HYDROGEN ION CONCENTRATION HYDROGEN ION CONCENTRATION IT& SIGNIFICANCE IN THE BIOLOGICAL SCIENCES AND METHODS FOR ITS DETERMINA TIONS BY LEONOR MICHAELIS, M.D. Professor in the University of Berlin, Resident Lecturer in Research Medicine in the Johns Hopkins University VOLUME I PRINCIPLES OF THE THEORY AUTHORIZED TRANSLATION FROM THE SECOND REVISED AND ENLARGED GERMAN EDITION BY WILLIAM A. PERLZWEIG, M.A., PH.D. Associate in Medicine and Chemist to the Medical Clinic in the Johns Hopkins University and Hospital BALTIMORE THE WILLIAMS & WILKINS COMPANY 1926 Copyright 1926 THE WILLIAMS & WILKINS COMPANY Made in United States of America PUBLISHED OCTOBER, 1926 Composed and Printed at the WAVERLY PRESS FOR The Williams & Wilkinb Compant Baltimore, Md. AUTHOR'S PREFACE TO THE SECOND EDITION When the first edition of this monograph appeared, the circle of those interested in this field was as yet small, and it gives me pleasure to have contributed to the widening of this circle by means of my little book. Since then, in spite of the external difficulties of the years of war, such an enormous amount of work has been performed in this field, that it would be beyond the power of a single individual to render a complete account of all the details. For better or worse I am obliged to sacrifice completeness of the literature and to present the subject in text-book manner. The scope had to be considerably enlarged, so that the second edition is planned in several volumes of which the present first volume com- prises the theoretical physico-chemical principles, while the follow- ing volumes will present the methodology and the colloid-chemical, physiological and medical applications. In this way a totally new book will appear which has outgrown the scope of its title, and which may be designated as a "second edition" only for the sake of continuity with its small predecessor. The subject has been covered in the interval in several text- books of physiology with various degrees of detail. The most nota- ble monographic account is found in the book by W. M, Clark, The Determination of Hydrogen Ions, Baltimore, 1920. This book naturally has had a significant influence upon my second edition, which influence will be evident in the volume on methods. The present first volume places the fundamental principles of theory upon a wider basis than was done in the corresponding section of the first edition. The cause for this lies in the first place in the broadened realm of pure physical chemistry, and secondly in the extraordinary growth of the multiplicity of applications of this branch to the other branches of science. Therefore I deemed it important to present to my biological readers the general theo- retical principles upon a broad foundation, before the details of their application are dealt with. Leonor Michaelis. Berlin, Christmas, 1921. AUTHOR'S PREFACE TO THE ENGLISH TRANSLATION It was not without hesitation that I yielded to the urging of my American friends and of the pubhshers in consenting to have this monograph appear in an Enghsh translation, for a number of signifi- cant advances have been made in this field since the appearance of the original German edition. I finally decided to make at least par- tial amends by pointing out in a number of addenda within the text the most significant of these recent advances in our science. It is hoped that these will suffice to call the reader's attention to them and to direct him to the original sources for further details and study. The subjects which are especially noted in these addenda are: the recent contributions to the activity theory of ionization by G. N, Lewis, Bjerrum, and Debye; the modification of the theory concern- ing the dissociation of the ampholytes by Bjerrum; the theory of oxidation-reduction potentials. The latter is sufficiently developed in the new text to give the reader a basis for understanding the appli- cation of this theory to the use of the quinhydrone electrode, and to prepare him for the more recent studies in this field, particularly those of W. M. Clark. I wish to express my gratitude to Dr. W. A. Perlzweig for his painstaking and careful work on this translation. L. MiCHAELIS. Baltimore, June 1926. Vll TRANSLATOR'S PREFACE Professor Michaelis's Wasserstoffionenkonzentration has become well enough known during the past twelve years to my English speak- ing fellow workers in the biological and related sciences to warrant further introduction or any further statement of the reasons moti- vating its translation into English, beyond the one of rendering this book more available and useful to them. I am particularly gratified in presenting this monograph with the additions