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AN INTRODUCTION
TO
ELECTROCHEMISTRY
BY
SAMUEL GLASSTONE, D.Sc., PH.D.
Contultanl, Untied Stales Atomic Energy Commirsion
TENTH PRINTING
(AN EASTWEST EDITION)
AFFILIATED EASTWEST PRESS PVT. LTD.
NEW DELHI.
Copyright 1942 by
LITTON EDUCATIONAL PUBLISHING, INC.
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First Published May 1942
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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without being burdened by much that is obsolete.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 protontransfer 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. OXIDATIONREDUCTION 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 northseeking 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 onetenth 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
crosssection, 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 voltcoulomb, and it is evident from Table I that
the absolute voltcoulomb 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 voltcoulomb 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
voltcoulomb 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 aform 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 mtrocompounds 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 socalled
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 nonelectrolytes, 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 biunivalent 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 freezingpoint, or analo
gous, measurements, the value of or, the socalled 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 Xray 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 waterrich 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 RsNHX 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 NHX + S ^ K 3 NHS+ + 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 nitrocompounds, 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., nitrocompounds 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 nonaqueous 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 (191516) at Harvard
University, and by Rosa and Vinal, 4 and others, at the National Bureau
of Standards in Washington, D. C. (191416). 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 socalled " 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 platinumindium
foil, joined by a Vshaped 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 coarsegrained 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
oxidationreduction 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 gramion,
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 alkalichlorine 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 crosssectional 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 righthand 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 socalled " 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 dirert 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 noripolarizable 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 platinumiridium, stretched along a meter scale, i.e., the
socalled " 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 selfinduction, and to use a
straightwire bridge, if a special noninductive 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 highfrequency
34
ELECTROLYTIC CONDUCTANCE
To bridge
Fia. 10. Vacuumtube 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 mercuryarc arrangement capable of giving a sym
metrical sinewave 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 vacuumtube 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 vacuumtube 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 vacuumtube (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 audiofrequency amplifier is shown in Fig. 11, in which
the conductance bridge is connected to the primary coil of an ironcored
transformer (P); T is a suitable vacuum tube and C is a condenser.
The use of a vacuumtube 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 vacuumtube, or other form
of A.C. rectifier, and a direct current galvanometer has been employed,
FIG. 11. Vacuumtube 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 finelydivided, 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 finelydivided 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 vacuumtube audioamplifier,
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 resistancecapacity 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
nonconductor.
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 vacuumtube
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 fortytwo 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 socalled " ultrapure " 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. Ultrapure 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 20liter 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 blocktin 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.
Nonaqueous 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 nonaqueous
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 ultrapure 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 ultrapure 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 ultrapure
water in conductance work with bases.
With nonaqueous 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 nonhydroxylic 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 nonpolarizable elec
trodes; the electrodes employed have been mercurymercurous chloride
in chloride solutions, mercurymercurous 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 nonpolarizable 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 nonpolarizable 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 nonpolarizable 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 nonpolarizable 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
finewire 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 IjtindoltBoriistein 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 uniunivalent
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 uniunivalont strong electrolyte,
and of nickel sulfatc, a hibivalent electrolyte, are plotted as functions of
the concentration. Electrolytes of an intermediate valence type, e.g.,
potassium sulfatc, a unibivalcnt electrolyte, and barium chloride, which
is a biuriivalcnt 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
Uniuni 0.98 0.93 O.S3
21} 0.05 0.87 0.75
Bibi 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 squareroot 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 nonaqueous 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
JSrn 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 nonaqueous 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. nonuniform 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 socalled 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. CONDUCTANCEVISCOSITY 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
ductanceviscosity product for infi s
nite dilution is, at least, approxi
mately independent of temperature
for a number of ions in water. The
data for nonaqueous 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 conductanceviscosity 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 ratedetermining 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 conductanceviscosity products
for a number of ions collected in Table XVIII; the data for hydrogen
TABLE XVIII. ION CONDUCTANCEVISCOSITY 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 conductanceviscosity 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 conductanceviscosity
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 conductanceviscosity 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 conductanceviscosity 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. CONDUCTANCEVISCOSITY 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 conductanceviscosity 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 nonhydroxylic solvents, the conductanceviscosity 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 righthand 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 selfionization
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 nonaqueous
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: 4Pyridine tricarboxylic acid 121.0 88.8 32.2 3
1:2:3: 4Pyridine 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 longchain 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 coordination 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 ztimes 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
Uniuni 120 ohms" 1 cm.*
Unibi or biuni 240
Uniter or teruni 360
Unitetra or tetrauni 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 nonclcctrolytc 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 equivalencepoint 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
equivalencepoint.
If the strong acid is titrated
with a weak base, e.g., an aqueous
solution of ammonia, the first part
of the conductancetitration 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 endpoint 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 endpoint, and
to draw two straight lines through them, as seen in Fig. 24; the inter
section of the lines gives the required endpoint. 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 conductancetitration 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 equivalencepoint 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 conductancetitration 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 equivalencepoint, however, there is no change in
conductance because of the small contribution of the free weak base.
The complete conductancetitration 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 endpoint.
