THE BOOK WAS DRENCHED [<OU_164128 OUP 875 2-U-6&-- 6,00<). OSMANIA UNIVERSITY LIBRARY Call No. J&l '0*3 Accession No. ( Author Title This book should be "returned on or before th date latumarked below. AN INTRODUCTION TO ELECTROCHEMISTRY BY SAMUEL GLASSTONE, D.Sc., PH.D. Contultanl, Untied Stales Atomic Energy Commirsion TENTH PRINTING (AN EAST-WEST EDITION) AFFILIATED EAST-WEST PRESS PVT. LTD. NEW DELHI. Copyright 1942 by LITTON EDUCATIONAL PUBLISHING, INC. No reproduction in any form of this book, in whole or in part (except for brief quotation in critical articles or reviews), may be made without written authorization from the publishers. First Published May 1942 AFFILIATED EAST-WEST PRESS PVT. LTD. East- West Student Edition - 1965 Second East-West Reprint - 1968 Third East-West Reprint 1971 Fourth East-West Reprint - 1974 Price in India Rs. 12 00 Sales Territory : India, Ban^la Desh, Burma, Ceylon, Malaysia, Pakistan and Singapore. Reprinted in India with the special permission of the original Publishers, Litton Educational Publishing, Inc. New York, U.S.A. and the copyright holders. This book has been published with the assistance of the joint Indian-American Textbook Programme Published by K.S. Padmanabhan for AFFILIATED EAST- WEST PRESS PVT. LTD., 9 Nizamuddm East, New Delhi 13, India, and printed by Mohan Makhijani at Rckha Printers, New Delhi. To V PREFACE The object of this book is to provide an introduction to electro- chemistry in its present state of development. An attempt has been made to explain the fundamentals of the subject as it stands today, de- voting little or no space to the consideration of theories and arguments that have been discarded or greatly modified. In this way it is hoped that the reader will acquire the modern point of view in electrochemistry without being burdened by much that is obsolete. In the opinion of the writer, there have been four developments in the past two decades that have had an important influence on electrochemistry. They are the ac- tivity concept, the interionic attraction theory, the proton-transfer theory of acids and bases, and the consideration of electrode reactions as rate processes. These ideas have been incorporated into the structure of the book, with consequent simplification and clarification in the treatment of many aspects of electrochemistry. This book differs from the au thor's earlier work, "The Electrochem- istry of Solutions/' in being less comprehensive and in giving less detail. While the latter is primarily a work of reference, the present book is more suited to the needs of students of physical chemistry, and to those of chemists, physicists and physiologists whose work brings them in con- tact with a variety of electrochemical problems. As the title implies, the book should also serve as an introductory text for those who in- tend to specialize in either the theoretical or practical applications of electrochemistry. In spite of some lack of detail, the main aspects of the subject have been covered, it is hoped impartially and adequately. There has been some tendency in recent electrochemical texts to pay scant attention to the phenomena at active electrodes, such as ovcrvoltage, passivity, cor- rosion, deposition of metals, and so on. These topics, vihich are of importance in applied electrochemistry, are treated here at Mich length as seems reasonable. In addition, in view of tho growing interest in electrophoresis, and its general acceptance as a branch of electrochem- istry, a chapter on clectrokinetic phenomena has boon included. No claim is made to anything approaching completeness in the matter of references to the scientific literature. Such reformers as arc given arc generally to the more recent publications, to review articles, and to papers that may, for one reason or another, have some special interest. References are also frequently included to indicate the sources from which data have been obtained for many of the diagrams and tables. Since no effort was made to be exhaustive in this connection, it was felt that an author index would be misleading. This has consequently been VI PREFACE omitted, but where certain theories, laws or equations are usually asso- ciated with the names of specific individuals, such names have been in- cluded in the general index. In conclusion, attention may be drawn to the problems which are to be found at the end of each chapter. These have been chosen with the object of illustrating particular points; very few are of the kind which involve mere substitution in a formula, and repetition of problems of the same type has been avoided as far as possible. Many of the problems are based on data taken directly from the literature, and their solution should provide both valuable exercise and instruction. The reference to the publication from which the material was taken has been given in the hope that when working the problem the student may become sufficiently interested to read the original paper and thus learn for himself something of the methods and procedures of electrochemical research. SAMUEL GLASSTONE NORMAN, OKLAHOMA March 1942 CONTENTS CHAPTER PAGE PREFACE v I. INTRODUCTION 1 II. ELECTROLYTIC CONDUCTANCE 29 III. THE THEORY OF ELECTROLYTIC CONDUCTANCE 79 t VI. THE MIGRATION OF IONS 107 i.V. FREE ENERGY AND ACTIVITY 131 REVERSIBLE CELLS 183 ELECTRODE POTENTIALS 226 VIII. OXIDATION-REDUCTION SYSTEMS 267 IX. ACIDS AND BASES 306 X. THE DETERMINATION OF HYDROGEN IONS 348 XI. NEUTRALIZATION AND HYDROLYSIS 370 XII. AMPHOTERIC ELECTROLYTES 418 XIII. POLARIZATION AND OVERVOLTAGE 435 XIV. THE DEPOSITION AND CORROSION OF METALS 482 < XV. ELECTROLYTIC OXIDATION AND REDUCTION 504 XVI. ELECTROKINETIC PHENOMENA 521 INDEX 547 CHAPTER I INTRODUCTION Properties of Electric Current. When plates of two dissimilar metals are placed in a conducting liquid, such as an aqueous solution of a salt or an acid, the resulting system becomes a source of electricity; this source is generally referred to as a voltaic cell or galvanic cell, in honor of Volta and Galvani, respectively, who made the classical discoveries in this field. If the plates of the cell are connected by a wire and a mag- netic needle placed near it, the needle will be deflected from its normal position ; it will be noted, at the same time, that the wire becomes warm. If the wire is cut and the two ends inserted in a conducting solution, chemical action will be observed where the wires come into contact with the liquid; this action may be in the form of gas evolution, or the libera- tion of a metal whose salt is present in the solution may be observed. These phenomena, viz., magnetic, heating and chemical effects, are said to be caused by the passage, or flow, of a current of electricity through the wire. Observation of the direction of the deflection of the magnetic needle and the nature of the chemical action, shows that it is possible to associate direction with the flow of electric current. The nature of this direction cannot be defined in absolute terms, and so it is desirable to adopt a convention and the one generally employed is the following: if a man were swimming with the electric current and watching a compass needle, the north-seeking pole of the needle would turn towards his left side. When electricity is passed through a solution, oxygen is generally liberated at the wire at which the positive current enters whereas hydro- gen or a metal is set free at the wire whereby the current leaves the solution. It is unfortunate that this particular convention was chosen, because when the electron was discovered it was observed that a flow of electrons produced a magnetic effect opposite in direction to that accompanying the flow of positive current in the same direction. It was necessary, therefore, to associate a negative charge with the electron, in order to be in harmony with the accepted convention concerning the direction of a current of electricity. Since current is carried through metals by means of electrons only, it means that the flow of electrons is opposite in direc- tion to that of the conventional current flow. It should be emphasized that there is nothing fundamental about this difference, for if the direc- tion of current flow had been defined in the opposite manner, the electron would have been defined as carrying a positive charge and the flow of electrons and of current would have been in the same direction. Al- 2 INTRODUCTION though a considerable simplification would result from the change in convention, it is too late in the development of the subject for any such change to be made. E.M.F., Current and Resistance: Ohm's Law. If two voltaic cells are connected together so that one metal, e.g., zinc, of one cell is con- nected to the other metal, e.g., copper, of the second cell, in a manner analogous to that employed by Volta in his electric pile, the magnetic and chemical effects of the current are seen to be increased, provided the same external circuit is employed. The two cells have a greater electrical driving force or pressure than a single one, and this force or pressure * which is regarded as driving the electric current through the wire is called the electromotive force, or E.M.F. Between any two points in the circuit carrying the current there is said to be a potential difference, the total E.M.F. being the algebraic sum of all the potential differences. By increasing the length of the wire connecting the plates of a given voltaic cell the effect on the magnetic needle and the chemical action are seen to be decreased: the greater length of the wire thus opposes the flow of current. This property of hindering the flow of electricity is called electrical resistance, the longer wire having a greater electrical resistance than the shorter one. It is evident that the current strength in a given circuit, as measured by its magnetic or chemical effect, is dependent on the E.M.F. of the cell producing the current and the resistance of the circuit. The relationship between these quantities is given by Ohm's law (1827), which states that the current strength (/) is directly proportional to the applied E.M.F. (E) and inversely proportional to the resistance (R) ; thus is the mathematical expression of Ohm's law. The accuracy of this law has been confirmed by many experiments with conductors of various types: it fails, apparently, for certain solutions when alternating currents of very high frequency are employed, or with very high voltages. The reasons for this failure of Ohm's law are of importance in connection with tho theory of solutions (see Chap. III). It is seen from equation (1) that the E.M.F. is equal to the product of the current and the resistance: a consequence of this result is that the potential difference between any two points in a circuit is given by the product of the resistance between those points and the current strength, the latter being the same through- out the circuit. This rule finds a number of applications in electro- chemical measurements, as will be evident in due course. * Electrical force or pressure does not have the dimensions of mechanical force or pressure; the terms are used, however, by analogy with the force or pressure required to produce the flow of a fluid through a pipe. ELECTRICAL DIMENSIONS AND UNITS 3 Electrical Dimensions and Units. The electrostatic force (F) be- tween two charges e and e' placed at a distance r apart is given by where K depends on the nature of the medium. Since force has the dimensions mlt~ 2 , where m represents a mass, I length and t time, it can be readily seen that the dimensions of electric charge are mWt* 1 **, the dimensions of K not being known. The strength of an electric current is defined by the rate at which an electric charge moves along a conductor, and so the dimensions of current are mWt-***. The electromagnetic force between two poles of strength p and p' separated by a distance r is pp'lnr*, where p, is a constant for the medium, and so the dimensions of pole strength must be mH*t~ l p*. It can be deduced theoretically that the work done in carrying a magnetic pole round a closed circuit is pro- portional to the product of the pole strength and the current, and since the dimensions of work are mPt~* 9 those of current must be m*W~V~~*- Since the dimensions of current should be the same, irrespective of the method used in deriving them, it follows that The dimensions l~ l t are those of a reciprocal velocity, and it has been shown, both experimentally and theoretically, that the velocity is that of light, i.e., 2.9977 X 10 10 cm. per sec., or, with sufficient accuracy for most purposes, 3 X 10 10 cm. per sec. In practice K and n are assumed to be unity in vacuum: they are then dimensionlcss and are called the dielectric constant and magnetic per- meability, respectively, of the medium. Since K and n cannot both be unity for the same medium, it is evident that the units based on the assumption that K is unity must be different from those obtained by taking /x as unity. The former are known as electrostatic (e.s.) and the latter as electromagnetic (e.m.) units, and according to the facts recorded above 1 e.m. unit of current - ------ - = 3 X 10 l crn. per sec. 1 e.s. unit of current It follows, therefore, that if length, mass and time are expressed in centi- meters, grams and seconds respectively, i.e., in the c.g.s. system, the e.m. unit of current is 3 X 10 10 times as great as the e.s. unit. The e.m. unit of current on this system is defined as that current which flowing through a wire in the form of an arc one cm. long and of one cm. radius exerts a force of one dyne on a unit magnetic pole at the center of the arc. The product of current strength and time is known as the quantity of electricity; it has the same dimensions as electric charge. The e.m. unit of charge or quantity of electricity is thus 3 X 10 10 larger than the corre- 4 INTRODUCTION spending e.s. unit. The product of quantity of electricity and potential or E.M.F. is equal to work, and if the same unit of work, or energy, is adopted in each case, the e.m. unit of potential must be smaller than the e.s. unit in the ratio of 1 to 3 X 10 10 . When one e.m. unit of potential difference exists between two points, one erg of work must be expended to transfer one e.m. unit of charge, or quantity of electricity, from one point to the other; the e.s. unit of potential is defined in an exactly analogous manner in terms of one e.s. unit of charge. The e.m. and e.s. units described above are not all of a convenient magnitude for experimental purposes, and so a set of practical units have been defined. The practical unit of current, the ampere, often abbrevi- ated to " amp.," is one-tenth the e.m. (c.g.s.) unit, and the corresponding unit of charge or quantity of electricity is the coulomb ; the latter is the quantity of electricity passing when one ampere flows for one second. The practical unit of potential or E.M.F. is the volt, defined as 10 8 e.m. units. Corresponding to these practical units of current and E.M.F. there is a unit of electrical resistance; this is called the ohm, and it is the re- sistance of a conductor through which a current of one ampere passes when the potential difference between the ends is one volt. With these units of current, E.M.F. and resistance it is possible to write Ohm's law in the form volts By utilizing the results given above for the relationships between e.m., e.s. and practical units, it is possible to draw up a table relating the various units to each other. Since the practical units are most fre- quently employed in electrochemistry, the most useful method of ex- pressing the connection between the various units is to give the number of e.m. or e.s. units corresponding to one practical unit: the values are recorded in Table I. TABLE I. CONVERSION OF ELECTRICAL UNITS Practical Equivalent in Unit e.m.u. e.s.u. Current Ampere 10" 1 3 X 10 fl Quantity or Charge Coulomb 10~ l 3 X 10' Potential or E.M.F. Volt 10 8 (300)~ l International Units. The electrical units described in the previous section are defined in terms of quantities which cannot be easily estab- lished in the laboratory, and consequently an International Committee (1908) laid down alternative definitions of the practical units of elec- tricity. The international ampere is defined as the quantity of electricity which flowing for one second will cause the deposition of 1.11800 milli- grams of silver from a solution of a silver salt, while the international ohm is the resistance at c. of a column of mercury 106.3 cm. long, of uniform cross-section, weighing 14.4521 g. The international volt is then the ELECTRICAL ENERGY 5 difference of electrical potential, or E.M.F., required to maintain a current of one international ampere through a system having a resistance of one international ohm. Since the international units were defined it has been found that they do not correspond exactly with those defined above in terms of the c.g.s. system; the latter are thus referred to as absolute units to distinguish them from the international units. The international ampere is 0.99986 times the absolute ampere, and the international ohm is 1.00048 times the absolute ohm, so that the international volt is 1.00034 times the absolute practical unit.* Electrical Energy. As already seen, the passage of electricity through a conductor is accompanied by the liberation of heat; according to the first law of thermodynamics, or the principle of conservation of energy, the heat liberated must be exactly equivalent to the electrical energy expended in the conductor. Since the heat can be measured, the value of the electrical energy can be determined and it is found, in agreement with anticipation, that the heat liberated by the current in a given con- ductor is proportional to the quantity of electricity passing and to the difference of potential at the extremities of the conductor. The practical unit of electrical energy is, therefore, defined as the energy developed when one coulomb is passed through a circuit by an E.M.F. of one volt; this unit is called the volt-coulomb, and it is evident from Table I that the absolute volt-coulomb is equal to 10 7 ergs, or one joule. It follows, therefore, that if a current of / amperes is passed for t seconds through a conductor under the influence of a potential of E volts, the energy liber- ated (Q) will be given by Q = Elt X 10 7 ergs, (3) or, utilizing Ohm's law, if R is the resistance of the conductor, Q = PRt X 10 7 ergs. (4) These results are strictly true only if the ampere, volt and ohm are in absolute units; there is a slight difference if international units are employed, the absolute volt-coulomb or joule being different from the international value. The United States Bureau of Standards has recom- mended that the unit of heat, the calorie, should be defined as the equiva- lent of 4.1833 international joules, and hence Elt Q = j^gjj calories, (5) where E and / are now expressed in international volts and amperes, respectively. Alternatively, it may be stated that one international volt-coulomb is equivalent to 0.2390 standard calorie. * These figures are obtained from the set of consistent fundamental constants recommended by Birge (1941); slightly different values are given in the International Critical Tables. 6 INTRODUCTION Classification of Conductors. All forms of matter appear to be able to conduct the electric current to some extent, but the conducting powers of different substances vary over a wide range; thus silver, one of the best conductors, is 10 24 times more effective than paraffin wax, which is one of the poorest conductors. It is not easy to distinguish sharply between good and bad conductors, but a rough division is possible; the systems studied in electrochemistry are generally good conductors. These may be divided into three main categories; they are: (a) gaseous, (6) metallic and (c) electrolytic. Gases conduct electricity with difficulty and only under the influence of high potentials or if exposed to the action of certain radiations. Metals are the best conductors, in general, and the passage of current is not accompanied by any movement of matter; it appears, therefore, that the electricity is carried exclusively by the electrons, the atomic nuclei re- maining stationary. This is in accordance with modern views which regard a metal as consisting of a relatively rigid lattice of ions together with a system of mobile electrons. Metallic conduction, or electronic conduction) as it is often called, is not restricted to pure metals, for it is a property possessed by most alloys, carbon and certain solid salts and oxides. Electrolytic conductors, or electrolytes, are distinguished by the fact that passage of an electric current through them results in an actual transfer of matter; this transfer is manifested by changes of concentration and frequently, in the case of electrolytic solutions, by the visible sepa- ration of material at the points where the current enters and leaves the solution. Electrolytic conductors are of two main types; there are, first, substances which conduct elcctrolytically in the pure state, such as fused salts and hydrides, the solid halides of silver, barium, lead and some other metals, and the a-form of silver sulfide. Water, alcohols, pure acids, and similar liquids are very poor conductors, but they must be placed in this category. The second class of electrolytic conductors consists of solutions of one or more substances; this is the type of con- ductor with which the study of electrochemistry is mainly concerned. The most common electrolytic solutions are made by dissolving a salt, acid or base in water; other solvents may be used, but the conducting power of the system depends markedly on their nature. Conducting sys- tems of a somewhat unusual type are lithium carbide and alkaline earth nitrides dissolved in the corresponding hydride, and organic acid amides and mtro-compounds in liquid ammonia or hydrazine. The distinction between electronic and electrolytic conductors is not sharp, for many substances behave as mixed conductors; that is, they conduct partly electronically and partly electrolytically. Solutions of the alkali and alkaline earth metals in liquid ammonia are apparently mixed conductors, and so also is the -form of silver sulfide. Fused cuprous sulfide conducts electronically, but a mixture with sodium or ferrous sulfide also exhibits electrolytic conduction; a mixture with nickel THE PHENOMENA AND MECHANISM OP ELECTROLYSIS 7 sulfide is, however, a pure electronic conductor. Although pure metals conduct electronically, conduction in certain liquid alloys involves the transfer of matter and appears to be partly electrolytic in nature. Some materials conduct electronically at one temperature and electrolytically at another; thus cuprous bromide changes its method of conduction between 200 and 300. The Phenomena and Mechanism of Electrolysis. The materials, generally small sheets of metal, which are employed to pass an electric current through an electrolytic solution, are called electrodes; the one at which the positive current enters is referred to as the positive electrode or anode, whereas the electrode at which current leaves is called the negative electrode, or cathode. The passage of current through solu- tions of salts of such metals as zinc, iron, nickel, cadmium, lead, copper, silver and mercury results in the liberation of these metals at the cathode; from solutions of salts of the very base metals, e.g., the alkali and alka- line earth metals, and from solutions of acids the substance set free is hydrogen gas. If the anode consists of an attackable metal, such as one of those just enumerated, the flow of the current is accompanied by the passage of the metal into solution. When the anode is made of an inert metal, e.g., platinum, an element is generally set free at this electrode; from solutions of nitrates, sulfates, phosphates, etc., oxygen gas is liber- ated, whereas from halide solutions, other than fluorides, the free halogen is produced. The decomposition of solutions by the electric current, resulting in the liberation of gases or metals, as described above, is known as electrolysis. The first definite proposals concerning the mechanism of electrolytic conduction and electrolysis were made by Grotthuss (1800) ; he suggested that the dissolved substance consisted of particles with positive and negative ends, these particles being _ _L distributed in a random manner throughout the solution. When a potential was applied it was believed q f+q 3- E3 E3 E3 E3 +E: ED ED E3 ED ED ii in IV that the particles (molecules) became oriented in the form of chains with the positive parts pointing in one di- In rection and the negative parts in the opposite direction (Fig. 1, I). It was supposed that the positive elec- T. * i. . * , , */. , ., '. , e FIQ. 1. Mechanism of trode attracts the negative part of Orotthuss conduction one end particle in the chain, resulting in the liberation of the corresponding material, e.g., oxygen in the elec- trolysis of water. Similarly, the negative electrode attracts the positive portion of the particle, e.g., the hydrogen of water, at the other end of the chain, and sets it free (Fig. 1, II). The residual parts of the end units were then imagined to exchange partners with adjacent molecules, this interchange being carried on until a complete series of new particles 8 INTRODUCTION is formed (Fig. 1, III). These are now rotated by the current to give the correct orientation (Fig. 1, IV), followed by their splitting up, and so on. The chief objection to the theory of Grotthuss is that it would require a relatively high B.M.F., sufficient to break up the molecules, before any appreciable current was able to flow, whereas many solutions can be electrolyzed by the application of quite small potentials. Al- though the proposed mechanism has been discarded, as far as most electrolytic conduction is concerned, it will be seen later (p. 66) that a type of Grotthuss conduction occurs in solutions of acids and bases. In order to account for the phenomena observed during the passage of an electric current through solutions, Faraday (1833) assumed that the flow of electricity was associated with the movement of particles of matter carrying either positive or negative charges. These charged particles were called ions ; the ions carrying positive charges and moving in the direction of the current, i.e., towards the cathode, were referred to as cations, and those carrying a negative charge and moving in the opposite direction, i.e., towards Cathode the anode, were called onions * (see Fig. 2). The function of the ap- plied E.M.F. is to direct the ions towards the appropriate electrodes where their charges are neutralized and they are set free as atoms or molecules. It may be noted that since hydrogen and metals are dis- charged at the cathode, the metal- lic part of a salt or base and the hydrogen of an acid form cations and carry positive charges. The acidic portion of a salt and the hydroxyl ion of a base consequently carry negative charges and constitute the anions. Although Faraday postulated the existence of charged material par- ticles, or ions, in solution, he offered no explanation of their origin: it was suggested, however, by Clausius (1857) that the positive and negative parts of the solute molecules were not firmly connected, but were each in a state of vibration that often became vigorous enough to cause the portions to separate. These separated charged parts, or ions, were be- lieved to have relatively short periods of free existence; while free they were supposed to carry the current. According to Clausius, a small fraction only of the total number of dissolved molecules was split into * The term "ion" is derived from a Greek word moaning "wanderer" or "traveler," the prefixes ana and cata meaning "up" and "down," respectively; the anion is thus the ion moving up, and the cation that moving down the potential gradient. These terms, as well as electrode, anode and cathode, were suggested to Faraday by Whewell (1834); see Oesper and Speter, Scientific Monthly, 45, 535 (1937). Direction of Election Flow FIG. 2. Illustration of electrochemical terms THE ELECTBOLYTIC DISSOCIATION THEORY 9 ions at any instant, but sufficient ions were always available for carrying the current and hence for discharge at the electrodes. Since no electrical energy is required to break up the molecules, this theory is in agreement with the fact that small E.M.P/S are generally adequate to cause elec- trolysis to occur; the applied potential serves merely to guide the ions to the electrodes where their charges are neutralized. The Electrolytic Dissociation Theory. 1 From his studies of the con- ductances of aqueous solutions of acids and their chemical activity, Arrhenius (1883) concluded that an electrolytic solution contained two kinds of solute molecules; these were supposed to be " active" molecules, responsible for electrical conduction and chemical action, and inactive molecules, respectively. It was believed that when an acid, base or salt was dissolved in water a considerable portion, consisting of the so-called active molecules, was spontaneously split up, or dissociated, into positive and negative ions; it was suggested that these ions are free to move independently and are directed towards the appropriate electrodes under the influence of an electric field. The proportion of active, or dissoci- ated, molecules to the total number of molecules, later called the "degree of dissociation," was considered to vary with the concentration of the electrolyte, and to be equal to unity in dilute solutions. This theory of electrolytic dissociation, or the ionic theory, attracted little attention until 1887 when van't Hoff's classical paper on the theory of solutions was published. The latter author had shown that the ideal gas law equation, with osmotic pressure in place of gas pressure, was applicable to dilute solutions of non-electrolytes, but that electrolytic solutions showed considerable deviations. For example, the osmotic effect, as measured by depression of the freezing point or in other ways, of hydrochloric acid, alkali chlorides and hydroxides was nearly twice as great as the value to be expected from the gas law equation; in some cases, e.g., barium hydroxide, and potassium sulfate and oxalate, the discrepancy was even greater. No explanation of these facts was offered by van't Iloff, but he introduced an empirical factor i into the gas law equation for electrolytic solutions, thus n = iRTc, where II is the observed osmotic pressure of the solution of concentra- tion c; the temperature is T, and R is the gas constant. According to this equation, the van't Hoff factor i is equal to the ratio of the experi- mental osmotic effect to the theoretical osmotic effect, based on the ideal gas laws, for the given solution. Since the osmotic effect is, at least approximately, proportional to the number of individual molecular par- ticles, a value of two for the van't Hoff factor means that the solution contains about twice the number of particles to be expected. This result 1 Arrhenius, J. Chem. Soc., 105, 1414 (1914); Walker, ibid., 1380 (1928). 10 INTRODUCTION is clearly in agreement with the views of Arrhenius, if the ions are re- garded as having the same osmotic effect as uncharged particles. The concept of "active molecules/' which was part of the original theory, was later discarded by Arrhenius as being unnecessary; he sug- gested that whenever a substance capable of yielding a conducting solution was dissolved in water, it dissociated spontaneously into ions, the extent of the dissociation being very considerable with salts and with strong acids and bases, especially in dilute solution. Thus, a molecule of potassium chloride should, according to the theory of electrolytic dissociation, be split up into potassium and chloride ions in the following manner: KC1 = K+ + Cl-.' If dissociation is complete, then each " molecular particle " of solid potas- sium chloride should give two particles in solution; the osmotic effect will thus approach twice the expected value, as has actually been found. A bi-univalent salt, such as barium chloride, will dissociate spontaneously according to the equation BaCl 2 = Ba++ + 2C1~, and hence the van't Hoff factor should be approximately 3, in agreement with experiment. Suppose a solution is made up by dissolving m molecules in a gr/en volume and a is the fraction of these molecules dissociated into ions; if each molecule produces v ions on dissociation, there will be present in the solution m(l a) undissociated molecules and vma ions, making a total of m ma + vma particles. If the van't Hoff factor is equal to the ratio of the number of molecular particles actually present to the number that would have been in the solution if there had been no dis- sociation, then m ma + vma i = - = 1 a + va: m ' (6) Since the van't Hoff factor is obtainable from freezing-point, or analo- gous, measurements, the value of or, the so-called degree of dissociation, in the given solution can be calculated from equation (6). An alterna- tive method of evaluating a, using conductance measurements (see p. 51), was proposed by Arrhenius (1887), and he showed that the results ob- tained by the two methods were in excellent agreement: this agreement was accepted as strong evidence for the theory of electrolytic dissocia- tion, which has played such an important role in the development of electrochemistry. It is now known that the agreement referred to above, which con- vinced many scientists of the value of the Arrhenius theory, was to a EVIDENCE FOR THE IONIC THEORY 11 great extent fortuitous; the conductance method for calculating the degree of dissociation is not applicable to salt solutions, and such solu- tions would, in any case, not be expected to obey the ideal gas law equation. Nevertheless, the theory of electrolytic dissociation, with certain modifications, is now universally accepted; it is believed that when a solute, capable of forming a conducting solution, is dissolved in a suitable solvent, it dissociates spontaneously into ions. If the solute is a salt or a strong acid or base the extent of dissociation is very con- siderable, it being almost complete in many cases provided the solution is not too concentrated; substances of this kind, which are highly dis- sociated and \\hich give good conducting solutions in water, are called strong electrolytes. Weak acids and weak bases, e.g., amines, phenols, most carboxylic acids and some inorganic acids and bases, such as hydro- cyanic acid and ammonia, and a few salts, e.g., mercuric chloride and cyanide, are dissociated only to a small extent at reasonable concentra- tions; these compounds constitute the weak electrolytes.* Salts of weak acids or bases, or of both, are generally strong electrolytes, in spite of the fact that one or both constituents are weak. These results are in harmony with modern developments of the ionic theory, as will be evident in later chapters. As is to be expected, it is impossible to classify all electrolytes as "strong" or "weak," although this forms a convenient rough division which is satisfactory for most purposes. Certain sub- stances, e.g., trichloroacetic acid, exhibit an intermediate behavior, but the number of intermediate electrolytes is not large, at least in aqueous solution. It may be noted, too, that the nature of the solvent is often important; a particular compound may be a strong electrolyte, being dissociated to a large extent, in one solvent, but may be only feebly dissociated, and hence is a weak electrolyte, in another medium (cf. p. 13). Evidence for the Ionic Theory. There is hardly any branch of elec- trochemistry, especially in its quantitative aspects, which does not pro- vide arguments in favor of the theory of electrolytic dissociation; without the ionic concept the remarkable systems tization of the experimental results which has been achieved during the past fifty years would cer- tainly not have been possible. It is of interest, however, to review briefly some of the lines of evidence which support the ionic theory. Although exception may be taken to the quantitative treatment given by Arrhenius, the fact of the abnormal osmotic properties of electrolytic solutions still remains; the simplest explanation of the high values can be given by postulating dissociation into ions. This, in conjunction with the ability of solutions to conduct the electric current, is one of the strongest arguments for the ionic theory. Another powerful argument is * Strictly speaking, the term "electrolyte" should refer to the conducting system as a whole, but it is also frequently applied to the solute; the word "ionogen," i.e., producer of ions, has been suggested for the latter [see, for example, Blum, Trans. Electrochem. Soc., 47, 125 (1925)], but this has not come into general use. 12 INTRODUCTION based on the realization in recent years, as a result of X-ray diffraction studies, that the structural unit of solid salts is the ion rather than the molecule. That is to say, salts are actually ionized in the solid state, and it is only the restriction to movement in the crystal lattice that prevents solid salts from being good electrical conductors. When fused or dis- solved in a suitable solvent, the ions, which are already present, can move relatively easily under the influence of an applied E.M.F., and conductance is observed. The concept that salts consist of ions held together by forces of electrostatic attraction is also in harmony with modern views concerning the nature of valence. Many properties of electrolytic solutions are additive functions of the properties of the respective ions; this is at once evident from the fact that the chemical properties of a salt solution are those of its constituent ions. For example, potassium chloride in solution has no chemical reac- tions which are characteristic of the compound itself, but only those of potassium and chloride ions. These properties are possessed equally by almost all potassium salts and all chlorides, respectively. Similarly, the characteristic chemical properties of acids and alkalis, in aqueous solu- tion, are those of hydrogen and hydroxyl ions, respectively. Certain physical properties of electrolytes are also additive in nature; the most outstanding example is the electrical conductance at infinite dilution. It will be seen in Chap. II that conductance values can be ascribed to all ions, and the appropriate conductance of any electrolyte is equal to the sum of the values for the individual ions. The densities of elec- trolytic solutions have also been found to be additive functions of the properties of the constituent ions. The catalytic effects of various acids and bases, and of mixtures with their salts, can be accounted for by associating a definite catalytic coefficient with each type of ion; since undissociated molecules often have appreciable catalytic properties due allowance must be made for their contribution. Certain thermal properties of electrolytes are in harmony with the theory of ionic dissociation; for example, the heat of neutralization of a strong acid by an equivalent amount of a strong base in dilute solution Is about 13.7 kcal. at 20 irrespective of the exact nature of the acid or base. 2 If the acid is hydrochloric acid and the base is sodium hydroxide, then according to the ionic theory the neutralization reaction should be written (H+ + C1-) + (Na+ + OH-) = (Na+ + Cl~) + H 2 O, the acid, base and the resulting salt being highly dissociated, whereas the water is almost completely undissociated. Since Na+ and Cl~ ap- pear on both sideb of this equation, the essential reaction is H+ + OH- = H 2 O, 8 Richards and Rowe, /. Am. Chem. Soc., *4, 684 (1922); see also, Lambert and Gillespie, ibid., 53, 2632 (1931); Rossini, /. Res. Nat. Bur. Standards, 6, 847 (1931); Pitzer, J. Am. Chem. Soc., 59, 2365 (1937). INFLUENCE OF THE SOLVENT ON DISSOCIATION 13 and this is obviously independent of the particular acid or base em- ployed: the heat of neutralization would thus be expected to be constant. It is of interest to mention that the heat of the reaction between hydro- gen and hydroxyl ions in aqueous solution has been calculated by an entirely independent method (see p. 344) and found to be almost identical with the value obtained from neutralization experiments. The heat of neutralization of a weak acid or a weak base is generally different from 13.7 kcal., since the acid or base must dissociate completely in order that it may be neutralized and the process of ionization is generally accom- panied by the absorption of heat. Influence of the Solvent on Dissociation. 8 The nature of the solvent often plays an important part in determining the degree of dissociation of a given substance, and hence in deciding whether the solution shall behave as a strong or as a weak electrolyte. Experiments have been made on solutions of tetraisoamylammonium nitrate in a series of mix- tures of water and dioxane (see p. 54). In the water-rich solvents the system behaves like a strong electrolyte, but in the solvents containing relatively large proportions of dioxane the properties are essentially those of a weak electrolyte. In this case, and in analogous cases where the solute consists of units which are held together by bonds that are almost exclusively electrovalent in character, it is probable that the dielectric constant is the particular property of the solvent that influences the dissociation (cf. Chaps. II and III). The higher the dielectric constant of the medium, the smaller is the electrostatic attraction between the ions and hence the greater is the probability of their existence in the free state. Since the dielectric constant of water at 25 is 78.6, compared with a value of about 2.2 for dioxane, the results described above can be readily understood. It should be noted, however, that there are many instances in which the dielectric constant of the solvent plays a secondary part: for example, hydrogen chloride dissolves in ethyl alcohol to form a solution which behaves as a strong electrolyte, but in nitrobenzene, having a dielectric constant differing little from that of alcohol, the solution is a weak elec- trolyte. As will be seen in Chap. IX the explanation of this difference lies in the ability of a molecule of ethyl alcohol to combine readily with a bare hydrogen ion, i.e., a proton, to form the ion C 2 H 6 OHt, and this represents the form in which the hydrogen ion exists in the alcohol solution. Nitrobenzene, however, does not form such a combination to any great extent; hence the degree of dissociation of the acid is small and the solution of hydrogen chloride behaves as a weak electrolyte. The ability of oxygen compounds, such as ethers, ketones and even sugars, to accept a proton from a strongly acidic substance, thus forming an ion, e.g., R 2 OH+ or R 2 COH+, accounts for the fact that solutions of such compounds in pure sulfuric acid or in liquid hydrogen fluoride are relatively strong electrolytes. See, Glasstone, "The Electrochemistry of Solutions/' 1937, p. 172. 14 INTRODUCTION Another aspect of the formation of compounds and its influence on electrolytic dissociation is seen in connection with substituted ammonium salts of the type RaNHX; although they are strong electrolytes in hy- droxylic solvents, e.g., in water and alcohols, they are dissociated to only a small extent in nitrobenzene, nitromethane, acetone and acetonitrile. It appears that in the salts under consideration the hydrogen atom can act as a link between the nitrogen atom and the acid radical X, so that the molecule RsN-H-X exists in acid solution. If the solvent S is of such a nature, however, that its molecules tend to form strong hydrogen bonds, it can displace the X~ ions, thus R 3 N-H-X + S ^ K 3 N-H-S+ + X~ so that ionization of the salt is facilitated. Hydroxylic solvents, in virtue of the type of oxygen atom which they contain, form hydrogen bonds more readily than do nitro-compounds, nitriles, etc.; the difference in behavior of the two groups of solvents can thus be understood. Salts of the type R 4 NX function as strong electrolytes in both groups of solvents, since the dielectric constants are relatively high, and the question of compound formation with the solvent is of secondary impor- tance. The fact that salts of different types show relatively little differ- ence of behavior in hydroxylic solvents has led to these substances being called levelling solvents. On the other hand, solvents of the other group, e.g., nitro-compounds and nitriles, are referred to as differentiating solvents because they bring out the differences between salts of different types. The characteristic properties of the levelling solvents are due partly to their high dielectric constants and partly to their ability to act both as electron donors and acceptors, so that they are capable of forming compounds with either anions or cations. The formation of a combination of some kind between the ion and a molecule of solvent, known as solvation, is an important factor in en- hancing the dissociation of a given electrolyte. The solvatcd ions are relatively large and hence their distance of closest approach is very much greater than the bare unsolvated ions. It will be seen in Chap. V that when the distance between the centers of two oppositely charged ions is less than a certain limiting value the system behaves as if it consisted of undissociated molecules. The effective degree of dissociation thus in- creases as the distance of closest approach becomes larger; hence solvation may be of direct importance in increasing the extent of dissociation of a salt in a particular solvent. It may be noted that solvation does riot necessarily involve a covalent bond, e.g., as is the case in CuCNTIs)^ and Cu(H 2 0)t 4 "; there is reason for believing that solvation is frequently electrostatic in character and is due to the orientation of solvent molecule dipoles about the ion. A solvent with a large dipole moment will thus tend to facilitate solvation and it will consequently increase the degree of dissociation. FARADAY'S LAWS OP ELECTROLYSIS 15 It was mentioned earlier in this chapter that acid amides and nitro- compounds form conducting solutions in liquid ammonia and hydrazine; the ionization in these cases is undoubtedly accompanied by, and is associated with, compound formation between solute and solvent. The same is true of triphenylmethyl chloride which is a fair electrolytic con- ductor when dissolved in liquid sulfur dioxide; it also conducts to some extent in nitromethane, nitrobenzene and acetone solutions. In chloro- form and benzene, however, there is no compound formation and no conductance. The electrolytic conduction of triphenylmethyl chloride in fused aluminum chloride, which is itself a poor conductor, appears to be due to the reaction Ph 3 CCl + A1C1 3 = Pb 3 C+ + AlClr; this process is not essentially different from that involved in the ioniza- tion of an acid, where the II f ion, instead of a Cl~ ion, is transferred from one molecule to another. Faraday's Laws of Electrolysis. During the years 1833 and 1834, Faraday published the results of an extended series of investigations on the relationship between the quantity of electricity passing through a solution and the amount of metal, or other substance, liberated at the electrodes: the conclusions may be expressed in the form of the two following laws. I. The amount of chemical decomposition produced by a current is proportional to the quantity of electricity passing through the electro- lytic solution. II. The amounts of different substances liberated by the same quan- tity of electricity are proportional to their chemical equivalent weights. The first law can be tested by passing a current of constant strength through a given electrolyte for various periods of time and determining the amounts of material deposited, on the cathode, for example; the weights should be proportional to the time in each case. Further, the time may be kept constant and the current varied; in these experiments the quantity of deposit should be proportional to the current strength. The second law of electrolysis may be confirmed by passing the same quantity of electricity through a number of different solutions, e.g., dilute sulfuric acid, silver nitrate and copper sulfate; if a current of one ampere flows for one hour the weights liberated at the respective cathodes should be 0.0379 gram of hydrogen, 4.0248 grams of silver and 1.186 grams of copper. These quantities are in the ratio of 1.008 to 107.88 to 31.78, which is the ratio of the equivalent weights. As the result of many experiments, in both aqueous and non-aqueous media, some of which will be described below, much evidence has been obtained for the accuracy of Faraday's laws of electrolysis within the limits of reasonable experimental error. Apart from small deviations, whic^gai^ J>e readily explained by the difficulty of obtaining pure deposit^ A^ WWi?* aiia ~ 16 INTRODUCTION lytical problems, there are a number of instances of more serious apparent exceptions to the laws of electrolysis. The amount of sodium liberated in the electrolysis of a solution of the metal in liquid ammonia is less than would be expected. It must be remembered, however, that Fara- day's laws are applicable only when the whole of the conduction is electrolytic in character; in the sodium solutions in liquid ammonia some of the conduction is electronic in nature. The quantities of metal de- posited from solutions of lead or antimony in liquid ammonia containing sodium are in excess of those required by the laws of electrolysis; in these solutions the motals exist in the form of complexes and the ions are quite different from those present in aqueous solution. It is consequently not possible to calculate the weights of the deposits to be expected from Faraday's laws. The applicability of the laws has been confirmed under extreme con- ditions: for example, Richards and Stull (1902) found that a given quan- tity of electricity deposited the same weight of silver, within 0.005 per cent, from an aqueous solution of silver nitrate at 20 and from a solution of this salt in a fused mixture of sodium and potassium nitrates at 260. The experimental results are quoted in Table II. TABLE II. COMPARISON OF SILVER DEPOSITS AT 20 AND 260 Deposit at 20 Deposit at 260 Difference 1.14916 g. 1.14919 g. 0.003 per cent 1.12185 1.12195 0.009 1.10198 1.10200 0.002 A solution of silver nitrate in pyridine at 55 also gives the same weight of silver on the cathode as does an aqueous solution of this salt at ordinary temperatures. Pressures up to 1500 atmospheres have no effect on the quantity of silver deposited from a solution of silver nitrate in water. Faraday's law holds for solid electrolytic conductors as well as for fused electrolytes and solutions; this is shown by the results of Tubandt and Eggert (1920) on the electrolysis of the cubic form of silver iodide quoted in Table III. The quantities of silver deposited in an ordinary TABLE III. APPLICATION OP FARADAY'S LAWS TO SOLID SILVER IODIDE Ag deposited Ag deposited Ag lost Temp. Current in coulometer on cathode from anode 150 0.1 amp. 0.8071 g. 0.8072 g. 0.8077 g. 150 0.1 0.9211 0.9210 0.9217 400 0.1 0.3997 0.3991 0.4004 400 0.4 0.4217 0.4218 0.4223 silver coulometer in the various experiments are recorded, together with the amounts of silver gained by the cathode and lost by the anode, respectively, when solid silver iodide was used as the electrolyte. The Faraday and its Determination. The quantity of electricity required to liberate 1 equiv. of any substance should, according to the THE FARADAY AND ITS DETERMINATION 17 second of Faraday's laws, be independent of its nature; this quantity is called the faraday; it is given the symbol F and, as will be seen shortly, is equal to 96,500 coulombs, within the limits of experimental error. If e is the equivalent weight of any material set free at an electrode, then 96,500 amperes flowing for one second liberate e grams of this substance; it follows, therefore, from the first of Faraday's laws, that 7 amperes flowing for t seconds will cause the deposition of w grams, where w = lie 96,500* (7) If the product It is unity, i.e., the quantity of electricity passed is 1 coulomb, the weight of substance deposited is e/96,500; the result is known as the electrochemical equivalent of the deposited element. If this quantity is given the symbol e, it follows that w = Ite. (7o) The electrochemical equivalents of some of the more common elements are recorded in Table IV; * since the value for any given element depends TABLE IV. ELECTROCHEMICAL EQUIVALENTS IN MILLIGRAMS PER COULOMB Element Valence Hydrogen Oxygen Chlorine Iron Cobalt Nickel e 0.01045 0.08290 0.36743 0.2893 0.3054 0.3041 Element Copper Bromine Cadmium Silver Iodine Mercury Valence 2 1 2 1 1 2 0.3294 0.8281 0.5824 1.1180 1.3152 1.0394 on the valence of the ions from which it is being deposited, the actual valence for which the results were calculated is given in each case. The results given above, and equation (7) or (7a), are the quantita- tive expression of Faraday's laws of electrolysis; they can be employed either to calculate the weight of any substance deposited by a given quantity of electricity, or to find the quantity of electricity passing through a circuit by determining the weight of a given metal set free by electrolysis. The apparatus used for the latter purpose was at one time referred to as a "voltameter," but the name coulometer, i.e., coulomb measurer, proposed by Richards and Heimrod (1902), is now widely employed. The most accurate determinations of the faraday have been made by means of the silver coulometer in which the amount of pure silver deposited from an aqueous solution of silver nitrate is measured. The first reliable observations with the silver coulometer were those of Kohlrausch in 1886, but the most accurate measurements in recent years were made by Smith, Mather and Lowry (1908) at the National Physical * For a complete list of electrochemical equivalents and for other data relating to Faraday's laws, see Roush, Trans. Electrochem. Soc., 73, 285 (1938). 18 INTRODUCTION Laboratory in England, by Richards and Anderegg (1915-16) at Harvard University, and by Rosa and Vinal, 4 and others, at the National Bureau of Standards in Washington, D. C. (1914-16). The conditions for ob- taining precise results have been given particularly by Rosa and Vinal (1914) : these are based on the necessity of insuring purity of the silver nitrate, of preventing particles of silver from the anode, often known as the " anode slime," from falling on to the cathode, and of avoiding the inclusion of water and silver nitrate in the deposited silver. The silver nitrate is purified by repeated crystallization from acidified solutions, followed by fusion. The purity of the salt is proved by the absence of the so-called " volume effect," the weight of silver deposited by a given quantity of electricity being independent of the volume of liquid in the coulometer: this moans that no extraneous impurities are included in the deposit. The solution of silver nitrate employed for the actual measurements should contain between 10 and 20 g. of the salt in 100 cc.; it should be neutral or slightly acid to methyl red indicator, after removal of the silver by neutral potassium chloride, both at the beginning and end of the electrolysis. The anode should be of pure silver with an area as large as the apparatus permits; the current density at the anode should not exceed 0.2 amp. per sq. cm. To prevent the anode slime II FIG. 3. Silver coulometers from reaching the cathode, the former electrode (A in Fig. 3), is inserted in a cup of porous porcelain, as shown at B in Fig. 3, 1 (Richards, 1900), or is surrounded by a glass vessel, B in Fig. 3, II (Smith, 1908). The cathode is a platinum dish or cup (C) and its area should be such as to make the cathodic current density less than 0.02 amp. per sq. cm. After electrolysis the solution is removed by a siphon, the deposited silver is washed thoroughly and then the platinum dish and deposit are dried at 150 and weighed. The gain in weight gives the amount of silver de- posited by the current; if the conditions described are employed, the impurities should not be more than 0.004 per cent. 4 Rosa and Vinal, Bur. Standards Bull, 13, 479 (1936); sec also, Vinal and Bovard, J. Am. Chem. Soc., 38, 496 (1916); Bovard arid Hulett, ibid., 39, 1077 (1917). THE FARADAY AND ITS DETERMINATION 19 If the observations are to be used for the determination of the faraday, it is necessary to know exactly the quantity of electricity passed or the current strength, provided it is kept constant during the experiment. In the work carried out at the National Physical Laboratory the absolute value of the current was determined by means of a magnetic balance, but at the Bureau of Standards the current strength was estimated from the known value of the applied E.M.F., based on the Weston standard cell as 1.01830 international volt at 20 (see p. 193), and the measured resistance of the circuit. According to the experiments of Smith, Mather and Lowry, one absolute coulomb deposits 1.11827 milligrams of silver, while Rosa and Vinal (1916) found that one international coulomb de- posits 1.1180 milligrams of silver. The latter figure is identical with the one used for the definition of the international coulomb (p. 4) and since it is based on the agreed value of the E.M.F. of the Weston cell it means that these definitions are consistent with one another within the limits of experimental accuracy. If the atomic weight of silver is taken as 107.88, it follows that 107.88 0.0011180 = 96,494 international coulombs If allowance is 8 5 D' yr are required to liberate one gram equivalent of silver. made for the 0.004 per cent of impurity in the deposit, this result becomes 96,498 coulombs. Since the atomic weight of silver is not known with an accuracy of more than about one part in 10,000, the figure is rounded off to 96,500 coulombs. It follows, therefore, that this quantity of electricity is required to liberate 1 gram equivalent of any substance: hence 1 faraday = 96,500 coulombs: The reliability of this value of the faraday has been confirmed by mea- surements with the iodine coulometer designed by Washburn and Bates, and employed by Bates and Vinal. 6 The ap- paratus is shown in Fig. 4; it consists of two vertical tubes, containing the anode (A) and cathode (C) of platinum-indium foil, joined by a V-shaped portion. A 10 per cent solution of potassium io- dide is first placed in the limbs and then 6 Washburn and Bates, /. Am. Chem. Soc., 34, 1341, 1515 (1912); Bates and Vinal, ibid., 36, 916 (1914). FIG. 4. Iodine coulometer (Wushburn and Bates) 20 INTRODUCTION by means of the filling tubes D and D' a concentrated solution of potassium iodide is introduced carefully beneath the dilute solution in the anode compartment, and a standardized solution of iodine in potas- sium iodide is similarly introduced into the cathode compartment. During the passage of current iodine is liberated at the anode while an equivalent amount is reduced to iodide ions at the cathode. After the completion of electrotysis the anode and cathode liquids are withdrawn, through D and D', and titrated with an accurately standardized solution of arsenious acid. In this way the amounts of iodine formed at one elec- trode and removed at the other can be determined; the agreement between the two results provides confirmation of the accuracy of the measurements. The results obtained by Bates and Vinal in a number of experiments, in which a silver and an iodine coulometer were in series, are given in Table V; the first column records the weight of silver de- TABLE V. DETERMINATION OF THE FARADAY BY THE IODINE COULOMETER Coulombs Passed From From Milligrams . Silver Iodine Silver E.M.F. and of Iodine mg. mg. deposited Resistance per Coulomb Faraday 4099.03 482224 3666.39 3666.65 1.31526 96,498 4397.11 5172.73 3933.01 .... 1.31521 96,502 4105.23 4828.51 3671.94 3671.84 1.31498 96,518 4123.10 4849.42 3687.92 .. . 1.31495 96,521 4104.75 4828.60 3671.51 3671.61 1.31515 96,506 4184.24 4921.30 3742.61 . 1.31494 96,521 4100.27 4822.47 3667.50 3667.65 1.31492 96,523 4105.16 4828.44 3671.88 3671.82 1.31498 96,519 Mean 1.31502 96,514 posited and the second the mean quantity of iodine liberated or removed; in the third column are the number of coulombs passed, calculated from the data in the first column assuming the faraday to be 96,494 coulombs, and in the fourth are the corresponding values derived from the E.M.F. of the cell employed, that of the Weston standard cell being 1.01830 volt at 25, and the resistance of the circuit. The agreement between the figures in these two columns shows that the silver coulometer was functioning satisfactorily. The fifth column gives the electrochemical equivalent of iodine in milligrams per coulomb, and the last column is the value of the faraday, i.e., the number of coulombs required to deposit 1 equiv. of iodine, the atomic weight being taken as 126.92. The faraday, calculated from the work on the iodine coulometer, is thus 96,514 coulombs compared with 96,494 coulombs from the silver coulometer; the agreement is within the limits of accuracy of the known atomic weights of silver and iodine. In view of the small difference between the two values of the faraday given above, the mean figure 96,500 coulombs is probably best for general use. Measurement of Quantities of Electricity. Since the magnitude of the faraday is known, it is possible, by means of equation (7), to deter- MEASUREMENT OP QUANTITIES OF ELECTRICITY 21 mine the quantity of electricity passing through any circuit by including in it a coulometer in which an element of known equivalent weight is deposited. Several coulometers, of varying degrees of accuracy and convenience of manipulation, have been described. Since the silver and iodine coulometers have been employed to determine the faraday, these are evidently capable of giving the most accurate results; the iodine coulometer is, however, rarely used in practice because of the difficulty of manipulation. One of the disadvantages of the ordinary form of the silver coulometer is that the deposits are coarse-grained and do not ad- here to the cathode; a method of overcoming this is to use an electrolyte made by dissolving silver oxide in a solution of hydrofluoric and boric acids. 6 In a simplified form of the silver coulometer, which is claimed to give results accurate to within 0.1 per cent, the amount of silver dissolved from the anode into a potassium nitrate solution during the passage of current is determined volumetrically. 7 For general laboratory purposes the copper coulometer is the one most frequently employed; 8 it contains a solution of copper sulfate, and the metallic copper deposited on the cathode is weighted. The chief sources of error are attack of the cathode in acid solution, especially in the presence of atmospheric oxygen, and formation of cuprous oxide in neutral solution. In practice slightly acid solutions are employed and the errors are minimized by using cathodes of small area and operating at relatively low temperatures; the danger of oxidation is obviated to a great extent by the presence of ethyl alcohol or of tartaric acid in the electrolyte. The cathode, which is a sheet of copper, is placed midway between two similar sheets which act as anodes; the current density at the cathode should be between 0.002 and 0.02 ampere per sq. cm. At the conclusion of the experiment the cathode is removed, washed with water and dried at 100. It can be calculated from equation (7) that one coulomb of electricity should deposit 0.3294 milligram of copper. In a careful study of the copper coulometer, in which electrolysis was carried out at about in an atmosphere of hydrogen, and allowance made for the. copper dissolved from the cathode by the acid solution, Richards, Collins arid Heimrod (1900) found the results to be within 0.03 per cent of those obtained from a silver coulometer in the same circuit. The electrolytic gas coulometer is useful for the approximate meas- urement of small quantities of electricity; the total volume of hydrogen and oxygen liberated in the electrolysis of an aqueous solution of sulfuric acid or of sodium, potassium or barium hydroxide can be measured, and from this the quantity of electricity passed can be estimated. If the electrolyte is dilute acid it is necessary to employ platinum electrodes, 6 von Wartenberg and Schutza, Z. Elektrochem., 36, 254 (1930). 7 Kisti&kowsky, Z. Elektrochem., 12, 713 (1906). 8 Datta and Dhar, J. Am. Chem. Soc., 38, 1156 (1916); Matthews and Wark, J. Phys. Chem., 35, 2345 (1931). 22 INTRODUCTION but with alkaline electrolytes nickel electrodes are frequently used. One faraday of electricity should liberate one gram equivalent of hydrogen at the cathode and an equivalent of oxygen at the anode, i.e., there should be produced 1 gram of hydrogen and 8 grams of oxygen. Allow- ing for the water vapor present in the liberated gases and for the decrease in volume of the solution as the water is electrolyzed, the passage of one coulomb of electricity should be accompanied by the formation of 0.174 cc. of mixed hydrogen and oxygen at S.T.P., assuming the gases to behave ideally. The mercury coulometer has been employed chiefly for the measurement of quantities of elec- tricity for commercial purposes, e.g., in electricity meters. 9 The form of apparatus used is shown in Fig. 5; the anode consists of an annular ring of mer- cury (A) surrounding the carbon cathode (C); the electrolyte is a solution of mercuric iodide in potas- sium iodide. The mercury liberated at the cathode falls off, under the influence of gravity, and is col- lected in the graduated tube D. From the height of the mercury in this tube the quantity of electricity passed may be read off directly. When the tube has become filled with mercury the apparatus is inverted and the mercury flows back to the reservoir J3. In actual practice a definite fraction only of the current to be measured is shunted through the meter, so that the life of the latter is prolonged. The accuracy of the mercury electricity meter is said to be within 1 to 2 per cent. A form of mercury coulometer suitable for the measurement of small currents of long duration has also been described. 10 An interesting form of coulometer, for which an accuracy of 0.01 per cent has been claimed, is the sodium coulometer; it involves the passage of sodium ions through glass. 11 The electrolyte is fused sodium nitrate at 340 and the electrodes are tubes of highly conducting glass, elec- trical contact being made by means of a platinum wire sealed through the glass and dipping into cadmium in the cathode, and cadmium containing some sodium in the anode (Fig. 6). When current is passed, sodium is deposited in the FIG. 5. Mercury coulometer electricity meter FIG. 6. Sodium roulometer (Stewart) Hatfield, Z. Elektrochem., 15, 728 (1909); Schulte, ibid., 27, 745 (1921). 10 Lehfeldt, Phil. Mag., 3, 158 (1902). Burt, Phys. Rev., 27, 813 (1926); Stewart, /. Am. Chem. Soc., 53 % 3366 (1931). GENERAL APPLICABILITY OP FARADAY'S LAWS 23 glass of the cathode and an equal amount moves out of the anode tube. From the change in weight the quantity of electricity passing may be determined; the anode gives the most reliable results, for with the cathode there is a possibility of the loss of silicate ions from the glass. In spite of the great accuracy that has been reported, it is doubtful if the sodium coulometer as described here will find any considerable application be- cause of experimental difficulties; its chief interest lies in the fact that it shows Faraday's laws hold under extreme conditions. General Applicability of Faraday's Laws. The discussion so far has been concerned mainly with the application of Faraday's laws to the material deposited at a cathode, but the laws are applicable to all types of processes occurring at both anode and cathode. The experiments on the iodine coulometer proved that the amount of iodine liberated at the anode was equal to that converted into iodide ions at the cathode, both quantities being in close agreement with the requirements of Faraday's laws. Similarly, provided there are no secondary processes to interfere, the volume of oxygen evolved at an anode in the electrolysis of a solution of dilute acid or alkali is half the volume of hydrogen set free at the cathode. In the cases referred to above, the anode consists of a metal which is not attacked during the passage of current, but if an attackable metal, e.g., zinc, silver, copper or mercury, is used as the anode, the latter dis- solves in amounts exactly equal to that which would be deposited on the cathode by the same quantity of electricity. The results obtained by Bovard and Hulett 12 for the loss in weight of a silver anode and for the amount of silver deposited on the cathode by the same current are given in Table VI; the agreement between the values in the eight experiments shows that Faraday's laws are applicable to the anode as well as to the cathode. TABLE VI. COMPARISON OP ANODIC AND CATHODIC PROCESSES Anode loss 4.18685 g. 4.13422 4.21204 4.08371 Cathode gain 4.18703 g. 4.13422 4.21240 4.08473 Anode loss 4.17651 g. 4.14391 4.08147 4.09386 Cathode gain 4.17741 g. 4.14320 4.08097 4.09478 The results obtained at the cathode in the iodine coulometer show that Faraday's laws hold for the reduction of iodine to iodide ions; the laws apply, in fact, to all types of electrolytic reduction occurring at the cathode, e.g., reduction of ferric to ferrous ions, ferri cyanide to ferro- cyanide, quinone to hydroquinone, etc. The laws are applicable simi- larly to the reverse process of electrolytic oxidation at the anode. The equivalent weight in these cases is based, of course, on the nature of the oxidation-reduction process. * Bovard and Hulett, JT. Am. Chem. Soc., 39, 1077 (1917). 24 INTRODUCTION In the discussion hitherto it has been supposed that only one process occurs at each electrode; there are numerous instances, however, of two or more reactions occurring simultaneously. For example, in the elec- trolysis of nickel salt solutions the deposition of the metal is almost invariably accompanied by the evolution of some hydrogen; when current is passed through a solution of a stannic salt there may be simultaneous reduction of the stannic ions to starinous ions, deposition of tin and liberation of hydrogen at the cathode. Similarly, the electrolysis of a dilute hydrochloric acid solution yields a mixture of oxygen and chlorine at the anode. The conditions which determine the possibility of two or more electrode processes occurring at the same time will be examined in later chapters; in the meantime, it must be pointed out that whenever simultaneous reactions occur, the total number of equivalents deposited or reduced at the cathode, or dissolved or oxidized at the anode, are equal to the amount required by Faraday's laws. The passage of one faraday of electricity through a solution of a nickel salt under certain conditions gave a deposit of 25.48 g. of the metal, instead of the theoretical amount 29.34 g. ; the number of equivalents of nickel deposited is thus 25.48/29.34, i.e., 0.8684, instead of unity. It follows, therefore, that 0.1316 equiv., i.e., 0.1326 g., of hydrogen is evolved at the same time. The ratio of the actual amount of material deposited, or, in general, the ratio of the actual extent of any electrode reaction, to that expected theoretically is called the current efficiency of the particular reaction. In the case under con- sideration the current efficiency for the deposition of nickel under the given conditions is 0.8684 or 86.84 per cent. Ions in Two Valence Stages. A special case of simultaneous elec- trode processes arises when a given ion can exist in two valence stages, e.g., mercuric (Hg++) and mercurous (HgJ+) ;* the passage of one faraday then results in the discharge at the cathode or the formation at the anode of a total of one gram equivalent of the two ions. An equilibrium exists between a metal and the ions of lower and higher valence; thus, for example, Hg + Hg++ and if the law of mass action is applicable to the system, it follows that Concn. of mercurous ions 7: - f - : -. - = constant, Concn. of mercuric ions the concentration of the metal being constant. By shaking a simple mercuric salt, e.g., the nitrate, with mercury until equilibrium was estab- lished and analyzing the solution, the constant was found to be 120, at room temperatures. When a mercury anode dissolves, the mercurous and mercuric ions are formed in amounts necessary to maintain the * There is much evidence in favor of the view that the mercurous ion has the for- mula H&*+ and not Hg+ (see p. 264). 25 equilibrium under consideration; that is, the proportion of mercurous ions is 120 to one part of mercuric ions. It would appear, therefore, that 99.166 per cent of the mercury which dissolves anodieally should form mercurous ions: this is true provided no secondary reactions take place in the solution. If the electrolyte is a chloride, the mercurous ions are removed in the form of insoluble mercurous chloride, and in order to maintain the equilibrium between mercury, mercuric and mercurous ions, the anode dissolves almost exclusively in the mercurous form. On the other hand, in a cyanide or iodide solution the mercuric ions are removed by the formation of complex ions, and hence a mercury anode dissolves mainly in the mercuric form. In each case the electrode mate- rial passes into solution in such a manner as to establish the theoretical equilibrium, but the existence of subsidiary equilibria in the electrolyte often results in the anode dissolving in the two valence stages in a ratio different from that of the concentrations of the simple ions at equilibrium. With a copper electrode, the equilibrium is greatly in favor of the cupric ions and so a copper anode normally dissolves virtually completely in the higher valence (cupric) state, i.e., as a bivalent metal. In a cyanide solution, however, cuprous ions are removed as complex cupro- cyanide ions; a copper anode then dissolves as a univalent element. Anodes of iron, lead and tin almost invariably dissolve in the lower valence state. Similar arguments to those given above will apply to the deposition at the cathodo; the proportion in which the higher and lower valence ions are discharged is identical with that in which an anode would dissolve in the same electrolyte. Thus, from a solution containing simple mer- curous and mercuric ions only, e.g., from a solution of the perchlorates or nitrates, the two ions would be discharged in the ratio of 120 to unity. From a complex cyanide or iodide electrolyte, however, mercuric ions are discharged almost exclusive!}'. Significance of Faraday's Laws. Since the discharge at a cathode or the formation at an anode of one gram equivalent of any ion requires the passage of one faraday, it is reasonable to suppose that this represents the charge * carried by a gram equivalent of any ion. If the ion has a valence z, then a molo of those ions, which is equivalent to z equiv., carries a charge of z faradays, i.e., zF coulombs, where F is 96,500. The number of individual ions in a mole is equal to the Avogadro number N, and so the electric charge carried by a single ion is zF/N coulombs. Since z is an integer, viz., one for a univalent ion, two for a bivalent ion, three for a tervalent ion, and so on, it follows that the charge of electricity carried by any single ion is a multiple of a fundamental unit charge equal to FIN. This result implies that electricity, like matter, is atomic in nature and that F/N is the unit or " atom" of electric charge. There arc many reasons for identifying this unit charge with the charge of an electron * It was seen on page 3 that quantity of electricity and electric charge have the same dimensions. 26 INTRODUCTION (*), so that F '-* W According to these arguments a univalcnt, i.e., singly charged, cation is formed when an atom loses a single electron, e.g., Na -> Na+ + . A bivalent cation results from the loss of two electrons, e.g., On -> ( 1 u++ + 26, and so on. Similarly, a univalent anion is formed when an atom gains an electron, e.g., Cl + -> C1-. In general an ion carries the number of charges equal to its valence, and it differs from the corresponding uncharged particle by a number of electrons equal in magnitude to the charge. Electrons in Electrolysis. The identification of the unit charge of a single ion with an electron permits a more complete picture to be given of the phenomena of electrolysis. It will be seen from Fig. 2 that the passage of current through a circuit is accompanied by a flow of electrons from anode to cathode, outside the electrolytic cell. If the current is to continue, some process must occur at the surface of the cathode in the electrolyte which removes electrons, while at the anode surface electrons must be supplied: these requirements are satisfied by the discharge and formation of positive ions, respectively, or in other ways. In general, a chemical reaction involving the formation or removal of electrons must always occur when current passes from an electronic to an electrolytic conductor. For example, at a cathode in a solution of silver nitrate, each silver ion takes an electron from the electrode, forming metallic silver; thus Ag+ + e -> Ag. At a silver anode it is necessary for electrons to be supplied, and this can be achieved by the atoms passing into solutions as ions; thus Ag - Ag+ + c. If the anode consisted of an unattackable metal, e.g., platinum, then the electrons must be supplied by the discharge of ariions, e.g., OH- -> OH + c, which is followed by 2OH = H 2 O + 2 , resulting in the liberation of oxygen; or Cl- -> Cl + 6, PROBLEMS 27 followed by 2C1 = C1 2 , which gives chlorine gas by the discharge of chloride ions. Since the same number of electrons is required by the anode as must be removed from the cathode, it is evident that equivalent amounts of chemical reaction, proportional to the quantity of electricity passing, i.e., to the number of electrons transferred, must take place at both electrodes. The electronic concept, in fact, provides a very simple interpretation of Faraday's laws of electrolysis. It should be clearly understood that although the current is carried through the metallic part of the circuit by the flow of electrons, it is carried through the electrolyte by the ions; the positive ions move in one direction and the negative ions in the opposite direction, the total charge of the moving ions being equivalent to the flow of electrons. This aspect of the subject of electrolytic conduction will be considered more fully in Chap. IV. Equations involving electron transfer, such as those given above, are frequently employed in electrochemistry to represent processes occurring at electrodes, either during electrolysis or in a voltaic cell capable of producing current. It is opportune, therefore, to emphasize their sig- nificance at this point: an equation such as means not only that an atom of copper gives up two electrons and be- comes a copper ion; it also implies that, two faradays are required to cause one gram atom of copper to go into solution forming a mole, or gram-ion, of cupric ions. In general, an electrode process written as involving z electrons requires the passage of z faradays for it to occur completely in terms of moles. PKOHLKMS 1. A constant current, which gave a reading of 25.0 inilliamp. on a milli- ainmeter, was passed through a solution of copper sulfate for exactly 1 hour; the deposit on the cathode weighed 0.0300 grain. What is the error of the meter at the 25 inilliamp. reading f 2. An average cell, in which aluminum is produced by the electrolysis of a solution of alumina in fused cryolite, takes about 20,000 amps. How much aluminum is produced per day in each cell, assuming a current efficiency of 92 per cent? 3. A current of 0.050 amp. was passed through a silver titration coulometer, and at the conclusion 23 8 cc. of 0.1 N sodium chloride solution were required to titrate the silver dissolved from the anode. How long was the current flowing? 4. What weights of sodium hydroxide and of sulfuric acid are produced at the cathode and anode, respectively, when 1,000 coulombs are passed through a solution of sodium sulfate? 5. Calculate the amount of iodine that would be liberated by a quantity of electricity which sets free 34.0 cc. of gas, at S.T.P., in an electrolytic gas coulometer. 28 INTRODUCTION 6. In the electrolysis of a solution containing copper (cuprous), nickel and zinc complex cyanides, Faust and Montillon [Trans. Electrochem. Soc., 65, 361 (1934)] obtained 0.175 g. of a deposit containing 72.8 per cent by weight of copper, 4.3 per cent of nickel and 22.9 per cent of zinc. Assuming no hydrogen was evolved, how many coulombs were passed through the solution? 7. Anthracene can be oxidized anodically to anthraquinone with an effi- ciency of 100 per cent, according to the reaction CuIIio + 30 = Ci4lI 8 02 + H 2 O. What weight of anthraquinone is produced by the passage of a current of 1 amp. for 1 hour? 8. A current of 0.10 amp. was passed for two hours through a solution of cuprocyanide and 0.3745 g. of copper was deposited on the cathode. Calcu- late the current efficiency for copper deposition and the volume of hydrogen, measured at S.T.P., liberated simultaneously. 9. The 140 liters of solution obtained from an alkali-chlorine cell, operating for 10 hours with a current of 1250 amps., contained on the average 116.5 g. of sodium hydroxide per liter. Determine the current efficiency with which the cells were operating. 10. In an experiment on the electrolytic reduction of sodium nitrate solution, Muller and Weber [_Z. Elektrochem., 9, 955 (1903)] obtained 0.0495 g. of sodium nitrite, 0.0173 g. ammonia and 695 cc. of hydrogen at S.T.P., while 2.27 g. of copper were deposited in a coulometer. Evaluate the current effi- ciency for each of the three products. 11. Oxygen at 25 atrn. pressure is reduced cathodically to hydrogen peroxide: from the data of Fischer and Priess \_Bcr., 46, 698 (1913)] the follow- ing results were calculated for the combined volume of hydrogen and oxygon, measured at S.T.P., liberated in an electrolytic gas coulometer (I) compared with the amount of hydrogen peroxide (II) obtained from the same quantity of electricity. I. 35.5 200 413 583 1,670 cc. of ft as. II. 34.7 150 265 334 596 mg. of H 2 O 2 . Calculate the current efficiency for the formation of hydrogen peroxide in each case, and plot the variation of the current efficiency with the quantity of electricity passed. 12 In the electrolysis of an alkaline sodium chloride solution at 52, Muller [Z. anory. Chcm., 22, 33 (1900)] obtained the following results: Active Oxygen as Copper in Hypochlonte Chlorate Coulometer 0.001 5 g. 0.0095 g. O.U5g. 0.0053 0.0258 0.450 0.105 0.2269 3.110 0.135 0.3185 4.SOO 0.139 0.4123 7.030 Plot curves showing the variation with the quantity of electricity passed of the current efficiencies for the formation of hypochlorite and of chlorate. CHAPTER II ELECTROLYTIC CONDUCTANCE Specific Resistance and Conductance. Consider a uniform bar of a conductor of length I cm. and cross-sectional area a sq. cm.; suppose, for simplicity, that the cross section is rectangular and that the whole is divided into cubes of one cm. side, as shown in Fig. 7, I. The resistance I II FIG. 7. Calculation of specific resistance of the bar is seen to be equivalent to that of / layers, such as the one depicted in Fig. 7, II, in series with one another; further, each layer is equivalent to a cubes, each of one cm. side, whoso resistances are in parallel. If p is the resistance, in ohms, of a centimeter cube, generally called the specific resistance of the substance constituting the conductor, the resistance r of the layer containing the a cubes is given by there being a terms on the right-hand side: it follows, therefore, that P r ~~~ If R is the resistance of the whole bar, which is equivalent to / layers each of resistance r in series, then R = Ir = p - ohms. (0 This equation is applicable to all conductors, electronic or electrolytic, and for uniform conductors of any cross section, not necessarily rec- tangular. 29 30 ELECTROLYTIC CONDUCTANCE The specific conductance of any conducting material is defined as the reciprocal of the specific resistance; it is given the symbol K and is stated in reciprocal ohm units, sometimes called " mhos."* Since, by defini- tion, K is equal to 1/p, it follows from equation (1) that 1 I R = ~ ohms. (2) K a ^ ' The conductance (C) is the reciprocal of resistance, i.e., C I//?, and hence C = K a ohms" 1 . (3) The physical meaning of the specific conductance may be understood by supposing an E.M.F. of one volt to be applied to a conductor; since E = 1, it follows, by Ohm's law, that the current 7 is equal to 1/R, and hence to the conductance (C). For a centimeter cube a and I are unity, and so C is equal to K. It is seen, therefore, that when a potential difference of one volt is applied to a centimeter cube of a conductor, the current in amperes flowing is equal in magnitude to the specific conductance in ohm" 1 cm.~ l units. Equivalent Conductance. For electrolytes it is convenient to define a quantity called the equivalent conductance (A), represent ing the con- ducting power of all the ions produced by 1 equiv. of electrolyte in a given solution. Imagine two large parallel electrodes set 1 cm. apart, and suppose the whole of the solution containing 1 equiv. is placed between these electrodes; the area of the electrodes covered will then be v sq. cm., where v cc. is the volume of solution containing the 1 equiv. of solute. The conductance of this system, which is the equivalent con- ductance A, may be derived from equation (3), where a is equal to v sq. cm. and I is 1 cm.; thus A = KV, (4) where v is the "dilution" of the solution in cc. per equiv. If c is the concentration of the solution, in equivalents per liter, then z> is equal to 1000/c, so that equation (4) becomes A = 1000 - - (5) The equivalent conductance of any solution can thus be readily derived from its specific conductance and concentration. Since the units of K are ohm" 1 cm." 1 , those of A are seen from equation (4) or (5) to be ohm" 1 cm. 2 * It will be apparent from equation (1) or (2) that if R is in ohms, and I and a are in cm. and sq. cm. respectively, the units of K are ohm" 1 cm" 1 . This exact notation will be used throughout the present book. DETERMINATION OF RESISTANCE 31 In some cases the molecular conductance (/x) is employed; it is the conductance of 1 mole of solute, instead of 1 equiv. If v m is the volume; in cc. containing a mole of solute, arid c is the corresponding concentra- tion in moles per liter* then = Kv m = 1000 (6) For an electrolyte consisting of two univalent ions, e.g., alkali halides, the values of A and /u are, of course, identical. Determination of Resistance. The measurement of resistance is most frequently carried out with some form of Wheats to no bridge circuit, the principle of which may be explained with the aid of Fig. 8. The four arms of the bridge, viz., ab, ac, bd and cd, have resistances Ri, Rt, Rz and /2 4 , respectively; a source of current S is connected across the bridge between b and c, and a cur- rent detector D is connected between a and d. Lot E\, E 2 , E^ and 7 4 be the fall of potential across the four arms, corresponding to the resistances Ri, Ri, Rs and / 4 , respectively, arid suppose the currents in these arms are I\, 1 2, Is and 7 4 , then by Ohm's hiw: FIG. 8. Wheatstone bridge circuit #4 = /4#4. If the resistances are adjusted so that there is no flow of current through the detector D, that is to say, when the bridge is " balanced," the poten- tial at a must be the same as that at d. Since the arms ab arid bd are joined at b and tho potentials are the same at a and d, it follows that the fall of potential across ab, i.e., E ly must equal that across bd, i.e., E*. Similarly, the fall of potential across ac must be the same as that across cd, i.e., E* and A T 4 are equal. Introducing the values of the various 7's given above, it is seen that IiRi = 7 3 /e 3 and I 2 R 2 = 7 4 72 4 , Since no current passes through ad when the bridge is balanced, it is * In accordance with the practice adopted by a number of writers, the symbol c is used to represent concentrations in equivalents and c in moles, per liter. 32 ELECTROLYTIC CONDUCTANCE evident that the current flowing in the arm ab must be the same as that in ac, i.e., /i = 7 2 , while that passing through bd must be identical with that in erf, i.e., /3 = /4. It follows, therefore, that at the balance point Kt Jt, and so if the resistances of three of the arms of the bridge are known, that of the fourth can be readily evaluated. In practice, R\ is generally the unknown resistance, and Ri is a resistance box which permits various known resistances to be used; the so-called " ratio arms" Its and R\ may be a uniform wire (bdc) on which the position of d is adjusted until the bridge is balanced, as shown by the absence of current in D. The ratio of the lengths of the two parts of the uniform wire, corresponding to bd and dc, gives the ratio Ra/R*. Resistance of Electrolytes: Introduction. In the earliest attempts to determine the resistance of electrolytic solutions the results were so erratic that it was considered possible that Ohm's law was not applicable to electrolytic conductors. The erratic behavior was shown to be due to the use of direct current in the measurement, and when the resulting errors were eliminated it became evident that Ohm's law held good for electrolytic as well as for metallic systems. The passage of direr-t cur- rent through an electrolyte is, as seen in Chap. I, accompanied by changes in composition of the solution and frequently by the liberation of gases at the electrodes. The former alter the conductance and the latter set. up an E.M.F. of" polarization" (see Chap. XIII) which tends to oppose the flow of current. The difficulties may be overcome by the use of nori-polarizable electrodes and the employment of such small currents that concentration changes are negligible; satisfactory conductance meas- urements have been made in this way with certain electrolytes by the use of direct current, as will be seen later (p. 47). The great majority of the work with solutions has, however, been carried out with a rapidly alternating current of low intensity, following the suggestion made by Kohlrausrh in 1868. The underlying principle of the use of an alternating current is that as a result of the reversal of the direction of the current about a thousand times per second, the polarization produced by each pulse of the current is completely neu- tralized by the next, provided the alternations are symmetrical. There is also exact compensation of any concentration changes which may occur. Kohlrausch used an induction coil as a source of alternating current (abbreviated to A.C.) and in his early work a bifllar galvanometer acted as detector; later (1880) he introduced the telephone earpiece, and this, with some improvements, is still the form of A.C. detector most frequently employed in electrolytic conductance measurements. The electrolyte was placed in a cell and its resistance measured by a Wheat- stone bridge arrangement shown schematically in Fig. 9. The cell C A.C. SOURCES AND DETECTORS 33 FIG. 9. Measurement of resistance of electrolyte is in the arm ab and a resistance box R constitutes the arm ac\ the source of A.C. is represented by S, and D is the telephone earpiece detector. In the simplest form of bridge, frequently employed for ordinary labora- tory purposes, the arms bd and dc are in the form of a uniform wire, preferably of platinum-iridium, stretched along a meter scale, i.e., the so-called " motor bridge," or suitably wound round a slate cylinder. The point d is a sliding contact which is moved back and forth until no sound can be heard in the detector; the bridge is then balanced. If the wire be is uniform, the ratio of the resistances of the two arms is equal to the ratio of the lengths, bd and dc, as seen above. If the resistance taken from the box R is adjusted so as to be approximately equal to that of the electrolyte in the cell C, the balance poiiiL d will be roughly midway between 6 and c; a small error in the set- ting of d will then cause the least discrepancy in the final value for the resistance of C. If somewhat greater accuracy is desired, two variable resistance boxes may be used for bd and dc, i.e., /?s and #4 (cf. Fig. 8), the resistance taken from each being adjusted until the bridge is balanced. Alternatively, two resistance boxes or coils may be joined by a wire, whose resistance is known in terms of that of the boxes or coils, for the purpose of making the final adjustment. It will be seen shortly that for precision measurements of electrolytic conductance it is necessary to take special precautions to obviate errors due to inductance and capacity in the bridge circuit. One immediate effect of these factors is to make the minimum sound in the telephone earpiece difficult to detect; for most general purposes this source of error can be overcome by using a good resistance box, in which the coils are wound in such a manner as to eliminate self-induction, and to use a straight-wire bridge, if a special non-inductive bridge is not available. Further, a variable condenser K is connected across the resistance box and adjusted until the telephone earpiece gives a sharply defined sound minimum; in this way the unavoidable capacity of the conductance cell may be balanced to some extent. A.C. Sources and Detectors. Although the induction coil suffers from being noisy in operation and does not give a symmetrical alter- nating current, it is still often employed in conductance measurements where great accuracy is not required. A mechanical high-frequency 34 ELECTROLYTIC CONDUCTANCE To bridge Fia. 10. Vacuum-tube oscillator A.C. generator was employed by Washburn (1913), and Taylor and Acree (1916) recommended the use of the Vreeland oscillator, which consists of a double mercury-arc arrangement capable of giving a sym- metrical sine-wave alternating current of constant frequency variable at will from 160 to 4,200 cycles per sec- ond. These costly instruments have been displaced in recent years by some kind of vacuum-tube oscillator, first employed in conductance work by Hall and Adams. 1 Several types of suitable oscillators have been de- scribed and others are available com- mercially; the essential circuit of one form of oscillator is shown in Fig. 10. The grid circuit of the thermionic vacuum tube T contains a grid coil LI of suitable inductance which is connected to the oscillator coil L 2 in parallel with the variable condenser C. The output coil L 3 , which is coupled inductively with L 2 , serves to convey the oscillations to the con- ductance bridge. The chief advantages of the vacuum-tube oscillator are that it is relatively inexpensive, it is silent in operation and gives a symmetrical sinusoidal alternating current of constant frequency; by suitable adjust- ment of inductance and capacity the frequency of the oscillations may be varied over the whole audible range, but for conductance work frequencies of 1,000 to 3,000 cycles per sec. are generally employed. The disad- vantage of this type of oscillator is that it is liable to introduce stray capacities into the bridge circuit which can be a serious source of error in precision work. The difficulty may be overcome, however, by the use of special grounding devices (see p. 42). If properly tuned to the frequency of the A.C., the telephone earpiece can be used to detect currents as small as 10" 9 amp. ; it is still regarded as the most satisfactory instrument for conductance measurements. The sensitivity of the telephone can be greatly increased by the addition of a vacuum-tube (low frequency) amplifier; this is particularly valuable when working with very dilute solutions having a high resistance, for it is then possible to determine the balance point of the bridge with greater precision than without the amplifier. The basic circuit of a simple type of audio-frequency amplifier is shown in Fig. 11, in which the conductance bridge is connected to the primary coil of an iron-cored transformer (P); T is a suitable vacuum tube and C is a condenser. The use of a vacuum-tube amplifier introduces the possibility of errors 1 Hall and Adams, J. Am. Chem. tfoc., 41, 1515 (1919); see also, Jones and Josephs, ibid., 50, 1049 (1928); Luder, ibid., 62, 89 (1940;; Jones, Mysels and Juda, ibid. t 62, 2919 (1940). ELECTRODES FOR CONDUCTANCE MEASUREMENTS 35 due to capacity and interaction effects, but these can be largely elimi- nated by suitable grounding and shielding (see p. 42). If results of a low order of accuracy are sufficient, as, for example, in conductance measurements for analytical or industrial purposes, the A.C. supply mains, of frequency about 60 cycles per sec., can be em- ployed as a source of current; in this case an A.C. galvanometer is a satisfactory detector. A combination of a vacuum-tube, or other form of A.C. rectifier, and a direct current galvanometer has been employed, FIG. 11. Vacuum-tube amplifier and in some cases the thermal effect of the alternating current has been used, in conjunction with a thermocouple and a sensitive galvanometer, for detection purposes. Electrodes for Conductance Measurements. For the determination of electrolytic conductance it is the general practice to use two parallel sheets of stout platinum foil, that do not bend readily; their relative positions are fixed by sealing the connecting tubes into the sides of the measuring cell (of. Fig. 12). In order to aid the elimination of polariza- tion effects by the alternating current, Kohlrausch (1875) coated the electrodes with a layer of finely divided platinum black; these are called platinized platinum electrodes. The platinization is carried out by elec- trolysis of a solution containing about 3 per cent of chloroplatinic acid and 0.02 to 0.03 per cent of lead acetate; the lead salt apparently favors the formation of the platinum deposit in a finely-divided, adherent form. The large surface area of the finely divided platinum appears to catalyze the union of the hydrogen and oxygen which tend to be liberated by the successive pulses of the current; the polarization E.M.F. is thus eliminated. In some cases the very properties which make the platinized platinum electrodes satisfactory for the reduction of polarization are a disadvan- tage. The finely-divided platinum may catalyze the oxidation of organic compounds, or it may adsorb appreciable quantities of the solute present 36 ELECTROLYTIC CONDUCTANCE in the electrolyte and so alter its concentration. Some workers have overcome this disadvantage of platinized electrodes by heating them to redness and so obtaining a gray surface; the resulting electrode is prob- ably not so effective in reducing polarization, but it adsorbs much less solute than does the black deposit. Others have employed electrodes covered with very thin layers of platinum black, and sometimes smooth electrodes have been used. By making measurements with smooth platinum electrodes at various frequencies and extrapolating the results to infinite frequency, conductance values have been obtained which are in agreement with those given by platinized electrodes; this method is thus available when platinum black must not be used. For the great majority of solutions of simple salts and of inorganic acids and bases it is the practice to employ electrodes coated with a thin layer of plati- num black obtained by electrolysis as already described. Conductance Cells: The Cell Constant. The cells for electrolytic conductance measurements are made of highly insoluble glass, such as Pyrex, or of quartz; they should be very carefully washed and steamed before use. For general laboratory requirements the sim- ple cell designed by Ostwald (Fig. 12, I)' is often employed, but for industrial purposes the "dipping cell" (Fig. 12, II) or the pipette- type of cell (Fig. 12, III) have been found convenient. By means of the two latter cells, samples ob- tained at various stages in a chemical process can be readily tested. The resistance (R) of the solu- tion in the cell can be measured, as already explained, and hence the specific conductance (K) is given by equation (2) as I K = = aR where I is the distance between the electrodes and a is the area of cross section of the electrolyte through which the current passes. For a given cell with fixed electrodes I/a is a constant, called the cell constant ; if this is given the symbol K cm.~ l , it follows that FIG. 12. Types of conductance cells K (8) It is neither convenient nor desirable, with the cells in general use, to measure / and a with any degree of accuracy, and so an indirect method DESIGN OP CELLS 37 is employed for the evaluation of the cell constant. If a solution whose specific conductance is known accurately, from other measurements, is placed in the experimental cell and its resistance R is measured, it is possible to obtain K for the given cell directly, by means of equation (8). The electrolyte almost invariably used for this purpose is potassium chloride, its specific conductance having been determined with high pre- cision in cells calibrated by measurement with a concentrated solution of sulfuric acid, the resistance of which has been compared in another cell with that of mercury; the specific conductance of the latter is known accurately from the definition of the international ohm as 10629.63 ohms" 1 cm.~ l at 0. The potassium chloride solutions employed in the most recent work contain 1.0, 0.1 or 0.01 mole in a cubic decimeter of solution at 0, i.e., 0.999973 liter; these solutions, designated as 1.0 D, 0.1 D and 0.01 D, where D stands for " demal," contain 76.627, 7.4789 and 0.74625 grams of potassium chloride to 1000 grams of water, respectively. The specific conductances of these solutions at 0, 18 and 25 are quoted in Table VII; 2 the particular solution chosen for calibrating a given cell depends TABLE VII. SPECIFIC CONDUCTANCES OF POTASSIUM CHLORIDE SOLUTIONS IN OHM" 1 CM.~ l Temp. I.OD 0.1 D 0.01 D 0.065176 0.0071379 0.00077364 18 0.097838 0.0111667 0.00122052 25 0.111342 0.0128560 0.00140877 on the range of conductances for which it is to be employed. The values recorded in this table do not include the conductance of the water; when carrying out a determination of the constant of a given conductance cell allowance must be made for this quantity. Design of Cells. In the design of conductance cells for precision measurements a number of factors must be taken into consideration. Kohlrausch showed theoretically that the error resulting from polariza- tion was determined by the quantity P 2 /o>/ 2 , where P is the polarization E.M.F., R is the resistance of the electrolyte in the cell and w is the fre- quency of the alternating current. It is evident that the error can be made small by adjusting the experimental conditions so that co/i! 2 is much greater than P 2 ; this can be done by making either or R, or both, as large as is reasonably possible. There is a limit to the increase in the frequency of the A.C. because the optimum range of audibility of the telephone earpiece is from 1,000 to 4,000 cycles per sec., and so it is desirable to make the resistance high. If this is too high, however, the current strength may fall below the limit of satisfactory audibility, and it is not possible to determine the balance point of the bridge. The Jones and Bradshaw, J. Am. CJiem. Soc., 55, 1780 (1933); see also, Jones and Prendergast, ibid., 59, 73> (1937); Bremner and Thompson, ibid., 59, 2371 (1937); Davies, J. Chem. Soc., 432, 1326 (1937). 38 ELECTROLYTIC CONDUCTANCE highest electrolytic resistances which can be measured with accuracy, taking advantage of the properties of the vacuum-tube audio-amplifier, are about 50,000 ohms. In order to measure low resistances the polari- zation P should be reduced by adequate platinization of the electrodes, but there is a limit to which this can be carried and experiments show that resistances below 1,000 ohms cannot be measured accurately. The resistances which can be determined in a given cell, therefore, cover a ratio of about 50 to unity. The observed specific conductances of electro- lytes in aqueous solution range from approximately 10" 1 to 10~ 7 ohms" 1 cm." 1 , and so it is evident that at least three cells of different dimensions, that is with different cell constants, must be available. Another matter which must be borne in mind in the design of a con- ductance cell is the necessity of preventing a rise of temperature in the electrolyte due to the heat liberated by the current. This can be achieved either by using a relatively large volume of solution or by making the cell in the form of a long narrow tube which gives good thermal contact with the liquid in the thermostat. Two main types of cell have been devised for the accurate measure- ment of electrolytic conductance; there is the " pipette " type, used by Washburn (1916), and the flask type, introduced by Hartley and Barrett (1913). In the course of a careful study of cells of the pipette form, Parker (1923) found that with solutions of high resistance, for which the polari- zation error is negligible, there was an apparent decrease of the cell con- stant with increasing resistance. This phenomenon, which became known as the " Parker effect," was confirmed by other workers; it was at first attributed to adsorption of the electrolyte by the platinized electrode, but its true nature was elucidated by Jones and Bollinger. 3 The pipette type of cell (Fig. 13, I) is electrically equiv- alent to the circuit depicted in Fig. 13, II; the resistance R is that of the solu- tion contained between the electrodes in the cell, and this is in parallel with the resistance (R p ) of the electrolyte in the filling tube at the right and a capacity (C p ). The latter is equiv- alent to the distributed capacity be- tween the electrolyte in the body of the cell and the mercury in the con- tact tube, on the one hand, and the solution in the filling tube, on the other hand; the glass walls of the tubes and the thermostat liquid act as the dielectric medium. An analysis of the effect of shunting the resistance R Q by a capacity C p and a resistance R p shows that, provided Jones and Bollinger, /. Am. Chem. Soc., 53, 411 (1931); cf., Washburn, ibid., 38, 2431 (1916). I H FIG. 13. Illustration of the "Parker effect" DESIGN OF CELLS 39 the cell is otherwise reasonably well designed, the error &R in the meas- ured resistance is given by - Aft (9) where o>, as before, is the frequency of the alternating current. Accord- ing to equation (9) the apparent cell constant will decrease with increasing resistance R , as found in the Parker effect. In order to reduce this source of error, it is necessary that RQ, w and C p should be small; as already seen, however, RQ and o> must be large to minimize the effect of polarization, and so the shunt capacity C p should be negligible if the Parker effect is to be eliminated. Since most of the shunt capacity lies between the filling tube and the portions of the cell of opposite polarity (cf. Fig. 13, I) it is desirable that these should be as far as possible from each other. This principle is em- bodied in the cells shown in Fig. 14, designed by Jones and Bollinger; the wider the tube and the closer the electrodes, the smaller the cell con- stant. These cells exhibit no appre- ciable Parker effect: the cell constants are virtually independent of the fre- quency of the A.C. and of the resist- ance of the electrolyte within reason- able limits. The Parker effect is absent from cells with dipping electrodes, such as in cells of the flask type; there are other sources, of error, however, as was pointed out by Shedlovsky. 4 In the cell represented diagram- matically in Fig. 15, I, the true resistance of the solution between the electrodes is R Qy and there is a capacity Ci between the contact tubes above the electrolyte, and a capacity C 2 in series with a resistance r between those parts immersed in the liquid; the equivalent electrical circuit is shown by Fig. 15, II. When the cell is placed in the arm of a Wheatstone bridge it is found necessary to insert a resistance R and a capacity C in parallel in the opposite arm in order to obtain a balance (cf . p. 33) ; it can be shown from the theory of alternating currents that FIG. 14. Cells for accurate con- ductance measurements (Jones and Bollinger) 1 R RQ \ a 4 Shedlovsky, J. Am. Chem. Soc., 54, 1411 (1932). (10) 40 ELECTROLYTIC CONDUCTANCE where r is taken as proportional to /2o, the constant a being equal to r//? . It follows, therefore, that if the cell is balanced by a resistance and a capacity in parallel, no error results if part of the current through the cell is shunted by a pure capacity such as Ci, since the quantity Ci does FIG. 15. Equivalent resistance and capacity of flask cell FIG. 16. Shedlovsky flask cell not appear in equation (10). On the other hand, parasitic currents resulting from a series resistance-capacity path, i.e., involving r and C%, will introduce errors, since the apparent resistance R will be different from the true resistance R . In order to eliminate parasitic currents, yet retaining the advantages of the flask type of cell for work with a series of solutions of different concentrations, Shedlovsky designed the cell depicted in Fig. 16; the experimental solution contained in the flask A is forced by gas pressure through the side tube into the bulb containing the electrodes B and B' . These consist of perforated platinum cones fused to the walls of the bulb; the contact tubes C and C" are kept apart in order to diminish the capacity between them. The Shedlovsky cell has been used particularly for accurate determination of the conductances of a series of dilute solutions of strong electrolytes. Temperature Control. The temperature coefficient of conductance of electrolytes is relatively high, viz., about 2 per cent per degree; in order to obtain an accuracy of two parts in 10,000, which is desirable for accurate work, the temperature should be kept constant within 0.01. The use of water in the thermostat is not recommended; this liquid has an appreciable conductance and there is consequently a danger of current leakage leading to errors in the measurement, as explained below. The thermostatic liquid should, therefore, be a hydrocarbon oil which is a non-conductor. Design of the A.C. Bridge. Strictly speaking the condition of balance of the Wheatstone bridge given by equation (7) is applicable for alter- nating current only if R iy /2 2 , #3 arid R* are pure resistances. It is un DESIGN OF THE A.C. BRIDGE 41 likely that the resistance coils will be entirely free from inductance and capacity and, in addition, the conductance cell and its connecting tubes are equivalent to a resistance shunted by a condenser. One consequence of this fact is that the alternating currents in the two arms (R\ and #2) of the bridge arc not in phase and it is found impossible to obtain any adjustment of the bridge which gives complete silence in the telephone earpiece. For most purposes, this difficulty may be overcome by the use of the condenser K in parallel with the resistance box / 2 , as suggested on page 33. For precision work it is necessary, however, to consider the problem in further detail. For alternating current, Ohm's law takes the form E = 7Z, where Z is the impedance of the circuit, i.e., Z 2 is equal to /j>2 _|_ x 2 , the quantities R and X being the resistance and reactance, respectively. The condition for balance of a Wheatstone bridge circuit with alternating current is, consequently, If there is no leakage of current from the bridge network to ground, or from one part of the bridge to any other part, and there is no mutual inductance between the arms, I\ is equal to /2, and /a to /4, so that at balance. It follows, therefore, that in a Wheatstone A.C. bridge, under the conditions specified, the impedances, rather than the resist- ances, are balanced. It can bo shown that if the resistances are also to be balanced, i.e., for RijRi to be equal to R^jR*, at the same time as Zi/Z 2 is equal to Z^/Z^ it is necessary that Xi X% A" 3 A 4 -Rl = ltl and R- 3 = R~<- The fraction X/R for any portion of an A.C\ circuit is equal to tan 0, where 6 is the phase angle between the voltage and current in the given conductor. It is soon, therefore, that the conditions for the simple Whcatstorie bridge relationship between resistance*, i.e., for equation (7), to be applicable when alternating current is used, are (a) that there should be no leakage currents, and (6) that the phase angles should be the same in the tw r o pairs of adjacent arms of the bridge. These requirements have been satisfied in the A.(\ bridge designed for electrolytic conductance measurements by Jones and Josephs; 5 the second condition is met by making the two ratio arms (A* 3 and 7tJ 4 , Fig. 8) as nearly as possible identical in resistance and construction, so that any 6 Jones and Josephs, J. Am. Chem. Sue., 50, 1049 (1928); see also, Luder, ilnd., 62, 89 (1940). 42 ELECTROLYTIC CONDUCTANCE reactance, which is deliberately kept small, is the same in each case. In this way X 3 /Ra is made equal to Xt/R*. It may be noted that this condition is automatically obtained when a straight bridge wire is em- ployed. The reactance of the measuring cell, i.e., X\, should be made small, but as it cannot be eliminated it should be balanced by a variable condenser in parallel with the resistance box R z ; in this way Xi/Ri can be made equal to X Z /R2. It has often been the practice in con- ductance work to ground certain parts of the bridge network for the purpose of improving the sharpness of the sound minimum in the detector at the balance point; unless this is done with care it is liable to introduce errors because of the existence of leakage currents to earth. The telephone earpiece must, however, be at ground potential, otherwise the capacity between the telephone coils and the observer will result in a leakage of current. Other sources of leakage are introduced by the use of vacuum-tube oscillator and amplifier, and by various unbalanced capacities to earth, etc. The special method of grounding proposed by Jones and Josephs is illus- trated in Fig. 17. The bridge circuit consists essentially of the resistances 7?i, # 2 , #3 and # 4 , as in Fig. 8; the re- sistances #6 and #6, with the movable contact g and the variable condenser C , constitute the earthing device, which is a modified form of the Wagner ground. By means of the switch Si the condenser C is connected either to A or to A', whichever is found to give better results. The bridge is first balanced by adjusting 7 2 in the usual manner;* the telephone detector D is then disconnected from R' and connected to ground by means of the switch /S 2 . The position of the contact g and the condenser C g are adjusted until there is silence in the telephone, thus bringing B to ground potential. The switch 82 is now returned to its original position, and 72 2 is again adjusted so as to balance the bridge. If the changes from the original positions arc appreciable, the process of adjusting (/.and C g should be repeated and the bridge again balanced. Shielding the A.C. Bridge. In order to eliminate the electrostatic influence between parts of the bridge on one another, and also that due * This adjustment includes that of a condenser (not shown) in parallel, as explained above; see also page 33 and Fig. 9. R 5 R 6 vwwywwwsA/ 4-> Fia. 17. Jones and Josephs bridge PREPARATION OF SOLVENT 43 to outside disturbances, grounded metallic shields have sometimes been placed between the various parts of the bridge, or the latter has been completely surrounded by such shields. It has been stated that this form of shielding may introduce more error than it eliminates, on account of the capacity between the shield and the bridge; it has been recom- mended, therefore, that the external origin of the disturbance, rather than the bridge, should be shielded. According to Shcdlovsky 6 the objection to the use of electrostatic screening is based on unsymmetrical shielding which introduces unbalanced capacity effects s to earth; further, it is pointed out that it is not always possible to shield the disturbing source. A bridge has, therefore, been designed in which the separate arms of each pair are screened symmetrically; the shields surrounding the cell and the variable resistance (#2) are grounded, while those around the ratio arms (R 3 and 72 4 ) are not. The leads connecting the oscillator and detector to the bridge arc also screened and grounded. In this way mutual and external electrostatic influences on the bridge are eliminated. By means of a special type of twin variable condenser, connected across Ri and # 2 , the reactances in these arms can be compensated so as to give a sharp minimum in the telephone detector and also the correct con- ditions for Ri/Rz to be equal to Rs/R*. It is probable that the screened bridge has advantages over the unscreened bridge when external dis- turbing influences are considerable. Preparation of Solvent: Conductance Water. Distilled water is a poor conductor of electricity, but owing to the presence of impurities such as ammonia, carbon dioxide and traces of dissolved substances derived from containing vessels, air and dust, it has a conductance suffi- ciently large to have an appreciable effect on the results in accurate work. This source of error is of greatest importance with dilute solutions or weak electrolytes, because the conductance of the water is then of the same order as that of the electrolyte itself. If the conductance of the solvent were merely superimposed on that of the electrolyte the correc- tion would be a comparatively simple matter. The conductance of the electrolyte would then be obtained by subtracting that of the solvent from the total; this is possible, however, for a limited number of solutes. In most cases the impurities in the water can influence the ionization of the electrolyte, or vice versa, or chemical reaction may occur, and the observed conductance of the solution is not the sum of the values of the constituents. It is desirable, therefore, to use water which is as free as possible from impurities; such water is called conductance water, or conductivity water. The purest water hitherto obtained was prepared by Kohlrausch and Heydweiller (1894) who distilled it forty-two times under reduced pres- sure; this water had a specific conductance of 0.043 X 10~ 6 ohmr 1 cm.~ l Shedlovsky, J. Am. Chem. Soc., 52, 1793 (1930). 44 ELECTROLYTIC CONDUCTANCE at 18.* Water of such a degree of purity is extremely tedious to pre- pare, but the so-called " ultra-pure " water, with a specific conductance of 0.05 to 0.06 X 10~ 6 ohm- 1 cm.- 1 at 18, can be obtained without serious difficulty. 7 The chief problem is the removal of carbon dioxide and two principles have been adopted to achieve this end; either a rapid stream of pure air is passed through the condenser in which the steam is being condensed in the course of distillation, or a small proportion only of the vapor obtained by heating ordinary distilled water is con- densed, the gaseous impurities being carried off by the uncondensed steam. Ultra-pure water will maintain its low conductance only if air is rigidly excluded, but as such water is not necessary except in special c&ses, it is the practice to allow the water to come to equilibrium with the carbon dioxide of the atmosphere. The resulting "equilibrium water" has a specific conductance of 0.8 X 10" 6 ohmr 1 cm." 1 and is quite satisfactory for most conductance measurements. The following brief outline will indicate the method 8 used for the J \_^\ ready preparation of water having a II __ specific conductance of 0.8 X 10~ 6 ' ' * " ohm~ l cm." 1 ; it utilizes both the air- stream and partial condensation meth- ods of purification. The 20-liter boiler A (Fig. 18) is of copper, while the remainder of the apparatus should be made of pure tin or of heavily tinned copper. Distilled water containing sodium hydroxide and potassium per- manganate is placed in the boiler and the steam passes first through the trap B t which collects spray, and then into the tube C. A current of purified air, drawn through the apparatus by connecting D and E to a water pump, enters at F; a suction of about 8 inches of water is employed. The tem- perature of the condenser G is so arranged (about 80) that approximately half as much water is condensed in // as in /; the best conductance * Calculations based on the known ionization product of water and the conductances of the hydrogen and hydroxyl ions at infinite dilution (see p. 340) show that the specific conductance of perfectly pure water should be 0.038 X 10~ ohm" 1 cm." 1 at 18. 7 Kraus and Dexter, J. Am. Chem. Soc., 44, 2468 (1922); Bencowitz and Hotchkiss, J. Phys. Chem., 29, 705 (1925); Stuart and Worm well, J. Chem. Hoc., 85 (1930). 