
i.e., A a. 

By means of a treatment analogous to that described above for cal- 
culating the association constant for the formation of ion-pairs, it is 
possible to derive an expression for k^ 1 which is analogous to equation 
(79) ; 18 the result may be put in the form 

'* (90) 



where 7(6, r) is a function of 6, which has the same significance as before, 
and of the distance r between the ions. In the region of the minimum 
conductance, the value of 7(6, r)/6 3 does not change appreciably, and 
equation (90) can be written as 



where A is a constant and D is the dielectric constant of the medium; 
the dissociation constant of the triple ions (& 3 ) is thus proportional to Z) 3 . 
Since the concentration c rn i n . at which the minimum equivalent conduct- 
ance is observed is proportional to & 3 , it follows that 

D 3 

- = constant; (91) 

Cmin. 

this is the rule derived empirically by Walden (p. 53). 19 

The fact that the concentration at which the conductance minimum 
occurs decreases with decreasing dielectric constant of the solvent is 
shown by the results in Fig. 21 (p. 53). In media of very low dielectric 

M Fuoss and Kraus, J. Am. Chem. Soc., 55, 2387 (1933). 

19 See also, Gross and Halpern, J. Chem. Phys., 2, 188 (1934); Fuoss and Kraus, 
ibid., 2, 386 (1934). 



EQUILIBRIA IN ELECTROLYTES 163 

constant, however, the minimum does not appear, but the conductance 
curves show inflections; these are attributed to mutual interactions be- 
tween two dipoles, i.e. , ion-pairs, as a result of which quadripoles are formed. 
The consequence of this is that the normal increase of conductance 
beyond the minimum, due to the formation of triple ions, is inhibited to 
so|me extent. If the dielectric constant of the solvent exceeds a certain 
value, depending on the mean diameter and valence of the ions, there is 
no appreciable formation of triple ions at any concentration, and hence 
there can be no conductance minimum. 

Equilibria in Electrolytes: The Dissociation Constant. When any 
electrolyte MA is dissolved in a suitable solvent, it yields M+ and A~ 
ions in solution to a greater or lesser extent depending on the nature of 
MA; even if ionization is complete, as is the case with simple salts in 
aqueous solution, there may still be a tendency for ion-pairs to form in 
relatively concentrated solution, so that dissociation is not necessarily 
complete. In general, therefore, there will be set up the equilibrium 

MA ^ M+ + A-, 

where M + and A~ represent the free ions and MA is the undissociated 
portion of the electrolyte which includes both un-ionized molecules and 
ion-pairs. Application of the law of mass action, in the form of equation 
(20), to this equilibrium gives 

(92) 



CtMA 

where the a terms are the activities of the indicated species; the equi- 
librium constant K is called the dissociation constant of the electrolyte. 
The term " ionization constant" is also frequently employed in the litera- 
ture of electrochemistry, but since the equilibrium is between free ions 
and undissociated molecules, the expression " dissociation constant" is 
preferred. Writing the activity terms in equation (92) as the product 
of the concentration and the activity coefficient, it becomes 



CMA JMA 



Further, if a is the degree of dissociation of the electrolyte (cf. p. 96) 
whose total concentration is c moles per liter, then CM+ and CA~ are each 
equal to ac, and CMA is equal to c(l a); it follows, therefore, that 



K = .L-. (94) 

1 a /MA 

If the solution is sufficiently dilute, the activity coefficients are approxi- 
mately unity, and so equation (94) reduces under these conditions to 



164 FREE ENERGY AND ACTIVITY 

which is the form of the so-called dilution law as originally deduced by 
Ostwald (1888). It will be noted that in the approximate equation (95) 
the symbol k has been used; this quantity is often called the "classical 
dissociation constant," but as it cannot be a true constant it is preferable 
to refer to it as the "classical dissociation function" or, in brief, as the 
"dissociation function." 

The relation between the function k and the true or " thermodynamic " 
dissociation constant K is obtained by combining equations (94) and 
(95); thus 

(96) 



JMA 

Provided the ionic strength of the medium is not too high, the activity 
coefficient of the undissociated molecules never differs greatly from unity; 
hence, equation (96) may be written as 

K = fc(/ M +/A-). (97) 

If the solution is sufficiently dilute for the Debye-Hiickel limiting law 
to be applicable, it follows from equation (54), assuming the ions M+ 
and A" to be univalent, for simplicity, that 



log/ M + = log /A- = - Ac, (98) 

the ionic strength, Zc t Z| 2 , being equal to |[(ac X I 2 ) + (<*c X I 2 )], i.e., 
to ac. Upon taking logarithms of equation (97) and substituting the 
values of log/M+ and log /A- as given by (98), the result is 

log K = log k - 2A Vac. (99) 

The plot of the values of log fc, obtained at various concentrations, 
against Vac should thus give a straight line of slope 2 A ; for water at 
25 the value of A is 0.509 (Table XXXV) and so the slope of the line 
should be - 1.018. 

In order to test the reliability of equation (99) it is necessary to know 
the value of the degree of dissociation at various concentrations of the 
electrolyte MA; in his classical studies of dissociation constants Ostwald, 
following Arrhenius, assumed that a at a given concentration was equal 
to the conductance ratio A/A , where A is the equivalent conductance of 
the electrolyte at that concentration and A is the value at infinite dilu- 
tion. As already seen (p. 95), this is approximately true for weak elec- 
trolytes, but it is more correct, for electrolytes of all types, to define a 
as A/A 7 where A' is the conductance of 1 equiv. of free ions at the same 
ionic concentration as in the given solution. It follows therefore, by 
substituting this value of a in equation (95), that 



EQUILIBRIA IN ELECTROLYTES 



165 



Since A for various concentrations can be obtained from conductance 
data and the Onsager equation, by one of the methods described in 
Chap. Ill, it is possible to derive the dissociation function k for the 
corresponding concentrations. The results obtained for acetic acid in 
agueous solution at 25 are given in Table XXXVIII, 20 and the values of 

TABLE XXXVIII. DISSOCIATION CONSTANT OF ACETIC ACID AT 25 



cX10 


A 


A' 


a 


*X10 


XX10 


0.028014 


210.38 


390.13 


0.5393 


1.768 


1.752 


0.11135 


127.75 


389.79 


0.3277 


1.779 


1.754 


0.21844 


96.49 


389.60 


0.2477 


1.781 


1.751 


1.02831 


48.15 


389.05 


0.1238 


1.797 


1.751 


2.41400 


32.22 


388.63 


0.08290 


1.809 


1.750 


5.91153 


20.96 


388.10 


0.05401 


1.823 


1.749 


9.8421 


16.37 


387.72 


0.04222 


1.832 


1.747 


20.000 


11.57 


387.16 


0.02987 


1.840 


1.737 


52.303 


7.202 


386.18 


0.01865 


1.854 


1.722 


119.447 


4.760 


385.18 


0.01236 


1.847 


1.688 


230.785 


3.392 


384.26 


0.008827 


1.814 


1.632 



log k are plotted against Vac in Fig. 56; the dotted line has the theoretical 
slope required by equation (99). It is clear that in the more dilute 
solutions the experimental results are in excellent agreement with theory, 



4.726 



4.786 



a* 

I 



4.746 



4.766 




0.01 



0.02 



0.03 



0.04 



FIQ. 56. Dissociation constant of acetic acid (Maclnnes and Shedlovsky) 

but at higher concentrations deviations become evident. The same con- 
clusion is reached from an examination of the last column in Table 

* Maclnnes and Shedlovsky, J. Am. Chem. Soc., 54, 1429 (1932); Maclnnes, 
J. Franklin Inst. t 225, 661 (1938). 



166 FREE ENERGY AND ACTIVITY 

XXXVIII which gives the results for K derived from equation (99) 
using the theoretical value of A, i.e., 0.51. The first figures are seen to 
be virtually constant, as is to be expected, the mean value of K being 
1.752 X 10"*. At infinite dilution the activity coefficient factor is unity 
and so the extrapolation of the dissociation functions k to infinite dilution 
should give the true dissociation constant K] the necessary extrapolation 
is carried out in Fig. 56, from which it is seen that the limiting value of 
log k is 4.7564, so that K is 1.752 X 10~ 6 , as given above. 

Similar results to those described for acetic acid in aqueous solution 
have been recorded for other weak acids in aqueous solution, and also 
for several^ acids in methyl alcohol. 21 In each case the plot of log k 
against Vac was found to be a straight line for dilute solutions, the slope 
being in excellent agreement with that required by the Debye-Huckel 
limiting law. The deviations observed with relatively concentrated 
solutions, such as those shown in Fig. 56, are partly due to the failure 
of the limiting law to apply under these conditions, and partly to the 
change in the nature, e.g., dielectric constant, of the medium resulting 
from the presence of appreciable amounts of an organic acid. 

Strong Electrolytes. The arguments presented above are readily 
applicable to weak electrolytes because the total concentration can be 
quite appreciable before the ionic strength becomes large enough for the 
Debye-Huckel limiting law to fail; for example, the results in Table 
XXXVIII extend up to a concentration of 0.2 N, but the ionic strength 
is then about 0.04. With relatively strong electrolytes, however, the 
procedure can be used only for very dilute solutions. In these circum- 
stances it is preferable to return to equation (97), which should hold for 
all types of electrolytes of the general formula MA, and to employ activity 
coefficients obtained by direct experimental measurement, instead of the 
values calculated from the Debye-Huckel equations. The product /M+/A~ 
in equation (97) may be replaced by the square of the mean activity 
coefficient of the electrolyte, i.e., by /, in accordance with the definition 
of equation (30); it follows, therefore, that equation (100) may be modi- 
fied so as to give 

AV 



The accuracy of this equation has been confirmed for a number of salts 
generally regarded as strong electrolytes, as the data in Table XXXIX 
serve to show. 22 It is evident from these results that the law of mass 
action holds for strong, as well as for weak electrolytes, provided it is 

"Maclnnes and Shedlovsky, /. Am. Chem. Soc., 57, 1705 (1935); Saxton et al t 
ibid., 55, 3638 (1933); 56, 1918 (1934); 59, 1048 (1937); Brock man and Kilpatrick, ibid., 
56, 1483 (1934); Martin and Tartar, ibid., 59, 2672 (1937); Belcher, ibid., 60, 2744 
(1938); see also, Maclnnes, "The Principles of Electrochemistry," 1939, Chap. 19. 

" Davies et al., Trans. Faraday Soc., 23, 351 (1927); 26, 592 (1930); 27, 621 (1931); 
28, 609 (1932); "The Conductivity of Solutions," 1933, Chap. IX. 



INTERMEDIATE AND WEAK ELECTROLYTES 167 

applied in the correct manner. The view expressed at one time that the 
law of mass action was not applicable to strong electrolytes was partly 
due to the employment of the Arrhenius method of calculating the degree 
of dissociation, and partly to the failure to make allowance for deviations 
from ideal behavior. 

TABLE XXXIX. APPLICATION OP LAW OF MASS ACTION TO STRONG ELECTROLYTES 

Salt c A/A' f K 

KNO, 0.01 0.994 0.916 1.40 

0.02 0.989 0.878 1.38 

0.05 0.975 0.806 1.32 

0.10 0.961 0.732 1.37 

AgNO, 0.01 0.993 0.902 1.10 

0.02 0.989 0.857 1.31 

0.05 0.973 0.783 1.12 

0.10 0.957 0.723 1.23 

0.50 0.883 0.526 1.18 

Intermediate and Weak Electrolytes. The calculation of the degree 
of dissociation by the methods given in Chap. Ill presuppose the availa- 
bility of suitable conductance data for electrolytes which are virtually 
completely dissociated at the appropriate concentrations. There is gen- 
erally no difficulty concerning this matter if the solvent is water, but for 
non-aqueous media, especially those of low dielectric constant, the pro- 
portion of undissociated molecules may be quite large even at small 
concentrations, and no direct method is available whereby the quantity 
A' can be evaluated from conductance data. For solvents of this type 
the following method, which can be used for any systems behaving as 
weak or intermediate electrolytes, may be employed. 23 The Onsager 
equation for incompletely dissociated electrolytes can be written (cf. 
p. 95) as 

A' = Ao - (A + A ) 



If a variable x is defined by 



xm (A + B ^ (1Q3) 



equation (102) becomes 



28 Fuoss and Kraus, J. Am. Chem. Soc., 55, 476 (1933); Fuoss, ibid., 57, 488 (1935); 
TVan*. Faraday Soc., 32, 594 (1936). 



168 FREE ENERGY AND ACTIVITY 

where F (x) is a function of x represented by the continued fraction 

F(x) = 1 - x(l - x(l - x(l )-)-)- 

= | cos 2 1 cos" 1 (- fsV3). 

Values of this function have been worked out and tabulated for values 
of x from zero to 0.209 in order to facilitate the calculations described 
below. 

Taking the activity coefficient of the undissociated molecules, as 
usual, to be equal to unity, and replacing /M+/A- by /, where f is the 
mean activity coefficient, equation (94) becomes 

(105) 
and if the value of a given by equation (104) is inserted, the result is 



which on multiplying out and rearranging gives 

*M = J_.^I + !. (106) 

A KA.Q F(x) AO 

It is seen from equation (106) that the plot of F(x)/\ against \cf/F(x) 
should be a straight line, the slope being equal to 1/KA.l and the inter- 
cept, for infinite dilution, giving 1/Ao. In this manner it should be 
possible to determine both the dissociation constant K of the electrolyte 
and the equivalent conductance at infinite dilution (A ) in one operation. 
In order to obtain the requisite plot, an approximate estimate of A 
is first made by extrapolating the experimental data of A against Vc, and 
from this a tentative result for x is derived by means of equation (103), 
since the Onsager constants A and B are presumably known (see Table 
XXIII). In this way a preliminary value of F(x) is obtained which is 
employed in equation (106) ; the activity coefficients required are calcu- 
lated from the Debye-Hiickel limiting law equation (98), using the value 
of a given by equation (104) from the rough estimates of A and F(x). 
The results are then plotted as required by equation (106), and the 
datum for AO so obtained may be employed to calculate F(x) and a more 
accurately; the plot whereby A and K are obtained may now be re- 
peated. The final results are apparently not greatly affected by a small 
error in the provisional value of A and so it is not often necessary to 
repeat the calculations. With A known accurately, it is possible to 
determine the degree of dissociation at any concentration, if required 
by means of equations (103) and (104), and the tabulated values of F(x). 



SOLUBILITY EQUILIBRIA 



169 



The work of Fuoss and Kraus and their collaborators and of others 
has shown that equation (106) is obeyed in a satisfactory manner by a 
number of electrolytes, both salts and acids, in solvents of low dielectric 
constant. 24 The results of plotting the values of F(x)/A against Acf 2 /F(x) 
for solutions of tetramethyl- and tetrabutyl-ammonium picrates in 
ethylene chloride are shown in Fig. 57; the intercepts are 0.013549 and 
0.17421, and the slopes of the straight lines are 5.638 and 1.3337, re- 



0.034 




i.o 



4.0X10' 



20 3.0 

Acf*/F(x) 
Fia. 57. Salts in media of low dielectric constant (Fuoss and Kraus) 

spectively. It follows, therefore, that for tetramethylammonium picrate 
K is 0.3256 X 10~ 4 and A is 73.81 ohms" 1 cm. 2 , whereas the correspond- 
ing values for tetrabutylammonium picrate are 2.276 X 10~ 4 and 57.40 
ohms" 1 cm. 2 , respectively. 

Solubility Equilibria : The Solubility Product Principle. It was seen 
on page 133 that the chemical potential of a solid is constant at a definite 
temperature and pressure; consequently, when a solution is saturated 
with a given salt M,, + A,_ the chemical potential of the latter in the solu- 
tion must also be constant, since the chemical potential of any substance 
present in two phases at equilibrium must be the same in each phase. 
It is immaterial whether this conclusion is applied to the undissociated 
molecules of the salt or to the ions, for the chemical potential is given by 

4 Kraus, Fuoss et al, Trans. Faraday Soc., 31, 749 (1935); 32, 594 (1936): J. Am. 
Chem. Soc., 58, 255 (1936); 61, 294 (1939); 62, 506, 2237 (1940); Owen and Waters, 
ibid., 60, 2371 (1938); see also, Maclnnes, "The Principles of Electrochemistry," 1939, 
Chap. 19. 



170 FREE ENERGY AND ACTIVITY 

either side of equation (26) ; thus, taking the left-hand side, it follows that 
M + + RT In a+) + v-(&- + RT In a_) = constant, 
v+ In a+ + v- In a_ = constant, 

- = constant (/.), (107) 



at a specified temperature and pressure. The constant K, as defined by 
equation (107) is the activity solubility product, and this equation ex- 
presses the solubility product principle, first enunciated in a less exact 
manner by Nernst (1889). If the activity of an ion is written as the 
product of its concentration, in moles (g.-ions) per liter, and the corre- 
sponding activity coefficient, equation (107) becomes 



=#., (108) 

and introducing the definition of the mean activity coefficient of the 
electrolyte M r+ A^_, it follows that 

t X A = K, t (109) 



where v is equal to v+ + i>_. If the ionic strength of the medium is low, 
the activity coefficient is approximately unity and equation (109) reduces 
to the approximate form 

= ft., (110) 



in which the solubility product principle is frequently employed. 

The significance of the solubility product principle is that when a 
solution is saturated with a given salt the product of the activities, or 
approximately the concentrations, of its constituent ions must be con- 
stant, irrespective of the nature of the other electrolytes present in the 
solution. If the latter contains an excess of one or other of the ions of 
the saturating salt, this must be taken into consideration in the activity 
product. Consider, for example, a solution saturated with silver chloride : 
then according to the solubility product principle, 



(111) 
or, approximately, 

CAg+Ccr = fc.<Agci). (112) 

If the solution which is being saturated with silver chloride already con- 
tains one of the ions of this salt, e.g., the chloride ion, then the term 
Ocr will represent the total activity of the chloride ion in the solution; 
since this is greater than that in a solution containing no excess of chloride 
ion, the value of a Ag + required according to equation (111) will be less in 
the former case. In its simplest terms, based on equation (112), the 
conclusion is that the silver ion concentration in a saturated solution of 
silver chloride containing an excess of chloride ions, e.g., due to the 
presence of potassium chloride in the solution, will be less than in a 
solution in pure water. Since the silver chloride in solution may be 



SOLUBILITY IN THE PRESENCE OF A COMMON ION 171 

regarded as completely ionized, the silver ion concentration is a measure 
of the solubility of the salt; it follows, therefore, that silver chloride is 
less soluble in the presence of excess of chloride ions than in pure water. 
In general, if there is no formation of complex ions to disturb the equi- 
librium (cf. p. 172), the solubility of any salt is less in a solution con- 
taining a common ion than in water alone; this fact finds frequent 
application in analytical chemistry. 

Solubility in the Presence of a Common Ion. If So is the solubility 
of any sparingly soluble salt M, + A,_ in moles per liter in pure water, then 
if the solution is sufficiently dilute for dissociation to be complete, c+ is 
equal to v+So and c_ is equal to v-8o] hence according to equation (109) 

= (j>>l-),S&A. (113) 

In the simple case of a uni-univalent sparingly soluble salt, this be- 
comes 

K. = 52/2=. (114) 

These equations relate the solubility product to the solubility in pure 
water and the activity coefficient in the saturated solution; for practical 
purposes it is convenient to take the activity coefficient to be approxi- 
mately unity, since the solutions are very dilute, so that equation (114) 
can be written 

if Q 2 

n/t OQ 

For a uni-univalent salt the saturation solubility in pure water is thus 
equal to the square-root of its solubility product; alternatively, it may be 
stated that the solubility product is equal to the square of the solubility 
in water. The solubility of silver chloride in water at 25 is 1.30 X 10"" 6 
mole per liter; the solubility product is consequently 1.69 X 10~~ 10 . 

Suppose the addition of x moles per liter of a completely dissociated 
salt containing a common ion, e.g., the anion, reduces the solubility of 
the sparingly soluble salt from So to S; for simplicity all the ions present 
may be assumed to be univalent. The concentrations of cations in the 
solution, resulting from the complete dissociation of the sparingly soluble 
salt, is S, while that of the anions is S + x; it follows, therefore, by the 
approximate solubility product principle that 

S(S + x) = k. = Si 

.-. S = - \x + Viz 2 + Si (115) 

Using this equation, or forms modified to allow for the valences of the 
ions which may differ from unity, it is possible to calculate the solubility 
(S) of a sparingly soluble salt in the presence of a known amount (x) of 
a common ion, provided the solubility in pure water (So) is known. An 
illustration of the application of equation (115) is provided by the results 
in Table XL for the solubility of silver nitrite in the presence of silver 



172 FREE ENERGY AND ACTIVITY 

nitrate (I), on the one hand, and of potassium nitrite (II) on the other 
hand; the calculated values are given in the last column. 25 The agree- 
ment between the observed and calculated results in these dilute solu- 

TABLE XL. SOLUBILITY OF SILVER NITRITE IN THE PRESENCE OF COMMON ION 

X S S 

moles/liter I II Calculated 

0.000 0.0269 0.0269 (0.0269 - So) 

0.00258 0.0260 0.0259 0.0259 

0.00588 0.0244 0.0249 0.0247 

0.01177 0.0224 0.0232 0.0227 

tions is seen to be good, perhaps better than would be expected in view 
of the neglect of activity coefficients ; in the presence of larger amounts of 
added electrolytes, however, deviations do occur. Much experimental 
work has been carried out with the object of verifying the solubility 
product principle in its approximate form, and the general conclusion 
reached is that it is satisfactory provided the total concentration of the 
solution is small; at higher concentrations discrepancies are observed, 
especially if ions of high valence are present. It was found, for example, 
that in the presence of lanthanum nitrate the solubility of the iodate 
decreases at first, in agreement with expectation, but as the concentration 
of the former salt is increased, the solubility of the lanthanum iodate, 
instead of decreasing steadily, passes through a minimum and then 
increases. Such deviations from the expected behavior are, of course, 
due to neglect of the activity coefficients in the application of the simple 
solubility product principle; the effect of this neglect becomes more 
evident with increasing concentration, especially if the solution contains 
ions of high valence. It is evident from the Debye-Huckel limiting law 
equation that the departure of the activity coefficients from unity is most 
marked with ions of high valence because the square of the valence 
appears not only in the factor preceding the square-root of the ionic 
strength but also in the ionic strength itself. The more exact treatment 
of solubility, taking the activity coefficients into consideration, is given 
later. 

Formation of Complex Ions. In certain cases the solubility of a 
sparingly soluble salt is greatly increased, instead of being decreased, 
by the addition of a common ion ; a familiar illustration of this behavior is 
provided by the high solubility of silver cyanide in a solution of cyanide 
ions. Similarly, mercuric iodide is soluble in the presence of excess of 
iodide ions and aluminum hydroxide dissolves in solutions of alkali 
hydroxides. In cases of this kind it is readily shown by transference 
measurements that the silver, mercury or other cation is actually present 
in the solution in the form of a complex ion. The solubility of a sparingly 
soluble salt can be increased by the addition of any substance, whether it 

Creighton and Ward, J. Am. Chem. Soc., 37, 2333 (1915). 



DETERMINATION OF INSTABILITY CONSTANT 173 

contains a common ion or not, which is able to remove the simple ions 
in the form of complex ions. For example, if either cyanide ion or am- 
monia is added to a slightly soluble silver compound, such as silver 
chloride, the silver ions are converted into the complex ions Ag(CN)i~ or 
Ag(NH 8 )i~, respectively. In either case the concentration of free silver 
ions is reduced and the product of the concentrations (activities) of the 
silver and chloride ions falls below the solubility product value: more 
silver chloride dissolves, therefore, in order to restore the condition 
requisite for a saturated solution. If sufficient complex forming material 
is present the removal of the silver ions will continue until the whole of 
the silver chloride has dissolved. 

Although by far the largest proportion of the silver in a complex 
cyanide solution is present in the form of argentocyanide ions, Ag(CN)F, 
there is reason for believing that a small concentration of simple silver 
ions is also present ; the addition of hydrogen sulfide, for example, causes 
the precipitation of silver sulfide which has a very low solubility product. 
It is probable, therefore, that an equilibrium of the type 

Ag(CN) 2 - ^ Ag+ + 2CN- 

dxists between complex and free ions in an argentocyanide solution and 
similar equilibria are established in other instances. For the general 
case of a complex ion M fl A*, the equilibrium is 



(116) 



rA~, 
and application of the law of mass action gives 



or, using concentrations in place of activities, 

(117) 

The constant Ki (or k t ) is called the instability constant of the complex 
ion ; it is apparent that the greater its value the greater the tendency of the 
complex to dissociate into simple ions, and hence the smaller its stability. 
The reciprocal of the instability constant is sometimes encountered; 
it is referred to as the stability constant of the complex ion. 

Determination of Instability Constant. Two methods have been 
mainly used for determining the instability constants of complex ions; 
one involves the measurement of the E.M.F.'S of suitable cells, which will 
be described in Chap. VII, and the other depends on solubility studies. 
The latter may be illustrated by reference to the silver-ammonia (argent- 
ammine) complex ion. 26 If the formula of the complex is Ag m (NHs);}~, the 

"See also, Edmonds and Birnbaum, J. Am. Chem. Soc. t 62, 2367 (1940); Lanford 
and Kiehl, ibid., 63, 667 (1941). 



174 FREE ENERGY AND ACTIVITY 

instability constant, using concentrations as in equation (117), is given 

by 

+ 

(118) 



ex 



where, for simplicity of representation, the concentration of the complex 
ion is given by ex- If a solution of ammonia is saturated with silver 
chloride, then by the solubility product principle, CAg+Ccr gives the 
solubility product A;., and hence CA B + is equal to fc./ccr ; for such a system 
equation (118) becomes 



The concentration c of the silver salt in the ammonia solution may be 
regarded as consisting entirely of the complex ion, since the normal 
solubility of silver chloride is very small, so that Cx is virtually equal to c ; 
the concentration of the chloride ion may be taken as me because of the 
reaction 

m AgCl + n NH 3 = Ag m (NH 3 )+ + m C1-, 

and so equation (119) may be written as 

*,- 

m ~v - 

. CNH, 
. . -^j = constant. 

By means of this equation it is possible to evaluate n/(m +1) from a 
number of measurements of the solubility (c) of silver chloride in solu- 
tions containing various concentrations (CNH,) of ammonia. 

In order to derive m it is necessary to determine the solubility of silver 
chloride in ammonia in the presence of an excess of chloride ions ; equa- 
tion (119) then takes the form 



If in a series of experiments the concentration of ammonia (CNH,) is 
kept constant, while the amount of excess chloride (ccr) is varied, equa- 
tion (120) becomes 

Cere = constant, 

so that if the solubility c is measured, the value of m may be determined. 
Alternatively, solubility measurements may be made in the presence of 
excess of silver ions; in this case ccr is set equal to &,/CA g + in equation 
(119), and the subsequent treatment is similar to that given above.* 

* For data obtained in an actual experiment, see Problem 11. 



ACTIVITY COEFFICIENTS FROM SOLUBILITY MEASUREMENTS 175 



Activity Coefficients from Solubility Measurements. The activity 
coefficient of a sparingly soluble salt can be determined in the presence 
of other electrolytes by making use of the solubility product principle. 27 
In addition to the equations already given, this principle may be stated 
in still another form by introducing the definition of the mean ionic con- 
centration, i.e., c, which is equal to c+c!r, into equation (109); this 
equation then becomes 

K., (121) 



(122) 



.-. f 



The mean activity coefficient of a sparingly soluble salt in any solution 
could thus be evaluated provided the solubility product (K 9 ) and the 
mean concentration of the 
ions of the salt in the given 
solution were known. In or- 
der to calculate K s the value of 
c-t is determined in solutions 
of different ionic strengths and 
the results are then extrapo- 
lated to infinite dilution ; un- 
der the latter conditions f 
is, of course, unity and hence 
K\ l9 is equal to the extrapo- 
lated value of c. 

The method of calculation 
will be described with refer- 
ence to thallous chloride, the 

FIG. 58. Extrapolation of solubility data for 
thallous chloride 



0.018 



0.016 



0,014 



<u 



0.2 



04 



solubility of which has been 
measured in the presence of 
various amounts of other elec- 
trolytes, with and without an ion in common with the saturating salt. 
By plotting the values of c for the thallium and chloride ions in solutions 
of different ionic strengths and extrapolating to zero, it is found that 
KU; which in this case is equal to V^, is 0.01428 at 25 (Fig. 58). 
It follows, therefore, from equation (122) that the mean activity coeffi- 
cient of thallous chloride in any saturated solution is given by 

0.01428 

c 
If the added electrolyte present contains neither thallous nor chloride 

"Lewis and Randall, J. Am. Chem. Soc. t 43, 1112 (1921); see also, Blagden and 
Davies, J. Chem. Soc. t 949 (1930); Davies, iWd., 2410, 2421 (1930); MacDougall and 
Hoffman, J. Phys. Chem. t 40, 317 (1936); Pearce and Oelke, iWd., 42, 95 (1938); 
Kolthoff and Lingane, ibid., 42, 133 (1938). 



176 FREE ENERGY AND ACTIVITY 

ions, the mean ionic concentration is merely the same as the molar con- 
centration of the thallous chloride in the saturated solution, for then 
CTI+ and c c r are both equal to the concentration of the salt. When 
another thallous salt or a chloride is present, however, appropriate 
allowance must be made for the ions introduced in this manner. For 
example, in a solution containing 0.025 mole of thallous sulfate per liter, 
the saturation solubility of thallous chloride is 0.00677 mole per liter at 
25; assuming both thallium salts to be completely dissociated at this 
low concentration, the total concentration of thallous ions is 2 X 0.025 
+ 0.00677, i.e., 0.05677 g.-ion per liter. The chloride ion concentration 
is 0.00677, and so the mean ionic concentration is (0.05677 X 0.00677)*, 
i.e., 0.01961 ; the mean activity coefficient is then 0.01428/0.01961, that is 
0.728. The ionic strength of the solution is 

tf = K(CTI* X I 2 ) + (ccr X I 2 ) + (c s o 4 -- X 2 2 )] 
= K0.05677 + 0.00677 + 0.10) 
= 0.0817, 

so that the mean activity coefficient of a saturated solution of thallous 
chloride in the presence of thallous sulfate at a total ionic strength of 
0.0817 is 0.728 at 25. 

The activity coefficients of thallous chloride at 25, obtained in the 
manner described above, in the presence of a number of salts are given in 
Table XLI; the data are recorded for solutions of various (total) ionic 

TABLE XLI. ACTIVITY COEFFICIENTS OF THALLOUS CHLORIDE IN THE PRESENCE OF 
VARIOUS ELECTROLYTES AT 25 

Added Electrolyte 

y KNO, KC1 HC1 TWO, Tl^SO* 

0.02 0.872 0.871 0.871 0.869 0.885 

0.05 0.809 0.797 0.798 0.784 0.726 

0.10 0.742 0.715 0.718 0.686 0.643 

0.20 0.676 0.613 0.630 0.546 

strengths. It is seen that at low ionic strengths the activity coefficient 
of the thallous chloride at a given ionic strength is almost independent of 
the nature of the added electrolyte; it has been claimed that if allowance 
is made for incomplete dissociation of the latter this independence per- 
sists to much higher concentrations. 

Solubility and the Debye-Hiickel Theory. The activity coefficients 
determined by the solubility method apply only to saturated solutions 
of the given salt in media of different ionic strengths; although their 
value is therefore limited, in many respects, they are of considerable 
interest as providing a means of testing the validity of the Debye-Htickel 
theory of electrolytes. It will be seen from equation (113), if the saturat- 
ing salt can be assumed to be completely dissociated, that the product 
Sf, where S is the solubility of the given salt in a solution not containing 
an ion in common with it, must be constant. It follows, therefore, that 



SOLUBILITY AND THE DEBYE-HttCKEL THEORY 177 

if S is the solubility of the salt in pure water and S the value in the 
presence of another electrolyte which has no ion in common with the 
salt, and / and / are the corresponding mean activity coefficients, then 

__ 
/ So' 

a 



Introducing the values of / and/ , as given by the Debye-Hiickel limiting 
law equation (54), it follows that 

logf--A+*-Ofi- V^), (123) 

OQ 

where yo and y are the ionic strengths of the solutions containing the 
sparingly soluble salt only and that to which other electrolytes have been 
added, respectively. Since vo is a constant for a given saturating salt, 
it follows that the plot of log S/S Q against Vji should be a straight line of 
slope AZ+Z-, where z+ and z_ are the valences of the two ions of the 
sparingly soluble substance. The constant A for water at 25 is 0.509, 
and so the linear slope in aqueous solutions should be 0.509 z+z_. 

For the purpose of verifying the conclusions derived from the Debye- 
Hiickel theory it is necessary to employ salts which are sufficiently soluble 
for their concentrations to be determined with accuracy, but not so 
soluble that the resulting solutions are too concentrated for the limiting 
law for activity coefficients to be applicable. A number of iodates, e.g., 
silver, thallous and barium iodates, and especially certain complex cobalt- 
ammines have been found to be particularly useful in this connection. 
The results, in general, are in very good agreement with the requirements 
of equation (123). The solubility measurements with the following 
four cobaltammines of different valence types, in the presence of such 
salts as sodium chloride, potassium nitrate, magnesium sulfate, barium 

Valence Theoretical 
Salt Type Slope 

I. [Co(NH,) 4 (NO,)(CNS)][Co(NH,) 2 (NO s ) 8 (C,04)] 1 : 1 0.509 

II. [Co(NHi) 4 (CiO 4 )]&Oi 1 : 2 1.018 

III. [Co(NH3)a][Co(NH 3 ) 2 (N0 2 MC 8 4 )]3 3 : 1 1.527 

IV. [Co(NH,)][Fe(CN) 6 ] 3 : 3 4.581 

chloride and potassium cobalticyanide, are of particular interest. 28 
The values of log S/So are plotted against the square-root of the ionic 
strength in Fig. 59 ; the experimental data are shown by the points and 
the theoretical slopes are indicated by the full lines in each case. In 
certain cases the agreement with theory is not as good as depicted in 

"Br0nsted and LaMer, J. Am. Chem. Soc., 46, 555 (1924); LaMer, King and 
Mason, ibid., 49, 363 (1927). 



178 



FREE ENERGY AND ACTIVITY 



Fig. 59; this is particularly true if both the saturating salt and the 
added electrolyte are of high valence types. 29 The deviations are often 
due to incomplete dissociation, and also to the approximations made in 
the derivation of the Debye-Hiickel equations; as already seen, both 
these factors become of importance with ions of high valence. 



0.10 - 




0.02 



0.04 



0.06 



0.08 



0.10 



Fia. 59. Dependence of solubility on ionic strength (LaMer, et al.) 



The factor A in equation (123) is proportional to 1/(DT)*, as shown 
on page 150; hence, a further test of this equation is to determine the 
slope of the plot of log S/So against Vp from Solubility data at different 
temperatures and in media of different dielectric constants. Such 
measurements have been made in water at 75 (D = 63.7), in mixtures 
of water and ethyl alcohol (D = 33.8 to 78.6), in methyl alcohol (D = 30), 
in acetone (D = 21), and in ethylene chloride (D = 10.4). The results 
have been found in all cases to be in very fair agreement with the re- 
quirements of the Debye-Huckel limiting law; as may be expected, ap- 
preciable discrepancies occur when the saturating salt is of a high valence 
type, especially in the presence of added ions of high valence. 30 

"LaMer and Cook, J. Am. Chem. Soc., 51, 2622 (1929); LaMer and Goldman, 
ibid., 51, 2632 (1929); Neuman, ibid., 54, 2195 (1933). 

Baxter, J. Am. Chem. Soc., 48, 626 (1926); Williams, ibid., 51, 1112 (1929); 
Hansen and Williams, ibid., 52, 2759 (1930); Scholl, Hutchison and Chandlee, ibid., 
55, 3081 (1933); Seward, ibid., 56, 2610 (1934); see, however, Anhorn and Hunt, J. Phys. 
Chem., 45, 351 (1941). 



THERMAL PROPERTIES OF STRONG ELECTROLYTES 179 

Thermal Properties of Strong Electrolytes. According to equation 
(42) the free energy of an ionic solution may be expressed in the form 

G = Go + Gei. 
and application of the Gibbs-Helmholtz equation (cf. p. 194) gives 



where H is the heat content of a solution of an electrolyte at an appre- 
ciable concentration. At infinite dilution the quantity in the second 
brackets on the right-hand side is zero, since the electrical contribution 
to the free energy is then zero ; the heat content of the solution under these 
conditions is consequently equal to the quantity in the first brackets. 
It follows, therefore, that the increase of heat content accompanying the 
dilution of a solution of an electrolyte from a concentration c to infinite 
dilution, i.e., A/J^oi which is the corresponding integral heat of dilution, 

is given by 

/ zn . \ 

(125) 



Utilizing the value of G e i., equal to W Wo given by equation (41), 
and remembering that K involves T~*, it is found that 

(126) 



where V is the volume of the system ; dD/dT and dV/dT refer to constant 
pressure. Since the heat of dilution is generally recorded for a mole of 
electrolyte, it follows that N % is equal to Nv % where N is the Avogadro 
number and v v is the number of ions of the iih kind produced by the ion- 
ization of a molecule of electrolyte. The expression I,N l Zt in equation 
(126) may therefore be replaced by N^v v z1 y and the result is 



o = -20- SF.#(T, D, 7), (127) 

where /( T, D, F) is the function included in the parentheses in equa- 
tion (126). 

The concentration c, of any ionic species is equal to i>,, where c is 
the concentration of the electrolyte in moles per liter; hence, the ionic 
strength may be written in an alternative form, thus 



It follows, therefore, using equation (51) to define x, that 

l " z?)l V '/ (T ' D > *> 

*?) V^ f(T, D, V) cal. per mole. 



180 FREE ENERGY AND ACTIVITY 

For water at 25 this can be written as 

Affo.0 = 503(S?,*?) Vc/(r, Z>, F) cal. per mole. (128) 

The temperature coefficient of the dielectric constant of water is not 
known with great accuracy, but utilizing the best data to evaluate 
f(T 9 D 9 V), equation (128) becomes, approximately, 

Aff^o = - 175(Zna?)* Vc, 

and lor a um-univalent electrolyte at 25, i.e., z+ = z_ = 1, and 
v+ = v- = 1, 

o = 495 Vc cal. per mole. 



It is seen from these equations that there should be a negative in- 
crease of heat content when an electrolyte solution is diluted; in other 
tfords, the theory of interionic attraction requires that heat should be 
evolved when a solution of an electrolyte is diluted. 31 Further, the in- 
tegral heat of dilution should be proportional to the square-root of the 
concentration, the slope of the plot of AH o^o against Vc should be about 
500 for an aqueous solution of a uni-univalent electrolyte at 25. It 
must be emphasized that the foregoing treatment presupposes a dilute 
solution, and in fact the slope mentioned should be the limiting value 
which is approached at infinite dilution. Accurate measurements of 
integral heats of dilution are difficult to make, but the careful work of the 
most recent investigators has given results in general agreement with 
theoretical expectation. The integral heat of dilution is actually nega- 
tive for dilute solutions, but at appreciable concentrations it becomes 
positive, so that heat is then absorbed when the solution is diluted. 

The limiting slope of the plot of A//<^ against Vc has been found to be 
approximately 500 for a number of uni-univalent electrolytes; the 
larger the effective size of the ion in solution, the closer the agreement 
between experiment and the requirements of the interionic attraction 
theory. By making allowance for the effective ionic diameter, either 
by the Debye-Hlickel method or by utilizing the treatment of Gronwall, 
LaMer and Sandved, fairly good agreement is obtained at appreciable 
concentrations. 32 

According to equation (128) the limiting slope of the plot of A#e_ 
against Vc for different electrolytes should vary in proportion to the 
factor (Sjs-z?)*; the results obtained with a number of uni-bivalent and 



"Bjerrum, Z. physik. Chem., 119, 145 (1926); Gatty, Phil. Mag., 11, 1082 (1931); 
18, 46 (1934); Scatchard, J. Am. Chem. Soc., 53, 2037 (1931); Falkenhagen, "Electro- 
lytes" (translated by Bell), 1934. 

"For summaries, with references, see Lange and Robinson, Chem. Revs., 9, 89 
(1931); Falkenhagen, "Electrolytes," 1934; Wolfenden, Ann. Rep. Chem. Soc., 29, 29 
(1932); Bell, ibid., 31, 58 (1934); for more recent work, see Robinson et al., J. Am. Chem. 
Soc., 56, 2312, 2637 (1934); 63, 958 (1941); Sturtevant, ibid., 62, 2171 (1940). 



PROBLEMS 181 

bi-univalent electrolytes are in harmony with this requirement of theory. 
In spite of the general agreement, the experimental data for integral 
heats of dilution, especially in non-aqueous solutions, show some dis- 
crepancies from the behavior postulated by the interionic attraction 
theory. It should be noted, however, that heat of dilution measure- 
ments provide an exceptionally stringent test of the theory, and the 
influence of such factors as ionic size, incomplete dissociation and ion- 
solvent interaction will produce relatively larger effects than is the case 
with activity coefficients. 

PROBLEMS 

1. The density of a 0.1 N solution of KI in ethyl alcohol at 17 is 0.8014 
while that of the pure solvent is 0.7919; calculate the ratio of the three activity 
coefficients, /*, f c and / TO , in the solution. 

2. Compare the mortalities and ionic strengths of uni-uni, uni-bi, bi-bi and 
uni-tervalent electrolytes in solutions of molality m. 

3. Use the values of the Debye-Huckel constants A and B at 25, given in 
Table XXXV, to plot log f for a uni-univalent electrolyte against Vy for 
ionic strengths 0.01, 0.1, 0.5 and 1.0, assuming in turn that the mean distance 
of approach of the ions, a, is either zero, or 1, 2, 4 and 8 A. Investigate, quali- 
tatively, the effect of increasing the valence of the ions. 

4. Evaluate the Debye-Hiickel constants A and B for ethyl alcohol at 
25, taking the dielectric constant to be 24.3. 

5. Utilize the results of the preceding problem, together wHh the known 
values of A and B for water, to calculate approximate activity coefficients for 
uni-uni, uni-bi, and bi-bi valent electrolytes in water and in ethyl alcohol, at 
ionic strengths 0.1 and 0.01, at 25. The mean ionic diameter may be taken 
as 3A in each case. 

6. The following values for the mean activity coefficients of potassium 
chloride were obtained by Maclnnes and Shedlovsky [J. Am. Chem. Soc., 59, 
503 (1937)]: 

c f c f 

0.005 0.9274 0.04 0.8320 

0.01 0.9024 0.06 0.8070 

0.02 0.8702 0.08 0.7872 

0.03 0.8492 0.10 0.7718 

Plot Vji/log/ against Vtf and determine the value of a which is in satisfactory 
agreement with these data. 

7. Kolthoff and Lingane [V. Phys. Chem., 42, 133 (1938)] determined the 
solubility of silver iodate in water and in the presence of various concentrations 
of potassium nitrate at 25. The solubility in pure water is 1.771 X 10~ 4 
mole per liter, and the following results were obtained in potassium nitrate 
solutions: 

KNO, AglO, KNO, AglO, 

mole/liter mole/liter mole/liter mole/liter 

0.1301 X 10-* 1.823 X 10~ 4 1.410 X 10-* 1.999 X 10~ 4 

0.3252 1.870 7.050 2.301 

0.6503 1.914 19.98 2.665 



182 FREE ENERGY AND ACTIVITY 

Calculate the activity coefficients of the silver iodate in the various solutions; 
plot the values of - log / against Vy to see how far the results agree with the 
Debye-Huckel limiting law. Determine the mean ionic diameter required to 
account for the deviations from the law at appreciable concentrations. 

8. Utilize the results obtained from the data of Saxton and Waters, given 
in Problem 7 of Chap. Ill, together with the activity coefficients derived from 
the Debye-Hiickel limiting equation, to evaluate the dissociation constant of 
a-crotonic acid. 

9. Apply the method of Fuoss and Kraus, described on page 167, to evalu- 
ate Ao and K for hydrochloric acid in a dioxane-water mixture, containing 70 
per cent of the former, at 25, utilizing the conductance data obtained by 
Owen and Waters [/. Am. Chem. Soc., 60, 2371 (1938)]: 

VcXlO* 1.160 2.037 2.420 2.888 3.919 

A 89.14 85.20 83.26 81.45 77.20 ohms- 1 cm. 2 

The dielectric constant of the solvent is 17.7 and its viscosity is 0.0192 poise. 
The required values of the function F(x) will be found in the paper by Fuoss, 
J. Am. Chem. Soc., 57, 488 (1935). 

10. By means of the value of K obtained in the preceding problem, calcu- 
late the mean ionic diameter, a, of hydrochloric acid in the given solvent. 
For this purpose, use equation (79) and the tabulation of Q(b) given by Fuoss 
and Kraus, /. Am. Chem. Soc., 55, 1019 (1933). 

11. In order to determine the formula of the complex argentammine ion, 
Ag,(NHs)"!:, Bodlander and Fittig [Z. physik. Chem., 39, 597 (1902)] measured 
the solubility (S) of silver chloride in ammonia solution at various concen- 
trations (CNH,) with the following results: 

CNH, 0.1006 0.2084 0.2947 0.4881 

S X 10 s 5.164 11.37 15.88 25.58 

In the presence of various concentrations (CKCI) of potassium chloride, the 
solubility (S) of silver chloride in 0.75 molal ammonia was as follows: 

CKCI 0.0102 0.0255 0.0511 

S 0.0439 0.0387 0.0333 

What is the formula of the silver-ammonia ion? 



CHAPTER VI 
REVERSIBLE CELLS 

Chemical Cells and Concentration Cells. A voltaic cell, or element, 
as it is sometimes called, consists essentially of two electrodes combined 
in such a manner that when they are connected by a conducting material, 
e.g., a metallic wire, an electric current will flow. Each electrode, in 
general, involves^ an electronic and an electrolytic pppjiuctof in contact 
(cf . p. 6) ; a^T tEe surface of separation between these two phases there 
exists a potential difference, called the electrode potential. Ifthere are 
no other potential differences in the cell, the E.M.P. of the latter is taken 
as equal to the algebraic sum of the two electrode potentials, allowance 
being made for the direction of the potential difference when assessing 
its sign. During the operation of a voltaic cell a^chemical^reaction takes 
place at each electrode7and it is the^ energy of these^feaHibns which 
provides the electrical energy oT the cell. In many cells there is an 
overall chemical reaction, when all the processes occurring within it are 
taken into consideration; such a cell is referred to as a chemical cell, to 
distinguish it from a voltaic element in which there is no resultant chemi- 
cal change. In the latter type of cell the reaction occurring at one 
electrode is exactly reversed at the other; there may, nevertheless, be a 
net change of energy because of a difference in concentration of one or 
other of the reactants concerned at the two electrodes. Such a source 
of E.M.F. is called a concentration cell, and the electrical energy arises 
from the energy change accompanying the transfer of material from one 
concentration to another. 

Irreversible and Reversible Cells. Apart from the differences men- 
tioned above, voltaic cells may, broadly speaking, be divided into two 
categories depending on whether a chemical reaction takes place at either 
electrode even when there is no flow of current, or whether there is no 
reaction until the electrodes are joined together by a conductor and 
current flows. An illustration of the former type is the simple cell con- 
sisting of zinc and copper electrodes immersed in dilute sulfuric acid, viz., 

Zn | Dilute H 2 S0 4 1 Cu; 

the zinc electrode reacts with the acid spontaneously, even if there is no 
passage of current. Cells of this type are always irreversible in the 
thermodynamic sense; thermodynamic reversibility implies a state of 
equilibrium at every stage, and the occurrence of a spontaneous reaction 
at the electrodes shows that the system is not in equilibrium. 

183 



184 REVERSIBLE CELLS 

In the Daniell cell, however, which is made up of a zinc electrode 
in zinc sulfate solution and a copper electrode in copper sulfate solu- 
tion, viz., 

Zn | ZnSO 4 soln. CuSO 4 soln. | Cu, 

the two solutions being usually separated by means of a porous partition, 
neither metal is attacked until the electrodes are connected and a current 
is allowed to flow. The extent of the chemical reaction occurring in 
such a cell is proportional to the quantity of electricity passing, in accord- 
ance with the requirements of Faraday's laws. Many, although not 
necessarily all, cells in this second category are, however, thermodynam- 
ically reversible cells, and the test of reversibility is as follows. If the 
cell under consideration is connected to an external source of E.M.P. which 
is adjusted so as exactly to balance the E.M.F. of the cell, i.e., so that no 
current flows, there should be no chemical change in the cell. If the 
external E.M.F. is decreased by an infinitesimally small amount, current 
will flow from the cell and a chemical change, proportional in extent to 
the quantity of electricity passing, should take place. On the other 
hand, if the external E.M.F. is increased by a small amount, the current 
should pass in the opposite direction and the cell reaction should be 
exactly reversed. The Daniell cell, mentioned above, satisfies these re- 
quirements and it is consequently a reversible cell. It should be noted 
that voltaic cells can only be expected to behave reversibly when the 
currents passing are infinitesimally small and the system is always vir- 
tually in equilibrium. If large currents flow, concentration gradients 
arise on account of diffusion being relatively slow, and the cell can no 
longer be regarded as being in a state of equilibrium. 

Reversible Electrodes. The electrodes constituting a reversible cell 
must themselves be reversible, and several types of such electrodes are 
known. The simplest, sometimes called "electrodes of the first kind/' 
consist of a luetal in contact with a solution of its own ions, e.g., zinc in 
zinc sulfate solution. In this category may be included hydrogen, 
oxygen and halogen electrodes in contact with solutions of hydrogen, 
hydroxyl or the appropriate halide ions, respectively; since the electrode 
material in these latter cases is a non-conductor, and often gaseous, finely 
divided platinum, or other unattackable metal, which comes rapidly into 
equilibrium with the hydrogen, oxygen, etc., is employed for the purpose 
of making electrical contact. Electrodes of the first kind are reversible 
with respect to the ions of the electrode material, e.g., metal, hydrogen, 
oxygen or halogen; the reaction occurring if the electrode material is a 
metal M may be represented by 

M ^ M+ + , 

the direction of the reaction depending on the direction of the flow of 
current. If the electrode is that of a non-metal, the corresponding 
reactions are 

A + ^ A-. 



REVERSIBLE ELECTRODES 185 

With an oxygen electrode, which is theoretically reversible with respect 
to hydroxyl ions, the reaction may be written 

O 2 + H 2 + 2c ^ 2OH-. 

Electrodes of the "second kind" involve a metal, a sparingly soluble 
salt of this metal, and a solution of a soluble salt of the same anion; 
a familiar example is the silver-silver chloride electrode consisting of 
silver, solid silver chloride and a solution of a soluble chloride, such as 
hydrochloric acid, viz., 

Ag | AgCl(s) HCl soln. 

These electrodes behave as if they were reversible with respect to the 
common anion, e.g., the chloride ion in the above electrode. The elec- 
trode reaction involves the passage of the electrode metal into solution 
as ions and their combination with the anions of the electrolyte to form 
the insoluble salt, or the reverse of these stages; thus, for the silver-silver 
chloride electrode, 

Ag(s) ^ Ag+ + , 
followed by 



so that the net reaction, writing it for convenience in the reverse order, is 
AgCl(s) + e ^ Ag(s) + C1-. 

This is virtually equivalent to the reaction at a chlorine gas electrode, 
viz., 

C1 2 + ^ 2C1-, 

except that the silver chloride can be regarded as the source of the 
chlorine. In fact the silver-silver chloride electrode is thermodynam- 
ically equivalent to a chlorine electrode with the chlorine at a pressure 
equal to the dissociation pressure of the silver chloride, into silver and 
chlorine, at the experimental temperature. Electrodes of the second 
kind are of great value in electrochemistry because they permit the ready 
establishment of an electrode reversible with respect to anions, e.g., 
sulfate, oxalate, etc., which could not be obtained in a direct manner. 
Even where it is possible, theoretically, to set up the electrode directly, 
as in the case of the halogens, it is more convenient, and advantageous 
in other ways, to employ an electrode of the second kind. 

Occasionally electrodes of the " third kind" are encountered; l these 
consist of a metal, one of its insoluble salts, another insoluble salt of the 
same anion, and a solution of a soluble salt having the same cation as the 
latter salt, e.g., 

Pb | PbC 2 4 (s) CaC 2 4 (s) CaCl 2 soln. 

1 Corten and Estermann, Z. physik. Chem., 136, 228 (1928); LeBlanc and Haraapp, 
, 166A, 321 (1933); Joseph, J. 'Biol. Chem., 130, 203 (1939). 



186 REVERSIBLE CELLS 

In this case the lead first dissolves to form lead ions, which combine 
with C^Oi" ions to form insoluble lead oxalate, thus 



Pb ^ Pb++ + 2< 
and 

Pb++ + C 2 ^ PbC 2 4 (s). 

The removal of the oxalate ions from the solution causes the calcium 
oxalate to dissolve and ionize in order that its solubility product may be 
maintained; thus 

CaC 2 O 4 (s) ^ Ca++ + C 2 04~, 

so that the net reaction is 

Pb(s) + CaC 2 4 00 ^ PbC 2 4 (s) + Ca++ + 2. 

The system thus behaves as an electrode reversible with respect to cal- 
cium ions. This result is of great interest since a reversible calcium 
electrode employing metallic calcium is difficult to realize experimentally. 
Another type of reversible electrode involves an unattackable metal, 
such as gold or platinum, immersed in a solution containing an appropri- 
ate oxidized and reduced form of an oxidation-reduction system, e.g., 
Sn++++ and Sn++, or Fe(CN)? and Fe(CN)? --- ; the metal merely 
acts as a conductor for making electrical contact, just as in the case of a 
gas electrode. The reaction at an oxidation-reduction electrode of this 
kind is either oxidation of the reduced state or reduction of the oxidized 
state, e.g., 

Sn++ ^ Sn++++ + 2c, 

depending on the direction of the current. In order that it may behave 
reversibly, the reaction being capable of occurring in either direction, 
a reversible oxidation-reduction system must contain both oxidized and 
reduced states. It is important to point out that there is no essential 
difference between an oxidation-reduction electrode and one of the first 
kind described above; for example, in a system consisting of a metal M 
and its ions M+, the former is the reduced state and the latter the oxidized 
state. Similarly the case of an anion electrode, e.g., chlorine-chloride 
ions, the anion is the reduced state and the uncharged material, e.g., 
chlorine, is the oxidized state. In all these instances the electrode process 
may be written in the general form : 

Reduced State ^ Oxidized State + n, 

where n is the number of electrons by which the oxidized and reduced 
states differ. It is a matter of convenience, however, to treat separately 
electrodes involving oxidation-reduction systems in the specialized sense 
of the terms oxidation and reduction. 

Direction of Current Flow and Sign of Reversible Cell. The com- 
bination of two reversible electrodes in a suitable manner will give a 



REACTIONS IN REVERSIBLE CELLS 187 

reversible cell; in this cell the reaction at one electrode is such that it 
yields electrons while at the other electrode the reaction removes elec- 
trons. The electrons are carried from the former electrode to the latter 
by the metallic conductor which connects them. The ability to supply 
or remove electrons is possessed by all reversible electrodes, as is evident 
from the discussion given above; the particular function which is manifest 
at any time, i.e., supplying or removing electrons, depends on the direc- 
tion of the current flow, and this is determined by the nature of the two 
electrodes combined to form the cell. The electrode Ag, AgCl(s) KC1 
soln., for example, acts as a remover of electrons when combined with 
Zn, ZnSO 4 soln., but it is a source of electrons in the cell obtained by 
coupling it with the Ag, AgNO 3 soln. electrode. Since it is not always 
possible to say a priori in which direction the current in a given cell will 
flow when the electrodes are connected by an external conductor, it is 
necessary to adopt a convention for describing the E.M.F. and the reaction 
occurring in a reversible cell. The convention most frequently employed 
by physical chemists in America is based on that proposed by Lewis and 
Randall; it may be stated as follows. 

The E.M.F. , including the sign, represents the tendency for posi- 
tive ions to pass spontaneously through the cell as written from left 
to right, or of negative ions to pass from right to left. 

Since a positive E.M.F. means the passage of positive ions through the 
cell from left to right, it can be readily seen that electrons must pass 
through the external conductor in the same direction (cf. Fig. 2). It 
follows, therefore, that when the E.M.F. of the cell is positive the left- 
hand electrode acts as a source of electrons while the right-hand elec- 
trode removes them; if the E.M.F. is negative, the reverse is true. When 
expressing the complete chemical reaction occurring in a cell the con- 
vention will be adopted of supposing that the condition is the one just 
derived for a positive E.M.F.* 

Reactions in Reversible Cells. It is of importance in many respects 
to know what is the reaction occurring in a reversible cell, and some 
different types of cells will be considered for the purpose of illustrating 
the procedure adopted in determining the cell reaction. The Daniell 
cell, for example, is 

Zn | ZnS0 4 aq. j CuSO 4 aq. | Cu, 

and taking the left-hand electrode as the electron source, i.e., the E.M.F. 
as stated is positive, the reaction here is 

Zn = Zn++ + 2, 

* Many physical chemists in Europe and practical electrochemists in America use 
a convention as to the sign of E.M.F. and electrode potential which is the opposite of 
that employed here. 



188 REVERSIBLE CELLS 

while at the right-hand electrode the electrons are removed by the 
process 

Cu++ + 2c = Cu. 

The complete reaction is thus 

Zn + Cu++ = Zn++ + Cu, 

and since two electrons are involved in each atomic act, the whole reac- 
tion as written, with quantities in gram-atoms or gram-ions, takes place 
for the passage of two faradays of electricity through the cell (cf. p. 27). 
Since the cupric ions originate from copper sulfate and the zinc ions form 
part of zinc sulfate, the reaction is sometimes written as 

Zn + CuSO 4 = ZnSO 4 + Cu. 

The E.M.F. of the cell depends on the concentrations of the zinc and 
cupric ions, respectively, in the two solutions, and so if the cell reaction 
is to be expressed more precisely, as is frequently necessary, the concen- 
tration of the electrolyte should be stated; thus 



Zn + CuSCMmi) = ZnSO^ms) + Cu, 

where mi and w 2 are the molalities of the copper sulfate and zinc sulfate, 
respectively, in the Daniell cell. 
In the cell 

Zn | ZnS0 4 aq. j KC1 aq. AgCl(s) | Ag, 

the left-hand electrode reaction is the same as above, i.e., 

Zn = Zn++ + 26, 

while at the right-hand electrode the removal of electrons occurs by 
means of the process described on page 185, i.e., 

AgCl(s) = Ag+ + Cl- 
and 

Ag+ + * = Ag, 
the net reaction being 

AgCl(s) + = Ag + CI-. 

The complete cell reaction for the passage of two faradays is thus 

Zn + 2AgCl(s) = Zn++ + 2C1~ + 2Ag, 
or 

Zn + 2AgCl(a) = ZnCl 2 + 2Ag. 

A specicl case of this type of cell arises when both electrodes arc of 
the same metal, viz., 

Ag ( AgCl(s) KC1 aq. j AgNO, aq. | Ag. 



MEASUREMENT OF E.M.F. 189 

By convention, the reaction at the left-hand electrode is the opposite of 
that at the right-hand electrode of the previous cell, viz., 

Ag + Cl- = AgCUs) + 6, 
and at the right-hand electrode the reaction is 

Ag+ + = Ag, 
so that the net reaction in the cell is 

Ag+ + Cl- = AgCl(s) 

for the passage of one faraday. 

Another type of cell in which the two electrodes are constituted of 
the same material is one involving two hydrogen gas electrodes, viz., 

H 2 | NaOH aq. j HC1 aq. | H 2 . 

If the E.M.F. is positive, the hydrogen passes into solution as ions at the 
left-hand electrode, i.e., 

JH,(0) = H+ + ,* 

but the hydrogen ions react immediately with the hydroxyl ions in the 
alkaline solution, viz., 

11+ + OH- = H 2 O, 

to form water. At the right-hand electrode electrons are removed by 
the discharge of hydrogen ions, thus 



so that the net reaction for the passage of one faraday is 

H+ + OH- = H,0, 

i.e., the neutralization of hydrogen ions by hydroxyl ions. Since the 
hydrogen ions are derived from hydrochloric acid and the hydroxyl ions 
from sodium hydroxide, the reaction can also be written (cf. p. 12) as 

IIC1 + NaOH = NaCl + H 2 0. 

Measurement of E.M.F. The principle generally employed in the 
measurement of the E.M.F.'S of voltaic cells is that embodied in the 
Poggendorff compensation method; it has the advantage of giving the 
E.M.F. of the cell on "open circuit," i.e., when it is producing no current. 
It has been already mentioned that a cell can be expected to behave 
reversibly only when it is producing an infinitesimally small current, and 
hence the condition of open circuit is the ideal one for determining the 
reversible E.M.F. 

* The hydrogen ion in aqueous solution is probably (H 2 O)H+, i e., H 3 O + , and not 
H+ UrfrpT308); this does not, however, affect the general nature of the results recorded 
here. 



190 



REVERSIBLE CELLS 



The potentiometer } as the apparatus for measuring E.M.F. 's is called, 
is shown schematically in Fig. 60; it consists of a working cell C, generally 
a storage battery, of constant E.M.F. which must be larger than that of 
the cell to be measured, connected across the ends of a uniform con- 



D 



FIG. 60. Measurement of E.M.F. 

ductor AB of high resistance. The cell X, which is being studied, is 
connected to A, with the poles in the same direction as the cell C, and 
then through a galvanometer G to a sliding contact D which can be 
moved along AB. The position of D is adjusted until no current flows 
through the galvanometer; the fall of potential between A and D due 
to the cell C is exactly compensated by the E.M.F. of X, that is Ex. By 
means of a suitable switch the cell X is now replaced by a standard cell S, 
of accurately known E.M.F. equal to Es, and the sliding contact is re- 
adjusted until a point of balance is reached at D'. The fall of potential 
between A and D' is consequently equal to Es, and since the conductor 
AB is supposed to be uniform, it follows that 



AD 

AD'' 
AD 



Ex 
E s 



Since E s is known, and AD and AD' can be measured, the E.M.F. of the 
unknown cell, Ex, can be evaluated. 

In its simplest form, the conductor AB may consist of a straight, 
uniform potentiometer wire of platinum, platinum-indium, or other re- 
sistant metal, stretched tightly along a meter scale; the position of the 
sliding contact can be read with an accuracy of about 0.5 mm., and if C 
is 2 volts and AB is 1 meter long, the corresponding error in the evalua- 
tion of the E.M.F. is 1 millivolt, i.e., 0.001 volt. Somewhat greater pre- 
cision can be achieved if the potentiometer wire is several meters in length 
wound on a slate cylinder. For more accurate work the wire may be 
replaced by two calibrated' resistance boxes; the contact D is fixed where 



CURRENT INDICATORS 



191 



the two boxes are joined, and the potential across AD is varied by 
changing the resistances in the boxes, keeping the total constant. If R x 
is the resistance between A and D with the cell X in circuit, when no 
current flows through the galvanometer G, then the fall of potential which 
is equal to Ex must be proportional to R x ; * further, if R is the resistance 
at the balance point when the standard cell S replaces X, it follows that 

Ex^Rx 

JJT D ' 

&s KS 



The unknown E.M.P. can thus be calculated from the two resistances. 
As a general rule the total resistance in the circuit is approximately 
11,000 ohms, and hence if the working cell has an E.M.P. of 2 volts, each 
ohm resistance represents about 0.2 millivolt. 

The majority of E.M.F. measurements are made at the present time 
by means of special potentiometers, operating on the Poggendorff prin- 
ciple, which are purchased from scientific instrument makers. They 
generally consist of a number of 
resistance coils with a movable 
contact, together with a slide wire 
for fine adjustment. A standard 
cell is used for calibration pur- 
poses, and the E.M.F. of the cell 
being measured can then be read 
off directly with an accuracy of 
0.1 millivolt, or better. 

For approximate purposes, as 
in electroanalytical work or in 
potentiometric titrations, a sim- 
ple procedure, known as the 



c 

1 


-AMA 
> 


A 
AAAAAA . 


B 

AAAAAAAA 


(^ 


VWYV YYVYVVV 

f T r \ 


1 




1 

X 


1 Cr J 



FIG. 61. 



Potentiometer-voltmeter 
arrangement 



potentiometer-voltmeter method, 
can be employed. The working 

cell C (Fig. 61) is connected across two continuously variable resistances 
A and B, as shown; one of these resistances is for coarse and the other 
for fine adjustment. The experimental cell is placed at X in series with a 
galvanometer (G), and a milli voltmeter (V) is connected across the vari- 
able resistances. The latter are adjusted until no current flows through 
G; the voltage then indicated on V gives the E.M.F. of the cell. 

Current Indicators. The best form of current detector for accurate 
work is a suitably Ha.mjftd mirror galvanometer of high megohm^ sensi-^ 
tivity ; for approximate purposes, however, a simple pointer galvanometer 

* Ex is actually equal to E e X RxIR, where E e is the B.M.F. of the working cell C, 
and R is the totpl resistance of the two boxes in the circuit; since E e and R are main- 
tained constant, EX is proportional to Rx- * * 



192 



REVERSIBLE CELLS 



is generally employed. At one time the capillary electrometer was widely 
used for the purpose of indicating the li^aillineTrt ^jf balance in the 
potenfSmeter circuit; it Has the advantage of being unaffected by elec- 
trical and magneTic^disturbances, and of not being damaged if large 
currents are inadvertently passed through it. On the other hand, the 
capillary electrometer is much less sensitive than most galvanometers 
and is liable to behave erratically in damp weather; for these and other 
reasons this form of detector has been discarded in recent years. 

An ordinary mirror galvanometer of good quality can detect a current 
of about 10~ 7 amp., and hence if an accuracy of 0.1 millivolt is -desired, 
as is the case in much work that is not of the highest precision, the re- 
sistance of the cell- should not exceed lO 3 ohms. Special high-sensitivity 
galvanometers are available which show' an observable deflection with a 
current of 10~ u amp., and so the E.M.F. of cells with resistances up to 
10 7 ohms can be measured with their aid; the quadrant electrometer, 
which detects actually differences of potential rather than current, has 
also been used for the study of high resistance cells. Another procedure 
which has been devised is to employ a condenser in series wjlh a ballistic 
galvanp.meter Jio determine the balance point of the potentiometer; the 
condenser is charged for a definite time by means of the cell being studied 
and is then discharged through the galvanometer with the aid of a suitable 
switch. When the potentiometer is balanced the ballistic galvanometer 
will undergo.no deflection when the cell is discharged through it. 

^or most measurements of E.M.F. of cells of high resistance some 
form of vacuum-tube potentiometer has been used ; 2 this instrument 
employs the amplifying properties of the vacuum tube, and the principle 

of operation may be illustrated by 
means of the simple circuit shown 

X.I a $ *<N PI'I 1 !' in Fig. 62. The tube is repre- 

'17! ' (wvyw) sented by T, and A, B and C in- 

dicate the filament, anode and 
grid batteries, respectively; Ri and 
Rz are variable resistances and G 
is a galvanometer. The cell X of 
unknown resistance is connected, 
as shown, to a potentiometer P 
from which any desired known 
voltage can be taken off; by means 
of the switch S the potentiom- 
eter and cell can be included, if 
required, in the grid circuit of the vacuum tube. The switch is first con- 
nected to b and the filament current is adjusted by means of Ri to provide 
the optimum sensitivity of the tube; the " compensating current " from 
A, which passes in the opposite direction to the anode current through 

2 See, for example, Garmauand Drusz, Ind. Eng. Chem. (Anal. Ed.), 11, 398 (1939); 
for review, see Glasstone, Ann. Rep. Chem. Soc., 30, 283 (1933). 



-Mh 




FIG. 62. 



Vacuum-tube potentiometer for 
cells of high resistance 



THE STANDARD CELL 193 

the galvanometer G, is then altered by means of the resistance ff 2 so as to 
give a suitable reading on G. The switch S is now turned to a, so that P 
and Xj as well as the battery C, are in the grid circuit; leaving R i and # 2 
unchanged, the potentiometer is adjusted until the deflection on G is the 
same as before. The potential on the grid of the tube must, therefore, 
be the same in both cases: hence the E.M.F. taken from the potentiometer 
P must be equal and opposite to that of the cell X. 

This simple type of vacuum-tube potentiometer is quite satisfactory 
for cells of not too high resistance, e.g., 10 7 ohms or less, but it is un- 
reliable for still higher resistances. Two sources of error then arise: 
first, the characteristics of the vacuum tube change as a result of intro- 
ducing the high resistance, so that a given anode current no longer 
corresponds to the same grid voltage; second, there is a fall of potential 
across the high resistance cell due to the flow of current in the grid 
circuit. With the best ordinary vacuum tubes the grid current may be 
about 10~ 10 amp., and so with a cell of resistance of 10 8 ohms, the error 
due to the fall of potential across the cell will be lO' 10 X 10 8 , i.e., 10~ 2 
volt. Several methods of varying complexity have been devised in order 
to overcome these sources of error; one of the simplest and most effective, 
which is employed in commercial potentiometers for the measurement of 
the E.M.F/S of cells involving the glass electrode (p. 356), is to use a 
special type of vacuum tube, known as an " electrometer tube." Al- 
though its amplification factor is generally smaller than that of the 
normal form of tube, the grid-circuit current is very small, 10~ 15 amp. 
or less, and the characteristics of the tube are not affected by high 
resistances. 

The Standard Cell. An essential feature of the Poggendorff method 
of measuring E.M.F. 's, and of all forms of apparatus employing the Poggen- 
dorff compensation principle, is a standard cell of accurately known 
E.M.F. The cell now invariably employed for this purpose is the Weston 
standard cell ; it is highly reproducible, its E.M.F. remains constant over 
long periods of time, and it has a small temperature coefficient. One 
electrode of the cell is a 12.5 per cent cadmium amalgam in a saturated 
solution of cadmium sulfate (3CdSO 4 -8H 2 O) and the other electrode 
consists of mercury and solid mercurous sulfate in the same solution, thus 

12.5% Cd in Hg | 3CdSO 4 -8H 2 O satd. soln. Hg 2 SO 4 (s) | Hg. 

The cell is set up in a H-shaped tube as shown in Fig. 63, the left-hand 
limb containing the cadmium amalgam and the right-hand the mercury; 
the amalgam is covered with crystals of 3CdSO 4 8H 2 O, and the mercury 
with solid mercurous sulfate, and the whole cell is filled with a saturated 
solution of cadmium sulfate. The E.M.F. of the Weston cell, in inter- 
national volts, over a range of temperatures is given by the expression 

E* = 1.018300 - 4.06 X 10~ 5 (* - 20) 

- 9.5 X I0"\t - 20) 2 + 1 X 10-"(* - 20), 



194 



EEVERSIBLE CELLS 



Cadmium-,, 
sulfate 
solution 



Cadmium-^ 
sulfate 

Cadmium 
amalgam-^ 




x Cadmium 
- x sulfate 

^-Mercurous 
sulfate 

^"-Mercury 



so that the value is 1.01830 volt at 20 and decreases about 4 X 10~ 2 
millivolt per degree in this region.* 

Although the so-called "saturated" Weston cell, containing a satu- 
rated solution of cadmium sul- 
fate, is the ultimate standard for 
E.M.F. measurement, a secondary 
standard for general laboratory 
use has been recommended; this 
is the "unsaturated" Weston 
cell, which has an even smaller 
temperature coefficient than the 
saturated cell. The form of un- 
saturated cell generally em- 
ployed contains a solution which 
has been saturated at 4 c., so 
that it is unsaturated at room 
Fia. 63. The Weston standard cell temperatures; its temperature 

coefficient is so small as to be 

negligible for all ordinary purposes and its E.M.F. may be taken as 
1.0186 volt. 3 

Free Energy and Heat Changes in Reversible Cells. Since the quan- 
titative consequences of the second law of thermodynamics are mainly 
applicable to reversible processes, the study of reversible cells is of par- 
ticular importance because it is possible to apply thermodynamic methods 
to the results. If the E.M.F. of a voltaic cell is E volts, and the process 
taking place in it is accompanied by the passage of n faradays, i.e., nF 
coulombs, where F represents 90,500 coulombs, the work done by the 
system in the cell is nFE volt-coulombs or joules (cf. p. 5). If the 
cell is a reversible one, this work represents " maximum work/' and since 
electrical work does not involve mechanical work resulting from a volume 
change, it may be taken as equal to the change of free energy accompany- 
ing the cell reaction. The increase of free energy of a process is equal to 
the reversible net work, i.e., excluding mechanical work, done on the 
system, and hence it follows that 



A(? = - nFE, 



(1) 



where A(? is the increase of free energy for the process taking place in the 
cell under consideration. According to the Gibbs-Hehnholtz equation, 
which is derived from the second law of thermodynamics applied to 
reversible changes, 



A(? 



Aff + r{^p), 

dl Jp 



(2) 



* It is important to note that the mercury electrode of a commercial Weston cell 
is always marked positive, while the cadmium amalgam electrode is marked negative. 

See Vinal, Trans. Electrochem. Soc., 68, 139 (1935). 



CONCENTRATION CELLS 195 

where AH is the increase of heat content * for the cell reaction, and 
introducing equation (1), the result is 



- nFE = A# - nFT > (3) 



(4) 

It is seen from equation (4) that if the E.M.F. of a reversible cell, i.e., E, 
and its temperature coefficient, dE/dT, at constant pressure are known, 
it is possible to evaluate the heat change of the reaction occurring in 
the cell. 

Some of the results obtained in the calculation of heat content changes 
from E.M.F. measurements are recorded in Table XLII; 4 the values de- 

TABLE XLII. HEAT CHANGES FROM E.M.F. MEASUREMENTS 

dE/dT A// kcal. 

Cell Reaction E X 10* E M F. Thermal 

Zn + 2Ag01 = ZnCl 2 -f 2Ag 1.015 (0) - 4.02 - 51.99 - 52.05 

Cd -f- PbCl 2 = OdCl 2 -f Pb 0.1880 (25) - 4.80 - lo.25 - 14.65 

Ag -f- $Hg 2 Cl 2 = AgCl -f Hg 0.0455 (25) + 3.38 + 1.275 -f 1.90 

Pb -f 2AgCl - PbCl 2 + 2Ag 0.4900 (25) - 1 86 - 25.17 - 24.17 

rived from thermochemical measurements are given in the last column 
for purposes of comparison. The agreement between the results for AH 
derived from E.M.F. measurements and from thermal data is seen to be 
satisfactory, especially when it is realized that an error of 1 X 10~ 5 in 
the temperature coefficient will mean an error of nearly 0.07 kcal. in 
AH at 298 K. It is probable, however, that the temperature coefficients 
are known with this degree of accuracy, and it is consequently believed 
that for many reactions the heat changes derived from E.M.F. data are 
more accurate than those obtained by direct thermal measurement. 