covering the latest developments which were so kindly prepared by Professor Michaelis for this translation. I am greatly indebted to him for this and for his interest and valuable suggestions. My further debt of gratitude is due to Dr. W. Mansfield Clark of the United States Hygienic Laboratory of Washington for his ex- tremely valuable criticism of the first draft of the manuscript; to my colleague Dr. Arthur Grollman of The Johns Hopkins Medical School for correcting the manuscript; to Dr. Irving H. Page of the Cornell Medical College and to Dr. R. M. Ellsworth of The Johns Hopkins Hospital for their generous help with the proofs; and to my wife, Dr. Olga Marx Perlzweig for her most valuable aid in the fur- nishing of adequate Enghsh equivalents for the more intricate Ger- man constructions, and for constant help on all possible occasions. To The Williams & Wilkins Company I express my appreciation of their generous cooperation in seeing this volume through the press in the most efficient manner. The Translator. Baltimore, June, 1926. /^'^l I library] a. IX CONTENTS PART I. THE CHEMICAL EQUILIBRIUM OF THE IONS Chapter I THE LAWS OF ELECTROLYTIC DISSOCIATION 1 . The law of mass action 3 2. Electrolytic dissociation 7 3. The dissociation of water 9 4. The properties of water and its ions. The definitions of acids, bases and ampholytes 13 5. The influence of dissolved substances upon the dissociation constant of water 19 6. The influence of dissolved electrolytes upon the state of dissociation of water 20 7. The hydrogen ion exponent and the pH scale 22 8. The numerical value of the dissociation constant of water 24 9. The general application of the mass law to the electrolytic dissociation of acids 29 10. True and apparent dissociation constants of acids. Pseudoacids 32 11. The dissociation of bases 37 12. The hydrogen ion concentration in pure acid solutions 39 13. Mixtures of weak acids and their alkaline salts: Regulators or buffers. 43 14. The two acids problem 46 15. The degree of dissociation and the dissociation residue of acids 48 16. The dissociation of polybasic acids 55 17. The dissociation of amphoteric electrolytes 60 18. The determination of the isoelectric point 69 19. The amphoteric ions 73 20. The hydrion concentration of a pure solution of ampholyte 74 21. The hydrion concentration and the solubility of weak acids 77 22. The hydrion concentration and the solubility of difficultly soluble acids. 82 23. The hydrion concentration and the solubility of difficultly soluble ampholytes 86 24. The hydrolysis of salts 87 25. Notes on methods for the determination of dissociation constants and of the isoelectric point 91 Chapter II THE theory of THE QUANTITATIVE DETERMINATION OF ACIDITY AND ALKALINITY 26. Definitions and statement of problem 94 27. The theory of indicators 96 xi 51530 XU CONTENTS 28. The determination of the hydrogen ion concentration, or the estima- tion of true acidity 102 29. The resistance of solutions to change in reaction and buffer value. . . 104 Chapter III THE DISSOCIATION OF STRONG ELECTROLYTES 30. The deviations from the mass law and their significance 110 31. The influence of the ionic charge on the conductivity 117 32. The influence of the ionic charge on the lowering of the freezing point. . 119 33. The three deviation coefiicients fo, i^ and fa (according to Bjerrum) . . . 121 34. Hydration of ions and its significance for the activity 123 35. The theory of activity and the hydrion concentration of buffers 127 36. The reduced constants of the physiologically important acids 129 37. Another exposition of the effect of strong electrolytes 131 Examples of the different methods of representation of the effect of strong electrolytes 133 Chapter IV THE state of dissociation OF ACIDS AND BASES DURING ACTUAL SALT-FORMATION 38. Introduction 136 39. The formation of the ionic equilibrium with true salt-formation 137 40. The behavior of ampholytes in the process of true salt-formation 141 41. The influence of salt-formation upon the isoelectric point 144 42. An extension of the conception of the reduced dissociation constant. . . 145 Chapter V electrolytic dissociation in NON-ACQUEOUS SOLUTIONS 43. Electrolytic dissociation 148 PART II. THE IONS, PARTICULARLY THE HYDROGEN IONS, AS SOURCES OF ELECTRIC POTENTIAL DIFFERENCES 44. Introduction 155 Chapter VI THE electrode POTENTIALS 45. The single potential of an electrode 156 46. The concentration chains 160 47. The numerical evaluation of Nernst's equation 161 48. Reversible electrodes for concentration chains 163 CONTENTS Xlll 49. The theory of gas chains 164 50. The electromotive series 167 50a. Oxidation-reduction potentials and the quinhydrone electrode.. .. 