8 Vogel and Jeffery, /. Chem. oc., 1201 (1931). I 6 fi^r Fia. 18. Apparatus for preparation of conductance water (Vogel and Jeffery) SOLVENT CORRECTIONS 45 water collects in the Pyrcx flask ./, while a somewhat inferior quality is obtained in larger amount at K. For general laboratory measurements water of specific conductance of about 1 X 10~ 6 ohm" 1 cm." 1 at 18 is satisfactory; this can be obtained by distilling good distilled water, to which a small quantity of permanga- nate or Nessler's solution is added. A distilling flask of resistance glass is used and the vapor is condensed either in a block-tin condenser or in one of resistance glass. If corks are used they should be covered with tin foil to prevent direct contact with water or steam. Non-aqueous solvents should be purified by careful distillation, special care being taken to eliminate all traces of moisture. Not only are con- ductances in water appreciably different from those in non-aqueous media, but in certain cases, particularly if the electrolytic solution con- tains hydrogen, hydroxyl or alkoxyl ions, small quantities of water have a very considerable effect on the conductance. Precautions should thus be taken to prevent access of water, as well as of carbon dioxide and ammonia from the atmosphere. Solvent Corrections. The extent of the correction which must be applied for the conductance of the solvent depends on the nature of the electrolyte; 9 although not all workers are in complete agreement on the subject, the following conclusions are generally accepted. If the solute is a neutral salt, i.e., the salt of a strong acid and a strong base, the ionization and conductance of the carbonic acid, which is the main im- purity in water, arc no* affected to any great extent; the whole of the conductance of the solvent should then be subtracted from that of the solution. With such electrolytes the particular kind of conductance water employed is not critical. Strictly speaking the change in ionic concentration due to the presence of the salt does affect the conductance of the carbonic acid to some extent; when the solvent correction is a small proportion of the total, e.g., in solutions of neutral salts more con- centrated than about 10~ 3 N, the alteration is negligible. For more dilute solutions, however, it is advisable to employ ultra-pure water, precautions being taken to prevent the access of carbon dioxide. Salts of weak bases or weak acids are hydrolyzed in aqueous solution (see Chap. XI) and they behave as if they contained excess of strong acid and strong base, respectively. According to the law of mass action the presence of one acid represses the ionization of a weaker one, so that the effective conductance of the water, which is due mainly to carbonic acid, is diminished. The solvent correction in the case of a salt of a weak base and a strong acid should thus be somewhat less than the total conductance of the water. For solutions of salts of a weak acid and a strong base, which react alkaline, the correction is uncertain, but methods of calculating it have been described; they are based on the assumption 'Kolthoff, Rec. trav. chim., 48, 664 (1929); Davies, Trans. Faraday Soc., 25, 129 (1929); "The Conductivity of Solutions," 1933, Chap. IV. 46 ELECTROLYTIC CONDUCTANCE that the impurity in the water is carbonic acid. 10 If ultra-pure water is used, the solvent correction can generally be ignored, provided the solu- tion is not too dilute. If the solution being studied is one of a strong acid of concentration greater than 10" 4 N, the ionization of the weak carbonic acid is depressed to such an extent that its contribution towards the total conductance is negligible. In these circumstances no water correction is necessary; at most, the value for pure water, i.e., about 0.04 X 10" 6 ohm" 1 cm." 1 at ordinary temperatures, may be subtracted from the total. If the con- centration of the strong acid is less than 10~ 4 N, a small correction is necessary and its magnitude may be calculated from the dissociation constant of carbonic acid. The specific conductance of a 10~ 4 N solution of a strong acid, which represents the lowest concentration for which the solvent correction may be ignored, is about 3.5 X 10~ 5 ohm" 1 cm." 1 Similarly, with weak acids the correction is unnecessary provided the specific conductance exceeds this value. For more dilute solutions the appropriate correction may be calculated, as mentioned above. The solvent correction to be applied to the results obtained with solutions of bases is very uncertain; the partial neutralization of the alkali by the carbonic acid of the conductance water results in a decrease of conductance, and so the solvent correction should be added, rather than subtracted. A method of calculating the value of the correction has been suggested, but it would appear to be best to employ ultra-pure water in conductance work with bases. With non-aqueous solvents of a hydroxylic type, such as alcohols, the corrections are probably similar to those for water; other solvents must be considered on their own merits. In general, the solvent should be as pure as possible, so that the correction is, in any case, small; as indi- cated above, access of atmospheric moisture, carbon dioxide and ammonia should be rigorously prevented. Since non-hydroxylic solvents such as acetone, acetonitrile, nitromethane, etc., have very small conductances when pure, the correction is generally negligible. Preparation of Solutions. When the conductances of a series of solutions of a given electrolyte are being measured, it is the custom to determine the conductance of the water first. Some investigators recom- mend that measurements should then commence with the most concen- trated solution of the series, in order to diminish the possibility of error resulting from the adsorption of solute from the more dilute solutions by the finely divided platinum on the electrodes. When working with cells of the flask type it is the general practice, however, to fill the cell with a known amount of pure solvent, and then to add successive small quantities of a concentrated solution of the electrolyte, of known cori- 10 Davies, Trans. Faraday Soc., 28, 607 (1932); Maclnnes and Shedlovsky, /. Am. Chem. Soc., 54, 1429 (1932); Jeff cry, Vogel and Lowry, J. Chem. Soc., 1637 (1933); 166 (1934); 21 (1935). DIRECT CURRENT METHODS 47 centration, from a weight burette. When rolls of other types are used it is necessary to prepare a separate solution for each measurement; this procedure must be adopted in any case if the solute is relatively insoluble. Direct Current Methods. A few measurements of electrolytic con- ductance have been made with direct current and non-polarizable elec- trodes; the electrodes employed have been mercury-mercurous chloride in chloride solutions, mercury-mercurous sulfate in sulfate solutions, and hydrogen electrodes in acid electrolytes.* Two main principles have been applied: in the first, the direct current is passed between two elec- trodes whose nature is immaterial; the two non-polarizable electrodes are then inserted at definite points in the electrolyte and the fall of potential between them is measured. The current strength is calculated by determining the potential difference between two ends of a wire of accurately known resistance placed in the circuit. Knowing the poten- tial difference between the two non-polarizable electrodes and the current passing, the resistance of the column of solution separating these elec- trodes is obtained immediately by means of Ohm's law. The second principle which has been employed is to use the non-polarizable electrodes for leading direct current into and out of the solution in the normal manner arid to determine the resistance of the electrolyte by means of a Wheatstone bridge network. A sensitive mirror galvanometer is used as the null instrument and no special precautions need be taken to avoid inductance, capacity and leakage effects, since these do not arise with direct current. 11 The cells used in the direct current measurements are quite different from those employed with alternating current; thore is nothing critical about their design, and they generally consist of horizontal tubes with the electrodes inserted either at the ends or at definite intermediate positions. The constants of the cells are determined either by direct measurement of the tubes, by means of mercury, or by using an electro- lyte whose specific conductance is known accurately from other sources. It is of interest to record that where data are available for both direct and alternating current methods, the agreement is very satisfactory, showing that the use of alternating current does not introduce any in- herent error. The direct current method has the disadvantage of being applicable only to those electrolytes for which non-polarizable electrodes can be found. The following simple method for measuring the resistance of solutions of very low specific conductance has been used. 12 A battery of storage * The nature of these electrodes will be understood better after Chap. VI has been studied. "Eastman, J. Am. Chem. Soc., 42, 1648 (1920); Br0nsted and Nielsen, Trans. Faraday Soc., 31, 1478 (1935); Andrews and Martin, J. Am. Chem. Soc., 60, 871 (1938). u LaMer and Downes, J. Am. Chem. Soc., 53, 888 (1931); Fuoss and Kraus, ibid.. 55, 21 (1933); Bent and Dorfman, ibid., 57, 1924 (1935). 48 ELECTROLYTIC CONDUCTANCE cells, having an E.M.F. of about 150 volts, is applied to the solution whose resistance exceeds 100,000 ohms; the strength of the current which passes is then measured on a calibrated mirror galvanometer. From a knowledge of the applied voltage and the current strength, the resistance is calculated with the aid of Ohm's law. In view of the high E.M.F. employed, relative to the polarization E.M.F., the error due to polarization is very small; further, since only minute currents flow, the influence of electrolysis and heating is negligible. Conductance Determinations at High Voltage and High Frequency. The electrolytic conductances of solutions with alternating current of very high frequency or of high voltage have acquired special interest in con- nection with modern theories of electrolytic solutions. Under these Hf- <il VU, K 7 o o o 1 Jl High | frequency 4H FIG. 19. Barretter bridge extreme conditions the simple Wheatstone bridge method cannot be used, and other experimental procedures have been described. The chief diffi- culty lies in the determination of the balance point, and in this connection the "barretter bridge" has been found to be particularly valuable. A form of this bridge is shown in Fig. 19, II; it is virtually a Wheatstone bridge, one arm containing the choke inductances Si and $ 3 , and a small fine-wire filament "barretter" lamp (Zi), across which is shunted a coup- ling inductance MI and a condenser Ci; the corresponding arm of the bridge contains the chokes S 2 and S 4 , and the barretter tube Z 2 , which is RESULTS OF CONDUCTANCE MEASUREMENTS 49 carefully matched with /i, shunted by the coupling inductance M^ and the condenser CY The ratio arms of the bridge consist of the variable resistances Jt 3 and R*. The actuating direct current voltage for the bridge is supplied by a direct current battery, and the detecting instru- ment is the galvanometer (7; an inductance in series with the latter prevents induced currents from passing through it. At the beginning of the experiment the resistances # 3 and R* are adjusted until the bridge is balanced. The actual resistance circuit is depicted in Fig. 19, I; K is the con- ductance cell and R is a variable resistance which are coupled to the barretter circuit by means of the inductances LI and L<>. The high fre- quency or high voltage is applied to the terminals of this circuit, and the currents induced in the bridge are restricted to the barretters li and Z 2 by the pairs of inductances NI *S 3 and *S f 2 -A>4, respectively. The heating effect of these currents causes a change of resistance of the barretters, and if the currents in L\ and L 2 are different, the bridge will be thrown out of balance. The resistance R is then adjusted until the bridge remains balanced when the current is applied to the cell circuit. The cell K is now replaced by a standard variable resistance and, keeping R constant, this is adjusted until the bridge is again balanced; the value of this re- sistance is then equal to that of the cell 7v. 13 Results of Conductance Measurements. --The results recorded here refer to measurements made at A.C. frequencies and voltages that are not too high, i.e., ono to four thousand cycles per sec. and a few volts per cm., respectively. Under these conditions the electrolytic conductances are independent of the voltage, i.e., Ohm's law is obeyed, and of fre- quency, provided polarization is eliminated. Although the property of a solution that is actually measured is the specific conductance at a given concentration, this quantity is not so useful for comparison purposes as is the equivalent conductance; the latter gives a measure of the con- ducting power of the ions produced by one equivalent of the electrolyte at the given concentration and is invariably employed in electrochemical work. The equivalent conductance is calculated from the measured specific conductance; by means of equation (5). A largo number of conductance measurements of varying degrees of accuracy have been reported; the most reliable* results for some electro- lytes in aqueous solution at 25 are recorded in Table; VIII, the concen- trations being expressed in equivalents per liter. 14 These data show that the equivalent conductance, and hence the con- ducting power o f the ions in one gram equivalent of any electrolyte, increases with decreasing concentration. The figures appear to approach 1J Malsch and Wien, Ann. Physik, 83, 305 (1927); Neese, ibid., 8, 929 (1931); Wien, i/m/., 11, 429 (1931); Srhicle, Physik. Z. t 35, 632 (1934). 14 For a critical compilation of recent accurate data, see Machines, "The Principles of Electrochemistry," 1939, p. 339; for other data International Critical Tables, Vol. VI, and the Ijtindolt-Boriistein Tabellen should be consulted. 50 ELECTROLYTIC CONDUCTANCE TABLE VIII. EQUIVALENT CONDUCTANCES AT 25 IX OHMS" 1 CM. 8 Concn. HCl KCl Nul NaOH AgNOa iBaClj iNiRO iLaCla iKFe(CN). 0.0005 N 422.74 147.81 125.36 246 131.36 135.96 118.7 139.6 0.001 421.36 140.95 124.25 245 130.51 134.34 113.1 137.0 167.24 0.005 415.80 143.55 121.25 240 127.20 128.02 93.2 127.5 146.09 0.01 412.00 141.27 119.24 237 124.76 123.94 82.7 121.8 134.83 0.02 407.24 138.34 llti.70 233 121.41 1 19.09 72.3 115.3 122.82 0.05 399.09 133.37 112.79 227 115.24 111.48 59.2 106.2 107.70 0.10 391.32 128.96 108.78 221 109.14 105.19 50.8 99.1 97.87 a limiting value in very dilute solutions; this quantity is known as the equivalent conductance at infinite dilution and is represented by the symbol A . An examination of the results of conductance measurements of many electrolytes of different kinds shows that the variation of the equivalent conductance with concentration depends to a great extent on the type of electrolyte, rather than on its actual nature. For strong uni-univalent electrolytes, i.e., with univalent cation and anion, such as hydrochloric acid, the alkali hydroxides and the alkali halides, the decrease of equiva- lent conductance with increasing concentration is not very large. As the valence of the ions increases, however, the falling off is more marked; this is shown by the curves in Fig. 20 in which the equivalent conduct- 160 120 90 i Potassium Chloride 0.01 0.02 0.05 0.1 Concentration in Equtv. per Liter Fio. 20. Conductances of electrolytes of different types THE CONDUCTANCE RATIO 51 anccs of potassium chloride, a typical uni-univalont strong electrolyte, and of nickel sulfatc, a hi-bivalent electrolyte, are plotted as functions of the concentration. Electrolytes of an intermediate valence type, e.g., potassium sulfatc, a uni-bivalcnt electrolyte, and barium chloride, which is a bi-uriivalcnt salt, behave in an intermediate manner. The substances referred to in Table VIII are all strong, or relatively strong, electrolytes, but weak electrolytes, such as weak acids and bases, exhibit an apparently different behavior. The results for acetic acid, a typical weak electrolyte, at 25 are given in Table IX. TABLE IX. EQUIVALENT CONDUCTANCE OF ACETIC ACID AT 25 Concn. 0.0001 0.001 0.005 0.01 0.02 0.05 0.10 N A 131.6 48.63 22.80 16.20 11.57 7.36 5.20 ohms- 1 cm. 2 It is seen that at the higher concentrations the equivalent conductance is very low, which is the characteristic of a weak electrolyte, but in the more dilute solutions the values rise with great rapidity; the limiting equivalent conductance of acetic acid is known from other sources to be 390.7 ohms" 1 cm. 2 at 25, and so there must be an increase from 131.6 to this value as the solution is made more dilute than 10 4 equiv. per liter. The plot of the results for acetic acid, shown in Fig. 20, may be regarded as characteristic of a weak electrolyte. As mentioned in Chap. I, it is not possible to make a sharp distinction between electro- lytes of different classes, and the variation of the equivalent conductance of an intermediate electrolyte, such as trichloroacetic, cyanoacetic and mandelic acids, lies between that for a weak electrolyte, e.g., acetic acid, and a moderately strong electrolyte, e.g., nickel sulfate (cf. Fig. 20). The Conductance Ratio. The ratio of the equivalent conductance (A) at any concentration to that at infinite dilution (A )* has played an important part in the development of electrochemistry; it is called the conductance ratio, and is given the symbol a, thus In the calculations referred to on page 10, Arrhenius assumed the con- ductance ratio to be equal to the degree of dissociation of the electro- lyte; this appears to be approximately true for weak electrolytes, but not for salts and strong acids and bases. Quite apart from any theoreti- cal significance which the conductance ratio may have, it is a useful empirical quantity because it indicates the extent to which the equivalent conductance at any specified concentration differs from the limiting value. The change of conductance ratio with concentration gives a measure of the corresponding falling off of the equivalent conductance. In accord- ance with the remarks made previously concerning the connection be- * For the methods of extrapolation of conductance data to give the limiting value, see p. 54. 52 ELECTROLYTIC CONDUCTANCE tween the variation of equivalent conductance with concentration and the valence type of the electrolyte, a similar relationship should hold for the conductance ratio. In dilute solutions of strong electrolytes, other than acids, the conductance ratio is in fact almost independent of the nature of the salt and is determined almost entirely by its valence type. Some mean values, derived from the study of a number of electrolytes at room temperatures, are given in Table X; the conductance ratio at any TABLE X. CONDUCTANCE RATIO AND VALENCE TYPE OP SALT Valence Type 0.001 0.01 0.1 N Uni-uni 0.98 0.93 O.S3 21} 0.05 0.87 0.75 Bi-bi 0.85 0.65 0.40 given concentration is seen to be smaller the higher the valence type. For weak electrolytes the conductance ratios are obviously very much less, as is immediately evident from the data in Table IX. As a general rule increase of temperature increases the equivalent conductance both at infinite dilution and at a definite concentration ; the conductance ratio, however, usually decreases with increasing tempera- ture, the effect being greater the higher the concentration. These con- clusions are supported by the results for potassium chloride solutions in Table XI taken from the extensive measurements of Noyes and his TABLE XI. VARIATION OF CONDUCTANCE RATIO OP POTASSIUM CHLORIDE SOLUTIONS WITH TEMPERATURE 18 100 150 21S 306 0.01 N 0.94 0.91 0.90 0.90 0.8 i 0.08 N 0.87 0.83 0.80 0.77 O.G4 collaborators. 16 The falling off is more marked for electrolytes of higher valence type, and especially for weak electrolytes. A few cases are known in which the conductance ratio passes through a maximum as the tem- perature is increased; this effect is probably due to changes in the extent of dissociation of relatively weak electrolytes. Equivalent Conductance Minima. Provided the dielectric constant of the medium is greater than about 30, the conductance behavior in that medium is usually similar to that of electrolytes in water; the differences are not fundamental and are generally differences of degree only. With solvents of low dielectric constant, however, the equivalent conductances often exhibit distinct abnormalities. It is frequently found, for example, that with decreasing concentration, the equivalent conductance decreases instead of increasing; at a certain concentration, however, the value passes through a minimum and the subsequent variation is normal. In other cases, e.g., potassium iodide in liquid sulfur dioxide and tetra- 14 Noyes et al., J. Am. Chem. Soc., 32, 159 (1910); sec also, Kraus, "Klectrioully Conducting Systems," 1922, Chap. VI. EQUIVALENT CONDUCTANCE MINIMA 53 propylammonium iodide in methylene chloride, the equivalent conduct- ances pass through a maximum and a minimum with decreasing concen- tration. The problem of the minimum equivalent conductance was investigated by Walden 16 who concluded that there was a definite rela- tionship between the concentration at which such a minimum could be observed and the dielectric constant of the solvent. If c m i n . is the con- centration for the minimum equivalent conductance, and D is the dielec- tric constant of the medium, then Walden's conclusion may be repre- sented as c mm . = kD\ (14) where k is a constant for the given electrolyte. It is evident from this equation that in solvents of high dielectric constant the minimum should be observed only at extremely high concentrations; even if such solutions could be prepared, it is probable that other factors would interfere under these conditions. It will be seen later that equation (14) has a theoreti- cal basis. -4.0 - -4.5 -3.5 -2.5 -1.5 logc Fia. 21. Influence of dielectric; constant on conductance (Kuoss and Kraus) M Walden, Z. physik. Chcm., 94, 263 (1920); 100, 512 (1922). 54 ELECTROLYTIC CONDUCTANCE The influence of dielectric constant on the variation of equivalent conductance with concentration has been demonstrated in a striking manner by the measurements made by Fuoss and Kraus 17 on tetra- isoamylammonium nitrate at 25 in a series of mixtures of water and dioxane, with dielectric constant varying from 78.6 to 2.2. The results obtained are depicted graphically in Fig. 21, the dielectric constant of the medium being indicated in each case; in view of the large range of conductances and concentrations the figure has been made more compact by plotting log A against log c. It is scon that as the dielectric constant becomes smaller, the falling off of equivalent conductance with increasing concentration is more marked. At sufficiently low dielectric constants the conductance minimum becomes evident; the concentration at which this occurs decreases with decreasing dielectric constant, in accordance with the Walden equation. The theoretical implication of these results will be considered more fully in Chap. V. Equivalent Conductance at Infinite Dilution. A number of methods have been proposed at various times for the extrapolation of experi- mental equivalent conductances to give the values at infinite dilution. Most of the procedures described for strong electrolytes are based on the use of a formula of the type A = Ao ac n , (15) where A is the equivalent conductance measured at concentration c; the quantities a and n are constants, the latter being approximately 0.5, as required by the modern theoretical treatment of electrolytes. If data for sufficiently dilute solutions are available, a reasonably satisfactory value for A may be obtained by plotting the experimental equivalent conductances against the square-root of the concentration and performing a linear extrapolation to zero concentration. It appears doubtful, from recent accurate work, if an equation of the form of (15) can represent completely the variation of equivalent conductance over an appreciable range of concentrations; it follows, therefore, that no simple extrapola- tion procedure can be regarded as entirely satisfactory. An improved method 18 is based on the theoretical Onsagcr equation (p. 90), i.e., A' - A "*" ^ r A ~~ i _ /W where A and B are constants which may be evaluated from known properties of the solvent. The results for \' derived from this equation for solutions of appreciable concentration are not constant, and hence the prime has been added to the symbol for the equivalent conductance. 17 Fuoss and Kraus, jr. Am. Chem. Soc., 55, 21 (1933). "Shedlovsky, J. Am. Chem. Soc., 54, 1405 (1932). THE INDEPENDENT MIGRATION OF IONS 56 For many strong electrolytes Aj is a linear function of the concentration, thus Ai = Ao + ac, so that if the values of AQ are plotted against the concentration c, the equivalent conductance at infinite dilution may be obtained by linear extrapolation. The data for sodium chloride and hydro- chloric acid at 25 are shown in Fig. 22; the limiting equiv- alent conductances at zero concentration are 126.45 and 426.16ohm- l cm. 2 ,respectively. For weak electrolytes, no form of extrapolation is sat- isfactory, as will be evident from an examination of Fig. 20. The equivalent conduct- ance at infinite dilution can then be obtained only from the values of the individ- ual ions, as will be described shortly. For electrolytes ex- hibiting intermediate behav- ior, e.g., solutions of salts in media of relatively low di- electric constant, an extrapo- lation method based on theo- retical considerations can bo employed (see p. 167). The Independent Migration of Ions. A survey of equivalent con- ductances at infinite dilution of a number of electrolytes having an ion in common will bring to light certain regularities; the data in Table XII, 128 0.02 0.04 0.06 0.08 Concentration in Equiv. per Liter FIG. 22. Extrapolation to infinite dilution TABLE XII. Electrolyte KC1 KNOj COMPARISON OF EQUIVALENT CONDUCTANCES AT INFINITE DILUTION A Electrolyte A Difference 130.0 NaCl 108.9 21.1 126.3 NaNO, 105.2 21.1 133.0 NajS0 4 111.9 21.1 for example, are for corresponding sodium and potassium salts at 18. The difference between the conductances of a potassium and a sodium salt of the same anion is seen to be independent of the nature of the latter. Similar results have been obtained for other pairs of salts with an anion or a cation in common, both in aqueous and non-aqueous solvents. Observations of this kind were first made by Kohlrausch (1879, 1885) by 56 ELECTROLYTIC CONDUCTANCE comparing equivalent conductances at high dilutions; he ascribed them to the fact that under these conditions every ion makes a definite con- tribution towards the equivalent conductance of the electrolyte, irre- spective of the nature of the other ion with which it is associated in tho solution. The value of the equivalent conductance at infinite dilution may thus'be regarded as made up of the sum of two independent factors, one characteristic of each ion; this result is known as Kohlrausch's law of independent migration of ions. The law may be expressed in the form Ao = \+ + Ai, (16) where X+ and X?. are known as the ion conductances, of cation and anion, respectively, at infinite dilution. The ion conductance is a definite con- stant for each ion, in a given solvent, its value depending only on the temperature. It will be seen later that the ion conductances at infinite dilution are related to the speeds with which the ions move under the influence of an applied potential gradient. Although it is possible to derive their values from the equivalent conductances of a number of electrolytes by a method of trial and error, a much more satisfactory procedure is based on the use of accurate transference number data; these transference numbers are deter- mined by the relative speeds of the ions present in the electrolyte and hence are related to the relative ion conductances. The determination of transference numbers will be described in Chap. IV and the method of evaluating ion conductances will be given there; the results will, however, be anticipated and some of the best values for ion conductances in water at 25 are quoted in Table XIII. 19 It should be noted that since these are TABLE XIII. ION CONDUCTANCES AT INFINITE DILUTION AT 25 IN OHMS" 1 CM. 2 Cation X^. a X 10 2 Anion Xi a X 10 2 H+ 349.82 1.42 OH~ 198 1.60 T1+ 74.7 1.87 Br~ 78.4 1.87 K+ 73.52 1.89 I~ 76.8 1.86 NH+ 73.4 1.92 Cl 76.34 1.88 A- 61.92 1.97 NO 3 ~ 71.44 1.80 Na + 50.11 2.09 ClOr 68.0 Li+ 38.69 2.26 HCO 3 ~ 44.5 JBa++ 63.64 2.06 }SO 4 79.8 1.96 JCa++ 59.50 2.11 iFe(CN)jf 101.0 JSr-n- 59.46 2.11 lFe(CN)<f 110.5 53.06 2.18 actually equivalent conductances, symbols such as JBa++and JFe(CN)if are employed. (The quantities recorded in the columns headed a are approximate temperature coefficients; their significance will be explained on page 61.) In the results recorded in Table XIII, there appears to be no con- nection between ionic size and conductance; for a number of ions be- 19 See Maclnnes, J. Franklin Iwt., 225, 661 (1938); "The Principles of Electro- chemistry." 1939, p. 342. APPLICATION OF ION CONDUCTANCES 57 longing to a homologous series, as for example the ions of normal fatty acids, a gradual decrease of conductance is observed and a limiting value appears to be approached with increasing chain length. The data for certain fatty acid anions are known accurately, but others are approxi- mate only; the values in Table XIV, nevertheless, show the definite trend TABLE XIV. ION CONDUCTANCES OP PATTY ACID IONS AT 25 Anion Formula X_ Formate HCOr ~52 ohms" 1 cm. 1 Acetate CH 8 COr 40.9 Propionate CH 8 CH,CO^ 35.8 Butyrate CH,(CH,)jCOf 32.6 Valerianate CH 8 (CH a )aCOf ~29 Caproate CH,(CH,)4COr ~28 towards a constant ion conductance. A similar tendency has been ob- served in connection with the conductances of alkylammonium ions. A large number of ion conductances, of more or less accuracy, have been determined in non-aqueous solvents; reference to these will be made shortly in the section dealing with the relationship between the con- ductance of a given ion in various solvents and the viscosities of the latter. Application of Ion Conductances. An important use of ion con- ductances is to determine the equivalent conductance at infinite dilution of certain electrolytes which cannot be, or have not been, evaluated from experimental data. For example, with a weak electrolyte the extrapo- lation to infinite dilution is very uncertain, and with sparingly soluble salts the number of measurements which can be made at appreciably different concentrations is very limited. The value of A can, however, bo obtained by adding the ion conductances. For example, the equiva- lent conductance of acetic acid at infinite dilution is the sum of the con- ductances of the hydrogen and acetate ions; the former is derived from a study of strong acids and the latter from measurements on acetates. It follows, therefore, that at 25 Ao(CH s co t H) = XH+ + XCH,CO;, = 349.8 + 40.9 = 390.7 ohms" 1 cm. 2 The same result can be derived in another manner which is often con- venient since it avoids the necessity of separating the conductance of an electrolyte into the contributions of its constituent ions. The equivalent conductance of any electrolyte MA at infinite dilution A (MA> is equal to XM* + X A -, where XM+ and X A - are the ion conductances of the ions M+ and A~ at infinite dilution; it follows, therefore, that Ao(MA) = Ao(MCl) + Ao(NaA) ~ Ao(NaCl), where A O <MCD, A (NaA) and AO(NCD are the equivalent conductances at infinite dilution of the chloride of the metal M, i.e., MCI, of the sodium salt of the anion A, i.e., NaA, and of sodium chloride, respectively. Any 58 ELECTROLYTIC CONDUCTANCE convenient anion may be used instead of the chloride ion, and similarly the sodium ion may be replaced by another metallic cation or by the hydrogen ion. For example, if M+ is the hydrogen ion and A~ is the acetate ion, it follows that Ao(CH s CO,H) = Ao(HCl) + AocCHjCOjN*) ~ A (NC1) = 426.16 + 91.0 - 126.45 = 390.71 ohms~ l cm. 2 at 25. In order to determine the equivalent conductance of a sparingly soluble salt it is the practice to add the conductances of the constituent ions; thus for silver chloride and barium sulfate the results are as follows: XA+ + Xcr 61.92 + 76.34 = 138.3 ohms" 1 cm. 2 at 25, 63.64 + 79.8 = 143.4 ohms~ l cm. 2 at 25. Absolute Ionic Velocities: Ionic Mobilities. The approach of the equivalent conductances of all electrolytes to a limiting value at very high dilutions may be ascribed to the fact that under these conditions all the ions that can be derived from one gram equivalent are taking part in conducting the current. At high dilutions, therefore, solutions con- taining one equivalent of various electrolytes will contain equivalent numbers of ions; the total charge carried by all the ions will thus be the same in every case. The ability of an electrolyte to transport current, and hence its conductance, is determined by the product of the number of ions and the charge carried by each, i.e., the total charge, and by the actual speeds of the ions. Since the total charge is constant for equiva- lent solutions at high dilution, the limiting equivalent conductance of an electrolyte must depend only on the ionic velocities: it is the difference in the speeds of the ions which is consequently responsible for the differ- ent values of ion conductances. The speed with which a charged particle moves is proportional to the potential gradient, i.e., the fall of potential per cm., directing the motion, and so the speeds of ions are specified under a potential gradient of unity, i.e., one volt per cm. These speeds arc known as the mobilities of the ions. If w+ and u*L are the actual velocities of positive and negative ions of a given electrolyte at infinite dilution under unit potential gradient, i.e., the respective mobilities, then the equivalent conductance at infinite dilution must be proportional to the sum of these quantities; thus Ao = k(u+ + u ) = fc< + ku-> (17) where k is the proportionality constant which must be the same for all electrolytes. The equivalent conductance, as seen above, is the sum of the ion conductances, i.e., Ao - 4 + X?., ABSOLUTE IONIC VELOCITIES 59 and since Xl and u+ are determined only by the nature of the positive ion, while X_ and u!L are determined only by the negative ion, it follows that X3_ = ku\ and X?. = fci. (18) Imagine a very dilute solution of an electrolyte, at a concentration c equiv. per liter, to be placed in a cube of 1 cm. side with square elec- trodes of 1 sq. cm. area at opposite faces, and suppose an E.M.F. of 1 volt to be applied. By definition, the specific conductance (*) is the con- ductance of a centimeter cube, and the equivalent conductance of the given dilute solution, which is virtually that at infinite dilution, is 1000 JC/G [see equation (5)], so that 1000 - = Ao = \\ + X?., c(XJ + X ) " 1000 It was shown on page 30 that when a potential difference of 1 volt is applied to a 1 cm. cube, the current in amperes is numerically equal to the specific conductance, i.e., 1000 and this represents the number of coulombs flowing through the cube per second. Since the mobilities u+ and u!L are the ionic velocities in cm. per sec. under a fall of potential of 1 volt per cm., all the cations within a length of u+ cm. will pass across a given plane in the direction of the current in 1 sec., while all the anions within a length of u*L cm. will pass in the opposite direction. If the plane has an area of 1 sq. cm., all the cations in a volume u+ cc. and all the anions in u*L cc. will move in opposite directions per sec.; since 1 cc. of the solution contains c/1000 equiv., it follows that a total of (u^ + w) c/1000 equiv. of cations and anions will be transported by the current in 1 sec. Each equivalent of any ion carries one faraday (F) of electricity; hence the total quantity carried per sec. will be F(u+ -f u?.) c/1000 coulombs. It has been seen that the quantity of electricity flowing per sec. through the 1 cm. cube is equal to 7 as given above; consequently, u!L)c c(X3. + X ) 1000 1000 /. F(4 + ui) - X$. + X?.. (19) It follows, therefore, that the constant k in equation (17) is equal to F, and hence by equation (18), X?. =* Fu+ and X?. = Fu. (20) 60 ELECTROLYTIC CONDUCTANCE The absolute velocity ef any ion in cm. per sec. under a potential gradient of 1 volt per cm. can thus be obtained by dividing the ion conductance in ohms~ l cm. 2 by the value of the faraday in coulombs, i.e., 96,500. Since the velocity is proportional to the potential gradient, as a conse- quence of the applicability of Ohm's law to electrolytes, the speed of an ion can be evaluated for any desired fall of potential. It should be pointed out that equation (20) gives the ionic velocity at infinite dilution; the values decrease with increasing concentration, especially for strong electrolytes. The ion conductances in Table XIII have been used to calculate the mobilities of a number of ions at infinite dilution at 25; the results are recorded in Table XV. It will be observed that, apart from hydrogen TABLE XV. CALCULATED IONIC MOBILITIES AT 25 Mobility Mobility Cation cm. /sec. Anion cm./sec. Hydrogen 36.2X10^ Hydroxyl 20.5X10^ Potassium 7.61 Sulfate 8.27 Barium 6.60 Chloride 7.91 Sodium 5.19 Nitrate 7.40 Lithium 4.01 Bicarbonate 4.61 and hydroxyl ions, most ions have velocities of about 5 X 10~ 4 cm. per sec. at 25 under a potential gradient of unity. The influence of tem- perature on ion conductance, and hence on ionic speeds, is discussed below. Experimental Determination of Ionic Velocities. An attempt to measure the speeds of ions directly was made by Lodge (1886) who made use of some characteristic property of the ion, e.g., production of color with an indicator or formation of a precipitate, to follow its movement under an applied field. In Lodge's apparatus the vessels containing the anode and cathode, respectively, were joined by a tube 40 cm. long filled with a conducting gelatin gel in which the indicating material was dissolved. For example, in determining the velocity of barium and chloride ions the gel contained acetic acid as conductor and a trace of silver sulfate as indicator; barium chloride was used in both anode and cathode vessels and the electrodes were of platinum. On passing current the barium and chloride ions moved into the gel, in opposite directions, producing visible precipitates of barium sulfate and silver chloride, re- spectively: the rates of forward movement of the precipitates gave the speeds of the respective ions under the particular potential gradient employed. Although the results obtained by Lodge in this manner were of the correct order of magnitude, they were generally two or three times less than those calculated from ion conductances by the method described above. The discrepancies were shown by Whetham (1893) to be due to a. non-uniform potential gradient and to lack of precautions to secure INFLUENCE OF TEMPERATURE ON ION CONDUCTANCES 61 sharp boundaries. Taking these factors into consideration, Whetham devised an apparatus for observing the movement of the boundary be- tween a colorless and a colored ion, or between two colored ions, without the use of a gel. The values for the velocities of ions obtained in this manner were in satisfactory agreement with those calculated, especially when allowance was made for the fact that the latter refer to infinite dilution. The principle employed by Whetham is almost identical with that used in the modern "moving boundary" method for determining transference numbers and this is described in Chap. IV. Influence of Temperature on Ion Conductances. Increase of tem- perature invariably results in an increase of ion conductance at infinite dilution; the variation with temperature may be expressed with fair accuracy by means of the equation X? = XS 5 [1 + (* - 25) + 0(i - 25) 2 ], (21) where X? is the ion conductance at infinite dilution at the temperature t, and X 5 is the value at 25. The factors a and are constants for a given ion in the particular solvent; for a narrow temperature range, e.g., about 10 on either side of 25, the constant ft may be neglected, and approxi- mate experimental values of a are recorded in Table XIII above. It vill be apparent that, except for hydrogen and hydroxyl ions, the tem- perature coefficients a. are all very close to 0.02 at 25. Since the conductance of an ion depends on its rate of movement, it seems reasonable to treat conductance in a manner analogous to that employed for other processes taking place at a definite rate which in- creases with temperature. If this is the case, it is possible to write X = Ae- E ' RT , (22) where A is a constant, which may be taken as being independent of temperature over a relatively small range; E is the activation energy of the process which determines the rate of movement of the ions, R is the gas constant and T is the absolute temperature. Differentiation of equation (22) with respect to temperature, assuming A to be constant, gives <HnX 1 d\ Q E dT ~ \' dT~ RT*' (23) Further, differentiation of equation (21) with respect to temperature, the factor being neglected, shows that for a narrow temperature range _ X ' dT " " and hence the activation energy is given by E - aRT*. 62 ELECTROLYTIC CONDUCTANCE Since a is approximately 0.02 for all ions, except hydrogen and hydroxyl ions, at 25, it is seen that for conductance in water the activation energy is about 3.60 kcal. in every case. Ion Conductance and Viscosity : Temperature and Pressure Effects. It is an interesting fact that the activation energy for electrolytic con- ductance is almost identical with that for the viscous flow of water, viz., 3.8 kcal. at 25; hence, it is probable that ionic conductance is related to the viscosity of the medium. Quite apart from any question of mecha- nism, however, equality of the so-called activation energies means that the positive temperature coefficient of ion conductance is roughly equal to the negative temperature coefficient of viscosity. In other words, the product of the conductance of a given ion and the viscosity of water at a series of temperatures should be approximately constant. The results in Table XVI give the product of the conductance of the acetate ion at TABLE XVI. CONDUCTANCE-VISCOSITY PRODUCT OP THE ACETATE ION Temperature 18 25 59 75 100 128 156 Xe*> 0.366 0.368 0.366 0.368 0.369 0.368 0.369 0.369 L05 Ap AT 0.95 infinite dilution (Xo) and the viscosity of water (ijo), i.e., A i?o, at tempera- tures between and 156; the re- sults are seen to be remarkably con- stant. It is true that such constancy is not always obtained, but the con- ductance-viscosity product for infi- s nite dilution is, at least, approxi- mately independent of temperature for a number of ions in water. The data for non-aqueous media are less complete, but it appears that in gen- eral the product of the ionic conduc- tance and the viscosity in such media is also approximately constant over a range of temperatures.* Another fact which points to a relationship between ionic mobility and viscosity is the effect of pressure on electrolytic conductance. Data are not available for infinite dilution, but the results of measurements on a number of electrolytes at a concentration of 0.01 N in water at 20 are shown in Fig. 23; the ordinates give the ratio of the equivalent conduct- ance at a pressure p to that at unit pressure, i.e., A p /Ai, while the ab- scissae represent the pressures in kg. per sq. cm. 20 The dotted line * It should be emphasized that the conductance-viscosity product constancy is, on the whole, not applicable to solutions of appreciable concentration. M Data mainly from Kdrber, Z. physik. Chem., 67, 212 (1909); see also, Adams and Hall, J. Phy 9 . Chem., 35, 2145 (1931); Zisman, Phys. Rev., 30, 151 (1932). 100 200 Pressure k./cm. 2 Fio. 23. Variation of conductance with pressure INFLUENCE OP SOLVENT ON ION CONDUCTANCE 63 indicates the variation with pressure of the fluidity, i.e., the reciprocal of the viscosity, of water relative to that at unit pressure. The existence of a maximum in both the conductance and fluidity curves suggests that there is some parallelism between these quantities: exact agreement would be expected only at infinite dilution, for other factors which are influenced by pressure may be important in solutions of appreciable concentration. The relationship between viscosity and ion conductance has been interpreted in at least two ways; some writers have suggested that the constancy of the product Xoi?o proves the applicability of Stokes's law to ions in solution. According to this law / - Gin,, (24) where u is the steady velocity with which a particle of radius r moves through a medium of viscosity 17 when a force / is applied. For a par- ticular ion, r may be regarded as constant, and since the conductance is proportional to the speed of the ion under the influence of a definite applied potential (see p. 58), it follows that according to Stokes's law X 7?o should be constant, as found experimentally. Another suggestion that has been made to explain this fact is that the ion in solution is so completely surrounded by solvent molecules which move with it, that is to say, it is so extensively "solvated," that its motion through the medium is virtually the same as the movement of solvent molecules past one another in viscous flow of the solvent. It is not certain, however, that either of these conclusions can be legitimately drawn from the results. Since the activation energies for ionic mobility and viscous flow are approximately equal, it is reasonable to suppose that the rate-determining stage in the movement of an ion under the influence of an applied electric field and that involved in the viscous flow of the medium are the same. It has been suggested that in the latter process the slow stage is the jump of a solvent molecule from one equilibrium position to another, and this must also be rate- determining for ionic conductance. It appears, therefore, that when an electric field is applied to a solution containing ions, the latter can move forward only if a solvent molecule standing in its path moves in the opposite direction. The actual rate of movement of an ion will depend to a great extent on its effective size in the given solvent, but the tem- perature coefficient should be determined almost entirely by the activa- tion energy for viscous flow. Influence of Solvent on Ion Conductance. In the course of his in- vestigation of the conductance of tetraethylammonium iodide in various solvents, Walden (1906) noted that the product of the equivalent con- ductance at infinite dilution and the viscosity of the solvent was approxi- 64 ELECTROLYTIC CONDUCTANCE mately constant and independent of the nature of the latter; 21 this conclusion, known as Walden's rule, may be expressed as constant, (25) for a given electrolyte in any solvent. The values of AOTJO for the afore- mentioned salt, obtained by Walden and others, in a variety of media are given in Table XVII; the viscosities are in poises, i.e., dynes per sq. cm. TABLE XVII. VALUES OF AQI?O FOR TETRAETHYLAMMONIUM IODIDE IN VARIOUS SOLVENTS Solvent CH 3 OH CHsCOCH, CH,CN 0.63 0.66 0.64 C 2 H 4 C1 2 0.60 CH 3 N0 2 0.69 C.H,N0 2 0.67 C 6 H 6 OH 0.63 The results were generally obtained at 25, but since X *?o is approximately independent of temperature, as seen above, it is evident that Aoijo will also not vary appreciably. If Walden's rule holds for other electrolytes, it follows, since A is the sum of the conductances of the constituent ions, that Xoijo should be approximately constant for a given ion in all solvents. The extent to which this is true may be seen from the conductance-viscosity products for a number of ions collected in Table XVIII; the data for hydrogen TABLE XVIII. ION CONDUCTANCE-VISCOSITY PRODUCTS IN VARIOUS SOLVENTS AT 25 Solvent Na+ K+ Ag+ N(CiHi)/ I- cio 4 - Picrate H S 0.460 0.670 0.563 0.295 0.685 0.606 0.276 CH a OH 0.250 0.293 0.274 0.338 0.334 0.387 0.255 C,H 6 OH 0.204 0.235 0.195 0.310 0.290 0.340 0.292 CH,COCH 8 0.253 0.259 0.284 0.366 0.366 0.275 CH 8 CN 0.241 0.296 0.296 0.347 0.359 0.268 CH,N0 2 0.364 0.383 0.326 0.310 0.403 0.276 C.H.NO, 0.330 0.322 0.366 0.277 NH, (-33) 0.333 0.430 0.297 0.437 and hydroxyl ions are deliberately excluded from Table XVIII, for reasons which will appear later. The results show that, for solvents other than water, the conductance-viscosity product of a given ion is approximately constant, thus confirming the approximate validity of Walden's rule. If Stokes's law were obeyed, the value of Xow would be constant only if the effective radius of the ion were the same in the different media; since there are reasons for believing that most ions are solvated in solution, the dimensions of the moving unit will undoubtedly a Walden et al. t Z. physik. Chem., 107, 219 (1923); 114, 297 (1925); 123, 429 (1926); "Salts, Acids and Bases," 1929; Ulich, Fortschritte der Chemie, Physik and phys. Chem., 18, No. 10 (1926); Trans. Faraday Soc., 23, 388 (1927); Barak and Hartley, Z. phys. Chem., 165, 273 (1933); Coates and Taylor, /. Chem. Soc., 1245, 1495 (1936); see also Longsworth and Maclnnes, J. Phys. Chem., 43, 239 (1939). ABNORMAL ION CONDUCTANCES 65 vary to some extent and exact constancy of the conductance-viscosity product is not to be expected. It should be pointed out, also, that the deduction of Stokes's law is based on the assumption of a spherical particle moving in a continuous medium, and this condition can be approximated only if the moving particle is large in comparison with the molecules of the medium. It is of interest to note in this connection that for large ions, such as the tetraethylammonium and picrate ions, the X O T;O values are much more nearly constant than is the case with other ions; further, the behavior of such ions in water is not exceptional. Stokes's law is presumably applicable to these large ions, and since they are probably solvated to a small extent only, they will have the same size in all solvents ; the constancy of the conductance-viscosity product is thus to be expected. For small ions the value of X O T?O will depend to some extent on the fundamental properties of the solvent, as well as on the effective size of the ion: for such ions, too, Stokes's law probably does not hold, and so exact constancy of the conductance-viscosity product is not to be expected. An interesting test of the validity of the Walden rule is provided by the conductance measurements, made by LaMer and his collaborators, of various salts in a series of mixtures of light water (H 2 0) and heavy water (D 2 0). The results indicate that, although the rule holds approximately, it is by no means exact. 22 Although no actual tabulation has been made here of the ion con- ductances of various ions in different solvents, it may be pointed out that these values are implicit in Table XVIII; knowing the viscosity of the solvent, the ion conductance at infinite dilution can be calculated. Abnormal Ion Conductances. An inspection of the conductance- viscosity products for the hydrogen ion recorded in Table XIX imme- TABLB XIX. CONDUCTANCE-VISCOSITY PRODUCT OF THE HYDROGEN ION Solvent H 2 CH,OH C,HOH CH,COCH 3 CH 3 NO 2 CH 6 NO, NH, Xoi?o 3.14 0.774 0.641 0.277 0.395 0.401 0.359 diately reveals the fact that the values in the hydroxylic solvents, and particularly in water, are abnormally high. It might appear, at first sight, that the high conductance-viscosity product of the hydrogen ion in water could be explained by its small size. In view of the high free energy of hydration of the proton (cf. p. 308), however, in aqueous solu- tion the reaction H+ + H 2 O * H 3 O+, where H+ represents a proton or "bare" hydrogen ion, must go to virtual completion. The hydrogen ion in water cannot, therefore, consist of a LaMer et a/., J. Chem. Phys., 3, 406 (1935); 9, 265 (1941); J. Am. Chcm. Soc., 58, 1642 (1936); 59, 2425 (1937); see also, Longsworth and Maclnnes, ibid., 59, 1666 (1937). 66 ELECTROLYTIC CONDUCTANCE bare ion, but must be combined with at least one molecule of water. The hydrogen ion in water is thus probably to be represented by H 8 O+, and its effective size and conducting power should then be approximately the same as that of the sodium ion; it is, however, actually many times greater, as the figures in Table XIX show. It is of interest to note that in acetone, nitromethane, nitrobenzene, liquid ammonia, and probably in other non-hydroxylic solvents, the conductance-viscosity product, and hence the conductance, of the hydrogen ion, which is undoubtedly sol- vated, is almost the same as that of the sodium ion. It is doubtful, therefore, if the high conductance of the hydrogen ion in hydroxylic solvents can be explained merely by its size. The suggestion has been frequently made that the high conductance is due to a type of Grotthuss conduction (p. 7), and this view has been developed by a number of workers in recent years. 23 It is supposed, as already indicated, that the hydrogen ion in water is H 8 0+ with three hydrogen atoms attached to the central oxygen atom. When a potential gradient is applied to an aqueous solution containing hydrogen ions, the latter travel to some extent by the same mechanism as do other ions, but there is in addition another mechanism which permits of a more rapid ionic movement. This second process is believed to involve the transfer of a proton (H+) from a H 3 O+ ion to an adjacent water molecule; thus H H H H -> I + I H O H Q H. The resulting H 3 f ion can now transfer a proton to another water molecule, and in this way the positive charge will be transferred a con- siderable distance in a short time. It has been calculated from the known structure of water that the proton has to jump a distance of 0.86 X 10~ 8 cm. from a HaO" 1 " ion to a water molecule, but as a result the positive charge is effectively transferred through 3.1 X 10~ 8 cm. The electrical conductance will thus be much greater than that due solely to the normal mechanism. It will be observed that after the proton has passed from the HaO" 1 " ion to the water molecule, the resulting water molecule, i.e., the one shown on the right-hand side, is oriented in a different manner from that to which the proton was transferred, i.e., the one on the left- hand side. If the process of proton jumping is to continue, each water molecule must rotate after the proton has passed on, so that it may be ready to receive another proton coming from the same direction. The combination of proton transfer and rotation of the water molecule, which has some features in common with the Grotthuss mechanism for conduc- Hiickel, Z. Ekktrochem., 34, 546 (1928); Bernal and Fowler, /. Chem. P%., 1, 515 (1933); Wannier, Ann. Physik, 24, 545, 569 (1935); Steam and Eyring, J. Chem. Phys., 5, 113 (1937); see also, Glasstone, Laidler and Eyring, "The Theory of Rate Processes," 1941, Chap. X. Jit/ H H ABNORMAL CONDUCTANCES OF HYDROXYL AND OTHER IONS 67 tion, is sufficient to account for the high conductance of the hydrogen ion in aqueous solution. The abnormal conductance of the hydrogen ion in methyl and ethyl alcohols, which is somewhat less than in water, can also be accounted for by a proton transfer analogous to that suggested for water; thus, if the hydrogen ion in an alcohol ROH is represented by ROHJ, the process is R R R R + 1 - I + I H H H O H, e e followed by rotation of the alcohol molecule. To account for the de- pendence of abnormal conductance on the nature of R, it must be sup- posed that the transfer of a proton from one alcohol molecule to another involves the passage over an energy barrier whose height increases as R is changed from hydrogen to methyl to ethyl. The Grotthuss type of conduction, therefore, diminishes in this order. It is probable that the effect decreases steadily with increasing chain length of the alcohol. Abnormal Conductances of Hydroxyl and Other Ions. The con- ductance of the hydroxyl ion in water is less than that of the hydrogen ion; it is nevertheless three times as great as that of most other anions (cf. Table XIII). It is probable that the abnormal conductance is here also due to the transfer of a proton, in this case from a water molecule to a hydroxyl ion, thus H H H H I + I -> I +1 O H O O H O, followed by rotation of the resulting water molecule. If this is the case, it might be expected that the anion RO~ should possess abnormal con- ductance in the corresponding alcohol ROH; such abnormalities, if they exist at all, are very small, for the conductances of the CPI 3 O~ and C 2 HsO~ ions in methyl and ethyl alcohol, respectively, are almost the same as that of the chloride ion which exhibits normal conductance only. The energy barriers involved in the abnormal mobility process must therefore be considerably higher than for water. These results emphasize the fact that ions produced by self-ionization of the solvent, e.g., H 3 0+ and OH~ in water, ROHt and R0~ in alcohols, and NHi" and NHJ" in liquid ammonia, do not of necessity possess ab- normal conductance, although they frequently do so. It is seen from Table XIX that the conductance of the hydrogen ion in liquid ammonia, i.e., NHi", is normal; the same is true for the NHjf ion. The anilinium and pyridinium ions also have normal conductances in the corresponding solvents. The conductance of the HSOr ion in sulfuric acid as solvent is, however, abnormally high; it is probable that a Grotthuss type of 68 ELECTROLYTIC CONDUCTANCE conduction, involving proton transfer, viz., HSOr + H 2 S0 4 = H 2 SO 4 + HSOr, is responsible for the abnormal conductance. 24 Influence of Traces of Water. The change in the equivalent con- ductance of a strong electrolyte, other than an acid, in a non-aqueous solvent resulting from the addition of small amounts of water, generally corresponds to the alteration in the viscosity. With strong acids, how- ever, there is an initial decrease of conductance in an alcoholic solvent which is much greater than is to be expected from the change in vis- cosity; this is subsequently followed by an increase towards the value in water. When acetone is the solvent, however, the conductance in the presence of water runs parallel with the viscosity of the medium. It should be noted that the abnormal behavior is observed in solvents in which the hydrogen ion manifests the Grotthuss type of conduction. The hydrogen ion in alcoholic solution is ROHt and the addition of water results in the occurrence of the reversible reaction ROUt + H 2 ^ ROH + H 3 0+. The equilibrium of this system lies well to the right, and so a large pro- portion of the ROH2" ions will be converted into H 3 0+ ions. Although the former possess abnormal conductance in the alcohol solution, the latter do not, since the proton must pass from H 3 O+ to ROH, and the position of the equilibrium referred to above shows that this process must be slow. The result of the addition of small quantities of water to an alcoholic solution of an acid is to replace an ion capable of abnormal conduction by one which is able to conduct in a normal manner only; the equivalent conductance of the system must consequently decrease markedly. As the amount of water present is increased it will become increasingly possible for the proton to pass from H 3 0+ to a molecule of water, and so there is some abnormal contribution to the conductance; the conductance thus eventually increases towards the usual value for the acid in pure water, which is higher than that in the alcohol. From the initial decrease in conductance accompanying the addition of small amounts of water to a solution of hydrochloric acid in ethyl alcohol, it is possible to evaluate the conductance of the H 3 O+ ion in the alcohol. The value has been found to be 16.8 ohms" 1 cm. 2 at 25, which may be compared with 18.7 ohms"" 1 cm. 2 for the sodium ion in the same solvent. It is evident, therefore, that the H 3 O+ ion poasesses only normal conductance in ethyl alcohol. 26 Determination of Solubilities of Sparingly Soluble Electrolytes. If a slightly soluble electrolyte dissociates in a simple manner, it is possible to calculate the saturation solubility from conductance measure- * Hammett and Lowenheim, J. Am. Chem. Soc., 56, 2620 (1934). Goldschmidt, Z. phyrik. Chem., 89, 129 (1914). DETERMINATION OF SOLUBILITIES 69 ments. If s is the solubility, in equivalents per liter, of a given substance and K is the specific conductance of the saturated solution, then the equivalent conductance of the solution is given by A = 1000 - (26) In general, the solution will be sufficiently dilute for the equivalent con- ductance to be little different from the value at infinite dilution: the latter can be obtained, as already seen, from the ion conductances of the constituent ions. It follows, therefore, since A is known and K for the saturated solution can be determined experimentally, that it is possible to evaluate the solubility s by means of equation (26). From Kohlrausch's measurements on the conductance of saturated solutions of pure silver chloride the specific conductance at 25 may be estimated as 3.41 X 10~ 6 ohm" 1 cm." 1 ; the specific conductance of the water used was 1.60 X 10~ 6 ohm" 1 cm." 1 , and so that due to the salt may be obtained by subtraction as 1.81 X 10~ 6 ohm" 1 cm." 1 This is the value of K to be employed in equation (26). From Table XIII the equivalent conductance of silver chloride at infinite dilution is 138.3 ohms" 1 cm. 2 at 25, and so if this is assumed to be the equivalent con- ductance in the saturated solution of the salt, it follows from equation (26) that 100 X L81 X 10" 6 = 1.31 X 10~ 6 equiv. per liter at 25. By means of this first approximation for the concentration of the satu- rated solution of silver chloride, it is possible to make a more exact estimate of the actual equivalent conductance by means of the Onsager equation (p. 89); a more precise value of the solubility may then be determined. In the particular case of silver chloride, however, the differ- ence is probably within the limits of the experimental error. It should be realized that the method described actually gives the ionic concentration in the saturated solution, and it is only when dis- sociation is virtually complete that the result is identical with the solu- bility. This fact is brought out by the data for thallous chloride: the solubility at 18 calculated from Kohlrausch's conductance measurements is 1.28 X 10~ 2 equiv. per liter, but the value obtained by direct solubility measurement is 1.32 X 10~ 2 equiv. per liter. The discrepancy, which is not very large in this instance, is probably to be ascribed to incomplete dissociation of the salt in the saturated solution; the degree of dissocia- tion appears to be 128/132, i.e., 0.97. If the sparingly soluble salt does not undergo simple dissociation, the solubility obtained by the conductance method may be seriously in error. For example, the value found for lanthanum oxalate in water at 25 is 70 ELECTROLYTIC CONDUCTANCE 6.65 X 10~* equiv. per liter, but direct determination gives 2.22 X 10~ 6 equiv. per liter. The difference is partly due to incomplete dissociation and partly to the formation of complex ions. In other words, the lantha- num oxalate does not ionize to yield simple La + ++ and C 2 O ions, as is assumed in the conductance method for determining the solubility; in addition complex ions, containing both lanthanum and oxalate, are present to an appreciable extent in the saturated solution. It is neces- sary, therefore, to exercise caution in the interpretation of the results obtained from conductance measurements with saturated solutions of sparingly soluble electrolytes. Determination of Basicity of Acids. From an examination of the conductances of the sodium salts of a number of acids, Ostwald (1887) discovered the empirical relation Aio24 - A 32 116, (27) where Aio24 and Aa 2 are the equivalent conductances of the salt at 25 at dilutions of 1024 and 32 liters per equivalent, respectively, and 6 is the basicity of the acid. The data in Table XX are taken from the work of TABLE XX. BASICITY OF ACID AND EQUIVALENT CONDUCTANCE OF SALT Sodium salt of: A 1024 A w Difference Basicity Nicotinic acid 85.0 73.8 11.2 1 Quinolinic acid 104.9 83.4 21.5 2 1:2: 4-Pyridine tricarboxylic acid 121.0 88.8 32.2 3 1:2:3: 4-Pyridine tetracarboxylic acid 131.1 87.3 43.8 4 Pyridine pentacarboxylic acid 138 1 83.9 54.2 5 Ostwald, recalculated so as to give the equivalent conductances in ohmr 1 cm. 2 units, instead of reciprocal Siemens units; they show that the equa- tion given above is approximately true, and hence it may be employed to determine the basicity of an acid. The method fails when applied to very weak acids whose salts are considerably hydrolyzed in solution. The results quoted in Table XX are perhaps exceptionally favorable, for the agreement with equation (27) is not always as good as these figures would imply. The Ostwald rule is, nevertheless, an expression of the facts already discussed, viz., that substances of the same valence type have approximately the same conductance ratios at equivalent concentrations and that the values diminish with increasing valence of one or both ions. The rule has been extended by Bredig (1894) to include electrolytes of various types. Mode of lonization of Salts. Most ions, with the exception of hydro- gen, hydroxyl and long-chain ions, have ion conductances of about 60 ohms" 1 cm. 2 at 25, and this fact may be utilized to throw light on the mode of ionization of electrolytes. It has been found of particular value, in connection with the Werner co-ordination compounds, to determine whether a halogen atom, or other negative group, is attached in a co valent or an electrovalent manner. CONDUCTOMETRIC TITRATION 71 Since the mode of ionization of the salt is not known, it is not possible to determine the equivalent weight and hence the equivalent conductance cannot be calculated ; it is necessary, therefore, to make use of the molar conductance, as defined on p. 31. In the simple case of a series of salts all of which have one univalent ion, either the cation or anion, whereas the other ion has a valence of z, the gram molecule contains z gram equivalents; the molar conductance is thus z-times the equivalent con- ductance. If the mean equivalent conductance of all ions is taken as 60, the equivalent conductance of any salt is 120 ohms~~ l cm. 2 , and the molar conductance is 120 z ohms" 1 cm. 2 The approximate results for a number of salts of different valence types with one univalent ion at 25 are given in Table XXI. The observed molar conductances of the platinosammine TABLE XXI. APPROXIMATE MOLAR CONDUCTANCES OF SALTS OF DIFFERENT VALENCE TYPES Type Molar Conductance Uni-uni 120 ohms" 1 cm.* Uni-bi or bi-uni 240 Uni-ter or ter-uni 360 Uni-tetra or tetra-uni 480 series, at a concentration of 0.001 M, arc in general agreement with expec- tation, as the following data show: [Pt(NH 8 ) 4 ]++2Cl- [Pt(NH 3 ) 3 Cl]+Cl- 260 116 K+[Pt(NH 3 )Cl 3 ]- 2K+[PtCl 4 ] 107 267 ohms~ l cm. 2 The other member of this group, Pt(NH 3 ) 2 Cl 2 , is a non-clcctrolytc and so produces no ions in solution; the two chlorine atoms are thus held to the central platinum atom by covalent forces. Conductometric Titration: (a) Strong Acids. When a strong alkali, e.g., sodium hydroxide, is added to a solution of a strong acid, e.g., hydro- chloric acid, the reaction (H+ + Cl~) + (Na+ + OH-) = Na+ + Cl~ + H 2 O occurs, so that the highly conducting hydrogen ions initially present in the solution are replaced by sodium ions having a much lower con- ductance. In other words, the salt formed has a smaller conductance than the strong acid from which it was made. The addition of the alkali to the acid solution will thus be accompanied by a decrease of conduct- ance. When neutralization is complete the further addition of alkali results in an increase of conductance, since the hydroxyl ions are no longer used up in the chemical reaction. At the neutral point, therefore, the conductance of the system will have a minimum value, from which the equivalence-point of the reaction can be estimated. When the 72 ELECTROLYTIC CONDUCTANCE I \ Jl I specific conductance of the acid solution is plotted against the volume of alkali added, the result will be of the form of Fig. 24. If the initial solution is relatively dilute and there is no appreciable change in volume in the course of the titration, the specific conductance will be approxi- mately proportional to the concentration of unneutralized acid or free alkali present at any instant. The specific conductance during the course of the titration of an acid by an al- kali under these conditions will conse- quently be linear with the amount of alkali added. It is seen, therefore, that Fig. 24 will consist of two straight linos which intersect at the equivalence-point. If the strong acid is titrated with a weak base, e.g., an aqueous solution of ammonia, the first part of the conductance-titration curve, representing the neutralization of the acid and its replacement by a salt, will be very similar to the first part of Fig. 24, since both salts are strong electrolytes. When .the equivalence- point is passed, however, the con- ductance will remain almost con- stant since the free base is a weak electrolyte and consequently has d very small conductance compared with that of the acid or salt. The determination of the end-point of a titration by means of con- ductance measurements is known as conductometric titration. 26 For practical purposes it is not necessary to know the actual specific con- ductance of the solution; any quantity proportional to it, as explained below, is satisfactory. The conductance readings corresponding to vari- ous added amounts of titrant are plotted against the latter, as in Fig. 24. The titrant should be at least ten times as concentrated as the solution being titrated, in order to keep the volume change small; if necessary the titrated solution may be diluted in order to satisfy this condition, for the method can be applied to solutions of strong acids as dilute as 0.0001 N. Since the variation of conductance is linear, it is sufficient to obtain six or eight readings covering the range before and after the end-point, and to draw two straight lines through them, as seen in Fig. 24; the inter- section of the lines gives the required end-point. The method of con- * Kolthoff, Ind. Eng. Chem. (Anal. Ed.), 2, 225 (1930); Davies, "The Conductivity of Solutions," 1933, Chap. XIX; Glasstone, Ann. Rep. Chem. Soc., 30, 294 (1933); Britton, "Conductometric Analysis," 1934; Jander and Pfundt, Bottger's "Physi- kalische Methoden der analytischen Xtoemie," 1935, Part II. Alka'i Added FIG. 24. Conductance titration of strong acid and alkali CONDUCTOMETRIC TITEATION 73 ductometric titration is capable of considerable accuracy provided there is good temperature control and a correction is applied for the volume change during titration. It can be used with very dilute solutions, as mentioned above, but in that case it is essential that extraneous electro- lytes should be absent; in the presence of such electrolytes the change of conductance would be a very small part of the total conductance and would be difficult to measure with accuracy. (b) Weak Acids. If a moder- ately weak acid, such as acetic acid, is titrated with a strong base, e.g., sodium hydroxide, the form of the conductance-titration curve is as shown in Fig. 25, 1. The initial solution of the weak acid has a low conductance and the addition of alkali may at first result in a fur- ther decrease, in spite of the for- mation of a salt, e.g., sodium ace- tate, with a high 'conducting power. The reason for this is that the common anion, i.e., the acetate ion, represses the dissociation of the acetic acid. With further addition of alkali, however, the conductance of the highly ionized salt soon ex- ceeds that of the weak acid which it replaces, and so the specific conduc- tance of the solution increases. After the equivalence-point there is a further increase of conductance because of the excess free alkali; the curve is then parallel to the corresponding part of Fig. 24. When a weak acid is titrated with a weak base the initial portion of the conductance-titration curve is similar to that for a strong ba*se, since the salt is a strong electrolyte in spite of the weakness of the acid and base. Beyond the equivalence-point, however, there is no change in conductance because of the small contribution of the free weak base. The complete conductance-titration curve is shown in Fig. 25, II. It will be observed that the intersection is sharper than in Fig. 25, I, for titration with a strong base; it is thus possible to determine the end- point of the titration of a moderately weak acid by the conductometric method if a moderately weak, rather than a strong, base is employed. As long as there is present an excess of acid or base the extent of hy- drolysis of the salt is repressed, but in the vicinity of the equivalence- point the salt of the weak acid and weak base is extensively split up by the water; the conductance measurements do not then fall on the lines shown, but these readings can be ignored in the graphical estimation of the end-point. Base Added FIG. 25. Conductance titration of weak acid 74 ELECTROLYTIC CONDUCTANCE If the acid is very weak, e.g., phenol or boric acid, or a very dilute solution of a moderately weak acid is employed, the initial conductance is extremely small and the addition of alkali is not accompanied by any decrease of conductance, such as is shown in Fig. 25. The conductance of the solution increases from the commencement of the neutralization as the very weak acid is replaced by its salt which is a strong electrolyte. After the equivalence-point the conductance shows a further increase if a strong base is used, and so the end-point can be found in the usual manner. Owing to the extensive hydrolysis of the salt of a weak base and a very weak acid, even when excess of acid is still present, the titra- tion by a weak base cannot be employed to give a conductometric end- point. One of the valuable features of the conductance method of analysis is that it permits the analysis of a mixture of a strong and a weak acid in one titration. The type of con- ductance-titration curve using a weak base is shown in Fig. 26; the initial decrease is due to the neutral- ization of the strong acid, and this is followed by an increase as the weak acid is replaced by its salt. When the neutralization is complete there is little further change of conduct- ance due to the excess weak base. The first point of intersection gives the amount of strong acid in the mixture and the difference between the first and second is equivalent to the amount of weak acid. (c) Strong and Weak Bases. The results obtained in the titration of a base by an acid are very similar to those just described for the reverse process. When a strong base is neutralized the highly conducting hy- droxyl ion is replaced by an anion with a smaller conductance; the con- ductance of the solution then decreases as the acid is added. When the end-point is passed, however, there is an increase of conductance, just as in Fig. 24, if a strong acid is used for titration purposes, but the value remains almost constant if a weak or very weak acid is employed. With an acid of intermediate strength there will be a small increase of con- ductance beyond the equivalence-point. In any case the intersections are relatively sharp and, provided carbon dioxide from the air can be ex- cluded, the best method of titrating acids of any degree of weakness conduc- tometrically is to add the acid solution to that of a standard strong alkali. The conductometric titration of weak bases and those of intermediate strength is analogous to the titration of the corresponding acida. Simi- Baac Added FIG. 26. Conductance titration of mixture of strong and weak acid CONDUCTOMETRIC TITRATION 75 larly, a mixture of a strong and a weak base can be titrated quantita- tively by means of a weak acid; the results are similar to those depicted in Fig. 26. (d) Displacement Reactions. The titration of the salt of a weak acid, e.g., sodium acetate, by a strong acid, e.g., hydrochloric acid, in which the weak acid is displaced by the strong acid, e.g., (CHaCOr + Na+) + (H+ + Cl~) = CH 3 C0 2 H + Na+ + C1-, can be followed conductometrically. In this reaction the highly ionized sodium acetate is replaced by highly ionized sodium chloride and almost un-ionized acetic acid. Since the chloride ion has a somewhat higher conductance than does the acetate ion, the conductance of the solution increases slowly at first, in this particular case, although in other in- stances the conductance may decrease somewhat or remain almost con- stant; in general, therefore, the change in conductance is small. After the end-point is passed, however, the free strong acid produces a marked increase, and its position can be determined by the intersection of the two straight lines. The salt of a weak base and a strong acid, e.g., ammonium chloride, may be titrated by a strong base, e.g., sodium hydroxide, in an analogous manner. It is also possible to carry out conductometrically the titration of a mixture of a salt of a weak acid, e.g., sodium acetate, and weak base, e.g., ammonia, by a strong acid; the first break corresponds to the neutralization of the base and the second to the completion of the displacement reaction. Similarly, it is possible to titrate a mixture of a weak acid and the salt of a weak base by means of a strong base. (e) Precipitation Reactions. In reactions of the type (K+ + C1-) + (Ag+ + NOr) = AgCl + K+ + NOr and (Mg++ + SO) + 2(Na+ + OH-) = Mg(OH) 2 + 2Na+ + S0i~, where a precipitate is formed, one salt is replaced by an equivalent amount of another, e.g., potassium chloride by potassium nitrate, and so the conductance remains almost constant in the early stages of the titration. After the equivalence-point is passed, however, the excess of the added salt causes a sharp rise in the conductance (Fig. 27, I) ; the end-point of the reaction can thus be determined. If both products of the reaction are sparingly soluble, as for example in the titration of sulfates by barium hydroxide, viz., (Mg++ + SO") + (Ba++ + 20H-) = Mg(OH) 2 + BaS0 4 , the conductance of the solution decreases right from the commencement, but increases after the end-point because of the free barium hydroxide (Fig. 27, II). 76 ELECTROLYTIC CONDUCTANCE Precipitant Added Fio. 27. Conductance titration of precipitation reactions Precipitation reactions cannot be carried out conductometrically with such accuracy as can the other reactions considered above; this is due to slow separation of the precipitate, with consequent supersaturation of the solution, to removal of titrated solute by adsorption on the precipi- tate, and to other causes. 27 The best results have been obtained by working with dilute solutions in the presence of a relatively large amount of alcohol; the latter causes a dimi- nution of the solubility of the precipi- tate and there is also less adsorption. Conductometric Titration: Ex- perimental Methods. The titration cell may take any convenient form, the electrodes being arranged verti- cally so as to permit mixing of the liquids being titrated (see Fig. 28). The conventional Wheatstone bridge, or other simple method of measur- ing conductance, may be employed. If the form of Fig. 9 is used and the resistance R is kept constant, the specific conductance of the solution in the measuring cell can be readily shown to be proportional to dc/bd. An alternative procedure is to make the ratio arms equal, i.e., Rz = R* in Fig. 8 or bd = dc in Fig. 9; the resistance of the cell is then equal to that taken from the box # 2 in Fig. 8 or ft in Fig. 9 when the bridge is balanced. If two boxes, or other standard resistances, one for coarse and the other for fine adjustment, are used in series, it is possible to read off directly the resistance of the cell; the reciprocal of this reading is proportional to the specific conductance and is plotted in the titration- conductance curve. Since for most titration purposes it is unnecessary to have results of high precision, a certain amount of accuracy has been sacrificed to con- venience in various forms of conductometric apparatus. 28 In some cases the Wheatstone bridge arrangement is retained, but a form of visual 27 van Suchtelen and Itano, J. Am. Chem. Soc., 36, 1793 (1914); Harned, ibid., 39, 252 (1917); Freak, /. Chem. Soc., 115, 55 (1919); Lucasse and Abrahams, J. Chem. Ed., 7, 341 (1930); Kolthoff and Kameda, Ind. Eng. Chem. (Anal Ed.), 3, 129 (1931). "Treadwell and Paoloni, Helu. Chim. Acta, 8, 89 (1925); Callan and Horrobin, J. Soc. Chem. Ind., 47, 329T (1928). FIG. 28. Vessel for con- ductometric titration PROBLEMS 77 detector replaces the telephone earpiece (see p. 35). In other simplified conductance-titration procedures the alternating current is passed directly through the cell and its magnitude measured by a suitable instrument in series; if the applied voltage is constant, then, by Ohm's law, the current is proportional to the conductance of the circuit. For analytical pur- poses all that is required is the change of conductance during the course of the titration, and this is equivalent to knowing the change of current at constant voltage. The type of apparatus employed is shown in Fig. 29; the source of current is the alternating-current supply mains A.C. C FIG. 29. Conductometric titration using A.C. supply mains (A.C.), which is reduced to about 3 to 5 volts by means of the trans- former T. The secondary of this transformer forms part of the circuit containing the titration cell and also a direct current galvanometer G and a rectifier Z); the 400-ohm resistances A and B are used as shunts for the purpose of adjusting the current to a value suitable for the meas- uring instrument. The rectifier D may be a rectifying crystal, a copper- copper oxide rectifier or a suitable vacuum-tube circuit giving rectifica- tion and amplification; alternatively, D and G may be combined in the form of a commercial A.C. microammeter. The solution to be titrated is placed in the vessel C, the resistances A and B are adjusted and then the current on G is noted: the titration is now carried out and the gal- vanometer readings are plotted against the volume of titrant added. The end-point is determined, as already explained, from the point of intersection of the two parts of the titration curve. PROBLEMS 1. A conductance cell has two parallel electrodes of 1.25 sq. cm. area placed 10.50 cm. apart; when filled with a solution of an electrolyte the resistance was found to be 1995.6 ohms. Calculate the cell constant of the cell and the specific conductance of the solution. 2. Jones and Bradshaw [J. Am. Chem. Soc. t 55, 1780 (1933)] found the resistance of a conductance cell (Z 4 ) when filled with mercury at to be 0.999076 ohm when compared with a standard ohm. The cell Z 4 and another cell YI were filled with sulfuric acid, and the ratio of the resistances Fi/Z 4 was 0.107812. The resistance of a third cell N* to that of Y lt i.e., Ni/Yi, was found to be 0.136564. Evaluate the cell constant of JV*, calculating the specific resistance of mercury at from the data on page 4. (It may be mentioned that the result is 0.014 per cent too high, because of a difference in the current lines in the cell Z 4 when filled with mercury and sulfuric acid, respectively.) 78 ELECTROLYTIC CONDUCTANCE 3. A conductance cell having a constant of 2.485 cm." 1 is filled with 0.01 N potassium chloride solution at 25; the value of A for this solution is 141.3 ohms" 1 cm. 2 If the specific conductance of the water employed as solvent is 1.0 X 10~* ohm" 1 cm." 1 , what is the measured resistance of the cell containing the solution? 4. The measured resistance of a cell containing a 0.1 demal solution of potassium chloride at 25, in water having a specific conductance of 0.8 X 10~ 6 ohm" 1 cm." 1 , was found to be 3468.86 ohms. A 0.1 N solution of another salt, dissolved in the same conductance water, had a resistance of 4573.42 ohms in the same cell. Calculate the specific conductance of the given solution at 25. 5. A conductance cell containing 0.01 N potassium chloride was found to have a resistance of 2573 ohms at 25. The same cell when filled with a solution of 0.2 N acetic acid had a resistance of 5085 ohms. Calculate (a) the cell constant, (b) the specific resistances of the potassium chloride and acetic acid solutions, (c) the conductance ratio of 0.2 N acetic acid, utilizing data given in Chap. II. (The conductance of the water may be neglected.) 6. Use the data in Tables X and XIII to estimate the equivalent conduct- ance of 0.1 N sodium chloride, 0.01 N barium nitrate and 0.001 N magnesium sulfate at 25. (Compare the results with the values in Table VIII.) 7. The following values for the resistance were obtained when 100 cc. of a solution of hydrochloric acid were titrated with 1.045 N sodium hydroxide: 1.0 2.0 3.0 4.0 5.0 cc. NaOH 2564 3521 5650 8065 4831 3401 ohms Determine the concentration of the acid solution. 8. A 0.01 N solution of hydrochloric acid (A = 412.0) was placed in a cell having a constant of 10.35 cm." 1 , and titrated with a more concentrated solution of sodium hydroxide. Assuming the equivalent conductance of each electrolyte to depend only on the total ionic concentration of the solution, plot the variation of the cell conductance resulting from the addition of 25, 50, 75, 100, 125 and 150 per cent of the amount of sodium hydroxide required for complete neutralization. The equivalent conductance of the sodium chloride may be taken as 118.5 ohms" 1 cm. 2 ; the change in volume of the solution during titration may be neglected. 9. The following values were obtained by Shedlovsky [V. Am. Chem. Soc., 54, 1405 (1932)] for the equivalent conductance of potassium chloride at various concentrations at 25: 0.1 0.05 0.02 0.01 0.005 0.001 N 128.96 133.37 138.34 141.27 143.55 146.95 ohms- 1 cm. 2 Evaluate the equivalent conductance of the salt at infinite dilution by the method described on page 54; the values of B and A may be taken as 0.229 and 60.2, respectively. 10. A potential of 5.6 volts is applied to two electrodes placed 9.8 cm. apart: how far would an ammonium ion be expected to move in 1 hour in a dilute solution of an ammonium salt at 25? 11. A saturated solution of silver chloride when placed in a conductance cell whose constant is 0.1802 had a resistance of 67,953 ohms at 25. The resistance of the water used as solvent was found to be 212,180 ohms in the same cell. Calculate the solubility of the salt at 25, assuming it to be com- pletely dissociated in its saturated solution in water. CHAPTER III THE THEORY OF ELECTROLYTIC CONDUCTANCE Variation of Ionic Speeds. It has been seen (p. 58) that the equiva- lent conductance of an electrolyte depends on the number of ions, on the charge carried by each ionic species and^on their speeds. For a given solute the charge is, of course, constant, and so the variation of equiva- lent conductance with concentration means that there is either a change in the number of ions present or in their velocities, or in both. In the early development of the theory of electrolytic dissociation, Arrhenius made the tacit assumption that the ionic speeds were independent of the concentration of the solution; the change of equivalent conductance would then be due to the change in the number of ions produced from the one equivalent of electrolyte as a result of the change of concen- tration. In other words, the change in the equivalent conductance should then be attributed to the change in the degree of dissociation. All electrolytes are probably completely dissociated into ions at infinite dilution, and so, if the speeds of the ions do not vary with the concentra- tion of the solution, it is seen that the ratio of the equivalent conductance A at any concentration to that (A ) at infinite dilution, i.e., A/A , should be equal to the degree of dissociation of the electrolyte. For many years, therefore, following the original work of Arrhenius, this quantity, which is now given the non-committal name of " conductance ratio" (p. 51), was identified with the degree of dissociation. There are good reasons for believing that the speeds of the ions do vary as the concentration of the solution of electrolyte is changed, and so the departure of the conductance ratio (A/A ) from unity with in- creasing concentration cannot be due merely to a decrease in the degree of dissociation. For strong electrolytes, in which the ionic concentration is high, the mutual interaction of the oppositely charged ions results in a considerable decrease in the velocities of the ions as the concentration of the solution is increased ; the fraction A/A under these conditions bears no relation to the degree of dissociation. In solutions of weak electro- lytes the number of ions in unit volume is relatively small, and hence so also is the interionic action which reduces the ionic speeds. The latter, consequently, do not change greatly with concentration, and the con- ductance ratio gives a reasonably good value of the degree of dissociation ; some correction should, however, be made for the influence of interionic forces, as will be seen later. The Degree of Dissociation. An expression for the degree of dis- sociation which will be found useful at a later stage is based on a con- 79 80 THE THEORY OF ELECTROLYTIC CONDUCTANCE sidcration of the relationship between the equivalent conductance of a solution and the speeds of the ions. It was deduced on page 59 that the speed of an ion at infinite dilution under a potential gradient of 1 volt per cm. is equal to X/^, the derivation being based on the assumption that the electrolyte is completely dissociated. A consid- eration of the arguments presented shows that they are of general appli- cability to solutions of any concentration; the only change is that if the electrolyte is not completely dissociated, an allowance must be made in calculating the actual ionic concentration. If a is the true degree of dissociation and c is the total (stoichiometric) concentration of the elec- trolyte, the ionic concentration ac equiv. per liter must be employed in evaluating the quantity of electricity carried by the ions; the total con- centration c is still used, however, for calculating the equivalent con- ductance. The result of making this change is that equation (19) on page 59 becomes aF(u+ + ii_) = X+ + X_ = A, (1) where A+ and X_ are the actual ion conductances and A the equivalent conductance of the solution; u+ and M_ are the mobilities of the ions in the same solution and a is the degree of dissociation at the given concen- tration. It follows, therefore, that -& <*> For a weak electrolyte the sum u*+ + ul, for infinite dilution, does not differ greatly from u+ + w_ in the actual solution, and so the degree of dissociation is approximately equal to the conductance ratio, as stated above. If equation (1) is divided into its constituent parts, for positive and negative ions, it is seen that aFu, = X, (3) for each ion ; hence X t where X< and Ui are the equivalent conductance and mobility of the ith ion in the actual solution. Interionic Attraction: The Ionic Atmosphere. The possibility that the attractive forces between ions might have some influence on electro- lytic conductance, especially with strong electrolytes, was considered by Noyes (1904), Sutherland (1906), Bjerrum (1909), and Milner (1912) INTERIONIC ATTRACTION 81 among others, but the modern quantitative treatment o? this concept is due mainly to the work of Debye and Hiickel and its extension chiefly by Onsager and by Falkenhagen. 1 The essential postulate of the Debye- Hiickel theory is that every ion may be considered as being surrounded by an ionic atmosphere of opposite sign: this atmosphere can be regarded as arising in the following manner. Imagine a positive ion situated at the point A in Fig. 30, and consider a small volume element dv at the end of a radius vector r; the distance r is supposed to be of the order of less than about one hundred times the diameter of an ion. As a result of thermal movements of the ions, there will sometimes be an excess of positive and sometimes an excess of negative ions in the volume element dv; if a time-average is taken, however, it will be found to have, as a consequence of electrostatic attraction by the positive charge at A, a negative charge density. In other words, the probability of finding ions of opposite sign in the space sur- rounding a given ion is greater than the prob- ability of finding ions of the same sign; every ion may thus be regarded as being associated with an ionic atmosphere of opposite sign. The net charge of the atmosphere is, of course, equal in magnitude but opposite in sign to that of the central ion: the charge density will FlG 30 Tlie ionic obviously be greater in the immediate vicinity atmosphere of the latter and will fall off with increasing distance. It is possible, nevertheless, to define an effective thickness of the ionic atmosphere, as will be explained shortly. Suppose the time-average of the electrical potential in the center of the volume element dv in Fig. 30 is $] the work required to bring a posi- tive ion from infinity up to this point is then z+ef/ and to bring up a negative ion it is z_c^, where z+ and z- are the numerical values of the valences of the positive and negative ions, respectively, and c is the unit charge, i.e., the electronic charge. If the Boltzmann law of the distri- bution of particles in a field of varying potential energy is applicable to ions, the time-average numbers of positive ions (dn+) and of negative ions (dnJ) present in the volume element dv are given by dn+ = and dn, = n-.e-<-'-'+ /kT >dv, where n+ and n_ are the total numbers of positive and negative ions, 1 Debye and Hiickel, Physik. Z., 24, 185, 305 (1923); 25, 145 (1924); for reviews, see Falkenhagen and Williams, Chem. Revs., 6, 317 (1929); Williams, ibid., 8, 303 (1931); Hartley et al., Ann. Rep. Chem. Soc., 27, 326 (1930); Falkenhagen, Rev. Modern Phys., 3, 412 (1931); "Electrolytes" (Translated by Bell), 1934; Maclnnes et al, Che.m. Revs., 13, 29 (1933); Trans. Electrochem. Soc., 66, 237 (1934); J. Franklin Inst., 225, 661 (1938). 82 THE THEORY OP ELECTROLYTIC CONDUCTANCE respectively, in unit volume of the solution; k is the Boltzmann constant, i.e., the gas constant per single molecule, and T is the absolute tempera- ture. The electrical density p, i.e., the net charge per unit volume, in the given volume element is therefore given by c(z+dn+ dv itikT __ ri-Z-e*-*+ lkT ). (5) For a uni-univalent electrolyte z+ and z_ are unity, and n+ and n_ must be equal, because of electrical neutrality; hence equation (5) becomes p = n<(e-'+' kT - e ikT ), (6) where n is the number of either kind of ion in unit volume. Expanding the two exponential series, and writing x in place of aff/kT, equation (6) becomes P == i. rp ^ and if it is assumed that x, i.e., c\fs/kT, is small in comparison with unity, all terms beyond the first in the parentheses may be neglected, so that In the general case, when z+ and 2_ are not necessarily unity, if the assumption is made that zel/jkT is much less than unity in each case, the corresponding expression for the electrical density is p = Sn2?, (8) where n,- and 2,- represent the number (per unit volume) and valence of the ions of the ith kind. The summation is taken over all the types of ions present in the solution, and equation (8) is applicable irrespective of the number of different kinds of ions. In order to solve for ^ it is necessary to have another relationship between p and ^, and this may be obtained by introducing Poisson's equation, which is equivalent to assuming that Coulomb's law of force between electrostatic charges also holds good for ions. This equation in rectangular coordinates is __ dx* "" dy* "" dz* ~ D ' x, y and z are the coordinates of the point in the given volume element, and D is the dielectric constant of the medium. Converting to polar coordinates, and making use of the fact that the terms containing ty/dO and d^/d^ will be zero, since the distribution of potential about any point INTERIONIC ATTRACTION 83 in the electrolyte must be spherically symmetrical, and consequently independent of the angles 8 and 0, equation (9) becomes If the value of p given by equation (8) is inserted, this becomes 1 d = *V, (11) where the quantity K (not to be confused with specific conductance) is defined by / A-~2 \ \ (12) The differential equation (11) can be solved, and the solution has the general form Ae~* r A'e"' where A and A' are constants which can be evaluated in the following manner. Since ^ must approach zero as r increases, because the poten- tial at an infinite distance from a given point in the solution must be zero, it follows that the constant A' must be zero; equation (13) conse- quently becomes Ap~ itr #-^ (14) For a very dilute solution 2n t z? is almost zero, and hence so also is K, as may be seen from equation (12) ; the value of the potential at the point under consideration will then be A/r, according to equation (14). In such a dilute solution the potential in the neighborhood of any ion will be due to that ion alone, since other ions are too far away to have any influence: further, if the ion is regarded as being a point charge, the potential at small distances will be z l /Dr. It follows, therefore, that A Z 7 = Wr' Zt * and insertion of this result in equation (14) gives 84 THE THEORY OF ELECTROLYTIC CONDUCTANCE This equation may be written in the form .*!!_*!(!_ e -.r) * Dr Dr^ e )j and if the solution is dilute, so that * is small and 1 tr* is practically equal to *r, this becomes *-- < The first term on the right of equation (16) is the potential at a distance r due to a given point ion when there are no surrounding ions; the second term must, therefore, represent the potential arising from the ionic atmosphere. It is seen, therefore, that ^, the potential due to the ionic atmosphere, is given by for a dilute solution. Since this expression is independent of r, it may be assumed to hold when r is zero, so that the potential on the ion itself, due to its surrounding atmosphere, is given by equation (17). If the whole of the charge of the ionic atmosphere which is e t c, since it is equal in magnitude and opposite in sign to that of the central ion itself, were placed at a distance I/K from the ion the potential produced at it would be z t K/D, which is identical with the value given by equation (17). It is seen, therefore, that the effect of the ion atmosphere is equiva- lent to that of a single charge, of the same magnitude, placed at a distance I/K from the ion; the quantity I/K can thus be regarded as a measure of the thickness of the ion atmosphere in a given solution. According to the definition of K, i.e., equation (12), the thickness of the ionic atmosphere will depend on the number of ions of each kind present in unit volume and on their valence. If c t is the concentration of the ions of the ith kind expressed in moles (gram-ions) per liter, then N where N is the Avogadro number; hence, from equation (12), after making a slight rearrangement, l_(DT 1000* \ -' The values of the universal constants are as follows: k is 1.38 X 10~ 18 erg per degree, e is 4.802 X 10~ 10 e.s. unit, and N is 6.025 X 10 23 ; hence - = 2.81 X 10- 10 | x ( DT V I ^ 2 I cm. \ ZcA 2 ) TIME OF RELAXATION OF IONIC ATMOSPHERE 85 For water as solvent at 25, D is 78.6 and T is 298, so that 1 4.31 X 10-* , 1ft . - = 2Ti cm. (19) K (2c t Z|)* The thickness of the ionic atmosphere is thus seen to be of the order of iO~~ 8 cm. ; it decreases with increasing concentration and increasing va- lence of the ions present in the electrolyte, and increases with increasing dielectric constant of the solvent and with increasing temperature. The value of I/K in Angstrom units for solutions of various types of electro- lytes at concentrations of 0.1, 0.01 and 0.001 moles per liter in water at 25 are given in Table XXII. TABLE XXII. THICKNESS OF THE IONIC ATMOSPHERE IN WATER AT 25 Concentration of Solution Valence Type 0.10 M 0.01 M 0.001 M Uni-uni 9.64A 30.5A 96.4A Uni-bi and bi-uni 5.58 19.3 55.8 Bi-bi 4.82 15.3 48.2 Uni-ter and ter-uni 3.94 13.6 39.4 Time of Relaxation of Ionic Atmosphere. As long as the ionic at- mosphere is "stationary," that is to say, it is not exposed to an applied electrical field or to a shearing force tending to cause movement of the ion with respect to the solvent, it has spherical symmetry. When the ion is made to move under the influence of an external force, however, e.g., by the application of an electrical field, the symmetry of the ionic atmosphere is disturbed. If a particular kind of ion moves to the right, for example, each ion will constantly have to build up its ionic atmos- phere to the right, while the charge density to the left gradually decays. The rate at which the atmosphere to the right forms and that to the left dies away is expressed in terms of a quantity called the time of relaxation of the ionic atmosphere. The decay of the ionic atmosphere occurs exponentially, and so the return to random distribution is asymptotic in natuie; it follows, therefore, that the time required for the ionic atmos- phere to fall actually to zero is, theoretically, infinite. It has been shown, however, that, after the removal of the central ion, the surrounding atmosphere falls virtually to zero in the time 4q6, where 9 is the time of relaxation of the ionic atmosphere and q is defined by g-r^-.Mh ; (20) z is the valence, excluding the sign, and X is the ion conductance, of the respective ions. For a binary electrolyte, i.e., one yielding only two ions, Zf and Z- are equal and q is 0.5; the time for the ionic atmosphere to decay virtually to zero is then 26. When an ion of valence z is moving with a steady velocity through a solution, under the influence of an electrical force tzV y where V is the 86 THE THEORY OF ELECTROLYTIC CONDUCTANCE applied potential gradient, this force must balance the force due to re- sistance represented by Ku\ K is the resultant coefficient of frictional resistance and u is the steady velocity of the ion. It follows, therefore, that zV = Ku, If the potential gradient is 1 volt per cm., then V is 1/300 e.s. unit; further the velocity u is then given, according to equation (20), Chap. II, by X/F, where F is 96,500, and hence since c is 4.802 X 10~ l e.s. unit. It has been shown by Debye and Falkenhagen 2 that the relaxation time is related to the frictional coeffi- cients K+ and /_ of the two ions constituting a binary electrolyte by the expression 6 - scc - ^ where K has the same significance as before. Utilizing equation (21) and remembering that z+ is equal to z_ for a binary electrolyte and that A+ + A_ is equal to A, the equivalent conductance of the electrolyte, equation (22) becomes = 30.8 X 10-' - - 2 scc. (23) Introducing the value of I/K for aqueous solutions at 25, given by equa- tion (19), into equation (23), the result is A 71.3 X 10- 10 6 = - : --- sec., (24) cz\ where c is the concentration of the solution in moles per liter. For most solutions other than acids and bases, A is about 120 ohms" 1 cm. 2 at 25, so that 0.6 X 10- 10 * - sec. cz The time of relaxation of the ionic atmosphere for a binary electrolyte is thus seen to be inversely proportional to the concentration of the solution and to the valence of the ions. The approximate relaxation times for 0.1, 0.01 and 0.001 N solutions of a uni-univalent electrolyte are 0.6 X 10"*, 0.6 X 10~ 8 and 0.6 X 10~ 7 sec., respectively. * Debye and Falkenhagen, Physik. Z., 29, 121, 401 (1928); Falkenhagen and Wil- liams, Z. phyrik. Chem., 137, 399 (1928); J. Phys. Chem., 33, 1121 (1929). MECHANISM OF ELECTROLYTIC CONDUCTANCE 87 Mechanism of Electrolytic Conductance. The existence of a finite time of relaxation means that the ionic atmosphere surrounding a moving ion is not symmetrical, the charge density being greater behind than in front; since the net charge of the atmosphere is opposite to that of the central ion, there will be an excess charge of the opposite sign behind the moving ion. The asymmetry of the ionic atmosphere, due to the time of relaxation, will thus result in a retardation of the ion moving under the influence of an applied field. This influence on the speed of an ion is called the relaxation effect or asymmetry effect. Another factor which tends to retard th,e motion of an ion in solution is the tendency of the applied potential to move the ionic atmosphere, with its associated solvent molecules, in a direction opposite to that in which the central ion, with its solvent molecules (cf. p. 114), is moving. An additional retarding influence, equivalent to an increase in the viscous resistance of the solvent, is thus exerted on the moving ion; this is known as the electrophoretic effect, since it is analogous to the resistance acting against the movement of a colloidal particle in an electrical field (cf. p. 530). An attempt to calculate the magnitude of the forces opposing the motion of an ion through a solution was made by Debye and Htickel: they assumed the applicability of Stokes's law and derived the following expression for the electrophoretic force on an ion of the ith kind: Electrophoretic Force = K t V, (25) where , z l and K have their usual significance, the latter being taken as equal to the reciprocal of the thickness of the ionic atmosphere; 77 is the viscosity of the medium, /< is the coefficient of frictional resistance of the solvent opposing the motion of the ion of the ith kind, and V is the applied potential gradient.* The same result was derived in an alter- native manner by Onsager, 3 who showed that it is not necessary for Stokes's law to be strictly applicable in the immediate vicinity of an ion. In the first derivation of the relaxation force Debye and Huckel did not take into account the natural Brownian movement of the ions; allow- ance for this was made by Onsager who deduced the equation: f^Z If Relaxation Force = n * wV, (26) * The coefficient Ki given here differs somewhat from that (K) employed on page 86; the latter is defined as the resultant frictional coefficient, based on the tacit assump- tion that all the forces opposing the motion of the ion in a solution of appreciable con- centration are frictional in nature. An attempt is made here to divide these forces into the true frictional force due to the solvent, for which the coefficient Ki is employed, and the electrophoretic and relaxation forces due to the presence of other ions. At infinite dilution, K and Ki are, of course, identical. 'Onsager, Physik. Z., 27, 388 (1926); 28, 277 (1927); Trans. Faraday Soc., 23, 341 (1927). 88 THE THEORY OF ELECTROLYTIC CONDUCTANCE where D is the dielectric constant of the medium and w is defined by the value of q being given by equation (20). It is now possible to equate the forces acting on an ion of the ith kind when it is moving through a solution with a steady velocity w,; the driving force due to the applied electrical field is zF, and this is opposed by the frictional force of the solvent, equal to X t w,-, together with the electrophoretic and relaxation forces; hence ~ wV. (28) ^ On dividing through by K V V and rearranging, this becomes U v Z t 2 t K 3 Z,K W V " K~> " 6^ ~ QDkT ' JT V ' If the field strength, or potential gradient, is taken as 1 volt per cm., i.e., V is 1/300, then J E !?i_ J^.\ KDkT ' KJ 1 300 A t 300 V GTnf ^ GDkT At infinite dilution K Ls zero, and so under these conditions this equa- tion becomes "* 300/v, and since Fifi Ls equal to X?, it follows that &->' w Further, according to equation (3), u l is equal to \ l /aF, where a is the degree of dissociation; and if this result and that of equation (30) are introduced into (29) the latter becomes X t X? ex / z t , tz l For simplicity, Me assumption is now made that the electrolyte is com- pletely dissociated, that is to say, a is assumed to be unity; this, as will be evident shortly, is true for solutions of strong electrolytes at quite appre- ciable concentrations. Equation (31) can then be put in the form MECHANISM OF ELECTROLYTIC CONDUCTANCE 89 making use of equation (30) to replace Z i /K l by 300\t/F. Introducing the expression for K given by equation (12), and utilizing the standard values of 6, k and N (p. 84), equation (32) becomes , [29.15z< 9.90X10* = x ~ ~ "- 1 ^ w J The quantities c f and c_ represent the concentrations of the ions in moles per liter; these may be replaced by the corresponding concentrations c in equivalents per liter, where c, which is the same for both ions, is equal to c t 2 t ; hence X, = X? - + - X " w V^TPT). (34) The equivalent conductance of an electrolyte is equal to the sum of the conductances of the constituent ions, and so it follows from equation (34) that 20.15(2, + O 9.90 X ----- - --- . . A = Ao - ----- j- --- + , Aow> Vc(z f + z_). (35) X 10* 1 j, } Aow> J In the simple case of a uni-univalent electrolyte, z+ and z_ are unity, and w is 2 A/2; equation (35) then reduces to I" 82.4 , 8.20 X 10 5 "l r A = Ao - [ pfft + -^r)T- A J VC ' (36) the concentration c, in equivalents, being replaced by c, in moles, since both are now identical. This equation and equations (33), (34) and (35) represent forms of the Debye-Hiickel-Onsager conductance equation; these relationships, based on the assumption that dissociation of the electrolyte is complete, attempt to account for the falling off of the equivalent conductance at appreciable concentrations in terms of a de- crease in ionic velocity resulting from interionic forces. The decrease of conductance due to those forces is represented by the quantities in the square brackets; the first term in the brackets gives the effect due to the olectrophoretic force and the second term represents the influence of the relaxation, or asymmetry, force. It will be apparent from equation (35) that, for a givon solvent at a definite temperature, the magnitude of the interionic forces increases, as is to be anticipated, with increasing valence of the ions and with increasing concentration of the electrolyte. Before proceeding with a description of the experiments that have been made to test the validity of the Onsager equation, attention may be called to the concentration term c (or c) which appears in the equa- tions (33) to (36). This quantity arises from the expression for K [equation (12)], and in the latter it represents strictly the actual ionic concentration. As long as dissociation is complete, as has been assumed above, this is equal to the stoichiometric concentration, but when cases 90 THE THEORY OF ELECTROLYTIC CONDUCTANCE of incomplete dissociation are considered it must be remembered that the actual ionic concentration is c, and this should be employed in the Onsager equation. Validity of the Debye-Huckel-Onsager Equation. For a uni-univa- lent electrolyte, the Onsager equation (36), assuming complete dissocia- tion, may be written in the form A = Ao - (A +Ao)Vc, (37) where A and B are constants dependent only on the nature of the solvent and the temperature; thus 82.4 A and 8.20 X 10 s B = (D2 1 ) 1 The values of A and B for a number of common solvents at 25 are given in Table XXIII. TABLE XXIII. VALUES OF THE ONSAGER CONSTANTS FOR UNI-UNIVALENT ELECTROLYTES AT 25 Solvent D 17 X 10* A B Water 78.5 8.95 60.20 0.229 Methyl alcohol 31.5 5.45 156.1 0.923 Ethyl alcohol 24.3 10.8 89.7 1.33 Acetone 21.2 3.16 32.8 1.63 Acetonitrile 36.7 3.44 22.9 0.716 Nitromethane 37.0 6.27 125.1 0.708 Nitrobenzene 34.8 18.3 44.2 0.776 (a) Aqueous Solutions. In testing the validity of equation (37), it is not sufficient to show that the equivalent conductance is a linear func- tion of the square-root of the concentration, as is generally found to be the case (cf. p. 54); the important point is that the slope of the line must be numerically equal to A + #A , where A and B have the values given in Table XXIII. It must be realized, further, that the Onsager equation is to be regarded as a limiting expression applicable to very dilute solutions only; the reason for this is that the identification of the ionic atmosphere with !/*, where K is defined by equation (12), involves simplifications resulting from the assumption of point charges and dilute solutions. It is necessary, therefore, to have reliable data of conduct- ances for solutions of low concentration in order that the accuracy of the Onsager equation may be tested. Such data have become available in recent years, particularly for aqueous solutions of a few uni-univalent electrolytes, e.g., hydrochloric acid, sodium and potassium chlorides and silver nitrate. The experimental results for these solutions at 25 are indicated by the points in Fig. 31, in which the observed equivalent VALIDITY OF THE DEBYE-HUCKEL-ONSAGER EQUATION 91 conductances are plotted against the square-roots of the corresponding concentrations. 4 The theoretical slopes of the straight lines to be ex- pected from the Onsager equation, calculated from the values of A and B in Table XXIII in conjunction with an estimated equivalent conductance NaCl 0.02 0.04 0.06 ^Concentration Fia. 31. Test of the Onsager equation at infinite dilution, are shown by the dotted lines. It is evident from Fig. 31 that for aqueous solutions of the uni-univalent electrolytes for which data are available, the Onsager equation is very closely obeyed at concentrations up to about 2 X 10~ 3 equiv. per liter. For electrolytes of unsymmetrical valence types, i.e., z. and z_ are different, the verification of the Debye-Hiickel-Onsager equation is more difficult since the evaluation of the factor w in equation (35) requires a knowledge of the mobilities of the individual ions at infinite dilution; for this purpose it is necessary to know the transference numbers of the 4 Shedlovsky, J. Am. Chem. Soc*, 54, 1411 (1032); Shedlovsky, Brown and Maclnncs, Trans. Ekttrockem. Soc., 66, 165 (1934); Krieger and Kilpatrick, J. Am. Chem. Soc., 59, 1878 (1937). 92 THE THEORY OP ELECTROLYTIC CONDUCTANCE Ions constituting the electrolyte (see Chap. IV). The requisite data for dilute aqueous solutions at 25 are available for calcium and lanthanum chlorides, i.e., CaCl 2 and LaCl 3 , and in both instances the results are in close agreement with the requirements of the theoretical equation at concentrations up to 4 X 10~ 6 equiv. per liter. 5 It is apparent that the higher the valence type of the electrolyte the lower is the limit of con- centration at which the Onsager equation is applicable. Less accurate measurements of the conductances of aqueous solutions of various electrolytes have been made, and in general the results bear out the validity of the Onsager equation. 6 A number of values of the experimental slopes arc compared in Table XXIV with those calculated TABLE XXIV. COMPARISON OF OBSERVED AND CALCULATED ONSAGER SLOPES IN AQUEOUS SOLUTIONS AT 25 Electrolyte Observed Slope Calculated Slope LiCl 81.1 72.7 NnNO 3 82.4 74.3 KBr 87.9 S0.2 KCNS 76.5 77.8 CsCl 76.0 80.5 MgCl 2 144.1 145.6 Ba(NO 3 ) 2 160.7 150.5 K 2 SO 4 140.3 159.5 theoretically; the agreement is seen to be fairly good, but it may be even better than would at first appear, owing to the lack of data in sufficiently dilute solutions. It is of interest to record in this connection that the experimental slope of the A versus Vc curve for silver nitrate was given at one time as 88.2, compared with the calculated value 76.5 at 18; more recent v, T ork on very dilute solutions has shown much better agreement than these results would imply (see Fig. 31). Further support for the Onsager theory is provided by conductance; measurements of a number of electrolytes made at and 100. At both temperatures the observed slope of the plot of A against Vc agrees with the calculated result within the limits of experimental error. The slope of the curve for potassium chloride changes from 47.3 to 313.4 within the temperature range studied. The data recorded above indicate that the Onsager equation repre- sents in a satisfactory manner the dependence on the concentration of the equivalent conductances of uni-univalent and uni-bi- (or bi-urii-) valent electrolytes. With bi-bivalent solutes, however, very marked discrep- ancies are observed; in the first place the plot of the equivalent con- 1 Jones and Bickford, /. Am. Chem. Soc., 56, 602 (1934); Shedlovsky and Brown, ibid., 56, 1066 (1934). See, Davies, "The Conductivity of Solutions," 1933, Chap. V; Hartley et a/., Ann. Rep. Chem. *S'oc., 27, 341 (1930); /. Chem. Soc., 1207 (1933); Z. physik. Chem., 165A, 272 (1933). VALIDITY OF THE DEBYK-HUCKEL-ONSAGKR EQUATION 93 ductance against the square-root of the concentration is not a straight lino, but is concave to the axis of the latter parameter (Fig. 32). Further, the slopes at appreciable concentrations are much greater than those calculated theoretically. It 144 r 128 112 IS probable that these results are to be explained by incomplete dissociation at the experimental concentrations : the shapes of the curves do in fact indicate that in sufficiently dilute solu- tions the slopes would probably be very close to the theoretical Onsager values. (b) N on- Aqueous Solutions. A number of cases of satisfactory agreement with theoretical re- quirements have been found in methyl alcohol solutions; this is particularly the case for the chlorides and thiocyanates of the alkali metals. 7 Other electro- lytes, such as nitrates, tetralkyl- ammoniurn salts and salts of higher valence types, however, exhibit appreciable deviations. These discrepancies become more marked the lower the dielectric constant of the medium, especially if the latter is non-hydroxylic in character. The conductance of potassium iodide has been determined in a number of solvents at 25 and the experimental and calculated slopes of the plots of A against Vc are quoted in Table XXV, TABLE XXV. OBSERVED AND CALCULATED OXSAGER SLOPES FOR POTASSIUM 0.02 0.04 0.06 Fm. 32. Deviation from Onsager equation Solvent Water Methyl alcohol Kthyl cyanoacetate Ethyl alcohol Benzomtrile Acetone IODIDE AT 25 D 7S.6 31.5 27.7 25.2 25.2 20.9 Onsager Slope Observed Calculated 73 260 115 209 263 1000 SO 268 63 153 142 638 together with the dielectric constant of the medium in each case. At still lower dielectric constants, and for other electrolytes, even greater discrepancies have been recorded : in many cases substances which are strong electrolytes, and hence almost completely dissociated in water, behave as weak, incompletely dissociated electrolytes in solvents of low 7 Hartley et al, Proc. Roy. Soc., 127A, 228 (1930); 132A, 427 (1931); J. Chun. Soc., 2488 (1930). 