Concentration Cells: Cells without Transference. In the operation 
of the cell 

H 2 (l atm.) | HC1 aq.(c) AgCl(s) | Ag, 

consisting of a hydrogen and a silver-silver chloride electrode in hydro- 
chloric acid,t the hydrogen at the left-hand electrode dissolves to form 
hydrogen ions, whereas at the right-hand electrode silver chloride passes 
into solution and silver is deposited; thus 

JH,(1 atm.) = H+ + e 

* The increase of heat content is equal to the heat absorbed in the reaction at con- 
stant pressure. 

4 Taylor and Perrott, J. Am. Chem. Soc., 43, 486 (1921); Gerke, ibid., 44, 1684 
(1922). 

t The construction of these electrodes is described later (pp. 234, 350). 



196 REVERSIBLE CELLS 

and 



+ - Ag + C1-, 
sc that the net reaction is represented by 

iH 2 (l atm.) + AgCl(s) = HCl(c) + Ag, 

since the hydrogen and chloride ions are formed in hydrochloric acid 
solution of concentration c moles per liter. If two stich cells containing 
hydrochloric acid at concentrations ci and c^ y and having E.M.F.'S of Ei 
and EZ, respectively, are connected in opposition, the result is the cell 

IT 2 (1 atm.) | HCl(ci) AgCl(s) | Ag | AgCl(s) HCl(c 2 ) | H 2 (l atm.), 

whose E.M.F. is equal to EI E z . The reaction in the left-hand cell for 
the passage of one faraday, as seen above, is 

JH 2 (1 atm.) + AgClOO = HCl(ci) + Ag, 

and that in the right-hand cell is the reverse of this, i.e., 

HCl(c 2 ) 4- Ag = iH 2 (l atm.) + AgCl(s). 

The net result of the passage of a faraday of electricity through the 
complete cell is the transfer (i) of hydrogen gas at 1 atm. pressure from 
the extreme left-hand to the extreme right-hand electrode, (ii) of solid 
silver chloride from left to right, and (iii) of hydrochloric^ acid from con- 
centration 02 to d. Since the chemical potentials of the hydrogen gas 
and solid silver chloride remain unchanged, the free energy change AG 
of the cell reaction is due only to that accompanying the removal of 
1 mole of hydrochloric acid, i.e., 1 g.-ion of hydrogen ions and 1 g.-ion 
of chloride ions, from the solution of concentration <% and its addition 
to c\. It follows, therefore, that 



where /*H+ and jeer are the chemical potentials of hydrogen and chloride 
ions, the suffixes 1 and 2 referring to the solutions of concentration Ci and 
C2, respectively. The quantities of solutions in the cells are assumed to 
be so large that the removal of hydrochloric acid from one and its trans- 
fer to the other brings about no appreciable change of concentration; 
the change of free energy is thus equal to the resultant change in the 
chemical potentials. 

If the chemical potentials are expressed by means of equation (10) 
on p. 133, the result is 



CONCENTRATION CELLS 197 

where an* and Ocr refer to the activities of the ions indicated by the sub- 
scripts. The electrical energy produced in the cell for the passage of one 
faraday is EF, where E, as already seen, is equal to E\ E*; it follows, 
therefore, from equation (6), since 

AC = - EF, 

that 

(7) 



2RT a* 
= In-, (8) 

where ai and a 2 are the mean activities of the hydrochloric acid in the 
two solutions (cf. p. 138). The activities may be replaced by the prod- 
ucts my or r/, so that 

2RT c,/ 2 



or 

(10) 
x 



A cell of the type described above is called a concentiation cell with- 
out transference, for the E.M.F. depends on the relative concentrations, 
or molalities, of the two solutions concerned, and the operation of the cell 
is not accompanied by the direct transfer of electrolyte from one solution 
to the other. The transfer occurs indirectly, as shown above, as the 
result of chemical reactions. In general, a concentration cell without 
transference results whenever two simple cells whose electrodes are re- 
versible with respect to each of the ions constituting the electrolyte are 
combined in opposition; in the case considered above, the electrolyte is 
hydrochloric acid, and one electrode is reversible with respect to hydro- 
gen ions and the other with respect to chloride ions. 

If a\ is the mean ionic activity of the electrolyte in the left-hand side 
of any concentration cell without transference, arid a, 2 is the value on the 
right-hand side, the E.M.F. of the complete cell can be expressed by means 
of the general equation 

*- '.-^ln^ (11) 

' V ZI< 0,1 

where v is the total number of ions, and v+ or *>_ is the number of positive 
or negative ions produced by the i'>nization of one molecule of electro- 
lyte; z+ or 2_ is the valence of the inn with respect to which the extreme 
electrodes are reversible. If this ion is positive, as in the cell alreadv 
discussed, the positive si^ns apply throughout, but if it is negative, as 



198 



REVERSIBLE CELLS 



in the cell 
Ag 



HCl(d) | H 2 (l atm.) | HCl(c 2 ) AgCl(s) | Ag, 



the negative signs are applicable. 

Amalgam Cells. If the electrolyte in the concentration cell without 
transference is a salt of an alkali metal, e.g., potassium chloride, it is 
necessary to set up some form of reversible alkali metal electrode. This 
is achieved by dissolving the metal in mercury, thus forming a dilute 
alkali metal amalgam which is attacked much less vigorously by water 
than is the metal in the pure stateA The amalgam nevertheless reacts 
with water to some extent, and also with traces of oxygen that may be 

present in the solution : the exposed 
surface of the amalgam is therefore 
continuously renewed by maintain- 
ing a flow from the end of a tube. 
For the cell 

Ag | AgCl(s) KCl(d) | KHg x 

| KCl(c 2 ) AgCl(s) | Ag, 

where KHg x represents the potas- 
sium amalgam, the apparatus is 
shown in Fig. 64; the reservoir A 
contains the dilute amalgam which 
flows slowly through the capillary 
tubes BI and B 2 , while Ci and 
C 2 represent the silver electrodes 
coated with silver chloride (see p. 
234). B The potassium chloride so- 
lutions of concentrations c\ and Cz 
respectively, from which all dis- 
solved oxygen has been removed, 
as far as possible, are introduced 
into the cells by means of the tubes 
A and D 2 . Although reproducible 
results can be obtained with the 
exercise of due care, the measurements are not reliable for solutions 
more dilute than about 0.1 N, because of interaction between the solution 
and the alkali metal. 

Amalgam cells are utilized for the study of alkali hydroxides, e.g., 

H,(l atm.) | NaOH(d) | NaHg x | NaOHfe) | H 2 (l atm.), 

where the hydrogen electrode is reversible with respect to hydroxyl ions, 
but equation (11) for the E.M.F. requires some modification in this case, 
because the cell reaction also involves the transfer of water. The reac- 

Machines and Parker, /. Am. Chem. Soc., 37, 1445 (1915). 




Fia. 64. Concentration cell with amalgam 
electrodes (Maclnnes and Parker) 



DETERMINATION OP ACTIVITY COEFFICIENTS 199 

tion in the left-hand cell for the passage of one faraday of electricity is 
iH 2 (l atm.) + NaOH( Cl ) = H 2 O + Na, 

and in the right-hand cell it is 

H 2 O + Na = H 2 (1 atm.) + NaOH(c 2 ), 

and consequently the net process is the transfer of a mole of sodium 
hydroxide, i.e., one g.-ion each of sodium and hydroxyl ions, from the 
solution of concentration ci to that of concentration c 2 , while at the same 
time a mole of water is transferred in the opposite direction. The in- 
crease of free energy accompanying the passage of one faraday is repre- 
sented by 

AC = [(MNa+)2 (MNaOlJ 



and hence, utilizing the equation on page 133 to give the chemical poten- 
tial of the water in terms of its vapor pressure, it follows that 



; 12) 

F 



where i and a 2 are the mean ionic activities of the sodium hydroxide 
in the two solutions, and (pH 2 o)i and (pn 2 o)2 are the respective aqueous 
vapor pressures.* 

Determination of Activity Coefficients. The E.M.F. of a concentra- 
tion cell without transference is equal to EI 7 2 , where EI and E 2 are 
determined by the concentrations Ci and c 2 , respectively, of the electro- 
lyte; then for a cell to which equation (8) is applicable, 

*. (13) 



If in one of the two solutions, e.g., C2, the activity is unity, and the 
corresponding E.M.F. of the half-cell containing that solution is Z, equa- 
tion (13) reduces to the general form 

E - E = In a. (14) 

If m is the molality of the electrolyte in the solution of activity a which 
gives an E.M.F. equal to E in the half-cell, then addition of (2RT/F) In m 
to both sides of equation (14) yields 

_ . 2#7\ _ 2/Zr. a 

~ (!) 



F 
2RT 



In 7, ' (16) 



* It should be noted that the H 2 (0), NaOH aq. electrode is to be regarded ae 
reversible with respect to OH~ ions; this accounts for the negative sign in equation (12). 



200 REVERSIBLE CELLS 

where 7 is the mean activity coefficient of the electrolyte in the solution 
of molality m. In order to convert the Naperian to Briggsian logarithms 
the corresponding terms are multiplied by 2.3026, and if at the same time 
the values of R, i.e., 8.313 joules per degree, and of F, i.e., 96,500 cou- 
lombs, are inserted, equation (16) can be written as 

E + 2 X 1.9835 X 10~ 4 T log m - E 

= - 2 X 1.9835 X 10- 4 T log y, (17) 
and, at 25, this becomes 

E + 0.1183 log m - E = - 0.1183 log y. (18) 

Since E can be measured for any molality m, it would be possible to 
evaluate the activity coefficient y if E were known. 6 One method of 
deriving E makes use of the fact that at infinite dilution, i.e., when m 
is zero, the activity coefficient 7 is unity ; under these conditions -B a will 
be equal to E + 0.1183 log m at 25. If this quantity, for various values 
of m, is plotted as ordinate against a function of the molality, generally 
Vm, as abscissa, and the curve extrapolated to m equal to zero, the 
limiting value of the ordinate is equal to E Q . To be accurate this extra- 
polation requires a precise knowledge of the E.M.F.'S of cells containing 
very dilute solutions, and the necessary data are not easy to obtain. Two 
alternative methods of extrapolation which avoid this difficulty may be 
employed ; only one of these will, however, be described here. 7 

According to the Debye-Hlickel-B rousted equation (63), p. 147, it is 
possible to express the variation of the activity coefficient of a uiii- 
univalent electrolyte with molality by the equation 

log 7 = - A Vm + Cm, (19) 

where A is a known constant, equal to 0.509 for water as solvent at 25. 
Combination of this with equation (18) then gives 

E + 0.1183 log m - 0.0602 Vm = E Q - 0.1183 Cm, 
/. E' - 0.0602 Vm = E Q - 0.1183 Cm, 

where E' is equal to E + 0.1183 log m. According to this result the 
quantity E f 0.0602 Vm should be a linear function of m, and extrapola- 
tion of the corresponding plot to m equal zero should give E. It is 
found in practice that the actual plot is not quite linear, as shown by the 
results in Fig. 65 for the cells 

H 2 (l atm.) | HCl(m) AgCl(s) | Ag, 
but reasonably accurate extrapolation is nevertheless possible. The 

Lewis and Randall, J. Am. Chem. Soc., 43, 1112 (1921); Randall and Young, 
ibid., 50, 989 (1928). 

7 Hitchcock, J. Am. Chem. Soc., 50, 2076 (1928); Hnrned et al, ibid., 54, 1350 (1932); 
55, 2179 (1933); 58, 989 (1936). 



CONCENTRATION CELLS WITH TRANSFERENCE 



201 



value of E for this cell at 25 is +0.2224 volt, and hence for solutions of 
hydrochloric acid 

E + 0.1183 log m - 0.2224 = - 0.1183 log 7. 

The activity coefficients can thus be determined directly from this equa- 
tion, using the measured values of the E.M.F. of the cell depicted above, 



0.223 
0.221 
0.219 
0.217 
215 
0.213 




0.02 



0.94 



0.06 0.08 0.10 

m 



FIG. 65. Extrapolation of E M.F. to infinite dilution 

for various molalitics of hydrochloric acid; the lesults obtained are given 
in Table XLI1I. 

TABLE XLIII. MEAN ACTIVITY COEFFICIENTS OF HYDROCHLORIC ACID FROM 
E M.F. MEASUREMENTS AT 25 

m 

0.1238 
0.0. r >391 
02563 
0013407 
000913S 
0005619 
0.003215 

Concentration Cells with Transference. When two solutions of the 
same electrolyte are brought into actual contact and if identical elec- 
trodes, reversible with respect to one or other of the ions of the electro- 
lyte, arc placed in each solution, the result is a concentration cell with 
transference; for example, the removal of the AgCl(s) | Ag | AgCl(s) 
system from the cell on page 196 gives 

H,(l atm.) | IICl(ci) j HCl(c 2 ) | H 2 (l atm.), 

in which the two solutions of hydrochloric acid are in contact, and direct 
transfer from one to the other is possible. The presence of a liquid 
junction, as the region where the two solutions are brought into contact 



E 


E + 1183 log m 


7 


34199 


0.23466 


0.788 


0.3X222 


23218 


0.827 


0.41S24 


22999 


0.863 


0.44974 


22820 


0.893 


0.46SOO 


22735 


0.908 


0.49257 


0.22636 


0926 


52053 


22562 


0.939 



202 



REVERSIBLE CELLS 



with one another is called, is represented by the vertical dotted line. 
When one faraday passes through the cell, 1 g.-atom of hydrogen dis- 
solves at the left-hand electrode to yield 1 g.-ion of hydrogen ions, and the 
same amount of hydrogen ions will be discharged and 1 g.-atom of 
hydrogen will be liberated at the right-hand electrode. While the current 
is passing, t+ g.-ion of hydrogen ions will migrate across the boundary 
between the two solutions in the direction of the current, i.e., from left 
to right, and _ g.-ion of chloride ions will move in the opposite direction; 
t+ and t- are the transference numbers of the hydrogen and chloride 
ions, respectively (see Fig. 66). Attention may be drawn to the fact 

H 2 |HC/( Cl ) i HC/(c 2 )|H 2 

I i i 



t- 

FIG. 66. Transference at liquid junction 

that the transference numbers involved are the Hittorf values, and not 
the so-called "true" transference numbers (p. 114); this allows for the 
transfer of water with the ions. 

The net result of the passage of one faraday is the transfer of 1 t+, 
i.e., t-, g.-ions of hydrogen ions and t^ g.-ions of chloride ions from right 
to left, so that the increase of free energy is 

A(? = /_[(MH+)I - G*H*)I] + *-[Ucr)i - (ncr)i]. (20) 

Since the transference numbers vary with concentration, it is convenient 
to consider two solutions whose concentrations differ by a small amount, 
viz., c and c + dc\ under these conditions equation (20) becomes 

AG = - L.(<W + <W) 

= - t.(RT d In a H - + RT d In a c r) 

= - 2t_RTd\na, (21) 

where a is the mean activity of the hydrochloric acid at the concentration 
c, and t- is the transference number of the anion at this concentration. 
The E.M.F. of the cell whose concentrations differ in amount by dc may 
be represented by dE, and the free energy increase FdE may be 
equated to the value given by equation (21) ; hence 

ft T 
dE = 2*_ -TT d In a. (22) 

r 

For a concentration cell with electrolytes of concentration Ci and c 2 , i.e., 
mean activities of a\ and a 2 , respectively, the E.M.F. is then obtained by 



ACTIVITY COEFFICIENTS FROM CELLS WITH TRANSFERENCE 203 

integrating equation (22) between these limits ; thus 

2RT C a * 
E = - t-d In a. (23) 

" Ja v 

In the general case this becomes 

E== ------ Mlna, (24) 



where PI, v-t- and z have the same significance as before (p. 197) ; the trans- 
ference number t^ refers to the ion other than that with respect to which 
the electrodes are reversible. 

If the transference number is taken as constant in the range of con- 
centration Ci to C2> equation (24) takes the form 

. (25) 

Q>i 

In the special case of the hydrogen-hydrochloric acid cell given above, v 
is 2, v- is 1, and z+ is 1, and the electrodes are reversible with respect to 
positive ions ; hence 

*-*. in*- (26) 

If the concentration cell is one of the type in which water is formed or 
removed in the cell reaction, e.g., 

H 2 1 NaOH(ci) j NaOH(c 2 ) | H 2 , 

in which a mole of water is transferred from c 2 to Ci for the passage of one 
faraday, due allowance must be made in the manner already described. 

Activity Coefficients from Cells With Transference. In order to set 
up a cell without transference it is necessary to have electrodes reversible 
with respect to each of the ions of the electrolyte ; this is not always pos- 
sible or convenient, and hence the use of cells with transference, which 
require electrodes reversible with respect to one ion only, has obvious 
advantages. In order that such cells may be employed for the purpose 
of determining activity coefficients, however, it is necessary to have 
accurate transference number data for the electrolyte being studied. 
Such data have become available in recent years, and in the method de- 
scribed below it will be assumed that the transference numbers are known 
over a range of concentrations. 8 

The E.M.F. of a cell of the type 

M | MA(c) j MA(c + dc) | M. 

8 Brown and Maclnnes, J. Am. Chem. Soc., 57, 1356 (1935); Shedlovsky and 
Maclnnes, iUd. y 58, 1970 (1936); 59, 503 (1937); 61, 200 (1939); Maclnnes and Brown, 
Chem. Revs., 18, 335 (1936). 



204 REVERSIBLE CELLS 

where M is a metal or hydrogen, yielding cations in solution, is given by 
equation (22), and since the activity a is equal to cf, this may be written 

27? T 



dE 



(27) 



The activity is expressed in terms of concentrations rather than molalities 
because the transference numbers are generally known as a function of 
the former; the procedure described here thus gives the activity coeffi- 
cient /, but the values can be readily converted into the corresponding 
7*s by means of the equations on page 135. 

The transference number at any concentration can be written as 

L 

where to is the value at some reference concentration c ; if this expression 
for 1/J_ is inserted in equation (27) and the latter multiplied out and 
rearranged, the result is 



Integrating between the limits c and c, the corresponding values of the 
mean activity coefficient of the electrolyte being /o and /, it follows, 
after converting the logarithms, that 

f FK r F 



2 " (28) 



The first two terms on the right-hand side of equation (28) may be evalu- 
ated directly from the experimental data, after deciding on the concen- 
tration c which is to represent the reference state. The third term is 
obtained by graphical integration of 8 against E, the value of 6 being 
derived from the known variation of the transference number with 
concentration. 

The method just described gives log ///o, and hence the activity 
coefficient / in the solution of concentration c is known in terms of an 
arbitrary reference scale, i.e., / at concentration c ; it is necessary now 
to convert the results to the usual standard state, i.e., the hypothetical 
ideal solution at unit concentration (see p. 137). For this purpose, use 
is made of the Debye-Huckol expression for uni-univalent electrolytes, 

1 (29) 



where A is the known Debye-Huckel constant for the solvent at the ex- 
perimental temperature, and J?', which is written in place of aB, is a 



DETERMINATION OP TRANSFERENCE NUMBERS 205 

constant for the electrolyte. The term log /// , i.e., log / log / , may 
be represented by log/ + a, where a is a constant, equal to log/ , and 
hence equation (29) may be rewritten as 

logf + A^c = a+ B'la- log^ V^ 
/ o \ /o / 

For solutions dilute enough for equation (29) to be applicable, the plot of 
log (///o) + A Vc against [a log (///o)]Vc should be a straight line with 
intercept equal to a. The value of a, which is required for the purpose 
of this plot, is obtained by a short series of approximations. Once a, 
which is equal to log / , is known, it is possible to derive log / for any 
solution from the values of log ///o obtained previously. The activity 
coefficient of the electrolyte can thus be evaluated from the E.M.F. 's of 
cells with transference, provided the required transference number 
information is available. 

Determination of Transference Numbers. Since activity coefficients 
can be derived from E.M.F. measurements if transference numbers are 
known, it is apparent that the procedure could be reversed so as to make 
it possible to calculate transference numbers from E.M.F. data. The 
method employed is based on measurements of cells containing the same 
electrolyte, with and without transference. The E.M.F. of a concentra- 
tion cell without transference (E) is given by equation (11), and if the 
intermediate electrodes are removed so as to form a concentration cell 
with transference, the E.M.F., represented by E t , is now determined by 
equation (25), provided the transference numbers may be taken as 
constant within the range of concentrations in the cells. It follows, 
therefore, on dividing equation (25) by (11), that 

Y = **> (30) 

where the transference number t^ refers to the negative ion if the ex- 
treme electrodes are reversible with respect to the positive ion, and 
vice versa. 9 

For example, if the amalgam cell without transference 

Ag | AgCl(s) LiCl(d) | LiHg, | LiCl(c 2 ) AgCl(s) | Ag 
is under consideration, the corresponding cell with transference is 
Ag | AgCl(s) LiCl(c,) j LiCl(c 2 ) AgCl(s) | Ag. 

The ratio of the E.M.F.'S of these cells then gives the transference number 
of the lithium ion, i.e., 



The method for determining transference numbers from E.M.F. measurements was 
first suggested by Helmholtz in 1878. 



206 REVERSIBLE CELLS 

since the extreme electrodes, i.e., Ag | AgCl(s) LiCl aq., are reversible with 
respect to the chloride ion. 

The use of equation (30) gives a mean transference number of the 
electrolyte within the range of concentrations from c\ to C2, but this is 
of little value because of the variation of transference numbers with 
concentration; a modified treatment, to give the results at a series of 
definite concentrations, may, however, be employed. If the concentra- 
tions of the solutions are c and c + dc, the E.M.F. of the cell with trans- 
ference is given by the general form of equation (22) as 

v RT 
dE t = =t < T --- = d In a, 



dE t v RT 

" 31 = *=F ---- > (31) 

d In a v zF ^ J 

where a is the mean activity of the electrolyte at the concentration c. 
The corresponding E.M.F. for the cell with transference, derived from 
equation (11), is 

v RT 

dE = 



d In a v 

It follows, therefore, from equations (31) and (32) that 

dB/alna" "^ 
or 

dEt/d log a ___ 
dlz]d log a = ^ T * 



(33) 



If the E.M.F.'S of the cells, with and without transference, in which the 
concentration of one of the solutions is varied while the other is kept at a 
constant low value, e.g., 0.001 molar, are plotted against log a of the 
variable solution, the slopes of the curves a dE t /d log a and dE/d log a, 
respectively. The transference number uf the appropriate ion may 
thus be determined at any concentration by taking the ratio of the slopes 
at the value of log a corresponding to this concentration. The activities 
at the different concentrations, from which the log a data are obtained, 
must be determined independently by E.M.F. or other methods. 

Since the exact measurement of the slopes of the curves is difficult, 
analytical procedures have been employed. In the simplest one of 
these, 10 the values of E t are expressed as a function of the logarithm of 
the activities of the electrolyte; from this dE t /d log a is readily derived 
by differentiation. Since dE/d log a is given directly by equation (32), 

M Maclnnes and Beattie, J. Am Chem. Soc., 42, 1117 (1920). 



LIQUID JUNCTION POTENTIALS 207 

t can also be written as a function of log a, and hence it may be evaluated 
at any desired concentration. 

A more rigid but laborious method, for deriving transference num- 
bers from E.M.P. data, makes use of the fact that the activity coefficient 
of an electrolyte can be expressed, by means of an extended form of the 
Debye-Huckel equation, as a function of the concentration and of two 
empirical constants. 11 When applied to the same data, however, this 
procedure gives results which are somewhat different from those obtained 
by the method just described. Since the values are in better agreement 
with the transference data derived from moving boundary and other 
measurements, they are probably more reliable. 

A number of determinations of transference numbers, in both aqueous 
and non-aqueous solutions, have been made by the E.M.F. method, and 
the results are in fair agreement with those obtained by other experi- 
mental procedures. The results in Table XLIV, for example, are for the 

TABLE XLIV. TRANSFERENCE NUMBER OF LITHIUM ION IN LITHIUM CHLORIDE AT 25 

Hittorf or 

E.M.F. Moving Boundary 

Cone. Method Method 

0.005 N 0.3351 0.3303 

0.01 0.3333 0.3289 

0.02 0.3308 0.3261 

0.05 0.3259 0.3211 

0.10 0.3203 0.3168 

0.20 0.3126 0.3112 

0.50 0.3067 0.3079 

1.00 0.2809 0.2873 

transference number of the lithium ion in lithium chloride at 25. The 
discrepancies between the two sets of values are often appreciable, how- 
ever, and since they are greater than the experimental errors of the best 
Hittorf or moving boundary measurements, it is probable that the E.M.F. 
results are in error. It must be concluded, therefore, that the E.M.F. 's of 
concentration cells cannot yet be obtained with sufficient precision for 
the transference numbers to be as accurate as the best results obtained 
by other methods. 

Liquid Junction Potentials: Solutions of the Same Electrolyte. The 
free energy change occurring in a concentration cell with transference may 
be divided into two parts ; these are (i) the contributions of the reactions 
at the electrodes, and (ii) that due to the transfer of ions across the 
boundary between the two solutions. It is evident, therefore, that when 
two solutions of the same or of different electrolytes are brought into 
contact, a difference of potential will be set up at the junction between 
them because of ionic transference. Potentials of this kind are called 
liquid junction potentials or diffusion potentials. 

11 Jones and Dole, J. Am. Chem. Soc., 51, 1073 (1929); Jones and Bradshaw, iWd., 
54, 138 (1932); see also, Hamer, i&id, 57, 66 (1935); Harned and Dreby, ibid., 61, 3113 
(1939). 



208 REVERSIBLE CELLS 

Consider the simplest case in which the junction is formed between 
two solutions of the same uni-univalent electrolyte at concentrations 
d and c 2 , e.g., 

KCl(ci) j KCl(c 2 ). 

Adopting the usual convention for a positive E.M.P. that the left-hand 
electrode is the source of electrons, so that positive current flows through 
the interior of the cell from left to right, it follows that the passage of one 
faraday of electricity through the cell results in the transfer of t+ g.-ion 
of cations, e.g., potassium ions, from left to right, i.e., from solution Ci 
to solution c 2 , and t- g.-ion of anions, e.g., chloride ions, in the opposite 
direction (cf. p. 202). If the Approximation is made of taking the trans- 
ference numbers to be independent of concentration, the free energy 
change accompanying the passage of one faraday across the liquid junc- 
tion may be expressed either as FEi> where E L is the liquid junction 
potential, or as 



Further, since t+ + t- is equal to unity, it follows that 

x- < 36) 

where ai and a 2 are the mean activities of the electrolyte in the two solu- 
tions. By making the further approximation of writing (a_) 2 /(a_)i as 
equal to a 2 /ai, equation (35) reduces to 

E L = (l -2 + )^ln^- (36) 

Since 1 t+ is equal to f_, this result may be expressed in the alternative 
form 

B L =(-- + )^ln^, (36a) 

which brings out clearly the dependence of the sign of the liquid junction 
potential on the relative values of the transference numbers of the anion 
and cation. 

If the liquid junction potential under consideration forms part of the 
concentration cell 

Ag | AgCl() KCl(d) i KCl(c 2 ) AgClW | Ag, 
the E.M.F. of the complete cell is given by equation (25) as 



LIQUID JUNCTION POTENTIALS 209 

and hence, from this and equation (36), it is seen that 

2J+ 1 
E L = -^ E t . (37) 

This approximate relationship can be tested by suitable measurements on 
concentration cells with transference. 

As indicated above, the E.M.F. of a cell with transference can be re- 
garded as made up of the potential differences at the two electrodes and 
the liquid junction potential. It will be seen shortly (p. 229) that each 
of the former may be regarded as determined by the activity of the re- 
versible ion in the solution contained in the particular electrode. In 
the cell depicted above, for example, the potential difference at the left- 
hand electrode is dependent on the activity of the chloride ions in the 
potassium chloride solution of concentration c\\ similarly the potential 
difference at the right-hand electrode depends on the chloride ion activity 
in the solution of concentration C2. For sufficiently dilute solutions the 
activity of a given ion, according to the simple Debye-Hiickel theory, is 
determined by the ionic strength of the solution and is independent of the 
nature of the other ions present. It follows, therefore, that the electrode 
potentials should be the same in all cells of the type 

Ag | AgCl() MCl(d) j MCl(c,) AgCl(s) | Ag, 

where c\ and c 2 represent dilute solutions of any uni-univalent chloride 
MCI, which must be a strong electrolyte. If E is the constant algebraic 
sum of these potentials, the E.M.F. of the complete cell with transference, 
which does vary with the nature of MCI, will be E + EL, i.e. 

Et = E + EL, 
.'. E = E t - E L . (38) 

The difference between E t and E L should thus be constant for given 
values of c\ and C2, irrespective of the nature of the uni-univalent chloride 
employed in the cell. Inserting the value of EL given by equation (37) 
into (38), the result is 



If the right-hand side is constant, for cells with transference contain- 
ing different chlorides at definite concentrations, it may be concluded 
that the approximate equation (36) gives a satisfactory measure of the 
liquid junction potential between two solutions of the same electrolyte. 
The results in Table XLV provide support for the reliability of this equa- 
tion, within certain limits; 12 the transference numbers employed are the 
mean values for the two solutions, the individual figures not differing 
greatly in the range of concentrations involved. 

u Maclnnes, "The Principles of Electrochemistry," 1939, p. 226; data mainly from 
Machines et al, J. Am. Chem. Soc., 57, 1356 (1935): 59. 503 (1937). 



210 REVERSIBLE CELLS 

TABLE XLV. TEST OP EQUATION FOR LIQUID JUNCTION POTENTIAL 

Electrolyte Ci c 2 t+ E t Etl%t+ E L 

NaCl 0.005 0.01 0.392 13.41 mv. 17.1 mv. - 3.7 mv. 

KC1 0.005 0.01 0.490 16.77 17.1 - 0.3 

HC1 0.005 0.01 0.824 28.29 17.2 +11.1 

NaCl 0.005 0.04 0.391 39.63 mv. 50.7 mv. - 11.1 mv. 

KC1 0.005 0.04 0.490 49.63 50.6 - 1.0 

HC1 0.005 0.04 0.826 84.16 50.9 +33.3 

In order to give some indication of the magnitude of the liquid junc- 
tion potential, the values of EL calculated from equation (37) are re- 
corded in the last column. In general, the larger the ratio of the con- 
centrations of the solutions and the more the transference number of 
either ion departs from 0.5, i.e., the larger the difference between the 
transference numbers of the two ions, the greater is the liquid junction 
potential. The sign is determined by the relative magnitudes of the 
transference numbers of cation and anion of the electrolyte, as seen from 
equation (36a). 

General Equation for Liquid Junction Potential. When the two 
solutions forming the junction contain different electrolytes, as in many 
chemical cells, the situation is more complicated ; it is convenient, there- 
fore, to consider here the most general case. Suppose a cell contains a 
solution in which there are several ions of concentration Ci, c 2 , , c,-, 
g.-ions per liter, and suppose this forms a junction with another solution 
in which the corresponding ionic concentrations are c\ + dci, c 2 + dc^ , 
c + dc ly ; the valences of the ions are zi, z 2 , , z t , and their 
transference numbers are /i, / 2 , , 2, , the latter being regarded as 
constant, since the differences of the ionic concentrations in the two solu- 
tions are small. If one faraday of electricity is passed through the cell, 
t l /z l g.-ion of each ionic species will be transferred across the boundary 
between the two solutions, the positive ions moving in one direction, 
i.e., left to right according to convention, and the negative ions moving 
in the opposite direction. The increase of free energy as a result of the 
transfer of an ion of the ith kind from the solution of concentration c t to 
that of concentration c + dc % is given by 

dG = ^ [(/i* + d/i.) - /*.] 
z 

*. , 
- ~ /*> 

* * 

where /i and m + dm are the chemical potentials of the particular ions 
in the two solutions. For the transfer of all the ions across the boundary 
when one faraday is passed, 

AG = S - dm. 



GENERAL EQUATION FOR LIQUID JUNCTION POTENTIAL 211 

and utilizing the familiar definition of /u t as /*? + RT In a,, it follows that 

AG = 2 - fir d Inc., (39) 

i Z t 

where a t is the activity of the zth ions at the concentration d. It should 
be remembered that in the summation the appropriate signs must be 
used when considering positive and negative ions, since they move in 
opposite directions. 

Provided the concentrations of any ion do not differ appreciably in 
the two solutions, the transfer of ions across the boundary when current 
passes may be regarded as reversible. If dEi, is the potential produced 
at the junction between the two solutions, then AG will also be equal to 
F &EL for the passage of one faraday ; combination of this result with 
equation (39) gives 



dE L = - Sdlna,- (40) 

for the liquid junction potential. Since in actual practice the concentra- 
tions of the two solutions differ by appreciable amounts, the liquid junc- 
tion potential can be regarded as being made up of a series of layers with 
infinitesimal concentration differences; the resultant potential EL is 
obtained by integrating equation (40) between the limits Ci and C2, 
representing the two solutions in the cell ; thus 

7? np /* C 2 / 
E L = --=r I 2 -din a,. (41) 

r t/fi z i 

This is the general form of the equation for the liquid junction potential 
between the two solutions ; 13 in order that the integration may be carried 
out, however, it is necessary to make approximations or to postulate 
certain properties of the boundary. 

For example, if the two solutions contain the same electrolyte, con- 
sisting of one cation and one anion, equation (41) becomes 

RT 
**-" 

If the approximation is made of taking the transference numbers to be 
independent of concentration, this relationship takes the form 

t + RT (o + ) 2 t- RT (o_), 

~ 



which is identical with equation (34) for a uni-univalent electrolyte. 

"Harned, J. Phys. Chem., 30, 433 (1926); Taylor, ibid., 31, 1478 (1927); see also, 
Guggenheim, Phil. Mag., 22, 983 (1936). 



212 



REVERSIBLE CELLS 



Type of Boundary and Liquid Junction Potential. When the two 
solutions forming the junction contain different electrolytes, the struc- 
ture of the boundary, and hence the concentrations of the ions at different 
points, will depend on the method used for bringing the solutions to- 
gether. It is evident that the transference number of each ionic species, 
and to some extent its activity, will be greatly dependent on the nature 
of the boundary; hence the liquid junction potential may vary with the 
type of junction employed. If the electrolyte is the same in both solu- 
tions, however, the potential should be independent of the manner in 
which the junction is formed. In these circumstances the solution at 
any point in the boundary layer will consist of only one electrolyte at a 
definite concentration; hence each ionic species should have a definite 
transference number and activity. When carrying out the integration 

of equation (41), the result will, there- 
fore, always be the same no matter 
what is the type of concentration 
gradient in the intermediate layer 
between the two solutions; this the- 
oretical expectation has been verified 
by experiment. 14 It is the fact that 
the liquid junction potential is in- 
dependent of the structure of the 
boundary, when the electrolyte is the 
same on both sides, that makes pos- 
sible accurate measurement of the 
E.M.F. of concentration cells with liq- 
uid junctions. In general, cells of this 
type are set up with simple "static 1 , 
junctions, as shown in Fig. 67; the 
more dilute solution is in the rela- 
tively narrow tube which is dipped 
into the somewhat wider vessel con- 
taining the more concentrated solution, so that the boundary is formed at 
the tip of the narrower tube. 

For solutions of different electrolytes four distinct forms of boundary 
have been described, 15 but only in two cases is anything like a satis- 
factory integration of equation (41) possible. 

I. The Continuous Mixture Boundary. This type of boundary, which 
is the one postulated by Henderson, 16 consists of a continuous series of 
mixtures of the two solutions, free from the effects of diffusion. If the two 
solutions are represented by the suffixes 1 and 2, and 1 x is the frac- 

" Scatchard and Buehrer, J. Am. Chem. Soc., 53, 574 (1931); Ferguson et al, ibid., 
54, 1285 (1932); Szabo, Z. physik. Chem., 174A, 33 (1935). 

"See Guggenheim, J. Am. Chem. Soc., 52, 1315 (1930). 

"Henderson, Z. physik. Chem., 59, 118 (1907); 63, 325 (1908); Hermans, Rec. trav. 
chim., 57, 1373 (1938); 58, 99 (1939). 




Dilute - 
solution 



FIQ. 67. 



Cell with static junction 
(Maclnnes) 



THE CONTINUOUS MIXTURE BOUNDARY 213 

tion of the former solution at a given point in the boundary, the fraction 
of solution 2 will be x, where x varies continuously from zero to unity; 
if d is the concentration of the zth kind of ion at this point, then 

Ci = (1 x)Ct(i) + C t (2), 

where C,(D and c t(2 ) are the concentrations of these ions in the bulk of the 
solutions 1 and 2, respectively. Making use of this expression, and re- 
placing activities in equation (41) by the corresponding concentrations, 
as an approximation, it is possible to integrate this equation ; the result, 
known as the Henderson equation for liquid junction potentials, is 

__RT_ (Ui - y,) - (t/2 - yj u[ + v( 
L " F ' (u{ + Fi) - (V* + Fi) ln # + F;' > (42) 

where U\, Fi, etc., are defined by 

Ui s 2(c+w+)i, Vi ss S(c_u_)i, 

U{ ss S(c+z+u+)i and Fi = 2(c_z_w_)i, 

where c+ and c_ refer to the concentrations of the cations and anions 
respectively, in g.-ions per liter, u+ and w_ are the corresponding ionic 
mobilities, and z+ and z_ their valences ; the suffix 1 refers to the ions in 
solution 1, and similar expressions hold for 1/2, V^, etc. in which the ions 
in solution 2 are concerned. 

The continuous mixture boundary presupposes the complete absence 
of diffusion; since diffusion of one solution into the other is inevitable, 
however, this type of boundary is probably unstable. It is possible that 
the flowing type of junction considered below may approximate in be- 
havior to the continuous mixture type of boundary. 

Two special cases of the Henderson equation are of interest. If the 
two solutions contain the same uni-univalent electrolyte at different 
concentrations, then 

and V\ = V{ = Ciu_, 
and F 2 = V*2 = c 2 M-. 



Insertion of these values in equation (42) gives 

RT .UtrJbta?!. (43 ) 

f U+ + U- C 2 

Since u+/(u+ + u_) is equal to the transference number of the cation, 
i.e., to t+, this result is equivalent to 



which is the same as the approximate equation (36), except that the ratio 
of the activities has been replaced by the ratio of the concentrations. 



214 REVERSIBLE CELLS 

Another interesting case is that in which two uni-univalent electro- 
lytes having an ion in common, e.g., sodium and potassium chlorides, 
are at the same concentration c; in these circumstances, assuming the 
anion to be common ion, 

Ui = U{ = cu+(v and V\ = V( = cu,-, 
/ 2 = C/2 = cu+ (2 ) and F 2 = V* = cu_, 
and substitution in equation (42) gives 

RT u+ (1 , + u- 
11, = r , in ; 

F M+<2) + M- 



= -^ In -, (44) 

where AI and A 2 are the equivalent conductances of the two solutions 
forming the junction. The resulting relationship is known as the Lewis 
and Sargent equation, 17 tests of which will be described shortly. 

II. The Constrained Diffusion Junction. The assumption made by 
Planck 18 in order to integrate the equation for the liquid junction poten- 
tial is equivalent to what has been called a "constrained diffusion junc- 
tion"; this is supposed to consist of two solutions of definite concentration 
separated by a layer of constant thickness in which a steady state is 
reached as a result of diffusion of the two solutions from opposite sides. 
The Planck type of junction could be set up by employing a membrane 
whose two surfaces are in contact with the two electrolytes which are 
continuously renewed; in this way the concentrations at the interfaces 
and the thickness of the intermediate layer are kept constant, and a 
steady state is maintained within the layer. The mathematical treat- 
ment of the constrained diffusion junction is complicated; for electrolytes 
consisting entirely of univalent ions, the result is the Planck equation, 

R T 
E L = -jr\n$, (45) 

where is defined by the relationship 

. Cj . 



* - 



.-k 

Cl 

Ui, [/ 2> V\ and Vz having the same significance as before. 

"Lewis and Sargent, /. Am. Chem. Soc., 31, 363 (1909); see also, Maclnnes and 
Yeh, ibid., 43, 2563 (1921); Martin and Newton, J. Phya. Chem., 39, 485 (1935). 

Planck, Ann. Physik, 40, 561 (1S90); see also, Fales and Vosburgh, /. Am. Chem. 
Soc., 40, 1291 (1918); Hermans, Rec. trav. Mm., 57, 1373 (1938). 



THE FLOWING JUNCTION 



215 



111 the two special cases considered above, first, two solutions of the 
same electrolyte at different concentrations, and second, two electrolytes 
with a common ion at the same concentration, the Planck equation 
reduces to the same form as does the Henderson equation, viz., equations 
(43) and (44), respectively. It appears, therefore, that in these par- 
ticular instances the value of the liquid junction potential does not 
depend on the type of boundary connecting the two solutions. 

III. Free Diffusion Junction. The free diffusion type of boundary is 
the simplest of all ir. practice, but it has not yet been possible to carry 
out an exact integration of equation (41) for such a junction. 19 In 
setting up a free diffusion boundary, an initially sharp junction is formed 
between the two solutions in a narrow tube and unconstrained diffusion 
is allowed to take place. The thickness of the transition layer increases 
steadily, but it appears that the liquid junction potential should be 
independent of time, within limits, provided that the cylindrical symme- 
try at the junction is maintained. The so-called " static " junction, 
formed at the tip of a relatively narrow tube immersed in a wider vessel 
(cf. p. 212), forms a free diffusion type of boundary, but it cannot retain 
its cylindrical symmetry for any appreciable time. Unless the two 
solutions contain the same electrolyte, therefore, the static type of junc- 
tion gives a variable potential. If the free diffusion junction is formed 
carefully within a tube, however, it can be made to give reproducible 
results. 20 

IV. The Flowing Junction. In order to obtain reproducible liquid 
junctions, in connection with the measurement of the E.M.F.'S of cells 
involving boundaries between two different electrolytes, Lamb and 
Larson devised the "flowing 

junction." 21 In the earlier 

forms of this type of junction 

(Fig. 68) an upward current 

of the more dense solution was 

allowed to meet a downward 

flow of the less dense solution 

at a point where a horizontal 

tube, leading to an overflow, 

joined the main tube. The 

levels of the liquids were so ar- 

ranged that they flowed at the 

same slow rate, and a sharp boundary was maintained within the hori- 

zontal portion of the overflow tube. Experiments with indicators have 

M Taylor, J. Phys. Chem., 31, 1478 (1927). 

10 Guggenheim, J. Am. Chem. Soc., 52, 1315 (1930). 

11 Lamb and Larson, J. Am. Chem. Soc., 42, 229 (1920); Maclnnes and Yeh, ibid., 
43, 2563 (1921); Scatchard, ibid., 47, 696 (1925); Scatchard and Buehrer, ibid., 53, 574 
(1931); see also, Roberts and Fenwick, ibid., 49, 2787 (1927); Lakhani, J. Chem. Soc., 
179 (1932); Ghosh, J. Indian Chem. Soc., 12, 15 (1935). 



e ieJ? 0( j e ^^f^ 

~~" " = 




FIQ. 68. The flowing junction 
(Lamb and Larson) 



216 



REVERSIBLE CELLS 



shown that the boundary between the two solutions in a good flowing 
junction is extremely thin. With such a junction the potentials between 
two electrolytes having an ion in common can be reproduced to 0.02 
millivolt. Simplified forms of flowing junction have been established 
by allowing the solutions to flow down opposite faces of a thin mica 
plate having a small hole in which the junction is formed (Fig. 69). The 



To electrode 




To electrode 



Fia. 69. Flowing junction (Roberts and Fenwick/ 

mica plate may even be eliminated and fine jets of the two liquids caused 
to impinge directly on one another. 

The problem of the flowing junction is too difficult to be treated 
theoretically; since the time of contact between the two solutions is so 
small, the extent of diffusion will probably be negligible, and hence it 
has been generally assumed that the flowing junction resembles a con- 
tinuous mixture (Henderson) type of boundary. On the other hand, it 
has been suggested that since the transition layer between the solutions 
is extremely thin, diffusion is of importance; the flowing junction would 
thus resemble the constrained diffusion (Planck) type of boundary. The 
only reasonably satisfactory experimental determinations of the potential 
of a flowing junction have been made with solutions of the same concen- 
tration and having an ion in common; as already seen, under these con- 
ditions the Henderson and Planck junctions lead to the same potentials. 

Measurement of Liquid Junction Potentials with Different Electro- 
lytes. If the same assumption is made as on page 209, that the potential 
of an electrode reversible with respect to a given ion depends only on the 
concentration of that ion, then in cells of the type 

Ag | AgCl(s) MCl(c) I M'Cl(c) AgCl() | Ag, 

where MCI and M'Cl, the chlorides of two different univalent cations, 
are present at the same concentration, the total E.M.F. is equal merely 
to the liquid junction potential. A number of measurements of cells of 
this form using 0.1 N and 0.01 N solutions of various chlorides have been 
made with a flowing junction of the type depicted in Fig. 68; the results 
are in fair agreement with those derived from the Lewis and Sargent 
equation (44), as shown by the data in Table XL VI. 22 The discrep- 

a Maclnnes and Yeh, J. Am. Chem. Soc., 43, 2563 (1921). 



ELIMINATION OF LIQUID JUNCTION POTENTIALS 217 

TABLE XLVI. CALCULATED AND OBSERVED FLOWING JUNCTION POTENTIALS AT 25 

Electrolytes Concentration Liquid Junction Potential 

Observed Calculated 

HC1 KC1 O.lN 26.78 mv. 28.52 mv. 

HCl NaCl 33.09 33.38 

KC1 NaCl 6.42 4.86 

KC1 LiCl 8.76 7.62 

NaCl NH 4 C1 -4.21 -4.81 

HCl NH 4 C1 0.01 N 27.02 mv. 27.50 mv. 

HCi LiCl 33.75 34.56 

KC1 NH 4 C1 1.31 0.02 

NaCl LiCl 2.63 2.53 

LiCl CsCl -7.80 -7.67 

ancles arc partly due to the assumption that the potentials of the two 
electrodes in the cell are the same, as well as to the neglect of activity 
coefficients in the derivation of equation (44). It is possible that the 
method of producing the flowing junction also has some influence on the 
observed results; for example, with 0.1 N solutions of hydrochloric acid 
and potassium chloride, a value of 28.00 mv. was obtained with the type 
of junction shown in Fig. 69, and 28.27 mv. when jets of the liquids were 
allowed to impinge on one another directly. 

Elimination of Liquid Junction Potentials. Electromotive force 
measurements are frequently used to determine thermodynamic quanti- 
ties of various kinds; in this connection the tendency in recent years has 
been to employ, as far as possible, cells without transference, so as to 
avoid liquid junctions, or, in certain cases, cells in which a junction is 
formed between two solutions of the same electrolyte. As explained 
above, the potential of the latter type of junction is, within reasonable 
limits, independent of the method of forming the boundary. 

In many instances, however, it has not yet been found possible to 
avoid a junction involving different electrolytes. If it is required to 
know the E.M.F. of the cell exclusive of the liquid junction potential, two 
alternatives are available: cither the junction may be set up in a repro- 
ducible manner and its potential calculated, approximately, by one of the 
methods already described, or an attempt may be made to eliminate 
entirely, or at least to minimize, the liquid junction potential. In order 
to achieve the latter objective, it is the general practice to place a salt 
bridge, consisting usually of a saturated solution of potassium chloride, 
between the two solutions that would normally constitute the junction 
(Fig. 70). An indication of the efficacy of potassium chloride in re- 
ducing the magnitude of the liquid junction potential is provided by the-- 
data in Table XL VII; 23 the values recorded are the E.M.F. 's of the cell 
with "free diffusion" junctions, 

Hg | H g2 Cl 2 (s) 0.1 N HCl I x N KC1 j 0.1 N KC1 Hg 2 Cl 2 (s) | Hg, 

28 Guggenheim, J. Am. Chem. Soc., 52, 1315 (1930); see also, Fales and Vosburgb, 
ibid., 40, 1291 (1918); Ferguson et al., ibid., 54, 1285 (1932). 



218 



REVERSIBLE CELLS 



TABLE ZLVH. EFFECT OF SATURATED POTASSIUM CHLORIDE SOLUTION ON 

LIQUID JUNCTION POTENTIALS 
X E.M.F. X E.M.F. 

0.2 19.95 mv. 1.75 5.15 mv. 

0.5 12.55 2.5 3.4 

1.0 8.4 3.5 1.1 

where x is varied from 0.2 to 3.5. When a; is 0.1 the E.M.F. of the cell is 
27.0 mv., and most of this represents the liquid junction potential be- 
tween 0.1 N hydrochloric acid and 0.1 N potassium chloride. As the 
concentration of the bridge solution is increased, the E.M.F. falls to a 
small value, which cannot be very different from that of the cell free from 
liquid junction potential. 




FIG. 70. Cell with salt bridge 

When it is not possible to employ potassium chloride solution, e.g., if 
one of the junction solutions contains a soluble silver, mercurous or 
thallous salt, satisfactory results can be obtained with a salt bridge con- 
taining a saturated solution of ammonium nitrate; the use of solutions of 
sodium nitrate and of lithium acetate has also been suggested. For 
non-aqueous solutions, sodium iodide in methyl alcohol and potassium 
thiocyanate in ethyl alcohol have been employed. 

The theoretical basis of the use of a bridge containing a concentrated 
salt solution to eliminate liquid junction potentials is that the ions of this 
salt are present in large excess at the junction, and they consequently 
carry almost the whole of the current across the boundary. The condi- 
tions will be somewhat similar to those existing when the electrolyte is 
the same on both sides of the junction. When the two ions have ap- 
proximately equal conductances, i.e., when their transference numbers are 
both about 0.5 in the given solution, the liquid junction potential will 
then be small [cf. equation (36a)]. The equivalent conductances at 
infinite dilution of the potassium and chloride ions are 73.5 and 76.3 
ohms" 1 cm. 2 at 25, and those of the ammonium and nitrate ions are 
73.4 and 71.4 ohms" 1 cm. 2 respectively; the approximate equality of the 
values for the cation and anion in each case accounts for the efficacy of 
potassium chloride and of ammonium nitrate in reducing liquid junction 
potentials. 



CONCENTRATION CELLS WITH A SINGLE ELECTROLYTE 219 

A procedure for the elimination of liquid junction potentials, sug- 
gested by Nernst (1897), is the addition of an indifferent electrolyte at 
the same concentration to both sides of the cell. If the concentration of 
this added substance is greater than that of any other electrolyte, the 
former will carry almost the whole of the current across the junction 
between the two solutions. Since its concentration is the same on both 
sides of the boundary, the liquid junction potential will be very small. 
This method of eliminating the potential between two solutions fell into 
disrepute when it was realized that the excess of the indifferent electro- 
lyte has a marked effect on the activities of the substances involved in the 
cell reaction. It has been revived, however, in recent years in a modified 
form: a series of cells are set up, each containing the indifferent electro- 
lyte at a different concentration, and the resulting E.M.F.'S are extrapo- 
lated to zero concentration of the added substance. 

Concentration Cells with a Single Electrolyte : Amalgam Concentra- 
tion Cells. In the concentration cells already described the E.M.P. is a 
result of the difference of activity or chemical potential, i.e., partial 
molal free energy, of the electrolyte in the two solutions; it is possible, 
however, to obtain concentration cells with only one solution, but the 
activities of the element with respect to which the ions in the solution 
are reversible are different in the two electrodes. A simple method of 
realizing such a cell is to employ two amalgams of a base metal at differ- 
ent concentrations as electrodes and a solution of a salt of the metal as 
electrolyte; thus 

Zn amalgam (zi) | ZnSO 4 soln. | Zn amalgam (rr 2 ), 

the mole fractions of zinc in the amalgams being x\ and ar 2 , as indicated. 
The passage of two faradays through this cell is accompanied by the 
reaction 

2 , 



at the left-hand electrode, and 

Zn^+ + 2c 

at the right-hand electrode. Since the concentration of zinc ions in the 
solution remains constant, the net change is the transfer of 1 g.-atom of 
zinc from the amalgam of concentration x\ to that of concentration x*\ 
the increase of free energy is thus 

AG = /iZn(2) ~ MZn(l) 



where a\ and a* are the activities of the zinc in the two amalgams. It 
should be noted that in this derivation it has been assumed that the 
molecule and atom of zinc are identical. 



220 REVERSIBLE CELLS 

The free energy change is also given by 2FE, where E is the E.M.F. 
of the cell, so that 



In the general case of an amalgam concentration cell in which the valence 
of the metal is z and there are m atoms in the molecule, the equation for 
the E.M.F. becomes 



, 

E = ^ In (47) 

zmF 0,2 

This result is of particular interest because it can be used to determine 
the activities of metals in amalgams or other alloys by E.M.F. measure- 
ments; such determinations have been carried out in a number of cases. 24 
If the amalgams are sufficiently dilute, the ratio of the activities may 
be taken as equal to that of their mole fractions, i.e., i/z 2 , or even to 
that of their concentrations Ci/C2j in the latter case equation (47) takes 
the approximate form 

-- (48) 

c 2 

Experiments with amalgams of a number of metals, e.g., zinc, lead, tin, 
copper and cadmium have given results in general agreement with equa- 
tion (48); the discrepancies observed are due to the approximation of 
taking the ratio of the concentrations to be equal to that of the activities. 
Gas Concentration Cells. Another form of concentration cell with 
electrodes of the same material at different activities, employing a single 
electrolyte, is obtained by using a gas, e.g., hydrogen, for the electrodes 
at two different pressures; thus 

IWpi) | Solution of hydrogen ions | H 2 (p 2 ), 

where p\ and p 2 are the partial pressures of hydrogen in the two elec- 
trodes. The passage of two faradays through this coll is accompanied, 
as may be readily shown, by the transfer of 1 mole of hydrogen gas from 
pressure p\ to pressure p 2 ; if the corresponding activities are a\ and a 2 , 
it is found, by using the same treatment as for amalgam concentration 
cells, that the E.M.F. is given by 



If the gas behaves ideally within the range of pressures employed, the 
ratio of activities may be replaced by the ratio of the pressures; hence 



24 Richards and Daniels, J. Am. Chan. Soc., 41, 1732 (1919). 



GAS CONCENTRATION CELLS 



221 



If one of the pressures, e.g., p 2 , is kept constant while the other is varied, 
equation (50) takes the general form 



RT 

E = In p + constant, 

&r 



(51) 



where p is the pressure that is varied. 

According to equation (51) the plot of the E.M.F. of the cell, in which 
one hydrogen electrode is kept at constant pressure while the other 
is changed, against the log p of the variable electrode should give a 
straight line. It is not convenient to test this equation by actual meas- 
urement of cells with two hydrogen electrodes, but an equivalent result 
should be obtained if the electrode of constant gas pressure is replaced 
by another not containing a gas, whose potential does not vary appre- 
ciably with pressure. Observations have thus been made on cells of 
the type 

H 2 (p) | HC1 (0.1 M) Hg 2 Cl 2 (s) | Hg, 

and the results for hydrogen pressures varying from a partial pressure 
of 0.00517 atm., obtained by admixture with nitrogen, up to 1000 a f m. 
are depicted in Fig. 71, in which the E.M.F.'S of the cells are plotted against 



0.48 



0.44 
b 

a 

w 

0.40 



0.36 



-2.0 



1.0 



FIG. 71. Hydrogen pressure and E.M p. 

the logarithm of the hydrogen pressure. 26 It is seen that the expected 
linear relationship holds up to pressures of about 100 atm. The devia- 
tions from linearity up to 600 atm. can be accounted for almost exactly 

26 Hainswoioh, Rowley and Maclnnes, ,/. Am. Chem. Soc., 46, 1437 (1924;; Ronraim 
and Chang, Butt. Soc. Mm., 51, 932 (1932). 



222 REVERSIBLE CELLS 

by making allowance for departure of the hydrogen gas from ideal be- 
havior. The discrepancies at still higher pressure must be attributed to 
the neglect of the influence of pressure on the mercury-mercurous chloride 
electrode. 

Since the passage of one mole of chlorine into solution requires two 
faradays, as is the case for a mole of hydrogen, the E.M.F. of a cell con- 
sisting of two chlorine electrodes at different pressures will be given by 
any of the equations derived above. It follows, therefore, that the 
E.M.F. 's of cells of the type 

Cl,(p) | HC1 soln. HfrCliW | Hg 

should be represented by equation (51) with the sign preceding the 
pressure term reversed, because the chlorine yields negative ions; the re- 
sulting equation may be put in the alternative form 

Tim 

E + -rrr In p = constant. (52) 

&r 

The data in Table XL VIII were obtained with a cell containing 0.1 N 

TABLE XLVIII. ELECTROMOTIVE FORCES OF CHLORINE GAS CELLS AT 25 
_ RT. v>* T * 

P E Inp E + ]np 

0.0492 atm. - 1.0509 - 0.0387 - 1.0896 

0.0247 - 1.0421 - 0.0475 - 1.0896 

0.0124 -1.0330 -0.0564 -1.0894 

0.0631 - 1.0243 - 0.0650 - 1.0893 

0.00293 - 1.0150 - 0.0749 - 1.0899 

hydrochloric acid, the pressure of the chlorine gas being reduced by ad- 
mixture with nitrogen; the constancy of the values in the last column 
confirm the accuracy of equation (52). 28 

In the case of an oxygen gas cell the electrode reactions may be rep- 
resented by 



so that the transfer of one mole of oxygen from one electrode to the other 
requires the passage of four faradays. The E.M.F. of the cell with two 
oxygen electrodes at different pressures is then 



or 

-!?* 

if the gas behaves ideally. The sign of the E.M.F. is opposite to that of 

" Lewis and Rupert, J. Am. Chem. Soc. t 33, 299 (1911); Kameyama ei al., J. Soc. 
Chem. Ind. (Japan), 29, 679 (1926). 



PROBLEMS 223 

the corresponding hydrogen cell [equations (49) and (50)] because of the 
opposite charges of the ions. Since the oxygen gas electrode does not 
normally function in a reversible manner (see p. 353), these equations 
cannot be tested by direct experiment. 

PROBLEMS 

1. Determine the reactions taking place at the separate electrodes and in 
the complete cell in the following reversible cells: 



(i) H,fo) 

(ii) Hg|HgO(s)NaOH|H 2 (<7); 

(iii) Ag | AgCl(*)KCl H g2 Cl 2 (s) | Hg; 
and 

(iv) Pb | PbCl 2 ()KCl j K 2 S0 4 PbS0 4 () | Pb. 

2. Devise reversible cells in which the over-all reactions are: 

(i) Hg + PbO(s) = Pb + HgO(s); 
(ii) Zn + Hg 2 S0 4 (s) = ZnS0 4 + 2Hg; 

(iii) Pb + 2HC1 = PbCl 2 (s) + H 2 (0); 
and 

(iv) H 2 (0) + J0,(f) = H 2 0(J). 

3. The following values for the E.M.F. of the cell 

Ag | AgBr(s) KBr aq. Hg 2 Br 2 (s) | Hg 

were obtained by Larson [J. Am. Chem. Soc., 62, 764 (1940)] at various tem- 
peratures: 

20 25 30 

0.06630 0.06839 0.07048 volt. 

State the reaction occurring in the cell for the passage of one faraday, and 
evaluate the heat content, free energy and entropy changes at 25. 

4. Harned and Donelson [J. Am. Chem. Soc., 59, 1280 (1937)] report that 
the variation of the E.M.F. of the cell 

H 2 (l atm.) | HBr(a = 1) AgBr(s) | Ag 
with temperature is represented by the equation 

E = 0.07131 - 4.99 X 10- 4 (* - 25) - 3.45 X 10-(* - 25) 2 . 
Calculate the change in heat content, in calories, accompanying the reaction 

H 2 (l atm.) + 2AgBr(s) = 2Ag + 2HBr(o = 1) 
at 25. 

5. The reversible cell 

Zn | ZnCl 2 (d) Hg 2 Cl 2 (s) | Hg | Hg 2 Cl 2 (s) ZnCl 2 (c 2 ) | Zn 

was found to have an E.M.F. of 0.09535 volt at 25. Determine the ratio of 
the mean ion activities of the zinc chloride in the two solutions. 



224 REVERSIBLE CELLS 

6. The E.M.F. of the cell 

H 2 (l atm.) | HBr(m) AgBr(s) | Ag 

with hydrobromic acid at various small molalities (m) was measured at 25 
by Keston [J. Am. Chem. Soc., 57, 1671 (1935)] who obtained the results 
given below: 



m 


E 


m 


E 


1.262 X 10~ 4 
1.775 
4.172 


0.53300 
0.51616 
0.47211 


10.994 X 10-* 
18.50 
37.19 


0.42280 
0.39667 
0.36173 



Use these data to evaluate E for the cell. 

7. The following results were derived from the measurements of Harned, 
Keston and Donclson [./. Am. Chem. Roc., 58, 989 (1936)] for the cell given 
in the preceding problem with more concentrated solutions of the acid: 

m E m E 

0.001 0.42770 0.05 0.23396 

0.0.5 0.34695 0.10 0.20043 

001 031262 0.20 0.16625 

0.02 0.27855 0.50 0.11880 

Using the value of E Q obtained above, determine the activity coefficients of 
hydrobromic acid at the various molalities. 

8. The following entropy values at 25 were obtained from thermal meas- 
urements: silver, 10.3 cal./deg. per g.-atom; silver chloride, 23.4 per mole; 
liquid mercury, 17.8 per g.-atom; and mercurous chloride, Hg2Cl 2 , 46.4 per 
mole. The increase in heat content of the reaction 



Ag(s) + Hg 2 Cl 2 (s) = AgCl(s) -f Hg(0 
is 1,900 cal. Calculate the E.M.F. of the cell 

Ag | AgCl(s) KC1 aq. Hg 2 Cl 2 (s) | Hg 

and its temperature coefficient at 25. 

9. Abegg and dimming [Z. Elektrochem., 13, 18 (1910)] found the E.M.F. 
of the cell with transference 

Ag | 0.1 N AgN0 3 j 0.01 N AgN0 3 1 Ag 

to be 0.0590 volt at 25. Compare the result with the calculated value 
using the following data: 

0.1 N AgNO 3 /i = 0.733 t+ = 0.468 

0.01 " 0.892 0.465. 

10. The E.M.F. 's of the cell with transference 

Ag | AgCl(s) 0.1 N HOI ; HCl(r) AgCl(s) | Ag 

at 25, and the transference numbers of the hydrogen ion in the hydrochloric 
acid of concentration c, are from the work of Shedlovsky and Maclnnes [ J. A m. 



PROBLEMS 225 

Chem. Soc., 58, 1970 (1936)] and of Longsworth [ibid., 54, 2741 (1932)]: 

c X 10 8 E * H + 

3.4468 0.136264 0.8234 

5.259 0.118815 0.8239 

10.017 0.092529 0.8251 

19.914 0.064730 0.8266 

40.492 0.036214 0.8286 

59.826 0.020600 0.8297 

78.076 0.009948 0.8306 

100.000 0.8314 

Utilize these data to calculate the activity coefficients of hydrochloric acid at 
the several concentrations. 

11. If the E.M.F. of the cell 

Hg | Hg 2 Cl 2 (s) 0.01 N KC1 j 0.01 N KOH j 0.01 N NaOH HgO(s) | Hg 

is E t calculate the value of the E.M.F. at 25 free from liquid junction poten- 
tials, using the Lewis and Sargent formula. 

12. The E.M.F.'S of the cells 

Zn in Hg(ci) | ZnSO 4 aq. | Zn in Hg(c 2 ) 

were measured by Meyer [Z. physik. Chem., 7, 447 (1891)] who obtained the 
ensuing results: 

Temp. ci c 2 E 

11.6 11.30 X 10~ 5 3.366 X lO" 8 0.0419 volt 

60.0 6.08 X 10-' 2.280 X 10~ 8 0.0520 

Assuming the amalgams are dilute enough to behave ideally, estimate the 
molecular weight of zinc in the amalgams. 

13. The E.M.F. of the cell 

C1 2 (1 atm.) | HC1 aq. AgCl(s) | Ag 

is 1.1364 volt at 25. The Ag, AgCl(s) electrode may be regarded as a 
chlorine electrode with the gas at a pressure equal to the dissociation pressure 
of silver chloride; calculate the value of this pressure at 25. 



CHAPTER VII 
ELECTRODE POTENTIALS 

Standard Potentials. When all the substances taking part in a reac- 
tion in a reversible cell are in their standard states, i.e., at unit activity, 
the E.M.F. is the standard value E for the given cell. If the reaction 
under consideration occurs for the passage of n faradays, then the stand- 
ard free energy change A(J is equal to nFE; hence by equation (23), 
page 137, with all the activities equal to unity, 

- AG = nFE = RT In K, (1) 

where K is the equilibrium constant of the cell reaction. If the reactants 
and resultants are at any arbitrary concentrations, or activities, the 
E.M.F. is E and the corresponding free energy change for the reaction AG 
is equal to nFE- it follows, therefore, from equation (22), page 136, 
that for the reaction 

aA + &B + - = IL + mM + 
occurring in the cell for the passage of n faradays, 



- AG = RTln K - RT In 



.'. nFE = nFE - RT In q a y 



This is the general equation for the E.M.F. of any reversible chemical cell 
in which the reactants and resultants are at any arbitrary activities 
O A , a B , and a L , a M , , respectively. 

Since E Q is related to the equilibrium constant of the reaction, it can 
clearly be regarded as equal to the difference between two constants 
Ei and E% characteristic of the separate electrode reactions which to- 
gether make up the process occurring in the cell as a whole. Further, 
the activity fraction may also be separated into two corresponding parts, 
so that equation (2) can be written as 

/ PT \ 

(3) 

where a\ and a* are the activity terms applicable to the two electrodes, 
and vi and v* are the numbers of molecules or ions of the corresponding 

226 



STANDARD POTENTIALS 227 

species involved in the ceil reaction. The actual B.M.P. of the cell can 
similarly be separated into the separate potentials of the electrodes; if 
these are represented by E\ and E^ it is evident that they may be identi- 
fied, respectively, with the quantities in the two sets of parentheses in 
equation (3). In general, therefore, it is possible to write 

RT 
JS? t = ?-2lna;< (4) 



for the potential of an electrode in terms of its standard potential 
and the activities of the species involved in the electrode process. It is 
evident from equation (4) that the standard potential is the potential of 
the electrode when all of these substances are at unit activity, i.e., in 
their standard states. 

The application of the procedure outlined above may be illustrated 
with reference to the reversible cell 

H,(l atm.) | HCl(c) AgCl(s) | Ag, 
in which the reaction is 

iH 2 (l atm.) + AgCl(s) = H+ + Cl~ + Ag(s) 

for the passage of one faraday. The appropriate form of equation (2) 
in this case is 



The individual electrode reactions (cf. p. 195) are 

(1) ^H 2 (l atm.) = H+ + e, 
and 

(2) AgCl + c = Ag(s) + C1-, 

so that equation (5) may be split up as follows 



(6a) 
and 

(66) 



The standard state of hydrogen is the ideal gas at 1 atm. pressure, and 
the standard states of silver and silver chloride are the solids; it follows, 
therefore, that in this particular case a H ,, a Ag ci and a Ag are unity, so that 

R T 

#H t ,H* = Eua+ -- ^r In H* (7a) 



228 ELECTRODE POTENTIALS 

and 

RT 

#Ag,Agci,cr = ^Ag,Agci,cr + ~TT In Ocr, (76) 

where the E Q terms are the standard potentials of the H 2 (l atm.), H+ 
and Ag(s), AgCl(s), Cl~ electrodes. It is seen, therefore, that in the cell 
under consideration the potential of each electrode depends only on the 
activity of one ionic species, apart from the standard potential of the 
system. 

The results given by equations (7a) and (76) may be expressed in a 
general form applicable to electrodes of all types; using the terms " oxi- 
dized " and "reduced" states in their most general sense (cf. p. 186), the 
potential of the electrode at which the reaction is 

Reduced Stated Oxidized State + n Electrons, 
is given by 



" nF (Reduced State) 

In the electrodes already considered the hydrogen ions and the silver 
chloride represent the respective oxidized states, whereas hydrogen gas, 
in the first case, and silver and chloride ions, in the second case, are the 
corresponding reduced states. For any electrode, therefore, at which 
the reaction occurring is 

aA + 6B + - - - = xX + i/Y + + nt, 
the general expression for the electrode potential is 

,.,_ 111 4^. 

nb a*a B 

If the electrode is one consisting of a metal M of valence z+, reversible 
with respect to M z + ions, so that the electrode reaction is 

M^ M'+ + *+, 
the equation for the potential takes the form 

L Sr j (8a) 

where OM is the activity of the solid metal and a M + is that of the cations 
in the solution with which the metal is in equilibrium. By convention, the 
solid state of thfe metal is taken as the standard state of unit activity; 
for an electrode consisting of the pure metal, therefore, OM may be re- 
placed by unity so that equation (8a) becomes 

_ RT . /QM 

i OM*. (80) 



INDIVIDUAL ION ACTIVITIES 229 

For an electrode involving a substance A which is reversible with 
respect to the anions A*-, the electrode reaction is 

A'-^ A + 2_e, 

the electrode material now being the oxidized state whereas the anions 
represent the reduced state; the equation for the electrode potential is 
then 

E- = E Q - -^In (9a) 

z-/' a A - v ' 

As before, the activity a\ of the substance A in the pure state, or if A is 
a gas then the activity at 1 atm. pressure, is taken as unity so that 
equation (9a) can be written as 

*_-*>--* In J- 

- 



The general form of equations (9a) and (9&) for any electrode revcr- 
ible with respect to a single ion of valence z is readily seen to be 

E = Eo-F~lna l , (10) 

wluTC a l is the activity of the particular ionic species; in this equation 
I lie upper signs apply throughout for a positive ion, while the lower signs 
are used for a negative ion. 

For practical purposes the value of 72, i.e., 8.313 joules, and F, i.e., 
96,500 coulombs, may be inserted in equation (10) and the factor 2.3026 
introduced to convert Naperian to Briggsian logarithms; the result is 

E = E* =F 1.9835 X 10~ 4 log a,. (10a)* 

z 

At 25 c., i.e., 298.16 K., which is the temperature most frequently 
employed for accurate electrochemical measurements, this equation 
becomes 

0.05915 t 

E = E Q T -- log a,. 

" 

Individual Ion Activities. The methods described in Chap. V for the 
determination of the activities or activity coefficients of electrolytes, r ^ 
well as those depending on vapor pressure, freezing-point or other osmotic 
measurements, give the mean values for b >th ions into which the solute 

* A convenient form of this equation for approximate purposes is 

E* = # ^ 0.0002 ~ log a t . 



230 ELECTRODE POTENTIALS 

dissociates. The question, therefore, arises as to whether it is possible 
to determine individual ion activities experimentally. An examination 
of the general equation (41), p. 211, or any of the other exact equations, 
for the liquid junction potential, shows that this potential is apparently 
determined by the activities of the individual ionic species; hence, if 
liquid junction potentials could be measured, a possible method would 
be available for the evaluation of single ion activities. It should be 
emphasized that the so-called experimental liquid junction potentials 
recorded in Chap. VI were based on an assumption concerning individual 
ion activities, e.g., that the activity of the chloride ion is the same in all 
solutions of univalent chlorides at the same concentration; they cannot, 
therefore, be used for the present purpose. 

The same point can be brought out in another manner. The E.M.F. 
of the cell with transference 

Ag | AgCl(s) KCl( Cl ) ; KCl(c 2 ) AgCl(s) | Ag 
is, according to equation (25), page 203, 

rt RT ^ a* 

E = - 2J+ -TT In > 
* F ai 

whereas the liquid junction potential, as given by equation (35), page 
208, is 

RT a* RT (ocr) 2 



If the ratio of the activities of the chloride ions were known, the value 
of the liquid junction potential could be derived precisely from equation 
(11), provided the E.M.F. of the complete cell, i.e., E, were measured. 
Although it is true, therefore, that the individual ion activities might be 
evaluated from a knowledge of the liquid junction potential, the latter 
can be obtained only if the single ion activities are known. 

A further possibility is that by a suitable device the liquid junction 
potential might be eliminated completely, i.e., EL might be made equal 
to zero; under these conditions, therefore, equation (11) would give 

RT (flcr)i /<rtx 

E = -;r\n~-> (12) 

P (acr)2 

and so the individual activities of the chloride ion at different concen- 
trations might be obtained by using an extrapolation procedure similar 
to that employed in Chap. VI to determine mean activities. It is doubt- 
ful, however, whether the results would have any real thermodynamic 
significance; the apparent individual ion activities obtained in this manner 
are actually complicated functions of the transference numbers and 



ARBITRARY POTENTIAL ZERO 231 

activities of all the ions present, including those contained in the salt 
bridge employed to eliminate the liquid junction potential. It is possible 
that, as a result of a cancellation of various factors, these activities are 
virtually equal numerically to the individual activities of the ions, but 
thermodynamically they cannot be the same quantities. 1 

Arbitrary Potential Zero: The Hydrogen Scale. Since the single 
electrode potential [cf. equation (10)] involves the activity of an indi- 
vidual ionic species, it has no strict thermodynamic significance; the use 
of such potentials is often convenient, however, and so the difficulty is 
overcome by defining an arbitrary zero of potential. The definition 
widely adopted, following on the original proposal by Nernst, is as 
follows : 

The potential of a reversible hydrogen electrode with gas at one 
atmosphere pressure in equilibrium with a solution of hydrogen ions at 
unit activity shall be taken as zero at all temperatures. 

According to this definition the standard potential of the hydrogen 
electrode is the arbitrary zero of potential [cf. equation (7a)]: electrode 
potentials based on this zero are thus said to refer to the hydrogen scale. 
Such a potential is actually the E.M.F. of a cell obtained by combining 
the given electrode with a standard hydrogen electrode; it has, conse- 
quently, a definite thermodynamic value. For example, the potential 
(E) on the hydrogen scale of the electrode M, M*+(aM + ), which is revers- 
ible with respect to the z-valent cations M>+, in a solution of activity GM+, 
is the E.M.F. of the cell 

M | M"(a M +) H+(a H + = 1) | H 2 (l atm.) 

free from liquid junction, or from which the liquid junction potential 
has been supposed to be completely eliminated. 
The reaction taking place in the cell is 



M + zH+(a n + = 1) = M"(a M +) + **IIi(l atm.), (13) 

and the change of free energy is equal to zFE volt-coulombs. If a\t + 
is equal to unity, the potential of the electrode is E and the free energy 
of the reaction is zFE; this quantity is called the standard free energy 
of formation of the M** ions, although it is really the increase of the free 
energy of the foregoing reaction with all substances in their standard 
states. 