169 Chapter VII DIFFUSION POTENTIALS (POTENTIAL DIFFERENCES AT LIQUID JUNCTIONS) 51. Diffusion potentials 174 52. The physiological significance of diffusion potentials 182 Chapter VIII POTENTIALS AT PHASE BOUNDARIES 53. The origin and calculation of phase boundary potentials 183 54. Phase boundary chains 191 55. The direction of the current in phase boundary chains 195 56. The concentration effect 196 57. The ionic series 205 58. The electromotive series of the oils 207 59. The physiological application of the theory of phase boundary potentials 209 60. Potentials at precipitation membranes 215 61. Polarization phenomena at phase boundaries 218 Chapter IX MEMBRANE POTENTIALS 62. The membrane as the cause of a potential 220 63. The relation of membrane potentials to phase boundary potentials 228 Chapter X ADSORPTION POTENTIALS AND ELECTROKINETIC PHENOMENA 64. Definition of adsorption 232 65. The adsorption of electrolytes by charcoal 234 66. The exceptional position of H- and OH-ions 236 67. Charcoal as an insoluble ampholyte; acidoids, basoids, ampholytoids, saloids 239 68. The equivalent adsorptions of the ions of a salt by charcoal ; the neu- tralization effect of charcoal 242 69. Other adsorbents 243 70. The formation of a charge upon phase boundary surfaces due to ad- sorption of ions 245 71. The earlier history of electroendosmosis 246 72. Helmholtz's theory of electroendosmosis 248 73. The theory of electrophoresis 250 XIV CONTENTS 74. The relation of electroendosmosis and electrophoresis to the ionic theory 254 75. Perrin's experiments 256 76. The isoelectric point of diaphragms 259 77. Application of the acidoid theory to the sign of charge on diaphragms . . 261 78. A consideration of the other ions 265 79. Summary of the theory of the formation of an electric charge on dia- phragms 269 80. Changes in concentration, especially of the H-ions in electroendos- mosis 271 81. The history of the theory of adsorption potentials 277 82. Hydrodynamic potentials 279 83. The influence of electrokinetic potentials upon surface tension 284 84. A summary of electrokinetic phenomena 284 85. The splitting of phase boundary potentials 285 86. Coehn's rule 288 87. The significance of adsorption potentials in physiology and colloidal chemistry 290 Index op Authors 293 Subject Index 297 PART I The Chemical Equilibrium of the Ions ■^c^ CHAPTER I The Laws of Electrolytic Dissociation SUMMARY The equilihrium of chemical reactions is governed hy the law of mass action. Only in the case of dilute solutions can this law be sim- ply and at the same time accurately formulated. This law can also he applied to the dissociation of electrolytes and to the particular case of the dissociation of water. The properties of the ions of water lead to peculiarly specific results. All electrolytes which split off one of the two ions of water change the state of dissociation of water; these are the acids and bases. The concentration of H- or OH-ions in an aqueous solution determines its acid or alkaline character respectively. The strength of the acids and bases depends closely upon their disso- ciation constants. The method of calculating the H-ion concentra- tion from the concentration and strength of acids and bases in solu- tion is described, and the development of the theory of regulators or buffers is traced. Thr'ough the introduction of special functions of H-ion concentration, of the degree of dissociation and of the dissocia- tion residue, the laws of dissociation are reduced to a practically usable form, leading to an explanation of the dissociation of amphoteric elec- trolytes and to a conception of the isoelectric point. A discussion of the dependence of the solubility of slightly soluble salts upon the H- ion concentration and of the hydrolysis of salts is presented. 1. The law of mass action The "law of mass action," as its name implies, is an expression of the effect of the masses of the substances involved in a chemical reaction upon this reaction. Its data relate to the establishment of a definitive state towards which the chemical system is tending, or, to the chemical equilibrium. This state of chemical equilibrium is characterized by the fact that the amounts of the various kinds of participating molecules remain unchanged with the time. It is not, however, a condition of chemical rest, but rather a stationary 3 4 HYDROGEN ION CONCENTRATION condition in which chemical changes actually occur, but in such manner that at any given instant as much of any given kind of molecule is being produced as is being destroyed. For the mainte- nance of this stationary condition no external work is required, and conversely no work can be gained from these constantly