94 THE THEORY OF ELECTROLYTIC CONDUCTANCE dielectric constant. It is not surprising, therefore, to find departures from the theoretical Onsager behavior. Deviations from the Onsager Equation. Two main types of devia- tion from the Onsager equation have been observed: the first type is exhibited by a number of salts in aqueous solution which give conduct- ances that are too large at relatively high concentrations, although the values are in excellent agreement with theory in the more dilute solutions. This effect can be seen from the results plotted in Fig. 31; it is probably to be ascribed to the approximations made in the derivation of the Onsager equation which, as already explained, can only be expected to hold for point ions in dilute solution. An empirical correction, involving c'and logc, has been applied to allow for these approximations in the following manner. Solving equation (37), for a uni-univalent electro- lyte, for AO it is found that *- according to the simple Onsager theory, and after applying the correc- tions proposed by Shedlovsky, 8 this becomes Ao = - p - Cc - DC log c + Ec\ (39) 1 J5Vc where C, D and E are empirical constants. In some cases D and E are very small and equation (39) reduces to the form * which was employed on page 55 to calculate equivalent conductances at infinite dilution. Its validity is confirmed by the results depicted in Fig. 22. In general, the Shedlovsky equation (39) adequately represents the behavior of a number of electrolytes in relatively concentrated solu- tions; it reduces to the simple Onsager equation at high dilutions when c is small. It is of interest to call attention to the fact that if the term in equation (39) involving log c is small, as it often is, and can be neg- lected, this equation can be written in the form of the power series A = Ao - 4'c* + B'c - C'c* + DV - #V, (41) where A' t B', etc., are constants for the given solute and solvent. For many electrolytes the plot of the equivalent conductance against the square-root of the concentration is linear, or slightly concave to the concentration axis, but the experimental slopes are numerically greater 'Shedlovsky, J. Am. Chem. Soc., 54, 1405 (1932); Shedlovsky and Brown, ibid., 56, 1066 (1934); cf., Onsager and Fuoss, J. Phys. Chem., 36, 2689 (1932). See, how- ever, Jones and Bickford, J. Am. Chem. Soc., 56, 602 (1934). DEVIATIONS FROM THE ON8AGER EQUATION 95 than those expected theoretically; this constitutes the second type of deviation from the Onsager equation, instances of which are given in Table XXV. In these cases the conductance is less than required by the theory and the explanation offered for the discrepant behavior, as indicated above, is that dissociation of the electrolyte is incomplete : the number of ions available for carrying the current is thus less than would be expected from the stoichiometric concentration. It will be seen from the treatment on page 89 that, strictly speaking, the left-hand side of equation (32), and hence of all other forms of the Onsager equation, should include a factor I/a, where a. is the degree of dissociation of the electrolyte; further, it was noted on page 90 that the concentration term should really be ac. It follows, therefore, that for a uni-univalent electrolyte the correct form of equation (37), which makes allowance for incomplete dissociation, is A = a[A - (A + A )Vac]. (42) This equation is sometimes written as A = aA', (43) where A', defined by A' s Ao - (A + #A )V^c, (44) is the equivalent conductance of 1 equiv. of free ions at the concentration ac equiv. per liter, i.e., at the actual ionic concentration in the solution^ It is not evident from equation (42) that the plot of A against Vc will be a straight line, since a varies with the concentration; but as a is less than unity, it is clear that the observed values of the equivalent conductance will be appreciably less than is to be expected from the simple Onsager equation. The second type of deviation, which occurs particularly with salts of high valence types and in media of low dielectric constant, can thus be accounted for by incomplete dissociation of the solute. It is seen from equation (43) that the degree of dissociation a. is numerically equal to A/A', instead of to A/A as proposed by Arrhenius. It is apparent from equation (44) that for all electrolytes, and especially those which are relatively strong, A' is considerably smaller than AoJ the true degree of dissociation (A/A') is thus appreciably closer to unity than is the value assumed to be equal to the conductance ratio (A/A ). For a weak electrolyte, the degree of dissociation is in any case small, and ac will also be small; the difference between A' and A is thus not large and the degree of dissociation will be approximately equal to the con- ductance ratio. The values for the degree of dissociation obtained in this way are, however, in all circumstances too small, the difference being greater the more highly ionized the electrolyte. The fact that the type of deviation from Onsager's equation under discussion is not observed, at least up to relatively high concentrations, with many simple electrolytes, e.g., the alkali halides in both aqueous 96 THE THEORY OF ELECTROLYTIC CONDUCTANCE and methyl alcohol solutions, shows that these substances are completely or almost completely dissociated under these conditions. At appreciable concentrations the degree of dissociation probably falls off from unity, but the value of a is undoubtedly much greater than the conductance ratio at the same concentration. Significance of the Degree of Dissociation. The quantity a, referred to as the degree of dissociation, represents the fraction of the solute which is free to carry current at a given concentration. The departure of the value of a from unity may be due to two causes which are, how- ever, indistinguishable as far as conductance is concerned. Although many salts probably exist in the ionic form even in the solid state, so that they are probably to be regarded as completely or almost completely ionized at all reasonable concentrations, the ions are not necessarily free to move independently. As a result of electrostatic attraction, ions of opposite sign may form a certain proportion of ion-pairs; although any particular ion-pair has a temporary existence only, for there is a con- tinual interchange between the various ions in the solution, nevertheless, at any instant a number of ions are made unavailable in this way for the transport of current. In cases of this kind the electrolyte may be com- pletely ionized, but riot necessarily completely dissociated. At high dilu- tions, when the simple Onsager equation is obeyed, the solute is both ionized and dissociated completely. In addition to the reason for incomplete dissociation just considered, there are some cases, e.g., weak acids and many salts of the transition arid other metals, in which the electrolyte is not wholly ionized. These substances exist to some extent in the form of un-ionizcd molecules; a weak acid, such as acetic acid, provides an excellent illustration of this type of behavior. The solution contains un-ionized, covalent molecules, quite apart from the possibility of ion-pairs. With sodium chloride, and similar electrolytes, on the other hand, there are probably no actual covalent molecules of sodium chloride in solution, although there may be ion-pairs in which the ions are held together by forces of electrostatic attraction. The quantity which has been called the " degree of dissociation 7 ' rep- resents the fraction of the electrolyte present as free ions capable of carrying the current, the remainder including both un-ionized and un- dissociated portions. Neither of the latter is able to transport current under normal conditions, and so the ordinary conductance treatment is unable to differentiate between them. The experimental data show that the deviations from the Onsager equation which may be attributed to incomplete dissociation occur more readily the smaller the ions, the higher their valence and the lower the dielectric constant of the medium. This generalization, as far as ionic size is concerned, appears at first sight not to hold for the salts of the alkali metals, for the deviations from the Onsager equation become more marked as the atomic weight of the metal increases; owing to the effect DETERMINATION OF THE DEGREE OF DISSOCIATION 97 of hydration, however, the effective size of the ion in solution decreases with increasing atomic weight. It is consequently the radius of the ion as it exists in solution, i.e. together with its associated solvent molecules, and not the size of the bare ion, that determines the extent of dissociation of the salt. According to the concept of ion association, developed by Bjerrum (see p. 155), small size and high valence of the ions and a medium of low dielectric constant are just the factors that would facilitate the formation of ion-pairs. The observed results are thus in general agreement with the theory of incomplete dissociation due to the association of ions in pairs held together by electrostatic forces. The theory of Bjerrum leads to the expectation that the extent of association of an electrolyte con- sisting of small or high-valence ions in a solvent of low dielectric constant would only become inappreciable, and hence the degree of dissociation becomes equal to unity, at very high dilutions. It follows, therefore, that the simple Onsager equation could only be expected to hold at very low concentrations; under these conditions, however, the experimental results would not be sufficiently accurate to provide an adequate test of the equation. Determination of the Degree of Dissociation. The determination of the degree of dissociation involves the evaluation of the quantity A' at the given concentration, as defined by equation (44) ; as seen previously, A' is the equivalent conductance the electrolyte would have if the solute were completely dissociated at the same ionic concentration as in the experimental solution. Since the definition of A' involves a, whereas A' is required in order to calculate a, it is evident that the former quantity can be obtained only as the result of a series of approximations. Two of the methods that have been used will be described here. If Kohlrausch's law of independent ionic migration is applicable to solutions of appreciable concentration, as well as to infinite dilution, as actually appears to be the case, the equivalent conductance of an electro- lyte MA may be represented by an equation similar to the one on page 57, viz., AM A = AMCI + ANHA ANUCI, (45) where the various equivalent conductances refer to solutions at the same ionic concentration. If MCI, NaA and NaCl are strong electrolytes, they may be regarded as completely dissociated, provided the solutions are not too concentrated; the equivalent conductances in equation (45) con- sequently refer to the same stoichiometric concentration in each case. If MA is a weak or intermediate uni-univalent electrolyte, however, the value of AMA derived from equation (45) will be equivalent to AMA, the corresponding ionic concentration being ac, where a is the degree of dis- sociation of MA at the total concentration c moles per liter. 9 f Maclnnes and Shedlovsky, J. Am. Chem. Soc., 54, 1429 (1932). 98 THE THEORY OF ELECTROLYTIC CONDUCTANCE The equivalent conductances of the three strong electrolytes may be written in the form of the power series [cf. equation (41)], A = Ao - A'c* + B'c - C'c* + -, (41a) where c is the actual ionic concentration, which in these instances is identical with the stoichiometric concentration. Combining the values of AMCI, A Na A and A Na ci expressed in this form, it is possible by adding AMCI and A Nft A and subtracting A Na ci to derive an equation for AM A; thus A MA = AO ( MA) + A"(ac) + JB"(ac) - C"(ac) + , (46) the c terms being replaced by ac to give the actual ionic concentration of the electrolyte MA. Since AO<MA) is known, and A", B", C", etc., are derived from the A', B', C', etc. values for MCI, NaA and NaCl, it follows that AMA could be calculated if a. were available. An approximate esti- mate is first made by taking a as equal to A/A for MA, and in this way a preliminary value for AMA is derived from equation (46) ; a. can now be obtained more accurately as AMA/AMA, and the calculations are repeated until there is no change in AMA. The method may be illustrated with special reference to the determination of the dissociation of acetic acid. The conductances of hydrochloric acid (MCI), sodium acetate (NaA) and sodium chloride (NaCl) can be expressed in the form of equation (4 la) : thus, at 25, ACHCD = 426.16 - 156.62A^ + 169.0c (1 - 0.2273>Tc), A (C H 3 co 2 Na) = 91.00 - 80.46V^ + 90.0c (1 - 0.2273 Vc), A(Naci) = 126.45 - 88.52Vc + 95.8c (1 - 0.2273Vc), .'. A' (C H 3 co,H) = 390.7 - 148.56V^ + 163.2c (1 - 0.2273\^). At a concentration of 1.0283 X 10~ 3 equiv. per liter, for example, the observed equivalent conductance of acetic acid is 48.15 ohms~ l cm. 2 and since A is 390.7 ohms" 1 cm. 2 , the value of a, as a first approximation, is 48.15/390.7, i.e., 0.1232; inserting this result in the expression for A(CH 3 co,H)> the latter is found to be 389.05. As a second approximation, a. is now taken as 48.15/389.05, i.e., 0.1238; repetition of the calculation produces no appreciable change in the value of A', and so 0.1238 may be taken as being the correct degree of dissociation of acetic acid at the given concentration. The difference between this result and the con- ductance ratio, 0.1232, is seen to be relatively small in this instance; for stronger electrolytes, however, the discrepancy is much greater. If there are insufficient data for the equivalent conductances to be expressed analytically in the form of equation (4 la), the calculations described above can be carried out in the following manner. 10 As a first approximation the value of a is taken as equal to the conduct- "Sherrffl and Noyes, J. Am. Chem. Soc., 48, 1861 (1926); Maclnnes, ibid., 48, 2068 (1926). CONDUCTANCE RATIO AND THE ONBAGER EQUATION 99 ance ratio and from this the ionic concentration ac is estimated. By graphical interpolation from the conductance data the equivalent con- ductances of MCI, NaA and NaCl are found at this stoichiometric con- centration, which in these cases is the same as the ionic concentration, and from them a preliminary result for A'MA) is obtained. With this a more accurate value of a is derived and the calculation of A' (M A) is re- peated; this procedure is continued u.itil the latter quantity remains unchanged. The final result is utilized to derive the correct degree of dissociation. This method of calculation is, of course, identical in prin- ciple with that described previously; the only difference lies in the fact that in the one case the interpolation to give the value of A' at the ionic concentration is carried out graphically while in the other it is achieved analytically. In the above procedure for determining the degree of dissociation, the correction for the change in ionic speeds due to interionic forces is made empirically by utilizing the experimental conductance data: the necessary correction can, however, also be applied with the aid of the Onsager equation. 11 Since A/A' is equal to a, equation (44) can be written as A' = Ao - A; VAc/A', (47) where k represents A + #A and is a constant for the given solute in a particular solvent at a definite temperature. The value of AO for the electrolyte under consideration can, in general, be obtained from the ion conductances at infinite dilution or from other conductance data (see p. 54); it may, therefore, be regarded as known. As a first approxi- mation, A' in the term VAc/A' is taken as equal to A , which is equivalent to identifying the degree of dissociation with the conductance ratio, and a preliminary value for A' can be derived from equation (47) by utilizing the experimental equivalent conductance A at the concentration c. This result for A' is inserted under the square-root sign, thus introducing a better value for a, and A' is again computed by means of equation (47). The procedure is continued until there is no further change in A' and this may be taken as the correct result from which the final value of a is calculated. Conductance Ratio and the Onsager Equation. Equation (42) can be written in the form (48) which is an expression for the conductance ratio, A/A ; the values, clearly, decrease steadily with increasing concentration. For weak electrolytes, the degree of dissociation decreases with increasing temperature, since "Davies, Trans. Faraday Soc., 23, 351 (1927); "The Conductivity of Solutions/' 1933, p. 101; see also, Banks, J. Chem. Soc., 3341 (1931). 100 THE THEORY OP ELECTROLYTIC CONDUCTANCE these substances generally possess a positive heat of ionization. It is apparent, therefore, from equation (48), that the conductance ratio will also decrease as the temperature is raised. For strong electrolytes, a being virtually unity, equation (48) becomes the influence of temperature on the conductance ratio is consequently determined by the quantity in the parentheses, viz., (.4/A ) + B. In general, this quantity increases with increasing temperature. That this is the case, at least with water as the solvent, is shown by the data in Table XXVI, for potassium chloride and tetraothylammonium picrate in TABLE XXVI. INFLUENCE OF TEMPERATURE ON CONDUCTANCE RATIO Temp. Potassium Chloride Tetraethylammonium Picratc Ao r-+* A T + B AO AO 81.8 0.54 31.2 1.16 18 129.8 0.61 53.2 1.17 100 406.0 0.77 196.5 1.30 aqueous solution. It follows, therefore, that the conductance ratio for strong electrolytes should decrease with increasing temperature, as found experimentally (p. 52). It will be evident from equation (49) that the decrease should be greater the more concentrated the solution, and this also is in agreement with observation. It may be noted that the quan- tity (-A/Ac) + B is equal to (A + J5Ao)/A , in which the numerator is a measure of the decrease in equivalent conductance due to the diminution of ionic speeds by interionic forces (p. 89) : it follows, therefore, that as a general rule the interionic forces increase with increasing temperature. Introducing the expressions for A and B given on page 90, it is seen that A . P _ 82 ' 4 4. 8 ' 20 * 1Q 6 A + and since ryAo is approximately constant for a given electrolyte in different solvents (cf. p. 64), this result may be written in the form (50) where a and 6 are numerical constants. It is at once evident, therefore, that the smaller the dielectric constant of the solvent, at constant tem- perature, the greater will be the value of (A/Ao) + B, and hence the smaller the conductance ratio. The increase of ion association which accompanies the decrease of dielectric constant will also result in a de- crease of the conductance ratio. DISPERSION OF CONDUCTANCE AT HIGH FREQUENCIES 101 The discussion so far has referred particularly to uni-univalent elec- trolytes; it is evident from equation (35) that the valences of the ions are important in determining the decrease of conductance due to inter- ionic forces and hcnco they must also affect the conductance ratio. The general arguments concerning the effect of concentration, temperature and dielectric constant apply to electrolytes of all valence types; in order to investigate the effect of valence, equation (35) for a strong electrolyte may be written in the general form + z_), (51) Q where A' and B' are constants for the solvent at a definite temperature. It is clear that for a given concentration the conductance ratio decreases with increasing valence of the ions, since the factors z+ + z_ and w both increase. It was seen in Chap. II that the equivalent conductances of most electrolytes, other than acids or bases, at infinite dilution are approx- imately the same; in this event it is apparent from equation (51) that for electrolytes of a given valence type the conductance ratio will depend only on the concentration of the solution (cf. p. 52). In the foregoing discussion the Onsager equation has been used for the purpose of drawing a number of qualitative conclusions which are in agreement with experiment. The equation could also be used for quantitative purposes, but the results would be expected to be correct only in very dilute solutions. At appreciable concentrations additional terms must be included, as in the Shedlovsky equation, to represent more exactly the variation of conductance with concentration; the general arguments presented above would, however, remain unchanged. Dispersion of Conductance at High Frequencies. An important con- sequence of the existence of the ionic atmosphere, with a finite time of relaxation, is the variation of conductance with frequency at high fre- quencies, generally referred to as the dispersion of conductance or the Debye-Falkenhagen effect. If an alternating potential of high fre- quency is applied to an electrolyte, so that the time of oscillation is small in comparison with the relaxation time of the ionic atmosphere, the un- symmetrical charge distribution generally formed around an ion in motion will not have time to form completely. In fact, if the oscillation frequency is high enough, the ion will be virtually stationary and its ionic atmosphere will be symmetrical. It follows, therefore, that the retarding force due to the relaxation or assymmetry effect will thus dis- appear partially or entirely as the frequency of the oscillations of the current is increased. At sufficiently high frequencies, therefore, the con- ductance of a solution should be greater than that observed with low- frequency alternating or with direct current. The frequency at which the increase of conductance might be expected will be approximately 1/0, where 6 is the relaxation time; according to equation (24) the relaxa- 102 THE THEORY OF ELECTROLYTIC CONDUCTANCE tion time for a binary electrolyte is 71.3 X 10~ lo /czA sec., and so the limiting frequency v above which abnormal conductance is to be expected is given by _ / l.o X 10 10 oscillations per second. The corresponding wave length in centimeters is obtained by dividing the velocity of light, i.e., 3 X 10 l cm. per sec. by this frequency; the result may be divided by 100 to give the value in meters, thus 2.14 r- czA meters. For most electrolytes, other than acids and bases, in aqueous solutions A is about 120 at 25, and hence 2 X cz meters. If the electrolyte is of the uni-univalent type and has a concentration of 0.001 molar, the Debye-Falkenhagen effect should become evident with high-frequency oscillations of wave length of about 20 meters or less. The higher the valence of the ions and the more concentrated the solution the smaller the wave length, and hence the higher the frequency, of the oscillations required for the effect to become apparent. 1 10 100 1.000 meters Wave Length FIG. 33. High frequency conductance dispersion of potassium chloride The dispersion of conductance at high frequencies was predicted by Debye and Falkenhagen, 12 who developed the theory of the subject; the phenomena were subsequently observed by Sack and others. 18 The "Debye and Falkenhagen, Phyaik. Z., 29, 121, 401 (1928); Falkenhagen and Williams, Z. phyrik. Chem., 137, 399 (1928); J. Phys. Chem., 33, 1121 (1929); Falken- hagen, Physik. Z., 39, 807 (1938). " Sack et al., Physik. Z., 29, 627 (1928); 30, 576 (1929); 31, 345, 811 (1930); Brendel, ibid., 32, 327 (1931); Debye and Sack, Z. Ekktrochem., 39, 512 (1933); Arnold and Williams, /. Am. Chem. Soc., 58, 2613, 2616 (1936). CONDUCTANCE WITH HIGH POTENTIAL GRADIENTS 103 nature of the results to be expected will be evident from an examination of Figs. 33 and 34, in which the calculated ratio of the decrease of con- ductance due to the relaxation effect * at a short wave length X, i.e., ), to that at long wave length A#, i.e., at low frequency, is plotted as 10 100 LOGO meters Wave Length FIG. 34. High frequency conductance dispersion of salts at 10~ 4 mole per liter ordinate against the wave length as abscissa. The values for potassium chloride at concentrations of 10~ 2 , 10~ 3 , and IQr 4 mole per liter are plotted in Fig. 33, and those for potassium chloride, magnesium sulfate, lanthanum chloride and potassium ferrocyanide at 10~ 4 mole per liter in water at 18 are shown in Fig. 34. It is seen that, in general, the decrease of conductance caused by the relaxation or asymmetry effect decreases with decreasing wave length or increasing frequency; the actual conductance of the solution thus increases correspondingly. The effect is not noticeable, however, until a certain low wave length is reached, which, as explained above, is smaller the higher the concentration. The influence of the valence of the ions is represented by the curves in Fig. 34; the higher the valence the smaller the relative conductance change at a given high frequency. The measurements of the Debye-Falkenhagen effect are generally made with reference to potassium chloride; the results for a number of electrolytes of different valence types have been found to be in satis- factory agreement with the theoretical requirements. Increase of tem- perature and decrease of the dielectric constant of the solvent necessitates the use of shorter wave lengths for the dispersion of conductance to be observed; these results are also in accordance with expectation from theory. Conductance with High Potential Gradients. When the applied potential is of the order of 20,000 volts per cm., an ion will move at a speed of about 1 meter per sec., and so it will travel several times the thickness of the effective ionic atmosphere in the time of relaxation. * At low frequencies this quantity is equal to the second term in the brackets in equation (35), multiplied by Vc(z+ -f z_). 104 THE THEORY OF ELECTROLYTIC CONDUCTANCE As a result, the moving ion is virtually free from an oppositely charged ion atmosphere, since there is never time for it to be built up to any extent. In these circumstances both asymmetry and electrophoretic effects will be greatly diminished and at sufficiently high voltages should 0.04 0.03 AA A 0.02 0.01 100.000 200,000 Volte per cm. FIG. 35. Wien effect for potassium ferncyariide disappear. Under the latter conditions the equivalent conductance at any appreciable concentration should be greater than the value at low voltages. The increase in conductance of an electrolyte at high potential gradients was observed by Wien 14 before any theoretical interpretation had been given, and it is consequently known as the Wien effect. It is to be expected that the Wien effect will be most marked under such conditions that the influence of the intcrionic forces resulting from the existence of an ionic atmosphere is abnormally large; this would be the case for concentrated solutions of high-valence ions. The experimental results shown in Figs. 35 and 36 confirm these expectations; those in Fig. 35 are for solutions of containing potassium ferricyanide at concen- trations of 7.5, 3.7 and 1.9 X 10~ 4 mole per liter, respectively, and the curves in Fig. 36 are for electrolytes of various valence types in solutions having equal low voltage conductances. The quantity AA is the increase * of equivalent conductance resulting from the application of a potential gradient represented by the abscissa. "Wien, Ann. Physik, 83, 327 (1927); 85, 795 (1928); 1, 400 (1929); Physik. Z., 32, 545 (1931); Falkenhagen, ibid., 32, 353 (1931); Schiele, Ann. Physik, 13, 811 (1932); Debye, Z. Elektrochem., 39, 478 (1933); Mead and Fuoss, J. Am. Chem. Soc., 61, 2047, 3257, 3589 (1939); 62, 1720 (1940); for review, nee Eckstrom and Schmelzer, Chem. Revs., 24, 367 (1939). PROBLEMS 105 o.io 0.05 100.000 200.000 Volte per cm. FIG. 36. Wien effect for salts of different valence types It will be observed that the values of AA tend towards a limit at very high potentials; the relaxation and electrophoretic effects are then virtually entirely eliminated. For an incompletely dissociated elec- trolyte the measured equivalent conductance under these conditions should be A , where a is the true degree of dissociation; since AO is known, determinations of con- ductance at high voltages would seem to provide a method of ob- taining the degree of dissociation at any concentration. It has been found, however, that the Wien effect for weak acids and bases, which are known to be dissociated to a relatively small extent, is several times greater than is to be expected; the dis- crepancy increases as the voltage is raised. It is very probable that in these cases the powerful electrical fields produce a temporary dissociation into ions of the molecules of weak acid or base; this phenomenon, referred to as the dissociation field effect, invalidates the proposed method for calculating the degree of dissociation. With strong electrolytes, which are believed to be completely dissociated, the conductances observed at very high potential gradients are close to the values for infinite dilution, in agree- ment with anticipation. It may be pointed out in conclusion that the conductance phenomena with very high frequency currents and at high potential gradients pro- vide striking evidence for the theory of electrolytic conductance, based on the existence of an ionic atmosphere surrounding every ion, proposed by Debye and Hiickel and described in this chapter. Not only does the theory account qualitatively for conductance results of all types, but it is also able to predict them quantitatively provided the solutions are not too concentrated. PROBLEMS 1. Calculate the thickness of the ionic atmosphere in 0.1 N solutions of a uni-univalent electrolyte in the following solvents: nitrobenzene (D = 34.8); ethyl alcohol (D = 24.3); and ethylene dichloride (D = 10.4). 2. Utilize the results obtained in the preceding problem to calculate the relaxation times of the ionic atmospheres and the approximate minimum fre- quencies at which the Debye-Falkenhagen effect is to be expected. It may be assumed that A O T?O has a constant value of 0.6. The viscosities of the sol- vents are as follows: nitrobenzene (0.0183 poise); ethyl alcohol (0.0109); and ethylene dichloride (0.00785). 106 THE THEORY OF ELECTROLYTIC CONDUCTANCE 3. The viscosity of water at is 0.01793 poise and at 100 it is 0.00284; the corresponding dielectric constants are 87.8 and 56. Calculate the values of the Onsager constants A and B for a uni-univalent electrolyte at these temperatures. Make an approximate comparison of the slopes of the plots of A against Vc at the two temperatures for an electrolyte for which A is 100 ohms" 1 cm. 2 at 0, assuming Walden's rule to be applicable. 4. Make an approximate comparison, by means of the Onsager equation, of the conductance ratios at 25 of 0.01 N solutions of a strong uni-univalent electrolyte in water and in ethyl alcohol; it may be assumed that A i?o has the constant value of 0.6 in each case. 5. The following values were obtained by Martin and Tartar [J. Am. Chem. Soc., 59, 2672 (1937)] for the equivalent conductance of sodium lactate at various concentrations at 25: c X 10 J 0.1539 0.3472 0.6302 1.622 2.829 4.762 A 87.89 87.44 86.91 85.80 84.87 83.78 Plot the values of A against Vc and determine the slope of the line; estimate AO and compare the experimental slope with that required by the Onsager equation. 6. Calculate the limiting theoretical slope for the plot of A against Vc for lanthanum chloride (Lads) in water at 25 ; A for this salt is 145.9 ohms" 1 cm. 9 and X- for the chloride ion is 76.3 ohms" 1 cm. 2 7. Saxton and Waters [V. Am. Chem. Soc., 59, 1048 (1937)] gave the ensuing expressions for the equivalent conductances in water at 25 of hydro- chloric acid, sodium chloride and sodium a-crotonate (Naa-C.) : AHCI - 426.28 - 156.84^ + 169.7c (1 - 0.2276^) ANECI = 126.47 - 88.65Vc + 94.8c (1 - 0.2276Vc) A N a-c. - 83.30 - 78.84Vc + 97.27c (1 - 0.2276Vc). The equivalent conductances of a-crotonic acid at various concentrations were as follows: c X 10 1 A c X 10* A 0.95825 51.632 7.1422 19.861 1.7050 39.473 14.511 14.053 3.2327 29.083 22.512 11.318 4.9736 23.677 33.246 9.317 Calculate the degree of dissociation of the crotonic acid at each concentration, making due allowance for interionic attraction. Compare the values obtained with the corresponding conductance ratios. (The results of this problem are required for Problem 8 of Chap. V.) 8. Employ the data of the preceding problem to calculate the degree of dissociation of a-crotonic acid at the various concentrations using the method of Davies described on page 99. CHAPTER IV THE MIGRATION OF IONS Transference Numbers. The quantity of electricity g carried through a certain volume of an electrolytic solution by ions of the ith kind is proportional to the number in unit volume, i.e., to the concen- tration d in gram-ions or moles per liter, to the charge z carried by each ion, and to the mobility w, i.e., the velocity under unit potential gradient (cf.'p. 58); thus , (1) where k is the proportionality constant, which includes the time. The total quantity of electricity Q carried by all the ions present in the elec- trolyte is thus the sum of the q % terms for each species; that is Q = kciziui + kc&tu* + fccjZsWs + - (2) (2a) the proportionality constant being the same for all the ions. It follows, therefore, that the fraction of the total current carried by an ion of the ith kind is given by This fraction is called the transference number, or transport number, of the given ion in the particular solution and is designated by the symbol ti', the sum of the transference numbers of all the ions present in the solution is clearly equal to unity. In the simplest case of a single electro- lyte yielding two ions, designated by the suffixes + and _, the corre- sponding transference numbers are given, according to equation (3), by C+Z+U+ , C-Z-U- t. = and <_ = ; C+2+U+ + C-Z-U- C+Z+U+ + C-Z-U- The quantities c+z+ and c~Z- 9 which represent the equivalent concentra- tions of the ions, are equal, and hence for this type of electrolyte, which has been most frequently studied, * +== u + + u- and ^"ut + uJ (4) and t+ + t. = 1. 107 108 THE MIGRATION OP IONS The speed of an ion in a solution at any concentration is proportional to the conductance of the ion at that concentration (p. 80), and so the transference number may be alternatively expressed in the form t - + t+ ~ A and (5) where the values of the ion conductances X+ and X_, and the equivalent conductance A of the solution, are those at the particular concentration to which the transference numbers are applicable. These values are, of course, different from those at infinite dilution, and so it is not surprising to find, as will be seen shortly, that transference numbers vary with the 1 concentration of the solution ; they approach a limiting value, however, at infinite dilution. Three methods have been generally employed for the experimental determination of transference numbers : the first, based on the procedure originally proposed by Hittorf (1853), involves measurement of changes of concentration in the vicinity of the electrodes; in the second, known as the "moving boundary " method, the rate of motion of the boundary between two solutions under the influence of current is studied (cf. p. 116); the third method, which will be considered in Chap. VI, is based on electromotive force measurements of suitable cells. Faraday's Laws and Ionic Velocities. It may appear surprising, at first sight, that equivalent quantities of different ions are liberated at the two electrodes in a given solution, as required by Faraday's Jaws, Anode Cathode II III + 4- + + + + + + + 4- + + V FIG. 37. Migration of ions in spite of the possible difference in the speeds of the ions moving towards the respective electrodes. The situation can, however, be understood by reference to the diagram in Fig. 37; this represents an electrolytic cell in which there are an equivalent number of positive and negative ions, THE HITTORP METHOD 109 indicated by plus and minus signs. The condition of the system at the commencement of electrolysis is shown in Fig. 37, I. Suppose that the cations only are able to move under the influence of an applied potential, and that two of these ions move from left to right; the condition attained will then be as at Fig. 37, II. At each electrode there are two ions un- paired and these must be considered to be discharged; the two electrons given up by the negative ions at the anode may be imagined to travel through the external circuit and discharge the two positive ions at the cathode. It is seen, therefore, that although only the positive ions are able to move, equivalent amounts of the two ions are discharged at the respective electrodes. A condition of this kind actually arises in cer- tain solid and fused electrolytes, where all the current is carried by the cations. If while the two cations are moving in one direction, three anions are carrying electricity in the opposite direction, so that the ionic velocities are in the ratio of 2 to 3, the result will be as in Fig. 37, III. Five ions are seen to be discharged at each electrode, in spite of the difference in speeds of the two ions. There is thus no difficulty in correlating Fara- day's laws with the fact that the oppositely charged ions in a solution may have different velocities. Incidentally it will be noted that the con- clusions to be drawn from Fig. 37 are in harmony with the results derived above, e.g., equation (4); the fraction of the total current carried by each ion, i.e., its transference number, is proportional to its speed. In the condition of Fig. 37, III, the total quantity of electricity passing may be taken as five faradays, since five ions are discharged; of these five faradays, two are carried by the cations in one direction and three by the anions in the opposite direction. Attention may be called here to a matter which will receive further discussion in Chap. XIII; the ions that carry the current through the solution are not necessarily those to be discharged at the electrodes. This is assumed to be the case here, however, for the sake of simplicity. The Hittorf Method. Suppose an electric current is passed through a solution of an electrolyte which yields the ions M+ and A~; these ions are not necessarily univalent, although a single + or sign is used for the sake of simplicity of representation. The fraction of the total cur- rent carried by the cations is t+ and that carried by the anions is L.; hence when one faraday of electricity is passed through the solution, t+ faradays are carried in one direction by t+ equivalents of M+ ions and J_ faradays are carried in the other direction by J_ equivalents of A" ions. At the same time one equivalent of each ion is discharged at the appro- priate electrode. The migration of the ions and their discharge under the influence of the current bring about changes of concentration in the vicinity of the electrodes, and from these changes it is possible to calcu- late the transference numbers. Imagine the cell containing the electrolyte to be divided into three compartments by means of two hypothetical partitions; one compart- 110 THE MIGRATION OP IONS ment surrounds the cathode, another the anode, and the third is a middle compartment in which there is no resultant change of concentration. The effect of passing one faraday of electricity through the solution of the electrolyte MA can then be represented in the following manner. Cathode Compartment (I) Middle Compartment Anode Compartment (II) 1 equiv. of M f is discharged f+ equiv. of M + migrate to I 1 equiv. of A" is discharged <+ equiv. of M+ migrate in *_ equiv. of A~ migrate from I J_ equiv. of A~ migrate in I- equiv. of A~ migrate out < h equiv. of M+ migrate from 1 1 t+ equiv. of M+ migrate out /_ equiv. of A" migrate to II Net Result: Loss of 1 1+ J.equiv. of M" 1 " No change of concentration LOBS of 1 <_ =/+equiv. of A" Loss of *_ equiv. of A~ Loss of t+ equiv. of M + .*. Net loss is /-equiv. of MA .'. Net loss is t+ equiv. of MA It follows, therefore, if the discharged ions may be regarded as being completely removed from the system and the electrodes are not attacked, as is tacitly assumed in the above tabulation, that Equiv. of electrolyte lost from anode compartment t+ Equiv. of electrolyte lost from cathode compartment ~~ t~ The total decrease in amount of the electrolyte MA in both compart- ments of the experimental cell is equal to the number of equivalents deposited on each electrode; if a coulometer (p. 17) is included in the circuit, then by Faraday's laws the same number of equivalents of ma- terial, no matter what its nature, will be deposited. It follows, therefore, that Equiv. of electrolyte lost from anode compartment Equiv. deposited on each electrode of cell or in coulometer + ' and Equiv. of electrolyte lost from cathode compartment __ Equiv. deposited on each electrode of cell or in coulometer ~~ By measuring the fall in concentration of electrolyte in the vicinity of anode and cathode of an electrolytic cell, and at the same time deter- mining the amount of material deposited on the cathode of the cell or of a coulometer in the circuit, it is possible to evaluate the transference numbers of the ions present in solution. Since the sum of t+ and _ must be unity, it is not necessary to measure the concentration changes in both anode and cathode compartments, except for confirmatory purposes; similarly, if the changes in both compartments are determined it is not strictly necessary to employ a coulometer in the circuit. It is, however, more accurate to evaluate the total amount of material deposited by the current by means of a coulometer than from the concentration changes. Chemical Changes at the Electrodes. Although the discharge of a cation generally leads to the deposition of metal on the cathode and its consequent removal from the system, this is not true for anions. If the anode consists of an attackable metal which does not form an insoluble HITTORF METHOD 111 compound with the anions present in the solution, these ions are not removed on discharge but an equivalent amount of the anode material passes into solution. In these circumstances the concentration of the anode solution actually increases instead of decreasing, but allowance can be readily made for the amount of dissolved material. In the sim- plest case the anode metal is the same as that of the cations in the electro- lyte, e.g., a silver anode in silver nitrate solution; the changes in the anode compartment resulting from the passage of one faraday of elec- tricity are as follows : 1 equiv. of M + dissolves from' the electrode t- equiv. of A~ migrate in t+ equiv. of M+ migrate out Net gain is t- equiv. of MA. It is thus possible to determine the transference number of the cation from the increase in concentration of the anode compartment. An alter- native way of treating the results is to subtract from the observed gain in amount of electrolyte the number of equivalents of M 4 " dissolved from the anode; the net result is a loss of 1 t-, i.e., t+, equiv. of MA per faraday, as would have been the case if the anions had been completely removed on discharge and the anode had not dissolved. It should be noted that the general results derived are applicable even if the anode material consists of a metal M' which differs from M; the increase or decrease of concentration now refers to the total number of equivalents of MA and M'A, but the presence of the extraneous ions will affect the transference numbers of the M+ and A~ ions. When working with a solution of an alkali or alkaline-earth halide, the anode is generally made of silver coated with the same metal in a finely-divided state, and the cathode is of silver covered with silver halide. In this case the discharged halogen at the anode combines with the silver to form the insoluble silver halide, and so is effectively removed from the anode compartment. At the cathode, however, the silver halide is reduced to metallic silver and halide ions pass into solution; there is con- sequently a gain in the concentration of the cathode compartment for which allowance must be made. Hittorf Method: Experimental Procedure. In Hittorfs original de- termination of transference numbers short, wide electrolysis tubes were used in order to reduce the electrical resistance, and porous partitions were inserted to prevent mixing by diffusion and convection. These partitions are liable to affect the results and so their use has been avoided in recent work, and other precautions have been taken to minimize errors due to mixing. Many types of apparatus have been devised for the determination of transference numbers by the Hittorf method. One form, which was favored by earlier investigators and is still widely used for ordinary laboratory purposes, consists of an H-shaped tube, as shown 112 THE MIGRATION OF IONS in Fig. 38, or a tube of this form in which the limbs are separated by a U-tube. The vertical tubes, about 1.5 to 2 cm. in width and 20 to 25 cm. approximately in length, contain the anode and cathode, respectively. If the electrolyte being studied is the salt of a metal, such as silver or copper, which is capable of being deposited on the II || cathode with 100 per cent efficiency, the metal itself "^ r "^ may be used as anode and cathode. Transference numbers can be calculated from the concentration changes in one electrode compartment only; if this procedure is adopted the nature of the electrolyte and of the electrode in the other compartment is imma- terial. With certain solutions, e.g., acids, alkali hydroxides and alkali halides, there is a possibility that gases may be liberated at one or both electrodes; the mixing thus caused and the acid or alkali set free will vitiate the experiment. Cadmium electrodes have been employed to avoid the liberation of chlorine at the anode, and cathodes of mercury covered with con- centrated solutions of zinc chloride or copper nitrate have been used to prevent the evolution of hydrogen. In the latter cases the change in the concentration of the experimental electrolyte in the anode compartment only can be utilized for the calculation of the trans- ference numbers, as indicated above. For alkali halides the best electrodes are finely divided silver as anode and silver coated with silver halide by electro- lysis (p. 234) as cathode; the behavior of these electrodes has been ex- plained previously. The apparatus is filled with the experimental solution whose weight concentration is known, and the electrodes are connected in series with a copper or silver voltameter; a current of 0.01 to 0.02 ampere is then passed for two to three hours. Too long a time must not be used, other- wise the results will be vitiated by diffusion, etc., and too large a current will produce mixing by convection due to heating. If both the time and current are too small, however, the concentration changes will not be appreciable. At the conclusion of the experiment a sufficient quantity of solution, believed to contain all that has changed in concentration during the electrolysis, is run off slowly from each limb, so as to avoid mixing, and analyzed. A further portion of liquid is removed from each limb; these represent the "middle compartment" and should have the same concentration as the original solution. The amount of metal de- posited in the coulometer during the electrolysis is determined and sufficient data are now available for the calculation of the transference numbers. Since the gain or loss of electrolyte near the electrode is accompanied by changes of density and hence in the volume of the solution, the con- FIG. 38. Simple apparatus for trans- ference numbers. IMPROVED APPARATUS FOR THE HITTORF METHOD 113 ccntration changes resulting from the passage of current must be deter- mined with reference to a definite weight of solvent present at the con- clusion of the electrolysis. Thus, if analysis of x grams of the anode solution showed it to contain y grams of the electrolyte at the end of the experiment, then the latter was associated with x y grams of water. The amount of electrolyte, say z grams, associated with this same amount of water at the beginning, is calculated from the known weight composi- tion of the original solution. The decrease of electrolyte in the anode compartment, assuming due allowance has been made for the amount, if any, of anode material that has dissolved, is thus z y grams or (z ~ y)/e equivalents, where e is the equivalent weight of the experi- mental substance. If c is the number of equivalents of material de- posited in the coulomcter during the electrolysis, it follows from equation (6) that the transference number of the cation (t+) is given by - y ec (8) The transference number of the anion (t.) is of course equal to 1 t+. Improved Apparatus for the Hittorf Method. Recent work on trans- ference number determinations of alkali and alkaline-earth chlorides by the Hittorf method has been made with a form of apparatus of which the principle is illustrated by Fig. 39. l It consists of two parts, each of which contains a stopcock of the same bore as the main tubes; the anode is inserted at A and the cathode at C, the parts of the apparatus being connected by the ground joint at B. The possibility of mixing between tho anode aiid cathode solutions is obviated by introducing right-angle bends below the anode, above the cathode and in the vertical tube between the two portions of the apparatus. For the study of alkali and alkaline-earth chlorides the anode is a coiled silver wire and the cathode is covered with silver chloride. In these cases the anode solution becomes more dilute and tends to rise, while the cathode solution in- creases in concentration during the course of the electrolysis and has a tendency to sink; the consequent danger of mixing is avoided by placing the anode at a higher level than the cathode, as shown in Fig. 39. 1 Jones and Dole, J. Am. Chem. A'oc., 51, 1073 (1929); Maclnnes and Dole, ibid., 53, 1357 (1931); Jones and Bradshaw, ibid., 54, 138 (1932). (fciS, u 6 8 [l = M ) i C FIG. 39. Apparatus for application of Hittorf method 114 THE MIGRATION OF IONS When carrying out a measurement the two parts of the apparatus, with the electrodes in position and the stopcocks open, are fitted to- gether, placed in a thermostat, and filled with the experimental solution. A silver coulometer is connected in series with each electrode to insure fche absence of leakage currents. A quantity of electricity, depending in amount on the concentration of the solution, is passed through the circuit, and the stopcocks are then closed. The liquid isolated above Si is the anode solution and that below 82 is the cathode solution; these are removed and analyzed. Quantities of liquid are withdrawn from the intermediate portion between Si and S z by inserting pipettes through the openings shown; these should have the same concentration as the original electrolyte. Although the Hittorf method is simple in principle, accurate results are difficult to obtain; it is almost impossible to avoid a certain amount of mixing as the result of diffusion, convection and vibration. Further, the concentration changes are relatively small and any attempt to increase them, by prolonged electrolysis or large currents, results in an enhance- ment of the sources of error just mentioned. In recent years, therefore, the Hittorf method for the determination of transference numbers has been largely displaced by the moving boundary method, to be described later. True and Apparent Transference Numbers. The fundamental as- sumption of the Hittorf method for evaluating transference numbers from concentration changes is that the water remains stationary. There is ample evidence, however, that ions are solvated in solution and hence they carry water molecules with them in their migration through the electrolyte; this will result in concentration changes which affect the measured or " apparent " transference number. Suppose that each cation and anion has associated with it w+ and w- molecules of water, respec- tively; let T+ arid T, be the "true" transference numbers, i.e., the actual fraction of current carried by cations and anions, respectively. For the passage of one faraday of electricity the cations will carry w+T+ moles of water in one direction and the anions will transport w-T- moles in the opposite direction; there will consequently be a resultant transfer of w+T+ - w-T- = x (9) moles of water from the anode to the cathode compartment. The trans- ference number t+ is equal to the apparent number of equivalents of electrolyte leaving the anode compartment, for the passage of one fara- day, whereas T+ is the true number of equivalents; the difference between these two quantities is equal to the change of concentration resulting from the transfer to the cathode compartment of x moles of water. If the original solution contained N 9 equiv. of salt associated with N w moles of water, then the removal of x moles of water from the anode compart- ment, for the passage of one faraday, will increase the amount of salt by TRUE AND APPARENT TRANSFERENCE NUMBERS 115 (N t /N w )x equiv. The apparent transference number of the cation will thus be smaller than the true value by this amount; that is, T, = t+ + jx. (10) In exactly the same way it may be shown that the water transported by the ions will cause a decrease of concentration in the cathode compart- ment; hence the transference number will be larger * than the true value, viz., If the net amount of water (x) transported were known, it would thus be possible to evaluate the true and apparent transference numbers from the results obtained by the Hittorf method. The suggestion was made by Nernst (1900) that the value of x could be determined by adding to the electrolyte solution an indifferent "reference substance," e.g., a sugar, which did not move with the current; if there were no resultant transfer of water by the ions, the concentration of the reference substance would remain unchanged, but if there were such a transfer, there would be a change in the concentration. From this change the amount of water transported could be calculated. The earliest attempts to apply this principle did not yield definite results, but later investigators, particularly Washburn, 2 were more successful. At one time the sugar raffinose was considered to be the best reference sub- stance, since its concentration could be readily determined from the optical rotation of the solution; more recently urea has been employed as the reference material, its amount being determined by chemical methods. 3 The mean values of x obtained for approximately 1.3 N solutions of a number of halidcs at 25 are quoted in Table XXVII, together with the TABLE XXVII. TRUE AND APPARENT TRANSFERENCE NUMBERS IN 1.3 N SOLUTIONS AT 25 Electrolyte x t+ T+ HC1 0.24 0.820 0.844 LiCl 1.5 0.278 0.304 NaCl 0.76 0.366 0.383 KC1 0.60 0.482 0.495 CsCl 0.33 0.485 0.491 apparent transference numbers (t+) of the cations and the corrected values (r+) derived from equation (10). The difference between the * The terms "smaller" and "larger" are used here in the algebraic sense; they also refer to the numerical values if x is positive. * Washburn, J. Am. Chem. Soc., 31, 322 (1909); Washburn and Millard, ibid., 37, 694 (1915). Taylor et al, J. Chem. Soc., 2095 (1929); 2497 (1932); 902 (1937). 116 THE MIGRATION OP IONS Fia. 40. Determination of transport of water transference numbers t+ and T+ in the relatively concentrated solutions employed is quite appreciable; it will be apparent from equation (10) that, provided x does not change greatly with concentration, the differ- ence between true and apparent transference numbers will be much less in the more dilute solutions, that is when N 9 is small. Another procedure for determining the net amount of water trans- ported during electrolysis is to separate the anode and cathode compart- ments by means of a parchment membrane and to measure the change in volume accompanying the passage of current. This is achieved by using closed ves- sels as anode and cathode compartments and observing the movement of the liquid in a capillary tube connected with each vessel (Fig. 40). After making corrections for the volume changes at the electrodes due to chemical reactions, the net change is attributed to the transport of water by the ions. 4 The results may bo affected to some extent by electro-osmosis (see p. 521) through the mem- brane separating the compartments, especially in the more concentrated solutions, but on the whole they are in fair agreement with those given in Table XXVII. The Moving Boundary Method. The moving boundary method for measuring transference numbers involves a modification and improve- ment of the idea employed by Lodge and by Whetham (cf. p. 60) for the study of the speeds of ions. On account of its relative simplicity and the accuracy of which it is capable, the method has been used in recent years for precision measurements. 5 If it is required to determine the transference numbers of the ions constituting the electrolyte MA, e.g., potassium chloride, by the moving boundary method, it may be supposed that two other electrolytes, desig- nated by M'A and MA', e.g., lithium chloride arid potassium acetate, each having an ion in common with the experimental solute MA, are available to act as "indicators." Imagine the solution of MA to be placed between the indicator solutions so as to form sharp boundaries at a and 6, as shown in Fig. 41; tho anode is inserted in the solution of M'A and the cathode in that of MA'. In order that the boundaries 4 Remy, Z. physik. Chem., 89, 529 (1915); 118, 161 (1925); 124, 394 (1926); Trans. Faraday Soc., 33, 381 (1927); BaborovskJ et al., Kec. trav. chim., 42, 229, 553 (1923); Z. physik. Chem., 120, 129 (1927); 131, 129 (1927); 163A, 122 (1933); Trans. Electrochem. Soc., 75, 283 (1939); Hepburn, Phil. Mag., 25, 1074 (193S). 6 Maclnnes and Longsworth, Chem. Revs., 11, 171 (1932); Longsworth, J. Am. Chem. Sor., 54, 2741 (1932); 57, 1185 (1935). THE MOVING BOUNDARY METHOD 117 between the solutions may remain distinct during the passage of the current, the first requirement is that the speed of the indicator ion M' shall be less than that of M, and that the speed of A' shall be less than that of the A ions. If these condi- tions hold, as well as another to be considered shortly, the M' ions do not overtake the M ions at a, and neither do the A' ions overtake the A ions at 6; the boundaries consequently do not become blurred. In view of the -slower speeds of the indicator ions, they v M A are sometimes referred to as "following ions." Under the influence of an electric field the boundary a moves to a', while at the same time 6 moves to 6'; the dis- tances aa f and 66' depend on the speeds of the ions M and A, and since there is a uniform potential gradient through the central solution MA, these will be proportional to the ionic velocities u+ and w_. It MA follows, therefore, from equation (4) that aa and aa' + 66' u+ + u_ "*" 66' u. aa' + 66' u f + w_ = <-, a' 6' MA' T FIG. 41. Prin- ciple of the moving boundary method so that the transference numbers can be determined from observations on the movements of the bounda- ries a and 6. In the practical application of the moving boundary method one boundary only is observed, and so the necessity of finding two indicator solutions is obvi- ated ; the method of calculation is as follows. If one faraday of electricity passes through the system, t+ equiv. of the cation must pass any given point in one direction; if c equiv. per unit volume is the concentration of the solution in the vicinity of the boundary formed by the M ions, this boundary must sweep through a volume t+fc while one faraday is passing. The volume <f> swept out by the cations for the passage of Q coulombs is thus * = ?,-' (12) r C where F is one faraday, i.e., 96,500 coulombs. If the cross section of the tube in which the boundary moves is a sq. cm., and the distance through which it moves during the passage of Q coulombs is I cm., then <t> is equal to /a, and hence from equation (12) laFc Q (13) 118 THE MIGRATION OF IONS Since the number of coulombs passing can be determined, the trans- ference number of the ion may be calculated from the rate of movement of one boundary. In accurate work a correction must be applied for the change in volume occurring as a result of chemical reactions at the electrodes and because of ionic migration. If At; is the consequent increase of volume of the cathode compartment for the passage of one faraday, equation (12) becomes . , Q Q'corr. - > v r F "" Fc .' korr. = Jobs. + CAtf, (14) where t corTm is the corrected transference number and / O b a . is the value given by equation (13); the difference is clearly only of importance in concentrated solutions. The Kohlrausch Regulating Function. An essential requirement for a sharp boundary is that the cations M and M', present on the two sides of the boundary, should move with exactly the same speed under the conditions of the experiment. It can be deduced that the essential requirement for this equality of speed is given by the Kohlrausch regulating function, viz., ^ = !/' (15) where t+ and c are the transference number and equivalent concentra- tion, respectively, of the ion M in the solution of MA, and t+ and c' are the corresponding quantities for the ion M' in the solution of M'A; the solutions are those constituting the two sides of the boundary. The equivalent concentration of each electrolyte at the boundary, i.e., of MA and M'A should be proportional to the transference number of its cation. Similarly, at the boundary between the salts MA and MA', the concentrations should be proportional to the transference numbers of the respective anions. The reason for this condition may be seen in an approximate way from equation (3) : the transference number divided by the equivalent concentration of the ion, which is equal to cz, is pro- portional to the speed of the ion; hence, when //c is the same for both ions the speeds will be equal. The indicator concentration at the boundary should, theoretically, adjust itself automatically during the passage of current so as to satisfy the requirement of the Kohlrausch regulating function. Suppose the indicator were more concentrated than is necessary according to equa- tion (15) ; the potential gradient in this solution would then be lower than is required to make the ion M' travel at the same speed as M. The M' ions would thus lag behind and their concentration at the boundary would fall; the potential gradient in this region would thus increase until the velocity of the M' ions was equal to that of the leading ion. Similar EXPERIMENTAL METHODS 119 automatic adjustment would be expected if the bulk of the indicator solution were more dilute than necessary to satisfy equation (15). It would appear, therefore, that the actual concentration of the indi- cator solution employed in transference measurements is immaterial: experiments show, however, that automatic attainment of the Kohl- rausch regulating condition is not quite complete, for the transference numbers have been found to be dependent to some extent on the concentration of the bulk of the indicator solution. This is shown by the results in Fig. 42 for the observed transference number of the potassium ion in 0.1 N potas- sium chloride, with lithium chlo- ride of various concentrations as indicator solution. The concen- tration of the latter required to satisfy equation (15) is 0.064 N, and hence it appears, from the constancy of the transference number over the range of 0.055 to 0.075 N lithium chloride, that automatic adjustment occurs only when the actual concentration of the indicator solution is not greatly different from the Kohl- rausch value. The failure of the adjustment to take place is probably due to the disturbing effects of convection resulting from temperature and density gradients in the electrolyte. 6 When carrying out a transference number measurement by the moving boundary method the bulk concentration of the indicator solution is chosen so as to comply with equation (15), as far as possible, using ap- proximate transference numbers for the purpose of evaluating c'. The experiment is then repeated with a somewhat different concentration of indicator solution until a constant value for the transference number is obtained; this value is found to be independent of the applied potential and hence of the current strength. Experimental Methods. One of the difficulties experienced in per- forming transference number measurements by the moving boundary method was the establishment of sharp boundaries; recent work, chiefly by Maclnnes and his collaborators, has resulted in such improvements of technique as to make this the most accurate method for the deter- mination of transference numbers. Since the earlier types of apparatus 6 Maclnnes and Smith, /. Am. Chem. Soc., 45, 2246 (1923); Maclnnes and Longs- worth, Chem. Revs., 11, 171 (1032); Hartley and Moilliet, Proc. Roy. Soc., 140A, 141 (1833). 0.607 0.604 ! 0.601 I 0.498 0.495 0.492 0.489 0.46 0.66 0.66 0.76 0.85 0.95 Concentration of Lithium Chloride FIG. 42. Variation of transference number with concentration of indicator solution 120 THE MIGRATION OF IONS FIG. 43. Sheared boundary apparatus (Maclnnes and Brighton) have been largely superseded, these will not be described here; reference will be made to the more modern forms only. The apparatus used in tho sheared boundary method is shown diagrammatically in Fig. 43. 7 The electrode vessel A is fitted into the upper of a pair of accurately ground discs, B and C, which can be rotated with respect to each other. Into the lower disc is fixed the graduated tube D in which the boundary is to move, and this is attached by a similar pair of discs, $and F, to the other electrode vessel G. The vessel A is filled with the indicator solution and a drop is allowed to protrude below the disc B, while the exper- imental solution is placed in the vessel G and the tube D so that a drop protrudes above the top of C Y ; the discs are so arranged that tho protruding drops d and d' are accommodated in the small holes, as shown in the enlarged diagram at the right of Fig. 43. The disc B is now rotated, with the result that the electrode vessel A fits exactly over />, as shown by the dotted lines at A'] in the process the protruding drops of liquid are sheared off and a sharp bound- ary is formed. The above procedure is employed for a falling boundary, moving down the tube D under the influence of current, i.e., when the indicator solution has a lower density than the experimental solution. If the reverse is the case, a rising boundary must be used, arid thin is formed in a similar manner between the two lower discs E and F\ the indicator solution is now placed in G and the experimental solution in A and D. If the ions of a metal, such as cadmium or silver, which forms an attackable anode, are suitable as indicator cations, it is possible to use the device of the autogenic boundary. 8 No special indicator solution is required, but a block of the metal serves as the anode and the experi- mental solution is placed in a vertical tube above it. For example, with nitrate solutions a silver anode can be used, and with chloride solutions one of cadmium can be employed; the silver nitrate or cadmium chloride, respectively, that is formed as the anode dissolves acts as indicator solution. It is claimed that there is automatic adjustment of the con- centration in accordance with the Kohlrausch regulating function, and a sharp boundary is formed and maintained throughout the experiment. 7 Maclnnes and Brighton, J. Am. Chem. Soc., 47, 994 (1925). Cady and Longsworth, J. Am. Chem. Soc., 51, 1656 (1929); Longsworth, ibid., 57, 1698 (1935); J. Chem. Ed., 11, 420 (1934). EXPERIMENTAL METHODS 121 The method is capable of giving results of considerable accuracy, although its application is limited to those cases for which a suitable anode mate- rial can be found. An alternative, somewhat simple- but less accurate, procedure for measuring transference numbers by the moving boundary principle, utilizes the air-lock method of estab- lishing the boundary. 9 The appara- tus for a rising boundary is shown in Fig. 44; the graduated measuring tube A has a bore of about 7 mm., whereas E and F are fine capillaries; the top of the latter is closed by rubber tubing with two pinchcocks. The electrodes are placed in the vessels B and C. With electrode B in position, and the upper pinchcock at the top -of F closed, the apparatus is filled with the experimental solution. By closing the lower pinchcock a small column of air G is forced into the tube where F joins A 9 thus separating the solutions A and D. The solution in CDE FIG. 44. Air-lock method for estab- lishing boundary (Hartley and Donald- son) in is then emptied by suction through C and E, care being taken not to disrupt the air column G. The tube CDE is now filled with the indicator solution, the electrode is inserted in C, and the lower pinchcock at the top of F is adjusted so that the air column G is withdrawn sufficiently to permit a boundary to form between the indicator solution in CDE and the experimental solution in A. Even if the boundary is not initially sharp, it is soon sharpened by the current. In following the movement of the boundary, no matter how it is formed, use is made of the difference in the refractive indices of the indicator and experimental solutions; if the boundary is to be clearly visible, this difference should be appreciable. If the distance (I) moved in a given time and the area of cross section (a) of the tube are measured, and the equivalent concentration (c) of the experimental solution is known, it is only necessary to determine the number of coulombs (Q) passed for the transference number to be calculated by equation (13). The quantity of electricity passing during the course of a moving bound- ary experiment is generally too small to be measured accurately in a coulometer. It is the practice, therefore, to employ a current of known strength for a measured period of time; the constancy of the current can be ensured by means of automatic devices which make use of the proper- ties of vacuum tubes. Hartley and Donaldson, Trans. Faraday Soc., 33, 457 (1937). 122 THE MIGRATION OP IONS Results of Transference Number Measurements. Provided the measurements are made with great precision, the results obtained by the Hittorf and moving boundary methods agree within the limits of experimental error; this is shown by the most accurate values for various solutions of potassium chloride at 25 as recorded in Table XXVIII. TABLE XXVIII. TRANSFERENCE NUMBERS OF POTASSIUM CHLORIDE SOLUTIONS AT 25 Concentration 0.02 0.05 0.10 0.50 l.ON Hittorf method 0.489 0.489 0.490 0.490 0.487 Moving boundary method 0.490 0.490 0.490 0.490 0.488 It is probable that, on the whole, transference numbers derived from moving boundary measurements are the more reliable. It may be noted that the values obtained by the moving boundary method, like those given by the Hittorf method, are the so-called "appar- ent" transference numbers (p. 114), because the transport of water by the ions will affect the volume through which the boundary moves. It is the practice, however, to record observed transference numbers without applying any correction, since much uncertainty is attached to the deter- mination of the transport of water during the passage of current. Fur- ther, in connection with the study of certain types of voltaic cell, it is the "apparent" rather than the "true" transference number that is involved (cf. p. 202). Some of the most recent data of the transference numbers of the cations of various salts at a number of concentrations at 25, mainly obtained by the moving boundary method, are given in Table XXIX; 10 TABLE XXIX. TRANSFERENCE NUMBERS OF CATIONS IN AQUEOUS SOLUTIONS AT 25 Concn. HC1 LiCl NaCl KCl KNOa AgNOs BaCU K 2 SO 4 LaCb 0.01 N 0.8251 0.3289 0.3918 0.4902 0.5084 0.4648 0.440 0.4829 0.4625 0.02 0.8266 0.3261 0.3902 0.4901 0.5087 0.4652 0.4375 0.4848 0.4576 0.05 0.8292 0.3211 0.3876 0.4899 0.5093 0.4664 0.4317 0.4870 0.4482 0.1 0.8314 0.3168 0.3854 0.4898 0.5103 4682 0.4253 0.4890 0.4375 0.2 0.8337 0.3112 0.3821 0.4894 0.5120 0.4162 0.4910 0.4233 0.5 0.300 0.4888 0.3986 0.4909 0.3958 1.0 0.287 0.4882 0.3792 the corresponding anion transference numbers may be obtained in each case by subtracting the cation transference number from unity. Influence of Temperature on Transference Numbers. The extent of the variation of transference numbers with temperature will be evident from the data for the cations of a number of chlorides at a concentration of 0.01 N recorded in Table XXX; these figures were obtained by the Hittorf method and, although they may be less accurate than those in Table XXIX, they are consistent among themselves. The transference Longsworth, J. Am. Chem. Soc., 57, 1185 (1935); 60, 3070 (1938). TRANSFERENCE NUMBER AND CONCENTRATION 123 TABLE XXX. INFLUENCE OF TEMPERATURE ON CATION TRANSFERENCE NUMBERS IN 0.01 N SOLUTIONS Temperature HC1 NaCl KC1 BaClj 0.846 0.387 0.493 0.437 18 0.833 0.397 0.496 30 0.822 0.404 0.498 0.444 50 0.801 0.475 numbers of the ions of potassium chloride vary little with temperature, but in sodium chloride solution, and particularly in hydrochloric acid, the change is appreciable. It has been observed, at least for uni-univa- lent electrolytes, that if the transference number of an ion is greater than 0.5, e.g., the hydrogen ion, there is a decrease as the temperature is raised. It appears, therefore, in general that transference numbers measured at appreciable concentrations tend to approach 0.5 as the temperature is raised; in other words, the ions tend towards equal speeds at high temperatures. Transference Number and Concentration : The Onsager Equation. It will be observed from the results in Table XXIX that transference numbers generally vary with the concentration of the electrolyte, and the following relationship was proposed to represent this variation, viz., t = to - AVc, (16) where t and / are the transference numbers of a given ion in a solution of concentration c and that extrapolated to infinite dilution, respectively, and A is a constant. Although this equation is applicable to dilute solutions, it does not represent the behavior of barium chloride and other electrolytes at appreciable concentrations. A better expression, which holds up to relatively high concentrations, is _ _ 'l + B+c ' where B is a constant for the given electrolyte. 11 This equation may be written in the form - 1 + Bc, (18) so that the plot of l/(t + 1) against Vc should be a straight line, as has been found to be true in a number of instances. Equation (17) can also be expressed as a power series, thus t = to ~ (to + l)#c + (<o + l)# 2 c + (fc + l)#'c + , and when c is small, i.e., for dilute solutions, so that all terms beyond that involving c* can be neglected, this reduces to equation (16) since (t Q + 1)5 is a constant. "Jones and Dole, J. Am. Chem. Soc., 51, 1073 (1929); Jones et al, ibid., 54, 138 (1932); 58, 1476 (1936); Dole, J. Phys. Chem., 35, 3647 (1931). 124 THE MIGRATION OF IONS The Onsager equation for the equivalent conductance X< of an ion may be written in the form [cf. equation (34), p. 89] X. = X? - A&, (19) where X? is the ion conductance at infinite dilution and A* is a constant. Introducing the expression for the transference number given by equa- tion (5), that is ti = X t /A, where A is the equivalent conductance of the electrolyte at the experimental concentration, it follows that The value of A can be expressed in terras of A by an equation similar to (19), and then equation (20) can be written in the form + D, (21) 1 XJ\C where A, B and D are constants. This equation derived from the Debye-Hiickel-Onsager theory of conductance is of the same form as the empirical equation (17), and hence is in general agreement with the facts; the constants A, B and D, however, which are required to satisfy the experimental results differ from those required by theory. This dis- crepancy is largely due to the fact that the transference measurements were made in solutions which are too concentrated for the simple Onsager equation to be applicable. Since the Onsager equation is, strictly speaking, a limiting equation, it is more justifiable to see if the variation of transference number with concentration approaches the theoretical behavior with increasing dilu- tion. The equivalent conductance of a univalent ion can be expressed in the form of equation (37), page 90, viz., A, = X?-QA+X?)c, (22) where .A and B as used here are the familiar Onsager values (Table XXIII, p. 90); the transference number (t+) of the cation in a uni- umvalent electrolyte can then be represented by X+ X + where A and B are the same for both ions. Differentiating equation (23) with respect to Vc, and introducing the condition that c approaches aero, it is found that .~ 2A, - It follows, therefore, that the slope of the plot of the transference number EQUIVALENT CONDUCTANCES OF IONS 125 of an ion against the square-root of the concentration should attain a limiting value, equal to (2J+ 1)4/2A as infinite dilution is approached; the results in Fig. 45, in which the full lines are drawn through the experi- mental cation transference numbers in aqueous solution at 25 and the 0.316 0.10 0.20 0.30 0.40 \J Concentration FIG. 45. Transference numbers and the Onsager equation (Longsworth) dotted lines represent the limiting slopes, are seen to be in good agree- ment with the requirements of the inter-ionic attraction theory, 1 - Equivalent Conductances of Ions. Since transference numbers and equivalent conductances at various concentrations are known, it should be possible, by utilizing the expression X = $,A, to extrapolate the re- "Longsworth, J. Am. Cham. Soc., 57, 1185 (1935); see also, Hartley and Donald- son, Trans. Faraday Soc., 33, 457 (1937); Samis, ibid., 33, 469 (1937). 126 THE MIGRATION OF IJNS suits to give ion conductances at infinite dilution. Two methods of extrapolating the data are possible. In the first place, the equivalent conductances and the transference numbers may be extrapolated sepa- rately to give the respective values at infinite dilution; the product of thase quantities would then be equal to the ionic conductance at infinite dilution. The data from which the conductance of the chloride ion can be evaluated are given in Table XXXI; the mean value of the conduct- TABLE XXXI. CALCULATION OP CHLORIDE ION CONDUCTANCE AT 25 Electrolyte &r A Afcr HC1 0.1790 426.16 76.28 LiCl 0.6633 115.03 76.30 NaCl 0.6035 126.45 76.31 KC1 0.5097 149.86 76.40 ance of the chloride ion at infinite dilution at 25, derived from measure- ments on solutions of the four chlorides, is thus found to be 76.32 ohms" 1 cm. 2 The results in the last column are seen to be virtually independent of the nature of the chloride, in agreement with Kohlrausch's law of the independent migration of ions. The second method of extrapolation is to obtain the values of X t at various concentrations and to extrapolate the results to infinite dilution. The equivalent conductances of the chloride ion at several concentrations obtained from transference and conductance measurements, on the four chlorides to which the data in Table XXXI refer, are given in Table XXXII. These results can be plotted against the square-root of the TABLE XXXII. EQUIVALENT CONDUCTANCES OP CHLORIDE ION AT 25 Electrolyte 0.01 0.02 0.05 0.10 N HC1 72.06 70.62 68.16 65.98 LiCl 72.02 70.52 67.96 65.49 NaCl 72.05 70.54 67.92 65.58 KC1 72.07 70.56 68.03 65.79 concentration and extrapolated to infinite dilution, thus giving 76.3 ohms~ l cm. 2 for the ion conductance, but a more precise method is similar to that described on page 54, based on the use of the Onsager equation. The conductance of a single univalent ion, assuming complete dissocia- tion of the electrolyte, is given by equation (22), the values of A and B being known; if the experimental data for \ v at various concentrations, as given in Table XXXII, are inserted in this equation, the corresponding results for X< can be obtained. If the solutions were sufficiently dilute for the Onsager equation to be strictly applicable, the values of X? would all be the same; on account of the incomplete nature of this equation in its simple form, however, they actually increase with increasing concen- tration (cf. p. 55). By plotting the results against the concentration and extrapolating to infinite dilution, the equivalent conductance of the chloride ion in aqueous solution has been found to be 76.34 ohms~ l cm. 2 TRANSFERENCE NUMBERS IN MIXTURES 127 at 25; this is the best available datum for the conductance of the chlo- ride ion. 13 Since the ion conductance of the chloride ion is now known accu- rately, that of the hydrogen, lithium, sodium, potassium and other cations can be derived by subtraction from the equivalent conductances at infinite dilution of the corresponding chloride solutions; from these results the values for other anions, and hence for further cations, can be obtained. The data recorded in Table XIII, page 56, were calculated in this manner. It is of interest to note from Table XXXII that the equivalent con- ductance of the chloride ion is almost the same in all four chloride solu- tions at equal concentrations, especially in the more dilute solutions. This fact supports the view expressed previously that Kohlrausch's law of the independent migration of ions is applicable to dilute solutions of strong electrolytes at equivalent concentrations, as well as at infinite dilution. Transference Numbers in Mixtures. Relatively little work has been done on the transference numbers of ions in mixtures, although both Hittorf and moving boundary methods have been employed. In the former case, it follows from equation (3) that the transference number of any ion in a mixture is equal to the number of equivalents of that ion migrating from the appropriate compartment divided by the total num- ber of equivalents deposited in a coulometer. It is possible, therefore, to derive the required transference numbers by analysis of the anode and cathode compartments before and after electrolysis. The moving boundary method has been used to study mixtures of alkali chlorides and hydrochloric acid, a cadmium anode being employed to form an "autogenic" boundary. After electrolysis has proceeded for some time two boundaries are observed; the leading boundary is due to the high mobility of the hydrogen ion and is formed between the mixture of hydrochloric acid and the alkali chloride on the one side, and a solution of the alkali chloride from which the hydrogen ion has completely mi- grated out on the other side. The rate of movement of this boundary gives the transference number of the hydrogen ion in the mixture of electrolytes. The slower boundary is formed between the pure alkali chloride solution and the cadmium chloride indicator solution, a,nd gives no information concerning transference numbers in the mixture. The transference number of the alkali metal ion cannot be determined directly from the movement of the boundaries, and so the transference number of the chloride ion in the mixed solution is obtained from a separate experiment with an anion boundary using a mixture of potassium iodate and iodic acid as indicator. Since the transference numbers of the three 13 Longsworth, J. Am. Chem. Soc., 54, 2741 (1932); Maclnnes, /. Franklin InsL, 225, 661 (1938); see also, Owen, J. Am. Chem. Soc., 57, 2441 (1935). 128 THE MIGRATION OF IONS ions must add up to unity, the value for the alkali metal can now be derived. 14 Abnormal Transference Numbers. In certain cases, particularly with solutions of cadmium iodide, the transference number varies mark- edly with concentration, and the values may become zero or even appar- ently negative; the results for aqueous solutions of cadmium iodide at 18 are quoted in Table XXXIII. At concentrations greater than 0.5 N, TABLE XXXIII. CATION TRANSFERENCE NUMBERS IN CADMIUM IODIDE AT 18 Concn. 0.0005 0.01 0.02 0.05 0.1 0.2 0.5 N t+ 0.445 0.444 0.442 0.396 0.296 0.127 0.003 the transference number of cadmium apparently becomes negative : this means that in relatively concentrated solutions of cadmium iodide, the cadmium is being carried by the current in a direction opposite to that in which positive electricity moves through the solution. In other words, cadmium must form part of the negative ion present in the electrolyte. A reasonable explanation of the results is that in dilute solution cadmium iodide ionizes to yield simple ions; thus CdI 2 ^ Cd++ + 2I-, and so the transference number of the cadmium ion, in solutions con- taining less than 0.02 equiv. per liter, is normal. As the concentration is increased, however, the iodide ions combine with unionized molecules of cadmium iodide to form complex Cdl ions, thus, CdI 2 + 21- ^ Cdli~, with the result that appreciable amounts of cadmium are present in the anions and hence are transferred in the direction opposite to that of the flow of positive current. The apparent transference number of the cadmium ion is thus observed to decrease; if equal quantities of elec- tricity are carried in opposite directions by Cd+ + and Cdlr~ ions the transference number will appear to be zero. The proportion of Cdl" ions increases with increasing concentration and eventually almost the whole of the iodine will be present as Cdl" ions; the current is then carried almost exclusively by Cd +4 ~ and Cdlr~ ions. If the speed of the latter is greater than that of the former, as appears actually to be the case, the apparent cation transference number will be negative. A simi- lar variation of the cation transference number with concentration has been observed in solutions of cadmium bromide and this may be attrib- uted to the existence of the analogous CdBr^" ion. Less marked changes of transference number have been observed with other electrolytes; these are also probably to be ascribed to the presence of complex ions in con- centrated solutions. " Longsworth, J. Am. Chem. Soc., 52, 1897 (1930). PROBLEMS 129 PROBLEMS 1. Maclnnes and Dole [/. Am. Chem. Soc., 53, 1357 (1931)] electrolyzed a 0.5 N solution of potassium chloride, containing 3.6540 g. of salt per 100 g. solution, at 25 using an anode of silver and a cathode of silver coated with silver chloride. After the passage of a current of about 0.018 amp. for ap- proximately 26 hours, 1.9768 g. of silver were deposited in a coulometer in the circuit and on analysis the 119.48 g. of anode solution were found to contain 3.1151 g. potassium chloride per 100 g. solution, while the 122.93 g. of cathode solution contained 4.1786 g. of salt per 100 g. Calculate the values of the transference number of the potassium ion obtained from the anode and cathode solutions, respectively. 2. Jones and Bradshaw [J. Am. Chem. /Soc., 54, 138 (1932)] passed a current of approximately 0.025 amp. for 8 hours through a solution of lithium chloride, using a silver anode and a silver chloride cathode; 0.73936 g. of silver was deposited in a coulometer. The original electrolyte contained 0.43124 g. of lithium chloride per 100 g. of water, and after electrolysis the anode portion, weighing 128.615 g., contained 0.35941 g. of salt per 100 g. water, while the cathode portion, weighing 123.074 g., contained 0.50797 g. of salt per 100 g. of water. Calculate the transference number of the chloride ion from the separate data for anode and cathode solutions. 3. In a moving boundary experiment with 0.1 N potassium chloride, using 0.065 N lithium chloride as indicator solution, Maclnnes and Smith [_J. Am. Chem. Soc., 45, 2246 (1923)] passed a constant current of 0.005893 amp. through a tube of 0.1142 sq. cm. uniform cross section and observed the boundary to pass the various scale readings at the following times: Scale reading 0.5 5.50 5.80 6.10 6.40 6.70 7.00cm. Time 1900 2016 2130 2243 2357 2472 sec. Calculate the mean transference number of the potassium ion. The potential gradient was 4 volts per cm.; evaluate the mobility of the potassium ion for unit potential gradient. 4. The following results were recorded by Jahn and his collaborators [Z. phyaik. Chem. t 37, 673 (1901)] in experiments on the transference number of cadmium in cadmium iodide solutions using a cadmium anode: Original Anode solution Silver solution after electrolysis deposited in Cd per cent * Weight Cd per cent coulometer 2.5974 138.073 2.8576 0.7521 g. 1.3565 395.023 1.4863 0.9538 0.8820 300.798 1.0096 0.9963 0.4500 289.687 0.5654 0.9978 0.2311 305.750 0.3264 0.9604 0.1390 301.700 0.1868 0.5061 * The expression "Cd per cent" refers to the number of grams of Cd per 100 g. of solution. Evaluate the apparent transference number of the cadmium ion at the different concentrations, and plot the results as a function of concentration. 5. A0.2 N solution of sodium chloride was found to have a specific con- ductance of 1.75 X 10~* ohm~ l cm." 1 at 18; the transference number of the 130 THE MIGRATION OF IONS cation in this solution is 0.385. Calculate the equivalent conductance of the sodium and chloride ions. 6. A solution contains 0.04 N sodium chloride, 0.02 N hydrochloric acid and 0.04 N potassium sulfate; calculate, approximately, the fraction of the current carried by each of the ionic species, Na+, K+, H+, Cl~ and S0r~, in this solution. Utilize the data in Tables X and XIII, and assume that the conductance of each ion is the same as in a solution of concentration equal to the total equivalent concentration of the given solution. 7. The equivalent conductances and cation transference numbers of ammo- nium chloride at several concentrations at 25 are as follows [Longsworth, /. Am. Chem. Soc., 57, 1185 (1935)]: c 0.01 0.02 0.05 0.10 N A 141.28 138.33 133.29 128.75 ohms' 1 cm. 2 t+ 0.4907 0.4906 0.4905 0.4907 Utilize the results to evaluate the equivalent conductance of the ammonium and chloride ions at infinite dilution by the method described on page 126. 8. Use the results of the preceding problem to calculate the limiting slope, according to the Onsager equation, of the plot of the transference number of the ammonium ion in ammonium chloride against the square-root of the concentration. 9. Hammett and Lowenheim [J. Am. Chem. Soc., 56, 2620 (1934)] electro- lyzed, with inert electrodes, a solution of Ba(HS0 4 )2 in sulfuric acid as solvent; 1 g. of this solution contained 0.02503 g. BaS0 4 before electrolysis. After the passage of 4956 coulombs, 41 cc. of the anode solution and 39 cc. of the cathode solution, each having a density of 1.9, were run off; they were found on analysis to contain 0.02411 and 0.02621 g. of BaS0 4 per gram of solution, respectively. Calculate the transference number of the cation. 10. A solution, 100 g. of which contained 2.9359 g. of sodium chloride and 0.58599 g. urea, was electrolyzed with a silver anode and a silver chloride cathode; after the passage of current which resulted in the deposition of 4.5025 g. of silver in a coulometer, Taylor and Sawyer [/. Chem. Soc., 2095 (1929)] found 141.984 g. of anode solution to contain 3.2871 g. sodium chloride and 0.84277 g. urea, whereas 57.712 g. of cathode solution contained 2.5775 g. sodium chloride and 0.32872 g. urea. Calculate the "true" and "apparent" transference numbers of the ions of sodium chloride in the experimental solution. CHAPTER V FREE ENERGY AND ACTIVITY Partial Molar Quantities. 1 The thermodynamic functions, such as heat content, free energy, etc., encountered in electrochemistry have the property of depending on the temperature; pressure and volume, i.e., the state of the system, and on the amounts of the various constituents present. For a given mass, the temperature, pressure and volume are not independent variables, and so it is, in general, sufficient to express the function in terms of two of these factors, e.g., temperature and pressure. If X represents any such extensive property, i.e., one whose magnitude is determined by the state of the system and the amounts, e.g., number of moles, of the constituents, then the partial molar value of that property, for any constituent i of the system, is defined by > and is indicated by writing a bar over the symbol for the property. The partial molar quantity is consequently the increase in the particular property X resulting from the addition, at constant temperature and pressure, of one mole of the constituent i to such a large quantity of the system that there is no appreciable change in its composition. If a small change is made in the system at constant temperature and pressure, such that the number of moles of the constituent 1 is increased by dn\, of 2 by dra 2 , or, in general, of the constituent i by dn, the total change dX in the value of the property X is given by (dX) T , P = Xidni + 2 dn 2 + - rfZn< + (2) In estimating dX from equation (2) it is, of course, necessary to insert a minus sign before the Xdn term for any constituent whose amount is decreased as a result of the change in the system. Partial Molar Free Energy: Chemical Potential. The partial molal free energy is an important thermodynamic property in connection with the study of electrolytes; it can be represented either as G, where G is employed for the Gibbs, or Lewis, free energy,* or by the symbol /i, when it is referred*to as the chemical potential; thus the appropriate form 1 Lewis and Randall, "Thermodynamics and the Free Energy of Substances," 1923, Chap. IV; Glasstone, "Text-book of Physical Chemistry," 1940, Chap. III. * Electrochemical processes are almost invariably carried out at constant tempera- ture and pressure; under these conditions G is the appropriate thermodynamic function. The symbol F has been generally used to represent the free energy, but in order to avoid confusion with the symbol for the faraday, many writers now adopt G instead. 131 132 FREE ENERGY AND ACTIVITY of equation (2), for the increase of free energy accompanying a change in a given system at constant temperature and pressure, is then (dG) T . p = mdni + p*dn* + /i<dn< + (3) One of the thermodynamic conditions of equilibrium is that (dG)r.p is zero; it follows, therefore, that for a system in equilibrium at constant temperature and pressure tJLidni + ndn 2 + /*dn< + - = S/i*dn< = 0. (4) The partial molal volume of the constituent i in a mixture of ideal gases, which do not react, is equal to its molar volume ;,- in the system, since there is no volume change on mixing; if p t is the partial pressure of the constituent, then t\ is equal to RT/pi, where R is the gas constant per mole and T is the absolute temperature. It can be shown by means of thermodynamics that the partial molal volume (D) is related to the chemical potential by the equation ,. ......... and so it follows that, for an ideal gas mixture, RT Integration of equation (6) then gives the chemical potential of the gas i in the mixture, thus /u-Mf + Brinp,, (7) where /i? is a constant depending only on the nature of the gas and on the temperature of the system. It is evident that M? is equal to the chem- ical potential of the ideal gas at unit partial pressure. Activity and Activity Coefficient. 2 When a pure liquid or a mixture is in equilibrium with its vapor, the chemical potential of any constituent in the liquid must be equal to that in the vapor; this is a consequence of the thermodynamic requirement that for a system at equilibrium a small change at constant temperature and pressure shall not be accompanied by any change of free energy, i.e., (dG)T. P is zero. It follows, therefore, that if the vapor can be regarded as behaving ideally, the chemical po- tential of the constituent i of a solution can be written in the same form as equation (7), where pt is now the partial pressure of the component in the vapor in equilibrium with the solution. If the vapor is not ideal, the partial pressure should be replaced by an ideal pressure, or " fugacity," but this correction need not be considered further. According to Raoult's 1 Lewis and Randall, J. Am. Chem. Soc., 43, 1112 (1921); "Thermodynamics and the Free Energy of Substances," 1923, Chaps. XXII to XXVIII; Glasstone, "Text- book of Physical Chemistry," 1940, Chap. IX. ACTIVITY AND ACTIVITY COEFFICIENT 133 law the partial vapor pressure of any constituent of an ideal solution is proportional to its mole fraction (z<) in the solution, and hence it follows that the chemical potential in the liquid is given by Xi. (8) The constant /z? for the particular constituent of the solution is inde- pendent of the composition, but depends on the temperature and pres- sure, for the relationship between the mole fraction and the vapor pressure is dependent on the total pressure of the system. If the solution under consideration is not ideal, as is generally the case, especially for solutions of electrolytes, equation (8) is not applicable, and it is modified arbitrarily by writing M i= d + RT Inzi/i, (9) where / is a correction factor known as the activity coefficient of the constituent i in the given solution. The product xf is called the activity of the particular component and is represented by the symbol a, so that n* - vt + RTlnat. (10) As may be seen from equations (8) and (10), the activity in this particular case may thus be regarded as an idealized mole fraction of the given constituent. A comparison of equations (8) and (9) shows that for an ideal solution the activity coefficient/ is unity; in general, the difference between unity and the actual value of the activity coefficient in a given solution is a measure of the departure from ideal behavior in that solution. For a system consisting of a solvent, designated by the suffix 1, and a solute, indicated by the suffix 2, the respective chemical potentials are lZi/i (11) and M2 = A) + RTlnxtf*. (12) It is known that a solution tends towards ideal behavior more closely the greater the dilution ; hence, it follows that / 2 approaches unity as x 2 approaches zero, and /i approaches unity as x\ attains unity. It is convenient, therefore, to adopt the definitions /i 1 as x\ 1 and /z > 1 as x z 0. Since /i and Xi become unity at infinite dilution, i.e., for the pure solvent, it follows from equation (11) that the chemical potential of a pure liquid becomes equal to /i2(n> and hence is a constant at a given temperature and pressure. By considering the equilibrium between a solid and its vapor, it can be readily shown that the same rule is applicable to a pure solid. x>^ \ n <K<. - 134 FREE ENERGY AND ACTIVITY Forms of the Activity Coefficient. The equations given above are satisfactory for representing the behavior of liquid solutes, but for solid solutes, especially electrolytes, a modified form is more convenient. In a very dilute solution the mole fraction of solute is proportional both to its concentration (c), i.e., moles per liter of solution, and to its molality (m), i.e., moles per 1000 g. of solvent; hence for such solutions, which are known to approach ideal behavior, it is possible to write either M = + RTlnx, (13a) or M = Mc + flZMnc, (136) or M = f& + RTlnm, (13c) where v&, M? and & are constants whose relationship to each other depends on the factors connecting x, c and m in dilute solutions. Since solutions of appreciable concentration do not behave ideally, it is neces- sary to include the appropriate activity coefficients ; thus M = M 2 + RTlnxf x = + RTlna I9 /* = /z? + RT In cf c = + RT In a e , ^ c ^(146) and v N v where the a terms are the respective activities. / x ^ It is evident from the equations (14) that the activity of a constituent of a solution can be expressed only in terms of a ratio * of two chemical potentials, viz., ju and /i, and so it is the practice to choose a reference state, or standard state, for each constituent in which the activity is arbitrarily taken as unity. It can be readily seen from the equations given above that in the standard state the chemical potential n is equal to the corresponding value of ff. The activity of a component in any solution is thus invariably expressed as the ratio of its value to that in the arbi- trary standard state. The actual standard state chosen differs, of course, according to which form of equation (14) is employed to define the activity. At infinite dilution, when a solution behaves ideally, the three activity coefficients of the solute, viz.,/ x ,/ c and/ m , are all unity, but at appreciable concentrations the values diverge from this figure and they are no longer equal. It is possible, however, to derive a relationship between them in the following manner. The mole fraction x, concentration c, and molality m of a solute can be readily shown to be related thus 0.001 cM ! 0.001 mM l X ~ P - 0.001 cM 2 + 0.001 cMi ~~ 1 + 0.001 mM l ' ( ' * It is a ratio, rather than a difference, because in equations (14) the activity appears in a loganthmic term. FORMS OF THE ACTIVITY COEFFICIENT 135 where A is the density of the solution, and MI and M 2 are the molecular weights of solvent and solute, respectively. In very dilute solutions the three related quantities are x , c and mo, and the density is po, which is virtually that of the pure solvent ; since the quantities 0.001 cM i, 0.001 cM 2 and 0.001 mM i are then negligibly small, it follows from equation (15) that Po Incidentally this relationship proves the statement made above that in very dilute solutions the mole fraction, concentration and molality are proportional to each other. If /no is the chemical potential of a given solute in a very dilute solu- tion, to which the terms Xo f CQ and mo apply, the three activity coefficients are all unity ; further, if ju is the chemical potential in some other solution, whose concentration is represented by x, c or m, it follows from the three forms of equation (14) that A* Mo may be written in three ways, thus XQ Co x Co mo Combination of equations (15), (16) and (17) then gives the relationship between the three activity coefficients for the solute in the given solution : _ ,- 0.00. cM, + 0.001 _ m Po It is evident from this expression that f c and f m must be almost identical in dilute solutions, and that f x cannot differ appreciably from the other coefficients for solutions more dilute than about 0.1 N. The arguments given above are applicable to a single molecular species as solute, but for electrolytes it is the common practice to em- ploy a mean activity coefficient (see p. 138) ; in this event it is necessary to introduce into the terms 0.001 cM\ and 0.001 mM\ the factor v which is equal to the number of ions produced by one molecule of electrolyte when it ionizes. The result is then . . p - 0.001 cM 2 + 0.001 , ,, , n Ani lf , /im f x = fc --------- = /(! + 0.001 vmMi). (19) Po The activity coefficient f x is sometimes called the rational activity coefficient, since it gives the most direct indication of the deviation from the ideal behavior required by Raoult's law. It is, however, not often used in connection with measurements on solutions of electrolytes, and so the coefficients f c and/*,, which are commonly employed, are described as the practical activity coefficients. The coefficient / c , from which the 136 FREE ENERGY AND ACTIVITY suffix is dropped, is generally used in the study of electrolytic equilibria to represent the activity of a particular ionic species; thus, the activity of ions of the ith kind is equal to c</,, where d is the actual ionic concen- tration, due allowance being made for incomplete dissociation if necessary. On the other hand / m , which is given the symbol 7, is almost invariably used in connection with the thermodynamics of voltaic cells; the activity of an ion is expressed as my , where m is the total molality of the ionic constituent of the electrolyte with no correction for incomplete dissocia- tion. For this reason 7 is sometimes called the stoichiometric activity coefficient. Equilibrium Constant and Free Energy Changes. If a system in- volving the reversible chemical process aA + 6B + ^ IL + mM + is in a state of equilibrium, it can be readily shown, by means of equations (4) and (14), that where K is the equilibrium constant for the system under consideration. Equation (20) is the exact form of the law of mass action applicable to any system, ideal or not. Writing fc or ym in place of the activity a, the following equations for the equilibrium constant, which are frequently employed in electrochemistry, are obtained, viz., v C LM ' * ' /L/M AC ~~ ~*J CA C B ' ' ' and /01 |v 1 (216) ' ' If the components of the system under consideration are at their equilibrium concentrations, or activities, the free energy change resulting from the transfer from reactants to resultants is zero. If, however, the various substances are present in arbitrary concentrations, or activities, the transfer process is accompanied by a definite change of free energy; thus, if a moles of A, b moles of B, etc., at arbitrary activities are trans- ferred to I moles of L, m moles of M, etc., under such conditions that the concentrations are not appreciably altered, the increase of free energy (AC?) at constant temperature is given by the expression - AG= RTln K- RTln*?? '" ; (22) a A a B this equation is a form of the familiar reaction isotherm. If the arbi- trary activities of reactants and resultants are chosen as the respective ACTIVITIES OF ELECTROLYTES 137 standard states, i.e., the a's in equation (22) are all unity, it follows that - A<? = RTln K, (23) where A(J is the standard free energy change of the process. Activities of Electrolytes. When the solute is an electrolyte, the standard states for the ions are chosen, in the manner previously indi- cated, as a hypothetical ideal solution of unit activity; in this solution the thermodynamic properties of the solute, e.g., the partial molal heat content, heat capacity, volume, etc., will be those of a real solution at infinite dilution, i.e., when it behaves ideally. With this definition of the standard state the activity of an ion becomes equal to its concentration at infinite dilution. For the undissociated part of the electrolyte it is convenient to define the standard state in such a way as to make its chemical potential equal to the sum of the values for the ions in their standard states. Consider, for example, the electrolyte M^A,. which ionizes thus to yield the number v+ of M + ions and v-. of A~ ions. The chemical potentials of these ions are given by the general equations MM+ = & + RT\na+ (24a) and MA- = + RT\na-, (246) where a+ and a_ are the activities of the ions M+ and A~ respectively. If /i2 is the chemical potential of the undissociated portion of the elec- trolyte in a given solution and /z is the value in the standard state, then by the definition given above, M? = *+/4 + "-M-. (25) When the system of undissociated molecules and free ions in solution is in equilibrium, a small change at constant temperature and pressure produces no change in the free energy of the system; since one molecule of electrolyte produces v+ positive and v- negative ions, it is seen that [cf. equation (4)] v+(\ + RT In a+) + _( M - + RT In a.) = + RT In a 2 . (26) Introducing equation (25) it follows, on the basis of the particular stand- ard states chosen, that v+RT In a+ + v-RT In a_ = RT In a 2 , /. afa- = a 2 . (27) If the total number of ions produced by a molecule of electrolyte, i.e., v+ + v~, is represented by v, then the mean activity a of the elec- 138 FREE ENERGY AND ACTIVITY trolyte is defined by a m (<#!-)'/', (28) and hence, according to equation (27), a = (a,) 1 " or a 2 = a v . (29) The activity of each ion may be written as the product of its activity coefficient and concentration, so that a+ = y+m+ and a_ = 7_w_, a+ , a- . . 7+ = and 7- = -- m+ m- The mean activity coefficient y of the electrolyte, defined by y s (7?7-) 1 ", (30) can consequently be represented by r If m is the molality of the electrolyte, m+ is equal to WP+ and m_ is equal to wy_, so that equation (31) may be written as (32) The mean molality m of the electrolyte is defined, in an analogous manner, by m* s (m^-ml-) 1 /" = m(v>!r) 1/ '', so that it is possible to write equation (32) as "*- (33) Relationships similar to those given above may, of course, be derived for the other activity coefficients. Values of Activity Coefficients. Without entering into details, it is evident from the foregoing discussion that activities and activity coefficients are related to chemical potentials or free energies; several methods, both direct and indirect, are available for determining the requisite differences of free energy so that activities, relative to the specified standard states, can be evaluated. In the study of the activity coefficients of electrolytes the procedures generally employed are based on measurements of either vapor pressure, freezing point, solubility or electromotive force. 3 The results obtained by the various methods are 1 See references on page 132, also pages 200 and 203. For a valuable summary of data and other information on activity coefficients, see Robinson and Harned, Chem. Revs., 28, 419 (1941). VALUES OP ACTIVITY COEFFICIENTS 130 TABLE XXXIV. MEAN ACTIVITY COEFFICIENTS OF ELECTROLYTES IN AQUEOUS SOLUTION AT 25 Molality HC1 NaCl KC1 HBr NaOH CaCli ZnCU HS04 ZnSO* LaCli Int(80)i 0.001 0.966 0.966 0.966 0.888 0.881 0.734 0.853 0.005 0.930 0.928 0.927 0.930 0.789 0.767 0.643 0.477 0.716 0.16 0.01 0.906 0.903 0.902 0.906 0.899 0.732 0.708 0.545 0.387 0.637 0.11 0.02 0.878 0.872 0.869 0.879 0.860 0.669 0.642 0.455 0.298 0.552 0.08 0.05 0.833 0.821 0.816 0.838 0.805 0.584 0.556 0.341 0.202 0.417 0.035 0.10 0.798 0.778 0.770 0.805 0.759 0.524 0.502 0.266 0.148 0.356 0.025 0.20 0.768 0.732 0.719 0.782 0.719 0.491 0.448 0.210 0.104 0.298 0.021 0.50 0.769 0.679 0.652 0.790 0.681 0.510 0.376 0.155 0.063 0.303 0.014 1.00 0.811 0.656 0.607 0.871 0.667 0.725 0.325 0.131 0.044 0.387 1.50 0.898 0.655 0.586 0.671 0.290 0.037 0.583 2.00 1.011 0.670 0.577 0.685 1.554 0.125 0.035 0.954 3.00 1.31 0.719 0.572 3.384 0.142 0.041 in good agreement with each other and hence they may be regarded aa reliable. Although the description of the principles on which the deter- minations of activity coefficients are based will be considered later, it will be convenient to summarize in Table XXXIV some actual values of the mean activity coefficients at 25 obtained for a number of electro- lytes of several valence types in aque- ous solution at various molalities. Some of the results are also depicted by the curves in Fig. 46; it will be observed that the activity coeffi- cients may deviate appreciably from unity. The values always decrease at first as the concentration is in- creased, but they generally pass through a minimum and then increase again. At high concentrations the activity coefficients often exceed unity, so that the mean activity of the electrolyte is actually greater than the concentration; the deviations from ideal behavior are now in the opposite direction to those which occur at low concentrations. An ex- FIQ. 46. Activity coefficients of electro- amination of Table XXXIV brings lytes of different valence types to light other important facts: it is seen, in the first place, that electrolytes of the same valence type, e.g , sodium and potassium chlorides, etc., or calcium and zinc chlorides, etc.. 0.60 1.60 2.0 Molality 140 FREE ENERGY AND ACTIVITY have almost identical activity coefficients in dilute solutions. Secondly, the deviation from ideal behavior at a given concentration is greater the higher the product of the valences of the ions constituting the electrolyte. The Ionic Strength. In order to represent the variation of activity coefficient with concentration, especially in the presence of added elec- trolytes, Lewis and Randall introduced the quantity called the ionic strength^ which is a measure of the intensity of the electrical field due to the ions in a solution. 4 It is given the symbol y and is defined as half the sum of the terms obtained by multiplying the molality, or concen- tration, of each ion present in the solution by the square of its valence; that is (34) In calculating the ionic strength it is necessary to use the actual ionic concentration or molality; for a weak electrolyte this would be obtained by multiplying its concentration by the degree of dissociation. Although the importance of the ionic strength was first realized from empirical considerations, it is now known to play an important part in the theory of electrolytes. It will be observed that equation (12) on page 83, which gives the reciprocal of the thickness of the ionic atmos- phere according to the theory of Debye and Hiickel, contains the quan- tity ^riiZi, where n,- is the number of ions of the zth kind in unit volume and hence is proportional to the concentration. This quantity is clearly related to the ionic strength of the solution as defined above; it will be seen shortly that it plays a part in the theoretical treatment of activity coefficients. It was pointed out by Lewis and Randall that, in dilute solutions, the activity coefficient of a given strong electrolyte is approximately the same in all solutions of a given ionic strength. The particular ionic strength may be due to the presence of other salts, but their nature does not affect the activity coefficient of the electrolyte under consideration. This generalization, to which further reference will be made later, holds only for solutions of relatively low ionic strength; as the concentration is increased the specific influence of the added electrolyte becomes manifest. The Debye-Hiickel Theory. The first successful attempt to account for the departure of electrolytes from ideal behavior was made by Milner (1912), but his treatment was very complicated; the ideas were essen- tially the same as those which were developed in a more elegant manner by Debye and Hiickel. The fundamental ideas have already been given on page 81 in connection with the theory of electrolytic conductance, and the application of the Dcbye-Hiickel theory to the problem of ac- tivity coefficients will be considered here. 6 4 Lewis and Randall, J. Am. Chem. Soc., 43, 1112 (1921). 6 Debye and Huckel, Physik. Z., 24, 185, 334 (1923); 25, 97 (1924); for reviews, see LaMer, T*ans. Electrochem. Soc., 51, 507 (1927); Falkenhagen, "Electrolytes" (trans- lated bj Bell), 1934; Williams, Chem. Revs., B, 303 (1931); Schingnitz, Z. Ekktrochem., 36, 861 (1930). THE DEBYE-HtfCKEL THEORY 141 According to equation (16), page 84, the potential $ due to ions of the ith kind may be represented by where the first term is the potential at a distance r from the central ion when there are no surrounding ions, and the second term is the contribu- tion of the ionic atmosphere; K is defined by equation (12), page 83. Suppose that all the ions are discharged and that successive small charges are brought up to the ions from infinity in such a way that at any instant all the ions have the same fraction X of their final charge z t . It follows, therefore, from equation (35), that at any stage during the charging process the potential fa due to ions of the ith kind is given by where K\ is the value of the quantity K at this stage. It can be seen, from the definition of *, that since the charge on the ion is then Xz, the value of KX will be a fraction X of the final value; the term KX in equa- tion (36) may thus be replaced by X*c. Making this substitution, equa- tion (36) becomes *x = gx- 2 fx> (37) If z#d\ is the magnitude of the small charge brought up to each ion of the ith kind, the corresponding work done is z t edX X ^x, and hence the total electrical work (Wi) done in charging completely, i.e., from X = to X = 1, an ion of the ith kind is p-i Wi = I z % Gl/\d\ Jx-o 2Dr 3D D (38) If Ni is the total number of ions of the ith kind,* the total electrical work (W) done in charging completely all the ions of the solution is obtained by multiplying equation (38) by Ni and summing over all the ions, thus * This should not be confused with n,, the number of these ions in unit volume. 142 FREE ENERGY AND ACTIVITY At infinite dilution there is no ionic atmosphere, and so K is zero and the second term on the right-hand side of equation (39) disappears; since the dielectric constant is that of the pure solvent, i.e., D , the electrical work (Wo) done in charging the ions at infinite dilution is Provided the solution is not too concentrated, D and Do are approxi- mately equal, and hence the difference in the electrical work of charging the same ions at a definite concentration and at infinite dilution is given by The volume change accompanying the charging process at constant pressure is negligible, and so W Wo may be identified with the differ- ence between the electrical free energy of an ionic solution at a definite concentration and at infinite dilution. The free energy (G) of a solution containing ions may be regarded as being made up of two parts: first, that corresponding to the value for an ideal solution at the same concentration as the ionic solution (G ), and second, an amount due to the electrical interaction of the ions (G e i.) ; thus G = Go + Gei., (42) where G e i. may be taken as being equal to W T7o, as given by equation (41). Differentiating with respect to N if the number of ions of the ith kind, at constant temperature and pressure, the result is dG dGo dGei. = ~~ ' or M = Mt(0) + Mt(el.). (43) According to the definition of the chemical potential /*, which now applies to a single ion y instead of to a g.-ion, M< = MI + kT In a t = d + kTlnx t + kTl*f>, (44) where k is the Boltzmann constant, i.e., the gas constant per single molecule. Further, since Go refers to an ideal solution, it follows that M.(O) = rf + kT\nx it (45) and hence from equations (43), (44) and (45), (46) Introducing the value of G e i. as given by equation (41), it is found on THE DEBYE-HtJCKEL LIMITING LAW 143 differentiating with respect to N { , remembering that K involves ^Ui and hence V#i, that (47) N being the Avogadro number and R, equal to kN, the gas constant per mole.* The Debye-Hiickel Limiting Law. The value of K as given on page 83 is (48) and if n,- is replaced by cJV/1000, where c< is the ionic concentration in moles per liter, and R/N is written for k, equation (48) becomes (49) The quantity 2c2j is seen to be analogous to twice the ionic strength as defined by Lewis and Randall [equation (34)]; the only difference is that the former involves volume concentrations whereas in the latter molalities are employed. For dilute aqueous solutions, such as were used in the work from which Lewis and Randall made the generalization given on page 140, the two values of the ionic strength are almost identical. It has been stated that if the Debye-Hiickel arguments are applied in a rigid manner the expression for K will actually involve molalities; never- theless, it is the practice in connection with the application of the equa- tions derived by the method of Debye and Hiickel to use an ionic strength defined in terms of molar concentrations, viz., (50) so that equation (49) can be written as / arjw V * = \IOOODRT* ) m (51) Introducing this value for K into equation (47) and at the same time dividing the right-hand side by 2.303 to convert natural to common logarithms, the result is i * JVV / 2* V z\ r log/ ' = " 303tf>V 1000/ (DT)* * ( } * It should be noted that in the differentiation the summation in equation (41) has been reduced to a single term. This is because the numbers of all the ions except of the ith kind remain constant, and so all the terms other than the one involving n will be zero. 144 PEBB ENERGY AND ACTIVITY The universal constants AT, c, v and R, as well as the numerical quantities, may be extracted from equation (52), and if the accepted values are employed, this equation becomes log/, = - 1.823 X 10' ~ V. (53) For a given solvent and temperature D and T have definite values which may be inserted; equation (53) then takes the general form log/. = - Az 2 ^, (54) where A is a constant for the solvent at the specified temperature. This equation, which represents what has been called the Debye- Hiickel limiting law, expresses the variation of the activity coefficient of an ion with the ionic strength of the medium. It is called the limiting law because, as seen previously, the approximations made in the deriva- tion of the potential at an ion due to its ionic atmosphere, can be expected to be justifiable only as infinite dilution is approached. The general conclusion may be drawn from equation (53) or (54) that the activity coefficient of an ion should decrease with increasing ionic strength of the solution: the decrease is greater the higher the valence of the ion and the lower the dielectric constant of the solvent. It will be seen later (p. 230) that there does not appear to be any experimental method of evaluating the activity coefficient of a single ionic species, so that the Debye-Hiickel equations cannot be tested in the forms given above. It is possible, however, to derive very readily an expression for the mean activity coefficient, this being the quantity that is obtained experimentally. The mean activity coefficient / of an electrolyte M+A~ is defined by an equation analogous to (30), and upon taking logarithms this becomes The values of log/+, which is equal to Ass+Jy, and of log/_, i.e., AzlVy, as given by equation (54) can now be inserted in (55); making use of the fact that z+v+ must be equal to Z-v-, it is found that log/ = ~ Az^.^, (56) which is the statement of the Debye-Huckel limiting law for the mean activity coefficient of an electrolyte whose ions have valences of z+ and z_, respectively. The values of the constant A for water at a number of temperatures are given in Table XXXV below. Attention should be drawn to the fact that the activity coefficients given by the Debye-Htickel treatment are the so-called rational coeffi- cients (p. 135) ; to express the values in the form of the practical activity coefficients, it is necessary to make use of equation (26). If the solvent DEBYE-HUCKEL EQUATION FOR APPRECIABLE CONCENTRATIONS 145 is water, so that M i is 18, it is seen that log 7 = log/ - log (1 + 0.018m), where y is the activity coefficient in terms of molalities, / is the value given by the Debye-Huckel equations, and v is the number of ions pro- duced by one molecule of electrolyte on dissociation. As already seen, however, the difference between the various coefficients is negligible in dilute solutions, and it is in such solutions that the most satisfactory tests of the Debye-Hiickel theory can be made. Debye-Hiickel Equation for Appreciable Concentrations. In the derivation of equation (12), page 83, the approximation was made of regarding the ion as being equivalent to a point charge; this will result in no serious error provided the radius of the ionic atmosphere is large in comparison with that of the ion. An examination of Table XXII, page 85, shows that this condition is satisfied in dilute solutions, but when the concentration approaches a value of about 0.1 molar the radius of the ionic atmosphere is about the same order as that of an ion, i.e., 2 X 10"" 8 cm. It follows, therefore, that in such solutions the approximation of a point charge is liable to lead to serious errors. A possible method of making the necessary correction has been pro- FlG 47 Mean distance of posed by Debye and Hiickel ; 6 it has been found approach of ions that if a is the mean distance of approach of other ions, e.g., B to the central ion A 9 as shown in Fig. 47, the potential due to ions of the zth kind is given by the expression 2 2,K 1 + -*--D'TTZ' (57) instead of by equation (35). The mean distance of approach a is often referred to as the "average effective diameter" of the ions, although its exact physical significance probably cannot be expressed precisely. It is seen that the correction term is (1 + KCL)~ I , which approaches unity in dilute solutions when K is small. By following through the derivation on page 141, using equation (57) instead of (35), the final result is I ,_ ' (58) in place of equation (47). It is apparent from equation (51) that, for a given solvent and a definite temperature, K is equivalent to #V|i, where B is a constant; hence 1 + KCI may be replaced by 1 + aB'fy. Making Debye and Huckel, Physik. Z. t 24, 185 (1923). 