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 equivalencepoint the conductance shows a further increase if
a strong base is used, and so the endpoint 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
ductancetitration 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
endpoint 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 equivalencepoint. 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
unionized 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 endpoint 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 equivalencepoint is passed, however, the excess of
the added salt causes a sharp rise in the conductance (Fig. 27, I) ; the
endpoint 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 endpoint 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
conductancetitration 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 alternatingcurrent 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 400ohm 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 vacuumtube 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 endpoint 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 noncommittal 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 timeaverage 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 timeaverage 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 timeaverage 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 __ riZe**+ lkT ). (5)
For a uniunivalent 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 (gramions) 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
Uniuni 9.64A 30.5A 96.4A
Unibi and biuni 5.58 19.3 55.8
Bibi 4.82 15.3 48.2
Uniter and teruni 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
gr^.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 uniunivalent 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 uniunivalent 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 DebyeHiickelOnsager 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 DebyeHuckelOnsager Equation. For a uniuniva
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 UNIUNIVALENT
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 squareroot 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 uniunivalent
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 DEBYEHUCKELONSAGER EQUATION
91
conductances are plotted against the squareroots 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 uniunivalent 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 DebyeHiickelOnsager 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 uniunivalent and unibi (or biurii) valent
electrolytes. With bibivalent 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 DEBYKHUCKELONSAGKR EQUATION
93
ductance against the squareroot 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
nonhydroxylic 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 uniunivalent 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 squareroot 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 lefthand 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 uniunivalent
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 ionpairs; although any
particular ionpair 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 unionizcd molecules;
a weak acid, such as acetic acid, provides an excellent illustration of this
type of behavior. The solution contains unionized, covalent molecules,
quite apart from the possibility of ionpairs. 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 ionpairs 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 unionized 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 ionpairs. 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 highvalence 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 uniunivalent 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 squareroot 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 uniunivalent 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
DebyeFalkenhagen 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 uniunivalent type and has a concentration
of 0.001 molar, the DebyeFalkenhagen effect should become evident
with highfrequency 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 DebyeFalkenhagen 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 highvalence 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
uniunivalent 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 DebyeFalkenhagen 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 uniunivalent 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 uniunivalent
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 acrotonate (NaaC.) :
AHCI  426.28  156.84^ + 169.7c (1  0.2276^)
ANECI = 126.47  88.65Vc + 94.8c (1  0.2276Vc)
A N ac.  83.30  78.84Vc + 97.27c (1  0.2276Vc).
The equivalent conductances of acrotonic 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 acrotonic 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 gramions 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+ , CZU
t. = and <_ = ;
C+2+U+ + CZU C+Z+U+ + CZU
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 alkalineearth halide,
the anode is generally made of silver coated with the same metal in a
finelydivided 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 Hshaped 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
Utube. 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 alkalineearth 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
rightangle 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 alkalineearth 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 wT moles in the
opposite direction; there will consequently be a resultant transfer of
w+T+  wT = 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 electroosmosis (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 airlock 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. Airlock 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 socalled "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 uniuniva
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
DebyeHiickelOnsager 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 squareroot 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 interionic 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 squareroot 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 squareroot 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, "Textbook 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
/uMf + 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+ + vRT 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 DebyeHiickel 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 DcbyeHiickel 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 DEBYEHtfCKEL 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
pi
Wi = I z % Gl/\d\
Jxo
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 righthand 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 DEBYEHtJCKEL 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 DebyeHiickel 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 DebyeHiickel 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 righthand 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 DebyeHiickel 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 Zv, it is found that
log/ = ~ Az^.^, (56)
which is the statement of the DebyeHuckel 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 DebyeHtickel treatment are the socalled 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
DEBYEHUCKEL 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 DebyeHuckel 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 DebyeHiickel theory can be made.
DebyeHiickel 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 #Vi, 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 DebyeHiickel 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. DEBYEHUCKEL 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
DebyeHiickel 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 DEBYEHUCKEL 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 righthand 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 socalled "saltingout effect," namely, the decrease in
solubility of neutral substances frequently observed in the presence of
salts; the constant C 1 is consequently called the saltingout constant.
The activity coefficient of a nonelectrolyte, 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 nonelectrolyte, 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 righthand
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 DebyeHiickelBr^nsted equation. In dilute solution,
when y is small, the term Cy can be neglected, and so this expression then
reduces to the DebyeHiickel limiting law.
Qualitative Verification of the DebyeHiickel 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 DebyeHiickel 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 DebyeHiickel 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
DebyeHuckel 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 saltingout 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 DebyeHiickel Limiting Equation. Al
though the DebyeHuckel 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) DebyeHuckel equations
QUANTITATIVE TESTS OF THE DEBYEHtfCKEL 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 DebyeHtickel 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 DebyeHuckel equation
The experimentally determined activity coefficients, based on vapor
pressure, freezingpoint 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 squareroot of the ionic strength; in these cases
the solutions contained no other electrolyte than the one under considera
tion. Since the DebyeHiickel 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 DebyeHiickel 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 dioxanewater mixtures, as well as in pure
water at 25. Some of the data are plotted in Fig. 50; the limiting slopes,
Fia. 50. Limiting DebyeHiickel 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 freezingpoint and other socalled osmotic measurements, the data
may be used directly to test the validity of the DebyeHiickel 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 freezingpoint
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 squareroot of the ionic strength, as it
undoubtedly is in dilute solutions, then
1  * =  iln/. (65)
Introducing the DebyeHiickel 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 freezingpoint 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 DebyeHuckel equation by freezingpoint 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 DebyeHuckel 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 freezingpoint
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 uniunivalent
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 lefthand 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 DebyeHuckel 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 DebyeHiickel 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 uniunivalent salt,
therefore, a should exceed 1.6 X 10~~ 8 cm., but for a bibivalent electro
lyte a must exceed 6.4 X 10~ 8 cm. if the DebyeHiickel 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 DebyeHiickel 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).