If the electrode is reversible with respect to an anion, e.g., X s ~, as in 
the cell 

X | X-(ax-) H*(a H * = 1) I H 2 (l atm.), 
the reaction is 

X'-(ax-) + *H+(a H + = 1) = X + JH(1 atm.), (14) 

1 Taylor, /. Phys. Chem., 31, 1478 (1927); Guggenheim, ibid., 33, 842, 1540, 1758 
(1929); see also, Phil Mag., 22, 983 (1936). 



232 ELECTRODE POTENTIALS 

and the standard free energy increase is zFE*. This is the standard 
free energy of discharge of the X*~ ions, and hence the standard free 
energy of formation of an anion is + zFE, where E is its standard 
potential. 

Sign of the Electrode Potential. The convention concerning the sign 
of the E.M.F. of a complete cell (p. 187), in conjunction with the inter- 
pretation of single electrode potentials just given, fixes the convention 
as to the sign of electrode potentials. The E.M.F. of the cell 

M | M+(a M +) H+(an- = 1) | H,(l atm.) 



will ciea/ly be equal and opposite to that of the cell 

H 2 (l atm.) | H+IOH* = 1) M+(a M *) I M, 

so that the sign of the potential of the electrode when written M, M + 
must be equal and opposite to that written M+, M. In accordance with 
the convention for E.M.F. 's, the positive sign as applied to an electrode 
potential represents the tendency for positive ions to pass spontaneously 
from left to right, or of negative ions from right to left, through a cell in 
which the electrode is combined with a hydrogen electrode. The poten- 
tial of the electrode M, M+ represents the tendency for the metal to 
pass into solution as ions, i.e., for the metal atoms to be oxidized, whereas 
that of the electrode M+, M is a measure of the tendency of the ions to 
be discharged, i.e., for the ions to be reduced. 

\ Subsidiary Reference Electrodes : The Calomel Electrode. The de- 
termination of electrode potentials involves, in principle, the combination 
of the given electrode with a standard hydrogen electrode and the meas- 
urement of the E.M.F. of the resulting cell. For various reasons, such 
as the difficulty in setting up a hydrogen gas electrode and the desire to 
avoid liquid junctions, several subsidiary reference electrodes, whose 
potentials are known on the hydrogen scale, have boon devised. The 
most common of these is the calomel electrode; it consists of mercury in 
contact with a solution of potassium chloride saturated with mercurous 
chloride. Three different concentrations of potassium chloride have 
been employed, viz., 0.1 or, 1.0 \ and a saturated solution. By making 
use of the standard poten f u:l of the Ag, A^Cl^s), Cl~ electrode described 
below, the following results have been obtained for the potentials on the 
hydrogen scale of the three calomel electrodes at temperatures in the 
vicinity of 25. 2 

Hg, Hg 2 Cl 2 (s) 0.1 N KC! - 0.3338 + 0.00007 (t - 25) 
Hg, Hg 2 Cl 2 (s) 1 .0 N K( '1 - 0.2800 + 0.00024 (t - 25) 
Hg, IIg a Cl 2 (s) Saturated KC1 - 0.2415 + 0.0007(3 (t - 25) 

These values cannot be regarded as exact, since in therr derivation it has 
been necessary to make allowance for liquid junction potentials or for 

2 Hamor, Trans. Eleclrochem. tim , 72, 45 (1937). 



SUBSIDIARY REFERENCE ELECTRODES 



233 



single ion activities; the calomel electrodes are, however, useful in con- 
nection with various aspects of electrochemical work, as will appear in 
this and later chapters (see p. 349). The electrode with 0.1 N potassium 
chloride is preferred for the more precise measurements because of its 
low temperature coefficient, but the calomel electrode with saturated 
potassium chloride is often employed because it is easily set up, and when 
used in conjunction with a saturated potassium chloride salt bridge one 
liquid junction, at least, is avoided. 

Various types of vessels have been described for the purpose of setting 
up calomel electrodes; the object of the special designs is generally to 
prevent diffusion of extraneous electrolytes into the potassium chloride 
solution. In order to obtain reproducible results the mercury and mer- 
curous chloride should be pure; the latter must be free from mercuric 
compounds arid from bromides, and must not be too finely divided. 
A small quantity of mercury is placed at the bottom of the vessel; it is 
then covered with a paste of pure mercurous chloride, mercury and 
potassium chloride solution. The vessel is then completely filled with 
the appropriate solution of potassium chloride which has been saturated 




FIG. 72. Forms of calomel electrode 



witli calomel. Electrical connection N made l>y moan- 01 phtnum \\irr 
sealed into a glass tube, or through the walls of fh< vessel. The method 
employed for connecting the calomel electrode to another electrode so as 
to make a cell whose E.M F. can he measured depends on the type of 
electrode vessel. In the special form used by some workers, Fig. 72, I, 
this purpose is served by a side tube, sealed into the main vessel, while 
in the simple apparatus, consisting of a 2 or 4 oz. bottle, often employed 
for laboratory work (Fig. 72, II), a siphon tube provides the means of 
connection. The compact calomel electrode of thn type used with many 
commercial potentiometers is dipped directly into the solution of the 



234 ELECTRODE POTENTIALS 

other electrode system; electrical connection between the two solutions 
occurs at the relatively loose ground joint (Fig. 72, III). 

The Silver-Silver Chloride Electrode. In recent years the silver- 
silver chloride electrode has been frequently employed as a reference 
electrode for accurate work, especially in connection with the determina- 
tion of standard potentials by the use of cells containing chloride which 
are thus free from liquid junction potentials. The standard potential 
of the Ag, AgCl(s), Cl~ electrode is obtained as follows: the E.M.F. of 
the cell 

H 2 (l atm.) | H+C1- AgCl(s) | Ag, 

where the activities of the hydrogen and chloride ions in the solution of 
hydrochloric acid have arbitrary values, is given by equation (5), as 

R T 
E = J5? - -jr In a H +ocr, (15) 

since the hydrogen, silver and silver chloride are in their standard states. 
Replacing the product aH+flcr by a 2 , where a is the mean activity of the 
hydrochloric acid, equation (15) becomes 

\na. (16) 



This equation is seen to be identical with equation (14) of Chap. VI, 
and in fact the E derived on page 201 by suitable extrapolation of the 
E.M.F. data of cells of the type shown above, containing hydrochloric 
acid at different concentrations, is identical with the E of equations (15) 
and (16). It follows, therefore, that the standard E.M.F. of the cell under 
consideration is + 0.2224 volt at 25, and hence the standard E.M.F. of 
the corresponding cell with the electrodes reversed, i.e., 

Ag | AgCl(s) H+C1- | H 2 (l atm.) 

is 0.2224 volt. 3 By the convention adopted here, this represents the 
standard potential of the silver-silver chloride electrode; hence 

Ag | AgCl(s), Cl-(ocr = 1): E = - 0.2224 volt at 25. 

If the potential of this electrode is required in any arbitrary chloride 
solution, an estimate must, be made of the chloride ion activity of the 
latter; the potential can then be calculated by means of equation (76). 

Several methods have been described for the preparation of silver- 
silver chloride electrodes: a small sheet or short coil of platinum is first 
coated with silver by electrolysis of an argentocyanide solution, and this 
is partly converted into silver chloride by using it as an anode in a chloride 
solution. Alternatively, a spiral of platinum wire may be covered with 
a paste of silver oxide which is reduced to finely divided silver by heating 

Earned and Enters, /. Am. Chem. Soc., 54, 1350 (1932); 55, 2179 (1933). 



DETERMINATION OF STANDARD POTENTIALS 235 

to about 400; the silver is then coated with silver chforide by electrolysis 
in a chloride solution as in the previous case. A third method is to 
decompose byvheat a paste of silver chlorate, silver oxide and water 
supported on a small spiral of platinum wire; in this way an intimate 
mixture of silver and silver chloride is obtained. It appears that if 
sufficient time is permitted for the electrodes to "age," the three methods 
of preparation give potentials which agree within 0.02 millivolt. 4 

Electrodes similar to that just described, but involving bromide or 
iodide instead of chloride, have been employed as subsidiary reference 
electrodes for measurements in bromide and iodide solutions, respec- 
tively. They are prepared and their standard potentials (see Table 
XLIX) are determined by methods precisely analogous to those em- 
ployed for the silver-silver chloride electrode. 5 

Sulfate Reference Electrodes. For measurements in sulfate solu- 
tions, the electrodes 

Pb(Hg) | PbS0 4 (s), SO 
and 

Hg | Hg 2 S0 4 (s), SO 

have been found useful; their standard potentials may be determined 
by suitable extrapolation, as in the case of the silver-silver chloride elec- 
trode, or by measuring one electrode against the other. 6 The best 
values are 

Pb(Hg) | PbS0 4 (s), SOr-faor = 1): # = + 0.3505 at 25 
and 

Hg | Hg 2 SO 4 (s), S04~(a8o 4 " = 1): # = - 0.6141 at 25. 

If the electrodes are required for use as reference electrodes of known 
potential in sulfate solutions of arbitrary activity, an estimate of this 
activity must be made. 

Determination of Standard Potentials : Zinc. The procedure adopted 
for determining the standard electrode potential of a given metal or 
non-metal depends on the nature of the substance concerned; a number 
of examples of different types will be described in order to indicate the 
different methods that have been employed. 

When a metal forms a soluble, highly dissociated chloride, e.g., zinc, 
the standard potential is best obtained from measurements on cells with- 
out liquid junction, viz., 

Zn | ZnCl 2 (m) AgCl(s) | Ag. 

* Smith and Taylor, J. Res. Nat. Bur. Standards, 20, 837 (1938); 22, 307 (1939). 

Keston, /. Am. Chem. Soc., 57, 1671 (1935); Harned, Keston and Donelson, ibid., 
58, 989 (1936); Owen, ibid., 57, 1526 (1935); Cann and Taylor, ibid., 59, 1841' (1937); 
Gould and Vosburgh, ibid., 62, 2280 (1940). 

Shrawder and Cowperthwaite, J. Am. Cheni. Soc., 56, 2340 (1934); Harned and 
Hamer, ibid. t 57, 33 (1935). 



236 ELECTRODE POTENTIALS 

The cell reaction for the passage of two faradays is 

Zn() + 2AgCl(s) = Ag(s) + Zn++ + 2C1-, 
and the E.M.P., according to equation (2), is 



- (17) 

Since the zinc, silver chloride and silver are present as solids, and hence? 
are in their standard states, their activities are unity; hence, equation 
(17) becomes 

r>7 T 

E = - -_v In az.**a?v. (18) 

&r 

The standard E.M.F. of the coll, i.e., E, is equal to the difference between 
the standard potentials of the Zn, Zn+ f and Ag, AgCl(s), Cl~ electrodes; 
the value of the latter is known, - 0.2224 volt at 25, and hence if E Q 
of equation (18) were obtained the standard potential Ez n ,zn++ would 
be available. The evaluation of E is carried out by one of the methods 
described in Chap. VI in connection with determination of activities and 
activity coefficients; the problem in the latter case is to evaluate E for 
a particular cell, and this is obviously identical with that involved in the 
estimation of standard potentials. 7 

Other Bivalent Metals.--The standard potentials of a number of 
bivalent metals forming highly dissociated soluble sulfatcs, e.g., cadmium, 
copper, nickel and cobalt, as well as zinc, have been obtained from (ells 
of the type 

M [ M^SOr-(w) PbS0 4 (s) i Ph(IIg) 
and 

M | M++SOr-(w) Hg 2 8O 4 (a) | Ifg. 

The extrapolation procedure is in principle identical with that noted 
above, and since the standard potentials of the electrodes Pb(Hg), 
PbS() 4 (.s')> S()f~ and Hg, IlgaSfV.s-), &O~4 ~ are known, the standard 
potential of the metal -M can be evaluated. In several cases the E M.F. 
data for dilute solutions are not easily obtainable and consequently the 
extrapolation i* not reliable. It is apparent, however, from measurements 
in moderately concentrated solutions that the sulfates of copper, nickel, 
cobalt and zinc behave in an exactly parallel manner, and hence the 
mean activity coefficients are probably the same in each case. The 
values for zinc MI! fate are known, since K.M.F. measurements have been 
made at sufficiently low concentrations for accurate extrapolation and 
the evaluation of E (] to be possible. The assumption is then made that 
the mean activity coefficients are equal in the four sulfate solutions at 
equal ionic strengths. It is thus possible to derive the appropriate values 

'Scatchard and Tefft, J. Am. i,litm. Nor., 52, 2272 (1930); Getman, J. Phys. Chem., 
35, 2749 (1931). 



THE ALKALI METALS 237. 

of E Q , for the cells involving copper, nickel or cobalt sulfate, directly 
from the E.M.F. measurements by means of the equations 

f? 7 1 

E = E "" 2F ln a 



D/TT JPT* 

= Jjo - __ ln m __ _ ln 7> 

which are applicable to the sulfate cells; in this instance m is equal to 
the molality m of the 911! fate solution. 

The Alkali Metals. The alkali metals present a special case in the 
determination of standard potentials since these substances attack water; 
the difficulty has been overcome by making measurements in aqueous 
solution with a dilute amalgam which reacts slowly with water (cf. p. 198), 
and then comparing the potential of the amalgam with that of the pure 
metal in a non-aqueous medium with which it does not react. 8 

The E.M.P. of the stable arid reproducible cell 

Na (metal) | Nal in ethylamine | 0.206% Na(Hg) 

is + 0.8449 volt at 25, independent of the concentration of the sodium 
iodide solution; since the process occurring in the cell is merely the 
transfer of sodium from the pure metal to the dilute amalgam, the poten- 
tial must also be independent of the nature of the solvent or solute. The 
E.M.F. of the cell 

0.206% Na(Hg) | NaCl aq. 1.022 M Hg 2 Cl 2 (s) | Hg 
is + 2.1582 volts at 25, and hence that of the combination 

Na | NaCl aq. 1.022 M Hg,Cl() | Hg 
is + 3.0031 volts. The reaction occurring in this cell is 

Na + JHg,CI,() = Hg(0 + Na+ + Cl~, 
for the passage of one faraday, and so the E.M.F. is represented by 

RT 

E = E - -T In aNa+acr 





= E* ~ -jr In m - -y- In y f (20) 

Lewis and Kraus, J. Am. Chem. Soe., 32, 1459 (1910); Armbruster and Crenshaw, 
ibid., 56, 2525 (1934); Bent and Swift, ibid., 58, 2216 (1936); Bent, Forbes and Forziati, 
ibid., 61, 709 (1939). 



238 ELECTRODE POTENTIALS 

where the mean molality m is equal to the molality m of the sodium 
chloride solution. The molality of the solution is 1.022, and at this con- 
centration the mean activity coefficient of sodium chloride is known from 
other measurements (Chap. VI) to be 0.655; it is thus readily found from 
equation (20) that E is + 2.9826 volt at 25. The standard potential 
of the electrode Hg, Hg 2 Cl 2 (s), Cl~ is - 0.2680 volt,* and so the stand- 
ard potential of the sodium electrode is given by 

Na | Na+(a Na + = 1): # = + 2.7146 volts at 25. 

Cells with Liquid Junction. In the cases described above it has been 
possible to utilize cells without liquid junctions, but this is not always 
feasible: the suitable salts may be sparingly soluble, they may hydrolyze 
in solution, their dissociation may be uncertain, or there may be other 
reasons which make it impossible, at least for the present, to avoid the 
use of cells with liquid junctions. In such circumstances it is desirable 
to choose, as far as possible, relatively simple junctions, e.g., between 
two electrolytes at the same concentration containing a common ion or 
between two solutions of the same electrolyte at different concentrations, 
so that their potentials can be calculated with fair accuracy, as shown in 
Chap. VI. 

The procedure may ,be illustrated with reference to the determination 
of the standard potential of silver, of which the only convenient salt for 
experimental purposes is the nitrate. Since the most reliable reference 
electrodes contain solutions of halides, it is necessary to interpose a bridge 
solution between them; the result is 

Ag | AgNO 3 (0.1 N) : KN0 3 (0.1 N) j KC1(0.1 N) Hg 2 Cl 2 (s) | Hg, 

in which the liquid junctions, indicated by the dotted lines, are both of 
the type to which the Lewis and Sargent equation is applicable. The 
E.M.F. of the complete cell is 0.3992 volt and the sum of the liquid 
junction potentials is calculated to be + 0.0007 volt, so that the E.M.F. 
of the cell 

Ag | AgN0 3 (0.1 N) || KC1(0.1 N) H g2 Cl 2 (s) | Hg, 

where the double vertical line between the two solutions is used to imply 
the complete elimination of the liquid junction potential, is 0.3992 
+ 0.0007, i.e., - 0.3985 volt at 25. The potential of the Hg, Hg 2 Cl 2 (s), 
KC1(0.1 N) electrode is known to be - 0.3338 volt (p. .232) and so that 
of the Ag, AgN0 3 (0.1 N) electrode is - 0.7323 volt. The potential of 
the silver electrode may be represented by means of equation (9) as 



E = Eft* At* - -jr In a Ag +, (21) 

*This value is obtained by utilizing the observation that the potentials of the 
Hg, HgiCli() and Ag, AgCl(a) electrodes in the same chloride solution differ by 0.0456 
volt at 25. 



HALOGEN ELECTRODES 239 

and although E is known, the activity of the silver ions in 0.1 N silver 
nitrate is, of course, not available. It is necessary, therefore, to make 
an assumption, and the one commonly employed is to take the activity 
of the silver ions in the silver nitrate solution as equal to the mean 
activity of the ions in that solution. The mean activity coefficient of 
0.1 N silver nitrate is 0.733, and so the mean activity which is used for 
ax,* in equation (21) is 0.0733. Since E is - 0.7323 volt, it is readily 
found that E AK , AK + is - 0.7994 volt at 25. 

Halogen Electrodes. The determination of the standard potentials 
of the halogens is simple in principle; it involves measurement of the 
potential of a platinum electrode, coated with a thin layer of platinum 
or indium black, dipping in a solution of the halogen acid or a halide, 
and surrounded by the free halogen. The uncertainty due to liquid 
junction can be avoided by employing the appropriate silver-silver halide 
or mercury-mercurous halide electrode as reference electrode. In prac- 
tice, however, difficulties arise because of the possibility of the reactions 

X 2 + H 2 0^ HXO + H+ + X~ 
and 

x 2 + x-- x^-, 

where X 2 is the halogen molecule; the former reaction occurs to an 
appreciable extent with chlorine and bromine, and the latter with bro- 
mine and iodine. The first of these disturbing effects is largely elimi- 
nated by using acid solutions as electrolytes, but due allowance for the 
removal of halide ions in the form of perhalide must be made from the 
known equilibrium constants. 

The electrode reaction for the system X 2 , X~ is 



X- = iX 2 + e, 

e arguments o 
the equation 



so that by the arguments on page 228 the electrode potential is given by 

^. (22) 

For chlorine and bromine the standard states may be chosen as the gas 
at 1 atm. pressure, and if the gases are assumed to behave ideally, as will 
be approximately true at low pressures, equation (22) can be written in 
the form 



E = *, x - - ^r In p x , + -j- In a x -, (23) 

where px, is the pressure of the gas in atmospheres. 
In the cell 

HC1 soln. Hg 2 Cl 2 | Hg 



240 ELECTRODE POTENTIALS 

the reaction for the passage of one f araday is 

= Hg + |Cl 2 (p 



so that the E.M.F., which is independent of the nature of the electrolyte, 
provided it is a chloride solution, is given by 

R T 

E = E*-\np C i t , (24) 



The standard E.M.F. of this cell as given by equation (25), with the 
pressure in atmospheres, is the difference between the standard potentials 
of the C1 2 (1 atm.), Cl~ and the Hg, Hg 2 Cl 2 (s), Cl~ electrodes; since the 
latter is known to be 0.2680 volt at 25, the value of the former could 
be obtained provided E of the cell under consideration were available. 
This cell is, in fact, identical with the one for which measurements are 
given on page 222, and the results in the last column of Table XL VI 1 1 
are actually the values of E Q required by equation (25) above. It follows, 
therefore, taking a mean result of 1.090 volts at 25 for E Q , that the 
standard potential of the chlorine electrode is 1.090 0.2680, i.e., 
- 1.358 volts at 25. 

The standard potentials of bromine and iodine have been determined 
by somewhat similar methods; with bromine the results are expressed 
in terms of two alternative standard states, viz., the gas at 1 atm. pressure 
or the pure liquid. The standard state adopted for iodine is the solid 
state, so that the solution is saturated with respect to the solid phase. 9 
The acandard potential of fluorine has not been determined by direct 
experiment, but its value has been calculated from free energies derived 
from thermal and entropy data. 10 

The Oxygen Electrode. The standard potential of the oxygen elec- 
trode cannot be determined directly from E.M.F. measurements on account 
of the irreversible behavior of this electrode (cf. p. 353); it is possible, 
however, to derive the value in an indirect manner. The problem is to 
determine the E.M.F. of the cell 



H 2 (l atm.) | H+(a H + = 1) || OH-(a ir = 1) | 2 (1 atm.), 
in which the reaction for the passage of two faradays is essentially 

H 2 (l atm.) + iO 2 (l atm.) = H 2 O(/). 
The object of the calculations is to evaluate the standard free energy 

Lewis and Storch, J. Am. Chem. Soc. t 39, 2544 (1917); Jones and Baeckstrom, 
ibid., 56, 1524 (1934); Jones and Kaplan, ibid., 50, 2066 (1928). 

10 Latimer, J. Am. Chem. Soc., 43, 2868 (1926); see also, Glasstone, "Text-Book of 
Physical Chemistry," 1940, p. 993. 



THE OXYGEN ELECTRODE 241 

(A(?) of this process, for this is equal to 2FE, where E is the standard 
E.M.P. of the cell. 

According to equation (1), 

A<7 = - RTlnK, 
where K for the given reaction is defined by 



The activity of liquid water is taken as unity, since this is the usual 
standard state, and the activities of the hydrogen and oxygen are repre- 
sented by their respective pressures, since the gases do not depart appre- 
ciably from ideal behavior at low pressure; hence, equation (26) may be 
written as 



From a study of the dissociation of water vapor into hydrogen and 
oxygen at high temperatures, it has been found that the variation with 
temperature of the equilibrium constant K' p , defined by 



can be represented, in terms of the free energy change, by 

AC ' = - 57,410 + 0.94 T In T + 1.65 X lO-'T 2 - 3.7 X 10~ 7 r 3 + 3.92T. 

If the relationship may be assumed to hold down to ordinary tempera- 
tures, then at 25, 

- KT In K' p = AC ' = - 54,600 cal., 

and this is the free energy increase accompanying the conversion of one 
mole of hydrogen gas and one-half mole of oxygen to one mole of water 
vapor, all at atmospheric pressure. For the present purpose, however, 
the free energy required is that of the conversion of hydrogen and oxygen 
at atmospheric pressure to liquid water, i.e., to water vapor at 23.7 mm. 
pressure at 25. The difference between these free energy quantities is 

23 7 
RT In -- = - 2,050 cal. at 25, 



and hence the AG required is - 54,600 - 2,050, i.e., - 56,650 cal. 

An entirely different method of arriving at this standard free energy 
change is based partly on E.M.F. measurements, and partly on equilibrium 
data. From the dissociation pressure of mercuric oxide at various tem- 
peratures it is possible to obtain the standard free energy of the reaction 

Hg(i) + \V*(g) = HgO(*), 



242 ELECTRODE POTENTIALS 

and when corrected to 25 the result is found to be - 13,940 cal. The 
E.M.F. of the reversible cell 

H 2 (l atm.) | KOH aq. HgO(s) | Hg 

is + 0.9264 volt at 25, and so the free energy of the reaction 
H 2 (l atm.) + HgO(s) = H 2 0(0 



which occurs in the cell for the passage of two faradays, is 2 X 96,500 
X 0.9264 volt-coulombs, i.e., 42,760 cal. Since all the reactants and 
resultants in this reaction are in their standard states, this is also the 
value of the standard free energy change.* Addition of tl\e two results 
gives the standard free energy of the reaction 



as - 56,700 cal. at 25. 

As a consequence of several different lines of approach, all of which 
give results in close agreement, it may be concluded that the standard 
free energy of this reaction is 56,700 cal. at 25, and since, as seen 
above, this is equal to 2FE Q , it follows that the standard E.M.F. of the 
oxygen-hydrogen cell is 

56,700 X 4.185 00ft 
2 X 96,500 " L229 V ltS 

at 25. It would appear, at first sight, that this is also the standard 
potential of the oxygen electrode, but such is not the case. The E.M.F. 
calculated is the standard value for the cell 

H 2 (l atm.) | Water | 2 (1 atm.) 

in which both oxygen and hydrogen electrodes are in contact with the 
same solution, the latter having the activity of pure water. If the hydro- 
gen ion activity in this solution is unity, the hydrogen electrode potential 
is zero, by convention, and hence 1.229 volts is the potential of the elec- 
trode 

H,0(0, H+(a H + = 1) | 2 (1 atm.). 



The standard potential of oxygen, as usually defined, refers to the elec- 
trode 

2 (1 atm.) | OH-(a H- = 1), H 2 0(Z), 

that is, in which the hydroxyl ions are at unit activity. It is known 
from the ionic product of water (see Chap. IX) that in pure water at 25, 

dH+aoH- = 1.008 X 10~ 14 , 

and so 1.229 volts is the potential of the oxygen electrode, at 1 atm. 

* A small correction may be necessary because the activity of the water in the 
KOH solution will be somewhat less than unity. 



STANDARD ELECTRODE POTENTIALS 



243 



pressure, when the activity of the hydroxyl ions is 1.008 X 10~ M . The 
standard potential for unit activity of the hydroxyl ions is then derived 
from equation (96) in the form 

R T 
E = Eo t ,oir + -TT In aoir, 

which, for a temperature of 25, becomes in this case 

- 1.229 = E^oir + 0.05915 log (1.008 X 10~ 14 ), 
r = - 0.401 volt. 



Standard Electrode Potentials. By the use of methods, such as those 
described above, involving either E.M.F. measurements or free energy and 
related calculations, the standard potentials of a number of electrodes 
have been determined; some of the results for a temperature of 25 are 
recorded in Table XLIX. It should be noted that the signs of the 

TABLE XLIX. STANDARD POTENTIALS AT 25 



Elec- 
trode 


Reaction 


Poten- 
tial 


Electrode 


Reaction 


Poten- 
tial 


Li, Li+ 


Li - Li+ + 


4-3024 


Ha, OH- 


iHi4-OH--HO4- 


4-0.828 


K, K + 


K - K + + i 


4- 2.924 


Oi, OH- 


20H-->|04-Hi04-2 


-0.401 


Na, Na+ 


Na - Na* + 


4-2.714 








Zn, Zn++ 


Zn -Zn + + +2 


4-0.761 


Clifo), Cl- 


Cl- -* iCli 4- f 


-1.358 


Fe, Fe++ 


Fe -+ Fe ++ + 2 


4-0.441 


Brj(0, Br- 


Br- - iBri 4- 


-1.066 


Cd, Cd++ 


Cd-*Cd++ + 2 


4- 0.402 


iiw, i- 


!--**!+ 


-0.636 


Co, Co++ 


Co ->Co** +2 


4-0283 








Ni, Ni+ + 


Ni-*Ni ++ +2. 


4-0.236 


A K| AgCl(), Cl- 


Ag J Cl- -* AgCl 4- 


- 0.2224 


Sn, 8n++ 


Sn -Sn+ + 4-2c 


4-0.140 


Ag, AgBr(), Br- 


Ag 4- Br~ - AgBr 4- 


- 0.0711 


Pb, Pb++ 


Pb -*Pb+ + 4- 2 


4-0.126 


Ag, Agl(.). I- 


Ag 4-1- -AgI4-f 


4-0.1522 


Hi, H+ 


JH - H+ 4- c 


db 0.000 


Hg, HgiCh(s), Cl~ 


Hg4-Cl- -*iH gI Cli4- 


-0.2680 


Cu, Cu + + 


Cu - Cu ++ -f 2e 


- 0.340 


Hg, HgjS04(a) f S0 4 


2Hg 4-SOj- -*HgjS044-2 


-0.6141 


Ag, Ag+ 


Ag - AK + + 


- 0.799 








Hg, HgJ+ 


H-*iHgf+ + 


-0.799 









potentials correspond to the tendency for positive electricity to pass from 
left to right, or negative electricity from right to left, in each case; in 
general, therefore, the potentials in Table XLIX when multiplied by 
nF give the standard free energy increase for the reaction 

Reduced State > Oxidized State + ne, 

the corresponding value for hydrogen being taken arbitrarily as zero. 
For the reverse process, the signs of the potentials would be reversed. 
Since the potentials in Table XLIX give the free energies of the oxi- 
dation reactions, using the term oxidation in its most general sense, they 
may be called oxidation potentials ; the potentials for the reverse proc- 
esses, i.e., with the signs reversed, are then reduction potentials (cf. 
p. 435). 



244 ELECTRODE POTENTIALS 

Potentials in Non-Aqueous Solutions. Many measurements of vary- 
ing accuracy have been made of voltaic cells containing solutions in non- 
aqueous media; in the earlier work efforts were made to correlate the 
results with the potentials of similar electrodes containing aqueous 
solutions. Any attempt to combine two electrodes each of which con- 
tains a different solvent is doomed to failure because of the large and 
uncertain potentials which exist at the boundary between the two liquids. 
It has been realized in recent years that the only satisfactory method of 
dealing with the situation is to consider each solvent as an entirely inde- 
pendent medium, and not to try to relate the results directly to those 
obtained in aqueous solutions. Since the various equations derived in 
this and the previous chapter are independent of the nature of the 
solvent, they may be applied to voltaic cells containing solutions in 
substances other than water. 

By adopting the convention that the potential of the standard hy- 
drogen electrode, i.e., with ideal gas at 1 atm. pressure in a solution of 
unit activity of hydrogen ions shall be zero in each solvent, and using 
methods essentially similar to those described above, the standard poten- 
tials of a number of electrodes have been evaluated in methyl alcohol, 
ethyl alcohol and liquid ammonia. These values represent therefore, in 
each case, the E.M.F. of the cell 

M | M+(a M * = 1) II H+(a H + = 1) | H 2 (l atm.), 
where M is a metal, or of 

A | A~(a A - = 1) || H+(a H + = 1) | H,(l atm.) 

if A is a system yielding anions. It would appear at first sight that since 
the cell reaction, as for example in the former case, 

M(s) + H+(a H + = 1) = M+(a M + = 1) + |H 2 (1 atm.) 

is the same in all solvents, the E.M.F. should be independent of the nature 
of the solvent. It must be remembered, however, that both M+ ions 
and hydrogen ions are solvated in solution, and since the ions which 
actually exist in the respective solvents are quite different in each case, 
the free energy of the reaction will depend on the nature of the solvent. 
This subject will be considered shortly in further detail. 

A number of standard potentials reported for three non-aqueous 
solvents are compared in Table L with the corresponding values for water 
as solvent; ll it should be emphasized that although the standard poten- 
tial of hydrogen is set arbitrarily at zero for each solvent, the actual 
potentials of these electrodes may bo quite different in the various media. 
The results in each solvent are, however, comparable with one another 
and it will be observed that there is a distinct parallelism between the 

"Buckley and Hartley, Phil. Mag., 8, 320 (1929); Macfarlane and Hartley, ibid., 
13, 425 (1932); 20, 611 (1935); Pleskow and Monossohn, Acta Physicochim. U.R.S.S., 1, 
871 (1935); 2, 615, 621, 679 (1935). 



FACTORS AFFECTING ELECTRODE POTENTIALS 245 

TABLE L. STANDARD ELECTRODE POTENTIALS IN DIFFERENT SOLVENTS 

Electrode H 2 O CH,OH C 2 HOH NH, 

(25) (25) (25) (-50) 

Li,Li+ +3.024 +3.095 +3.042 

K,K+ +2.924 +1.98 

Na,Na+ +2.714 +2.728 +2.657 +1.84 

Zn,Zn++ +0.761 +0.52 

Cd,Cd++ +0.402 +0.18 

T1/T1+ +0.338 +0.379 +0.343 

Pb,Pb^ +0.126 -0.33 

H 2 ,H+ 0.000 0.000 0.000 0.000 

Cu,Cu++ -0.340 -0.43 

Ag,Ag+ -0.799 -0.764 -0.749 -0.83 

C1 2 ,C1- -1.358 -1.116 -1.048 -1.28* 

Br 2 ,Br- - 1.066 - 0.837 - 0.777 - 1.08 * 

I,,I- -0.536 -0.357 -0.305 -0.70* 

* Calculated from free energy data at about c. 

standard potentials of the various electrodes in the four solvents. The 
tendency for the reaction M > M+ + to occur, as indicated by a high 
positive value of the potential, is always greatest with the alkali metals 
and least with the more noble metals, e.g., copper and silver. The order 
of the halogens is also the same in each case. 

Factors Affecting Electrode Potentials. If E is the standard poten- 
tial of a metal in a given solvent, then it is evident from the arguments 
given above that zFE is equal to the standard free energy of the 
reaction 

M + zH+ = M'* 



This reaction, which is the displacement of hydrogen ions from the solu- 
tion and their liberation as hydrogen gas, is virtually that occurring 
when a metal dissolves in a dilute acid solution, provided there are no 
accompanying complications, e.g., formation of complex ions. It follows, 
therefore, that zFE Q may be regarded as the standard free energy of 
solution of the metal. 

According to thermodynamics 



and experiments have shown that the standard entropy change AS re- 
sulting from the solution of a metal in dilute acid is relatively small 
compared with the heat change A#; it is possible, therefore, to write as 
a very approximate relationship 

- zFE* A//, 

where AH is heat of solution of the metal. In general, therefore, a 
parallelism is to be expected between the latter quantity and the stand- 



246 ELECTRODE POTENTIALS 

ard potential of the metal; hence the factors determining the heat of 
solution may be regarded as those influencing the standard potential. 11 

In order to obtain some information concerning these factors the 
reaction involved in the solution of the metal may be imagined to take 
place in a series of stages, as shown in Fig. 73; the reactants, M and 




FIQ. 73. Theoretical stages in solution of a metal in acid 

solvated hydrogen ions, are shown at the left, and the products, hydrogen 
gas and solvated M*+ ions, at the right. The stages are as follows: 

I. An atom of the metal is vaporized; the heat supplied is equal to the 
heat of sublimation, S; hence, 

Affi = + S. 

II. The atom of vapor is ionized to form metal ions M*+ and z elec- 
trons; the energy which must be supplied is determined by the ionization 
potential of the metal, the various stages of ionization being taken into 
consideration if the ion has more than one charge. If /M is the sum of the 
ionization potentials, the energy of ionization is /MC, and if it is supposed 
that this is converted into the standard units of energy used throughout 
these calculations, then 



III. The gaseous metal ion is dissolved in the solvent, when energy 
equal to the heat of solvation WM+ is evolved; hence, 



IV. An equivalent quantity of solvated hydrogen ions (z ions) are 
removed from the solvent; the energy of solvation WH+ per ion is ab- 
sorbed, so that 

AHiv = + zW*+. 

V. The unsolvated (gaseous) hydrogen ions are combined with the 
electrons removed from the metal to form atomic hydrogen; if JH is the 
ionization potential of the hydrogen atom, then 



since z electrons are added. 

"Butler, "Electrocapillarity," 1940, Chap. III. 



ABSOLUTE SINGLE ELECTRODE POTENTIALS 247 

VI. The hydrogen atoms are combined in pairs to form hydrogen 
molecules; if DH S is the heat of dissociation of a hydrogen molecule into 
atoms, then 



The net result of these six stages is the same as the solution of a 
metal in a dilute acid; hence Aflf for this process is given by the sum of 
the six heat changes recorded for the separate stages, thus, assuming 
constant pressure, 



S + J M 6 - WM+ - 2(iDH, + /H+ - FH+). (28) 

The quantity in parentheses is characteristic of the hydrogen electrode 
in the given solvent, and so the factors which determine the heat of 
solution of a particular metal, and consequently (approximately) its 
standard potential, may be represented by the expression 

A# M S + 7 M - WM+. 

The standard potential of a metal in a given solvent thus apparently 
depends on the sublimation energy of the metal, its ionization potential 
and the energy of solvation of the ions. Calculations have shown that 
of these factors the heat of sublimation is much the smallest, but since 
the other two quantities generally do not differ very greatly, all three 
factors must play an important part in determining the actual electrode 
potential. 

When comparing the heat changes accompanying the solution of a 
given metal in different media, it is seen that the factors S, IM, DH, and 
JH are independent of the nature of the solvent. The standard potential 
of the metal in different solvents is thus determined by the quantity 
TFn+ WM+, where FH+ and TF M + are the energies, strictly the free 
energies, of solvation of the hydrogen and M+ ions, respectively; this 
result is in agreement with the general conclusion reached previously 
(p. 244). For a series of similar solvents, such as water and alcohols, 
the values of WH+ WM+ for a number of metals will follow much the 
same order in each solvent; in that case the standard potentials will show 
the type of parallelism observed in Table L. On the other hand it would 
not be surprising if for dissimilar solvents, e.g., water and acetonitrile, 
the order followed by the potentials of a number of electrodes was quite 
different in the two solvents. 

Absolute Single Electrode Potentials. The electrode potentials dis- 
cussed hitherto are actually the E.M.F.'S of cells resulting from the com- 
bination of the electrode with a standard hydrogen electrode. A single 
electrode potential, as already seen, involves individual ion activities and 
hence has no thermodynamic significance; " the absolute potential differ- 
ence at an electrode is nevertheless a quantity of theoretical interest. 
Many attempts have been made to set up so-called "null electrodes " 

See, for example, Guggenheim, J. Phys. Chem., 33, 842 (1929). 



248 ELECTRODE POTENTIALS 

in which there is actually no differenfee of potential between the metal 
and the solution; if such an electrode were available it would be possible 
by combining it with another electrode to derive the absolute potential 
of the latter. It appears doubtful, however, whether the "null elec- 
trodes" so far prepared actually have the significance attributed to them, 
since they generally involve relative movement of the metal and the 
solution (cf. Chap. XVI). A possible approach to the problem is based 
on a treatment similar to that used in the previous section. 

The absolute single potential of a metal is a measure of the standard 
free energy of the reaction 

M + solvent = M*+ (solvated) + zt, 

and this process may be imagined to occur by the series of stages de- 
picted in Fig. 74. These, with the accompanying free energy changes, 
are vaporization of the metal (+ *S); ionization of the atom in the vapor 



M+(solvated) 




d+ (vapor) 
FIG. 74. Theoretical stages in formation of ions in solution 

state (/M*); solvation of the gaseous ion ( WM+)} and finally return 
of the electrons produced in the ionization stage to the metal ( 2<e), 
where <t> is the electronic work function of the metal.* It follows, there- 
fore, that 

AG = 8 + IM - WM+ - z<t*. (29) 



Since S, /M and <, as well as z and e, may be regarded as being known 
for a given metal, it should be feasible to evaluate A(? for the ionization 
process, provided the free energy of solvation of the M+ ions, i.e., WM + , 
were known. 

The sum of the energies of solvation of the ions of a salt can be esti- 
mated, at least approximately, from the heat of solution of the salt and 
its lattice energy in the crystalline form. There is, unfortunately, no 
direct method of dividing this sum into the contributions for the separate 
ions; it is of interest, however, to consider the theoretical approach to 
this problem as outlined in the following section. 

Free Energy of Solvation of Ions. If the solvent medium is con- 
sidered as a continuous dielectric, the free energy of solvation may be 

*The electronic work function, or thermionic work function, generally expressed 
in volts, is a measure of the amount of energy required to remove an electron from the 
metal; zfc is, therefore, the free energy change, in electron-volts, accompanying the 
return of the z electrons to the metal. 



FREE ENERGY OF SOLVATION OF IONS 249 

regarded as equivalent to the difference in the electrostatic energy of a 
gaseous ion and that of an ion in the medium of dielectric constant D. 
In order to evaluate this quantity, use is made of the method proposed 
by Born: 14 the free energy increase accompanying the charging of a 
single gaseous ion, i.e., in a medium of dielectric constant unity, is zV/2r, 
where ze is the charge carried by the ion and r is its effective radius, the 
ion being treated as a conducting sphere. If the same ion is charged in 
a medium of dielectric constant D, the free energy change is zV/2Dr, 
and so the increase of free energy accompanying the transfer of the 
gaseous ion to the particular medium, which may be equated to the free 
energy of solvation, is given by the Born equation as 



where AT, the Avogadro number, is introduced to give the free energy 
change per mole. 

One of the difficulties in applying the Born equation is that the 
effective radius of the ion is not known; further, the calculations assume 
the dielectric constant of the solvent to be constant in the neighborhood 
of the ion. The treatment has boon modified by Webb 15 who allowed 
for the variation of dielectric constant and also for the work required to 
compress the solvent in the vicinity of the ion; further, by expressing the 
effective ionic radius as a function of the partial molal volume of the 
ion, it was possible to derive values of the free energy of solvation without 
making any other assumptions concerning the effective ionic radius. 

Another approach to the problem of ionic solvation has been made by 
Latimer and his collaborators; 16 by taking the effective radii of negative 
halogen ions as 0.1 A greater than the corresponding crystal radii and 
those of positive alkali metal ions as 0.85A greater than the crystal radii, 
it has been found possible to divide up the experimental free energies of 
hydration of alkali halides into the separate values for the individual 
ions. The results so obtained are in agreement with the requirements 
of the original form of the Born equation with the dielectric constant 
equal to the normal value for water. 

The free energies of hydration of single ions derived by the different 
methods of computation show general agreement. For univalent ions 
the values are approximately 70 to 100 kcal. per g.-ion; the hydrogen ion 
is exceptional in this respect, its free energy of hydration being about 
250 kcal. In any series of ions, e.g., alkali metal ions or halide ions, the 
hydration free energy usually decreases with increasing mass of the ion. 

In spite of the fact that the different treatments yield similar values, 
it must be emphasized that there is considerable doubt if the results are 

14 Born, Z. Physik, 1, 45 (1920). 

15 Webb, /. Am. Chem. Hoc., 48, 2,"589 (1926). 

16 Latimor, Pitzcr and Slansky, J. Chem. Phys., 7, 108 (1939). 



250 ELECTRODE POTENTIALS 

of sufficient significance to permit of their use in the determination of 
absolute potentials. 17 The problem of single potentials must, therefore, 
still be regarded as incompletely solved. 

Rates of Electrode Processes. When a metal M is inserted in a 
solution of its ions M(H 2 O)2", the solvent being assumed for simplicity to 
be water, there will be a tendency for the metal to pass into solution as 
ions and also for the ions from the solution to discharge on to the metal; 
in other words the two processes represented by the reversible reaction 

*i 
M(H 2 0)+ + e^ M + zH 2 O 

will occur simultaneously, the ions M(H 2 O)j~ being in solution and the 
electrons on the metal. When equilibrium is attained, and the revers- 
ible potential of the electrode is established, the two reactions take place 
at equal rates. 

According to modern views, 18 the rate of a process is equal to the 
specific rate, defined in terms of the accepted standard states, multiplied 
by the activities of the reacting species;* if ki and & 2 are the specific rates 
of the direct and reverse processes represented above, in the absence of 
any potential difference, then, since a+ is the activity of the solvated ions 
in solution and the activity of the solid metal is unity, by convention, 
the rates of the reactions are k\a+ and & 2 , respectively. If k 2 is greater 
than k\a+, that is to say, if the reverse reaction in the absence of a poten- 
tial difference at the electrode, i.e., the passage of ions from the metal 
into the solution, is more rapid than the direct reaction, i.e., the discharge 
of ions, the cations will pass into solution from the metal more rapidly 
than they can return. As a result, therefore, free electrons will be left 
on the metal and positive ions will accumulate on the solution side of the 
electrode, thus building up what is known as an electrical double layer 
(see Chap. XVI); the potential difference across this double layer is the 
single electrode potential. The setting up of the double layer, with its 
associated potential difference, makes it more difficult for ions to leave 
the negatively charged metal and enter the solution, while the transfer 
of ions to the metal, i.e., the direct reaction, is facilitated. When equi- 
librium is established the two processes are occurring at the same rate 
and the electrode exhibits its reversible potential. 

If E is the actual potential difference across the double layer, formed 
by the electrons on the metal and the ions in solution, it may be supposed 
that a fraction a of this potential facilitates the discharge of ions, while 
the remainder, 1 a, hinders the reverse process, i.e., the passage of ions 

" Fnimkin, J. Chem. Phys., 7, 552 (1939). 

" Glasstone, Laidler and Eyring, "The Theory of Rate Processes," 1941, Chap. X. 

* Strictly speaking, the result should be divided by the activity coefficient of the 
"activated state 1 ' for the reaction; in any case this factor cancels out when equilibrium 
processes are considered. 



ELECTRODE POTENTIALS AND EQUILIBRIUM CONSTANTS 251 

from the metal into the solution. The actual value of a, which lies be- 
tween zero and unity, is immaterial for present purposes, since it cancels 
out at a later stage. In its transfer across the double layer, therefore, 
the free energy of the discharging ion is increased by azFE, where z is 
the valence of the ion, while the free energy of the atom which passes 
into .solution is diminished by an amount (1 a)zFE. The result of 
these free energy changes Is that, in the presence of the double layer 
potential E, the rates of the forward and reverse reactions under con- 
sideration are : 

Rate of discharge of ions from solution = kia + e azFE/RT 
Rate of passage of ions into solution = k^e~ (l ~ a)zFEIRT J 

where, as already seen, the corresponding rates in the absence of the 
potential are k\a+ and fe, respectively. At the equilibrium (reversible) 
potential the rates of the two processes must be equal; hence 



'. # = ~m---,lna+. (30) 

zF k 2 zb * 

Since ki/kz is a constant at definite temperature, this equation is obvi- 
ously of the same form as the electrode potential equations derived by 
thermodynamic methods, e.g., equation (86) for an electrode reversible 
with respect to positive ions. The first term on the right-hand side of 
equation (30) is clearly the absolute single standard potential of the 
electrode; it is equal to the standard free energy of the conversion of solid 
metal to solvated ions in solution divided by zF, and its physical signifi- 
cance has been already discussed. 

Electrode Potentials and Equilibrium Constants. According to equa- 
tion (1) the standard E.M.F., i.e., B, of any reversible cell can be related 
to f]ie equilibrium constant of the reaction occurring in the cell by the 
expression 

E' = !~]nK, (31) 

and hence a knowledge of the standard E.M.F. permits the equilibrium 
constant to be calculated, or vice versa. 
The reaction occurring in the cell 

Zn | ZnS0 4 aq. || CuS0 4 aq. | Zn, 
for example, for the passage of two faradays is 

Zn + Cu++aq. = Zn ++ aq. + Cu, 



252 ELECTRODE POTENTIALS 

and if E is the standard E.M.F., it follows from equation (31) that 



,-0 RT /azn"0cu\ 

EZ&.C* = -^r In I I i 

*r \aznOcu* */ 



the suffix e indicating that the activities involved are the equilibrium 
values. Since the solid zinc and copper constituting the electrodes are 
in their standard states, their respective activities are unity; hence, 



(32) 



If Ezn,zn++ and Jcu,cu++ represent the standard electrode potentials on the 
hydrogen scale of the zinc and copper electrodes, as recorded in Table 
XLIX, then Ezn.z*++ is actually the E.M.F. of the cell 



Zn | Zn++(a Za " = 1) || H+(a H + = 1) | H 2 (l atm.), 
while JScu.cu** is the E.M.F. of the cell 

Cu | Cu+ + (ocu" = 1) || H+(a H + = 1) | H 2 (l atm.). 
Hence the E.M.F. of the cell 

Zn | Zn++(am*+ = 1) || Cu ++ (acu+* = 1) I Cu, 

which has been defined above as #zn,cu, is also equal to Ez*. zn + + 
It follows, therefore, from equation (32) that 

(33) 



and inserting the standard potentials from Table XLIX, the result is, 
at 25, 

+ 0.- (-0.340, 



-1.7X10". 



The ratio of the activities of the zinc and copper ions at equilibrium will 
be approximately equal to the ratio of the concentrations under the same 
conditions; it follows, therefore, that when the system consisting of zinc, 
copper and their bivalent ions attains equilibrium the ratio of the zinc 
ion to the copper ion concentration is extremely large. If zinc is placed 
in contact with a solution of cupric ions, e.g., copper sulfatc, the zinc will 
displace the cupric ions from solution until the Cz n + *Ateu* f ratio is about 
10 37 ; in other words the zinc will replace the copper in solution until the 
quantity of cupric ions remaining is too small to be detected. 

It is thus possible from a knowledge of the standard electrode poten- 
tials of two metals to determine the extent to which one metal will re- 
place another, or hydrogen, from a solution of its ions. In the general 



ELECTRODE POTENTIALS AND EQUILIBRIUM CONSTANTS 253 

case of two metals MI and M 2 , of valence z\ and 2 2 , respectively, the 
reaction which occurs for the passage of z& 2 faradays is 



and the corresponding general form of equation (33) is 

*- (34) 



It can be seen from this equation that the greater the difference between 
the standard potentials of the two metals MI and M 2 , the larger will be 
the equilibrium ratio of activities (or concentrations) of the respective 
ions. The greater the difference between the standard potentials, there- 
fore, the more completely will one metal displace another from a solution 
of its ions. The metal with the more positive (oxidation) potential, as 
recorded in Table XLIX, will, in general, pass into solution and displace 
the metal with the less positive potential. The series of standard poten- 
tials, or electromotive series, as it is sometimes called, thus gives the 
order in which metals are able to displace each other from solution; the 
further apart the metals are in the series the more completely will the 
higher one displace the lower one. It is not true, however, to say that 
a metal lower in the series will not displace one higher in the series; some 
displacement must always occur until the required equilibrium is estab- 
lished, and the equilibrium amounts of both ions are present in the 
solution. 

By re-arranging equation (34) the result is 

zprn 7? / 7 7 

tfV.Mt - ^ In (a M j). = BM..M; - p In (a M j).. (35) 

The left-hand side of this equation clearly represents the reversible poten- 
tial of the metal Mi in the equilibrium solution and the right-hand side 
is that of the metal M 2 . It must be concluded, therefore, that when the 
metal Mi is placed in contact with a solution of Mt ions, or M 2 is placed 
in a solution of Mi" ions, or in general whenever the conditions are such 
that the equilibrium 

Mi + Mt^ Mt + M 2 

is established, the reversible potential of the system MI, Mt is equal to 
that of M 2 , Mt . It is clear from equation (35) that the more positive 
the standard potential of a given metal, the greater the activity of the 
corresponding ions which must be present at equilibrium, and hence t[u 
more completely will it displace the other metal. 

Although equations (34) and (35) are exact, the qualitative conclu- 
sions drawn from them are not always strictly correct; for example, sincv 
copper has a standard potential of 0.340 on the hydrogen scale, u 
would be expected, as is true in the majority of cases, that copper should 
be unable to displace hydrogen from solution. It must be recorded, 



254 ELECTRODE POTENTIALS 

however, that copper dissolves in hydrobromic acid, and even in potas- 
sium cyanide solution, with the liberation of hydrogen. The reason for 
this surprising behavior is to be found in the fact that in both instances 
complex ions are formed whereby the cupric ions are removed from the 
solution. It is true that when equilibrium is attained the concentration 
(or activity) of cupric ions is very small in comparison with that of the 
hydrogen ions, but in order to attain even this small concentration it is 
necessary for a considerable amount of copper to pass into solution; most of 
this dissolved copper is present in the form of complex ions, and it is the 
amount of free cupric ions in equilibrium with these complexes that must 
be inserted in equation (34) or (35). 

It is of interest to note that if the equilibrium constant of the system 
consisting of two metals and their simple ions could be determined experi- 
mentally and the standard potential of one of them were known, the 
standard potential of the other metal could be evaluated by means of 
equation (34). This method was actually used to obtain the standard 
potential of tin recorded in Table XLIX. Finely divided tin and lead 
were shaken with a solution containing lead and tin perchlorates until 
equilibrium was attained; the ratio of the concentrations of lead and 
stannous ions in the solution was then determined by analysis. The 
standard potential of lead being known, that of tin could be calculated. 

Electrode Potentials and Solubility Product. The solubility product 
is an equilibrium constant, namely for the equilibrium between the solid 
salt on the one hand and the ions in solution on the other hand, arid 
methods are available for the evaluation of this property from E.M.F. 
measurements. 

The reaction taking place in the cell 

C1 2 (1 atm.) | HC1 AgCl() | Ag 
for the passage of one faraday is readily seen to be 
AgCl(s) = Ag + JC1,(1 atm.), 

but since the solid silver chloride is in equilibrium with silver and chloride 
ions in the solution, the reaction can be considered to be 

AgCl(s) - Ag+ + Cl- = Ag + 1C1,(1 atm.). 
The E.M.F. of the cell is then written as 



, (36) 

aAg+Ocr 

where a^+ and Ocr refer to the activities in the saturated solution. The 
value of E in this equation is the E.M.F. of the cell in which the activity 
of the chlorine gas and of chloride ions on the one hand, an J of solid 
silver and silver ions on the other hand, are unity; these conditions arise 
for the standard C1 2 , Cl~ and Ag, Ag 4 " electrodes, respectively, so that U 



ELECTRODE POTENTIALS AND SOLUBILITY PRODUCT 255 

in equation (36) is defined by 



Since solid silver and chlorine gas at atmospheric pressure are the re- 
spective standard states, i.e., the activity is unity, equation (36) can be 
written as 

ffT 1 
E - Z$i t .cr - #Ag,A g + + -y In a A .nicr. (37) 

The product a AK + acr in the saturated solution may be replaced by the 
solubility product of silver chloride, i.e., A^Agco, and so equation (37) 
becomes 

D/Tt 

E = Eci t .ci ^A,Ag* + ~rr In /(Agci). 

It is seen from Table XLIX that #ci 2 .cr and E/ig.^ are respectively 
- 1.358 and - 0.799 volt at 25; hence", 

E = - 1.358 + 0.799 + 0.05915 log /C. (A ci). 

From measurements on the cell depicted at the head of this section, it is 
found that E is 1.136 volt at 25, and consequently it follows that 

AW,) = 1.78 X lO- 10 . 

The value derived from the solubility of silver chloride obtained by the 
conductance method is 1.71 X 10~ 10 . 

In general, the above procedure can be applied to any sparingly 
soluble salt, provided an electrode can be obtainable which is reversible 
with respect to each ion, viz., 

A | Soluble salt of A~ ions MA() | M, 

although for a hydroxide, the oxygen electrode may be replaced by a 
hydrogen electrode. 

A less accurate method for the determination of solubility products, 
but which is of wider applicability, is the following. If MA is the 
sparingly soluble salt, and NaA is a soluble salt of the same anion, then 
the potential of the electrode M, MA(s), NaA aq. may be obtained by 
combining it with a reference electrode, e.g., a calomel electrode, thus 

M | MA() NaA aq. || KC1 aq. Hg 2 Cl 2 (s) | Hg, 

with a suitable salt bridge to minimize the liquid junction potential, and 
measuring the E.M.F. of the resulting cell. Since the potential of the 
calomel electrode is known, that of the other electrode may be evaluated, 
on the hydrogen scale. The potential of the M | MA(s) NaA electrode 
which can be treated as reversible with respect to M+ ions as well as to 



256 ELECTRODE POTENTIALS 

A~ ions, may be written as 

E = J&M.M+ ~~p* In OM*, 

and if #M,M* is known, the activity of the M+ ions in the solution satu- 
rated with MA can be calculated. The activity of the A~ ions may be 
taken as approximately equal to the mean activity of the salt MA whose 
concentration is known; the product of a\i + and a A - in the solution then 
gives the solubility product of MA. 

Electrometric Titration: Precipitation Reactions. One of the most 
important practical applications of electrode potentials is to the deter- 
mination of the end-points of various types of titration; 19 the subject 
will be treated here from the standpoint of precipitation reactions, while 
neutralization and oxidation-reduction processes are described more con- 
veniently in later chapters. 

Suppose a solution of the soluble salt MX, e.g., silver nitrate, is titrated 
with a solution of another soluble salt BA, e.g., potassium chloride, with 
the result that the sparingly soluble salt MA, e.g., silver chloride, is pre- 
cipitated. Let c moles per liter be the initial concentration of the salt 
MX, and suppose that at any instant during the titration x moles of BA 
have been added per liter; further, let y moles per liter be the solubility 
of the sparingly soluble salt MA at that instant. The value of y will 
vary throughout the course of the titration since the concentration of M 
ions is being continuously altered. If the salts are assumed to be com- 

pletely dissociated, the concentration of M+ at any instant is given by 

A Vt<f- *> ? 
CM* = c x + y,* 

where c x is due to unchanged MX and y to the amount of the sparingly 
soluble MA remaining in solution. The simultaneous concentration of 
A~ ions is then 



because the A" ions in solution arise solely from the solubility of MA, 
the remainder having been removed in the precipitate. Since the solu- 
tion is saturated with MA, it follows from the approximate solubility 
product principle, assuming activity coefficients to be unity, that 

k. = CM* X C A - = (c - x + y)y, (38) 

where k is the concentration solubility product. 

"For reviews, see Kolthoff and Furman, " Potentiometric Titrations," 1931; 
Furman, Ind. Eng. Chem. (Anal. Ed.), 2, 213 (1930); Trans. Ekctrochem. Soc., 76, 45 
(1939); Gladstone, Ann. Rep. Chem. Soc., 30, 283 (1933); Glasstone, "Button's Volu- 
metric Analysis/ 1 1935, Part V. 

* The change of volume during titration is neglected since its effect is relatively 
small. 



ELECTROMETRIC TITRATION 257 

If an electrode of the metal M, reversible with respect to M+ ions, 
were placed in the solution of MX during the titration, its potential 
would be given by 

E = #M,M+ -- Erhia M + 

Zr 

RT 
SW-^rln(c-* + 0), (39) 

where the activity of M+ ions, i.e., a M +, has been replaced by the concen- 
tration as derived above. If the solubility product A;, is available, then 
since c and x are known for any point in the titration, it is possible to 
calculate y by means of equation (38) ; the values of c x + y can now 
be inserted in equation (39) and the variation of electrode potential 
during the course of titration can be determined. At the equivalence- 
point, i.e., the ideal end-point of the titration, when the amount of BA 
added is equivalent to that of MX initially present, c and x are equal; 
equation (39) then reduces to 

E #8t,M+ 



Jk.. (40) 

r 

Should the titration be carried beyond the end-point, the value of 
CA~ now becomes x c + y, while that of CM+ is y, since the solution now 
contains excess of A*~ ions; x c arises from the excess of BA over MX, 
and y from the solubility of the sparingly soluble MA. The solubility 
product is given by 

k. = y(x - c + y), 

and equation (39) becomes 

E E2f, M +-^lnt,. (41) 

The value of y can be calculated as before, if the solubility product is 
known, and hence the electrode potential of M can be determined. 

By means of equations (38), (39) and (41) it is thus possible to calcu- 
late the potential of an electrode of the metal M during the course of the 
whole precipitation titration, from the beginning to beyond the equiva- 
lence-point, provided the solubility product of the precipitated salt is 
known. The calculations show that there is at first a gradual change of 
potential, but a very rapid increase occurs as the equivalence-point is 
approached; the change of potential for a given increase in the amount 
of the titrant added, i.e., dE/dx, is found to be a maximum at the theo- 
retical equivalence-point. This result immediately suggests a method 



258 ELECTRODE POTENTIALS 

for determining experimentally the end-point of a precipitation titration 
by E.M.F. measurement. The reversible potential (E) of an M electrode 
during the course of the titration is plotted against the amount of titrant 
added (x); the point at which the potential rises most sharply, i.e., the 
point of inflection where dE/dx is a maximum, is the required end-point. 
This procedure constitutes the fundamei.cal basis of potentiometric 
titration. 

The same general conclusion may be reached without going through 
the Jetciiled calculations just described. If equation (39) is differentiated 
twice with respect to x and the resulting expression for d~E/dx* equated 
to zero, the condition for dE/dx to be a maximum can be obtained. This 
is found to be that x should be equal to c, which is, of course, the con- 
dition for the equivalence-point, in agreement with the conclusion already 
reached. 

By differentiating equation (39) with respect to x it is seen that at 
the equivalence-point the value of dE/dx is inversely proportional to Vfc,. 
The potential jump observed at the end-point is thus greater the smallei 
the solubility product /;. of the precipitate. The sharpness of a particular 
titration can thus often be improved by the addition of alcohol to the 
solution being titrated in order to reduce the solubility of the precipi- 
tated salt. 

In the treatment given here it has been assumed that the precipitate 
MA is a salt of symmetrical valence type; if it is an unsymmetrical salt, 
e.g., MaA or MA2, the potential-titratioii curve, i.e., the plot of the 
potential (E) against the amount (x) of titrant added, is not symmetrical 
and tho maximum value of dE/dx does not occur exactly at the equiva- 
lence-point. The deviations are, however, relatively small if the solu- 
bility product of the precipitate is small and the titrated solutions are 
not too dilute. 

Potentiometric Titration: Experimental Methods. Since the silver 
electrode generally behaves in a satisfactory manner, the potentiometric 
method of titration can be applied particularly to the estimation of 
anions which yieid insoluble silver salts, ?e.g., halides, cyanides, thio- 
cyanates, phosphates, etc. In its simplest form, the experimental pro- 
cedure is to take a known volume of the solution containing the aniou 
to be titrated and to insert a clean silver sheet or wire, preferably coated 
with silver by the electrolysis of an argentooyanide solution; this con- 
stitutes the "indicator" electrode, and its potential is measured by 
connecting it, through a salt bridge, with a reference electrode, e.g., a 
calomel electrode. Since the actual electrode potential is not required, 
but merely the point at which it undergoes a rapid change, the E.M.F. 
of the resulting cell is recorded after the addition of known amounts of 
the silver nitrate solution. The values obtained in the course of the 
titration of 10 cc. of approximately 0.1 N sodium chloride with 0.1 N 
silver nitrate, using a silver indicator electrode and a calomel reference 



POTENTIOMETRIC TITRATION 



259 



TABLE LI. POTENTIOMETRIC TITRATION OF SODIUM CHLORIDE WITH SILVER NITRATE 



ANOi (t) 


E 


*E 


AP 


A/Av 


0.1 cc. 
5.0 
8.0 
10.0 
11.0 


114 mv. 
130 
145 
168 
202 


16 
15 
23 
34 


4.9 
3.0 
2.0 
1.0 


3.3 

5.0 
11.5 
34 


11.10 
11.20 
11.30 
11.35 
11.40 
11.45 
11.50 
12.0 
13.0 
14.0 


210 
224 
250 
277 
303 
318 
328 
361 
389 
401 


8 
14 
26 
27 
26 
15 
10 
36 
23 
12 


0.1 
0.1 
0.1 
0.05 
0.05 
0.05 
0.05 
0.5 
1.0 
1.0 


80 
140 
260 
540* 
520 
300 
200 
72 
25 
12 



electrode, are recorded in Table LI and plotted in Fig. 75; the first 
column of the table gives the volume v of standard silver nitrate added, 



0.88 
0.84 
0.80 

I 

[o.26 
0.22 
0.18 
0.11 



I 



\ y 



TOO 
600 
600 

400 
300 
200 




Fio. 75. 



4 6 8 10 12 14 
CC.AgNO, 

Potentioinetric titration 



1U 11.4 11.5 
cc. AgNOs 

FIG 70. Determination 
of end-point in potentio- 
motric tit rat ion 



which is equivalent to x in the treatment given above, and hence A/?/Ar, 
in the last column, is an approximation to dE/dx. It is clear from the 
data that A/?/ At; is a maximum when v is about 11.35 cc., and this must 
represent the end-point of the titratiou. 



260 ELECTRODE POTENTIALS 

It is not always possible to estimate the end-point directly by inspec- 
tion of the data and the following method, which is always to be preferred, 
should be used. The values of AJE/At; in the vicinity of the end-point 
are plotted against v + iAt>, i.e., the volume of titrant corresponding to 
the middle of each titration interval, as in Fig. 76; the volume of titrant 
corresponding to the maximum value of AU/Av can now be determined 
very precisely. This graphical method is particularly useful when the 
inflection in the potential-titration curve at the end-point is relatively 
small. 

Differential Titration. The object of potentiometric titration is to 
determine the point at which AJ/At> is a maximum, and this can be 
achieved directly, without the use of graphical methods, by utilizing the 
principle of differential titration. If to two identical solutions, e.g., of 
sodium chloride, are added v and v + 0.1 cc. respectively of titrant, e.g., 
silver nitrate, the difference of potential between similar electrodes placed 
in the two solutions gives a direct measure of AS/Av, where At; is 0.1 cc., 
at the point in the titration corresponding to the addition of v + 0.05 cc. 
of silver nitrate. The E.M.F. of the cell made up of these two electrodes 
will thus be a maximum at the end-point. 

In the earliest applications of the method of differential titration the 
solution to be titrated was divided into two equal parts; similar elec- 
trodes were placed in each and electrical connection between the two 
solutions was made with wet filter-paper. The electrodes were con- 
nected through a suitable high resistance to a galvanometer. Titrant 
was then added to the two solutions from two separate burettes, one 
being always kept a small amount, e.g., 0.1 cc., in advance of the other. 
The point of maximum potential difference, and hence that at which 
AE/Av was a maximum, was indicated by the largest deflection of the 
galvanometer; the total titrant added at this point was then equivalent 
to the total solution titrated. By this means the end-point of the titra- 
tion was obtained without the use of a reference electrode or a poten- 
tiometer, and the necessity for graphical estimation of the titration 
corresponding to the maximum Al?/Av was avoided. 20 

The method of differential titration has been modified so that the 
process can be carried out in one vessel with one burette; by means of 
special devices, a small quantity of the titrated solution surrounding one 
of the two identical electrodes is kept temporarily from mixing with the 
bulk of the solution before each addition of titrant. The difference of 
potential between the two electrodes after the addition of an amount Aw 
of titrant gives a measure of A#/Ay. The form of apparatus devised by 
Maclnncs and Dole, 21 which is capable of giving results of great accuracy, 
is depicted in Fig. 77. One of the two identical indicator electrodes, 

10 Cox, J. Am. Chem. Soc., 47, 2138 (1925). 

21 Maclnnes et al, J. Am. Chem. Sac., 48, 2831 (1926); 51, 1119 (1929); S3, 555 
(1931); Z. physik. Chem., 130, 217 (1927). 



COMPLEX IONS 



261 





I 


-EL 




mm 


J 


1 


^ 

^ 


i 


A 




HJ 





Fia. 77. Apparatus for 
differential titration (Mac- 
Innes and Dole) 



viz., E\, is placed directly in the titration vessel, and the other, J? 2 , is 
inserted in the tube A, which should be as small as convenient; at the 
bottom of this tube there is a small hole B, and a "gas-lift" C is sealed 
into its side. The hole D in the tube A permits the overflow of liquid 
when the gas-lift is in operation. In order to carry out a titration, a 
known volume of solution is placed in a beaker 
and the two electrodes are inserted; the liquid is 
allowed to enter A, but the gas-stream is turned 
off. Titrant is added from the burette, with con- 
stant stirring, until there is a large increase in the 
E.M.F. of the cell formed by the two electrodes ; this 
may be indicated by a potentiometer, for precision 
work, or by means of a galvanometer with a resist- 
ance in series. The solution in the beaker is actu- 
ally somewhat over-titrated, but when the gas- 
stream is started the reserve solution in the tube A , 
which normally mixes only slowly with the bulk of 
the liquid, because of the smallness of the hole JB, 
is forced out; in this way the titration is brought 
back, although the end-point is near. The differ- 
ence of potential between the two electrodes is 
now zero, since the same solution surrounds both of them. The gas- 
stream is stopped, and a drop (At;) of titrant is added to the bulk of the 
solution in the beaker; the galvanometer deflection, or potential differ- 
ence, is then a measure of A^E/At;, since one electrode, J5? 2 , is immersed in 
a solution to which v cc. of titrant have been added, while the other, E\, 
is surrounded by one to which v + At; cc. have been added. The gas- 
stream is started once more so as to obtain complete mixing of the 
solutions; it is then stopped, another drop of titrant added, and the 
potential reading again noted. This procedure is continued until the 
end-point is passed, the end-point itself being characterized by the maxi- 
mum potential difference between the two electrodes. 

Many simplified potentiometric titration methods have been de- 
scribed from time to time, and various forms of apparatus have been 
devised to facilitate the performance of these tit rations; for reference to 
these matters the more specialized literature should be consulted. 22 

Complex Ions. The formula of a relatively stable complex ion can 
be determined by means of E.M.F, measurements; in the general case 
already considered on page 173, viz., 

MA r ; ^ 0M+ + rA~, 

* See the books and review articles to which reference is made on page 256; the 
subject of potentiometric titrations, among others, is also treated in Koltho? and 
Laitinen, "pH and Electro-Titrations," 1941. 



262 ELECTRODE POTENTIALS 

it was seen that the instability constant K l can be represented by 



If an electrode of the metal M is inserted in the solution of the complex 
ion, the reversible potential should be given by 

r>/r? 

E = /?M,M+ - ~ In a M * 





For two solutions containing different total amounts of the complex ion, 
but the same relatively large excess of tin; anion A~, it follows from 
equation (42) that 



Bl -K. , (43) 

qzF (a Mf A,*)i V ' 

where the suffixes 1 and 2 refer to the two solutions; the value of a\- is 
assumed to be the same for the two cases. If the complex ion M q \f 
is relatively stable, then in the presence of excess of A~ ions, virtually 
the whole of the M present in solution will be in the form of complex ions. 
As an approximation, therefore, tho ratio of the activities of the M^A* 
ions in the two solutions in equation (43) may bo replaced by the ratio 
of the total concentrations of M ; hence 



,, } (c M ) 2 , AA . 

E l E 2 = 7; In 7 ;- (44) 

qzF (CM) i v 

If (CM)I and (c\i)2, the total concentrations of the species M in the respec- 
tive solutions, are known, and the potentials EI and 7? 2 arc measured, 
it is possible to evaluate q by means of equation (41). 

If the solutions are made up with same concentration of M, i.e., 
approximately the same concentration, or activity, of the complex ions 
M q A.f y but with different amounts of the anion A~, it follows from equa- 
tion (42) that in this case 



The ratio of the activities of the A~ ions may be replaced, as an approxi- 
mation, by the ratio of the concentrations; hence*, from equation (45), 

ft-ft.,. (46) 

-- v ' 



ELECTRODE POTENTIAL AND VALENCE 



263 



Since q has been already determined, the value of r can be derived from 
equation (46) so that the formula of the complex ion has been found. 

Another method for deriving the ratio r/q involves the same principle 
as is used in potentiometric titration; for simplicity of explanation a 
definite case, namely the formation of the argentocyanide ion, Ag(CN)J, 
will be considered. If a solution of potassium cyanide is titrated with 
silver nitrate, che potential of a silver electrode in the titrated solution 
will be found to undergo a sudden change of potential when the whole 
of the cyanide has been converted into argentocyanide ions. From the 
relative amounts of silver and cyanide ions at the point where dE/dx is 
a maximum the formula of the complex ion can be calculated. An 
analogous titration method can be used to determine the formula of any 
stable complex ion; the procedure actually gives the ratio of M+ to A" 
in the complex ion M q Af, but if this ratio is known there is generally no 
difficulty, from valence and other chemical considerations, in deriving the 
molecular formula. 

By expressing the concentration, or activity, of the M+ ions in the 
titrated solution, and hence the potential of an M electrode, in terms of c, 
the initial concentration of the solution, x, the amount of titrant added, 
and fc t , the instability constant of the complex ion, it is possible, utilizing 
the method of differentiation described in connection with precipitation 
titrations (page 258), to show that dE/dx is a maximum at the point 
corresponding to complete formation 
of the complex ion. Further, the 
value of dE/dx at this point, and 
hence the sharpness of the inflection 
in the titration curve, can be shown 
to be greater the smaller the insta- 
bility constant. 

The potential of a silver electrode 
during the course of the titration 
of silver nitrate with potassium cyan- 
ide is shown in Fig. 78; the first 
marked change of potential occurs 
when one equivalent of cyanide has 
been added to one of silver, so that 
the whole of the silver cyanide is pre- 
cipitated, and the second, when two 
equivalents of cyanide have been 
added, corresponds to the complete 
formation of the Ag(CN)J ion. It 
will be seen that the changes of potential occur very sharply in each oaso; 
this means that the silver cyanide is very slightly soluble and thai. t4vo 
complex ion is very stable. 

Electrode Potential and Valence. The equation (86) for the poten- 
tial of an electrode reversible with respect to positive ions may be written 



f 0.2 



*0.0 



-0.2 



-0.6 




1 2 3 

Equivalents of Cyanide 

Fia. 78. Formula of complex 
argentocyanide ion 



264 ELECTRODE POTENTIALS 

in the approximate form 

E E -0.0002- log c,, 


where the activity of the ionic species is replaced by the concentration. 
At ordinary laboratory temperatures, about 20c., i.e., T is 293 K., this 
equation becomes 



E ~ E* - logc t . (47) 

z 

It follows, therefore, that a ten-fold change of concentration of the ions 
will produce a change of 0.058/2, volt in the electrode potential, where 
Zi is the valence of the ions with respect to which the electrode is revers- 
ible. It is possible, therefore, to utilize equation (47) to determine the 
valence of an ion. 23 For example, the result of a ten-fold change in the 
concentration of a mercurous nitrate solution was found to cause a change 
of 0.029 volt in the potential of a mercury electrode at 17; it is evident, 
therefore, that z must be 2, so that the mercurous ions are bivalent. 
These ions are therefore written as Hgf + and mercurous chloride and 
nitrate are represented by Hg 2 Cl 2 and Hg2(NOa)2. 

PROBLEMS 

1. Work out the expressions for the E.M.F.'S and single potentials of the 
cells and electrodes given in Problem 1 of Chap. VI in terms of the variable 
activities. 