146 FREE ENERGY AND ACTIVITY this substitution in equation (58), 1 /KQ . ' (59) and hence the Debye-Hiickel limiting law, corresponding to equation (54), now becomes ' (60) where A has the same significance as before. The expression for the mean activity coefficient of an electrolyte is then ' (61) a\v Both the constants A and B depend on the nature of the solvent and the temperature; the values for water at several temperatures arc given in Table XXXV; the corresponding dielectric constants arc also recorded. TABLE XXXV. DEBYE-HUCKEL CONSTANTS AND DIELECTRIC CONSTANT OF WATER Temp. DAB 88.15 0.488 0.325 X 10 8 15 82.23 0.500 0.328 25 78.54 0.509 0.330 30 76.76 0.514 0.331 40 73.35 0.524 0.333 50 70.10 0.535 0.335 It will be observed from Table XXXV that at ordinary temperatures the value of B with water as solvent is approximately 0.33 X 10 8 ; for most electrolytes the mean ionic diameter a is about 3 to 4 X 10~ 8 cm. (see Table XXXVI), and hence aB does not differ greatly from unity. A reasonably satisfactory and simple approximation of equation (61) is therefore The Hiickel and Breasted Equations. A further correction to the Debye-Hiickel equation has been proposed in order to allow for the polari- zation of the solvent molecules by the central ion; since these molecules are, in general, more polarizable than the ions themselves, there will be a tendency for the solvent molecules to displace the other ions from the vicinity of a particular ion. The dipolar nature of the solvent molecules will also facilitate the tendency for these molecules to orient themselves about the central ion. It has been suggested that the result of this orientation is equivalent to an increase in the dielectric constant in the immediate vicinity of the ion above that in the bulk of the solvent. By QUALITATIVE VERIFICATION OF THE DEBYE-HUCKEL EQUATIONS 147 assuming the increase to be proportional to the ionic concentration of the solution, it has been deduced that an additional term CV> where C" is an empirical constant, should be added to the right-hand side of equa- tions (60) and (61) ; hence, the latter now becomes + c '*- (62) This result has sometimes been called the Hiickel equation. 7 It is not certain that the theoretical arguments, which led to the introduction of the term C't*> are completely satisfactory, but it seems to be established that the experimental data require a term of this type. The aggregation of solvent molecules in the vicinity of an ion is the factor responsible for the so-called "salting-out effect," namely, the decrease in solubility of neutral substances frequently observed in the presence of salts; the constant C 1 is consequently called the salting-out constant. The activity coefficient of a non-electrolyte, as measured by its solubility in the presence of electrolytes, is often given by an expression of the form log/ = C"|i; this is the result to which equation (62) would reduce for the activity of a non-electrolyte, i.e., when z+ and z_ are zero, in a salt solu- tion of ionic strength y. By dividing through tfce numerator of the fraction on the right-hand side of equation (62) by the denominator, and neglecting all terms in the power series beyond that involving p, the result is + (aABz+Z- + C")u + CV, (63) where C is a constant for the given electrolyte, equal to aABz+z_ + C'. This relationship is of the same form as an empirical equation proposed by Br0nsted, 8 and hence is in general agreement with experiment; it has been called the Debye-Hiickel-Br^nsted equation. In dilute solution, when y is small, the term Cy can be neglected, and so this expression then reduces to the Debye-Hiickel limiting law. Qualitative Verification of the Debye-Hiickel Equations. The gen- eral agreement of the limiting law equation (54) with experiment is shown by the empirical conclusion of Lewis and Randall (p. 140) that the activity coefficient of an electrolyte is the same in all solutions of a given ionic strength. Apart from the valence of the ions constituting the particular electrolyte under consideration, the Debye-Hiickel limiting equation contains no reference to the specific properties of the salts that may be present in the solution. It is of interest to record that the 7 Hiickel, Physik. Z., 26, 93 (1925); see also, Butler, /. Phys. Chem., 33, 1015 (1929); Scatchard, Physik. Z., 33, 22 (1932). 8 Br0nsted, J. Am. Chem. Soc., 44, 938 (1922); Br0nsted and LaMer, ibid., 46, 555 (1924). 148 FREE ENERGY AND ACTIVITY empirical equation proposed by Lewis and Linhart 9 to account for their results on the freezing points of dilute solutions of various electrolytes is of the form log/ = ftc a , where a was found to be about 0.4 to 0.5 for several salts and ft depended on their valence type. Further, as already mentioned, Br0nsted's empirical equation for more concentrated solutions is in agreement with the extended equation (62). It can be seen from the Debye-Hiickel limiting law equation that at a definite ionic strength the departure of the activity coefficient of a given electrolyte from unity should be greater the higher the valences of the ions con- stituting the electrolyte; this conclusion is in harmony with the results given in Table XXXIV which have been already discussed. It was noted on page 139 that although activity coefficients generally decrease with increasing concentration in dilute solutions, in accordance with the requirement of equation (58), the values frequently pass through a minimum at higher concentrations. It is of interest, therefore, to see how far this fact can be explained, at least qualitatively, by means of the Debye-Huckel theory. According to the limiting law equation, the plot of log/ against Vy should be a straight line of slope AZ+Z-] for a uni- univalent electrolyte in water at 25 this is equal to approximately 0.51, as shown in Fig. 48, 1. If the ionic size factor is introduced, as in equation (61), the plot of log/ against Vy becomes of the form of Fig. 48, II, representing a type of curve which is often obtained experimentally. Finally, the addition of the salting-out fac- tor, as in equation (62), results in a further increase of the activity coefficient by an amount propor- 0.2 o.4 0.6 0.8 1.0 1.2 tional to the ionic strength; the result is that the log/ against Vjt curve becomes similar to Fig. 48, III. It may be mentioned that the latter curve duplicates closely the variation of the activity coefficient of sodium chloride with concen- tration up to relatively high values of the latter. Quantitative Tests of the Debye-Hiickel Limiting Equation. Al- though the Debye-Huckel equations are generally considered as applying to solutions of strong electrolytes, it is important to emphasize that they are by no means restricted to such solutions; they are of general applica- bility and the only point that must be noted is that in the calculation of the ionic strength the actual ionic concentrations must be employed. 9 Lewis and Linhart, J. Am. Chem. Soc., 41, 1951 (1919). -0.2 -0.4 -0.6 Fia. 48. Simple (I) and extended (II and III) Debye-Huckel equations QUANTITATIVE TESTS OF THE DEBYE-HtfCKEL LIMITING EQUATIONS 149 For incompletely dissociated electrolytes this involves a knowledge of the degree of dissociation, which may not always be available with sufficient accuracy. It is for this reason that the Debye-Htickel equations are generally tested by means of data obtained with strong electrolytes, since they can be assumed to be completely dissociated. It is probable that some of the discrepancies observed with certain electrolytes of high valence types are due to incomplete dissociation for which adequate allow- ance has not been made. 1.0 0.9 0.8 0.7 0.6 0.5 0.4 J_ 0.10 0.40 0.20 0.30 vr FIG. 49. Test of the limiting Debye-Huckel equation The experimentally determined activity coefficients, based on vapor pressure, freezing-point and electromotive force measurements, for a number of typical electrolytes of different valence types in aqueous solution at 25, are represented in Fig. 49, in which the values of log / are plotted against the square-root of the ionic strength; in these cases the solutions contained no other electrolyte than the one under considera- tion. Since the Debye-Hiickel constant A for water at 25 is seen from Table XXXV to be 0.509, the limiting slopes of the plots in Fig. 49 should be equal to -0.509 z+z_; the results to be expected theoretically, cal- culated in this manner, are shown by the dotted lines. It is evident that the experimental results approach the values required by the Debye- Hiickel limiting law as infinite dilution is attained. The influence of valence on the dependence of the activity coefficient on concentration is evidently in agreement with theoretical expectation. Another verifica- tion of the valence factor in the Debye-Hiickel equation will be given later (p. 177). A comparison of equations (52) and (53) shows that, for electrolytes of the same valence type, the limiting slope of the plot of log/ against Vy at constant temperature should be inversely proportional to Z> f , where D 160 FREE ENERGY AND ACTIVITY is the dielectric constant of the medium. A stringent test of the Debye- Hiickel equation is, therefore, to determine the activity coefficients of a given electrolyte in a number of different media of varying dielectric constant; the results are available for hydrochloric acid in methyl and ethyl alcohols, in a number of dioxane-water mixtures, as well as in pure water at 25. Some of the data are plotted in Fig. 50; the limiting slopes, Fia. 50. Limiting Debye-Hiickel equation at different dielectric constants (Earned, et al) marked with the appropriate value of the dielectric constant, are indi- cated by the dotted lines in each case. The agreement with expectation, over a range of dielectric constant from about 10 to 78.6, is very striking. 10 The influence of one other variable, namely, the temperature, re- mains to be considered. It is not an easy matter to vary the temperature without changing the dielectric constant, and so these factors may be considered together. From equations (55) and (56) it is evident that the limiting slope of the plot of log/ against Vy should vary as 1/(DT)*, where T is the absolute temperature at which the activity coefficients are measured. The experimental results obtained under a wide variety of conditions, e.g., in liquid ammonia at 75 and in water at the boiling w Earned et al., J. Am. Chem. Soc., 61, 49 (1939). THE OSMOTIC COEFFICIENT 151 point, are generally in satisfactory agreement with theoretical require- ments. 11 Where discrepancies are observed they can probably be ex- plained by incomplete dissociation in media of low dielectric constant. The Osmotic Coefficient. Instead of calculating activity coefficients from freezing-point and other so-called osmotic measurements, the data may be used directly to test the validity of the Debye-Hiickel treatment. If is the depression of the freezing point of a solution of molality m of an electrolyte which dissociates into v ions, and X is the molal freezing-point depression, viz., 1.858 for water, a quantity <, called the osmotic coefficient, may be defined by the expression e (64) This coefficient is equivalent to the van't Hoff factor i (see p. 9) di- vided by v. It can be shown by means of thermodynamics that if log / is proportional to the square-root of the ionic strength, as it undoubtedly is in dilute solutions, then 1 - * = - iln/. (65) Introducing the Debye-Hiickel limiting law for log /, it is seen that 2.303 (66) where A has the same significance as before. Since <t> can be determined directly from freezing-point measurements, by means of equation (64), 0.5 0.4 0.3 0.2 0.1 x Lithium Chloride o Lithium Bromide + Lithium Perchlorate o Guanidine Nitrate 0.05 0.10 0.15 FIG. 51. Test of Debye-Huckel equation by freezing-point measurements in cyclohexanol (Schreiner and Frivold) "Saxton and Smith, /. Am. Chem. Soc., 54, 2626 (1932); Webb, /. Am. Chem. Soc., 48, 2263 (1926). 152 FREE ENERGY AND ACTIVITY it is possible to test the Debye-Huckel theory in the form of equation (66); the plot of 1 against Vy should approach a limiting value of 0.768 A z+z_. The experimental results for electrolytes of different valence types in aqueous solutions are in agreement with expectation; since the data are in principle similar to many that were used in the compilation of Fig. 49, they need not be considered further. It is of interest, however, to examine the values derived from freezing-point measurements in a solvent of low dielectric constant, viz., cyclohexanol, whose dielectric constant is 15.0 and freezing point 23.6; the full curve in Fig. 51 is drawn through the results for a number of uni-univalent electrolytes, while the dotted curve shows the limiting slope required by equation (66). 12 Activities at Appreciable Concentrations. A comparison of the ex- perimental curves in Figs. 49 and 50 with the general form of curve II in Fig. 48 suggests that equation (61) might represent the variation of activity coefficient with concentration in solutions of electrolytes that 1.4 1.3 1.1 I I I I 0.05 0.20 0.10 0.15 vr FIG. 52. Determination of mean ionic diameter were not too concentrated; by a slight rearrangement this equation can be put in the form A 9 . 9 A/M . (67) log/ ^ so that if the left-hand side of equation (67) is plotted against Vy the result should be a straight line of slope aB. Since the value of B is "Schreiner and Frivold, Z. physik. Chem., 124, 1 (1926). ACTIVITIES IN CONCENTRATED SOLUTIONS 153 known (cf. Table XXXV), the magnitude of the mean ionic diameter required to satisfy the experimental results can be obtained. The data for aqueous solutions of hydrochloric acid at 25 are shown in Fig. 52; the points are seen to fall approximately on a straight line so that an equation of the form of (61) and (67) is obeyed. The slope of this line is about 1.75 and since B is 0.33 X 10 8 , it follows that for hydrochloric acid a is equal to 5.3 X 10~~ 8 . It has been found in a number of cases that by using values of a that appear to be of a reasonable magnitude it is possible to represent quantitatively the activity coefficients of a num- ber of electrolytes up to ionic strengths of ^about 0.1. Some of the mean values, collected from those reported in the literature, are given in Table XXXVI. It must be pointed out, however, that such satisfactory results TABLE XXXVI. MEAN EFFECTIVE IONIC DIAMETERS Electrolyte a Electrolyte a HC1 5.3 X 10- cm. CaCl 2 5.2 X lO" 8 cm. NaCl 4.4 MgS0 4 3.4 KC1 4.1 K 2 SO 4 3.0 CsNO, 3.0 La 2 (S0 4 ) 3.0 are not always obtained; in order to satisfy the experimental data in the case of silver nitrate, for example, a should be 2.3 X 10~ 8 cm., and for potassium nitrate 0.43 X 10~ 8 cm., both of which values are lower than would be expected. It is nevertheless of interest that the figures are at least of the correct order of magnitude for an ionic radius, namely about 10~~ 8 cm. In some instances, particularly with salts of high valence types, it is found necessary to employ variable or even negative values of a; this may be attributed either to incomplete dissociation or to the ap- proximations made in the Debye-Huckel derivation. Activities in Concentrated Solutions. For relatively concentrated solutions it is necessary to use the complete Hlickel equation (62); by choosing suitable values for the two adjustable parameters a and C", it has been found possible to represent the variation of activity coeffi- cients with concentration of several electrolytes from 0.001 to 1 molal, and sometimes up to 3 molal. The values of C' seem to lie approximately between 0.05 and 0.15 in aqueous solution. At the higher concentrations it is necessary to make allowance for the difference between the rational and stoichiometric activity coefficients; the latter, which is the experi- mentally determined quantity, is represented by an extension of equa- tion (62); thus (cf. p. 135), log 7 = - - + C"Y - 1<* (1 + - 001 where v is the number of ions produced by one molecule of electrolyte on dissociation, m is the molality of the solution and M i is the molecular 154 FREE ENERGY AND ACTIVITY weight of the solvent. This equation has been employed for the purpose of extrapolating activity coefficient data to dilute solutions from ac- curate measurements made at relatively high concentrations. It is not certain that this procedure is altogether justifiable, for the value of a obtained from activity data at high concentrations is often different from that derived from measurements on the same electrolyte in dilute solutions. Extension of the Debye-Hiickel Theory. In the calculation of the electrical density in the vicinity of an ion (p. 82), it was assumed that ZiGp/kT was negligible in comparison with unity, so that all terms beyond the first in the exponential series could be neglected. According to calculations made by Miiller (1927), the neglect of the additional terms is justifiable provided that a > that is, if the mean ionic diameter a is greater than about 1.4 X 10- 8 z 2 /D at 25. It follows, therefore, that the additional terms are negligible in aqueous solution if a/2 2 exceeds 1.6 X 10~ 8 ; for a uni-univalent salt, therefore, a should exceed 1.6 X 10~~ 8 cm., but for a bi-bivalent electro- lyte a must exceed 6.4 X 10~ 8 cm. if the Debye-Hiickel approximation is to be valid. Since ionic diameters are rarely as high as the latter figure, it is seen that salts of high valence type might be expected to exhibit dis- crepancies from the simple Debye-Hiickel behavior. Since the limiting values of a are larger the smaller the dielectric constant D of the medium, the deviations become more marked and will occur with electrolytes of lower valence type in media of low dielectric constant. The potential ^ is given approximately by equation (15) on page 83, and hence the assumption, made by Debye and Hiickel, that 2c^//cT is small compared with unity, is equivalent to stating that - D r and this is less likely to be true the higher the valence of the ion and the smaller its radius, and the smaller the dielectric constant of the medium. In order to avoid the approximation involved in neglecting the higher terms in the exponential series, Gronwall, LaMer and Sandved w used the complete expansion for the electrical density, and solved the differen- tial equation, following the introduction of the Poisson equation, in the form of a power series. The result obtained for a symmetrical valence type electrolyte, that is one with both ions of the same valence, is given by the following expression, which should be compared with equation "Gronwall, LaMer and Sandved, Physik. Z., 29, 358 (1928); see also, LaMer, Gronwall and Greiff, /. Phys. Chem., 35, 2345 (1031). ION-ASSOCIATION 155 (58), viz., Nft* 1 2DRT I + *a 00 / Nz 2 * 2 \* m+1 - 2m y 2m+1 (Ka)], (69) where X(KCL) and 7(ica) are known, but complicated, functions of *a. The summation in equation (69) should be carried over all integral values of m from unity to infinity, but it is found that successive terms in the series decrease rapidly and it is sufficient, in general, to include only two terms. In the application of equation (69) an arbitrary value of a is chosen so as to give calculated activity coefficients which agree with those de- rived by direct experiment; the proper choice of a is made by a process of trial and error until a value is found that is satisfactory over a range of concentrations. There is no doubt that the Gronwall-LaMer-Sandved extension represents an important advance over the simple Debye- Hlickel treatment, for it frequently leads to more reasonable values of the mean ionic diameter. 14 The validity of equation (69) has been tested by a variety of activity measurements and the results have been found satisfactory; were it not for the tedious nature of the calculations it would probably be more widely used. It is necessary to call attention to the fact that equation (69) was deduced for symmetrical valence electrolytes; for unsymmetrical types the corresponding equation is of a still more complicated nature. Ion-Association. A device, proposed by Bjerrum, 15 for avoiding the difficulty of integrating the Poisson equation when it is not justifiable to assume that z^lkT is much smaller than unity, involves the concept of the association of ions to form ion-pairs (cf. p. 96). It may be remarked that, in a sense, a solution, such as that of Gronwall, Sandved and LaMer, of the differential equation resulting from the use of the complete expres- sion for the electrical density, makes the Bjerrum treatment unnecessary. The results obtained are, nevertheless, of interest, especially in connection with their application to media of low dielectric constant. According to the Boltzmann distribution law, the number drii of ions of the iih kind in a spherical shell of radius r and thickness dr, sur- rounding a specified ion, is given by dm = n % 4an*c~ w i kT dr, (70) " LaMer et al., /. Phys. Chem., 35, 1953 (1931); 40, 287 (1936); /. Am. Chem. Soc. t 53, 2040, 4333 (1931); 54, 2763 (1932); 56, 544 (1934); Partington et al., Trans. Faraday Soc., 30, 1134 (1934); Phil Mag., 22, 857 (1936). Bjerrum, K. Danske Vidensk. Selsk. Mat.-fys. Medd., 7, No. 9 (1926); Fuoss and Kraus, J. Am. Chem. Soc., 55, 1019 (1933); Fuoss, Trans. Faraday Soc., 30, 967 (1934); Chem. Revs., 17, 227 (1935). 156 FREE ENEROT AND ACTIVITY where n t is the number of ions of the z'th kind in unit volume and W is the work required to separate one of these ions from the central ion; k is the Boltzmann constant and T is the absolute temperature. The central ion, supposed to be positive, carries a charge z+e and that of the ith ion, which is of opposite sign, is 2_c; if Coulomb's law is assumed to hold at small interionic distances and the ions are regarded as point charges separated by a medium with an effective dielectric constant (D) equal to that of the solvent, then the work required to separate the ions from a distance r to infinity, and hence the value of W, is given by W (71) The influence of ions other than the pair under consideration is neglected in this derivation. Substituting this result for W in equation (70), it follows that dni = n % 4wr*e-+'-'*' DrkT dr. (72) The fraction dn^dr is a measure of the probability P(r) of finding an ion of charge opposite to that of the central ion at a distance r from the latter; thus P(r) P(r) . (73) If the right-hand side of this equa- tion, for various values of r, is plotted against r, the result is a curve of the type shown in Fig. 53, the actual form depending on the valences z+ and Z-. of the oppositely charged ions, and also on the dielectric con- stant of the medium. It will be observed that at small distances of approach there is a very high prob- ability of finding the two ions to- gether, but this probability falls rapidly, passes through a minimum and then increases somewhat for in- creasing distances between the ions. The interionic distance r m i n ., for which the probability of finding two oppositely charged ions together is a minimum, can be obtained by differentiating equation (73) with respect to r and setting the result equal to zero ; in this way it is found that (74) FIG. 53. Distribution of oppositely charged ions about a central ion (Bjerrum) min * 2DkT The suggestion was made by Bjerrum that all ions lying within a sphere of radius r m i n . should be regarded as associated to form ion-pairs, THE FRACTION OP ASSOCIATION 157 whereas those outside this sphere may be considered to be free. The higher the value of r m [ n . the greater the volume round a given ion in which the oppositely charged ions can be found, and hence the greater the probability of the occurrence of the ion-pairs. It is evident, there- fore, from equation (74) that ion association will take place more readily the higher the valences, z+ and z_, of the ions of the electrolyte and the smaller the dielectric constant of the medium. This conclusion is in general agreement with experiment concerning the deviations from the behavior to be expected from the Debye-Hiickel treatment based on the assumption of complete dissociation. Attention may be called to the fact, the exact significance of which is not altogether clear, that the value of r min . given by equation (74) is about twice the mean ionic diameter a which must be exceeded if the additional terms in the Debye-Hiickel expansion may be neglected (see p. 154). The Fraction of Association. If equation (72) is integrated between r = a, where a is the effective mean diameter of the ions, or their dis- tance of closest approach, and r = r m i n ., the result should give the num- ber, which will be less than unity, of oppositely charged ions that may be regarded as associated with a given ion. In other words, this quantity is equal to the fraction of association (6) of the strong electrolyte into ion- pairs; thus rmin. rV-'+'-^^r. (75) If JVc/1000, where c is the concentration in moles per liter, is written in place of n,, and if both ions are assumed to be univalent, equation (75) may be expressed in the form where and The values of Q(b) as defined above have been tabulated for various values of b from 1 to 80, and so by means of equation (76) it is possible to estimate the extent of association of a uni-univalent electrolyte con- sisting of ions of any required mean diameter a, at a concentration c in a medium of dielectric constant D. It will be seen from equation (76) that in general B increases as b increases, i.e., 6 increases as the mean diameter a of the ions and the dielectric constant of the solvent decrease. The values for the fraction of association of a uni-univalent electrolyte in water at 18 have been calculated by Bjerrum for various concentra- tions for four assumed ionic diameters ; the results are recorded in Table XXXVII. The extent of association is seen to increase markedly with decreasing ionic diameter and increasing concentration. The values are 158 FREE ENERGY AND ACTIVITY appreciably greater in solutions of low dielectric constant, as is apparent from the factor 1/D 8 in equation (76). TABLE XXXVII. FRACTION OP ASSOCIATION (0) OF UNI-UNIVALBNT ELECTROLYTE IN WATER AT 18 Concentration a 0.001 0.005 0.01 0.05 0.1 0.5 1.0 H 2.82A 0.002 0.005 0.017 0.029 0.090 0.138 2.35 0.001 0.004 0.008 0.028 0.048 0.140 0.206 1.76 0.001 0.007 0.012 0.046 0.072 0.204 0.286 The Association Constant. Suppose that a salt MA is completely ionized in solution and that a certain fraction of the ions are associated as ion-pairs; an equilibrium may be supposed to exist between the free M+ and A~ ions, on the one hand, and ion-pairs on the other hand. If the law of mass action [cf. equation (20)] is applied to this equilibrium, the result is _ Activity of M+ X Activity of A~ Activity of ion-pairs where K is the dissociation constant (cf. p. 163). If c is the concentra- tion of the salt MA, the concentration of associated ions is 6c while that of each of the free ions is (1 0)c; further, if fi represents the mean activity coefficient of the ions and /2 is that of the ion-pairs, then (1 - 8)c X (1 - fl)c ft (1 - OYc /! *~ fe u = ~~e -- 7T (77) For very dilute solutions, i.e., when c is small, the activity coefficients are almost unity, while 8 is negligible in comparison with unity (see Table XXXVII); equation (77) then reduces to *-! .'. K- - j. (78) where Jf" 1 , the reciprocal of the dissociation constant, is called the association constant of the completely ionized electrolyte. Introducing the value of given by equation (76), the result is and so the dissociation constant K can be calculated for any assumed value of the distance of closest approach of the ions in a medium of known dielectric constant. TRIPLE IONS 159 15 10 $ I . I 0.5 L5 A test of equation (79), based on the theory of ion association, is provided by the measurements of Fuoss and Kraus 16 of the conductance of tetraisoamylammonium nitrate in a series of dioxane-water mixtures of dielectric constant ranging from 2.2 to 78.6 (cf. Fig. 21) at 25. From the results in dilute solution the dissociation constants were calculated by the method described on page 158. The values of log K, plotted against log D of the medium, are indicated by the points in Fig. 54, whereas the full curve is that to be expected from equa- tion (79) if a is taken as 6.4A. The agree- ment between the experimental and theor- etical results is very striking. It will be observed that as the dielectric constant increases the curve turns sharply down- wards and crosses the log D axis at a value of the dielectric constant of approximately 41. The significance of this result is that for ions of mean effective diameter equal to 6.4A, the dissociation constant of the electrolyte is very large, and hence the extent of association becomes negligible when the dielectric constant of the solvent exceeds a value of about 41.* For smaller ions or for ions of higher valence, the di- electric constant would have to attain a larger value before an almost completely ionized electrolyte would be also completely dissociated. Triple Ions. The concept of ion-pairs has been extended to include the possibility of the presence in solution of groups of three ions, viz., H h or | , i.e., triple ions, held together by electrostatic forces. 17 Such triplets might be expected to form most readily in solvents of low dielectric constant, for it is in such media that the forces of electrostatic attraction would be greatest. Consider an electrolyte MA in a medium of low dielectric constant ; there will be an equilibrium between the ions M+ and A~ and the ion-pairs, as described above. In this case, however, the ion-pair formation will be considerable and will approach unity. If 1 6 is replaced by , the fraction of the electrolyte present as free ions, and if both 6 and the activity coefficient factor are assumed to be 18 Fuoss and Kraus, J. Am. Chem. Soc., 55, 1019 (1933). * According to the simple calculations on page 156 the dielectric constant necessary for the solvent in which a uni-univalent electrolyte whose mean ionic diameter is 6.4 X 10~* cm. should be dissolved in order that there may be no appreciable association is 2.79 X 10-/6.4 X Ifr* i.e., 42. "Fuoss and Kraus, J. Am. Chem. Soc., 55, 2387 (1933); Fuoss, ibid., 57, 2604 (1935); Chem. Revs., 17, 227 (1935); for reviews, see Kraus, /. Franklin Inst., 225, 687 (1938); Science, 90, 281 (1939). 1.0 lo? D FIG. 54. Association constant and dielectric constant (Fuoss and Kraus) 160 FREE ENERGY AND ACTIVITY unity, equation (77) can be written as k 2 c, (80) where k is the approximate dissociation constant and c is the total elec- trolyte concentration. If in addition to ion-pairs (dual ions) there are present triple ions, viz., MAM+ and AMA", the following equilibria. MAM+ ^ MA + M+, AMA- ^ MA + A-, also exist. If the formation of MAM+ and AMA~ is due to electrical effects only, there will be an equal tendency for both these ions to form ; the mass action constant & 8 of the two equilibria may thus be expected to be the same. Hence, neglecting activity coefficients, , CMACM+ CMAM+ CAMA" .-.-^--J-. (82) CMAM+ CAMA- The triple ions should consequently be formed in the same ratio as that in which the simple ions are present in the solution. If ot 3 is the fraction of the total electrolyte existing as either of the triple ions, e.g., MAM+, then CMAM+ is equal to <x 3 c. Since the amount of these ions will be small, CMA may be taken as approximately equal to the total concentration c, and CM* can be assumed to remain as ac. Substituting these results in equation (81), it follows that k, = -> (83) s and since k, by equation (80), is equal to 2 c, i.e., a is Vfc/c, it is found that Vfo 3 = -r-- (84) A/3 Although dual ions have no conducting power, since they are elec- trically neutral, triple ions are able to carry current and contribute to the conductance of the solution. If A is the sum of the equivalent con- ductances of the simple ions at infinite dilution, and Xo is the sum of the values for the two kinds of triple ions, then since the latter are formed in the same ratio as the simple ions, it follows that the observed equiva- lent conductance is given by A = AO + 0(3X0, interionic effects being neglected. Substituting Vfc/c for a, and Vfcc/fc* TRIPLE IONS AND CONDUCTANCE MINIMA for as, it is seen that A A /*.* ^ A = A \h + A -r~ * C A/3 .'. AVc = , + c. 161 (85) (86) If AVc is plotted against c for media of low dielectric constant, in which triple ions can form to an appreciable extent, the result should be a straight line; this expectation has been cpnfirmed by experiment, as 1.66 1.55 0.25 0.50 C X 10* 0.75 Fia. 55. Test of triple-ion theory (Fuoss and Kraus) shown by the points in Fig. 55 which are for tetrabutylammonium picrate in anisole. The deviation from the straight line becomes evident only at high concentrations. Triple Ions and Conductance Minima. Since equation (85) is of the form A-A + Vc (87) where A and B are constants, it is evident that the first term on the right- hand side decreases and the second term increases as the concentration is increased; it is possible, therefore, for a minimum in the equivalent conductance to occur, as has been found experimentally (p. 52). The physical significance of this result is that with increasing concentration the single ions are replaced by electrically neutral ion-pairs, and so the conductance falls; at still higher concentrations, however, the ion-pairs are replaced by triple ions having a relatively high conducting power, and so the equivalent conductance of the solution tends to increase. 162 FREE ENERGY AND ACTIVITY The condition for the conductance minimum is found by differentiating equation (87) with respect to c and setting the result equal to zero; this procedure gives _ A min. n = * (88) AO By substituting this value in equation (85), and utilizing the relation- ships given above for a and 3 , it is found that A min . = 2(A a) min . = 2(Xoa 3 ) mi n.. (89) It is seen from equation (88) that the concentration for the minimum conductance is proportional to fc 3 , and so is inversely proportional to the stability of the triple ions. The minimum occurs when the conductance due to these ions, i.e., Xoa, is equal to that due to the single ions, i.e., A a. By means of a treatment analogous to that described above for cal- culating the association constant for the formation of ion-pairs, it is possible to derive an expression for k^ 1 which is analogous to equation (79) ; 18 the result may be put in the form '* (90) where 7(6, r) is a function of 6, which has the same significance as before, and of the distance r between the ions. In the region of the minimum conductance, the value of 7(6, r)/6 3 does not change appreciably, and equation (90) can be written as where A is a constant and D is the dielectric constant of the medium; the dissociation constant of the triple ions (& 3 ) is thus proportional to Z) 3 . Since the concentration c rn i n . at which the minimum equivalent conduct- ance is observed is proportional to & 3 , it follows that D 3 - = constant; (91) Cmin. this is the rule derived empirically by Walden (p. 53). 19 The fact that the concentration at which the conductance minimum occurs decreases with decreasing dielectric constant of the solvent is shown by the results in Fig. 21 (p. 53). In media of very low dielectric M Fuoss and Kraus, J. Am. Chem. Soc., 55, 2387 (1933). 19 See also, Gross and Halpern, J. Chem. Phys., 2, 188 (1934); Fuoss and Kraus, ibid., 2, 386 (1934). EQUILIBRIA IN ELECTROLYTES 163 constant, however, the minimum does not appear, but the conductance curves show inflections; these are attributed to mutual interactions be- tween two dipoles, i.e. , ion-pairs, as a result of which quadripoles are formed. The consequence of this is that the normal increase of conductance beyond the minimum, due to the formation of triple ions, is inhibited to so|me extent. If the dielectric constant of the solvent exceeds a certain value, depending on the mean diameter and valence of the ions, there is no appreciable formation of triple ions at any concentration, and hence there can be no conductance minimum. Equilibria in Electrolytes: The Dissociation Constant. When any electrolyte MA is dissolved in a suitable solvent, it yields M+ and A~ ions in solution to a greater or lesser extent depending on the nature of MA; even if ionization is complete, as is the case with simple salts in aqueous solution, there may still be a tendency for ion-pairs to form in relatively concentrated solution, so that dissociation is not necessarily complete. In general, therefore, there will be set up the equilibrium MA ^ M+ + A-, where M + and A~ represent the free ions and MA is the undissociated portion of the electrolyte which includes both un-ionized molecules and ion-pairs. Application of the law of mass action, in the form of equation (20), to this equilibrium gives (92) CtMA where the a terms are the activities of the indicated species; the equi- librium constant K is called the dissociation constant of the electrolyte. The term " ionization constant" is also frequently employed in the litera- ture of electrochemistry, but since the equilibrium is between free ions and undissociated molecules, the expression " dissociation constant" is preferred. Writing the activity terms in equation (92) as the product of the concentration and the activity coefficient, it becomes CMA JMA Further, if a is the degree of dissociation of the electrolyte (cf. p. 96) whose total concentration is c moles per liter, then CM+ and CA~ are each equal to ac, and CMA is equal to c(l a); it follows, therefore, that K = .L-. (94) 1 a /MA If the solution is sufficiently dilute, the activity coefficients are approxi- mately unity, and so equation (94) reduces under these conditions to 164 FREE ENERGY AND ACTIVITY which is the form of the so-called dilution law as originally deduced by Ostwald (1888). It will be noted that in the approximate equation (95) the symbol k has been used; this quantity is often called the "classical dissociation constant," but as it cannot be a true constant it is preferable to refer to it as the "classical dissociation function" or, in brief, as the "dissociation function." The relation between the function k and the true or " thermodynamic " dissociation constant K is obtained by combining equations (94) and (95); thus (96) JMA Provided the ionic strength of the medium is not too high, the activity coefficient of the undissociated molecules never differs greatly from unity; hence, equation (96) may be written as K = fc(/ M +/A-). (97) If the solution is sufficiently dilute for the Debye-Hiickel limiting law to be applicable, it follows from equation (54), assuming the ions M+ and A" to be univalent, for simplicity, that log/ M + = log /A- = - Ac, (98) the ionic strength, Zc t Z| 2 , being equal to |[(ac X I 2 ) + (<*c X I 2 )], i.e., to ac. Upon taking logarithms of equation (97) and substituting the values of log/M+ and log /A- as given by (98), the result is log K = log k - 2A Vac. (99) The plot of the values of log fc, obtained at various concentrations, against Vac should thus give a straight line of slope 2 A ; for water at 25 the value of A is 0.509 (Table XXXV) and so the slope of the line should be - 1.018. In order to test the reliability of equation (99) it is necessary to know the value of the degree of dissociation at various concentrations of the electrolyte MA; in his classical studies of dissociation constants Ostwald, following Arrhenius, assumed that a at a given concentration was equal to the conductance ratio A/A , where A is the equivalent conductance of the electrolyte at that concentration and A is the value at infinite dilu- tion. As already seen (p. 95), this is approximately true for weak elec- trolytes, but it is more correct, for electrolytes of all types, to define a as A/A 7 where A' is the conductance of 1 equiv. of free ions at the same ionic concentration as in the given solution. It follows therefore, by substituting this value of a in equation (95), that EQUILIBRIA IN ELECTROLYTES 165 Since A for various concentrations can be obtained from conductance data and the Onsager equation, by one of the methods described in Chap. Ill, it is possible to derive the dissociation function k for the corresponding concentrations. The results obtained for acetic acid in agueous solution at 25 are given in Table XXXVIII, 20 and the values of TABLE XXXVIII. DISSOCIATION CONSTANT OF ACETIC ACID AT 25 cX10 A A' a *X10 XX10 0.028014 210.38 390.13 0.5393 1.768 1.752 0.11135 127.75 389.79 0.3277 1.779 1.754 0.21844 96.49 389.60 0.2477 1.781 1.751 1.02831 48.15 389.05 0.1238 1.797 1.751 2.41400 32.22 388.63 0.08290 1.809 1.750 5.91153 20.96 388.10 0.05401 1.823 1.749 9.8421 16.37 387.72 0.04222 1.832 1.747 20.000 11.57 387.16 0.02987 1.840 1.737 52.303 7.202 386.18 0.01865 1.854 1.722 119.447 4.760 385.18 0.01236 1.847 1.688 230.785 3.392 384.26 0.008827 1.814 1.632 log k are plotted against Vac in Fig. 56; the dotted line has the theoretical slope required by equation (99). It is clear that in the more dilute solutions the experimental results are in excellent agreement with theory, 4.726 4.786 a* I 4.746 4.766 0.01 0.02 0.03 0.04 FIQ. 56. Dissociation constant of acetic acid (Maclnnes and Shedlovsky) but at higher concentrations deviations become evident. The same con- clusion is reached from an examination of the last column in Table * Maclnnes and Shedlovsky, J. Am. Chem. Soc., 54, 1429 (1932); Maclnnes, J. Franklin Inst. t 225, 661 (1938). 166 FREE ENERGY AND ACTIVITY XXXVIII which gives the results for K derived from equation (99) using the theoretical value of A, i.e., 0.51. The first figures are seen to be virtually constant, as is to be expected, the mean value of K being 1.752 X 10"*. At infinite dilution the activity coefficient factor is unity and so the extrapolation of the dissociation functions k to infinite dilution should give the true dissociation constant K] the necessary extrapolation is carried out in Fig. 56, from which it is seen that the limiting value of log k is 4.7564, so that K is 1.752 X 10~ 6 , as given above. Similar results to those described for acetic acid in aqueous solution have been recorded for other weak acids in aqueous solution, and also for several^ acids in methyl alcohol. 21 In each case the plot of log k against Vac was found to be a straight line for dilute solutions, the slope being in excellent agreement with that required by the Debye-Huckel limiting law. The deviations observed with relatively concentrated solutions, such as those shown in Fig. 56, are partly due to the failure of the limiting law to apply under these conditions, and partly to the change in the nature, e.g., dielectric constant, of the medium resulting from the presence of appreciable amounts of an organic acid. Strong Electrolytes. The arguments presented above are readily applicable to weak electrolytes because the total concentration can be quite appreciable before the ionic strength becomes large enough for the Debye-Huckel limiting law to fail; for example, the results in Table XXXVIII extend up to a concentration of 0.2 N, but the ionic strength is then about 0.04. With relatively strong electrolytes, however, the procedure can be used only for very dilute solutions. In these circum- stances it is preferable to return to equation (97), which should hold for all types of electrolytes of the general formula MA, and to employ activity coefficients obtained by direct experimental measurement, instead of the values calculated from the Debye-Huckel equations. The product /M+/A~ in equation (97) may be replaced by the square of the mean activity coefficient of the electrolyte, i.e., by /, in accordance with the definition of equation (30); it follows, therefore, that equation (100) may be modi- fied so as to give AV The accuracy of this equation has been confirmed for a number of salts generally regarded as strong electrolytes, as the data in Table XXXIX serve to show. 22 It is evident from these results that the law of mass action holds for strong, as well as for weak electrolytes, provided it is "Maclnnes and Shedlovsky, /. Am. Chem. Soc., 57, 1705 (1935); Saxton et al t ibid., 55, 3638 (1933); 56, 1918 (1934); 59, 1048 (1937); Brock man and Kilpatrick, ibid., 56, 1483 (1934); Martin and Tartar, ibid., 59, 2672 (1937); Belcher, ibid., 60, 2744 (1938); see also, Maclnnes, "The Principles of Electrochemistry," 1939, Chap. 19. " Davies et al., Trans. Faraday Soc., 23, 351 (1927); 26, 592 (1930); 27, 621 (1931); 28, 609 (1932); "The Conductivity of Solutions," 1933, Chap. IX. INTERMEDIATE AND WEAK ELECTROLYTES 167 applied in the correct manner. The view expressed at one time that the law of mass action was not applicable to strong electrolytes was partly due to the employment of the Arrhenius method of calculating the degree of dissociation, and partly to the failure to make allowance for deviations from ideal behavior. TABLE XXXIX. APPLICATION OP LAW OF MASS ACTION TO STRONG ELECTROLYTES Salt c A/A' f K KNO, 0.01 0.994 0.916 1.40 0.02 0.989 0.878 1.38 0.05 0.975 0.806 1.32 0.10 0.961 0.732 1.37 AgNO, 0.01 0.993 0.902 1.10 0.02 0.989 0.857 1.31 0.05 0.973 0.783 1.12 0.10 0.957 0.723 1.23 0.50 0.883 0.526 1.18 Intermediate and Weak Electrolytes. The calculation of the degree of dissociation by the methods given in Chap. Ill presuppose the availa- bility of suitable conductance data for electrolytes which are virtually completely dissociated at the appropriate concentrations. There is gen- erally no difficulty concerning this matter if the solvent is water, but for non-aqueous media, especially those of low dielectric constant, the pro- portion of undissociated molecules may be quite large even at small concentrations, and no direct method is available whereby the quantity A' can be evaluated from conductance data. For solvents of this type the following method, which can be used for any systems behaving as weak or intermediate electrolytes, may be employed. 23 The Onsager equation for incompletely dissociated electrolytes can be written (cf. p. 95) as A' = Ao - (A + A ) If a variable x is defined by xm (A + B ^ (1Q3) equation (102) becomes 28 Fuoss and Kraus, J. Am. Chem. Soc., 55, 476 (1933); Fuoss, ibid., 57, 488 (1935); TVan*. Faraday Soc., 32, 594 (1936). 168 FREE ENERGY AND ACTIVITY where F (x) is a function of x represented by the continued fraction F(x) = 1 - x(l - x(l - x(l )-)-)- = | cos 2 1 cos" 1 (- fsV3). Values of this function have been worked out and tabulated for values of x from zero to 0.209 in order to facilitate the calculations described below. Taking the activity coefficient of the undissociated molecules, as usual, to be equal to unity, and replacing /M+/A- by /, where f is the mean activity coefficient, equation (94) becomes (105) and if the value of a given by equation (104) is inserted, the result is which on multiplying out and rearranging gives *M = J_.^I + !. (106) A KA.Q F(x) AO It is seen from equation (106) that the plot of F(x)/\ against \cf/F(x) should be a straight line, the slope being equal to 1/KA.l and the inter- cept, for infinite dilution, giving 1/Ao. In this manner it should be possible to determine both the dissociation constant K of the electrolyte and the equivalent conductance at infinite dilution (A ) in one operation. In order to obtain the requisite plot, an approximate estimate of A is first made by extrapolating the experimental data of A against Vc, and from this a tentative result for x is derived by means of equation (103), since the Onsager constants A and B are presumably known (see Table XXIII). In this way a preliminary value of F(x) is obtained which is employed in equation (106) ; the activity coefficients required are calcu- lated from the Debye-Hiickel limiting law equation (98), using the value of a given by equation (104) from the rough estimates of A and F(x). The results are then plotted as required by equation (106), and the datum for AO so obtained may be employed to calculate F(x) and a more accurately; the plot whereby A and K are obtained may now be re- peated. The final results are apparently not greatly affected by a small error in the provisional value of A and so it is not often necessary to repeat the calculations. With A known accurately, it is possible to determine the degree of dissociation at any concentration, if required by means of equations (103) and (104), and the tabulated values of F(x). SOLUBILITY EQUILIBRIA 169 The work of Fuoss and Kraus and their collaborators and of others has shown that equation (106) is obeyed in a satisfactory manner by a number of electrolytes, both salts and acids, in solvents of low dielectric constant. 24 The results of plotting the values of F(x)/A against Acf 2 /F(x) for solutions of tetramethyl- and tetrabutyl-ammonium picrates in ethylene chloride are shown in Fig. 57; the intercepts are 0.013549 and 0.17421, and the slopes of the straight lines are 5.638 and 1.3337, re- 0.034 i.o 4.0X10' 20 3.0 Acf*/F(x) Fia. 57. Salts in media of low dielectric constant (Fuoss and Kraus) spectively. It follows, therefore, that for tetramethylammonium picrate K is 0.3256 X 10~ 4 and A is 73.81 ohms" 1 cm. 2 , whereas the correspond- ing values for tetrabutylammonium picrate are 2.276 X 10~ 4 and 57.40 ohms" 1 cm. 2 , respectively. Solubility Equilibria : The Solubility Product Principle. It was seen on page 133 that the chemical potential of a solid is constant at a definite temperature and pressure; consequently, when a solution is saturated with a given salt M,, + A,_ the chemical potential of the latter in the solu- tion must also be constant, since the chemical potential of any substance present in two phases at equilibrium must be the same in each phase. It is immaterial whether this conclusion is applied to the undissociated molecules of the salt or to the ions, for the chemical potential is given by 4 Kraus, Fuoss et al, Trans. Faraday Soc., 31, 749 (1935); 32, 594 (1936): J. Am. Chem. Soc., 58, 255 (1936); 61, 294 (1939); 62, 506, 2237 (1940); Owen and Waters, ibid., 60, 2371 (1938); see also, Maclnnes, "The Principles of Electrochemistry," 1939, Chap. 19. 170 FREE ENERGY AND ACTIVITY either side of equation (26) ; thus, taking the left-hand side, it follows that M + + RT In a+) + v-(&- + RT In a_) = constant, v+ In a+ + v- In a_ = constant, - = constant (/.), (107) at a specified temperature and pressure. The constant K, as defined by equation (107) is the activity solubility product, and this equation ex- presses the solubility product principle, first enunciated in a less exact manner by Nernst (1889). If the activity of an ion is written as the product of its concentration, in moles (g.-ions) per liter, and the corre- sponding activity coefficient, equation (107) becomes =#., (108) and introducing the definition of the mean activity coefficient of the electrolyte M r+ A^_, it follows that t X A = K, t (109) where v is equal to v+ + i>_. If the ionic strength of the medium is low, the activity coefficient is approximately unity and equation (109) reduces to the approximate form = ft., (110) in which the solubility product principle is frequently employed. The significance of the solubility product principle is that when a solution is saturated with a given salt the product of the activities, or approximately the concentrations, of its constituent ions must be con- stant, irrespective of the nature of the other electrolytes present in the solution. If the latter contains an excess of one or other of the ions of the saturating salt, this must be taken into consideration in the activity product. Consider, for example, a solution saturated with silver chloride : then according to the solubility product principle, (111) or, approximately, CAg+Ccr = fc.<Agci). (112) If the solution which is being saturated with silver chloride already con- tains one of the ions of this salt, e.g., the chloride ion, then the term Ocr will represent the total activity of the chloride ion in the solution; since this is greater than that in a solution containing no excess of chloride ion, the value of a Ag + required according to equation (111) will be less in the former case. In its simplest terms, based on equation (112), the conclusion is that the silver ion concentration in a saturated solution of silver chloride containing an excess of chloride ions, e.g., due to the presence of potassium chloride in the solution, will be less than in a solution in pure water. Since the silver chloride in solution may be SOLUBILITY IN THE PRESENCE OF A COMMON ION 171 regarded as completely ionized, the silver ion concentration is a measure of the solubility of the salt; it follows, therefore, that silver chloride is less soluble in the presence of excess of chloride ions than in pure water. In general, if there is no formation of complex ions to disturb the equi- librium (cf. p. 172), the solubility of any salt is less in a solution con- taining a common ion than in water alone; this fact finds frequent application in analytical chemistry. Solubility in the Presence of a Common Ion. If So is the solubility of any sparingly soluble salt M, + A,_ in moles per liter in pure water, then if the solution is sufficiently dilute for dissociation to be complete, c+ is equal to v+So and c_ is equal to v-8o] hence according to equation (109) = (j>>l-),S&A. (113) In the simple case of a uni-univalent sparingly soluble salt, this be- comes K. = 52/2=. (114) These equations relate the solubility product to the solubility in pure water and the activity coefficient in the saturated solution; for practical purposes it is convenient to take the activity coefficient to be approxi- mately unity, since the solutions are very dilute, so that equation (114) can be written if Q 2 n/t OQ For a uni-univalent salt the saturation solubility in pure water is thus equal to the square-root of its solubility product; alternatively, it may be stated that the solubility product is equal to the square of the solubility in water. The solubility of silver chloride in water at 25 is 1.30 X 10"" 6 mole per liter; the solubility product is consequently 1.69 X 10~~ 10 . Suppose the addition of x moles per liter of a completely dissociated salt containing a common ion, e.g., the anion, reduces the solubility of the sparingly soluble salt from So to S; for simplicity all the ions present may be assumed to be univalent. The concentrations of cations in the solution, resulting from the complete dissociation of the sparingly soluble salt, is S, while that of the anions is S + x; it follows, therefore, by the approximate solubility product principle that S(S + x) = k. = Si .-. S = - \x + Viz 2 + Si (115) Using this equation, or forms modified to allow for the valences of the ions which may differ from unity, it is possible to calculate the solubility (S) of a sparingly soluble salt in the presence of a known amount (x) of a common ion, provided the solubility in pure water (So) is known. An illustration of the application of equation (115) is provided by the results in Table XL for the solubility of silver nitrite in the presence of silver 172 FREE ENERGY AND ACTIVITY nitrate (I), on the one hand, and of potassium nitrite (II) on the other hand; the calculated values are given in the last column. 25 The agree- ment between the observed and calculated results in these dilute solu- TABLE XL. SOLUBILITY OF SILVER NITRITE IN THE PRESENCE OF COMMON ION X S S moles/liter I II Calculated 0.000 0.0269 0.0269 (0.0269 - So) 0.00258 0.0260 0.0259 0.0259 0.00588 0.0244 0.0249 0.0247 0.01177 0.0224 0.0232 0.0227 tions is seen to be good, perhaps better than would be expected in view of the neglect of activity coefficients ; in the presence of larger amounts of added electrolytes, however, deviations do occur. Much experimental work has been carried out with the object of verifying the solubility product principle in its approximate form, and the general conclusion reached is that it is satisfactory provided the total concentration of the solution is small; at higher concentrations discrepancies are observed, especially if ions of high valence are present. It was found, for example, that in the presence of lanthanum nitrate the solubility of the iodate decreases at first, in agreement with expectation, but as the concentration of the former salt is increased, the solubility of the lanthanum iodate, instead of decreasing steadily, passes through a minimum and then increases. Such deviations from the expected behavior are, of course, due to neglect of the activity coefficients in the application of the simple solubility product principle; the effect of this neglect becomes more evident with increasing concentration, especially if the solution contains ions of high valence. It is evident from the Debye-Huckel limiting law equation that the departure of the activity coefficients from unity is most marked with ions of high valence because the square of the valence appears not only in the factor preceding the square-root of the ionic strength but also in the ionic strength itself. The more exact treatment of solubility, taking the activity coefficients into consideration, is given later. Formation of Complex Ions. In certain cases the solubility of a sparingly soluble salt is greatly increased, instead of being decreased, by the addition of a common ion ; a familiar illustration of this behavior is provided by the high solubility of silver cyanide in a solution of cyanide ions. Similarly, mercuric iodide is soluble in the presence of excess of iodide ions and aluminum hydroxide dissolves in solutions of alkali hydroxides. In cases of this kind it is readily shown by transference measurements that the silver, mercury or other cation is actually present in the solution in the form of a complex ion. The solubility of a sparingly soluble salt can be increased by the addition of any substance, whether it Creighton and Ward, J. Am. Chem. Soc., 37, 2333 (1915). DETERMINATION OF INSTABILITY CONSTANT 173 contains a common ion or not, which is able to remove the simple ions in the form of complex ions. For example, if either cyanide ion or am- monia is added to a slightly soluble silver compound, such as silver chloride, the silver ions are converted into the complex ions Ag(CN)i~ or Ag(NH 8 )i~, respectively. In either case the concentration of free silver ions is reduced and the product of the concentrations (activities) of the silver and chloride ions falls below the solubility product value: more silver chloride dissolves, therefore, in order to restore the condition requisite for a saturated solution. If sufficient complex forming material is present the removal of the silver ions will continue until the whole of the silver chloride has dissolved. Although by far the largest proportion of the silver in a complex cyanide solution is present in the form of argentocyanide ions, Ag(CN)F, there is reason for believing that a small concentration of simple silver ions is also present ; the addition of hydrogen sulfide, for example, causes the precipitation of silver sulfide which has a very low solubility product. It is probable, therefore, that an equilibrium of the type Ag(CN) 2 - ^ Ag+ + 2CN- dxists between complex and free ions in an argentocyanide solution and similar equilibria are established in other instances. For the general case of a complex ion M fl A*, the equilibrium is (116) rA~, and application of the law of mass action gives or, using concentrations in place of activities, (117) The constant Ki (or k t ) is called the instability constant of the complex ion ; it is apparent that the greater its value the greater the tendency of the complex to dissociate into simple ions, and hence the smaller its stability. The reciprocal of the instability constant is sometimes encountered; it is referred to as the stability constant of the complex ion. Determination of Instability Constant. Two methods have been mainly used for determining the instability constants of complex ions; one involves the measurement of the E.M.F.'S of suitable cells, which will be described in Chap. VII, and the other depends on solubility studies. The latter may be illustrated by reference to the silver-ammonia (argent- ammine) complex ion. 26 If the formula of the complex is Ag m (NHs);}~, the "See also, Edmonds and Birnbaum, J. Am. Chem. Soc. t 62, 2367 (1940); Lanford and Kiehl, ibid., 63, 667 (1941). 174 FREE ENERGY AND ACTIVITY instability constant, using concentrations as in equation (117), is given by + (118) ex where, for simplicity of representation, the concentration of the complex ion is given by ex- If a solution of ammonia is saturated with silver chloride, then by the solubility product principle, CAg+Ccr gives the solubility product A;., and hence CA B + is equal to fc./ccr ; for such a system equation (118) becomes The concentration c of the silver salt in the ammonia solution may be regarded as consisting entirely of the complex ion, since the normal solubility of silver chloride is very small, so that Cx is virtually equal to c ; the concentration of the chloride ion may be taken as me because of the reaction m AgCl + n NH 3 = Ag m (NH 3 )+ + m C1-, and so equation (119) may be written as *,- m ~v - . CNH, . . -^j = constant. By means of this equation it is possible to evaluate n/(m +1) from a number of measurements of the solubility (c) of silver chloride in solu- tions containing various concentrations (CNH,) of ammonia. In order to derive m it is necessary to determine the solubility of silver chloride in ammonia in the presence of an excess of chloride ions ; equa- tion (119) then takes the form If in a series of experiments the concentration of ammonia (CNH,) is kept constant, while the amount of excess chloride (ccr) is varied, equa- tion (120) becomes Cere = constant, so that if the solubility c is measured, the value of m may be determined. Alternatively, solubility measurements may be made in the presence of excess of silver ions; in this case ccr is set equal to &,/CA g + in equation (119), and the subsequent treatment is similar to that given above.* * For data obtained in an actual experiment, see Problem 11. ACTIVITY COEFFICIENTS FROM SOLUBILITY MEASUREMENTS 175 Activity Coefficients from Solubility Measurements. The activity coefficient of a sparingly soluble salt can be determined in the presence of other electrolytes by making use of the solubility product principle. 27 In addition to the equations already given, this principle may be stated in still another form by introducing the definition of the mean ionic con- centration, i.e., c, which is equal to c+c!r, into equation (109); this equation then becomes K., (121) (122) .-. f The mean activity coefficient of a sparingly soluble salt in any solution could thus be evaluated provided the solubility product (K 9 ) and the mean concentration of the ions of the salt in the given solution were known. In or- der to calculate K s the value of c-t is determined in solutions of different ionic strengths and the results are then extrapo- lated to infinite dilution ; un- der the latter conditions f is, of course, unity and hence K\ l9 is equal to the extrapo- lated value of c. The method of calculation will be described with refer- ence to thallous chloride, the FIG. 58. Extrapolation of solubility data for thallous chloride 0.018 0.016 0,014 <u 0.2 04 solubility of which has been measured in the presence of various amounts of other elec- trolytes, with and without an ion in common with the saturating salt. By plotting the values of c for the thallium and chloride ions in solutions of different ionic strengths and extrapolating to zero, it is found that KU; which in this case is equal to V^, is 0.01428 at 25 (Fig. 58). It follows, therefore, from equation (122) that the mean activity coeffi- cient of thallous chloride in any saturated solution is given by 0.01428 c If the added electrolyte present contains neither thallous nor chloride "Lewis and Randall, J. Am. Chem. Soc. t 43, 1112 (1921); see also, Blagden and Davies, J. Chem. Soc. t 949 (1930); Davies, iWd., 2410, 2421 (1930); MacDougall and Hoffman, J. Phys. Chem. t 40, 317 (1936); Pearce and Oelke, iWd., 42, 95 (1938); Kolthoff and Lingane, ibid., 42, 133 (1938). 176 FREE ENERGY AND ACTIVITY ions, the mean ionic concentration is merely the same as the molar con- centration of the thallous chloride in the saturated solution, for then CTI+ and c c r are both equal to the concentration of the salt. When another thallous salt or a chloride is present, however, appropriate allowance must be made for the ions introduced in this manner. For example, in a solution containing 0.025 mole of thallous sulfate per liter, the saturation solubility of thallous chloride is 0.00677 mole per liter at 25; assuming both thallium salts to be completely dissociated at this low concentration, the total concentration of thallous ions is 2 X 0.025 + 0.00677, i.e., 0.05677 g.-ion per liter. The chloride ion concentration is 0.00677, and so the mean ionic concentration is (0.05677 X 0.00677)*, i.e., 0.01961 ; the mean activity coefficient is then 0.01428/0.01961, that is 0.728. The ionic strength of the solution is tf = K(CTI* X I 2 ) + (ccr X I 2 ) + (c s o 4 -- X 2 2 )] = K0.05677 + 0.00677 + 0.10) = 0.0817, so that the mean activity coefficient of a saturated solution of thallous chloride in the presence of thallous sulfate at a total ionic strength of 0.0817 is 0.728 at 25. The activity coefficients of thallous chloride at 25, obtained in the manner described above, in the presence of a number of salts are given in Table XLI; the data are recorded for solutions of various (total) ionic TABLE XLI. ACTIVITY COEFFICIENTS OF THALLOUS CHLORIDE IN THE PRESENCE OF VARIOUS ELECTROLYTES AT 25 Added Electrolyte y KNO, KC1 HC1 TWO, Tl^SO* 0.02 0.872 0.871 0.871 0.869 0.885 0.05 0.809 0.797 0.798 0.784 0.726 0.10 0.742 0.715 0.718 0.686 0.643 0.20 0.676 0.613 0.630 0.546 strengths. It is seen that at low ionic strengths the activity coefficient of the thallous chloride at a given ionic strength is almost independent of the nature of the added electrolyte; it has been claimed that if allowance is made for incomplete dissociation of the latter this independence per- sists to much higher concentrations. Solubility and the Debye-Hiickel Theory. The activity coefficients determined by the solubility method apply only to saturated solutions of the given salt in media of different ionic strengths; although their value is therefore limited, in many respects, they are of considerable interest as providing a means of testing the validity of the Debye-Htickel theory of electrolytes. It will be seen from equation (113), if the saturat- ing salt can be assumed to be completely dissociated, that the product Sf, where S is the solubility of the given salt in a solution not containing an ion in common with it, must be constant. It follows, therefore, that SOLUBILITY AND THE DEBYE-HttCKEL THEORY 177 if S is the solubility of the salt in pure water and S the value in the presence of another electrolyte which has no ion in common with the salt, and / and / are the corresponding mean activity coefficients, then __ / So' a Introducing the values of / and/ , as given by the Debye-Hiickel limiting law equation (54), it follows that logf--A+*-Ofi- V^), (123) OQ where yo and y are the ionic strengths of the solutions containing the sparingly soluble salt only and that to which other electrolytes have been added, respectively. Since vo is a constant for a given saturating salt, it follows that the plot of log S/S Q against Vji should be a straight line of slope AZ+Z-, where z+ and z_ are the valences of the two ions of the sparingly soluble substance. The constant A for water at 25 is 0.509, and so the linear slope in aqueous solutions should be 0.509 z+z_. For the purpose of verifying the conclusions derived from the Debye- Hiickel theory it is necessary to employ salts which are sufficiently soluble for their concentrations to be determined with accuracy, but not so soluble that the resulting solutions are too concentrated for the limiting law for activity coefficients to be applicable. A number of iodates, e.g., silver, thallous and barium iodates, and especially certain complex cobalt- ammines have been found to be particularly useful in this connection. The results, in general, are in very good agreement with the requirements of equation (123). The solubility measurements with the following four cobaltammines of different valence types, in the presence of such salts as sodium chloride, potassium nitrate, magnesium sulfate, barium Valence Theoretical Salt Type Slope I. [Co(NH,) 4 (NO,)(CNS)][Co(NH,) 2 (NO s ) 8 (C,04)] 1 : 1 0.509 II. [Co(NHi) 4 (CiO 4 )]&Oi 1 : 2 1.018 III. [Co(NH3)a][Co(NH 3 ) 2 (N0 2 MC 8 4 )]3 3 : 1 1.527 IV. [Co(NH,)][Fe(CN) 6 ] 3 : 3 4.581 chloride and potassium cobalticyanide, are of particular interest. 28 The values of log S/So are plotted against the square-root of the ionic strength in Fig. 59 ; the experimental data are shown by the points and the theoretical slopes are indicated by the full lines in each case. In certain cases the agreement with theory is not as good as depicted in "Br0nsted and LaMer, J. Am. Chem. Soc., 46, 555 (1924); LaMer, King and Mason, ibid., 49, 363 (1927). 178 FREE ENERGY AND ACTIVITY Fig. 59; this is particularly true if both the saturating salt and the added electrolyte are of high valence types. 29 The deviations are often due to incomplete dissociation, and also to the approximations made in the derivation of the Debye-Hiickel equations; as already seen, both these factors become of importance with ions of high valence. 0.10 - 0.02 0.04 0.06 0.08 0.10 Fia. 59. Dependence of solubility on ionic strength (LaMer, et al.) The factor A in equation (123) is proportional to 1/(DT)*, as shown on page 150; hence, a further test of this equation is to determine the slope of the plot of log S/So against Vp from Solubility data at different temperatures and in media of different dielectric constants. Such measurements have been made in water at 75 (D = 63.7), in mixtures of water and ethyl alcohol (D = 33.8 to 78.6), in methyl alcohol (D = 30), in acetone (D = 21), and in ethylene chloride (D = 10.4). The results have been found in all cases to be in very fair agreement with the re- quirements of the Debye-Huckel limiting law; as may be expected, ap- preciable discrepancies occur when the saturating salt is of a high valence type, especially in the presence of added ions of high valence. 30 "LaMer and Cook, J. Am. Chem. Soc., 51, 2622 (1929); LaMer and Goldman, ibid., 51, 2632 (1929); Neuman, ibid., 54, 2195 (1933). Baxter, J. Am. Chem. Soc., 48, 626 (1926); Williams, ibid., 51, 1112 (1929); Hansen and Williams, ibid., 52, 2759 (1930); Scholl, Hutchison and Chandlee, ibid., 55, 3081 (1933); Seward, ibid., 56, 2610 (1934); see, however, Anhorn and Hunt, J. Phys. Chem., 45, 351 (1941). THERMAL PROPERTIES OF STRONG ELECTROLYTES 179 Thermal Properties of Strong Electrolytes. According to equation (42) the free energy of an ionic solution may be expressed in the form G = Go + Gei. and application of the Gibbs-Helmholtz equation (cf. p. 194) gives where H is the heat content of a solution of an electrolyte at an appre- ciable concentration. At infinite dilution the quantity in the second brackets on the right-hand side is zero, since the electrical contribution to the free energy is then zero ; the heat content of the solution under these conditions is consequently equal to the quantity in the first brackets. It follows, therefore, that the increase of heat content accompanying the dilution of a solution of an electrolyte from a concentration c to infinite dilution, i.e., A/J^oi which is the corresponding integral heat of dilution, is given by / zn . \ (125) Utilizing the value of G e i., equal to W Wo given by equation (41), and remembering that K involves T~*, it is found that (126) where V is the volume of the system ; dD/dT and dV/dT refer to constant pressure. Since the heat of dilution is generally recorded for a mole of electrolyte, it follows that N % is equal to Nv % where N is the Avogadro number and v v is the number of ions of the iih kind produced by the ion- ization of a molecule of electrolyte. The expression I,N l Zt in equation (126) may therefore be replaced by N^v v z1 y and the result is o = -20- SF.#(T, D, 7), (127) where /( T, D, F) is the function included in the parentheses in equa- tion (126). The concentration c, of any ionic species is equal to i>,, where c is the concentration of the electrolyte in moles per liter; hence, the ionic strength may be written in an alternative form, thus It follows, therefore, using equation (51) to define x, that l " z?)l V '/ (T ' D > *> *?) V^ f(T, D, V) cal. per mole. 180 FREE ENERGY AND ACTIVITY For water at 25 this can be written as Affo.0 = 503(S?,*?) Vc/(r, Z>, F) cal. per mole. (128) The temperature coefficient of the dielectric constant of water is not known with great accuracy, but utilizing the best data to evaluate f(T 9 D 9 V), equation (128) becomes, approximately, Aff^o = - 175(Zna?)* Vc, and lor a um-univalent electrolyte at 25, i.e., z+ = z_ = 1, and v+ = v- = 1, o = 495 Vc cal. per mole. It is seen from these equations that there should be a negative in- crease of heat content when an electrolyte solution is diluted; in other tfords, the theory of interionic attraction requires that heat should be evolved when a solution of an electrolyte is diluted. 31 Further, the in- tegral heat of dilution should be proportional to the square-root of the concentration, the slope of the plot of AH o^o against Vc should be about 500 for an aqueous solution of a uni-univalent electrolyte at 25. It must be emphasized that the foregoing treatment presupposes a dilute solution, and in fact the slope mentioned should be the limiting value which is approached at infinite dilution. Accurate measurements of integral heats of dilution are difficult to make, but the careful work of the most recent investigators has given results in general agreement with theoretical expectation. The integral heat of dilution is actually nega- tive for dilute solutions, but at appreciable concentrations it becomes positive, so that heat is then absorbed when the solution is diluted. The limiting slope of the plot of A//<^ against Vc has been found to be approximately 500 for a number of uni-univalent electrolytes; the larger the effective size of the ion in solution, the closer the agreement between experiment and the requirements of the interionic attraction theory. By making allowance for the effective ionic diameter, either by the Debye-Hlickel method or by utilizing the treatment of Gronwall, LaMer and Sandved, fairly good agreement is obtained at appreciable concentrations. 32 According to equation (128) the limiting slope of the plot of A#e_ against Vc for different electrolytes should vary in proportion to the factor (Sjs-z?)*; the results obtained with a number of uni-bivalent and "Bjerrum, Z. physik. Chem., 119, 145 (1926); Gatty, Phil. Mag., 11, 1082 (1931); 18, 46 (1934); Scatchard, J. Am. Chem. Soc., 53, 2037 (1931); Falkenhagen, "Electro- lytes" (translated by Bell), 1934. "For summaries, with references, see Lange and Robinson, Chem. Revs., 9, 89 (1931); Falkenhagen, "Electrolytes," 1934; Wolfenden, Ann. Rep. Chem. Soc., 29, 29 (1932); Bell, ibid., 31, 58 (1934); for more recent work, see Robinson et al., J. Am. Chem. Soc., 56, 2312, 2637 (1934); 63, 958 (1941); Sturtevant, ibid., 62, 2171 (1940). PROBLEMS 181 bi-univalent electrolytes are in harmony with this requirement of theory. In spite of the general agreement, the experimental data for integral heats of dilution, especially in non-aqueous solutions, show some dis- crepancies from the behavior postulated by the interionic attraction theory. It should be noted, however, that heat of dilution measure- ments provide an exceptionally stringent test of the theory, and the influence of such factors as ionic size, incomplete dissociation and ion- solvent interaction will produce relatively larger effects than is the case with activity coefficients. PROBLEMS 1. The density of a 0.1 N solution of KI in ethyl alcohol at 17 is 0.8014 while that of the pure solvent is 0.7919; calculate the ratio of the three activity coefficients, /*, f c and / TO , in the solution. 2. Compare the mortalities and ionic strengths of uni-uni, uni-bi, bi-bi and uni-tervalent electrolytes in solutions of molality m. 3. Use the values of the Debye-Huckel constants A and B at 25, given in Table XXXV, to plot log f for a uni-univalent electrolyte against Vy for ionic strengths 0.01, 0.1, 0.5 and 1.0, assuming in turn that the mean distance of approach of the ions, a, is either zero, or 1, 2, 4 and 8 A. Investigate, quali- tatively, the effect of increasing the valence of the ions. 4. Evaluate the Debye-Hiickel constants A and B for ethyl alcohol at 25, taking the dielectric constant to be 24.3. 5. Utilize the results of the preceding problem, together wHh the known values of A and B for water, to calculate approximate activity coefficients for uni-uni, uni-bi, and bi-bi valent electrolytes in water and in ethyl alcohol, at ionic strengths 0.1 and 0.01, at 25. The mean ionic diameter may be taken as 3A in each case. 6. The following values for the mean activity coefficients of potassium chloride were obtained by Maclnnes and Shedlovsky [J. Am. Chem. Soc., 59, 503 (1937)]: c f c f 0.005 0.9274 0.04 0.8320 0.01 0.9024 0.06 0.8070 0.02 0.8702 0.08 0.7872 0.03 0.8492 0.10 0.7718 Plot Vji/log/ against Vtf and determine the value of a which is in satisfactory agreement with these data. 7. Kolthoff and Lingane [V. Phys. Chem., 42, 133 (1938)] determined the solubility of silver iodate in water and in the presence of various concentrations of potassium nitrate at 25. The solubility in pure water is 1.771 X 10~ 4 mole per liter, and the following results were obtained in potassium nitrate solutions: KNO, AglO, KNO, AglO, mole/liter mole/liter mole/liter mole/liter 0.1301 X 10-* 1.823 X 10~ 4 1.410 X 10-* 1.999 X 10~ 4 0.3252 1.870 7.050 2.301 0.6503 1.914 19.98 2.665 182 FREE ENERGY AND ACTIVITY Calculate the activity coefficients of the silver iodate in the various solutions; plot the values of - log / against Vy to see how far the results agree with the Debye-Huckel limiting law. Determine the mean ionic diameter required to account for the deviations from the law at appreciable concentrations. 8. Utilize the results obtained from the data of Saxton and Waters, given in Problem 7 of Chap. Ill, together with the activity coefficients derived from the Debye-Hiickel limiting equation, to evaluate the dissociation constant of a-crotonic acid. 9. Apply the method of Fuoss and Kraus, described on page 167, to evalu- ate Ao and K for hydrochloric acid in a dioxane-water mixture, containing 70 per cent of the former, at 25, utilizing the conductance data obtained by Owen and Waters [/. Am. Chem. Soc., 60, 2371 (1938)]: VcXlO* 1.160 2.037 2.420 2.888 3.919 A 89.14 85.20 83.26 81.45 77.20 ohms- 1 cm. 2 The dielectric constant of the solvent is 17.7 and its viscosity is 0.0192 poise. The required values of the function F(x) will be found in the paper by Fuoss, J. Am. Chem. Soc., 57, 488 (1935). 10. By means of the value of K obtained in the preceding problem, calcu- late the mean ionic diameter, a, of hydrochloric acid in the given solvent. For this purpose, use equation (79) and the tabulation of Q(b) given by Fuoss and Kraus, /. Am. Chem. Soc., 55, 1019 (1933). 11. In order to determine the formula of the complex argentammine ion, Ag,(NHs)"!:, Bodlander and Fittig [Z. physik. Chem., 39, 597 (1902)] measured the solubility (S) of silver chloride in ammonia solution at various concen- trations (CNH,) with the following results: CNH, 0.1006 0.2084 0.2947 0.4881 S X 10 s 5.164 11.37 15.88 25.58 In the presence of various concentrations (CKCI) of potassium chloride, the solubility (S) of silver chloride in 0.75 molal ammonia was as follows: CKCI 0.0102 0.0255 0.0511 S 0.0439 0.0387 0.0333 What is the formula of the silver-ammonia ion? CHAPTER VI REVERSIBLE CELLS Chemical Cells and Concentration Cells. A voltaic cell, or element, as it is sometimes called, consists essentially of two electrodes combined in such a manner that when they are connected by a conducting material, e.g., a metallic wire, an electric current will flow. Each electrode, in general, involves^ an electronic and an electrolytic pppjiuctof in contact (cf . p. 6) ; a^T tEe surface of separation between these two phases there exists a potential difference, called the electrode potential. Ifthere are no other potential differences in the cell, the E.M.P. of the latter is taken as equal to the algebraic sum of the two electrode potentials, allowance being made for the direction of the potential difference when assessing its sign. During the operation of a voltaic cell a^chemical^reaction takes place at each electrode7and it is the^ energy of these^feaHibns which provides the electrical energy oT the cell. In many cells there is an overall chemical reaction, when all the processes occurring within it are taken into consideration; such a cell is referred to as a chemical cell, to distinguish it from a voltaic element in which there is no resultant chemi- cal change. In the latter type of cell the reaction occurring at one electrode is exactly reversed at the other; there may, nevertheless, be a net change of energy because of a difference in concentration of one or other of the reactants concerned at the two electrodes. Such a source of E.M.F. is called a concentration cell, and the electrical energy arises from the energy change accompanying the transfer of material from one concentration to another. Irreversible and Reversible Cells. Apart from the differences men- tioned above, voltaic cells may, broadly speaking, be divided into two categories depending on whether a chemical reaction takes place at either electrode even when there is no flow of current, or whether there is no reaction until the electrodes are joined together by a conductor and current flows. An illustration of the former type is the simple cell con- sisting of zinc and copper electrodes immersed in dilute sulfuric acid, viz., Zn | Dilute H 2 S0 4 1 Cu; the zinc electrode reacts with the acid spontaneously, even if there is no passage of current. Cells of this type are always irreversible in the thermodynamic sense; thermodynamic reversibility implies a state of equilibrium at every stage, and the occurrence of a spontaneous reaction at the electrodes shows that the system is not in equilibrium. 183 184 REVERSIBLE CELLS In the Daniell cell, however, which is made up of a zinc electrode in zinc sulfate solution and a copper electrode in copper sulfate solu- tion, viz., Zn | ZnSO 4 soln. CuSO 4 soln. | Cu, the two solutions being usually separated by means of a porous partition, neither metal is attacked until the electrodes are connected and a current is allowed to flow. The extent of the chemical reaction occurring in such a cell is proportional to the quantity of electricity passing, in accord- ance with the requirements of Faraday's laws. Many, although not necessarily all, cells in this second category are, however, thermodynam- ically reversible cells, and the test of reversibility is as follows. If the cell under consideration is connected to an external source of E.M.P. which is adjusted so as exactly to balance the E.M.F. of the cell, i.e., so that no current flows, there should be no chemical change in the cell. If the external E.M.F. is decreased by an infinitesimally small amount, current will flow from the cell and a chemical change, proportional in extent to the quantity of electricity passing, should take place. On the other hand, if the external E.M.F. is increased by a small amount, the current should pass in the opposite direction and the cell reaction should be exactly reversed. The Daniell cell, mentioned above, satisfies these re- quirements and it is consequently a reversible cell. It should be noted that voltaic cells can only be expected to behave reversibly when the currents passing are infinitesimally small and the system is always vir- tually in equilibrium. If large currents flow, concentration gradients arise on account of diffusion being relatively slow, and the cell can no longer be regarded as being in a state of equilibrium. Reversible Electrodes. The electrodes constituting a reversible cell must themselves be reversible, and several types of such electrodes are known. The simplest, sometimes called "electrodes of the first kind/' consist of a luetal in contact with a solution of its own ions, e.g., zinc in zinc sulfate solution. In this category may be included hydrogen, oxygen and halogen electrodes in contact with solutions of hydrogen, hydroxyl or the appropriate halide ions, respectively; since the electrode material in these latter cases is a non-conductor, and often gaseous, finely divided platinum, or other unattackable metal, which comes rapidly into equilibrium with the hydrogen, oxygen, etc., is employed for the purpose of making electrical contact. Electrodes of the first kind are reversible with respect to the ions of the electrode material, e.g., metal, hydrogen, oxygen or halogen; the reaction occurring if the electrode material is a metal M may be represented by M ^ M+ + , the direction of the reaction depending on the direction of the flow of current. If the electrode is that of a non-metal, the corresponding reactions are A + ^ A-. REVERSIBLE ELECTRODES 185 With an oxygen electrode, which is theoretically reversible with respect to hydroxyl ions, the reaction may be written O 2 + H 2 + 2c ^ 2OH-. Electrodes of the "second kind" involve a metal, a sparingly soluble salt of this metal, and a solution of a soluble salt of the same anion; a familiar example is the silver-silver chloride electrode consisting of silver, solid silver chloride and a solution of a soluble chloride, such as hydrochloric acid, viz., Ag | AgCl(s) HCl soln. These electrodes behave as if they were reversible with respect to the common anion, e.g., the chloride ion in the above electrode. The elec- trode reaction involves the passage of the electrode metal into solution as ions and their combination with the anions of the electrolyte to form the insoluble salt, or the reverse of these stages; thus, for the silver-silver chloride electrode, Ag(s) ^ Ag+ + , followed by so that the net reaction, writing it for convenience in the reverse order, is AgCl(s) + e ^ Ag(s) + C1-. This is virtually equivalent to the reaction at a chlorine gas electrode, viz., C1 2 + ^ 2C1-, except that the silver chloride can be regarded as the source of the chlorine. In fact the silver-silver chloride electrode is thermodynam- ically equivalent to a chlorine electrode with the chlorine at a pressure equal to the dissociation pressure of the silver chloride, into silver and chlorine, at the experimental temperature. Electrodes of the second kind are of great value in electrochemistry because they permit the ready establishment of an electrode reversible with respect to anions, e.g., sulfate, oxalate, etc., which could not be obtained in a direct manner. Even where it is possible, theoretically, to set up the electrode directly, as in the case of the halogens, it is more convenient, and advantageous in other ways, to employ an electrode of the second kind. Occasionally electrodes of the " third kind" are encountered; l these consist of a metal, one of its insoluble salts, another insoluble salt of the same anion, and a solution of a soluble salt having the same cation as the latter salt, e.g., Pb | PbC 2 4 (s) CaC 2 4 (s) CaCl 2 soln. 1 Corten and Estermann, Z. physik. Chem., 136, 228 (1928); LeBlanc and Haraapp, , 166A, 321 (1933); Joseph, J. 'Biol. Chem., 130, 203 (1939). 186 REVERSIBLE CELLS In this case the lead first dissolves to form lead ions, which combine with C^Oi" ions to form insoluble lead oxalate, thus Pb ^ Pb++ + 2< and Pb++ + C 2 ^ PbC 2 4 (s). The removal of the oxalate ions from the solution causes the calcium oxalate to dissolve and ionize in order that its solubility product may be maintained; thus CaC 2 O 4 (s) ^ Ca++ + C 2 04~, so that the net reaction is Pb(s) + CaC 2 4 00 ^ PbC 2 4 (s) + Ca++ + 2. The system thus behaves as an electrode reversible with respect to cal- cium ions. This result is of great interest since a reversible calcium electrode employing metallic calcium is difficult to realize experimentally. Another type of reversible electrode involves an unattackable metal, such as gold or platinum, immersed in a solution containing an appropri- ate oxidized and reduced form of an oxidation-reduction system, e.g., Sn++++ and Sn++, or Fe(CN)? and Fe(CN)? --- ; the metal merely acts as a conductor for making electrical contact, just as in the case of a gas electrode. The reaction at an oxidation-reduction electrode of this kind is either oxidation of the reduced state or reduction of the oxidized state, e.g., Sn++ ^ Sn++++ + 2c, depending on the direction of the current. In order that it may behave reversibly, the reaction being capable of occurring in either direction, a reversible oxidation-reduction system must contain both oxidized and reduced states. It is important to point out that there is no essential difference between an oxidation-reduction electrode and one of the first kind described above; for example, in a system consisting of a metal M and its ions M+, the former is the reduced state and the latter the oxidized state. Similarly the case of an anion electrode, e.g., chlorine-chloride ions, the anion is the reduced state and the uncharged material, e.g., chlorine, is the oxidized state. In all these instances the electrode process may be written in the general form : Reduced State ^ Oxidized State + n, where n is the number of electrons by which the oxidized and reduced states differ. It is a matter of convenience, however, to treat separately electrodes involving oxidation-reduction systems in the specialized sense of the terms oxidation and reduction. Direction of Current Flow and Sign of Reversible Cell. The com- bination of two reversible electrodes in a suitable manner will give a REACTIONS IN REVERSIBLE CELLS 187 reversible cell; in this cell the reaction at one electrode is such that it yields electrons while at the other electrode the reaction removes elec- trons. The electrons are carried from the former electrode to the latter by the metallic conductor which connects them. The ability to supply or remove electrons is possessed by all reversible electrodes, as is evident from the discussion given above; the particular function which is manifest at any time, i.e., supplying or removing electrons, depends on the direc- tion of the current flow, and this is determined by the nature of the two electrodes combined to form the cell. The electrode Ag, AgCl(s) KC1 soln., for example, acts as a remover of electrons when combined with Zn, ZnSO 4 soln., but it is a source of electrons in the cell obtained by coupling it with the Ag, AgNO 3 soln. electrode. Since it is not always possible to say a priori in which direction the current in a given cell will flow when the electrodes are connected by an external conductor, it is necessary to adopt a convention for describing the E.M.F. and the reaction occurring in a reversible cell. The convention most frequently employed by physical chemists in America is based on that proposed by Lewis and Randall; it may be stated as follows. The E.M.F. , including the sign, represents the tendency for posi- tive ions to pass spontaneously through the cell as written from left to right, or of negative ions to pass from right to left. Since a positive E.M.F. means the passage of positive ions through the cell from left to right, it can be readily seen that electrons must pass through the external conductor in the same direction (cf. Fig. 2). It follows, therefore, that when the E.M.F. of the cell is positive the left- hand electrode acts as a source of electrons while the right-hand elec- trode removes them; if the E.M.F. is negative, the reverse is true. When expressing the complete chemical reaction occurring in a cell the con- vention will be adopted of supposing that the condition is the one just derived for a positive E.M.F.* Reactions in Reversible Cells. It is of importance in many respects to know what is the reaction occurring in a reversible cell, and some different types of cells will be considered for the purpose of illustrating the procedure adopted in determining the cell reaction. The Daniell cell, for example, is Zn | ZnS0 4 aq. j CuSO 4 aq. | Cu, and taking the left-hand electrode as the electron source, i.e., the E.M.F. as stated is positive, the reaction here is Zn = Zn++ + 2, * Many physical chemists in Europe and practical electrochemists in America use a convention as to the sign of E.M.F. and electrode potential which is the opposite of that employed here. 188 REVERSIBLE CELLS while at the right-hand electrode the electrons are removed by the process Cu++ + 2c = Cu. The complete reaction is thus Zn + Cu++ = Zn++ + Cu, and since two electrons are involved in each atomic act, the whole reac- tion as written, with quantities in gram-atoms or gram-ions, takes place for the passage of two faradays of electricity through the cell (cf. p. 27). Since the cupric ions originate from copper sulfate and the zinc ions form part of zinc sulfate, the reaction is sometimes written as Zn + CuSO 4 = ZnSO 4 + Cu. The E.M.F. of the cell depends on the concentrations of the zinc and cupric ions, respectively, in the two solutions, and so if the cell reaction is to be expressed more precisely, as is frequently necessary, the concen- tration of the electrolyte should be stated; thus Zn + CuSCMmi) = ZnSO^ms) + Cu, where mi and w 2 are the molalities of the copper sulfate and zinc sulfate, respectively, in the Daniell cell. In the cell Zn | ZnS0 4 aq. j KC1 aq. AgCl(s) | Ag, the left-hand electrode reaction is the same as above, i.e., Zn = Zn++ + 26, while at the right-hand electrode the removal of electrons occurs by means of the process described on page 185, i.e., AgCl(s) = Ag+ + Cl- and Ag+ + * = Ag, the net reaction being AgCl(s) + = Ag + CI-. The complete cell reaction for the passage of two faradays is thus Zn + 2AgCl(s) = Zn++ + 2C1~ + 2Ag, or Zn + 2AgCl(a) = ZnCl 2 + 2Ag. A specicl case of this type of cell arises when both electrodes arc of the same metal, viz., Ag ( AgCl(s) KC1 aq. j AgNO, aq. | Ag. MEASUREMENT OF E.M.F. 189 By convention, the reaction at the left-hand electrode is the opposite of that at the right-hand electrode of the previous cell, viz., Ag + Cl- = AgCUs) + 6, and at the right-hand electrode the reaction is Ag+ + = Ag, so that the net reaction in the cell is Ag+ + Cl- = AgCl(s) for the passage of one faraday. Another type of cell in which the two electrodes are constituted of the same material is one involving two hydrogen gas electrodes, viz., H 2 | NaOH aq. j HC1 aq. | H 2 . If the E.M.F. is positive, the hydrogen passes into solution as ions at the left-hand electrode, i.e., JH,(0) = H+ + ,* but the hydrogen ions react immediately with the hydroxyl ions in the alkaline solution, viz., 11+ + OH- = H 2 O, to form water. At the right-hand electrode electrons are removed by the discharge of hydrogen ions, thus so that the net reaction for the passage of one faraday is H+ + OH- = H,0, i.e., the neutralization of hydrogen ions by hydroxyl ions. Since the hydrogen ions are derived from hydrochloric acid and the hydroxyl ions from sodium hydroxide, the reaction can also be written (cf. p. 12) as IIC1 + NaOH = NaCl + H 2 0. Measurement of E.M.F. The principle generally employed in the measurement of the E.M.F.'S of voltaic cells is that embodied in the Poggendorff compensation method; it has the advantage of giving the E.M.F. of the cell on "open circuit," i.e., when it is producing no current. It has been already mentioned that a cell can be expected to behave reversibly only when it is producing an infinitesimally small current, and hence the condition of open circuit is the ideal one for determining the reversible E.M.F. * The hydrogen ion in aqueous solution is probably (H 2 O)H+, i e., H 3 O + , and not H+ UrfrpT308); this does not, however, affect the general nature of the results recorded here. 190 REVERSIBLE CELLS The potentiometer } as the apparatus for measuring E.M.F. 's is called, is shown schematically in Fig. 60; it consists of a working cell C, generally a storage battery, of constant E.M.F. which must be larger than that of the cell to be measured, connected across the ends of a uniform con- D FIG. 60. Measurement of E.M.F. ductor AB of high resistance. The cell X, which is being studied, is connected to A, with the poles in the same direction as the cell C, and then through a galvanometer G to a sliding contact D which can be moved along AB. The position of D is adjusted until no current flows through the galvanometer; the fall of potential between A and D due to the cell C is exactly compensated by the E.M.F. of X, that is Ex. By means of a suitable switch the cell X is now replaced by a standard cell S, of accurately known E.M.F. equal to Es, and the sliding contact is re- adjusted until a point of balance is reached at D'. The fall of potential between A and D' is consequently equal to Es, and since the conductor AB is supposed to be uniform, it follows that AD AD'' AD Ex E s Since E s is known, and AD and AD' can be measured, the E.M.F. of the unknown cell, Ex, can be evaluated. In its simplest form, the conductor AB may consist of a straight, uniform potentiometer wire of platinum, platinum-indium, or other re- sistant metal, stretched tightly along a meter scale; the position of the sliding contact can be read with an accuracy of about 0.5 mm., and if C is 2 volts and AB is 1 meter long, the corresponding error in the evalua- tion of the E.M.F. is 1 millivolt, i.e., 0.001 volt. Somewhat greater pre- cision can be achieved if the potentiometer wire is several meters in length wound on a slate cylinder. For more accurate work the wire may be replaced by two calibrated' resistance boxes; the contact D is fixed where CURRENT INDICATORS 191 the two boxes are joined, and the potential across AD is varied by changing the resistances in the boxes, keeping the total constant. If R x is the resistance between A and D with the cell X in circuit, when no current flows through the galvanometer G, then the fall of potential which is equal to Ex must be proportional to R x ; * further, if R is the resistance at the balance point when the standard cell S replaces X, it follows that Ex^Rx JJT D ' &s KS The unknown E.M.P. can thus be calculated from the two resistances. As a general rule the total resistance in the circuit is approximately 11,000 ohms, and hence if the working cell has an E.M.P. of 2 volts, each ohm resistance represents about 0.2 millivolt. The majority of E.M.F. measurements are made at the present time by means of special potentiometers, operating on the Poggendorff prin- ciple, which are purchased from scientific instrument makers. They generally consist of a number of resistance coils with a movable contact, together with a slide wire for fine adjustment. A standard cell is used for calibration pur- poses, and the E.M.F. of the cell being measured can then be read off directly with an accuracy of 0.1 millivolt, or better. For approximate purposes, as in electroanalytical work or in potentiometric titrations, a sim- ple procedure, known as the c 1 -AMA > A AAAAAA . B AAAAAAAA (^ VWYV YYVYVVV f T r \ 1 1 X 1 Cr J FIG. 61. Potentiometer-voltmeter arrangement potentiometer-voltmeter method, can be employed. The working cell C (Fig. 61) is connected across two continuously variable resistances A and B, as shown; one of these resistances is for coarse and the other for fine adjustment. The experimental cell is placed at X in series with a galvanometer (G), and a milli voltmeter (V) is connected across the vari- able resistances. The latter are adjusted until no current flows through G; the voltage then indicated on V gives the E.M.F. of the cell. Current Indicators. The best form of current detector for accurate work is a suitably Ha.mjftd mirror galvanometer of high megohm^ sensi-^ tivity ; for approximate purposes, however, a simple pointer galvanometer * Ex is actually equal to E e X RxIR, where E e is the B.M.F. of the working cell C, and R is the totpl resistance of the two boxes in the circuit; since E e and R are main- tained constant, EX is proportional to Rx- * * 192 REVERSIBLE CELLS is generally employed. At one time the capillary electrometer was widely used for the purpose of indicating the li^aillineTrt ^jf balance in the potenfSmeter circuit; it Has the advantage of being unaffected by elec- trical and magneTic^disturbances, and of not being damaged if large currents are inadvertently passed through it. On the other hand, the capillary electrometer is much less sensitive than most galvanometers and is liable to behave erratically in damp weather; for these and other reasons this form of detector has been discarded in recent years. An ordinary mirror galvanometer of good quality can detect a current of about 10~ 7 amp., and hence if an accuracy of 0.1 millivolt is -desired, as is the case in much work that is not of the highest precision, the re- sistance of the cell- should not exceed lO 3 ohms. Special high-sensitivity galvanometers are available which show' an observable deflection with a current of 10~ u amp., and so the E.M.F. of cells with resistances up to 10 7 ohms can be measured with their aid; the quadrant electrometer, which detects actually differences of potential rather than current, has also been used for the study of high resistance cells. Another procedure which has been devised is to employ a condenser in series wjlh a ballistic galvanp.meter Jio determine the balance point of the potentiometer; the condenser is charged for a definite time by means of the cell being studied and is then discharged through the galvanometer with the aid of a suitable switch. When the potentiometer is balanced the ballistic galvanometer will undergo.no deflection when the cell is discharged through it. ^or most measurements of E.M.F. of cells of high resistance some form of vacuum-tube potentiometer has been used ; 2 this instrument employs the amplifying properties of the vacuum tube, and the principle of operation may be illustrated by means of the simple circuit shown X.I a $ *<N PI'I 1 !' in Fig. 62. The tube is repre- '17! ' (wvyw) sented by T, and A, B and C in- dicate the filament, anode and grid batteries, respectively; Ri and Rz are variable resistances and G is a galvanometer. The cell X of unknown resistance is connected, as shown, to a potentiometer P from which any desired known voltage can be taken off; by means of the switch S the potentiom- eter and cell can be included, if required, in the grid circuit of the vacuum tube. The switch is first con- nected to b and the filament current is adjusted by means of Ri to provide the optimum sensitivity of the tube; the " compensating current " from A, which passes in the opposite direction to the anode current through 2 See, for example, Garmauand Drusz, Ind. Eng. Chem. (Anal. Ed.), 11, 398 (1939); for review, see Glasstone, Ann. Rep. Chem. Soc., 30, 283 (1933). -Mh FIG. 62. Vacuum-tube potentiometer for cells of high resistance THE STANDARD CELL 193 the galvanometer G, is then altered by means of the resistance ff 2 so as to give a suitable reading on G. The switch S is now turned to a, so that P and Xj as well as the battery C, are in the grid circuit; leaving R i and # 2 unchanged, the potentiometer is adjusted until the deflection on G is the same as before. The potential on the grid of the tube must, therefore, be the same in both cases: hence the E.M.F. taken from the potentiometer P must be equal and opposite to that of the cell X. This simple type of vacuum-tube potentiometer is quite satisfactory for cells of not too high resistance, e.g., 10 7 ohms or less, but it is un- reliable for still higher resistances. Two sources of error then arise: first, the characteristics of the vacuum tube change as a result of intro- ducing the high resistance, so that a given anode current no longer corresponds to the same grid voltage; second, there is a fall of potential across the high resistance cell due to the flow of current in the grid circuit. With the best ordinary vacuum tubes the grid current may be about 10~ 10 amp., and so with a cell of resistance of 10 8 ohms, the error due to the fall of potential across the cell will be lO' 10 X 10 8 , i.e., 10~ 2 volt. Several methods of varying complexity have been devised in order to overcome these sources of error; one of the simplest and most effective, which is employed in commercial potentiometers for the measurement of the E.M.F/S of cells involving the glass electrode (p. 356), is to use a special type of vacuum tube, known as an " electrometer tube." Al- though its amplification factor is generally smaller than that of the normal form of tube, the grid-circuit current is very small, 10~ 15 amp. or less, and the characteristics of the tube are not affected by high resistances. The Standard Cell. An essential feature of the Poggendorff method of measuring E.M.F. 's, and of all forms of apparatus employing the Poggen- dorff compensation principle, is a standard cell of accurately known E.M.F. The cell now invariably employed for this purpose is the Weston standard cell ; it is highly reproducible, its E.M.F. remains constant over long periods of time, and it has a small temperature coefficient. One electrode of the cell is a 12.5 per cent cadmium amalgam in a saturated solution of cadmium sulfate (3CdSO 4 -8H 2 O) and the other electrode consists of mercury and solid mercurous sulfate in the same solution, thus 12.5% Cd in Hg | 3CdSO 4 -8H 2 O satd. soln. Hg 2 SO 4 (s) | Hg. The cell is set up in a H-shaped tube as shown in Fig. 63, the left-hand limb containing the cadmium amalgam and the right-hand the mercury; the amalgam is covered with crystals of 3CdSO 4 8H 2 O, and the mercury with solid mercurous sulfate, and the whole cell is filled with a saturated solution of cadmium sulfate. The E.M.F. of the Weston cell, in inter- national volts, over a range of temperatures is given by the expression E* = 1.018300 - 4.06 X 10~ 5 (* - 20) - 9.5 X I0"\t - 20) 2 + 1 X 10-"(* - 20), 194 EEVERSIBLE CELLS Cadmium-,, sulfate solution Cadmium-^ sulfate Cadmium amalgam-^ x Cadmium - x sulfate ^-Mercurous sulfate ^"-Mercury so that the value is 1.01830 volt at 20 and decreases about 4 X 10~ 2 millivolt per degree in this region.* Although the so-called "saturated" Weston cell, containing a satu- rated solution of cadmium sul- fate, is the ultimate standard for E.M.F. measurement, a secondary standard for general laboratory use has been recommended; this is the "unsaturated" Weston cell, which has an even smaller temperature coefficient than the saturated cell. The form of un- saturated cell generally em- ployed contains a solution which has been saturated at 4 c., so that it is unsaturated at room Fia. 63. The Weston standard cell temperatures; its temperature coefficient is so small as to be negligible for all ordinary purposes and its E.M.F. may be taken as 1.0186 volt. 3 Free Energy and Heat Changes in Reversible Cells. Since the quan- titative consequences of the second law of thermodynamics are mainly applicable to reversible processes, the study of reversible cells is of par- ticular importance because it is possible to apply thermodynamic methods to the results. If the E.M.F. of a voltaic cell is E volts, and the process taking place in it is accompanied by the passage of n faradays, i.e., nF coulombs, where F represents 90,500 coulombs, the work done by the system in the cell is nFE volt-coulombs or joules (cf. p. 5). If the cell is a reversible one, this work represents " maximum work/' and since electrical work does not involve mechanical work resulting from a volume change, it may be taken as equal to the change of free energy accompany- ing the cell reaction. The increase of free energy of a process is equal to the reversible net work, i.e., excluding mechanical work, done on the system, and hence it follows that A(? = - nFE, (1) where A(? is the increase of free energy for the process taking place in the cell under consideration. According to the Gibbs-Hehnholtz equation, which is derived from the second law of thermodynamics applied to reversible changes, A(? Aff + r{^p), dl Jp (2) * It is important to note that the mercury electrode of a commercial Weston cell is always marked positive, while the cadmium amalgam electrode is marked negative. See Vinal, Trans. Electrochem. Soc., 68, 139 (1935). CONCENTRATION CELLS 195 where AH is the increase of heat content * for the cell reaction, and introducing equation (1), the result is - nFE = A# - nFT > (3) (4) It is seen from equation (4) that if the E.M.F. of a reversible cell, i.e., E, and its temperature coefficient, dE/dT, at constant pressure are known, it is possible to evaluate the heat change of the reaction occurring in the cell. Some of the results obtained in the calculation of heat content changes from E.M.F. measurements are recorded in Table XLII; 4 the values de- TABLE XLII. HEAT CHANGES FROM E.M.F. MEASUREMENTS dE/dT A// kcal. Cell Reaction E X 10* E M F. Thermal Zn + 2Ag01 = ZnCl 2 -f 2Ag 1.015 (0) - 4.02 - 51.99 - 52.05 Cd -f- PbCl 2 = OdCl 2 -f Pb 0.1880 (25) - 4.80 - lo.25 - 14.65 Ag -f- $Hg 2 Cl 2 = AgCl -f Hg 0.0455 (25) + 3.38 + 1.275 -f 1.90 Pb -f 2AgCl - PbCl 2 + 2Ag 0.4900 (25) - 1 86 - 25.17 - 24.17 rived from thermochemical measurements are given in the last column for purposes of comparison. The agreement between the results for AH derived from E.M.F. measurements and from thermal data is seen to be satisfactory, especially when it is realized that an error of 1 X 10~ 5 in the temperature coefficient will mean an error of nearly 0.07 kcal. in AH at 298 K. It is probable, however, that the temperature coefficients are known with this degree of accuracy, and it is consequently believed that for many reactions the heat changes derived from E.M.F. data are more accurate than those obtained by direct thermal measurement. Concentration Cells: Cells without Transference. In the operation of the cell H 2 (l atm.) | HC1 aq.(c) AgCl(s) | Ag, consisting of a hydrogen and a silver-silver chloride electrode in hydro- chloric acid,t the hydrogen at the left-hand electrode dissolves to form hydrogen ions, whereas at the right-hand electrode silver chloride passes into solution and silver is deposited; thus JH,(1 atm.) = H+ + e * The increase of heat content is equal to the heat absorbed in the reaction at con- stant pressure. 4 Taylor and Perrott, J. Am. Chem. Soc., 43, 486 (1921); Gerke, ibid., 44, 1684 (1922). t The construction of these electrodes is described later (pp. 234, 350). 196 REVERSIBLE CELLS and + - Ag + C1-, sc that the net reaction is represented by iH 2 (l atm.) + AgCl(s) = HCl(c) + Ag, since the hydrogen and chloride ions are formed in hydrochloric acid solution of concentration c moles per liter. If two stich cells containing hydrochloric acid at concentrations ci and c^ y and having E.M.F.'S of Ei and EZ, respectively, are connected in opposition, the result is the cell IT 2 (1 atm.) | HCl(ci) AgCl(s) | Ag | AgCl(s) HCl(c 2 ) | H 2 (l atm.), whose E.M.F. is equal to EI E z . The reaction in the left-hand cell for the passage of one faraday, as seen above, is JH 2 (1 atm.) + AgClOO = HCl(ci) + Ag, and that in the right-hand cell is the reverse of this, i.e., HCl(c 2 ) 4- Ag = iH 2 (l atm.) + AgCl(s). The net result of the passage of a faraday of electricity through the complete cell is the transfer (i) of hydrogen gas at 1 atm. pressure from the extreme left-hand to the extreme right-hand electrode, (ii) of solid silver chloride from left to right, and (iii) of hydrochloric^ acid from con- centration 02 to d. Since the chemical potentials of the hydrogen gas and solid silver chloride remain unchanged, the free energy change AG of the cell reaction is due only to that accompanying the removal of 1 mole of hydrochloric acid, i.e., 1 g.-ion of hydrogen ions and 1 g.-ion of chloride ions, from the solution of concentration <% and its addition to c\. It follows, therefore, that where /*H+ and jeer are the chemical potentials of hydrogen and chloride ions, the suffixes 1 and 2 referring to the solutions of concentration Ci and C2, respectively. The quantities of solutions in the cells are assumed to be so large that the removal of hydrochloric acid from one and its trans- fer to the other brings about no appreciable change of concentration; the change of free energy is thus equal to the resultant change in the chemical potentials. If the chemical potentials are expressed by means of equation (10) on p. 133, the result is CONCENTRATION CELLS 197 where an* and Ocr refer to the activities of the ions indicated by the sub- scripts. The electrical energy produced in the cell for the passage of one faraday is EF, where E, as already seen, is equal to E\ E*; it follows, therefore, from equation (6), since AC = - EF, that (7) 2RT a* = In-, (8) where ai and a 2 are the mean activities of the hydrochloric acid in the two solutions (cf. p. 138). The activities may be replaced by the prod- ucts my or r/, so that 2RT c,/ 2 or (10) x A cell of the type described above is called a concentiation cell with- out transference, for the E.M.F. depends on the relative concentrations, or molalities, of the two solutions concerned, and the operation of the cell is not accompanied by the direct transfer of electrolyte from one solution to the other. The transfer occurs indirectly, as shown above, as the result of chemical reactions. In general, a concentration cell without transference results whenever two simple cells whose electrodes are re- versible with respect to each of the ions constituting the electrolyte are combined in opposition; in the case considered above, the electrolyte is hydrochloric acid, and one electrode is reversible with respect to hydro- gen ions and the other with respect to chloride ions. If a\ is the mean ionic activity of the electrolyte in the left-hand side of any concentration cell without transference, arid a, 2 is the value on the right-hand side, the E.M.F. of the complete cell can be expressed by means of the general equation *- '.-^ln^ (11) ' V ZI< 0,1 where v is the total number of ions, and v+ or *>_ is the number of positive or negative ions produced by the i'>nization of one molecule of electro- lyte; z+ or 2_ is the valence of the inn with respect to which the extreme electrodes are reversible. If this ion is positive, as in the cell alreadv discussed, the positive si^ns apply throughout, but if it is negative, as 198 REVERSIBLE CELLS in the cell Ag HCl(d) | H 2 (l atm.) | HCl(c 2 ) AgCl(s) | Ag, the negative signs are applicable. Amalgam Cells. If the electrolyte in the concentration cell without transference is a salt of an alkali metal, e.g., potassium chloride, it is necessary to set up some form of reversible alkali metal electrode. This is achieved by dissolving the metal in mercury, thus forming a dilute alkali metal amalgam which is attacked much less vigorously by water than is the metal in the pure stateA The amalgam nevertheless reacts with water to some extent, and also with traces of oxygen that may be present in the solution : the exposed surface of the amalgam is therefore continuously renewed by maintain- ing a flow from the end of a tube. For the cell Ag | AgCl(s) KCl(d) | KHg x | KCl(c 2 ) AgCl(s) | Ag, where KHg x represents the potas- sium amalgam, the apparatus is shown in Fig. 64; the reservoir A contains the dilute amalgam which flows slowly through the capillary tubes BI and B 2 , while Ci and C 2 represent the silver electrodes coated with silver chloride (see p. 234). B The potassium chloride so- lutions of concentrations c\ and Cz respectively, from which all dis- solved oxygen has been removed, as far as possible, are introduced into the cells by means of the tubes A and D 2 . Although reproducible results can be obtained with the exercise of due care, the measurements are not reliable for solutions more dilute than about 0.1 N, because of interaction between the solution and the alkali metal. Amalgam cells are utilized for the study of alkali hydroxides, e.g., H,(l atm.) | NaOH(d) | NaHg x | NaOHfe) | H 2 (l atm.), where the hydrogen electrode is reversible with respect to hydroxyl ions, but equation (11) for the E.M.F. requires some modification in this case, because the cell reaction also involves the transfer of water. The reac- Machines and Parker, /. Am. Chem. Soc., 37, 1445 (1915). Fia. 64. Concentration cell with amalgam electrodes (Maclnnes and Parker) DETERMINATION OP ACTIVITY COEFFICIENTS 199 tion in the left-hand cell for the passage of one faraday of electricity is iH 2 (l atm.) + NaOH( Cl ) = H 2 O + Na, and in the right-hand cell it is H 2 O + Na = H 2 (1 atm.) + NaOH(c 2 ), and consequently the net process is the transfer of a mole of sodium hydroxide, i.e., one g.-ion each of sodium and hydroxyl ions, from the solution of concentration ci to that of concentration c 2 , while at the same time a mole of water is transferred in the opposite direction. The in- crease of free energy accompanying the passage of one faraday is repre- sented by AC = [(MNa+)2 (MNaOlJ and hence, utilizing the equation on page 133 to give the chemical poten- tial of the water in terms of its vapor pressure, it follows that ; 12) F where i and a 2 are the mean ionic activities of the sodium hydroxide in the two solutions, and (pH 2 o)i and (pn 2 o)2 are the respective aqueous vapor pressures.* Determination of Activity Coefficients. The E.M.F. of a concentra- tion cell without transference is equal to EI 7 2 , where EI and E 2 are determined by the concentrations Ci and c 2 , respectively, of the electro- lyte; then for a cell to which equation (8) is applicable, *. (13) If in one of the two solutions, e.g., C2, the activity is unity, and the corresponding E.M.F. of the half-cell containing that solution is Z, equa- tion (13) reduces to the general form E - E = In a. (14) If m is the molality of the electrolyte in the solution of activity a which gives an E.M.F. equal to E in the half-cell, then addition of (2RT/F) In m to both sides of equation (14) yields _ . 2#7\ _ 2/Zr. a ~ (!) F 2RT In 7, ' (16) * It should be noted that the H 2 (0), NaOH aq. electrode is to be regarded ae reversible with respect to OH~ ions; this accounts for the negative sign in equation (12). 200 REVERSIBLE CELLS where 7 is the mean activity coefficient of the electrolyte in the solution of molality m. In order to convert the Naperian to Briggsian logarithms the corresponding terms are multiplied by 2.3026, and if at the same time the values of R, i.e., 8.313 joules per degree, and of F, i.e., 96,500 cou- lombs, are inserted, equation (16) can be written as E + 2 X 1.9835 X 10~ 4 T log m - E = - 2 X 1.9835 X 10- 4 T log y, (17) and, at 25, this becomes E + 0.1183 log m - E = - 0.1183 log y. (18) Since E can be measured for any molality m, it would be possible to evaluate the activity coefficient y if E were known. 6 One method of deriving E makes use of the fact that at infinite dilution, i.e., when m is zero, the activity coefficient 7 is unity ; under these conditions -B a will be equal to E + 0.1183 log m at 25. If this quantity, for various values of m, is plotted as ordinate against a function of the molality, generally Vm, as abscissa, and the curve extrapolated to m equal to zero, the limiting value of the ordinate is equal to E Q . To be accurate this extra- polation requires a precise knowledge of the E.M.F.'S of cells containing very dilute solutions, and the necessary data are not easy to obtain. Two alternative methods of extrapolation which avoid this difficulty may be employed ; only one of these will, however, be described here. 7 According to the Debye-Hlickel-B rousted equation (63), p. 147, it is possible to express the variation of the activity coefficient of a uiii- univalent electrolyte with molality by the equation log 7 = - A Vm + Cm, (19) where A is a known constant, equal to 0.509 for water as solvent at 25. Combination of this with equation (18) then gives E + 0.1183 log m - 0.0602 Vm = E Q - 0.1183 Cm, /. E' - 0.0602 Vm = E Q - 0.1183 Cm, where E' is equal to E + 0.1183 log m. According to this result the quantity E f 0.0602 Vm should be a linear function of m, and extrapola- tion of the corresponding plot to m equal zero should give E. It is found in practice that the actual plot is not quite linear, as shown by the results in Fig. 65 for the cells H 2 (l atm.) | HCl(m) AgCl(s) | Ag, but reasonably accurate extrapolation is nevertheless possible. The Lewis and Randall, J. Am. Chem. Soc., 43, 1112 (1921); Randall and Young, ibid., 50, 989 (1928). 