occurring processes within this stationary state. Furthermore, the algebraic sum of all the work which may be derived from the various com- ponent processes of this state, as well as tlie smn of work expended for these processes, is equal to zero. This differentiates the sta- tionary state of chemical equiUbrium from the so called dynamic equilibrium as represented in a living organism. The latter is also a chemical system which may frequently keep its chemical com- position unaltered for a long period of time, but for the maintenance of this condition it requires a constant supply of energy. The apparently resting condition of a living organism does not come within the scope of the above definition of the law of mass action for a closed system. Nevertheless this does not preclude the validity of the application of the mass law to many particular processes occurring in the living organism. Thus processes which progress with a great velocity, such as ionic interreactions, take place also in the living cell in such a manner that a true chemical equi- librium is established. Thus, when free carbonic acid reaches the alkaline blood, a part of it is at once bound, i.e., it produces ions and salts, and exactly the same condition of equihbrium is established, as if the blood were not a component of a living organism. On the other hand the slowly progressing reactions in an organism do not result in the true chemical equihbrium, but at best reach a dynamic equihbrium. For example, the true condition of equilibrium for the combustion of sugar in the blood with the aid of the oxidizing ferments and oxygen is the practically complete destruction of the sugar. Tlds state of true equilibrium is, however, never reached during life, for before it is reached new sugar is brought in from the food or from the glycogen depots, so that the concentration of sugar in the blood is constantly maintained at a fairly fixed level. In this book we shall deal with ionic reactions only. These occur so rapidly that they always reach true equihbrium. Given two different blood samples containing different concentrations of hydrogen ions, the difference in these two concentrations never signifies that the reaction between the hydrogen ions and the other LAWS OF ELECTROLYTIC DISSOCIATION 5 ions has progressed to unequal extents. This reaction occurs with such prodigious velocity that we can have no conception of the intermediate stages; the cause lies in the fact that the two blood samples have actually a different chemical composition. The first chapter of this book deals with the laws of this true ionic equilib- rium, and its basis is, therefore, the law of mass action. The meaning of the law of mass action is as follows: 1. A simple special case: When any molecular species A decom- poses (''dissociates") into two molecular species Bi and B2, then the equihbrium, i.e., the apparent standstill of the reaction, is reached as soon as [A] [Bi] . [B,] = k The brackets denote the molar concentrations of the inclosed sub- stances. The constant k depends upon the nature of the reaction and the temperature. It is called the affinity constant of the reac- tion. Its reciprocal value [Bil [B2] lA] is called the dissociation constant of the substance A. 2. General case. When 1 molecule Ai + 1 molecule A2 + .... are transformed into 1 molecule Bi + 1 molecule B2 + . . . . , equilibrium is reached when [Ai] • [A,] . ■ . . ^ [Bi] . [B,].... When two or more of the reacting molecular species are identical, the rule is applied in the same way as above; as for instance in the oxy-hydrogen gas reaction: 1 mol. H2 + 1 mol. H2 + 1 mol. Oo ^ 1 mol. H2O + 1 mol. H5O hydrogen oxygen water vapor equilibrium is attained when IH2] • [H2] . IO2] [H2O] . [H2O] = k 6 HYDROGEN ION CONCENTRATION or abbreviated, the reaction 2 mol, H2 + 1 mol. O2 ;::i 2 mol. H2O is in equilibrium when [H2J2 ■ [O2] [H20]2 = k This law of mass action holds good only under the assumption that all of the interreacting molecular complexes as well as those newly formed coexist in a homogeneous solution (or in gaseous state). If one of the substances involved in a reaction is partially precipitated (removed from solution) because of oversaturation, then the term "concentration" of this substance should comprise only the concentration of its part still remaining in solution. Should approximate accuracy be accepted as satisfactory, then it could be stated that the law of mass action is confirmed by so large a number of cases that it is to be regarded as a general law also valid for those cases in which the theoretical proof of its cor- rectness could be not obtained. But in attempt at a higher degree of accuracy it must be