2. From the standard potential data in Table XLIX determine (i) the 
standard free energies at 25 of the reactions 

Ag+aq. + Cl-aq. = AgCl(s) 
and 

Ag + JC1 2 (1 atm.) = AgCl(s), 

and (ii) the solubility product of silver chloride. 

3. It is known from thermal measurements that the entropy of aluminum 
at 25 is 6.7 cal./deg. per g.-atom, and that of hydrogen gas at 1 atm. pressure 
is 31.2 per mole. The heat of solution of aluminum in dilute acid shows that 
AH for the reaction 

Al + 3H+aq. = Al+++aq. + f H t (l atm.) 

is 127,000 cal. From measurements on the entropy of solid cesium alum 
and its solubility, etc., Latimer and his collaborators [J. Am. Chem. Soc., 60, 
1829 (1938)] have estimated the entropy of the Al+++aq. ion to be - 76 
cal./deg. per g.-ion. Calculate the standard potential of aluminum on the 
usual hydrogen scale. 

" Ogg, Z. physik. Chem., 27, 285 (1898); see also, Reichinstein, ibid., 97, 257 (1921); 
Kasarnowsky, Z. anorg. Chem., 128, 117 (1923). 



PROBLEMS 265 

4. Jones and Baeckstrom [J. Am. Chem. Soc., 56, 1524 (1934)] found the 
E.M.P. of the cell 

Pt | Br 2 (J) KBr aq. AgBr(a) | Ag 

to be 0.9940 volt at 25. The vapor pressure of the saturated solution of 
bromine in the potassium bromide solution is 159.45 mm. of mercury; calculate 
the standard potential of the Br 2 (gr, 1 atm.), Br~ electrode. 

5. The standard free energy of the process 

JH 2 (1 atm.) + id 2 (l atm.) = HC1(1 atm.), 

is given in International Critical Tables, VII, 233, by the expression 
AG = - 21,870 + 0.45Tln T - 0.25 X 1Q~ 6 T 2 - 5.31T. 

The partial pressure of hydrogen chloride over 1.11 N hydrochloric acid solu- 
tion is 4.03 X 10- 4 mm. at 25. Calculate the E.M.F. of the cell 

H 2 (l atm.) | 1.11 N HC1 aq. | C1 2 (1 atm.) 

at this temperature. Use the result to determine the standard potential of 
the chlorine electrode, the mean activity coefficient of the hydrochloric acid 
being estimated from the data in Table XXXIV. 

6. Calculate from the standard potentials of cadmium and thallium the 
ratio of the activities of Cd++ and Tl~ ions when metallic cadmium is shaken 
with thallous perchlorate solution until equilibrium is attained. 

7. Knuppfer [Z. physik. Chem., 26, 255 (1898)] found the E.M.F. of the cell 

Tl (Hg) | TlCl(s) KCl(d) j KCNS(c 2 ) TlCNS(s) | Tl (Hg) 

to be - 0.0175 volt at 0.8 and - 0.0105 volt at 20 with Ci/c 2 equal to 0.84. 
Assuming the solutions to behave ideally, calculate the equilibrium ratios of 
Ci/c 2 at the two temperatures and estimate the temperature at which the arbi- 
trary ratio, i.e., 0.84, will become the equilibrium value. 

8. The E.M.F. of the cell 

Pb | Pb(OH) 2 (s) N NaOH HgO(s) | Hg 

is 0.554 volt at 20; the potential of the Hg, HgO(s) N NaOH electrode is 
0.114 volt. Calculate the approximate solubility product of lead hydroxide. 

9. In the potentiometric titration of 25 cc. of a potassium cyanide solution 
with 0.1 N silver nitrate, using a silver indicator electrode and a calomel refer- 
ence electrode, the following results were obtained: 



cc. AgNO, (v) 

E.M.F. (E) 


2.20 
0.550 


11.70 
0.481 


15.50 
0.445 


18.00 
0.422 


19.60 
0.392 


20.90 
0.363 


cc. AgNO, (v) 

E.M.F. (E) 


21.50 
0.343 


21.75 
0.309 


21.95 
0.259 


22.15 
0.187 


22.35 
- 0.255 


22.55 
- 0.319 



Plot E against v, and A# against At; in the vicinity of the end-point; from the 
results determine the concentration of the potassium cyanide solution. 

10. When studying the behavior of a tin anode in potassium oxalate 
solution, Jeffery [Trans. Faraday Soc. t 20, 390 (1924)] noted that a complex 
anion, having the general formula Sn^CaOOr* was formed. In order to deter- 



266 ELECTRODE POTENTIALS 

mine its constitution, measurements of the cell 

Sn | Sn,(Ct0 4 ) r - K 2 C 2 4 aq. j KC1 (satd.) Hg 2 Cl 2 (a) | Hg 

were made: in one series of experiments (A) the concentration of potassium 
oxalate was large and approximately constant while the total amount of tin 
in solution (cs n ) was varied; in the second series (B) t cs n was kept constant 
at 0.01 g.-atom per liter, while the concentration of potassium oxalate (c ox .) 
was varied. The results were as follows: 

A B 

CSn E Cox. E 

1.00 X 10-* 0.7798 2.0 0.7866 

0.833 0.7823 2.5 0.7937 

0.714 07842 3.0 0.7990 

0.625 0.7859 3.5 0.8002 

0.556 0.7877 4.0 0.8052 

Devise a graphical method, based on equations (44) and (46), to evaluate q 
and r; activity corrections may be neglected, and the whole of the tin present 
in solution may be assumed to be in the form of the complex anion. 



CHAPTER VIII 
OXIDATION-REDUCTION SYSTEMS 

Oxidation-Reduction Potentials. It was seen on page 186 that a 
reversible electrode can be obtained by inserting an inert electrode in a 
solution containing the oxidized and reduced forms of a given system; 
such electrodes are called oxidation-reduction electrodes. It has been 
pointed out, and it should be emphasized strongly, that there is no 
essential difference between electrodes of this type and those already 
considered involving a metal and its cations, or a non-metal and its 
corresponding anions. This lack of distinction is brought out by the 
fact that the iodine-iodide ion system is frequently considered from the 
oxidation-reduction standpoint. Nevertheless, certain oxidation-reduc- 
tion systems, using the expression in its specialized meaning, have inter- 
esting features and they possess properties in common which make it 
desirable to consider them separately. 

According to the general arguments at the beginning of Chap. VI, 
which are applicable to reactions of all types, including those involving 
oxidation and reduction, the potential of an electrode containing the 
system 

Reduced Stated Oxidized State + n Electrons 

is given by the general equation 



_ ro __ (Oxidized State) 

h "nF R (Reduced State) ' (1) 

where n is the number of electrons difference between the two states, and 
the parentheses represent activities. 

Oxidation-reduction potentials, like the other types discussed in the 
preceding chapter, are generally expressed on the hydrogen scale, so that 
for the system 

e, 



for example, the electrode potential as usually recorded is really the 
JO.M.F. of the cell 



Pt | Fe+ f , Fe + ++ || H+(a H + = 1) | H 2 (l atm.). 



Using the familiar convention that a positive E.M.F. represents the tend- 
ency of positive current to flow from left to right through the cell, the 
reaction at the left-hand electrode may evidently be written as 

Fe++ = Fe++ + + , 
267 



268 OXIDATION-REDUCTION SYSTEMS 

for the passage of one faraday. This result may be obtained directly 
by analogy with the process occurring at the electrode M, M+, namely 
M = M+ + c. At the right-hand electrode, the reaction is 

H+ + 6 = *H 2 , 
so that the net cell reaction, for one faraday of electricity, is 

Fe++ + H+ = Fe+++ + H 2 . 
The E.M.F. of the complete cell is then given in the usual manner by 



and since, by convention, the activities of the hydrogen gas and the 
hydrogen ions are taken as unity, it follows that 

E = JBk** Pe "+ - ~ In ^ (2) 

The oxidation-reduction potential is thus seen to be determined by the 
ratio of the activities of the oxidized and reduced states, in agreement 
with the general equation (1). The standard potential E is evidently 
that for a system in which both states are at unit activity. 

In the most general case of an oxidation-reduction system repre- 
sented by 

aA + 6B + - ^ xX + i/Y + + n , 

for which there is a difference of n electrons between the reduced state, 
involving A, B, etc., and the oxidized state, involving X, Y, etc., the 
potential is given by (cf . page 228) 



When all the species concerned, viz., A, B, , X, Y, etc., are in their 
standard states, i.e., at unit activity, the potential is equal to E , the 
standard oxidation-reduction potential of the system. It is important 
to remember that in order that a stable reversible potential may be 
obtained, all the substances involved in the system must be present; 
the actual potential will, according to equation (3), depend on their 
respective activities. 

Types of Reversible Oxidation-Reduction Systems. Various types 
of reversible oxidation-reduction systems have been studied: the simplest 
consist of ions of the same metal in two stages of valence, e.g., ferrous 
and ferric ions. If M* 1 * and M n + are two cations of the metal M, carry- 
ing charges z\ and 22, respectively, where z 2 is greater than z\ 9 the elec- 
trode reaction is 



TYPES OF REVERSIBLE OXIDATION-REDUCTION SYSTEMS 269 

and the potential is given by 



where a* and a\ are the activities of the oxidized and reduced forms, 
respectively. 

Another type of system consists of two anions carrying different 
charges, e.g., ferro- and ferri-cyanide, i.e., 

Fe(CN)e --- ^ Fe(CN)e + e, 
and the electrode potential for this system is 



In certain cases both anions and cations of the same metal are con- 
cerned; for such systems the equilibria, and hence the equations for the 
electrode potential, involve hydrogen ions. An instance of this kind is 
the permanganate-manganous ion system, viz., 

Mn++ + 4H 2 ^ MnOr + 8H+ + 5c, 
for which the electrode potential is 



the activity of the water being unity provided the solutions are rela- 
tively dilute. 

In some important oxidation-reduction systems one or more solids 
are concerned; for example, in the case of the equilibrium 

Mn++ + 2H 2 ^ Mn0 2 (s) + 4H+ + 2, 
the potential is 



f 

= _ m . 



since the activity of the solid manganese dioxide is taken as unity, in 
accordance with the usual convention as to standard states. 
In the equilibrium 

PbS0 4 (s) + 2H 2 O ? Pb0 2 (s) + 4H+ + SO?- + 2 , 

which is of importance in connection with the lead storage battery, two 
solids are involved, namely lead sulfate and lead dioxide, and hence 

_. _ A RT , 4 

E = Z? - In aH*a 8 o 4 - -. 



270 OXIDATION-REDUCTION SYSTEMS 

The potential thus depends on the fourth power of the activity of the 
hydrogen ions and also on that of the sulfate ions in the solution. 

A large number of reversible oxidation-reduction systems involving 
organic compounds are known; most of these, although not all, are of 
the quinone-hydroquinone type. The simplest example is 

OH 




OH 

and such systems may be represented by the general equation 
H 2 Q ^ Q + 2H+ + 2e, 

where H 2 Q is the reduced, i.e., hydroquinone, form and Q is the oxidized, 
i.e., quinone, form. The potential of such a system is given by 

(4) 

For many purposes it is convenient to maintain the hydrogen ion activity 
constant and to include the corresponding term in the standard potential; 
equation (4) then becomes 



where E*' is a subsidiary standard potential applicable to the system at 
the specified hydrogen ion activity. 

Determination of Standard Oxidation-Reduction Potentials. In prin- 
ciple, the determination of the standard potential of an oxidation-reduc- 
tion system involves setting up electrodes containing the oxidized and 
reduced states at known activities and measuring the potential E by 
combination with a suitable reference electrode; insertion of the value 
of E in the appropriate form of equation (3) then permits E T to be calcu- 
lated. The inert metal employed in the oxidation-reduction electrode 
is frequently of smooth platinum, plthough platinized platinum, nercury 
and particularly gold are often used. 

In the actual evaluation of the standard potential from the experi- 
mental data a numbe** of difficulties arise, and, as a result of the failure 
to overcome or to make adequate allowance for them, most of the meas- 
urements of oxidation-reduction potentials carried out prior to about 
1925 must be regarded as lacking m accuracy. In the first case, it w 
rarely possible to avoid a liquid junction potential in setting up the cell 
for measuring the oxidation-reduction potential; secondly, there is often 



DETERMINATION OF STANDARD OXIDATION-REDUCTION POTENTIALS 271 

uncertainty concerning the actual concentrations of the various species, 
because of complex ion formation and because of incomplete dissociation 
and hydrolysis of the salts present; finally, activity coefficients, which 
were neglected in th3 earlier work, have an important influence, as will 
be apparent from the following considerations. 

In the simple case of a system consisting of two ions carrying different 
charges, e.g., Fe++, Fe+++ or Fe(CN)e --- , FeCCN) , designated by 
the suffixes 1 and 2, respectively, the equation for the potential is 



where the activity has been replaced by the product of the concentration 
and the activity coefficient. Utilizing the Debye-Huckel limiting equa- 
tion (p. 144), viz., 



it follows that 

log - A(z\ - 

and insertion in equation (6) gives 



If water i.^> the solvent, then at 25 the constant A is 0.509; hence, this 
equation becomes 

^ 0.05915, c 2 0.0301 , N r 

E = E Q ---- log - -- (z\ - 2|)\v- (7) 

* ^ ' 



For most oxidation-reduction systems z\ z\ is relatively high, e.g., 7 for 
the Fe(CN)e" , Fe(CN)o" system, and so the last term in equation 
(7), which represents the activity coefficient factor, may be quite con- 
siderable; further, the terms in the ionic strength involve the square of 
the valence and hence y will be large even for relatively dilute solutions. 1 
In any case, the presence of neutral salts, which were frequently added 
to the solution in the earlier studies of oxidation-reduction potentials, 
increases the ionic strength; they will consequently have an appreciable 
influence on the potential, although the ratio of the amounts of oxidized 
to reduced forms remains constant. 

A striking illustration of the effect of neglecting the activity coeffi- 
cient is provided by the results obtained by Peters (1898) in one of the 

1 Kolthoff and Tomsicck, J. Phys. Chern., 39, 945 (1935); Glasstone, "The Electro- 
chemistry of Solutions," 1937, p. 346. 



272 OXIDATION-REDUCTION SYSTEMS 

earliest quantitative studies of reversible oxidation-reduction electrodes. 
From measurements made in solutions containing various proportions 
of ferrous and ferric chloride chloride in 0.1 N hydrochloric acid, an 
approximately constant value of 0.713 volt at 17 was calculated for 
the standard potential of the ferric-ferrous system, using the ratio of 
concentratibns instead of activities. This result was accepted as correct 
for some years, but it differs from the most recent values by about 0.07 
volt; the discrepancy is close to that estimated from equation (5) on the 
basis of an ionic strength of 0.25, which is approximately that existing 
in the experimental solutions. Actually, of course, the Debye-Huckel 
limiting equation would not hold with any degree of exactness at such a 
high ionic strength, but it is of interest to observe that it gives an activity 
correction of the right order. 

In recent years care has been taken to eliminate, or reduce, as far as 
possible the sources of error in the evaluation of standard oxidation- 
reduction potentials; highly dissociated salts, such as perchlorates, are 
employed wherever possible, and corrections are applied for hydrolysis 
if it occurs. The cells are made up so as to have liquid junction poten- 
tials whose values are small and which can be determined if necessary, 
and the results are extrapolated to infinite dilution to avoid activity 
corrections. One type of procedure adopted is illustrated by the case 
described below.* 

In order to determine the oxidation-reduction potential of the system 
involving penta- (VOt) and tetra-valent (VO++) vanadium, viz., 

VO++ + H 2 = VOt + 2H+ + , 
measurements were made with cells of the form 

Pt | V0 2 C1, VOC1 2 , HC1 j HC1 H&C1.W | Hg 

containing the three constituents, VO 2 C1, VOC1 2 and hydrochloric acid 
at various concentrations. 2 By employing acid of the same concentra- 
tion in both parts of the cell, the liquid junction potential was reduced 
to a negligible amount. The reaction taking place in the cell for the 
passage of one f araday is 

VO++ + H 2 + JH&C1.W = VOt + 2H+ + Cl- + Hg(0, 
so that the E.M.F. is given by 



RT , 

E = EO - -TT In j - (8) 

r ++ 



where the standard potential for the cell (J?) is equal to the difference 
between the standard potentials of the V 6 , V 4 system and that of the 

* See also, Problem 4, page 304. 

Carpenter, /. Am. Chem. Soc., 56, 1847 (1934); Hart and Partington. /. Chem. 

, 1532 (1940). 



DETERMINATION OF STANDARD OXIDATION-REDUCTION POTENTIALS 27C 

Hg, Hg 2 Cl 2 (s), Cl- electrode, the latter being - 0.2680 volt at 25. Re- 
placing the activities of the VO++ and VO2~ ions by the products of their 
respective concentrations and activity coefficients, represented by / 2 and 
/i, respectively, equation (8) becomes, after rearrangement, 



Since the hydrochloric acid may be regarded as being completely ionized, 
C H + and Ccr may each be taken as equal to CHCI, the concentration of this 
acid in the cell; further, the product of /H + and/cr is equal to /HCI, where 
/HCI is the mean activity coefficient of the hydrochloric acid. It follows, 
therefore, that the quantity an+ocr, which is equal to (cH+Ccr)/iV/cr> 
may be replaced by CHCI/HCI/H+; upon inserting this result in equation (9) 
and rearranging, it is found that 



_, _ _ fl 

E + -p- In CHCI + -jjrln = E - _l n ^-. (1 ) 

The activity coefficient term in this equation becomes zero at infinite 
dilution; it follows, therefore, that extrapolation of the left-hand side 
to zero concentration, using the results obtained with cells containing 
various concentrations of the three constituents, should give E for the 
cell. The value obtained in this manner, by plotting the left-hand side 
of equation (10) against a suitable function of the ionic strength, was 
0.7303 volt; it follows, therefore, that the standard potential of the 
VO++, VOt + 2H+ system is - 0.730 + (- 0.268), i.e., - 0.998 volt. 

An alternative extrapolation procedure is based on the approximation 
of taking fn + to be equal to / nc i; equation (9) can then be written as 



_ , ZRT. , 3flr RT, cvo" - RT, f, 

E + -y- In CHCI + -p~ In /HCI + y In -^ = # - y In ~ (11) 

The values of the activity coefficients of hydrochloric acid at the ionic 
strengths existing in the cell are obtained from tabulated data, and hence 
the left-hand side of this equation, for various concentrations, may be 
extrapolated to zero ionic strength, thus giving E. A further possi- 
bility is to replace log /HCI by the Debye-Hiickel expression A Vy, and 
to extrapolate, as before, by plotting against a suitable function of the 
ionic strength. As a general rule, several methods of extrapolation are 
possible; the procedure preferred is the one giving an approximate straight 
line plot, for this will probably give the most reliable result when ex- 
trapolating to infinite dilution. 

Another method of evaluating standard oxidation-reduction potentials 
is to make use of chemical determinations of equilibrium constants. 3 

' Schumb and Sweetser, J. Am. Chem. Soc., 57, 871 (1035). 



274 OXIDATION-REDUCTION SYSTEMS 

The chemical reaction occurring in the hypothetical cell, free from liquid 
junction, 

Ag | Ag+ || Fe++, FO-H-+ | Pt, 

for the passage of one faraday is 

Ag + Fe+++ = Ag+ 



The standard E.M.F. of this cell (#) with all reactants at unit activity 
is given by (cf. p. 251) 



F F 

where the activities are those at equilibrium, indicated by the suffix e; 
the activity of the solid silver is equal to unity, and so is omitted from 
the equilibrium constant. The standard E.M.F. is also equal to the differ- 
ence of the standard potentials of the silver and ferrous-ferric electrodes, 
thus 

JE' = fii*,^- JEFe^Fo"*, (13) 

and hence if the equilibrium constant of the coll reaction could be deter- 
mined by chemical analysis, the value of Ev e +\ Vo +++ could be calculated, 
since the standard potential of silver is known (Table XLIX). 

A solution of ferric pcrchlorate, containing free perchloric acid in 
order to repress hydrolysis, was shaken with finely divided silver until 
equilibrium of the system 

Ag + Fe(C10 4 ) 3 ^ AgC10 4 + Fe(ClO 4 ) 2 

was attained. Since perchlorates are very strong electrolytes, they are 
generally regarded as being completely dissociated at not too high con- 
centrations; this reaction is, therefore, equivalent to that of the hypo- 
thetical cell considered above. By analyzing the solution at equilibrium, 
a concentration equilibrium "constant" (fc), for various total ionic 
strengths, was calculated; this function k is related to the true equilib- 
rium constant in the following manner: 



j r 

= k - f --- > 

/Fe +++ 

and if the activity coefficients are expressed in terms of the ionic strength 
by means of the extended form of the Debye-Huckel equation (p. 147), 
it is found that 

log K = log k + log/ Ag + + 

= log k - 



APPROXIMATE DETERMINATION OF STANDARD POTENTIALS 275 

The value of A is known to be 0.509 for water at 25, and that of C is 
found empirically; another term, Z)y 2 , with an empirical value of D, may 
be added if necessary, and the true dissociation constant K can then be 
calculated from the experimental data. In this manner, it was found 
that K is 0.531 at 25, and hence from equations (12) and (13), making 
use of the fact that the standard potential of silver is 0.799, it follows 
that at 25 

- 0.799 - J3?-e+*.Fe + ** = 0.05915 log 0.531 
= - 0.016, 

.'. #Fe+ + ,Fe 4 + f = - 0.783 Volt. 

Direct measurements of the potential of the ferric-ferrous system have 
also been made; after allowing for hydrolysis and activity effects, the 
standard potential at 25 was found to be 0.772 volt, but so many 
corrections were involved in arriving at this result that the value based 
on equilibrium measurements is probably more accurate. 4 

Approximate Determination of Standard Potentials. Many studies 
have been made of oxidation-reduction systems with which, for one 
reason or another, it is not possible to obtain accurate results: this may 
be due to the difficulty of applying activity corrections, uncertainty as 
to the exact concentrations of the substances involved, or to the slowness 
of the establishment of equilibrium with the inert metal of the electrode. 
It is probable that whenever the difference in the number of electrons 
between the oxidized and reduced states, i.e., the value of n for the 
oxidation-reduction system, is relatively large the processes of oxidation 
and reduction occur in stages, one or more of which may be slow. In 
that event equilibrium between the system in the solution and the elec- 
trode will be established slowly, and the measured potential may be in 
error. To expedite the attainment of the equilibrium a potential medi- 
ator may be employed; 5 this is a substance that undergoes reversible 
oxidation-reduction and rapidly reaches equilibrium with the electrode. 

Consider, for example, a system of two ions M + and M++ which is 
slow in the attainment of equilibrium with the electrode, and suppose a 
very small amount of a eerie salt (Ce ++++ ^ is added to act as potential 
mediator; the reaction 

M+ + Ce+ +++ ^ M++ + Ce+++ 

takes place until equilibrium is attained. At this point the potential of 
the M+, M-n system must be identical with that of the Ce+++, Ce++++ 
system (cf. p. 284). The cerio-cerous system comes to equilibrium rap- 
idly with the inert metal, e.g., platinum, electrode and the potential 
registered is consequently both that of the Ce+ ++, Ce++ ++ and M+, M++ 

4 Popoff and Kunz, J. Am. Chem. Soc., 51, 382 (1929); Bray and Hershey, ibid., 56, 
1889 (1934). 

Loimaranta, Z. Elektrochem., 13, 33 (1907); F6erster and Pressprich, ibid., 33, 176 
(1927); Goard and Rideal, Trans. Faraday Soc., 19, 740 (1924). 



276 



OXIDATION-REDUCTION SYSTEMS 



systems in the experimental solution. If the potential mediator is added 
in very small amount, a negligible quantity of M+ is used up and M++ 
formed in the establishment of the chemical equilibrium represented 
above : the measured potential in the presence of the mediator may thus 
be regarded as the value for the original system. In addition to eerie 
salts, iodine has been used as a potential mediator; the platinum elec- 
trode then measures the potential of the iodine-iodide ion system. If the 
results obtained in the presence of a mediator are to have definite thermo- 
dynamic significance they should be independent of the nature of the 
mediator and of the electrode material, provided the latter is not attacked 
in any way. 

Standard Potentials from Titration Curves. A method of studying 
oxidation-reduction systems involving the determination of potentials 
during the course of titration with a suitable substance, which frequently 
acts as a potential mediator, has been emplo v to a considerable extent 
in work on systems containing organic compounds. The pure oxidized 
form of the system, e.g., a quinone or related substance, is dissolved in a 
solution of definite hydrogen ion concentration, viz., a buffer solution 
(see Chap. XI); known amounts of a reducing solution, e.g., titanous 
chloride or sodium hydrosulfite, are added, in the absence of air, and the 
solution is kept agitated by means of a current of nitrogen. The poten- 
tial of an inert electrode, e.g., 
Per Cent Reduction platinum, gold or mercury, im- 

mersed in the reacting solution is 

measured after each addition of 
the titrant, by combination with 
a reference electrode such as a 
form of calomel electrode. The 
results obtained are of the type 
shown in Fig. 79, in which the 
electrode potentials observed 
during the course of the addi- 
tion of various amounts of 
titanous chloride to a buffered 
(pll 6.98) solution of 1-naph- 
thol-2-sulfonate indophenol at 
30 are plotted as ordinates 
against the volumes of added 
reagent as abscissae. 8 The point 
at which the potential undergoes a rapid change is that corresponding to 
complete reduction (cf. p. 286), and the quantity of reducing solution 
then added is equivalent to the whole of the oxidized organic compound 
originally present. From the amounts of reducing agent added at various 

Clark et al, "Studies on Oxidation-Reduction," Hygienic Laboratory Bulletin, 
No. 151, 1928; see also, Conant et al., J. Am. Chem. Soc.. t 44, 1382, 2480 (1922); LaMer 
and Baker, ibid., 44, 1954 (1922). 




10 20 

ee. Reducing Agent 

Fia. 79. Reduction of l-naphthol-2- 
sulfonate indophenol (Clark) 



32.8 



STANDARD POTENTIALS FROM TITRATION CURVES 277 

stages the corresponding ratios of the concentrations of the oxidized form 
(o) to the reduced form (r) may be calculated without any knowledge of 
the initial amount of the former or of the concentration of the titrating 
agent. If t c is the volume of titrant added when the sudden change of 
potential occurs, i.e., when the reduction is complete, and t is the amount 
of titrant added at any point in the titration, then at this point o is 
equivalent to t c t, and r is equivalent to t, provided the titrant em- 
ployed is a powerful reducing agent.* According to equation (1), re- 
placing the ratio of the activities by the ratio of concentrations, it follows 
that 



-* -* <> 

where E*' is the standard potential of the system for the hydrogen ion 
concentration employed in the experiment. Values of E Q/ can thus be 
obtained for a series of points on the titration curve; if the system is 
behaving in a satisfactory manner these values should be approximately 
constant. The results obtained by applying equation (14) to the data 
in Fig. 79 are recorded ; n Table LII. 

TABLE LII. EVALUATION OP APPROXIMATE STANDARD POTENTIAL AT 30 OF 
l-NAPHTHOL-2-SULFONATE INDOPHENOL AT pH 6.98 



Per cent 



_ i 



/ Reduction E 2F t E*' 

4.0 12.2 - 0.1479 - 0.0258 - 0.1221 

8.0 24.4 -0.1368 -0.0148 -0.1220 

12.0 36.6 -0.1292 -0.0072 -0.1220 

16.0 48.8 -0.1224 -0.0006 -0.1218 

20.0 61.0 -0.1159 + 0.0058 -0.1217 

24.0 73.2 -0.1085 +0.0131 -0.1216 

28.0 85.4 -0.0985 +0.0230 -0,1215 

32.8 (O 100.0 -0.036 

The experiment described above can also be carried out by starting 
with the reduced form of the system and titrating it with an oxidizing 
agent, e.g., potassium dichromate. The standard potentials obtained in 
this manner agree with those derived from the titration of the oxidized 
form with a reducing agent, and also with the potentials measured in 
mixtures made up from known amounts of oxidized and reduced forms. 
The presence of the inorganic oxidizing or reducing system, which often 
has the advantage of serving as a potential mediator, does not affect the 
results to any appreciable extent- 

It will be seen shortly that the value of n 9 the number of electrons 
involved in the oxidation-reduction system, is of some interest; if this is 

* The precise conditions for efficient reduction are discussed on page 280. 



278 



OXIDATION-REDUCTION SYSTEMS 



not known, it can be evaluated from the slope of the flat portion of the 
titration curve such as that in Fig. 79. This slope is determined by the 
value of n only, and is independent of the chemical nature of the system; 
the larger is n the flatter is the curve. An exact estimate of n may be 
made by plotting the measured potential E against log o/r, or its equiva- 
lent log (t c t)lt\ the plot, according to equation (14), should be a 
straight line of slope - 2.303RT/nF, i.e., - 0.059/n at 25 or - 0.060/n 
at 30. The results derived from Fig. 79 are plotted in this manner in 
Fig. 80; the points are seen to fall approximately on a straight line, in 



-0.16 



3-0.14 



*-0.13 



3-0.12 



g-0.11 



0.10 




+0.8 



+ 0.4 



-0.4 



-0.8 



lo, 



FIG. 80. Determination of n and E' 

agreement with expectation, and the slope is 0.03 at 30, so that n is 
equal to 2. The standard potential of the system at the given hydrogen 
ion concentration, i.e., E**', is given by the point at which the ratio o/r is 
unity, i.e., log o/r is zero; this is seen to be 0.122 volt, in agreement 
with the values in Table LIL 

Standard Oxidation-Reduction Potentials. Some values of standard 
oxidation-reduction potentials at 25 are given in Table LI 1 1. 7 The sign 
of the potential is based on the usual convention (p. 187), and the assump- 
tion that an inert material precedes the system mentioned in each case; 
for example, for Pt | Fe++, Fe+++ the standard potential is - 0.783 volt. 
A positive sign would indicate the tendency for negative electricity, e.g., 
electrons, to pass from solution to the metal, i.e., 

Fe++(+ Pt) - Fe+++ + e(Pt), 

so that in this particular case the standard free energy change of the 
process 

Fe+ + = Fe+ ++ + e 

'For further data, see International Critical Tables, Vol. VI, and Latimer, "The 
Oxidation States of the Elements and their Potentials in Aqueous Solutions, 11 1938. 



VARIATION OF OXIDATION-REDUCTION POTENTIAL 



279 



is given by 



A(? = - nFE = + 0.783F. 



If the electrode had been represented by Fe++, Fe+++ | Pt, i.e., with the 
inert metal succeeding the system, the sign of the potential would be 
reversed, i.e., + 0.783 volt. A positive potential in this case means a 
tendency for the process 

Fe+++ + (pt) = Fe++ + (Pt) 

to occur, which is the reverse of that just given. The order of writing 
the components present in the solution, viz., Fe++, Fe 4 ++ or Fe+++, Fe++ 
is immaterial, although the usual convention is to employ the former 
method of representation. 

TABLE LI II. STANDARD OXIDATION-REDUCTION POTENTIALS AT 25 



Electrode 


Reaction 


Potential 


Co* +. Oo + * 


Co- 1 "*- -Co-*- + + 


-1^2 


Pb + +, Pb++++ 


l'b* + -Pb* + + 4 -f 2 


- 1.75 


PbS()4(), PbOjGO.SO" 


PbSO + 2HjQ - PbOj -1- 4H + + SO^ - + 2 


- 1 68-> 


CV + +, Ce+ +++ 


Ce* + + -Ce* + + * -f c 


- 1.61 


Mn++. MnO, H+ 


Mn+* -|- 4H?0 - Mn0 4 - -f 8ir -h ,^ 


- 1.62 


Tl 4 , T!*"* * 


Tl + -Tl + ^ + -f2 


-1.22 


HtfMlR** 


llK^ + -* 2Hg ++ -f 2 


-0.006 


Fc*+. Fe'+ + 


Fo^ -Fe +< -* -f 


-0.783 


MnO", MnO; 


MnO;- - -> MnO- + * 


-0.64 


Fe(CN) e ----, I't-(CN) fl 


Ke(CN),- -Fe(C\), -- -f e 


-0356 


Cu-'-.Cu*-* 


Cu* Cu** + 


-0.16 


Sn^^.Sn-*** 


Sn ++ -Sn** + + +2e 


-0 15 


Ti+*vn + ** + 


Ti f + * -^Ti^ 4 * + 


-0.06 


Cr++. C'r* + * 


Cr f< " ->Cr + + + {- 


+ 0.41 



The potentials recorded in Table LIII may be called "oxidation 
potentials" (cf. p. 243) since they give a measure of the free energies of 
the oxidation processes; for the reverse reactions, the potentials, with the 
signs reversed, are the co responding " reduction potentials." 

Variation of Oxidation-Reduction Potential. From a knowledge of 
the standard oxidation-reduction potential of a given system it is possible 
to calculate, with the aid of the appropriate form 'of equation (3), the 
potential of any mixture of oxidized and reduced forms. For approxi- 
mate purposes it is sufficient to substitute concentrations for activities; 
the results are then more strictly applicable to dilute solutions, but they 
serve to illustrate certain general points. A number of curves, obtained 
in this manner, for the dependence of the oxidation-reduction poten- 
tial on the proportion of the system present in the oxidized form, are 



280 



OXIDATION-REDUCTION SYSTEMS 



depicted in Fig. 81 ; these curves are obviously of the same form as the 
experimental curve in Fig. 79. The position of the curve on the oxida- 
tion-reduction scale depends on the standard potential of the system, 
which corresponds approximately to 50 per cent oxidation, while its slope 
is determined by the number of electrons by which the oxidized and 

reduced states differ. The in- 
fluence of hydrogen ion con- 
centration in the case of the 
permanganate-manganous ion 
system is shown by the curves 
for an* equal to 1 and 0.1, re- 
spectively. 

It is seen from the curves in 
Fig. 81 that the potential rises 
rapidly at first as the amount 
of oxidized form is increased: 
this is due to the fact that when 
the proportion of the latter is 
small a relatively small actual 
increase in its amount brings 
about- a large relative change. 
For example, if the solution 
contained 0.1 per cent of oxi- 
dized form and 99.9 per cent 
of reduced form, the potential 
would be 
^ . . ^ A 

---0.2 



1.8 



- 1.6 



- 1.4 



-1.2 



I- 0.3 



'-0.6 



0.0 



0.2 




0.059 . 



25 50 

Per Cent Oxidation 



75 



100 



+ 



0.177 



Fia. 81. Oxidation-reduction potentials 



at 25. A change of 1 per cent 
in the proportion of oxidized 
form in the system would make the actual proportion M per cent, while 
there would be 98.9 per cent of oxidized form: the oxidation-reduction 
potential would then be given by 



0.059, 



0.118 



1 



indicating a change of potential of about 0.059/n volt. As the amounts 



IONIZATION IN STAGES 281 

of oxidized and reduced states become of the same order, the potential 
changes only slowly, since an increase or decrease in either brings about 
little change in the ratio which determines the oxidation-reduction poten- 
tial. Thus a change of 1 per cent in the amount of the oxidized form 
from 40 to 50 per cent, for example, alters the ratio of oxidized to reduced 
forms from 49/51 to 50/50; this will correspond to a change of 0.0052/n volt 
in potential. Solutions in this latter condition are said to be "poised" :* 
the addition of appreciable amounts of an oxidizing or reducing agent to 
such a solution produces relatively little change in the oxidation-reduction 
potential. Finally, when the system consists almost exclusively of the 
oxidized form, i.e., at the right-hand side of Fig. 81, the potential again 
changes rapidly ; the amount of reduced form is now very small, and con- 
sequently a small actual change means a large change in the ratio of 
oxidized to reduced forms in the solution. 

lonization in Stages. When a metal yields two positive ions, M* l + 
and M** 4 ", there are three standard potentials of the system; these are 
the potentials of the electrodes M, M z i + and M, M 2 *+ in addition to the 
oxidation-reduction potential M*i+, M** 4 ". If the values of these standard 
potentials are Eft, El and #?, 2 , respectively, then the free energy changes 
for the following process are as indicated below: 



M = M*i+ + 2 l , AC? = - 

M = M** 4 - + 2 2 , ACS = - z 2 FE 2 ] 

and 

M z t + = M * 2 + + fe _ 2l ) > A( JO 2 = 

It follows from these three equations that 



so that the three potentials are not independent. If any two of the three 
potentials are known, the third can be evaluated directly. For example, 
the standard potentials for ( 1 u, Cu 4 " 4 " and Cu 4 ", Cu 4 " 4 ", which are equiva- 
lent to El and #i, 2 , respectively, are - 0.340 and - 0.160 volt at 25. 
It follows, therefore, since z\ is equal to 1 and 22 to 2, that 

- 2 X 0.340 - ? = - 0.160, 

/. EI = - 0.520 volt. 

When a metal M is placed in contact with a solution containing either 
or M* + ions, or both, reaction will occur until the equilibrium 



(2 2 ~ 2i)M 

is established; in this condition, it follows from the law of mass action 

* This is the equivalent of the term "buffered" as applied to hydrogen ion poten- 
tials (cf. p. 410). 



282 OXIDATION-REDUCTION SYSTEMS 

that 



where a\ and 02 are the activities of the M*i+ and M** + ions, respectively, 
at equilibrium. The activity of the solid metal M is taken as unity. 

The value of this equilibrium constant can be calculated from the 
standard potentials derived above. It can be deduced, although it is 
obvious from general considerations, that when equilibrium is attained 
the potential of the metal M must be the same with respect to both 
M'I* and M**"*" ions; hence, 



(15) 

It has been seen above that for the copper-copper ion system, U? is 
- 0.520 and E 2 is - 0.340, and so at 25, 



. K = ^ = 8 22 x 10- 7 . 

Ocu + * 

When metallic copper comes to equilibrium with a solution containing 
its ions, therefore, the concentration of cuprous ions will be very much 
smaller than that of cupric ions. For mercury on the other hand, ff{ 
for Hg, Hgf + is 0.799 volt, while E lt2 is - 0.906; from these data it is 
found that at equilibrium aH g jV^H g ++ is 91. The ratio of the activity 
of the mercurous ions to that of the mercuric ions is thus 91, and hence 
the system in equilibrium with metallic mercury consists mainly of mer- 
curous ions, although mercuric ions are also present to an appreciable 
extent. It can be seen from equation (15) that the equilibrium constant 
between the two ions of a given metal in the presence of that metal is 
greater the larger the difference of the standard potentials with respect 
to the two ions; the ions giving the less negative standard potential are 
present in excess at equilibrium (cf. p. 253). 

Attention may be called to the fact that if the equilibrium constant 
could be determined by chemical methods, and if one of the three stand- 
ard potentials of a particular metal-ion system is known, the other two 
could be evaluated. This procedure was actually used for copper, 
the calculations given above being carried out in the reverse direction. 8 

Oxidation-Reduction Equilibria. When two reversible oxidation- 
reduction systems are mixed a definite equilibrium is attained which is 

Heinerth, Z. Ekktrochem., 37, 61 (1931). 



OXIDATION-REDUCTION EQUILIBRIA 283 

determined largely by the standard potentials of the systems. For ex- 
ample, for the reaction between the ferrous-ferric and stannous-stannic 
systems, the equilibrium can be represented by 



2Fe++ + Sn++++ ^ 2Fe+++ + Sn++, 
and when equilibrium is attained, the law of mass action gives 



The reaction takes place in the cell 

Pt | Fe++, Fe+ ++ || Sn++, Sn++++ | Ft, 

for the passage of two faradays, and so it follows that the standard E.M.F. 
is given by 

K -' ' " ^"* 
2F 



' , / aan++al> e +++ \ 
In I ^7^7; 1 J 



where E Q is equal to the difference in the standard potentials of the 
ferrous-ferric and stannous-stannic systems, i.e., 

E = JBj-e** Pa*** - JS8n**.Sn****- (18) 

It is evident, therefore, from equations (17) and (18), that the equilibrium 
constant depends on the difference of the standard potentials of the 
interacting systems; if the equilibrium constant were determined experi- 
mentally it would be possible to calculate the difference of standard 
potentials, exactly as in the case of the replacement of one metal by 
another (cf. p. 254). Alternatively, if the difference in standard poten- 
tials is known, the equilibrium constant can be evaluated. 

The value of EF^\ F ^^ is - 0.783 and that of JE 8 n** s n * f ** is - 0.15 at 
25; hence making use of the relationship, from equations (17) and (18), 

r/TT 
JEra** Pa*** ~ JESn*+ 8n**** = gjp ^ K, (19) 

it is readily found that 



This low value of the equilibrium constant means that when equilibrium 
is attained in the ferrous-ferric and stannous-stannic mixture, the con- 
centrations (activities) of ferric and stannous ions must be negligibly 
small in comparison with those of the ferrous and stannic ions. In 
other words, when these two systems are mixed, reaction occurs so that 
the ferric ions are virtually completely reduced to ferrous ions while the 
stannous are oxidized to stannic ions. This fact is utilized in analytical 
work for the reduction of ferric to ferrous ions prior to the estimation of 
the latter by means of dichromate. 



284 OXIDATION-REDUCTION SYSTEMS 

Inserting the expression for K, given by equation (16), into equation 
(19) and rearranging, the result is 

_ RT (a r .+++) f RT (OB,****), 

*** ^ - -jr in ^-^ = EL-sn - ^in-^-^-, (20) 

the left-hand side of this equation boing the potential of the ferrous- 
ferric system and the right-hand side that of the stannous-stannic system 
at equilibrium. When this condition is attained, therefore, both systems 
must exhibit the same oxidation-reduction potential; this fact has been 
already utilized in connection with the employment of potential mediators. 

Oxidation-Reduction Systems in Analytical Chemistry. An exami- 
nation of the calculation just made shows that the very small equilibrium 
constant,* and hence the virtually complete interaction of one system 
with the other, is due to the large difference in the standard potentials 
of the two systems. The system with the more negative standard poten- 
tial as recorded in Table LIII, e.g., Pt | Fe ++ , Fe+ ++ in the case con- 
sidered above, always oxidizes the system with the less negative standard 
potential, e.g., Pt | Sn 1 " 1 ", Sn+++ + , the extent of the oxidation being 
greater the larger the difference between the standard potentials. The 
same conclusion may be stated in the alternative manner: the system 
with the less negative potential reduces the one with the more negative 
potential, the extent being greater the farther the systems are apart in 
the table of standard potentials. It is of interest to call attention to the 
fact that as a consequence of these arguments the terms "oxidizing 
agent" and "reducing agent" are to be regarded as purely relative. 
A given system, e.g., ferrous-ferric, will reduce a system above it in 
Table LIII, e.g., cerous-ceric, but it will oxidize one below it, e.g., stan- 
nous-stannic. 

The question of the extent to which one system oxidizes or reduces 
another is of importance in connection with oxidation-reduction titra- 
tions in analytical chemistry. The reason why eerie sulfate and acidified 
potassium permanganate are such useful reagents in volumetric analysis 
is because they have large negative standard potentials and are conse- 
quently able to bring about virtually complete oxidation of many other 
systems. If the permanganate system had a standard potential which 
did not differ greatly from that of the system being titrated, the equilib- 
rium constant might be of the order of unity; free permanganate, indi- 
cated by its pink color, would then be present in visible amount long 
before oxidation of the other system was complete. The titration values 
would thus have no analytical validity. In order that oxidation or 
reduction of a system should be "complete," within the limits of accuracy 
of ordinary volumetric analysis, it is necessary that the concentration of 
one form at the end-point should be at least 10 3 times that of the other; 
that is to say, oxidation or reduction is complete within 0.1 per cent or 

* If the reaction were considered in the opposite direction the equilibrium constant 
would be the reciprocal jf the value given, and hence would be very large. 



POTENTIOMETRIC OXIDATION-REDUCTION TITHATIONS 



285 



better. The equilibrium constant should thus be smaller than 10" 6 if n 
is the same for both interacting oxidation-reduction systems, or 10~ 9 if n 
is unity for one system and two for the other. By making use of equations 
similar to (19), it can be readily shown that if two oxidation-reduction 
systems are to react completely in the ordinary analytical sense, the 
standard potentials should differ by at least 0.35 volt if n is unity for 
both systems, 0.26 volt if n is unity for one and two for the other, or 
0.18 volt if n is two for both. 

Potentiometric Oxidation-Reduction Titrations. The variation of 
potential during the course of the conversion of the completely reduced 
state of any system to the completely oxidized state is represented by a 
curve of the type shown in Figs. 79 and 81; these curves are, therefore, 
equivalent to potential-titration curves, the end-point of the titration in 
each case being marked by a relatively rapid change of potential. The 
question arises as to whether this end-point could be estimated with 
sufficient accuracy in any given case by measuring the potential of an 
inert electrode, e.g., platinum, inserted in the titration system. An 
answer can be obtained by considering the further change in potential 
after the end-point has been passed; before the equivalence-point the 
potentials are determined by the titrated system, since this is present in 



,67 




Titrated 
System 



Titrant 
Syatem 



E 



E Q 



Titrated 
System 



Titrant 
System 



Fia. 82. Potential-titration curve; deter- 
mination of end-point is possible 



FIG. 83. Potential-titration curve; de- 
termination of the end-point is not satis- 
factory 



excess, while after the equivalent point they are determined by the titrant 
system. The potential-titration curve from one extreme to the other can 
then be derived by placing side by side the curves for the two separate 
systems and joining them by a tangent. Two examples are shown in 
Figs. 82 and 83; in the former the standard potentials, represented by the 



286 OXIDATION-REDUCTION SYSTEMS 

respective mid-points, are reasonably far apart, but in the latter they 
are close together. In Fig. 82 there is a rapid increase of potential at 
the titration end-point, and so its position can be determined accurately; 
systems of this type, therefore, lend themselves to potentiometric titra- 
tion. When the standard potentials of the titrated and titraiit systems 
are close together, however, the change of potential at the equivalence- 
point is not marked to any appreciable extent ; satisfactory potentiometric 
detection of the end-point in such a titration is therefore not possible. 

It will be recalled that the condition for reliable potentiometric titra- 
tion is just that required for one system to reduce or oxidize another 
completely within the normal limits of analytical accuracy. It follows, 
therefore, that when the standard potentials of the two interacting sys- 
tems are such as to make them suitable for analytical work, the reaction 
is also one whose end-point can be derived reasonably accurately poten- 
tiometrically. The minimum differences between the standard potentials 
given on page 285 for an analytical accuracy of about 0.1 per cent, with 
systems of different types, may also be taken as those requisite for satis- 
factory potentiometric titration. The greater the actual difference, of 
course, the more precisely can the end-point be estimated. 

The method of carrying out oxidation-reduction titrations potentio- 
metrically is essentially similar to that for precipitation reactions, except 
that the indicator electrode now consists merely of an inert metal. The 
determination of the end-point graphically or by some form of differential 
titration procedure is carried out in a manner exactly analogous to that 
described in Chap. VII; various forms of simplified methods of oxidation- 
reduction titration have also been described. 9 

Potential at the Equivalence-Point. Since the potentials of the two 
oxidation-reduction systems, represented by the subscripts I and II, 
involved in a titration must be the same, it follows that 



where E is the actual potential and Ei and En are the respective standard 
potentials. Consider the case in which the reduced form of the system I, 
i.e., Ri, is titrated with the oxidized form of the system II, i.e., OH, so 
that the reaction 

Hi + On = Oi + RII 

occurs during the titration. At the equivalence-point, not only are the 
concentrations of Oi and Rn equal, as at any point in the titration, but 
Ri and On are also equal to each other; hence Oi/Ri is then equal to 
Rii/On. Substitution of this result into equation (21) immediately gives 
for Uequiv., the potential at the equivalence-point, 



^equi 



9 See general references to potentiometric titration on page 256. 



OXIDATION-REDUCTION INDICATORS 287 

This result holds for the special case in which each oxidation-reduction 
system involves the transfer of the same number of electrons, i.e., the 
value of n is the same in each case. If they are different, however, the 
equation for the reaction between the two systems becomes 

miRi + niOn = nnOi + niRn, 

where ni and n\\ refer to the systems I and II, respectively. By using 
the same general arguments as were employed above, it is found that the 
potential at the equivalence-point is given by 



Oxidation-Reduction Indicators. A reversible oxidation-reduction 
indicator is a substance or, more correctly, an oxidation-reduction sys- 
tem, exhibiting different colors in the oxidized and reduced states, 
generally colored and colorless, respectively. Mixtures of the two states 
in different proportions, and hence corresponding to different oxidation- 
reduction potentials, will have different colors, or depths of color; every 
color thus corresponds to a definite potential which depends on the 
standard potential of the system, and frequently on the hydrogen ion 
concentration of the solution. If a small amount of an indicator is placed 
in another oxidation-reduction system, the former, acting as a potential 
mediator, will come to an equilibrium in which its oxidation-reduction 
potential is the same as that of the system under examination. The 
potential of the given indicator can be estimated from its color in the 
solution, and hence the potential of the system under examination will 
have the same value. 

Since the eye, or even mechanical devices, are capable of detecting 
color variations within certain limits only, any given oxidation-reduction 
indicator can be effectively employed only in a certain range of potential. 
Consider, for example, the simple case of an indicator system for which 
n is unity; the oxidation-reduction potential at constant hydrogen ion 
concentration is given approximately by 



Suppose the limits within which color changes can be detected are 9 per 
cent of oxidized form, i.e., o/r is 9/91 1/10, at one extreme, to 91 per 
cent of oxidized form, i.e., o/r is 91/9 10; the corresponding potential 
limits at ordinary temperatures are then given by the foregoing equation 
as E Q + 0.058, and E* - 0.058, respectively. If n for the indicator sys- 
tem had been 2, the limits of potential would have been E Q + 0.029 and 
E* 0.029. It is seen, therefore, that an oxidation-reduction indicator 
can be used for determining the potentials of unknown systems only if 
the values lie relatively close to the standard potential E Q of the indi- 



288 -OXIDATION-REDUCTION SYSTEMS 

cator. In other words, it is only in the vicinity of its standard potential, 
at the particular hydrogen ion concentration of the medium, that an 
oxidation-reduction indicator undergoes detectable color changes. In 
order to cover an appreciable range of potentials, it is clearly necessary to 
have a range of indicators with different standard potentials. 

Indicators for Biological Systems. 10 Many investigations have been 
carried out of substances which have the properties necessary for a 
suitable oxidation-reduction indicator. As a result of this work it is 
convenient for practical purposes to divide such indicators into two 
categories: there are those of relatively low potential, viz., 0.3 to 
+ 0.5 volt in neutral solution, which are especially useful for the study 
of biological systems, and those of more negative standard potentials 
that are employed in volumetric analysis. The majority of substances 
proposed as oxidation-reduction indicators for biological purposes are 
also acid-base indicators, exhibiting different colors in acid and alkaline 
solutions. They are frequently reddish-brown in acid media, i.e., at high 
hydrogen ion concentrations, arid blue in alkaline solutions, i.e., at low 
hydrogen ion concentrations, and since the former color is less intense 
than the latter it is desirable to use the indicator in its blue form. In 
biological systems it is generally not possible to alter the hydrogen ion 
concentration from the vicinity of the neutral point, i.e., pH 7,* and so 
indicators are required with relatively strong acidic, or weakly basic, 
groups so that they exhibit their alkaline colors at relatively high hydro- 
gen ion concentrations (cf. Chap. X). A number of such indicators have 
been synthesized by Clark and his co-workers, by introducing halogen 
atoms into one of the phenolic groups of phenol-indophenol, e.g., 2 : 6- 
dichlorophenol-indophenol. In addition to the members of this series, 
other indicators of biological interest are indamines, e.g., Bindschedler's 
green and toluylene blue; thiazines, e.g., Lauth's violet and methylene 
blue; oxazines, e.g., cresyl blue and ethyl Capri blue; and certain indigo- 
sulfonates, safranines and rosindulines. A group of oxidation-reduction 
indicators of special interest are the so-called " viologens," introduced by 
Michaelis; they are NN'-di-substitutcd-4 : 4-dipyridilium chlorides which 
are deeply colored in the reduced state, and have the most positive 
standard potentials of any known indicators. A few typical oxidation- 
reduction indicators used in biological work, together with their standard 
potentials (E') at pH 7, determined by direct measurement, are given 
in Table LIV; it will be observed that these cover almost the whole range 
of potentials from 0.3 to + 0.45 volt, with but few gaps. 

It is rarely feasible in biological investigations to determine the actual 
potential from the color of the added indicator, although this should be 
possible theoretically, because the indicators are virtually of the one 

10 Clark et al., "Studies on Oxidation-Reduction/' 1928 el seq.\ Michaolis, "Oxy- 
dat ions-Reductions Potentiate," 1933; for review, see Glasstone, Ann. Rep. Chem. &oc., 
31, 305 (1934). 

* For a discussion of pH and its significance, see Chap. X; see also, page 292. 



INDICATORS FOR VOLUMETRIC ANALYSIS 



289 



TABLE LIY. OXIDATION-REDUCTION INDICATORS FOR BIOLOGICAL WORK 

Indicator E 9 ' Indicator 



Phenol-m-sulfonate indo- 

2 : 6-dibromophenol - 0.273 

w-Bromophenol indophenol 0.248 

2 : 6-Dichlorophenol indophenol 0.217 

2 : 6-Dichlorophenol indo-o-cresol 0.181 

2 : 6-Dibromophenol indoguaiacol 0.159 

Toluylene blue -0.115 

Cresyl blue - 0.047 

Methylene blue -0.011 

Indigo tetrasulfonate -f 0.046 



E" 

Ethyl Capri blue + 0.072 

Indigo trisulfonate + 0.081 

Indigo disulfonate + 0.125 

Cresyl violet -f 0.173 

Phenosafranine -j- 0.252 

Tetramethyl phenosafranine -f 0.273 

Rosinduline scarlet + 0.296 

Neutral red + 0.325 

Sulfonated rosindone -f 0.380 

Methyl viologen + 0.445 



color type. For most purposes, therefore, it is the practice to take a 
number of samples of the solution under examination, to add different 
indicators to each and to observe which are reduced; if one indicator is 
decolorized and the other not, the potential must lie between the standard 
potentials of these two indicators at the hydrogen ion concentration (pH) 
of the solution. Similarly, indicators may be used in the reduced state 
and their oxidation observed. Indicators are also often employed as 
potential mediators in solutions for which equilibrium with the electrode 
is established slowly; the potential is then measured electrometrically. 
When employing an oxidation-reduction indicator it is essential that the 
solution to which it is added should be well poised (p. 281), so that in 
oxidizing or reducing the indicator the ratio of oxidized to reduced states 
of the experimental system should not be appreciably altered. The 
amount of indicator added must, of course, be relatively small. 

Indicators for Volumetric Analysis. The indicators described above 
are frequently too unstable for use in volumetric analysis and, in addition, 
they show only feeble color changes in acid solution. The problem of 
suitable indicators for detecting the end-points of oxidation-reduction 
titrations is, however, in some senses, simpler than that of finding a series 
of indicators for use over a wide range of potentials. It has been seen 
that if two oxidation-reduction systems interact sufficiently completely 
to be of value for analytical purposes, there is a marked change of poten- 
tial of the system at the equivalence-point (cf. Fig. 82). Ideally, the 
standard potential of the indicator should coincide with the equivalence- 
point potential of the titration; actually it is sufficient, however, for the 
former to lie somewhere in the region of the rapidly changing potential 
of the titration system. When the end-point is reached, therefore, and 
the oxidation-reduction potential undergoes a rapid alteration, the color 
of the indicator system will change sharply from one extreme to the other. 
If the standard potential of the indicator is either below or above the 
region in which the potential inflection occurs, the color change will take 
place either before or after the equivalence-point, and in any case will be 
gradual rather than sharp. Such indicators would be of no value for the 
particular titration under consideration. It has been found (p. 285) that 
if two systems are to interact sufficiently for analytical purposes their 



290 OXIDATION-REDUCTION SYSTEMS 

standard potentials must differ by about 0.3 volt, and hence the standard 
potential of a suitable oxidation-reduction indicator must be about 0.15 
volt below that of one system and 0.15 volt above that of the other. 
Since the most important volumetric oxidizing agents have high negative 
potentials, however, a large number of indicators is not necessary for 
most purposes. 

The interest in the application of indicators in oxidation-reduction 
titrations has followed on the discovery that the familiar color change 
undergone by diphenylamine on oxidation could be used to determine the 
end-point of the titration of ferrous ion by dichromate in acid solution. 
Diphenylamine, preferably in the form of its soluble sulfonic acid, at first 
undergoes irreversible oxidation to diphenylbenzidine, and it is this sub- 
stance, with its oxidation product diphenylamine violet, that constitutes 
the real indicator. 11 

The standard potential of the indicator system is not known exactly, 
but experiments have shown that in not too strongly acid solutions the 
sharp color change from colorless to violet, with green as a possible 
intermediate, occurs at a potential of about 0.75 volt. The standard 
potential of the ferrous-ferric system is 0.78 whereas that of the di- 
chromate-chromic ion system in an acid medium is approximately 1.2 
volt; hence a suitable oxidation-reduction indicator might be expected to 
have a standard potential of about 0.95 volt. It would thus appear 
that diphenylamine would not be satisfactory for the titration of ferrous 
ions by acid dichromate, and this is actually true if a simple ferrous salt 
is employed. In actual practice, for titration purposes, phosphoric acid 
or a fluoride is added to the solution ; these substances form complex ions 
with the ferric ions with the result that the effective standard potential 
of the ferrous-ferric system is lowered (numerically) to about 0.5 volt. 
The change of potential at the end-point of the titration is thus from 
about 0.6 to 1.1 volt, and hence diphenylamine, changing color in 
the vicinity of 0.75 volt, is a satisfactory indicator. 

Ceric sulfate is a valuable oxidizing agent, the employment of which 
in volumetric work was limited by the difficulty of detecting the end-point 
unless a potentiometric method was used. A number of indicators are 
now available, however, which permit direct titration with eerie sulfate 
solution to be carried out. One of the most interesting and useful of 
these is o-phenanthroline ferrous sulfate, the cations of which, viz., 
FeCCuHsNi) +, with the corresponding ferric ions, viz., Fe(Ci 2 H 8 N 2 ) ++, 
form a reversible oxidation-reduction system; the reduced state has an 
intense red color and the oxidized state a relatively feeble blue color, so 
that there is a marked change in the vicinity of the standard potential 
which is about 1.1 volt. 12 The high potential of the phenanthroline- 

" Kolthoff and Sarver, J. Am. Chem. Soc., 52, 4179 (1930); S3, 2902 (1931); 59, 23 
(1937); for review, see Glasstone, Ann. Rep. Chem. Soc., 31, 309 (1934); also, Whitehead 
and Wills, Chem. Revs., 29, 69 (1941). 

11 Walden, Hammett and Chapman, /. Am. Chem. Soc., 53, 3908 (1931); 55, 2649 
(1933); Walden and Edmonds, Chem. Revs., 16, 81 (1935). 



QUINONE-HYDROQUINONE SYSTEMS 291 

ferrous ion indicator permits it to be used in connection with the titration 
of ferrous ions without the addition of phosphoric acid or fluoride ions. 
The indicator has been employed for a number of titrations with eerie 
sulfate and also with acid dichromate, and even with very dilute solutions 
of permanganate when the color of the latter was too feeble to be of any 
value for indicator purposes. Another indicator having a high standard 
potential is phenylanthranilic acid; this is a diphenylamine derivative 
which changes color in the vicinity of 1.08 volt. It has been recom- 
mended for use with eerie sulfate as the oxidizing titrant. 13 

Although there are now several useful indicators for titrations in- 
volving strongly oxidizing reactants, the situation is not so satisfactory 
in connection with reducing reagents, e.g., titanous salts. The standard 
potential of the titanous-titanic system is approximately 0.05 volt, 
and hence a useful indicator should show a color change at a potential 
of about 0.2 volt or somewhat more negative. The only substance 
that is reasonably satisfactory for this purpose, as far as is known at 
present, is methylene blue which changes color at about 0.3 volt in 
acid solution. 

Quinone-Hydroquinone Systems. In the brief treatment of the 
quinone-hydroquinone system on page 270 no allowance was made for 
the possibility of the hydroquinone ionizing as an acid; actually such 
ionization occurs in alkaline solutions and has an important effect on the 
oxidation-reduction potential of the system. Hydroquinone, or any of 
its substituted derivatives, can function as a dibasic acid. It ionizes in 
two stages, viz., 

H 2 Q ^ 11+ + HQ- 
and 

HQ- ^ H+ + Q , 

and the dissociation constants corresponding to these two equilibria 
(cf. p. 318) are given by 

- 

and A 2 = 



The hydroquinone in solution thus exists partly as undissociated H 2 Q, 
and also as HQ~ and Q ions formed in the two stages of ionization; 
the total stoichiometric concentration h of the hydroquinone is equal to 
the sum of the concentrations of these three species, i.e., 

h = CH Z Q + CHQ~ + CQ--, 

and if the values of CHQ- and CQ derived from the expressions for K\ 
and KZ are inserted in this equation, the approximation being made of 
taking the activity coefficients of H a Q, HQ~ and Q to be equal to 

"Syrokomsky and Stiepin, J. Am. Chem. Soc., 58, 928 (1936). 



292 OXIDATION-REDUCTION SYSTEMS 

unity, the result is 

, , CH Q , , CH Q , , 

h = C H ,Q + - ki + --*- kik 2 , 
1 a+ a+ 



In view of the neglect of the activity coefficients, the constants KI and K* 
have been replaced by k\ and kz which become identical with the former 
at infinite dilution. If q is the concentration of the quinone form, which 
is supposed to be a neutral substance exhibiting neither acidic nor basic 
properties, the oxidation-reduction potential, which according to equa- 
tion (4) may be written as 



is given by 

7?T n RT 

---Ino 2 ^, (24) 



the ratio of the activities of Q and H 2 Q being taken as equal to the ratio 
of their concentrations. Introduction of the value of CH 2 q from equation 
(22) into (24) now gives 

t> r r* xv z> r n 

rCl q it 1 o 

E = E -i." In vTrT In (OH* -f- ki(in+ + fci& 2 ). (25) 

Zr fl 2ib 

If ki and fc 2 are small, the terms k\a^ and k\k 2 may be neglected in com- 
parison with afi+, and equation (25) then reduces to 

(26) 

AT H, 1' 

which is the conventional form for the quinone-hydroquinone system, 
q and h representing the total concentrations of the two constituents. 

According to equation (26) the variation of the oxidation-reduction 
potential with hydrogen ion concentration is relatively simple, but if the 
acidic dissociation functions ki and kz of the hydroquinone are appre- 
ciable, equation (25) must be employed, and the situation becomes 
somewhat more complicated. The method of studying this problem is 
to maintain the ratio q/h constant, i.e., the stoichiometric composition 
of the quinone-hydroquinone mixture is unchanged, but to suppose the 
hydrogen ion concentration is altered. For this purpose the equations 
for the electrode potential are differentiated with respect to log OH*; 
this quantity is a very useful function of the hydrogen ion concentration, 
designated by the symbol pH and referred to as the hydrogen ion expo- 
nent. Differentiation of equation (25) thus gives 

o wvj RT 

d\og< 



QUINONE-HYDROQUINONE SYSTEMS 



293 



as applicable over the whole pH range. If an* is large in comparison 
with k\ and &2, i.e., in relatively acid solutions, this equation reduces to 



JET 



- 2 - 303 - for 



(28) 



which can also be derived directly from equation (26). 

If, however, k\ is much greater than OH* and this is much greater than 
&2, the terms 2an+ in the numerator and OH+ and kik z in the denominator 
of equation (27) may be neglected; the result is 



d(pH) 



RT 
2F 



7TFT for 



(29) 



Finally, when an + becomes very small, i.e., in alkaline solutions, both 
terms in the numerator of equation (27) may be disregarded, and so 



dE 
d(pH) 



= 



for 



(30) 



The slope of the plot of the oxidation-reduction potential, for con- 
stant quinone-hydroquinone ratio, against the pH, i.e., against log a H +, 
thus undergoes changes, as shown in Fig. 84; the temperature is 30 



so 



-0.1 - 




Fia. 84. Variation of a quinone-hydroquinone (anthraquinone sulfonate) 
potential with pH 

that the slopes corresponding to equations (28), (29) and (30) are 0.060, 
0.030 and zero, respectively. The position and length of the intermediate 
portion of slope 0.030 depend on the actual values and ratio of the acidic 



294 OXIDATION-REDUCTION SYSTEMS 

dissociation functions k\ and A: 2 ; this may be seen by investigating the 
conditions for which an + is equal to k\ and k* respectively. 
If OH* in equation (27) is set equal to ki, the result is 

dE RT 3^ 

565) = 2 ' 303 w ' afT+lS for aH * = * lf 

and since fa is generally much smaller than k\ 9 this becomes 



= 2.303 - ~ = 0.045 at 30 



d(pH) " >wu 2F 2 

When the hydrogen ion activity air is equal to the acidic function k\ 9 
i.e., when the pH is equal to log k\ 9 the latter quantity being repre- 
sented by pki, the slope of the pH-potential curve is thus seen to be 
intermediate between 0.060 and 0.030. Such a slope corresponds, in 
general, to a point on the first bend of the curve in Fig. 84; the exact 
position for a slope of 0.045 is 0-btained by finding the point of inter- 
section of the two lines of slope 0.060 and 0.030, as shown. At this point, 
therefore, the pH is equal to pki. 

To find the slope of the pH potential curve when OH + is equal to fa, 
i.e., when the pH is equal to pfc 2 , the values of <JH+ in equation (27) are 
replaced by & 2 ; hence 

dE nM RT 2h + ki 

S5) = 2 ' 3 3 2F ' k^+U[ f r * H * - k *> 
and since, as before, k 2 may be regarded as being much smaller than k\ 9 

dE 



d(pH) - 2 ' 303 W 2 - ' 015 at 30 ' 

The pH is thus equal to p& 2 when the slope of the pH-potential curve is 
midway between 0.030 and zero ; the value of p& 2 can be found by extend- 
ing the lines of slopes 0.030 and zero until they intersect, as shown in Fig. 
84. An examination of the pH-potential curve thus gives the values of 
the acidic dissociation functions for the particular hydroquinone as 7.9 
and 10.6 for pki and pfa, respectively, at 30. 

The case considered here is relatively simple, but more complex be- 
havior is frequently encountered: the reduced form may have more than 
two stages of acidic dissociation and in addition the oxidized form may 
exhibit one or more acidic dissociations. There is also the possibility of 
basic dissociation occurring, but this can be readily treated as equivalent 
to an acidic ionization (cf. p. 362). The method of treatment given 
above can, however, be applied to any case, no matter how complex, and 
the following general rules have been derived which facilitate the analysis 
of pH-potential curves for oxidation-reduction systems of constant 
stoichiometric composition. 14 

14 Clark, "Studies on Oxidation-Reduction/ 1 Hygienic Laboratory Bulletin, 1928. 



TWO STAGE OXIDATION-REDUCTION 295 

(1) Each bend in the curve may be correlated with an acidic dis- 
sociation constant; if the curve becomes steeper with increasing pH, i.e., 
as the solution is made more alkaline, the dissociation has occurred in 
the oxidized form, but if it becomes flatter it has occurred in the reduced 
form (cf. Fig. 84). 

(2) The intersection of the extensions of adjacent linear parts of the 
curve occurs at the pH equal to pfc for the particular dissociation function 
responsible for the bend. 

(3) Each dissociation constant changes the slope by 2.3Q3RT/nF 
volt per pH unit, where n is the number of electrons difference between 
oxidized and reduced states. 

Two Stage Oxidation-Reduction. The completely oxidized, i.e., holo- 
quinone, form of a quinone differs from the completely reduced, i.e., 
hydroquinone, form by two hydrogen atoms, involving the addition or 
removal, respectively, of two electrons and two protons in one stage, viz., 

H 2 Q ^ Q 4- 2H+ + 2c. 

It is known from chemical studies, however, that in many cases there is 
an intermediate stage between the hydroquinone (H 2 Q) and the quinone 
(Q) ; this may be a meriquinone, which may be regarded as a molecular 
compound (Q-H 2 Q), or it may be a semiquinone (HQ). The latter is a 
true intermediate with a molecular weight of the same order as that of 
the quinone, instead of double, as it is for the meriquinone. The possi- 
bility that oxidation and reduction of quinonoid compounds might take 
place in two stages, each involving one electron, i.e., n is unity, with the 
intermediate formation of a semiquinone was considered independently 
by Michaelis and by Elema. 16 If the two stages of oxidation-reduction 
do not interfere, a ready distinction between meriquinone and semi- 
quinone formation as intermediate is possible by means of E.M.F. meas- 
urements. 

For meriquinone formation the stages of oxidation-reduction may be 
written 

(1) 2H 2 Q - H 2 Q Q + 2H+ + 2, 
and 

(2) H 2 Q-Q^2Q 



so that if EI represents the standard potential of the first stage at a 
definite hydrogen ion concentration, 

RT (H,Q.Q) 
~ 



where the parentheses represent activities. If the original amount of 
the reduced form (H 2 Q) in a given solution is a, and x equiv. of a strong 

" Friedheim and Michaelis, J. Biol Ghent., 91, 355 (1931); Michaelis, ibid., 92, 211 
(1931); 96, 703 (1932); Elema, Rec. trav. chim., 50, 807 (1931); 52, 569 (1933); /. Biol 
Chem., 100, 149 (1933). 



296 OXIDATION-REDUCTION SYSTEMS 

oxidizing agent are added, %x moles of Q are formed, and these combine 
with an equivalent amount of H 2 Q to form \x moles of meriquinone, 
J^Q-Q; an amount a x moles of HQ remains unchanged. It follows, 
therefore, neglecting activity coefficients, that in a solution of volume t;, 
equation (31) becomes 



_ RT , x RT, v 

__ ET __ ^_ 1- _ _ _ \Yl I ^X I 

ir ( Qf x) &r & 

The potential thus depends on the volume of the solution, and hence the 
position of the curve showing the variation of the oxidation-reduction 
potential during the course of the titration of H2Q by a strong oxidizing 
agent varies with the concentration of the solution. At constant volume 
equation (32) becomes 

_ RT f B7 1 , , 

E = E l -lnx + In (a-*), 

so that in the early stages of oxidation, i.e., when x is small, the last term 
on the right-hand side may be regarded as constant, and the slope of the 
titration curve will correspond to a process in which two electrons are 
involved, i.e., n is 2. In the later stages, however, the change of potential 
is determined mainly by the last term, and the slope of the curve will 
change to that of a one-electron system, i.e., n is effectively unity. 

When a true semiquinone is formed, the two stages of oxidation- 
reduction are 

(1) H 2 Q ^ HQ + H + + , 
and 

(2) HQ ^ Q + 11+ + | 
so that 



(33) 



for a definite hydrogen ion concentration. The value of the potential 
is seen to depend on the ratio of x to a re, and not on the actual con- 
centration of the solution; the position of the titration curve is thus 
independent of tho volume. Further, it is evident from equation (33) 
that the type of slope is the same throughout the curve, and corresponds 
to a one-electron process, i.e., n is unity. 

If the two stages of oxidation are fairly distinct, it is thus possible to 
distinguish between meriquinone and semiquinone formation. In the 
former case the position of the titration curve will depend on the volume 
of the solution and it will be unsymmetrical, the earlier part correspond- 



SEMIQUINONE FORMATION CONSTANT 297 

ing to an n value of 2, and the later part to one of unity. If semiquinone 
formation occurs, however, the curve will be symmetrical, with n equal 
to unity over the whole range, and its position will not be altered by 
changes in the total volume of the solution. A careful investigation 
along these lines has shown that many oxidation-reduction systems satisfy 
the conditions for semiquinone formation; in one way or another, this 
has been found to be true for a-oxyphenazine and some of its derivatives, 
e.g., Wurster's red, and for a number of anthraquinones. 

Semiquinone Formation Constant. It was assumed in the foregoing 
treatment that the two stages of oxidation are fairly distinct, but when 
this is not the case the whole system behaves as a single two-electron 
process, as in Fig. 79. In view of the interest associated with the forma- 
tion of semiquinone intermediates in oxidation-reduction reactions, 
methods have been developed for the study of systems in which the two 
stages may or may not overlap. The treatment is somewhat compli- 
cated, and so the outlines only will be given here. 16 

If R represents the completely reduced form (H 2 Q), S the semi- 
quinone (HQ), and T the totally oxidized form (Q), the electrical equi- 
libria, assuming a constant hydrogen ion concentration, are 

(1) R ^ S + e and (2) S ^ T + , 
so that if r, s and t are the concentrations of the three forms, 

RT s 
E = E!- In-, (34) 

and 

RT t 
= 2 --jrln-> (35) 

during the first and second stages, respectively; EI and E z are the stand- 
ard potentials of these stages. The potential can also be formulated in 
terms of the equilibrium between initial arid final states, viz., 

R ^ T + 2c, 
so that 



- r , (36) 

where E m is the usual standard potential for the system as a whole at 
some definite hydrogen ion concentration. It can be seen from equa- 
tions (34), (35) and (36) that 



"For reviews, see Michaelis, " Oxydations-Reductions Potentiate," 1933; Trans. 
Electrochem. Soc., 71, 107 (1937); Chem. Revs., 16, 243 (1935); Michaelis and Schubert, 
ibid., 22, 437 (1938); Michaelis, Ann. New York Acad. Sci., 40, 39 (1940); Miiller, ibid., 
40, 91 (1940). 



298 



OXIDATION-REDUCTION SYSTEMS 



and since E\ and E z will be in the centers of the first and second parts of 
the titration curves, i.e., when s/r and t/s are unity, respectively, it 
follows that E m will be the potential in the middle of the whole curve. 
In addition to the electrical equilibria, there will be a chemical equi- 
librium between R, S and T, viz., 

R + T ^ 2S, 
so that by the approximate form of the law of mass action 



where k is known as the semiquinone formation constant. 