IONASSOCIATION 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 GronwallLaMerSandved
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.
IonAssociation. 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 ionpairs (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 righthand 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 ionpairs,
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 ionpairs. 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 DebyeHiickel 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 DebyeHiickel
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 uniunivalent 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 uniunivalent 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 UNIUNIVALBNT 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
ionpairs; an equilibrium may be supposed to exist between the free
M+ and A~ ions, on the one hand, and ionpairs 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 ionpairs
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 ionpairs, 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 dioxanewater 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 ionpairs 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 ionpairs, as described above. In this case, however,
the ionpair 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 uniunivalent 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 ionpairs (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 tripleion 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
AA +
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 ionpairs, and so the
conductance falls; at still higher concentrations, however, the ionpairs
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 ionpairs, 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. , ionpairs, 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
some 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 ionpairs 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 unionized molecules and
ionpairs. 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 socalled 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 DebyeHiickel 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 DebyeHuckel
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
DebyeHuckel 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 DebyeHuckel 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
nonaqueous 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 DebyeHiickel 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 tetrabutylammonium 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 lefthand 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 v8o] hence according to equation (109)
= (j>>l),S&A. (113)
In the simple case of a uniunivalent 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 uniunivalent salt the saturation solubility in pure water is thus
equal to the squareroot 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 DebyeHuckel 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 squareroot 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 silverammonia (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
ct 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 DebyeHiickel 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 DebyeHtickel
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 DEBYEHttCKEL 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 DebyeHiickel limiting
law equation (54), it follows that
logfA+*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 squareroot 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 DebyeHiickel 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 DebyeHuckel 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 GibbsHelmholtz 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 righthand 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 umunivalent 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 squareroot of the
concentration, the slope of the plot of AH o^o against Vc should be about
500 for an aqueous solution of a uniunivalent 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 uniunivalent 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 DebyeHlickel 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 (Sjsz?)*; the results obtained with a number of unibivalent 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
biunivalent 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 nonaqueous 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 uniuni, unibi, bibi and
unitervalent electrolytes in solutions of molality m.
3. Use the values of the DebyeHuckel constants A and B at 25, given in
Table XXXV, to plot log f for a uniunivalent 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 DebyeHiickel 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
uniuni, unibi, and bibi 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
DebyeHuckel 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 DebyeHiickel limiting equation, to evaluate the dissociation constant of
acrotonic acid.
9. Apply the method of Fuoss and Kraus, described on page 167, to evalu
ate Ao and K for hydrochloric acid in a dioxanewater 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 silverammonia 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 nonconductor, 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 nonmetal, 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 silversilver 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 silversilver
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 silversilver 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 oxidationreduction 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 oxidationreduction 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 oxidationreduction system must contain both oxidized and
reduced states. It is important to point out that there is no essential
difference between an oxidationreduction 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., chlorinechloride
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 oxidationreduction 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 righthand 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 lefthand 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 righthand 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 gramatoms or gramions, 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 lefthand electrode reaction is the same as above, i.e.,
Zn = Zn++ + 26,
while at the righthand 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 lefthand electrode is the opposite of
that at the righthand electrode of the previous cell, viz.,
Ag + Cl = AgCUs) + 6,
and at the righthand 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
lefthand 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 righthand 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, platinumindium, 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.
Potentiometervoltmeter
arrangement
potentiometervoltmeter 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 highsensitivity
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 vacuumtube 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.
Vacuumtube 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 vacuumtube 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 gridcircuit 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 Hshaped tube as shown in Fig. 63, the lefthand
limb containing the cadmium amalgam and the righthand 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 socalled "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 voltcoulombs 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 GibbsHehnholtz 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 silversilver chloride electrode in hydro
chloric acid,t the hydrogen at the lefthand electrode dissolves to form
hydrogen ions, whereas at the righthand 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 lefthand cell for
the passage of one faraday, as seen above, is
JH 2 (1 atm.) + AgClOO = HCl(ci) + Ag,
and that in the righthand 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 lefthand to the extreme righthand 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 lefthand side
of any concentration cell without transference, arid a, 2 is the value on the
righthand 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 lefthand cell for the passage of one faraday of electricity is
iH 2 (l atm.) + NaOH( Cl ) = H 2 O + Na,
and in the righthand 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 halfcell 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 halfcell, 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 DebyeHlickelB 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 lefthand 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 righthand 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 socalled "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 =  td In a. (23)
" Ja v
In the general case this becomes
E==  Mlna, (24)
where PI, vt 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 hydrogenhydrochloric 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 righthand 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 DebyeHuckol expression for uniunivalent electrolytes,
1 (29)
where A is the known DebyeHuckel 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
DebyeHuckel 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 nonaqueous 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 uniunivalent 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 lefthand
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 righthand 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 DebyeHiickel 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 uniunivalent 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 uniunivalent chloride
employed in the cell. Inserting the value of EL given by equation (37)
into (38), the result is
If the righthand 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 uniunivalent 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 uniunivalent 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 uniunivalent 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 socalled " 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
nonaqueous 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 lefthand electrode, and
Zn^+ + 2c
at the righthand 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 mercurymercurous 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) HgHgO(s)NaOHH 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 overall 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 = EoF~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, freezingpoint 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 socalled 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 zvalent 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 voltcoulombs. 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 SilverSilver 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 silversilver 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 silversilver 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 silversilver chloride elec
trode, or by measuring one electrode against the other. 6 The best
values are
Pb(Hg)  PbS0 4 (s), SOrfaor = 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
nonmetal 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 nonaqueous 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 silversilver halide
or mercurymercurous 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, "TextBook 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 onehalf 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 voltcoulombs, 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
oxygenhydrogen 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+ +
43024
Ha, OH
iHi4OHHO4
40.828
K, K +
K  K + + i
4 2.924
Oi, OH
20H>04Hi042
0.401
Na, Na+
Na  Na* +
42.714
Zn, Zn++
Zn Zn + + +2
40.761
Clifo), Cl
Cl * iCli 4 f
1.358
Fe, Fe++
Fe + Fe ++ + 2
40.441
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
40283
Ni, Ni+ +
Ni*Ni ++ +2.