7 Hitchcock, J. Am. Chem. Soc., 50, 2076 (1928); Hnrned et al, ibid., 54, 1350 (1932); 55, 2179 (1933); 58, 989 (1936). CONCENTRATION CELLS WITH TRANSFERENCE 201 value of E for this cell at 25 is +0.2224 volt, and hence for solutions of hydrochloric acid E + 0.1183 log m - 0.2224 = - 0.1183 log 7. The activity coefficients can thus be determined directly from this equa- tion, using the measured values of the E.M.F. of the cell depicted above, 0.223 0.221 0.219 0.217 215 0.213 0.02 0.94 0.06 0.08 0.10 m FIG. 65. Extrapolation of E M.F. to infinite dilution for various molalitics of hydrochloric acid; the lesults obtained are given in Table XLI1I. TABLE XLIII. MEAN ACTIVITY COEFFICIENTS OF HYDROCHLORIC ACID FROM E M.F. MEASUREMENTS AT 25 m 0.1238 0.0. r >391 02563 0013407 000913S 0005619 0.003215 Concentration Cells with Transference. When two solutions of the same electrolyte are brought into actual contact and if identical elec- trodes, reversible with respect to one or other of the ions of the electro- lyte, arc placed in each solution, the result is a concentration cell with transference; for example, the removal of the AgCl(s) | Ag | AgCl(s) system from the cell on page 196 gives H,(l atm.) | IICl(ci) j HCl(c 2 ) | H 2 (l atm.), in which the two solutions of hydrochloric acid are in contact, and direct transfer from one to the other is possible. The presence of a liquid junction, as the region where the two solutions are brought into contact E E + 1183 log m 7 34199 0.23466 0.788 0.3X222 23218 0.827 0.41S24 22999 0.863 0.44974 22820 0.893 0.46SOO 22735 0.908 0.49257 0.22636 0926 52053 22562 0.939 202 REVERSIBLE CELLS with one another is called, is represented by the vertical dotted line. When one faraday passes through the cell, 1 g.-atom of hydrogen dis- solves at the left-hand electrode to yield 1 g.-ion of hydrogen ions, and the same amount of hydrogen ions will be discharged and 1 g.-atom of hydrogen will be liberated at the right-hand electrode. While the current is passing, t+ g.-ion of hydrogen ions will migrate across the boundary between the two solutions in the direction of the current, i.e., from left to right, and _ g.-ion of chloride ions will move in the opposite direction; t+ and t- are the transference numbers of the hydrogen and chloride ions, respectively (see Fig. 66). Attention may be drawn to the fact H 2 |HC/( Cl ) i HC/(c 2 )|H 2 I i i t- FIG. 66. Transference at liquid junction that the transference numbers involved are the Hittorf values, and not the so-called "true" transference numbers (p. 114); this allows for the transfer of water with the ions. The net result of the passage of one faraday is the transfer of 1 t+, i.e., t-, g.-ions of hydrogen ions and t^ g.-ions of chloride ions from right to left, so that the increase of free energy is A(? = /_[(MH+)I - G*H*)I] + *-[Ucr)i - (ncr)i]. (20) Since the transference numbers vary with concentration, it is convenient to consider two solutions whose concentrations differ by a small amount, viz., c and c + dc\ under these conditions equation (20) becomes AG = - L.(<W + <W) = - t.(RT d In a H - + RT d In a c r) = - 2t_RTd\na, (21) where a is the mean activity of the hydrochloric acid at the concentration c, and t- is the transference number of the anion at this concentration. The E.M.F. of the cell whose concentrations differ in amount by dc may be represented by dE, and the free energy increase FdE may be equated to the value given by equation (21) ; hence ft T dE = 2*_ -TT d In a. (22) r For a concentration cell with electrolytes of concentration Ci and c 2 , i.e., mean activities of a\ and a 2 , respectively, the E.M.F. is then obtained by ACTIVITY COEFFICIENTS FROM CELLS WITH TRANSFERENCE 203 integrating equation (22) between these limits ; thus 2RT C a * E = - t-d In a. (23) " Ja v In the general case this becomes E== ------ Mlna, (24) where PI, v-t- and z have the same significance as before (p. 197) ; the trans- ference number t^ refers to the ion other than that with respect to which the electrodes are reversible. If the transference number is taken as constant in the range of con- centration Ci to C2> equation (24) takes the form . (25) Q>i In the special case of the hydrogen-hydrochloric acid cell given above, v is 2, v- is 1, and z+ is 1, and the electrodes are reversible with respect to positive ions ; hence *-*. in*- (26) If the concentration cell is one of the type in which water is formed or removed in the cell reaction, e.g., H 2 1 NaOH(ci) j NaOH(c 2 ) | H 2 , in which a mole of water is transferred from c 2 to Ci for the passage of one faraday, due allowance must be made in the manner already described. Activity Coefficients from Cells With Transference. In order to set up a cell without transference it is necessary to have electrodes reversible with respect to each of the ions of the electrolyte ; this is not always pos- sible or convenient, and hence the use of cells with transference, which require electrodes reversible with respect to one ion only, has obvious advantages. In order that such cells may be employed for the purpose of determining activity coefficients, however, it is necessary to have accurate transference number data for the electrolyte being studied. Such data have become available in recent years, and in the method de- scribed below it will be assumed that the transference numbers are known over a range of concentrations. 8 The E.M.F. of a cell of the type M | MA(c) j MA(c + dc) | M. 8 Brown and Maclnnes, J. Am. Chem. Soc., 57, 1356 (1935); Shedlovsky and Maclnnes, iUd. y 58, 1970 (1936); 59, 503 (1937); 61, 200 (1939); Maclnnes and Brown, Chem. Revs., 18, 335 (1936). 204 REVERSIBLE CELLS where M is a metal or hydrogen, yielding cations in solution, is given by equation (22), and since the activity a is equal to cf, this may be written 27? T dE (27) The activity is expressed in terms of concentrations rather than molalities because the transference numbers are generally known as a function of the former; the procedure described here thus gives the activity coeffi- cient /, but the values can be readily converted into the corresponding 7*s by means of the equations on page 135. The transference number at any concentration can be written as L where to is the value at some reference concentration c ; if this expression for 1/J_ is inserted in equation (27) and the latter multiplied out and rearranged, the result is Integrating between the limits c and c, the corresponding values of the mean activity coefficient of the electrolyte being /o and /, it follows, after converting the logarithms, that f FK r F 2 " (28) The first two terms on the right-hand side of equation (28) may be evalu- ated directly from the experimental data, after deciding on the concen- tration c which is to represent the reference state. The third term is obtained by graphical integration of 8 against E, the value of 6 being derived from the known variation of the transference number with concentration. The method just described gives log ///o, and hence the activity coefficient / in the solution of concentration c is known in terms of an arbitrary reference scale, i.e., / at concentration c ; it is necessary now to convert the results to the usual standard state, i.e., the hypothetical ideal solution at unit concentration (see p. 137). For this purpose, use is made of the Debye-Huckol expression for uni-univalent electrolytes, 1 (29) where A is the known Debye-Huckel constant for the solvent at the ex- perimental temperature, and J?', which is written in place of aB, is a DETERMINATION OP TRANSFERENCE NUMBERS 205 constant for the electrolyte. The term log /// , i.e., log / log / , may be represented by log/ + a, where a is a constant, equal to log/ , and hence equation (29) may be rewritten as logf + A^c = a+ B'la- log^ V^ / o \ /o / For solutions dilute enough for equation (29) to be applicable, the plot of log (///o) + A Vc against [a log (///o)]Vc should be a straight line with intercept equal to a. The value of a, which is required for the purpose of this plot, is obtained by a short series of approximations. Once a, which is equal to log / , is known, it is possible to derive log / for any solution from the values of log ///o obtained previously. The activity coefficient of the electrolyte can thus be evaluated from the E.M.F. 's of cells with transference, provided the required transference number information is available. Determination of Transference Numbers. Since activity coefficients can be derived from E.M.F. measurements if transference numbers are known, it is apparent that the procedure could be reversed so as to make it possible to calculate transference numbers from E.M.F. data. The method employed is based on measurements of cells containing the same electrolyte, with and without transference. The E.M.F. of a concentra- tion cell without transference (E) is given by equation (11), and if the intermediate electrodes are removed so as to form a concentration cell with transference, the E.M.F., represented by E t , is now determined by equation (25), provided the transference numbers may be taken as constant within the range of concentrations in the cells. It follows, therefore, on dividing equation (25) by (11), that Y = **> (30) where the transference number t^ refers to the negative ion if the ex- treme electrodes are reversible with respect to the positive ion, and vice versa. 9 For example, if the amalgam cell without transference Ag | AgCl(s) LiCl(d) | LiHg, | LiCl(c 2 ) AgCl(s) | Ag is under consideration, the corresponding cell with transference is Ag | AgCl(s) LiCl(c,) j LiCl(c 2 ) AgCl(s) | Ag. The ratio of the E.M.F.'S of these cells then gives the transference number of the lithium ion, i.e., The method for determining transference numbers from E.M.F. measurements was first suggested by Helmholtz in 1878. 206 REVERSIBLE CELLS since the extreme electrodes, i.e., Ag | AgCl(s) LiCl aq., are reversible with respect to the chloride ion. The use of equation (30) gives a mean transference number of the electrolyte within the range of concentrations from c\ to C2, but this is of little value because of the variation of transference numbers with concentration; a modified treatment, to give the results at a series of definite concentrations, may, however, be employed. If the concentra- tions of the solutions are c and c + dc, the E.M.F. of the cell with trans- ference is given by the general form of equation (22) as v RT dE t = =t < T --- = d In a, dE t v RT " 31 = *=F ---- > (31) d In a v zF ^ J where a is the mean activity of the electrolyte at the concentration c. The corresponding E.M.F. for the cell with transference, derived from equation (11), is v RT dE = d In a v It follows, therefore, from equations (31) and (32) that dB/alna" "^ or dEt/d log a ___ dlz]d log a = ^ T * (33) If the E.M.F.'S of the cells, with and without transference, in which the concentration of one of the solutions is varied while the other is kept at a constant low value, e.g., 0.001 molar, are plotted against log a of the variable solution, the slopes of the curves a dE t /d log a and dE/d log a, respectively. The transference number uf the appropriate ion may thus be determined at any concentration by taking the ratio of the slopes at the value of log a corresponding to this concentration. The activities at the different concentrations, from which the log a data are obtained, must be determined independently by E.M.F. or other methods. Since the exact measurement of the slopes of the curves is difficult, analytical procedures have been employed. In the simplest one of these, 10 the values of E t are expressed as a function of the logarithm of the activities of the electrolyte; from this dE t /d log a is readily derived by differentiation. Since dE/d log a is given directly by equation (32), M Maclnnes and Beattie, J. Am Chem. Soc., 42, 1117 (1920). LIQUID JUNCTION POTENTIALS 207 t can also be written as a function of log a, and hence it may be evaluated at any desired concentration. A more rigid but laborious method, for deriving transference num- bers from E.M.P. data, makes use of the fact that the activity coefficient of an electrolyte can be expressed, by means of an extended form of the Debye-Huckel equation, as a function of the concentration and of two empirical constants. 11 When applied to the same data, however, this procedure gives results which are somewhat different from those obtained by the method just described. Since the values are in better agreement with the transference data derived from moving boundary and other measurements, they are probably more reliable. A number of determinations of transference numbers, in both aqueous and non-aqueous solutions, have been made by the E.M.F. method, and the results are in fair agreement with those obtained by other experi- mental procedures. The results in Table XLIV, for example, are for the TABLE XLIV. TRANSFERENCE NUMBER OF LITHIUM ION IN LITHIUM CHLORIDE AT 25 Hittorf or E.M.F. Moving Boundary Cone. Method Method 0.005 N 0.3351 0.3303 0.01 0.3333 0.3289 0.02 0.3308 0.3261 0.05 0.3259 0.3211 0.10 0.3203 0.3168 0.20 0.3126 0.3112 0.50 0.3067 0.3079 1.00 0.2809 0.2873 transference number of the lithium ion in lithium chloride at 25. The discrepancies between the two sets of values are often appreciable, how- ever, and since they are greater than the experimental errors of the best Hittorf or moving boundary measurements, it is probable that the E.M.F. results are in error. It must be concluded, therefore, that the E.M.F. 's of concentration cells cannot yet be obtained with sufficient precision for the transference numbers to be as accurate as the best results obtained by other methods. Liquid Junction Potentials: Solutions of the Same Electrolyte. The free energy change occurring in a concentration cell with transference may be divided into two parts ; these are (i) the contributions of the reactions at the electrodes, and (ii) that due to the transfer of ions across the boundary between the two solutions. It is evident, therefore, that when two solutions of the same or of different electrolytes are brought into contact, a difference of potential will be set up at the junction between them because of ionic transference. Potentials of this kind are called liquid junction potentials or diffusion potentials. 11 Jones and Dole, J. Am. Chem. Soc., 51, 1073 (1929); Jones and Bradshaw, iWd., 54, 138 (1932); see also, Hamer, i&id, 57, 66 (1935); Harned and Dreby, ibid., 61, 3113 (1939). 208 REVERSIBLE CELLS Consider the simplest case in which the junction is formed between two solutions of the same uni-univalent electrolyte at concentrations d and c 2 , e.g., KCl(ci) j KCl(c 2 ). Adopting the usual convention for a positive E.M.P. that the left-hand electrode is the source of electrons, so that positive current flows through the interior of the cell from left to right, it follows that the passage of one faraday of electricity through the cell results in the transfer of t+ g.-ion of cations, e.g., potassium ions, from left to right, i.e., from solution Ci to solution c 2 , and t- g.-ion of anions, e.g., chloride ions, in the opposite direction (cf. p. 202). If the Approximation is made of taking the trans- ference numbers to be independent of concentration, the free energy change accompanying the passage of one faraday across the liquid junc- tion may be expressed either as FEi> where E L is the liquid junction potential, or as Further, since t+ + t- is equal to unity, it follows that x- < 36) where ai and a 2 are the mean activities of the electrolyte in the two solu- tions. By making the further approximation of writing (a_) 2 /(a_)i as equal to a 2 /ai, equation (35) reduces to E L = (l -2 + )^ln^- (36) Since 1 t+ is equal to f_, this result may be expressed in the alternative form B L =(-- + )^ln^, (36a) which brings out clearly the dependence of the sign of the liquid junction potential on the relative values of the transference numbers of the anion and cation. If the liquid junction potential under consideration forms part of the concentration cell Ag | AgCl() KCl(d) i KCl(c 2 ) AgClW | Ag, the E.M.F. of the complete cell is given by equation (25) as LIQUID JUNCTION POTENTIALS 209 and hence, from this and equation (36), it is seen that 2J+ 1 E L = -^ E t . (37) This approximate relationship can be tested by suitable measurements on concentration cells with transference. As indicated above, the E.M.F. of a cell with transference can be re- garded as made up of the potential differences at the two electrodes and the liquid junction potential. It will be seen shortly (p. 229) that each of the former may be regarded as determined by the activity of the re- versible ion in the solution contained in the particular electrode. In the cell depicted above, for example, the potential difference at the left- hand electrode is dependent on the activity of the chloride ions in the potassium chloride solution of concentration c\\ similarly the potential difference at the right-hand electrode depends on the chloride ion activity in the solution of concentration C2. For sufficiently dilute solutions the activity of a given ion, according to the simple Debye-Hiickel theory, is determined by the ionic strength of the solution and is independent of the nature of the other ions present. It follows, therefore, that the electrode potentials should be the same in all cells of the type Ag | AgCl() MCl(d) j MCl(c,) AgCl(s) | Ag, where c\ and c 2 represent dilute solutions of any uni-univalent chloride MCI, which must be a strong electrolyte. If E is the constant algebraic sum of these potentials, the E.M.F. of the complete cell with transference, which does vary with the nature of MCI, will be E + EL, i.e. Et = E + EL, .'. E = E t - E L . (38) The difference between E t and E L should thus be constant for given values of c\ and C2, irrespective of the nature of the uni-univalent chloride employed in the cell. Inserting the value of EL given by equation (37) into (38), the result is If the right-hand side is constant, for cells with transference contain- ing different chlorides at definite concentrations, it may be concluded that the approximate equation (36) gives a satisfactory measure of the liquid junction potential between two solutions of the same electrolyte. The results in Table XLV provide support for the reliability of this equa- tion, within certain limits; 12 the transference numbers employed are the mean values for the two solutions, the individual figures not differing greatly in the range of concentrations involved. u Maclnnes, "The Principles of Electrochemistry," 1939, p. 226; data mainly from Machines et al, J. Am. Chem. Soc., 57, 1356 (1935): 59. 503 (1937). 210 REVERSIBLE CELLS TABLE XLV. TEST OP EQUATION FOR LIQUID JUNCTION POTENTIAL Electrolyte Ci c 2 t+ E t Etl%t+ E L NaCl 0.005 0.01 0.392 13.41 mv. 17.1 mv. - 3.7 mv. KC1 0.005 0.01 0.490 16.77 17.1 - 0.3 HC1 0.005 0.01 0.824 28.29 17.2 +11.1 NaCl 0.005 0.04 0.391 39.63 mv. 50.7 mv. - 11.1 mv. KC1 0.005 0.04 0.490 49.63 50.6 - 1.0 HC1 0.005 0.04 0.826 84.16 50.9 +33.3 In order to give some indication of the magnitude of the liquid junc- tion potential, the values of EL calculated from equation (37) are re- corded in the last column. In general, the larger the ratio of the con- centrations of the solutions and the more the transference number of either ion departs from 0.5, i.e., the larger the difference between the transference numbers of the two ions, the greater is the liquid junction potential. The sign is determined by the relative magnitudes of the transference numbers of cation and anion of the electrolyte, as seen from equation (36a). General Equation for Liquid Junction Potential. When the two solutions forming the junction contain different electrolytes, as in many chemical cells, the situation is more complicated ; it is convenient, there- fore, to consider here the most general case. Suppose a cell contains a solution in which there are several ions of concentration Ci, c 2 , , c,-, g.-ions per liter, and suppose this forms a junction with another solution in which the corresponding ionic concentrations are c\ + dci, c 2 + dc^ , c + dc ly ; the valences of the ions are zi, z 2 , , z t , and their transference numbers are /i, / 2 , , 2, , the latter being regarded as constant, since the differences of the ionic concentrations in the two solu- tions are small. If one faraday of electricity is passed through the cell, t l /z l g.-ion of each ionic species will be transferred across the boundary between the two solutions, the positive ions moving in one direction, i.e., left to right according to convention, and the negative ions moving in the opposite direction. The increase of free energy as a result of the transfer of an ion of the ith kind from the solution of concentration c t to that of concentration c + dc % is given by dG = ^ [(/i* + d/i.) - /*.] z *. , - ~ /*> * * where /i and m + dm are the chemical potentials of the particular ions in the two solutions. For the transfer of all the ions across the boundary when one faraday is passed, AG = S - dm. GENERAL EQUATION FOR LIQUID JUNCTION POTENTIAL 211 and utilizing the familiar definition of /u t as /*? + RT In a,, it follows that AG = 2 - fir d Inc., (39) i Z t where a t is the activity of the zth ions at the concentration d. It should be remembered that in the summation the appropriate signs must be used when considering positive and negative ions, since they move in opposite directions. Provided the concentrations of any ion do not differ appreciably in the two solutions, the transfer of ions across the boundary when current passes may be regarded as reversible. If dEi, is the potential produced at the junction between the two solutions, then AG will also be equal to F &EL for the passage of one faraday ; combination of this result with equation (39) gives dE L = - Sdlna,- (40) for the liquid junction potential. Since in actual practice the concentra- tions of the two solutions differ by appreciable amounts, the liquid junc- tion potential can be regarded as being made up of a series of layers with infinitesimal concentration differences; the resultant potential EL is obtained by integrating equation (40) between the limits Ci and C2, representing the two solutions in the cell ; thus 7? np /* C 2 / E L = --=r I 2 -din a,. (41) r t/fi z i This is the general form of the equation for the liquid junction potential between the two solutions ; 13 in order that the integration may be carried out, however, it is necessary to make approximations or to postulate certain properties of the boundary. For example, if the two solutions contain the same electrolyte, con- sisting of one cation and one anion, equation (41) becomes RT **-" If the approximation is made of taking the transference numbers to be independent of concentration, this relationship takes the form t + RT (o + ) 2 t- RT (o_), ~ which is identical with equation (34) for a uni-univalent electrolyte. "Harned, J. Phys. Chem., 30, 433 (1926); Taylor, ibid., 31, 1478 (1927); see also, Guggenheim, Phil. Mag., 22, 983 (1936). 212 REVERSIBLE CELLS Type of Boundary and Liquid Junction Potential. When the two solutions forming the junction contain different electrolytes, the struc- ture of the boundary, and hence the concentrations of the ions at different points, will depend on the method used for bringing the solutions to- gether. It is evident that the transference number of each ionic species, and to some extent its activity, will be greatly dependent on the nature of the boundary; hence the liquid junction potential may vary with the type of junction employed. If the electrolyte is the same in both solu- tions, however, the potential should be independent of the manner in which the junction is formed. In these circumstances the solution at any point in the boundary layer will consist of only one electrolyte at a definite concentration; hence each ionic species should have a definite transference number and activity. When carrying out the integration of equation (41), the result will, there- fore, always be the same no matter what is the type of concentration gradient in the intermediate layer between the two solutions; this the- oretical expectation has been verified by experiment. 14 It is the fact that the liquid junction potential is in- dependent of the structure of the boundary, when the electrolyte is the same on both sides, that makes pos- sible accurate measurement of the E.M.F. of concentration cells with liq- uid junctions. In general, cells of this type are set up with simple "static 1 , junctions, as shown in Fig. 67; the more dilute solution is in the rela- tively narrow tube which is dipped into the somewhat wider vessel con- taining the more concentrated solution, so that the boundary is formed at the tip of the narrower tube. For solutions of different electrolytes four distinct forms of boundary have been described, 15 but only in two cases is anything like a satis- factory integration of equation (41) possible. I. The Continuous Mixture Boundary. This type of boundary, which is the one postulated by Henderson, 16 consists of a continuous series of mixtures of the two solutions, free from the effects of diffusion. If the two solutions are represented by the suffixes 1 and 2, and 1 x is the frac- " Scatchard and Buehrer, J. Am. Chem. Soc., 53, 574 (1931); Ferguson et al, ibid., 54, 1285 (1932); Szabo, Z. physik. Chem., 174A, 33 (1935). "See Guggenheim, J. Am. Chem. Soc., 52, 1315 (1930). "Henderson, Z. physik. Chem., 59, 118 (1907); 63, 325 (1908); Hermans, Rec. trav. chim., 57, 1373 (1938); 58, 99 (1939). Dilute - solution FIQ. 67. Cell with static junction (Maclnnes) THE CONTINUOUS MIXTURE BOUNDARY 213 tion of the former solution at a given point in the boundary, the fraction of solution 2 will be x, where x varies continuously from zero to unity; if d is the concentration of the zth kind of ion at this point, then Ci = (1 x)Ct(i) + C t (2), where C,(D and c t(2 ) are the concentrations of these ions in the bulk of the solutions 1 and 2, respectively. Making use of this expression, and re- placing activities in equation (41) by the corresponding concentrations, as an approximation, it is possible to integrate this equation ; the result, known as the Henderson equation for liquid junction potentials, is __RT_ (Ui - y,) - (t/2 - yj u[ + v( L " F ' (u{ + Fi) - (V* + Fi) ln # + F;' > (42) where U\, Fi, etc., are defined by Ui s 2(c+w+)i, Vi ss S(c_u_)i, U{ ss S(c+z+u+)i and Fi = 2(c_z_w_)i, where c+ and c_ refer to the concentrations of the cations and anions respectively, in g.-ions per liter, u+ and w_ are the corresponding ionic mobilities, and z+ and z_ their valences ; the suffix 1 refers to the ions in solution 1, and similar expressions hold for 1/2, V^, etc. in which the ions in solution 2 are concerned. The continuous mixture boundary presupposes the complete absence of diffusion; since diffusion of one solution into the other is inevitable, however, this type of boundary is probably unstable. It is possible that the flowing type of junction considered below may approximate in be- havior to the continuous mixture type of boundary. Two special cases of the Henderson equation are of interest. If the two solutions contain the same uni-univalent electrolyte at different concentrations, then and V\ = V{ = Ciu_, and F 2 = V*2 = c 2 M-. Insertion of these values in equation (42) gives RT .UtrJbta?!. (43 ) f U+ + U- C 2 Since u+/(u+ + u_) is equal to the transference number of the cation, i.e., to t+, this result is equivalent to which is the same as the approximate equation (36), except that the ratio of the activities has been replaced by the ratio of the concentrations. 214 REVERSIBLE CELLS Another interesting case is that in which two uni-univalent electro- lytes having an ion in common, e.g., sodium and potassium chlorides, are at the same concentration c; in these circumstances, assuming the anion to be common ion, Ui = U{ = cu+(v and V\ = V( = cu,-, / 2 = C/2 = cu+ (2 ) and F 2 = V* = cu_, and substitution in equation (42) gives RT u+ (1 , + u- 11, = r , in ; F M+<2) + M- = -^ In -, (44) where AI and A 2 are the equivalent conductances of the two solutions forming the junction. The resulting relationship is known as the Lewis and Sargent equation, 17 tests of which will be described shortly. II. The Constrained Diffusion Junction. The assumption made by Planck 18 in order to integrate the equation for the liquid junction poten- tial is equivalent to what has been called a "constrained diffusion junc- tion"; this is supposed to consist of two solutions of definite concentration separated by a layer of constant thickness in which a steady state is reached as a result of diffusion of the two solutions from opposite sides. The Planck type of junction could be set up by employing a membrane whose two surfaces are in contact with the two electrolytes which are continuously renewed; in this way the concentrations at the interfaces and the thickness of the intermediate layer are kept constant, and a steady state is maintained within the layer. The mathematical treat- ment of the constrained diffusion junction is complicated; for electrolytes consisting entirely of univalent ions, the result is the Planck equation, R T E L = -jr\n$, (45) where is defined by the relationship . Cj . * - .-k Cl Ui, [/ 2> V\ and Vz having the same significance as before. "Lewis and Sargent, /. Am. Chem. Soc., 31, 363 (1909); see also, Maclnnes and Yeh, ibid., 43, 2563 (1921); Martin and Newton, J. Phya. Chem., 39, 485 (1935). Planck, Ann. Physik, 40, 561 (1S90); see also, Fales and Vosburgh, /. Am. Chem. Soc., 40, 1291 (1918); Hermans, Rec. trav. Mm., 57, 1373 (1938). THE FLOWING JUNCTION 215 111 the two special cases considered above, first, two solutions of the same electrolyte at different concentrations, and second, two electrolytes with a common ion at the same concentration, the Planck equation reduces to the same form as does the Henderson equation, viz., equations (43) and (44), respectively. It appears, therefore, that in these par- ticular instances the value of the liquid junction potential does not depend on the type of boundary connecting the two solutions. III. Free Diffusion Junction. The free diffusion type of boundary is the simplest of all ir. practice, but it has not yet been possible to carry out an exact integration of equation (41) for such a junction. 19 In setting up a free diffusion boundary, an initially sharp junction is formed between the two solutions in a narrow tube and unconstrained diffusion is allowed to take place. The thickness of the transition layer increases steadily, but it appears that the liquid junction potential should be independent of time, within limits, provided that the cylindrical symme- try at the junction is maintained. The so-called " static " junction, formed at the tip of a relatively narrow tube immersed in a wider vessel (cf. p. 212), forms a free diffusion type of boundary, but it cannot retain its cylindrical symmetry for any appreciable time. Unless the two solutions contain the same electrolyte, therefore, the static type of junc- tion gives a variable potential. If the free diffusion junction is formed carefully within a tube, however, it can be made to give reproducible results. 20 IV. The Flowing Junction. In order to obtain reproducible liquid junctions, in connection with the measurement of the E.M.F.'S of cells involving boundaries between two different electrolytes, Lamb and Larson devised the "flowing junction." 21 In the earlier forms of this type of junction (Fig. 68) an upward current of the more dense solution was allowed to meet a downward flow of the less dense solution at a point where a horizontal tube, leading to an overflow, joined the main tube. The levels of the liquids were so ar- ranged that they flowed at the same slow rate, and a sharp boundary was maintained within the hori- zontal portion of the overflow tube. Experiments with indicators have M Taylor, J. Phys. Chem., 31, 1478 (1927). 10 Guggenheim, J. Am. Chem. Soc., 52, 1315 (1930). 11 Lamb and Larson, J. Am. Chem. Soc., 42, 229 (1920); Maclnnes and Yeh, ibid., 43, 2563 (1921); Scatchard, ibid., 47, 696 (1925); Scatchard and Buehrer, ibid., 53, 574 (1931); see also, Roberts and Fenwick, ibid., 49, 2787 (1927); Lakhani, J. Chem. Soc., 179 (1932); Ghosh, J. Indian Chem. Soc., 12, 15 (1935). e ieJ? 0( j e ^^f^ ~~" " = FIQ. 68. The flowing junction (Lamb and Larson) 216 REVERSIBLE CELLS shown that the boundary between the two solutions in a good flowing junction is extremely thin. With such a junction the potentials between two electrolytes having an ion in common can be reproduced to 0.02 millivolt. Simplified forms of flowing junction have been established by allowing the solutions to flow down opposite faces of a thin mica plate having a small hole in which the junction is formed (Fig. 69). The To electrode To electrode Fia. 69. Flowing junction (Roberts and Fenwick/ mica plate may even be eliminated and fine jets of the two liquids caused to impinge directly on one another. The problem of the flowing junction is too difficult to be treated theoretically; since the time of contact between the two solutions is so small, the extent of diffusion will probably be negligible, and hence it has been generally assumed that the flowing junction resembles a con- tinuous mixture (Henderson) type of boundary. On the other hand, it has been suggested that since the transition layer between the solutions is extremely thin, diffusion is of importance; the flowing junction would thus resemble the constrained diffusion (Planck) type of boundary. The only reasonably satisfactory experimental determinations of the potential of a flowing junction have been made with solutions of the same concen- tration and having an ion in common; as already seen, under these con- ditions the Henderson and Planck junctions lead to the same potentials. Measurement of Liquid Junction Potentials with Different Electro- lytes. If the same assumption is made as on page 209, that the potential of an electrode reversible with respect to a given ion depends only on the concentration of that ion, then in cells of the type Ag | AgCl(s) MCl(c) I M'Cl(c) AgCl() | Ag, where MCI and M'Cl, the chlorides of two different univalent cations, are present at the same concentration, the total E.M.F. is equal merely to the liquid junction potential. A number of measurements of cells of this form using 0.1 N and 0.01 N solutions of various chlorides have been made with a flowing junction of the type depicted in Fig. 68; the results are in fair agreement with those derived from the Lewis and Sargent equation (44), as shown by the data in Table XL VI. 22 The discrep- a Maclnnes and Yeh, J. Am. Chem. Soc., 43, 2563 (1921). ELIMINATION OF LIQUID JUNCTION POTENTIALS 217 TABLE XLVI. CALCULATED AND OBSERVED FLOWING JUNCTION POTENTIALS AT 25 Electrolytes Concentration Liquid Junction Potential Observed Calculated HC1 KC1 O.lN 26.78 mv. 28.52 mv. HCl NaCl 33.09 33.38 KC1 NaCl 6.42 4.86 KC1 LiCl 8.76 7.62 NaCl NH 4 C1 -4.21 -4.81 HCl NH 4 C1 0.01 N 27.02 mv. 27.50 mv. HCi LiCl 33.75 34.56 KC1 NH 4 C1 1.31 0.02 NaCl LiCl 2.63 2.53 LiCl CsCl -7.80 -7.67 ancles arc partly due to the assumption that the potentials of the two electrodes in the cell are the same, as well as to the neglect of activity coefficients in the derivation of equation (44). It is possible that the method of producing the flowing junction also has some influence on the observed results; for example, with 0.1 N solutions of hydrochloric acid and potassium chloride, a value of 28.00 mv. was obtained with the type of junction shown in Fig. 69, and 28.27 mv. when jets of the liquids were allowed to impinge on one another directly. Elimination of Liquid Junction Potentials. Electromotive force measurements are frequently used to determine thermodynamic quanti- ties of various kinds; in this connection the tendency in recent years has been to employ, as far as possible, cells without transference, so as to avoid liquid junctions, or, in certain cases, cells in which a junction is formed between two solutions of the same electrolyte. As explained above, the potential of the latter type of junction is, within reasonable limits, independent of the method of forming the boundary. In many instances, however, it has not yet been found possible to avoid a junction involving different electrolytes. If it is required to know the E.M.F. of the cell exclusive of the liquid junction potential, two alternatives are available: cither the junction may be set up in a repro- ducible manner and its potential calculated, approximately, by one of the methods already described, or an attempt may be made to eliminate entirely, or at least to minimize, the liquid junction potential. In order to achieve the latter objective, it is the general practice to place a salt bridge, consisting usually of a saturated solution of potassium chloride, between the two solutions that would normally constitute the junction (Fig. 70). An indication of the efficacy of potassium chloride in re- ducing the magnitude of the liquid junction potential is provided by the-- data in Table XL VII; 23 the values recorded are the E.M.F. 's of the cell with "free diffusion" junctions, Hg | H g2 Cl 2 (s) 0.1 N HCl I x N KC1 j 0.1 N KC1 Hg 2 Cl 2 (s) | Hg, 28 Guggenheim, J. Am. Chem. Soc., 52, 1315 (1930); see also, Fales and Vosburgb, ibid., 40, 1291 (1918); Ferguson et al., ibid., 54, 1285 (1932). 218 REVERSIBLE CELLS TABLE ZLVH. EFFECT OF SATURATED POTASSIUM CHLORIDE SOLUTION ON LIQUID JUNCTION POTENTIALS X E.M.F. X E.M.F. 0.2 19.95 mv. 1.75 5.15 mv. 0.5 12.55 2.5 3.4 1.0 8.4 3.5 1.1 where x is varied from 0.2 to 3.5. When a; is 0.1 the E.M.F. of the cell is 27.0 mv., and most of this represents the liquid junction potential be- tween 0.1 N hydrochloric acid and 0.1 N potassium chloride. As the concentration of the bridge solution is increased, the E.M.F. falls to a small value, which cannot be very different from that of the cell free from liquid junction potential. FIG. 70. Cell with salt bridge When it is not possible to employ potassium chloride solution, e.g., if one of the junction solutions contains a soluble silver, mercurous or thallous salt, satisfactory results can be obtained with a salt bridge con- taining a saturated solution of ammonium nitrate; the use of solutions of sodium nitrate and of lithium acetate has also been suggested. For non-aqueous solutions, sodium iodide in methyl alcohol and potassium thiocyanate in ethyl alcohol have been employed. The theoretical basis of the use of a bridge containing a concentrated salt solution to eliminate liquid junction potentials is that the ions of this salt are present in large excess at the junction, and they consequently carry almost the whole of the current across the boundary. The condi- tions will be somewhat similar to those existing when the electrolyte is the same on both sides of the junction. When the two ions have ap- proximately equal conductances, i.e., when their transference numbers are both about 0.5 in the given solution, the liquid junction potential will then be small [cf. equation (36a)]. The equivalent conductances at infinite dilution of the potassium and chloride ions are 73.5 and 76.3 ohms" 1 cm. 2 at 25, and those of the ammonium and nitrate ions are 73.4 and 71.4 ohms" 1 cm. 2 respectively; the approximate equality of the values for the cation and anion in each case accounts for the efficacy of potassium chloride and of ammonium nitrate in reducing liquid junction potentials. CONCENTRATION CELLS WITH A SINGLE ELECTROLYTE 219 A procedure for the elimination of liquid junction potentials, sug- gested by Nernst (1897), is the addition of an indifferent electrolyte at the same concentration to both sides of the cell. If the concentration of this added substance is greater than that of any other electrolyte, the former will carry almost the whole of the current across the junction between the two solutions. Since its concentration is the same on both sides of the boundary, the liquid junction potential will be very small. This method of eliminating the potential between two solutions fell into disrepute when it was realized that the excess of the indifferent electro- lyte has a marked effect on the activities of the substances involved in the cell reaction. It has been revived, however, in recent years in a modified form: a series of cells are set up, each containing the indifferent electro- lyte at a different concentration, and the resulting E.M.F.'S are extrapo- lated to zero concentration of the added substance. Concentration Cells with a Single Electrolyte : Amalgam Concentra- tion Cells. In the concentration cells already described the E.M.P. is a result of the difference of activity or chemical potential, i.e., partial molal free energy, of the electrolyte in the two solutions; it is possible, however, to obtain concentration cells with only one solution, but the activities of the element with respect to which the ions in the solution are reversible are different in the two electrodes. A simple method of realizing such a cell is to employ two amalgams of a base metal at differ- ent concentrations as electrodes and a solution of a salt of the metal as electrolyte; thus Zn amalgam (zi) | ZnSO 4 soln. | Zn amalgam (rr 2 ), the mole fractions of zinc in the amalgams being x\ and ar 2 , as indicated. The passage of two faradays through this cell is accompanied by the reaction 2 , at the left-hand electrode, and Zn^+ + 2c at the right-hand electrode. Since the concentration of zinc ions in the solution remains constant, the net change is the transfer of 1 g.-atom of zinc from the amalgam of concentration x\ to that of concentration x*\ the increase of free energy is thus AG = /iZn(2) ~ MZn(l) where a\ and a* are the activities of the zinc in the two amalgams. It should be noted that in this derivation it has been assumed that the molecule and atom of zinc are identical. 220 REVERSIBLE CELLS The free energy change is also given by 2FE, where E is the E.M.F. of the cell, so that In the general case of an amalgam concentration cell in which the valence of the metal is z and there are m atoms in the molecule, the equation for the E.M.F. becomes , E = ^ In (47) zmF 0,2 This result is of particular interest because it can be used to determine the activities of metals in amalgams or other alloys by E.M.F. measure- ments; such determinations have been carried out in a number of cases. 24 If the amalgams are sufficiently dilute, the ratio of the activities may be taken as equal to that of their mole fractions, i.e., i/z 2 , or even to that of their concentrations Ci/C2j in the latter case equation (47) takes the approximate form -- (48) c 2 Experiments with amalgams of a number of metals, e.g., zinc, lead, tin, copper and cadmium have given results in general agreement with equa- tion (48); the discrepancies observed are due to the approximation of taking the ratio of the concentrations to be equal to that of the activities. Gas Concentration Cells. Another form of concentration cell with electrodes of the same material at different activities, employing a single electrolyte, is obtained by using a gas, e.g., hydrogen, for the electrodes at two different pressures; thus IWpi) | Solution of hydrogen ions | H 2 (p 2 ), where p\ and p 2 are the partial pressures of hydrogen in the two elec- trodes. The passage of two faradays through this coll is accompanied, as may be readily shown, by the transfer of 1 mole of hydrogen gas from pressure p\ to pressure p 2 ; if the corresponding activities are a\ and a 2 , it is found, by using the same treatment as for amalgam concentration cells, that the E.M.F. is given by If the gas behaves ideally within the range of pressures employed, the ratio of activities may be replaced by the ratio of the pressures; hence 24 Richards and Daniels, J. Am. Chan. Soc., 41, 1732 (1919). GAS CONCENTRATION CELLS 221 If one of the pressures, e.g., p 2 , is kept constant while the other is varied, equation (50) takes the general form RT E = In p + constant, &r (51) where p is the pressure that is varied. According to equation (51) the plot of the E.M.F. of the cell, in which one hydrogen electrode is kept at constant pressure while the other is changed, against the log p of the variable electrode should give a straight line. It is not convenient to test this equation by actual meas- urement of cells with two hydrogen electrodes, but an equivalent result should be obtained if the electrode of constant gas pressure is replaced by another not containing a gas, whose potential does not vary appre- ciably with pressure. Observations have thus been made on cells of the type H 2 (p) | HC1 (0.1 M) Hg 2 Cl 2 (s) | Hg, and the results for hydrogen pressures varying from a partial pressure of 0.00517 atm., obtained by admixture with nitrogen, up to 1000 a f m. are depicted in Fig. 71, in which the E.M.F.'S of the cells are plotted against 0.48 0.44 b a w 0.40 0.36 -2.0 1.0 FIG. 71. Hydrogen pressure and E.M p. the logarithm of the hydrogen pressure. 26 It is seen that the expected linear relationship holds up to pressures of about 100 atm. The devia- tions from linearity up to 600 atm. can be accounted for almost exactly 26 Hainswoioh, Rowley and Maclnnes, ,/. Am. Chem. Soc., 46, 1437 (1924;; Ronraim and Chang, Butt. Soc. Mm., 51, 932 (1932). 222 REVERSIBLE CELLS by making allowance for departure of the hydrogen gas from ideal be- havior. The discrepancies at still higher pressure must be attributed to the neglect of the influence of pressure on the mercury-mercurous chloride electrode. Since the passage of one mole of chlorine into solution requires two faradays, as is the case for a mole of hydrogen, the E.M.F. of a cell con- sisting of two chlorine electrodes at different pressures will be given by any of the equations derived above. It follows, therefore, that the E.M.F. 's of cells of the type Cl,(p) | HC1 soln. HfrCliW | Hg should be represented by equation (51) with the sign preceding the pressure term reversed, because the chlorine yields negative ions; the re- sulting equation may be put in the alternative form Tim E + -rrr In p = constant. (52) &r The data in Table XL VIII were obtained with a cell containing 0.1 N TABLE XLVIII. ELECTROMOTIVE FORCES OF CHLORINE GAS CELLS AT 25 _ RT. v>* T * P E Inp E + ]np 0.0492 atm. - 1.0509 - 0.0387 - 1.0896 0.0247 - 1.0421 - 0.0475 - 1.0896 0.0124 -1.0330 -0.0564 -1.0894 0.0631 - 1.0243 - 0.0650 - 1.0893 0.00293 - 1.0150 - 0.0749 - 1.0899 hydrochloric acid, the pressure of the chlorine gas being reduced by ad- mixture with nitrogen; the constancy of the values in the last column confirm the accuracy of equation (52). 28 In the case of an oxygen gas cell the electrode reactions may be rep- resented by so that the transfer of one mole of oxygen from one electrode to the other requires the passage of four faradays. The E.M.F. of the cell with two oxygen electrodes at different pressures is then or -!?* if the gas behaves ideally. The sign of the E.M.F. is opposite to that of " Lewis and Rupert, J. Am. Chem. Soc. t 33, 299 (1911); Kameyama ei al., J. Soc. Chem. Ind. (Japan), 29, 679 (1926). PROBLEMS 223 the corresponding hydrogen cell [equations (49) and (50)] because of the opposite charges of the ions. Since the oxygen gas electrode does not normally function in a reversible manner (see p. 353), these equations cannot be tested by direct experiment. PROBLEMS 1. Determine the reactions taking place at the separate electrodes and in the complete cell in the following reversible cells: (i) H,fo) (ii) Hg|HgO(s)NaOH|H 2 (<7); (iii) Ag | AgCl(*)KCl H g2 Cl 2 (s) | Hg; and (iv) Pb | PbCl 2 ()KCl j K 2 S0 4 PbS0 4 () | Pb. 2. Devise reversible cells in which the over-all reactions are: (i) Hg + PbO(s) = Pb + HgO(s); (ii) Zn + Hg 2 S0 4 (s) = ZnS0 4 + 2Hg; (iii) Pb + 2HC1 = PbCl 2 (s) + H 2 (0); and (iv) H 2 (0) + J0,(f) = H 2 0(J). 3. The following values for the E.M.F. of the cell Ag | AgBr(s) KBr aq. Hg 2 Br 2 (s) | Hg were obtained by Larson [J. Am. Chem. Soc., 62, 764 (1940)] at various tem- peratures: 20 25 30 0.06630 0.06839 0.07048 volt. State the reaction occurring in the cell for the passage of one faraday, and evaluate the heat content, free energy and entropy changes at 25. 4. Harned and Donelson [J. Am. Chem. Soc., 59, 1280 (1937)] report that the variation of the E.M.F. of the cell H 2 (l atm.) | HBr(a = 1) AgBr(s) | Ag with temperature is represented by the equation E = 0.07131 - 4.99 X 10- 4 (* - 25) - 3.45 X 10-(* - 25) 2 . Calculate the change in heat content, in calories, accompanying the reaction H 2 (l atm.) + 2AgBr(s) = 2Ag + 2HBr(o = 1) at 25. 5. The reversible cell Zn | ZnCl 2 (d) Hg 2 Cl 2 (s) | Hg | Hg 2 Cl 2 (s) ZnCl 2 (c 2 ) | Zn was found to have an E.M.F. of 0.09535 volt at 25. Determine the ratio of the mean ion activities of the zinc chloride in the two solutions. 224 REVERSIBLE CELLS 6. The E.M.F. of the cell H 2 (l atm.) | HBr(m) AgBr(s) | Ag with hydrobromic acid at various small molalities (m) was measured at 25 by Keston [J. Am. Chem. Soc., 57, 1671 (1935)] who obtained the results given below: m E m E 1.262 X 10~ 4 1.775 4.172 0.53300 0.51616 0.47211 10.994 X 10-* 18.50 37.19 0.42280 0.39667 0.36173 Use these data to evaluate E for the cell. 7. The following results were derived from the measurements of Harned, Keston and Donclson [./. Am. Chem. Roc., 58, 989 (1936)] for the cell given in the preceding problem with more concentrated solutions of the acid: m E m E 0.001 0.42770 0.05 0.23396 0.0.5 0.34695 0.10 0.20043 001 031262 0.20 0.16625 0.02 0.27855 0.50 0.11880 Using the value of E Q obtained above, determine the activity coefficients of hydrobromic acid at the various molalities. 8. The following entropy values at 25 were obtained from thermal meas- urements: silver, 10.3 cal./deg. per g.-atom; silver chloride, 23.4 per mole; liquid mercury, 17.8 per g.-atom; and mercurous chloride, Hg2Cl 2 , 46.4 per mole. The increase in heat content of the reaction Ag(s) + Hg 2 Cl 2 (s) = AgCl(s) -f Hg(0 is 1,900 cal. Calculate the E.M.F. of the cell Ag | AgCl(s) KC1 aq. Hg 2 Cl 2 (s) | Hg and its temperature coefficient at 25. 9. Abegg and dimming [Z. Elektrochem., 13, 18 (1910)] found the E.M.F. of the cell with transference Ag | 0.1 N AgN0 3 j 0.01 N AgN0 3 1 Ag to be 0.0590 volt at 25. Compare the result with the calculated value using the following data: 0.1 N AgNO 3 /i = 0.733 t+ = 0.468 0.01 " 0.892 0.465. 10. The E.M.F. 's of the cell with transference Ag | AgCl(s) 0.1 N HOI ; HCl(r) AgCl(s) | Ag at 25, and the transference numbers of the hydrogen ion in the hydrochloric acid of concentration c, are from the work of Shedlovsky and Maclnnes [ J. A m. PROBLEMS 225 Chem. Soc., 58, 1970 (1936)] and of Longsworth [ibid., 54, 2741 (1932)]: c X 10 8 E * H + 3.4468 0.136264 0.8234 5.259 0.118815 0.8239 10.017 0.092529 0.8251 19.914 0.064730 0.8266 40.492 0.036214 0.8286 59.826 0.020600 0.8297 78.076 0.009948 0.8306 100.000 0.8314 Utilize these data to calculate the activity coefficients of hydrochloric acid at the several concentrations. 11. If the E.M.F. of the cell Hg | Hg 2 Cl 2 (s) 0.01 N KC1 j 0.01 N KOH j 0.01 N NaOH HgO(s) | Hg is E t calculate the value of the E.M.F. at 25 free from liquid junction poten- tials, using the Lewis and Sargent formula. 12. The E.M.F.'S of the cells Zn in Hg(ci) | ZnSO 4 aq. | Zn in Hg(c 2 ) were measured by Meyer [Z. physik. Chem., 7, 447 (1891)] who obtained the ensuing results: Temp. ci c 2 E 11.6 11.30 X 10~ 5 3.366 X lO" 8 0.0419 volt 60.0 6.08 X 10-' 2.280 X 10~ 8 0.0520 Assuming the amalgams are dilute enough to behave ideally, estimate the molecular weight of zinc in the amalgams. 13. The E.M.F. of the cell C1 2 (1 atm.) | HC1 aq. AgCl(s) | Ag is 1.1364 volt at 25. The Ag, AgCl(s) electrode may be regarded as a chlorine electrode with the gas at a pressure equal to the dissociation pressure of silver chloride; calculate the value of this pressure at 25. CHAPTER VII ELECTRODE POTENTIALS Standard Potentials. When all the substances taking part in a reac- tion in a reversible cell are in their standard states, i.e., at unit activity, the E.M.F. is the standard value E for the given cell. If the reaction under consideration occurs for the passage of n faradays, then the stand- ard free energy change A(J is equal to nFE; hence by equation (23), page 137, with all the activities equal to unity, - AG = nFE = RT In K, (1) where K is the equilibrium constant of the cell reaction. If the reactants and resultants are at any arbitrary concentrations, or activities, the E.M.F. is E and the corresponding free energy change for the reaction AG is equal to nFE- it follows, therefore, from equation (22), page 136, that for the reaction aA + &B + - = IL + mM + occurring in the cell for the passage of n faradays, - AG = RTln K - RT In .'. nFE = nFE - RT In q a y This is the general equation for the E.M.F. of any reversible chemical cell in which the reactants and resultants are at any arbitrary activities O A , a B , and a L , a M , , respectively. Since E Q is related to the equilibrium constant of the reaction, it can clearly be regarded as equal to the difference between two constants Ei and E% characteristic of the separate electrode reactions which to- gether make up the process occurring in the cell as a whole. Further, the activity fraction may also be separated into two corresponding parts, so that equation (2) can be written as / PT \ (3) where a\ and a* are the activity terms applicable to the two electrodes, and vi and v* are the numbers of molecules or ions of the corresponding 226 STANDARD POTENTIALS 227 species involved in the ceil reaction. The actual B.M.P. of the cell can similarly be separated into the separate potentials of the electrodes; if these are represented by E\ and E^ it is evident that they may be identi- fied, respectively, with the quantities in the two sets of parentheses in equation (3). In general, therefore, it is possible to write RT JS? t = ?-2lna;< (4) for the potential of an electrode in terms of its standard potential and the activities of the species involved in the electrode process. It is evident from equation (4) that the standard potential is the potential of the electrode when all of these substances are at unit activity, i.e., in their standard states. The application of the procedure outlined above may be illustrated with reference to the reversible cell H,(l atm.) | HCl(c) AgCl(s) | Ag, in which the reaction is iH 2 (l atm.) + AgCl(s) = H+ + Cl~ + Ag(s) for the passage of one faraday. The appropriate form of equation (2) in this case is The individual electrode reactions (cf. p. 195) are (1) ^H 2 (l atm.) = H+ + e, and (2) AgCl + c = Ag(s) + C1-, so that equation (5) may be split up as follows (6a) and (66) The standard state of hydrogen is the ideal gas at 1 atm. pressure, and the standard states of silver and silver chloride are the solids; it follows, therefore, that in this particular case a H ,, a Ag ci and a Ag are unity, so that R T #H t ,H* = Eua+ -- ^r In H* (7a) 228 ELECTRODE POTENTIALS and RT #Ag,Agci,cr = ^Ag,Agci,cr + ~TT In Ocr, (76) where the E Q terms are the standard potentials of the H 2 (l atm.), H+ and Ag(s), AgCl(s), Cl~ electrodes. It is seen, therefore, that in the cell under consideration the potential of each electrode depends only on the activity of one ionic species, apart from the standard potential of the system. The results given by equations (7a) and (76) may be expressed in a general form applicable to electrodes of all types; using the terms " oxi- dized " and "reduced" states in their most general sense (cf. p. 186), the potential of the electrode at which the reaction is Reduced Stated Oxidized State + n Electrons, is given by " nF (Reduced State) In the electrodes already considered the hydrogen ions and the silver chloride represent the respective oxidized states, whereas hydrogen gas, in the first case, and silver and chloride ions, in the second case, are the corresponding reduced states. For any electrode, therefore, at which the reaction occurring is aA + 6B + - - - = xX + i/Y + + nt, the general expression for the electrode potential is ,.,_ 111 4^. nb a*a B If the electrode is one consisting of a metal M of valence z+, reversible with respect to M z + ions, so that the electrode reaction is M^ M'+ + *+, the equation for the potential takes the form L Sr j (8a) where OM is the activity of the solid metal and a M + is that of the cations in the solution with which the metal is in equilibrium. By convention, the solid state of thfe metal is taken as the standard state of unit activity; for an electrode consisting of the pure metal, therefore, OM may be re- placed by unity so that equation (8a) becomes _ RT . /QM i OM*. (80) INDIVIDUAL ION ACTIVITIES 229 For an electrode involving a substance A which is reversible with respect to the anions A*-, the electrode reaction is A'-^ A + 2_e, the electrode material now being the oxidized state whereas the anions represent the reduced state; the equation for the electrode potential is then E- = E Q - -^In (9a) z-/' a A - v ' As before, the activity a\ of the substance A in the pure state, or if A is a gas then the activity at 1 atm. pressure, is taken as unity so that equation (9a) can be written as *_-*>--* In J- - The general form of equations (9a) and (9&) for any electrode revcr- ible with respect to a single ion of valence z is readily seen to be E = Eo-F~lna l , (10) wluTC a l is the activity of the particular ionic species; in this equation I lie upper signs apply throughout for a positive ion, while the lower signs are used for a negative ion. For practical purposes the value of 72, i.e., 8.313 joules, and F, i.e., 96,500 coulombs, may be inserted in equation (10) and the factor 2.3026 introduced to convert Naperian to Briggsian logarithms; the result is E = E* =F 1.9835 X 10~ 4 log a,. (10a)* z At 25 c., i.e., 298.16 K., which is the temperature most frequently employed for accurate electrochemical measurements, this equation becomes 0.05915 t E = E Q T -- log a,. " Individual Ion Activities. The methods described in Chap. V for the determination of the activities or activity coefficients of electrolytes, r ^ well as those depending on vapor pressure, freezing-point or other osmotic measurements, give the mean values for b >th ions into which the solute * A convenient form of this equation for approximate purposes is E* = # ^ 0.0002 ~ log a t . 230 ELECTRODE POTENTIALS dissociates. The question, therefore, arises as to whether it is possible to determine individual ion activities experimentally. An examination of the general equation (41), p. 211, or any of the other exact equations, for the liquid junction potential, shows that this potential is apparently determined by the activities of the individual ionic species; hence, if liquid junction potentials could be measured, a possible method would be available for the evaluation of single ion activities. It should be emphasized that the so-called experimental liquid junction potentials recorded in Chap. VI were based on an assumption concerning individual ion activities, e.g., that the activity of the chloride ion is the same in all solutions of univalent chlorides at the same concentration; they cannot, therefore, be used for the present purpose. The same point can be brought out in another manner. The E.M.F. of the cell with transference Ag | AgCl(s) KCl( Cl ) ; KCl(c 2 ) AgCl(s) | Ag is, according to equation (25), page 203, rt RT ^ a* E = - 2J+ -TT In > * F ai whereas the liquid junction potential, as given by equation (35), page 208, is RT a* RT (ocr) 2 If the ratio of the activities of the chloride ions were known, the value of the liquid junction potential could be derived precisely from equation (11), provided the E.M.F. of the complete cell, i.e., E, were measured. Although it is true, therefore, that the individual ion activities might be evaluated from a knowledge of the liquid junction potential, the latter can be obtained only if the single ion activities are known. A further possibility is that by a suitable device the liquid junction potential might be eliminated completely, i.e., EL might be made equal to zero; under these conditions, therefore, equation (11) would give RT (flcr)i /<rtx E = -;r\n~-> (12) P (acr)2 and so the individual activities of the chloride ion at different concen- trations might be obtained by using an extrapolation procedure similar to that employed in Chap. VI to determine mean activities. It is doubt- ful, however, whether the results would have any real thermodynamic significance; the apparent individual ion activities obtained in this manner are actually complicated functions of the transference numbers and ARBITRARY POTENTIAL ZERO 231 activities of all the ions present, including those contained in the salt bridge employed to eliminate the liquid junction potential. It is possible that, as a result of a cancellation of various factors, these activities are virtually equal numerically to the individual activities of the ions, but thermodynamically they cannot be the same quantities. 1 Arbitrary Potential Zero: The Hydrogen Scale. Since the single electrode potential [cf. equation (10)] involves the activity of an indi- vidual ionic species, it has no strict thermodynamic significance; the use of such potentials is often convenient, however, and so the difficulty is overcome by defining an arbitrary zero of potential. The definition widely adopted, following on the original proposal by Nernst, is as follows : The potential of a reversible hydrogen electrode with gas at one atmosphere pressure in equilibrium with a solution of hydrogen ions at unit activity shall be taken as zero at all temperatures. According to this definition the standard potential of the hydrogen electrode is the arbitrary zero of potential [cf. equation (7a)]: electrode potentials based on this zero are thus said to refer to the hydrogen scale. Such a potential is actually the E.M.F. of a cell obtained by combining the given electrode with a standard hydrogen electrode; it has, conse- quently, a definite thermodynamic value. For example, the potential (E) on the hydrogen scale of the electrode M, M*+(aM + ), which is revers- ible with respect to the z-valent cations M>+, in a solution of activity GM+, is the E.M.F. of the cell M | M"(a M +) H+(a H + = 1) | H 2 (l atm.) free from liquid junction, or from which the liquid junction potential has been supposed to be completely eliminated. The reaction taking place in the cell is M + zH+(a n + = 1) = M"(a M +) + **IIi(l atm.), (13) and the change of free energy is equal to zFE volt-coulombs. If a\t + is equal to unity, the potential of the electrode is E and the free energy of the reaction is zFE; this quantity is called the standard free energy of formation of the M** ions, although it is really the increase of the free energy of the foregoing reaction with all substances in their standard states. If the electrode is reversible with respect to an anion, e.g., X s ~, as in the cell X | X-(ax-) H*(a H * = 1) I H 2 (l atm.), the reaction is X'-(ax-) + *H+(a H + = 1) = X + JH(1 atm.), (14) 1 Taylor, /. Phys. Chem., 31, 1478 (1927); Guggenheim, ibid., 33, 842, 1540, 1758 (1929); see also, Phil Mag., 22, 983 (1936). 232 ELECTRODE POTENTIALS and the standard free energy increase is zFE*. This is the standard free energy of discharge of the X*~ ions, and hence the standard free energy of formation of an anion is + zFE, where E is its standard potential. Sign of the Electrode Potential. The convention concerning the sign of the E.M.F. of a complete cell (p. 187), in conjunction with the inter- pretation of single electrode potentials just given, fixes the convention as to the sign of electrode potentials. The E.M.F. of the cell M | M+(a M +) H+(an- = 1) | H,(l atm.) will ciea/ly be equal and opposite to that of the cell H 2 (l atm.) | H+IOH* = 1) M+(a M *) I M, so that the sign of the potential of the electrode when written M, M + must be equal and opposite to that written M+, M. In accordance with the convention for E.M.F. 's, the positive sign as applied to an electrode potential represents the tendency for positive ions to pass spontaneously from left to right, or of negative ions from right to left, through a cell in which the electrode is combined with a hydrogen electrode. The poten- tial of the electrode M, M+ represents the tendency for the metal to pass into solution as ions, i.e., for the metal atoms to be oxidized, whereas that of the electrode M+, M is a measure of the tendency of the ions to be discharged, i.e., for the ions to be reduced. \ Subsidiary Reference Electrodes : The Calomel Electrode. The de- termination of electrode potentials involves, in principle, the combination of the given electrode with a standard hydrogen electrode and the meas- urement of the E.M.F. of the resulting cell. For various reasons, such as the difficulty in setting up a hydrogen gas electrode and the desire to avoid liquid junctions, several subsidiary reference electrodes, whose potentials are known on the hydrogen scale, have boon devised. The most common of these is the calomel electrode; it consists of mercury in contact with a solution of potassium chloride saturated with mercurous chloride. Three different concentrations of potassium chloride have been employed, viz., 0.1 or, 1.0 \ and a saturated solution. By making use of the standard poten f u:l of the Ag, A^Cl^s), Cl~ electrode described below, the following results have been obtained for the potentials on the hydrogen scale of the three calomel electrodes at temperatures in the vicinity of 25. 2 Hg, Hg 2 Cl 2 (s) 0.1 N KC! - 0.3338 + 0.00007 (t - 25) Hg, Hg 2 Cl 2 (s) 1 .0 N K( '1 - 0.2800 + 0.00024 (t - 25) Hg, IIg a Cl 2 (s) Saturated KC1 - 0.2415 + 0.0007(3 (t - 25) These values cannot be regarded as exact, since in therr derivation it has been necessary to make allowance for liquid junction potentials or for 2 Hamor, Trans. Eleclrochem. tim , 72, 45 (1937). SUBSIDIARY REFERENCE ELECTRODES 233 single ion activities; the calomel electrodes are, however, useful in con- nection with various aspects of electrochemical work, as will appear in this and later chapters (see p. 349). The electrode with 0.1 N potassium chloride is preferred for the more precise measurements because of its low temperature coefficient, but the calomel electrode with saturated potassium chloride is often employed because it is easily set up, and when used in conjunction with a saturated potassium chloride salt bridge one liquid junction, at least, is avoided. Various types of vessels have been described for the purpose of setting up calomel electrodes; the object of the special designs is generally to prevent diffusion of extraneous electrolytes into the potassium chloride solution. In order to obtain reproducible results the mercury and mer- curous chloride should be pure; the latter must be free from mercuric compounds arid from bromides, and must not be too finely divided. A small quantity of mercury is placed at the bottom of the vessel; it is then covered with a paste of pure mercurous chloride, mercury and potassium chloride solution. The vessel is then completely filled with the appropriate solution of potassium chloride which has been saturated FIG. 72. Forms of calomel electrode witli calomel. Electrical connection N made l>y moan- 01 phtnum \\irr sealed into a glass tube, or through the walls of fh< vessel. The method employed for connecting the calomel electrode to another electrode so as to make a cell whose E.M F. can he measured depends on the type of electrode vessel. In the special form used by some workers, Fig. 72, I, this purpose is served by a side tube, sealed into the main vessel, while in the simple apparatus, consisting of a 2 or 4 oz. bottle, often employed for laboratory work (Fig. 72, II), a siphon tube provides the means of connection. The compact calomel electrode of thn type used with many commercial potentiometers is dipped directly into the solution of the 234 ELECTRODE POTENTIALS other electrode system; electrical connection between the two solutions occurs at the relatively loose ground joint (Fig. 72, III). The Silver-Silver Chloride Electrode. In recent years the silver- silver chloride electrode has been frequently employed as a reference electrode for accurate work, especially in connection with the determina- tion of standard potentials by the use of cells containing chloride which are thus free from liquid junction potentials. The standard potential of the Ag, AgCl(s), Cl~ electrode is obtained as follows: the E.M.F. of the cell H 2 (l atm.) | H+C1- AgCl(s) | Ag, where the activities of the hydrogen and chloride ions in the solution of hydrochloric acid have arbitrary values, is given by equation (5), as R T E = J5? - -jr In a H +ocr, (15) since the hydrogen, silver and silver chloride are in their standard states. Replacing the product aH+flcr by a 2 , where a is the mean activity of the hydrochloric acid, equation (15) becomes \na. (16) This equation is seen to be identical with equation (14) of Chap. VI, and in fact the E derived on page 201 by suitable extrapolation of the E.M.F. data of cells of the type shown above, containing hydrochloric acid at different concentrations, is identical with the E of equations (15) and (16). It follows, therefore, that the standard E.M.F. of the cell under consideration is + 0.2224 volt at 25, and hence the standard E.M.F. of the corresponding cell with the electrodes reversed, i.e., Ag | AgCl(s) H+C1- | H 2 (l atm.) is 0.2224 volt. 3 By the convention adopted here, this represents the standard potential of the silver-silver chloride electrode; hence Ag | AgCl(s), Cl-(ocr = 1): E = - 0.2224 volt at 25. If the potential of this electrode is required in any arbitrary chloride solution, an estimate must, be made of the chloride ion activity of the latter; the potential can then be calculated by means of equation (76). Several methods have been described for the preparation of silver- silver chloride electrodes: a small sheet or short coil of platinum is first coated with silver by electrolysis of an argentocyanide solution, and this is partly converted into silver chloride by using it as an anode in a chloride solution. Alternatively, a spiral of platinum wire may be covered with a paste of silver oxide which is reduced to finely divided silver by heating Earned and Enters, /. Am. Chem. Soc., 54, 1350 (1932); 55, 2179 (1933). DETERMINATION OF STANDARD POTENTIALS 235 to about 400; the silver is then coated with silver chforide by electrolysis in a chloride solution as in the previous case. A third method is to decompose byvheat a paste of silver chlorate, silver oxide and water supported on a small spiral of platinum wire; in this way an intimate mixture of silver and silver chloride is obtained. It appears that if sufficient time is permitted for the electrodes to "age," the three methods of preparation give potentials which agree within 0.02 millivolt. 4 Electrodes similar to that just described, but involving bromide or iodide instead of chloride, have been employed as subsidiary reference electrodes for measurements in bromide and iodide solutions, respec- tively. They are prepared and their standard potentials (see Table XLIX) are determined by methods precisely analogous to those em- ployed for the silver-silver chloride electrode. 5 Sulfate Reference Electrodes. For measurements in sulfate solu- tions, the electrodes Pb(Hg) | PbS0 4 (s), SO and Hg | Hg 2 S0 4 (s), SO have been found useful; their standard potentials may be determined by suitable extrapolation, as in the case of the silver-silver chloride elec- trode, or by measuring one electrode against the other. 6 The best values are Pb(Hg) | PbS0 4 (s), SOr-faor = 1): # = + 0.3505 at 25 and Hg | Hg 2 SO 4 (s), S04~(a8o 4 " = 1): # = - 0.6141 at 25. If the electrodes are required for use as reference electrodes of known potential in sulfate solutions of arbitrary activity, an estimate of this activity must be made. Determination of Standard Potentials : Zinc. The procedure adopted for determining the standard electrode potential of a given metal or non-metal depends on the nature of the substance concerned; a number of examples of different types will be described in order to indicate the different methods that have been employed. When a metal forms a soluble, highly dissociated chloride, e.g., zinc, the standard potential is best obtained from measurements on cells with- out liquid junction, viz., Zn | ZnCl 2 (m) AgCl(s) | Ag. * Smith and Taylor, J. Res. Nat. Bur. Standards, 20, 837 (1938); 22, 307 (1939). Keston, /. Am. Chem. Soc., 57, 1671 (1935); Harned, Keston and Donelson, ibid., 58, 989 (1936); Owen, ibid., 57, 1526 (1935); Cann and Taylor, ibid., 59, 1841' (1937); Gould and Vosburgh, ibid., 62, 2280 (1940). Shrawder and Cowperthwaite, J. Am. Cheni. Soc., 56, 2340 (1934); Harned and Hamer, ibid. t 57, 33 (1935). 236 ELECTRODE POTENTIALS The cell reaction for the passage of two faradays is Zn() + 2AgCl(s) = Ag(s) + Zn++ + 2C1-, and the E.M.P., according to equation (2), is - (17) Since the zinc, silver chloride and silver are present as solids, and hence? are in their standard states, their activities are unity; hence, equation (17) becomes r>7 T E = - -_v In az.**a?v. (18) &r The standard E.M.F. of the coll, i.e., E, is equal to the difference between the standard potentials of the Zn, Zn+ f and Ag, AgCl(s), Cl~ electrodes; the value of the latter is known, - 0.2224 volt at 25, and hence if E Q of equation (18) were obtained the standard potential Ez n ,zn++ would be available. The evaluation of E is carried out by one of the methods described in Chap. VI in connection with determination of activities and activity coefficients; the problem in the latter case is to evaluate E for a particular cell, and this is obviously identical with that involved in the estimation of standard potentials. 7 Other Bivalent Metals.--The standard potentials of a number of bivalent metals forming highly dissociated soluble sulfatcs, e.g., cadmium, copper, nickel and cobalt, as well as zinc, have been obtained from (ells of the type M [ M^SOr-(w) PbS0 4 (s) i Ph(IIg) and M | M++SOr-(w) Hg 2 8O 4 (a) | Ifg. The extrapolation procedure is in principle identical with that noted above, and since the standard potentials of the electrodes Pb(Hg), PbS() 4 (.s')> S()f~ and Hg, IlgaSfV.s-), &O~4 ~ are known, the standard potential of the metal -M can be evaluated. In several cases the E M.F. data for dilute solutions are not easily obtainable and consequently the extrapolation i* not reliable. It is apparent, however, from measurements in moderately concentrated solutions that the sulfates of copper, nickel, cobalt and zinc behave in an exactly parallel manner, and hence the mean activity coefficients are probably the same in each case. The values for zinc MI! fate are known, since K.M.F. measurements have been made at sufficiently low concentrations for accurate extrapolation and the evaluation of E (] to be possible. The assumption is then made that the mean activity coefficients are equal in the four sulfate solutions at equal ionic strengths. It is thus possible to derive the appropriate values 'Scatchard and Tefft, J. Am. i,litm. Nor., 52, 2272 (1930); Getman, J. Phys. Chem., 35, 2749 (1931). THE ALKALI METALS 237. of E Q , for the cells involving copper, nickel or cobalt sulfate, directly from the E.M.F. measurements by means of the equations f? 7 1 E = E "" 2F ln a D/TT JPT* = Jjo - __ ln m __ _ ln 7> which are applicable to the sulfate cells; in this instance m is equal to the molality m of the 911! fate solution. The Alkali Metals. The alkali metals present a special case in the determination of standard potentials since these substances attack water; the difficulty has been overcome by making measurements in aqueous solution with a dilute amalgam which reacts slowly with water (cf. p. 198), and then comparing the potential of the amalgam with that of the pure metal in a non-aqueous medium with which it does not react. 8 The E.M.P. of the stable arid reproducible cell Na (metal) | Nal in ethylamine | 0.206% Na(Hg) is + 0.8449 volt at 25, independent of the concentration of the sodium iodide solution; since the process occurring in the cell is merely the transfer of sodium from the pure metal to the dilute amalgam, the poten- tial must also be independent of the nature of the solvent or solute. The E.M.F. of the cell 0.206% Na(Hg) | NaCl aq. 1.022 M Hg 2 Cl 2 (s) | Hg is + 2.1582 volts at 25, and hence that of the combination Na | NaCl aq. 1.022 M Hg,Cl() | Hg is + 3.0031 volts. The reaction occurring in this cell is Na + JHg,CI,() = Hg(0 + Na+ + Cl~, for the passage of one faraday, and so the E.M.F. is represented by RT E = E - -T In aNa+acr = E* ~ -jr In m - -y- In y f (20) Lewis and Kraus, J. Am. Chem. Soe., 32, 1459 (1910); Armbruster and Crenshaw, ibid., 56, 2525 (1934); Bent and Swift, ibid., 58, 2216 (1936); Bent, Forbes and Forziati, ibid., 61, 709 (1939). 238 ELECTRODE POTENTIALS where the mean molality m is equal to the molality m of the sodium chloride solution. The molality of the solution is 1.022, and at this con- centration the mean activity coefficient of sodium chloride is known from other measurements (Chap. VI) to be 0.655; it is thus readily found from equation (20) that E is + 2.9826 volt at 25. The standard potential of the electrode Hg, Hg 2 Cl 2 (s), Cl~ is - 0.2680 volt,* and so the stand- ard potential of the sodium electrode is given by Na | Na+(a Na + = 1): # = + 2.7146 volts at 25. Cells with Liquid Junction. In the cases described above it has been possible to utilize cells without liquid junctions, but this is not always feasible: the suitable salts may be sparingly soluble, they may hydrolyze in solution, their dissociation may be uncertain, or there may be other reasons which make it impossible, at least for the present, to avoid the use of cells with liquid junctions. In such circumstances it is desirable to choose, as far as possible, relatively simple junctions, e.g., between two electrolytes at the same concentration containing a common ion or between two solutions of the same electrolyte at different concentrations, so that their potentials can be calculated with fair accuracy, as shown in Chap. VI. The procedure may ,be illustrated with reference to the determination of the standard potential of silver, of which the only convenient salt for experimental purposes is the nitrate. Since the most reliable reference electrodes contain solutions of halides, it is necessary to interpose a bridge solution between them; the result is Ag | AgNO 3 (0.1 N) : KN0 3 (0.1 N) j KC1(0.1 N) Hg 2 Cl 2 (s) | Hg, in which the liquid junctions, indicated by the dotted lines, are both of the type to which the Lewis and Sargent equation is applicable. The E.M.F. of the complete cell is 0.3992 volt and the sum of the liquid junction potentials is calculated to be + 0.0007 volt, so that the E.M.F. of the cell Ag | AgN0 3 (0.1 N) || KC1(0.1 N) H g2 Cl 2 (s) | Hg, where the double vertical line between the two solutions is used to imply the complete elimination of the liquid junction potential, is 0.3992 + 0.0007, i.e., - 0.3985 volt at 25. The potential of the Hg, Hg 2 Cl 2 (s), KC1(0.1 N) electrode is known to be - 0.3338 volt (p. .232) and so that of the Ag, AgN0 3 (0.1 N) electrode is - 0.7323 volt. The potential of the silver electrode may be represented by means of equation (9) as E = Eft* At* - -jr In a Ag +, (21) *This value is obtained by utilizing the observation that the potentials of the Hg, HgiCli() and Ag, AgCl(a) electrodes in the same chloride solution differ by 0.0456 volt at 25. HALOGEN ELECTRODES 239 and although E is known, the activity of the silver ions in 0.1 N silver nitrate is, of course, not available. It is necessary, therefore, to make an assumption, and the one commonly employed is to take the activity of the silver ions in the silver nitrate solution as equal to the mean activity of the ions in that solution. The mean activity coefficient of 0.1 N silver nitrate is 0.733, and so the mean activity which is used for ax,* in equation (21) is 0.0733. Since E is - 0.7323 volt, it is readily found that E AK , AK + is - 0.7994 volt at 25. Halogen Electrodes. The determination of the standard potentials of the halogens is simple in principle; it involves measurement of the potential of a platinum electrode, coated with a thin layer of platinum or indium black, dipping in a solution of the halogen acid or a halide, and surrounded by the free halogen. The uncertainty due to liquid junction can be avoided by employing the appropriate silver-silver halide or mercury-mercurous halide electrode as reference electrode. In prac- tice, however, difficulties arise because of the possibility of the reactions X 2 + H 2 0^ HXO + H+ + X~ and x 2 + x-- x^-, where X 2 is the halogen molecule; the former reaction occurs to an appreciable extent with chlorine and bromine, and the latter with bro- mine and iodine. The first of these disturbing effects is largely elimi- nated by using acid solutions as electrolytes, but due allowance for the removal of halide ions in the form of perhalide must be made from the known equilibrium constants. The electrode reaction for the system X 2 , X~ is X- = iX 2 + e, e arguments o the equation so that by the arguments on page 228 the electrode potential is given by ^. (22) For chlorine and bromine the standard states may be chosen as the gas at 1 atm. pressure, and if the gases are assumed to behave ideally, as will be approximately true at low pressures, equation (22) can be written in the form E = *, x - - ^r In p x , + -j- In a x -, (23) where px, is the pressure of the gas in atmospheres. In the cell HC1 soln. Hg 2 Cl 2 | Hg 240 ELECTRODE POTENTIALS the reaction for the passage of one f araday is = Hg + |Cl 2 (p so that the E.M.F., which is independent of the nature of the electrolyte, provided it is a chloride solution, is given by R T E = E*-\np C i t , (24) The standard E.M.F. of this cell as given by equation (25), with the pressure in atmospheres, is the difference between the standard potentials of the C1 2 (1 atm.), Cl~ and the Hg, Hg 2 Cl 2 (s), Cl~ electrodes; since the latter is known to be 0.2680 volt at 25, the value of the former could be obtained provided E of the cell under consideration were available. This cell is, in fact, identical with the one for which measurements are given on page 222, and the results in the last column of Table XL VI 1 1 are actually the values of E Q required by equation (25) above. It follows, therefore, taking a mean result of 1.090 volts at 25 for E Q , that the standard potential of the chlorine electrode is 1.090 0.2680, i.e., - 1.358 volts at 25. The standard potentials of bromine and iodine have been determined by somewhat similar methods; with bromine the results are expressed in terms of two alternative standard states, viz., the gas at 1 atm. pressure or the pure liquid. The standard state adopted for iodine is the solid state, so that the solution is saturated with respect to the solid phase. 9 The acandard potential of fluorine has not been determined by direct experiment, but its value has been calculated from free energies derived from thermal and entropy data. 10 The Oxygen Electrode. The standard potential of the oxygen elec- trode cannot be determined directly from E.M.F. measurements on account of the irreversible behavior of this electrode (cf. p. 353); it is possible, however, to derive the value in an indirect manner. The problem is to determine the E.M.F. of the cell H 2 (l atm.) | H+(a H + = 1) || OH-(a ir = 1) | 2 (1 atm.), in which the reaction for the passage of two faradays is essentially H 2 (l atm.) + iO 2 (l atm.) = H 2 O(/). The object of the calculations is to evaluate the standard free energy Lewis and Storch, J. Am. Chem. Soc. t 39, 2544 (1917); Jones and Baeckstrom, ibid., 56, 1524 (1934); Jones and Kaplan, ibid., 50, 2066 (1928). 10 Latimer, J. Am. Chem. Soc., 43, 2868 (1926); see also, Glasstone, "Text-Book of Physical Chemistry," 1940, p. 993. THE OXYGEN ELECTRODE 241 (A(?) of this process, for this is equal to 2FE, where E is the standard E.M.P. of the cell. According to equation (1), A<7 = - RTlnK, where K for the given reaction is defined by The activity of liquid water is taken as unity, since this is the usual standard state, and the activities of the hydrogen and oxygen are repre- sented by their respective pressures, since the gases do not depart appre- ciably from ideal behavior at low pressure; hence, equation (26) may be written as From a study of the dissociation of water vapor into hydrogen and oxygen at high temperatures, it has been found that the variation with temperature of the equilibrium constant K' p , defined by can be represented, in terms of the free energy change, by AC ' = - 57,410 + 0.94 T In T + 1.65 X lO-'T 2 - 3.7 X 10~ 7 r 3 + 3.92T. If the relationship may be assumed to hold down to ordinary tempera- tures, then at 25, - KT In K' p = AC ' = - 54,600 cal., and this is the free energy increase accompanying the conversion of one mole of hydrogen gas and one-half mole of oxygen to one mole of water vapor, all at atmospheric pressure. For the present purpose, however, the free energy required is that of the conversion of hydrogen and oxygen at atmospheric pressure to liquid water, i.e., to water vapor at 23.7 mm. pressure at 25. The difference between these free energy quantities is 23 7 RT In -- = - 2,050 cal. at 25, and hence the AG required is - 54,600 - 2,050, i.e., - 56,650 cal. An entirely different method of arriving at this standard free energy change is based partly on E.M.F. measurements, and partly on equilibrium data. From the dissociation pressure of mercuric oxide at various tem- peratures it is possible to obtain the standard free energy of the reaction Hg(i) + \V*(g) = HgO(*), 242 ELECTRODE POTENTIALS and when corrected to 25 the result is found to be - 13,940 cal. The E.M.F. of the reversible cell H 2 (l atm.) | KOH aq. HgO(s) | Hg is + 0.9264 volt at 25, and so the free energy of the reaction H 2 (l atm.) + HgO(s) = H 2 0(0 which occurs in the cell for the passage of two faradays, is 2 X 96,500 X 0.9264 volt-coulombs, i.e., 42,760 cal. Since all the reactants and resultants in this reaction are in their standard states, this is also the value of the standard free energy change.* Addition of tl\e two results gives the standard free energy of the reaction as - 56,700 cal. at 25. As a consequence of several different lines of approach, all of which give results in close agreement, it may be concluded that the standard free energy of this reaction is 56,700 cal. at 25, and since, as seen above, this is equal to 2FE Q , it follows that the standard E.M.F. of the oxygen-hydrogen cell is 56,700 X 4.185 00ft 2 X 96,500 " L229 V ltS at 25. It would appear, at first sight, that this is also the standard potential of the oxygen electrode, but such is not the case. The E.M.F. calculated is the standard value for the cell H 2 (l atm.) | Water | 2 (1 atm.) in which both oxygen and hydrogen electrodes are in contact with the same solution, the latter having the activity of pure water. If the hydro- gen ion activity in this solution is unity, the hydrogen electrode potential is zero, by convention, and hence 1.229 volts is the potential of the elec- trode H,0(0, H+(a H + = 1) | 2 (1 atm.). The standard potential of oxygen, as usually defined, refers to the elec- trode 2 (1 atm.) | OH-(a H- = 1), H 2 0(Z), that is, in which the hydroxyl ions are at unit activity. It is known from the ionic product of water (see Chap. IX) that in pure water at 25, dH+aoH- = 1.008 X 10~ 14 , and so 1.229 volts is the potential of the oxygen electrode, at 1 atm. * A small correction may be necessary because the activity of the water in the KOH solution will be somewhat less than unity. STANDARD ELECTRODE POTENTIALS 243 pressure, when the activity of the hydroxyl ions is 1.008 X 10~ M . The standard potential for unit activity of the hydroxyl ions is then derived from equation (96) in the form R T E = Eo t ,oir + -TT In aoir, which, for a temperature of 25, becomes in this case - 1.229 = E^oir + 0.05915 log (1.008 X 10~ 14 ), r = - 0.401 volt. Standard Electrode Potentials. By the use of methods, such as those described above, involving either E.M.F. measurements or free energy and related calculations, the standard potentials of a number of electrodes have been determined; some of the results for a temperature of 25 are recorded in Table XLIX. It should be noted that the signs of the TABLE XLIX. STANDARD POTENTIALS AT 25 Elec- trode Reaction Poten- tial Electrode Reaction Poten- tial Li, Li+ Li - Li+ + 4-3024 Ha, OH- iHi4-OH--HO4- 4-0.828 K, K + K - K + + i 4- 2.924 Oi, OH- 20H-->|04-Hi04-2 -0.401 Na, Na+ Na - Na* + 4-2.714 Zn, Zn++ Zn -Zn + + +2 4-0.761 Clifo), Cl- Cl- -* iCli 4- f -1.358 Fe, Fe++ Fe -+ Fe ++ + 2 4-0.441 Brj(0, Br- Br- - iBri 4- -1.066 Cd, Cd++ Cd-*Cd++ + 2 4- 0.402 iiw, i- !--**!+ -0.636 Co, Co++ Co ->Co** +2 4-0283 Ni, Ni+ + Ni-*Ni ++ +2. 4-0.236 A K| AgCl(), Cl- Ag J Cl- -* AgCl 4- - 0.2224 Sn, 8n++ Sn -Sn+ + 4-2c 4-0.140 Ag, AgBr(), Br- Ag 4- Br~ - AgBr 4- - 0.0711 Pb, Pb++ Pb -*Pb+ + 4- 2 4-0.126 Ag, Agl(.). I- Ag 4-1- -AgI4-f 4-0.1522 Hi, H+ JH - H+ 4- c db 0.000 Hg, HgiCh(s), Cl~ Hg4-Cl- -*iH gI Cli4- -0.2680 Cu, Cu + + Cu - Cu ++ -f 2e - 0.340 Hg, HgjS04(a) f S0 4 2Hg 4-SOj- -*HgjS044-2 -0.6141 Ag, Ag+ Ag - AK + + - 0.799 Hg, HgJ+ H-*iHgf+ + -0.799 potentials correspond to the tendency for positive electricity to pass from left to right, or negative electricity from right to left, in each case; in general, therefore, the potentials in Table XLIX when multiplied by nF give the standard free energy increase for the reaction Reduced State > Oxidized State + ne, the corresponding value for hydrogen being taken arbitrarily as zero. For the reverse process, the signs of the potentials would be reversed. Since the potentials in Table XLIX give the free energies of the oxi- dation reactions, using the term oxidation in its most general sense, they may be called oxidation potentials ; the potentials for the reverse proc- esses, i.e., with the signs reversed, are then reduction potentials (cf. p. 435). 244 ELECTRODE POTENTIALS Potentials in Non-Aqueous Solutions. Many measurements of vary- ing accuracy have been made of voltaic cells containing solutions in non- aqueous media; in the earlier work efforts were made to correlate the results with the potentials of similar electrodes containing aqueous solutions. Any attempt to combine two electrodes each of which con- tains a different solvent is doomed to failure because of the large and uncertain potentials which exist at the boundary between the two liquids. It has been realized in recent years that the only satisfactory method of dealing with the situation is to consider each solvent as an entirely inde- pendent medium, and not to try to relate the results directly to those obtained in aqueous solutions. Since the various equations derived in this and the previous chapter are independent of the nature of the solvent, they may be applied to voltaic cells containing solutions in substances other than water. By adopting the convention that the potential of the standard hy- drogen electrode, i.e., with ideal gas at 1 atm. pressure in a solution of unit activity of hydrogen ions shall be zero in each solvent, and using methods essentially similar to those described above, the standard poten- tials of a number of electrodes have been evaluated in methyl alcohol, ethyl alcohol and liquid ammonia. These values represent therefore, in each case, the E.M.F. of the cell M | M+(a M * = 1) II H+(a H + = 1) | H 2 (l atm.), where M is a metal, or of A | A~(a A - = 1) || H+(a H + = 1) | H,(l atm.) if A is a system yielding anions. It would appear at first sight that since the cell reaction, as for example in the former case, M(s) + H+(a H + = 1) = M+(a M + = 1) + |H 2 (1 atm.) is the same in all solvents, the E.M.F. should be independent of the nature of the solvent. It must be remembered, however, that both M+ ions and hydrogen ions are solvated in solution, and since the ions which actually exist in the respective solvents are quite different in each case, the free energy of the reaction will depend on the nature of the solvent. This subject will be considered shortly in further detail. A number of standard potentials reported for three non-aqueous solvents are compared in Table L with the corresponding values for water as solvent; ll it should be emphasized that although the standard poten- tial of hydrogen is set arbitrarily at zero for each solvent, the actual potentials of these electrodes may bo quite different in the various media. The results in each solvent are, however, comparable with one another and it will be observed that there is a distinct parallelism between the "Buckley and Hartley, Phil. Mag., 8, 320 (1929); Macfarlane and Hartley, ibid., 13, 425 (1932); 20, 611 (1935); Pleskow and Monossohn, Acta Physicochim. U.R.S.S., 1, 871 (1935); 2, 615, 621, 679 (1935). FACTORS AFFECTING ELECTRODE POTENTIALS 245 TABLE L. STANDARD ELECTRODE POTENTIALS IN DIFFERENT SOLVENTS Electrode H 2 O CH,OH C 2 HOH NH, (25) (25) (25) (-50) Li,Li+ +3.024 +3.095 +3.042 K,K+ +2.924 +1.98 Na,Na+ +2.714 +2.728 +2.657 +1.84 Zn,Zn++ +0.761 +0.52 Cd,Cd++ +0.402 +0.18 T1/T1+ +0.338 +0.379 +0.343 Pb,Pb^ +0.126 -0.33 H 2 ,H+ 0.000 0.000 0.000 0.000 Cu,Cu++ -0.340 -0.43 Ag,Ag+ -0.799 -0.764 -0.749 -0.83 C1 2 ,C1- -1.358 -1.116 -1.048 -1.28* Br 2 ,Br- - 1.066 - 0.837 - 0.777 - 1.08 * I,,I- -0.536 -0.357 -0.305 -0.70* * Calculated from free energy data at about c. standard potentials of the various electrodes in the four solvents. The tendency for the reaction M > M+ + to occur, as indicated by a high positive value of the potential, is always greatest with the alkali metals and least with the more noble metals, e.g., copper and silver. The order of the halogens is also the same in each case. Factors Affecting Electrode Potentials. If E is the standard poten- tial of a metal in a given solvent, then it is evident from the arguments given above that zFE is equal to the standard free energy of the reaction M + zH+ = M'* This reaction, which is the displacement of hydrogen ions from the solu- tion and their liberation as hydrogen gas, is virtually that occurring when a metal dissolves in a dilute acid solution, provided there are no accompanying complications, e.g., formation of complex ions. It follows, therefore, that zFE Q may be regarded as the standard free energy of solution of the metal. According to thermodynamics and experiments have shown that the standard entropy change AS re- sulting from the solution of a metal in dilute acid is relatively small compared with the heat change A#; it is possible, therefore, to write as a very approximate relationship - zFE* A//, where AH is heat of solution of the metal. In general, therefore, a parallelism is to be expected between the latter quantity and the stand- 246 ELECTRODE POTENTIALS ard potential of the metal; hence the factors determining the heat of solution may be regarded as those influencing the standard potential. 11 In order to obtain some information concerning these factors the reaction involved in the solution of the metal may be imagined to take place in a series of stages, as shown in Fig. 73; the reactants, M and FIQ. 73. Theoretical stages in solution of a metal in acid solvated hydrogen ions, are shown at the left, and the products, hydrogen gas and solvated M*+ ions, at the right. The stages are as follows: I. An atom of the metal is vaporized; the heat supplied is equal to the heat of sublimation, S; hence, Affi = + S. II. The atom of vapor is ionized to form metal ions M*+ and z elec- trons; the energy which must be supplied is determined by the ionization potential of the metal, the various stages of ionization being taken into consideration if the ion has more than one charge. If /M is the sum of the ionization potentials, the energy of ionization is /MC, and if it is supposed that this is converted into the standard units of energy used throughout these calculations, then III. The gaseous metal ion is dissolved in the solvent, when energy equal to the heat of solvation WM+ is evolved; hence, IV. An equivalent quantity of solvated hydrogen ions (z ions) are removed from the solvent; the energy of solvation WH+ per ion is ab- sorbed, so that AHiv = + zW*+. V. The unsolvated (gaseous) hydrogen ions are combined with the electrons removed from the metal to form atomic hydrogen; if JH is the ionization potential of the hydrogen atom, then since z electrons are added. "Butler, "Electrocapillarity," 1940, Chap. III. ABSOLUTE SINGLE ELECTRODE POTENTIALS 247 VI. The hydrogen atoms are combined in pairs to form hydrogen molecules; if DH S is the heat of dissociation of a hydrogen molecule into atoms, then The net result of these six stages is the same as the solution of a metal in a dilute acid; hence Aflf for this process is given by the sum of the six heat changes recorded for the separate stages, thus, assuming constant pressure, S + J M 6 - WM+ - 2(iDH, + /H+ - FH+). (28) The quantity in parentheses is characteristic of the hydrogen electrode in the given solvent, and so the factors which determine the heat of solution of a particular metal, and consequently (approximately) its standard potential, may be represented by the expression A# M S + 7 M - WM+. The standard potential of a metal in a given solvent thus apparently depends on the sublimation energy of the metal, its ionization potential and the energy of solvation of the ions. Calculations have shown that of these factors the heat of sublimation is much the smallest, but since the other two quantities generally do not differ very greatly, all three factors must play an important part in determining the actual electrode potential. When comparing the heat changes accompanying the solution of a given metal in different media, it is seen that the factors S, IM, DH, and JH are independent of the nature of the solvent. The standard potential of the metal in different solvents is thus determined by the quantity TFn+ WM+, where FH+ and TF M + are the energies, strictly the free energies, of solvation of the hydrogen and M+ ions, respectively; this result is in agreement with the general conclusion reached previously (p. 244). For a series of similar solvents, such as water and alcohols, the values of WH+ WM+ for a number of metals will follow much the same order in each solvent; in that case the standard potentials will show the type of parallelism observed in Table L. On the other hand it would not be surprising if for dissimilar solvents, e.g., water and acetonitrile, the order followed by the potentials of a number of electrodes was quite different in the two solvents. Absolute Single Electrode Potentials. The electrode potentials dis- cussed hitherto are actually the E.M.F.'S of cells resulting from the com- bination of the electrode with a standard hydrogen electrode. A single electrode potential, as already seen, involves individual ion activities and hence has no thermodynamic significance; " the absolute potential differ- ence at an electrode is nevertheless a quantity of theoretical interest. Many attempts have been made to set up so-called "null electrodes " See, for example, Guggenheim, J. Phys. Chem., 33, 842 (1929). 248 ELECTRODE POTENTIALS in which there is actually no differenfee of potential between the metal and the solution; if such an electrode were available it would be possible by combining it with another electrode to derive the absolute potential of the latter. It appears doubtful, however, whether the "null elec- trodes" so far prepared actually have the significance attributed to them, since they generally involve relative movement of the metal and the solution (cf. Chap. XVI). A possible approach to the problem is based on a treatment similar to that used in the previous section. The absolute single potential of a metal is a measure of the standard free energy of the reaction M + solvent = M*+ (solvated) + zt, and this process may be imagined to occur by the series of stages de- picted in Fig. 74. These, with the accompanying free energy changes, are vaporization of the metal (+ *S); ionization of the atom in the vapor M+(solvated) d+ (vapor) FIG. 74. Theoretical stages in formation of ions in solution state (/M*); solvation of the gaseous ion ( WM+)} and finally return of the electrons produced in the ionization stage to the metal ( 2<e), where <t> is the electronic work function of the metal.* It follows, there- fore, that AG = 8 + IM - WM+ - z<t*. (29) Since S, /M and <, as well as z and e, may be regarded as being known for a given metal, it should be feasible to evaluate A(? for the ionization process, provided the free energy of solvation of the M+ ions, i.e., WM + , were known. The sum of the energies of solvation of the ions of a salt can be esti- mated, at least approximately, from the heat of solution of the salt and its lattice energy in the crystalline form. There is, unfortunately, no direct method of dividing this sum into the contributions for the separate ions; it is of interest, however, to consider the theoretical approach to this problem as outlined in the following section. Free Energy of Solvation of Ions. If the solvent medium is con- sidered as a continuous dielectric, the free energy of solvation may be *The electronic work function, or thermionic work function, generally expressed in volts, is a measure of the amount of energy required to remove an electron from the metal; zfc is, therefore, the free energy change, in electron-volts, accompanying the return of the z electrons to the metal. FREE ENERGY OF SOLVATION OF IONS 249 regarded as equivalent to the difference in the electrostatic energy of a gaseous ion and that of an ion in the medium of dielectric constant D. In order to evaluate this quantity, use is made of the method proposed by Born: 14 the free energy increase accompanying the charging of a single gaseous ion, i.e., in a medium of dielectric constant unity, is zV/2r, where ze is the charge carried by the ion and r is its effective radius, the ion being treated as a conducting sphere. If the same ion is charged in a medium of dielectric constant D, the free energy change is zV/2Dr, and so the increase of free energy accompanying the transfer of the gaseous ion to the particular medium, which may be equated to the free energy of solvation, is given by the Born equation as where AT, the Avogadro number, is introduced to give the free energy change per mole. One of the difficulties in applying the Born equation is that the effective radius of the ion is not known; further, the calculations assume the dielectric constant of the solvent to be constant in the neighborhood of the ion. The treatment has boon modified by Webb 15 who allowed for the variation of dielectric constant and also for the work required to compress the solvent in the vicinity of the ion; further, by expressing the effective ionic radius as a function of the partial molal volume of the ion, it was possible to derive values of the free energy of solvation without making any other assumptions concerning the effective ionic radius. Another approach to the problem of ionic solvation has been made by Latimer and his collaborators; 16 by taking the effective radii of negative halogen ions as 0.1 A greater than the corresponding crystal radii and those of positive alkali metal ions as 0.85A greater than the crystal radii, it has been found possible to divide up the experimental free energies of hydration of alkali halides into the separate values for the individual ions. The results so obtained are in agreement with the requirements of the original form of the Born equation with the dielectric constant equal to the normal value for water. The free energies of hydration of single ions derived by the different methods of computation show general agreement. For univalent ions the values are approximately 70 to 100 kcal. per g.-ion; the hydrogen ion is exceptional in this respect, its free energy of hydration being about 250 kcal. In any series of ions, e.g., alkali metal ions or halide ions, the hydration free energy usually decreases with increasing mass of the ion. In spite of the fact that the different treatments yield similar values, it must be emphasized that there is considerable doubt if the results are 14 Born, Z. Physik, 1, 45 (1920). 15 Webb, /. Am. Chem. Hoc., 48, 2,"589 (1926). 16 Latimor, Pitzcr and Slansky, J. Chem. Phys., 7, 108 (1939). 250 ELECTRODE POTENTIALS of sufficient significance to permit of their use in the determination of absolute potentials. 17 The problem of single potentials must, therefore, still be regarded as incompletely solved. Rates of Electrode Processes. When a metal M is inserted in a solution of its ions M(H 2 O)2", the solvent being assumed for simplicity to be water, there will be a tendency for the metal to pass into solution as ions and also for the ions from the solution to discharge on to the metal; in other words the two processes represented by the reversible reaction *i M(H 2 0)+ + e^ M + zH 2 O will occur simultaneously, the ions M(H 2 O)j~ being in solution and the electrons on the metal. When equilibrium is attained, and the revers- ible potential of the electrode is established, the two reactions take place at equal rates. According to modern views, 18 the rate of a process is equal to the specific rate, defined in terms of the accepted standard states, multiplied by the activities of the reacting species;* if ki and & 2 are the specific rates of the direct and reverse processes represented above, in the absence of any potential difference, then, since a+ is the activity of the solvated ions in solution and the activity of the solid metal is unity, by convention, the rates of the reactions are k\a+ and & 2 , respectively. If k 2 is greater than k\a+, that is to say, if the reverse reaction in the absence of a poten- tial difference at the electrode, i.e., the passage of ions from the metal into the solution, is more rapid than the direct reaction, i.e., the discharge of ions, the cations will pass into solution from the metal more rapidly than they can return. As a result, therefore, free electrons will be left on the metal and positive ions will accumulate on the solution side of the electrode, thus building up what is known as an electrical double layer (see Chap. XVI); the potential difference across this double layer is the single electrode potential. The setting up of the double layer, with its associated potential difference, makes it more difficult for ions to leave the negatively charged metal and enter the solution, while the transfer of ions to the metal, i.e., the direct reaction, is facilitated. When equi- librium is established the two processes are occurring at the same rate and the electrode exhibits its reversible potential. If E is the actual potential difference across the double layer, formed by the electrons on the metal and the ions in solution, it may be supposed that a fraction a of this potential facilitates the discharge of ions, while the remainder, 1 a, hinders the reverse process, i.e., the passage of ions " Fnimkin, J. Chem. Phys., 7, 552 (1939). " Glasstone, Laidler and Eyring, "The Theory of Rate Processes," 1941, Chap. X. * Strictly speaking, the result should be divided by the activity coefficient of the "activated state 1 ' for the reaction; in any case this factor cancels out when equilibrium processes are considered. ELECTRODE POTENTIALS AND EQUILIBRIUM CONSTANTS 251 from the metal into the solution. The actual value of a, which lies be- tween zero and unity, is immaterial for present purposes, since it cancels out at a later stage. In its transfer across the double layer, therefore, the free energy of the discharging ion is increased by azFE, where z is the valence of the ion, while the free energy of the atom which passes into .solution is diminished by an amount (1 a)zFE. The result of these free energy changes Is that, in the presence of the double layer potential E, the rates of the forward and reverse reactions under con- sideration are : Rate of discharge of ions from solution = kia + e azFE/RT Rate of passage of ions into solution = k^e~ (l ~ a)zFEIRT J where, as already seen, the corresponding rates in the absence of the potential are k\a+ and fe, respectively. At the equilibrium (reversible) potential the rates of the two processes must be equal; hence '. # = ~m---,lna+. (30) zF k 2 zb * Since ki/kz is a constant at definite temperature, this equation is obvi- ously of the same form as the electrode potential equations derived by thermodynamic methods, e.g., equation (86) for an electrode reversible with respect to positive ions. The first term on the right-hand side of equation (30) is clearly the absolute single standard potential of the electrode; it is equal to the standard free energy of the conversion of solid metal to solvated ions in solution divided by zF, and its physical signifi- cance has been already discussed. Electrode Potentials and Equilibrium Constants. According to equa- tion (1) the standard E.M.F., i.e., B, of any reversible cell can be related to f]ie equilibrium constant of the reaction occurring in the cell by the expression E' = !~]nK, (31) and hence a knowledge of the standard E.M.F. permits the equilibrium constant to be calculated, or vice versa. The reaction occurring in the cell Zn | ZnS0 4 aq. || CuS0 4 aq. | Zn, for example, for the passage of two faradays is Zn + Cu++aq. = Zn ++ aq. + Cu, 252 ELECTRODE POTENTIALS and if E is the standard E.M.F., it follows from equation (31) that ,-0 RT /azn"0cu\ EZ&.C* = -^r In I I i *r \aznOcu* */ the suffix e indicating that the activities involved are the equilibrium values. Since the solid zinc and copper constituting the electrodes are in their standard states, their respective activities are unity; hence, (32) If Ezn,zn++ and Jcu,cu++ represent the standard electrode potentials on the hydrogen scale of the zinc and copper electrodes, as recorded in Table XLIX, then Ezn.z*++ is actually the E.M.F. of the cell Zn | Zn++(a Za " = 1) || H+(a H + = 1) | H 2 (l atm.), while JScu.cu** is the E.M.F. of the cell Cu | Cu+ + (ocu" = 1) || H+(a H + = 1) | H 2 (l atm.). Hence the E.M.F. of the cell Zn | Zn++(am*+ = 1) || Cu ++ (acu+* = 1) I Cu, which has been defined above as #zn,cu, is also equal to Ez*. zn + + It follows, therefore, from equation (32) that (33) and inserting the standard potentials from Table XLIX, the result is, at 25, + 0.- (-0.340, -1.7X10". The ratio of the activities of the zinc and copper ions at equilibrium will be approximately equal to the ratio of the concentrations under the same conditions; it follows, therefore, that when the system consisting of zinc, copper and their bivalent ions attains equilibrium the ratio of the zinc ion to the copper ion concentration is extremely large. If zinc is placed in contact with a solution of cupric ions, e.g., copper sulfatc, the zinc will displace the cupric ions from solution until the Cz n + *Ateu* f ratio is about 10 37 ; in other words the zinc will replace the copper in solution until the quantity of cupric ions remaining is too small to be detected. It is thus possible from a knowledge of the standard electrode poten- tials of two metals to determine the extent to which one metal will re- place another, or hydrogen, from a solution of its ions. In the general ELECTRODE POTENTIALS AND EQUILIBRIUM CONSTANTS 253 case of two metals MI and M 2 , of valence z\ and 2 2 , respectively, the reaction which occurs for the passage of z& 2 faradays is and the corresponding general form of equation (33) is *- (34) It can be seen from this equation that the greater the difference between the standard potentials of the two metals MI and M 2 , the larger will be the equilibrium ratio of activities (or concentrations) of the respective ions. The greater the difference between the standard potentials, there- fore, the more completely will one metal displace another from a solution of its ions. The metal with the more positive (oxidation) potential, as recorded in Table XLIX, will, in general, pass into solution and displace the metal with the less positive potential. The series of standard poten- tials, or electromotive series, as it is sometimes called, thus gives the order in which metals are able to displace each other from solution; the further apart the metals are in the series the more completely will the higher one displace the lower one. It is not true, however, to say that a metal lower in the series will not displace one higher in the series; some displacement must always occur until the required equilibrium is estab- lished, and the equilibrium amounts of both ions are present in the solution. By re-arranging equation (34) the result is zprn 7? / 7 7 tfV.Mt - ^ In (a M j). = BM..M; - p In (a M j).. (35) The left-hand side of this equation clearly represents the reversible poten- tial of the metal Mi in the equilibrium solution and the right-hand side is that of the metal M 2 . It must be concluded, therefore, that when the metal Mi is placed in contact with a solution of Mt ions, or M 2 is placed in a solution of Mi" ions, or in general whenever the conditions are such that the equilibrium Mi + Mt^ Mt + M 2 is established, the reversible potential of the system MI, Mt is equal to that of M 2 , Mt . It is clear from equation (35) that the more positive the standard potential of a given metal, the greater the activity of the corresponding ions which must be present at equilibrium, and hence t[u more completely will it displace the other metal. Although equations (34) and (35) are exact, the qualitative conclu- sions drawn from them are not always strictly correct; for example, sincv copper has a standard potential of 0.340 on the hydrogen scale, u would be expected, as is true in the majority of cases, that copper should be unable to displace hydrogen from solution. It must be recorded, 254 ELECTRODE POTENTIALS however, that copper dissolves in hydrobromic acid, and even in potas- sium cyanide solution, with the liberation of hydrogen. The reason for this surprising behavior is to be found in the fact that in both instances complex ions are formed whereby the cupric ions are removed from the solution. It is true that when equilibrium is attained the concentration (or activity) of cupric ions is very small in comparison with that of the hydrogen ions, but in order to attain even this small concentration it is necessary for a considerable amount of copper to pass into solution; most of this dissolved copper is present in the form of complex ions, and it is the amount of free cupric ions in equilibrium with these complexes that must be inserted in equation (34) or (35). It is of interest to note that if the equilibrium constant of the system consisting of two metals and their simple ions could be determined experi- mentally and the standard potential of one of them were known, the standard potential of the other metal could be evaluated by means of equation (34). This method was actually used to obtain the standard potential of tin recorded in Table XLIX. Finely divided tin and lead were shaken with a solution containing lead and tin perchlorates until equilibrium was attained; the ratio of the concentrations of lead and stannous ions in the solution was then determined by analysis. The standard potential of lead being known, that of tin could be calculated. Electrode Potentials and Solubility Product. The solubility product is an equilibrium constant, namely for the equilibrium between the solid salt on the one hand and the ions in solution on the other hand, arid methods are available for the evaluation of this property from E.M.F. measurements. The reaction taking place in the cell C1 2 (1 atm.) | HC1 AgCl() | Ag for the passage of one faraday is readily seen to be AgCl(s) = Ag + JC1,(1 atm.), but since the solid silver chloride is in equilibrium with silver and chloride ions in the solution, the reaction can be considered to be AgCl(s) - Ag+ + Cl- = Ag + 1C1,(1 atm.). The E.M.F. of the cell is then written as , (36) aAg+Ocr where a^+ and Ocr refer to the activities in the saturated solution. The value of E in this equation is the E.M.F. of the cell in which the activity of the chlorine gas and of chloride ions on the one hand, an J of solid silver and silver ions on the other hand, are unity; these conditions arise for the standard C1 2 , Cl~ and Ag, Ag 4 " electrodes, respectively, so that U ELECTRODE POTENTIALS AND SOLUBILITY PRODUCT 255 in equation (36) is defined by Since solid silver and chlorine gas at atmospheric pressure are the re- spective standard states, i.e., the activity is unity, equation (36) can be written as ffT 1 E - Z$i t .cr - #Ag,A g + + -y In a A .nicr. (37) The product a AK + acr in the saturated solution may be replaced by the solubility product of silver chloride, i.e., A^Agco, and so equation (37) becomes D/Tt E = Eci t .ci ^A,Ag* + ~rr In /(Agci). It is seen from Table XLIX that #ci 2 .cr and E/ig.