If a is the initial amount of reduced form H 2 Q which is being titrated, 
and x equiv. of strong oxidizing agent are added, then x/a is equal to 1 
in the middle of the complete titration curve and to 2 at the end. By 
making use of the relationships given above, it is possible to derive an 
equation of some complexity giving the variation of E E m with x/a, 




Fia. 85. Titration curves for semiquinone formation 

i.e., during the course of the titration, for any value of k, the semiquinone 
formation constant. Some of the results obtained in this manner are 
shown in Fig. 85; as long as k is small, the titration curve throughout has 
the shape of a normal two-electron oxidation-reduction system, there 
being no break at the midpoint where x/a is unity. As k increases, the 
slope changes until it corresponds to that of a one-electron process; iu 



STORAGE BATTERIES (SECONDARY CELLS) 299 

fact when the value of k lies between 4 and 16, the slope is that for a 
system with n equal to unity, but there is no break at the midpoint. 
The presence of a semiquinone is often indicated in these cases, however, 
by the appearance of a color which differs from that of either the com- 
pletely oxidized or the completely reduced forms. When the semiqui- 
none formation constant k exceeds 16, a break appears at the midpoint, 
and the extent of this break becomes more marked as k increases. The 
detection of semiquinone formation by the shape of the titration curve 
is only possible, therefore, when the semiquinone formation constant 
is large. 

If actual oxidation-reduction measurements are made on a particular 
system during the course of a titration, it is possible, by utilizing the 
equation from which the data in Fig. 85 were calculated, to evaluate the 
semiquinone formation constant for that system. The standard poten- 
tials Ely E 2 and E m can also be obtained for the hydrogen ion concen- 
tration existing in the experimental solution. 

Influence of Hydrogen Ion Concentration. The values of EI, E 2 and 
E m will depend on the pH of the solution, and since the forms R, S and T 
may possess acidic or basic functions, the slopes of the curves of these 
three standard potentials against pH may change direction at various 
points and crossings may occur. A system for which E 2 is above EI at 
one pH, i.e., the semiquinone formation constant is large, may thus be- 
have in a reverse manner, i.e., EI is above E 2 , and the semiquinone 
formation is very small, at another pH. It is apparent, therefore, that 
although a given system may show distinct semiquinone formation at 
one hydrogen ion concentration, there may be no definite indication of 
such formation at another hydrogen ion concentration. If the oxidized 
form of the system consists of a positive ion, e.g., anthraquinone sulfonic 
acid, semiquinone formation is readily observed in alkaline solutions 
only, but if it is a negative ion, e.g., a-oxyphenazine, the situation is 
reversed and the semiquinone formation can be detected most easily in 
acid solution. 

Storage Batteries (Secondary Cells). 17 When an electric current is 
passed through an electrolytic cell chemical changes are produced and 
electrical energy is converted into chemical energy. If the cell is revers- 
ible, then on removing the source of current and connecting the elec- 
trodes of the cell by means of a conductor, electrical energy will be 
produced at the expense of the stored chemical energy and current will 
flow through the conductor. Such a device is a form of storage battery, 
or secondary cell; * certain chemical changes occur when the cell is 
"charged" with electricity, and these changes are reversed during dis- 

" Vinal, "Storage Batteries," 1940. 

* A primary cell is one which acts as a source of electricity without being previously 
charged up by an electric current from an external source; in the most general sense, 
every voltaic cell is a primary cell, although the latter term is usually restricted to cells 
which can function as practical sources of current, e.g., the Leclanchl cell. 



300 OXIDATION-REDUCTION SYSTEMS 

charge. Theoretically, any reversible cell should be able to store elec- 
trical energy, but for practical purposes most of them are unsuitable 
because of low electrical capacity, incomplete reversibility as to the 
physical form of the substances involved, chemical action or other changes 
when idle, etc. Only two types of storage battery have hitherto found 
any wide application, and since they both involve oxidation-reduction 
systems their theoretical aspects will be considered here. 

The Acid Storage Cell. The so-called "acid" or "lead" storage cell 
consists essentially of two lead electrodes, one of which is covered with 
lead dioxide, with approximately 20 per cent sulfuric acid, i.e., with a 
specific gravity of about 1.15 at 25, as the electrolyte. The charged 
cell is generally represented simply as Pb, H 2 SC>4, PbO 2 , but it is more 
correct to consider it as 

Pb | PbSO 4 (s) H 2 S0 4 aq. PbS0 4 (s), Pb0 2 (s) | Pb, 

the right-hand lead electrode acting as an inert electrode for an oxidation- 
reduction system. The reactions occurring in the cell when it produces 
current, i.e., on discharge, are as follows. 
Left-hand electrode: 

Pb = Pb++ + 2 
Pb++ + SO?- = PbS0 4 (s). 

.". Net reaction for two faradays is 

Pb + SOr- = PbSO 4 (s) + 2 . 
Right-hand electrode: 



PbO 2 (s) + 2H 2 ^ Pb++++ + 4OH- 
Pb++++ + 2 = Pb++, 
Pb++ + SO?- = PbSO 4 (s), 
4OH- + 4H+ = 4H 2 O, 

.". Net reaction for two faradays is 

Pb0 2 (s) + 4H+ + SO + 2 = PbS0 4 (s) + 2H 2 0. 

Since both electrodes are reversible, the processes occurring when elec- 
tricity is passed through the cell, i.e., on charge, are the reverse of those 
given above; it follows, therefore, that the complete cell reaction in both 
directions may be written as the sum of the individual electrode processes, 
thus 

discharge 

Pb + Pb0 2 + 2H 2 S0 4 ^ 2PbS0 4 + 2H 2 

charge 

for two faradays. The mechanism of the operation of the lead storage 
battery as represented by this equation was first proposed by Gladstone 
and Tribe (1883) before the theory of electrode processes in general was 
well understood; it is known as the "double sulfation" theory, because it 



THE ACID STORAGE CELL 301 

postulates the formation of lead sulfate at both electrodes. Various 
alternative theories concerning the lead cell have been proposed from 
time to time but these appear to have little to recommend them; apart 
from certain processes which occur to a minor extent, e.g., formation of 
oxides higher than PbO 2 , there is no doubt that the reactions given here 
represent essentially the processes occurring at the electrodes of an acid 
storage battery. 

It will be observed that according to the suggested cell reaction, two 
molecules of sulfuric acid should be removed from the electrolyte and 
two molecules of water formed for the discharge of two faradays of elec- 
tricity from the charged cell. This expectation has been confirmed 
experimentally. Further, it is possible to calculate the free energy of 
this change thermodynamically in terms of the aqueous vapor pressure of 
sulfuric acid solutions; the values should be equal to 2FE, where E is 
the E.M.F. of the cell and this has been found to be the case. 

A striking confirmation of the validity of the double sulfation theory 
is provided by thermal measurements; since the E.M.F. of the storage cell 
and its temperature coefficient are known, it is possible to calculate the 
heat change of the reaction taking place in the cell by means of the Gibbs- 
Helmholtz equation (p. 194). The value of the heat of the reaction 
believed to occur can be derived from direct thermochemical measure- 
ments, and the results can be compared. The data obtained in this 
manner for lead storage cells containing sulfuric acid at various concen- 
trations, given in the first column with the density in the second, are 
quoted in Table LV; 18 the agreement between the values in the last two 

TABLE LV. HEAT CHANGE OP REACTION IN LEAD STORAGE BATTERY 

H2SO 4 E u * dE/dT A// 

per cent dl volts X 10 4 E.M.F. Thermal 

4.55 1.030 1.876 

7.44 1.050 1.905 +1.5 - 85.83 - 86.53 

14.72 1.100 1.962 +2.9 -86.54 -87.44 

21.38 1.150 2.005 +3.3 -87.97 -87.37 

27.68 1.200 2.050 +3.0 -90.46 -90.32 

33.80 1.250 2.098 +2.2 -93.77 -93.08 

39.70 1.300 2.148 +1.8 -96.63 -96.22 

columns is very striking, and appears to provide conclusive proof of the 
suggested mechanism. 

It is evident from the data in Table LV that the E.M.F. of the lead 
storage cell increases with increasing concentration of sulfuric acid; this 
result is, of course, to be expected from the cell reactions. According 
to the reaction occurring at the Pb, PbSO 4 electrode, generally referred 
to as the negative electrode of the battery, its potential (EJ) is given by 

r>rn 

E- = #pb,pbso 4 ,so;~ + ~2p I* 1 a so;~- (37) 

Since the activity, or concentration, of sulfate ions depends on the con- 
18 Craig and Vinal, /. Res. Nat. Bur. Standards, 24, 475 (1940). 



302 OXIDATION-REDUCTION SYSTEMS 

centration of sulfuric acid, it is clear that the potential of this electrode 
will vary accordingly. The standard potential in equation (37) is + 0.350 
volt at 25, and if the activity of the sulfate ion is taken as equal to the 
mean activity of sulfuric acid, it is readily calculated that for a storage 
battery containing acid of the usual concentration, i.e., 4 to 5 N, in which 
the mean activity coefficient is about 0.18 to 0.2, the actual potential of 
the negative electrode is about + 0.33 volt. The so-called negative 
electrode potential may also be represented by 

~n RT 

E- = $ b ,pb" - ^p In a Pb ++, (38) 

but since the solution is saturated with lead sulfate, a Pb ++ will be inversely 
proportional to aso;-; equations (37) and (38) are thus consistent. 

The potential of the PbSO 4 , PbO 2 electrode, usually called the positive 
electrode, can be represented by (cf. p. 269) 




j? * E*> , , Hso /on\ 

E+* = #pbso 4 , Pbo,, sor + TTTT In - 2 - (39) 



and hence will be very markedly dependent on the concentration of 
sulfuric acid, since this affects a n + , asor and an,o. The standard 
potential required for equation (39) is 1.68 volts at 25 (see Table LIII) ; 
making the assumption that the activities of the hydrogen and sulfate 
ions are equal to the mean activity of sulfuric acid in which the activity 
of water from vapor pressure data is 0.3, it is found that, for 4 to 5 N acid, 
the potential E+ of the positive electrode is about 1.70 volts. 

The positive electrode may also be regarded as a simple oxidation- 
reduction electrode involving the plumbous-plumbic system; thus 

-* + --^. a + ln. (40) 



The activity of plumbic ions in a solution saturated with lead dioxide 
(or plumbic hydroxide) will be inversely proportional to the fourth power 
of the hydroxyl ion activity, and hence it is directly proportional to the 
fourth power of the hydrogen ion activity (cf. p. 339), in agreement with 
the requirements of equation (39). 

The Alkaline Storage Battery. The alkaline or Edison battery is 
made up of an iron (negative) and a nickel sesquioxide (positive) elec- 
trode in potassium hydroxide solution; it may be represented as 

Fe | FeO(s) KOH aq. NiO(s), Ni 2 O 3 (s) | Ni, 

the nickel acting virtually as an inert electrode material. The reactions 
taking place in the charged cell during discharge are as follows. 

*The negative sign is used because the potential of the electrode as written, viz., 
PbSO 4 (a), PbOiW, Pb, is opposite in direction to that corresponding to the convention 
on which the standard potentials in Tables XLIX and LIII are based. 



THE ALKALINE STORAGE BATTERY 303 

Left-hand electrode: 

Fe = Fe++ + 2 , 

Fe++ + 20H- = FeO(s) + H 2 0, 
.". Net reaction for two faradays is 

Fe + 20H- = FeO(s) + H 2 O + 2. 
Right-hand electrode: 

Ni 2 O 3 (s) + 3H 2 O ^ 2Ni+++ + 60H-, 

2Ni+++ + 2e = 2Ni++, 
2Ni++ + 4OII- = 2NiO(s) + 2H 2 0, 
.". Net reaction for two faradays is 

Ni 2 O 3 (s) + H 2 O + 2 = 2NiO(s) + 2OH~. 

The complete cell reaction during charge and discharge, respectively, 
may be represented by 

discharge 

Fe + Ni 2 O 3 ^ FeO + 2NiO. 

charge 

The potential of the iron ("negative") electrode, which is about + 0.8 
volt in practice, is given by the expression 



. 

&- = -CTe,FeO,OH~ ~T TTIT In - > 

*r an t o 

and similarly that of the nickel sesquioxide ("positive") electrode, which 
is approximately + 0.55 volt, is represented by 

T RT 



The potentials of both individual electrodes are dependent on the hy- 
droxyl ion activity (or concentration) of the potassium hydroxide solution 
employed as electrolyte. It is evident, however, that in theory the 
E.M.F. of the complete cell, which is equal to E- E+, should be inde- 
pendent of the concentration of the hydroxide solution. In practice a 
small variation is observed, viz., 1.35 to 1.33 volts for N to 5 N potassium 
hydroxide; this is attributed to the fact that the oxides involved in the 
cell reactions are all in a "hydrous" or "hydrated" form, with the result 
that a number of molecules of water are transferred in the reaction. The 
equations for the potentials of the separate electrodes should then con- 
tain different terms for the activity of the water in each case: the E.M.F. 
of the complete cell thus depends on the activity of the water in the 
electrolyte, and hence on the concentration of the potassium hydroxide. 



304 OXIDATION-REDUCTION SYSTEMS 

PROBLEMS 

1. Write down the electrochemical equations for the oxidation-reduction 
systems involving (i) ClOj and C1 2 , and (ii) Cr 2 07~ and Cr+++. Use the 
results to derive the complete equations for the reactions of each of these with 
the Sn++++, Sn++ system. 

2. According to Br0nsted and Pedersen [Z. physik. Chem., 103, 307 (1924)] 
the equilibrium constant of the reaction 

I" = Fe++ + JI 2 



at 25 is approximately 21, after allowing for the tri-iodide equilibrium. The 
standard potential of the I 2 (s), I" electrode is 0.535 volt and the solubility 
of iodine in water is 0.00132 mole per liter; calculate the approximate standard 
potential of the (Pt)Fe++, Fe+++ system. 

3. From the measurements of Sammet \_Z. physik. Chem., 53, 678 (1905)] 
the standard potential of the system (PtJIOi" -f 6H+, JI 2 has been estimated 
as 1.197 volt. Determine the theoretical equilibrium constant of the 
reaction 

10? + 51- + 6H+ = 3I 2 + 3H 2 0. 

What conclusion may be drawn concerning the quantitative determination of 
iodate by the addition of acidified potassium iodide followed by titration with 
thiosulfate? 

4. Kolthoff and Tomsicek [J. Phys. Chem., 39, 945 (1935)] measured the 
potentials of the electrode (Pt)Fe(CN)e --- , Fe(CN)e~~ at 25; the concen- 
trations of potassium ferro- and ferri-cyanide were varied, but the ratio was 
unity in every case. The concentrations (c) of each of the salts, in moles per 
liter, and the corresponding electrode potentials (Eo), on the hydrogen scale, 
are given below: 

c E' Q c Ei 

0.04 -0.4402 0.0004 -0.3754 

0.02 -0.4276 0.0002 -0.3714 

0.01 -0.4154 0.0001 -0.3664 

0.004 -0.4011 0.00008 -0.3652 

0.002 -0.3908 0.00006 -0.3642 

0.001 -0.3834 0.00004 -0.3619 

Plot the values of E'o against Vjji and extrapolate the results to infinite dilution 
to obtain the standard potential of the ferrocyanide-ferricyanide system. 
Alternatively, derive the value of E Q from each E' Q by applying the activity 
correction given by the Debye-Hiickel limiting law. 

5. The oxidation-reduction system involving 5- and 4-valent vanadium 
may be represented by the general equation 



- z V,OJ*+ + (y - z)H 2 = V x O< 5 *- 2 *>+ + 2(y - z)H+ + X. 

Using the symbol V 5 to represent the oxidized form V X O V and V 4 for the re- 
duced form V0, write the equation for the E.M.F. of the cell consisting of the 
V 4 , V 5 and H+, Kfo electrodes. Derive the expressions to which this equa- 
tion reduces (i) when V 4 and H+ are kept constant, (ii) when V 5 and V 4 are 
constant, and (iii) when V 6 and H+ are constant. The experimental results of 
Carpenter [J. Am. Chem. Soc., 56, 1847 (1934)] are as follows: 



PROBLEMS 305 

d) (ii) (iii) 

V 6 E H+ E V 4 E 

0.529 X 10~ 3 - 0.9031 0.0240 - 0.9098 4.42 X 10~ - 0.9554 

2.489 -0.9395 0.1077 -0.9554 35.11 -0.9048 

9.855 -0.9723 0.4442 -0.9974 

19.67 - 0.9875 0.9000 - 1.0198 

Using the expressions already derived, show that the values of z, (2y 3x)/x 
and z can be obtained by plotting E against log V 5 , log H+ and log V 4 , respec- 
tively. Insert the values of x\ y and z in the expression given above and so 
derive the actual equation for the oxidation-reduction system. 

6. In an investigation of the oxidation-reduction potentials of the system 
in which the oxidized form was anthraquinone 2 : 6-disulfonate, Conant and 
his collaborators [J. Am. Chem. Soc., 44, 1382 (1922)] obtained the following 
values for E Qf at various pH's: 

pH 6.90 7.64 9.02 9.63 10.49 11.27 11.88 12.20 
W 0.181 0.220 0.275 0.292 0.311 0.324 0.326 0.326 

Plot E ' against the pH and interpret the results. 

7. By extrapolating the E.M.F.'S to infinite dilution, Andrews and Brown 
[J. Am. Chem. Soc., 57, 254 (1935)] found E Q for the cell 

Pt | KMn0 4 , Mn0 2 (s) KOH aq. HgO(s) | Hg 

to be - 0.489 at 25. The standard potential of the Hg, HgO(s), OH~ elec- 
trode is 0.098, and the equilibrium constant of the system 



+ 2H,0 = 2MnOi + MnO 2 (s) + 40H~ 

is 16 at this temperature. Calculate the standard potential of the (Pt)MnO4, 
MnOf ~ electrode. 

8. The standard potential of the (Pt) | PbS0 4 (s), PbO 2 (s), SOi electrode 
is 1.685 volts at 25; calculate the E.M.F.'S of the cell 

Pt | PbS0 4 (s), Pb0 2 (s) H 2 S0 4 (c) | H 2 (l atm.) 

for 1.097 and 6.83 molal sulfuric acid solutions. The mean activity coeffi- 
cients (y) and aqueous vapor pressures (p) of the solutions are: 

m y p 

1.097 0.146 22.76 mm. 

6.83 0.386 12.95 

The vapor pressure of water at 25 is 23.76 mm. of mercury. 

9. From the standard potentials of the systems (Pt)Cu+, Cu+ + and I 2 , I~ 
evaluate the equilibrium constant of the reaction 

Cu++ + I- = Cu+ + *I 2 , 

and show that it is entirely owing to the low solubility product of cuprous 
iodide, Cul, i.e., approximately 10~ 12 , that this reaction can be used for the 
analytical determination of cupric ions. 

10. The solubility products of cupric and cuprous hydroxides, Cu(OH) 2 
and CuOH, respectively, are approximately 10~ 19 and 10~ 14 at ordinary tem- 
peratures [Allmand, J. Chem. Soc., 95, 2151 (1909)]; show that the solid 
cupric hydroxide is unstable in contact with metallic copper and tends to be 
reduced to cuprous hydroxide. 



CHAPTER IX 
ACIDS AND BASES 

Definition of Acids and Bases.* The old definitions of an acid as a 
substance which yields hydrogen ions, of a base as one giving hydroxyl 
ions, and of neutralization as the formation of a salt and water from an 
acid and a base, are reasonably satisfactory for aqueous solutions, but 
there are serious limitations when non-aqueous media, such as ethers, 
nitro-compounds, ketones, etc., are involved. As a result of various 
studies, particularly those on the catalytic influence of un-ionized mole- 
cules of acids and bases and of certain ions, a new concept of acids and 
bases, generally associated with the names of Brjzfnsted and of Lowry, 
has been developed in recent years. 1 According to this point of view 
an acid is defined as a substance with a tendency to lose a proton, while 
a base is any substance with a tendency to gain a proton ; the relationship 
between an acid and a base may then be written in the form 

A ^ H+ + B. (1) 

acid proton base 

The acid and base which differ by a proton according to this relationship 
are said to be conjugate to one another; every acid must, in fact, have 
its conjugate base, and every base its conjugate acid. It is unlikely that 
free protons exist to any extent in solution, and so the acidic or basic 
properties of any species cannot become manifest unless the solvent 
molecules are themselves able to act as proton acceptors or donors, 
respectively : that is to say, the medium must itself have basic or acidic 
properties. The interaction between an acid or base and the solvent, 
and in fact almost all types of acid-base reactions, may be represented 
as an equilibrium between two acid-base systems, viz., 

A! + B 2 ^ B! + A 2 , (2) 

acidi bascz basei acid 2 

where Ai and BI are the conjugate acid and base of one system, and 

* G. N. Lewis [/. Franklin Inst., 226, 293 (1938); see also, /. Am. Chem. Soc., 61, 
1886, 1894 (1939); 62, 2122 (1940)] proposes to define a base as a substance capable of 
furnishing a pair of electrons to a bond, i.e., an electron donor, whereas an acid is able 
to accept a pair of electrons, i.e., an electron acceptor. The somewhat restricted defini- 
tions employed in this book are, however, more convenient from the electrochemical 
standpoint. 

1 Lowry, Chem. and Ind., 42, 43 (1923); Br0nsted, Rec. trav. chim., 42, 718 (1923); 
J. Phys. Chem., 30, 377 (1926); for reviews, see Br0nsted, Chem. Revs., 3, 231 (1928); 
Hall, ibid., 8, 191 (1931); Bjerrum, ibid., 16, 287 (1935); Bell, Ann. Rep. Chem. Soc., 
31, 71 (1934). 

306 



ACIDS 307 

A 2 and B 2 are those of the other system, e.g., the solvent. Actually A! 
possesses a proton in excess of BI, while A 2 has a proton more than B 2 ; 
the reaction, therefore, involves the transfer of a proton from AI to B 2 in 
one direction, or from A 2 to BI in the other direction. 

Types of Solvent. In order that a particular solvent may permit a 
substance dissolved in it to behave as an acid, the solvent itself must 
be a base, or proton acceptor. A solvent of this kind is said to be proto- 
philic in character; instances of protophilic solvents are water and alco- 
hols, acetone, ether, liquid ammonia, amines and, to some extent, formic 
and acetic acids. On the other hand, solvents which permit the mani- 
festation of basic properties by a dissolved substance must be proton 
donors, or acidic; such solvents are protogenic in nature. Water and 
alcohols arc examples of such solvents, but the most marked protogenic 
solvents are those of a strongly acidic character, e.g., pure acetic, formic 
and sulfuric acids, and liquid hydrogen chloride and fluoride. Certain 
solvents, water arid alcohols, in particular, are amphiprotic, for they can 
act both as proton donors and acceptors; these solvents permit sub- 
stances to show both acidic and basic properties, whereas a purely proto- 
philic solvent, e.g., ether, or a completely protogenic one, e.g., hydrogen 
fluoride, would permit the manifestation of either acidic or basic functions 
only. In addition to the types of solvent already considered, there is 
another class which can neither supply nor take up protons: these are 
called aprotic solvents, and their neutral character makes them especially 
useful when it is desired to study the interaction of an acidic and a basic 
substance without interference by the solvent. 

Acids. Since an acid must possess a labile proton it can be repre- 
sented by HA, and if S is a protophilic, i.e., basic, solvent, the equilibrium 
existing in the solution, which is of the type represented by equation (2),' 
may be written as 

HA + S ^ HS+ + A-, (3) 

acidi bases aeid 2 basei 

where HS+ is the form of the hydrogen ion in the particular solvent and 
A~ is the conjugate base of the acid HA. There arc a number of impor- 
tant consequences of this representation which must be considered. In 
the first place, it is seen that the anion A~~ of every acid HA must be 
regarded as the conjugate base of the latter. If the acid is a strong one, 
it will tend to give up its proton very readily; this is, in fact, what is 
meant by a "strong acid." For such an acid, e.g., hydrochloric v acid, 
the equilibrium between acid and solvent, represented by equation (3), 
lies considerably to the right; that is to say, the reverse process occurs 
to a small extent only. This means that the anion of a strong acid, 
e.g., the chloride ion, will not have a great affinity for a proton, and 
hence it must be regarded as a "weak base." On the other hand, if HA 
is a very weak acid, e.g., phenol, the equilibrium of equation (3) lies well 



308 ACIDS AND BASES 

to the left, so that the process 

A- + HS+ ^ HA 4- S 

will take place to an appreciable extent; the anion A~, e.g., the phenoxide 
ion, will be a moderately strong base. 

Another consequence of the interaction between the acid and the 
solvent is that the hydrogen ion in solution is not to be regarded as a 
bare proton, but as a combination of a proton with, at least, one molecule 
of solvent; the hydrogen ion thus depends on the nature of the solvent. 
In water, for example, there are good reasons for believing that the 
hydrogen ion is actually H 3 0+, sometimes called the "oxonium" or 
"hydronium" ion: the free energy of hydration of the proton is so high, 
approximately 250 kcal. (see p. 249), that the concentration of free pro- 
tons in water must be quite negligible, and hence almost all the protons 
must have united with water molecules to form H 3 O+ ions. Further 
hydration of the H 3 O+ ions probably occurs in aqueous solution, but this 
is immaterial for present purposes. 

Striking evidence of the part played by the water in connection with 
the manifestation of acidic properties is provided by observations on the 
properties of hydrogen bromide solutions in liquid sulfur dioxide. 2 The 
latter is only feebly basic and, although it dissolves hydrogen bromide, 
the solution is a poor conductor; there is consequently little or no ioniza- 
tion under these conditions. The solution of hydrogen bromide in sulfur 
dioxide is able, however, to dissolve a mole of water for every mole of 
hydrogen bromide present, and the resulting solution is an excellent con- 
ductor. Since water is sparingly soluble in sulfur dioxide alone, it is 
clear that the reaction 

HBr + H 2 O = H 3 0+ + Br~ 

must take place between the hydrogen bromide and water. Confirma- 
tion of this view is to be found in the observation that on electrolysis of 
the solution one mole of water is liberated at the cathode for each faraday 
passing; the discharge of the H 3 O + ion clearly results in the formation of 
an atom, or half a molecule, of hydrogen and a molecule of water. 

It is of interest to note in connection with the question of the nature 
of the hydrogen ion in solution that the crystalline hydrate of perchloric 
acid, HC104-H 2 O, has been shown by X-ray diffraction methods to have 
the same fundamental structure as ammonium perchlorate. Since the 
latter consists of interpenetrating lattices of NHj and C1OJ" ions, it is 
probable that the former is built up of H 3 O+ and ClOr ions. 

A third conclusion to be drawn from the equilibrium represented by 
equation (3) is that since the solvent S is to be regarded as a base, the 
corresponding hydrogen ion SH+ is an acid. The hydronium ion H 3 O+ 
is thus an acid, and in fact the acidity of the strong acids, e.g., perchloric, 

2 Bagster and Cooling, J. Chem. Soc., 117, 693 (1920). 



ACIDS 



309 



hydrobromic, sulfuric, hydrochloric and nitric acids, in water is due 
almost exclusively to the H 3 O+ ion. It is because the process 



HA + H 2 
acidi bases 



H,0+ 

acids 



basei 



where HA is a strong acid, goes almost completely to the right, that the 
aforementioned acids appear to be 
equally strong in aqueous solution, 
provided the latter is not too concen- 
trated. In solutions more concen- 
trated than about 2 N, however, these 
acids do show differences in cata- 
lytic behavior for the inversion of 
sucrose; the results indicate that the 
strengths decrease in the order given . 
(Fig. 86). | 

In order that it may be possible 
to distinguish in strength between , 
the so-called strong acids, it is evi- 2 
dently necessary to employ a solvent 
which is less strongly pro tophilic than 
water; the equilibrium of equation 
(3) will then not lie completely to the 
right, but its position will be deter- 
mined by the relative proton-donat- 
ing tendencies, i.e., strengths, of the 
various acids. A useful solvent for 
this purpose is pure acetic acid ; this FIG. 86. 
is primarily a protogenic (acidic) 
solvent, but it has slight basic properties, so that the reaction 




2 3 

Normality of Acio 

Catalytic activity of strong acids 



HA + CH 3 C0 2 H ^ 



A~ 



occurs to some extent, although the equilibrium cannot lie far to the 
right. Even acids, such as perchloric and hydrochloric, which are re- 
garded as strong acids, will interact to a small extent only with the 
solvent, and the number of ions in solution will be relatively small; the 
extent of ionization will, therefore, depend on the strength of the acid in 
a manner not observed in aqueous solution. The curves in Fig. 87 show 
the variation of the conductance of a number of acids in pure acetic acid 
at 25; the very low equivalent conductances recorded arc due to the 
very small degrees of ionization. It is seen, therefore, that acids which 
appear to be equally strong in aqueous solution behave as weak acids 
when dissolved in acetic acid; moreover, it is possible to distinguish 
between their relative strengths, the order being as follows: 

HC10 4 > HBr > H 2 S0 4 > HC1 > HNO 8 . 



310 



ACIDS AND BASES 



This order agrees with that found by catalytic methods and also by 

potentiometric titration. 8 

In spite of the small extent of ionization of acids in a strongly proto- 

genic medium such as acetic acid, the activity of the resulting hydrogen 

ions is very high; this may be at- 
tributed to the strong tendency 
of the CHaCC^Hj ion to lose a 
proton, so that the ion will behave 
as an acid of exceptional strength. 
The intense acidity of these solu- 
tions, as shown by hydrogen elec- 
trode measurements, by their cat- 
alytic activity, and in other ways, 
has led to them being called super- 
acid solutions. 4 The property of 
superacidity can, of course, be 
observed only with solvents which 
are strongly protogenic, but which 
still possess some protophilic na- 
ture. Hydrogen fluoride, for ex- 
ample, has no protophilic proper- 
ties, and so it cannot be used to 
exhibit superacidity; in fact no 
known substance exhibits acidic 



2.0 



1.6 



1.0 



0.5 




0.02 



0.04 



0.06 



0.08 



FIG. 87. Conductance of acids in glacial 
acetic acid (Kolthoff and Willman) 



behavior in this solvent, as ex- 
plained below. 

It is an obvious corollary, from 
the discussiom given here concern- 
ing the influence of the solvent, that in a highly basic, i.e., protophiiic, 
medium, even acids that are normally regarded as weak would be highly 
ionized. It is probable that in liquid ammonia interaction with a weak 
acid, such as acetic acid, would occur to such an extent that it would 
appear to be as strong as hydrochloric acid. 

Bases. The equilibrium between an acidic, i.e., protogenic, solvent and 
a base may be represented by another form of thegeneral equation (2), viz., 



B + SH 
base acid 



BH+ + S-, 
acid base 



(4) 



where the solvent is designated by SH to indicate its acidic property. 
It is seen from this equilibrium that the cation BH+ corresponding to 
the base B is to be regarded as an acid; for example, if the base is NH 8 , 

> HalI and Conant, J. Am. Chem. Soc., 49, 3047, 3062 (1927); Hall and Werner, 
ibid., 50, 2367 (1928); Hantzsch and Langbein, Z. anorg. Chem., 204, 193 (1932); Kolthoff 
and Willman, J. Am. Chem. Soc., 56, 1007 (1934); Weidner, Hutchison and Chandlee, 
ibid., 56, 1285 (1934). 

* Hall and Conant, J. Am. Chem. Soc., 40, 3047, 3062 (1927); Hall and Werner, 
ibid, 50, 2367 (1928); Conant and Werner, ibid., 52, 436 (1930). 



BABES 311 

the corresponding cation is NHt, and so the ammonium ion and, in fact, 
all mono-, di-, and tri-substituted ammonium ions are to be regarded as 
the conjugate acids of the corresponding amine (anhydro-) bases. It 
can be readily shown, by arguments analogous to those used in connec- 
tion with acids, that when the base is a strong one, e.g., hydroxyl ions, 
its conjugate acid, i.e., water, will be a weak acid; similarly, the conjugate 
acid to a very weak base will be moderately strong. 

The strength of a base like that of an acid must depend on the nature 
of the solvent : in a strongly protogenic medium, such as acetic acid or 
other acid, the ionization process 

B + CH 3 CO 2 H = BH+ + CHaCOl 
base acid acid base 

will take place to a very considerable extent even with bases which are 
weak in aqueous solution. Just as it is impossible to distinguish between 
the strengths of weak acids in liquid ammonia, weak bases are indis- 
tinguishable in strength when dissolved in acetic acid; it has been found 
experimentally, by measurement of dissociation constants, that all bases 
stronger than aniline, which is a very weak base in water, are equally 
strong in acetic acid solution. 6 To arrange a series of weak bases in the 
order of their strengths, it would be necessary to use a protophilic solvent, 
such as liquid ammonia: water is obviously better than acetic acid for 
this purpose, but it is not possible to distinguish between the strong bases 
in the former medium, since they all produce OH~ ions almost completely. 
Substances which are normally weak bases in water exhibit con- 
siderable basicity in strongly acid media; the results in Table LVI, for 

TABLE LVI. EQUIVALENT CONDUCTANCES IN HYDROGEN FLUORIDE SOLUTIONS 
AT 15 IN OHMS" 1 CM.* 

Concentration Methyl alcohol Acetone Glucose 

0.026 N 243 244 279 

0.115 200 190 208 

0.24 164 181 165 

0.50 139 176 114 

example, show that methyl alcohol, acetone and glucose, which are non- 
conductors in aqueous solution, are excellent conductors when dissolved 
in hydrogen fluoride. 6 These, and other oxygen compounds, behave as 
bases and ionize in the following manner: 

\) + HF = \DH + F-. 

base acid acid base 

A number of substances which are acids in aqueous solution function as 

Hall, /. Am. Chem. Soc., 52, 5115 (1930); Chem. Revs., 8, 191 (1931). 

Fredenhagen et a/., Z. phyrik. Chem., 146A, 245 (1930); 164A, 176 (1933); Simons, 
Chem. Revs., 8, 213 (1931). 



312 ACIDS AND BASES 

bases in hydrogen fluoride, e.g., 

CHsC0 2 H + HF = CH 8 C0 2 Itf + F-. 
acid acid base 



This reaction occurs because the acid possesses some protophilic proper- 
ties, and these become manifest in the presence of the very strongly 
protogenic solvent. As may be expected, the stronger the acid is in 
water, the weaker does it behave as a base in hydrogen fluoride. 

Dissociation Constants of Acids and Bases. If the law of mass action 
is applied to the equilibrium between an acid HA and the basic solvent 
S, i.e., to the equilibrium 

HA + S ^ HS+ + A-, 
the result is 

(5) 

If the concentration of dissolved substances in the solvent is not large, 
the activity of the latter, i.e., a s , may be regarded as unity, as for the 
pure solvent; equation (5) then becomes 

(6) 

The substance HS+ is the effective hydrogen ion in the solvent 8, so that 
OHS* is equivalent to the quantity conventionally written in previous 
chapters as OH+, it being understood that the symbol H+ does not refer 
to a proton but to the appropriate hydrogen ion in the given solvent; it 
follows, therefore, that equation (6) may be written in the form 



' (7) 

which is identical with that obtained by regarding the acid as HA ionizing 
into H+ and A~, in accordance with the general treatment on page 163. 
The constant as defined by equation (6) or (7) is thus identical with the 
familiar dissociation constant of the acid HA in the given solvent as 
obtained by the methods described in Chap. V; further reference to the 
determination of dissociation constants is made below. 

Application of the law of mass action to the general base-solvent 
equilibrium 

B + SH ^ BH+ + S-, 
gives 

aBH+Qs- 
K = -^~> (8) 

the activity of the Solvent SH being regarded as constant; the quantity 
Kb is the dissociation constant of the base. If the base is an amine, 



DETERMINATION OF DISSOCIATION CONSTANTS 313 

, in aqueous solution, then the equilibrium 

RNH 2 + H 2 O ^ RNHjj + Oil- 
is established, and the dissociation constant is given by 

K b = 

The result is therefore the same as would be obtained by means of the 
general treatment given in Chap. V for an electrolyte MA, if the un- 
dissociated base were regarded as having the formula RNH 3 OH in 
aqueous solution. 

The dissociation constants of acids, and bases, are of importance as 
giving a measure of the relative strengths of the acids, and bases, in the 
given medium. The strength of an acid is measured by its tendency to 
give up a proton, and hence the position of the equilibrium with a given 
solvent, as determined by the dissociation constant, is an indication of 
the strength of the acid. Similarly, the strength of a base, which depends 
on its ability to take up a proton, is also measured by its dissociation 
constant, since this is the equilibrium constant for the reaction in which 
the solvent molecule transfers a proton to the base. 

Determination of Dissociation Constants: The Conductance Method. 
As seen in Chap. V, equation (7) may be written in the form 



_ 

A a 



and if a is the true degree of dissociation of the solution of acid whose 
stoichiometric concentration is c, then 

a' 2 c /n*/ A - 

A = - -- -- (9) 

1 a JHA 

Accurate methods for evaluating K a based on this equation, involving 
the use of conductance measurements, have been already described in 
Chap. V; these require a lengthy experimental procedure, but if carried 
out carefully the results are of high precision. For solvents of high 
dielectric constant the calculation based on the Onsager equation may 
be employed (p. 165), but for low dielectric constant media the method 
of Fuoss and Kraus (p. 167) should be used. 

Many of the dissociation constants in the older literature have been 
determined by the procedure originally employed by Ostwald (1888), 
which is now known to be approximate in nature; if the activity coeffi- 
cient factor in equation (9) is neglected, and the degree of dissociation a 
is set equal to the conductance ratio (A/A ), the result is 

A 2 c 
*~ 



Ao(A.-A) 



314 ACIDS AND BASES 

An approximate dissociation function k was thus calculated from the 
measured equivalent conductance of the solution of weak acid, or weak 
base, at the concentration c, and the known value at infinite dilution. 
For moderately weak acids, of dissociation constant of 10~ 6 or less, the 
degree of dissociation is not greatly different from the conductance ratio, 
provided the solutions are relatively dilute; under these conditions, too, 
the activity coefficient factor will be approximately unity. If the acid 
solutions are sufficiently dilute, therefore, the dissociation constants given 
by equation (10) are not seriously in error. For example, if the data in 
Table XXXVIII on page 165 for acetic acid solutions are treated by the 
Ostwald method they give k a values varying from about 1.74 X 10~ 5 in 
the most dilute solutions to 1.82 X 10~ 5 in the more concentrated. The 
results in dilute solution do not differ appreciably from those obtained 
by the more complicated but more accurate method of treating the data. 
It may be mentioned, however, that the earlier determinations of dis- 
sociation constants were generally based on conductance measurements 
with solutions which were rarely more dilute than 0.001 N, whereas those 
in Table XXXVIII refer to much less concentrated solutions. For acids 
whose dissociation constants arc greater than about 10~ 5 the Ostwald 
method would give reasonably accurate results for the dissociation con- 
stant only at dilutions which are probably too great to yield reliable 
conductance measurements. 

Electromotive Force Method. An alternative procedure for the 
evaluation of dissociation constants, which also leads to very accurate 
results, involves the study of cells without liquid junction. 7 The chemi- 
cal reaction occurring in the cell 

H 2 (l aim.) | HA(wii) NaAK) NaCl(m 3 ) AgCl(s) | Ag, 

where HA is an acid, whose molality is m\ in the solution, and NaA is 
its sodium salt, of molality w 2 , is 

|H,(1 atm.) + AgCl(s) = Ag + H+ + Cl- 

for the passage of one faraday. The E.M.F. of the coll is therefore given 
by (cf. p. 226) 

RT 
E = E'--prlna H *acr, (11) 

\\here E is the standard E.M.F. of the hydrogen-silver chloride cell, i.e., 
of the hypothetical cell 

H,(l atm.) | H+(a H + = 1) II Cl-(ocr = 1) AgCl() | Ag. 

The E.M.F. of this cell is clearly equal in magnitude but opposite in sign 
to the standard potential of the Ag, AgCl(s) Cl~ electrode, and hence 
E in equation (11) is + 0.2224 volt at 25. The subscripts H+ and Cl~ 

7 Earned and Ehlere, J. Am. Chem. Soc. t 54, 1350 (1932); for reviews, see Harned, 
J. Franklin Inst., 225, 623 (1938); Harned and Owen, Chem. Revs., 25, 31 (1939). 



ELECTROMOTIVE FORCE METHOD 315 

in equation (11), etc., refer to the hydrogen and chloride ions, respec- 
tively, it being understood that the former is really H 3 0+ in aqueous 
solution, and the corresponding oxonium ion in other solvents. The 
activities in equation (11) may be replaced by the product of the respec- 
tive molalities (ra) and the stoichiometric activity coefficients (7), so that 

RT RT 

E = E -- In WH+WCI- - -jr In yn+ycr. (12) 

The activities in equation (7) for the dissociation constant may be 
expressed in a similar manner, so that 



= > 

HA WHA THA 

and combination of this expression with equation (12) gives 

E = E - ^- ]n ^^- - ^inliffil _ ^intf, (13) 

- - 



F(E - E Q ) , , v 

-- + log log - ~ log K > 







or, at 25 

(E E Q ) WHAfWci" 7HA7C1" ,r /"I -\ 

n Acni c~~ ^8 == *^S *O? ** v^W 

U.uoyio w\~ 7A~ 

The right-hand side of equation (14) may be set equal to log K', where 
K' becomes identical with K at infinite dilution, for then the activity 
coefficient factor 7TL\7cr/7A~ becomes unity and the term log 7HA7cr/7A~ 
in equation (15) is zero. 

Since E Q is known, and the E.M.F. of the cell (E) can be measured 
with various concentrations of acid, sodium salt and sodium chloride, 
i.e., for various values of Wi, mz and w 3 in the cell depicted above, it is 
possible to evaluate the left-hand side of equation (14) or (15). In dilute 
solution, the sodium chloride may be assumed to be completely dis- 
sociated so that the molality of the chloride ion can be taken as equal to 
that of the sodium chloride, i.e., mcr is equal to ra 3 . The acid HA will 
be partly in the undissociated form and partly dissociated into hydrogen 
and A~~ ions; the stoichiometric molality of HA is m\ 9 and if WH+ is the 
molality of the hydrogen ions resulting from dissociation, the molality of 
undissociated HA molecules, i.e., WHA in equation (15), is equal to 
wii WH+. Finally, it is required to known WA-: the A~ ions are pro- 
duced by the dissociation of NaA, which may be assumed to be complete, 
and also by the small dissociation of the acid HA; it follows, therefore, 
that m A - is equal to mz + IH*. Since m^*, the hydrogen ion concen- 
tration, is required for these calculations, a sufficiently accurate value is 
estimated from the approximate dissociation constant (cf. p. 390); this 



316 



ACIDS AND BASES 



procedure is satisfactory provided the dissociation constant of the acid is 
about IQr 4 or less, as is generally the case. If the values of the left-hand 
side of equation (14) or (15) are plotted against the ionic strength of the 
solution and extrapolated to infinite dilution, the intercept gives log K, 
from which the dissociation constant K can be readily obtained. The 
general practice is to keep the ratio of acid to salt, i.e., m\ to m*, constant, 
approximately unity, in a series of experiments, and to vary the ionic 
strength by using different concentrations of sodium chloride. The re- 
sults obtained for acetic acid are shown in Fig. 88; the value of log K a 
is seen to be - 4.756, so that K a is 1.754 X 10~ 5 at 25. 

When comparing the dissociation constant obtained by the con- 
ductance method with that derived from E.M.F. measurements, it must 
be remembered that the former is based on volume concentrations, i.e., 
g.-ions or moles per liter, while the latter involves molalities. This 
difference arises because it is more convenient to treat conductance data 
in terms of volume concentrations, whereas the standard states for E.M.F. 
studies are preferably chosen in terms of molalities. If K c and K m are 
the dissociation constants based on volume concentrations and molalities, 

respectively, then it can be 
readily seen that K c is equal 
to K m p, where p is the density 
of the solvent at the experimen- 
tal temperature. For water 
at 25, p is 0.9971, and hence 
K c for acetic acid, calculated 
from E.M.F. measurements, is 
1.749 X 10~ 5 , compared with 
1.753 X 10~ 5 from conductance 
data. Considering the differ- 
ence in principle involved in 
the two methods, the agree- 
FIG. 88. Dissociation constant of acetic ment is ver y striking. Almost 

acid (Harned and Eklers) as good correspondence has 

been found for other acids with 

which accurate conductance and E.M.F. studies have been made; this 
may be regarded as providing strong support for the theoretical treat- 
ments involved, especially in the case of the conductance method. 

The procedure described here may be regarded as typical of that 
adopted for any moderately weak acid, i.e., of dissociation constant 10~ 8 
to 10~~ 5 ; for weaker acids, however, some modification is necessary. In 
addition to the acid dissociation 

HA + H 2 ^ H 3 0+ + A- 
allowance must be made for the equilibrium 

A- + H 2 ^ OH- + HA, 



4.762 



4.760 



,4,768 



4.756 



0.04 



o.oa 



0.12 



o.ie 



DISSOCIATION CONSTANTS OF BASES 317 

which is due to the water, i.e., the solvent, functioning as an acid to 
some extent; this corresponds to the phenomenon of hydrolysis to be 
discussed in Chap. XI. It follows, therefore, that if the stoichiometric 
molality of HA is Wi, then 

TttHA = Wi WH* + WOH~, 

since HA is used up in the dissociation process while it is formed in the 
hydrolysis reaction, in amounts equivalent to the hydrogen and hydroxyl 
ions, respectively. Further, if the molality of the salt NaA is W2, then 



since A~ ions are formed in the dissociation process but are used up in 
the hydrolysis. If the dissociation constant is greater than 10~ 6 and the 
ratio of acid to salt, i.e., m\\m^ is approximately unity, WOH~ is found by 
calculation to be less than 10~ 9 , and so this term can be neglected in the 
expressions for WHA and rn A -, as was done above. If the dissociation 
constant lies between 10~ 5 and 10~ 9 , and mi/m^ is about unity, m H + WOH- 
is negligibly small, so that WHA and m\- may be taken as equal to mi and 
mz t respectively. For still weaker acids, m\i+ is so small that it may be 
ignored in comparison with WOH~; WUA is now equal to mi + W?OH", and 
m A - is m2 moH~. The values of m ir required for determining T/IHA 
and m^~ are obtained by utilizing the fact that mammon" is equal to 
10- u at ?5. 

Dissociation Constants of Bases. The dissociation constants of bases 
can be determined, in principle, by methods which are essentially similar 
to those employed for acids. Replacing activities in equation (8) by the 
product of molalities and activity coefficients, it is seen that for a base 



WB 7B 

and this may be replaced by 



1 a 7n 



(Lt) 



where a represents the degree of dissociation of the hypothetical solvated 
base, e.g., BH OH in water. By neglecting the activity coefficient factor 
in equation (17) and replacing a by the conductance ratio, an approxi- 
mate equation identical in form with (10) is obtained; the value of Ao in 
this equation is the sum of the equivalent conductances of the BH+ and 
OH~ ions, e.g., of NHi" and OH~ if the base is ammonia. 

Very little accurate E.M.F. work has been done on the dissociation 
constants of bases, chiefly because moderately w r eak bases are very vola- 
tile, while the non-volatile bases, e.g., anilines, are usually very weak. 
An exception to this generalization is to be found in the aliphatic amino- 
acids which will be considered in connection with the subject of ampho- 



318 ACIDS AND BABES 

teric electrolytes. Since silver chloride is soluble in aqueous solutions of 
ammonia and of many amines, it is not possible to use silver-silver 
chloride electrodes with such bases; the employment of sodium amalgam 
has been proposed, but it is probable that the silver-silver iodide electrode 
will prove most useful for the purpose of the accurate determination of 
the dissociation constants of bases by the E.M.F. method. 

Apart from determinations of dissociation constants made from con- 
ductance data, most values derived from E.M.F. measurements have been 
obtained by an approximate procedure which will be described later. 

Dissociation Constants of Polybasic Acids: Conductance Method. 
A polybasic acid ionizes in stages, each stage having its own characteristic 
dissociation constant : for example, the ionization of a tribasic acid HsA, 
such as phosphoric acid, may be represented by: 

flij + {JfT A~ 

1. H 3 A + H 2 ^ H 3 0+ + H 2 A~, K, = - (18a) 

a H 3 A 
ttn + flTIA~~ 

2. H 2 A- + H 2 ^ H 3 0+ + HA, K 2 = (186) 

flHjA- 

3. HA + H 2 ? H 3 0+ + A --- , K 3 = an * aA "" 



The fact that ionization occurs in these three stages successively with 
increasing dilution shows that KI > K 2 > KZ] this is always true, be- 
cause the presence of a negative charge on H 2 A~ and of two such charges 
on HA makes it increasingly difficult for a proton to be lost. 

If the dissociation constants for any two successive stages are suffi- 
ciently different it is sometimes feasible to apply the methods employed 
for monobasic acids; the conditions under which this is possible will be 
considered with reference to a dibasic acid, but the general conclusions 
can be extended to more complex cases. If H 2 A is a dibasic acid for 
which KI, the dissociation constant of the first stage,* 

H 2 A + H 2 O ^ H 3 O+ + HA-, 

is of the order of 10~ 3 to 10~ 5 , while the constant K% of the second stage of 
dissociation, 

HA- + H 2 ^ H 3 0+ + A, 

* The first stage dissociation constant of a dibasic acid is actually the sum of two 
constants; consider, for example, the unsymmetrical dibasic acid HX-X'H, where X 
and X' are different. This acid can dissociate in two ways, viz., 

HX-X'H + H 2 O ;= H 3 O+ + -X-X'H, 
and 

HX-X'H + H 2 ^ H 8 0+ + HX-X'- 

and if K{ and K" are the corresponding dissociation constants, the experimental first 
stage dissociation constant KI is actually equal to K( -f K('. If the acid is a sym- 
metrical one, e.g., of the type CO 2 H(CH) n COjH, the constants K{ and K" are identical, 
BO that KI is equal to 2K{. Similar considerations apply to all polybasic acids. 



DISSOCIATION CONSTANTS OF POLYBASIC ACIDS 319 

is very small, i.e., the acid is moderately weak in the first stage and very 
weak in the second stage, then it may be treated virtually as a monobasic 
acid. The value of K\ may be determined in the usual manner from 
conductance measurements on the acid H 2 A and its salt NaHA at various 
concentrations, together with the known values for hydrochloric acid and 
sodium chloride (cf. p. 164). Provided the dissociation constant K 2 of 
the acid HA~ is very small, the extent of the second stage dissociation 
will be negligible in the solutions of both H2A and NallA. This method 
has been applied to the determination of the first dissociation constant 
of phosphoric acid; 8 for this acid KI is 7.5 X 10~ 3 at 25, whereas K 2 is 
6.2 X 10- 8 . 

If the dissociation constant of the second stage is relatively large, 
e.g., about 10~ 5 or more, it is not possible to carry out the normal con- 
ductance procedure for evaluating K\; this is because the HA~ ion in the 
solution of the completely ionized salt NaHA dissociates to an appreciable 
extent to form H 3 0+ and A ions, and the measured conductance is 
much too large. As a result of this further dissociation, it is not possible 
to derive the equivalent conductances of NaHA required for the calcu- 
lation of the dissociation constant. An attempt has been made to over- 
come this difficulty by estimating the equivalent conductance of the ion 
HA~ in an indirect manner, so that the value for the salt NaHA may be 
calculated. By assuming that the intermediate ion of an organic dibasic 
acid, viz., OII-CORCOiT, has the same equivalent conductance at infinite 
dilution as the anion of the corresponding amic acid, viz., NH 2 -CORCO", 
which can be obtained by direct measurement, it has been concluded that 
the equivalent conductance XHA~ of the intermediate ion is equal to 
0.53XA", where XA~~ is the conductance of the A ion, i.e., of ~~C0 2 RCO2~ 
in the case under consideration. Since the latter quantity can be deter- 
mined without great difficulty by conductance measurements with the 
salt Na^A, the value of Xn A - for the given acid at infinite dilution can be 
obtained. The known equivalent conductance of sodium is now added 
to that of the HA~, thus giving the value of A for the salt NaHA; the 
variation of the equivalent conductance with concentration can now be 
expressed by assuming the Onsagcr equation to be applicable. Since 
the conductance of the acid H 2 A at various concentrations is known, as 
well as that of HC1 and NaCl, all the information is available for calcu- 
lating the dissociation constant of H 2 A as a monobasic acid. This method 
cannot be regarded as accurate, however, for the identification of XHA- 
with 0.53XA" is known to be an approximation. 9 

The determination of the second dissociation constant (/ 2 ) of a di- 
basic acid also requires a knowledge of the equivalent conductance of the 
intermediate ion HA~, and if the value of K z is large enough to be deter- 
mined from conductance measurements, the further dissociation of HA"" 

Sherrill and Noyes, J. Am. Chem. Soc., 48, 1861 (1926). 

Jeffery and Vogel, /. Chem. Soc., 21 (1935); 1756 (1936); Davies, ibid., 1850 
(1939). 



320 ACIDS AND BASES 

is too great for the equivalent conductance to be derived accurately from 
the experimental data for the salt NaHA. In the earlier attempts to 
evaluate K% the assumption was made of a constant ratio of XHA~ to XA~, 
as described above; this, however, leads to results that are too uncertain 
to have any serious worth. If transference data are available, it is 
possible in certain cases to determine the required value of XHA~ and 
hence to calculate the second dissociation constant of the acid. The 
method has been used to evaluate K^ for sulfuric acid : in its first stage of 
dissociation this is a very strong acid, but the second stage dissociation, 
although very considerable, is much smaller. 10 

Dissociation Constants of Dibasic Acids by E.M.F. Measurement. 
If the ratio of the dissociation constants of a dibasic acid, or of any two 
successive stages of ionization of a polybasic acid, is greater than about 
10 2 or 10 3 , it is possible to treat each stage as a separate acid and to 
determine its dissociation constant by means of cells without liquid junc- 
tion in the manner already described. In a mixture of the free dibasic 
acid H 2 A with its salt NaHA, the essential equilibria are 

1. H 2 A + H 2 ^ H 3 0+ + HA-, 
and 

2. HA- + H 2 ^ II 3 0+ + A, 

and from these, by subtraction, may be obtained the equilibrium 

3. 2HA- ^ H 2 A + A. 

If K\ and KZ are the dissociation constants for the stages 1 and 2, it can 
be readily shown that the equilibrium constant for the process 3 is equal 
to KtlKi. 

If the stoichiometric molality of H 2 A is m\ in a given solution and 
that of the salt NaHA, assumed to be completely dissociated into HA"" 
ions, is w 2 , then 

Wii 2 A m\ WH+ + flix", (19) 

since H 2 A is removed to form hydrogen ions in process 1, while it is 
formed in process 3 in an amount equivalent to A ; further, 

WHA- = m* + mn+ - 2m A ", (20) 

since HA~ is formed in reaction 1 and removed in 3, in amounts equiva- 
lent to H 3 0+ and 2A respectively. It has been seen that the equilib- 
rium constant of process 3 is equal to K 2 IKi t and the smaller this ratio 
the less will be the tendency of the reaction to take place from left to 
right; if Kt/Ki is smaller than about 10~ 3 , i.e., Ki/K* > 10 8 , the extent 
of the reaction will be negligible, and then the W A terms in equations 
(19) and (20) can be ignored. The expressions for mn^ and WHA- then 
reduce to the same form as do the corresponding ones for WHA and m\- t 

10 Sherrill and Noyes, J. Am. Chem. Soc., 48, 1861 (1926). 



DISSOCIATION CONSTANTS OF DIBASIC ACIDS BY E.M.F. MEASUREMENT 321 



respectively, for a monobasic acid. If KI lies between 10~ B and 
and Wi/W2 is approximately unity, WH+ may be neglected, as explained 
on page .317; for weaker acids, however, the term m ir, arising on account 
of hydrolysis, must be included. 

It follows, therefore, that when K 2 /Ki is small, or Ki/K 2 is large, the 
value of KI can be readily determined by measurements on cells of the 
type 

H 2 (l atm.) | H 2 A(mO NaHA(wii) NaCl^) AgCl(s) | Ag, 



the equation for the E.M.F. being, by analogy with equation (14), 



F(E - E) mummer 7H,A7cr 

- + log ~ = - log ~~ ~ log 



The values of w H2 A and WHA~ are derived as explained above, and wcr is 
taken as equal to w 3 ; the method of extrapolation, which yields log KI, 
is the same as described for a monobasic acid. 

In order to investigate the second stage dissociation constant, the 
system studied consists of a mixture of highly ionized NaHA, which is 
equivalent to the acid IIA~, of molality m\, and its salt NasA, of molality 
m*. In this case, it follows from the three processes given above, that 



A~ = mi WH+ ~ 
and 



If Kz/Ki is small the WH 2 A terms may be neglected, just as the m^ 
terms were neglected in the previous case, since process 3 occurs to a 
small extent only; under these conditions the expressions for mnA~ and 
WA" are equivalent to those applicable to a monobasic acid. The deter- 
mination of K 2 can then be carried out by means of the cell 

II 2 (1 atm.) | NaHA(?wO Na 2 A(m 2 ) NaCl(m 3 ) AgCl(s) | Ag, 
the E.M.F. of which is given by the expression 



, , , 

+ log " 7nT~ = ~ log ^^ ~ log Az ' (22) 

The values of mn\-, ?n\ and mcr are determined in the usual manner, 
but since the activity coefficient factor THA'TCI ly\~~ involves two uni- 
valent ions in the numerator with a bivalent ion in the denominator, it 
will differ more from unity than does the corresponding factor in equa- 
tions (14) and (21); the usual extrapolation procedure is consequently 
liable to be less accurate. Utilizing the form 

log 7. = - Az]^v + Cy 
of the extended Debye-Huckel equation, however, it is seen that equation 



322 



ACIDS AND BASES 



(22) may be written as 

r (A A ) 
2.303BT + 



log 



(23) 



a 



7.30 



7.26 



7.20 




0.06 



0.10 



0.16 



The plot of the left-hand side of this expression, where A is 0.509 at 25, 
against the ionic strength y should thus be a straight line, at least ap- 
proximately; the intercept for zero ionic strength gives the value of 
log / 2 . The results obtained in the determination of the second dis- 
sociation constant of phosphoric acid are shown in Fig. 89; the upper 
curve is for cells containing the salts KH 2 P0 4 and Na 2 HP0 4 , and the 
lower for the two corresponding sodium salts in a different proportion. 

In this case the acid is H 2 POr, 
and its dissociation constant 
is seen to be antilog 7.206, 
i.e., K 2 is 6.223 X 10" at 25. u 
If the ratio Ki/K^ for two 
successive stages is smaller 
than 10 3 , it would be neces- 
sary to include the m\ and 
fftH,A terms, which were ne- 
glected previously, in the de- 
termination of Ki and K 2l 
respectively. The evaluation 
of these quantities, as well 

FIG. 89. Second dissociation constant of as of m H + or WOET, would re- 
phosphoric acid (Nims) quire preliminary values of 

Ki and K 2 , and the calcula- 

tions, although feasible, would be tedious. No complete determina- 
tion by means of cells without liquid junction appears yet to have been 
made of the dissociation constants of a dibasic acid for which Ki/Ki is 
less than 10 3 . 

Dissociation Constants by Approximate E.M.F. Methods. When, 

for various reasons, it is not convenient or desirable to carry out the 
lengthy series of measurements required for the determination of accurate 
dissociation constants by the conductance method or by means of cells 
without liquid junction, approximate E.M.F. methods, utilizing cells with 
liquid junctions, can be applied. These methods involve the determina- 
tion of the hydrogen ion concentration, or activity, in solutions con- 
taining a series of mixtures of the acid and its salt with a strong base, 
generally obtained by adding definite quantities of the latter to a known 
amount of acid. The procedures used for actual measurement of hydro- 
gen ion activities are described in Chap. X, but the theoretical basis of 
the evaluation of dissociation constants will be considered here. 



/. Am. Chan. Soc., 55, 1946 (1033); for application to malonic acid, see 
Hamer, Burton and Acree, /. Res. Nat. Bur. Standards, 24, 269 (1940). 



DISSOCIATION CONSTANTS BY APPROXIMATE E.M.F. METHODS 323 

If a is the initial concentration (molality) of the weak or moderately 
weak acid HA, and 6 is the amount of strong, monoacid base MOH 
added at any instant, then 6 is also equal to W M +, the molality of M+ ions 
at that instant, since the salt MA produced on neutralization may be 
taken as being completely dissociated. The acid HA is only partially 
neutralized to form A~ ions, and so 

a = WHA + W A -. (24) 

Further, as the solution must be electrically neutral, the sum of all the 
positive charges will be equal to the sum of the negative charges; hence 

WM+ + WH+ = WA~ + WOH~, 
or 

b + win* = w A - + WOH-. (25) 

The dissociation constant K a of the acid HA may be expressed in the 
form 

A a 
0H 

W A - TA- 
= an+ - - - > 
WHA THA 

and if W A - and WHA are eliminated by means of equations (24) and (25), 
it is found that 



a - 6 - W H * + WQH- THA f . 

= A a r ; - -- (26) 

-f- WH* ~ Won" 7A~ 



If the quantity B is defined by 

B =s b + W H * ~ WOH-, 
then equation (26) may be written as 

ir a ~ ^ THA 
or, taking logarithms, 

i i v i i a "~ ^ i i THA 

log a H + = log K a + log B + log 7 

It was seen on page 292 that the pH, or hydrogen ion exponent, of a 
solution may be defined as log a H +; in an analogous manner the symbol 
pX, called the dissociation exponent, may be substituted for log K a ; 
hence 

pH = pK a + log ^-g + log^- (27) 



324 ACIDS AND BASES 

According to the extended Debye-Hiickel theory, it is possible to write 
log = - A^ + C, (28) 

i HA 

remembering that A~ is a univalent ion and HA an undissociated mole- 
cule, and so equation (27) becomes 

pH = pK a + log ^-^ - A Vtf + C V , (29) 

D _ 

.'. pH - log -^^ + A V v = p# + C. 

If the left-hand side of this equation for a series of acid-base mixtures is 
plotted against the ionic strength of the solution, the intercept for y 
equal to zero would give the value of pK a , i.e., log K a . 

The methods used for the determination of the pH of the solution 
will be described in the following chapter, but in the meantime the evalua- 
tion of B and y will be considered. If the hydrogen ion concentration of 
the solution is greater than 10~~ 4 g.-ion per liter, i.e., for an acid of medium 
strength, the hydroxyl ion concentration woir will be less than 10~ 10 and 
so can be neglected in comparison with W H +; B then becomes equal to 
b + WH+. On the other hand, for a very weak acid, when the hydrogen 
ion concentration is less than 10~ 10 g.-ion per liter, the quantity WH+ may 
be ignored, so that B is equal to 6 + m ir. For solutions of inter- 
mediate hydrogen ion concentration, i.e., between 10~ 4 and 10~ 10 g.-ion 
per liter, WH+ moH~ is negligibly small and so B may be taken as equal 
to b. The values of a and b are known from the amounts of acid and 
base, respectively, employed to make up the given mixture, and WH+ and 
raoir are readily determined by the aid of the relationships WH+ = an + /yn+ 
and WH+WOH- = k lo , which is 10~ 14 at 25 (cf. p. 339). The quantity a tt + 
is derived from the measured pH, and 711+ is calculated with sufficient 
accuracy by means of the simple Debye-Hlickel equation. The ionic 
strength ft of the solution is given by 6 + WH + Woir; except at the 
beginning of the neutralization, however, when b is small, the value of y 
may be taken as equal to 6. 

The data obtained for acetic acid at 25 are plotted in Fig. 90; 12 
the results are seen to fall approximately on a straight line, and from 
the intercept at zero ionic strength pK a is seen to be 4.72. The difference 
between this value and that given previously is to be attributed to an 
incorrect standardization of the pH scale (cf. footnote, p. 349). 

Instead of employing the graphical method described above, the 
general practice is to make use of equation (27) ; the quantities pH and 
B are obtained for each solution and the corresponding pk a evaluated. 
The activity correction may be applied by means of equation (28) since 

12 Walpole, J. Chem. Soc., 105, 2501 (1914). 



DIBASIC ACIDS 



325 



A is known, and C can be guessed approximately or neglected as being 
small; alternatively, the tentative pk a values obtained by neglecting the 
activity coefficients may be plotted against a function of the ionic strength 
and extrapolated to infinite dilution. 



4.86 




0.04 



0.08 



0.12 



Fia. 90. Dissociation constant of acetic acid 

If B is equal to ^a, and the solution is relatively dilute, so that the 
terms involving the ionic strength are small, equation (29) reduces to 

pH = pfc a . 

Provided the pH of the system lies between 4 and 10, the quantity B 
is virtually equal to 6, and hence it follows that when b is equal to \a 
the pll of the solution is (approximately) equal to the pk a of the acid. 
In other words, the pH of a half-neutralized solution of an acid, i.e., of a 
solution containing equivalent amounts of the acid and its salt, is equal 
to pfc a . This fact is frequently utilized for the approximate determina- 
tion of dissociation functions. 

Dibasic Acids. The treatment given above is applicable to any stage 
of ionization of a polybasic acid, provided its dissociation constant differs 
by a factor of at least 10 3 from those of the stages immediately preceding 
and following it: the activity correction, equivalent to equation (28), 
will however depend on the charges carried by the undissociatod acid 
and the corresponding anion. If these are r 1 and r, respectively, then 
according to the extended Dobye-Hiickel equation 



- (r - 



log -^- = - 



so that equation (29) for the rth dissociation constant of a polybasic 



326 ACIDS AND BASES 

acid becomes 

pH = pK r + log ^ - A(2r - 1) VJ + C V . (30) 

When the dissociation constants of successive stages are relatively 
close together, a more complicated treatment becomes necessary. 13 The 
dissociation constants of the first and second stages of a dibasic acid 
H 2 A may be written in a form analogous to that given above, viz., 

, v m A TA" /ot . 

and K 2 = a H - - ---- (31) 

~ 7HA" 



If to a solution containing the acid H 2 A at molality a there are added 6 
equivalents of a strong monoacid base, MOH, the solution will contain 
H+, M+, HA~, A and OH~ ions; for electrical neutrality therefore, 

WM* + mH+ = mHA~ + 2mA h #k)H~> 

the term 2mA" arising because the A ions carry two negative charges. 
Replacing the concentration of M+ ions, i.e., WM*, by 6, as in the previous 
case, this equation becomes 

b + mn+ = WHA" + 2mA h moir. (32) 



Further, the initial amount of the acid a will be equivalent to the total 
quantity of un-neutralized H 2 A and of HA~ and A ions present at any 
instant; that is 

a = m Hj A + m H A- + m A -. (33) 

If a quantity B is defined, as before, by 



it can be shown that equations (31), (32) and (33) lead to the result 

H+ o^ ' I = aH * o^ S ' I KI + KiK 2 . (34) 

2a n 7H 2 A *a li 7iu~ 

It follows, therefore, that if the left-hand side of this expression (X) is 
plotted against the coefficient of KI in the first term on the right-hand 
side (F), a straight line of slope K\ and intercept K\K* should result. 
The evaluation of B involves the same principles as described in connec- 
tion with monobasic acids. In the first stage of neutralization, i.e., when 
a > 6, the ionic strength may be taken as b + mn+, as before, but in the 

" Auerbach and Smolczyk, Z. physik. Chem., 110, 83 (1924); Britton, J. Chem. Soc., 
125, 423 (1924); 127, 1896 (1925); Morton, Trans. Faraday Soc., 24, 14 (1928); Parting- 
ton et al, t'Wd., 30, 598 (1934); 31, 922 (1935); Gane and Ingold, J. Chem. Soc., 2151 
(1931); German, Jeffery and Vogel, ibid., 1624 (1935); German and Vogel, J. Am. Chem 
Soc., 58, 1546 (1936); Jones and Soper, J. Chem. Soc., 133 (1936); see also, Simms, 
J. Am. Chem. Soc., 48, 1239 (1926); Muralt, ibid., 52, 3518 (1930). 



DIBASIC ACIDS 



327 



second stage, i.e., when b > a, a sufficient approximation is 26 a. 
Provided the solutions are reasonably dilute the limiting law of Debye 
and Hiickel may be used to derive y\ and the ratio TA" /THA-, the 
activity coefficient of the undissociated molecules 7 H ,A being taken as 



6.0 



4.0 



s 
S 



,2.0 



1.0 




-0.5 



LO 



1.5 



0.5 

r x io 6 

Fia. 91. Dissociation constants of adipic acid (Speakman) 

unity. The experimental results obtained in this manner for adipic 
acid are shown in Fig. 91 ; 14 the plot is seen to approximate very closely 
to a straight line, the values of K\ and K\K^ being 3.80 X 10~ 6 and 
1.43 X 10~ 10 respectively, so that K 2 is 3.76 X 1Q- 6 . 

An alternative treatment of equation (34) is to write it in the form 



X = 
where X and Y are defined by 

*Y 2 

and 

Vj" _ 



(35) 



B 



TA 



2a - B 

a- B 
2a- B 



The solutions of equation (35) are 
X 



#1 = ^ 



and 



X - 



If two points during the neutralization are chosen, such that the quan- 
tities X and Y have the values X' and Y' and X" and Y", respectively, 
then it is readily found that 



X' - X" 



and 



*M Y f v ri 

14 Speakman, J. Chem. Soc. t 855 (1940). 



X'Y" - X"Y' 
X" - X' 



328 ACIDS AND BASES 

Since the X's and Y's can be evaluated, as already described, the two 
dissociation constants of a dibasic acid can be determined from pairs of 
pH measurements. 

The methods just described can be extended so as to be applicable 
to acids of higher basicity, irrespective of the ratio of successive dissocia- 
tion constants. 

Colorimetric Determination of Dissociation Constants. The colori- 
metric method for determining or comparing dissociation constants has 
been chiefly applied in connection with non-aqueous solvents, but it has 
also been used to study certain acids in aqueous solution. It can be 
employed, in general, whenever the ionized and non-ionized forms of an 
acid, or base, have different absorption spectra in the visible, i.e., they 
have different visible colors, or in the near ultra-violet regions of the 
spectrum. If the acid is a moderately strong one, e.g., picric acid, it will 
dissociate to a considerable extent when dissolved in water, and the 
amounts of un-ionized form HA and of ions A~ will be of the same order; 
under these conditions an accurate determination of the dissociation 
constant is possible. By means of preliminary studies on solutions which 
have been made either definitely acid, so as to suppress the ionization 
entirely, or definitely alkaline, so that the salt only is present and ioniza- 
tion is complete, the "extinction coefficient" for light of a given wave 
length of the form HA or A~ can be determined. As a general rule the 
ions A~ have a more intense color and it is the extinction coefficient of 
this species which is actually measured. Once this quantity is known, 
the amount of A~ in any system, such as the solution of the acid in water, 
can be found, provided Beer's law is applicable.* In a solution of the 
pure acid of concentration a in pure water, C H + is equal to C A -, while 
the concentration of undissociated acid CHA is equal to a C H + or to 
a CA~; hence if C A - is determined colorimetrically, it is possible to evalu- 
ate directly the concentration dissociation function CH+CA-/ C HA. This 
function, as already seen, depends on the ionic strength of the medium, 
but extrapolation to infinite dilution should give the true dissociation 
constant. 

If the acid is too weak to yield an appreciable amount of A~ ions 
when dissolved in pure water, e.g., p-riitrophenol, it is necessary to employ 
a modified procedure which is probably less accurate. A definite quan- 
tity of the acid being studied is added to excess of a "buffer solution" 
(see Chap. XI) of known pH; the pll chosen should be close to the 
expected pK a of the acid, for under these conditions the resulting solution 
will contain approximately equal amounts of the undissociated acid HA 
and of A~ ions. The amount of either HA or A", whichever is the more 
convenient, is then determined by studying the absorption of light of 

* According to Beer's law, log /<>// = *cd, where 7 is the intensity of the incident 
light and / is that of the emergent light for a given wave length, for which the extinction 
coefficient is e, d is the thickness of the layer of solution, and c is its concentration. If 6 
is known, the value of c can be estimated from the experimental value of /o//. 



APROTJC SOLVENTS 329 

suitable wave length, the corresponding extinction coefficient having been 
obtained from separate experiments, as explained previously. If CA~ is 
determined in this manner, CHA is known, since it is equal to c C A ~, 
where c is the stoichiometric concentration of the acid. In this way it is 
possible to calculate the ratio C\-/CHA, and since a H + is known from the pH 
of the solution, the function a H + c A -/CHA can be evaluated. For many 
purposes this is sufficiently close to the dissociation constant to be em- 
ployed where great accuracy is not required. Alternatively, the values 
of the function in different solutions may be extrapolated to zero ionic 
strength. This method has been used to study acids which exhibit visible 
color changes in alkaline solutions, e.g., nitrophenols, 16 as well as for 
substances that are colorless in both acid and alkaline media but have 
definite absorption spectra in the ultra-violet region of the spectrum, 
e.g., benzoic and phcnylacetic acids. 18 

Approximate Methods for Bases. The procedures described for de- 
termining the dissociation constants of acids can also be applied, in 
principle, to bases; analogous equations are applicable except that hy- 
droxyl ions replace hydrogen ions, and vice versa, in all the expressions. 
Since the value of the product of an+ and OOH~ is known to have a definite 
value at every temperature (cf. Table LXI), it is possible to derive OOIT 
from an* obtained experimentally. 

Dissociation Constant Data. The dissociation constants at 25 of a 
number of acids and bases obtained by the methods described above are 
recorded in Table LVII; the varying accuracy of the results is indicated, 
to some extent, by the number of significant figures quoted. 17 The pK a 
and pK b values are given in each case, since these are more frequently 
employed in calculations than are the dissociation constants themselves. 
Acids and bases having dissociation constants of about 10~ 5 , i.e., pK is in 
the vicinity of 5, are generally regarded as "weak," but if the values are 
in the region of 10~ 9 , i.e., pK is about 9, they are referred to as "very 
weak." If the dissociation constant is about 10~ 2 or 10~ 3 , the acid or base 
is said to be "moderately strong/! and at the other extreme, when the 
dissociation constant is 10~ 12 or less, the term "extremely weak" is em- 
ployed. 

Aprotic Solvents. The colorimetric method of studying dissociation 
constants has found a special application in aprotic solvents such as 
benzene; these solvents exhibit neither acidic nor basic properties, and 
so they do not have the levelling effects observed with acids in proto- 

w von Halban and Kortiim, Z. physik. Chem., 170A, 351 (1934); 173A, 449 (1935); 
Kilpatrick et al, J. Am. Chem. Soc., 59, 572 (1937); 62, 3047 (1940); J. Phys. Chem., 
43, 259 (1939). 

"Flexser, Hammett and Dingwall, J. Am. Chem. Sec., 57, 2103 (1935); Martin 
and Butler, J. Chem. Soc. t 1366 (1939). 

17 For further data, see Harned and Owen, Chem. Revs., 25, 31 (1939); Dippy, ibid., 
25, 151 (1939); Gane arid Ingold, J. Chem. Soc., 2153 (1931); Jeffery and Vogel, ibid., 21 
(1935); 1756 (1936); German, Jeffery and Vogel, ibid., 1624 (1935); 1604 (1937). 