40.236
A K AgCl(), Cl
Ag J Cl * AgCl 4
 0.2224
Sn, 8n++
Sn Sn+ + 42c
40.140
Ag, AgBr(), Br
Ag 4 Br~  AgBr 4
 0.0711
Pb, Pb++
Pb *Pb+ + 4 2
40.126
Ag, Agl(.). I
Ag 41 AgI4f
40.1522
Hi, H+
JH  H+ 4 c
db 0.000
Hg, HgiCh(s), Cl~
Hg4Cl *iH gI Cli4
0.2680
Cu, Cu + +
Cu  Cu ++ f 2e
 0.340
Hg, HgjS04(a) f S0 4
2Hg 4SOj *HgjS0442
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 NonAqueous 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 nonaqueous
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 socalled "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 electronvolts, 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 righthand 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 rearranging equation (34) the result is
zprn 7? / 7 7
tfV.Mt  ^ In (a M j). = BM..M;  p In (a M j).. (35)
The lefthand side of this equation clearly represents the reversible poten
tial of the metal Mi in the equilibrium solution and the righthand 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 endpoints of various types of titration; 19 the subject
will be treated here from the standpoint of precipitation reactions, while
neutralization and oxidationreduction 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 endpoint 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 endpoint, 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
lencepoint, 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 equivalencepoint 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 equivalencepoint. This result immediately suggests a method
258 ELECTRODE POTENTIALS
for determining experimentally the endpoint 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 endpoint.
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 equivalencepoint, in agreement with the conclusion already
reached.
By differentiating equation (39) with respect to x it is seen that at
the equivalencepoint the value of dE/dx is inversely proportional to Vfc,.
The potential jump observed at the endpoint 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 potentialtitratioii 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
lencepoint. 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 endpoint 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 endpoint of the titratiou.
260 ELECTRODE POTENTIALS
It is not always possible to estimate the endpoint 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 endpoint
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 potentialtitration curve at the endpoint 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 endpoint.
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 filterpaper. 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 endpoint 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 "gaslift" C is sealed
into its side. The hole D in the tube A permits the overflow of liquid
when the gaslift 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 gasstream 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 overtitrated, 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 endpoint 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
endpoint is passed, the endpoint 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 ElectroTitrations," 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),
ftft.,. (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 tenfold 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 tenfold 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. + Claq. = 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 endpoint; 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
OXIDATIONREDUCTION SYSTEMS
OxidationReduction 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 oxidationreduction 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 nonmetal and its
corresponding anions. This lack of distinction is brought out by the
fact that the iodineiodide ion system is frequently considered from the
oxidationreduction standpoint. Nevertheless, certain oxidationreduc
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.
Oxidationreduction 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 lefthand electrode may evidently be written as
Fe++ = Fe++ + + ,
267
268 OXIDATIONREDUCTION 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 righthand 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 oxidationreduction 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 oxidationreduction 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 oxidationreduction 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 OxidationReduction Systems. Various types
of reversible oxidationreduction 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 OXIDATIONREDUCTION 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 ferricyanide, 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 permanganatemanganous 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 oxidationreduction 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 OXIDATIONREDUCTION 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 oxidationreduction systems involving
organic compounds are known; most of these, although not all, are of
the quinonehydroquinone 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 OxidationReduction Potentials. In prin
ciple, the determination of the standard potential of an oxidationreduc
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 oxidationreduction 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 oxidationreduction 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 oxidationreduction potential; secondly, there is often
DETERMINATION OF STANDARD OXIDATIONREDUCTION 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 DebyeHuckel 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 oxidationreduction 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 oxidationreduction 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 OXIDATIONREDUCTION SYSTEMS
earliest quantitative studies of reversible oxidationreduction 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 ferricferrous 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 DebyeHuckel
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 oxidationreduction potential of the system
involving penta (VOt) and tetravalent (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 OXIDATIONREDUCTION 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 lefthand 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 lefthand 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 lefthand 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 DebyeHiickel 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 oxidationreduction potentials
is to make use of chemical determinations of equilibrium constants. 3
' Schumb and Sweetser, J. Am. Chem. Soc., 57, 871 (1035).
274 OXIDATIONREDUCTION SYSTEMS
The chemical reaction occurring in the hypothetical cell, free from liquid
junction,
Ag  Ag+  Fe++, FOH+  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 ferrousferric 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 DebyeHuckel 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 ferricferrous 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 oxidationreduction 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
oxidationreduction 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
oxidationreduction 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+, Mn system must be identical with that of the Ce+++, Ce++++
system (cf. p. 284). The ceriocerous 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
OXIDATIONREDUCTION 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 iodineiodide 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
oxidationreduction 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 1naph
thol2sulfonate 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 OxidationReduction," 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 lnaphthol2
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
lNAPHTHOL2SULFONATE 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 oxidationreduction system, is of some interest; if this is
* The precise conditions for efficient reduction are discussed on page 280.