recognized that the law in the form stated above is only approximately correct. The smaller the concentra- tion of the reacting substances the closer the approximation, and it is closest for ideal gases and for substances reacting in extremely dilute solutions. In fact, the strict thermodynamic test permits the application of the law to these latter extreme cases only. This test leads to a really general law which differs from the above stated law insofar as the term ''concentration" is everywhere re- placed by the term "pressure." It may either be gas pressure or osmotic pressure. Since, in the state of very great dilution, pres- sure and concentration are proportional to each other, it follows that under this condition one may simply substitute concentration for pressure. The thermodynamic form of the law is vahd for any order of magnitude of pressure. But at high pressures concentra- tion and pressure are no longer proportional. Hence the theo- retical explanation of the fact that at the higher concentrations the law of mass action, expressed in terms of concentration, cannot be absolutely correct. In order to render it in an absolutely valid form it would be necessary to multiply every factor involving concen- tration by an appropriately chosen factor. This factor is generally less than 1. The values so corrected are called active masses. LAWS OF ELECTROLYTIC DISSOCIATION 7 The question may be raised: within what limits of concentra- tions may the mass law be used without corrections? In solutions of non-electrolytes of 0.1 molar concentration the error is insig- nificant, while it is within allowable limits even in solutions of molar concentration. It has been recently found that these errors or deviations become much greater as soon as one deals with ionized solutions. Since relatively large deviations have been discovered between the ionic equilibrium and the mass law, this fact has to be taken into consideration. As unportant as these corrections are whenever the determina- tion of exact numerical figures is involved, nevertheless the fact cannot be over-emphasized that in most cases they are very small. It is, therefore, essentially justifiable to develop first the compli- cated system of ionic equilibrium on the basis of the simple law of mass action, and then subsequently to explain when and how the necessary corrections are to be applied. 2. Electrolj^ic dissociation The publication in 1887 by Svante Arrhenius^ of the theory of the dissociation of the electrolytes marked the beginning of a new era in physical chemistry. A great mass of facts which had been awaiting a proper explanation became at once comprehensible. The astounding range of importance of this theory is best illustrated by the fact that the number of its consequences is still growing even at present, and all of this part of the book represents but one series of such consequences. The context of the Arrhenius hypothesis is briefly this: The conductivity of an electrolyte in solution does not depend, as it had been hitherto assumed, on its decomposition by the current into ions, but upon the spontaneous splitting of at least part of the electrolyte into ions, when it is brought into solu- tion. These ions may be regarded as independent molecules; their interreactions obey the usual law of mass action, which was first elaborated by Guldberg and Waage and then chiefly by van't Hoff. Until recently the origin of the opposite free electric charges in dissociated solutions was not clear. The following explanation had been offered-: Space was imagined to be filled with electrically ^ Svante Arrhenius, Zeitschr. f. physikal. Chem. 1, 631 (1887). * W. Nernst, Theoretische Chemie (the older German editions). 8 HYDROGEN ION CONCENTRATION neutral elementary corpuscles of electricity, each of which in turn consisted of one negatively and one positively charged particles. These particles were supposed to serve as an inexhaustible source for electrical charges arising in dissociation. Thus in the dissocia- tion of water an H-atom removes from this reservoir of electricity a positive charge and the OH-residue removes a negative charge. This assumption has become superfluous with the introduction of the Rutherford-Bohr atomic model. The reservoir of these free charges is in the atoms themselves. Each atom consists of a posi- tively charged nucleus, the charge consisting of a definite number of elementary charges. The nucleus is surrounded by negatively charged electrons. In an electrically neutral atom the number of electrons is equal to the number of positive charges upon the nu- cleus. Thus in the dissociation of an electrically neutral molecule of H2O into H+ and 0H~ what happens