^ are respectively - 1.358 and - 0.799 volt at 25; hence", E = - 1.358 + 0.799 + 0.05915 log /C. (A ci). From measurements on the cell depicted at the head of this section, it is found that E is 1.136 volt at 25, and consequently it follows that AW,) = 1.78 X lO- 10 . The value derived from the solubility of silver chloride obtained by the conductance method is 1.71 X 10~ 10 . In general, the above procedure can be applied to any sparingly soluble salt, provided an electrode can be obtainable which is reversible with respect to each ion, viz., A | Soluble salt of A~ ions MA() | M, although for a hydroxide, the oxygen electrode may be replaced by a hydrogen electrode. A less accurate method for the determination of solubility products, but which is of wider applicability, is the following. If MA is the sparingly soluble salt, and NaA is a soluble salt of the same anion, then the potential of the electrode M, MA(s), NaA aq. may be obtained by combining it with a reference electrode, e.g., a calomel electrode, thus M | MA() NaA aq. || KC1 aq. Hg 2 Cl 2 (s) | Hg, with a suitable salt bridge to minimize the liquid junction potential, and measuring the E.M.F. of the resulting cell. Since the potential of the calomel electrode is known, that of the other electrode may be evaluated, on the hydrogen scale. The potential of the M | MA(s) NaA electrode which can be treated as reversible with respect to M+ ions as well as to 256 ELECTRODE POTENTIALS A~ ions, may be written as E = J&M.M+ ~~p* In OM*, and if #M,M* is known, the activity of the M+ ions in the solution satu- rated with MA can be calculated. The activity of the A~ ions may be taken as approximately equal to the mean activity of the salt MA whose concentration is known; the product of a\i + and a A - in the solution then gives the solubility product of MA. Electrometric Titration: Precipitation Reactions. One of the most important practical applications of electrode potentials is to the deter- mination of the end-points of various types of titration; 19 the subject will be treated here from the standpoint of precipitation reactions, while neutralization and oxidation-reduction processes are described more con- veniently in later chapters. Suppose a solution of the soluble salt MX, e.g., silver nitrate, is titrated with a solution of another soluble salt BA, e.g., potassium chloride, with the result that the sparingly soluble salt MA, e.g., silver chloride, is pre- cipitated. Let c moles per liter be the initial concentration of the salt MX, and suppose that at any instant during the titration x moles of BA have been added per liter; further, let y moles per liter be the solubility of the sparingly soluble salt MA at that instant. The value of y will vary throughout the course of the titration since the concentration of M ions is being continuously altered. If the salts are assumed to be com- pletely dissociated, the concentration of M+ at any instant is given by A Vt<f- *> ? CM* = c x + y,* where c x is due to unchanged MX and y to the amount of the sparingly soluble MA remaining in solution. The simultaneous concentration of A~ ions is then because the A" ions in solution arise solely from the solubility of MA, the remainder having been removed in the precipitate. Since the solu- tion is saturated with MA, it follows from the approximate solubility product principle, assuming activity coefficients to be unity, that k. = CM* X C A - = (c - x + y)y, (38) where k is the concentration solubility product. "For reviews, see Kolthoff and Furman, " Potentiometric Titrations," 1931; Furman, Ind. Eng. Chem. (Anal. Ed.), 2, 213 (1930); Trans. Ekctrochem. Soc., 76, 45 (1939); Gladstone, Ann. Rep. Chem. Soc., 30, 283 (1933); Glasstone, "Button's Volu- metric Analysis/ 1 1935, Part V. * The change of volume during titration is neglected since its effect is relatively small. ELECTROMETRIC TITRATION 257 If an electrode of the metal M, reversible with respect to M+ ions, were placed in the solution of MX during the titration, its potential would be given by E = #M,M+ -- Erhia M + Zr RT SW-^rln(c-* + 0), (39) where the activity of M+ ions, i.e., a M +, has been replaced by the concen- tration as derived above. If the solubility product A;, is available, then since c and x are known for any point in the titration, it is possible to calculate y by means of equation (38) ; the values of c x + y can now be inserted in equation (39) and the variation of electrode potential during the course of titration can be determined. At the equivalence- point, i.e., the ideal end-point of the titration, when the amount of BA added is equivalent to that of MX initially present, c and x are equal; equation (39) then reduces to E #8t,M+ Jk.. (40) r Should the titration be carried beyond the end-point, the value of CA~ now becomes x c + y, while that of CM+ is y, since the solution now contains excess of A*~ ions; x c arises from the excess of BA over MX, and y from the solubility of the sparingly soluble MA. The solubility product is given by k. = y(x - c + y), and equation (39) becomes E E2f, M +-^lnt,. (41) The value of y can be calculated as before, if the solubility product is known, and hence the electrode potential of M can be determined. By means of equations (38), (39) and (41) it is thus possible to calcu- late the potential of an electrode of the metal M during the course of the whole precipitation titration, from the beginning to beyond the equiva- lence-point, provided the solubility product of the precipitated salt is known. The calculations show that there is at first a gradual change of potential, but a very rapid increase occurs as the equivalence-point is approached; the change of potential for a given increase in the amount of the titrant added, i.e., dE/dx, is found to be a maximum at the theo- retical equivalence-point. This result immediately suggests a method 258 ELECTRODE POTENTIALS for determining experimentally the end-point of a precipitation titration by E.M.F. measurement. The reversible potential (E) of an M electrode during the course of the titration is plotted against the amount of titrant added (x); the point at which the potential rises most sharply, i.e., the point of inflection where dE/dx is a maximum, is the required end-point. This procedure constitutes the fundamei.cal basis of potentiometric titration. The same general conclusion may be reached without going through the Jetciiled calculations just described. If equation (39) is differentiated twice with respect to x and the resulting expression for d~E/dx* equated to zero, the condition for dE/dx to be a maximum can be obtained. This is found to be that x should be equal to c, which is, of course, the con- dition for the equivalence-point, in agreement with the conclusion already reached. By differentiating equation (39) with respect to x it is seen that at the equivalence-point the value of dE/dx is inversely proportional to Vfc,. The potential jump observed at the end-point is thus greater the smallei the solubility product /;. of the precipitate. The sharpness of a particular titration can thus often be improved by the addition of alcohol to the solution being titrated in order to reduce the solubility of the precipi- tated salt. In the treatment given here it has been assumed that the precipitate MA is a salt of symmetrical valence type; if it is an unsymmetrical salt, e.g., MaA or MA2, the potential-titratioii curve, i.e., the plot of the potential (E) against the amount (x) of titrant added, is not symmetrical and tho maximum value of dE/dx does not occur exactly at the equiva- lence-point. The deviations are, however, relatively small if the solu- bility product of the precipitate is small and the titrated solutions are not too dilute. Potentiometric Titration: Experimental Methods. Since the silver electrode generally behaves in a satisfactory manner, the potentiometric method of titration can be applied particularly to the estimation of anions which yieid insoluble silver salts, ?e.g., halides, cyanides, thio- cyanates, phosphates, etc. In its simplest form, the experimental pro- cedure is to take a known volume of the solution containing the aniou to be titrated and to insert a clean silver sheet or wire, preferably coated with silver by the electrolysis of an argentooyanide solution; this con- stitutes the "indicator" electrode, and its potential is measured by connecting it, through a salt bridge, with a reference electrode, e.g., a calomel electrode. Since the actual electrode potential is not required, but merely the point at which it undergoes a rapid change, the E.M.F. of the resulting cell is recorded after the addition of known amounts of the silver nitrate solution. The values obtained in the course of the titration of 10 cc. of approximately 0.1 N sodium chloride with 0.1 N silver nitrate, using a silver indicator electrode and a calomel reference POTENTIOMETRIC TITRATION 259 TABLE LI. POTENTIOMETRIC TITRATION OF SODIUM CHLORIDE WITH SILVER NITRATE ANOi (t) E *E AP A/Av 0.1 cc. 5.0 8.0 10.0 11.0 114 mv. 130 145 168 202 16 15 23 34 4.9 3.0 2.0 1.0 3.3 5.0 11.5 34 11.10 11.20 11.30 11.35 11.40 11.45 11.50 12.0 13.0 14.0 210 224 250 277 303 318 328 361 389 401 8 14 26 27 26 15 10 36 23 12 0.1 0.1 0.1 0.05 0.05 0.05 0.05 0.5 1.0 1.0 80 140 260 540* 520 300 200 72 25 12 electrode, are recorded in Table LI and plotted in Fig. 75; the first column of the table gives the volume v of standard silver nitrate added, 0.88 0.84 0.80 I [o.26 0.22 0.18 0.11 I \ y TOO 600 600 400 300 200 Fio. 75. 4 6 8 10 12 14 CC.AgNO, Potentioinetric titration 1U 11.4 11.5 cc. AgNOs FIG 70. Determination of end-point in potentio- motric tit rat ion which is equivalent to x in the treatment given above, and hence A/?/Ar, in the last column, is an approximation to dE/dx. It is clear from the data that A/?/ At; is a maximum when v is about 11.35 cc., and this must represent the end-point of the titratiou. 260 ELECTRODE POTENTIALS It is not always possible to estimate the end-point directly by inspec- tion of the data and the following method, which is always to be preferred, should be used. The values of AJE/At; in the vicinity of the end-point are plotted against v + iAt>, i.e., the volume of titrant corresponding to the middle of each titration interval, as in Fig. 76; the volume of titrant corresponding to the maximum value of AU/Av can now be determined very precisely. This graphical method is particularly useful when the inflection in the potential-titration curve at the end-point is relatively small. Differential Titration. The object of potentiometric titration is to determine the point at which AJ/At> is a maximum, and this can be achieved directly, without the use of graphical methods, by utilizing the principle of differential titration. If to two identical solutions, e.g., of sodium chloride, are added v and v + 0.1 cc. respectively of titrant, e.g., silver nitrate, the difference of potential between similar electrodes placed in the two solutions gives a direct measure of AS/Av, where At; is 0.1 cc., at the point in the titration corresponding to the addition of v + 0.05 cc. of silver nitrate. The E.M.F. of the cell made up of these two electrodes will thus be a maximum at the end-point. In the earliest applications of the method of differential titration the solution to be titrated was divided into two equal parts; similar elec- trodes were placed in each and electrical connection between the two solutions was made with wet filter-paper. The electrodes were con- nected through a suitable high resistance to a galvanometer. Titrant was then added to the two solutions from two separate burettes, one being always kept a small amount, e.g., 0.1 cc., in advance of the other. The point of maximum potential difference, and hence that at which AE/Av was a maximum, was indicated by the largest deflection of the galvanometer; the total titrant added at this point was then equivalent to the total solution titrated. By this means the end-point of the titra- tion was obtained without the use of a reference electrode or a poten- tiometer, and the necessity for graphical estimation of the titration corresponding to the maximum Al?/Av was avoided. 20 The method of differential titration has been modified so that the process can be carried out in one vessel with one burette; by means of special devices, a small quantity of the titrated solution surrounding one of the two identical electrodes is kept temporarily from mixing with the bulk of the solution before each addition of titrant. The difference of potential between the two electrodes after the addition of an amount Aw of titrant gives a measure of A#/Ay. The form of apparatus devised by Maclnncs and Dole, 21 which is capable of giving results of great accuracy, is depicted in Fig. 77. One of the two identical indicator electrodes, 10 Cox, J. Am. Chem. Soc., 47, 2138 (1925). 21 Maclnnes et al, J. Am. Chem. Sac., 48, 2831 (1926); 51, 1119 (1929); S3, 555 (1931); Z. physik. Chem., 130, 217 (1927). COMPLEX IONS 261 I -EL mm J 1 ^ ^ i A HJ Fia. 77. Apparatus for differential titration (Mac- Innes and Dole) viz., E\, is placed directly in the titration vessel, and the other, J? 2 , is inserted in the tube A, which should be as small as convenient; at the bottom of this tube there is a small hole B, and a "gas-lift" C is sealed into its side. The hole D in the tube A permits the overflow of liquid when the gas-lift is in operation. In order to carry out a titration, a known volume of solution is placed in a beaker and the two electrodes are inserted; the liquid is allowed to enter A, but the gas-stream is turned off. Titrant is added from the burette, with con- stant stirring, until there is a large increase in the E.M.F. of the cell formed by the two electrodes ; this may be indicated by a potentiometer, for precision work, or by means of a galvanometer with a resist- ance in series. The solution in the beaker is actu- ally somewhat over-titrated, but when the gas- stream is started the reserve solution in the tube A , which normally mixes only slowly with the bulk of the liquid, because of the smallness of the hole JB, is forced out; in this way the titration is brought back, although the end-point is near. The differ- ence of potential between the two electrodes is now zero, since the same solution surrounds both of them. The gas- stream is stopped, and a drop (At;) of titrant is added to the bulk of the solution in the beaker; the galvanometer deflection, or potential differ- ence, is then a measure of A^E/At;, since one electrode, J5? 2 , is immersed in a solution to which v cc. of titrant have been added, while the other, E\, is surrounded by one to which v + At; cc. have been added. The gas- stream is started once more so as to obtain complete mixing of the solutions; it is then stopped, another drop of titrant added, and the potential reading again noted. This procedure is continued until the end-point is passed, the end-point itself being characterized by the maxi- mum potential difference between the two electrodes. Many simplified potentiometric titration methods have been de- scribed from time to time, and various forms of apparatus have been devised to facilitate the performance of these tit rations; for reference to these matters the more specialized literature should be consulted. 22 Complex Ions. The formula of a relatively stable complex ion can be determined by means of E.M.F, measurements; in the general case already considered on page 173, viz., MA r ; ^ 0M+ + rA~, * See the books and review articles to which reference is made on page 256; the subject of potentiometric titrations, among others, is also treated in Koltho? and Laitinen, "pH and Electro-Titrations," 1941. 262 ELECTRODE POTENTIALS it was seen that the instability constant K l can be represented by If an electrode of the metal M is inserted in the solution of the complex ion, the reversible potential should be given by r>/r? E = /?M,M+ - ~ In a M * For two solutions containing different total amounts of the complex ion, but the same relatively large excess of tin; anion A~, it follows from equation (42) that Bl -K. , (43) qzF (a Mf A,*)i V ' where the suffixes 1 and 2 refer to the two solutions; the value of a\- is assumed to be the same for the two cases. If the complex ion M q \f is relatively stable, then in the presence of excess of A~ ions, virtually the whole of the M present in solution will be in the form of complex ions. As an approximation, therefore, tho ratio of the activities of the M^A* ions in the two solutions in equation (43) may bo replaced by the ratio of the total concentrations of M ; hence ,, } (c M ) 2 , AA . E l E 2 = 7; In 7 ;- (44) qzF (CM) i v If (CM)I and (c\i)2, the total concentrations of the species M in the respec- tive solutions, are known, and the potentials EI and 7? 2 arc measured, it is possible to evaluate q by means of equation (41). If the solutions are made up with same concentration of M, i.e., approximately the same concentration, or activity, of the complex ions M q A.f y but with different amounts of the anion A~, it follows from equa- tion (42) that in this case The ratio of the activities of the A~ ions may be replaced, as an approxi- mation, by the ratio of the concentrations; hence*, from equation (45), ft-ft.,. (46) -- v ' ELECTRODE POTENTIAL AND VALENCE 263 Since q has been already determined, the value of r can be derived from equation (46) so that the formula of the complex ion has been found. Another method for deriving the ratio r/q involves the same principle as is used in potentiometric titration; for simplicity of explanation a definite case, namely the formation of the argentocyanide ion, Ag(CN)J, will be considered. If a solution of potassium cyanide is titrated with silver nitrate, che potential of a silver electrode in the titrated solution will be found to undergo a sudden change of potential when the whole of the cyanide has been converted into argentocyanide ions. From the relative amounts of silver and cyanide ions at the point where dE/dx is a maximum the formula of the complex ion can be calculated. An analogous titration method can be used to determine the formula of any stable complex ion; the procedure actually gives the ratio of M+ to A" in the complex ion M q Af, but if this ratio is known there is generally no difficulty, from valence and other chemical considerations, in deriving the molecular formula. By expressing the concentration, or activity, of the M+ ions in the titrated solution, and hence the potential of an M electrode, in terms of c, the initial concentration of the solution, x, the amount of titrant added, and fc t , the instability constant of the complex ion, it is possible, utilizing the method of differentiation described in connection with precipitation titrations (page 258), to show that dE/dx is a maximum at the point corresponding to complete formation of the complex ion. Further, the value of dE/dx at this point, and hence the sharpness of the inflection in the titration curve, can be shown to be greater the smaller the insta- bility constant. The potential of a silver electrode during the course of the titration of silver nitrate with potassium cyan- ide is shown in Fig. 78; the first marked change of potential occurs when one equivalent of cyanide has been added to one of silver, so that the whole of the silver cyanide is pre- cipitated, and the second, when two equivalents of cyanide have been added, corresponds to the complete formation of the Ag(CN)J ion. It will be seen that the changes of potential occur very sharply in each oaso; this means that the silver cyanide is very slightly soluble and thai. t4vo complex ion is very stable. Electrode Potential and Valence. The equation (86) for the poten- tial of an electrode reversible with respect to positive ions may be written f 0.2 *0.0 -0.2 -0.6 1 2 3 Equivalents of Cyanide Fia. 78. Formula of complex argentocyanide ion 264 ELECTRODE POTENTIALS in the approximate form E E -0.0002- log c,, where the activity of the ionic species is replaced by the concentration. At ordinary laboratory temperatures, about 20c., i.e., T is 293 K., this equation becomes E ~ E* - logc t . (47) z It follows, therefore, that a ten-fold change of concentration of the ions will produce a change of 0.058/2, volt in the electrode potential, where Zi is the valence of the ions with respect to which the electrode is revers- ible. It is possible, therefore, to utilize equation (47) to determine the valence of an ion. 23 For example, the result of a ten-fold change in the concentration of a mercurous nitrate solution was found to cause a change of 0.029 volt in the potential of a mercury electrode at 17; it is evident, therefore, that z must be 2, so that the mercurous ions are bivalent. These ions are therefore written as Hgf + and mercurous chloride and nitrate are represented by Hg 2 Cl 2 and Hg2(NOa)2. PROBLEMS 1. Work out the expressions for the E.M.F.'S and single potentials of the cells and electrodes given in Problem 1 of Chap. VI in terms of the variable activities. 2. From the standard potential data in Table XLIX determine (i) the standard free energies at 25 of the reactions Ag+aq. + Cl-aq. = AgCl(s) and Ag + JC1 2 (1 atm.) = AgCl(s), and (ii) the solubility product of silver chloride. 3. It is known from thermal measurements that the entropy of aluminum at 25 is 6.7 cal./deg. per g.-atom, and that of hydrogen gas at 1 atm. pressure is 31.2 per mole. The heat of solution of aluminum in dilute acid shows that AH for the reaction Al + 3H+aq. = Al+++aq. + f H t (l atm.) is 127,000 cal. From measurements on the entropy of solid cesium alum and its solubility, etc., Latimer and his collaborators [J. Am. Chem. Soc., 60, 1829 (1938)] have estimated the entropy of the Al+++aq. ion to be - 76 cal./deg. per g.-ion. Calculate the standard potential of aluminum on the usual hydrogen scale. " Ogg, Z. physik. Chem., 27, 285 (1898); see also, Reichinstein, ibid., 97, 257 (1921); Kasarnowsky, Z. anorg. Chem., 128, 117 (1923). PROBLEMS 265 4. Jones and Baeckstrom [J. Am. Chem. Soc., 56, 1524 (1934)] found the E.M.P. of the cell Pt | Br 2 (J) KBr aq. AgBr(a) | Ag to be 0.9940 volt at 25. The vapor pressure of the saturated solution of bromine in the potassium bromide solution is 159.45 mm. of mercury; calculate the standard potential of the Br 2 (gr, 1 atm.), Br~ electrode. 5. The standard free energy of the process JH 2 (1 atm.) + id 2 (l atm.) = HC1(1 atm.), is given in International Critical Tables, VII, 233, by the expression AG = - 21,870 + 0.45Tln T - 0.25 X 1Q~ 6 T 2 - 5.31T. The partial pressure of hydrogen chloride over 1.11 N hydrochloric acid solu- tion is 4.03 X 10- 4 mm. at 25. Calculate the E.M.F. of the cell H 2 (l atm.) | 1.11 N HC1 aq. | C1 2 (1 atm.) at this temperature. Use the result to determine the standard potential of the chlorine electrode, the mean activity coefficient of the hydrochloric acid being estimated from the data in Table XXXIV. 6. Calculate from the standard potentials of cadmium and thallium the ratio of the activities of Cd++ and Tl~ ions when metallic cadmium is shaken with thallous perchlorate solution until equilibrium is attained. 7. Knuppfer [Z. physik. Chem., 26, 255 (1898)] found the E.M.F. of the cell Tl (Hg) | TlCl(s) KCl(d) j KCNS(c 2 ) TlCNS(s) | Tl (Hg) to be - 0.0175 volt at 0.8 and - 0.0105 volt at 20 with Ci/c 2 equal to 0.84. Assuming the solutions to behave ideally, calculate the equilibrium ratios of Ci/c 2 at the two temperatures and estimate the temperature at which the arbi- trary ratio, i.e., 0.84, will become the equilibrium value. 8. The E.M.F. of the cell Pb | Pb(OH) 2 (s) N NaOH HgO(s) | Hg is 0.554 volt at 20; the potential of the Hg, HgO(s) N NaOH electrode is 0.114 volt. Calculate the approximate solubility product of lead hydroxide. 9. In the potentiometric titration of 25 cc. of a potassium cyanide solution with 0.1 N silver nitrate, using a silver indicator electrode and a calomel refer- ence electrode, the following results were obtained: cc. AgNO, (v) E.M.F. (E) 2.20 0.550 11.70 0.481 15.50 0.445 18.00 0.422 19.60 0.392 20.90 0.363 cc. AgNO, (v) E.M.F. (E) 21.50 0.343 21.75 0.309 21.95 0.259 22.15 0.187 22.35 - 0.255 22.55 - 0.319 Plot E against v, and A# against At; in the vicinity of the end-point; from the results determine the concentration of the potassium cyanide solution. 10. When studying the behavior of a tin anode in potassium oxalate solution, Jeffery [Trans. Faraday Soc. t 20, 390 (1924)] noted that a complex anion, having the general formula Sn^CaOOr* was formed. In order to deter- 266 ELECTRODE POTENTIALS mine its constitution, measurements of the cell Sn | Sn,(Ct0 4 ) r - K 2 C 2 4 aq. j KC1 (satd.) Hg 2 Cl 2 (a) | Hg were made: in one series of experiments (A) the concentration of potassium oxalate was large and approximately constant while the total amount of tin in solution (cs n ) was varied; in the second series (B) t cs n was kept constant at 0.01 g.-atom per liter, while the concentration of potassium oxalate (c ox .) was varied. The results were as follows: A B CSn E Cox. E 1.00 X 10-* 0.7798 2.0 0.7866 0.833 0.7823 2.5 0.7937 0.714 07842 3.0 0.7990 0.625 0.7859 3.5 0.8002 0.556 0.7877 4.0 0.8052 Devise a graphical method, based on equations (44) and (46), to evaluate q and r; activity corrections may be neglected, and the whole of the tin present in solution may be assumed to be in the form of the complex anion. CHAPTER VIII OXIDATION-REDUCTION SYSTEMS Oxidation-Reduction Potentials. It was seen on page 186 that a reversible electrode can be obtained by inserting an inert electrode in a solution containing the oxidized and reduced forms of a given system; such electrodes are called oxidation-reduction electrodes. It has been pointed out, and it should be emphasized strongly, that there is no essential difference between electrodes of this type and those already considered involving a metal and its cations, or a non-metal and its corresponding anions. This lack of distinction is brought out by the fact that the iodine-iodide ion system is frequently considered from the oxidation-reduction standpoint. Nevertheless, certain oxidation-reduc- tion systems, using the expression in its specialized meaning, have inter- esting features and they possess properties in common which make it desirable to consider them separately. According to the general arguments at the beginning of Chap. VI, which are applicable to reactions of all types, including those involving oxidation and reduction, the potential of an electrode containing the system Reduced Stated Oxidized State + n Electrons is given by the general equation _ ro __ (Oxidized State) h "nF R (Reduced State) ' (1) where n is the number of electrons difference between the two states, and the parentheses represent activities. Oxidation-reduction potentials, like the other types discussed in the preceding chapter, are generally expressed on the hydrogen scale, so that for the system e, for example, the electrode potential as usually recorded is really the JO.M.F. of the cell Pt | Fe+ f , Fe + ++ || H+(a H + = 1) | H 2 (l atm.). Using the familiar convention that a positive E.M.F. represents the tend- ency of positive current to flow from left to right through the cell, the reaction at the left-hand electrode may evidently be written as Fe++ = Fe++ + + , 267 268 OXIDATION-REDUCTION SYSTEMS for the passage of one faraday. This result may be obtained directly by analogy with the process occurring at the electrode M, M+, namely M = M+ + c. At the right-hand electrode, the reaction is H+ + 6 = *H 2 , so that the net cell reaction, for one faraday of electricity, is Fe++ + H+ = Fe+++ + H 2 . The E.M.F. of the complete cell is then given in the usual manner by and since, by convention, the activities of the hydrogen gas and the hydrogen ions are taken as unity, it follows that E = JBk** Pe "+ - ~ In ^ (2) The oxidation-reduction potential is thus seen to be determined by the ratio of the activities of the oxidized and reduced states, in agreement with the general equation (1). The standard potential E is evidently that for a system in which both states are at unit activity. In the most general case of an oxidation-reduction system repre- sented by aA + 6B + - ^ xX + i/Y + + n , for which there is a difference of n electrons between the reduced state, involving A, B, etc., and the oxidized state, involving X, Y, etc., the potential is given by (cf . page 228) When all the species concerned, viz., A, B, , X, Y, etc., are in their standard states, i.e., at unit activity, the potential is equal to E , the standard oxidation-reduction potential of the system. It is important to remember that in order that a stable reversible potential may be obtained, all the substances involved in the system must be present; the actual potential will, according to equation (3), depend on their respective activities. Types of Reversible Oxidation-Reduction Systems. Various types of reversible oxidation-reduction systems have been studied: the simplest consist of ions of the same metal in two stages of valence, e.g., ferrous and ferric ions. If M* 1 * and M n + are two cations of the metal M, carry- ing charges z\ and 22, respectively, where z 2 is greater than z\ 9 the elec- trode reaction is TYPES OF REVERSIBLE OXIDATION-REDUCTION SYSTEMS 269 and the potential is given by where a* and a\ are the activities of the oxidized and reduced forms, respectively. Another type of system consists of two anions carrying different charges, e.g., ferro- and ferri-cyanide, i.e., Fe(CN)e --- ^ Fe(CN)e + e, and the electrode potential for this system is In certain cases both anions and cations of the same metal are con- cerned; for such systems the equilibria, and hence the equations for the electrode potential, involve hydrogen ions. An instance of this kind is the permanganate-manganous ion system, viz., Mn++ + 4H 2 ^ MnOr + 8H+ + 5c, for which the electrode potential is the activity of the water being unity provided the solutions are rela- tively dilute. In some important oxidation-reduction systems one or more solids are concerned; for example, in the case of the equilibrium Mn++ + 2H 2 ^ Mn0 2 (s) + 4H+ + 2, the potential is f = _ m . since the activity of the solid manganese dioxide is taken as unity, in accordance with the usual convention as to standard states. In the equilibrium PbS0 4 (s) + 2H 2 O ? Pb0 2 (s) + 4H+ + SO?- + 2 , which is of importance in connection with the lead storage battery, two solids are involved, namely lead sulfate and lead dioxide, and hence _. _ A RT , 4 E = Z? - In aH*a 8 o 4 - -. 270 OXIDATION-REDUCTION SYSTEMS The potential thus depends on the fourth power of the activity of the hydrogen ions and also on that of the sulfate ions in the solution. A large number of reversible oxidation-reduction systems involving organic compounds are known; most of these, although not all, are of the quinone-hydroquinone type. The simplest example is OH OH and such systems may be represented by the general equation H 2 Q ^ Q + 2H+ + 2e, where H 2 Q is the reduced, i.e., hydroquinone, form and Q is the oxidized, i.e., quinone, form. The potential of such a system is given by (4) For many purposes it is convenient to maintain the hydrogen ion activity constant and to include the corresponding term in the standard potential; equation (4) then becomes where E*' is a subsidiary standard potential applicable to the system at the specified hydrogen ion activity. Determination of Standard Oxidation-Reduction Potentials. In prin- ciple, the determination of the standard potential of an oxidation-reduc- tion system involves setting up electrodes containing the oxidized and reduced states at known activities and measuring the potential E by combination with a suitable reference electrode; insertion of the value of E in the appropriate form of equation (3) then permits E T to be calcu- lated. The inert metal employed in the oxidation-reduction electrode is frequently of smooth platinum, plthough platinized platinum, nercury and particularly gold are often used. In the actual evaluation of the standard potential from the experi- mental data a numbe** of difficulties arise, and, as a result of the failure to overcome or to make adequate allowance for them, most of the meas- urements of oxidation-reduction potentials carried out prior to about 1925 must be regarded as lacking m accuracy. In the first case, it w rarely possible to avoid a liquid junction potential in setting up the cell for measuring the oxidation-reduction potential; secondly, there is often DETERMINATION OF STANDARD OXIDATION-REDUCTION POTENTIALS 271 uncertainty concerning the actual concentrations of the various species, because of complex ion formation and because of incomplete dissociation and hydrolysis of the salts present; finally, activity coefficients, which were neglected in th3 earlier work, have an important influence, as will be apparent from the following considerations. In the simple case of a system consisting of two ions carrying different charges, e.g., Fe++, Fe+++ or Fe(CN)e --- , FeCCN) , designated by the suffixes 1 and 2, respectively, the equation for the potential is where the activity has been replaced by the product of the concentration and the activity coefficient. Utilizing the Debye-Huckel limiting equa- tion (p. 144), viz., it follows that log - A(z\ - and insertion in equation (6) gives If water i.^> the solvent, then at 25 the constant A is 0.509; hence, this equation becomes ^ 0.05915, c 2 0.0301 , N r E = E Q ---- log - -- (z\ - 2|)\v- (7) * ^ ' For most oxidation-reduction systems z\ z\ is relatively high, e.g., 7 for the Fe(CN)e" , Fe(CN)o" system, and so the last term in equation (7), which represents the activity coefficient factor, may be quite con- siderable; further, the terms in the ionic strength involve the square of the valence and hence y will be large even for relatively dilute solutions. 1 In any case, the presence of neutral salts, which were frequently added to the solution in the earlier studies of oxidation-reduction potentials, increases the ionic strength; they will consequently have an appreciable influence on the potential, although the ratio of the amounts of oxidized to reduced forms remains constant. A striking illustration of the effect of neglecting the activity coeffi- cient is provided by the results obtained by Peters (1898) in one of the 1 Kolthoff and Tomsicck, J. Phys. Chern., 39, 945 (1935); Glasstone, "The Electro- chemistry of Solutions," 1937, p. 346. 272 OXIDATION-REDUCTION SYSTEMS earliest quantitative studies of reversible oxidation-reduction electrodes. From measurements made in solutions containing various proportions of ferrous and ferric chloride chloride in 0.1 N hydrochloric acid, an approximately constant value of 0.713 volt at 17 was calculated for the standard potential of the ferric-ferrous system, using the ratio of concentratibns instead of activities. This result was accepted as correct for some years, but it differs from the most recent values by about 0.07 volt; the discrepancy is close to that estimated from equation (5) on the basis of an ionic strength of 0.25, which is approximately that existing in the experimental solutions. Actually, of course, the Debye-Huckel limiting equation would not hold with any degree of exactness at such a high ionic strength, but it is of interest to observe that it gives an activity correction of the right order. In recent years care has been taken to eliminate, or reduce, as far as possible the sources of error in the evaluation of standard oxidation- reduction potentials; highly dissociated salts, such as perchlorates, are employed wherever possible, and corrections are applied for hydrolysis if it occurs. The cells are made up so as to have liquid junction poten- tials whose values are small and which can be determined if necessary, and the results are extrapolated to infinite dilution to avoid activity corrections. One type of procedure adopted is illustrated by the case described below.* In order to determine the oxidation-reduction potential of the system involving penta- (VOt) and tetra-valent (VO++) vanadium, viz., VO++ + H 2 = VOt + 2H+ + , measurements were made with cells of the form Pt | V0 2 C1, VOC1 2 , HC1 j HC1 H&C1.W | Hg containing the three constituents, VO 2 C1, VOC1 2 and hydrochloric acid at various concentrations. 2 By employing acid of the same concentra- tion in both parts of the cell, the liquid junction potential was reduced to a negligible amount. The reaction taking place in the cell for the passage of one f araday is VO++ + H 2 + JH&C1.W = VOt + 2H+ + Cl- + Hg(0, so that the E.M.F. is given by RT , E = EO - -TT In j - (8) r ++ where the standard potential for the cell (J?) is equal to the difference between the standard potentials of the V 6 , V 4 system and that of the * See also, Problem 4, page 304. Carpenter, /. Am. Chem. Soc., 56, 1847 (1934); Hart and Partington. /. Chem. , 1532 (1940). DETERMINATION OF STANDARD OXIDATION-REDUCTION POTENTIALS 27C Hg, Hg 2 Cl 2 (s), Cl- electrode, the latter being - 0.2680 volt at 25. Re- placing the activities of the VO++ and VO2~ ions by the products of their respective concentrations and activity coefficients, represented by / 2 and /i, respectively, equation (8) becomes, after rearrangement, Since the hydrochloric acid may be regarded as being completely ionized, C H + and Ccr may each be taken as equal to CHCI, the concentration of this acid in the cell; further, the product of /H + and/cr is equal to /HCI, where /HCI is the mean activity coefficient of the hydrochloric acid. It follows, therefore, that the quantity an+ocr, which is equal to (cH+Ccr)/iV/cr> may be replaced by CHCI/HCI/H+; upon inserting this result in equation (9) and rearranging, it is found that _, _ _ fl E + -p- In CHCI + -jjrln = E - _l n ^-. (1 ) The activity coefficient term in this equation becomes zero at infinite dilution; it follows, therefore, that extrapolation of the left-hand side to zero concentration, using the results obtained with cells containing various concentrations of the three constituents, should give E for the cell. The value obtained in this manner, by plotting the left-hand side of equation (10) against a suitable function of the ionic strength, was 0.7303 volt; it follows, therefore, that the standard potential of the VO++, VOt + 2H+ system is - 0.730 + (- 0.268), i.e., - 0.998 volt. An alternative extrapolation procedure is based on the approximation of taking fn + to be equal to / nc i; equation (9) can then be written as _ , ZRT. , 3flr RT, cvo" - RT, f, E + -y- In CHCI + -p~ In /HCI + y In -^ = # - y In ~ (11) The values of the activity coefficients of hydrochloric acid at the ionic strengths existing in the cell are obtained from tabulated data, and hence the left-hand side of this equation, for various concentrations, may be extrapolated to zero ionic strength, thus giving E. A further possi- bility is to replace log /HCI by the Debye-Hiickel expression A Vy, and to extrapolate, as before, by plotting against a suitable function of the ionic strength. As a general rule, several methods of extrapolation are possible; the procedure preferred is the one giving an approximate straight line plot, for this will probably give the most reliable result when ex- trapolating to infinite dilution. Another method of evaluating standard oxidation-reduction potentials is to make use of chemical determinations of equilibrium constants. 3 ' Schumb and Sweetser, J. Am. Chem. Soc., 57, 871 (1035). 274 OXIDATION-REDUCTION SYSTEMS The chemical reaction occurring in the hypothetical cell, free from liquid junction, Ag | Ag+ || Fe++, FO-H-+ | Pt, for the passage of one faraday is Ag + Fe+++ = Ag+ The standard E.M.F. of this cell (#) with all reactants at unit activity is given by (cf. p. 251) F F where the activities are those at equilibrium, indicated by the suffix e; the activity of the solid silver is equal to unity, and so is omitted from the equilibrium constant. The standard E.M.F. is also equal to the differ- ence of the standard potentials of the silver and ferrous-ferric electrodes, thus JE' = fii*,^- JEFe^Fo"*, (13) and hence if the equilibrium constant of the coll reaction could be deter- mined by chemical analysis, the value of Ev e +\ Vo +++ could be calculated, since the standard potential of silver is known (Table XLIX). A solution of ferric pcrchlorate, containing free perchloric acid in order to repress hydrolysis, was shaken with finely divided silver until equilibrium of the system Ag + Fe(C10 4 ) 3 ^ AgC10 4 + Fe(ClO 4 ) 2 was attained. Since perchlorates are very strong electrolytes, they are generally regarded as being completely dissociated at not too high con- centrations; this reaction is, therefore, equivalent to that of the hypo- thetical cell considered above. By analyzing the solution at equilibrium, a concentration equilibrium "constant" (fc), for various total ionic strengths, was calculated; this function k is related to the true equilib- rium constant in the following manner: j r = k - f --- > /Fe +++ and if the activity coefficients are expressed in terms of the ionic strength by means of the extended form of the Debye-Huckel equation (p. 147), it is found that log K = log k + log/ Ag + + = log k - APPROXIMATE DETERMINATION OF STANDARD POTENTIALS 275 The value of A is known to be 0.509 for water at 25, and that of C is found empirically; another term, Z)y 2 , with an empirical value of D, may be added if necessary, and the true dissociation constant K can then be calculated from the experimental data. In this manner, it was found that K is 0.531 at 25, and hence from equations (12) and (13), making use of the fact that the standard potential of silver is 0.799, it follows that at 25 - 0.799 - J3?-e+*.Fe + ** = 0.05915 log 0.531 = - 0.016, .'. #Fe+ + ,Fe 4 + f = - 0.783 Volt. Direct measurements of the potential of the ferric-ferrous system have also been made; after allowing for hydrolysis and activity effects, the standard potential at 25 was found to be 0.772 volt, but so many corrections were involved in arriving at this result that the value based on equilibrium measurements is probably more accurate. 4 Approximate Determination of Standard Potentials. Many studies have been made of oxidation-reduction systems with which, for one reason or another, it is not possible to obtain accurate results: this may be due to the difficulty of applying activity corrections, uncertainty as to the exact concentrations of the substances involved, or to the slowness of the establishment of equilibrium with the inert metal of the electrode. It is probable that whenever the difference in the number of electrons between the oxidized and reduced states, i.e., the value of n for the oxidation-reduction system, is relatively large the processes of oxidation and reduction occur in stages, one or more of which may be slow. In that event equilibrium between the system in the solution and the elec- trode will be established slowly, and the measured potential may be in error. To expedite the attainment of the equilibrium a potential medi- ator may be employed; 5 this is a substance that undergoes reversible oxidation-reduction and rapidly reaches equilibrium with the electrode. Consider, for example, a system of two ions M + and M++ which is slow in the attainment of equilibrium with the electrode, and suppose a very small amount of a eerie salt (Ce ++++ ^ is added to act as potential mediator; the reaction M+ + Ce+ +++ ^ M++ + Ce+++ takes place until equilibrium is attained. At this point the potential of the M+, M-n system must be identical with that of the Ce+++, Ce++++ system (cf. p. 284). The cerio-cerous system comes to equilibrium rap- idly with the inert metal, e.g., platinum, electrode and the potential registered is consequently both that of the Ce+ ++, Ce++ ++ and M+, M++ 4 Popoff and Kunz, J. Am. Chem. Soc., 51, 382 (1929); Bray and Hershey, ibid., 56, 1889 (1934). Loimaranta, Z. Elektrochem., 13, 33 (1907); F6erster and Pressprich, ibid., 33, 176 (1927); Goard and Rideal, Trans. Faraday Soc., 19, 740 (1924). 276 OXIDATION-REDUCTION SYSTEMS systems in the experimental solution. If the potential mediator is added in very small amount, a negligible quantity of M+ is used up and M++ formed in the establishment of the chemical equilibrium represented above : the measured potential in the presence of the mediator may thus be regarded as the value for the original system. In addition to eerie salts, iodine has been used as a potential mediator; the platinum elec- trode then measures the potential of the iodine-iodide ion system. If the results obtained in the presence of a mediator are to have definite thermo- dynamic significance they should be independent of the nature of the mediator and of the electrode material, provided the latter is not attacked in any way. Standard Potentials from Titration Curves. A method of studying oxidation-reduction systems involving the determination of potentials during the course of titration with a suitable substance, which frequently acts as a potential mediator, has been emplo v to a considerable extent in work on systems containing organic compounds. The pure oxidized form of the system, e.g., a quinone or related substance, is dissolved in a solution of definite hydrogen ion concentration, viz., a buffer solution (see Chap. XI); known amounts of a reducing solution, e.g., titanous chloride or sodium hydrosulfite, are added, in the absence of air, and the solution is kept agitated by means of a current of nitrogen. The poten- tial of an inert electrode, e.g., Per Cent Reduction platinum, gold or mercury, im- mersed in the reacting solution is measured after each addition of the titrant, by combination with a reference electrode such as a form of calomel electrode. The results obtained are of the type shown in Fig. 79, in which the electrode potentials observed during the course of the addi- tion of various amounts of titanous chloride to a buffered (pll 6.98) solution of 1-naph- thol-2-sulfonate indophenol at 30 are plotted as ordinates against the volumes of added reagent as abscissae. 8 The point at which the potential undergoes a rapid change is that corresponding to complete reduction (cf. p. 286), and the quantity of reducing solution then added is equivalent to the whole of the oxidized organic compound originally present. From the amounts of reducing agent added at various Clark et al, "Studies on Oxidation-Reduction," Hygienic Laboratory Bulletin, No. 151, 1928; see also, Conant et al., J. Am. Chem. Soc.. t 44, 1382, 2480 (1922); LaMer and Baker, ibid., 44, 1954 (1922). 10 20 ee. Reducing Agent Fia. 79. Reduction of l-naphthol-2- sulfonate indophenol (Clark) 32.8 STANDARD POTENTIALS FROM TITRATION CURVES 277 stages the corresponding ratios of the concentrations of the oxidized form (o) to the reduced form (r) may be calculated without any knowledge of the initial amount of the former or of the concentration of the titrating agent. If t c is the volume of titrant added when the sudden change of potential occurs, i.e., when the reduction is complete, and t is the amount of titrant added at any point in the titration, then at this point o is equivalent to t c t, and r is equivalent to t, provided the titrant em- ployed is a powerful reducing agent.* According to equation (1), re- placing the ratio of the activities by the ratio of concentrations, it follows that -* -* <> where E*' is the standard potential of the system for the hydrogen ion concentration employed in the experiment. Values of E Q/ can thus be obtained for a series of points on the titration curve; if the system is behaving in a satisfactory manner these values should be approximately constant. The results obtained by applying equation (14) to the data in Fig. 79 are recorded ; n Table LII. TABLE LII. EVALUATION OP APPROXIMATE STANDARD POTENTIAL AT 30 OF l-NAPHTHOL-2-SULFONATE INDOPHENOL AT pH 6.98 Per cent _ i / Reduction E 2F t E*' 4.0 12.2 - 0.1479 - 0.0258 - 0.1221 8.0 24.4 -0.1368 -0.0148 -0.1220 12.0 36.6 -0.1292 -0.0072 -0.1220 16.0 48.8 -0.1224 -0.0006 -0.1218 20.0 61.0 -0.1159 + 0.0058 -0.1217 24.0 73.2 -0.1085 +0.0131 -0.1216 28.0 85.4 -0.0985 +0.0230 -0,1215 32.8 (O 100.0 -0.036 The experiment described above can also be carried out by starting with the reduced form of the system and titrating it with an oxidizing agent, e.g., potassium dichromate. The standard potentials obtained in this manner agree with those derived from the titration of the oxidized form with a reducing agent, and also with the potentials measured in mixtures made up from known amounts of oxidized and reduced forms. The presence of the inorganic oxidizing or reducing system, which often has the advantage of serving as a potential mediator, does not affect the results to any appreciable extent- It will be seen shortly that the value of n 9 the number of electrons involved in the oxidation-reduction system, is of some interest; if this is * The precise conditions for efficient reduction are discussed on page 280. 278 OXIDATION-REDUCTION SYSTEMS not known, it can be evaluated from the slope of the flat portion of the titration curve such as that in Fig. 79. This slope is determined by the value of n only, and is independent of the chemical nature of the system; the larger is n the flatter is the curve. An exact estimate of n may be made by plotting the measured potential E against log o/r, or its equiva- lent log (t c t)lt\ the plot, according to equation (14), should be a straight line of slope - 2.303RT/nF, i.e., - 0.059/n at 25 or - 0.060/n at 30. The results derived from Fig. 79 are plotted in this manner in Fig. 80; the points are seen to fall approximately on a straight line, in -0.16 3-0.14 *-0.13 3-0.12 g-0.11 0.10 +0.8 + 0.4 -0.4 -0.8 lo, FIG. 80. Determination of n and E' agreement with expectation, and the slope is 0.03 at 30, so that n is equal to 2. The standard potential of the system at the given hydrogen ion concentration, i.e., E**', is given by the point at which the ratio o/r is unity, i.e., log o/r is zero; this is seen to be 0.122 volt, in agreement with the values in Table LIL Standard Oxidation-Reduction Potentials. Some values of standard oxidation-reduction potentials at 25 are given in Table LI 1 1. 7 The sign of the potential is based on the usual convention (p. 187), and the assump- tion that an inert material precedes the system mentioned in each case; for example, for Pt | Fe++, Fe+++ the standard potential is - 0.783 volt. A positive sign would indicate the tendency for negative electricity, e.g., electrons, to pass from solution to the metal, i.e., Fe++(+ Pt) - Fe+++ + e(Pt), so that in this particular case the standard free energy change of the process Fe+ + = Fe+ ++ + e 'For further data, see International Critical Tables, Vol. VI, and Latimer, "The Oxidation States of the Elements and their Potentials in Aqueous Solutions, 11 1938. VARIATION OF OXIDATION-REDUCTION POTENTIAL 279 is given by A(? = - nFE = + 0.783F. If the electrode had been represented by Fe++, Fe+++ | Pt, i.e., with the inert metal succeeding the system, the sign of the potential would be reversed, i.e., + 0.783 volt. A positive potential in this case means a tendency for the process Fe+++ + (pt) = Fe++ + (Pt) to occur, which is the reverse of that just given. The order of writing the components present in the solution, viz., Fe++, Fe 4 ++ or Fe+++, Fe++ is immaterial, although the usual convention is to employ the former method of representation. TABLE LI II. STANDARD OXIDATION-REDUCTION POTENTIALS AT 25 Electrode Reaction Potential Co* +. Oo + * Co- 1 "*- -Co-*- + + -1^2 Pb + +, Pb++++ l'b* + -Pb* + + 4 -f 2 - 1.75 PbS()4(), PbOjGO.SO" PbSO + 2HjQ - PbOj -1- 4H + + SO^ - + 2 - 1 68-> CV + +, Ce+ +++ Ce* + + -Ce* + + * -f c - 1.61 Mn++. MnO, H+ Mn+* -|- 4H?0 - Mn0 4 - -f 8ir -h ,^ - 1.62 Tl 4 , T!*"* * Tl + -Tl + ^ + -f2 -1.22 HtfMlR** llK^ + -* 2Hg ++ -f 2 -0.006 Fc*+. Fe'+ + Fo^ -Fe +< -* -f -0.783 MnO", MnO; MnO;- - -> MnO- + * -0.64 Fe(CN) e ----, I't-(CN) fl Ke(CN),- -Fe(C\), -- -f e -0356 Cu-'-.Cu*-* Cu* Cu** + -0.16 Sn^^.Sn-*** Sn ++ -Sn** + + +2e -0 15 Ti+*vn + ** + Ti f + * -^Ti^ 4 * + -0.06 Cr++. C'r* + * Cr f< " ->Cr + + + {- + 0.41 The potentials recorded in Table LIII may be called "oxidation potentials" (cf. p. 243) since they give a measure of the free energies of the oxidation processes; for the reverse reactions, the potentials, with the signs reversed, are the co responding " reduction potentials." Variation of Oxidation-Reduction Potential. From a knowledge of the standard oxidation-reduction potential of a given system it is possible to calculate, with the aid of the appropriate form 'of equation (3), the potential of any mixture of oxidized and reduced forms. For approxi- mate purposes it is sufficient to substitute concentrations for activities; the results are then more strictly applicable to dilute solutions, but they serve to illustrate certain general points. A number of curves, obtained in this manner, for the dependence of the oxidation-reduction poten- tial on the proportion of the system present in the oxidized form, are 280 OXIDATION-REDUCTION SYSTEMS depicted in Fig. 81 ; these curves are obviously of the same form as the experimental curve in Fig. 79. The position of the curve on the oxida- tion-reduction scale depends on the standard potential of the system, which corresponds approximately to 50 per cent oxidation, while its slope is determined by the number of electrons by which the oxidized and reduced states differ. The in- fluence of hydrogen ion con- centration in the case of the permanganate-manganous ion system is shown by the curves for an* equal to 1 and 0.1, re- spectively. It is seen from the curves in Fig. 81 that the potential rises rapidly at first as the amount of oxidized form is increased: this is due to the fact that when the proportion of the latter is small a relatively small actual increase in its amount brings about- a large relative change. For example, if the solution contained 0.1 per cent of oxi- dized form and 99.9 per cent of reduced form, the potential would be ^ . . ^ A ---0.2 1.8 - 1.6 - 1.4 -1.2 I- 0.3 '-0.6 0.0 0.2 0.059 . 25 50 Per Cent Oxidation 75 100 + 0.177 Fia. 81. Oxidation-reduction potentials at 25. A change of 1 per cent in the proportion of oxidized form in the system would make the actual proportion M per cent, while there would be 98.9 per cent of oxidized form: the oxidation-reduction potential would then be given by 0.059, 0.118 1 indicating a change of potential of about 0.059/n volt. As the amounts IONIZATION IN STAGES 281 of oxidized and reduced states become of the same order, the potential changes only slowly, since an increase or decrease in either brings about little change in the ratio which determines the oxidation-reduction poten- tial. Thus a change of 1 per cent in the amount of the oxidized form from 40 to 50 per cent, for example, alters the ratio of oxidized to reduced forms from 49/51 to 50/50; this will correspond to a change of 0.0052/n volt in potential. Solutions in this latter condition are said to be "poised" :* the addition of appreciable amounts of an oxidizing or reducing agent to such a solution produces relatively little change in the oxidation-reduction potential. Finally, when the system consists almost exclusively of the oxidized form, i.e., at the right-hand side of Fig. 81, the potential again changes rapidly ; the amount of reduced form is now very small, and con- sequently a small actual change means a large change in the ratio of oxidized to reduced forms in the solution. lonization in Stages. When a metal yields two positive ions, M* l + and M** 4 ", there are three standard potentials of the system; these are the potentials of the electrodes M, M z i + and M, M 2 *+ in addition to the oxidation-reduction potential M*i+, M** 4 ". If the values of these standard potentials are Eft, El and #?, 2 , respectively, then the free energy changes for the following process are as indicated below: M = M*i+ + 2 l , AC? = - M = M** 4 - + 2 2 , ACS = - z 2 FE 2 ] and M z t + = M * 2 + + fe _ 2l ) > A( JO 2 = It follows from these three equations that so that the three potentials are not independent. If any two of the three potentials are known, the third can be evaluated directly. For example, the standard potentials for ( 1 u, Cu 4 " 4 " and Cu 4 ", Cu 4 " 4 ", which are equiva- lent to El and #i, 2 , respectively, are - 0.340 and - 0.160 volt at 25. It follows, therefore, since z\ is equal to 1 and 22 to 2, that - 2 X 0.340 - ? = - 0.160, /. EI = - 0.520 volt. When a metal M is placed in contact with a solution containing either or M* + ions, or both, reaction will occur until the equilibrium (2 2 ~ 2i)M is established; in this condition, it follows from the law of mass action * This is the equivalent of the term "buffered" as applied to hydrogen ion poten- tials (cf. p. 410). 282 OXIDATION-REDUCTION SYSTEMS that where a\ and 02 are the activities of the M*i+ and M** + ions, respectively, at equilibrium. The activity of the solid metal M is taken as unity. The value of this equilibrium constant can be calculated from the standard potentials derived above. It can be deduced, although it is obvious from general considerations, that when equilibrium is attained the potential of the metal M must be the same with respect to both M'I* and M**"*" ions; hence, (15) It has been seen above that for the copper-copper ion system, U? is - 0.520 and E 2 is - 0.340, and so at 25, . K = ^ = 8 22 x 10- 7 . Ocu + * When metallic copper comes to equilibrium with a solution containing its ions, therefore, the concentration of cuprous ions will be very much smaller than that of cupric ions. For mercury on the other hand, ff{ for Hg, Hgf + is 0.799 volt, while E lt2 is - 0.906; from these data it is found that at equilibrium aH g jV^H g ++ is 91. The ratio of the activity of the mercurous ions to that of the mercuric ions is thus 91, and hence the system in equilibrium with metallic mercury consists mainly of mer- curous ions, although mercuric ions are also present to an appreciable extent. It can be seen from equation (15) that the equilibrium constant between the two ions of a given metal in the presence of that metal is greater the larger the difference of the standard potentials with respect to the two ions; the ions giving the less negative standard potential are present in excess at equilibrium (cf. p. 253). Attention may be called to the fact that if the equilibrium constant could be determined by chemical methods, and if one of the three stand- ard potentials of a particular metal-ion system is known, the other two could be evaluated. This procedure was actually used for copper, the calculations given above being carried out in the reverse direction. 8 Oxidation-Reduction Equilibria. When two reversible oxidation- reduction systems are mixed a definite equilibrium is attained which is Heinerth, Z. Ekktrochem., 37, 61 (1931). OXIDATION-REDUCTION EQUILIBRIA 283 determined largely by the standard potentials of the systems. For ex- ample, for the reaction between the ferrous-ferric and stannous-stannic systems, the equilibrium can be represented by 2Fe++ + Sn++++ ^ 2Fe+++ + Sn++, and when equilibrium is attained, the law of mass action gives The reaction takes place in the cell Pt | Fe++, Fe+ ++ || Sn++, Sn++++ | Ft, for the passage of two faradays, and so it follows that the standard E.M.F. is given by K -' ' " ^"* 2F ' , / aan++al> e +++ \ In I ^7^7; 1 J where E Q is equal to the difference in the standard potentials of the ferrous-ferric and stannous-stannic systems, i.e., E = JBj-e** Pa*** - JS8n**.Sn****- (18) It is evident, therefore, from equations (17) and (18), that the equilibrium constant depends on the difference of the standard potentials of the interacting systems; if the equilibrium constant were determined experi- mentally it would be possible to calculate the difference of standard potentials, exactly as in the case of the replacement of one metal by another (cf. p. 254). Alternatively, if the difference in standard poten- tials is known, the equilibrium constant can be evaluated. The value of EF^\ F ^^ is - 0.783 and that of JE 8 n** s n * f ** is - 0.15 at 25; hence making use of the relationship, from equations (17) and (18), r/TT JEra** Pa*** ~ JESn*+ 8n**** = gjp ^ K, (19) it is readily found that This low value of the equilibrium constant means that when equilibrium is attained in the ferrous-ferric and stannous-stannic mixture, the con- centrations (activities) of ferric and stannous ions must be negligibly small in comparison with those of the ferrous and stannic ions. In other words, when these two systems are mixed, reaction occurs so that the ferric ions are virtually completely reduced to ferrous ions while the stannous are oxidized to stannic ions. This fact is utilized in analytical work for the reduction of ferric to ferrous ions prior to the estimation of the latter by means of dichromate. 284 OXIDATION-REDUCTION SYSTEMS Inserting the expression for K, given by equation (16), into equation (19) and rearranging, the result is _ RT (a r .+++) f RT (OB,****), *** ^ - -jr in ^-^ = EL-sn - ^in-^-^-, (20) the left-hand side of this equation boing the potential of the ferrous- ferric system and the right-hand side that of the stannous-stannic system at equilibrium. When this condition is attained, therefore, both systems must exhibit the same oxidation-reduction potential; this fact has been already utilized in connection with the employment of potential mediators. Oxidation-Reduction Systems in Analytical Chemistry. An exami- nation of the calculation just made shows that the very small equilibrium constant,* and hence the virtually complete interaction of one system with the other, is due to the large difference in the standard potentials of the two systems. The system with the more negative standard poten- tial as recorded in Table LIII, e.g., Pt | Fe ++ , Fe+ ++ in the case con- sidered above, always oxidizes the system with the less negative standard potential, e.g., Pt | Sn 1 " 1 ", Sn+++ + , the extent of the oxidation being greater the larger the difference between the standard potentials. The same conclusion may be stated in the alternative manner: the system with the less negative potential reduces the one with the more negative potential, the extent being greater the farther the systems are apart in the table of standard potentials. It is of interest to call attention to the fact that as a consequence of these arguments the terms "oxidizing agent" and "reducing agent" are to be regarded as purely relative. A given system, e.g., ferrous-ferric, will reduce a system above it in Table LIII, e.g., cerous-ceric, but it will oxidize one below it, e.g., stan- nous-stannic. The question of the extent to which one system oxidizes or reduces another is of importance in connection with oxidation-reduction titra- tions in analytical chemistry. The reason why eerie sulfate and acidified potassium permanganate are such useful reagents in volumetric analysis is because they have large negative standard potentials and are conse- quently able to bring about virtually complete oxidation of many other systems. If the permanganate system had a standard potential which did not differ greatly from that of the system being titrated, the equilib- rium constant might be of the order of unity; free permanganate, indi- cated by its pink color, would then be present in visible amount long before oxidation of the other system was complete. The titration values would thus have no analytical validity. In order that oxidation or reduction of a system should be "complete," within the limits of accuracy of ordinary volumetric analysis, it is necessary that the concentration of one form at the end-point should be at least 10 3 times that of the other; that is to say, oxidation or reduction is complete within 0.1 per cent or * If the reaction were considered in the opposite direction the equilibrium constant would be the reciprocal jf the value given, and hence would be very large. POTENTIOMETRIC OXIDATION-REDUCTION TITHATIONS 285 better. The equilibrium constant should thus be smaller than 10" 6 if n is the same for both interacting oxidation-reduction systems, or 10~ 9 if n is unity for one system and two for the other. By making use of equations similar to (19), it can be readily shown that if two oxidation-reduction systems are to react completely in the ordinary analytical sense, the standard potentials should differ by at least 0.35 volt if n is unity for both systems, 0.26 volt if n is unity for one and two for the other, or 0.18 volt if n is two for both. Potentiometric Oxidation-Reduction Titrations. The variation of potential during the course of the conversion of the completely reduced state of any system to the completely oxidized state is represented by a curve of the type shown in Figs. 79 and 81; these curves are, therefore, equivalent to potential-titration curves, the end-point of the titration in each case being marked by a relatively rapid change of potential. The question arises as to whether this end-point could be estimated with sufficient accuracy in any given case by measuring the potential of an inert electrode, e.g., platinum, inserted in the titration system. An answer can be obtained by considering the further change in potential after the end-point has been passed; before the equivalence-point the potentials are determined by the titrated system, since this is present in ,67 Titrated System Titrant Syatem E E Q Titrated System Titrant System Fia. 82. Potential-titration curve; deter- mination of end-point is possible FIG. 83. Potential-titration curve; de- termination of the end-point is not satis- factory excess, while after the equivalent point they are determined by the titrant system. The potential-titration curve from one extreme to the other can then be derived by placing side by side the curves for the two separate systems and joining them by a tangent. Two examples are shown in Figs. 82 and 83; in the former the standard potentials, represented by the 286 OXIDATION-REDUCTION SYSTEMS respective mid-points, are reasonably far apart, but in the latter they are close together. In Fig. 82 there is a rapid increase of potential at the titration end-point, and so its position can be determined accurately; systems of this type, therefore, lend themselves to potentiometric titra- tion. When the standard potentials of the titrated and titraiit systems are close together, however, the change of potential at the equivalence- point is not marked to any appreciable extent ; satisfactory potentiometric detection of the end-point in such a titration is therefore not possible. It will be recalled that the condition for reliable potentiometric titra- tion is just that required for one system to reduce or oxidize another completely within the normal limits of analytical accuracy. It follows, therefore, that when the standard potentials of the two interacting sys- tems are such as to make them suitable for analytical work, the reaction is also one whose end-point can be derived reasonably accurately poten- tiometrically. The minimum differences between the standard potentials given on page 285 for an analytical accuracy of about 0.1 per cent, with systems of different types, may also be taken as those requisite for satis- factory potentiometric titration. The greater the actual difference, of course, the more precisely can the end-point be estimated. The method of carrying out oxidation-reduction titrations potentio- metrically is essentially similar to that for precipitation reactions, except that the indicator electrode now consists merely of an inert metal. The determination of the end-point graphically or by some form of differential titration procedure is carried out in a manner exactly analogous to that described in Chap. VII; various forms of simplified methods of oxidation- reduction titration have also been described. 9 Potential at the Equivalence-Point. Since the potentials of the two oxidation-reduction systems, represented by the subscripts I and II, involved in a titration must be the same, it follows that where E is the actual potential and Ei and En are the respective standard potentials. Consider the case in which the reduced form of the system I, i.e., Ri, is titrated with the oxidized form of the system II, i.e., OH, so that the reaction Hi + On = Oi + RII occurs during the titration. At the equivalence-point, not only are the concentrations of Oi and Rn equal, as at any point in the titration, but Ri and On are also equal to each other; hence Oi/Ri is then equal to Rii/On. Substitution of this result into equation (21) immediately gives for Uequiv., the potential at the equivalence-point, ^equi 9 See general references to potentiometric titration on page 256. OXIDATION-REDUCTION INDICATORS 287 This result holds for the special case in which each oxidation-reduction system involves the transfer of the same number of electrons, i.e., the value of n is the same in each case. If they are different, however, the equation for the reaction between the two systems becomes miRi + niOn = nnOi + niRn, where ni and n\\ refer to the systems I and II, respectively. By using the same general arguments as were employed above, it is found that the potential at the equivalence-point is given by Oxidation-Reduction Indicators. A reversible oxidation-reduction indicator is a substance or, more correctly, an oxidation-reduction sys- tem, exhibiting different colors in the oxidized and reduced states, generally colored and colorless, respectively. Mixtures of the two states in different proportions, and hence corresponding to different oxidation- reduction potentials, will have different colors, or depths of color; every color thus corresponds to a definite potential which depends on the standard potential of the system, and frequently on the hydrogen ion concentration of the solution. If a small amount of an indicator is placed in another oxidation-reduction system, the former, acting as a potential mediator, will come to an equilibrium in which its oxidation-reduction potential is the same as that of the system under examination. The potential of the given indicator can be estimated from its color in the solution, and hence the potential of the system under examination will have the same value. Since the eye, or even mechanical devices, are capable of detecting color variations within certain limits only, any given oxidation-reduction indicator can be effectively employed only in a certain range of potential. Consider, for example, the simple case of an indicator system for which n is unity; the oxidation-reduction potential at constant hydrogen ion concentration is given approximately by Suppose the limits within which color changes can be detected are 9 per cent of oxidized form, i.e., o/r is 9/91 1/10, at one extreme, to 91 per cent of oxidized form, i.e., o/r is 91/9 10; the corresponding potential limits at ordinary temperatures are then given by the foregoing equation as E Q + 0.058, and E* - 0.058, respectively. If n for the indicator sys- tem had been 2, the limits of potential would have been E Q + 0.029 and E* 0.029. It is seen, therefore, that an oxidation-reduction indicator can be used for determining the potentials of unknown systems only if the values lie relatively close to the standard potential E Q of the indi- 288 -OXIDATION-REDUCTION SYSTEMS cator. In other words, it is only in the vicinity of its standard potential, at the particular hydrogen ion concentration of the medium, that an oxidation-reduction indicator undergoes detectable color changes. In order to cover an appreciable range of potentials, it is clearly necessary to have a range of indicators with different standard potentials. Indicators for Biological Systems. 10 Many investigations have been carried out of substances which have the properties necessary for a suitable oxidation-reduction indicator. As a result of this work it is convenient for practical purposes to divide such indicators into two categories: there are those of relatively low potential, viz., 0.3 to + 0.5 volt in neutral solution, which are especially useful for the study of biological systems, and those of more negative standard potentials that are employed in volumetric analysis. The majority of substances proposed as oxidation-reduction indicators for biological purposes are also acid-base indicators, exhibiting different colors in acid and alkaline solutions. They are frequently reddish-brown in acid media, i.e., at high hydrogen ion concentrations, arid blue in alkaline solutions, i.e., at low hydrogen ion concentrations, and since the former color is less intense than the latter it is desirable to use the indicator in its blue form. In biological systems it is generally not possible to alter the hydrogen ion concentration from the vicinity of the neutral point, i.e., pH 7,* and so indicators are required with relatively strong acidic, or weakly basic, groups so that they exhibit their alkaline colors at relatively high hydro- gen ion concentrations (cf. Chap. X). A number of such indicators have been synthesized by Clark and his co-workers, by introducing halogen atoms into one of the phenolic groups of phenol-indophenol, e.g., 2 : 6- dichlorophenol-indophenol. In addition to the members of this series, other indicators of biological interest are indamines, e.g., Bindschedler's green and toluylene blue; thiazines, e.g., Lauth's violet and methylene blue; oxazines, e.g., cresyl blue and ethyl Capri blue; and certain indigo- sulfonates, safranines and rosindulines. A group of oxidation-reduction indicators of special interest are the so-called " viologens," introduced by Michaelis; they are NN'-di-substitutcd-4 : 4-dipyridilium chlorides which are deeply colored in the reduced state, and have the most positive standard potentials of any known indicators. A few typical oxidation- reduction indicators used in biological work, together with their standard potentials (E') at pH 7, determined by direct measurement, are given in Table LIV; it will be observed that these cover almost the whole range of potentials from 0.3 to + 0.45 volt, with but few gaps. It is rarely feasible in biological investigations to determine the actual potential from the color of the added indicator, although this should be possible theoretically, because the indicators are virtually of the one 10 Clark et al., "Studies on Oxidation-Reduction/' 1928 el seq.\ Michaolis, "Oxy- dat ions-Reductions Potentiate," 1933; for review, see Glasstone, Ann. Rep. Chem. &oc., 31, 305 (1934). * For a discussion of pH and its significance, see Chap. X; see also, page 292. INDICATORS FOR VOLUMETRIC ANALYSIS 289 TABLE LIY. OXIDATION-REDUCTION INDICATORS FOR BIOLOGICAL WORK Indicator E 9 ' Indicator Phenol-m-sulfonate indo- 2 : 6-dibromophenol - 0.273 w-Bromophenol indophenol 0.248 2 : 6-Dichlorophenol indophenol 0.217 2 : 6-Dichlorophenol indo-o-cresol 0.181 2 : 6-Dibromophenol indoguaiacol 0.159 Toluylene blue -0.115 Cresyl blue - 0.047 Methylene blue -0.011 Indigo tetrasulfonate -f 0.046 E" Ethyl Capri blue + 0.072 Indigo trisulfonate + 0.081 Indigo disulfonate + 0.125 Cresyl violet -f 0.173 Phenosafranine -j- 0.252 Tetramethyl phenosafranine -f 0.273 Rosinduline scarlet + 0.296 Neutral red + 0.325 Sulfonated rosindone -f 0.380 Methyl viologen + 0.445 color type. For most purposes, therefore, it is the practice to take a number of samples of the solution under examination, to add different indicators to each and to observe which are reduced; if one indicator is decolorized and the other not, the potential must lie between the standard potentials of these two indicators at the hydrogen ion concentration (pH) of the solution. Similarly, indicators may be used in the reduced state and their oxidation observed. Indicators are also often employed as potential mediators in solutions for which equilibrium with the electrode is established slowly; the potential is then measured electrometrically. When employing an oxidation-reduction indicator it is essential that the solution to which it is added should be well poised (p. 281), so that in oxidizing or reducing the indicator the ratio of oxidized to reduced states of the experimental system should not be appreciably altered. The amount of indicator added must, of course, be relatively small. Indicators for Volumetric Analysis. The indicators described above are frequently too unstable for use in volumetric analysis and, in addition, they show only feeble color changes in acid solution. The problem of suitable indicators for detecting the end-points of oxidation-reduction titrations is, however, in some senses, simpler than that of finding a series of indicators for use over a wide range of potentials. It has been seen that if two oxidation-reduction systems interact sufficiently completely to be of value for analytical purposes, there is a marked change of poten- tial of the system at the equivalence-point (cf. Fig. 82). Ideally, the standard potential of the indicator should coincide with the equivalence- point potential of the titration; actually it is sufficient, however, for the former to lie somewhere in the region of the rapidly changing potential of the titration system. When the end-point is reached, therefore, and the oxidation-reduction potential undergoes a rapid alteration, the color of the indicator system will change sharply from one extreme to the other. If the standard potential of the indicator is either below or above the region in which the potential inflection occurs, the color change will take place either before or after the equivalence-point, and in any case will be gradual rather than sharp. Such indicators would be of no value for the particular titration under consideration. It has been found (p. 285) that if two systems are to interact sufficiently for analytical purposes their 290 OXIDATION-REDUCTION SYSTEMS standard potentials must differ by about 0.3 volt, and hence the standard potential of a suitable oxidation-reduction indicator must be about 0.15 volt below that of one system and 0.15 volt above that of the other. Since the most important volumetric oxidizing agents have high negative potentials, however, a large number of indicators is not necessary for most purposes. The interest in the application of indicators in oxidation-reduction titrations has followed on the discovery that the familiar color change undergone by diphenylamine on oxidation could be used to determine the end-point of the titration of ferrous ion by dichromate in acid solution. Diphenylamine, preferably in the form of its soluble sulfonic acid, at first undergoes irreversible oxidation to diphenylbenzidine, and it is this sub- stance, with its oxidation product diphenylamine violet, that constitutes the real indicator. 11 The standard potential of the indicator system is not known exactly, but experiments have shown that in not too strongly acid solutions the sharp color change from colorless to violet, with green as a possible intermediate, occurs at a potential of about 0.75 volt. The standard potential of the ferrous-ferric system is 0.78 whereas that of the di- chromate-chromic ion system in an acid medium is approximately 1.2 volt; hence a suitable oxidation-reduction indicator might be expected to have a standard potential of about 0.95 volt. It would thus appear that diphenylamine would not be satisfactory for the titration of ferrous ions by acid dichromate, and this is actually true if a simple ferrous salt is employed. In actual practice, for titration purposes, phosphoric acid or a fluoride is added to the solution ; these substances form complex ions with the ferric ions with the result that the effective standard potential of the ferrous-ferric system is lowered (numerically) to about 0.5 volt. The change of potential at the end-point of the titration is thus from about 0.6 to 1.1 volt, and hence diphenylamine, changing color in the vicinity of 0.75 volt, is a satisfactory indicator. Ceric sulfate is a valuable oxidizing agent, the employment of which in volumetric work was limited by the difficulty of detecting the end-point unless a potentiometric method was used. A number of indicators are now available, however, which permit direct titration with eerie sulfate solution to be carried out. One of the most interesting and useful of these is o-phenanthroline ferrous sulfate, the cations of which, viz., FeCCuHsNi) +, with the corresponding ferric ions, viz., Fe(Ci 2 H 8 N 2 ) ++, form a reversible oxidation-reduction system; the reduced state has an intense red color and the oxidized state a relatively feeble blue color, so that there is a marked change in the vicinity of the standard potential which is about 1.1 volt. 12 The high potential of the phenanthroline- " Kolthoff and Sarver, J. Am. Chem. Soc., 52, 4179 (1930); S3, 2902 (1931); 59, 23 (1937); for review, see Glasstone, Ann. Rep. Chem. Soc., 31, 309 (1934); also, Whitehead and Wills, Chem. Revs., 29, 69 (1941). 11 Walden, Hammett and Chapman, /. Am. Chem. Soc., 53, 3908 (1931); 55, 2649 (1933); Walden and Edmonds, Chem. Revs., 16, 81 (1935). QUINONE-HYDROQUINONE SYSTEMS 291 ferrous ion indicator permits it to be used in connection with the titration of ferrous ions without the addition of phosphoric acid or fluoride ions. The indicator has been employed for a number of titrations with eerie sulfate and also with acid dichromate, and even with very dilute solutions of permanganate when the color of the latter was too feeble to be of any value for indicator purposes. Another indicator having a high standard potential is phenylanthranilic acid; this is a diphenylamine derivative which changes color in the vicinity of 1.08 volt. It has been recom- mended for use with eerie sulfate as the oxidizing titrant. 13 Although there are now several useful indicators for titrations in- volving strongly oxidizing reactants, the situation is not so satisfactory in connection with reducing reagents, e.g., titanous salts. The standard potential of the titanous-titanic system is approximately 0.05 volt, and hence a useful indicator should show a color change at a potential of about 0.2 volt or somewhat more negative. The only substance that is reasonably satisfactory for this purpose, as far as is known at present, is methylene blue which changes color at about 0.3 volt in acid solution. Quinone-Hydroquinone Systems. In the brief treatment of the quinone-hydroquinone system on page 270 no allowance was made for the possibility of the hydroquinone ionizing as an acid; actually such ionization occurs in alkaline solutions and has an important effect on the oxidation-reduction potential of the system. Hydroquinone, or any of its substituted derivatives, can function as a dibasic acid. It ionizes in two stages, viz., H 2 Q ^ 11+ + HQ- and HQ- ^ H+ + Q , and the dissociation constants corresponding to these two equilibria (cf. p. 318) are given by - and A 2 = The hydroquinone in solution thus exists partly as undissociated H 2 Q, and also as HQ~ and Q ions formed in the two stages of ionization; the total stoichiometric concentration h of the hydroquinone is equal to the sum of the concentrations of these three species, i.e., h = CH Z Q + CHQ~ + CQ--, and if the values of CHQ- and CQ derived from the expressions for K\ and KZ are inserted in this equation, the approximation being made of taking the activity coefficients of H a Q, HQ~ and Q to be equal to "Syrokomsky and Stiepin, J. Am. Chem. Soc., 58, 928 (1936). 292 OXIDATION-REDUCTION SYSTEMS unity, the result is , , CH Q , , CH Q , , h = C H ,Q + - ki + --*- kik 2 , 1 a+ a+ In view of the neglect of the activity coefficients, the constants KI and K* have been replaced by k\ and kz which become identical with the former at infinite dilution. If q is the concentration of the quinone form, which is supposed to be a neutral substance exhibiting neither acidic nor basic properties, the oxidation-reduction potential, which according to equa- tion (4) may be written as is given by 7?T n RT ---Ino 2 ^, (24) the ratio of the activities of Q and H 2 Q being taken as equal to the ratio of their concentrations. Introduction of the value of CH 2 q from equation (22) into (24) now gives t> r r* xv z> r n rCl q it 1 o E = E -i." In vTrT In (OH* -f- ki(in+ + fci& 2 ). (25) Zr fl 2ib If ki and fc 2 are small, the terms k\a^ and k\k 2 may be neglected in com- parison with afi+, and equation (25) then reduces to (26) AT H, 1' which is the conventional form for the quinone-hydroquinone system, q and h representing the total concentrations of the two constituents. According to equation (26) the variation of the oxidation-reduction potential with hydrogen ion concentration is relatively simple, but if the acidic dissociation functions ki and kz of the hydroquinone are appre- ciable, equation (25) must be employed, and the situation becomes somewhat more complicated. The method of studying this problem is to maintain the ratio q/h constant, i.e., the stoichiometric composition of the quinone-hydroquinone mixture is unchanged, but to suppose the hydrogen ion concentration is altered. For this purpose the equations for the electrode potential are differentiated with respect to log OH*; this quantity is a very useful function of the hydrogen ion concentration, designated by the symbol pH and referred to as the hydrogen ion expo- nent. Differentiation of equation (25) thus gives o wvj RT d\og< QUINONE-HYDROQUINONE SYSTEMS 293 as applicable over the whole pH range. If an* is large in comparison with k\ and &2, i.e., in relatively acid solutions, this equation reduces to JET - 2 - 303 - for (28) which can also be derived directly from equation (26). If, however, k\ is much greater than OH* and this is much greater than &2, the terms 2an+ in the numerator and OH+ and kik z in the denominator of equation (27) may be neglected; the result is d(pH) RT 2F 7TFT for (29) Finally, when an + becomes very small, i.e., in alkaline solutions, both terms in the numerator of equation (27) may be disregarded, and so dE d(pH) = for (30) The slope of the plot of the oxidation-reduction potential, for con- stant quinone-hydroquinone ratio, against the pH, i.e., against log a H +, thus undergoes changes, as shown in Fig. 84; the temperature is 30 so -0.1 - Fia. 84. Variation of a quinone-hydroquinone (anthraquinone sulfonate) potential with pH that the slopes corresponding to equations (28), (29) and (30) are 0.060, 0.030 and zero, respectively. The position and length of the intermediate portion of slope 0.030 depend on the actual values and ratio of the acidic 294 OXIDATION-REDUCTION SYSTEMS dissociation functions k\ and A: 2 ; this may be seen by investigating the conditions for which an + is equal to k\ and k* respectively. If OH* in equation (27) is set equal to ki, the result is dE RT 3^ 565) = 2 ' 303 w ' afT+lS for aH * = * lf and since fa is generally much smaller than k\ 9 this becomes = 2.303 - ~ = 0.045 at 30 d(pH) " >wu 2F 2 When the hydrogen ion activity air is equal to the acidic function k\ 9 i.e., when the pH is equal to log k\ 9 the latter quantity being repre- sented by pki, the slope of the pH-potential curve is thus seen to be intermediate between 0.060 and 0.030. Such a slope corresponds, in general, to a point on the first bend of the curve in Fig. 84; the exact position for a slope of 0.045 is 0-btained by finding the point of inter- section of the two lines of slope 0.060 and 0.030, as shown. At this point, therefore, the pH is equal to pki. To find the slope of the pH potential curve when OH + is equal to fa, i.e., when the pH is equal to pfc 2 , the values of <JH+ in equation (27) are replaced by & 2 ; hence dE nM RT 2h + ki S5) = 2 ' 3 3 2F ' k^+U[ f r * H * - k *> and since, as before, k 2 may be regarded as being much smaller than k\ 9 dE d(pH) - 2 ' 303 W 2 - ' 015 at 30 ' The pH is thus equal to p& 2 when the slope of the pH-potential curve is midway between 0.030 and zero ; the value of p& 2 can be found by extend- ing the lines of slopes 0.030 and zero until they intersect, as shown in Fig. 84. An examination of the pH-potential curve thus gives the values of the acidic dissociation functions for the particular hydroquinone as 7.9 and 10.6 for pki and pfa, respectively, at 30. The case considered here is relatively simple, but more complex be- havior is frequently encountered: the reduced form may have more than two stages of acidic dissociation and in addition the oxidized form may exhibit one or more acidic dissociations. There is also the possibility of basic dissociation occurring, but this can be readily treated as equivalent to an acidic ionization (cf. p. 362). The method of treatment given above can, however, be applied to any case, no matter how complex, and the following general rules have been derived which facilitate the analysis of pH-potential curves for oxidation-reduction systems of constant stoichiometric composition. 14 14 Clark, "Studies on Oxidation-Reduction/ 1 Hygienic Laboratory Bulletin, 1928. TWO STAGE OXIDATION-REDUCTION 295 (1) Each bend in the curve may be correlated with an acidic dis- sociation constant; if the curve becomes steeper with increasing pH, i.e., as the solution is made more alkaline, the dissociation has occurred in the oxidized form, but if it becomes flatter it has occurred in the reduced form (cf. Fig. 84). (2) The intersection of the extensions of adjacent linear parts of the curve occurs at the pH equal to pfc for the particular dissociation function responsible for the bend. (3) Each dissociation constant changes the slope by 2.3Q3RT/nF volt per pH unit, where n is the number of electrons difference between oxidized and reduced states. Two Stage Oxidation-Reduction. The completely oxidized, i.e., holo- quinone, form of a quinone differs from the completely reduced, i.e., hydroquinone, form by two hydrogen atoms, involving the addition or removal, respectively, of two electrons and two protons in one stage, viz., H 2 Q ^ Q 4- 2H+ + 2c. It is known from chemical studies, however, that in many cases there is an intermediate stage between the hydroquinone (H 2 Q) and the quinone (Q) ; this may be a meriquinone, which may be regarded as a molecular compound (Q-H 2 Q), or it may be a semiquinone (HQ). The latter is a true intermediate with a molecular weight of the same order as that of the quinone, instead of double, as it is for the meriquinone. The possi- bility that oxidation and reduction of quinonoid compounds might take place in two stages, each involving one electron, i.e., n is unity, with the intermediate formation of a semiquinone was considered independently by Michaelis and by Elema. 16 If the two stages of oxidation-reduction do not interfere, a ready distinction between meriquinone and semi- quinone formation as intermediate is possible by means of E.M.F. meas- urements. For meriquinone formation the stages of oxidation-reduction may be written (1) 2H 2 Q - H 2 Q Q + 2H+ + 2, and (2) H 2 Q-Q^2Q so that if EI represents the standard potential of the first stage at a definite hydrogen ion concentration, RT (H,Q.Q) ~ where the parentheses represent activities. If the original amount of the reduced form (H 2 Q) in a given solution is a, and x equiv. of a strong " Friedheim and Michaelis, J. Biol Ghent., 91, 355 (1931); Michaelis, ibid., 92, 211 (1931); 96, 703 (1932); Elema, Rec. trav. chim., 50, 807 (1931); 52, 569 (1933); /. Biol Chem., 100, 149 (1933). 296 OXIDATION-REDUCTION SYSTEMS oxidizing agent are added, %x moles of Q are formed, and these combine with an equivalent amount of H 2 Q to form \x moles of meriquinone, J^Q-Q; an amount a x moles of HQ remains unchanged. It follows, therefore, neglecting activity coefficients, that in a solution of volume t;, equation (31) becomes _ RT , x RT, v __ ET __ ^_ 1- _ _ _ \Yl I ^X I ir ( Qf x) &r & The potential thus depends on the volume of the solution, and hence the position of the curve showing the variation of the oxidation-reduction potential during the course of the titration of H2Q by a strong oxidizing agent varies with the concentration of the solution. At constant volume equation (32) becomes _ RT f B7 1 , , E = E l -lnx + In (a-*), so that in the early stages of oxidation, i.e., when x is small, the last term on the right-hand side may be regarded as constant, and the slope of the titration curve will correspond to a process in which two electrons are involved, i.e., n is 2. In the later stages, however, the change of potential is determined mainly by the last term, and the slope of the curve will change to that of a one-electron system, i.e., n is effectively unity. When a true semiquinone is formed, the two stages of oxidation- reduction are (1) H 2 Q ^ HQ + H + + , and (2) HQ ^ Q + 11+ + | so that (33) for a definite hydrogen ion concentration. The value of the potential is seen to depend on the ratio of x to a re, and not on the actual con- centration of the solution; the position of the titration curve is thus independent of tho volume. Further, it is evident from equation (33) that the type of slope is the same throughout the curve, and corresponds to a one-electron process, i.e., n is unity. If the two stages of oxidation are fairly distinct, it is thus possible to distinguish between meriquinone and semiquinone formation. In the former case the position of the titration curve will depend on the volume of the solution and it will be unsymmetrical, the earlier part correspond- SEMIQUINONE FORMATION CONSTANT 297 ing to an n value of 2, and the later part to one of unity. If semiquinone formation occurs, however, the curve will be symmetrical, with n equal to unity over the whole range, and its position will not be altered by changes in the total volume of the solution. A careful investigation along these lines has shown that many oxidation-reduction systems satisfy the conditions for semiquinone formation; in one way or another, this has been found to be true for a-oxyphenazine and some of its derivatives, e.g., Wurster's red, and for a number of anthraquinones. Semiquinone Formation Constant. It was assumed in the foregoing treatment that the two stages of oxidation are fairly distinct, but when this is not the case the whole system behaves as a single two-electron process, as in Fig. 79. In view of the interest associated with the forma- tion of semiquinone intermediates in oxidation-reduction reactions, methods have been developed for the study of systems in which the two stages may or may not overlap. The treatment is somewhat compli- cated, and so the outlines only will be given here. 16 If R represents the completely reduced form (H 2 Q), S the semi- quinone (HQ), and T the totally oxidized form (Q), the electrical equi- libria, assuming a constant hydrogen ion concentration, are (1) R ^ S + e and (2) S ^ T + , so that if r, s and t are the concentrations of the three forms, RT s E = E!- In-, (34) and RT t = 2 --jrln-> (35) during the first and second stages, respectively; EI and E z are the stand- ard potentials of these stages. The potential can also be formulated in terms of the equilibrium between initial arid final states, viz., R ^ T + 2c, so that - r , (36) where E m is the usual standard potential for the system as a whole at some definite hydrogen ion concentration. It can be seen from equa- tions (34), (35) and (36) that "For reviews, see Michaelis, " Oxydations-Reductions Potentiate," 1933; Trans. Electrochem. Soc., 71, 107 (1937); Chem. Revs., 16, 243 (1935); Michaelis and Schubert, ibid., 22, 437 (1938); Michaelis, Ann. New York Acad. Sci., 40, 39 (1940); Miiller, ibid., 40, 91 (1940). 298 OXIDATION-REDUCTION SYSTEMS and since E\ and E z will be in the centers of the first and second parts of the titration curves, i.e., when s/r and t/s are unity, respectively, it follows that E m will be the potential in the middle of the whole curve. In addition to the electrical equilibria, there will be a chemical equi- librium between R, S and T, viz., R + T ^ 2S, so that by the approximate form of the law of mass action where k is known as the semiquinone formation constant. If a is the initial amount of reduced form H 2 Q which is being titrated, and x equiv. of strong oxidizing agent are added, then x/a is equal to 1 in the middle of the complete titration curve and to 2 at the end. By making use of the relationships given above, it is possible to derive an equation of some complexity giving the variation of E E m with x/a, Fia. 85. Titration curves for semiquinone formation i.e., during the course of the titration, for any value of k, the semiquinone formation constant. Some of the results obtained in this manner are shown in Fig. 85; as long as k is small, the titration curve throughout has the shape of a normal two-electron oxidation-reduction system, there being no break at the midpoint where x/a is unity. As k increases, the slope changes until it corresponds to that of a one-electron process; iu STORAGE BATTERIES (SECONDARY CELLS) 299 fact when the value of k lies between 4 and 16, the slope is that for a system with n equal to unity, but there is no break at the midpoint. The presence of a semiquinone is often indicated in these cases, however, by the appearance of a color which differs from that of either the com- pletely oxidized or the completely reduced forms. When the semiqui- none formation constant k exceeds 16, a break appears at the midpoint, and the extent of this break becomes more marked as k increases. The detection of semiquinone formation by the shape of the titration curve is only possible, therefore, when the semiquinone formation constant is large. If actual oxidation-reduction measurements are made on a particular system during the course of a titration, it is possible, by utilizing the equation from which the data in Fig. 85 were calculated, to evaluate the semiquinone formation constant for that system. The standard poten- tials Ely E 2 and E m can also be obtained for the hydrogen ion concen- tration existing in the experimental solution. Influence of Hydrogen Ion Concentration. The values of EI, E 2 and E m will depend on the pH of the solution, and since the forms R, S and T may possess acidic or basic functions, the slopes of the curves of these three standard potentials against pH may change direction at various points and crossings may occur. A system for which E 2 is above EI at one pH, i.e., the semiquinone formation constant is large, may thus be- have in a reverse manner, i.e., EI is above E 2 , and the semiquinone formation is very small, at another pH. It is apparent, therefore, that although a given system may show distinct semiquinone formation at one hydrogen ion concentration, there may be no definite indication of such formation at another hydrogen ion concentration. If the oxidized form of the system consists of a positive ion, e.g., anthraquinone sulfonic acid, semiquinone formation is readily observed in alkaline solutions only, but if it is a negative ion, e.g., a-oxyphenazine, the situation is reversed and the semiquinone formation can be detected most easily in acid solution. Storage Batteries (Secondary Cells). 17 When an electric current is passed through an electrolytic cell chemical changes are produced and electrical energy is converted into chemical energy. If the cell is revers- ible, then on removing the source of current and connecting the elec- trodes of the cell by means of a conductor, electrical energy will be produced at the expense of the stored chemical energy and current will flow through the conductor. Such a device is a form of storage battery, or secondary cell; * certain chemical changes occur when the cell is "charged" with electricity, and these changes are reversed during dis- " Vinal, "Storage Batteries," 1940. * A primary cell is one which acts as a source of electricity without being previously charged up by an electric current from an external source; in the most general sense, every voltaic cell is a primary cell, although the latter term is usually restricted to cells which can function as practical sources of current, e.g., the Leclanchl cell. 300 OXIDATION-REDUCTION SYSTEMS charge. Theoretically, any reversible cell should be able to store elec- trical energy, but for practical purposes most of them are unsuitable because of low electrical capacity, incomplete reversibility as to the physical form of the substances involved, chemical action or other changes when idle, etc. Only two types of storage battery have hitherto found any wide application, and since they both involve oxidation-reduction systems their theoretical aspects will be considered here. The Acid Storage Cell. The so-called "acid" or "lead" storage cell consists essentially of two lead electrodes, one of which is covered with lead dioxide, with approximately 20 per cent sulfuric acid, i.e., with a specific gravity of about 1.15 at 25, as the electrolyte. The charged cell is generally represented simply as Pb, H 2 SC>4, PbO 2 , but it is more correct to consider it as Pb | PbSO 4 (s) H 2 S0 4 aq. PbS0 4 (s), Pb0 2 (s) | Pb, the right-hand lead electrode acting as an inert electrode for an oxidation- reduction system. The reactions occurring in the cell when it produces current, i.e., on discharge, are as follows. Left-hand electrode: Pb = Pb++ + 2 Pb++ + SO?- = PbS0 4 (s). .". Net reaction for two faradays is Pb + SOr- = PbSO 4 (s) + 2 . Right-hand electrode: PbO 2 (s) + 2H 2 ^ Pb++++ + 4OH- Pb++++ + 2 = Pb++, Pb++ + SO?- = PbSO 4 (s), 4OH- + 4H+ = 4H 2 O, .". Net reaction for two faradays is Pb0 2 (s) + 4H+ + SO + 2 = PbS0 4 (s) + 2H 2 0. Since both electrodes are reversible, the processes occurring when elec- tricity is passed through the cell, i.e., on charge, are the reverse of those given above; it follows, therefore, that the complete cell reaction in both directions may be written as the sum of the individual electrode processes, thus discharge Pb + Pb0 2 + 2H 2 S0 4 ^ 2PbS0 4 + 2H 2 charge for two faradays. The mechanism of the operation of the lead storage battery as represented by this equation was first proposed by Gladstone and Tribe (1883) before the theory of electrode processes in general was well understood; it is known as the "double sulfation" theory, because it THE ACID STORAGE CELL 301 postulates the formation of lead sulfate at both electrodes. Various alternative theories concerning the lead cell have been proposed from time to time but these appear to have little to recommend them; apart from certain processes which occur to a minor extent, e.g., formation of oxides higher than PbO 2 , there is no doubt that the reactions given here represent essentially the processes occurring at the electrodes of an acid storage battery. It will be observed that according to the suggested cell reaction, two molecules of sulfuric acid should be removed from the electrolyte and two molecules of water formed for the discharge of two faradays of elec- tricity from the charged cell. This expectation has been confirmed experimentally. Further, it is possible to calculate the free energy of this change thermodynamically in terms of the aqueous vapor pressure of sulfuric acid solutions; the values should be equal to 2FE, where E is the E.M.F. of the cell and this has been found to be the case. A striking confirmation of the validity of the double sulfation theory is provided by thermal measurements; since the E.M.F. of the storage cell and its temperature coefficient are known, it is possible to calculate the heat change of the reaction taking place in the cell by means of the Gibbs- Helmholtz equation (p. 194). The value of the heat of the reaction believed to occur can be derived from direct thermochemical measure- ments, and the results can be compared. The data obtained in this manner for lead storage cells containing sulfuric acid at various concen- trations, given in the first column with the density in the second, are quoted in Table LV; 18 the agreement between the values in the last two TABLE LV. HEAT CHANGE OP REACTION IN LEAD STORAGE BATTERY H2SO 4 E u * dE/dT A// per cent dl volts X 10 4 E.M.F. Thermal 4.55 1.030 1.876 7.44 1.050 1.905 +1.5 - 85.83 - 86.53 14.72 1.100 1.962 +2.9 -86.54 -87.44 21.38 1.150 2.005 +3.3 -87.97 -87.37 27.68 1.200 2.050 +3.0 -90.46 -90.32 33.80 1.250 2.098 +2.2 -93.77 -93.08 39.70 1.300 2.148 +1.8 -96.63 -96.22 columns is very striking, and appears to provide conclusive proof of the suggested mechanism. It is evident from the data in Table LV that the E.M.F. of the lead storage cell increases with increasing concentration of sulfuric acid; this result is, of course, to be expected from the cell reactions. According to the reaction occurring at the Pb, PbSO 4 electrode, generally referred to as the negative electrode of the battery, its potential (EJ) is given by r>rn E- = #pb,pbso 4 ,so;~ + ~2p I* 1 a so;~- (37) Since the activity, or concentration, of sulfate ions depends on the con- 18 Craig and Vinal, /. Res. Nat. Bur. Standards, 24, 475 (1940). 302 OXIDATION-REDUCTION SYSTEMS centration of sulfuric acid, it is clear that the potential of this electrode will vary accordingly. The standard potential in equation (37) is + 0.350 volt at 25, and if the activity of the sulfate ion is taken as equal to the mean activity of sulfuric acid, it is readily calculated that for a storage battery containing acid of the usual concentration, i.e., 4 to 5 N, in which the mean activity coefficient is about 0.18 to 0.2, the actual potential of the negative electrode is about + 0.33 volt. The so-called negative electrode potential may also be represented by ~n RT E- = $ b ,pb" - ^p In a Pb ++, (38) but since the solution is saturated with lead sulfate, a Pb ++ will be inversely proportional to aso;-; equations (37) and (38) are thus consistent. The potential of the PbSO 4 , PbO 2 electrode, usually called the positive electrode, can be represented by (cf. p. 269) j? * E*> , , Hso /on\ E+* = #pbso 4 , Pbo,, sor + TTTT In - 2 - (39) and hence will be very markedly dependent on the concentration of sulfuric acid, since this affects a n + , asor and an,o. The standard potential required for equation (39) is 1.68 volts at 25 (see Table LIII) ; making the assumption that the activities of the hydrogen and sulfate ions are equal to the mean activity of sulfuric acid in which the activity of water from vapor pressure data is 0.3, it is found that, for 4 to 5 N acid, the potential E+ of the positive electrode is about 1.70 volts. The positive electrode may also be regarded as a simple oxidation- reduction electrode involving the plumbous-plumbic system; thus -* + --^. a + ln. (40) The activity of plumbic ions in a solution saturated with lead dioxide (or plumbic hydroxide) will be inversely proportional to the fourth power of the hydroxyl ion activity, and hence it is directly proportional to the fourth power of the hydrogen ion activity (cf. p. 339), in agreement with the requirements of equation (39). The Alkaline Storage Battery. The alkaline or Edison battery is made up of an iron (negative) and a nickel sesquioxide (positive) elec- trode in potassium hydroxide solution; it may be represented as Fe | FeO(s) KOH aq. NiO(s), Ni 2 O 3 (s) | Ni, the nickel acting virtually as an inert electrode material. The reactions taking place in the charged cell during discharge are as follows. *The negative sign is used because the potential of the electrode as written, viz., PbSO 4 (a), PbOiW, Pb, is opposite in direction to that corresponding to the convention on which the standard potentials in Tables XLIX and LIII are based. THE ALKALINE STORAGE BATTERY 303 Left-hand electrode: Fe = Fe++ + 2 , Fe++ + 20H- = FeO(s) + H 2 0, .". Net reaction for two faradays is Fe + 20H- = FeO(s) + H 2 O + 2. Right-hand electrode: Ni 2 O 3 (s) + 3H 2 O ^ 2Ni+++ + 60H-, 2Ni+++ + 2e = 2Ni++, 2Ni++ + 4OII- = 2NiO(s) + 2H 2 0, .". Net reaction for two faradays is Ni 2 O 3 (s) + H 2 O + 2 = 2NiO(s) + 2OH~. The complete cell reaction during charge and discharge, respectively, may be represented by discharge Fe + Ni 2 O 3 ^ FeO + 2NiO. charge The potential of the iron ("negative") electrode, which is about + 0.8 volt in practice, is given by the expression . &- = -CTe,FeO,OH~ ~T TTIT In - > *r an t o and similarly that of the nickel sesquioxide ("positive") electrode, which is approximately + 0.55 volt, is represented by T RT The potentials of both individual electrodes are dependent on the hy- droxyl ion activity (or concentration) of the potassium hydroxide solution employed as electrolyte. It is evident, however, that in theory the E.M.F. of the complete cell, which is equal to E- E+, should be inde- pendent of the concentration of the hydroxide solution. In practice a small variation is observed, viz., 1.35 to 1.33 volts for N to 5 N potassium hydroxide; this is attributed to the fact that the oxides involved in the cell reactions are all in a "hydrous" or "hydrated" form, with the result that a number of molecules of water are transferred in the reaction. The equations for the potentials of the separate electrodes should then con- tain different terms for the activity of the water in each case: the E.M.F. of the complete cell thus depends on the activity of the water in the electrolyte, and hence on the concentration of the potassium hydroxide. 304 OXIDATION-REDUCTION SYSTEMS PROBLEMS 1. Write down the electrochemical equations for the oxidation-reduction systems involving (i) ClOj and C1 2 , and (ii) Cr 2 07~ and Cr+++. Use the results to derive the complete equations for the reactions of each of these with the Sn++++, Sn++ system. 2. According to Br0nsted and Pedersen [Z. physik. Chem., 103, 307 (1924)] the equilibrium constant of the reaction I" = Fe++ + JI 2 at 25 is approximately 21, after allowing for the tri-iodide equilibrium. The standard potential of the I 2 (s), I" electrode is 0.535 volt and the solubility of iodine in water is 0.00132 mole per liter; calculate the approximate standard potential of the (Pt)Fe++, Fe+++ system. 3. From the measurements of Sammet \_Z. physik. Chem., 53, 678 (1905)] the standard potential of the system (PtJIOi" -f 6H+, JI 2 has been estimated as 1.197 volt. Determine the theoretical equilibrium constant of the reaction 10? + 51- + 6H+ = 3I 2 + 3H 2 0. What conclusion may be drawn concerning the quantitative determination of iodate by the addition of acidified potassium iodide followed by titration with thiosulfate? 4. Kolthoff and Tomsicek [J. Phys. Chem., 39, 945 (1935)] measured the potentials of the electrode (Pt)Fe(CN)e --- , Fe(CN)e~~ at 25; the concen- trations of potassium ferro- and ferri-cyanide were varied, but the ratio was unity in every case. The concentrations (c) of each of the salts, in moles per liter, and the corresponding electrode potentials (Eo), on the hydrogen scale, are given below: c E' Q c Ei 0.04 -0.4402 0.0004 -0.3754 0.02 -0.4276 0.0002 -0.3714 0.01 -0.4154 0.0001 -0.3664 0.004 -0.4011 0.00008 -0.3652 0.002 -0.3908 0.00006 -0.3642 0.001 -0.3834 0.00004 -0.3619 Plot the values of E'o against Vjji and extrapolate the results to infinite dilution to obtain the standard potential of the ferrocyanide-ferricyanide system. Alternatively, derive the value of E Q from each E' Q by applying the activity correction given by the Debye-Hiickel limiting law. 5. The oxidation-reduction system involving 5- and 4-valent vanadium may be represented by the general equation - z V,OJ*+ + (y - z)H 2 = V x O< 5 *- 2 *>+ + 2(y - z)H+ + X. Using the symbol V 5 to represent the oxidized form V X O V and V 4 for the re- duced form V0, write the equation for the E.M.F. of the cell consisting of the V 4 , V 5 and H+, Kfo electrodes. Derive the expressions to which this equa- tion reduces (i) when V 4 and H+ are kept constant, (ii) when V 5 and V 4 are constant, and (iii) when V 6 and H+ are constant. The experimental results of Carpenter [J. Am. Chem. Soc., 56, 1847 (1934)] are as follows: PROBLEMS 305 d) (ii) (iii) V 6 E H+ E V 4 E 0.529 X 10~ 3 - 0.9031 0.0240 - 0.9098 4.42 X 10~ - 0.9554 2.489 -0.9395 0.1077 -0.9554 35.11 -0.9048 9.855 -0.9723 0.4442 -0.9974 19.67 - 0.9875 0.9000 - 1.0198 Using the expressions already derived, show that the values of z, (2y 3x)/x and z can be obtained by plotting E against log V 5 , log H+ and log V 4 , respec- tively. Insert the values of x\ y and z in the expression given above and so derive the actual equation for the oxidation-reduction system. 6. In an investigation of the oxidation-reduction potentials of the system in which the oxidized form was anthraquinone 2 : 6-disulfonate, Conant and his collaborators [J. Am. Chem. Soc., 44, 1382 (1922)] obtained the following values for E Qf at various pH's: pH 6.90 7.64 9.02 9.63 10.49 11.27 11.88 12.20 W 0.181 0.220 0.275 0.292 0.311 0.324 0.326 0.326 Plot E ' against the pH and interpret the results. 7. By extrapolating the E.M.F.'S to infinite dilution, Andrews and Brown [J. Am. Chem. Soc., 57, 254 (1935)] found E Q for the cell Pt | KMn0 4 , Mn0 2 (s) KOH aq. HgO(s) | Hg to be - 0.489 at 25. The standard potential of the Hg, HgO(s), OH~ elec- trode is 0.098, and the equilibrium constant of the system + 2H,0 = 2MnOi + MnO 2 (s) + 40H~ is 16 at this temperature. Calculate the standard potential of the (Pt)MnO4, MnOf ~ electrode. 8. The standard potential of the (Pt) | PbS0 4 (s), PbO 2 (s), SOi electrode is 1.685 volts at 25; calculate the E.M.F.'S of the cell Pt | PbS0 4 (s), Pb0 2 (s) H 2 S0 4 (c) | H 2 (l atm.) for 1.097 and 6.83 molal sulfuric acid solutions. The mean activity coeffi- cients (y) and aqueous vapor pressures (p) of the solutions are: m y p 1.097 0.146 22.76 mm. 6.83 0.386 12.95 The vapor pressure of water at 25 is 23.76 mm. of mercury. 9. From the standard potentials of the systems (Pt)Cu+, Cu+ + and I 2 , I~ evaluate the equilibrium constant of the reaction Cu++ + I- = Cu+ + *I 2 , and show that it is entirely owing to the low solubility product of cuprous iodide, Cul, i.e., approximately 10~ 12 , that this reaction can be used for the analytical determination of cupric ions. 10. The solubility products of cupric and cuprous hydroxides, Cu(OH) 2 and CuOH, respectively, are approximately 10~ 19 and 10~ 14 at ordinary tem- peratures [Allmand, J. Chem. Soc., 95, 2151 (1909)]; show that the solid cupric hydroxide is unstable in contact with metallic copper and tends to be reduced to cuprous hydroxide. CHAPTER IX ACIDS AND BASES Definition of Acids and Bases.* The old definitions of an acid as a substance which yields hydrogen ions, of a base as one giving hydroxyl ions, and of neutralization as the formation of a salt and water from an acid and a base, are reasonably satisfactory for aqueous solutions, but there are serious limitations when non-aqueous media, such as ethers, nitro-compounds, ketones, etc., are involved. As a result of various studies, particularly those on the catalytic influence of un-ionized mole- cules of acids and bases and of certain ions, a new concept of acids and bases, generally associated with the names of Brjzfnsted and of Lowry, has been developed in recent years. 1 According to this point of view an acid is defined as a substance with a tendency to lose a proton, while a base is any substance with a tendency to gain a proton ; the relationship between an acid and a base may then be written in the form A ^ H+ + B. (1) acid proton base The acid and base which differ by a proton according to this relationship are said to be conjugate to one another; every acid must, in fact, have its conjugate base, and every base its conjugate acid. It is unlikely that free protons exist to any extent in solution, and so the acidic or basic properties of any species cannot become manifest unless the solvent molecules are themselves able to act as proton acceptors or donors, respectively : that is to say, the medium must itself have basic or acidic properties. The interaction between an acid or base and the solvent, and in fact almost all types of acid-base reactions, may be represented as an equilibrium between two acid-base systems, viz., A! + B 2 ^ B! + A 2 , (2) acidi bascz basei acid 2 where Ai and BI are the conjugate acid and base of one system, and * G. N. Lewis [/. Franklin Inst., 226, 293 (1938); see also, /. Am. Chem. Soc., 61, 1886, 1894 (1939); 62, 2122 (1940)] proposes to define a base as a substance capable of furnishing a pair of electrons to a bond, i.e., an electron donor, whereas an acid is able to accept a pair of electrons, i.e., an electron acceptor. The somewhat restricted defini- tions employed in this book are, however, more convenient from the electrochemical standpoint. 1 Lowry, Chem. and Ind., 42, 43 (1923); Br0nsted, Rec. trav. chim., 42, 718 (1923); J. Phys. Chem., 30, 377 (1926); for reviews, see Br0nsted, Chem. Revs., 3, 231 (1928); Hall, ibid., 8, 191 (1931); Bjerrum, ibid., 16, 287 (1935); Bell, Ann. Rep. Chem. Soc., 31, 71 (1934). 306 ACIDS 307 A 2 and B 2 are those of the other system, e.g., the solvent. Actually A! possesses a proton in excess of BI, while A 2 has a proton more than B 2 ; the reaction, therefore, involves the transfer of a proton from AI to B 2 in one direction, or from A 2 to BI in the other direction. Types of Solvent. In order that a particular solvent may permit a substance dissolved in it to behave as an acid, the solvent itself must be a base, or proton acceptor. A solvent of this kind is said to be proto- philic in character; instances of protophilic solvents are water and alco- hols, acetone, ether, liquid ammonia, amines and, to some extent, formic and acetic acids. On the other hand, solvents which permit the mani- festation of basic properties by a dissolved substance must be proton donors, or acidic; such solvents are protogenic in nature. Water and alcohols arc examples of such solvents, but the most marked protogenic solvents are those of a strongly acidic character, e.g., pure acetic, formic and sulfuric acids, and liquid hydrogen chloride and fluoride. Certain solvents, water arid alcohols, in particular, are amphiprotic, for they can act both as proton donors and acceptors; these solvents permit sub- stances to show both acidic and basic properties, whereas a purely proto- philic solvent, e.g., ether, or a completely protogenic one, e.g., hydrogen fluoride, would permit the manifestation of either acidic or basic functions only. In addition to the types of solvent already considered, there is another class which can neither supply nor take up protons: these are called aprotic solvents, and their neutral character makes them especially useful when it is desired to study the interaction of an acidic and a basic substance without interference by the solvent. Acids. Since an acid must possess a labile proton it can be repre- sented by HA, and if S is a protophilic, i.e., basic, solvent, the equilibrium existing in the solution, which is of the type represented by equation (2),' may be written as HA + S ^ HS+ + A-, (3) acidi bases aeid 2 basei where HS+ is the form of the hydrogen ion in the particular solvent and A~ is the conjugate base of the acid HA. There arc a number of impor- tant consequences of this representation which must be considered. In the first place, it is seen that the anion A~~ of every acid HA must be regarded as the conjugate base of the latter. If the acid is a strong one, it will tend to give up its proton very readily; this is, in fact, what is meant by a "strong acid." For such an acid, e.g., hydrochloric v acid, the equilibrium between acid and solvent, represented by equation (3), lies considerably to the right; that is to say, the reverse process occurs to a small extent only. This means that the anion of a strong acid, e.g., the chloride ion, will not have a great affinity for a proton, and hence it must be regarded as a "weak base." On the other hand, if HA is a very weak acid, e.g., phenol, the equilibrium of equation (3) lies well 308 ACIDS AND BASES to the left, so that the process A- + HS+ ^ HA 4- S will take place to an appreciable extent; the anion A~, e.g., the phenoxide ion, will be a moderately strong base. Another consequence of the interaction between the acid and the solvent is that the hydrogen ion in solution is not to be regarded as a bare proton, but as a combination of a proton with, at least, one molecule of solvent; the hydrogen ion thus depends on the nature of the solvent. In water, for example, there are good reasons for believing that the hydrogen ion is actually H 3 0+, sometimes called the "oxonium" or "hydronium" ion: the free energy of hydration of the proton is so high, approximately 250 kcal. (see p. 249), that the concentration of free pro- tons in water must be quite negligible, and hence almost all the protons must have united with water molecules to form H 3 O+ ions. Further hydration of the H 3 O+ ions probably occurs in aqueous solution, but this is immaterial for present purposes. Striking evidence of the part played by the water in connection with the manifestation of acidic properties is provided by observations on the properties of hydrogen bromide solutions in liquid sulfur dioxide. 2 The latter is only feebly basic and, although it dissolves hydrogen bromide, the solution is a poor conductor; there is consequently little or no ioniza- tion under these conditions. The solution of hydrogen bromide in sulfur dioxide is able, however, to dissolve a mole of water for every mole of hydrogen bromide present, and the resulting solution is an excellent con- ductor. Since water is sparingly soluble in sulfur dioxide alone, it is clear that the reaction HBr + H 2 O = H 3 0+ + Br~ must take place between the hydrogen bromide and water. Confirma- tion of this view is to be found in the observation that on electrolysis of the solution one mole of water is liberated at the cathode for each faraday passing; the discharge of the H 3 O + ion clearly results in the formation of an atom, or half a molecule, of hydrogen and a molecule of water. It is of interest to note in connection with the question of the nature of the hydrogen ion in solution that the crystalline hydrate of perchloric acid, HC104-H 2 O, has been shown by X-ray diffraction methods to have the same fundamental structure as ammonium perchlorate. Since the latter consists of interpenetrating lattices of NHj and C1OJ" ions, it is probable that the former is built up of H 3 O+ and ClOr ions. A third conclusion to be drawn from the equilibrium represented by equation (3) is that since the solvent S is to be regarded as a base, the corresponding hydrogen ion SH+ is an acid. The hydronium ion H 3 O+ is thus an acid, and in fact the acidity of the strong acids, e.g., perchloric, 2 Bagster and Cooling, J. Chem. Soc., 117, 693 (1920). ACIDS 309 hydrobromic, sulfuric, hydrochloric and nitric acids, in water is due almost exclusively to the H 3 O+ ion. It is because the process HA + H 2 acidi bases H,0+ acids basei where HA is a strong acid, goes almost completely to the right, that the aforementioned acids appear to be equally strong in aqueous solution, provided the latter is not too concen- trated. In solutions more concen- trated than about 2 N, however, these acids do show differences in cata- lytic behavior for the inversion of sucrose; the results indicate that the strengths decrease in the order given . (Fig. 86). | In order that it may be possible to distinguish in strength between , the so-called strong acids, it is evi- 2 dently necessary to employ a solvent which is less strongly pro tophilic than water; the equilibrium of equation (3) will then not lie completely to the right, but its position will be deter- mined by the relative proton-donat- ing tendencies, i.e., strengths, of the various acids. A useful solvent for this purpose is pure acetic acid ; this FIG. 86. is primarily a protogenic (acidic) solvent, but it has slight basic properties, so that the reaction 2 3 Normality of Acio Catalytic activity of strong acids HA + CH 3 C0 2 H ^ A~ occurs to some extent, although the equilibrium cannot lie far to the right. Even acids, such as perchloric and hydrochloric, which are re- garded as strong acids, will interact to a small extent only with the solvent, and the number of ions in solution will be relatively small; the extent of ionization will, therefore, depend on the strength of the acid in a manner not observed in aqueous solution. The curves in Fig. 87 show the variation of the conductance of a number of acids in pure acetic acid at 25; the very low equivalent conductances recorded arc due to the very small degrees of ionization. It is seen, therefore, that acids which appear to be equally strong in aqueous solution behave as weak acids when dissolved in acetic acid; moreover, it is possible to distinguish between their relative strengths, the order being as follows: HC10 4 > HBr > H 2 S0 4 > HC1 > HNO 8 . 310 ACIDS AND BASES This order agrees with that found by catalytic methods and also by potentiometric titration. 8 In spite of the small extent of ionization of acids in a strongly proto- genic medium such as acetic acid, the activity of the resulting hydrogen ions is very high; this may be at- tributed to the strong tendency of the CHaCC^Hj ion to lose a proton, so that the ion will behave as an acid of exceptional strength. The intense acidity of these solu- tions, as shown by hydrogen elec- trode measurements, by their cat- alytic activity, and in other ways, has led to them being called super- acid solutions. 