278
OXIDATIONREDUCTION 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
30.14
*0.13
30.12
g0.11
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 OxidationReduction Potentials. Some values of standard
oxidationreduction 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 OXIDATIONREDUCTION 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 OXIDATIONREDUCTION 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 OxidationReduction Potential. From a knowledge of
the standard oxidationreduction 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 oxidationreduction poten
tial on the proportion of the system present in the oxidized form, are
280
OXIDATIONREDUCTION 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
tionreduction 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
permanganatemanganous 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. Oxidationreduction 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 oxidationreduction
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 oxidationreduction 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 oxidationreduction
potential. Finally, when the system consists almost exclusively of the
oxidized form, i.e., at the righthand 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
oxidationreduction 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 OXIDATIONREDUCTION 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 coppercopper 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 metalion 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
OxidationReduction Equilibria. When two reversible oxidation
reduction systems are mixed a definite equilibrium is attained which is
Heinerth, Z. Ekktrochem., 37, 61 (1931).
OXIDATIONREDUCTION EQUILIBRIA 283
determined largely by the standard potentials of the systems. For ex
ample, for the reaction between the ferrousferric and stannousstannic
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
ferrousferric and stannousstannic systems, i.e.,
E = JBje** 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 ferrousferric and stannousstannic 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 OXIDATIONREDUCTION 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 ^^ = ELsn  ^in^^, (20)
the lefthand side of this equation boing the potential of the ferrous
ferric system and the righthand side that of the stannousstannic system
at equilibrium. When this condition is attained, therefore, both systems
must exhibit the same oxidationreduction potential; this fact has been
already utilized in connection with the employment of potential mediators.
OxidationReduction 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., ferrousferric, will reduce a system above it in
Table LIII, e.g., cerousceric, but it will oxidize one below it, e.g., stan
nousstannic.
The question of the extent to which one system oxidizes or reduces
another is of importance in connection with oxidationreduction 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 endpoint 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 OXIDATIONREDUCTION TITHATIONS
285
better. The equilibrium constant should thus be smaller than 10" 6 if n
is the same for both interacting oxidationreduction 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 oxidationreduction
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 OxidationReduction 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 potentialtitration curves, the endpoint of the titration in
each case being marked by a relatively rapid change of potential. The
question arises as to whether this endpoint 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 endpoint has been passed; before the equivalencepoint 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. Potentialtitration curve; deter
mination of endpoint is possible
FIG. 83. Potentialtitration curve; de
termination of the endpoint is not satis
factory
excess, while after the equivalent point they are determined by the titrant
system. The potentialtitration 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 OXIDATIONREDUCTION SYSTEMS
respective midpoints, 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 endpoint, 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 endpoint 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 endpoint 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 endpoint be estimated.
The method of carrying out oxidationreduction 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 endpoint 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 EquivalencePoint. Since the potentials of the two
oxidationreduction 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 equivalencepoint, 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 equivalencepoint,
^equi
9 See general references to potentiometric titration on page 256.
OXIDATIONREDUCTION INDICATORS 287
This result holds for the special case in which each oxidationreduction
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 equivalencepoint is given by
OxidationReduction Indicators. A reversible oxidationreduction
indicator is a substance or, more correctly, an oxidationreduction 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 oxidationreduction system, the former, acting as a potential
mediator, will come to an equilibrium in which its oxidationreduction
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 oxidationreduction
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 oxidationreduction 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 oxidationreduction 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 OXIDATIONREDUCTION 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
oxidationreduction 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 oxidationreduction 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 oxidationreduction indicators for biological purposes are
also acidbase indicators, exhibiting different colors in acid and alkaline
solutions. They are frequently reddishbrown 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 coworkers, by introducing halogen
atoms into one of the phenolic groups of phenolindophenol, e.g., 2 : 6
dichlorophenolindophenol. 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 oxidationreduction
indicators of special interest are the socalled " viologens," introduced by
Michaelis; they are NN'disubstitutcd4 : 4dipyridilium 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 OxidationReduction/' 1928 el seq.\ Michaolis, "Oxy
dat ionsReductions 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. OXIDATIONREDUCTION INDICATORS FOR BIOLOGICAL WORK
Indicator E 9 ' Indicator
Phenolmsulfonate indo
2 : 6dibromophenol  0.273
wBromophenol indophenol 0.248
2 : 6Dichlorophenol indophenol 0.217
2 : 6Dichlorophenol indoocresol 0.181
2 : 6Dibromophenol 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 oxidationreduction 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 endpoints of oxidationreduction
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 oxidationreduction systems interact sufficiently completely
to be of value for analytical purposes, there is a marked change of poten
tial of the system at the equivalencepoint (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 endpoint is reached, therefore, and
the oxidationreduction 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 equivalencepoint, 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 OXIDATIONREDUCTION SYSTEMS
standard potentials must differ by about 0.3 volt, and hence the standard
potential of a suitable oxidationreduction 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 oxidationreduction
titrations has followed on the discovery that the familiar color change
undergone by diphenylamine on oxidation could be used to determine the
endpoint 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 ferrousferric system is 0.78 whereas that of the di
chromatechromic ion system in an acid medium is approximately 1.2
volt; hence a suitable oxidationreduction 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 ferrousferric system is lowered (numerically) to about 0.5 volt.