330 



ACIDS AND BASES 



TABLE LVII. DISSOCIATION CONSTANT EXPONENTS OF ACIDS AND BASES AT 25 

Monobasic Organic Acids 



Acid 


pA'a 


Acid 


P/C. 


Formic 


3.751 


Benzoic 


4.20 


Acetic 


4.756 


o-Chlorobenzoic 


2.92 


Propionic 


4.874 


w-Phlorobenzoic 


3.82 


n-Butyric 


4.820 


p-Chlorobenzoic 


3.98 


wo-Butyric 


4.821 


p-Bromobenzoic 


3.97 


n-Valeric 


4.86 


p-Hydroxybenzoic 


4.52 


Trimethylacetic 


5.05 


p-Nitrobenzoic 


3.42 


Diethylacetic 


4.75 


p-Toluic 


4.37 


Chloroacetic 


2.870 


Phenylacetic 


4.31 


Lactic 


3.862 


Cinnamic (cis) 


3.88 


Gycolic 


3.831 


Cinnamic (trans) 


4.44 


Acrylic 


4.25 


Phenol 


9.92 



Dibasic Organic Acids 



Acid 


pA'i 


pA'i 


Acid 


P Ki 


ptft 


Oxalic 


1.30 


4.2S6 


Pimelic 


4.51 


5.42 


Malonic 


2.84 


5.695 


Suberic 


4.53 


5.40 


Succinic 


4.20 


5.60 


Maleic 


2.00 


6.27 


Glutaric 


4.35 


5.42 


Fu marie 


3.03 


4.48 


Adipic 


4.42 


5.41 


Phthalic 


2.S9 


5.42 



Bases 



Base 


pA'& 


Base 


pA'6 


Ammonia 


4.76 


Triethylamine 


3.20 


Methylamine 


3.30 


Aniline 


9.39 


Dimethylamme 


3.13 


Benzylamine 


4.63 


Tnmethylarnine 


4.13 


Diphonylamine 


13.16 


Ethylamine 


3.25 


Pyridine 


8.80 


Diethylamine 


2.90 


Piperidine 


2.88 



Inorganic Acids 



Sulfuric (2nd stage) 




1.02 


Hydrogen sulfide 


7.2, 11.9 


Phosphoric 2.124, 


7.206, 


1232 


Hydrogen cyanide 


9.14 


Carbonic 


6.35, 


10.25 


Boric 


9.24 



philic and with bases in protogenic media (cf. pp. 309, 311). It is thus 
possible to make a comparison of the strengths of acids and bases without 
any interfering influence of the solvent. Suppose a certain amount of 
an acid HA is dissolved in an aprotic solvent and a known quantity of a 
base B is added; although neither acid nor base can function alone, they 
can exercise their respective functions when present together, so that an 



APROTIC SOLVENTS 331 

acid-base equilibrium of the familiar form 

HA + B ^ BH+ + A- 

acid base acid base 

is established. Application of the law of mass action to the equilibrium 
then gives 



n, (36) 

JHA/B 

If the essential dissociations of the acids HA and BH + , to yield protons, 
i.e., 

HA ^ 11+ 4- A- and BH+ ^ H+ + B, 

where H+ represents a proton, are considered, the fundamental dissocia- 
tion constants are 



^ . v 

AHA = - and ABH+ = - (37)* 

+ 



respectively; comparison of these quantities with the equilibrium con- 
stant of equation (36) shows that 



ABH+ 

and hence is equal to the ratio of the fundamental dissociation constants 
of the acids HA and BH+, the latter being the conjugate acid of the 
added base B. 

If the color of the base B differs from that of its conjugate acid BH+, 
it is possible by light absorption experiments to estimate the value of 
either CB or CBH+," since the stoichiometric composition of the solution is 
known, the concentrations of all the four species CHA, CA-, CB and CBH+ can 
be thus estimated, and value of A in equation (36), apart from the activity 
coefficient factor, can be calculated. In this way the approximate ratio 
of the dissociation constant of the acid HA to that of BH+ is obtained. 
The procedure is now repeated with an acid HA' using the same base B, 
and from the two values of A the ratio of the dissociation constants of 
HA and HA' can be found. This method can be carried through for a 
number of acids, new bases being used as the series is extended. 18 

On account of the low dielectric constants of aprotic solvents, con- 
siderable proportions of ion-pairs and triple ions are present, but spectro- 
metric methods are unable to distinguish between these and single ions; 
the determinations of the amounts of free ions, which are required by 
the calculations, will thus be in error. The activity coefficient factor, 
neglected in the above treatment, will also be of appreciable magnitude, 
but this can be diminished if the base is a negatively charged ion B~; 

* In these expressions <Z H + stands for the activity of protons. 
" LaMer and Downes, J. Am. Chem. Soc., 53, 888 (1931); 55, 1840 (1933); Chem. 
Revs., 13, 47 (1933). 



332 ACIDS AND BASES 

the activity factor will then be /HA/B-//A~/BH which involves a neutral 
molecule and a singly charged ion in both numerator and denominator, 
and hence will not differ greatly from unity. 

The Acidity Function. A property of highly acid solutions, which is 
of some interest in connection with catalysis, is the acidity function H<>: 
it is defined with reference to an added electrically neutral base B, and 
measures the tendency of the solution to transfer a proton to the base; 19 
thus 

#0= -log a f^- (38) 

JBH* 

There are reasons for believing that the fraction /B//BH+ is practically 
constant for all bases of the same electrical type, and so the acidity func- 
tion may be regarded as being independent of the nature of the base B. 
Combination of equation (38) with the usual definition of K a , the con- 
ventional dissociation constant of the acid BH+, gives 

#o = p#o + log (39) 

CBH+ 

This equation provides a method for evaluating the acidity function of 
any acid solution; a small amount of a base B, for which P/BH+ is known, 
is added to the given solution and the ratio CB/CBH+ is estimated colori- 
metrically. The acidity functions of a number of mixtures of perchloric, 
sulfuric and formic acids with water have been determined in this manner. 
By reversing the procedure, equation (39) may be used, in conjunc- 
tion with the known acidity functions of strongly acid media, to deter- 
mine the dissociation constants of the conjugate acids BH+ of a series of 
extremely weak bases. The relative amounts of B and BH+ can be 
determined by suitable light-absorption measurements. The method 
has been applied to the study of a number of bases which are much too 
weak to exhibit basic properties in water. The results obtained in certain 
cases are given in Table LVIII; the figures in parentheses are the reference 
points for each solvent medium. 20 It is soon, therefore, that all the dis- 
sociation constants recorded are based on the pK a value of 2.80 for the 
acid conjugate to aminoazobcnzene, this being the normal result in 
aqueous solution. The results in Table LVIII, which are seen to be inde- 
pendent of the acidic medium usod as the solvent, thus refer to dis- 
sociation constants of the various conjugate acids BH" 1 " in aqueous solu- 
tions. The dissociation exponents pX& of the bases (B) themselves can 
be derived by subtracting the corresponding pK a values, for BH+, from 
pK w , i.e., from 14. It is evident that many of the bases included in 

19 Hammett and Dcyrup, J. Am. Chem. Xoc., 54, 2721, 4239 (1932); Hammett and 
Paul, ibid., 56, 827 (1934); Hall et al., ibid., 62, 2487, 2493 (1940). 

10 Hammett and Paul, J. Am. Chem. Soc., 56, 827 (1934); Hammett, Chem. Revs., 
16, 67 (1935). 



EFFECT OF SOLVENT ON DISSOCIATION CONSTANTS 



333 



TABLE LVIII. DISSOCIATION CONSTANTS (pK a ) OP CONJUGATE ACIDS 



RnAA 


Solvent Medium 


DBSO 


HC1 - HaO 


HtSO< - H0 


HC1O4 - HO 


HCOjH 


Aminoazobenzene 


(2.80) 











Benzeneazodiphenylamine 


1.52 











p-Nitroaniline 


1.11 


(1.11) 


(1.11) 





o-Nitroaniline 


-0.17 


-0.13 


-0.19 


(- 0.17) 


p-Chloronitroaniline 


-0.91 


-0.85 


-0.91 


-0.94 


p-Nitrodiphenylamine 





-2.38 





-2.51 


2 : 4-Dichloro-6-nitroaniline 





-3.22 


-3.18 


-3.31 


p-Nitroazobenzene 


- 


-3.35 


-3.35 


-3.29 


2 : 4-Dinitroanilinc 





-4.38 


-4.43 





Benzalacctophenone 





-5.61 








Anthraquinone 


-- 


-8.15 








2:4: 6-Trinitroaniline 





-9.29 









Table LVIII arc extremely weak; the dissociation constant of 2 : 4 : 6- 
trmitroaniline, for example, is as low as 5 X 10~ 24 . 

Effect of Solvent on Dissociation Constants. The dissociation equi- 
librium of an uncharged acid HA in the solvent S can be represented as 

HA + S;=SH+ + A-; 

the dissociation process consequently involves the formation of a positive 
and a negative ion from two uncharged molecules. Since the electro- 
static attraction between two oppositely charged particles decreases with 
increasing dielectric constant of the medium, it is to be expected that, 
other factors being more or less equal, an increase of the dielectric con- 
stant of the solvent will result in an increase in the dissociation constant 
of an electrically neutral acid. It has been found experimentally, in 
agreement with expectation, that the dissociation constant of an un- 
charged carboxylic acid decreases by a factor of about 10 5 or 10 6 on 
passing from water to ethyl alcohol as solvent. In the same way, the 
dissociation constant of an uncharged base is diminished by a factor of 
approximately 10 3 to 10 4 for the same change of solvent. 

If the acid is a positive ion, e.g., NH 4 f , or the base is a negative ion, 
e.g., CHsCOj", the process of dissociation does not involve the separation 
of charges, viz., 

NIIJ + S = SI1+ + NH 3 , 
or 

+ HS = CHCOiH + S-. 



The effect of changing the dielectric constant of the medium would thus 
be expected to be small, and in fact the dissociation constants do not 
differ very greatly in water and in ethyl alcohol. The value of pK a for 
the ammonium ion acid, for example, is about 9.3 in water and 11.0 in 



334 ACIDS AND BASES 

methyl alcohol. It should be noted that the foregoing arguments do 
not take into consideration the different tendencies of the solvent mole- 
cule to take up a proton; the conclusions arrived at are consequently 
more likely to be applicable to a series of similar solvents, e.g., hydroxylic 
substances. 

A quantitative approach to the problem of the influence of the medium 
on the dissociation constants of acids, which eliminates the proton accept- 
ing tendency of the solvent, involves a comparison of the dissociation 
constants of a series of acids with the value for a reference acid. Con- 
sider the acid HA in the solvent S; the dissociation constant is given by 



whereas that for the reference acid IIA in the same solvent is 



so that, since SH+ is the same in both cases, 

K._ a 

Ko c 



where K is the equilibrium constant of the reaction between the two 
acid-base systems, viz., 

HA + AQ ^ A- + HAo. 
The standard free energy change of this process is then given by 

- AG = RT In K 

= 2.30 RT log X, 

where log K, equal to log (K a /Ko), is equivalent to pK pK a . 

This free energy change may be regarded as consisting of a non- 
electrostatic term A(? n and an electrostatic term AGvi. equivalent to the 
gain in electrostatic free energy resulting from the charging up of the ion 
A~ and the discharge of A^" in the medium of dielectric constant D. 
According to the Born equation (see p. 249), the electrostatic free energy 
increase per mole accompanying the charging of a spherical univalent 
ion is given by 



and so in the case under consideration, for charge and discharge of the 
ions A~ and AJT, respectively, 



EFFECT OF SOLVENT ON DISSOCIATION CONSTANTS 335 

where r A - and r A o arc the radii of the corresponding spherical ions. It 
follows, therefore, that 



- 
-~ R T 

If the effective radii of the two ions remain approximately constant in a 
series of solvents, it follows that 



where a is a constant. The plot of log K, that is, of 
against 1/D, i.e., the reciprocal of the dielectric constant of the solvent, 
should thus be a straight line; the intercept for 1/D equal to zero, i.e., 
for infinite dielectric constant, should give a measure of the dissociation 
constant of the acid HA free from electrostatic effects. 

Measurements of dissociation constants of carboxylic acids, e.g., of 
substituted acetic and benzoic acids, using either acetic or benzoic acid 
as the reference substance HA, made in water, methyl and ethyl alcohols 
and ethylene glycol, are in good agreement with expectation. 11 The 
plot of the values of log (K a /K<>) against 1/D is very close to a straight 
line for each acid, provided D is greater than about 25. The slope of the 
line, however, varies with the nature of the acid, so that an acid which is 
stronger than another in one solvent may be weaker in a second solvent. 
The comparison of the dissociation constants of a scries of acids in a 
given solvent may consequently be misleading, since a different order of 
strengths would be obtained in another solvent. It has been suggested, 
therefore, that when comparing the dissociation constants of acids the 
values employed should be those extrapolated to infinite dielectric con- 
stant; in this way the electrostatic effect, at least, of the solvent would be 
eliminated. 

Attempts to verify the linear relationship between log K and 1/D 
by means of a series of dioxane-water mixtures have brought to light 
considerable discrepancies. 22 The addition of dioxane to water results in 
a much greater decrease in the dissociation constant than would be 
expected from the change in the dielectric constant of the medium. 
Since the organic acids studied are more soluble in dioxane than in water, 
it is probable that molecules of the former solvent are preferentially 
oriented about the acid anion; the effective dielectric constant would 
then be less than in the bulk of the solution. It is thus possible to 

11 Wynne-Jones, Proc. Roy. Soc. t 140A, 440 (1933); Kilpatrick et oJ., /. Am. Chem. 
Soc. t 59, 572 (1937); 62, 3051 (1940); /. Phys. Chem., 43, 259 (1939); 45, 454, 466, 472 
(1941); Lynch and LaMer, J. Am. Chem. Soc., 60, 1252 (1938); see also, Hammett, 
ibid., 59, 96 (1937); J. Chem. Phys., 4, 618 (1986). 

Elliott and Kilpatrick, J. Phys. Chem., 45, 472 (1941); see also, Earned, ibid., 43, 
275 (1939). 



336 ACIDS AND BASES 

account for the unexpectedly low dissociation constants in the dioxane- 
water mixtures. 

Dissociation Constant and Temperature. The dissociation constants 
of uncharged acids do not vary greatly with temperature, as may be 
seen from the results recorded in Table LIX for a number of simple fatty 

TABLE LIX. INFLUENCE OF TEMPERATURE ON DISSOCIATION CONSTANT 



Acid 





10 


20 


30 


40 


50 


60 


Formic acid 


1.638 


1.728 


1.765 


1.768 


1.716 


1.650 


1.551 X ID' 4 


Acetic acid 


1.657 


1.729 


1.753 


1.750 


1.703 


1.633 


1.542 X 10-* 


Propionic acid 


1.274 


1.326 


1.338 


1.326 


1.280 


1.229 


1.160 X 10-' 


n-Butyric acid 


1.563 


1.576 


1.542 


1.484 


1.395 


1.302 


1.199 X 10' 5 



acids. A closer examination of the figures, however, reveals the fact that 
in each case the dissociation constant at first increases and then decreases 
as the temperature is raised; this type of behavior has been found to be 
quite general, and Harned and Embree 23 showed that the temperature 
variation of dissociation constants could be represented by the general 
equation 

log K a = log A', - p(t - 0) 2 , 

where K a is the dissociation constant of the acid at the temperature t, 
Ke is the maximum value, attained at the temperature 6, and p is a 
constant. It is an interesting fact that for a number of acids p has the 
same value, viz., 5 X 10~ 6 ; this means that if log K a log KB for a num- 
ber of acids is plotted against the corresponding value of t 6, the 
results all fall on a single parabolic curve. The actual temperature at 
which the maximum value of the dissociation constant is attained de- 
pends on the nature of the acid; for acetic acid it it, 22.6, but higher 
and lower values have been found for other acids. For some acids, e.g., 
chloroacetic acid and the first stage of phosphoric acid, the maximum 
dissociation constant would be reached only at temperatures below the 
freezing point of water. 

An alternative relationship 24 

n 

log K = A + - - 20 log T, 

where A and B are constants, has been proposed by Pitzer to represent 
the dependence of dissociation constant on the absolute temperature T. 
This equation has a semi-theoretical basis, involving the empirical facts 
that the entropy change and the change in heat capacity accompanying 
the dissociation of a monobasic acid are approximately constant. 

Some attempts have been made to account for the observed maximum 
in the dissociation constant. It was seen on page 334 that the division 

Harned and Kmbree, ./. Am. Chem. Soc., 56, 1050, 2797 (1934); see also, Harned, 
J. Franklin Inst., 225, 623 (1938); Harned and Owen, Chem. Revs., 25, 131 (1939). 

84 Pitzer, /. Am. Chem. Soc., 59, 2365 (1937); see also, Walde, J. Phys. Chem., 39, 
477 (1935); Wynne-Jones and Everett, Trans. Faraday Soc., 35, 1380 (1939). 



AMPHIPROTIC SOLVENTS 337 

of the free energy of dissociation of an acid into non-electrostatic and 
electrostatic terms leads to the expectation that log K a is related to the 
reciprocal of the dielectric constant of the solvent. Since l/D for water 
increases with increasing temperature, the value of log K a should de- 
crease; in addition to this effect there is the normal tendency for the 
dissociation constant, regarded as the equilibrium constant of an endo- 
thermic reaction, to increase with increasing temperature. The simul- 
taneous operation of these two factors will lead to a maximum dissocia- 
tion constant at a particular temperature. 26 

Amphiprotic Solvents: The Ionic Product. In an amphiprotic solvent 
both an acid and its conjugate base can function independently; for 
example, if the acid is HA the conjugate base is A~, and if the amphi- 
protic solvent is SH, the acidic and basic equilibria are 



HA + SH ^ SUt + A- 

and 

SII + A- ^ HA + S-, 
acid base acid base 

respectively. The ion SHt is the hydrogen ion, sometimes called the 
lyonium ion, in the given medium, arid S~ is the anion, or lyate ion, of the 
solvent. The conventional dissociation constants of the acid HA and of 
its conjugate base A~ are then written as 

asn 2 f a A - 
A a = - ana Kb 

Qll\ CijC 

and the product is thus 

K a K b = asHjas-, (40) 

which is evidently a specific property of the solvent. Since the solvent 
is amphiprotic and can itself function as either an acid or a base, the 
equilibrium 

SH + SH ^= Slit + S- 

acid base acid base 

must always exist, and if the activity of the undissociated molecules of 
solvent is taken as unity, it follows that the equilibrium constant KS of 
this process IK given by 

K s = flsujfls-, (41) 

the constant A~? defined in this manner being called the ionic product 
or ionization constant of the solvent. It is sometimes referred to as the 
autoprotolysis constant, since it is a measure of the spontaneous tendency 
for the transfer of a proton from one molecule of solvent to another to 

*Gurney, J. Chem. Phys., 6, 499 (1938); Baughan, ibid., 7, 951 (1939); see also, 
Magee, Ri and Eyring, ibid., 9, 419 (1941); LaMer and Brescia, /. Am. Chem. Soc., 62, 
617 (1940). 



338 ACIDS AND BASES 

take place. Comparison of equations (40) and (41) shows that 

K a K b = K s , (42) 

and so the dissociation constant of a base is inversely proportional to 
that of its conjugate acid, and vice versa; the proportionality constant 
is the ionic product of the solvent. This is the quantitative expression 
of the conclusion reached earlier that the anion of a strong acid, which 
is its conjugate base, will be weak, while the anion of a weak acid will 
be a moderately strong base, and similarly for the conjugate acids of 
strong and weak bases. 

For certain purposes it is useful to define the dissociation constant of 
the solvent itself as an acid or base; by analogy with the conventional 
method of writing the dissociation constant of any acid or base, the 
activity of the solvent molecule taking part in the equilibrium is assumed 
to be unity. In the equilibrium 

SH + SH ^ SUt + S- 

one molecule of SH may be regarded as functioning as the acid or base, 
while the other is the solvent molecule; the conventional dissociation 
constant of either acid or base is then 



J\.a -L**b 



(43) 



For most purposes <ZSH may be replaced by the molecular concentration 
of solvent molecules in the pure solvent; with water, for example, the 
concentration of water molecules in moles per liter is 1000/18, i.e., 55.5, 
so that the dissociation constant of H 2 as an acid or base is equal to the 
ionic product of water divided by 55.5. 

The Ionic Product of Water. An ionic product of particular interest 
is that of water: the autoprotolytic equilibrium is 

H 2 O + H 2 O ^ H 3 O+ + OH-, 

and hence the ionic product K w may be defined by either of the following 
equivalent expressions, viz., 

(44) 



= CH 3 o+coH--/H,o + /oir. (446) 

By writing the ionic product in this manner it is tacitly assumed that 
the activity of the water is always unity; in solutions containing dissolved 
substances, however, the activity is diminished and K w as defined above 
will not be constant but will increase. The activity of water in any 



THE TONIC PRODUCT OF WATER 339 

solution may be taken as equal to p/po, where p is the vapor pressure of 
the solution and p that of the pure water at the same temperature; in 
a solution containing 1 g.-ion per liter of solute, which is to be regarded 
as relatively concentrated, the activity of the water is about 0.98. The 
effect on K w of the change in the activity of the water is thus not large 
in most cases. 

The equilibrium between HaO* and OH~ ions will exist in pure water 
and in all aqueous solutions: if the ionic strength of the medium is low, 
the ionic activity coefficients may be taken as unity, and hence the ionic 
product of water, now represented by k w) is given by 

k w = C H ,O+COH- (or c H +c ir). (45) 

As will be seen later, the value of k w is approximately 10~ 14 at ordinary 
temperatures, and this figure will be adopted for the present. 

In an exactly neutral solution, or in perfectly pure water, the con- 
centrations of hydrogen (H 3 O+) and hydroxyl ions must be equal; hence 
under these conditions, 

CH+ = COH- = 10~ 7 g.-ion per liter, 

the product being 10~ 14 as required. The question of the exact signifi- 
cance of the experimental value of pll will be considered in Chap. X, 
but for the present the pH of a solution may be defined, approximately, by 

pH log CH+. 

It follows, therefore, that in pure water or in a neutral solution at ordi- 
nary temperatures, the pH is 7. If the quantity pOH is defined in an 
analogous approximate manner, as log COIT, the value must also be 7 
in water. 

By taking logarithms of equation (45), it can be shown that for any 
dilute aqueous solution 

pH + pOH = pfc, = 14 (46) 

at ordinary temperatures, where pk w is written for log k w . If the 
hydrogen ion concentration of a solution exceeds 10~ 7 g.-ion per liter, the 
pH is less than 7 and the solution is said to be acid; the pOH is corre- 
spondingly greater than 7. Similarly, in an alkaline solution, the hydro- 
gen ion concentration is less than 10~ 7 g.-ion per liter, but the hydroxyl 
ion concentration is greater than this value; the pH is greater than 7, 
but the pOH is smaller than this figure. The relationships between pH, 
pOH, CH+ and coir, at about 25, may be summarized in the manner 
represented below. 

CH+ i io-> io- io- io- 10-' io- io~ 7 io- io- io- l io~ 11 io~ u io- 
COH- 10-" 10-" 10-" io- 10-" io- io- io- 7 io- 10-* io- io- 10-' lo- 1 i 

pH 1 2 3 4 5 6 7 8 9 10 11 12 13 14 
pOH 14 13 12 11 10 987654 3 2 1 

Neu- 
. Acid * tral - Alkaline 



340 ACIDS AND BASES 

It is seen that the range of pH from zero to 14 covers the range of hydro- 
gen and hydroxyl ion concentrations from a N solution of strong acid on 
the one hand to a N solution of a strong base on the other hand. A solu- 
tion of hydrogen ion concentration, or activity, exceeding 1 g.-ion per 
liter would have a negative pH, but values less than about 1 in water 
are uncommon. 

Determination of Ionic Product: Conductance Method. Since it 
contains a certain proportion of hydrogen and hydroxyl ions, even per- 
fectly pure water may be expected to have a definite conductance; the 
purest water hitherto reported was obtained by Kohlrausch and Heyd- 
weiller 26 after forty-eight distillations under reduced pressure. The 
specific conductance of this water was found to be 0.043 X 10~ fl ohm" 1 
cm." 1 at 18, but it was believed that this still contained some impurity 
and the conductance of a 1 cm. cube of perfectly pure water was esti- 
mated to be 0.0384 X 10" 6 ohm" 1 cm." 1 at 18. The equivalent con- 
ductances of hydrogen and hydroxyl ions at the very small concentra- 
tions existing in pure water may be taken as equal to the accepted values 
at infinite dilution; these are 315.2 and 173.8 ohms" 1 cm. 2 , respectively, 
at 18, and hence the total conductance of 1 equiv. of hydrogen and 
1 equiv. of hydroxyl ions, at infinite dilution, should be 489.0 ohms" 1 cm. 2 
It follows, therefore, that 1 cc. of water contains 

0.0384 X 10~ 6 

- = - 78 X 10 ~ 10 equiv. per cc. 



of hydrogen and hydroxyl ions; the concentrations in g.-ion per liter are 
thus 0.78 X 10" 7 , and hence 

k w = CH+COH- - (0.78 X 10~ 7 ) 2 
= 0.61 X 10" 14 . 

Since the activity coefficients of the ions in pure water cannot differ 
appreciably from unity, this result is probably very close to K w , the 
activity ionic product, at 18. The results in Table LX give the ob- 

TABLE LX. SPECIFIC CONDUCTANCE AND IONIC PRODUCT OF WATER 

Temp. 18 25 34 50 

ic 0015 0.043 0.062 0095 0.187 X 10~ ohm-' cm." 1 

K u 0.12 0.61 1.04 2.05 5.66 X 10~ 14 

served specific conductances and the values of K w at several tempera- 
tures from to 50. 

Conductance measurements have been used to determine the ionic 
products of the amphiprotic solvents ethyl alcohol, formic acid and 
acetic acid. 

"Kohlrausch and Heydweiller, Z. physik. Chem., 14, 317 (1894); Heydweiller, 
Ann. Physik, 28, 503 (1909). 



ELECTROMOTIVE FORCE METHODS 341 

Electromotive Force Methods. The earliest E.M.F. methods for 
evaluating the ionic product of water employed cells with liquid junc- 
tion; 27 the E.M.F. of the cell 

H.(l atm.) | KOH(0.01 N) || HC1(0.01 N) | H 2 (l atm.), 

from which it is supposed that the liquid junction potential has been 
completely eliminated, is given by 

_ RT . a' H + ,_ 

E = -grin -77;. (47) 

r a\i+ 

where a'n + and OH* represent the hydrogen ion activities in the right-hand 
and left-hand solutions, i.e., in the 0.01 N hydrochloric acid and 0.01 N 
potassium hydroxide, respectively. If aoir is the hydroxyl ion activity 
in the latter solution, then 



and substitution of K w /adn- for OH* in equation (47) gives 

RT. a'H+aoir , 4Q . 

T" K w ' ( 8) 

By measuring each of the electrodes separately against a calomel refer- 
ence electrode containing 0.1 N potassium chloride, and estimating the 
magnitude of the liquid junction potential in each case, the E.M.F. of the 
complete cell under consideration was found to be + 0.5874 volt at 25. 
The ionic activity coefficients were assumed to be 0.903 in the 0.01 N 
solutions, so that a' H + and ao'ir, representing the activities of hydrogen 
and hydroxyl ions in 0.01 N hydrochloric acid and 0.01 N potassium hy- 
droxide, respectively, were both taken to be equal to 0.0093; insertion 
of these figures in equation (48) gives K w as 1.01 X 10~ 14 at 25. This 
result is almost identical with some of the best later data, but the close 
agreement is probably partly fortuitous. 

The most satisfactory method for determining the ionic product of 
water makes use of cells without liquid junction, similar to those em- 
ployed for the evaluation of dissociation constants (cf. p. 314). 28 The 
E.M.F. of the cell 

H 2 (l atm.) | MOH(mO MCl(m) AgCl(s) | Ag, 
where M is an alkali metal, e.g., lithium, sodium or potassium, is 

RT 

E = E - -=- In a H *ocr. (49) 

r 

87 Lewis, Brighton and Sebastian, /. Am. Chem. Soc., 39, 2245 (1917); Wynne-Jones, 
Trans. Faraday Soc., 32, 1397 (1936). 

Roberts, J. Am. Chem. Soc., 52, 3877 (1930); Harned and Hamer, ibid., 55, 2194 
(1933); for reviews, with full references, see Harned, /. Franklin Inst., 225, 623 (1938); 
Harned and Owen, Chem. Revs., 25, 31 (1939). 



342 



ACIDS AND BABES 



Since O H *OOH- is equal to K u , the activity of the water being assumed 
constant, it follows that 



^1-^ln 



yon" 



and rearrangement gives 



E - E + 
F(E - Jg) 

2.303/er 



RT 



, 
In 



RT . 



^J- = - ^ In K a - ^ln -> 
TOOK- r ** Ton" 



+ log ~ = - log K w - log 



7cr 

70H" 



(50) 



The activity coeflBicient fraction 7cr/7oir is unity at infinite dilution, 

and so the value of the right-hand side of equation (50) becomes equal 

to log Kw under these conditions. 
It follows, therefore, that if the left- 
hand side of this equation, for var- 
ious concentrations of alkali hydrox- 
ide and chloride, is plotted against 
the ionic strength, the intercept for 
infinite dilution gives log K w . The 
value of is known to be + 0.2224 
volt at 25, and by making the as- 
sumption that MOH and MCI are 
completely dissociated, as wil 1 be the 
case in relatively dilute solutions, 
men- and mcr may be identified with 
mi and m 2 , respectively. The results 
shown in Fig. 92 are for a series of 
cells containing cesium (I), potassium 
(II), sodium (III), barium (IV), and 
lithium (V) chlorides together with 
the corresponding hydroxides; the 
agreement between the values extra- 
polated to infinite dilution is very 

striking. The value of - log K w is found to be 13.9965 at 25, so that 

K w is 1.008 X 10~ 14 . 

Another method of obtaining the ionic product of water is to combine 

the E.M.F. of the cell 




0.05 



0.10 



FIG. 92. Determination of the ionic 
product of water (Harned, et al.) 



H 2 (l atm.) | HCl(w',) MCl(it^l) AgCl(s) | Ag 
with that just considered; the E.M.F. of this cell is given by the same 



ELECTROMOTIVE FORCE METHODS 343 

general equation, 

RT 
' = #o- lnaW:i. (51) 

Combination of equations (49) and (51) gives 



RT m'n+rr&r , RT , 

= - In --- h t In - > (52 ; 

+ P 7n + 7cr 



where the primed quantities refer to the cell containing hydrochloric acid 
whereas those without primes refer to the alkali hydroxide cell. 

If the ionic strengths in the two cells are kept equal, then provided 
the solutions are relatively dilute the activity coefficient .actor will be 
virtually unity, and the second term on the right-hand side of equation 
(52) is zero; hence under these conditions 

RT 
E E f = -=- In 



and making use of the fact that K w is equal to mn+?noir"yn+7on-, this 
becomes 

RT WH+rocrWoH- RT * RT 



E. LV ^ * ^ /.ON 

E - E ' = -TT In - -- - + -=r In TH^TOH- - -jr In K w . (53) 
" "*cr ^ /* 

According to the extended Debye-Huckol equation, the value of log TH^OFT 
may be represented by A Vp + C|i, where A is a known constant for 
water at the experimental temperature; hence, equation (53), after re- 
arrangement, becomes 



_ ^ 

F 7ttcr F 



RT 
-jr\nK u + 2. 

F 



. (54) 

The plot of the left-hand side of equation (54) against the ionic strength 
y should be, at least approximately, a straight line whose intercept for y 
equal to zero gives log K w . As before, the values of WaS ^cr, ttk>H~ 
and mcr are estimated on the assumption that the electrolytes HC1, MCI 
and MOH are completely dissociated. 

A large number of measurements of cells of the types described, con- 
taining different halides, have been made by Harned and his collab- 
orators over a series of temperatures from to 50; the excellent agree- 
ment between the results obtained in different cases may be taken as 



344 ACIDS AND BASES 

evidence of their accuracy. A selection of the values of the ionic product 
of water, derived from measurements of cells without liquid junction, 
is quoted in Table LXI; the data in the last column may be taken as 
the most reliable values of the ionic product of water. 

TABLE LXI. IONIC PRODUCT FROM CELLS CONTAINING VARIOUS HALIDE8 

t NaCl KC1 LiBr BaCl 2 Mean 

0.113 0.115 0.113 0.112 0.113 X 10~ M 

10 0.292 0.293 0.292 0.280 0.292 

20 0.681 0.681 0.681 0.681 0.681 

25 1.007 1.008 1.007 1.009 1.008 

30 1.470 1.471 1.467 1.466 1.468 

40 2.914 2.916 2.920 2.917 

50 5.482 5.476 5.465 5.474 

Effect of Temperature on the Ionic Product of Water. The values of 
the ionic product in Table LXI are seen to increase with increasing 
temperature; at 100, the ionic product of water is about 50 X 10~ 14 . 
According to Harned and Hamer 29 the values between and 35 may 
be expressed accurately by means of the equation 

4787 3 
log K w = y^- - 7.1321 log T - 0.0103657* + 22.801. 

From this expression it is possible, by making use of the reaction iso- 
chore, i.e., 

dlnK _ A// 

dT " RT*' 

to derive the heat change accompanying the ionization of water; the 
results at 0, 20 and 25 are as follows: 

20 25 

14.51 13.69 13.48 kcal. 

These values are strictly applicable at infinite dilution, i.e., in pure water. 
It was seen on page 12, and it is obvious from the considerations 
discussed in the present chapter, that the neutralization of a strong acid 
by a strong base in aqueous solution is to be represented as 

H 3 0+ + OH- = H 2 + H 2 0, 

which is the same reaction as is involved in the ionization of water, except 
that it is in the opposite direction. The heats of neutralization obtained 
experimentally are 14.71, 13.69 and 13.41 kcal. at 0, 20 and 25, re- 
spectively; the agreement with the values derived from K w is excellent. 
Although the relationship given above for the dependence of K w on 
temperature is only intended to hold over a limited temperature range, 

Harned and Hamer, /. Am. Chem. Soc., 55, 4496 (1933); see also, Harned and 
Geary, ibid., 59, 2032 (1937). 



THE IONIZATION OF WATER IN HALIDE SOLUTIONS 



345 



it shows nevertheless that the ionic product of water, like the dissociation 
constants of acids, to which reference has already been made, should pass 
through a maximum at a relatively high temperature and then decrease. 
Although the temperature at which the maximum value of K w is to be 
expected lies beyond the range of the recent accurate work on the ionic 
product of water, definite evidence for the existence of this maximum had 
been obtained several years ago by Noyes (1910). The temperature at 
which the maximum ionic product was observed is about 220, the value 
of K w being then about 460 X 10~ 14 . 

The lonization of Water in Halide Solutions. The cells employed 
for the determination of the ionic product of water have also been used 
to study the extent of dissociation of water in halide solutions. 30 Since 
K w is equal to a H + a H- and a H + a H-/7n + 7oH- is equal to memoir, equation 
(53) becomes, after rearrangement, 



RT 

~ = & & -- ^- in 
r 



RT 

~- In ni 
r 

and so the molal ionization product WH+WOH- in the halide solution present 
in the cells may be evaluated directly from the E.M.F.'S E and E', and 
the molalities of the electrolytes. The amounts of hydrogen and hy- 
droxyl ions are equal in the pure halide solution; consequently, the 
square-root of WH+WOH" gives the concentration of these ions, in g.-ions 



LiCl 
'LLBr 




1.0 



Fia. 93. Variation of molal ionization product of water (Harned, et al.) 

per 1000 g. of water, produced by the ionization of the water in the halide 
solution. The results for a number of alkali halides at 25 are shown 
in Fig. 93; it will be seen that, in general, the extent of the ionization of 
water increases at first, then reaches a maximum and decreases with 

M For reviews with full references, see Harned, /. Franklin Inst., 225, 623 (1938); 
Harned and Owen, Chem. Revs., 23, 31 (1939). 



346 \CIDS AND BASES 

increasing ionic strength of the medium. With lithium salts the maxi- 
mum is attained at a higher concentration than is shown in the diagram. 
The explanation of this variation is not difficult to find: the quantity 
an f aoH-/ a H 2 o, i.e., WH+WOH- X 7H+7ojr/ a H 2 o, which includes the activity 
of the water, must remain constant in all aqueous solutions, and since 
the activity coefficients always decrease and then increase as the ionic 
strength of the medium is increased (cf. Fig. 40), while an 2 o, i.e., Wpo,* 
decreases steadily, it follows that the variation of WH+WOH- must be *:* the 
form shown in Fig. 93. In spite of the dependence of WH+WOIT on tho 
ionic strength of the solution, it is still satisfactory, for purposes of 
approximate computation, to take the ionic concentration product of 
water (A: u .) to be about 10~ n at ordinary temperatures, provided the con- 
centration of electrolyte in tho solution is not too great. 

PROBLEMS 

1. Show that according to equation (10) the plot of Ac against I/A should 
be a straight line; test the accuracy of this (approximate) result by means of 
the data for acetic acid on page 105 and for a-crotonic acid in Piohlcm 7 of 
Chap. III. 

2. Utilize the data referred to in Problem 1 to calculate the dissociation 
functions of acetic and a-crotonic acids at several concentrations by means of 
equation (10); compare the results with the thermodynamic dissociation con- 
stants obtained in Chap. V. 

3. In their measurements of the cell 

H,a atm.) | HP(mi) NaP(m 2 ) NaCl(m,) AgCl(s) | Ag, 

where HP represents propionic acid, Harned and Ehlers [J. Am. Chcm. Soc. t 
55, 2379 (1933)] made mi. w 2 and m 3 equal and obtained the following K.M.F.'S 
at 25: 

m E m E 

4.899 X 10~ 3 0.64758 18.669 X 10" 3 0.61311 

8.716 063275 25.546 0.60522 

12.812 0.62286 31 7'J3 59958 

Evaluate the dissociation constant of propionic acid. 

4. Walpole [./. Chem. *SV., 105, 2501 (1914)] measured tho pll's of a series 
of mixtures of x cc. of 0.2 N acetic acid with 10 x cc. of 0.2 N sodium acetate, 
and obtained the following results: 

x 8.0 7.0 6.0 5.0 4.0 30 20 cc. 

pH 4.05 4.27 4.45 4.63 4.80 4.99 5.23 

Calculate the dissociation constant of acetic acid by the use of equation (27), 
the activity coefficients of the acetate ions being obtained by means of the 
simple Debye-Huckel equation. Derive the dissociation constant by means 
of the graphical method described on page 324. 

* Since pure water, vapor pressure p , is takwi as the standard state, the activity of 
water i- any solution of aqueous vapor pressure p will be p/p Q . 



PROBLEMS 347 

5. Bennett, Brooks and Glasstonc [</. Chem. Sac., 1821 (1935)] obtained 
the following results in the titration of o-fluorophenol in 30 per cent alcohol 
at 25; when x cc. of 0.01 N sodium hydroxide was added to 50 cc. of a 0.01 N 
solution of the phenol the pll's were: 

x 10 15 20 25 30 40 cc. 

pll 8.73 9.01 9.20 9.37 9.56 10.00 

Calculate the dissociation constant of o-fluorophenol, using the expression 
log/ = 0.683 Vp 4- 2.0y to obtain the activity coefficient of the anion. 
The activity coefficient of the undissociated acid may be taken as unity. 

6. The following pH values were obtained by German arid Vogel [</. Am. 
Chem. Roc., 58, 1546 (1936) J in the titration of 100 cc. of 0.005 molar succinic 
acid with x cc. of 0.01 N sodium hydroxide at 25: 

x pH x pH 

20.0 400 60.0 5.11 

300 428 70.0 5.39 

40.0 4.56 80.0 5.68 

50.0 4.84 90.0 6.03 

Determine the two dissociation constants of succinic acid by the graphical 
method described on page 320. 

7. The E.M.F. .,f the cell 

H,(l atni.) | NaOH(ro) NaCl(m) AgCl(s) | Ag, 

with the sodium hydroxide and chloride at equal mobilities, was found by 
Roberts [,/. Am. Chem. tioc., 52, 3877 (1930)] to have a constant value of 
1.0508 volt at 25 when the solutions were dilute. Calculate the ionic product 
of water from this result. 

8. The following E.M.F.'S were obtained at 25 by Harned and Copson 
[/. Am. Chem. Svc., 55, 2206 (1933j] for the cells 

(A) II 2 (1 iitm.) | LiOH (0.01) LiCl(wi) AgCl(s) | Ag 

(B) H 2 (l atm.) 1 HC1 (0.01) LiCl(m) AgCl(s) 1 Ag. 

m E A E B 

0.01 104979 0.4 1 779 

002 1.03175 043S.V> 

0.0:> 1.00755 0.422S2 

0.10 0.9SSS3 0.40017 

0.20 0V'">7 039453 

0.50 94277 0.37235 

1.00 0.91992 0.35191 

2.00 0.89203 32352 

3.00 0.87151 0.2V9:>9 

4 00 0.85407 0.27754 

Utilize the method given on page 343 to derive the ionic product of water from 
these data. Plot the variation of the molai ionization product with the ionic 
strength of the solution. 



CHAPTER X 
THE DETERMINATION OF HYDROGEN IONS 

Standardization of pH Values. The hydrogen ion exponent, pH, was 

originally defined by S0rensen (1909) as the " negative logarithm of the 
hydrogen ion concentration,' 1 i.e., as log CH+; most determinations of 
pH are, however, based ultimately on E.M.F. measurements with hydro- 
gen electrodes, and the values obtained are, theoretically, an indication 
of the hydrogen ion activity rather than of the concentration. For this 
reason, it has become the practice in recent years to regard the pH as 
defined by 

pH 5= - log a H +, (1) 

where H+ stands for the hydrogen ion, i.e., lyonium ion, in the particular 
solvent. This definition, however, involves the activity of a single ionic 
species and so can have no strict thermodynamic significance; it follows, 
therefore, that there is no method available for the precise determination 
of pH defined in this manner. It is desirable, nevertheless, to establish, 
if possible, an arbitrary pll scale that shall be reasonably consistent with 
certain thermodynamic quantities, such as dissociation constants, which 
are known exactly, within the limits of experimental error. The values 
obtained with the aid of this scale will not, of course, be actual pH's, 
since such quantities cannot be determined, but they will at least be data 
which if inserted in equations involving pH, i.e., log a H +, will give 
results consistent with those determined by strict thermodyriamic meth- 
ods not involving individual ion activities. 

The E.M.F. of a cell free from liquid junction potential, consisting of 
a hydrogen electrode and a reference electrode, should be given by 

zprn 

E = J^ref. pT~ In an*, 

or, introducing the definition of pll according to equation (1), 

RT 
E = E&. + 2.303 -ypH 

where E nf . is the potential of the reference electrode on the hydrogen 
scale. It follows, therefore, that 

F(E - E nl .) 



348 



STANDARDIZATION OF PH VALUES 349 

If the usual value for E ro t. of the reference electrode is employed in this 
equation to derive pH's, the results are found to be inconsistent with 
other determinations that arc thermodynamically exact. A possible way 
out of this difficulty is to find a value for E ref . such that its use in equation 
(2) gives pH values which are consistent with known thermodynamic 
dissociation constants. For this purpose use is made of equation (29) of 
Chap. IX, viz., 

pll = pK a + log - fl -~ - A^ + C V , (3) 

which combined with equation (2) gives 

F(E - ffref.) . . R . I' , r 

= pA - + l *~ ~ A ^ + C *> 



_ 2.3Q3RT/ B A ,-\ yi , 2.303/2 T , A . 

'. E - -- j, -- (pK a + log ^g - A V v j = 1U + - j - C v . (4) 

A series of mixtures, at different total concentrations, of an acid, 
whose dissociation constant is known exactly, e.g., from observations on 
cells without liquid junction, and its salt are made up, thus giving a 
series of values for B and a B. The K.M.F/S of the cells consisting of 
a hydrogen electrode in this solution combined with a reference electrode 
are measured; a saturated solution of potassium chloride is used as a salt 
bridge between the experimental solution and the one contained in the 
reference electrode. The E values obtained in this manner, together 
with B and a B, calculated from the known composition of the acid- 
salt mixture (cf. p. 324), and the pK a of the acid, permit the left-hand 
side of equation (4) to be evaluated for a number of solutions of different 
ionic strengths. The results plotted against the ionic strength should 
fall on a straight line, the intercept for zero ionic strength giving the 
required quantity E T ^ m . In order for this result to have any significance 
it should be approximately constant for a number of solutions covering 
a range of pH values and involving different acids; this has in fact been 
found to be the case in the pH range of 4 to 9, and hence a pH scale 
consistent with the known pA" values for a number of acids is possible. 1 

The conclusions reached from this work may be stated in terms of 
the potentials of the reference electrodes; for example, the value of 
E ro t. of the 0.1 N KC1 calomel electrode for the purpose of determining 
pH's by means of equation (2) is 0.3358 volt * at 25. In view of possible 
variations in the salt bridge from one set of experiments to another, it is 
preferable to utilize these potentials to determine the pll values of a 
number of reproducible buffer solutions (cf. p. 410) which can form a 

i Hitchcock and Taylor, J. Am. Chem. Soc., 59, 1812 (1937); 60, 2710 (1938); 
Maclnnes, Belcher and Shedlovsky, ibid., 60, 1094 (1938); see also, Cohn, Heyroth 
and Menkin, ibid., 50, 696 (1928). 

* This may be compared with 0.3338 volt, given on page 232, employed in earlier 
pH work. 



350 THE DETERMINATION OF HYDROGEN IONS 

scale of reference. The results obtained in this manner are recorded in 
Table LXII for temperatures of 25 and 38; they are probably correct 

TABLE LXII. STANDARDIZATION OF pH VALUES OF REFERENCE SOLUTIONS 

Solution 25 38 

O.lNHCl 1 .085 1.082 

0.1 M Potassium totroxalate 1.480 1.495 

0.01 N HC1 and 0.09 N KC1 2.075 2.075 

0.05 M Potassium and phthalate 4.005 4.020 

0.1 N Acetic acid and 0.1 N Sodium acetate 4.640 4.650 

0.025 M KH 2 PO 4 and 0.025 M Na 2 HPO 4 6.855 6.835 

0.05 M Na 2 B 4 O 7 10H 2 O 9.180 9.070 

to db 0.01 pH unit. With this series of reference solutions it is possible 
to standardize a convenient combination of hydrogen and reference elec- 
trodes; the required pH of any solution may thus be determined. The 
pH's obtained in this way arc such that if inserted in equation (3), they 
will give a pA' value which should not differ greatly from one obtained 
by a completely thermodynamic procedure. Those pi I values can then 
be used in connection with equations (29) and (34) of Chap. IX to give 
reasonably accurate dissociation constants. 

Reversible Hydrogen Electrodes. In previous references to the hy- 
drogen electrode it has been stated briefly that it consists of a platinum 
electrode in contact with hydrogen gas; the details of the construction 
of this electrode will be considered here. In addition to the hydrogen 
gas electrode, a number of other electrodes are known which behave 
reversibly with respect to hydrogen ions. Any one of these can be used 
for the determination of pil values, although the electrode involving 
hydrogen gas at 1 atm. pressure is the standard to which others are 
referred. 

I. The Hydrogen Gas Electrode. The hydrogen gas electrode con- 
sists of a small platinum sheet or wire coated with finely divided platinum 
black by electrolysis of a solution of chloroplatinic acid containing a 
small proportion of lead acetate (cf. p. 35). The platinum foil or wire, 
attached to a suitable connecting wire, is inserted in the experimental 
solution through which a stream of hydrogen is passed at atmospheric 
pressure. The position of the electrode in the solution is arranged so 
that it is partly in the solution and partly in tho atmosphere of hydrogen 
gas. A number of forms of electrode vessel, suitable for a variety of 
uses, have been employed for the purpose of setting up hydrogen gas 
electrodes; some of these are depicted in Fig. 94. A simple and con- 
venient type of hydrogen electrode is that, usually associated with the 
name of Hildebrand, 2 shown in Fig. 95; a rectangular sheet of platinum, 

*Hildebrand, J. Am. Chem. Soc., 35, 847 (1913); for further details concerning 
hydrogen electrodes, see Clark, "The Determination of Hydrogen Ions/ 1 1928; Britton, 
"Hydrogen Ions," 1932; Glasstone, "The Electrochemistry of Solutions," 1937, p. 375. 
See also, Hamer and Acree, J. Res. Nat. Bur. Standards, 23, 647 (1939). 



THE HYDROGEN GAS ELECTRODE 



351 



of about 1 to 3 sq. cm. exposed area, which is subsequently platinized, is 
welded to a short length of platinum wire sealed into a glass tube con- 
taining mercury. This tube is sealed into another, closed at the top, 
but widening out into a bell shape in the region surrounding the platinum 



Hydrogen 




Hydrogen 



Hydrogen 



Fio. 94. Forms of hydrogen electrode 



FIG. 95. Hydrogen electrode: 
Hildebrand type 



shoot; a sido connection is provided for the inlet of hydrogen gas. A 
number of holes, or slits, are mado in. the boll-shaped portion of the tube 
at a level midway up the platinum, so that when the electrode is inserted 
in a solution and hydrogen passed in through the side-tube the platinum 
shoot is half immersed in liquid and half surrounded by gas. This 
arrangement permits the rapid attainment of equilibrium between the 
electrode material, the hydrogen gas and the solution. The time taken 
to reach this state of equilibrium depends, among other factors, on tho 
nature of the solution, the thickness of the deposit, and on the pre\5ous 
history of the electrode. As a general rule, an electrode that is func- 
tioning in a satisfactory manner will give a steady potential within five 
or ten minutes of commencing the passage of hydrogen. The use of a 
platinum shoot in the Ilildebrand electrode is not essential, and many 
workers prefer to use a simple wire of 2 or 3 cm. in length, straight or 
coiled, for such an electrode attains equilibrium rapidly, although it has 
a somewhat higher resistance than the form represented in Fig. 95. The 
hydrogen gas should be purified by bubbling it through alkaline per- 



352 THE DETERMINATION OF HYDROGEN IONS 

manganate and alkaline pyrogallol solutions to remove oxygen and other 
impurities which may influence the functioning of the hydrogen electrode. 
Whatever form of electrode vessel is employed, the fundamental 
principle of the operation is always the same. The hydrogen gas is 
adsorbed by the finely divided platinum and this permits the rapid 
establishment of equilibrium between molecular hydrogen on the one 
hand, and hydrogen ions in solution and electrons, on the other hand, 
thus 

}H,fo) ^ JH,(Pt) + H 2 ^ H 3 0+ + . 

This equilibrium can be attained rapidly from either direction, and so 
the electrode behaves as one that is reversible with respect to hydrogen 
ions. 

The hydrogen gas electrode behaves erratically in the presence of 
arsenic, mercury and sulfur compounds, which are known to be catalytic 
poisons; they probably function by being preferentially adsorbed on the 
platinum, thus preventing the establishment of equilibrium. An elec- 
trode whose operation is affected in this manner is said to be "poisoned"; 
if it cannot be regenerated by heating with concentrated hydrochloric 
acid, the platinum black should be removed by means of aqua regia and 
the electrode should be roplatinized. The hydrogen gas electrode cannot 
be employed in solutions containing oxidizing agents, such as nitrates, 
chlorates, permanganates and ferric salts, or other substances capable of 
reduction, e.g., unsaturated and other reducible organic compounds, 
alkaloids, etc. The electrode does not function in a satisfactory manner 
in solutions containing noble metals, e.g., gold, silver and mercury, since 
they tend to be replaced by hydrogen (cf. p. 253), neither can it be used 
in the presence of lead, cadmium and thallous salts. In spite of these 
limitations the hydrogen gas electrode has been extensively employed 
for precise measurements in cells with or without liquid junction, such 
as those mentioned in Chaps. VI and IX. The electrode has also been 
found to give fairly satisfactory results iii non-aqueous solvents such as 
alcohols, acetone, benzene and liquid ammonia. 

Since the standard state of hydrogen is the gas at 760 mm. pressure, 
it would be desirable to employ the gas at this pressure; even if the 
hydrogen were actually passed in at this pressure, which would not be 
easy to arrange, the partial pressure in the electrode vessel would be 
somewhat less because of the vapor pressure of the water. A correction 
for the pressure difference should therefore be made in accordance with 
equation (50) of Chap. VI; the correction is, however, small as is shown 
by the values calculated from this equation and recorded in Table LXIII. 
The results are given for a series of temperatures and for three gas 
pressures; the corrections are those which must be added, or subtracted 
if marked by a negative sign, to give the potential of the electrode with 
hydrogen gas at a partial pressure of 760 mm. 



THE OXYGEN ELECTRODE 353 

TABLE LXIII. PRESSURE CORRECTIONS FOR HYDROGEN ELECTRODE IN MILLIVOLTS 

Temperature 15 20 25 30 

Vapor Pressure 12.8 15.5 23.7 31.7mm. 
Gas Pressure 

740mm. 0.54 0.61 0.75 0.92 

760mm. 0.20 0.26 0.38 0.56 

780mm. -0.13 -0.08 0.04 0.20 

II. The Oxygen Electrode. The potential of an oxygen electrode, 
expressed in the form of equation (96) of Chap. VII, is 

r>m 

E = #o 2 ,oH- + -y ^ aom (5) 

and since OOH~ may be replaced by K u ,/au+, where K w is the ionic product 
of water, it follows that 

71 rn 

E = #o 2 ,n+- ylnem*. (6) 

The oxygen electrode should thus, in theory, function as if it were re- 
versible with respect to hydrogen ions. 

Attempts have been made to set up oxygen electrodes in a manner 
similar to that adopted for the hydrogen gas electrode, as described 
above; the results, however, have been found to be unreliable. The 
potential rises rapidly at first but this is followed by a drift lasting several 
days. The value reached finally is lower than that expected from the 
calculated standard potential of oxygen (cf. p. 243) and the known pH 
of the solution. The use of either iridium or smooth platinum instead of 
platinized platinum does not bring the potential appreciably nearer the 
theoretical reversible value, although the use of platinized gold has been 
recommended. It is evident that the oxygen gas electrode in its usual 
form does not function reversibly; the difference of potential when the 
equilibrium 

K> 2 + H 2 O + 2 ^ 2OH- 

is attained is less than would be expected, and this means that the direct 
reaction, as represented by this equation, is retarded in some manner not 
yet clearly understood. 

In spite of its irreversibility, the oxygen electrode was at one time 
used for the approximate comparison of pH values in solutions containing 
oxidizing substances, in which the hydrogen gas electrode would not 
function satisfactorily. In order for the results to have any significance 
the particular oxygen electrode employed was standardized by means of 
a hydrogen electrode in a solution in which the latter could be employed. 
The oxygen electrode, with air as the source of oxygen, has also been 
used for potentiometric titration purposes; in work of this kind the actual 
potential or pH is immaterial, for all that is required is an indication of 



354 THE DETERMINATION OF HYDROGEN IONS 

the point at which the potential undergoes rapid change. 3 In recent 
years the difficulty of measuring pH's in solutions containing reducible 
substances has been largely overcome by the wide adoption of the glass 
electrode which is described below. 

HI. The Quinhydrone Electrode. It was seen in Chap. VIII that a 
mixture of quinone (Q) and hydroquinonc (1I 2 Q) in the presence of 
hydrogen ions constitutes a reversible oxidation-reduction system, and 
the potential of such a system is given by equation (4), page 270, as 

^lna H *. (7) 

r 

It is seen, therefore, that the potential of the quirione-hydroquinone 
system depends on the hydrogen ion activity of the system. For the 
purpose of pH determination the solution is saturated with quinhydrone, 
which is an cquimolecular compound of quinone and hydroquinone; in 
this manner the ratio of the concentrations CQ to CH Z Q is maintained at 
unity, and if the ionic strength of the solution is relatively low the ratio 
of the activities, i.e., aq/aH 2 Q, may be regarded as constant. The first two 
terms on the right-hand side of equation (7) may thus be combined to 
give 

RT 
E = E% - -- In a l{ - (8) 

r 

RT 

= E Q - 2.303 ~v log an- (8a) 

r 

RT 

= E Q Q + 2.303 -TT PH- (86) 

r 

By using the method of standardization described at the beginning of 
this chapter, the value of EQ is found at to be 

E Q Q = - 0.6994 + 0.00074 (t - 25). 

This method of expressing the results is of little value for practical pur- 
poses; the particular reference electrode and salt bridge employed should 
be standardized by means of equation (2) using one of the reference 
solutions in Table LXII. If the reference electrode is a calomel electrode 
with 0.1 N potassium chloride, and a bridge of a saturated solution of this 
electrolyte is employed, it has been found possible to express the experi- 
mental data by means of the equation 

#Q<cai.) = - 0.363(5 + 0.0070(J - 25). 
This is the potential of the quinhydrone electrode against the Hg, Hg 2 Cl 2 , 

8 Furman, J. Am. Chem. Soc., 44, 12 (1922); Trans. Electrochem. tfoc., 43, 79 (1923); 
Britton, /. Chem. Soc., 127, 1896, 2148 (1925); Richards, J. Phys. Chem., 32, 990 (1928). 



THE QUINHYDRONE ELECTRODE 355 

KC1(0.1 N) ejectrode when the former contains a solution of hydrogen 
ions of unit activity, i.e., its pH is zero. 4 

The quinhydrone electrode is easily set up by adding a small quantity 
of the sparingly soluble quinhydrone, which can be obtained commer- 
cially, to the experimental solution so as to saturate it; this solution is 
shaken gently and then an indicating electrode of platinum or gold is 
inserted. The surface of the electrode metal should be clean and free 
from grease; it is first treated with hot chromic acid mixture, washed 
well with distilled water, and finally dried by heating in an alcohol flame. 
Gentle agitation of the solution by means of a stream of nitrogen gas is 
sometimes advantageous. The electrode gives accurate results in solu- 
tions of pH less than 8; in more alkaline solutions errors arise, first, 
because of oxidation of the hydroquinone by oxygen of the air, and 
second, on account of the ionization of the hydroquinone as an acid 
(rf. p. 291). Oxidizing or reducing agents capable of reacting rapidly 
with quinone or hydroquinone are liable to disturb the normal ratio of 
these su bstances, and so will affect the potential. The quinhydrone 
electrode can bo used in the presence of the ions of many metals which 
have a deleterious effect on the hydrogen gas electrode, but ammonium 
salts exert a harmful influence. The potential of the quinhydrone elec- 
trode is affected to some extent by all salts and even by non-electrolytes; 
this "salt effect" is to be attributed to the varying influence of the salts, 
etc., on the activities of the quinone and hydroquinone; although the 
ratio CQ/cn 2 q remains constant, therefore, this is not necessarily true for 
aQ/aii 2 Q upon which the electrode potential actually depends. The "salt 
error" is proportional to the concentration of electrolyte, within reason- 
able limits; its value, which may be positive or negative, according to the 
nature of the "salt." is about + 0.02 to 0.05 pll unit per equiv. per 
liter of electrolyte. Provided the solution is more dilute than about 
0.1 x, the "salt error" is therefore negligible for most purposes. The 
quinhydrone electrode has an appreciable "protein error," and so cannot 
be employed to give reliable pH values in solutions containing proteins 
or certain of thoir degradation products. 6 

The quinhydrone electrode has been adapted for pH measurements 
in non-aqueous media, such as alcohols, acetone, formic acid, benzene 
and liquid ammonia. For the determination of hydrogen ion activities 
in solutions in pure acetic acid a form of quinhydrone electrode involving 
tetrachloroquinone (chloranil) and its hydroquinone has been used. 6 

4 Harned and Wright, /. Am. Chem. Soc., 55, 4849 (1933); Hovorka and Dearing, 
ibid., 57, 446 (1935). 

5 For general references, see Glasstone, "The Electrochemistry of Solutions," 1937, 
p. 378. 

Conant et al, J. Am. Chem. Soc., 47, 1959 (1925); 49, 3047 (1927); Heston and 
Hall, ibid., 56, 1462 (1934). 



356 THE DETERMINATION OF HYDROGEN IONS 

IV. The Antimony Electrode. The so-called " antimony electrode" 
is really an electrode consisting of antimony and its trioxide, the reaction 
being 

2Sb(s) + 3H 2 O = Sb 2 O 3 (s) + 611+ + 6c, 

so that the potential is given by 

DAT! 

E = tfgb.sb^.H* - -jr In a H +, (9) 

the activities of the solid antimony and antimony trioxide, and of the 
water, being taken as unity. The potential of the Sb, Sb 2 O 3 electrode 
should thus depend on the hydrogen ion activity of the solution in which 
it is placed. The electrode is generally prepared by casting a stick of 
antimony in the presence of air; in this way it becomes sufficiently oxi- 
dized for the further addition of oxide to be unnecessary. A wire is 
attached to one end of the rod of antimony obtained in this manner, 
while the other is inserted in the experimental solution; its potential is 
then measured against a convenient reference electrode. As the poten- 
tials differ from one electrode to another, it is necessary that each anti- 
mony electrode should be standardized by means of one of the solutions 
in Table LXII. The antimony electrode behaves, at least approximately, 
according to equation (0) over the range of pll from 2 to 7, but in more 
acid or more alkaline solutions deviations occur; these 4 discrepancies are 
probably connected with the solubility of the antimony oxide in such 
solutions. Since no special technique is required for setting up or meas- 
uring the potential of the antimony electrode, and it is not easily poisoned, 
it has advantages over other forms of hydrogen electrode. It is, there- 
fore, very convenient where approximate results are adequate, but it is 
not recommended for precision work. 7 

V. The Glass Electrode. One of the most important advances of 
recent years in connection with the determination of pll's is the develop- 
ment wm-h has taken place in the use of the glass electrode. It has long 
been known that a potential difference is set up at the interface between 
glass arid a solution in contact with it which is dependent on the pll of 
the latter; * this dependence has been found to correspond to the familiar 
equation for a reversible hydrogen electrode, viz., 

IJrn 

E - /ft - -jjrlnaii*, (">) 

7 Kolthoff arid Hartong, Rec. trav. chirn., 44, 113 (1925); Roberta and Fenwirk, 
J. Am. Chem. floe., 50, 2125 (1928); Parka and Hoard, ibid., 54, 850 (1932); Pcrlcy, 
Ind. Eng. Chem. (Anal. Kd.), 11, 316 (1939); Hovorka and Chapman, ,/. Am. Chem. fior., 
63, 955 (1941). 

8 For references to experimental methods, see GlaHstono, Ann. Rep. ('hem. S'oc., 30, 
283 (1933); Muller and Diirichen, Z. Elektrochem., 41, 559 (1935); 42, 31, 730 (1936); 
Schwabe, ibid., 41, 681 (1935). For complete review, see Dole, "The Glass Electrode," 
1941. 



THE GLASS ELECTRODE 



357 




where $?> is the " standard potential " for the particular glass employed, 
i.e., the potential when in contact with a solution of hydrogen ions at unit 
activity. It is evident, therefore, that measurements of the potential 
of the so-called "glass electrode" can be utilized for the determination of 
pH values. 

In its simplest form the glass electrode consists of a tube terminating 
in a thin-walled bulb, as shown at A, in Fig. 96; the glass most suitable 
for the purpose (Corning 015) 
contains about 72 per cent SiO 2 , 
22 per cent Na^O and 6 per cent 
CaO; it has a relatively low melt- 
ing point and a high electrical con- 
ductivity. The bulb contains a 
solution of constant hydrogen ion 
concentration and an electrode of 
definite potential; a silver chloride 
electrode in 0.1 N hydrochloric 
acid or a platinum wire inserted 
in a buffer solution, e.g., 0.05 
molar potassium acid phthalate, FIG. 96. Glass electrode cell 

saturated with quinhydrone, is 

generally used. The bulb is inserted in the experimental solution (B) 
so that the glass electrode consists of the system 

Ag | AgCl(s) 0.1 N HC1 1 glass | experimental solution, 

if silver-silver chloride is the inner electrode of constant potential. The 
potential of the glass electrode is then measured by combining it with 
a suitable reference electrode, such as the calomel electrode C in Fig. 96, 
the inner electrode of the glass electrode system serving to make elec- 
trical connection. 

Owing to the very high resistance of the glass, viz., 10 to 100 million 
ohms, special methods have to be employed for determining the E.M.F. 
of the cell; these generally involve the use of an electrometer or of vacuum- 
tube circuits, as described on page 192. Some workers have successfully 
prepared thin-walled glass electrodes of relatively large area and hence of 
comparatively low resistance; in these cases it has been found possible 
to make E.M.F. measurements without special apparatus, by using a 
reasonably sensitive galvanometer as the indicating instrument in the 
potentiometer circuit. Various forms of glass electrode have been em- 
ployed for different purposes, but the simple bulb type described above 
can easily be made in a form that is not too fragile and yet has not a 
veiy high resistance. Several commercial forms of apparatus are now 
available which employ robust glass electrodes; by using some form of 
electrometer triode vacuum tube (p. 193), it is possible to measure the 
potential to about 0.0005 volt, i.e., 0.01 pH unit, without difficulty. An 
accuracy of rb 0.002 pH unit has been claimed for special measuring 



358 THE DETERMINATION OF HYDROGEN IONS 

circuits, but it is doubtful whether the pH scale has been established with 
this degree of precision. 

If both internal and external surfaces of the glass electrode were 
identical, it is obvious from equation (10) that the potential of the elec- 
trode system would be determined simply by the difference of pH of the 
solutions on the two sides of the glass membrane, apart from the potential 
of the inner electrode, e.g., Ag, AgCl. This expectation can be tested 
by measuring the E.M.F. of a cell in which the solution is the same inside 
and outside the glass bulb and the reference electrode is the same as the 
inner electrode; thus 

Ag | AgCl(s) 0.1 N HC1 1 glass | 0.1 N IIC1 AgCl(s) | Ag. 

The E.M.F. of this cell should be zero, but in practice the value is found 
to be of the order of 2 millivolts, for a good electrode. This small 
difference is called the asymmetry potential of the glass electrode; it is 
probably due to differences in the strain of the inner and outer surfaces 
of the glass membrane. It is necessary, therefore, to standardize each 
glass electrode by means of a series of buffer solutions of known pH; in 
this way the value of JQ in equation (10) for the particular electrode is 
found. 

Before use the glass electrode should be allowed to soak in water for 
some time, following its preparation, and should not be allowed to become 
dry subsequently; if treated in this manner equilibrium with the solution 
in which it is placed is attained rapidly. The potential satisfies equation 
(10) for a reversible hydrogen electrode very closely in the pi I range of 
1 to 9, and with fair accuracy up to pH 12,* provided there is no large 
concentration of salts in the solution. At pll's greater than 9 appreciable 
salt effects become evident which increase with increasing pH, i.e., 
increasing alkalinity, of the solution; the magnitude of the salt effects in 
such solutions depends primarily on the nature of thr cations present, 
but it is of the order of 0.1 to 0.2 unit in the vicinity of pll 11 for 0.1 to 
1 N solutions of the salt. In very acid solutions, of pi I less than unity, 
other salt effects, determined mainly by the unions, are observed. Apart 
from these limitations, the glass electrode has the outstanding advantage 
that it can be employed in aqueous solutions of almost any kind; the 
electrode cannot be poisoned, neither is it affected by oxidizing or re- 
ducing substances or by organic compounds. It can be used in un- 
buffered solutions and can be adapted for measurements with very small 
quantities of liquid. The glass electrode does not function satisfactorily 
in pure ethyl alcohol or in acetic arid, but it has been employed in mix- 
tures of these substances with water. 9 

* The accuracy may be improved by the use of a special glass now available. 

9 Hughes, J. Chun. /S'or., 401 (1928); Machines and Dole, J. Aw. Chem. tfoe., 52, 29 
(1930); Maclnnes and Belcher, ibid , 53, 3315 (1931); Dole, ibid., 53, 4260 (1931); 
54, 3095 (1932); for reviews with references, see Schwabe, Z. Elektrochem., 41, 681 
(1935); Kratz, ibid., 46, 259 (1940). 



ACID-BABE INDICATORS 359 

There is no completely satisfactory explanation of why a glass elec- 
trode functions as a reversible hydrogen electrode, but it is probable that 
the hydrogen ions in the solution exchange, to some extent, with the 
sodium ions on the surface of the glass membrane. The result is that a 
potential, similar to a liquid junction potential, is set up at each surface 
of the glass; if no ions other than hydrogen ions, and their associated 
water molecules, are able to enter the glass, the free energy change accom- 
panying the transfer of 1 g.-ion of hydrogen ion from the solution on one 
side of the membrane, where the activity is an + , to the other side, where 
the activity is aii% is then 



A(? = RTln 



where x is the number of molecules of water associated with each hydro- 
gen ion in the transfer; an 2 o and a!i 2 o are the activities of the water in 
the two solutions. The potential across the glass membrane is conse- 
quently given by 

EG = -TT In -777 + TT~ In ~Tf- - (11) 

/' a H + r 



If the solutions are sufficiently dilute, the activities of the water are the 
same on both sides of the membrane; the second term on the right-hand 
side of equation (11) then becomes zero. By retaining the hydrogen 
ion activity, e.g., ali + , constant on one side of the membrane, equation 
(11) reduces to the same form as (10). If the activity of the water is 
altered by the addition of alcohol or of appreciable amounts of salts or 
acids, equation (10) is no longer applicable, and deviations from the ideal 
reversible behavior of the glass electrode are observed. The salt errors 
found in relatively alkaline solutions, of pll greater than 9, are probably 
due to the fact that at these low hydrogen ion concentrations other 
cations present in the solution are transferred across the glass membrane 
to some extent. Under these conditions equation (11) is no longer valid, 
and so the glass electrode cannot behave in accordance with the require- 
ments of equation (10). 10 

Acid-Base Indicators. An arid-base indicator is a substance, which, 
within certain limits, varies in color according to the hydrogen ion con- 
centration, or activity, of its environment; it is thus possible to determine 
the pll of a solution by observing the color of a suitable indicator when 
placed in that solution. Investigation into the chemistry of substances 
which function as acid-base indicators has shown that they are capable 
of existing in two or more tautomeric forms having different structures 
and different colors. In one or other of these; forms the molecule is 
capable of functioning as a weak acid or base, and it is this property, 

10 Dole, J. Am. Chcm. flor , 53, 4260 (1930); 54, 2120, 3095 (1932); "Experimental 
and Theoretical Electrochemistry/' 1935, Chap. XXV; "The Glass Electrode/ 7 1941; 
Haugaard, J. Phys. Chem., 45, 148 (1941). 



360 THE DETERMINATION OF HYDROGEN IONS 

together with the difference in color of the tautomoric states, that permits 
the use of the given compound as an acid-base indicator. 11 

If HIni represents the un-ionized, colorless form of an indicator that 
is acidic in character, its ionization will be represented by 



II 2 O ^ H 3 + + In?, 
colorless colorless 

the anion Inr having the same structure and color as the molecule HIni. 
Application of the law of mass action to this equilibrium gives the dis- 
sociation constant of the acid as 

A',-^. (12) 



The colorless ion Ini will be in equilibrium with its tautomeric form Iii2 , 
thus 

In? ^ Inif, 
colorless colored 

but the latter, having a different structure from that of Inr, will have a 
different color, and the constant of the tautomeric equilibrium (K t ) will 
be given by 

Kt = ^T-' (13) 

Finally, the colored In^ ions will be in equilibrium with hydrogen ions 
and the colored un-ionized molecules HIn 2 , thus 

HIn 2 + IW) ^ H 3 O+ + Injf; 
colored colored 

the dissociation constant of the acid HIn 2 is then 

K 2 = ^^- (14) 

Combination of equations (12), (1.3) and (14) gives 

^^ n +^f)~ = "It V KiJrT = Kln ' (15) 

where K\ n is a composite constant involving A'i, K 2 and K t \ it follows, 
therefore, from equation (15) that 



mf + am.;) 



MO) 

v ' 



If the ionic strength of the medium is relatively low, the activities of 
HIni, Hln 2 , InT and In-J may be replaced by their respective concen- 

11 For a full discussion of the properties of indicators, sec Kolthoff and Kosenbhun, 
" Acid-Base Indicators," 1937. 



ACID-BASE INDICATORS 361 

trations, so that equation (16) becomes 

7 Clllrii + CHIn- 

an* = tin - . - - > (17) 

Clni ~T Ci u ~ 

where the approximate " constant " ki tl , known as the indicator constant, 
replaces K\ n . 

If a particular compound is to be satisfactory as an acid-base or pH 
indicator, the numerator and denominator in equation (17) must corre- 
spond to two distinct colors: a change in the hydrogen ion activity must 
clearly be accompanied by an alteration in the ratio of numerator to 
denominator, and unless these represent two markedly different colors 
the system as a whole will undergo no noticeable change of color. Since 
HIni and HIn 2 have different colors, on the one hand, and Inr and In^ 
are also different, but the same as HIni and HIn 2 , respectively, it is 
evident that in order to satisfy the condition given above it is necessary 
that the un-ionized molecules must be almost completely in the form 
HIni, or HIn 2 , and the ions must be almost exclusively in the other form. 
It follows from equation (13) that if the tautomeric constant K t is small 
the ions Inf will predominate over InjT ; further, if Ki/K* is large, so that 
HIni is a much stronger acid than HIn 2 , it follows that the un-ionized 
molecules HIn 2 will greatly exceed those of HIni. These are, in fact, the 
conditions required to make the substance under consideration a satis- 
factory indicator. An alternative possibility which is equally satisfactory 
is that K t should be large while Ki/K* is small; the ionized form will then 
consist mainly of Injf while the un-ionized molecules will be chiefly in the 
HIni form. For a satisfactory indicator, therefore, equation (17) may 
be written as 

Un-ionized form 
a * = *" Ionized form (18) 



where a is the fraction of the total indicator present in the ionized form. 
The actual color exhibited by the indicator will, of course, depend on the 
ratio of the un-ionized to the ionized form, since those have different 
colors; hence it follows from equation (18) that it will be directly related 
to the hydrogen ion activity, or concentration, of the medium. In an 
acid solution, i.e., a H f is high, the concentration of un-ionized form must 
increase, according to equation (18), and the indicator will exhibit the 
color associated with the main Hln form; in an alkaline medium, on the 
other hand, the ionized form must predominate and the color will be that 
of the chief In~ species. 