4 The property of superacidity can, of course, be observed only with solvents which are strongly protogenic, but which still possess some protophilic na- ture. Hydrogen fluoride, for ex- ample, has no protophilic proper- ties, and so it cannot be used to exhibit superacidity; in fact no known substance exhibits acidic 2.0 1.6 1.0 0.5 0.02 0.04 0.06 0.08 FIG. 87. Conductance of acids in glacial acetic acid (Kolthoff and Willman) behavior in this solvent, as ex- plained below. It is an obvious corollary, from the discussiom given here concern- ing the influence of the solvent, that in a highly basic, i.e., protophiiic, medium, even acids that are normally regarded as weak would be highly ionized. It is probable that in liquid ammonia interaction with a weak acid, such as acetic acid, would occur to such an extent that it would appear to be as strong as hydrochloric acid. Bases. The equilibrium between an acidic, i.e., protogenic, solvent and a base may be represented by another form of thegeneral equation (2), viz., B + SH base acid BH+ + S-, acid base (4) where the solvent is designated by SH to indicate its acidic property. It is seen from this equilibrium that the cation BH+ corresponding to the base B is to be regarded as an acid; for example, if the base is NH 8 , > HalI and Conant, J. Am. Chem. Soc., 49, 3047, 3062 (1927); Hall and Werner, ibid., 50, 2367 (1928); Hantzsch and Langbein, Z. anorg. Chem., 204, 193 (1932); Kolthoff and Willman, J. Am. Chem. Soc., 56, 1007 (1934); Weidner, Hutchison and Chandlee, ibid., 56, 1285 (1934). * Hall and Conant, J. Am. Chem. Soc., 40, 3047, 3062 (1927); Hall and Werner, ibid, 50, 2367 (1928); Conant and Werner, ibid., 52, 436 (1930). BABES 311 the corresponding cation is NHt, and so the ammonium ion and, in fact, all mono-, di-, and tri-substituted ammonium ions are to be regarded as the conjugate acids of the corresponding amine (anhydro-) bases. It can be readily shown, by arguments analogous to those used in connec- tion with acids, that when the base is a strong one, e.g., hydroxyl ions, its conjugate acid, i.e., water, will be a weak acid; similarly, the conjugate acid to a very weak base will be moderately strong. The strength of a base like that of an acid must depend on the nature of the solvent : in a strongly protogenic medium, such as acetic acid or other acid, the ionization process B + CH 3 CO 2 H = BH+ + CHaCOl base acid acid base will take place to a very considerable extent even with bases which are weak in aqueous solution. Just as it is impossible to distinguish between the strengths of weak acids in liquid ammonia, weak bases are indis- tinguishable in strength when dissolved in acetic acid; it has been found experimentally, by measurement of dissociation constants, that all bases stronger than aniline, which is a very weak base in water, are equally strong in acetic acid solution. 6 To arrange a series of weak bases in the order of their strengths, it would be necessary to use a protophilic solvent, such as liquid ammonia: water is obviously better than acetic acid for this purpose, but it is not possible to distinguish between the strong bases in the former medium, since they all produce OH~ ions almost completely. Substances which are normally weak bases in water exhibit con- siderable basicity in strongly acid media; the results in Table LVI, for TABLE LVI. EQUIVALENT CONDUCTANCES IN HYDROGEN FLUORIDE SOLUTIONS AT 15 IN OHMS" 1 CM.* Concentration Methyl alcohol Acetone Glucose 0.026 N 243 244 279 0.115 200 190 208 0.24 164 181 165 0.50 139 176 114 example, show that methyl alcohol, acetone and glucose, which are non- conductors in aqueous solution, are excellent conductors when dissolved in hydrogen fluoride. 6 These, and other oxygen compounds, behave as bases and ionize in the following manner: \) + HF = \DH + F-. base acid acid base A number of substances which are acids in aqueous solution function as Hall, /. Am. Chem. Soc., 52, 5115 (1930); Chem. Revs., 8, 191 (1931). Fredenhagen et a/., Z. phyrik. Chem., 146A, 245 (1930); 164A, 176 (1933); Simons, Chem. Revs., 8, 213 (1931). 312 ACIDS AND BASES bases in hydrogen fluoride, e.g., CHsC0 2 H + HF = CH 8 C0 2 Itf + F-. acid acid base This reaction occurs because the acid possesses some protophilic proper- ties, and these become manifest in the presence of the very strongly protogenic solvent. As may be expected, the stronger the acid is in water, the weaker does it behave as a base in hydrogen fluoride. Dissociation Constants of Acids and Bases. If the law of mass action is applied to the equilibrium between an acid HA and the basic solvent S, i.e., to the equilibrium HA + S ^ HS+ + A-, the result is (5) If the concentration of dissolved substances in the solvent is not large, the activity of the latter, i.e., a s , may be regarded as unity, as for the pure solvent; equation (5) then becomes (6) The substance HS+ is the effective hydrogen ion in the solvent 8, so that OHS* is equivalent to the quantity conventionally written in previous chapters as OH+, it being understood that the symbol H+ does not refer to a proton but to the appropriate hydrogen ion in the given solvent; it follows, therefore, that equation (6) may be written in the form ' (7) which is identical with that obtained by regarding the acid as HA ionizing into H+ and A~, in accordance with the general treatment on page 163. The constant as defined by equation (6) or (7) is thus identical with the familiar dissociation constant of the acid HA in the given solvent as obtained by the methods described in Chap. V; further reference to the determination of dissociation constants is made below. Application of the law of mass action to the general base-solvent equilibrium B + SH ^ BH+ + S-, gives aBH+Qs- K = -^~> (8) the activity of the Solvent SH being regarded as constant; the quantity Kb is the dissociation constant of the base. If the base is an amine, DETERMINATION OF DISSOCIATION CONSTANTS 313 , in aqueous solution, then the equilibrium RNH 2 + H 2 O ^ RNHjj + Oil- is established, and the dissociation constant is given by K b = The result is therefore the same as would be obtained by means of the general treatment given in Chap. V for an electrolyte MA, if the un- dissociated base were regarded as having the formula RNH 3 OH in aqueous solution. The dissociation constants of acids, and bases, are of importance as giving a measure of the relative strengths of the acids, and bases, in the given medium. The strength of an acid is measured by its tendency to give up a proton, and hence the position of the equilibrium with a given solvent, as determined by the dissociation constant, is an indication of the strength of the acid. Similarly, the strength of a base, which depends on its ability to take up a proton, is also measured by its dissociation constant, since this is the equilibrium constant for the reaction in which the solvent molecule transfers a proton to the base. Determination of Dissociation Constants: The Conductance Method. As seen in Chap. V, equation (7) may be written in the form _ A a and if a is the true degree of dissociation of the solution of acid whose stoichiometric concentration is c, then a' 2 c /n*/ A - A = - -- -- (9) 1 a JHA Accurate methods for evaluating K a based on this equation, involving the use of conductance measurements, have been already described in Chap. V; these require a lengthy experimental procedure, but if carried out carefully the results are of high precision. For solvents of high dielectric constant the calculation based on the Onsager equation may be employed (p. 165), but for low dielectric constant media the method of Fuoss and Kraus (p. 167) should be used. Many of the dissociation constants in the older literature have been determined by the procedure originally employed by Ostwald (1888), which is now known to be approximate in nature; if the activity coeffi- cient factor in equation (9) is neglected, and the degree of dissociation a is set equal to the conductance ratio (A/A ), the result is A 2 c *~ Ao(A.-A) 314 ACIDS AND BASES An approximate dissociation function k was thus calculated from the measured equivalent conductance of the solution of weak acid, or weak base, at the concentration c, and the known value at infinite dilution. For moderately weak acids, of dissociation constant of 10~ 6 or less, the degree of dissociation is not greatly different from the conductance ratio, provided the solutions are relatively dilute; under these conditions, too, the activity coefficient factor will be approximately unity. If the acid solutions are sufficiently dilute, therefore, the dissociation constants given by equation (10) are not seriously in error. For example, if the data in Table XXXVIII on page 165 for acetic acid solutions are treated by the Ostwald method they give k a values varying from about 1.74 X 10~ 5 in the most dilute solutions to 1.82 X 10~ 5 in the more concentrated. The results in dilute solution do not differ appreciably from those obtained by the more complicated but more accurate method of treating the data. It may be mentioned, however, that the earlier determinations of dis- sociation constants were generally based on conductance measurements with solutions which were rarely more dilute than 0.001 N, whereas those in Table XXXVIII refer to much less concentrated solutions. For acids whose dissociation constants arc greater than about 10~ 5 the Ostwald method would give reasonably accurate results for the dissociation con- stant only at dilutions which are probably too great to yield reliable conductance measurements. Electromotive Force Method. An alternative procedure for the evaluation of dissociation constants, which also leads to very accurate results, involves the study of cells without liquid junction. 7 The chemi- cal reaction occurring in the cell H 2 (l aim.) | HA(wii) NaAK) NaCl(m 3 ) AgCl(s) | Ag, where HA is an acid, whose molality is m\ in the solution, and NaA is its sodium salt, of molality w 2 , is |H,(1 atm.) + AgCl(s) = Ag + H+ + Cl- for the passage of one faraday. The E.M.F. of the coll is therefore given by (cf. p. 226) RT E = E'--prlna H *acr, (11) \\here E is the standard E.M.F. of the hydrogen-silver chloride cell, i.e., of the hypothetical cell H,(l atm.) | H+(a H + = 1) II Cl-(ocr = 1) AgCl() | Ag. The E.M.F. of this cell is clearly equal in magnitude but opposite in sign to the standard potential of the Ag, AgCl(s) Cl~ electrode, and hence E in equation (11) is + 0.2224 volt at 25. The subscripts H+ and Cl~ 7 Earned and Ehlere, J. Am. Chem. Soc. t 54, 1350 (1932); for reviews, see Harned, J. Franklin Inst., 225, 623 (1938); Harned and Owen, Chem. Revs., 25, 31 (1939). ELECTROMOTIVE FORCE METHOD 315 in equation (11), etc., refer to the hydrogen and chloride ions, respec- tively, it being understood that the former is really H 3 0+ in aqueous solution, and the corresponding oxonium ion in other solvents. The activities in equation (11) may be replaced by the product of the respec- tive molalities (ra) and the stoichiometric activity coefficients (7), so that RT RT E = E -- In WH+WCI- - -jr In yn+ycr. (12) The activities in equation (7) for the dissociation constant may be expressed in a similar manner, so that = > HA WHA THA and combination of this expression with equation (12) gives E = E - ^- ]n ^^- - ^inliffil _ ^intf, (13) - - F(E - E Q ) , , v -- + log log - ~ log K > or, at 25 (E E Q ) WHAfWci" 7HA7C1" ,r /"I -\ n Acni c~~ ^8 == *^S *O? ** v^W U.uoyio w\~ 7A~ The right-hand side of equation (14) may be set equal to log K', where K' becomes identical with K at infinite dilution, for then the activity coefficient factor 7TL\7cr/7A~ becomes unity and the term log 7HA7cr/7A~ in equation (15) is zero. Since E Q is known, and the E.M.F. of the cell (E) can be measured with various concentrations of acid, sodium salt and sodium chloride, i.e., for various values of Wi, mz and w 3 in the cell depicted above, it is possible to evaluate the left-hand side of equation (14) or (15). In dilute solution, the sodium chloride may be assumed to be completely dis- sociated so that the molality of the chloride ion can be taken as equal to that of the sodium chloride, i.e., mcr is equal to ra 3 . The acid HA will be partly in the undissociated form and partly dissociated into hydrogen and A~~ ions; the stoichiometric molality of HA is m\ 9 and if WH+ is the molality of the hydrogen ions resulting from dissociation, the molality of undissociated HA molecules, i.e., WHA in equation (15), is equal to wii WH+. Finally, it is required to known WA-: the A~ ions are pro- duced by the dissociation of NaA, which may be assumed to be complete, and also by the small dissociation of the acid HA; it follows, therefore, that m A - is equal to mz + IH*. Since m^*, the hydrogen ion concen- tration, is required for these calculations, a sufficiently accurate value is estimated from the approximate dissociation constant (cf. p. 390); this 316 ACIDS AND BASES procedure is satisfactory provided the dissociation constant of the acid is about IQr 4 or less, as is generally the case. If the values of the left-hand side of equation (14) or (15) are plotted against the ionic strength of the solution and extrapolated to infinite dilution, the intercept gives log K, from which the dissociation constant K can be readily obtained. The general practice is to keep the ratio of acid to salt, i.e., m\ to m*, constant, approximately unity, in a series of experiments, and to vary the ionic strength by using different concentrations of sodium chloride. The re- sults obtained for acetic acid are shown in Fig. 88; the value of log K a is seen to be - 4.756, so that K a is 1.754 X 10~ 5 at 25. When comparing the dissociation constant obtained by the con- ductance method with that derived from E.M.F. measurements, it must be remembered that the former is based on volume concentrations, i.e., g.-ions or moles per liter, while the latter involves molalities. This difference arises because it is more convenient to treat conductance data in terms of volume concentrations, whereas the standard states for E.M.F. studies are preferably chosen in terms of molalities. If K c and K m are the dissociation constants based on volume concentrations and molalities, respectively, then it can be readily seen that K c is equal to K m p, where p is the density of the solvent at the experimen- tal temperature. For water at 25, p is 0.9971, and hence K c for acetic acid, calculated from E.M.F. measurements, is 1.749 X 10~ 5 , compared with 1.753 X 10~ 5 from conductance data. Considering the differ- ence in principle involved in the two methods, the agree- FIG. 88. Dissociation constant of acetic ment is ver y striking. Almost acid (Harned and Eklers) as good correspondence has been found for other acids with which accurate conductance and E.M.F. studies have been made; this may be regarded as providing strong support for the theoretical treat- ments involved, especially in the case of the conductance method. The procedure described here may be regarded as typical of that adopted for any moderately weak acid, i.e., of dissociation constant 10~ 8 to 10~~ 5 ; for weaker acids, however, some modification is necessary. In addition to the acid dissociation HA + H 2 ^ H 3 0+ + A- allowance must be made for the equilibrium A- + H 2 ^ OH- + HA, 4.762 4.760 ,4,768 4.756 0.04 o.oa 0.12 o.ie DISSOCIATION CONSTANTS OF BASES 317 which is due to the water, i.e., the solvent, functioning as an acid to some extent; this corresponds to the phenomenon of hydrolysis to be discussed in Chap. XI. It follows, therefore, that if the stoichiometric molality of HA is Wi, then TttHA = Wi WH* + WOH~, since HA is used up in the dissociation process while it is formed in the hydrolysis reaction, in amounts equivalent to the hydrogen and hydroxyl ions, respectively. Further, if the molality of the salt NaA is W2, then since A~ ions are formed in the dissociation process but are used up in the hydrolysis. If the dissociation constant is greater than 10~ 6 and the ratio of acid to salt, i.e., m\\m^ is approximately unity, WOH~ is found by calculation to be less than 10~ 9 , and so this term can be neglected in the expressions for WHA and rn A -, as was done above. If the dissociation constant lies between 10~ 5 and 10~ 9 , and mi/m^ is about unity, m H + WOH- is negligibly small, so that WHA and m\- may be taken as equal to mi and mz t respectively. For still weaker acids, m\i+ is so small that it may be ignored in comparison with WOH~; WUA is now equal to mi + W?OH", and m A - is m2 moH~. The values of m ir required for determining T/IHA and m^~ are obtained by utilizing the fact that mammon" is equal to 10- u at ?5. Dissociation Constants of Bases. The dissociation constants of bases can be determined, in principle, by methods which are essentially similar to those employed for acids. Replacing activities in equation (8) by the product of molalities and activity coefficients, it is seen that for a base WB 7B and this may be replaced by 1 a 7n (Lt) where a represents the degree of dissociation of the hypothetical solvated base, e.g., BH OH in water. By neglecting the activity coefficient factor in equation (17) and replacing a by the conductance ratio, an approxi- mate equation identical in form with (10) is obtained; the value of Ao in this equation is the sum of the equivalent conductances of the BH+ and OH~ ions, e.g., of NHi" and OH~ if the base is ammonia. Very little accurate E.M.F. work has been done on the dissociation constants of bases, chiefly because moderately w r eak bases are very vola- tile, while the non-volatile bases, e.g., anilines, are usually very weak. An exception to this generalization is to be found in the aliphatic amino- acids which will be considered in connection with the subject of ampho- 318 ACIDS AND BABES teric electrolytes. Since silver chloride is soluble in aqueous solutions of ammonia and of many amines, it is not possible to use silver-silver chloride electrodes with such bases; the employment of sodium amalgam has been proposed, but it is probable that the silver-silver iodide electrode will prove most useful for the purpose of the accurate determination of the dissociation constants of bases by the E.M.F. method. Apart from determinations of dissociation constants made from con- ductance data, most values derived from E.M.F. measurements have been obtained by an approximate procedure which will be described later. Dissociation Constants of Polybasic Acids: Conductance Method. A polybasic acid ionizes in stages, each stage having its own characteristic dissociation constant : for example, the ionization of a tribasic acid HsA, such as phosphoric acid, may be represented by: flij + {JfT A~ 1. H 3 A + H 2 ^ H 3 0+ + H 2 A~, K, = - (18a) a H 3 A ttn + flTIA~~ 2. H 2 A- + H 2 ^ H 3 0+ + HA, K 2 = (186) flHjA- 3. HA + H 2 ? H 3 0+ + A --- , K 3 = an * aA "" The fact that ionization occurs in these three stages successively with increasing dilution shows that KI > K 2 > KZ] this is always true, be- cause the presence of a negative charge on H 2 A~ and of two such charges on HA makes it increasingly difficult for a proton to be lost. If the dissociation constants for any two successive stages are suffi- ciently different it is sometimes feasible to apply the methods employed for monobasic acids; the conditions under which this is possible will be considered with reference to a dibasic acid, but the general conclusions can be extended to more complex cases. If H 2 A is a dibasic acid for which KI, the dissociation constant of the first stage,* H 2 A + H 2 O ^ H 3 O+ + HA-, is of the order of 10~ 3 to 10~ 5 , while the constant K% of the second stage of dissociation, HA- + H 2 ^ H 3 0+ + A, * The first stage dissociation constant of a dibasic acid is actually the sum of two constants; consider, for example, the unsymmetrical dibasic acid HX-X'H, where X and X' are different. This acid can dissociate in two ways, viz., HX-X'H + H 2 O ;= H 3 O+ + -X-X'H, and HX-X'H + H 2 ^ H 8 0+ + HX-X'- and if K{ and K" are the corresponding dissociation constants, the experimental first stage dissociation constant KI is actually equal to K( -f K('. If the acid is a sym- metrical one, e.g., of the type CO 2 H(CH) n COjH, the constants K{ and K" are identical, BO that KI is equal to 2K{. Similar considerations apply to all polybasic acids. DISSOCIATION CONSTANTS OF POLYBASIC ACIDS 319 is very small, i.e., the acid is moderately weak in the first stage and very weak in the second stage, then it may be treated virtually as a monobasic acid. The value of K\ may be determined in the usual manner from conductance measurements on the acid H 2 A and its salt NaHA at various concentrations, together with the known values for hydrochloric acid and sodium chloride (cf. p. 164). Provided the dissociation constant K 2 of the acid HA~ is very small, the extent of the second stage dissociation will be negligible in the solutions of both H2A and NallA. This method has been applied to the determination of the first dissociation constant of phosphoric acid; 8 for this acid KI is 7.5 X 10~ 3 at 25, whereas K 2 is 6.2 X 10- 8 . If the dissociation constant of the second stage is relatively large, e.g., about 10~ 5 or more, it is not possible to carry out the normal con- ductance procedure for evaluating K\; this is because the HA~ ion in the solution of the completely ionized salt NaHA dissociates to an appreciable extent to form H 3 0+ and A ions, and the measured conductance is much too large. As a result of this further dissociation, it is not possible to derive the equivalent conductances of NaHA required for the calcu- lation of the dissociation constant. An attempt has been made to over- come this difficulty by estimating the equivalent conductance of the ion HA~ in an indirect manner, so that the value for the salt NaHA may be calculated. By assuming that the intermediate ion of an organic dibasic acid, viz., OII-CORCOiT, has the same equivalent conductance at infinite dilution as the anion of the corresponding amic acid, viz., NH 2 -CORCO", which can be obtained by direct measurement, it has been concluded that the equivalent conductance XHA~ of the intermediate ion is equal to 0.53XA", where XA~~ is the conductance of the A ion, i.e., of ~~C0 2 RCO2~ in the case under consideration. Since the latter quantity can be deter- mined without great difficulty by conductance measurements with the salt Na^A, the value of Xn A - for the given acid at infinite dilution can be obtained. The known equivalent conductance of sodium is now added to that of the HA~, thus giving the value of A for the salt NaHA; the variation of the equivalent conductance with concentration can now be expressed by assuming the Onsagcr equation to be applicable. Since the conductance of the acid H 2 A at various concentrations is known, as well as that of HC1 and NaCl, all the information is available for calcu- lating the dissociation constant of H 2 A as a monobasic acid. This method cannot be regarded as accurate, however, for the identification of XHA- with 0.53XA" is known to be an approximation. 9 The determination of the second dissociation constant (/ 2 ) of a di- basic acid also requires a knowledge of the equivalent conductance of the intermediate ion HA~, and if the value of K z is large enough to be deter- mined from conductance measurements, the further dissociation of HA"" Sherrill and Noyes, J. Am. Chem. Soc., 48, 1861 (1926). Jeffery and Vogel, /. Chem. Soc., 21 (1935); 1756 (1936); Davies, ibid., 1850 (1939). 320 ACIDS AND BASES is too great for the equivalent conductance to be derived accurately from the experimental data for the salt NaHA. In the earlier attempts to evaluate K% the assumption was made of a constant ratio of XHA~ to XA~, as described above; this, however, leads to results that are too uncertain to have any serious worth. If transference data are available, it is possible in certain cases to determine the required value of XHA~ and hence to calculate the second dissociation constant of the acid. The method has been used to evaluate K^ for sulfuric acid : in its first stage of dissociation this is a very strong acid, but the second stage dissociation, although very considerable, is much smaller. 10 Dissociation Constants of Dibasic Acids by E.M.F. Measurement. If the ratio of the dissociation constants of a dibasic acid, or of any two successive stages of ionization of a polybasic acid, is greater than about 10 2 or 10 3 , it is possible to treat each stage as a separate acid and to determine its dissociation constant by means of cells without liquid junc- tion in the manner already described. In a mixture of the free dibasic acid H 2 A with its salt NaHA, the essential equilibria are 1. H 2 A + H 2 ^ H 3 0+ + HA-, and 2. HA- + H 2 ^ II 3 0+ + A, and from these, by subtraction, may be obtained the equilibrium 3. 2HA- ^ H 2 A + A. If K\ and KZ are the dissociation constants for the stages 1 and 2, it can be readily shown that the equilibrium constant for the process 3 is equal to KtlKi. If the stoichiometric molality of H 2 A is m\ in a given solution and that of the salt NaHA, assumed to be completely dissociated into HA"" ions, is w 2 , then Wii 2 A m\ WH+ + flix", (19) since H 2 A is removed to form hydrogen ions in process 1, while it is formed in process 3 in an amount equivalent to A ; further, WHA- = m* + mn+ - 2m A ", (20) since HA~ is formed in reaction 1 and removed in 3, in amounts equiva- lent to H 3 0+ and 2A respectively. It has been seen that the equilib- rium constant of process 3 is equal to K 2 IKi t and the smaller this ratio the less will be the tendency of the reaction to take place from left to right; if Kt/Ki is smaller than about 10~ 3 , i.e., Ki/K* > 10 8 , the extent of the reaction will be negligible, and then the W A terms in equations (19) and (20) can be ignored. The expressions for mn^ and WHA- then reduce to the same form as do the corresponding ones for WHA and m\- t 10 Sherrill and Noyes, J. Am. Chem. Soc., 48, 1861 (1926). DISSOCIATION CONSTANTS OF DIBASIC ACIDS BY E.M.F. MEASUREMENT 321 respectively, for a monobasic acid. If KI lies between 10~ B and and Wi/W2 is approximately unity, WH+ may be neglected, as explained on page .317; for weaker acids, however, the term m ir, arising on account of hydrolysis, must be included. It follows, therefore, that when K 2 /Ki is small, or Ki/K 2 is large, the value of KI can be readily determined by measurements on cells of the type H 2 (l atm.) | H 2 A(mO NaHA(wii) NaCl^) AgCl(s) | Ag, the equation for the E.M.F. being, by analogy with equation (14), F(E - E) mummer 7H,A7cr - + log ~ = - log ~~ ~ log The values of w H2 A and WHA~ are derived as explained above, and wcr is taken as equal to w 3 ; the method of extrapolation, which yields log KI, is the same as described for a monobasic acid. In order to investigate the second stage dissociation constant, the system studied consists of a mixture of highly ionized NaHA, which is equivalent to the acid IIA~, of molality m\, and its salt NasA, of molality m*. In this case, it follows from the three processes given above, that A~ = mi WH+ ~ and If Kz/Ki is small the WH 2 A terms may be neglected, just as the m^ terms were neglected in the previous case, since process 3 occurs to a small extent only; under these conditions the expressions for mnA~ and WA" are equivalent to those applicable to a monobasic acid. The deter- mination of K 2 can then be carried out by means of the cell II 2 (1 atm.) | NaHA(?wO Na 2 A(m 2 ) NaCl(m 3 ) AgCl(s) | Ag, the E.M.F. of which is given by the expression , , , + log " 7nT~ = ~ log ^^ ~ log Az ' (22) The values of mn\-, ?n\ and mcr are determined in the usual manner, but since the activity coefficient factor THA'TCI ly\~~ involves two uni- valent ions in the numerator with a bivalent ion in the denominator, it will differ more from unity than does the corresponding factor in equa- tions (14) and (21); the usual extrapolation procedure is consequently liable to be less accurate. Utilizing the form log 7. = - Az]^v + Cy of the extended Debye-Huckel equation, however, it is seen that equation 322 ACIDS AND BASES (22) may be written as r (A A ) 2.303BT + log (23) a 7.30 7.26 7.20 0.06 0.10 0.16 The plot of the left-hand side of this expression, where A is 0.509 at 25, against the ionic strength y should thus be a straight line, at least ap- proximately; the intercept for zero ionic strength gives the value of log / 2 . The results obtained in the determination of the second dis- sociation constant of phosphoric acid are shown in Fig. 89; the upper curve is for cells containing the salts KH 2 P0 4 and Na 2 HP0 4 , and the lower for the two corresponding sodium salts in a different proportion. In this case the acid is H 2 POr, and its dissociation constant is seen to be antilog 7.206, i.e., K 2 is 6.223 X 10" at 25. u If the ratio Ki/K^ for two successive stages is smaller than 10 3 , it would be neces- sary to include the m\ and fftH,A terms, which were ne- glected previously, in the de- termination of Ki and K 2l respectively. The evaluation of these quantities, as well FIG. 89. Second dissociation constant of as of m H + or WOET, would re- phosphoric acid (Nims) quire preliminary values of Ki and K 2 , and the calcula- tions, although feasible, would be tedious. No complete determina- tion by means of cells without liquid junction appears yet to have been made of the dissociation constants of a dibasic acid for which Ki/Ki is less than 10 3 . Dissociation Constants by Approximate E.M.F. Methods. When, for various reasons, it is not convenient or desirable to carry out the lengthy series of measurements required for the determination of accurate dissociation constants by the conductance method or by means of cells without liquid junction, approximate E.M.F. methods, utilizing cells with liquid junctions, can be applied. These methods involve the determina- tion of the hydrogen ion concentration, or activity, in solutions con- taining a series of mixtures of the acid and its salt with a strong base, generally obtained by adding definite quantities of the latter to a known amount of acid. The procedures used for actual measurement of hydro- gen ion activities are described in Chap. X, but the theoretical basis of the evaluation of dissociation constants will be considered here. /. Am. Chan. Soc., 55, 1946 (1033); for application to malonic acid, see Hamer, Burton and Acree, /. Res. Nat. Bur. Standards, 24, 269 (1940). DISSOCIATION CONSTANTS BY APPROXIMATE E.M.F. METHODS 323 If a is the initial concentration (molality) of the weak or moderately weak acid HA, and 6 is the amount of strong, monoacid base MOH added at any instant, then 6 is also equal to W M +, the molality of M+ ions at that instant, since the salt MA produced on neutralization may be taken as being completely dissociated. The acid HA is only partially neutralized to form A~ ions, and so a = WHA + W A -. (24) Further, as the solution must be electrically neutral, the sum of all the positive charges will be equal to the sum of the negative charges; hence WM+ + WH+ = WA~ + WOH~, or b + win* = w A - + WOH-. (25) The dissociation constant K a of the acid HA may be expressed in the form A a 0H W A - TA- = an+ - - - > WHA THA and if W A - and WHA are eliminated by means of equations (24) and (25), it is found that a - 6 - W H * + WQH- THA f . = A a r ; - -- (26) -f- WH* ~ Won" 7A~ If the quantity B is defined by B =s b + W H * ~ WOH-, then equation (26) may be written as ir a ~ ^ THA or, taking logarithms, i i v i i a "~ ^ i i THA log a H + = log K a + log B + log 7 It was seen on page 292 that the pH, or hydrogen ion exponent, of a solution may be defined as log a H +; in an analogous manner the symbol pX, called the dissociation exponent, may be substituted for log K a ; hence pH = pK a + log ^-g + log^- (27) 324 ACIDS AND BASES According to the extended Debye-Hiickel theory, it is possible to write log = - A^ + C, (28) i HA remembering that A~ is a univalent ion and HA an undissociated mole- cule, and so equation (27) becomes pH = pK a + log ^-^ - A Vtf + C V , (29) D _ .'. pH - log -^^ + A V v = p# + C. If the left-hand side of this equation for a series of acid-base mixtures is plotted against the ionic strength of the solution, the intercept for y equal to zero would give the value of pK a , i.e., log K a . The methods used for the determination of the pH of the solution will be described in the following chapter, but in the meantime the evalua- tion of B and y will be considered. If the hydrogen ion concentration of the solution is greater than 10~~ 4 g.-ion per liter, i.e., for an acid of medium strength, the hydroxyl ion concentration woir will be less than 10~ 10 and so can be neglected in comparison with W H +; B then becomes equal to b + WH+. On the other hand, for a very weak acid, when the hydrogen ion concentration is less than 10~ 10 g.-ion per liter, the quantity WH+ may be ignored, so that B is equal to 6 + m ir. For solutions of inter- mediate hydrogen ion concentration, i.e., between 10~ 4 and 10~ 10 g.-ion per liter, WH+ moH~ is negligibly small and so B may be taken as equal to b. The values of a and b are known from the amounts of acid and base, respectively, employed to make up the given mixture, and WH+ and raoir are readily determined by the aid of the relationships WH+ = an + /yn+ and WH+WOH- = k lo , which is 10~ 14 at 25 (cf. p. 339). The quantity a tt + is derived from the measured pH, and 711+ is calculated with sufficient accuracy by means of the simple Debye-Hlickel equation. The ionic strength ft of the solution is given by 6 + WH + Woir; except at the beginning of the neutralization, however, when b is small, the value of y may be taken as equal to 6. The data obtained for acetic acid at 25 are plotted in Fig. 90; 12 the results are seen to fall approximately on a straight line, and from the intercept at zero ionic strength pK a is seen to be 4.72. The difference between this value and that given previously is to be attributed to an incorrect standardization of the pH scale (cf. footnote, p. 349). Instead of employing the graphical method described above, the general practice is to make use of equation (27) ; the quantities pH and B are obtained for each solution and the corresponding pk a evaluated. The activity correction may be applied by means of equation (28) since 12 Walpole, J. Chem. Soc., 105, 2501 (1914). DIBASIC ACIDS 325 A is known, and C can be guessed approximately or neglected as being small; alternatively, the tentative pk a values obtained by neglecting the activity coefficients may be plotted against a function of the ionic strength and extrapolated to infinite dilution. 4.86 0.04 0.08 0.12 Fia. 90. Dissociation constant of acetic acid If B is equal to ^a, and the solution is relatively dilute, so that the terms involving the ionic strength are small, equation (29) reduces to pH = pfc a . Provided the pH of the system lies between 4 and 10, the quantity B is virtually equal to 6, and hence it follows that when b is equal to \a the pll of the solution is (approximately) equal to the pk a of the acid. In other words, the pH of a half-neutralized solution of an acid, i.e., of a solution containing equivalent amounts of the acid and its salt, is equal to pfc a . This fact is frequently utilized for the approximate determina- tion of dissociation functions. Dibasic Acids. The treatment given above is applicable to any stage of ionization of a polybasic acid, provided its dissociation constant differs by a factor of at least 10 3 from those of the stages immediately preceding and following it: the activity correction, equivalent to equation (28), will however depend on the charges carried by the undissociatod acid and the corresponding anion. If these are r 1 and r, respectively, then according to the extended Dobye-Hiickel equation - (r - log -^- = - so that equation (29) for the rth dissociation constant of a polybasic 326 ACIDS AND BASES acid becomes pH = pK r + log ^ - A(2r - 1) VJ + C V . (30) When the dissociation constants of successive stages are relatively close together, a more complicated treatment becomes necessary. 13 The dissociation constants of the first and second stages of a dibasic acid H 2 A may be written in a form analogous to that given above, viz., , v m A TA" /ot . and K 2 = a H - - ---- (31) ~ 7HA" If to a solution containing the acid H 2 A at molality a there are added 6 equivalents of a strong monoacid base, MOH, the solution will contain H+, M+, HA~, A and OH~ ions; for electrical neutrality therefore, WM* + mH+ = mHA~ + 2mA h #k)H~> the term 2mA" arising because the A ions carry two negative charges. Replacing the concentration of M+ ions, i.e., WM*, by 6, as in the previous case, this equation becomes b + mn+ = WHA" + 2mA h moir. (32) Further, the initial amount of the acid a will be equivalent to the total quantity of un-neutralized H 2 A and of HA~ and A ions present at any instant; that is a = m Hj A + m H A- + m A -. (33) If a quantity B is defined, as before, by it can be shown that equations (31), (32) and (33) lead to the result H+ o^ ' I = aH * o^ S ' I KI + KiK 2 . (34) 2a n 7H 2 A *a li 7iu~ It follows, therefore, that if the left-hand side of this expression (X) is plotted against the coefficient of KI in the first term on the right-hand side (F), a straight line of slope K\ and intercept K\K* should result. The evaluation of B involves the same principles as described in connec- tion with monobasic acids. In the first stage of neutralization, i.e., when a > 6, the ionic strength may be taken as b + mn+, as before, but in the " Auerbach and Smolczyk, Z. physik. Chem., 110, 83 (1924); Britton, J. Chem. Soc., 125, 423 (1924); 127, 1896 (1925); Morton, Trans. Faraday Soc., 24, 14 (1928); Parting- ton et al, t'Wd., 30, 598 (1934); 31, 922 (1935); Gane and Ingold, J. Chem. Soc., 2151 (1931); German, Jeffery and Vogel, ibid., 1624 (1935); German and Vogel, J. Am. Chem Soc., 58, 1546 (1936); Jones and Soper, J. Chem. Soc., 133 (1936); see also, Simms, J. Am. Chem. Soc., 48, 1239 (1926); Muralt, ibid., 52, 3518 (1930). DIBASIC ACIDS 327 second stage, i.e., when b > a, a sufficient approximation is 26 a. Provided the solutions are reasonably dilute the limiting law of Debye and Hiickel may be used to derive y\ and the ratio TA" /THA-, the activity coefficient of the undissociated molecules 7 H ,A being taken as 6.0 4.0 s S ,2.0 1.0 -0.5 LO 1.5 0.5 r x io 6 Fia. 91. Dissociation constants of adipic acid (Speakman) unity. The experimental results obtained in this manner for adipic acid are shown in Fig. 91 ; 14 the plot is seen to approximate very closely to a straight line, the values of K\ and K\K^ being 3.80 X 10~ 6 and 1.43 X 10~ 10 respectively, so that K 2 is 3.76 X 1Q- 6 . An alternative treatment of equation (34) is to write it in the form X = where X and Y are defined by *Y 2 and Vj" _ (35) B TA 2a - B a- B 2a- B The solutions of equation (35) are X #1 = ^ and X - If two points during the neutralization are chosen, such that the quan- tities X and Y have the values X' and Y' and X" and Y", respectively, then it is readily found that X' - X" and *M Y f v ri 14 Speakman, J. Chem. Soc. t 855 (1940). X'Y" - X"Y' X" - X' 328 ACIDS AND BASES Since the X's and Y's can be evaluated, as already described, the two dissociation constants of a dibasic acid can be determined from pairs of pH measurements. The methods just described can be extended so as to be applicable to acids of higher basicity, irrespective of the ratio of successive dissocia- tion constants. Colorimetric Determination of Dissociation Constants. The colori- metric method for determining or comparing dissociation constants has been chiefly applied in connection with non-aqueous solvents, but it has also been used to study certain acids in aqueous solution. It can be employed, in general, whenever the ionized and non-ionized forms of an acid, or base, have different absorption spectra in the visible, i.e., they have different visible colors, or in the near ultra-violet regions of the spectrum. If the acid is a moderately strong one, e.g., picric acid, it will dissociate to a considerable extent when dissolved in water, and the amounts of un-ionized form HA and of ions A~ will be of the same order; under these conditions an accurate determination of the dissociation constant is possible. By means of preliminary studies on solutions which have been made either definitely acid, so as to suppress the ionization entirely, or definitely alkaline, so that the salt only is present and ioniza- tion is complete, the "extinction coefficient" for light of a given wave length of the form HA or A~ can be determined. As a general rule the ions A~ have a more intense color and it is the extinction coefficient of this species which is actually measured. Once this quantity is known, the amount of A~ in any system, such as the solution of the acid in water, can be found, provided Beer's law is applicable.* In a solution of the pure acid of concentration a in pure water, C H + is equal to C A -, while the concentration of undissociated acid CHA is equal to a C H + or to a CA~; hence if C A - is determined colorimetrically, it is possible to evalu- ate directly the concentration dissociation function CH+CA-/ C HA. This function, as already seen, depends on the ionic strength of the medium, but extrapolation to infinite dilution should give the true dissociation constant. If the acid is too weak to yield an appreciable amount of A~ ions when dissolved in pure water, e.g., p-riitrophenol, it is necessary to employ a modified procedure which is probably less accurate. A definite quan- tity of the acid being studied is added to excess of a "buffer solution" (see Chap. XI) of known pH; the pll chosen should be close to the expected pK a of the acid, for under these conditions the resulting solution will contain approximately equal amounts of the undissociated acid HA and of A~ ions. The amount of either HA or A", whichever is the more convenient, is then determined by studying the absorption of light of * According to Beer's law, log /<>// = *cd, where 7 is the intensity of the incident light and / is that of the emergent light for a given wave length, for which the extinction coefficient is e, d is the thickness of the layer of solution, and c is its concentration. If 6 is known, the value of c can be estimated from the experimental value of /o//. APROTJC SOLVENTS 329 suitable wave length, the corresponding extinction coefficient having been obtained from separate experiments, as explained previously. If CA~ is determined in this manner, CHA is known, since it is equal to c C A ~, where c is the stoichiometric concentration of the acid. In this way it is possible to calculate the ratio C\-/CHA, and since a H + is known from the pH of the solution, the function a H + c A -/CHA can be evaluated. For many purposes this is sufficiently close to the dissociation constant to be em- ployed where great accuracy is not required. Alternatively, the values of the function in different solutions may be extrapolated to zero ionic strength. This method has been used to study acids which exhibit visible color changes in alkaline solutions, e.g., nitrophenols, 16 as well as for substances that are colorless in both acid and alkaline media but have definite absorption spectra in the ultra-violet region of the spectrum, e.g., benzoic and phcnylacetic acids. 18 Approximate Methods for Bases. The procedures described for de- termining the dissociation constants of acids can also be applied, in principle, to bases; analogous equations are applicable except that hy- droxyl ions replace hydrogen ions, and vice versa, in all the expressions. Since the value of the product of an+ and OOH~ is known to have a definite value at every temperature (cf. Table LXI), it is possible to derive OOIT from an* obtained experimentally. Dissociation Constant Data. The dissociation constants at 25 of a number of acids and bases obtained by the methods described above are recorded in Table LVII; the varying accuracy of the results is indicated, to some extent, by the number of significant figures quoted. 17 The pK a and pK b values are given in each case, since these are more frequently employed in calculations than are the dissociation constants themselves. Acids and bases having dissociation constants of about 10~ 5 , i.e., pK is in the vicinity of 5, are generally regarded as "weak," but if the values are in the region of 10~ 9 , i.e., pK is about 9, they are referred to as "very weak." If the dissociation constant is about 10~ 2 or 10~ 3 , the acid or base is said to be "moderately strong/! and at the other extreme, when the dissociation constant is 10~ 12 or less, the term "extremely weak" is em- ployed. Aprotic Solvents. The colorimetric method of studying dissociation constants has found a special application in aprotic solvents such as benzene; these solvents exhibit neither acidic nor basic properties, and so they do not have the levelling effects observed with acids in proto- w von Halban and Kortiim, Z. physik. Chem., 170A, 351 (1934); 173A, 449 (1935); Kilpatrick et al, J. Am. Chem. Soc., 59, 572 (1937); 62, 3047 (1940); J. Phys. Chem., 43, 259 (1939). "Flexser, Hammett and Dingwall, J. Am. Chem. Sec., 57, 2103 (1935); Martin and Butler, J. Chem. Soc. t 1366 (1939). 17 For further data, see Harned and Owen, Chem. Revs., 25, 31 (1939); Dippy, ibid., 25, 151 (1939); Gane arid Ingold, J. Chem. Soc., 2153 (1931); Jeffery and Vogel, ibid., 21 (1935); 1756 (1936); German, Jeffery and Vogel, ibid., 1624 (1935); 1604 (1937). 330 ACIDS AND BASES TABLE LVII. DISSOCIATION CONSTANT EXPONENTS OF ACIDS AND BASES AT 25 Monobasic Organic Acids Acid pA'a Acid P/C. Formic 3.751 Benzoic 4.20 Acetic 4.756 o-Chlorobenzoic 2.92 Propionic 4.874 w-Phlorobenzoic 3.82 n-Butyric 4.820 p-Chlorobenzoic 3.98 wo-Butyric 4.821 p-Bromobenzoic 3.97 n-Valeric 4.86 p-Hydroxybenzoic 4.52 Trimethylacetic 5.05 p-Nitrobenzoic 3.42 Diethylacetic 4.75 p-Toluic 4.37 Chloroacetic 2.870 Phenylacetic 4.31 Lactic 3.862 Cinnamic (cis) 3.88 Gycolic 3.831 Cinnamic (trans) 4.44 Acrylic 4.25 Phenol 9.92 Dibasic Organic Acids Acid pA'i pA'i Acid P Ki ptft Oxalic 1.30 4.2S6 Pimelic 4.51 5.42 Malonic 2.84 5.695 Suberic 4.53 5.40 Succinic 4.20 5.60 Maleic 2.00 6.27 Glutaric 4.35 5.42 Fu marie 3.03 4.48 Adipic 4.42 5.41 Phthalic 2.S9 5.42 Bases Base pA'& Base pA'6 Ammonia 4.76 Triethylamine 3.20 Methylamine 3.30 Aniline 9.39 Dimethylamme 3.13 Benzylamine 4.63 Tnmethylarnine 4.13 Diphonylamine 13.16 Ethylamine 3.25 Pyridine 8.80 Diethylamine 2.90 Piperidine 2.88 Inorganic Acids Sulfuric (2nd stage) 1.02 Hydrogen sulfide 7.2, 11.9 Phosphoric 2.124, 7.206, 1232 Hydrogen cyanide 9.14 Carbonic 6.35, 10.25 Boric 9.24 philic and with bases in protogenic media (cf. pp. 309, 311). It is thus possible to make a comparison of the strengths of acids and bases without any interfering influence of the solvent. Suppose a certain amount of an acid HA is dissolved in an aprotic solvent and a known quantity of a base B is added; although neither acid nor base can function alone, they can exercise their respective functions when present together, so that an APROTIC SOLVENTS 331 acid-base equilibrium of the familiar form HA + B ^ BH+ + A- acid base acid base is established. Application of the law of mass action to the equilibrium then gives n, (36) JHA/B If the essential dissociations of the acids HA and BH + , to yield protons, i.e., HA ^ 11+ 4- A- and BH+ ^ H+ + B, where H+ represents a proton, are considered, the fundamental dissocia- tion constants are ^ . v AHA = - and ABH+ = - (37)* + respectively; comparison of these quantities with the equilibrium con- stant of equation (36) shows that ABH+ and hence is equal to the ratio of the fundamental dissociation constants of the acids HA and BH+, the latter being the conjugate acid of the added base B. If the color of the base B differs from that of its conjugate acid BH+, it is possible by light absorption experiments to estimate the value of either CB or CBH+," since the stoichiometric composition of the solution is known, the concentrations of all the four species CHA, CA-, CB and CBH+ can be thus estimated, and value of A in equation (36), apart from the activity coefficient factor, can be calculated. In this way the approximate ratio of the dissociation constant of the acid HA to that of BH+ is obtained. The procedure is now repeated with an acid HA' using the same base B, and from the two values of A the ratio of the dissociation constants of HA and HA' can be found. This method can be carried through for a number of acids, new bases being used as the series is extended. 18 On account of the low dielectric constants of aprotic solvents, con- siderable proportions of ion-pairs and triple ions are present, but spectro- metric methods are unable to distinguish between these and single ions; the determinations of the amounts of free ions, which are required by the calculations, will thus be in error. The activity coefficient factor, neglected in the above treatment, will also be of appreciable magnitude, but this can be diminished if the base is a negatively charged ion B~; * In these expressions <Z H + stands for the activity of protons. " LaMer and Downes, J. Am. Chem. Soc., 53, 888 (1931); 55, 1840 (1933); Chem. Revs., 13, 47 (1933). 332 ACIDS AND BASES the activity factor will then be /HA/B-//A~/BH which involves a neutral molecule and a singly charged ion in both numerator and denominator, and hence will not differ greatly from unity. The Acidity Function. A property of highly acid solutions, which is of some interest in connection with catalysis, is the acidity function H<>: it is defined with reference to an added electrically neutral base B, and measures the tendency of the solution to transfer a proton to the base; 19 thus #0= -log a f^- (38) JBH* There are reasons for believing that the fraction /B//BH+ is practically constant for all bases of the same electrical type, and so the acidity func- tion may be regarded as being independent of the nature of the base B. Combination of equation (38) with the usual definition of K a , the con- ventional dissociation constant of the acid BH+, gives #o = p#o + log (39) CBH+ This equation provides a method for evaluating the acidity function of any acid solution; a small amount of a base B, for which P/BH+ is known, is added to the given solution and the ratio CB/CBH+ is estimated colori- metrically. The acidity functions of a number of mixtures of perchloric, sulfuric and formic acids with water have been determined in this manner. By reversing the procedure, equation (39) may be used, in conjunc- tion with the known acidity functions of strongly acid media, to deter- mine the dissociation constants of the conjugate acids BH+ of a series of extremely weak bases. The relative amounts of B and BH+ can be determined by suitable light-absorption measurements. The method has been applied to the study of a number of bases which are much too weak to exhibit basic properties in water. The results obtained in certain cases are given in Table LVIII; the figures in parentheses are the reference points for each solvent medium. 20 It is soon, therefore, that all the dis- sociation constants recorded are based on the pK a value of 2.80 for the acid conjugate to aminoazobcnzene, this being the normal result in aqueous solution. The results in Table LVIII, which are seen to be inde- pendent of the acidic medium usod as the solvent, thus refer to dis- sociation constants of the various conjugate acids BH" 1 " in aqueous solu- tions. The dissociation exponents pX& of the bases (B) themselves can be derived by subtracting the corresponding pK a values, for BH+, from pK w , i.e., from 14. It is evident that many of the bases included in 19 Hammett and Dcyrup, J. Am. Chem. Xoc., 54, 2721, 4239 (1932); Hammett and Paul, ibid., 56, 827 (1934); Hall et al., ibid., 62, 2487, 2493 (1940). 10 Hammett and Paul, J. Am. Chem. Soc., 56, 827 (1934); Hammett, Chem. Revs., 16, 67 (1935). EFFECT OF SOLVENT ON DISSOCIATION CONSTANTS 333 TABLE LVIII. DISSOCIATION CONSTANTS (pK a ) OP CONJUGATE ACIDS RnAA Solvent Medium DBSO HC1 - HaO HtSO< - H0 HC1O4 - HO HCOjH Aminoazobenzene (2.80) Benzeneazodiphenylamine 1.52 p-Nitroaniline 1.11 (1.11) (1.11) o-Nitroaniline -0.17 -0.13 -0.19 (- 0.17) p-Chloronitroaniline -0.91 -0.85 -0.91 -0.94 p-Nitrodiphenylamine -2.38 -2.51 2 : 4-Dichloro-6-nitroaniline -3.22 -3.18 -3.31 p-Nitroazobenzene - -3.35 -3.35 -3.29 2 : 4-Dinitroanilinc -4.38 -4.43 Benzalacctophenone -5.61 Anthraquinone -- -8.15 2:4: 6-Trinitroaniline -9.29 Table LVIII arc extremely weak; the dissociation constant of 2 : 4 : 6- trmitroaniline, for example, is as low as 5 X 10~ 24 . Effect of Solvent on Dissociation Constants. The dissociation equi- librium of an uncharged acid HA in the solvent S can be represented as HA + S;=SH+ + A-; the dissociation process consequently involves the formation of a positive and a negative ion from two uncharged molecules. Since the electro- static attraction between two oppositely charged particles decreases with increasing dielectric constant of the medium, it is to be expected that, other factors being more or less equal, an increase of the dielectric con- stant of the solvent will result in an increase in the dissociation constant of an electrically neutral acid. It has been found experimentally, in agreement with expectation, that the dissociation constant of an un- charged carboxylic acid decreases by a factor of about 10 5 or 10 6 on passing from water to ethyl alcohol as solvent. In the same way, the dissociation constant of an uncharged base is diminished by a factor of approximately 10 3 to 10 4 for the same change of solvent. If the acid is a positive ion, e.g., NH 4 f , or the base is a negative ion, e.g., CHsCOj", the process of dissociation does not involve the separation of charges, viz., NIIJ + S = SI1+ + NH 3 , or + HS = CHCOiH + S-. The effect of changing the dielectric constant of the medium would thus be expected to be small, and in fact the dissociation constants do not differ very greatly in water and in ethyl alcohol. The value of pK a for the ammonium ion acid, for example, is about 9.3 in water and 11.0 in 334 ACIDS AND BASES methyl alcohol. It should be noted that the foregoing arguments do not take into consideration the different tendencies of the solvent mole- cule to take up a proton; the conclusions arrived at are consequently more likely to be applicable to a series of similar solvents, e.g., hydroxylic substances. A quantitative approach to the problem of the influence of the medium on the dissociation constants of acids, which eliminates the proton accept- ing tendency of the solvent, involves a comparison of the dissociation constants of a series of acids with the value for a reference acid. Con- sider the acid HA in the solvent S; the dissociation constant is given by whereas that for the reference acid IIA in the same solvent is so that, since SH+ is the same in both cases, K._ a Ko c where K is the equilibrium constant of the reaction between the two acid-base systems, viz., HA + AQ ^ A- + HAo. The standard free energy change of this process is then given by - AG = RT In K = 2.30 RT log X, where log K, equal to log (K a /Ko), is equivalent to pK pK a . This free energy change may be regarded as consisting of a non- electrostatic term A(? n and an electrostatic term AGvi. equivalent to the gain in electrostatic free energy resulting from the charging up of the ion A~ and the discharge of A^" in the medium of dielectric constant D. According to the Born equation (see p. 249), the electrostatic free energy increase per mole accompanying the charging of a spherical univalent ion is given by and so in the case under consideration, for charge and discharge of the ions A~ and AJT, respectively, EFFECT OF SOLVENT ON DISSOCIATION CONSTANTS 335 where r A - and r A o arc the radii of the corresponding spherical ions. It follows, therefore, that - -~ R T If the effective radii of the two ions remain approximately constant in a series of solvents, it follows that where a is a constant. The plot of log K, that is, of against 1/D, i.e., the reciprocal of the dielectric constant of the solvent, should thus be a straight line; the intercept for 1/D equal to zero, i.e., for infinite dielectric constant, should give a measure of the dissociation constant of the acid HA free from electrostatic effects. Measurements of dissociation constants of carboxylic acids, e.g., of substituted acetic and benzoic acids, using either acetic or benzoic acid as the reference substance HA, made in water, methyl and ethyl alcohols and ethylene glycol, are in good agreement with expectation. 11 The plot of the values of log (K a /K<>) against 1/D is very close to a straight line for each acid, provided D is greater than about 25. The slope of the line, however, varies with the nature of the acid, so that an acid which is stronger than another in one solvent may be weaker in a second solvent. The comparison of the dissociation constants of a scries of acids in a given solvent may consequently be misleading, since a different order of strengths would be obtained in another solvent. It has been suggested, therefore, that when comparing the dissociation constants of acids the values employed should be those extrapolated to infinite dielectric con- stant; in this way the electrostatic effect, at least, of the solvent would be eliminated. Attempts to verify the linear relationship between log K and 1/D by means of a series of dioxane-water mixtures have brought to light considerable discrepancies. 22 The addition of dioxane to water results in a much greater decrease in the dissociation constant than would be expected from the change in the dielectric constant of the medium. Since the organic acids studied are more soluble in dioxane than in water, it is probable that molecules of the former solvent are preferentially oriented about the acid anion; the effective dielectric constant would then be less than in the bulk of the solution. It is thus possible to 11 Wynne-Jones, Proc. Roy. Soc. t 140A, 440 (1933); Kilpatrick et oJ., /. Am. Chem. Soc. t 59, 572 (1937); 62, 3051 (1940); /. Phys. Chem., 43, 259 (1939); 45, 454, 466, 472 (1941); Lynch and LaMer, J. Am. Chem. Soc., 60, 1252 (1938); see also, Hammett, ibid., 59, 96 (1937); J. Chem. Phys., 4, 618 (1986). Elliott and Kilpatrick, J. Phys. Chem., 45, 472 (1941); see also, Earned, ibid., 43, 275 (1939). 336 ACIDS AND BASES account for the unexpectedly low dissociation constants in the dioxane- water mixtures. Dissociation Constant and Temperature. The dissociation constants of uncharged acids do not vary greatly with temperature, as may be seen from the results recorded in Table LIX for a number of simple fatty TABLE LIX. INFLUENCE OF TEMPERATURE ON DISSOCIATION CONSTANT Acid 10 20 30 40 50 60 Formic acid 1.638 1.728 1.765 1.768 1.716 1.650 1.551 X ID' 4 Acetic acid 1.657 1.729 1.753 1.750 1.703 1.633 1.542 X 10-* Propionic acid 1.274 1.326 1.338 1.326 1.280 1.229 1.160 X 10-' n-Butyric acid 1.563 1.576 1.542 1.484 1.395 1.302 1.199 X 10' 5 acids. A closer examination of the figures, however, reveals the fact that in each case the dissociation constant at first increases and then decreases as the temperature is raised; this type of behavior has been found to be quite general, and Harned and Embree 23 showed that the temperature variation of dissociation constants could be represented by the general equation log K a = log A', - p(t - 0) 2 , where K a is the dissociation constant of the acid at the temperature t, Ke is the maximum value, attained at the temperature 6, and p is a constant. It is an interesting fact that for a number of acids p has the same value, viz., 5 X 10~ 6 ; this means that if log K a log KB for a num- ber of acids is plotted against the corresponding value of t 6, the results all fall on a single parabolic curve. The actual temperature at which the maximum value of the dissociation constant is attained de- pends on the nature of the acid; for acetic acid it it, 22.6, but higher and lower values have been found for other acids. For some acids, e.g., chloroacetic acid and the first stage of phosphoric acid, the maximum dissociation constant would be reached only at temperatures below the freezing point of water. An alternative relationship 24 n log K = A + - - 20 log T, where A and B are constants, has been proposed by Pitzer to represent the dependence of dissociation constant on the absolute temperature T. This equation has a semi-theoretical basis, involving the empirical facts that the entropy change and the change in heat capacity accompanying the dissociation of a monobasic acid are approximately constant. Some attempts have been made to account for the observed maximum in the dissociation constant. It was seen on page 334 that the division Harned and Kmbree, ./. Am. Chem. Soc., 56, 1050, 2797 (1934); see also, Harned, J. Franklin Inst., 225, 623 (1938); Harned and Owen, Chem. Revs., 25, 131 (1939). 84 Pitzer, /. Am. Chem. Soc., 59, 2365 (1937); see also, Walde, J. Phys. Chem., 39, 477 (1935); Wynne-Jones and Everett, Trans. Faraday Soc., 35, 1380 (1939). AMPHIPROTIC SOLVENTS 337 of the free energy of dissociation of an acid into non-electrostatic and electrostatic terms leads to the expectation that log K a is related to the reciprocal of the dielectric constant of the solvent. Since l/D for water increases with increasing temperature, the value of log K a should de- crease; in addition to this effect there is the normal tendency for the dissociation constant, regarded as the equilibrium constant of an endo- thermic reaction, to increase with increasing temperature. The simul- taneous operation of these two factors will lead to a maximum dissocia- tion constant at a particular temperature. 26 Amphiprotic Solvents: The Ionic Product. In an amphiprotic solvent both an acid and its conjugate base can function independently; for example, if the acid is HA the conjugate base is A~, and if the amphi- protic solvent is SH, the acidic and basic equilibria are HA + SH ^ SUt + A- and SII + A- ^ HA + S-, acid base acid base respectively. The ion SHt is the hydrogen ion, sometimes called the lyonium ion, in the given medium, arid S~ is the anion, or lyate ion, of the solvent. The conventional dissociation constants of the acid HA and of its conjugate base A~ are then written as asn 2 f a A - A a = - ana Kb Qll\ CijC and the product is thus K a K b = asHjas-, (40) which is evidently a specific property of the solvent. Since the solvent is amphiprotic and can itself function as either an acid or a base, the equilibrium SH + SH ^= Slit + S- acid base acid base must always exist, and if the activity of the undissociated molecules of solvent is taken as unity, it follows that the equilibrium constant KS of this process IK given by K s = flsujfls-, (41) the constant A~? defined in this manner being called the ionic product or ionization constant of the solvent. It is sometimes referred to as the autoprotolysis constant, since it is a measure of the spontaneous tendency for the transfer of a proton from one molecule of solvent to another to *Gurney, J. Chem. Phys., 6, 499 (1938); Baughan, ibid., 7, 951 (1939); see also, Magee, Ri and Eyring, ibid., 9, 419 (1941); LaMer and Brescia, /. Am. Chem. Soc., 62, 617 (1940). 338 ACIDS AND BASES take place. Comparison of equations (40) and (41) shows that K a K b = K s , (42) and so the dissociation constant of a base is inversely proportional to that of its conjugate acid, and vice versa; the proportionality constant is the ionic product of the solvent. This is the quantitative expression of the conclusion reached earlier that the anion of a strong acid, which is its conjugate base, will be weak, while the anion of a weak acid will be a moderately strong base, and similarly for the conjugate acids of strong and weak bases. For certain purposes it is useful to define the dissociation constant of the solvent itself as an acid or base; by analogy with the conventional method of writing the dissociation constant of any acid or base, the activity of the solvent molecule taking part in the equilibrium is assumed to be unity. In the equilibrium SH + SH ^ SUt + S- one molecule of SH may be regarded as functioning as the acid or base, while the other is the solvent molecule; the conventional dissociation constant of either acid or base is then J\.a -L**b (43) For most purposes <ZSH may be replaced by the molecular concentration of solvent molecules in the pure solvent; with water, for example, the concentration of water molecules in moles per liter is 1000/18, i.e., 55.5, so that the dissociation constant of H 2 as an acid or base is equal to the ionic product of water divided by 55.5. The Ionic Product of Water. An ionic product of particular interest is that of water: the autoprotolytic equilibrium is H 2 O + H 2 O ^ H 3 O+ + OH-, and hence the ionic product K w may be defined by either of the following equivalent expressions, viz., (44) = CH 3 o+coH--/H,o + /oir. (446) By writing the ionic product in this manner it is tacitly assumed that the activity of the water is always unity; in solutions containing dissolved substances, however, the activity is diminished and K w as defined above will not be constant but will increase. The activity of water in any THE TONIC PRODUCT OF WATER 339 solution may be taken as equal to p/po, where p is the vapor pressure of the solution and p that of the pure water at the same temperature; in a solution containing 1 g.-ion per liter of solute, which is to be regarded as relatively concentrated, the activity of the water is about 0.98. The effect on K w of the change in the activity of the water is thus not large in most cases. The equilibrium between HaO* and OH~ ions will exist in pure water and in all aqueous solutions: if the ionic strength of the medium is low, the ionic activity coefficients may be taken as unity, and hence the ionic product of water, now represented by k w) is given by k w = C H ,O+COH- (or c H +c ir). (45) As will be seen later, the value of k w is approximately 10~ 14 at ordinary temperatures, and this figure will be adopted for the present. In an exactly neutral solution, or in perfectly pure water, the con- centrations of hydrogen (H 3 O+) and hydroxyl ions must be equal; hence under these conditions, CH+ = COH- = 10~ 7 g.-ion per liter, the product being 10~ 14 as required. The question of the exact signifi- cance of the experimental value of pll will be considered in Chap. X, but for the present the pH of a solution may be defined, approximately, by pH log CH+. It follows, therefore, that in pure water or in a neutral solution at ordi- nary temperatures, the pH is 7. If the quantity pOH is defined in an analogous approximate manner, as log COIT, the value must also be 7 in water. By taking logarithms of equation (45), it can be shown that for any dilute aqueous solution pH + pOH = pfc, = 14 (46) at ordinary temperatures, where pk w is written for log k w . If the hydrogen ion concentration of a solution exceeds 10~ 7 g.-ion per liter, the pH is less than 7 and the solution is said to be acid; the pOH is corre- spondingly greater than 7. Similarly, in an alkaline solution, the hydro- gen ion concentration is less than 10~ 7 g.-ion per liter, but the hydroxyl ion concentration is greater than this value; the pH is greater than 7, but the pOH is smaller than this figure. The relationships between pH, pOH, CH+ and coir, at about 25, may be summarized in the manner represented below. CH+ i io-> io- io- io- 10-' io- io~ 7 io- io- io- l io~ 11 io~ u io- COH- 10-" 10-" 10-" io- 10-" io- io- io- 7 io- 10-* io- io- 10-' lo- 1 i pH 1 2 3 4 5 6 7 8 9 10 11 12 13 14 pOH 14 13 12 11 10 987654 3 2 1 Neu- . Acid * tral - Alkaline 340 ACIDS AND BASES It is seen that the range of pH from zero to 14 covers the range of hydro- gen and hydroxyl ion concentrations from a N solution of strong acid on the one hand to a N solution of a strong base on the other hand. A solu- tion of hydrogen ion concentration, or activity, exceeding 1 g.-ion per liter would have a negative pH, but values less than about 1 in water are uncommon. Determination of Ionic Product: Conductance Method. Since it contains a certain proportion of hydrogen and hydroxyl ions, even per- fectly pure water may be expected to have a definite conductance; the purest water hitherto reported was obtained by Kohlrausch and Heyd- weiller 26 after forty-eight distillations under reduced pressure. The specific conductance of this water was found to be 0.043 X 10~ fl ohm" 1 cm." 1 at 18, but it was believed that this still contained some impurity and the conductance of a 1 cm. cube of perfectly pure water was esti- mated to be 0.0384 X 10" 6 ohm" 1 cm." 1 at 18. The equivalent con- ductances of hydrogen and hydroxyl ions at the very small concentra- tions existing in pure water may be taken as equal to the accepted values at infinite dilution; these are 315.2 and 173.8 ohms" 1 cm. 2 , respectively, at 18, and hence the total conductance of 1 equiv. of hydrogen and 1 equiv. of hydroxyl ions, at infinite dilution, should be 489.0 ohms" 1 cm. 2 It follows, therefore, that 1 cc. of water contains 0.0384 X 10~ 6 - = - 78 X 10 ~ 10 equiv. per cc. of hydrogen and hydroxyl ions; the concentrations in g.-ion per liter are thus 0.78 X 10" 7 , and hence k w = CH+COH- - (0.78 X 10~ 7 ) 2 = 0.61 X 10" 14 . Since the activity coefficients of the ions in pure water cannot differ appreciably from unity, this result is probably very close to K w , the activity ionic product, at 18. The results in Table LX give the ob- TABLE LX. SPECIFIC CONDUCTANCE AND IONIC PRODUCT OF WATER Temp. 18 25 34 50 ic 0015 0.043 0.062 0095 0.187 X 10~ ohm-' cm." 1 K u 0.12 0.61 1.04 2.05 5.66 X 10~ 14 served specific conductances and the values of K w at several tempera- tures from to 50. Conductance measurements have been used to determine the ionic products of the amphiprotic solvents ethyl alcohol, formic acid and acetic acid. "Kohlrausch and Heydweiller, Z. physik. Chem., 14, 317 (1894); Heydweiller, Ann. Physik, 28, 503 (1909). ELECTROMOTIVE FORCE METHODS 341 Electromotive Force Methods. The earliest E.M.F. methods for evaluating the ionic product of water employed cells with liquid junc- tion; 27 the E.M.F. of the cell H.(l atm.) | KOH(0.01 N) || HC1(0.01 N) | H 2 (l atm.), from which it is supposed that the liquid junction potential has been completely eliminated, is given by _ RT . a' H + ,_ E = -grin -77;. (47) r a\i+ where a'n + and OH* represent the hydrogen ion activities in the right-hand and left-hand solutions, i.e., in the 0.01 N hydrochloric acid and 0.01 N potassium hydroxide, respectively. If aoir is the hydroxyl ion activity in the latter solution, then and substitution of K w /adn- for OH* in equation (47) gives RT. a'H+aoir , 4Q . T" K w ' ( 8) By measuring each of the electrodes separately against a calomel refer- ence electrode containing 0.1 N potassium chloride, and estimating the magnitude of the liquid junction potential in each case, the E.M.F. of the complete cell under consideration was found to be + 0.5874 volt at 25. The ionic activity coefficients were assumed to be 0.903 in the 0.01 N solutions, so that a' H + and ao'ir, representing the activities of hydrogen and hydroxyl ions in 0.01 N hydrochloric acid and 0.01 N potassium hy- droxide, respectively, were both taken to be equal to 0.0093; insertion of these figures in equation (48) gives K w as 1.01 X 10~ 14 at 25. This result is almost identical with some of the best later data, but the close agreement is probably partly fortuitous. The most satisfactory method for determining the ionic product of water makes use of cells without liquid junction, similar to those em- ployed for the evaluation of dissociation constants (cf. p. 314). 28 The E.M.F. of the cell H 2 (l atm.) | MOH(mO MCl(m) AgCl(s) | Ag, where M is an alkali metal, e.g., lithium, sodium or potassium, is RT E = E - -=- In a H *ocr. (49) r 87 Lewis, Brighton and Sebastian, /. Am. Chem. Soc., 39, 2245 (1917); Wynne-Jones, Trans. Faraday Soc., 32, 1397 (1936). Roberts, J. Am. Chem. Soc., 52, 3877 (1930); Harned and Hamer, ibid., 55, 2194 (1933); for reviews, with full references, see Harned, /. Franklin Inst., 225, 623 (1938); Harned and Owen, Chem. Revs., 25, 31 (1939). 342 ACIDS AND BABES Since O H *OOH- is equal to K u , the activity of the water being assumed constant, it follows that ^1-^ln yon" and rearrangement gives E - E + F(E - Jg) 2.303/er RT , In RT . ^J- = - ^ In K a - ^ln -> TOOK- r ** Ton" + log ~ = - log K w - log 7cr 70H" (50) The activity coeflBicient fraction 7cr/7oir is unity at infinite dilution, and so the value of the right-hand side of equation (50) becomes equal to log Kw under these conditions. It follows, therefore, that if the left- hand side of this equation, for var- ious concentrations of alkali hydrox- ide and chloride, is plotted against the ionic strength, the intercept for infinite dilution gives log K w . The value of is known to be + 0.2224 volt at 25, and by making the as- sumption that MOH and MCI are completely dissociated, as wil 1 be the case in relatively dilute solutions, men- and mcr may be identified with mi and m 2 , respectively. The results shown in Fig. 92 are for a series of cells containing cesium (I), potassium (II), sodium (III), barium (IV), and lithium (V) chlorides together with the corresponding hydroxides; the agreement between the values extra- polated to infinite dilution is very striking. The value of - log K w is found to be 13.9965 at 25, so that K w is 1.008 X 10~ 14 . Another method of obtaining the ionic product of water is to combine the E.M.F. of the cell 0.05 0.10 FIG. 92. Determination of the ionic product of water (Harned, et al.) H 2 (l atm.) | HCl(w',) MCl(it^l) AgCl(s) | Ag with that just considered; the E.M.F. of this cell is given by the same ELECTROMOTIVE FORCE METHODS 343 general equation, RT ' = #o- lnaW:i. (51) Combination of equations (49) and (51) gives RT m'n+rr&r , RT , = - In --- h t In - > (52 ; + P 7n + 7cr where the primed quantities refer to the cell containing hydrochloric acid whereas those without primes refer to the alkali hydroxide cell. If the ionic strengths in the two cells are kept equal, then provided the solutions are relatively dilute the activity coefficient .actor will be virtually unity, and the second term on the right-hand side of equation (52) is zero; hence under these conditions RT E E f = -=- In and making use of the fact that K w is equal to mn+?noir"yn+7on-, this becomes RT WH+rocrWoH- RT * RT E. LV ^ * ^ /.ON E - E ' = -TT In - -- - + -=r In TH^TOH- - -jr In K w . (53) " "*cr ^ /* According to the extended Debye-Huckol equation, the value of log TH^OFT may be represented by A Vp + C|i, where A is a known constant for water at the experimental temperature; hence, equation (53), after re- arrangement, becomes _ ^ F 7ttcr F RT -jr\nK u + 2. F . (54) The plot of the left-hand side of equation (54) against the ionic strength y should be, at least approximately, a straight line whose intercept for y equal to zero gives log K w . As before, the values of WaS ^cr, ttk>H~ and mcr are estimated on the assumption that the electrolytes HC1, MCI and MOH are completely dissociated. A large number of measurements of cells of the types described, con- taining different halides, have been made by Harned and his collab- orators over a series of temperatures from to 50; the excellent agree- ment between the results obtained in different cases may be taken as 344 ACIDS AND BASES evidence of their accuracy. A selection of the values of the ionic product of water, derived from measurements of cells without liquid junction, is quoted in Table LXI; the data in the last column may be taken as the most reliable values of the ionic product of water. TABLE LXI. IONIC PRODUCT FROM CELLS CONTAINING VARIOUS HALIDE8 t NaCl KC1 LiBr BaCl 2 Mean 0.113 0.115 0.113 0.112 0.113 X 10~ M 10 0.292 0.293 0.292 0.280 0.292 20 0.681 0.681 0.681 0.681 0.681 25 1.007 1.008 1.007 1.009 1.008 30 1.470 1.471 1.467 1.466 1.468 40 2.914 2.916 2.920 2.917 50 5.482 5.476 5.465 5.474 Effect of Temperature on the Ionic Product of Water. The values of the ionic product in Table LXI are seen to increase with increasing temperature; at 100, the ionic product of water is about 50 X 10~ 14 . According to Harned and Hamer 29 the values between and 35 may be expressed accurately by means of the equation 4787 3 log K w = y^- - 7.1321 log T - 0.0103657* + 22.801. From this expression it is possible, by making use of the reaction iso- chore, i.e., dlnK _ A// dT " RT*' to derive the heat change accompanying the ionization of water; the results at 0, 20 and 25 are as follows: 20 25 14.51 13.69 13.48 kcal. These values are strictly applicable at infinite dilution, i.e., in pure water. It was seen on page 12, and it is obvious from the considerations discussed in the present chapter, that the neutralization of a strong acid by a strong base in aqueous solution is to be represented as H 3 0+ + OH- = H 2 + H 2 0, which is the same reaction as is involved in the ionization of water, except that it is in the opposite direction. The heats of neutralization obtained experimentally are 14.71, 13.69 and 13.41 kcal. at 0, 20 and 25, re- spectively; the agreement with the values derived from K w is excellent. Although the relationship given above for the dependence of K w on temperature is only intended to hold over a limited temperature range, Harned and Hamer, /. Am. Chem. Soc., 55, 4496 (1933); see also, Harned and Geary, ibid., 59, 2032 (1937). THE IONIZATION OF WATER IN HALIDE SOLUTIONS 345 it shows nevertheless that the ionic product of water, like the dissociation constants of acids, to which reference has already been made, should pass through a maximum at a relatively high temperature and then decrease. Although the temperature at which the maximum value of K w is to be expected lies beyond the range of the recent accurate work on the ionic product of water, definite evidence for the existence of this maximum had been obtained several years ago by Noyes (1910). The temperature at which the maximum ionic product was observed is about 220, the value of K w being then about 460 X 10~ 14 . The lonization of Water in Halide Solutions. The cells employed for the determination of the ionic product of water have also been used to study the extent of dissociation of water in halide solutions. 30 Since K w is equal to a H + a H- and a H + a H-/7n + 7oH- is equal to memoir, equation (53) becomes, after rearrangement, RT ~ = & & -- ^- in r RT ~- In ni r and so the molal ionization product WH+WOH- in the halide solution present in the cells may be evaluated directly from the E.M.F.'S E and E', and the molalities of the electrolytes. The amounts of hydrogen and hy- droxyl ions are equal in the pure halide solution; consequently, the square-root of WH+WOH" gives the concentration of these ions, in g.-ions LiCl 'LLBr 1.0 Fia. 93. Variation of molal ionization product of water (Harned, et al.) per 1000 g. of water, produced by the ionization of the water in the halide solution. The results for a number of alkali halides at 25 are shown in Fig. 93; it will be seen that, in general, the extent of the ionization of water increases at first, then reaches a maximum and decreases with M For reviews with full references, see Harned, /. Franklin Inst., 225, 623 (1938); Harned and Owen, Chem. Revs., 23, 31 (1939). 346 \CIDS AND BASES increasing ionic strength of the medium. With lithium salts the maxi- mum is attained at a higher concentration than is shown in the diagram. The explanation of this variation is not difficult to find: the quantity an f aoH-/ a H 2 o, i.e., WH+WOH- X 7H+7ojr/ a H 2 o, which includes the activity of the water, must remain constant in all aqueous solutions, and since the activity coefficients always decrease and then increase as the ionic strength of the medium is increased (cf. Fig. 40), while an 2 o, i.e., Wpo,* decreases steadily, it follows that the variation of WH+WOH- must be *:* the form shown in Fig. 93. In spite of the dependence of WH+WOIT on tho ionic strength of the solution, it is still satisfactory, for purposes of approximate computation, to take the ionic concentration product of water (A: u .) to be about 10~ n at ordinary temperatures, provided the con- centration of electrolyte in tho solution is not too great. PROBLEMS 1. Show that according to equation (10) the plot of Ac against I/A should be a straight line; test the accuracy of this (approximate) result by means of the data for acetic acid on page 105 and for a-crotonic acid in Piohlcm 7 of Chap. III. 2. Utilize the data referred to in Problem 1 to calculate the dissociation functions of acetic and a-crotonic acids at several concentrations by means of equation (10); compare the results with the thermodynamic dissociation con- stants obtained in Chap. V. 3. In their measurements of the cell H,a atm.) | HP(mi) NaP(m 2 ) NaCl(m,) AgCl(s) | Ag, where HP represents propionic acid, Harned and Ehlers [J. Am. Chcm. Soc. t 55, 2379 (1933)] made mi. w 2 and m 3 equal and obtained the following K.M.F.'S at 25: m E m E 4.899 X 10~ 3 0.64758 18.669 X 10" 3 0.61311 8.716 063275 25.546 0.60522 12.812 0.62286 31 7'J3 59958 Evaluate the dissociation constant of propionic acid. 4. Walpole [./. Chem. *SV., 105, 2501 (1914)] measured tho pll's of a series of mixtures of x cc. of 0.2 N acetic acid with 10 x cc. of 0.2 N sodium acetate, and obtained the following results: x 8.0 7.0 6.0 5.0 4.0 30 20 cc. pH 4.05 4.27 4.45 4.63 4.80 4.99 5.23 Calculate the dissociation constant of acetic acid by the use of equation (27), the activity coefficients of the acetate ions being obtained by means of the simple Debye-Huckel equation. Derive the dissociation constant by means of the graphical method described on page 324. * Since pure water, vapor pressure p , is takwi as the standard state, the activity of water i- any solution of aqueous vapor pressure p will be p/p Q . PROBLEMS 347 5. Bennett, Brooks and Glasstonc [</. Chem. Sac., 1821 (1935)] obtained the following results in the titration of o-fluorophenol in 30 per cent alcohol at 25; when x cc. of 0.01 N sodium hydroxide was added to 50 cc. of a 0.01 N solution of the phenol the pll's were: x 10 15 20 25 30 40 cc. pll 8.73 9.01 9.20 9.37 9.56 10.00 Calculate the dissociation constant of o-fluorophenol, using the expression log/ = 0.683 Vp 4- 2.0y to obtain the activity coefficient of the anion. The activity coefficient of the undissociated acid may be taken as unity. 6. The following pH values were obtained by German arid Vogel [</. Am. Chem. Roc., 58, 1546 (1936) J in the titration of 100 cc. of 0.005 molar succinic acid with x cc. of 0.01 N sodium hydroxide at 25: x pH x pH 20.0 400 60.0 5.11 300 428 70.0 5.39 40.0 4.56 80.0 5.68 50.0 4.84 90.0 6.03 Determine the two dissociation constants of succinic acid by the graphical method described on page 320. 7. The E.M.F. .,f the cell H,(l atni.) | NaOH(ro) NaCl(m) AgCl(s) | Ag, with the sodium hydroxide and chloride at equal mobilities, was found by Roberts [,/. Am. Chem. tioc., 52, 3877 (1930)] to have a constant value of 1.0508 volt at 25 when the solutions were dilute. Calculate the ionic product of water from this result. 8. The following E.M.F.'S were obtained at 25 by Harned and Copson [/. Am. Chem. Svc., 55, 2206 (1933j] for the cells (A) II 2 (1 iitm.) | LiOH (0.01) LiCl(wi) AgCl(s) | Ag (B) H 2 (l atm.) 1 HC1 (0.01) LiCl(m) AgCl(s) 1 Ag. m E A E B 0.01 104979 0.4 1 779 002 1.03175 043S.V> 0.0:> 1.00755 0.422S2 0.10 0.9SSS3 0.40017 0.20 0V'">7 039453 0.50 94277 0.37235 1.00 0.91992 0.35191 2.00 0.89203 32352 3.00 0.87151 0.2V9:>9 4 00 0.85407 0.27754 Utilize the method given on page 343 to derive the ionic product of water from these data. Plot the variation of the molai ionization product with the ionic strength of the solution. CHAPTER X THE DETERMINATION OF HYDROGEN IONS Standardization of pH Values. The hydrogen ion exponent, pH, was originally defined by S0rensen (1909) as the " negative logarithm of the hydrogen ion concentration,' 1 i.e., as log CH+; most determinations of pH are, however, based ultimately on E.M.F. measurements with hydro- gen electrodes, and the values obtained are, theoretically, an indication of the hydrogen ion activity rather than of the concentration. For this reason, it has become the practice in recent years to regard the pH as defined by pH 5= - log a H +, (1) where H+ stands for the hydrogen ion, i.e., lyonium ion, in the particular solvent. This definition, however, involves the activity of a single ionic species and so can have no strict thermodynamic significance; it follows, therefore, that there is no method available for the precise determination of pH defined in this manner. It is desirable, nevertheless, to establish, if possible, an arbitrary pll scale that shall be reasonably consistent with certain thermodynamic quantities, such as dissociation constants, which are known exactly, within the limits of experimental error. The values obtained with the aid of this scale will not, of course, be actual pH's, since such quantities cannot be determined, but they will at least be data which if inserted in equations involving pH, i.e., log a H +, will give results consistent with those determined by strict thermodyriamic meth- ods not involving individual ion activities. The E.M.F. of a cell free from liquid junction potential, consisting of a hydrogen electrode and a reference electrode, should be given by zprn E = J^ref. pT~ In an*, or, introducing the definition of pll according to equation (1), RT E = E&. + 2.303 -ypH where E nf . is the potential of the reference electrode on the hydrogen scale. It follows, therefore, that F(E - E nl .) 348 STANDARDIZATION OF PH VALUES 349 If the usual value for E ro t. of the reference electrode is employed in this equation to derive pH's, the results are found to be inconsistent with other determinations that arc thermodynamically exact. A possible way out of this difficulty is to find a value for E ref . such that its use in equation (2) gives pH values which are consistent with known thermodynamic dissociation constants. For this purpose use is made of equation (29) of Chap. IX, viz., pll = pK a + log - fl -~ - A^ + C V , (3) which combined with equation (2) gives F(E - ffref.) . . R . I' , r = pA - + l *~ ~ A ^ + C *> _ 2.3Q3RT/ B A ,-\ yi , 2.303/2 T , A . '. E - -- j, -- (pK a + log ^g - A V v j = 1U + - j - C v . (4) A series of mixtures, at different total concentrations, of an acid, whose dissociation constant is known exactly, e.g., from observations on cells without liquid junction, and its salt are made up, thus giving a series of values for B and a B. The K.M.F/S of the cells consisting of a hydrogen electrode in this solution combined with a reference electrode are measured; a saturated solution of potassium chloride is used as a salt bridge between the experimental solution and the one contained in the reference electrode. The E values obtained in this manner, together with B and a B, calculated from the known composition of the acid- salt mixture (cf. p. 324), and the pK a of the acid, permit the left-hand side of equation (4) to be evaluated for a number of solutions of different ionic strengths. The results plotted against the ionic strength should fall on a straight line, the intercept for zero ionic strength giving the required quantity E T ^ m . In order for this result to have any significance it should be approximately constant for a number of solutions covering a range of pH values and involving different acids; this has in fact been found to be the case in the pH range of 4 to 9, and hence a pH scale consistent with the known pA" values for a number of acids is possible. 1 The conclusions reached from this work may be stated in terms of the potentials of the reference electrodes; for example, the value of E ro t. of the 0.1 N KC1 calomel electrode for the purpose of determining pH's by means of equation (2) is 0.3358 volt * at 25. In view of possible variations in the salt bridge from one set of experiments to another, it is preferable to utilize these potentials to determine the pll values of a number of reproducible buffer solutions (cf. p. 410) which can form a i Hitchcock and Taylor, J. Am. Chem. Soc., 59, 1812 (1937); 60, 2710 (1938); Maclnnes, Belcher and Shedlovsky, ibid., 60, 1094 (1938); see also, Cohn, Heyroth and Menkin, ibid., 50, 696 (1928). * This may be compared with 0.3338 volt, given on page 232, employed in earlier pH work. 350 THE DETERMINATION OF HYDROGEN IONS scale of reference. The results obtained in this manner are recorded in Table LXII for temperatures of 25 and 38; they are probably correct TABLE LXII. STANDARDIZATION OF pH VALUES OF REFERENCE SOLUTIONS Solution 25 38 O.lNHCl 1 .085 1.082 0.1 M Potassium totroxalate 1.480 1.495 0.01 N HC1 and 0.09 N KC1 2.075 2.075 0.05 M Potassium and phthalate 4.005 4.020 0.1 N Acetic acid and 0.1 N Sodium acetate 4.640 4.650 0.025 M KH 2 PO 4 and 0.025 M Na 2 HPO 4 6.855 6.835 0.05 M Na 2 B 4 O 7 10H 2 O 9.180 9.070 to db 0.01 pH unit. With this series of reference solutions it is possible to standardize a convenient combination of hydrogen and reference elec- trodes; the required pH of any solution may thus be determined. The pH's obtained in this way arc such that if inserted in equation (3), they will give a pA' value which should not differ greatly from one obtained by a completely thermodynamic procedure. Those pi I values can then be used in connection with equations (29) and (34) of Chap. IX to give reasonably accurate dissociation constants. Reversible Hydrogen Electrodes. In previous references to the hy- drogen electrode it has been stated briefly that it consists of a platinum electrode in contact with hydrogen gas; the details of the construction of this electrode will be considered here. In addition to the hydrogen gas electrode, a number of other electrodes are known which behave reversibly with respect to hydrogen ions. Any one of these can be used for the determination of pil values, although the electrode involving hydrogen gas at 1 atm. pressure is the standard to which others are referred. I. The Hydrogen Gas Electrode. The hydrogen gas electrode con- sists of a small platinum sheet or wire coated with finely divided platinum black by electrolysis of a solution of chloroplatinic acid containing a small proportion of lead acetate (cf. p. 35). The platinum foil or wire, attached to a suitable connecting wire, is inserted in the experimental solution through which a stream of hydrogen is passed at atmospheric pressure. The position of the electrode in the solution is arranged so that it is partly in the solution and partly in tho atmosphere of hydrogen gas. A number of forms of electrode vessel, suitable for a variety of uses, have been employed for the purpose of setting up hydrogen gas electrodes; some of these are depicted in Fig. 94. A simple and con- venient type of hydrogen electrode is that, usually associated with the name of Hildebrand, 2 shown in Fig. 95; a rectangular sheet of platinum, *Hildebrand, J. Am. Chem. Soc., 35, 847 (1913); for further details concerning hydrogen electrodes, see Clark, "The Determination of Hydrogen Ions/ 1 1928; Britton, "Hydrogen Ions," 1932; Glasstone, "The Electrochemistry of Solutions," 1937, p. 375. See also, Hamer and Acree, J. Res. Nat. Bur. Standards, 23, 647 (1939). THE HYDROGEN GAS ELECTRODE 351 of about 1 to 3 sq. cm. exposed area, which is subsequently platinized, is welded to a short length of platinum wire sealed into a glass tube con- taining mercury. This tube is sealed into another, closed at the top, but widening out into a bell shape in the region surrounding the platinum Hydrogen Hydrogen Hydrogen Fio. 94. Forms of hydrogen electrode FIG. 95. Hydrogen electrode: Hildebrand type shoot; a sido connection is provided for the inlet of hydrogen gas. A number of holes, or slits, are mado in. the boll-shaped portion of the tube at a level midway up the platinum, so that when the electrode is inserted in a solution and hydrogen passed in through the side-tube the platinum shoot is half immersed in liquid and half surrounded by gas. This arrangement permits the rapid attainment of equilibrium between the electrode material, the hydrogen gas and the solution. The time taken to reach this state of equilibrium depends, among other factors, on tho nature of the solution, the thickness of the deposit, and on the pre\5ous history of the electrode. As a general rule, an electrode that is func- tioning in a satisfactory manner will give a steady potential within five or ten minutes of commencing the passage of hydrogen. The use of a platinum shoot in the Ilildebrand electrode is not essential, and many workers prefer to use a simple wire of 2 or 3 cm. in length, straight or coiled, for such an electrode attains equilibrium rapidly, although it has a somewhat higher resistance than the form represented in Fig. 95. The hydrogen gas should be purified by bubbling it through alkaline per- 352 THE DETERMINATION OF HYDROGEN IONS manganate and alkaline pyrogallol solutions to remove oxygen and other impurities which may influence the functioning of the hydrogen electrode. Whatever form of electrode vessel is employed, the fundamental principle of the operation is always the same. The hydrogen gas is adsorbed by the finely divided platinum and this permits the rapid establishment of equilibrium between molecular hydrogen on the one hand, and hydrogen ions in solution and electrons, on the other hand, thus }H,fo) ^ JH,(Pt) + H 2 ^ H 3 0+ + . This equilibrium can be attained rapidly from either direction, and so the electrode behaves as one that is reversible with respect to hydrogen ions. The hydrogen gas electrode behaves erratically in the presence of arsenic, mercury and sulfur compounds, which are known to be catalytic poisons; they probably function by being preferentially adsorbed on the platinum, thus preventing the establishment of equilibrium. An elec- trode whose operation is affected in this manner is said to be "poisoned"; if it cannot be regenerated by heating with concentrated hydrochloric acid, the platinum black should be removed by means of aqua regia and the electrode should be roplatinized. The hydrogen gas electrode cannot be employed in solutions containing oxidizing agents, such as nitrates, chlorates, permanganates and ferric salts, or other substances capable of reduction, e.g., unsaturated and other reducible organic compounds, alkaloids, etc. The electrode does not function in a satisfactory manner in solutions containing noble metals, e.g., gold, silver and mercury, since they tend to be replaced by hydrogen (cf. p. 253), neither can it be used in the presence of lead, cadmium and thallous salts. In spite of these limitations the hydrogen gas electrode has been extensively employed for precise measurements in cells with or without liquid junction, such as those mentioned in Chaps. VI and IX. The electrode has also been found to give fairly satisfactory results iii non-aqueous solvents such as alcohols, acetone, benzene and liquid ammonia. Since the standard state of hydrogen is the gas at 760 mm. pressure, it would be desirable to employ the gas at this pressure; even if the hydrogen were actually passed in at this pressure, which would not be easy to arrange, the partial pressure in the electrode vessel would be somewhat less because of the vapor pressure of the water. A correction for the pressure difference should therefore be made in accordance with equation (50) of Chap. VI; the correction is, however, small as is shown by the values calculated from this equation and recorded in Table LXIII. The results are given for a series of temperatures and for three gas pressures; the corrections are those which must be added, or subtracted if marked by a negative sign, to give the potential of the electrode with hydrogen gas at a partial pressure of 760 mm. THE OXYGEN ELECTRODE 353 TABLE LXIII. PRESSURE CORRECTIONS FOR HYDROGEN ELECTRODE IN MILLIVOLTS Temperature 15 20 25 30 Vapor Pressure 12.8 15.5 23.7 31.7mm. Gas Pressure 740mm. 0.54 0.61 0.75 0.92 760mm. 0.20 0.26 0.38 0.56 780mm. -0.13 -0.08 0.04 0.20 II. The Oxygen Electrode. The potential of an oxygen electrode, expressed in the form of equation (96) of Chap. VII, is r>m E = #o 2 ,oH- + -y ^ aom (5) and since OOH~ may be replaced by K u ,/au+, where K w is the ionic product of water, it follows that 71 rn E = #o 2 ,n+- ylnem*. (6) The oxygen electrode should thus, in theory, function as if it were re- versible with respect to hydrogen ions. Attempts have been made to set up oxygen electrodes in a manner similar to that adopted for the hydrogen gas electrode, as described above; the results, however, have been found to be unreliable. The potential rises rapidly at first but this is followed by a drift lasting several days. The value reached finally is lower than that expected from the calculated standard potential of oxygen (cf. p. 243) and the known pH of the solution. The use of either iridium or smooth platinum instead of platinized platinum does not bring the potential appreciably nearer the theoretical reversible value, although the use of platinized gold has been recommended. It is evident that the oxygen gas electrode in its usual form does not function reversibly; the difference of potential when the equilibrium K> 2 + H 2 O + 2 ^ 2OH- is attained is less than would be expected, and this means that the direct reaction, as represented by this equation, is retarded in some manner not yet clearly understood. In spite of its irreversibility, the oxygen electrode was at one time used for the approximate comparison of pH values in solutions containing oxidizing substances, in which the hydrogen gas electrode would not function satisfactorily. In order for the results to have any significance the particular oxygen electrode employed was standardized by means of a hydrogen electrode in a solution in which the latter could be employed. The oxygen electrode, with air as the source of oxygen, has also been used for potentiometric titration purposes; in work of this kind the actual potential or pH is immaterial, for all that is required is an indication of 354 THE DETERMINATION OF HYDROGEN IONS the point at which the potential undergoes rapid change. 3 In recent years the difficulty of measuring pH's in solutions containing reducible substances has been largely overcome by the wide adoption of the glass electrode which is described below. HI. The Quinhydrone Electrode. It was seen in Chap. VIII that a mixture of quinone (Q) and hydroquinonc (1I 2 Q) in the presence of hydrogen ions constitutes a reversible oxidation-reduction system, and the potential of such a system is given by equation (4), page 270, as ^lna H *. (7) r It is seen, therefore, that the potential of the quirione-hydroquinone system depends on the hydrogen ion activity of the system. For the purpose of pH determination the solution is saturated with quinhydrone, which is an cquimolecular compound of quinone and hydroquinone; in this manner the ratio of the concentrations CQ to CH Z Q is maintained at unity, and if the ionic strength of the solution is relatively low the ratio of the activities, i.e., aq/aH 2 Q, may be regarded as constant. The first two terms on the right-hand side of equation (7) may thus be combined to give RT E = E% - -- In a l{ - (8) r RT = E Q - 2.303 ~v log an- (8a) r RT = E Q Q + 2.303 -TT PH- (86) r By using the method of standardization described at the beginning of this chapter, the value of EQ is found at to be E Q Q = - 0.6994 + 0.00074 (t - 25). This method of expressing the results is of little value for practical pur- poses; the particular reference electrode and salt bridge employed should be standardized by means of equation (2) using one of the reference solutions in Table LXII. If the reference electrode is a calomel electrode with 0.1 N potassium chloride, and a bridge of a saturated solution of this electrolyte is employed, it has been found possible to express the experi- mental data by means of the equation #Q<cai.) = - 0.363(5 + 0.0070(J - 25). This is the potential of the quinhydrone electrode against the Hg, Hg 2 Cl 2 , 8 Furman, J. Am. Chem. Soc., 44, 12 (1922); Trans. Electrochem. tfoc., 43, 79 (1923); Britton, /. Chem. Soc., 127, 1896, 2148 (1925); Richards, J. Phys. Chem., 32, 990 (1928). THE QUINHYDRONE ELECTRODE 355 KC1(0.1 N) ejectrode when the former contains a solution of hydrogen ions of unit activity, i.e., its pH is zero. 4 The quinhydrone electrode is easily set up by adding a small quantity of the sparingly soluble quinhydrone, which can be obtained commer- cially, to the experimental solution so as to saturate it; this solution is shaken gently and then an indicating electrode of platinum or gold is inserted. The surface of the electrode metal should be clean and free from grease; it is first treated with hot chromic acid mixture, washed well with distilled water, and finally dried by heating in an alcohol flame. Gentle agitation of the solution by means of a stream of nitrogen gas is sometimes advantageous. The electrode gives accurate results in solu- tions of pH less than 8; in more alkaline solutions errors arise, first, because of oxidation of the hydroquinone by oxygen of the air, and second, on account of the ionization of the hydroquinone as an acid (rf. p. 291). Oxidizing or reducing agents capable of reacting rapidly with quinone or hydroquinone are liable to disturb the normal ratio of these su bstances, and so will affect the potential. The quinhydrone electrode can bo used in the presence of the ions of many metals which have a deleterious effect on the hydrogen gas electrode, but ammonium salts exert a harmful influence. The potential of the quinhydrone elec- trode is affected to some extent by all salts and even by non-electrolytes; this "salt effect" is to be attributed to the varying influence of the salts, etc., on the activities of the quinone and hydroquinone; although the ratio CQ/cn 2 q remains constant, therefore, this is not necessarily true for aQ/aii 2 Q upon which the electrode potential actually depends. The "salt error" is proportional to the concentration of electrolyte, within reason- able limits; its value, which may be positive or negative, according to the nature of the "salt." is about + 0.02 to 0.05 pll unit per equiv. per liter of electrolyte. Provided the solution is more dilute than about 0.1 x, the "salt error" is therefore negligible for most purposes. The quinhydrone electrode has an appreciable "protein error," and so cannot be employed to give reliable pH values in solutions containing proteins or certain of thoir degradation products. 6 The quinhydrone electrode has been adapted for pH measurements in non-aqueous media, such as alcohols, acetone, formic acid, benzene and liquid ammonia. For the determination of hydrogen ion activities in solutions in pure acetic acid a form of quinhydrone electrode involving tetrachloroquinone (chloranil) and its hydroquinone has been used. 6 4 Harned and Wright, /. Am. Chem. Soc., 55, 4849 (1933); Hovorka and Dearing, ibid., 57, 446 (1935). 5 For general references, see Glasstone, "The Electrochemistry of Solutions," 1937, p. 378. Conant et al, J. Am. Chem. Soc., 47, 1959 (1925); 49, 3047 (1927); Heston and Hall, ibid., 56, 1462 (1934). 356 THE DETERMINATION OF HYDROGEN IONS IV. The Antimony Electrode. The so-called " antimony electrode" is really an electrode consisting of antimony and its trioxide, the reaction being 2Sb(s) + 3H 2 O = Sb 2 O 3 (s) + 611+ + 6c, so that the potential is given by DAT! E = tfgb.sb^.H* - -jr In a H +, (9) the activities of the solid antimony and antimony trioxide, and of the water, being taken as unity. The potential of the Sb, Sb 2 O 3 electrode should thus depend on the hydrogen ion activity of the solution in which it is placed. The electrode is generally prepared by casting a stick of antimony in the presence of air; in this way it becomes sufficiently oxi- dized for the further addition of oxide to be unnecessary. A wire is attached to one end of the rod of antimony obtained in this manner, while the other is inserted in the experimental solution; its potential is then measured against a convenient reference electrode. As the poten- tials differ from one electrode to another, it is necessary that each anti- mony electrode should be standardized by means of one of the solutions in Table LXII. The antimony electrode behaves, at least approximately, according to equation (0) over the range of pll from 2 to 7, but in more acid or more alkaline solutions deviations occur; these 4 discrepancies are probably connected with the solubility of the antimony oxide in such solutions. Since no special technique is required for setting up or meas- uring the potential of the antimony electrode, and it is not easily poisoned, it has advantages over other forms of hydrogen electrode. It is, there- fore, very convenient where approximate results are adequate, but it is not recommended for precision work. 7 V. The Glass Electrode. One of the most important advances of recent years in connection with the determination of pll's is the develop- ment wm-h has taken place in the use of the glass electrode. It has long been known that a potential difference is set up at the interface between glass arid a solution in contact with it which is dependent on the pll of the latter; * this dependence has been found to correspond to the familiar equation for a reversible hydrogen electrode, viz., IJrn E - /ft - -jjrlnaii*, (">) 7 Kolthoff arid Hartong, Rec. trav. chirn., 44, 113 (1925); Roberta and Fenwirk, J. Am. Chem. floe., 50, 2125 (1928); Parka and Hoard, ibid., 54, 850 (1932); Pcrlcy, Ind. Eng. Chem. (Anal. Kd.), 11, 316 (1939); Hovorka and Chapman, ,/. Am. Chem. fior., 63, 955 (1941). 8 For references to experimental methods, see GlaHstono, Ann. Rep. ('hem. S'oc., 30, 283 (1933); Muller and Diirichen, Z. Elektrochem., 41, 559 (1935); 42, 31, 730 (1936); Schwabe, ibid., 41, 681 (1935). For complete review, see Dole, "The Glass Electrode," 1941. THE GLASS ELECTRODE 357 where $?> is the " standard potential " for the particular glass employed, i.e., the potential when in contact with a solution of hydrogen ions at unit activity. It is evident, therefore, that measurements of the potential of the so-called "glass electrode" can be utilized for the determination of pH values. In its simplest form the glass electrode consists of a tube terminating in a thin-walled bulb, as shown at A, in Fig. 96; the glass most suitable for the purpose (Corning 015) contains about 72 per cent SiO 2 , 22 per cent Na^O and 6 per cent CaO; it has a relatively low melt- ing point and a high electrical con- ductivity. The bulb contains a solution of constant hydrogen ion concentration and an electrode of definite potential; a silver chloride electrode in 0.1 N hydrochloric acid or a platinum wire inserted in a buffer solution, e.g., 0.05 molar potassium acid phthalate, FIG. 96. Glass electrode cell saturated with quinhydrone, is generally used. The bulb is inserted in the experimental solution (B) so that the glass electrode consists of the system Ag | AgCl(s) 0.1 N HC1 1 glass | experimental solution, if silver-silver chloride is the inner electrode of constant potential. The potential of the glass electrode is then measured by combining it with a suitable reference electrode, such as the calomel electrode C in Fig. 96, the inner electrode of the glass electrode system serving to make elec- trical connection. Owing to the very high resistance of the glass, viz., 10 to 100 million ohms, special methods have to be employed for determining the E.M.F. of the cell; these generally involve the use of an electrometer or of vacuum- tube circuits, as described on page 192. Some workers have successfully prepared thin-walled glass electrodes of relatively large area and hence of comparatively low resistance; in these cases it has been found possible to make E.M.F. measurements without special apparatus, by using a reasonably sensitive galvanometer as the indicating instrument in the potentiometer circuit. Various forms of glass electrode have been em- ployed for different purposes, but the simple bulb type described above can easily be made in a form that is not too fragile and yet has not a veiy high resistance. Several commercial forms of apparatus are now available which employ robust glass electrodes; by using some form of electrometer triode vacuum tube (p. 193), it is possible to measure the potential to about 0.0005 volt, i.e., 0.01 pH unit, without difficulty. An accuracy of rb 0.002 pH unit has been claimed for special measuring 358 THE DETERMINATION OF HYDROGEN IONS circuits, but it is doubtful whether the pH scale has been established with this degree of precision. If both internal and external surfaces of the glass electrode were identical, it is obvious from equation (10) that the potential of the elec- trode system would be determined simply by the difference of pH of the solutions on the two sides of the glass membrane, apart from the potential of the inner electrode, e.g., Ag, AgCl. This expectation can be tested by measuring the E.M.F. of a cell in which the solution is the same inside and outside the glass bulb and the reference electrode is the same as the inner electrode; thus Ag | AgCl(s) 0.1 N HC1 1 glass | 0.1 N IIC1 AgCl(s) | Ag. The E.M.F. of this cell should be zero, but in practice the value is found to be of the order of 2 millivolts, for a good electrode. This small difference is called the asymmetry potential of the glass electrode; it is probably due to differences in the strain of the inner and outer surfaces of the glass membrane. It is necessary, therefore, to standardize each glass electrode by means of a series of buffer solutions of known pH; in this way the value of JQ in equation (10) for the particular electrode is found. Before use the glass electrode should be allowed to soak in water for some time, following its preparation, and should not be allowed to become dry subsequently; if treated in this manner equilibrium with the solution in which it is placed is attained rapidly. The potential satisfies equation (10) for a reversible hydrogen electrode very closely in the pi I range of 1 to 9, and with fair accuracy up to pH 12,* provided there is no large concentration of salts in the solution. At pll's greater than 9 appreciable salt effects become evident which increase with increasing pH, i.e., increasing alkalinity, of the solution; the magnitude of the salt effects in such solutions depends primarily on the nature of thr cations present, but it is of the order of 0.1 to 0.2 unit in the vicinity of pll 11 for 0.1 to 1 N solutions of the salt. In very acid solutions, of pi I less than unity, other salt effects, determined mainly by the unions, are observed. Apart from these limitations, the glass electrode has the outstanding advantage that it can be employed in aqueous solutions of almost any kind; the electrode cannot be poisoned, neither is it affected by oxidizing or re- ducing substances or by organic compounds. It can be used in un- buffered solutions and can be adapted for measurements with very small quantities of liquid. The glass electrode does not function satisfactorily in pure ethyl alcohol or in acetic arid, but it has been employed in mix- tures of these substances with water. 9 * The accuracy may be improved by the use of a special glass now available. 9 Hughes, J. Chun. /S'or., 401 (1928); Machines and Dole, J. Aw. Chem. tfoe., 52, 29 (1930); Maclnnes and Belcher, ibid , 53, 3315 (1931); Dole, ibid., 53, 4260 (1931); 54, 3095 (1932); for reviews with references, see Schwabe, Z. Elektrochem., 41, 681 (1935); Kratz, ibid., 46, 259 (1940). ACID-BABE INDICATORS 359 There is no completely satisfactory explanation of why a glass elec- trode functions as a reversible hydrogen electrode, but it is probable that the hydrogen ions in the solution exchange, to some extent, with the sodium ions on the surface of the glass membrane. The result is that a potential, similar to a liquid junction potential, is set up at each surface of the glass; if no ions other than hydrogen ions, and their associated water molecules, are able to enter the glass, the free energy change accom- panying the transfer of 1 g.-ion of hydrogen ion from the solution on one side of the membrane, where the activity is an + , to the other side, where the activity is aii% is then A(? = RTln where x is the number of molecules of water associated with each hydro- gen ion in the transfer; an 2 o and a!i 2 o are the activities of the water in the two solutions. The potential across the glass membrane is conse- quently given by EG = -TT In -777 + TT~ In ~Tf- - (11) /' a H + r If the solutions are sufficiently dilute, the activities of the water are the same on both sides of the membrane; the second term on the right-hand side of equation (11) then becomes zero. By retaining the hydrogen ion activity, e.g., ali + , constant on one side of the membrane, equation (11) reduces to the same form as (10). If the activity of the water is altered by the addition of alcohol or of appreciable amounts of salts or acids, equation (10) is no longer applicable, and deviations from the ideal reversible behavior of the glass electrode are observed. The salt errors found in relatively alkaline solutions, of pll greater than 9, are probably due to the fact that at these low hydrogen ion concentrations other cations present in the solution are transferred across the glass membrane to some extent. Under these conditions equation (11) is no longer valid, and so the glass electrode cannot behave in accordance with the require- ments of equation (10). 10 Acid-Base Indicators. An arid-base indicator is a substance, which, within certain limits, varies in color according to the hydrogen ion con- centration, or activity, of its environment; it is thus possible to determine the pll of a solution by observing the color of a suitable indicator when placed in that solution. Investigation into the chemistry of substances which function as acid-base indicators has shown that they are capable of existing in two or more tautomeric forms having different structures and different colors. In one or other of these; forms the molecule is capable of functioning as a weak acid or base, and it is this property, 10 Dole, J. Am. Chcm. flor , 53, 4260 (1930); 54, 2120, 3095 (1932); "Experimental and Theoretical Electrochemistry/' 1935, Chap. XXV; "The Glass Electrode/ 7 1941; Haugaard, J. Phys. Chem., 45, 148 (1941). 360 THE DETERMINATION OF HYDROGEN IONS together with the difference in color of the tautomoric states, that permits the use of the given compound as an acid-base indicator. 11 If HIni represents the un-ionized, colorless form of an indicator that is acidic in character, its ionization will be represented by II 2 O ^ H 3 + + In?, colorless colorless the anion Inr having the same structure and color as the molecule HIni. Application of the law of mass action to this equilibrium gives the dis- sociation constant of the acid as A',-^. (12) The colorless ion Ini will be in equilibrium with its tautomeric form Iii2 , thus In? ^ Inif, colorless colored but the latter, having a different structure from that of Inr, will have a different color, and the constant of the tautomeric equilibrium (K t ) will be given by Kt = ^T-' (13) Finally, the colored In^ ions will be in equilibrium with hydrogen ions and the colored un-ionized molecules HIn 2 , thus HIn 2 + IW) ^ H 3 O+ + Injf; colored colored the dissociation constant of the acid HIn 2 is then K 2 = ^^- (14) Combination of equations (12), (1.3) and (14) gives ^^ n +^f)~ = "It V KiJrT = Kln ' (15) where K\ n is a composite constant involving A'i, K 2 and K t \ it follows, therefore, from equation (15) that mf + am.;) MO) v ' If the ionic strength of the medium is relatively low, the activities of HIni, Hln 2 , InT and In-J may be replaced by their respective concen- 11 For a full discussion of the properties of indicators, sec Kolthoff and Kosenbhun, " Acid-Base Indicators," 1937. ACID-BASE INDICATORS 361 trations, so that equation (16) becomes 7 Clllrii + CHIn- an* = tin - . - - > (17) Clni ~T Ci u ~ where the approximate " constant " ki tl , known as the indicator constant, replaces K\ n . If a particular compound is to be satisfactory as an acid-base or pH indicator, the numerator and denominator in equation (17) must corre- spond to two distinct colors: a change in the hydrogen ion activity must clearly be accompanied by an alteration in the ratio of numerator to denominator, and unless these represent two markedly different colors the system as a whole will undergo no noticeable change of color. Since HIni and HIn 2 have different colors, on the one hand, and Inr and In^ are also different, but the same as HIni and HIn 2 , respectively, it is evident that in order to satisfy the condition given above it is necessary that the un-ionized molecules must be almost completely in the form HIni, or HIn 2 , and the ions must be almost exclusively in the other form. It follows from equation (13) that if the tautomeric constant K t is small the ions Inf will predominate over InjT ; further, if Ki/K* is large, so that HIni is a much stronger acid than HIn 2 , it follows that the un-ionized molecules HIn 2 will greatly exceed those of HIni. These are, in fact, the conditions required to make the substance under consideration a satis- factory indicator. An alternative possibility which is equally satisfactory is that K t should be large while Ki/K* is small; the ionized form will then consist mainly of Injf while the un-ionized molecules will be chiefly in the HIni form. For a satisfactory indicator, therefore, equation (17) may be written as Un-ionized form a * = *" Ionized form (18) where a is the fraction of the total indicator present in the ionized form. The actual color exhibited by the indicator will, of course, depend on the ratio of the un-ionized to the ionized form, since those have different colors; hence it follows from equation (18) that it will be directly related to the hydrogen ion activity, or concentration, of the medium. In an acid solution, i.e., a H f is high, the concentration of un-ionized form must increase, according to equation (18), and the indicator will exhibit the color associated with the main Hln form; in an alkaline medium, on the other hand, the ionized form must predominate and the color will be that of the chief In~ species. A few indicators are bases in the state in which they are normally employed; an example is methyl orange, which is the sodium salt of p-dimethylaminoazobenzcne sulfouic acid, the indicator action being due 362 THE DETERMINATION OP HYDROGEN IONS to the basic dimethylamino-group, i.e., N(CH 3 ) 2 . There is no reason, however, why the conjugate acid, viz., NH(CH 3 )j!~, should not be con- sidered as the indicator, although this is not the form in which it is usually supplied. In view of the fact that the properties of aqueous solutions are invariably expressed in terms of pH, and not of pOH, it is convenient to treat all indicators as acids. If the indicator in its familiar form happens to be a base, then the system is treated as if it consisted of its conjugate acid. All indicator systems, of course, consist of conjugate acid and base, e.g., HIn and In~, and it is in a sense somewhat arbitrary to refer to certain indicators as acids and to others as bases. The par- ticular term employed refers to the nature of the substance in the form in which it is usually encountered; methyl orange is generally employed as the sodium salt of the sulfonic acid of the free base, and hence it is called a basic indicator; but if it were used as the hydrochloride, or other salt, of the base, it would be called an acid indicator. In the subsequent treatment all indicators will for simplicity and uniformity be treated as acids. Indicator Range. If, as on page 287, it is assumed that the color of the ionized form In~~ is barely visible when 9 per cent of the total indi- cator is in this form, i.e., when a is 0.09, it follows from equation (19) that the limiting hydrogen ion activity at which the indicator will show its acid color, due to HIn, will be given by 7 - 91 tm. OH* = kin 0-gg 10/bin, /. pH pki n - 1, (20) where pki n is the indicator exponent, denned in the usual manner as log ki n . On the other hand when 91 per cent of the indicator is in the ionized form, i.e., a. is 0.91, the color of the un-ionized form will be virtually undetcetable in the mixture, and so the color will be that of the alkaline form; the pH at which the indicator shows its full alkaline color is then obtained from equation (19), thus i, - 09 1 i a * + = fcln o79i ~ To /Cln ' /. P H l>km + 1. (21) It is seen, therefore, that as the pH of a solution is increased by the addition of alkali, the color of an indicator begins to change visibly at a pH approximately equal to pki n 1, and is completely changed, as far as the eye can detect, at a pH of about pki n + 1. The effective transi- tion interval of an indicator is thus very roughly two pll units, one on each side of the pH equal to pki n of the indicator. Since various indi- cators have different values of fri,,, the range of pH over which the color changes will vary from one indicator to another. DETERMINATION OF INDICATOR CONSTANTS 363 When the indicator is ionized to an extent of 50 per cent, i.e., a is 0.5, it is seen from equation (19) that an+ = kin, .'