The change of potential at the endpoint 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 endpoint
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 ophenanthroline ferrous sulfate, the cations of which, viz.,
FeCCuHsNi) +, with the corresponding ferric ions, viz., Fe(Ci 2 H 8 N 2 ) ++,
form a reversible oxidationreduction 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).
QUINONEHYDROQUINONE 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 titanoustitanic 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.
QuinoneHydroquinone Systems. In the brief treatment of the
quinonehydroquinone 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
oxidationreduction 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 OXIDATIONREDUCTION 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 oxidationreduction 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 quinonehydroquinone system,
q and h representing the total concentrations of the two constituents.
According to equation (26) the variation of the oxidationreduction
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 quinonehydroquinone 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<
QUINONEHYDROQUINONE 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 oxidationreduction potential, for con
stant quinonehydroquinone 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 quinonehydroquinone (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 OXIDATIONREDUCTION 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 pHpotential 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 0btained 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 pHpotential 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 pHpotential 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 pHpotential curves for oxidationreduction systems of constant
stoichiometric composition. 14
14 Clark, "Studies on OxidationReduction/ 1 Hygienic Laboratory Bulletin, 1928.
TWO STAGE OXIDATIONREDUCTION 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 OxidationReduction. 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 (QH 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 oxidationreduction
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 oxidationreduction may be
written
(1) 2H 2 Q  H 2 Q Q + 2H+ + 2,
and
(2) H 2 QQ^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 OXIDATIONREDUCTION 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^QQ; 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 oxidationreduction
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 righthand 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 oneelectron 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 oneelectron 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 oxidationreduction systems satisfy
the conditions for semiquinone formation; in one way or another, this
has been found to be true for aoxyphenazine 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 twoelectron
process, as in Fig. 79. In view of the interest associated with the forma
tion of semiquinone intermediates in oxidationreduction 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, " OxydationsReductions 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
OXIDATIONREDUCTION 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 twoelectron oxidationreduction 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 oneelectron 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 oxidationreduction 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., aoxyphenazine, 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 OXIDATIONREDUCTION 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 oxidationreduction
systems their theoretical aspects will be considered here.
The Acid Storage Cell. The socalled "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 righthand 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.
Lefthand electrode:
Pb = Pb++ + 2
Pb++ + SO? = PbS0 4 (s).
.". Net reaction for two faradays is
Pb + SOr = PbSO 4 (s) + 2 .
Righthand 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 OXIDATIONREDUCTION 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 socalled 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 plumbousplumbic 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
Lefthand 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.
Righthand 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 OXIDATIONREDUCTION SYSTEMS
PROBLEMS
1. Write down the electrochemical equations for the oxidationreduction
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 triiodide 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 ferricyanide 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 ferrocyanideferricyanide system.
Alternatively, derive the value of E Q from each E' Q by applying the activity
correction given by the DebyeHiickel limiting law.
5. The oxidationreduction system involving 5 and 4valent 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 oxidationreduction system.
6. In an investigation of the oxidationreduction potentials of the system
in which the oxidized form was anthraquinone 2 : 6disulfonate, 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 nonaqueous media, such as ethers,
nitrocompounds, ketones, etc., are involved. As a result of various
studies, particularly those on the catalytic influence of unionized 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 acidbase reactions, may be represented
as an equilibrium between two acidbase 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, HC104H 2 O, has been shown by Xray 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 socalled 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 protondonat
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 trisubstituted 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 basesolvent
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 hydrogensilver 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 righthand 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 lefthand 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 lefthand
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 nonvolatile 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 silversilver
chloride electrodes with such bases; the employment of sodium amalgam
has been proposed, but it is probable that the silversilver 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 HXX'H, where X
and X' are different. This acid can dissociate in two ways, viz.,
HXX'H + H 2 O ;= H 3 O+ + XX'H,
and
HXX'H + H 2 ^ H 8 0+ + HXX'
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., OIICORCOiT, 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 DebyeHuckel 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 lefthand 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 DebyeHiickel 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 lefthand side of this equation for a series of acidbase 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 DebyeHlickel 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 halfneutralized 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 DobyeHiickel 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 unneutralized 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 lefthand side of this expression (X) is
plotted against the coefficient of KI in the first term on the righthand
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 nonaqueous solvents, but it has
also been used to study certain acids in aqueous solution. It can be
employed, in general, whenever the ionized and nonionized forms of an
acid, or base, have different absorption spectra in the visible, i.e., they
have different visible colors, or in the near ultraviolet 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 unionized 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., priitrophenol, 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 ultraviolet 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
oChlorobenzoic
2.92
Propionic
4.874
wPhlorobenzoic
3.82
nButyric
4.820
pChlorobenzoic
3.98
woButyric
4.821
pBromobenzoic
3.97
nValeric
4.86
pHydroxybenzoic
4.52
Trimethylacetic
5.05
pNitrobenzoic
3.42
Diethylacetic
4.75
pToluic
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
acidbase 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 ionpairs 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 lightabsorption 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
pNitroaniline
1.11
(1.11)
(1.11)
oNitroaniline
0.17
0.13
0.19
( 0.17)
pChloronitroaniline
0.91
0.85
0.91
0.94
pNitrodiphenylamine
2.38
2.51
2 : 4Dichloro6nitroaniline
3.22
3.18
3.31
pNitroazobenzene

3.35
3.35
3.29
2 : 4Dinitroanilinc
4.38
4.43
Benzalacctophenone
5.61
Anthraquinone

8.15
2:4: 6Trinitroaniline
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
acidbase 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 dioxanewater 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 WynneJones, 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'
nButyric 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 semitheoretical 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); WynneJones and Everett, Trans. Faraday Soc., 35, 1380 (1939).