A few indicators are bases in the state in which they are normally 
employed; an example is methyl orange, which is the sodium salt of 
p-dimethylaminoazobenzcne sulfouic acid, the indicator action being due 



362 THE DETERMINATION OP HYDROGEN IONS 

to the basic dimethylamino-group, i.e., N(CH 3 ) 2 . There is no reason, 
however, why the conjugate acid, viz., NH(CH 3 )j!~, should not be con- 
sidered as the indicator, although this is not the form in which it is usually 
supplied. In view of the fact that the properties of aqueous solutions 
are invariably expressed in terms of pH, and not of pOH, it is convenient 
to treat all indicators as acids. If the indicator in its familiar form 
happens to be a base, then the system is treated as if it consisted of its 
conjugate acid. All indicator systems, of course, consist of conjugate 
acid and base, e.g., HIn and In~, and it is in a sense somewhat arbitrary 
to refer to certain indicators as acids and to others as bases. The par- 
ticular term employed refers to the nature of the substance in the form 
in which it is usually encountered; methyl orange is generally employed 
as the sodium salt of the sulfonic acid of the free base, and hence it is 
called a basic indicator; but if it were used as the hydrochloride, or other 
salt, of the base, it would be called an acid indicator. In the subsequent 
treatment all indicators will for simplicity and uniformity be treated 
as acids. 

Indicator Range. If, as on page 287, it is assumed that the color of 
the ionized form In~~ is barely visible when 9 per cent of the total indi- 
cator is in this form, i.e., when a is 0.09, it follows from equation (19) 
that the limiting hydrogen ion activity at which the indicator will show 
its acid color, due to HIn, will be given by 

7 - 91 tm. 
OH* = kin 0-gg 10/bin, 

/. pH pki n - 1, (20) 

where pki n is the indicator exponent, denned in the usual manner as 
log ki n . On the other hand when 91 per cent of the indicator is in the 
ionized form, i.e., a. is 0.91, the color of the un-ionized form will be 
virtually undetcetable in the mixture, and so the color will be that of the 
alkaline form; the pH at which the indicator shows its full alkaline color 
is then obtained from equation (19), thus 

i, - 09 1 i 

a * + = fcln o79i ~ To /Cln ' 

/. P H l>km + 1. (21) 

It is seen, therefore, that as the pH of a solution is increased by the 
addition of alkali, the color of an indicator begins to change visibly at a 
pH approximately equal to pki n 1, and is completely changed, as far 
as the eye can detect, at a pH of about pki n + 1. The effective transi- 
tion interval of an indicator is thus very roughly two pll units, one on 
each side of the pH equal to pki n of the indicator. Since various indi- 
cators have different values of fri,,, the range of pH over which the color 
changes will vary from one indicator to another. 



DETERMINATION OF INDICATOR CONSTANTS 363 

When the indicator is ionized to an extent of 50 per cent, i.e., a is 0.5, 
it is seen from equation (19) that 

an+ = kin, 
.'. pH = pfcm. (22) 

The indicator will thus consist of equal amounts of the ionized and un- 
ionized forms, arid hence will show its exact intermediate color, when 
the hydrogen ion activity, or concentration, is equal to the indicator 
constant. 

Determination of Indicator Constants. A simple method of evalu- 
ating the constant of an indicator is to make use of equation (22). Two 
solutions, containing the same amount of indicator, one in the completely 
acid form and the other in the alkaline form, are superimposed; the net 
color is equivalent to that of the total amount of indicator with equal 
portions in the ionized and un-ionized forms. A series of buffer solutions 
of known pll (see Chap. XI) are then prepared and a quantity of indi- 
cator, twice that present in each of the two superimposed solutions, is 
added; the colors are then compared with that of the latter until a match 
is obtained. The matching buffer solution consequently contains equal 
amounts of ionized and un-ionized indicator and so its pH is equal to the 
required pki n . 

The general procedure is to utilize equation (19) and to determine 
the proportion of un-ionized to ionized form of the indicator in a solution 
of known pH; the most accurate method is to measure this ratio by a 
spcotrophotometric method similar to that described on page 328. If 
the substance is a one color indicator, that is to say it is colored in one 
(ionized) form and colorless in the other (un-ionized) form, e.g., phenol- 
phthalein and p-nitrophenol, it generally has one sharp absorption band 
in the visible spectrum; by measuring the extinction coefficient when the 
substance is completely in its colored form, e.g., in alkaline solution, it is 
possible, by utilizing Boer's law (cf. p. 328, footnote), to determine the 
concentration of colored form in any solution of known pH from the 
extent of light absorption by the indicator in that solution (cf. Fig. 100). 
From the total amount of indicator present, the ratio (1 a) /a can be 
evaluated and hence ki n can be obtained. The principle of this method 
of determining the indicator constant is identical with that described 
on pago 329 for the dissociation constant of an acid; A*i n is in fact the 
apparent dissociation constant of the indicator, assuming it to consist 
of a single un-ionized form II In and an ionized form In~ with a different 
color. 

A two color indicator will, in general, have two absorption bands, 
one for each colored form; by studying the extent of absorption in these 
bands in a solution of definite pll, as compared with the values in a com- 
pletely acid and a completely alkaline solution, it is possible to calculate 
directly the ratio of the amounts of un-ionized and ionized forms in the 
given solution. 



364 



THE DETERMINATION OF HYDROGEN IONS 



Instead of utilizing spectrophotometric devices, the ratio of the 
amounts of ionized to un-ionized indicator can be estimated, although 
less accurately, by visual means. With a one color indicator the fraction 
of ionized, generally colored, form is determined by comparing the color 
intensity with that of a solution containing various known amounts of 
indicator which have been completely transformed by the addition of 
alkali. With a two-color indicator it is necessary to superimpose the 
acid and alkaline colors in different amounts until a match is obtained. 
The precision of the measurements can be greatly improved by the use 
of a commercial form of colorimeter specially designed for the matching 
of colors. 

The values of pki n for a number of useful indicators, together with 
the pH ranges in which they can be employed and their characteristic 
colors in acid and alkaline solutions, are recorded in Table LXIV. 

TABLE LXIV. USEFUL INDICATORS AND THEIR CHARACTERISTIC PROPERTIES 



Indicator pki n pH Range 

Thymol blue 1.51 1.2- 2.8 

Methyl orange 3.7 3.1-4.4 

Bromphenol blue 3.98 3.0- 4.6 

Bromcresol green 4.67 3.8- 5.4 

Methyl red 5.1 4.2- 6.3 

Chlorphenol red 5.98 4.8- 6.4 

Bromphenol red 6.16 5.2- 6.8 

Bromcresol purple 6.3 5.2- 6.8 

Bromthymol blue 7.0 6.0- 7.6 

p-Nitrophenol 7.1 5.6- 7.6 

Phenol red 7.9 6.8- 8.4 

Cresol red 8.3 7.2- 8.8 

Metacresol purple 8.32 7.4- 9.0 

Thymol blue 8.9 8.0- 9.6 

Cresolphthalein 9.4 8.2- 9.8 

Phenolphthalein 9.4 8.3-10.0 

Thymolphthalein 9.4 9.2-10.6 

Alizarine yellow - 10.0-12.0 

Nitramine 11.0-13,0 



Color Change 
Acid Alkaline 



Red 

Red 

Yellow 

Yellow 

Red 

Yellow 

Yellow 

Yellow 

Yellow 

Colorless 

Yellow 

Yellow 

Yellow 

Yellow 

Colorless 

Colorless 

Colorless 

Yellow 

Colorless 



Yellow 

Yellow 

Blue 

Blue 

Yellow 

Red 

Red 

Purple 

Blue 

Yellow 

Red 

Red 

Purple 

Blue 

Red 

Red 

Blue 

Lilac 

Orange-brown 



Determination of pH: With Buffer Solutions. If a series of buffer 
solutions of known pH, which must lie in the region of the pH to be 
determined, is available the estimation of the unknown pH is a relatively 
simple matter. It is first necessary to choose, by preliminary experi- 
ments, an indicator that exhibits a definite intermediate color in the 
solution under examination. The color produced is then compared with 
that given by the same amount of the indicator in the various solutions 
of known pH. In the absence of a "salt error," to which reference will 
be made later, the pH of the unknown solution will be the same as that 
of the buffer solution in which the indicator exhibits the same color. 
Provided a sufficient number of solutions of known pH are available, this 
method can give results which are correct to about 0.05 pH unit. 



BJERRUM'S WEDGE METHOD 



365 








When colored solutions are being studied, allowance must be made 
for the superimposition of the color on to that of the indicator; this may 
be done by means of the arrangement shown in 
plan in Fig. 97. The colored experimental solu- 
tion, to which a definite amount of indicator has 
been added, is placed in the tube A and pure 
water is placed in B\ the tube C contains the test 
solution without indicator, and D contains the 
buffer solution of known pH together with the 
same amount of indicator as in A. The solution 
in D is varied until the color of C and D super- 
imposed is the same as that of A and B super- 
imposed. The pH of the solution in A is then the 
same as that in D. 

Determination of pH: Without Buffer Solutions. Provided the con- 
stant of an indicator is known, it is possible to determine the pH of an 
unknown solution without the use of buffer solutions; the methods are 
the same in principle as those employed for the evaluation of the indi- 
cator constant, except that in one case the pH of the solution is supposed 
to be known while pfcj n is determined and in the other the reverse is true. 
For this purpose, equation (18), after taking logarithms, may be written as 

TT , , . Ionized form 
pH = pfci n + log 



FIG. 97. Indicator 
measurements with 
colored solutions 



+ 



Un-ionized form 

Color due to alkaline form 



(23) 



i.o 



Alkaline form (Ia) 

0.75 0.60 0.25 
[ l I 



Color due to acid form 

The problem of determining pH values thus reduces to that of measuring 
the ratio of the two extreme colors exhibited by a particular indicator in 
the given solution. 

L Bjerrum's Wedge Method. 12 A rectangular glass box is divided 
into two wedge-shaped compartments by the insertion of a sheet of glass 

diagonally, or two separate wedges are 
cemented together by Canada balsam to 
give a vessel of the form shown in pluri 
in Fig. 98. A solution of the indicator 
which has been made definitely acid is 
placed in one wedge, and one that is 
definitely alkaline is placed in the other. 
By viewing the combination from the 
front a gradation of colors, from the acid 
to the alkaline forms of the indicator, 
can be seen as a result of the superposi- 
tion of steadily decreasing amounts of acid color on increasing amounts 

"Hjerrum, Ahren's SammlunK, 1914, No. 21; Kolthoff, Rec. trav. chim., 43, 144 
(1924); McCrae, Analyst, 51, 287 (1926). 



!5 0,50 
Acid form 



T 
0.75 



0- 



1.0 



Fia. 98. Representation of 
Bjerrum wedge 



366 



THE DETERMINATION OP HYDROGEN IONS 



B 



* 2 



of the alkaline color. The test solution is placed in a narrow glass 
box of the same thickness as the combined wedges (Fig. 98, A) and the 
indicator is added so that its concentration is the same as in the 
wedges. A position is then found at which the color of the test solution 
matches that of the superimposed acid and alkaline colors; the ratio of 
the depths of the wedge solutions at this point thus gives the ratio of the 
colors required for equation (23). If the sides of the box are graduated, 
as shown, the depths of the two solutions can be obtained and the corre- 
sponding pH evaluated. The double-wedge can of course be calibrated 
so that the logarithmic term, i.e., the second term on the right-hand side, 
of equation (23) can be read off directly. 

II. Colorimeter Method. One of the simplest forms of colorimeter 
is shown in Fig. 99; the experimental solution is placed in the vessel A 
and an amount of indicator, giving a known concentration, is added; the 

fixed flat-bottomed tube B contains ^ater 
to a definite height. The fixed tube 0, ar- 
ranged at the same level as B, also contains 
water to the same height as in B. Surround- 
ing C is a movable tube D in which is placed 
the acid form of the indicator, and this is 
surrounded by the vessel E containing the in- 
dicator in its alkaline form; the concentra- 
tion of the indicator in D and E is the same 
as in the test solution in A. The inner tube 
D is moved up and down until the color as 
seen through (7, D and E is the same as that 

seen through B and A ; the ratio of the alkaline color to the acid color in 
A is then given by the ratio of the heights Ai/fe, so that the pll can be 
calculated if these heights are measured. If the test solution is colored, 
the water in C is replaced by the test solution to an equal depth; its 
color is then superimposed on that of the indicator in each case. By 
the use of special colorimeters it is possible to match the colors with 
such precision that pll values can be estimated with an accuracy of 
0.01 unit. 

HI. Spectrophotometric Method. 13 The use of absorption spectra 
permits an accurate estimate to be made of the ratio of the amounts of 
the two colors in a given solution; the method is the same in principle 
as that already referred to on pages 328 and 363. In order to show the 
magnitude of the effect on the absorption of light resulting from a change 
of pH, the transmission curves obtained for bromcresol green in solutions 
of various pH's are shown in Fig. 100. It is evident that once the extent 
of the absorption produced by the completely alkaline form of the indi- 

" Erode, J. Am. Chem. Soc., 46, 581 (1924); Holmes, ibid., 46, 2232 (1924); Holmes 
and Snyder, ibid., 47, 221, 226 (1925); Vies, Compt. rend., 180, 584 (1925); Fortune and 
Mellon, J. Am. Chem. Soc., 60, 2607 (1938). 





C 
D 




E 



FIG. 99. Colorimeter for 
pH determinations 



ERRORS IN MEASUREMENTS WITH INDICATORS 



367 



cator is known, the proportion present in a given solution, and hence the 
pH, can be estimated with fair accuracy. 



60 




6400 A 
Fia. 100. Light absorption of bromoresol green (Fortune and Mellon) 



4800 5600 

Wave Length 



Errors in Measurements with Indicators. Three chief sources of 
error in connection with pi I determinations by means of indicators may 
be mentioned. 14 In the first place, if the test solution is not buffered, 
eg., solutions of very weak acids or bases or of neutral salts of strong 
acids and bases, the addition of the indicator may produce an appreciable 
change of pH; this source of error may be minimized by employing small 
amounts of indicator which have been previously adjusted, as a result of 
preliminary experiments, to have approximately the same pH as the test 
solution. Such indicator solutions are said to be isohydric with the test 
solution. 

The second possible cause of erroneous results is the presence of 
proteins; as a general rule, indicator methods are not satisfactory for the 
determination of pH in protein solutions. The error varies with the 
nature of the indicator; it is usually less for low molecular weight com- 
pounds than for complex molecules. 

Appreciable quantities of neutral salts produce color changes in an 
indicator that are not due to an alteration of pH and hence lead to erro- 
neous results. This effect of neutral salts is due to two factors, at least; 
in the first place, the salt may affect the light absorbing properties of one 
or both forms of the indicator; and, in the second place, the altered 
ionic strength changes the activity of the indicator species. In deriving 
equation (17) the activities of the un-ionized and ionized forms of the 
indicator were taken to be the same as the respective concentrations; 
this can only be reasonably true if the ionic strength of the solution is 

14 McCrumb and Kenny, J. floe. Chem. Ind. t 49, 425T (1930); Kolthoff and Rosen- 
blum, "Acid- Base Indicators," 1937, Chap. X. 



368 THE DETERMINATION OF HYDROGEN IONS 

low, otherwise equation (19) should be written 

/HIn 



OH + = Kin 



/In" 



where /Hin and /i n - are the activity coefficients of the un-ionized and 
ionized species, respectively. Taking logarithms, this equation can be 
put in the form 



P H - pK In + log 



and the use of the extended Debye-Hiickel equation for log (/i n -//mn) 
then gives 



P H = ptfm + log T- - A + C v . (24) 

1 a 

For a given color tint, i.e., corresponding to a definite value of a/(l a), 
the actual pH will clearly depend on the value of the ionic strength of 
the solution; at low ionic strengths, e.g., less than about 0.01, the neutral 
salt error \s negligible for most purposes. 

The actual neutral salt error is less than estimated by equation (24) 
because the experimental values of pK i n are generally based on deter- 
minations made in buffer solutions of appreciable ionic strength. For 
equation (24) to be strictly applicable the indicator exponent p/f i n should 
be the true thermodynamic value obtained by extrapolation to infinite 
dilution. 

Universal Indicators. Since the pH range over which a given indi- 
cator can be employed is limited, it is always necessary, as mentioned 
above, to carry out preliminary measurements with an unknown solution 
in order to find the approximate pH; with this information available 
the most suitable indicator can be chosen. For the purpose of making 
these preliminary observations the so-called universal indicators have 
been found useful: 1B they consist of mixtures of four or five indicators, 
suitably chosen so that they do not interfere with each other to any 
extent, which show a series of color changes over a range of pH from 
about 3 to 11. A convenient and simple form of universal indicator can 
be prepared by mixing equal volumes of 0.1 per cent solutions of methyl 
red, a-naphtholphthalein, thymolphthalein, phenolphthalein and brom- 
thymol blue; the colors at different pH values are given below. 

pH 4 5 6 78 9 10 11 

Color Red Orange- Yellow Green- Green Blue- Blue- Red- 
red yellow green violet violet 

Carr, Andy*, 47, 196 (1922); Clark, "The Determination of Hydrogen Ions," 
1928, p. 97; Britton, "Hydrogen Ions/' 1932, p. 286; Kolthoff and Rosenblum, "Acid- 
Base Indicators/' 1937, p. 170. 



PROBLEMS 369 

The addition of a small quantity of this universal indicator to an un- 
known solution permits the pH of the latter to be determined very 
approximately; it is then possible to choose the most suitable indicator 
from Table LXIV in order to make a more precise determination of the 
pH. Universal indicators are frequently employed when approximate 
pH values only are required, as, for example, in certain processes in 
qualitative and gravimetric analysis, and for industrial purposes. 

PROBLEMS 

1. What are the pH values of solutions whose hydrogen ion concentrations 
(activities) are 2.50, 4.85 X 10" 4 and 0.79 X 10~ 10 g.-ion per liter? Assuming 
complete dissociation and ideal behavior, evaluate the pH of 0.0095 N sodium 
hydroxide at 25. 

2. A hydrogen gas electrode in 0.05 molar potassium acid phthalate, when 
combined with a calomel electrode containing saturated potassium chloride, 
gave a cell with an E.M.F. of 0.4765 volt at 38. Calculate the pH of the 
solution which gave an E.M.F. of 0.7243 volt in a similar cell. 

3. The glass electrode cell 

Pt | Quinhydrone pH 4.00 Buffer | Glass | pH 7.63 Buffer Quinhydrone | Pt 

gave an E.M.F. of 0.2265 volt at 25; calculate the asymmetry potential of 
the glass electrode. 

4. What are the hydrogen ion activities of solutions of pH 13.46, 5.94 and 
0.5? What are the corresponding hydroxyl ion activities at 25, assuming 
the activity of the water to be unity in each case? 

5. A quinhydrone electrode in a solution of unknown pH was combined 
with a KC1(0.1 N), Hg2Cl 2 (s), Hg electrode through a saturated potassium 
chloride salt bridge; the E.M.F. of the resulting cell was 0.3394 volt at 30. 
Calculate the pH of the solution. 

6. If the oxygen electrode were reversible, what change of E.M.F. would be 
expected when an oxygen gas electrode at 1 atm. pressure replaced (i) a hydro- 
gen electrode at 1 atm., and (ii) a quinhydrone electrode, in a given cell at 25? 

7. An indicator is yellow in the acid form and red in its alkaline form; 
when placed in a buffer solution of pH 6.35 it was found by spectrophoto- 
metric measurements that the extent of absorption in the yellow region of the 
spectrum was 0.82 of the value in a-solution of pH 3.0. Evaluate p&i n for the 
given indicator. 



CHAPTER XI 
NEUTRALIZATION AND HYDROLYSIS 

Types of Neutralization. The term neutralization is generally ap- 
plied to the reaction of one equivalent of an acid with one equivalent of 
base; if the terms "acid" and "base" are employed in the sense of the 
general definitions given in Chap. IX, the products are not necessarily a 
salt and water, as in the classical concept of acids and bases, but they 
are the conjugate base and acid, respectively, of the reacting acid and 
base. For the reaction between conventional acids, such as hydro- 
chloric, acetic, etc., and strong bases, such as hydroxides in water or 
alkyloxides in alcohols, there is no difference between the new and the old 
points of view; it is, however, preferable to discuss all neutralizations 
from the general standpoint provided by the modern theory of acids and 
bases. According to this the following reactions are all examples of 
neutralization: 

HC1 + (Na+)OC,Hr = (Na+)Cl- + C,II 6 OH 

CHaCOjH + (Na+)OH- = (Na+)CII 3 CO^ + H 2 O 
HOI + RNIT 2 = a- + RNIIJ 

IIC1 + (Na+)CH 3 COj = (Na+)Cl- + CH 3 CO 2 H 

RNHf(Cl-) + (Na*-)OH- = RNH 2 + H 2 O + (Na+Cl-). 

Acidi Bases Basei AcicU 

The last two reactions are of special interest, since they belong to the 
category usually known as "displacement reactions"; in the first of the 
two a strong acid, hydrochloric acid, displaces a weak acid, acetic acid, 
from its salt, while in the second a weak base, e.g., ammonia or an aminn, 
is displaced from its hydrochloridc by a strong base. It will be seen 
later that a much better understanding of these processes can be obtained 
by treating them as neutralizations, which in fact they are if this term 
is used in its widor sense. 

Incomplete Neutralization: Lyolysis. The extent to which neutrali- 
zation occurs, when one equivalent of arid and base are mixed, depends 
on the nature of the acid, the base and the solvent. If the acid is HA, 
the base is B and SH Is an amphiprotic solvent, i.e., one which can 
function either as an acid or as a base, the neutralization reaction 

HA + B ^ BH* + A- 

takes place, but in addition, since the solvent is amphiprotic, two proc- 
esses involving it can occur, thus 

(a) BH+ + SH ^ Silt + B, (la) 

Acid Base Acid Base 

370 



CONDITIONS FOR COMPLETE NEUTRALIZATION 371 

and 

(6) SH + A- ^ HA + S- (16) 

Acid Base Acid Base 

In the first of these the free base B is re-formed while in the second the 
free acid HA is regenerated; it follows, therefore, that the processes 
(a) and (6) militate against complete neutralization. This partial re- 
versal of neutralization, or the prevention of complete neutralization, is 
called by the general name of lyolysis or solvolysis; in the particular case 
of water as solvent, the term used is hydrolysis. 

Conditions for Complete Neutralization. In order that neutraliza- 
tion may be virtually complete it is necessary that the lyolysis reactions 
should be reduced as far as possible. For reaction (a) to be suppressed 
it is necessary that B should be a much stronger base than the solvent 
SH, so that the equilibrium lies to the left. Further, the actual neutrali- 
zation reaction equilibrium must lie to the right if it is to be practically 
complete; this means that B must be a stronger base than the anion A~. 
For the complete neutralization, therefore, the order of basic strengths 
must be 

A- < B > SH. 

If B is a weak base, it is necessary that A~ should be still weaker; it has 
been seen (p. 307) that a strong acid will have a very weak conjugate 
base, and hence this condition is satisfied if HA is a very strong acid. 
It is also necessary that the solvent should be a very weak base, and this 
can be achieved by using a strongly protogenic, i.e., acidic, medium. 
It has been found, in agreement with these conclusions, that extremely 
weak bases, e.g., acetoxime, can be neutralized completely by means of 
perchloric acid, the strongest known acid, in acetic acid as solvent. In 
water, hydrolysis of the type (a) is so considerable that neutralization of 
acetoxime, even by means of a strong acid, occurs to a negligible extent 
only. 

By similar arguments it can be shown, from a consideration of the 
lyolytic equilibrium (6), that if an acid HA is to be neutralized com- 
pletely, the condition is that the order of acid strengths must be 

BH+ < HA > SH. 

To neutralize completely a weak acid HA it is necessary, therefore, to 
use a very strong base, so that its conjugate acid BH+ is extremely weak, 
and to work in a protophilic medium, such as ether, acetonitrile or, 
preferably, liquid ammonia. 

It will be evident from the conclusions reached that the lyolysis 
process (a) is due primarily to the weakness of the base B, whereas the 
process (fe) results from the weakness of the acid HA. If both acid and 
base are weak in the particular solvent, then both types of lyolysis can 
occur, and complete neutralization is only possible in an aprotic solvent, 



372 NEUTRALIZATION AND HYDROLYSIS 

provided the proton donating tendency of the acid HA is considerably 
greater than that of BH+, or the proton affinity of the base B is greater 
than that of A~ (cf. p. 331). If the medium is exclusively protophilic, 
e.g., acetonitrile, then only the (a) type of lyolysis, namely that involving 
a weak base, is possible; weak acids should be completely neutralized 
provided a strong base is used. Similarly, in an exclusively protogenic 
solvent, e.g., hydrogen fluoride, the (6) type of lyolysis only can occur; 
a weak base can thus be completely neutralized in such a medium if a 
sufficiently strong acid is employed. 

Hydrolysis of Salts. The subject of lyolysis, or hydrolysis, in the 
event of water being the solvent, can be treated from two angles; in the 
general treatment already given it has been considered from the point of 
view of incomplete neutralization, and a return will be made later to this 
aspect of the subject. Another approach to the phenomena of hydrolysis 
is to study the equilibria resulting when a salt is dissolved in the given 
solvent; the situation is, of course, exactly the same as that which arises 
when an equivalent of the particular acid constituting the salt is neu- 
tralized by an equivalent of the base. This particular aspect of the 
subject of hydrolysis will be treated here; it is convenient to consider the 
material with special reference to the salt of (a) a weak acid, (b) a weak 
base, and (c) a weak acid and weak base. The first two of these are 
often referred to as salts of "one-sided" weakness, and the latter as a 
salt of "two-sided" weakness. Salts of strong acids and strong bases 
do not undergo hydrolytic reaction with the solvent, because the con- 
jugate base and acid, respectively, arc extremely weak; such salts, there- 
fore, will not be discussed in this section, but reference will be made 
below to the neutralization of a strong acid by a strong base. 

I. Salt of Weak Acid and Strong Base. When a salt, e.g., NaA, of a 
weak acid HA is dissolved in water, it may be regarded as undergoing 
complete dissociation into Na + and A~ ions, provided the solution is not 
too concentrated. Since HA is a weak acid the conjugate base A~ will 
be moderately strong; hence the latter will react with the solvent mole- 
cules (II 2 O) giving the type of hydrolytic equilibrium represented by 
equation (16); in the particular case of water as solvent, this may be 
written 

A- + H 2 ^ HA + OH- 

Unhydro- Free Free 

lyzed salt acid base 

The hydrolysis of the salt thus results in the partial reformation of the free 
weak acid HA and of the strong base (Na+)OH- from which the salt 
was constituted. As a consequence of the weakness of the acid HA, there- 
fore, there is a partial reversal of neutralization, and the term hydrolysis 
is often defined in this sense. It will be observed that the hydrolytic 
process results in the formation of OH~ ions, and this must obviously be 
accompanied by a decrease of hydrogen ion concentration (cf. p. 339); 



SALT OF WEAK ACID AND STRONG BASE 373 

the salt of a weak acid and a strong base thus reacts alkaline on account 
of hydrolysis. This accounts for the well-known fact that such salts as 
the cyanides, acetates, borates, phosphates, etc., of the alkali metals are 
definitely alkaline in solution. 

Application of the law of mass action to the hydrolytic equilibrium 
gives the hydrolysis constant (K h ) of the salt as 

(2) 



the activity of the water being, as usual, taken as unity. The ionic 
product of water (K w ) and the dissociation constant (K a ) of the acid and 
HA are defined by 

TS OH^A- . 

K a = - > 



hence, it follows immediately from these expressions and equation (2) 
that 

K* = TF' (3) 

A 

The hydrolysis constant is thus inversely proportional to the dissociation 
constant of the weak acid; * the weaker the acid the greater is the hy- 
drolysis constant of the salt. 

If the activities are replaced by the product of the concentration and 
activity coefficient in each case, equation (2) becomes 

" /HA/OH" ,.. 



CA~ 



In solutions of low ionic strength the activity coefficient /HA of the un- 
dissociated molecules is very close to unity, and, further, the ratio of the 
activity coefficients of the two univalont ions, i.e., /OH-//A-, is then also 
unity, by the Debye-IIuckel limiting law; equation (4), therefore, reduces 
to the less exact form 



which is particularly applicable to dilute solutions. As in other cases, 
the thermodynamic constant K h has been replaced by the approximate 
"constant," kh* 

The degree of hydrolysis (z) is defined as the fraction of each mole 
of salt that is hydrolyzed when equilibrium is attained. If c is the 
stoichiometric, i.e., total, concentration of the salt NaA in the solution, 
the concentration of unhydrolyzed salt will be c(l x) ; since this may 
be regarded as completely dissociated into Na+ and A~ ions, it is possible 

* It should be noted that the hydrolysis constant is equal to the dissociation con- 
stant of the base A~ which is conjugate to the acid HA. 



374 NEUTRALIZATION AND HYDROLYSIS 

to write 

CA- = c(l - x). 

In the hydrolytic reaction, equivalent amounts of OH~ and HA are 
formed, and if the dissociation of the latter is neglected, since it is likely 
to be very small especially in the presence of the large concentration of 
A~ ions, it follows that COH~ and CHA must be equal; further, both of these 
must be equal to ex, where x is the fraction of the salt hydrolyzed; hence, 

Coir = CHA == ex. 
Substitution of these values for CA~ and Coir in equation (5) gives 



From equation (7) it is possible to calculate the degree of hydrolysis at 
any desired concentration, provided the hydrolysis constant of the salt, 
or the dissociation constant of the acid [cf. equation (3)], is known. 
If kh is small, e.g., for the salt of a moderately strong acid, at not too small 
a concentration equation (7) reduces to 

' (8) 

so that the degree of hydrolysis is approximately proportional to the 
square-root of the hydrolysis constant and inversely proportional to the 
square-root of the concentration of the salt solution. The result of equa- 
tion (8) may be expressed in a slightly different form by making use of 
equation (3) which may be written, for the present purpose, as k h = k u lk a ] 
thus, 

' " 

If two salts of different weak acids are compared at the same concen- 
tration, it is seen that 



so that the degree of hydrolysis of each is inversely proportional to the 
square-root of the dissociation constant of the acid; hence the weaker 
the acid the greater the degree of hydrolysis at a particular concentra- 
tion. For a given salt, equation (9) shows the degree of hydrolysis to 
increase with decreasing concentration. 

By making use of equation (7) or (8) it is possible to calculate the 
degree of hydrolysis of the gait of a strong base and a weak acid of known 



SALT OF WEAK ACID AND STRONG BASE 375 

dissociation constant at any desired concentration. The results of such 
calculations are given in Table LXV; the temperature is assumed to be 

TABLE LXV. DEQIIEE OF HYDROLYSIS OF SALTS OF WEAK ACIDS AND STRONG BASES 

AT 25 

Concentration of Solution 

k a k h 0.001 N 0.01 N 0.1 N 1.0 N 

10 < 10 10 3.3 X 10 4 10~ 4 3.2 X 10 B 10~ 6 

10 - 10- 8 3.2 X 10~ 3 10- 3 3.2 X10" 4 10~ 4 

10' 8 10" 8 3.2 X 10 2 lO^ 2 3.2 X 10 3 10~ 3 

10 10~ 4 0.27 0.095 3.2 X 10 2 lO" 8 

about 25, so that k w can be taken as 10~~ 14 . It is seen that the degree of 
hydrolysis increases with decreasing strength of the acid and decreasing 
concentration of the solution. In a 0.001 N solution, the sodium salt of 
an acid of dissociation constant equal to 10~ 10 , e.g., a phenol, is hydro- 
lyzed to an extent of 27 per cent. It may be noted that equations (7) 
and (8) give almost identical values for the degree of hydrolysis in 
Table LXV, except for the two most dilute solutions of the salt of the 
acid of k a equal to 10~ 10 . In these eases the approximate equation (8) 
would give 0.32 and 0.10, instead of 0.27 and 0.095 given in the table. 
It has been seen above that COH is equal to ex, and since the product 
of fa f and Coir is k w , it follows that 



and introducing the value of x from equation (9), the result is 

CH+ = 

Taking logarithms and changing the signs throughout, this becomes 

- log c u + = ~ i log k w log k a + % log c. (12) 

As an approximation, log C H + may be replaced by pH, and using the 
analogous exponent forms for log k w arid log A: , it follow* that 

pH = \ pk w + \ pfc + \ log c. (12a) 

It is seen, therefore, that the pH, or alkalinity, of a solution of the salt 
of a weak acid and strong base increases with decreasing acid strength, 
i.e., increasing pk a , and increasing concentration. Attention may be 
called to the fact that although the degree of hydrolysis decreases with 
increasing concentration of the salt, the pH, or alkalinity, increases. 
The pH values in Table LXVI have been calculated for dissociation con- 
stants and salt concentrations corresponding to those in Table LXV; 
equation (12) is satisfactory in all cases for which (8) is applicable, but 



376 NEUTRALIZATION AND HYDROLYSIS 

TABLE LXVI. VALUES OP pH IN SOLUTIONS OP SALTS OP WEAK ACIDS AND STRONG 

BASES AT 25 

Concentration of Solution 
* k h 

10-4 10-w 



O.OOt N 


0.01 N 


0.1 N 


1.0 N 


7.5 


8.0 


8.5 


9.0 


8.5 


9.0 


9.5 


10.0 


9.5 


10.0 


10.5 


11.0 


10.4 


11.0 


11.5 


12.0 



!0-w 10-4 

in the others use has been made of the x values in Table LXV together 
with equation (11). Since the pH of a neutral solution is about 7.0 at 
25, it follows that the solutions of salts of weak acids can be considerably 
alkaline in reaction. 

It was seen in Chap. IX that the dissociation constant of an acid 
undergoes relatively little change with temperature between and 100; 
on the other hand the ionic product of water increases nearly five hundred- 
fold. It is evident, therefore, from equation (3) that the hydrolysis 
constant will increase markedly with increasing temperature; the degree 
of hydrolysis and the pH at any given concentration of salt will thus in- 
crease at the same time. 

EL Salt of Weak Base and Strong Acid. When the base B is weak, 
the conjugate acid BH+ will have appreciable strength and hence it will 
tend to react with the solvent in accordance with the hydrolytic equi- 
librium (la). It follows, therefore, that if the salt of a weak base and 
a strong acid is dissolved in water there will be a partial reversal of 
neutralization, some of the acid II 3 O+ and the weak base B being re- 
generated; in other words, the salt is hydrolyzed in solution. If the 
weak base is of the type RNH 2 , e.g., ammonia or an amine, the con- 
jugate acid is RNHiJ~, and when the salt, e.g., RNH 3 C1, is dissolved in 
water it dissociates virtually completely to yield RNHf and Cl~" ions, 
the former of which establish the hydrolytic equilibrium 

RNHt + H 2 ^ H 3 0+ + RNH 2 . 

When the weak base is a metallic hydroxide, it is probable that the 
conjugate acid is the hydrated ion of the metal, e.g., Fe(H 2 O)j" H+ or 
Cu(H 2 O)t + , which may be represented in general by M(H 2 0)mJ the 
hydrolysis must then be expressed by 

M(H 2 0)+ + H 2 ^ H 3 0+ + M(H 2 0) m _ 1 OII, 

where M(H 2 0) m _i(OH) is the weak base. The formation of H 3 0+ ions 
shows that the solutions react acid in each case. 

Writing the hydrolytic equilibrium in the general form 

BH+ + H 2 ^ H 3 0+ + B, 
Unhydro- Free Free 

lyzed salt acid 



SALT OP WEAK BASE AND STRONG ACID 377 

application of the law of mass action gives, for the hydrolytic constant, 



and since 

, 

and 



a B 
it follows that 



(14) 



where K b is the dissociation constant of the base B. It is seen that 
equation (14) is exactly analogous to (3), except that KI now replaces K a . 
By making the same assumptions as before, concerning the neglect of 
activity coefficients in dilute solution, equation (13) reduces to 

i CH * CB ,, K , 

k h = i (15) 

CBH+ 

and from this, since CH+ is now equal to CB, both of which are equal to ex, 
while CBH+ is equal to c(l x), it follows that 



J. 



(16) 



which is identical in form with equation (6). The degree of hydrolysis 
in this case is, consequently, also given by equation (7) which reduces to 
(8) provided the base is not too weak or the solution too dilute. Re- 
placing k h now by fc,/fc&, by the approximate form of equation (14), it 
follows that 

(17) 

The same general conclusions concerning the effect of the dissociation 
constant of the weak base and the concentration of the salt on the degree 
of hydrolysis are applicable as for the salt of a weak acid. The results 
in Table LXV would hold for the present case provided the column 
headed k a were replaced by fc&. Further, since the dissociation constants 
of bases do not vary greatly with temperature, the influence of increasing 
temperature on the hydrolysis of the salt of a weak base will be very 
similar to that on the salt of a weak acid. 

The hydrogen ion concentration CH+ in the solution of a salt of a weak 
base is given by ex, as mentioned above, and if the value of x from equa- 
tion (17) is employed, it follows that 



378 NEUTRALIZATION AND HYDROLYSIS 

This result may he expressed in the logarithmic form 

pH-ipfc.- Jrfr fc -*logc. (18) 

It is evident that the pTI of the solution must be less than $pk w , i.e., 
less than 7.0, and so solutions of salts of the type under consideration 
will exhibit an acid reaction. It was seen on page 339 that in any 
aqueous solution 

pH + pOII = pk w , 

hence in this particular case 

pOH - lpk u + Jpfa + 1 log c, (19) 

which is exactly analogous to equation (12a), except that pOII and pkt> 
replace pH and pfc,, respectively. It follows, therefore, that the results 
in Table LXVI give the pOH values in solutions of salts of a weak base, 
provided the column headed pA" a is replaced by pA&. 

III. Salt of Weak Acid and Weak Base. If both the acid and base 
from which a given salt is made are weak, the respective conjugate base 
and acid will have appreciable strength and consequently will tend to 
interact with the amphiprotic solvent water. When a salt such as 
ammonium acetate is dissolved in water, it dissociates almost completely 
into NH^ and Ac~ ions, and these acting as acid and base, respectively, 
take part in the hydrolytic equilibria 

NI1| + II a O -^ II 3 0< + NII 3 , 
and 

Ac- + 11,0 ^ II Ac + Oil-. 

Combining the two equations, the complete equilibrium is 

Nllf -I- Ac- + 2II 2 ^ 11,0 * + OH" + Nil, + HAc, 
or, representing the? weak base hi general by B and the acid by TIA, 
BH+ + A" + 2H,() ;-^ H 3 0+ + OH - + B + ITA. 

Since the normal equilibrium between water molecules and hydrogen and 
hydroxyl ions, viz., 

2H 2 O ^ H 3 O+ + Oil', 

xi*ts in any event, this may be subtracted from the hydrolytic equi- 
librium; the result may thus be represented by 

NHt + Ac- ^ NH S + HAc 
for ammonium acetate or, in the general case, by 



BH+ + A 
Unhydro- Free Fne 
lyzed salt acid base 



SALT OF WEAK ACID AND WEAK BASE 379 

The law of mass action then gives for the hydrolysis constant 

(20) 



and introduction of the expressions for K a and K b leads to the result 



The hydrolysis constant equation (20) may also be written as 

/HA/B 
--- - 



, N 
(22) 
BHA- BHVA- 

and since this expression involves the product of the activity coefficients 
of two univalent ions, instead of their ratio as in the previous cases, it is 
less justifiable than before to assume that the activity coefficient fraction 
will become unity in dilute solution. Nevertheless, this approximation 
can be made without introducing any serious error, and the result is 

(23) 
BHA- ' 

If the original, i.e., stoichiometric, concentration of the salt is c moles 
per liter, and x is tho degree of hydrolysis, then CHA and CB may both be 
set equal to c.r, whereas f BH + and CA~ are both equal to the concentration 
of unhydrolyxcd salt c(l -- x), the salt being regarded as completely 
dissociated. Insertion of these values in equation (23) then gives 



If V/c/, is small in comparison with unity, it may be neglected in the 
denominator so that equation (25) becomes 

x VA-,~, (26) 

or, introducing the approximate form of equation (21) for kh, 



It appears from equations (25), (26) and (27) that the degree of hydroly- 
sis of a given salt of two-^ided weakness is independent of the concen- 
tration of the solution; this conclusion is only approximately true, as 
will be seen shortly. 



380 NEUTRALIZATION AND HYDROLYSIS 

The hydrogen ion concentration of the solution of hydrolyzed salt 
may be calculated by using the expression for the dissociation function 
of the acid, k a ', thus, 