. pH = pfcm. (22) The indicator will thus consist of equal amounts of the ionized and un- ionized forms, arid hence will show its exact intermediate color, when the hydrogen ion activity, or concentration, is equal to the indicator constant. Determination of Indicator Constants. A simple method of evalu- ating the constant of an indicator is to make use of equation (22). Two solutions, containing the same amount of indicator, one in the completely acid form and the other in the alkaline form, are superimposed; the net color is equivalent to that of the total amount of indicator with equal portions in the ionized and un-ionized forms. A series of buffer solutions of known pll (see Chap. XI) are then prepared and a quantity of indi- cator, twice that present in each of the two superimposed solutions, is added; the colors are then compared with that of the latter until a match is obtained. The matching buffer solution consequently contains equal amounts of ionized and un-ionized indicator and so its pH is equal to the required pki n . The general procedure is to utilize equation (19) and to determine the proportion of un-ionized to ionized form of the indicator in a solution of known pH; the most accurate method is to measure this ratio by a spcotrophotometric method similar to that described on page 328. If the substance is a one color indicator, that is to say it is colored in one (ionized) form and colorless in the other (un-ionized) form, e.g., phenol- phthalein and p-nitrophenol, it generally has one sharp absorption band in the visible spectrum; by measuring the extinction coefficient when the substance is completely in its colored form, e.g., in alkaline solution, it is possible, by utilizing Boer's law (cf. p. 328, footnote), to determine the concentration of colored form in any solution of known pH from the extent of light absorption by the indicator in that solution (cf. Fig. 100). From the total amount of indicator present, the ratio (1 a) /a can be evaluated and hence ki n can be obtained. The principle of this method of determining the indicator constant is identical with that described on pago 329 for the dissociation constant of an acid; A*i n is in fact the apparent dissociation constant of the indicator, assuming it to consist of a single un-ionized form II In and an ionized form In~ with a different color. A two color indicator will, in general, have two absorption bands, one for each colored form; by studying the extent of absorption in these bands in a solution of definite pll, as compared with the values in a com- pletely acid and a completely alkaline solution, it is possible to calculate directly the ratio of the amounts of un-ionized and ionized forms in the given solution. 364 THE DETERMINATION OF HYDROGEN IONS Instead of utilizing spectrophotometric devices, the ratio of the amounts of ionized to un-ionized indicator can be estimated, although less accurately, by visual means. With a one color indicator the fraction of ionized, generally colored, form is determined by comparing the color intensity with that of a solution containing various known amounts of indicator which have been completely transformed by the addition of alkali. With a two-color indicator it is necessary to superimpose the acid and alkaline colors in different amounts until a match is obtained. The precision of the measurements can be greatly improved by the use of a commercial form of colorimeter specially designed for the matching of colors. The values of pki n for a number of useful indicators, together with the pH ranges in which they can be employed and their characteristic colors in acid and alkaline solutions, are recorded in Table LXIV. TABLE LXIV. USEFUL INDICATORS AND THEIR CHARACTERISTIC PROPERTIES Indicator pki n pH Range Thymol blue 1.51 1.2- 2.8 Methyl orange 3.7 3.1-4.4 Bromphenol blue 3.98 3.0- 4.6 Bromcresol green 4.67 3.8- 5.4 Methyl red 5.1 4.2- 6.3 Chlorphenol red 5.98 4.8- 6.4 Bromphenol red 6.16 5.2- 6.8 Bromcresol purple 6.3 5.2- 6.8 Bromthymol blue 7.0 6.0- 7.6 p-Nitrophenol 7.1 5.6- 7.6 Phenol red 7.9 6.8- 8.4 Cresol red 8.3 7.2- 8.8 Metacresol purple 8.32 7.4- 9.0 Thymol blue 8.9 8.0- 9.6 Cresolphthalein 9.4 8.2- 9.8 Phenolphthalein 9.4 8.3-10.0 Thymolphthalein 9.4 9.2-10.6 Alizarine yellow - 10.0-12.0 Nitramine 11.0-13,0 Color Change Acid Alkaline Red Red Yellow Yellow Red Yellow Yellow Yellow Yellow Colorless Yellow Yellow Yellow Yellow Colorless Colorless Colorless Yellow Colorless Yellow Yellow Blue Blue Yellow Red Red Purple Blue Yellow Red Red Purple Blue Red Red Blue Lilac Orange-brown Determination of pH: With Buffer Solutions. If a series of buffer solutions of known pH, which must lie in the region of the pH to be determined, is available the estimation of the unknown pH is a relatively simple matter. It is first necessary to choose, by preliminary experi- ments, an indicator that exhibits a definite intermediate color in the solution under examination. The color produced is then compared with that given by the same amount of the indicator in the various solutions of known pH. In the absence of a "salt error," to which reference will be made later, the pH of the unknown solution will be the same as that of the buffer solution in which the indicator exhibits the same color. Provided a sufficient number of solutions of known pH are available, this method can give results which are correct to about 0.05 pH unit. BJERRUM'S WEDGE METHOD 365 When colored solutions are being studied, allowance must be made for the superimposition of the color on to that of the indicator; this may be done by means of the arrangement shown in plan in Fig. 97. The colored experimental solu- tion, to which a definite amount of indicator has been added, is placed in the tube A and pure water is placed in B\ the tube C contains the test solution without indicator, and D contains the buffer solution of known pH together with the same amount of indicator as in A. The solution in D is varied until the color of C and D super- imposed is the same as that of A and B super- imposed. The pH of the solution in A is then the same as that in D. Determination of pH: Without Buffer Solutions. Provided the con- stant of an indicator is known, it is possible to determine the pH of an unknown solution without the use of buffer solutions; the methods are the same in principle as those employed for the evaluation of the indi- cator constant, except that in one case the pH of the solution is supposed to be known while pfcj n is determined and in the other the reverse is true. For this purpose, equation (18), after taking logarithms, may be written as TT , , . Ionized form pH = pfci n + log FIG. 97. Indicator measurements with colored solutions + Un-ionized form Color due to alkaline form (23) i.o Alkaline form (Ia) 0.75 0.60 0.25 [ l I Color due to acid form The problem of determining pH values thus reduces to that of measuring the ratio of the two extreme colors exhibited by a particular indicator in the given solution. L Bjerrum's Wedge Method. 12 A rectangular glass box is divided into two wedge-shaped compartments by the insertion of a sheet of glass diagonally, or two separate wedges are cemented together by Canada balsam to give a vessel of the form shown in pluri in Fig. 98. A solution of the indicator which has been made definitely acid is placed in one wedge, and one that is definitely alkaline is placed in the other. By viewing the combination from the front a gradation of colors, from the acid to the alkaline forms of the indicator, can be seen as a result of the superposi- tion of steadily decreasing amounts of acid color on increasing amounts "Hjerrum, Ahren's SammlunK, 1914, No. 21; Kolthoff, Rec. trav. chim., 43, 144 (1924); McCrae, Analyst, 51, 287 (1926). !5 0,50 Acid form T 0.75 0- 1.0 Fia. 98. Representation of Bjerrum wedge 366 THE DETERMINATION OP HYDROGEN IONS B * 2 of the alkaline color. The test solution is placed in a narrow glass box of the same thickness as the combined wedges (Fig. 98, A) and the indicator is added so that its concentration is the same as in the wedges. A position is then found at which the color of the test solution matches that of the superimposed acid and alkaline colors; the ratio of the depths of the wedge solutions at this point thus gives the ratio of the colors required for equation (23). If the sides of the box are graduated, as shown, the depths of the two solutions can be obtained and the corre- sponding pH evaluated. The double-wedge can of course be calibrated so that the logarithmic term, i.e., the second term on the right-hand side, of equation (23) can be read off directly. II. Colorimeter Method. One of the simplest forms of colorimeter is shown in Fig. 99; the experimental solution is placed in the vessel A and an amount of indicator, giving a known concentration, is added; the fixed flat-bottomed tube B contains ^ater to a definite height. The fixed tube 0, ar- ranged at the same level as B, also contains water to the same height as in B. Surround- ing C is a movable tube D in which is placed the acid form of the indicator, and this is surrounded by the vessel E containing the in- dicator in its alkaline form; the concentra- tion of the indicator in D and E is the same as in the test solution in A. The inner tube D is moved up and down until the color as seen through (7, D and E is the same as that seen through B and A ; the ratio of the alkaline color to the acid color in A is then given by the ratio of the heights Ai/fe, so that the pll can be calculated if these heights are measured. If the test solution is colored, the water in C is replaced by the test solution to an equal depth; its color is then superimposed on that of the indicator in each case. By the use of special colorimeters it is possible to match the colors with such precision that pll values can be estimated with an accuracy of 0.01 unit. HI. Spectrophotometric Method. 13 The use of absorption spectra permits an accurate estimate to be made of the ratio of the amounts of the two colors in a given solution; the method is the same in principle as that already referred to on pages 328 and 363. In order to show the magnitude of the effect on the absorption of light resulting from a change of pH, the transmission curves obtained for bromcresol green in solutions of various pH's are shown in Fig. 100. It is evident that once the extent of the absorption produced by the completely alkaline form of the indi- " Erode, J. Am. Chem. Soc., 46, 581 (1924); Holmes, ibid., 46, 2232 (1924); Holmes and Snyder, ibid., 47, 221, 226 (1925); Vies, Compt. rend., 180, 584 (1925); Fortune and Mellon, J. Am. Chem. Soc., 60, 2607 (1938). C D E FIG. 99. Colorimeter for pH determinations ERRORS IN MEASUREMENTS WITH INDICATORS 367 cator is known, the proportion present in a given solution, and hence the pH, can be estimated with fair accuracy. 60 6400 A Fia. 100. Light absorption of bromoresol green (Fortune and Mellon) 4800 5600 Wave Length Errors in Measurements with Indicators. Three chief sources of error in connection with pi I determinations by means of indicators may be mentioned. 14 In the first place, if the test solution is not buffered, eg., solutions of very weak acids or bases or of neutral salts of strong acids and bases, the addition of the indicator may produce an appreciable change of pH; this source of error may be minimized by employing small amounts of indicator which have been previously adjusted, as a result of preliminary experiments, to have approximately the same pH as the test solution. Such indicator solutions are said to be isohydric with the test solution. The second possible cause of erroneous results is the presence of proteins; as a general rule, indicator methods are not satisfactory for the determination of pH in protein solutions. The error varies with the nature of the indicator; it is usually less for low molecular weight com- pounds than for complex molecules. Appreciable quantities of neutral salts produce color changes in an indicator that are not due to an alteration of pH and hence lead to erro- neous results. This effect of neutral salts is due to two factors, at least; in the first place, the salt may affect the light absorbing properties of one or both forms of the indicator; and, in the second place, the altered ionic strength changes the activity of the indicator species. In deriving equation (17) the activities of the un-ionized and ionized forms of the indicator were taken to be the same as the respective concentrations; this can only be reasonably true if the ionic strength of the solution is 14 McCrumb and Kenny, J. floe. Chem. Ind. t 49, 425T (1930); Kolthoff and Rosen- blum, "Acid- Base Indicators," 1937, Chap. X. 368 THE DETERMINATION OF HYDROGEN IONS low, otherwise equation (19) should be written /HIn OH + = Kin /In" where /Hin and /i n - are the activity coefficients of the un-ionized and ionized species, respectively. Taking logarithms, this equation can be put in the form P H - pK In + log and the use of the extended Debye-Hiickel equation for log (/i n -//mn) then gives P H = ptfm + log T- - A + C v . (24) 1 a For a given color tint, i.e., corresponding to a definite value of a/(l a), the actual pH will clearly depend on the value of the ionic strength of the solution; at low ionic strengths, e.g., less than about 0.01, the neutral salt error \s negligible for most purposes. The actual neutral salt error is less than estimated by equation (24) because the experimental values of pK i n are generally based on deter- minations made in buffer solutions of appreciable ionic strength. For equation (24) to be strictly applicable the indicator exponent p/f i n should be the true thermodynamic value obtained by extrapolation to infinite dilution. Universal Indicators. Since the pH range over which a given indi- cator can be employed is limited, it is always necessary, as mentioned above, to carry out preliminary measurements with an unknown solution in order to find the approximate pH; with this information available the most suitable indicator can be chosen. For the purpose of making these preliminary observations the so-called universal indicators have been found useful: 1B they consist of mixtures of four or five indicators, suitably chosen so that they do not interfere with each other to any extent, which show a series of color changes over a range of pH from about 3 to 11. A convenient and simple form of universal indicator can be prepared by mixing equal volumes of 0.1 per cent solutions of methyl red, a-naphtholphthalein, thymolphthalein, phenolphthalein and brom- thymol blue; the colors at different pH values are given below. pH 4 5 6 78 9 10 11 Color Red Orange- Yellow Green- Green Blue- Blue- Red- red yellow green violet violet Carr, Andy*, 47, 196 (1922); Clark, "The Determination of Hydrogen Ions," 1928, p. 97; Britton, "Hydrogen Ions/' 1932, p. 286; Kolthoff and Rosenblum, "Acid- Base Indicators/' 1937, p. 170. PROBLEMS 369 The addition of a small quantity of this universal indicator to an un- known solution permits the pH of the latter to be determined very approximately; it is then possible to choose the most suitable indicator from Table LXIV in order to make a more precise determination of the pH. Universal indicators are frequently employed when approximate pH values only are required, as, for example, in certain processes in qualitative and gravimetric analysis, and for industrial purposes. PROBLEMS 1. What are the pH values of solutions whose hydrogen ion concentrations (activities) are 2.50, 4.85 X 10" 4 and 0.79 X 10~ 10 g.-ion per liter? Assuming complete dissociation and ideal behavior, evaluate the pH of 0.0095 N sodium hydroxide at 25. 2. A hydrogen gas electrode in 0.05 molar potassium acid phthalate, when combined with a calomel electrode containing saturated potassium chloride, gave a cell with an E.M.F. of 0.4765 volt at 38. Calculate the pH of the solution which gave an E.M.F. of 0.7243 volt in a similar cell. 3. The glass electrode cell Pt | Quinhydrone pH 4.00 Buffer | Glass | pH 7.63 Buffer Quinhydrone | Pt gave an E.M.F. of 0.2265 volt at 25; calculate the asymmetry potential of the glass electrode. 4. What are the hydrogen ion activities of solutions of pH 13.46, 5.94 and 0.5? What are the corresponding hydroxyl ion activities at 25, assuming the activity of the water to be unity in each case? 5. A quinhydrone electrode in a solution of unknown pH was combined with a KC1(0.1 N), Hg2Cl 2 (s), Hg electrode through a saturated potassium chloride salt bridge; the E.M.F. of the resulting cell was 0.3394 volt at 30. Calculate the pH of the solution. 6. If the oxygen electrode were reversible, what change of E.M.F. would be expected when an oxygen gas electrode at 1 atm. pressure replaced (i) a hydro- gen electrode at 1 atm., and (ii) a quinhydrone electrode, in a given cell at 25? 7. An indicator is yellow in the acid form and red in its alkaline form; when placed in a buffer solution of pH 6.35 it was found by spectrophoto- metric measurements that the extent of absorption in the yellow region of the spectrum was 0.82 of the value in a-solution of pH 3.0. Evaluate p&i n for the given indicator. CHAPTER XI NEUTRALIZATION AND HYDROLYSIS Types of Neutralization. The term neutralization is generally ap- plied to the reaction of one equivalent of an acid with one equivalent of base; if the terms "acid" and "base" are employed in the sense of the general definitions given in Chap. IX, the products are not necessarily a salt and water, as in the classical concept of acids and bases, but they are the conjugate base and acid, respectively, of the reacting acid and base. For the reaction between conventional acids, such as hydro- chloric, acetic, etc., and strong bases, such as hydroxides in water or alkyloxides in alcohols, there is no difference between the new and the old points of view; it is, however, preferable to discuss all neutralizations from the general standpoint provided by the modern theory of acids and bases. According to this the following reactions are all examples of neutralization: HC1 + (Na+)OC,Hr = (Na+)Cl- + C,II 6 OH CHaCOjH + (Na+)OH- = (Na+)CII 3 CO^ + H 2 O HOI + RNIT 2 = a- + RNIIJ IIC1 + (Na+)CH 3 COj = (Na+)Cl- + CH 3 CO 2 H RNHf(Cl-) + (Na*-)OH- = RNH 2 + H 2 O + (Na+Cl-). Acidi Bases Basei AcicU The last two reactions are of special interest, since they belong to the category usually known as "displacement reactions"; in the first of the two a strong acid, hydrochloric acid, displaces a weak acid, acetic acid, from its salt, while in the second a weak base, e.g., ammonia or an aminn, is displaced from its hydrochloridc by a strong base. It will be seen later that a much better understanding of these processes can be obtained by treating them as neutralizations, which in fact they are if this term is used in its widor sense. Incomplete Neutralization: Lyolysis. The extent to which neutrali- zation occurs, when one equivalent of arid and base are mixed, depends on the nature of the acid, the base and the solvent. If the acid is HA, the base is B and SH Is an amphiprotic solvent, i.e., one which can function either as an acid or as a base, the neutralization reaction HA + B ^ BH* + A- takes place, but in addition, since the solvent is amphiprotic, two proc- esses involving it can occur, thus (a) BH+ + SH ^ Silt + B, (la) Acid Base Acid Base 370 CONDITIONS FOR COMPLETE NEUTRALIZATION 371 and (6) SH + A- ^ HA + S- (16) Acid Base Acid Base In the first of these the free base B is re-formed while in the second the free acid HA is regenerated; it follows, therefore, that the processes (a) and (6) militate against complete neutralization. This partial re- versal of neutralization, or the prevention of complete neutralization, is called by the general name of lyolysis or solvolysis; in the particular case of water as solvent, the term used is hydrolysis. Conditions for Complete Neutralization. In order that neutraliza- tion may be virtually complete it is necessary that the lyolysis reactions should be reduced as far as possible. For reaction (a) to be suppressed it is necessary that B should be a much stronger base than the solvent SH, so that the equilibrium lies to the left. Further, the actual neutrali- zation reaction equilibrium must lie to the right if it is to be practically complete; this means that B must be a stronger base than the anion A~. For the complete neutralization, therefore, the order of basic strengths must be A- < B > SH. If B is a weak base, it is necessary that A~ should be still weaker; it has been seen (p. 307) that a strong acid will have a very weak conjugate base, and hence this condition is satisfied if HA is a very strong acid. It is also necessary that the solvent should be a very weak base, and this can be achieved by using a strongly protogenic, i.e., acidic, medium. It has been found, in agreement with these conclusions, that extremely weak bases, e.g., acetoxime, can be neutralized completely by means of perchloric acid, the strongest known acid, in acetic acid as solvent. In water, hydrolysis of the type (a) is so considerable that neutralization of acetoxime, even by means of a strong acid, occurs to a negligible extent only. By similar arguments it can be shown, from a consideration of the lyolytic equilibrium (6), that if an acid HA is to be neutralized com- pletely, the condition is that the order of acid strengths must be BH+ < HA > SH. To neutralize completely a weak acid HA it is necessary, therefore, to use a very strong base, so that its conjugate acid BH+ is extremely weak, and to work in a protophilic medium, such as ether, acetonitrile or, preferably, liquid ammonia. It will be evident from the conclusions reached that the lyolysis process (a) is due primarily to the weakness of the base B, whereas the process (fe) results from the weakness of the acid HA. If both acid and base are weak in the particular solvent, then both types of lyolysis can occur, and complete neutralization is only possible in an aprotic solvent, 372 NEUTRALIZATION AND HYDROLYSIS provided the proton donating tendency of the acid HA is considerably greater than that of BH+, or the proton affinity of the base B is greater than that of A~ (cf. p. 331). If the medium is exclusively protophilic, e.g., acetonitrile, then only the (a) type of lyolysis, namely that involving a weak base, is possible; weak acids should be completely neutralized provided a strong base is used. Similarly, in an exclusively protogenic solvent, e.g., hydrogen fluoride, the (6) type of lyolysis only can occur; a weak base can thus be completely neutralized in such a medium if a sufficiently strong acid is employed. Hydrolysis of Salts. The subject of lyolysis, or hydrolysis, in the event of water being the solvent, can be treated from two angles; in the general treatment already given it has been considered from the point of view of incomplete neutralization, and a return will be made later to this aspect of the subject. Another approach to the phenomena of hydrolysis is to study the equilibria resulting when a salt is dissolved in the given solvent; the situation is, of course, exactly the same as that which arises when an equivalent of the particular acid constituting the salt is neu- tralized by an equivalent of the base. This particular aspect of the subject of hydrolysis will be treated here; it is convenient to consider the material with special reference to the salt of (a) a weak acid, (b) a weak base, and (c) a weak acid and weak base. The first two of these are often referred to as salts of "one-sided" weakness, and the latter as a salt of "two-sided" weakness. Salts of strong acids and strong bases do not undergo hydrolytic reaction with the solvent, because the con- jugate base and acid, respectively, arc extremely weak; such salts, there- fore, will not be discussed in this section, but reference will be made below to the neutralization of a strong acid by a strong base. I. Salt of Weak Acid and Strong Base. When a salt, e.g., NaA, of a weak acid HA is dissolved in water, it may be regarded as undergoing complete dissociation into Na + and A~ ions, provided the solution is not too concentrated. Since HA is a weak acid the conjugate base A~ will be moderately strong; hence the latter will react with the solvent mole- cules (II 2 O) giving the type of hydrolytic equilibrium represented by equation (16); in the particular case of water as solvent, this may be written A- + H 2 ^ HA + OH- Unhydro- Free Free lyzed salt acid base The hydrolysis of the salt thus results in the partial reformation of the free weak acid HA and of the strong base (Na+)OH- from which the salt was constituted. As a consequence of the weakness of the acid HA, there- fore, there is a partial reversal of neutralization, and the term hydrolysis is often defined in this sense. It will be observed that the hydrolytic process results in the formation of OH~ ions, and this must obviously be accompanied by a decrease of hydrogen ion concentration (cf. p. 339); SALT OF WEAK ACID AND STRONG BASE 373 the salt of a weak acid and a strong base thus reacts alkaline on account of hydrolysis. This accounts for the well-known fact that such salts as the cyanides, acetates, borates, phosphates, etc., of the alkali metals are definitely alkaline in solution. Application of the law of mass action to the hydrolytic equilibrium gives the hydrolysis constant (K h ) of the salt as (2) the activity of the water being, as usual, taken as unity. The ionic product of water (K w ) and the dissociation constant (K a ) of the acid and HA are defined by TS OH^A- . K a = - > hence, it follows immediately from these expressions and equation (2) that K* = TF' (3) A The hydrolysis constant is thus inversely proportional to the dissociation constant of the weak acid; * the weaker the acid the greater is the hy- drolysis constant of the salt. If the activities are replaced by the product of the concentration and activity coefficient in each case, equation (2) becomes " /HA/OH" ,.. CA~ In solutions of low ionic strength the activity coefficient /HA of the un- dissociated molecules is very close to unity, and, further, the ratio of the activity coefficients of the two univalont ions, i.e., /OH-//A-, is then also unity, by the Debye-IIuckel limiting law; equation (4), therefore, reduces to the less exact form which is particularly applicable to dilute solutions. As in other cases, the thermodynamic constant K h has been replaced by the approximate "constant," kh* The degree of hydrolysis (z) is defined as the fraction of each mole of salt that is hydrolyzed when equilibrium is attained. If c is the stoichiometric, i.e., total, concentration of the salt NaA in the solution, the concentration of unhydrolyzed salt will be c(l x) ; since this may be regarded as completely dissociated into Na+ and A~ ions, it is possible * It should be noted that the hydrolysis constant is equal to the dissociation con- stant of the base A~ which is conjugate to the acid HA. 374 NEUTRALIZATION AND HYDROLYSIS to write CA- = c(l - x). In the hydrolytic reaction, equivalent amounts of OH~ and HA are formed, and if the dissociation of the latter is neglected, since it is likely to be very small especially in the presence of the large concentration of A~ ions, it follows that COH~ and CHA must be equal; further, both of these must be equal to ex, where x is the fraction of the salt hydrolyzed; hence, Coir = CHA == ex. Substitution of these values for CA~ and Coir in equation (5) gives From equation (7) it is possible to calculate the degree of hydrolysis at any desired concentration, provided the hydrolysis constant of the salt, or the dissociation constant of the acid [cf. equation (3)], is known. If kh is small, e.g., for the salt of a moderately strong acid, at not too small a concentration equation (7) reduces to ' (8) so that the degree of hydrolysis is approximately proportional to the square-root of the hydrolysis constant and inversely proportional to the square-root of the concentration of the salt solution. The result of equa- tion (8) may be expressed in a slightly different form by making use of equation (3) which may be written, for the present purpose, as k h = k u lk a ] thus, ' " If two salts of different weak acids are compared at the same concen- tration, it is seen that so that the degree of hydrolysis of each is inversely proportional to the square-root of the dissociation constant of the acid; hence the weaker the acid the greater the degree of hydrolysis at a particular concentra- tion. For a given salt, equation (9) shows the degree of hydrolysis to increase with decreasing concentration. By making use of equation (7) or (8) it is possible to calculate the degree of hydrolysis of the gait of a strong base and a weak acid of known SALT OF WEAK ACID AND STRONG BASE 375 dissociation constant at any desired concentration. The results of such calculations are given in Table LXV; the temperature is assumed to be TABLE LXV. DEQIIEE OF HYDROLYSIS OF SALTS OF WEAK ACIDS AND STRONG BASES AT 25 Concentration of Solution k a k h 0.001 N 0.01 N 0.1 N 1.0 N 10 < 10 10 3.3 X 10 4 10~ 4 3.2 X 10 B 10~ 6 10 - 10- 8 3.2 X 10~ 3 10- 3 3.2 X10" 4 10~ 4 10' 8 10" 8 3.2 X 10 2 lO^ 2 3.2 X 10 3 10~ 3 10 10~ 4 0.27 0.095 3.2 X 10 2 lO" 8 about 25, so that k w can be taken as 10~~ 14 . It is seen that the degree of hydrolysis increases with decreasing strength of the acid and decreasing concentration of the solution. In a 0.001 N solution, the sodium salt of an acid of dissociation constant equal to 10~ 10 , e.g., a phenol, is hydro- lyzed to an extent of 27 per cent. It may be noted that equations (7) and (8) give almost identical values for the degree of hydrolysis in Table LXV, except for the two most dilute solutions of the salt of the acid of k a equal to 10~ 10 . In these eases the approximate equation (8) would give 0.32 and 0.10, instead of 0.27 and 0.095 given in the table. It has been seen above that COH is equal to ex, and since the product of fa f and Coir is k w , it follows that and introducing the value of x from equation (9), the result is CH+ = Taking logarithms and changing the signs throughout, this becomes - log c u + = ~ i log k w log k a + % log c. (12) As an approximation, log C H + may be replaced by pH, and using the analogous exponent forms for log k w arid log A: , it follow* that pH = \ pk w + \ pfc + \ log c. (12a) It is seen, therefore, that the pH, or alkalinity, of a solution of the salt of a weak acid and strong base increases with decreasing acid strength, i.e., increasing pk a , and increasing concentration. Attention may be called to the fact that although the degree of hydrolysis decreases with increasing concentration of the salt, the pH, or alkalinity, increases. The pH values in Table LXVI have been calculated for dissociation con- stants and salt concentrations corresponding to those in Table LXV; equation (12) is satisfactory in all cases for which (8) is applicable, but 376 NEUTRALIZATION AND HYDROLYSIS TABLE LXVI. VALUES OP pH IN SOLUTIONS OP SALTS OP WEAK ACIDS AND STRONG BASES AT 25 Concentration of Solution * k h 10-4 10-w O.OOt N 0.01 N 0.1 N 1.0 N 7.5 8.0 8.5 9.0 8.5 9.0 9.5 10.0 9.5 10.0 10.5 11.0 10.4 11.0 11.5 12.0 !0-w 10-4 in the others use has been made of the x values in Table LXV together with equation (11). Since the pH of a neutral solution is about 7.0 at 25, it follows that the solutions of salts of weak acids can be considerably alkaline in reaction. It was seen in Chap. IX that the dissociation constant of an acid undergoes relatively little change with temperature between and 100; on the other hand the ionic product of water increases nearly five hundred- fold. It is evident, therefore, from equation (3) that the hydrolysis constant will increase markedly with increasing temperature; the degree of hydrolysis and the pH at any given concentration of salt will thus in- crease at the same time. EL Salt of Weak Base and Strong Acid. When the base B is weak, the conjugate acid BH+ will have appreciable strength and hence it will tend to react with the solvent in accordance with the hydrolytic equi- librium (la). It follows, therefore, that if the salt of a weak base and a strong acid is dissolved in water there will be a partial reversal of neutralization, some of the acid II 3 O+ and the weak base B being re- generated; in other words, the salt is hydrolyzed in solution. If the weak base is of the type RNH 2 , e.g., ammonia or an amine, the con- jugate acid is RNHiJ~, and when the salt, e.g., RNH 3 C1, is dissolved in water it dissociates virtually completely to yield RNHf and Cl~" ions, the former of which establish the hydrolytic equilibrium RNHt + H 2 ^ H 3 0+ + RNH 2 . When the weak base is a metallic hydroxide, it is probable that the conjugate acid is the hydrated ion of the metal, e.g., Fe(H 2 O)j" H+ or Cu(H 2 O)t + , which may be represented in general by M(H 2 0)mJ the hydrolysis must then be expressed by M(H 2 0)+ + H 2 ^ H 3 0+ + M(H 2 0) m _ 1 OII, where M(H 2 0) m _i(OH) is the weak base. The formation of H 3 0+ ions shows that the solutions react acid in each case. Writing the hydrolytic equilibrium in the general form BH+ + H 2 ^ H 3 0+ + B, Unhydro- Free Free lyzed salt acid SALT OP WEAK BASE AND STRONG ACID 377 application of the law of mass action gives, for the hydrolytic constant, and since , and a B it follows that (14) where K b is the dissociation constant of the base B. It is seen that equation (14) is exactly analogous to (3), except that KI now replaces K a . By making the same assumptions as before, concerning the neglect of activity coefficients in dilute solution, equation (13) reduces to i CH * CB ,, K , k h = i (15) CBH+ and from this, since CH+ is now equal to CB, both of which are equal to ex, while CBH+ is equal to c(l x), it follows that J. (16) which is identical in form with equation (6). The degree of hydrolysis in this case is, consequently, also given by equation (7) which reduces to (8) provided the base is not too weak or the solution too dilute. Re- placing k h now by fc,/fc&, by the approximate form of equation (14), it follows that (17) The same general conclusions concerning the effect of the dissociation constant of the weak base and the concentration of the salt on the degree of hydrolysis are applicable as for the salt of a weak acid. The results in Table LXV would hold for the present case provided the column headed k a were replaced by fc&. Further, since the dissociation constants of bases do not vary greatly with temperature, the influence of increasing temperature on the hydrolysis of the salt of a weak base will be very similar to that on the salt of a weak acid. The hydrogen ion concentration CH+ in the solution of a salt of a weak base is given by ex, as mentioned above, and if the value of x from equa- tion (17) is employed, it follows that 378 NEUTRALIZATION AND HYDROLYSIS This result may he expressed in the logarithmic form pH-ipfc.- Jrfr fc -*logc. (18) It is evident that the pTI of the solution must be less than $pk w , i.e., less than 7.0, and so solutions of salts of the type under consideration will exhibit an acid reaction. It was seen on page 339 that in any aqueous solution pH + pOII = pk w , hence in this particular case pOH - lpk u + Jpfa + 1 log c, (19) which is exactly analogous to equation (12a), except that pOII and pkt> replace pH and pfc,, respectively. It follows, therefore, that the results in Table LXVI give the pOH values in solutions of salts of a weak base, provided the column headed pA" a is replaced by pA&. III. Salt of Weak Acid and Weak Base. If both the acid and base from which a given salt is made are weak, the respective conjugate base and acid will have appreciable strength and consequently will tend to interact with the amphiprotic solvent water. When a salt such as ammonium acetate is dissolved in water, it dissociates almost completely into NH^ and Ac~ ions, and these acting as acid and base, respectively, take part in the hydrolytic equilibria NI1| + II a O -^ II 3 0< + NII 3 , and Ac- + 11,0 ^ II Ac + Oil-. Combining the two equations, the complete equilibrium is Nllf -I- Ac- + 2II 2 ^ 11,0 * + OH" + Nil, + HAc, or, representing the? weak base hi general by B and the acid by TIA, BH+ + A" + 2H,() ;-^ H 3 0+ + OH - + B + ITA. Since the normal equilibrium between water molecules and hydrogen and hydroxyl ions, viz., 2H 2 O ^ H 3 O+ + Oil', xi*ts in any event, this may be subtracted from the hydrolytic equi- librium; the result may thus be represented by NHt + Ac- ^ NH S + HAc for ammonium acetate or, in the general case, by BH+ + A Unhydro- Free Fne lyzed salt acid base SALT OF WEAK ACID AND WEAK BASE 379 The law of mass action then gives for the hydrolysis constant (20) and introduction of the expressions for K a and K b leads to the result The hydrolysis constant equation (20) may also be written as /HA/B --- - , N (22) BHA- BHVA- and since this expression involves the product of the activity coefficients of two univalent ions, instead of their ratio as in the previous cases, it is less justifiable than before to assume that the activity coefficient fraction will become unity in dilute solution. Nevertheless, this approximation can be made without introducing any serious error, and the result is (23) BHA- ' If the original, i.e., stoichiometric, concentration of the salt is c moles per liter, and x is tho degree of hydrolysis, then CHA and CB may both be set equal to c.r, whereas f BH + and CA~ are both equal to the concentration of unhydrolyxcd salt c(l -- x), the salt being regarded as completely dissociated. Insertion of these values in equation (23) then gives If V/c/, is small in comparison with unity, it may be neglected in the denominator so that equation (25) becomes x VA-,~, (26) or, introducing the approximate form of equation (21) for kh, It appears from equations (25), (26) and (27) that the degree of hydroly- sis of a given salt of two-^ided weakness is independent of the concen- tration of the solution; this conclusion is only approximately true, as will be seen shortly. 380 NEUTRALIZATION AND HYDROLYSIS The hydrogen ion concentration of the solution of hydrolyzed salt may be calculated by using the expression for the dissociation function of the acid, k a ', thus, CHA / CHA cx - - ~~~ A/a / ^ v ~ o -i c(l x) 1 x By equation (24), the fraction x/(l x) is equal to or, expressed logarithmically, pH = pfc w + $pk a - P fc 6 . (29) If the dissociation constants of the weak base and acid are approximately equal, i.e., pk a is equal to pk b) it follows that pH is %pk w ; the solution will thus be neutral, in spite of hydrolysis. If, on the other hand, k a is greater than k b , the salt solution will have an acid reaction; if k a is less than kb the solution has an alkaline reaction. As a first approximation the pH of a solution of a salt of a weak acid and weak base is seen to be independent of the concentration. The conclusion that the degree of hydrolysis and pH of a solution of a salt of double-sided weakness is independent of the concentration is only strictly true if CBH+ is equal to C A - and if CB is equal to CHA, as assumed above. This condition is only realized if k a and kb are equal, but not otherwise. If the dissociation constants of HA and B are differ- ent, so also will be those of the conjugate base and acid, i.e., A~ and BH+, respectively. The separate hydrolytic reactions A- + H 2 ^ HA + OH- and BH+ + H 2 ^ H 3 0+ + B, will, therefore, take place to different extents, so that the equilibrium concentrations of A- and BH+, on the one hand, and of HA and B, on the other hand, will not be equal. The assumptions made above, that CBH* is equal to CA- and that CB is equal to CHA, are consequently not justifiable, and the conclusions drawn are not strictly correct. The problem may be solved in principle by writing c = CA- + COH- = CBH+ + CB, HYDROLYSIS OF ACID SALTS 381 where the total concentration c is divided into the unhydrolyzed part, i.e., CA~ or CBH+, and the hydrolyzed part, i.e., COIT or C B , respectively. Further, by the condition of electrical neutrality, CH+ + CBH+ = COBT + CA~, and if these equations are combined with the usual expressions for k a , kb and k w , it is possible to eliminate CA-, COH~, CB and CBH+, and to derive an equation for CH + in terms of c and k a , kb and k w . Unfortunately, the resulting expression is of the fourth order, and can be solved only by a process of trial and error. The calculations have been carried out for aniline acetate (k a = 1.75 X 10~ 5 , k b = 4.00 X 10~ 10 ): at concentrations greater than about 0.01 N the result for the hydrogen ion concentration is practically the same as that obtained by the approximate method given previously. In more dilute solutions, however, the values differ some- what, the differences increasing with increasing dilution. 1 Hydrolysis of Acid Salts. The acid salt of a strong base and a weak dibasic acid, e.g., NaHA, will be hydrolyzed in solution because of the interaction between the ion HA~", functioning here as a base, and the solvent, thus HA- + H 2 O ^ H 2 A + OH". The ion HA~ can also act as an acid, HA- + H 2 ^ H 8 0+ + A~, and the H 3 0+ ions formed in this manner may interact with HA" to form H 2 A, thus HA- + H 3 0+ = H 2 A + H 2 O. If it were not for this latter reaction CA~~ would have been equal to CH+I but since some of the hydrogen ions are removed in the formation of an equivalent amount of H 2 A, it follows that CA" = C H + + C H ,A. Further, if the salt NaHA is hydrolyzed to a small extent only, CHA~ will be almost equal to c, the stoichiometric concentration of the salt. With these expressions for CA~ and CHA-, together with the equations for ki and & 2 , the dissociation functions of the first and second stages of the acid H 2 A, viz., , CH+CHA- , , C H *C A " fa ._. - an( l 2 _ - , CH,A CHA~ it is readily possible to derive the result 1 Griffith, Trans. Faraday Soc., 17, 525 (1922). 382 NEUTRALIZATION AND HYDROLYSIS If ki is small in comparison with the concentration, so that it may be neglected in the denominator, equation (30) reduces to the simple form c H * = VA^, (31) .'. pll = -JpA-! + Jpfe. (32) In this case, therefore, tho pll of the solution is independent of the con- centration of the acid salt. The difference between the results given by equations (30) and (31) increases with increasing dilution, as is to be expected. If ki is less than about 0.01 c, however, the discrepancy is negligible. Displacement of Hydrolytic Equilibrium. When a salt is hydrolyzed, the equilibrium Unhydrolyzed salt + Water ^ Free acid -f Free base is always established; this -equilibrium can be displaced in either direction by altering the concentrations of the products of hydrolysis. The addi- tion of either the free acid or the free base, for example, will increase the concentration of unhydrolyzed salt and so repress the hydrolysis; this fact is utilized in a method for investigating hydrolytic equilibria (p. 383). If, on the other hand, the free acid or base is removed in some manner, the extent of hydrolysis of the salt must increase in order to maintain the hydrolytic equilibrium. For example, if a solution of potassium cyanide is heated or if a current of air is passed through it, the hydrogen cyanide formed by hydrolysis can be volatilized; as it is removed, however, more is regenerated by the continued hydrolysis of the potassium cyanide. When a solution of ferric chloride is heated, the hydrogen chloride is removed and hence the hydrolytic process continues; the hydra ted ferric oxide which is formed remains in colloidal solution and imparts a dark brown color to the system. Determination of Hydrolysis Constants : I. Hydrogen Ion Methods. A number of methods of varying degrees of accuracy have been proposed for the estimation of the degree of hydrolysis in salt solutions or of the hydrolysis constant of tho salt. One principle which can be used is to evaluate the hydrogen ion concentration of the solution; for a salt of a weak acid en 4 - is equal to k w /cx, where c is the stoichiometric concentration of the salt, and hence it follows from equation (16) that (33) If CH+ is known, the hydrolysis constant can be calculated. For a salt of a weak base, on the other hand, C H + is equal to ex; hence CONDUCTANCE METHOD 383 If the salt is one of two-sided weakness the hydrogen ion concentration alone is insufficient to permit k h to be evaluated; it is necessary to know, in addition, k a or fo>. The hydrogen ion concentration of a hydrolyzed salt solution can be determined by one of the E.M.F. or indicator methods described in Chap. X; it is true that the results obtained in this manner are not actual concentrations, but in view of the approximate nature of equations (33) and (34), the k h values are approximate in any case. n. Conductance Method. 2 In a solution containing c equiv. per liter of a salt of a weak base and a strong acid, for example, there will be present c(l x) equiv. of unhydrolyzed salt and ex equiv. of both free acid and base. If the base is very weak, it may be regarded as com- pletely un-ionized, and so it will contribute nothing towards the total conductance of the solution of the salt. The conductance of 1 equiv. of a salt of a very weak base is thus made up of the conductance of 1 x equiv. of unhydrolyzed salt and x equiv. of free acid, i.e., A = (1 - z)A c + zA HA . (35) In this equation A is the apparent equivalent conductance of the solution, which is equal to 1000 K/C, where K is the observed specific conductance and c is the stoichiometric concentration of the salt in the solution; A c is the hypothetical equivalent conductance of the unhydrolyzed salt, and AHA is the equivalent conductance of the free acid in the salt solution. It follows from equation (35) that (36) and so the calculation of x involves a knowledge of A, AHA and A c . As mentioned above, A is derived from direct measurement of the specific conductance of the hydrolyzed salt solution; the value of AHA is generally taken as the equivalent conductance of the strong acid at infinite dilution, since its concentration is small, but it is probably more correct to use the equivalent conductance at the same total ionic strength as exists in tho salt solution. The method is, however, approximate only, and this re- finement is hardly necessary. The evaluation of A c for the unhydrolyzed salt presents a special problem. As already seen, the addition of excess of free base will repress the hydrolysis of the salt, and in the method employed sufficient of the almost non-conducting free base is added to the salt solution until the hydrolysis of the latter is almost zero. For example, with aniline hydro- chloride, free aniline is added until the conductance of the solution reaches a constant value; at this point hydrolysis is reduced to a negligible 1 Bredig, Z. physik. Chem., 13, 213, 221 (1894); Kanolt, ,/. Am. Chem. Soc., 29, 1402 (1907); Noyes, Sosman and Kato, ibid., 32, 159 (1910) ; Kameyama, Trans. Ekctrochem. Soc., 40, 131 (1921); Gulezian and Mtiller, J. Am. Chem. Soc., 54, 3151 (1932). 384 NEUTRALIZATION AND HYDROLYSIS amount. The conductance of the solution is virtually that of the unhydrolyzed salt, and so A c can be calculated. The data in Table LXVII are taken from the work of Bredig (1894) on a series of solutions TABLE LXVII. HYDROLYSIS OP ANILINE HYDBOCHLOBIDB AT 18 FROM CONDUCTANCE MEASUREMENTS * c A Aj \' e ' x k H X 10 6 0.01563 106.2 96.0 95.9 0.036 2.1 0.00781 113.7 98.2 98.1 0.055 2.5 0.00391 122.0 100.3 100.1 0.077 2.5 0.00195 131.8 101.5 101.4 0.109 2.6 0.000977 144.0 103.3 103.3 0.147 2.5 * Bredig's measurements are not accurate because they were based on an incorrect conductance standard; the values of x and kn derived from them are, however, not affected. of aniline hydrochloride of concentration c equiv. per liter and observed equivalent conductance A; the columns headed A c and A" give the meas- ured equivalent conductances in the presence of N/64 and N/32, respec- tively, added free aniline. Since the values in the two columns do not differ appreciably, it is evident that N/64 free aniline is sufficient to repress the hydrolysis of the aniline hydrochloride almost to zero; hence either AC or AC' may be taken as equal to the required value of A c . Taking AHA for hydrochloric acid as 380 at 18, the degree of hydrolysis x has been calculated in each case; from these the results for kh in the last column has been derived. The values are seen to be approximately con- stant at about 2.5 X 10~ 5 . For the salt of a weak acid, the method would be exactly similar to that described above except that excess of the free acid would be added to repress hydrolysis. The equation for the degree of hydrolysis is then A - A c x = AMOH A c where AMOH is the equivalent conductance of the strong base. The con- ductance method has also been used to study the hydrolysis of salts of weak acids and bases, but the calculations involved are somewhat com- plicated. The determinations of hydrolysis constants from conductance meas- urements cannot be regarded as accurate; the assumption has to be made that the added free acid or free base has a negligible conductance. This is reasonably satisfactory if the acid or base is very weak, e.g., a phenol or an aniline derivative, but for somewhat stronger acids or bases, e.g., acetic acid, an appreciable error would be introduced; it is sometimes possible, however, to make an allowance for the conductance of the added acid or base. DISTRIBUTION METHOD 385 HI. Distribution Method. 8 Another approximate method for study- ing hydrolysis is applicable if one constituent of the salt, generally the weak acid or base, is soluble in a liquid that is not miscible with water, while the salt itself and the other constituent are not soluble in that liquid. Consider, for example, the salt of a weak base, e.g., aniline hydrochloride ; the free base is soluble in benzene, in which it has a normal molecular weight, whereas the salt and the free hydrochloric acid are insoluble in benzene. A definite volume (vi) of an aqueous solution of the salt at a known concentration (c) is shaken with a given volume (t> 2 ) of benzene, and the amount of free aniline in the latter is determined by analysis. If m is the concentration in equiv. per liter of the aniline in benzene found in this manner, then the concentration of free aniline in the aqueous solution (CB) should be m/D, where D is the "distribution coefficient" of aniline between benzene and water; the value of D must be found by separate experiments on the manner in which pure aniline distributes itself between benzene and water, in the absence of salts, etc. The amounts of free aniline in the benzene and aqueous layers are mv 2 and mvi/D respectively; hence, the amount of free acid in the aqueous solution, assuming none to have dissolved in the benzene, must be the sum of these two quantities, i.e., mv* + mvi/D. Since this amount is present in a volume v\, it follows that the concentration of free acid in the aqueous solution (c n +) is mv^/vi + m/D. The concentration of un- hydrolyzed salt (CBH+) is equal to the stoiohiometric concentration (c) less the concentration of free acia, since the latter is equivalent to the salt that has been hydrolyzed; hence, CBH+ is equal to c mv 2 /vi m/D. The results derived above may then be summarized thus: mvz m CH+ = h 7: > vi D _ !? and mv% m ~~ vi /)' and so it follows from equation (15) that m \m C ~ Vl D By determining m y therefore, all the quantities required for the evalua- tion of fa by means of equation (38) are available, provided D is known 8 Farmer, J. Cham. Soc. y 79, 863 (1901); Farmer and Warth, t&id., 85, 1713 (1904); Williams and Soper, ibid., 2469 (1930). 386 NEUTRALIZATION AND HYDROLYSIS from separate experiments. The results in Table LXVIII, taken from the work of Farmer and Warth (1904), illustrate the application of the method to the determination of the hydrolysis of aniline hydrochloride; the non-aqueous solvent employed was benzene, for which D is 10.1, arid the volumes v\ and v z were 1000 cc. and 59 cc., respectively. The value of kh is seen to be in satisfactory agreement with that obtained for aniline hydrochloride by the conductance method (Table LXVII). TABLE LXVIII. HYDROLYSIS OF ANILINE HYDROCHLORIDE FROM DISTRIBUTION MEASUREMENTS c m CB = ^ cn+ TBII+ k k X 10* 0.0997 0.0124 000123 19.6 X 10 4 0.0978 2.4 0.0314 0.00628 0.000622 9.9 X 10~ 0304 2.0 The distribution method for studying hydrolysis can be applied to salts of a weak acid, provided a suitable solvent for the acid is available; the hydrolysis constant is given by an equation identical with (38), except that m now represents the concentration of free acid in the non- aqueous liquid. The same principle can be applied to the investigation of salts of two-sided weakness provided a solvent can be found which dissolves either the weak acid or the weak base, but not both. IV. Vapor Pressure Method. 4 If the free weak acid or weak base is appreciably volatile, it is possible to determine its concentration or, more correctly, its activity, from vapor pressure measurements. In practice the actual vapor pressure is not measured, but the volatility of the sub- stance in the hydrolyzed salt solution is compared with that in a series of solutions of known concentration. In the case of an alkali cyanide, for example, the free hydrogen cyanide produced by hydrolysis is appreciably volatile. A current of air is passed at a definite rate through the alkali cyanide solution and at exactly the same rate through a hydrogen cyanide solution; the free acid vaporizing with the air in each case is then ab- sorbed in a suitable reagent and the amounts are compared. The con- centration of the hydrogen cyanide solution is altered until one is found that vaporizes at the same rate as does the alkali cyanide solution. It may be assumed that the concentrations, or really activities, of the free acid are the same in both solutions. The concentration of free acid CHA in the solution of the hydrolyzed salt of the weak acid may be put equal to ex (cf. p. 374) and hence x and kh can be calculated. V. Dissociation Constant Method. All the methods described above give approximate values only of the so-called hydrolysis "constant" of the salt; the most accurate method for obtaining the true hydrolysis constant is to make use of the thermodynamic dissociation constants of the weak acid or base, or both, and the ionic product of water. For this < Worley et al., J. Chem. Soc., Ill, 1057 (1917) ; Trans. Faraday Soc., 20, 502 (1925) ; Britton and Dodd, J. Chem. Soc., 2332 (1931). STRONG ACID AND STRONG BASE 387 purpose equations (3), (4) and (21) are employed. The results derived in this manner are, of course, strictly applicable to infinite dilution, but allowance can be made for the influence of the ionic strength of the medium by making use of the Debye-Huckel equations. The methods I to IV are of interest, in so far as they provide definite experimental evidence for hydrolysis, but they would not be used in modern work unless it were not possible, for some reason or other, to determine the dissociation constant of the weak acid or base. It is of interest to note that some of the earlier measurements of kh were used, together with the known dissociation constant of the acid or base, to evaluate k w for water. For example, k h for aniline hydrochloride has been found by the conductance method (Table LXVII) to be about 2.5 X 10~ 6 , and k b for aniline is 4.0 X 10~ 10 ; it follows, therefore, that k w , which is equal to ktJth, should be about 1.0 X 10~ 14 , in agreement with the results recorded in Chap. IX. Neutralization Curves. The variation of the pH of a solution of acid or base during the course of neutralization, and especially in the vicinity of the equivalence-point, i.e., when equivalent amounts of acid and base are present, is of great practical importance in connection with analytical and other problems. It is, of course, feasible to measure the pH experi- mentally at various points of the neutralization process, but a theoretical study of the subject is possible and the results are of considerable interest. For this purpose it is convenient to consider the behavior of different types of acid, viz., strong and weak, with different bases, viz., strong and weak. For the present the discussion will be restricted to neutralization involving a conventional acid and base in aqueous solution, but it will be shown that the results can be extended to all forms of acids and bases in aqueous as well as non-aqueous solvents. L Strong Acid and Strong Base. The changes in hydrogen ion con- centration occurring when a strong base is added to a solution of a strong acid can be readily calculated, provided the acid may be assumed to be completely dissociated. The concentration of hydrogen ion (CH+) at any instant is then equal to the concentration of un-neutralized strong acid at that instant. If a is the initial concentration of the acid in equiv. per liter, and 6 equiv. per liter is the amount of base added at any instant, the concentration of un-neutralized acid is a b equiv. per liter, and this is equal to the hydrogen ion concentration. The results obtained in this manner when 100 cc. of 0.1 N hydrochloric acid, i.e., a is 0.1, are titrated with 0.1 N sodium hydroxide are given in Table LXIX. In order to simplify the calculations it is assumed that the volume of the system remains constant at 100 cc. ; this simplification involves a slight error, but it will not affect the main conclusions which will be reached here. The values of pH in the last column are derived from the approximate defini- tion of pH as Jog CH+. When the solution contains equivalent amounts of acid and alkali the method of calculation given above fails, for a b is then zero; the 388 NEUTRALIZATION AND HYDROLYSIS TABLE LXIX. NEUTRALIZATION OP 100 CC. O.I N HCL BY 0.1 N N*OH NaOH added b CH+ pH 0.0 cc. 0.00 10-* 1.0 50.0 0.05 5 X 10-' 1.3 90.0 0.09 10~* 2.0 99.0 0.099 10~ 8 3.0 99.9 0.0999 10~ 4 4.0 100.0 0.1000 10~ 7 7.0 100.1 0.1001 lO" 10 10.0 system is now, however, identical with one containing the neutral salt sodium chloride, and so the value of CH+ is 10~ 7 g.-ion per liter and the pH is 7.0 at ordinary temperatures. If the addition of base is continued beyond the equivalence-point, the solution will contain free alkali; the pH of the system can then be calculated by assuming that COH~ is equal to the concentration of the excess alkali and that the ionic product CH+COIT is 10~ 14 . For example, in Table LXIX the addition of 100.1 cc. of 0.1 N sodium hydroxide means an excess of 0.1 cc. of 0.1 N alkali, i.e., 10~ 6 equiv. in 100 cc. of solution; the concentration of free alkali, and hence of hydroxyl ions, is thus 10~ 4 equiv. per liter. If COH" is 10~ 4 , it follows that CH+ must be 10~ 10 and hence the solution has a pH of 10.0. If the titration is carried out in the opposite direction, i.e., the addi- tion of strong acid to a solution of a strong base, the variation of pH may be calculated in a similar manner to that used above. The hy- droxyl ion concentration is now taken as equal to the concentration of un-neutralized base, i.e., b a, and the hydrogen ion concentration is then derived from the ionic product of water. The results calculated for the neutralization of 100 cc. of 0.1 N sodium hydroxide by 0.1 N hy- drochloric acid, the volume being assumed constant, are recorded in Table LXX. TABLE LXX. NEUTRALIZATION OF 100 CC. OF 0.1 N NAOH BY 0.1 N HCL HC'l added a <*OH~ pi I 0.0 re. 0.00 10- 1 13.0 50.0 0.05 5 X 10-* 12.7 90.0 0.090 10~ 2 12.0 99.0 0.099 10- 11.0 99.9 0.0999 10~ 4 10.0 100.0 0.1000 10~ 7 7.0 100.1 0.1001 10 - 10 4.0 The data in Tables LXIX and LXX are plotted in Fig. 101, in which curve I shows the variation of pH with the extent of neutralization of 0.1 N solutions of strong acid and strong base; the two portions of the curve may be regarded as parts of one continuous curve representing the change of pH as a solution of a strong acid is titrated with a strong base until the system contains a large excess of the latter, or vice versa. At- WEAK ACID AND STRONG BASE 389 tention may be called here to the sudden change of pH, from approxi- mately 4 to 10, as the equivalence-point, marked by an arrow, is attained; further reference to this subject will be made later. 25 50 75 100 75 50 25 Per cent Acid Per cent Base Neutralized Neutralized Fia. 101. Neutralization of strong acid and strong base Similar calculations can be made and analogous pH-neutralization curves can be plotted for solutions of strong acid and base at other con- centrations; curve II represents the results obtained for 10~ 4 N solutions. The pH at the equivalence-point is, of course, independent of the con- centration, since the pH of the neutral salt is always 7.0. The change of pH at the equivalence-point in curve II is seen to be much less sharp, however, than is the case with the more concentrated solutions. II. Weak Acid and Strong Base. The determination of the pH in the course of the neutralization of a weak acid is riot so simple as for a strong acid, but the calculations can nevertheless be made with the aid of equations derived in Chap. IX. It was seen on page 323 that if a weak acid, whose initial concentration is a equiv. per liter, is partially neutralized by the addition of 6 equiv. per liter of base, the activity of the hydrogen ions is given by n - K flH Ka B ' 7A - which may be written in the logarithmic form pH log log (39) (40) 390 NEUTRALIZATION AND HYDROLYSIS or, utilizing the Debye-Hvickel equations, pll = P K a + log ^-^ - A ^ + C. (41) The quantity B is defined in this case by B = b + CH* ~ coir, (42) using volume concentrations instead of molalities: since this involves both CH+ and. COH", the latter being equivalent to k w /cn + , equation (39) and those derived from it are cubic equations in CH+, and an exact solution is difficult. The problem is therefore simplified by considering certain special cases. If the pH of the solution lies between 4 and 10, i.e., CH+ is between 10~ 4 and 10~ l , the quantity C H + COH- in equation (42) is negligibly small; under these conditions B is equal to 6, and equation (41) becomes P H = pK a + log ~ - A Vtf + C tf . (43) The partly-neutralized acid system is equivalent to a mixture of un- neutralized acid and its salt, the concentration of the former being a 6 and that of the latter 6; equation (43) can consequently be written as pH = P K a + log - A + C V . (44) This relationship, without the activity correction, is equivalent to one derived by L. J. Henderson (1908) and is generally known as the Hender- son equation. The equation, omitting the activity terms, gives reason- ably good results for the pH during the neutralization of a weak base by a strong acid over a range of pH from 4 to 10, but it fails at the beginning and end of the process: under these latter conditions the approximation of setting B equal to 6 is not justifiable. For these extreme cases the general equations (39) to (41) are still applicable, and suitable approximations can be made in order to simplify the calculations. At the very beginning of the titration, i.e., when the weak acid is alone present, b is zero and since the solution is relatively acid COH~ may be neglected; the quantity B is then equal to CH+, and equation (39) becomes a ~~ CH+ If the solution has a sufficiently low ionic strength for the activity co- efficients to be taken as unity, which is approximately true for the weak acid solution, this equation may be written in the form , a - c H + CH+ = CH+ ak a . WEAK ACID AND STRONG BASE 391 If CH+ or k a is small, that is for a very weak acid, these equations reduce to At the equivalence-point, which represents the other extreme of the titration, a and b are equal, and CH + may be neglected in comparison with Com since the solution is alkaline owing to hydrolysis of the salt of the weak acid and strong base. It is seen, from equation (42), therefore, that B is now equivalent to a coir, and, neglecting the activity coeffi- cients, equation (39) becomes , = k a This is a quadratic in C H +, since foir is equal to k w /cji+, and so it can be solved without difficulty, thus If A/to /to , / a ,u7 (46) Since fc /P /2a is generally very small, it may usually be neglected and so this equation reduces to the form lK w K, a CH* = \ ' (47) or pH = ipfc* + %pk a + % log a, (47a) which is identical, as it should be, with the approximate equation (12a) for the hydrogen ion concentration in a solution of a salt of a weak acid and strong base; at the equivalence-point the acid-base system under consideration is, of course, equivalent to such a solution. It is thus possible to calculate the whole of the pH-neutralization curve of a weak acid by a strong base: equations (45) and (47) are used for the beginning and end, respectively, and equation (43), without the activity corrections, for the intermediate points. The pH values ob- tained in this manner for the titration of 100 cc. of 0.1 N acetic acid, for which k a is taken on 1.75 X 10~ 5 , with 0.1 N sodium hydroxide are quoted in Table LXXI. When the titration is carried out in the reverse direction, i.e., a strong base is titrated with a weak acid, the pH changes in the early stages of neutralization are almost identical with those obtained when a strong acid is employed. It is true that the salt formed, being one of a weak acid and a strong base, is liable to hydrolyze, but as long as excess of the strong base is present this hydrolysis is quite negligible (cf. p. 382). The hydroxyl ion concentration is then equal to the stoichiometric concen- tration of un-neutralized base, i.e., c ir is equal to b a where b and a are the concentrations of base and acid which make up the solution, just 392 NEUTRALIZATION AND HYDROLYSIS TABLE LXXI. NEUTRALIZATION OP 100 CC. OP 0.1 N ACETIC ACID BY 0.1 N NAOH NaOH added b a-b CH+ pH 0.0 cc. 0.0 0.10 1.32 X 10-* 2.88 10.0 0.01 0.09 1. 60X10' 4 3.80 20.0 0.02 0.08 6.93 X 10 ~ 5 4.16 40.0 0.04 0.06 2.63 X 10~ 6 4.58 50.0 0.05 0.05 1.75 X 10- 5 4.76 70.0 0.07 0.03 7.42 X 10~ 5.13 90.0 0.09 0.01 1.95 X !Q- 5.71 99.0 0.099 0.001 1.75 X 10 ~ 7 6.76 99.9 0.0999 0.0001 1.75 X 10~ 8 7.76 100.0 0.10 1.32 X 10- 8.88 as if the salt were not hydrolyzed. As- the equivalence-point is ap- proached closely, however, the concentration of base is greatly reduced and so the hydrolysis of the salt becomes appreciable. The form of the pH curve is then determined by the fact that the hydrogen ion concen- tration at the equivalence-point is given by equation (47). The complete curve for the neutralization of 0.1 N acetic acid by 0.1 N sodium hydroxide and vice versa, is shown in Fig. 102, I; the right- 2 - 25 50 75 100 75 50 25 Per cent Acid Per cent Base Neutralized Neutralized Fia. 102. Neutralization of weak (I) and very weak (II) acid by strong base hand side is almost identical with that of Fig. 101, I, for a strong base neutralized by a strong acid. It is observed that in this instance there is also a rapid change of pH at the equivalence-point, but it is not so marked as for a strong acid at the same concentration. The equivalence- point itself, indicated by an arrow, now occurs at pH 8.88, the solution of sodium acetate being alkaline because of hydrolysis. If a more dilute VERY WEAK ACID AND STRONG BASE 393 acetic acid solution, e.g., 0.01 N, is titrated with a strong base, the main position of the pH-neutralization curve is not affected, as may be seen from an examination of the Henderson equation (44); the pH depends on the ratio of salt to un-neutralized acid, and this will be the same at a given stage of neutralization irrespective of the actual concentration. When the neutralization has occurred to the extent of 50 per cent, i.e., at the midpoint of the curve, the ratio of salt to acid is always unity; the pH is then equal to pk a for the given acid (cf. p. 325), and this does not change markedly with the concentration of the solution. At the be- ginning and end of the neutralization, when the Henderson equation is not applicable, the pH's, given by equations (45) and (47), are seen to be dependent on the concentration; for 0.01 N acetic acid the values are 3.38 and 8.38, respectively, instead of 2.88 and 8.88 for the 0.1 N solution. HI. Moderately Strong Acid and Strong Base. If the acid is a mod- erately strong one, the pH may be less than 4 for an appreciable part of the early stages of the neutralization. The quantity CH+ COH- which appears in the term B cannot then be neglected, but it is more accurate to neglect COH- only, so that B becomes b + CH + ; under these conditions equation (39), neglecting activity coefficients, becomes <-- This is a quadratic equation which can be readily solved for C H +. The pH values for the beginning and end of the titration are derived from equations (45) and (47), as before. The pH-neutralization curve for a moderately strong acid lies between that of a strong acid (Fig. 101) and that of a weak acid (Fig. 102). IV. Very Weak Acid and Strong Base. For very weak acids, whose dissociation constants are less than about 10~ 7 , or for very dilute solu- tions, e.g., more dilute than 0.001 N, of weak acids, the pH of the solution exceeds 10 before the equivalence-point is reached. It is then necessary to include COH- in B, although C H + can be neglected; equation (39) then takes the form a b + COU- CH* = K a - 7 - o coir o-6 + fc./cn* " ka b- ( } This equation is also a quadratic in CH+, and so it can be solved and C H + evaluated. The results for the neutralization of a 0.1 N solution of an acid of k a equal to 10~ 9 by a strong base are shown in Fig. 102, II: the equivalence-point, indicated by an arrow, occurs at a pH of 11.0. The inflexion at the equivalence-point is seen to be small, and it is even less marked for more dilute solutions of the acid. It has been calculated that 394 NEUTRALIZATION AND HYDROLYSIS if ak a is less than about 27 k w there is no appreciable change in the slope of the pH-neutralization curve as the equivalence-point is attained. V. Weak Base and Strong Acid. The equations applicable to the neutralization of weak bases are similar to those for weak acids; the only alterations necessary are that the terms for 11+ and OH~ are exchanged, a and b are interchanged, and kb replaces k a . The appropriate form of equation (39), which is fundamental to the whole subject, is (50) u run' where B is now defined by B = a + COH CH + . The Henderson equation, omitting the activity correction, can be written as a pOH = pkb + log 7 > CL or i salt pOH = pA'b + log 7 9 base salt .'. pll = pk w pOH = pk w pkb log T (51) This equation is applicable over the same pll range as before, viz., 4 to 10; outside this range COH~ may be neglected in more acid solutions, while CH+ can be ignored in more alkaline solutions. At the extremes of the neutralization, i.e., for the pure base and the salt, respectively, the pH values can be obtained by making the appropriate simplifications of equation (50) ; alternatively, they may be derived from considerations of the dissociation of the base and of the hydrolyzed salt (cf. p. 390). A little consideration will show that the pll-neutralization curves for weak bases are exactly analogous to those for weak acids, except that they appear at the top right-hand corner of the diagram, with the mid- point, at pH 7, as a center of symmetry. The weaker the base and the less concentrated the solution, the smaller is the change of potential at the equivalence-point, just as in the neutralization of a weak base. VI. Weak Acid and Weak Base. The exact treatment of the neu- tralization of a weak acid by a weak base is somewhat complicated; it is analogous to that for the hydrolysis of a salt of a weak acid and weak base to which brief reference was made on page 381. The result is an equation of the fourth order in C H K and so cannot be solved easily. The course of the pH-neutralization curve can, however, be obtained, with sufficient accuracy for most purposes, by the use of approximate equa- tions. For the pure weak acid, the pH is given by equation (45) and the values up to about 90 per cent neutralization are obtained by the DISPLACEMENT REACTIONS 395 same equations as were used for the titration of a weak acid by a strong base; as long as there is at least 10 per cent of free excess acid the effect of hydrolysis is negligible. The pll at the equivalence-point is derived from equation (29), based on considerations of the hydrolysis of a salt of a weak acid and weak base. The complete treatment of the region between 90 and 100 per cent neutralization is somewhat complicated, but the general form of the curve can be obtained without difficulty by joining the available points. The variation of the pH in the neutraliza- tion of a weak base by a weak acid is derived in an analogous manner; up to about 90 per cent neutralization the behavior is virtually identical 12 10 8 pH 6 4 pHT.O I I 25 50 75 100 75 GO 25 Per cent Acid Per cent Base Neutralized Neutralized FIG. 103. Neutralization of acetic arid by ammonia with that obtained for a strong acid. The complete pll-neutralization curve for a 0.1 N solution of acetic acid and 0.1 N ammonia, for which A; rt arid k b are both taken to be equal to 1.75 X 10~ 5 , is shown in Fig. 103; the change of pll is seen to be very gradual throughout the neutralization and is not very marked at the equivalence-point. Displacement Reactions. In a displacement reaction a strong acid, or strong base, displaces a weak acid, or weak base, respectively, from one of its salts; an instance which will be considered is the displacement of acetic acid from sodium acetate by hydrochloric acid. Since this process is the opposite of the neutralization of acetic acid by sodium hydroxide, the variation of pH during the displacement reaction will be practically identical with that for the neutralization, except that it is in the reverse direction. In this particular case, therefore, the pll curve is represented by Fig. 102, 1, starting from the midpoint, which represents 396 NEUTRALIZATION AND HYDROLYSIS sodium acetate, and finishing at the left-hand end, representing an equiva- lent amount of free acetic acid. It is evident that there is no sharp change of potential when the equivalence-point is attained. On the other hand, if the salt of a very weak acid, e.g., k a equal to 10~ 9 , is titrated with hydrochloric acid, the variation of pH is given by Fig. 102, II, also starting from the center and proceeding to the left; a relatively marked inflexion is now observed at the equivalence-point, i.e., at the extreme left of the figure. The foregoing conclusions are in complete harmony with the con- cept of acids and bases developed in Chap. IX and of neutralization, in its widest sense, to which reference was made at the beginning of the present chapter. The reaction between sodium acetate and hydrochloric acid, i.e., (Na+)Ac- + H 3 0+(C1-) = HAc + H 2 + (Na+Cl-), Base Acid Acid Base is really the neutralization of the acetate ion base by a strong acid. It was seen on page 338 that the dissociation constant of a conjugate base, such as Ac~, is equal to k w /k a , where k a is the dissociation constant of the acid HAc; in this case k a is 1.75 X 10~ 5 and since k w is 10~ 14 , it follows that kb for the acetate ion base is about 5.7 X 10~ 10 . This represents a relatively weak base and its neutralization would not be expected to be marked by a sharp pH inflexion; this is in agreement with the result derived previously. If the acid is a very weak one, however, the con- jugate base is relatively strong; for example, if k a is 10~ 9 then k b for the anion base is 10~ 5 . The displacement reaction, which is effectively the neutralization of the anion base by a strong acid, should therefore be accompanied by a change of pH similar to that observed in the neutrali- zation of ammonia by-a^stpong acid. The arguments given above may be applied equally to the displace- ment of a weak base, such as ammonia or an amine, from a solution of its salt, e.g., ammonium chloride, by means of a strong base. If the amine RNH 2 has a dissociation constant of about 10~ 6 , its conjugate acid RNH^~ will be extremely weak, since k a will be 10~ 14 /10~ 5 , i.e., 10~ 9 , and the equivalence-point of the displacement titration will not be marked by an appreciable inflexion. On the other hand, if the base is a very weak one, such as aniline (fa equal to 10~ 10 ), the conjugate ariilinium ion acid will be moderately strong, k a about 10~ 4 , and the equivalence- point will be associated with a definite pH change. It follows, therefore, that only with salts of very weak acids or bases is there any considerable inflexion in the pH curve at the theoretical end-point of the displacement reaction. Neutralization in Non-Aqueous Media. As already seen, the mag- nitude of the inflexion in a pll-neutralization curve depends on the dis- sociation constant of the acid or base being neutralized; concentration is also important, but for the purposes of the present discussion this will NEUTRALIZATION IN NON-AQUEOUS MEDIA 397 be assumed to be constant. Another important factor, which is less evident at first sight, is the magnitude of k w ; an examination of the equations derived in the previous sections shows that the value of k w does not affect the pH during the neutralization of an acid, but it has an important influence at the equivalence-point. A decrease of k w will result in a decrease of hydrogen ion concentration, i.e., the pH is in- creased, at the equivalence point. When a base is being neutralized, the value of k w is important, as may be deduced from equation (51); a de- crease of k w , i.e., an increase of pA:^, will be accompanied by a corre- sponding increase of pH. It may be concluded, therefore, that if the ionic product of water is decreased in some manner, the acid and base parts of the neutralization curve are drawn apart and the inflexion at the equivalence-point is more marked. The two results derived above may be combined in the statement that the smaller k w /k y where k is the dissociation constant of the acid or base, the greater will be the change of pH as the equivalence-point of a neutralization is approached. The quantity k w /k is the hydrolysis constant of the salt formed in the reaction; hence, as may be expected, the smaller the extent of hydrolysis the more distinct is the pH inflexion at the end-point of the neutralization. There are thus two possibilities for increasing the sharpness of the approach to the equivalence-point; either k w may be decreased, while k a or k* is approximately unchanged, or k a or kb may be increased. The same general conclusions will, of course, be applicable to any other amphi- protic solvent, the quantity k w being replaced by the corresponding ionic product. For cation acids, e.g., NHt or RNHj, or for anion bases, e.g., CH 8 C05", the dissociation constants in ethyl alcohol are only slightly less than in water (cf. p. 333), but the ionic product is diminished by a factor of approximately 10 6 . It is clear, therefore, from the arguments given above that neutralization of such charged acids and bases will be much more complete in alcoholic solution than in water. The equivalence- points in the neutralization of the anions of acids and of the cations of substituted ammonium salts in alcohol have consequently been found to be accompanied by more marked inflexions than are obtained in aqueous solution. The dissociation constants of uncharged acids and bases are dimin- ished in the presence of alcohol, and since the ionic product of the solvent is decreased to a somewhat similar extent, the inflexion at the equivalence- point for these substances is similar to that in water. It was seen on page 371 that lyolysis could be avoided and neutraliza- tion made more complete when a weak base was neutralized in a strongly protogenic medium, such as acetic acid. The use of a solvent with a marked proton donating tendency is, effectively, to increase the dis- sociation constant of the weak base; hence a sharper change of pH is to be expected at the equivalence-point in a strongly protogenic solvent than in water. This argument applies to bases of all types, i.e., charged or NEUTRALIZATION AND HYDROLYSIS uncharged, and the experimental results have been shown to be in ac- cordance with anticipation ; the curves in Fig. 104, for example, show the change of pH, as measured by a form of hydrogen electrode, during the course of the neutralization of the extremely weak bases urea and acet- oxirne by perchloric acid in acetic acid solution. 8 In aqueous solu- tions these bases would show no detectable change of pH at the equivalence-point. In order to increase the magnitude of the inflexion in the neutralization of a very weak acid it would be nec- essary to employ a strongly pro- tophilic medium, such as liquid ammonia, or one having no proto- genic properties, e.g., acetonitrile. Neutralization of Mixture of Two Monobasic Acids. An ex- pression for the variation of the pH during the whole course of the neutralization of a mixture of two monobasic acids by a strong base can bo derived, but as it is somewhat complicated, simplifi- cations are made which are ap- plicable to certain specific conditions. Let ai and an be the initial con- centrations of the two acids HAi arid II AH, whose dissociation constants are fci and &n; suppose that at a certain stage of the neutralization a concentration 6 of strong base MOH has been addeH to the mixture of acids. If the salts formed when the acids are neutralized are completely dissociated, then at any instant 0.70 0.60 0.60 0.2 0.4 0.6 0.8 1.0 1.2 of Perchloric Acid Fio. 104. Neutralization of very weak based in glacial acetic acid solution and (52) (53) where CHA represents in each case the concentration of un-neutralized acid while C A - is that of the neutralized acid, the total adding up to the initial acid concentration. Since the solution must be electrically neu- tral, the sum of the positive charges must equal that of the negative charges, i.e., CM* + C H + = C A + C A -J -f COH'. (54) The salts MAi and MAn are completely dissociated and so CM* may be Hall and Werner, /. Am. Chem. Soc., 50, 2367 (1928); Hall, Chem. Revs., 8, 191 (1031); see also, Nadeau and Branchen, J. Am. Chem. Soc., 57, 1363 (1935). NEUTRALIZATION OF MIXTURE OF TWO MONOBASIC ACIDS 399 identified with 6, the concentration of added base; further, except towards the end of the neutralization, COR- may be neglected, and so equation (54) becomes b + c H + = CAJ + CAJ,. (55) The approximate dissociation constants of the two acids are given by *. CH * CA J ^ i CH * CA " Id = - and ku = - > CHA r CHA II and if these expressions together with equations (52) and (53) are used to eliminate the concentration terms involving Af, AH, as well as HAi and II An, from (55), the result is , t . - b - ( 56 ) This is a cubic equation which can be solved to give the value of the hydrogen ion concentration at any point of the titration of the mixture of acids. A special case of interest is that arising when the amount of base added is equivalent to the concentration of the stronger of the two acids, e.g., HAi; under these conditions b may be replaced by ai, and if all terms of the third order in equation (56) are neglected, since they are likely to be small, it is found that aiCn + + kn(ai - aii)c n + aukiku = 0. If ai and an ire not greatly different and ku is small, the second term on the left-hand side in this equation can be omitted, so that fankikii * ~ \ Ql ' .'. pll = Jpfri + pfcn + \ log ai \ log an. (58) This relationship gives the pH at the theoretical first equivalence-point in the neutralization of a mixture of two monobasic acids. If the two acids have the same initial concentration, i.e., ai is equal to an, then equation (57) for the first equivalence-point becomes CH+ Vfcifcn, (59) .'. pH = |pfci + Jpfax. (60) The pH at the equivalence-point for the acid HAi in the absence of HAn is given by equation (12) as pll = Jpfc w + Ipki + % log ai, (61) and comparison of this with the value for the mixture at the first equiva- 400 NEUTRALIZATION AND HYDROLYSIS lence-point, the latter being designated by (pH) m , shows that in the general case pH - (pH) m = Jpfcu, - Jpfcn + 4 log an. (62) Since pfcn is generally less than pfc*,, the quantity pH (pH) m is posi- tive; the pH at the first equivalence-point of a mixture is thus less than that for the stronger acid alone. This result indicates a flattening of the pH curve in the vicinity of the first equivalence-point, the extent of the flattening being, according to equation (62), more marked the smaller pfcn, i.e., the stronger the acid HAn, and the greater its concen- tration. If the acid HAn is very weak or its concentration small, or both, the flattening at the first equivalence-point will be negligible, and the neutralization curve of the mixture will differ little from that of the single acid HAi. Equations for the variation of pH during the course of neutralization beyond the first equivalence-point, similar to those already given, could be derived if necessary, but for most requirements a simpler treatment of the whole neutralization curve is adequate. At the very commence- ment of the titration the pH is little different from that of the solu- tion of the stronger acid, and during the early stages of neutralization the pH is close to that which would be given by this acid alone. In the vicinity of the first equivalence-point deviations occur, but these can be inferred with sufficient accuracy from the pH at that point, as given by equations (58) or (60). At a short distance beyond the first equivalence-point the pH is close to that for the neutralization of the second acid alone; the pH at the final equivalence-point is the same as that of the salt NaAn and is consequently given by equation (12). A satisfactory idea of the com- plete neutralization curve can thus be obtained by plotting the curves for the two acids separately side by side, the curve for the stronger acid (HAi) being at the left; the two curves are then joined by a tangent (Fig. 105). The re- gion between the two curves may be fixed more exactly by making use of equation (58) or (60) for the first equiva- lence-point. The figure shows clearly that if the weaker acid HAn is moderately weak, as at HA, the inflexion at the first equiva- lence-point will be negligible, but if it is very weak, as at II B, the inflexion will not differ appreciably from that given by the acid HAi alone. A decrease in the concentration of HAu makes the pH higher PH ii Neutralization of HA K Neutralization of HA u FIG. 105. Neutralization of mixture of acids NEUTRALIZATION OF DIBASIC ACID BY A STRONG BASE 401 at the beginning of the HAn curve and so increases the inflexion to some extent, in agreement with the conclusion already reached. If HAi is a strong acid, e.g., hydrochloric acid, and HAii is a weak acid, the pH follows that for the neutralization of the strong acid alone almost exactly up to the first equivalence-point. Neutralization of Dibasic Acid by a Strong Base. If the first stage of the dissociation of the dibasic acid corresponds to that of a strong acid while the second is relatively weak, e.g., chromic acid, the system behaves virtually as two separate acids. The first stage is neutralized as a normal strong acid, then the second stage becomes neutralized independently as a weak acid. When both stages are relatively weak, however, there is some interference between them, and the variation of pH during the course of neutralization may be calculated by means of equations de- rived in Chap. IX. For the present purpose, equation (34), page 326, for the hydrogen ion activity of a solution of a dibasic acid, of initial concentration a moles per liter, to which has been added a concentration of b equiv. per liter of strong base, may be written as n _ D cfi * 2^Te = CH * to~=~B kl + klk2 ' (63) The activity coefficients have been omitted and the approximate func- tions ki and fe, for the two stages of dissociation of the dibasic acid, have replaced the corresponding thermodynamic constants. The quantity B is defined as B s b + c n + - COH-, and insertion of this value in equation (63) gives a quartic equation for CH+, which can be solved if necessary. For a considerable range of the neutralization it is possible to neglect COH~ in the expression for B } and so the equation reduces to a cubic. At the first equivalence-point, a is equal to b and if COET is neglected, as just suggested, it follows from equation (63) that Since CH+ is generally small in comparison with o, this equation reduces to 2 kik 2 a c ^-k~Ta' (65) which is identical, as it should be, with equation (30), for at the first equivalence-point in the neutralization of the dibasic acid H 2 A the system is identical with a solution of NaHA. If k\ is small, equation (65) be- comes, as before, _ CH+ = V*S, (66) .'. pH = ipfe! + ipfe. (66a) 402 NEUTRALIZATION AND HYDROLYSIS It will be noted that this result is the same as equation (59) for the first equivalence-point in the neutralization of a mixture of equivalent amounts of two weak acids. For a dibasic acid with a very weak first stage dis- sociation, it may not be justifiable to neglect COH~; under these conditions, however, CH+ may be ignored, and the corresponding equations can be derived. The form of the pll-neutralization curve for a dibasic acid can be represented in an adequate manner by the method used for a mixture of acids; the curves for the two stages are drawn side by side, from the individual dissociation constants k\ and k% treated separately, and then joined by a tangent. The general conclusions drawn concerning the inflexion at the first equivalence-point are similar to those for a mixture of acids; the essential requirement for a dibasic acid to show an appre- ciable inflexion at the first equivalence-point is that k\/kz should be large. Under these conditions the individual pH-neutralization curves for the two stages of the dibasic acids are relatively far apart and the tangent joining them approaches a vertical direction. Distribution of Strong Base between the Stages of a Dibasic Acid. During the course of neutralization of a dibasic acid, the system will con- sist of undissociatcd molecules HkA and of the ions HA~ and A ; the fraction of the total present as HA~ ions, i.e., a\ 9 is then JA + CHA- + C A - while that present as A ions, i.e., 2 , is . . r Cn 2 A -r CHA~ ~r c\~~ Since the HA~ ions arise almost entirely from NaHA, assuming the base to be sodium hydroxide, while the A ions originate mainly from NaA, it follows that i represents, approximately, the fraction of the dibasic acid neutralized in the first stage only, while 0.1 is the fraction neutralized in both stages. By using the familiar expressions for the first and second stage dissociation functions (p. 381) to eliminate CA~ from equation (67) and CHA~ from equation (68), the results are and 2 = - - - -27- (70) 5! . i i li! b ^ "*" kfa It is thus possible, by means of equations (69) and (70), to evaluate the NEUTRALIZATION OF POLYBASIC ACIDS AND MIXTURES OF ACIDS 403 fractions of HA" and of A present at any pTI for a given dibasic acid, provided ki and 7c 2 are known. As is to be expected, the fraction present as HA~, i.e., i, increases at first as neutralization proceeds; the value then reaches a maximum and falls off to zero when both stages of the acid are completely neutralized. The fraction of A , on the other hand, increases slowly at first and then more rapidly and finally approaches unity when neutralization is complete and the system consists entirely of Na2A. Many interesting conclusions can be drawn from the curves for different values of ki and fc> concerning the pi I at whi^h the second stage neutralization becomes appreciable, and so on; the main results have, however, already been obtained from a consideration of the pH-neutrali- zation curves. 6 The point at which the fraction i attains a maximum can be derived by writing 1/ai by means of equation ((59) as tti /:i CH* differentiating with respect to CH+, thus !_ _ A- 2 _ C/CH+ A*i CH* and equating to zero, since l/i must be a minimum when i is a maxi- mum. It follows, therefore, that _ _ /. CH+ = VA^, (71) under these conditions. According to equation (06) this is, approxi- mately, the hydrogen ion concentration at the first equivalence-point; hence the fraction of the total acid in the form of II A" ions is greatest at this point. Neutralization of Polybasic Acids and Mixtures of Acids. The treat- ment of a system consisting of a tribasic or higher acid, or of a mixture of three or more simple acids is complicated, but the general nature of the results can be obtained in the manner already described. The pH-neutralization curve for the whole system is obtained with a fair degree of accuracy by drawing the separate curves for the individual stages of neutralization of the polybasic acid, or for the individual acids in a mixture of acids, in the order of decreasing dissociation constants, and connecting them by means of tangents in the usual way. The pH's at the various equivalence-points can be fixed by using a relationship similar to equation (66); the pH at the nth equivalence-point, i.e., when 6 Michaelis, "Hydrogen Ion Concentration," translated by Perlzweig, 1926, p. 55. 404 NEUTRALIZATION AND HYDROLYSIS sufficient strong base has been added to neutralize the first n stages, or n acids, is given by pH - ipfc w + ipfcn+i, (72) where pfc and pfc+i are the dissociation exponents for the nth and (n + l)th stages, respectively, of a polybasic acid, or of the nth and (n + l)th acids in a mixture arranged in order of decreasing strength. Another useful method for considering the neutralization of polybasic acids or mixtures of acids, which avoids the necessity of plotting curves, is the following. In general, when an acid is neutralized to the extent of 0.1 per cent, i.e., salt/acid is 1/999, the pH, according to the approxi- mate Henderson equation, is pH = pk a + log 7 fg- Pk a - 3. It follows, therefore, that in a polybasic acid system, or in a mixture of approximately equivalent amounts of different acids, the neutralization of a particular stage or of a particular acid may be regarded as commencing effectively when the pH is equal to pfc n +i 3, where pfc n +i is the dis- sociation of the (n + l)th stage or acid; at this point the pH-neutraliza- tion curve for the mixture will commence to diverge from that of the previous stage of neutralization. Similarly, when an acid is 99.9 per cent neutralized, pk a + 3. The neutralization of any stage may, therefore, be regarded as approxi- mately complete when the pH of the system is equal to pA; n + 3, wherf pfc n is the dissociation exponent for the nth stage of a polybasic acid or for the nth acid in a mixture. If this pH is less than pA; n +i 3, the neutralization of the nth stage will be substantially complete before that of the (n + l)th stage commences; if this condition holds, i.e., if pfc n +i 3 > pfc n + 3, the neutralization of the weaker acid, or stage, will have no appreciable effect on that of the stronger. It is seen, therefore, that if pfcn+i pk n is greater than 6, or fc n /fc n +i is greater than 10 6 , the pH- neutralization curve for the mixture will show no appreciable divergence, at the n-h equivalence-point, from that of the nth acid alone. The inflexion at the nth equivalence-point will then be as definite as for the single nth acid. If pfc+i pk n is less than 6, the neutralization of the (n + l}th acid, or stage, commences before that of the nth acid is com- plete, and the result will be a flattening of the pH-curve at the nth equivalence-point; if fc n /fc n +i is less than 16, there is no detectable in- flexion in the pH-neutralization curve. Potentiometric Titrations. 7 The general conclusions drawn from the treatment in the foregoing sections provide the basis for potentiometric, 7 See general references to potentiometric titrationa on page 256. POTENTIOMETRIC TITRATIONS 405 as well as ordinary volumetric, titrations of acids and bases. The poten- tial E of any iorm of hydrogen electrode, measured against any con- venient reference electrode, is related to the pH of the solution by the general equation RT E = E nft + -TT pH, or, at ordinary temperatures, i.e., about 22, E = # r ef. + 0.059 pH, where E re i. is a constant. It is apparent, therefore, that the curves rep- resenting the variation of pH during neutralization are identical in form with those giving the change of hydrogen electrode potential. It should thus be possible to determine the end-point of an acid-base titration by measuring the potential of any convenient form of hydrogen electrode at various points and finding the amount of titrant at which the potential undergoes a sharp inflexion. The underlying principle of the poten- tiometric titration of a neutralization process is thus fundamentally the same as that involved in precipitation (p. 256) and oxidation-reduction titrations (p. 285). The position of the end-point is found either by graphical determination of the volume of titrant corresponding to the maximum value of A/?/ At;, where A# is the change of hydrogen electrode potential resulting from the addition of Ay of titrant, or it can be deter- mined by a suitable adaptation of the principle of differential titration. The apparatus described on page 261 (Fig. 77) can, of course, be em- ployed without modification with glass or quinhydrone electrodes; if hydrogen gas electrodes are used, however, the electrodes are of platinized platinum and the hydrogen must be used for operating the gas-lift, the stream being shut off before each addition of titrant so as to avoid mixing. Any form of hydrogen electrode can be used for carrying out a potentiometric neutralization titration, and even oxygen gas and air electrodes have been employed; since all that is required to be known is the point at which the potential undergoes a rapid change, the irre- versibility of these electrodes is not a serious disadvantage. Potentio- metric determinations of the end-point of neutralization reactions can be carried out with colored solutions, and often with solutions that are too dilute to be titrated in any other manner. The accuracy with which the end-point can be estimated obviously depends on the magnitude of the inflexion in the hydrogen potential- neutralization curve at the equivalence-point, and this depends on the dissociation constant of the acid and base, and on the concentration of the solution, as already seen. When a strong acid is titrated with a strong base, the change of potential at the equivalence-point is large, even with relatively dilute solutions (cf. Fig. 101), and the end-point can be obtained accurately. If a weak acid and a strong base, or vice versa, are employed the end-point is generally satisfactory provided the 406 NEUTRALIZATION AND HYDROLYSIS solutions are not too dilute or the acid or base too weak (cf . Fig. 102, I) ; if c is the concentration of the titrated solution and k a or ki the dissocia- tion constant of the weak acid or base being titrated, by a strong base or acid, respectively, then an appreciable break occurs in the neutraliza- tion curve at the end-point provided ck a or ck b is greater than 10~ 8 . Titrations can be carried out potentiometrically even if ck a or ckb is less than 10~ 8 , but the results are less accurate (cf. Fig. 102, II). The poten- tiometric titration of very weak bases can, of course, be carried out satisfactorily in a strongly protogenic medium (cf. Fig. 104). When a weak acid and weak base are titrated against one another the change of pH at the end-point is never very marked (Fig. 103), but if potential measurements are made carefully, an accuracy of about 1 per cent may be obtained with 0.1 N solutions by determining graphical ly the position at which AE/Av is a maximum. The principles outlined above apply, of course, to displacement reactions, which are to be regarded as involving neutralization in its widest sense. Such titrations can be performed accurately in aqueous solution if the acid or base that is being displaced is very weak; in other cases satisfactory end-points may be obtained in alcoholic solution. The separate acids in a mixture of acids, or bases, can often be titrated potentiometrically, provided there is an appreciable difference in their strengths: this condition is realized if one of the acids is strung, e.g., a mineral acid, and the other is weak, e.g., an organic tyul. It has been seen that if the ratio of the dissociation constants of two acids exceeds about 10 6 , the weaker does not interfere with the neutralization of the stronger acid in the mixture; this conclusion does not take into account the influence of differences of concentration, and it is more correct to say that Ciki/ciikn should be greater than 10 6 whore ci and k\ are the concen- tration and dissociation constant of one acid and CH and k\\ that of the other. If this condition is combined with that previously given for obtaining a satisfactory end-point with a single arid, ihe following con- clusions may be drawn: if Ciki and CH/TH both exceed 10~ 8 and cjtilcnku is greater than 10 6 , accurate titration of the separate acids in the mixture is possible. If Ciki/cuku is less than 10 6 the firot equivalence-point cannot be very accurate even if c\ki is greater than 10~ 8 , but an accuracy of about 1 per cent can be achieved by careful titration even if ciki/cukn is as low as 10 4 . When the fir&t equivalence-point is not detectable, the second equivalence-point, representing neutralization of both acids, may still be obtained provided cufcn exceeds 10" 8 . The general relationship applicable to mixtures of acids can be extended to polybasic acids, al- though in the latter case a and CH are equal. 8 In the titration of a strong acid and a strong base the cquivaleiije- point corresponds exactly to the point on the pH-neutralizatiori curve, or the potential-titration curve, at which the slope is a maximum. This Noyes, J. Am. Chem. Soc., 32, 815 (1910); see also, Tizard and Boeree, J. Chem. Soc., 121, 132 (1922); Koltboff and Furman, "Indicators/* 1926, p. 121. NEUTRALIZATION TITRATIONS WITH INDICATORS 407 is not strictly true, however, in the case of the neutralization of a weak acid or a weak base; if (CH + ) P is the hydrogen ion concentration at the potentiometric end-point, i.e., where AE/Av is a maximum, and (C H +) is thg, value at the theoretical, or stoichiometric, equivalence-point, it can be shown that ! c J^i , 3 (CH*). ~ * Provided ak a is greater than 10~ 8 , which is the condition for a satisfactory point of inflexion in the titration curve, the ratio of the two hydrogen ion concentrations differs from unity by about one part in 700; this would be equivalent to a potential difference of 0.016 millivolt and h^nce is well within the limits of experimental error. Neutralization Titrations with Indicators. Since, as seen on page 362, an acid-base indicator changes color within a range of approximately one unit of pH on either .side of a pH value equal to the indicator exponent (pfcin), such indicators are frequently used to determine the end-points of neutralization titrations. 9 The choice of the indicator for a particular titration can best be determined from an examination of the pH-neutrali- zation curve. Before proceeding to consider this aspect of the problem it is useful to define the titration exponent (pkr) of ari indicator; this is the pTl of a solution at which the indicator shows the color usually associated with the end-point when that indicator is employed in a neutralization titration. It is the general practice in such work to titrate from the lighter to the darker color, e g., colorless to pink with phenol- phthalein and yellow to red with methyl orange; as a general rule a 20 per cent conversion is necessary before the color change can be defi- nitely detected visually, and so if the darker colored form is the one existing in alkaline solution, it follows from the simple Henderson equa- tion (cf. p. 390) that 20 pH = pk T = phn + log This approximate relationship between the titration exponent and pki n is applicable to phenolphthalein and to many of the sulfonephthalein indicators introduced by Clark and Lubs (sec Table LXIV, page 364). If the darker color is obtained in acid solution, as is the case with methyl orange and methyl red, then it is approximately true that 80 pH = pk T = pfcin + log = pki n + 0.6. The results quoted in Table LXX1I give the titration exponents based Kolthoff and Furman, "Indicators," 1926, Chap. IV. 408 NEUTRALIZATION AND HYDROLYSIS TABLE LXXII. TITRATION EXPONENTS OF USEFUL INDICATORS Indicator Bromphenol blue Methyl orange Methyl red Bromcresol purple Bromthymol blue Phenol red Cresol red Thymol blue Phenolphthalein Thymolphthalein pkr 4 4 5 6 6.8 7.5 8 8.8 9 10 End-point Color Purplish-green Orange Yellowish-red Purplish-green Green Rose-red Red Blue-violet Pale rose Pale blue on actual experimental observations, together with practical information, for a number of indicators which may be useful for neutralization titra- tions; they cover the pH range of from about 4 to 10, since titration indicators are seldom employed outside this range. In order that a particular indicator may be of use for a given acid- base titration, it is necessary that its exponent should correspond to a pH on the almost vertical portion of the pH-neutralization curve. When the end-point of the titration is approached the pH changes rapidly, and the correct indicator will undergo a sharp color change. The choice of indicator may be readily facilitated by means of Fig. 106 in which the Indicators PH I Alizarine yellow Thymol phthalein Phenol phthalein Phenol red Bromthymol blue Bromcreeol purple Methyl red Methyl orange Bromphenol blue Thymol blue 26 60 75 100 76 60 25 Per cent Add Per cent Base Neutralized Neutralized Fio. 106. Neutralization curves for various acids and bases pH-neutralization curves for a number of acids and bases of different strengths are plotted, while at the right-hand side a series of indicators are arranged at the pH levels corresponding to their titration exponents. The positions of the equivalence-points for the various types of neutrali- zation are marked by arrows. The curves IA, HA and IIlA show pH changes during the course of neutralization of 0.1 N solutions of a strong acid, a normally weak acid (k a = 10~*) and a very weak acid (k a = 10"'), NEUTRALIZATION TITRATIONB WITH INDICATORS 409 respectively; curves IB, HB and Ills refer to 0.1 N solutions of a strong base, a normally weak base (kb = 10~ 6 ) and a very weak base (k b = 10~ 9 ), respectively. The complete titration curve for any particular acid and base is obtained by joining the appropriate individual curves. In the titration of 0.1 N strong acid by 0.1 N strong base (curve IA-!B), the pH of the solution undergoes a very sharp change from pH 4 to pH 10 within 0.1 per cent of the equivalence-point (see Table LXIX); any indicator changing color in this range can, therefore, be used to give a reliable indication when the end-point is reached. Consequently, both phenolphthalein, pfcr equal to 9, and methyl orange, p&r equal to 4, may be employed to give almost identical results in this particular titration. If the solutions are diluted to 0.01 N, however, the change of pH at the equivalence-point is less sharp, viz., from 5 to 9; methyl orange will, therefore, undergo its color change before the end-point is attained, and the titration value would consequently be somewhat too low. When a 0.1 N solution of an acid of k a equal to 10~ 5 is titrated with a strong base, the equivalence-point is at pH 9, and there is a fairly sharp increase from pH 8 to 10 (curve HA-!B); of the common indicators phenolphthalein is the only one that is satisfactory. The less familiar cresolphthalein or thymol blue (second range) could also be used. Any indicator having a titration exponent below 8 is, of course, quite unsatisfactory. In the titration of 0.1 N base of k b equal to 10~ 6 , the equivalence-point is at pH 5, and the change of potential between pH 4 to 6 is rapid (curve lA-IIs). Methyl orange is frequently used for such titrations, e.g., ammonia with hydrochloric acid, but it is obvious that the results cannot be too reliable, especially if the solutions are more dilute than 0.1 N; methyl red is a better indicator for a base whose dissociation constant is about 10~ 6 . It will be evident that if the indicator color is to change sharply at the required end-point, the pH-neutralization curve must rise rapidly at this point. If this curve is not almost vertical, the pH changes slowly and the indicator will show a gradual transition from one color to the other; under these conditions, even if the correct indicator has been chosen, it will be impossible to detect the end-point with any degree of accuracy. In general, the condition requisite for the accurate estima- tion of a potentiometric end-point, i.e., that ck a or ck* should exceed 10~ 8 , is also applicable to titration with an indicator; if ck is less than this value, the results are liable to be in error. They can, however, be im- proved by using a suitable indicator and titrating to the pH of the theoretical equivalence-point by means of a comparison flask containing a solution of the salt formed at the end-point, together with the same amount of indicator. This procedure may be adopted if it is necessary to titrate a very weak acid or base (curves IIlA-Is and lA-IIIs) or a moderately weak acid by a weak base (curve IlA-IIs); in none of these instances is there a sharp change of pH at the equivalence-point. 410 NEUTRALIZATION AND HYDROLYSIS Displacement reactions may be treated as neutralizations from the standpoint of the foregoing discussion. If the acid or base displaced is moderately weak, i.e., k a or /r& is about 10~ 5 , the displacement reaction is equivalent to the neutralization of a very weak base or acid, with kb or k a equal to 10~ 9 , respectively; no indicator is likely to give a satisfactory end-point in aqueous solution, although one may possibly be obtained in an alcoholic medium (cf. p. 396). If the acid or base being displaced is very weak, e.g., carbonic acid from a carbonate or boric acid from a borate, there is a marked pH inflexion at the equivalence-point which can be detected with fair accuracy by means of an indicator. The problem of the detection of the various equivalence-points in a mixture of acids of different concentrations or in a solution of a polybasic acid is essentially the same as that already discussed on page 406 in connection with potentiometric titration, and need not be treated further here. Where the conditions are such that the determination of an accu- rate end-point appears feasible, the appropriate indicator is the one whose pfcm value lies close to the pH at the required equivalence-point. Buffer Solutions. It is evident from a consideration of pH-neutrali- zation curves that there are some solutions in which the addition of a small amount of acid or base produces a marked change of pH, whereas in others the corresponding change is very small. A system of the latter type, generally consisting of a mixture of approximately similar amounts of a conjugate weak acid and base, is said to be a buffer solution; the resistance to change in the hydrogen ion concentration on the addition of acid or alkali is known as buffer action. The magnitude of the buffer action of a given solution is determined by its buffer capacity ; 10 it is measured by the amount of strong base required to produce unit change of pH in the solution, thus : db Buffer capacity (ft) = ., v^: An indication of the buffer capacity of any acid-base system can thus be obtained directly from the pH-neutralization curve; if the curve is flat, d(pH)/db is obviously small and the buffer capacity, which is the recipro- cal of this slope, is large. An examination of curves I A and IB, Fig. 106, shows that a relatively concentrated solution of strong acid or base is a buffer in regions of low or high pH, respectively. A solution of a weak acid or a weak base alone is not a good buffer, but when an appre- ciable amount of salt is present, i.e., towards the middle of the individual neutralization curves HA, III A, II B or Ills, the buffer capacity of the system is very marked. As the equivalence-point is approached the pH changes rapidly and so the buffer capacity of the salt solution is small. If the acid or base is very weak, or if both are moderately weak, the slope "van Slyke, /. Biol Chem., 52, 525 (1922); Kilpi, Z. physikal. Chem., 173, 223 (1935). BUFFER SOLUTIONS 411 of the pH curve at the equivalence-point is not very great and hence the corresponding salts have moderate buffer capacity. The buffer action of a solution of a weak acid (HA) and its salt (A~~), i.e., its conjugate base, is explained by the fact that the added hydrogen ions are "neutralized" by the anions of the salt acting as a base, thus H 3 0+ + A- = H 2 + HA, whereas added hydroxyl ions are removed by the neutralization OH- + HA = H 2 O + A-. According to the Henderson equation the pH of the solution is deter- mined by the logarithm of the ratio of the concentrations of salt to acid; if this ratio is of the order of unity, it will not be greatly changed by the removal of A~ or HA in one or other of these neutralizations, and so its logarithm will be hardly affected. The pH of the solution will conse- quently not alter very greatly, and the system will exert buffer action. If the buffer is a mixture of a weak base (B) and its salt, i.e., its conjugate acid (BH+), the corresponding equations are H 8 0+ + B = H 2 O + BH+ and OH- + BH+ = H 2 O + B. In this case the pH depends on the logarithm of the ratio of B to BH+, and this will not be changed to any great extent if the buffer contains the weak base and its salt in approximately equivalent amounts. By the treatment on page 323, the initial concentration of acid, a moles per liter, is equal, at any instant, to the sum of the concentrations of HA and A", i.e., a = CHA + C A -, (73) and according to the condition for electrical neutrality, b + C H + = c A - + COH-, (74) where 6 is the concentration of base added at that instant; since the salt MA is completely dissociated the concentration of M+ ions, CM*, has been replaced by 6 in equation (74). Writing k a for the dissociation function of the acid, in the usual manner, __ C H* CA " CH\ and utilizing the value of CHA as a CA~ given by equation (73), it is found that Substitution of this expression for CA~, and k w /CR+ for COH~, in equation 412 NEUTRALIZATION AND HYDROLYSIS (74), yields the result _ dk/a tow Ilemembering that pH is defined, for present purposes, as log CH+, differentiation of this equation with respect to pH gives the buffer capacity of the system, thus ft = = 2.303 Tr, + ** + ' (75) In the effective buffer region the buffer capacity is determined almost exclusively by the first term in the brackets; hence, neglecting the other terms, it follows that - (76) The quantity a represent^ the total concentration of free acid and salt, and so the buffer capacity is proportional to the total concentration of the solution. To find the pH at which ft is a maximum this expression should be differentiated with respect to pH and the result equated to zero; thus /. fc = C H +. (77) It follows, therefore, that the buffer capacity is a maximum when the hydrogen ion concentration of the buffer solution is equal to the dis- sociation constant of the acid. This condition, i.e., pH is equal to pk a , arises when the solution contains equivalent amounts of the acid and its s.ilt; such a system, which corresponds to the middle of the neutrali- zation curve of the acid, has the maximum buffer capacity. The actual value of j3 at this point is found by inserting the condition given by (77) into equation (76) ; the result is 2.303 0mux. =-J-> (78) and so it is independent of the actual dissociation constant. Exactly analogous results can, of course, be deduced for buffer systems consisting of weak bases and their salts, although it is convenient to consider them as involving the cation acid (BH+) and its conjugate base (B). The conclusions reached above then hold exactly; the dissociation constant k a refers to that of the acid BH+, and is equal to k w /k b , where fa is that of the base B. Buffer Capacity of Water. According to equation (74), the condition for electrical neutrality, when a strong base of concentration 6 has been PREPARATION OP BUFFER SOLUTIONS 413 added to water or to a solution containing a strong acid HA, is b = C A -- CH+ + COH- = C A -- CH+ + A^/CH*, and differentiation with respect to pH, i.e., log C H +, gives the buffer capacity 0H 2 o of water as = 2.303(c H + + COH-). (79) It should be noted that the further addition of base does not affect the concentration of A~ and so its derivative with respect to pH is zero. The buffer capacity of water, as given by equation (79), is negligible between pH values of 2.4 and 11.6, but in more strongly acid, or more strongly alkaline, solutions the buffer capacity of "water" is evidently quite considerable. This conclusion is in harmony with the fact that the pH-neutralization curve of a strong acid or strong base is relatively flat in its early stages. Preparation of Buffer Solutions. The buffer capacity of a given acid- base system is a maximum, according to equation (77), when there are present equivalent amounts of acid and salt; the hydrogen ion concen- tration is then equal to k a and the pH is equal to pk a . If the ratio of acid to salt is increased or decreased ten-fold, i.e., to 10 : 1 or 1 : 10, the hydrogen ion concentration is then 10k a or Q.lk a , and the pH is pk a 1 or pfc a + 1, respectively. If these values for CH+ are inserted in equation (76), it is found that the buffer capacity is then which is only about one-third of the value at the maximum. If the pH lies within the range of pk a 1 to pk a + 1 the buffer capacity is appre- ciable, but outside this range it falls off to such an extent as to be of relatively little value. It follows, therefore, that a given acid-base buffer system has useful buffer action in a range of one pll unit on either side of the pk a of the acid. In order to cover the whole range of pH, say from 2.4 to 11.6, i.e., between the range of strong acids and bases, it is necessary to have a series of weak acids whose pk a values differ by not more than 2 units. To make a buffer solution of a given pH, it is first necessary to choose an acid with a pk a value as near as possible to the required pH, so as to obtain the maximum buffer capacity. The actual ratio of acid to salt necessary can then be found from the simple Henderson equation TT >ii 8alt ' P H = pfc a + log 414 NEUTRALIZATION AND HYDROLYSIS provided the pH lies within the range of 4 to 10. If the required pH is less than 4 or greater than 10, it is necessary to use the appropriate form of equation (40), where B is defined by (42). Sometimes a buffer solu- tion is made up of two salts representing different stages of neutralization of a polybasic acid, e.g., NaH 2 PO 4 and Na 2 HPO 4 ; in this case the former provides the acid H^POr while the latter is the corresponding salt, or conjugate base HPO". In view of the importance of buffer mixtures in various aspects of scientific work a number of such solutions have been made up and their pH values carefully checked by direct experiment with the hydrogen gas electrode. By following the directions given in each case a solution of any desired pH can be prepared with rapidity and precision. A few of the mixtures studied, and their effective ranges, are recorded in Table LXXIII; 11 for further details the original literature or special mono- graphs should be consulted. TABLE LXXIiI. BUFFER SOLUTIONS Composition Hydrochloric acid and Potassium chloride Glycine and Hydrochloric acid Potassium acid phthalate and Hydrochloric acid Sodium phenylacetate and Phenylacetic acid Succinic acid and Borax Acetic acid and Sodium acetate Potassium acid phthalate and Sodium hydroxide Disodium hydrogen citrate and Sodium hydroxide pH Range 1.0-2.2 1.0-3.7 2.2-3.8 3.2-4.9 3.0-5.8 3.7-5.6 4.0-6.2 5.0-6.3 Composition Potassium dihydrogen phosphate and Sodium hydroxide Boric acid and Borax Diethylbarbituric acid and Sodium salt Borax and Hydrochloric acid Boric acid and Sodium hydroxide Glycine and Sodium hydroxide Borax and Sodium hydroxide Disodium hydrogen phosphate and Sodium hydroxide PH Range 5.8- 8.0 6.8- 9 2 7.0- 9.2 7.6- 9.2 7.8-10.0 8.2-10.1 9.2-11.0 11.0-12.0 Each buffer system is generally applicable over a limited range, viz., about 2 units of pH, but by making suitable mixtures of acids and acid salts, whose pk a values differ from one another by 2 units or less, it is possible to prepare a universal buffer mixture; by adding a pre-deter- mined amount of alkali, a buffer solution of any desired pH from 2 to 12 can be obtained. An example of this type of mixture is a system of citric acid, diethylbarbituric acid (veronal), boric acid and potassium dihydrogen phosphate; this is virtually a system of seven acids whose exponents are given below. 11 For details concerning the preparation of buffer solutions, see Clark, "The De- termination of Hydrogen Ions," 1928, Chap. IX; Britton, "Hydrogen Ions," 1932, Chap. XI; Kolthoff and Rosenblum, "Acid-Base Indicators," 1937, Chap. VIII. INFLUENCE OP IONIC STRENGTH 415 Citric acid Citric acid Citric acid H 2 POr Veronal Boric acid 1st stage 2nd stage 3rd stage pka 3.06 4.74 5.40 7.21 7.43 9.24 12.32 Apart from the last two acids, the successive pfc values differ by less than 2 units, and so the system, when appropriately neutralized, is capable of exhibiting appreciable buffer capacity over a range of from pH 2 to 12. Influence of Ionic Strength. In the discussion so far the activity factor has been omitted from the Henderson equation, and so the results may be regarded as applicable to dilute solutions only. Further, the pH values recorded in the literature for given buffer solutions apply to systems of exactly the concentrations employed in the experiments; if the solution is diluted or if a neutral salt is added, the pH will change because of the alteration of the