AMPHIPROTIC SOLVENTS 337
of the free energy of dissociation of an acid into nonelectrostatic 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 fortyeight 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 righthand
and lefthand 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); WynneJones,
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 righthand 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 righthand 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 DebyeHuckol equation, the value of log TH^OFT
may be represented by A Vp + Ci, 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 lefthand 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
squareroot 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 acrotonic acid in Piohlcm 7 of
Chap. III.
2. Utilize the data referred to in Problem 1 to calculate the dissociation
functions of acetic and acrotonic 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 DebyeHuckel 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 ofluorophenol 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 ofluorophenol, 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 lefthand
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 bollshaped 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 sidetube 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 nonaqueous 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 oxidationreduction 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 quirionehydroquinone
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 righthand 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 nonelectrolytes;
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 nonaqueous 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 socalled " 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 wmh 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 socalled "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 thinwalled 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 silversilver 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 thinwalled 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).
ACIDBABE 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 righthand
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
AcidBase Indicators. An aridbase 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 acidbase 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 acidbase indicator. 11
If HIni represents the unionized, 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 unionized 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 InJ may be replaced by their respective concen
11 For a full discussion of the properties of indicators, sec Kolthoff and Kosenbhun,
" AcidBase Indicators," 1937.
ACIDBASE 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 acidbase 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 unionized 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 unionized
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 unionized molecules will be chiefly in the
HIni form. For a satisfactory indicator, therefore, equation (17) may
be written as
Unionized 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 unionized 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 unionized 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
pdimethylaminoazobenzcne sulfouic acid, the indicator action being due
362 THE DETERMINATION OP HYDROGEN IONS
to the basic dimethylaminogroup, 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 0gg 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 unionized 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 unionized 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 unionized 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 unionized 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 (unionized) form, e.g., phenol
phthalein and pnitrophenol, 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 unionized 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 unionized 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 unionized 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 twocolor 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 3.14.4
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
pNitrophenol 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 8.310.0
Thymolphthalein 9.4 9.210.6
Alizarine yellow  10.012.0
Nitramine 11.013,0
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
Orangebrown
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
+
Unionized 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 wedgeshaped 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 doublewedge can of course be calibrated
so that the logarithmic term, i.e., the second term on the righthand 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 flatbottomed 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 unionized 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 unionized 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 DebyeHiickel 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 socalled 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, anaphtholphthalein, 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 asolution 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 reformed 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 "onesided" weakness, and the latter as a
salt of "twosided" 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 wellknown 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 DebyeIIuckel 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
squareroot of the hydrolysis constant and inversely proportional to the
squareroot 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
squareroot 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
104 10w
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
!0w 104
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
pHipfc. 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 doublesided 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 twosided 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 unionized, 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 nonconducting 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 nonaqueous 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 twosided 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 socalled 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 DebyeHuckel 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 equivalencepoint, 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 nonaqueous 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 unneutralized 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 unneutralized 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 equivalencepoint, 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
unneutralized 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 equivalencepoint, 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 pHneutralization
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 equivalencepoint 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 equivalencepoint 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 DebyeHvickel 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 partlyneutralized 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 equivalencepoint, 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 equivalencepoint the acidbase system under
consideration is, of course, equivalent to such a solution.
It is thus possible to calculate the whole of the pHneutralization
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 unneutralized 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 ab 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 equivalencepoint 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 equivalencepoint 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 equivalencepoint, 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 pHneutralization 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 unneutralized 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 pHneutralization 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 equivalencepoint 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
o6 + 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
equivalencepoint, indicated by an arrow, occurs at a pH of 11.0. The
inflexion at the equivalencepoint 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 pHneutralization curve as the equivalencepoint 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 pllneutralization curves for
weak bases are exactly analogous to those for weak acids, except that
they appear at the top righthand 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 equivalencepoint, 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 pHneutralization 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 equivalencepoint 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 pllneutralization
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 equivalencepoint.
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 lefthand end, representing an equiva
lent amount of free acetic acid. It is evident that there is no sharp
change of potential when the equivalencepoint 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 equivalencepoint, 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 bya^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 equivalencepoint 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 endpoint of the displacement
reaction.
Neutralization in NonAqueous Media. As already seen, the mag
nitude of the inflexion in a pllneutralization 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 NONAQUEOUS 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 equivalencepoint. 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 equivalencepoint 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 equivalencepoint 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 endpoint of the neutralization. There
are thus two possibilities for increasing the sharpness of the approach to
the equivalencepoint; 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 equivalencepoint 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
equivalencepoint. 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 unneutralized
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 lefthand 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 equivalencepoint
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 equivalencepoint becomes
CH+ Vfcifcn, (59)
.'. pH = pfci + Jpfax. (60)
The pH at the equivalencepoint 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
lencepoint, 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 equivalencepoint 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 equivalencepoint, 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 equivalencepoint 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 equivalencepoint, 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 equivalencepoint 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
equivalencepoint the pH is close to that for the neutralization of the
second acid alone; the pH at the final equivalencepoint 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
lencepoint. The figure shows
clearly that if the weaker acid
HAn is moderately weak, as at HA, the inflexion at the first equiva
lencepoint 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 equivalencepoint.