CHA 

/ CHA 


cx 



- - 

~~~ A/a / ^ v ~ o -i 

c(l x) 1 x 

By equation (24), the fraction x/(l x) is equal to 



or, expressed logarithmically, 

pH = pfc w + $pk a - P fc 6 . (29) 



If the dissociation constants of the weak base and acid are approximately 
equal, i.e., pk a is equal to pk b) it follows that pH is %pk w ; the solution will 
thus be neutral, in spite of hydrolysis. If, on the other hand, k a is 
greater than k b , the salt solution will have an acid reaction; if k a is less 
than kb the solution has an alkaline reaction. As a first approximation 
the pH of a solution of a salt of a weak acid and weak base is seen to 
be independent of the concentration. 

The conclusion that the degree of hydrolysis and pH of a solution 
of a salt of double-sided weakness is independent of the concentration 
is only strictly true if CBH+ is equal to C A - and if CB is equal to CHA, as 
assumed above. This condition is only realized if k a and kb are equal, 
but not otherwise. If the dissociation constants of HA and B are differ- 
ent, so also will be those of the conjugate base and acid, i.e., A~ and BH+, 
respectively. The separate hydrolytic reactions 

A- + H 2 ^ HA + OH- 
and 

BH+ + H 2 ^ H 3 0+ + B, 

will, therefore, take place to different extents, so that the equilibrium 
concentrations of A- and BH+, on the one hand, and of HA and B, 
on the other hand, will not be equal. The assumptions made above, 
that CBH* is equal to CA- and that CB is equal to CHA, are consequently not 
justifiable, and the conclusions drawn are not strictly correct. The 
problem may be solved in principle by writing 

c = CA- + COH- = CBH+ + CB, 



HYDROLYSIS OF ACID SALTS 381 

where the total concentration c is divided into the unhydrolyzed part, 
i.e., CA~ or CBH+, and the hydrolyzed part, i.e., COIT or C B , respectively. 
Further, by the condition of electrical neutrality, 

CH+ + CBH+ = COBT + CA~, 

and if these equations are combined with the usual expressions for 
k a , kb and k w , it is possible to eliminate CA-, COH~, CB and CBH+, and to derive 
an equation for CH + in terms of c and k a , kb and k w . Unfortunately, the 
resulting expression is of the fourth order, and can be solved only by a 
process of trial and error. The calculations have been carried out for 
aniline acetate (k a = 1.75 X 10~ 5 , k b = 4.00 X 10~ 10 ): at concentrations 
greater than about 0.01 N the result for the hydrogen ion concentration 
is practically the same as that obtained by the approximate method given 
previously. In more dilute solutions, however, the values differ some- 
what, the differences increasing with increasing dilution. 1 

Hydrolysis of Acid Salts. The acid salt of a strong base and a weak 
dibasic acid, e.g., NaHA, will be hydrolyzed in solution because of the 
interaction between the ion HA~", functioning here as a base, and the 
solvent, thus 

HA- + H 2 O ^ H 2 A + OH". 

The ion HA~ can also act as an acid, 

HA- + H 2 ^ H 8 0+ + A~, 

and the H 3 0+ ions formed in this manner may interact with HA" to 
form H 2 A, thus 

HA- + H 3 0+ = H 2 A + H 2 O. 

If it were not for this latter reaction CA~~ would have been equal to CH+I 
but since some of the hydrogen ions are removed in the formation of an 
equivalent amount of H 2 A, it follows that 

CA" = C H + + C H ,A. 

Further, if the salt NaHA is hydrolyzed to a small extent only, CHA~ will 
be almost equal to c, the stoichiometric concentration of the salt. With 
these expressions for CA~ and CHA-, together with the equations for ki 
and & 2 , the dissociation functions of the first and second stages of the 
acid H 2 A, viz., 

, CH+CHA- , , C H *C A " 

fa ._. - an( l 2 _ - , 

CH,A CHA~ 

it is readily possible to derive the result 




1 Griffith, Trans. Faraday Soc., 17, 525 (1922). 



382 NEUTRALIZATION AND HYDROLYSIS 

If ki is small in comparison with the concentration, so that it may be 
neglected in the denominator, equation (30) reduces to the simple form 

c H * = VA^, (31) 

.'. pll = -JpA-! + Jpfe. (32) 

In this case, therefore, tho pll of the solution is independent of the con- 
centration of the acid salt. 

The difference between the results given by equations (30) and (31) 
increases with increasing dilution, as is to be expected. If ki is less than 
about 0.01 c, however, the discrepancy is negligible. 

Displacement of Hydrolytic Equilibrium. When a salt is hydrolyzed, 
the equilibrium 

Unhydrolyzed salt + Water ^ Free acid -f Free base 

is always established; this -equilibrium can be displaced in either direction 
by altering the concentrations of the products of hydrolysis. The addi- 
tion of either the free acid or the free base, for example, will increase the 
concentration of unhydrolyzed salt and so repress the hydrolysis; this 
fact is utilized in a method for investigating hydrolytic equilibria (p. 383). 
If, on the other hand, the free acid or base is removed in some manner, 
the extent of hydrolysis of the salt must increase in order to maintain the 
hydrolytic equilibrium. For example, if a solution of potassium cyanide 
is heated or if a current of air is passed through it, the hydrogen cyanide 
formed by hydrolysis can be volatilized; as it is removed, however, more 
is regenerated by the continued hydrolysis of the potassium cyanide. 
When a solution of ferric chloride is heated, the hydrogen chloride is 
removed and hence the hydrolytic process continues; the hydra ted ferric 
oxide which is formed remains in colloidal solution and imparts a dark 
brown color to the system. 

Determination of Hydrolysis Constants : I. Hydrogen Ion Methods. 
A number of methods of varying degrees of accuracy have been proposed 
for the estimation of the degree of hydrolysis in salt solutions or of the 
hydrolysis constant of tho salt. One principle which can be used is to 
evaluate the hydrogen ion concentration of the solution; for a salt of a 
weak acid en 4 - is equal to k w /cx, where c is the stoichiometric concentration 
of the salt, and hence it follows from equation (16) that 



(33) 



If CH+ is known, the hydrolysis constant can be calculated. For a salt 
of a weak base, on the other hand, C H + is equal to ex; hence 



CONDUCTANCE METHOD 383 

If the salt is one of two-sided weakness the hydrogen ion concentration 
alone is insufficient to permit k h to be evaluated; it is necessary to know, 
in addition, k a or fo>. 

The hydrogen ion concentration of a hydrolyzed salt solution can 
be determined by one of the E.M.F. or indicator methods described in 
Chap. X; it is true that the results obtained in this manner are not 
actual concentrations, but in view of the approximate nature of equations 
(33) and (34), the k h values are approximate in any case. 

n. Conductance Method. 2 In a solution containing c equiv. per liter 
of a salt of a weak base and a strong acid, for example, there will be 
present c(l x) equiv. of unhydrolyzed salt and ex equiv. of both free 
acid and base. If the base is very weak, it may be regarded as com- 
pletely un-ionized, and so it will contribute nothing towards the total 
conductance of the solution of the salt. The conductance of 1 equiv. of 
a salt of a very weak base is thus made up of the conductance of 1 x 
equiv. of unhydrolyzed salt and x equiv. of free acid, i.e., 

A = (1 - z)A c + zA HA . (35) 

In this equation A is the apparent equivalent conductance of the solution, 
which is equal to 1000 K/C, where K is the observed specific conductance 
and c is the stoichiometric concentration of the salt in the solution; A c is 
the hypothetical equivalent conductance of the unhydrolyzed salt, and 
AHA is the equivalent conductance of the free acid in the salt solution. 
It follows from equation (35) that 



(36) 



and so the calculation of x involves a knowledge of A, AHA and A c . As 
mentioned above, A is derived from direct measurement of the specific 
conductance of the hydrolyzed salt solution; the value of AHA is generally 
taken as the equivalent conductance of the strong acid at infinite dilution, 
since its concentration is small, but it is probably more correct to use the 
equivalent conductance at the same total ionic strength as exists in tho 
salt solution. The method is, however, approximate only, and this re- 
finement is hardly necessary. 

The evaluation of A c for the unhydrolyzed salt presents a special 
problem. As already seen, the addition of excess of free base will repress 
the hydrolysis of the salt, and in the method employed sufficient of the 
almost non-conducting free base is added to the salt solution until the 
hydrolysis of the latter is almost zero. For example, with aniline hydro- 
chloride, free aniline is added until the conductance of the solution reaches 
a constant value; at this point hydrolysis is reduced to a negligible 

1 Bredig, Z. physik. Chem., 13, 213, 221 (1894); Kanolt, ,/. Am. Chem. Soc., 29, 1402 
(1907); Noyes, Sosman and Kato, ibid., 32, 159 (1910) ; Kameyama, Trans. Ekctrochem. 
Soc., 40, 131 (1921); Gulezian and Mtiller, J. Am. Chem. Soc., 54, 3151 (1932). 



384 NEUTRALIZATION AND HYDROLYSIS 

amount. The conductance of the solution is virtually that of the 
unhydrolyzed salt, and so A c can be calculated. The data in Table 
LXVII are taken from the work of Bredig (1894) on a series of solutions 

TABLE LXVII. HYDROLYSIS OP ANILINE HYDBOCHLOBIDB AT 18 FROM CONDUCTANCE 

MEASUREMENTS * 

c A Aj \' e ' x k H X 10 6 

0.01563 106.2 96.0 95.9 0.036 2.1 

0.00781 113.7 98.2 98.1 0.055 2.5 

0.00391 122.0 100.3 100.1 0.077 2.5 

0.00195 131.8 101.5 101.4 0.109 2.6 

0.000977 144.0 103.3 103.3 0.147 2.5 

* Bredig's measurements are not accurate because they were based on an incorrect 
conductance standard; the values of x and kn derived from them are, however, not 
affected. 

of aniline hydrochloride of concentration c equiv. per liter and observed 
equivalent conductance A; the columns headed A c and A" give the meas- 
ured equivalent conductances in the presence of N/64 and N/32, respec- 
tively, added free aniline. Since the values in the two columns do not 
differ appreciably, it is evident that N/64 free aniline is sufficient to 
repress the hydrolysis of the aniline hydrochloride almost to zero; hence 
either AC or AC' may be taken as equal to the required value of A c . Taking 
AHA for hydrochloric acid as 380 at 18, the degree of hydrolysis x has 
been calculated in each case; from these the results for kh in the last 
column has been derived. The values are seen to be approximately con- 
stant at about 2.5 X 10~ 5 . 

For the salt of a weak acid, the method would be exactly similar to 
that described above except that excess of the free acid would be added 
to repress hydrolysis. The equation for the degree of hydrolysis is then 

A - A c 

x = 



AMOH A c 

where AMOH is the equivalent conductance of the strong base. The con- 
ductance method has also been used to study the hydrolysis of salts of 
weak acids and bases, but the calculations involved are somewhat com- 
plicated. 

The determinations of hydrolysis constants from conductance meas- 
urements cannot be regarded as accurate; the assumption has to be made 
that the added free acid or free base has a negligible conductance. This 
is reasonably satisfactory if the acid or base is very weak, e.g., a phenol 
or an aniline derivative, but for somewhat stronger acids or bases, e.g., 
acetic acid, an appreciable error would be introduced; it is sometimes 
possible, however, to make an allowance for the conductance of the added 
acid or base. 



DISTRIBUTION METHOD 385 

HI. Distribution Method. 8 Another approximate method for study- 
ing hydrolysis is applicable if one constituent of the salt, generally the 
weak acid or base, is soluble in a liquid that is not miscible with water, 
while the salt itself and the other constituent are not soluble in that 
liquid. Consider, for example, the salt of a weak base, e.g., aniline 
hydrochloride ; the free base is soluble in benzene, in which it has a 
normal molecular weight, whereas the salt and the free hydrochloric acid 
are insoluble in benzene. A definite volume (vi) of an aqueous solution 
of the salt at a known concentration (c) is shaken with a given volume 
(t> 2 ) of benzene, and the amount of free aniline in the latter is determined 
by analysis. If m is the concentration in equiv. per liter of the aniline 
in benzene found in this manner, then the concentration of free aniline 
in the aqueous solution (CB) should be m/D, where D is the "distribution 
coefficient" of aniline between benzene and water; the value of D must 
be found by separate experiments on the manner in which pure aniline 
distributes itself between benzene and water, in the absence of salts, etc. 
The amounts of free aniline in the benzene and aqueous layers are mv 2 
and mvi/D respectively; hence, the amount of free acid in the aqueous 
solution, assuming none to have dissolved in the benzene, must be the 
sum of these two quantities, i.e., mv* + mvi/D. Since this amount is 
present in a volume v\, it follows that the concentration of free acid in 
the aqueous solution (c n +) is mv^/vi + m/D. The concentration of un- 
hydrolyzed salt (CBH+) is equal to the stoiohiometric concentration (c) 
less the concentration of free acia, since the latter is equivalent to the 
salt that has been hydrolyzed; hence, CBH+ is equal to c mv 2 /vi m/D. 
The results derived above may then be summarized thus: 

mvz m 

CH+ = h 7: > 

vi D 

_ !? 

and 

mv% m 

~~ vi /)' 

and so it follows from equation (15) that 

m \m 






C ~ Vl D 

By determining m y therefore, all the quantities required for the evalua- 
tion of fa by means of equation (38) are available, provided D is known 

8 Farmer, J. Cham. Soc. y 79, 863 (1901); Farmer and Warth, t&id., 85, 1713 (1904); 
Williams and Soper, ibid., 2469 (1930). 



386 NEUTRALIZATION AND HYDROLYSIS 

from separate experiments. The results in Table LXVIII, taken from 
the work of Farmer and Warth (1904), illustrate the application of the 
method to the determination of the hydrolysis of aniline hydrochloride; 
the non-aqueous solvent employed was benzene, for which D is 10.1, arid 
the volumes v\ and v z were 1000 cc. and 59 cc., respectively. The value 
of kh is seen to be in satisfactory agreement with that obtained for aniline 
hydrochloride by the conductance method (Table LXVII). 

TABLE LXVIII. HYDROLYSIS OF ANILINE HYDROCHLORIDE FROM DISTRIBUTION 

MEASUREMENTS 

c m CB = ^ cn+ TBII+ k k X 10* 

0.0997 0.0124 000123 19.6 X 10 4 0.0978 2.4 

0.0314 0.00628 0.000622 9.9 X 10~ 0304 2.0 

The distribution method for studying hydrolysis can be applied to 
salts of a weak acid, provided a suitable solvent for the acid is available; 
the hydrolysis constant is given by an equation identical with (38), 
except that m now represents the concentration of free acid in the non- 
aqueous liquid. The same principle can be applied to the investigation 
of salts of two-sided weakness provided a solvent can be found which 
dissolves either the weak acid or the weak base, but not both. 

IV. Vapor Pressure Method. 4 If the free weak acid or weak base is 
appreciably volatile, it is possible to determine its concentration or, more 
correctly, its activity, from vapor pressure measurements. In practice 
the actual vapor pressure is not measured, but the volatility of the sub- 
stance in the hydrolyzed salt solution is compared with that in a series of 
solutions of known concentration. In the case of an alkali cyanide, for 
example, the free hydrogen cyanide produced by hydrolysis is appreciably 
volatile. A current of air is passed at a definite rate through the alkali 
cyanide solution and at exactly the same rate through a hydrogen cyanide 
solution; the free acid vaporizing with the air in each case is then ab- 
sorbed in a suitable reagent and the amounts are compared. The con- 
centration of the hydrogen cyanide solution is altered until one is found 
that vaporizes at the same rate as does the alkali cyanide solution. It 
may be assumed that the concentrations, or really activities, of the free 
acid are the same in both solutions. The concentration of free acid CHA 
in the solution of the hydrolyzed salt of the weak acid may be put equal 
to ex (cf. p. 374) and hence x and kh can be calculated. 

V. Dissociation Constant Method. All the methods described above 
give approximate values only of the so-called hydrolysis "constant" of 
the salt; the most accurate method for obtaining the true hydrolysis 
constant is to make use of the thermodynamic dissociation constants of 
the weak acid or base, or both, and the ionic product of water. For this 

< Worley et al., J. Chem. Soc., Ill, 1057 (1917) ; Trans. Faraday Soc., 20, 502 (1925) ; 
Britton and Dodd, J. Chem. Soc., 2332 (1931). 



STRONG ACID AND STRONG BASE 387 

purpose equations (3), (4) and (21) are employed. The results derived 
in this manner are, of course, strictly applicable to infinite dilution, but 
allowance can be made for the influence of the ionic strength of the 
medium by making use of the Debye-Huckel equations. The methods 
I to IV are of interest, in so far as they provide definite experimental 
evidence for hydrolysis, but they would not be used in modern work 
unless it were not possible, for some reason or other, to determine the 
dissociation constant of the weak acid or base. 

It is of interest to note that some of the earlier measurements of kh 
were used, together with the known dissociation constant of the acid or 
base, to evaluate k w for water. For example, k h for aniline hydrochloride 
has been found by the conductance method (Table LXVII) to be about 
2.5 X 10~ 6 , and k b for aniline is 4.0 X 10~ 10 ; it follows, therefore, that 
k w , which is equal to ktJth, should be about 1.0 X 10~ 14 , in agreement with 
the results recorded in Chap. IX. 

Neutralization Curves. The variation of the pH of a solution of acid 
or base during the course of neutralization, and especially in the vicinity 
of the equivalence-point, i.e., when equivalent amounts of acid and base 
are present, is of great practical importance in connection with analytical 
and other problems. It is, of course, feasible to measure the pH experi- 
mentally at various points of the neutralization process, but a theoretical 
study of the subject is possible and the results are of considerable interest. 
For this purpose it is convenient to consider the behavior of different 
types of acid, viz., strong and weak, with different bases, viz., strong and 
weak. For the present the discussion will be restricted to neutralization 
involving a conventional acid and base in aqueous solution, but it will be 
shown that the results can be extended to all forms of acids and bases 
in aqueous as well as non-aqueous solvents. 

L Strong Acid and Strong Base. The changes in hydrogen ion con- 
centration occurring when a strong base is added to a solution of a strong 
acid can be readily calculated, provided the acid may be assumed to be 
completely dissociated. The concentration of hydrogen ion (CH+) at any 
instant is then equal to the concentration of un-neutralized strong acid 
at that instant. If a is the initial concentration of the acid in equiv. per 
liter, and 6 equiv. per liter is the amount of base added at any instant, 
the concentration of un-neutralized acid is a b equiv. per liter, and 
this is equal to the hydrogen ion concentration. The results obtained in 
this manner when 100 cc. of 0.1 N hydrochloric acid, i.e., a is 0.1, are 
titrated with 0.1 N sodium hydroxide are given in Table LXIX. In order 
to simplify the calculations it is assumed that the volume of the system 
remains constant at 100 cc. ; this simplification involves a slight error, but 
it will not affect the main conclusions which will be reached here. The 
values of pH in the last column are derived from the approximate defini- 
tion of pH as Jog CH+. 

When the solution contains equivalent amounts of acid and alkali 
the method of calculation given above fails, for a b is then zero; the 



388 NEUTRALIZATION AND HYDROLYSIS 

TABLE LXIX. NEUTRALIZATION OP 100 CC. O.I N HCL BY 0.1 N N*OH 

NaOH 

added b CH+ pH 

0.0 cc. 0.00 10-* 1.0 

50.0 0.05 5 X 10-' 1.3 

90.0 0.09 10~* 2.0 

99.0 0.099 10~ 8 3.0 

99.9 0.0999 10~ 4 4.0 

100.0 0.1000 10~ 7 7.0 

100.1 0.1001 lO" 10 10.0 

system is now, however, identical with one containing the neutral salt 
sodium chloride, and so the value of CH+ is 10~ 7 g.-ion per liter and the 
pH is 7.0 at ordinary temperatures. If the addition of base is continued 
beyond the equivalence-point, the solution will contain free alkali; the 
pH of the system can then be calculated by assuming that COH~ is equal 
to the concentration of the excess alkali and that the ionic product CH+COIT 
is 10~ 14 . For example, in Table LXIX the addition of 100.1 cc. of 0.1 N 
sodium hydroxide means an excess of 0.1 cc. of 0.1 N alkali, i.e., 10~ 6 
equiv. in 100 cc. of solution; the concentration of free alkali, and hence 
of hydroxyl ions, is thus 10~ 4 equiv. per liter. If COH" is 10~ 4 , it follows 
that CH+ must be 10~ 10 and hence the solution has a pH of 10.0. 

If the titration is carried out in the opposite direction, i.e., the addi- 
tion of strong acid to a solution of a strong base, the variation of pH 
may be calculated in a similar manner to that used above. The hy- 
droxyl ion concentration is now taken as equal to the concentration of 
un-neutralized base, i.e., b a, and the hydrogen ion concentration is 
then derived from the ionic product of water. The results calculated for 
the neutralization of 100 cc. of 0.1 N sodium hydroxide by 0.1 N hy- 
drochloric acid, the volume being assumed constant, are recorded in 
Table LXX. 

TABLE LXX. NEUTRALIZATION OF 100 CC. OF 0.1 N NAOH BY 0.1 N HCL 
HC'l 

added a <*OH~ pi I 

0.0 re. 0.00 10- 1 13.0 

50.0 0.05 5 X 10-* 12.7 

90.0 0.090 10~ 2 12.0 

99.0 0.099 10- 11.0 

99.9 0.0999 10~ 4 10.0 

100.0 0.1000 10~ 7 7.0 

100.1 0.1001 10 - 10 4.0 

The data in Tables LXIX and LXX are plotted in Fig. 101, in which 
curve I shows the variation of pH with the extent of neutralization of 
0.1 N solutions of strong acid and strong base; the two portions of the 
curve may be regarded as parts of one continuous curve representing the 
change of pH as a solution of a strong acid is titrated with a strong base 
until the system contains a large excess of the latter, or vice versa. At- 



WEAK ACID AND STRONG BASE 



389 



tention may be called here to the sudden change of pH, from approxi- 
mately 4 to 10, as the equivalence-point, marked by an arrow, is attained; 
further reference to this subject will be made later. 




25 50 75 100 75 50 25 

Per cent Acid Per cent Base 

Neutralized Neutralized 

Fia. 101. Neutralization of strong acid and strong base 

Similar calculations can be made and analogous pH-neutralization 
curves can be plotted for solutions of strong acid and base at other con- 
centrations; curve II represents the results obtained for 10~ 4 N solutions. 
The pH at the equivalence-point is, of course, independent of the con- 
centration, since the pH of the neutral salt is always 7.0. The change 
of pH at the equivalence-point in curve II is seen to be much less sharp, 
however, than is the case with the more concentrated solutions. 

II. Weak Acid and Strong Base. The determination of the pH in 
the course of the neutralization of a weak acid is riot so simple as for a 
strong acid, but the calculations can nevertheless be made with the aid 
of equations derived in Chap. IX. It was seen on page 323 that if a 
weak acid, whose initial concentration is a equiv. per liter, is partially 
neutralized by the addition of 6 equiv. per liter of base, the activity of 
the hydrogen ions is given by 



n - K 

flH Ka B ' 7A - 
which may be written in the logarithmic form 



pH 



log 



log 



(39) 



(40) 



390 NEUTRALIZATION AND HYDROLYSIS 

or, utilizing the Debye-Hvickel equations, 

pll = P K a + log ^-^ - A ^ + C. (41) 

The quantity B is defined in this case by 

B = b + CH* ~ coir, (42) 

using volume concentrations instead of molalities: since this involves 
both CH+ and. COH", the latter being equivalent to k w /cn + , equation (39) 
and those derived from it are cubic equations in CH+, and an exact solution 
is difficult. The problem is therefore simplified by considering certain 
special cases. 

If the pH of the solution lies between 4 and 10, i.e., CH+ is between 
10~ 4 and 10~ l , the quantity C H + COH- in equation (42) is negligibly 
small; under these conditions B is equal to 6, and equation (41) becomes 

P H = pK a + log ~ - A Vtf + C tf . (43) 

The partly-neutralized acid system is equivalent to a mixture of un- 
neutralized acid and its salt, the concentration of the former being a 6 
and that of the latter 6; equation (43) can consequently be written as 



pH = P K a + log - A + C V . (44) 



This relationship, without the activity correction, is equivalent to one 
derived by L. J. Henderson (1908) and is generally known as the Hender- 
son equation. The equation, omitting the activity terms, gives reason- 
ably good results for the pH during the neutralization of a weak base by 
a strong acid over a range of pH from 4 to 10, but it fails at the beginning 
and end of the process: under these latter conditions the approximation 
of setting B equal to 6 is not justifiable. 

For these extreme cases the general equations (39) to (41) are still 
applicable, and suitable approximations can be made in order to simplify 
the calculations. At the very beginning of the titration, i.e., when the 
weak acid is alone present, b is zero and since the solution is relatively 
acid COH~ may be neglected; the quantity B is then equal to CH+, and 
equation (39) becomes 

a ~~ CH+ 



If the solution has a sufficiently low ionic strength for the activity co- 
efficients to be taken as unity, which is approximately true for the weak 
acid solution, this equation may be written in the form 



, a - c H + 
CH+ = 



CH+ 



ak a . 



WEAK ACID AND STRONG BASE 391 

If CH+ or k a is small, that is for a very weak acid, these equations reduce to 



At the equivalence-point, which represents the other extreme of the 
titration, a and b are equal, and CH + may be neglected in comparison with 
Com since the solution is alkaline owing to hydrolysis of the salt of the 
weak acid and strong base. It is seen, from equation (42), therefore, 
that B is now equivalent to a coir, and, neglecting the activity coeffi- 
cients, equation (39) becomes 



, 
= k a 



This is a quadratic in C H +, since foir is equal to k w /cji+, and so it can be 
solved without difficulty, thus 



If 

A/to /to , / a ,u7 

(46) 

Since fc /P /2a is generally very small, it may usually be neglected and so 
this equation reduces to the form 

lK w K, a 

CH* = \ ' (47) 

or 



pH = ipfc* + %pk a + % log a, (47a) 

which is identical, as it should be, with the approximate equation (12a) 
for the hydrogen ion concentration in a solution of a salt of a weak acid 
and strong base; at the equivalence-point the acid-base system under 
consideration is, of course, equivalent to such a solution. 

It is thus possible to calculate the whole of the pH-neutralization 
curve of a weak acid by a strong base: equations (45) and (47) are used 
for the beginning and end, respectively, and equation (43), without the 
activity corrections, for the intermediate points. The pH values ob- 
tained in this manner for the titration of 100 cc. of 0.1 N acetic acid, for 
which k a is taken on 1.75 X 10~ 5 , with 0.1 N sodium hydroxide are quoted 
in Table LXXI. 

When the titration is carried out in the reverse direction, i.e., a strong 
base is titrated with a weak acid, the pH changes in the early stages of 
neutralization are almost identical with those obtained when a strong 
acid is employed. It is true that the salt formed, being one of a weak 
acid and a strong base, is liable to hydrolyze, but as long as excess of the 
strong base is present this hydrolysis is quite negligible (cf. p. 382). The 
hydroxyl ion concentration is then equal to the stoichiometric concen- 
tration of un-neutralized base, i.e., c ir is equal to b a where b and a 
are the concentrations of base and acid which make up the solution, just 



392 



NEUTRALIZATION AND HYDROLYSIS 



TABLE LXXI. NEUTRALIZATION OP 100 CC. OP 0.1 N ACETIC ACID BY 0.1 N NAOH 

NaOH 

added b a-b CH+ pH 

0.0 cc. 0.0 0.10 1.32 X 10-* 2.88 

10.0 0.01 0.09 1. 60X10' 4 3.80 

20.0 0.02 0.08 6.93 X 10 ~ 5 4.16 

40.0 0.04 0.06 2.63 X 10~ 6 4.58 

50.0 0.05 0.05 1.75 X 10- 5 4.76 

70.0 0.07 0.03 7.42 X 10~ 5.13 

90.0 0.09 0.01 1.95 X !Q- 5.71 

99.0 0.099 0.001 1.75 X 10 ~ 7 6.76 

99.9 0.0999 0.0001 1.75 X 10~ 8 7.76 

100.0 0.10 1.32 X 10- 8.88 

as if the salt were not hydrolyzed. As- the equivalence-point is ap- 
proached closely, however, the concentration of base is greatly reduced 
and so the hydrolysis of the salt becomes appreciable. The form of the 
pH curve is then determined by the fact that the hydrogen ion concen- 
tration at the equivalence-point is given by equation (47). 

The complete curve for the neutralization of 0.1 N acetic acid by 
0.1 N sodium hydroxide and vice versa, is shown in Fig. 102, I; the right- 




2 - 



25 50 75 100 75 50 25 

Per cent Acid Per cent Base 

Neutralized Neutralized 

Fia. 102. Neutralization of weak (I) and very weak (II) acid by strong base 

hand side is almost identical with that of Fig. 101, I, for a strong base 
neutralized by a strong acid. It is observed that in this instance there 
is also a rapid change of pH at the equivalence-point, but it is not so 
marked as for a strong acid at the same concentration. The equivalence- 
point itself, indicated by an arrow, now occurs at pH 8.88, the solution 
of sodium acetate being alkaline because of hydrolysis. If a more dilute 



VERY WEAK ACID AND STRONG BASE 393 

acetic acid solution, e.g., 0.01 N, is titrated with a strong base, the main 
position of the pH-neutralization curve is not affected, as may be seen 
from an examination of the Henderson equation (44); the pH depends 
on the ratio of salt to un-neutralized acid, and this will be the same at a 
given stage of neutralization irrespective of the actual concentration. 
When the neutralization has occurred to the extent of 50 per cent, i.e., 
at the midpoint of the curve, the ratio of salt to acid is always unity; the 
pH is then equal to pk a for the given acid (cf. p. 325), and this does not 
change markedly with the concentration of the solution. At the be- 
ginning and end of the neutralization, when the Henderson equation is 
not applicable, the pH's, given by equations (45) and (47), are seen to be 
dependent on the concentration; for 0.01 N acetic acid the values are 
3.38 and 8.38, respectively, instead of 2.88 and 8.88 for the 0.1 N solution. 

HI. Moderately Strong Acid and Strong Base. If the acid is a mod- 
erately strong one, the pH may be less than 4 for an appreciable part of 
the early stages of the neutralization. The quantity CH+ COH- which 
appears in the term B cannot then be neglected, but it is more accurate 
to neglect COH- only, so that B becomes b + CH + ; under these conditions 
equation (39), neglecting activity coefficients, becomes 



<-- 



This is a quadratic equation which can be readily solved for C H +. The 
pH values for the beginning and end of the titration are derived from 
equations (45) and (47), as before. The pH-neutralization curve for a 
moderately strong acid lies between that of a strong acid (Fig. 101) and 
that of a weak acid (Fig. 102). 

IV. Very Weak Acid and Strong Base. For very weak acids, whose 
dissociation constants are less than about 10~ 7 , or for very dilute solu- 
tions, e.g., more dilute than 0.001 N, of weak acids, the pH of the solution 
exceeds 10 before the equivalence-point is reached. It is then necessary 
to include COH- in B, although C H + can be neglected; equation (39) then 
takes the form 



a b + COU- 

CH* = K a - 7 - 

o coir 
o-6 + fc./cn* 
" ka b- 



( } 

This equation is also a quadratic in CH+, and so it can be solved and C H + 
evaluated. The results for the neutralization of a 0.1 N solution of an 
acid of k a equal to 10~ 9 by a strong base are shown in Fig. 102, II: the 
equivalence-point, indicated by an arrow, occurs at a pH of 11.0. The 
inflexion at the equivalence-point is seen to be small, and it is even less 
marked for more dilute solutions of the acid. It has been calculated that 



394 NEUTRALIZATION AND HYDROLYSIS 

if ak a is less than about 27 k w there is no appreciable change in the slope 
of the pH-neutralization curve as the equivalence-point is attained. 

V. Weak Base and Strong Acid. The equations applicable to the 
neutralization of weak bases are similar to those for weak acids; the only 
alterations necessary are that the terms for 11+ and OH~ are exchanged, 
a and b are interchanged, and kb replaces k a . The appropriate form of 
equation (39), which is fundamental to the whole subject, is 



(50) 
u run' 

where B is now defined by 

B = a + COH CH + . 

The Henderson equation, omitting the activity correction, can be written 
as 

a 
pOH = pkb + log 7 > 

CL 

or 

i salt 

pOH = pA'b + log 7 9 

base 

salt 
.'. pll = pk w pOH = pk w pkb log T (51) 

This equation is applicable over the same pll range as before, viz., 4 to 
10; outside this range COH~ may be neglected in more acid solutions, while 
CH+ can be ignored in more alkaline solutions. At the extremes of the 
neutralization, i.e., for the pure base and the salt, respectively, the pH 
values can be obtained by making the appropriate simplifications of 
equation (50) ; alternatively, they may be derived from considerations of 
the dissociation of the base and of the hydrolyzed salt (cf. p. 390). 

A little consideration will show that the pll-neutralization curves for 
weak bases are exactly analogous to those for weak acids, except that 
they appear at the top right-hand corner of the diagram, with the mid- 
point, at pH 7, as a center of symmetry. The weaker the base and the 
less concentrated the solution, the smaller is the change of potential at 
the equivalence-point, just as in the neutralization of a weak base. 

VI. Weak Acid and Weak Base. The exact treatment of the neu- 
tralization of a weak acid by a weak base is somewhat complicated; it is 
analogous to that for the hydrolysis of a salt of a weak acid and weak 
base to which brief reference was made on page 381. The result is an 
equation of the fourth order in C H K and so cannot be solved easily. The 
course of the pH-neutralization curve can, however, be obtained, with 
sufficient accuracy for most purposes, by the use of approximate equa- 
tions. For the pure weak acid, the pH is given by equation (45) and 
the values up to about 90 per cent neutralization are obtained by the 



DISPLACEMENT REACTIONS 



395 



same equations as were used for the titration of a weak acid by a strong 
base; as long as there is at least 10 per cent of free excess acid the effect 
of hydrolysis is negligible. The pll at the equivalence-point is derived 
from equation (29), based on considerations of the hydrolysis of a salt of 
a weak acid and weak base. The complete treatment of the region 
between 90 and 100 per cent neutralization is somewhat complicated, 
but the general form of the curve can be obtained without difficulty by 
joining the available points. The variation of the pH in the neutraliza- 
tion of a weak base by a weak acid is derived in an analogous manner; 
up to about 90 per cent neutralization the behavior is virtually identical 



12 
10 

8 

pH 

6 

4 



pHT.O 



I 



I 



25 50 75 100 75 GO 25 
Per cent Acid Per cent Base 

Neutralized Neutralized 

FIG. 103. Neutralization of acetic arid by ammonia 

with that obtained for a strong acid. The complete pll-neutralization 
curve for a 0.1 N solution of acetic acid and 0.1 N ammonia, for which 
A; rt arid k b are both taken to be equal to 1.75 X 10~ 5 , is shown in Fig. 103; 
the change of pll is seen to be very gradual throughout the neutralization 
and is not very marked at the equivalence-point. 

Displacement Reactions. In a displacement reaction a strong acid, 
or strong base, displaces a weak acid, or weak base, respectively, from 
one of its salts; an instance which will be considered is the displacement 
of acetic acid from sodium acetate by hydrochloric acid. Since this 
process is the opposite of the neutralization of acetic acid by sodium 
hydroxide, the variation of pH during the displacement reaction will be 
practically identical with that for the neutralization, except that it is in 
the reverse direction. In this particular case, therefore, the pll curve 
is represented by Fig. 102, 1, starting from the midpoint, which represents 



396 NEUTRALIZATION AND HYDROLYSIS 

sodium acetate, and finishing at the left-hand end, representing an equiva- 
lent amount of free acetic acid. It is evident that there is no sharp 
change of potential when the equivalence-point is attained. On the 
other hand, if the salt of a very weak acid, e.g., k a equal to 10~ 9 , is titrated 
with hydrochloric acid, the variation of pH is given by Fig. 102, II, also 
starting from the center and proceeding to the left; a relatively marked 
inflexion is now observed at the equivalence-point, i.e., at the extreme 
left of the figure. 

The foregoing conclusions are in complete harmony with the con- 
cept of acids and bases developed in Chap. IX and of neutralization, 
in its widest sense, to which reference was made at the beginning of the 
present chapter. The reaction between sodium acetate and hydrochloric 
acid, i.e., 

(Na+)Ac- + H 3 0+(C1-) = HAc + H 2 + (Na+Cl-), 
Base Acid Acid Base 

is really the neutralization of the acetate ion base by a strong acid. It 
was seen on page 338 that the dissociation constant of a conjugate base, 
such as Ac~, is equal to k w /k a , where k a is the dissociation constant of the 
acid HAc; in this case k a is 1.75 X 10~ 5 and since k w is 10~ 14 , it follows 
that kb for the acetate ion base is about 5.7 X 10~ 10 . This represents a 
relatively weak base and its neutralization would not be expected to be 
marked by a sharp pH inflexion; this is in agreement with the result 
derived previously. If the acid is a very weak one, however, the con- 
jugate base is relatively strong; for example, if k a is 10~ 9 then k b for the 
anion base is 10~ 5 . The displacement reaction, which is effectively the 
neutralization of the anion base by a strong acid, should therefore be 
accompanied by a change of pH similar to that observed in the neutrali- 
zation of ammonia by-a^stpong acid. 

The arguments given above may be applied equally to the displace- 
ment of a weak base, such as ammonia or an amine, from a solution of 
its salt, e.g., ammonium chloride, by means of a strong base. If the 
amine RNH 2 has a dissociation constant of about 10~ 6 , its conjugate acid 
RNH^~ will be extremely weak, since k a will be 10~ 14 /10~ 5 , i.e., 10~ 9 , and 
the equivalence-point of the displacement titration will not be marked 
by an appreciable inflexion. On the other hand, if the base is a very 
weak one, such as aniline (fa equal to 10~ 10 ), the conjugate ariilinium 
ion acid will be moderately strong, k a about 10~ 4 , and the equivalence- 
point will be associated with a definite pH change. It follows, therefore, 
that only with salts of very weak acids or bases is there any considerable 
inflexion in the pH curve at the theoretical end-point of the displacement 
reaction. 

Neutralization in Non-Aqueous Media. As already seen, the mag- 
nitude of the inflexion in a pll-neutralization curve depends on the dis- 
sociation constant of the acid or base being neutralized; concentration 
is also important, but for the purposes of the present discussion this will 



NEUTRALIZATION IN NON-AQUEOUS MEDIA 397 

be assumed to be constant. Another important factor, which is less 
evident at first sight, is the magnitude of k w ; an examination of the 
equations derived in the previous sections shows that the value of k w 
does not affect the pH during the neutralization of an acid, but it has 
an important influence at the equivalence-point. A decrease of k w will 
result in a decrease of hydrogen ion concentration, i.e., the pH is in- 
creased, at the equivalence point. When a base is being neutralized, the 
value of k w is important, as may be deduced from equation (51); a de- 
crease of k w , i.e., an increase of pA:^, will be accompanied by a corre- 
sponding increase of pH. It may be concluded, therefore, that if the 
ionic product of water is decreased in some manner, the acid and base 
parts of the neutralization curve are drawn apart and the inflexion at 
the equivalence-point is more marked. The two results derived above 
may be combined in the statement that the smaller k w /k y where k is the 
dissociation constant of the acid or base, the greater will be the change 
of pH as the equivalence-point of a neutralization is approached. The 
quantity k w /k is the hydrolysis constant of the salt formed in the reaction; 
hence, as may be expected, the smaller the extent of hydrolysis the more 
distinct is the pH inflexion at the end-point of the neutralization. There 
are thus two possibilities for increasing the sharpness of the approach to 
the equivalence-point; either k w may be decreased, while k a or k* is 
approximately unchanged, or k a or kb may be increased. The same 
general conclusions will, of course, be applicable to any other amphi- 
protic solvent, the quantity k w being replaced by the corresponding ionic 
product. 

For cation acids, e.g., NHt or RNHj, or for anion bases, e.g., CH 8 C05", 
the dissociation constants in ethyl alcohol are only slightly less than in 
water (cf. p. 333), but the ionic product is diminished by a factor of 
approximately 10 6 . It is clear, therefore, from the arguments given 
above that neutralization of such charged acids and bases will be much 
more complete in alcoholic solution than in water. The equivalence- 
points in the neutralization of the anions of acids and of the cations of 
substituted ammonium salts in alcohol have consequently been found to 
be accompanied by more marked inflexions than are obtained in aqueous 
solution. 

The dissociation constants of uncharged acids and bases are dimin- 
ished in the presence of alcohol, and since the ionic product of the solvent 
is decreased to a somewhat similar extent, the inflexion at the equivalence- 
point for these substances is similar to that in water. 

It was seen on page 371 that lyolysis could be avoided and neutraliza- 
tion made more complete when a weak base was neutralized in a strongly 
protogenic medium, such as acetic acid. The use of a solvent with a 
marked proton donating tendency is, effectively, to increase the dis- 
sociation constant of the weak base; hence a sharper change of pH is to 
be expected at the equivalence-point in a strongly protogenic solvent than 
in water. This argument applies to bases of all types, i.e., charged or 



NEUTRALIZATION AND HYDROLYSIS 



uncharged, and the experimental results have been shown to be in ac- 
cordance with anticipation ; the curves in Fig. 104, for example, show the 
change of pH, as measured by a form of hydrogen electrode, during the 
course of the neutralization of the extremely weak bases urea and acet- 

oxirne by perchloric acid in acetic 
acid solution. 8 In aqueous solu- 
tions these bases would show no 
detectable change of pH at the 
equivalence-point. In order to 
increase the magnitude of the 
inflexion in the neutralization of 
a very weak acid it would be nec- 
essary to employ a strongly pro- 
tophilic medium, such as liquid 
ammonia, or one having no proto- 
genic properties, e.g., acetonitrile. 
Neutralization of Mixture of 
Two Monobasic Acids. An ex- 
pression for the variation of the 
pH during the whole course of 
the neutralization of a mixture of 
two monobasic acids by a strong 
base can bo derived, but as it is 
somewhat complicated, simplifi- 
cations are made which are ap- 
plicable to certain specific conditions. Let ai and an be the initial con- 
centrations of the two acids HAi arid II AH, whose dissociation constants 
are fci and &n; suppose that at a certain stage of the neutralization a 
concentration 6 of strong base MOH has been addeH to the mixture of 
acids. If the salts formed when the acids are neutralized are completely 
dissociated, then at any instant 



0.70 



0.60 



0.60 




0.2 0.4 



0.6 0.8 1.0 1.2 
of Perchloric Acid 



Fio. 104. Neutralization of very weak 
based in glacial acetic acid solution 



and 



(52) 
(53) 



where CHA represents in each case the concentration of un-neutralized 
acid while C A - is that of the neutralized acid, the total adding up to the 
initial acid concentration. Since the solution must be electrically neu- 
tral, the sum of the positive charges must equal that of the negative 
charges, i.e., 

CM* + C H + = C A + C A -J -f COH'. (54) 

The salts MAi and MAn are completely dissociated and so CM* may be 

Hall and Werner, /. Am. Chem. Soc., 50, 2367 (1928); Hall, Chem. Revs., 8, 191 
(1031); see also, Nadeau and Branchen, J. Am. Chem. Soc., 57, 1363 (1935). 



NEUTRALIZATION OF MIXTURE OF TWO MONOBASIC ACIDS 399 

identified with 6, the concentration of added base; further, except towards 
the end of the neutralization, COR- may be neglected, and so equation (54) 
becomes 

b + c H + = CAJ + CAJ,. (55) 

The approximate dissociation constants of the two acids are given by 

*. CH * CA J ^ i CH * CA " 

Id = - and ku = - > 

CHA r CHA II 

and if these expressions together with equations (52) and (53) are used 
to eliminate the concentration terms involving Af, AH, as well as HAi and 
II An, from (55), the result is 

, t . 

- b - ( 56 ) 



This is a cubic equation which can be solved to give the value of the 
hydrogen ion concentration at any point of the titration of the mixture 
of acids. 

A special case of interest is that arising when the amount of base 
added is equivalent to the concentration of the stronger of the two acids, 
e.g., HAi; under these conditions b may be replaced by ai, and if all 
terms of the third order in equation (56) are neglected, since they are 
likely to be small, it is found that 

aiCn + + kn(ai - aii)c n + aukiku = 0. 

If ai and an ire not greatly different and ku is small, the second term on 
the left-hand side in this equation can be omitted, so that 



fankikii 
* ~ \ Ql ' 



.'. pll = Jpfri + pfcn + \ log ai \ log an. (58) 

This relationship gives the pH at the theoretical first equivalence-point 
in the neutralization of a mixture of two monobasic acids. If the two 
acids have the same initial concentration, i.e., ai is equal to an, then 
equation (57) for the first equivalence-point becomes 

CH+ Vfcifcn, (59) 

.'. pH = |pfci + Jpfax. (60) 

The pH at the equivalence-point for the acid HAi in the absence of 
HAn is given by equation (12) as 

pll = Jpfc w + Ipki + % log ai, (61) 

and comparison of this with the value for the mixture at the first equiva- 



400 



NEUTRALIZATION AND HYDROLYSIS 



lence-point, the latter being designated by (pH) m , shows that in the 
general case 

pH - (pH) m = Jpfcu, - Jpfcn + 4 log an. (62) 



Since pfcn is generally less than pfc*,, the quantity pH (pH) m is posi- 
tive; the pH at the first equivalence-point of a mixture is thus less 
than that for the stronger acid alone. This result indicates a flattening 
of the pH curve in the vicinity of the first equivalence-point, the extent 
of the flattening being, according to equation (62), more marked the 
smaller pfcn, i.e., the stronger the acid HAn, and the greater its concen- 
tration. If the acid HAn is very weak or its concentration small, or 
both, the flattening at the first equivalence-point will be negligible, and 
the neutralization curve of the mixture will differ little from that of the 
single acid HAi. 

Equations for the variation of pH during the course of neutralization 
beyond the first equivalence-point, similar to those already given, could 
be derived if necessary, but for most requirements a simpler treatment 
of the whole neutralization curve is adequate. At the very commence- 
ment of the titration the pH is little different from that of the solu- 
tion of the stronger acid, and during the early stages of neutralization 
the pH is close to that which would be given by this acid alone. In 
the vicinity of the first equivalence-point deviations occur, but these 
can be inferred with sufficient accuracy from the pH at that point, as 
given by equations (58) or (60). At a short distance beyond the first 
equivalence-point the pH is close to that for the neutralization of the 
second acid alone; the pH at the final equivalence-point is the same as 
that of the salt NaAn and is consequently given by equation (12). A 

satisfactory idea of the com- 
plete neutralization curve can 
thus be obtained by plotting 
the curves for the two acids 
separately side by side, the 
curve for the stronger acid 
(HAi) being at the left; the 
two curves are then joined by 
a tangent (Fig. 105). The re- 
gion between the two curves 
may be fixed more exactly 
by making use of equation 
(58) or (60) for the first equiva- 
lence-point. The figure shows 
clearly that if the weaker acid 
HAn is moderately weak, as at HA, the inflexion at the first equiva- 
lence-point will be negligible, but if it is very weak, as at II B, the 
inflexion will not differ appreciably from that given by the acid HAi 
alone. A decrease in the concentration of HAu makes the pH higher 



PH 



ii 




Neutralization of HA K Neutralization of HA u 
FIG. 105. Neutralization of mixture of acids 



NEUTRALIZATION OF DIBASIC ACID BY A STRONG BASE 401 

at the beginning of the HAn curve and so increases the inflexion to some 
extent, in agreement with the conclusion already reached. If HAi is a 
strong acid, e.g., hydrochloric acid, and HAii is a weak acid, the pH 
follows that for the neutralization of the strong acid alone almost exactly 
up to the first equivalence-point. 

Neutralization of Dibasic Acid by a Strong Base. If the first stage 
of the dissociation of the dibasic acid corresponds to that of a strong acid 
while the second is relatively weak, e.g., chromic acid, the system behaves 
virtually as two separate acids. The first stage is neutralized as a normal 
strong acid, then the second stage becomes neutralized independently 
as a weak acid. When both stages are relatively weak, however, there 
is some interference between them, and the variation of pH during the 
course of neutralization may be calculated by means of equations de- 
rived in Chap. IX. 

For the present purpose, equation (34), page 326, for the hydrogen 
ion activity of a solution of a dibasic acid, of initial concentration a moles 
per liter, to which has been added a concentration of b equiv. per liter 
of strong base, may be written as 

n _ D 

cfi * 2^Te = CH * to~=~B kl + klk2 ' (63) 

The activity coefficients have been omitted and the approximate func- 
tions ki and fe, for the two stages of dissociation of the dibasic acid, have 
replaced the corresponding thermodynamic constants. The quantity B 
is defined as 

B s b + c n + - COH-, 

and insertion of this value in equation (63) gives a quartic equation for 
CH+, which can be solved if necessary. For a considerable range of the 
neutralization it is possible to neglect COH~ in the expression for B } and 
so the equation reduces to a cubic. 

At the first equivalence-point, a is equal to b and if COET is neglected, 
as just suggested, it follows from equation (63) that 



Since CH+ is generally small in comparison with o, this equation reduces to 

2 kik 2 a 
c ^-k~Ta' (65) 

which is identical, as it should be, with equation (30), for at the first 
equivalence-point in the neutralization of the dibasic acid H 2 A the system 
is identical with a solution of NaHA. If k\ is small, equation (65) be- 
comes, as before, _ 

CH+ = V*S, (66) 

.'. pH = ipfe! + ipfe. (66a) 



402 NEUTRALIZATION AND HYDROLYSIS 

It will be noted that this result is the same as equation (59) for the first 
equivalence-point in the neutralization of a mixture of equivalent amounts 
of two weak acids. For a dibasic acid with a very weak first stage dis- 
sociation, it may not be justifiable to neglect COH~; under these conditions, 
however, CH+ may be ignored, and the corresponding equations can be 
derived. 

The form of the pll-neutralization curve for a dibasic acid can be 
represented in an adequate manner by the method used for a mixture of 
acids; the curves for the two stages are drawn side by side, from the 
individual dissociation constants k\ and k% treated separately, and then 
joined by a tangent. The general conclusions drawn concerning the 
inflexion at the first equivalence-point are similar to those for a mixture 
of acids; the essential requirement for a dibasic acid to show an appre- 
ciable inflexion at the first equivalence-point is that k\/kz should be large. 
Under these conditions the individual pH-neutralization curves for the 
two stages of the dibasic acids are relatively far apart and the tangent 
joining them approaches a vertical direction. 

Distribution of Strong Base between the Stages of a Dibasic Acid. 
During the course of neutralization of a dibasic acid, the system will con- 
sist of undissociatcd molecules HkA and of the ions HA~ and A ; the 
fraction of the total present as HA~ ions, i.e., a\ 9 is then 







JA + CHA- + C A - 
while that present as A ions, i.e., 2 , is 



. . r 

Cn 2 A -r CHA~ ~r c\~~ 

Since the HA~ ions arise almost entirely from NaHA, assuming the base 
to be sodium hydroxide, while the A ions originate mainly from NaA, 
it follows that i represents, approximately, the fraction of the dibasic 
acid neutralized in the first stage only, while 0.1 is the fraction neutralized 
in both stages. By using the familiar expressions for the first and second 
stage dissociation functions (p. 381) to eliminate CA~ from equation (67) 
and CHA~ from equation (68), the results are 



and 

2 = - - - -27- (70) 

5! . i i li! 
b ^ "*" kfa 

It is thus possible, by means of equations (69) and (70), to evaluate the 



NEUTRALIZATION OF POLYBASIC ACIDS AND MIXTURES OF ACIDS 403 

fractions of HA" and of A present at any pTI for a given dibasic acid, 
provided ki and 7c 2 are known. As is to be expected, the fraction present 
as HA~, i.e., i, increases at first as neutralization proceeds; the value 
then reaches a maximum and falls off to zero when both stages of the 
acid are completely neutralized. The fraction of A , on the other hand, 
increases slowly at first and then more rapidly and finally approaches 
unity when neutralization is complete and the system consists entirely of 
Na2A. Many interesting conclusions can be drawn from the curves for 
different values of ki and fc> concerning the pi I at whi^h the second stage 
neutralization becomes appreciable, and so on; the main results have, 
however, already been obtained from a consideration of the pH-neutrali- 
zation curves. 6 

The point at which the fraction i attains a maximum can be derived 
by writing 1/ai by means of equation ((59) as 



tti /:i CH* 

differentiating with respect to CH+, thus 



!_ _ A- 2 _ 

C/CH+ A*i CH* 

and equating to zero, since l/i must be a minimum when i is a maxi- 
mum. It follows, therefore, that 



_ _ 



/. CH+ = VA^, (71) 

under these conditions. According to equation (06) this is, approxi- 
mately, the hydrogen ion concentration at the first equivalence-point; 
hence the fraction of the total acid in the form of II A" ions is greatest 
at this point. 

Neutralization of Polybasic Acids and Mixtures of Acids. The treat- 
ment of a system consisting of a tribasic or higher acid, or of a mixture 
of three or more simple acids is complicated, but the general nature of 
the results can be obtained in the manner already described. The 
pH-neutralization curve for the whole system is obtained with a fair 
degree of accuracy by drawing the separate curves for the individual 
stages of neutralization of the polybasic acid, or for the individual acids 
in a mixture of acids, in the order of decreasing dissociation constants, 
and connecting them by means of tangents in the usual way. The pH's 
at the various equivalence-points can be fixed by using a relationship 
similar to equation (66); the pH at the nth equivalence-point, i.e., when 

6 Michaelis, "Hydrogen Ion Concentration," translated by Perlzweig, 1926, p. 55. 



404 NEUTRALIZATION AND HYDROLYSIS 

sufficient strong base has been added to neutralize the first n stages, or 
n acids, is given by 

pH - ipfc w + ipfcn+i, (72) 



where pfc and pfc+i are the dissociation exponents for the nth and 
(n + l)th stages, respectively, of a polybasic acid, or of the nth and 
(n + l)th acids in a mixture arranged in order of decreasing strength. 

Another useful method for considering the neutralization of polybasic 
acids or mixtures of acids, which avoids the necessity of plotting curves, 
is the following. In general, when an acid is neutralized to the extent 
of 0.1 per cent, i.e., salt/acid is 1/999, the pH, according to the approxi- 
mate Henderson equation, is 

pH = pk a + log 7 fg- 

Pk a - 3. 

It follows, therefore, that in a polybasic acid system, or in a mixture of 
approximately equivalent amounts of different acids, the neutralization of 
a particular stage or of a particular acid may be regarded as commencing 
effectively when the pH is equal to pfc n +i 3, where pfc n +i is the dis- 
sociation of the (n + l)th stage or acid; at this point the pH-neutraliza- 
tion curve for the mixture will commence to diverge from that of the 
previous stage of neutralization. Similarly, when an acid is 99.9 per cent 
neutralized, 



pk a + 3. 

The neutralization of any stage may, therefore, be regarded as approxi- 
mately complete when the pH of the system is equal to pA; n + 3, wherf 
pfc n is the dissociation exponent for the nth stage of a polybasic acid or 
for the nth acid in a mixture. If this pH is less than pA; n +i 3, the 
neutralization of the nth stage will be substantially complete before that 
of the (n + l)th stage commences; if this condition holds, i.e., if pfc n +i 3 
> pfc n + 3, the neutralization of the weaker acid, or stage, will have no 
appreciable effect on that of the stronger. It is seen, therefore, that if 
pfcn+i pk n is greater than 6, or fc n /fc n +i is greater than 10 6 , the pH- 
neutralization curve for the mixture will show no appreciable divergence, 
at the n-h equivalence-point, from that of the nth acid alone. The 
inflexion at the nth equivalence-point will then be as definite as for the 
single nth acid. If pfc+i pk n is less than 6, the neutralization of the 
(n + l}th acid, or stage, commences before that of the nth acid is com- 
plete, and the result will be a flattening of the pH-curve at the nth 
equivalence-point; if fc n /fc n +i is less than 16, there is no detectable in- 
flexion in the pH-neutralization curve. 

Potentiometric Titrations. 7 The general conclusions drawn from the 
treatment in the foregoing sections provide the basis for potentiometric, 

7 See general references to potentiometric titrationa on page 256. 



POTENTIOMETRIC TITRATIONS 405 

as well as ordinary volumetric, titrations of acids and bases. The poten- 
tial E of any iorm of hydrogen electrode, measured against any con- 
venient reference electrode, is related to the pH of the solution by the 
general equation 

RT 
E = E nft + -TT pH, 

or, at ordinary temperatures, i.e., about 22, 

E = # r ef. + 0.059 pH, 

where E re i. is a constant. It is apparent, therefore, that the curves rep- 
resenting the variation of pH during neutralization are identical in form 
with those giving the change of hydrogen electrode potential. It should 
thus be possible to determine the end-point of an acid-base titration by 
measuring the potential of any convenient form of hydrogen electrode 
at various points and finding the amount of titrant at which the potential 
undergoes a sharp inflexion. The underlying principle of the poten- 
tiometric titration of a neutralization process is thus fundamentally the 
same as that involved in precipitation (p. 256) and oxidation-reduction 
titrations (p. 285). The position of the end-point is found either by 
graphical determination of the volume of titrant corresponding to the 
maximum value of A/?/ At;, where A# is the change of hydrogen electrode 
potential resulting from the addition of Ay of titrant, or it can be deter- 
mined by a suitable adaptation of the principle of differential titration. 
The apparatus described on page 261 (Fig. 77) can, of course, be em- 
ployed without modification with glass or quinhydrone electrodes; if 
hydrogen gas electrodes are used, however, the electrodes are of platinized 
platinum and the hydrogen must be used for operating the gas-lift, the 
stream being shut off before each addition of titrant so as to avoid 
mixing. Any form of hydrogen electrode can be used for carrying out a 
potentiometric neutralization titration, and even oxygen gas and air 
electrodes have been employed; since all that is required to be known 
is the point at which the potential undergoes a rapid change, the irre- 
versibility of these electrodes is not a serious disadvantage. Potentio- 
metric determinations of the end-point of neutralization reactions can be 
carried out with colored solutions, and often with solutions that are too 
dilute to be titrated in any other manner. 

The accuracy with which the end-point can be estimated obviously 
depends on the magnitude of the inflexion in the hydrogen potential- 
neutralization curve at the equivalence-point, and this depends on the 
dissociation constant of the acid and base, and on the concentration of 
the solution, as already seen. When a strong acid is titrated with a 
strong base, the change of potential at the equivalence-point is large, 
even with relatively dilute solutions (cf. Fig. 101), and the end-point 
can be obtained accurately. If a weak acid and a strong base, or vice 
versa, are employed the end-point is generally satisfactory provided the 



406 NEUTRALIZATION AND HYDROLYSIS 

solutions are not too dilute or the acid or base too weak (cf . Fig. 102, I) ; 
if c is the concentration of the titrated solution and k a or ki the dissocia- 
tion constant of the weak acid or base being titrated, by a strong base 
or acid, respectively, then an appreciable break occurs in the neutraliza- 
tion curve at the end-point provided ck a or ck b is greater than 10~ 8 . 
Titrations can be carried out potentiometrically even if ck a or ckb is less 
than 10~ 8 , but the results are less accurate (cf. Fig. 102, II). The poten- 
tiometric titration of very weak bases can, of course, be carried out 
satisfactorily in a strongly protogenic medium (cf. Fig. 104). When a 
weak acid and weak base are titrated against one another the change of 
pH at the end-point is never very marked (Fig. 103), but if potential 
measurements are made carefully, an accuracy of about 1 per cent may 
be obtained with 0.1 N solutions by determining graphical ly the position 
at which AE/Av is a maximum. The principles outlined above apply, 
of course, to displacement reactions, which are to be regarded as involving 
neutralization in its widest sense. Such titrations can be performed 
accurately in aqueous solution if the acid or base that is being displaced 
is very weak; in other cases satisfactory end-points may be obtained in 
alcoholic solution. 

The separate acids in a mixture of acids, or bases, can often be titrated 
potentiometrically, provided there is an appreciable difference in their 
strengths: this condition is realized if one of the acids is strung, e.g., a 
mineral acid, and the other is weak, e.g., an organic tyul. It has been 
seen that if the ratio of the dissociation constants of two acids exceeds 
about 10 6 , the weaker does not interfere with the neutralization of the 
stronger acid in the mixture; this conclusion does not take into account 
the influence of differences of concentration, and it is more correct to say 
that Ciki/ciikn should be greater than 10 6 whore ci and k\ are the concen- 
tration and dissociation constant of one acid and CH and k\\ that of the 
other. If this condition is combined with that previously given for 
obtaining a satisfactory end-point with a single arid, ihe following con- 
clusions may be drawn: if Ciki and CH/TH both exceed 10~ 8 and cjtilcnku 
is greater than 10 6 , accurate titration of the separate acids in the mixture 
is possible. If Ciki/cuku is less than 10 6 the firot equivalence-point cannot 
be very accurate even if c\ki is greater than 10~ 8 , but an accuracy of 
about 1 per cent can be achieved by careful titration even if ciki/cukn is 
as low as 10 4 . When the fir&t equivalence-point is not detectable, the 
second equivalence-point, representing neutralization of both acids, may 
still be obtained provided cufcn exceeds 10" 8 . The general relationship 
applicable to mixtures of acids can be extended to polybasic acids, al- 
though in the latter case a and CH are equal. 8 

In the titration of a strong acid and a strong base the cquivaleiije- 
point corresponds exactly to the point on the pH-neutralizatiori curve, 
or the potential-titration curve, at which the slope is a maximum. This 

Noyes, J. Am. Chem. Soc., 32, 815 (1910); see also, Tizard and Boeree, J. Chem. 
Soc., 121, 132 (1922); Koltboff and Furman, "Indicators/* 1926, p. 121. 



NEUTRALIZATION TITRATIONS WITH INDICATORS 407 

is not strictly true, however, in the case of the neutralization of a weak 
acid or a weak base; if (CH + ) P is the hydrogen ion concentration at the 
potentiometric end-point, i.e., where AE/Av is a maximum, and (C H +) is 
thg, value at the theoretical, or stoichiometric, equivalence-point, it can 
be shown that 

! c J^i , 3 
(CH*). ~ * 

Provided ak a is greater than 10~ 8 , which is the condition for a satisfactory 
point of inflexion in the titration curve, the ratio of the two hydrogen 
ion concentrations differs from unity by about one part in 700; this would 
be equivalent to a potential difference of 0.016 millivolt and h^nce is well 
within the limits of experimental error. 

Neutralization Titrations with Indicators. Since, as seen on page 362, 
an acid-base indicator changes color within a range of approximately one 
unit of pH on either .side of a pH value equal to the indicator exponent 
(pfcin), such indicators are frequently used to determine the end-points 
of neutralization titrations. 9 The choice of the indicator for a particular 
titration can best be determined from an examination of the pH-neutrali- 
zation curve. Before proceeding to consider this aspect of the problem 
it is useful to define the titration exponent (pkr) of ari indicator; this is 
the pTl of a solution at which the indicator shows the color usually 
associated with the end-point when that indicator is employed in a 
neutralization titration. It is the general practice in such work to titrate 
from the lighter to the darker color, e g., colorless to pink with phenol- 
phthalein and yellow to red with methyl orange; as a general rule a 
20 per cent conversion is necessary before the color change can be defi- 
nitely detected visually, and so if the darker colored form is the one 
existing in alkaline solution, it follows from the simple Henderson equa- 
tion (cf. p. 390) that 

20 

pH = pk T = phn + log 



This approximate relationship between the titration exponent and pki n 
is applicable to phenolphthalein and to many of the sulfonephthalein 
indicators introduced by Clark and Lubs (sec Table LXIV, page 364). 
If the darker color is obtained in acid solution, as is the case with methyl 
orange and methyl red, then it is approximately true that 

80 
pH = pk T = pfcin + log 

= pki n + 0.6. 

The results quoted in Table LXX1I give the titration exponents based 
Kolthoff and Furman, "Indicators," 1926, Chap. IV. 



408 



NEUTRALIZATION AND HYDROLYSIS 



TABLE LXXII. TITRATION EXPONENTS OF USEFUL INDICATORS 



Indicator 
Bromphenol blue 
Methyl orange 
Methyl red 
Bromcresol purple 
Bromthymol blue 
Phenol red 
Cresol red 
Thymol blue 
Phenolphthalein 
Thymolphthalein 



pkr 

4 

4 

5 

6 

6.8 
7.5 

8 
8.8 

9 

10 



End-point Color 
Purplish-green 
Orange 
Yellowish-red 
Purplish-green 
Green 
Rose-red 
Red 

Blue-violet 
Pale rose 
Pale blue 



on actual experimental observations, together with practical information, 
for a number of indicators which may be useful for neutralization titra- 
tions; they cover the pH range of from about 4 to 10, since titration 
indicators are seldom employed outside this range. 

In order that a particular indicator may be of use for a given acid- 
base titration, it is necessary that its exponent should correspond to a 
pH on the almost vertical portion of the pH-neutralization curve. When 
the end-point of the titration is approached the pH changes rapidly, and 
the correct indicator will undergo a sharp color change. The choice of 
indicator may be readily facilitated by means of Fig. 106 in which the 

Indicators 



PH 




I 



Alizarine yellow 
Thymol phthalein 
Phenol phthalein 
Phenol red 
Bromthymol blue 
Bromcreeol purple 
Methyl red 
Methyl orange 
Bromphenol blue 
Thymol blue 



26 60 75 100 76 60 25 

Per cent Add Per cent Base 

Neutralized Neutralized 

Fio. 106. Neutralization curves for various acids and bases 

pH-neutralization curves for a number of acids and bases of different 
strengths are plotted, while at the right-hand side a series of indicators 
are arranged at the pH levels corresponding to their titration exponents. 
The positions of the equivalence-points for the various types of neutrali- 
zation are marked by arrows. The curves IA, HA and IIlA show pH 
changes during the course of neutralization of 0.1 N solutions of a strong 
acid, a normally weak acid (k a = 10~*) and a very weak acid (k a = 10"'), 



NEUTRALIZATION TITRATIONB WITH INDICATORS 409 

respectively; curves IB, HB and Ills refer to 0.1 N solutions of a strong 
base, a normally weak base (kb = 10~ 6 ) and a very weak base (k b = 10~ 9 ), 
respectively. The complete titration curve for any particular acid and 
base is obtained by joining the appropriate individual curves. 

In the titration of 0.1 N strong acid by 0.1 N strong base (curve 
IA-!B), the pH of the solution undergoes a very sharp change from pH 4 
to pH 10 within 0.1 per cent of the equivalence-point (see Table LXIX); 
any indicator changing color in this range can, therefore, be used to give 
a reliable indication when the end-point is reached. Consequently, both 
phenolphthalein, pfcr equal to 9, and methyl orange, p&r equal to 4, may 
be employed to give almost identical results in this particular titration. 
If the solutions are diluted to 0.01 N, however, the change of pH at the 
equivalence-point is less sharp, viz., from 5 to 9; methyl orange will, 
therefore, undergo its color change before the end-point is attained, and 
the titration value would consequently be somewhat too low. When a 
0.1 N solution of an acid of k a equal to 10~ 5 is titrated with a strong base, 
the equivalence-point is at pH 9, and there is a fairly sharp increase from 
pH 8 to 10 (curve HA-!B); of the common indicators phenolphthalein is 
the only one that is satisfactory. The less familiar cresolphthalein or 
thymol blue (second range) could also be used. Any indicator having a 
titration exponent below 8 is, of course, quite unsatisfactory. In the 
titration of 0.1 N base of k b equal to 10~ 6 , the equivalence-point is at 
pH 5, and the change of potential between pH 4 to 6 is rapid (curve 
lA-IIs). Methyl orange is frequently used for such titrations, e.g., 
ammonia with hydrochloric acid, but it is obvious that the results cannot 
be too reliable, especially if the solutions are more dilute than 0.1 N; 
methyl red is a better indicator for a base whose dissociation constant is 
about 10~ 6 . 

It will be evident that if the indicator color is to change sharply at 
the required end-point, the pH-neutralization curve must rise rapidly at 
this point. If this curve is not almost vertical, the pH changes slowly 
and the indicator will show a gradual transition from one color to the 
other; under these conditions, even if the correct indicator has been 
chosen, it will be impossible to detect the end-point with any degree of 
accuracy. In general, the condition requisite for the accurate estima- 
tion of a potentiometric end-point, i.e., that ck a or ck* should exceed 10~ 8 , 
is also applicable to titration with an indicator; if ck is less than this 
value, the results are liable to be in error. They can, however, be im- 
proved by using a suitable indicator and titrating to the pH of the 
theoretical equivalence-point by means of a comparison flask containing 
a solution of the salt formed at the end-point, together with the same 
amount of indicator. This procedure may be adopted if it is necessary 
to titrate a very weak acid or base (curves IIlA-Is and lA-IIIs) or a 
moderately weak acid by a weak base (curve IlA-IIs); in none of these 
instances is there a sharp change of pH at the equivalence-point. 



410 NEUTRALIZATION AND HYDROLYSIS 

Displacement reactions may be treated as neutralizations from the 
standpoint of the foregoing discussion. If the acid or base displaced is 
moderately weak, i.e., k a or /r& is about 10~ 5 , the displacement reaction is 
equivalent to the neutralization of a very weak base or acid, with kb or 
k a equal to 10~ 9 , respectively; no indicator is likely to give a satisfactory 
end-point in aqueous solution, although one may possibly be obtained 
in an alcoholic medium (cf. p. 396). If the acid or base being displaced 
is very weak, e.g., carbonic acid from a carbonate or boric acid from a 
borate, there is a marked pH inflexion at the equivalence-point which can 
be detected with fair accuracy by means of an indicator. 

The problem of the detection of the various equivalence-points in a 
mixture of acids of different concentrations or in a solution of a polybasic 
acid is essentially the same as that already discussed on page 406 in 
connection with potentiometric titration, and need not be treated further 
here. Where the conditions are such that the determination of an accu- 
rate end-point appears feasible, the appropriate indicator is the one whose 
pfcm value lies close to the pH at the required equivalence-point. 

Buffer Solutions. It is evident from a consideration of pH-neutrali- 
zation curves that there are some solutions in which the addition of a 
small amount of acid or base produces a marked change of pH, whereas 
in others the corresponding change is very small. A system of the latter 
type, generally consisting of a mixture of approximately similar amounts 
of a conjugate weak acid and base, is said to be a buffer solution; the 
resistance to change in the hydrogen ion concentration on the addition 
of acid or alkali is known as buffer action. The magnitude of the buffer 
action of a given solution is determined by its buffer capacity ; 10 it is 
measured by the amount of strong base required to produce unit change 
of pH in the solution, thus : 

db 
Buffer capacity (ft) = ., v^: 

An indication of the buffer capacity of any acid-base system can thus be 
obtained directly from the pH-neutralization curve; if the curve is flat, 
d(pH)/db is obviously small and the buffer capacity, which is the recipro- 
cal of this slope, is large. An examination of curves I A and IB, Fig. 
106, shows that a relatively concentrated solution of strong acid or base 
is a buffer in regions of low or high pH, respectively. A solution of 
a weak acid or a weak base alone is not a good buffer, but when an appre- 
ciable amount of salt is present, i.e., towards the middle of the individual 
neutralization curves HA, III A, II B or Ills, the buffer capacity of the 
system is very marked. As the equivalence-point is approached the pH 
changes rapidly and so the buffer capacity of the salt solution is small. 
If the acid or base is very weak, or if both are moderately weak, the slope 

"van Slyke, /. Biol Chem., 52, 525 (1922); Kilpi, Z. physikal. Chem., 173, 223 
(1935). 



BUFFER SOLUTIONS 411 

of the pH curve at the equivalence-point is not very great and hence the 
corresponding salts have moderate buffer capacity. 

The buffer action of a solution of a weak acid (HA) and its salt (A~~), 
i.e., its conjugate base, is explained by the fact that the added hydrogen 
ions are "neutralized" by the anions of the salt acting as a base, thus 

H 3 0+ + A- = H 2 + HA, 

whereas added hydroxyl ions are removed by the neutralization 
OH- + HA = H 2 O + A-. 

According to the Henderson equation the pH of the solution is deter- 
mined by the logarithm of the ratio of the concentrations of salt to acid; 
if this ratio is of the order of unity, it will not be greatly changed by the 
removal of A~ or HA in one or other of these neutralizations, and so its 
logarithm will be hardly affected. The pH of the solution will conse- 
quently not alter very greatly, and the system will exert buffer action. 
If the buffer is a mixture of a weak base (B) and its salt, i.e., its conjugate 
acid (BH+), the corresponding equations are 

H 8 0+ + B = H 2 O + BH+ 
and 

OH- + BH+ = H 2 O + B. 

In this case the pH depends on the logarithm of the ratio of B to BH+, 
and this will not be changed to any great extent if the buffer contains the 
weak base and its salt in approximately equivalent amounts. 

By the treatment on page 323, the initial concentration of acid, 
a moles per liter, is equal, at any instant, to the sum of the concentrations 
of HA and A", i.e., 

a = CHA + C A -, (73) 

and according to the condition for electrical neutrality, 

b + C H + = c A - + COH-, (74) 

where 6 is the concentration of base added at that instant; since the salt 
MA is completely dissociated the concentration of M+ ions, CM*, has 
been replaced by 6 in equation (74). Writing k a for the dissociation 
function of the acid, in the usual manner, 

__ C H* CA " 
CH\ 

and utilizing the value of CHA as a CA~ given by equation (73), it is 
found that 



Substitution of this expression for CA~, and k w /CR+ for COH~, in equation 



412 NEUTRALIZATION AND HYDROLYSIS 

(74), yields the result 

_ dk/a tow 



Ilemembering that pH is defined, for present purposes, as log CH+, 
differentiation of this equation with respect to pH gives the buffer 
capacity of the system, thus 



ft = = 2.303 Tr, + ** + ' (75) 





In the effective buffer region the buffer capacity is determined almost 
exclusively by the first term in the brackets; hence, neglecting the other 
terms, it follows that 

- (76) 



The quantity a represent^ the total concentration of free acid and salt, 
and so the buffer capacity is proportional to the total concentration of 
the solution. 

To find the pH at which ft is a maximum this expression should be 
differentiated with respect to pH and the result equated to zero; thus 



/. fc = C H +. (77) 

It follows, therefore, that the buffer capacity is a maximum when the 
hydrogen ion concentration of the buffer solution is equal to the dis- 
sociation constant of the acid. This condition, i.e., pH is equal to pk a , 
arises when the solution contains equivalent amounts of the acid and 
its s.ilt; such a system, which corresponds to the middle of the neutrali- 
zation curve of the acid, has the maximum buffer capacity. The actual 
value of j3 at this point is found by inserting the condition given by (77) 
into equation (76) ; the result is 

2.303 

0mux. =-J-> (78) 

and so it is independent of the actual dissociation constant. Exactly 
analogous results can, of course, be deduced for buffer systems consisting 
of weak bases and their salts, although it is convenient to consider them 
as involving the cation acid (BH+) and its conjugate base (B). The 
conclusions reached above then hold exactly; the dissociation constant 
k a refers to that of the acid BH+, and is equal to k w /k b , where fa is that 
of the base B. 

Buffer Capacity of Water. According to equation (74), the condition 
for electrical neutrality, when a strong base of concentration 6 has been 



PREPARATION OP BUFFER SOLUTIONS 413 

added to water or to a solution containing a strong acid HA, is 
b = C A -- CH+ + COH- 
= C A -- CH+ + A^/CH*, 

and differentiation with respect to pH, i.e., log C H +, gives the buffer 
capacity 0H 2 o of water as 



= 2.303(c H + + COH-). (79) 

It should be noted that the further addition of base does not affect the 
concentration of A~ and so its derivative with respect to pH is zero. 
The buffer capacity of water, as given by equation (79), is negligible 
between pH values of 2.4 and 11.6, but in more strongly acid, or more 
strongly alkaline, solutions the buffer capacity of "water" is evidently 
quite considerable. This conclusion is in harmony with the fact that the 
pH-neutralization curve of a strong acid or strong base is relatively flat 
in its early stages. 

Preparation of Buffer Solutions. The buffer capacity of a given acid- 
base system is a maximum, according to equation (77), when there are 
present equivalent amounts of acid and salt; the hydrogen ion concen- 
tration is then equal to k a and the pH is equal to pk a . If the ratio of 
acid to salt is increased or decreased ten-fold, i.e., to 10 : 1 or 1 : 10, the 
hydrogen ion concentration is then 10k a or Q.lk a , and the pH is pk a 1 
or pfc a + 1, respectively. If these values for CH+ are inserted in equation 
(76), it is found that the buffer capacity is then 



which is only about one-third of the value at the maximum. If the pH 
lies within the range of pk a 1 to pk a + 1 the buffer capacity is appre- 
ciable, but outside this range it falls off to such an extent as to be of 
relatively little value. It follows, therefore, that a given acid-base buffer 
system has useful buffer action in a range of one pll unit on either side 
of the pk a of the acid. In order to cover the whole range of pH, say from 
2.4 to 11.6, i.e., between the range of strong acids and bases, it is necessary 
to have a series of weak acids whose pk a values differ by not more than 
2 units. 

To make a buffer solution of a given pH, it is first necessary to choose 
an acid with a pk a value as near as possible to the required pH, so as to 
obtain the maximum buffer capacity. The actual ratio of acid to salt 
necessary can then be found from the simple Henderson equation 

TT >ii 8alt ' 
P H = pfc a + log 



414 



NEUTRALIZATION AND HYDROLYSIS 



provided the pH lies within the range of 4 to 10. If the required pH is 
less than 4 or greater than 10, it is necessary to use the appropriate form 
of equation (40), where B is defined by (42). Sometimes a buffer solu- 
tion is made up of two salts representing different stages of neutralization 
of a polybasic acid, e.g., NaH 2 PO 4 and Na 2 HPO 4 ; in this case the former 
provides the acid H^POr while the latter is the corresponding salt, or 
conjugate base HPO". 

In view of the importance of buffer mixtures in various aspects of 
scientific work a number of such solutions have been made up and their 
pH values carefully checked by direct experiment with the hydrogen gas 
electrode. By following the directions given in each case a solution of 
any desired pH can be prepared with rapidity and precision. A few of 
the mixtures studied, and their effective ranges, are recorded in Table 
LXXIII; 11 for further details the original literature or special mono- 
graphs should be consulted. 



TABLE LXXIiI. BUFFER SOLUTIONS 



Composition 



Hydrochloric acid and 

Potassium chloride 
Glycine and Hydrochloric acid 
Potassium acid phthalate and 

Hydrochloric acid 
Sodium phenylacetate and 

Phenylacetic acid 
Succinic acid and 

Borax 
Acetic acid and 

Sodium acetate 
Potassium acid phthalate and 

Sodium hydroxide 
Disodium hydrogen citrate and 

Sodium hydroxide 



pH 
Range 



1.0-2.2 
1.0-3.7 

2.2-3.8 
3.2-4.9 
3.0-5.8 
3.7-5.6 
4.0-6.2 
5.0-6.3 



Composition 



Potassium dihydrogen phosphate 

and Sodium hydroxide 
Boric acid and Borax 
Diethylbarbituric acid and 

Sodium salt 
Borax and Hydrochloric 

acid 
Boric acid and 

Sodium hydroxide 
Glycine and 

Sodium hydroxide 
Borax and 

Sodium hydroxide 
Disodium hydrogen phosphate 

and Sodium hydroxide 



PH 
Range 



5.8- 8.0 
6.8- 9 2 

7.0- 9.2 
7.6- 9.2 
7.8-10.0 
8.2-10.1 
9.2-11.0 
11.0-12.0 



Each buffer system is generally applicable over a limited range, viz., 
about 2 units of pH, but by making suitable mixtures of acids and acid 
salts, whose pk a values differ from one another by 2 units or less, it is 
possible to prepare a universal buffer mixture; by adding a pre-deter- 
mined amount of alkali, a buffer solution of any desired pH from 2 to 
12 can be obtained. An example of this type of mixture is a system of 
citric acid, diethylbarbituric acid (veronal), boric acid and potassium 
dihydrogen phosphate; this is virtually a system of seven acids whose 
exponents are given below. 

11 For details concerning the preparation of buffer solutions, see Clark, "The De- 
termination of Hydrogen Ions," 1928, Chap. IX; Britton, "Hydrogen Ions," 1932, 
Chap. XI; Kolthoff and Rosenblum, "Acid-Base Indicators," 1937, Chap. VIII. 



INFLUENCE OP IONIC STRENGTH 415 



Citric acid Citric acid Citric acid H 2 POr Veronal Boric acid 
1st stage 2nd stage 3rd stage 
pka 3.06 4.74 5.40 7.21 7.43 9.24 12.32 

Apart from the last two acids, the successive pfc values differ by less 
than 2 units, and so the system, when appropriately neutralized, is 
capable of exhibiting appreciable buffer capacity over a range of from 
pH 2 to 12. 

Influence of Ionic Strength. In the discussion so far the activity 
factor has been omitted from the Henderson equation, and so the results 
may be regarded as applicable to dilute solutions only. Further, the 
pH values recorded in the literature for given buffer solutions apply to 
systems of exactly the concentrations employed in the experiments; if 
the solution is diluted or if a neutral salt is added, the pH will change 
because of the alteration of the activity coefficients which are neglected 
in the simple Henderson equation. In order to make allowance for 
changes in the ionic strength of the medium, and of the accompanying 
changes in the activity coefficients, it is convenient to use the complete 
form of the Henderson equation with the activity coefficients expressed 
in terms of the ionic strength by means of the Debye-Hlickel relation- 
ship; as shown on page 326, this may be written as 



pH = pK n + log - (2n - 1) A + C v , (81) 

a> Jo 

where pK n is the exponent for the nth stage of ionization of the acid, 
and B has the same significance as before [cf. equation (42)]. If the 
pH lies between 4 and 10, the fraction B/(a B) may be replaced by 
the ratio of "salt" to "acid," as on page 390. For a monobasic acid, 
e.g., acetic or boric acid, n is unity, and equation (81) reduces to equation 
(41), but if the acid has a higher basicity, the result is somewhat different. 
For example, if the buffer consists of KH 2 PO 4 and Na 2 HP0 4 , the con- 
centration of "acid," i.e., H 2 POi~, may be put equal to that of KH 2 PO 4 , 
while that of its "salt" is equal to the concentration of Na 2 HPO 4 ; the 
dissociation constant of the acid H 2 POJ" is that for the second stage of 
phosphoric acid, i.e., J 2 , and n is equal to 2; equation (81) thus becomes, 
in this particular case, 



pH = pX 2 + log - 3A + (V 



The value of A is known to be 0.509 at 25 (cf. p. 146), but that of C 
must be determined by experiment; to do this two or more measurements 
of the pH are made in solutions containing a constant ratio of "acid" 
to "salt" at different ionic strengths. Once C is known, an interpolation 
formula is available which permits the pH to be calculated at any desired 
ionic strength. 12 

Cohn et al., J. Am. Chem. Soc., 49, 173 (1927); 50, 696 (1928); Green, iWd., 55, 
2331 (1933). 



416 NEUTRALIZATION AND HYDROLYSIS 

It can be readily seen from equation (81) that the effect of ionic 
strength is greater the higher the basicity of the "acid" constituent of 
the buffer solution. 

The effect of varying the ionic strength of a buffer solution of con- 
stant composition may be expressed quantitatively by differentiating 
equation (81) with respect to Vy, thus 

= - (2n - 



It follows therefore that a change in the ionic strength, resulting from 
a change in the concentration of the buffer solution or from the addition 
of neutral salts, results in a greater change in the pH the higher the value 
of n, i.e., the higher the stage of dissociation of the acid whose salts con- 
stitute the buffer system. The change of pll may be positive or nega- 
tive, depending on the conditions. 13 

PROBLEMS 

1. Calculate the degree of hydrolysis and pH of (i) 0.01 N sodium formate, 
(ii) 0.1 N sodium phenoxide, (iii) N ammonium chloride, and (iv) 0.01 N aniline 
hydrochloride at 25. The following dissociation constants may be employed: 
formic acid, 1.77 X 10~ 4 ; phenol, 1.20 X 10" 10 ; ammonia, 1.8 X 10~ r '; aniline, 
4.00 X 10- 10 . 

2. If equivalent amounts of aniline and phenol are mixed, what propor- 
tion, approximately, of salt formation may be expected in aqueous solution? 
What would be the pH of the resulting mixture? 

3. A 0.046 N solution of the potassium salt of a weak monobasic acid was 
found to have a pH of 9.07 at 25; calculate the hydrolysis constant and degree 
of hydrolysis of the salt, in the given solution, and the dissociation constant 
of the acid. 

4. It was found by Williams and Soper [/. Chem. 800., 24G9 (1930)] that 
when 1 liter of a solution containing 0.03086 mole of o-nitraniline and 0.05040 
mole of hydrochloric acid was shaken with 60 cc. of heptane until equilibrium 
was established at 25 that 50 cc. of the heptnne layer contained 0.0989 g. 
of the free base. The distribution coefficient of o-nitraniline between heptane 
and water is 1.790. Determine the hydrolysis constant of the amine hydro- 
chloride. 

5. The equivalent conductance of a 0.025 N solution of sodium hydroxide 
was found by Kameyama [Trans. Electrochem. Soc., 40, 131 (1921)] to be 
228.4 ohms" 1 cm. 2 The addition of various amounts of cyanamide to the 
solution, so that the molecular ratio of cyanamide to sodium hydroxide was x, 
gave the following equivalent conductances: 

x 1.0 1.5 2.0 4.0 

A 105.8 94.4 94.1 93.3 

Calculate the hydrolysis constant of sodium cyanamide, NaHCN-2. 

"Morton, J. Chem. Soc., 1401 (1928); see also, Kolthoff and Rosenblum, "Acid- 
Base Indicators," 1937, p. 269. 



PROBLEMS 417 

6. Hattox and De Vries [J. Am. Chem. Soc., 58, 2126 (1936)] determined 
the hydrogen ion activities in solutions of indium sulfate, I^CSO^j, at various 
molalities (m) at 25; the results were: 

m X 10 2 9.99 5.26 2.81 1.58 1.00 

pH 2.01 2.20 2.36 2.57 2.69 

Evaluate the hydrolytic constants for the two reactions 

H 2 = InO + + 2H+ 



and 

In+++ + H 2 = In(OH)++ + H+, 

and determine from the results which is the more probable. Allowance may 
be made for the activity of the ions by using the Debye-Hiickel equation in 
the approximate form log/; = 0.50? V^/(l + Vtf). 

7. The pH of a 0.05 molar solution of acid potassium phthalate is 4.00; 
the first stage dissociation constant of phthalic acid is 1.3 X 10" 3 ; what is 
p&2 for this acid? 

8. Plot the pH-neutralization curves for 0.1 N solutions of (i) formic acid 
and (ii) phenol, by a strong base. Use the dissociation constants given in 
Problem 1. 

9. Plot the pH-neutralization curves for a mixture of (i) N hydrochloric 
acid and 0.1 N acetic acid, and (ii) 0.01 N hydrochloric acid and 0.1 N acetic 
acid. What are the possibilities of estimating the amount of each acid sepa- 
rately by titration? 

10. Use the data on page 415 to plot the complete pH-neutralization curve 
of citric acid in a 0.1 molar solution. Over what range of pH could partially 
neutralized citric acid be expected to have appreciable buffer capacity? 

11. Plot the pH-buffer capacity curve for mixtures of acetic acid and 
sodium acetate of total concentration 0.2 N. Points should be obtained for 
mixtures containing 10, 20, 30, 40, 50, 60, 70, 80 and 90 per cent of sodium 
acetate, the pH's being estimated by the approximate form of the Henderson 
equation. Plot the buffer capacity curve for water at pH's 1, 2, 3 and 4, and 
superimpose the result on the curve for acetic acid. 

12. Utilize the general form of the acetic acid-acetate buffer capacity curve 
obtained in Problem 11 to draw an approximate curve for the buffer capacity 
over the range of pH from 2 to 13 of the universal buffer mixture described 
on page 415. It may be assumed that the total concentration of each acid 
and its salt is always 0.2 molar. 

13. It is desired to prepare a buffer solution of pH 4.50 having a buffer 
capacity of 0.18 equiv. per pH; suggest how such a solution would be prepared, 
using phenylacetic acid (pK a = 4.31) and sodium hydroxide. 



CHAPTER XII 
AMPHOTERIC ELECTROLYTES 

Dipolar Ions. The term " amphoteric " is applied to all substances 
which are capable of exhibiting both acidic and basic functions; among 
these must, therefore, be included water, alcohols and other amphiprotic 
solvents and a number of metallic hydroxides, e.g., lead and aluminum 
hydroxides. In these compounds it is generally the same group, viz., 
OH, which is responsible for the acidic and basic properties; the dis- 
cussion in the present chapter will, however, be devoted to those ampho- 
teric electrolytes, or ampholytes, that contain separate acidic and basic 
groups. The most familiar examples of this type of ampholyte are pro- 
vided by the ammo-acids, which may be represented by the general 
formula NH 2 RCO 2 IL Until relatively recent times these substances 
were usually regarded as having this particular structure in the neutral 
state, and it was assumed that addition of acid resulted in the neutrali- 
zation of the NII 2 group, viz., 

NH,RC0 2 H + H 3 0+ = +NH 3 RCO 2 H + H 2 0, 

whereas a strong base was believed to react with the CO 2 H group, viz., 
NH 2 RC0 2 H + OH- = NH 2 RCOJ + H 2 0. 



It has been long realized, however, that in addition to the uncharged 
molecules NH 2 RCO2H, a solution of an amino-acid might contain mole- 
cules carrying a positive charge at one end and a negative charge at the 
other, thus constituting an electrically neutral system, viz., ^NHaRCOj. 
These particles have been variously called zwitterions, i.e., hermaphro- 
dite (or hybrid) ions, amphions, ampholyte ions, dual ions and dipolar 
ions. The existence of these dual ions was postulated by Kiister (1897) 
to explain the behavior of methyl orange which, in its neutral form, is an 
amino-sulfonic acid, but their importance in connection with ampholytic 
equilibria in amino-carboxylic acids was not clearly realized. The sug- 
gestion was made by Bjerrum, 1 however, that nearly the whole of a 
neutral aliphatic amino-acid is present in solution in the form of the 
dipolar ion, and that reaction with acids and bases is of a different type 
from that represented above. A solution of glycine, for example, i.e., 
NII 2 CH 2 CO2H, is compared with one of ammonium acetate; if a strong 
acid is added to the latter, the reaction is with the basic CHaCOlz" ion 
and CH 3 CO 2 H is formed, but a strong base reacts with the acidic NHi" 

1 Bjerrum, Z. physik. Chem., 104, 417 (1923); see also, Adams, /. Am. Chem. Soc. t 
38, 1503 (1916). 

418 



EVIDENCE FOR THE EXISTENCE OF DIPOLAR IONS 419 

ion to yield NH 3 . In the same way, the addition of strong acid to glycine, 
consisting mainly of the dual ions + NHsCH 2 CO2"~, results in the reaction 



H 3 0+ - +NH 8 CH 2 C0 2 H + H 2 O, 
while reaction with alkali is 

+ OH- = NH 2 CH 2 COr + H 2 0. 



The products are, of course, the same as in the alternative representa- 
tion, since there is no doubt that in acid solution the amino-acid forms 
* f NH 3 CH 2 CO 2 H ions while in alkaline solution the anions NH 2 CH 2 COi~" 
are formed. It should be noted, however, that the groups exhibiting the 
acidic and basic functions are the reverse of those accepted in the original 
treatment of amino-acids; the basic property of the ampholyte is due to 
the CO? group whereas the acidic property is that of the NHjj" 
group. 

Evidence for the Existence of Dipolar Ions. The evidence for the 
presence of large proportions of dipolar ions in solutions of aliphatic 
amino-acids is very convincing. According to the older treatment the 
dissociation constants of the NH 2 and CO 2 H groups were ex- 
tremely small, viz., about 10~ 8 to 10~ 12 ; such low values were difficult to 
understand if they referred to these particular groups, but they are not 
at all unexpected if they really apply, as just suggested, to the conjugate 
groups NHjJ" and COi", respectively. The ammonium ion acids, 
e.g., RNHi}", and anion bases, e.g., RCOjf, are known, from the facts 
mentioned in previous chapters, to have very low dissociation constants. 
In changing from water to a medium of lower dielectric constant, such 
as ethyl alcohol, the dissociation constants of cation acids and of anion 
bases are not affected appreciably, although the values for carboxylic 
acids are greatly decreased and those of amines are diminished to a lesser 
extent (cf. p. 333). It is therefore significant that the acidic and basic 
dissociation constants of aliphatic amino-acids, as determined from pH 
measurements in the course of neutralization by alkali and acid, respec- 
tively (see Chap. IX), are apparently slightly larger in ethyl alcohol than 
in water. It is evident that the groups being neutralized cannot be 
C0 2 H and NH 2 , but are probably NH^ and COjf , respectively. 
Further, if the neutral amino-acid has the structure NH 2 RCO 2 H, it is 
to be expected that the basic dissociation constant would be almost the 
same as that of the corresponding methyl ester NH 2 RCO 2 Me; actually 
the two values are of an entirely different order, and hence it appears 
that the basic groups are not the same in the acid and the ester. 

The addition of formaldehyde to an aqueous solution of an amino- 
acid results in no change in the curve showing the variation of pH in the 
course of the neutralization by acid, but that for the neutralization by 
alkali is shifted in the direction of increased acid strength, as shown in 
Fig. 107. It is known that the formaldehyde reacts with the ammo- 



420 



AMPHOTERIC ELECTROLYTES 



PH 





Acid Added 



Alkali Added 



Fia. 107. Titration of amino-arid with 
and without formaldehyde 



portion of the amino-acid, and it is evidently this part of the molecule 
which is neutralized by the alkali. The acidic portion of the electrically 
neutral ampholyte must consequently be the NH^" group. 2 

Important evidence for the dual-ion structure of aliphatic amino- 
acids has been provided by a study of their Raman spectra; in these 
spectra ach group, or, more exactly, each type of linkage, exhibits a 

characteristic line. It has been 
found that neutral amino-acids 
do not show the line which is 
characteristic of the carboxylic 
acid group in aqueous solution, 
and so the former presumably do 
not possess this group. When 
alkali is added to an ordinary car- 
boxylic acid, e.g., acetic acid, the 
characteristic line of the CO 2 H 
group disappears, but it appears 
when a strong acid is added to an 
amino-acid solution. This is strik- 
ing evidence for the argument that 
the basic function of the latter is 
exercised by the COif group, for 
the addition of acid would con- 
vert this into C0 2 H, in harmony with the findings from the Raman 
spectra. Similarly, froo amines have a characteristic Raman line which 
is absent from the spectrum of an aliphatic amino-acid; the line appears, 
however, when the latter is neutralized by alkali, implying that reac- 
tion takes place with the NHJ group. 3 

There are several other properties of amino-acids which are in agree- 
ment with the dipolar-ion type of structure: these are the high melting 
point, the sparing solubility in alcohol and acetone, and increased solu- 
bility in the presence of neutral salts, all of which are properties associated 
with ionized substances. Examination of crystals of glycine by the 
method of X-ray diffraction indicates that the substance has the struc- 
ture +NH 3 CH 2 COj in the solid state. The high dielectric constants of 
aqueous solutions of aliphatic amino-acids lead to the conclusion that the 
molecules have very large dipole moments; such large values can only 
be explained by the presence within the molecule of unit charges of 
opposite sign separated by several atomic diameters, as would be ex- 
pected for dipolar ions. 4 

Attention should be called to the fact that the arguments given above 
apply primarily to aliphatic amino-acids; it is true that aromatic amino- 

Harris, Biochem. J. t 24, 1080, 1086 (1930). 
Edsall, J. Chem. Phys., 4, 1 (1936); 5, 225 (1937). 

4 For summaries of evidence, see Richardson, Proc. Roy. Sac., 115B, 121 (1934); 
Neuberger, ibid., 11 SB, 180 (1934). 



DISSOCIATION CONSTANTS OF AMINO-ACID8 421 

sulfonic acids also exist largely in the dual-ion form, but amino-benzoic 
acids and amino-phenols consist almost exclusively of neutral, uncharged 
molecules in aqueous solutions. The properties of these substances are 
quite different from those of the aliphatic acids. 

Dissociation Constants of Amino-Acids. A very considerable simpli- 
fication in the treatment of amino-acids can be achieved by regarding 
them as dibasic acids. Consider, for example, the hydrochloride of 
glycine, i.e., Cl~ +NH 3 CH 2 CO 2 H ; when this is neutralized by an alkali 
hydroxide, there are two stages of the reaction, corresponding in principle 
to the two stages of neutralization of a dibasic acid, thus 

(1) +NH 3 CH 2 CO 2 H + OH- = + NH 8 CH 2 COr + H 2 O 
and 

(2) +NH 3 CH 2 CO2- + OH- = NH 2 CH 2 C02- + H 2 O. 

The two acidic groups are C0 2 H and NHt , and since the former is 
undoubtedly the stronger of the two, it will be neutralized first. 

It will be noted that the first stage produces the so-called neutral 
form of the ammo-acid which, in this instance, consists almost exclusively 
of the dual-ion form. If, in the most general case, the dipolar ion, i.e., 
+NH 3 RCC>2~, is represented by RH , the positive ion existing in acid 
solution, i.e., +NH 3 RCO 2 H, by RH^", and the negative ion present in 
alkaline solution, i.e., NH 2 RCOiF, by R~, the two stages of ionization of 
the dibasic acid +NH 3 RCO 2 H may be written as 



(1) RH + H 2 O ^ H 3 O+ 
and 

(2) RH* + H 2 O ^ H 3 O+ + R- 

In the first stage the dissociation of +NH 3 RCO 2 H is that of the carboxylic 
acid, while in the second stage the ammonium ion acid dissociates. 
Applying the law of mass action to these ionization equilibria, the dis- 
sociation constants of the two stages are 

H* , T . OH^OR- ,, x 

and K* = -- (1) 

* 



respectively. 

The values of these dissociation constants may be determined by 
means of cells without liquid junction in a manner similar to that de- 
scribed in Chap. IX. 6 For the first stage the acid is the hydrochloride 
C1--+NH 3 RCO 2 H, i.e., RHC1~ and the corresponding "salt" is the 
electrically neutral form "^NHsRCOi", i.e., RH*, and so the appropriate 
cell without liquid junction is 



H 2 (l atm.) | RHCl-(wi) RH^mO AgCl(s) | Ag. 

Nims and Smith, J. Biol Chem., 101, 401 (1933); Owen, J. Am. Chem. Soe., 56, 
24 (1934); Smith, Taylor and Smith, J. Biol. Chem., 122, 109 (1937). 



422 AMPHOTERIC ELECTROLYTES 

The E.M.F. of this cell is written in the usual manner, as 

J>rp 

E = E - In a H *Ocr, (2) 

and introducing the definition of K\ given by equation (1), this becomes 



a RH * 



which on rearranging, and replacing the activities by the product of the 
molalities and the activity coefficients, gives 

RT 



- lf 
F(E - g) . , mRHjmcr 7nn*7cr 

log ~~ + log ~- - - lQ g *- (3) 



The variation of the activity coefficient of the dipolar ion with ionic 
strength is given by an expression of the form log YRH* = C'y (cf. 
p. 432) and since the values of log 7 RH J and log 701- for the univalent 
ions RHj" and Cl" can be written, with the aid of the extended Debye- 
Hlickel equation, as A Vy + C"i*> it follows that equation (3) may be 
put in the form 

9 _ 

- 10 **-^ (4) 

The plot of the left-hand side of equation (4) against the ionic strength 
should thus be a straight line and the intercept for zero ionic strength 
should give the value of log K\. As in the case treated on page 315, 
the salt may be taken as completely dissociated so that mcr is equal 
to m\] WRH* is equal to m^ + WHS and WRHJ is m\ WH+. The value of 
win*, the hydrogen ion concentration, required for this purpose is best 
obtained from equation (2) which may be written in the form 



c>m 
-p- In W H * = E - E Q + ~jr In mcr + "y In 7H*7cr- 

The product of the activity coefficients can be estimated from the Debye- 
Hiickel equations, and mcr and 7 are known; hence m H f in the given 
solution can be derived from the measured E.M.F. of the coll. 

In order to determine KI a series of cells of the type depicted above, 
in which the ratio of mi to m^ is kept constant but the amounts of RH^Cl"" 
and of RH* are varied, are set up and the E.M.F.'S (E) measured. The 
value of E Q for the hydrogen-silver chloride cell is known, and so the 
left-hand side of equation (4) can be evaluated; the Debye-Hiickel factor 
A is 0.509 at 25. In calculating the ionic strength of the solution the 



APPROXIMATE METHODS FOR DISSOCIATION CONSTANTS 423 

dipolar ion RH* is treated as a neutral molecule so that it may be re- 
garded as making no contribution to the total. The plot of the left-hand 
side of equation (4) against the ionic strength is not exactly linear, but 
it is sufficiently close for an accurate value of K\ to be obtained by 
extrapolation. 

In the determination of the second dissociation constant (K the 
"acid" is the neutral form +NH 3 RCOr, i.e., RH*, whereas the corre- 
sponding "salt" is the sodium salt NH 2 RCOr-Na+, i.e., Na+R~; the cell 
without liquid junction will thus be 

H 2 (l atm.) | RH^roi) Na+R-(m2) NaCl(m 8 ) AgCl(s) | Ag. 



The E.M.F. is given by the general equation (2), and introduction of the 
value of -K a from equation (1) results in the expression, 



nrrt _ ,_ D/P 

. ** . &RH *flci~ *t* . 

E = E* ^r In =r In Jt, 

b a R - r 

and hence, using the same procedure as before, 

E* f IP B^0\ 4- 

ono DT i *^8 IZ ~"~ *^8 ~~" *^8 **! W/ 

.oUo/v I WlR- 7R- 

The activity coefficient term in equation (5) involves a univalent ion in 
the numerator and denominator, in addition to the dual ion; it follows, 
therefore, that in dilute solution this term is proportional to the ionic 
strength. The plot of the left-hand side of equation (5) against y will 
thus be linear at low ionic strengths, and the intercept for p equal to 
zero gives log A 2 . The experimental procedure is similar to that de- 
scribed for the evaluation of K\. 

Approximate Methods for Dissociation Constants. Approximate, 
but more rapid, methods, similar to those used for simple monobasic 
acids and monoacid bases, have been frequently employed to determine 
dissociation constants of ampholytes. 6 Upon taking logarithms, the 
equation for K\ may be written as 

log Ki = log a H * + log 
.'. pAi = pH log log 7^- (6) 

CRH, JRH 2 

If a solution is made up of c equiv. of neutral ammo-acid and a eqtiiv. of 
a strong acid, CRH* is equal to c a + C H + and C R + to a CH+ (cf . p. 422) ; 
inserting these values in equation (6), the result is 



8 Schmidt, Appleman and Kirk, J. Biol. Chem., 81, 723 (1929) ; Edsall and Blanchard, 
J. Am. Chem. Soc., 55, 2337 (1933); Glasstone and Hammel, ibid., 63, 243 (1941). 



424 AMPHOTERIC ELECTROLYTES 

For the second dissociation constant (7 2 ) the equation analogous to 
(6) is 

rr TT i C R~ i /R~ 

P#2 = pH - log ~ log f > 

CRH* JRH* 

and if the solution consists of c equiv. of neutral amino-acid and 6 equiv. 
of strong base, CRH* is equal to c 6 -f- COH~ and CR- to 6 CQ H -, this 
becomes 



RH* 



(8) 



In order to determine pKi or p/ 2 a solution is made up of known 
amounts of the neutral amino-acid (c) and either strong acid (a) or strong 
base (6), and the pH of the solution is determined by means of some form 
of hydrogen electrode. The values of CH+ or coir are derived from the 
pH by assuming the activity coefficient of the hydrogen or hydroxyl ions 
to be equal to the mean values for hydrochloric acid or sodium hydroxide, 
respectively, at the same ionic strength. Within the pH range of about 
4 to 10, however, the terms CH+ and COM" may be neglected in equations 
(7) and (8) respectively, provided the solution is not too dilute. The 
estimation of the activity coefficient factor presents some difficulty since 
~ log/RH* is proportional to y while log/ RH + or log/ R - is related to Vy; 
for most purposes, however, the last term in equations (7) and (8) may 
be taken as zero, provided the ionic strength of the solution is not large. 
In this event it is necessary to use the symbols pki and pA; 2 for the dis- 
sociation exponents, or to add a prime, thus pK [ and pKi. 

The results of measurements made in this manner with glycine at 20 
are given in Table LXXIV; the values of pki and pk 2 are seen to be 2.33 

TABLE LXXIV. DETERMINATION OF DISSOCIATION CONSTANTS OF GLYCINE AT 20 

Mixtures of Glycine (c) and Hydrochloric acid (a) 

c a pH CH+ X 10 3 - -- 1 pki 

a - CH + 

0.0769 0.0231 2.76 2.00 2.650 2.34 

0.0714 0.0286 2.58 3.02 1.786 2.33 

0.0667 0.0333 2.45 4.17 1.283 2.33 

0.0625 0.0375 2.31 5.75 0.972 2.32 

0.0588 0.0412 221 7.41 0742 2.34 

0.0555 0.0445 2.10 9.55 0.590 2.33 

Mixtures of Glycine (c) and Sodium hydroxide (b) 

c b pH con- X 10-* - - -- 1 pk t 

b - coir 

0.0833 0.0167 9.22 1.29 3.878 9.82 

0.0769 0.0231 9.42 2.09 2.333 9.79 

0.0714 0.0286 9.63 3.47 1.500 9.81 

0.0667 0.0333 9.78 4.90 1.000 9.78 

0.0625 0.0375 9.98 7.95 0.667 9.81 

0.0588 0.0412 10.14 10.14 0.250 9.78 



APPROXIMATE METHODS FOR DISSOCIATION CONSTANTS 



425 



and 9.80, which may be compared with 2.37 and 9.75, respectively, de- 
rived from cells without liquid junction. 

In the methods described above the tacit assumption has been made 
that the neutralizations of RH^ and of RH^ do not overlap; this is 
always true in the early stages of the neutralization of RHjj" and in the 
later stages for RH^, but it is not necessarily the case in the region of the 
first equivalence-point, i.e., at RH*. The problem is, of course, iden- 
tical with that of an ordinary dibasic acid; provided Ki/K* is greater than 
about 10 6 , i.e., pK 2 p^i is greater than 6, the two stages may be 
regarded as independent. If this condition does not hold, the system 
may be treated as a conventional dibasic acid in the manner described 
on page 326. 

The dissociation constant exponents at 25 of a number of physio- 
logically important amino-acids are recorded in Table LXXV; 7 those 

TABLE LXXV. DISSOCIATION CONSTANTS OP AMIXO-ACIDR AT 23 



Ainino-ftcid 


P/Ci 


pA'z 


Ammo-acid 


pKi 


pJvz 


Alariine 


2.340 


9.S70 


Diglycine 


3.15 


8.10 


Argmine 


2.02 


f 9.04 
\12.48 


Histidine 
Hydroxyproline 


1.77 
1.92 


9.18 
9.73 


Aspartic acid 


/2.09 
13.87 


9.82 


Isoleucine 
Leucine 


2.318 
2.328 


9.758 
9.744 


Glutamic and 


/2.19 
1 4.28 


96C 


Norleucine 
Valinc 


2.335 
2287 


9833 
9.719 


Glycine 


2 350 


9778 


Tryptophane 


2.38 


939 



given by four significant figures are thermodynamic values, but the 
others are approximate. The data for KI show that the carboxylic acid 
+NH3RCO2II is a moderately strong acid; the reason is that the positive 
charge on the nitrogen atom facilitates the departure of the proton from 
the CO2H group, thus increasing the acid strength of the latter. The 
ammonium ion acid + NH 3 RCO^ is relatively weak, however, because 
the negative charge on the COjf group has the opposite effect. As 
the distance of separation increases, the influence of the electrostatic 
charges becomes less marked. From an examination of the dissociation 
constants of glycine and diglycine it has been found possible to calculate 
the distances between the terminal groups. 

According to the older ideas concerning amino-acids, neutralization 
of the electrically neutral form by a strong acid gave the basic dissocia- 
tion constant k b of the NH 2 gr