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 equivalencepoint, 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
equivalencepoint 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
equivalencepoint 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 pllneutralization 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 equivalencepoint 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 equivalencepoint is that k\/kz should be large.
Under these conditions the individual pHneutralization 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 pHneutrali
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 equivalencepoint;
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
pHneutralization 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 equivalencepoints can be fixed by using a relationship
similar to equation (66); the pH at the nth equivalencepoint, 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 pHneutraliza
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 nh equivalencepoint, from that of the nth acid alone. The
inflexion at the nth equivalencepoint 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 pHcurve at the nth
equivalencepoint; if fc n /fc n +i is less than 16, there is no detectable in
flexion in the pHneutralization 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 endpoint of an acidbase 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 oxidationreduction
titrations (p. 285). The position of the endpoint 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 gaslift, 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 endpoint 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 endpoint can be estimated obviously
depends on the magnitude of the inflexion in the hydrogen potential
neutralization curve at the equivalencepoint, 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 equivalencepoint is large,
even with relatively dilute solutions (cf. Fig. 101), and the endpoint
can be obtained accurately. If a weak acid and a strong base, or vice
versa, are employed the endpoint 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 endpoint 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 endpoint 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 endpoints 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 endpoint 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 equivalencepoint 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 equivalencepoint is not detectable, the
second equivalencepoint, 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 pHneutralizatiori curve,
or the potentialtitration 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 endpoint, i.e., where AE/Av is a maximum, and (C H +) is
thg, value at the theoretical, or stoichiometric, equivalencepoint, 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 acidbase 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 endpoints
of neutralization titrations. 9 The choice of the indicator for a particular
titration can best be determined from an examination of the pHneutrali
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 endpoint 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
Endpoint Color
Purplishgreen
Orange
Yellowishred
Purplishgreen
Green
Rosered
Red
Blueviolet
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 pHneutralization curve. When
the endpoint 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
pHneutralization curves for a number of acids and bases of different
strengths are plotted, while at the righthand side a series of indicators
are arranged at the pH levels corresponding to their titration exponents.
The positions of the equivalencepoints 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 equivalencepoint (see Table LXIX);
any indicator changing color in this range can, therefore, be used to give
a reliable indication when the endpoint 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
equivalencepoint is less sharp, viz., from 5 to 9; methyl orange will,
therefore, undergo its color change before the endpoint 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 equivalencepoint 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 equivalencepoint is at
pH 5, and the change of potential between pH 4 to 6 is rapid (curve
lAIIs). 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 endpoint, the pHneutralization 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 endpoint with any degree of
accuracy. In general, the condition requisite for the accurate estima
tion of a potentiometric endpoint, 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 equivalencepoint by means of a comparison flask containing
a solution of the salt formed at the endpoint, 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 IIlAIs and lAIIIs) or a
moderately weak acid by a weak base (curve IlAIIs); in none of these
instances is there a sharp change of pH at the equivalencepoint.
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
endpoint 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 equivalencepoint which can
be detected with fair accuracy by means of an indicator.
The problem of the detection of the various equivalencepoints 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 endpoint appears feasible, the appropriate indicator is the one whose
pfcm value lies close to the pH at the required equivalencepoint.
Buffer Solutions. It is evident from a consideration of pHneutrali
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 acidbase system can thus be
obtained directly from the pHneutralization 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 equivalencepoint 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 equivalencepoint 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
pHneutralization 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 tenfold, 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 onethird 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 acidbase 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
1.02.2
1.03.7
2.23.8
3.24.9
3.05.8
3.75.6
4.06.2
5.06.3
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
7.810.0
8.210.1
9.211.0
11.012.0
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 predeter
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, "AcidBase 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 DebyeHlickel 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 onitraniline 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 onitraniline 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, NaHCN2.
"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 DebyeHiickel 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 pHneutralization 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 pHneutralization 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 pHneutralization 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 pHbuffer 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 acidacetate 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 ammoacids, 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 aminoacid 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
aminosulfonic acid, but their importance in connection with ampholytic
equilibria in aminocarboxylic acids was not clearly realized. The sug
gestion was made by Bjerrum, 1 however, that nearly the whole of a
neutral aliphatic aminoacid 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 aminoacid 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 aminoacids; 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
aminoacids 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 aminoacids, 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 aminoacid 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 aminoarid with
and without formaldehyde
portion of the aminoacid, 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 dualion 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 aminoacids
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
aminoacid 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 aminoacid; 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 aminoacids which are in agree
ment with the dipolarion 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 Xray 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 aminoacids 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 aminoacids; 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 AMINOACID8 421
sulfonic acids also exist largely in the dualion form, but aminobenzoic
acids and aminophenols 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 AminoAcids. A very considerable simpli
fication in the treatment of aminoacids 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 socalled neutral
form of the ammoacid which, in this instance, consists almost exclusively
of the dualion 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 lefthand 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 hydrogensilver chloride cell is known, and so the
lefthand side of equation (4) can be evaluated; the DebyeHiickel 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 lefthand
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 RCOrNa+, 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 lefthand 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 ammoacid 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 aminoacid 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 aminoacid (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 equivalencepoint, 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 aminoacids are recorded in Table LXXV; 7 those
TABLE LXXV. DISSOCIATION CONSTANTS OP AMIXOACIDR AT 23
Aininoftcid
P/Ci
pA'z
Ammoacid
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 aminoacids, neutralization
of the electrically neutral form by a strong acid gave the basic dissocia
tion constant k b of the NH 2 gr