is that the dissociated OH radical carries along with it one electron in excess of its electro- neutral state, this extra electron having been derived from the H-atom which now becomes the positively charged H-ion, or H+. The electrons which determine the state of an atom are held by the latter with varying degrees of firmness. If one of the electrons happens to be held in the atom by a loose bond, then it may be easily thrown off, as it were, and, as a result of this loss of a nega- tive charge, the rest of the atom becomes a positively charged ion. This electron is designated as the valence-electron. If two such are present in an atom, it forms bivalent positive ions, etc. On the other hand, there are atoms in which the saturation with their full quota of electrons is not complete, when given in their usual electroneutral state. In these the positively charged atomic nu- cleus is capable of combining with one or more electrons, and thus to become a uni- or polyvalent negative ion. According to their capacity to give up or to take on electrons the positive and nega- tive elements are differentiated. The Berzelius theory is thus, in a modified form, revived again. When a "positive" (Na) and "negative" (CI) atom are found together, a competition arises between the two for the labile electron, A nmnber of these pairs of atoms arrange themselves so that the electron leaves the sphere of attraction of the positive atom and is bound by the negative atom; another part becomes so arranged that the valence-electron becomes attached to the negative atom, but without becoming completely de- LAWS OF ELECTROLYTIC DISSOCIATION 9 tached from the positive atom. The first arrangement represents the electrolytically dissociated portion, or the ions, while the second represents the undissociated molecules (of NaCl). According to the predominance of the first or second mode of atomic arrangement stronger or weaker degrees of electrolj^tic dissociation are designated. The condition of dissociation of an electrolyte is not a resting state of equilibrimn; for one and the same molecule may in the course of time pass back and forth between the dissociated and undissociated state repeatedly. The "degree of dissociation" is a static quantity only in the sense that it represents the ratio of the dissociated portion to the total amount of electrolyte at a given instant. Since even in the most dilute solutions the number of individual molecules is enormous, the probability that a statistical table composed of figures estimated at different times would yield varying results is very slight. It is in this sense that in a solution of an electrolyte a dynamic equilibrium exists, just as it does in every solution in which an irreversible reaction has run its course to a state of equilibrium. Radicals, such as OH, CN, etc., behave in the above respects exactly as the ions of the individual elements do. Since the atom of hydrogen possesses but a single electron, it follows that the hydrogen ion consists of a bare atomic nucleus carrying a single positive charge. The theory of the hydrogen-atomic model admits of the possibil- ity of the H-atom taking up a second electron and thus becoming a negative ion. Indeed Moers^ (with Nernst) has recently demon- strated that the definite crystalUne compound lithium hydride, LiH, on electrolysis when in a molten state produces positive Li-ions and negative H-ions; i.e., the hydrogen is evolved at the anode instead of at the cathode, as is usual. Here it behaves as a halogen, just as it is frequently simply substituted for a halogen in organic com- pounds. Otherwise it behaves as a metal. In aqueous solution negative H-ions cannot occur. 3. The dissociation of water Water belongs to that group of substances which undergo electro- lytic dissociation. The laws of electrolysis have been worked out ' W. Nernst, Dtsch. Bunsen-Ges. 1920; K. Moers, Zeitschr. f. anorg. u. allg. Chem. 113, 179 (1920). 10 HYDROGEN ION CONCENTRATION best for aqueous solutions in which water is the solvent medium. Whenever any electrolyte is dissolved in water, the resulting solu- tion contains not only the ions of this electrolyte, but also the ions yielded by the water, namely, H+ and 0H~ ions. While the degree of the dissociation of water is indeed very slight, still, the purest water shows a definite capacity to conduct an electric current. This fact can only be explained on the basis of its electrolytic disso- ciation, be it ever so slight, if Arrhenius's theory is to be accepted as being correct. F. Kohlrausch and A. Heydeweyller* obtained after repeated distillations of water under rigorous precautions a final value for the conductivity of water, which value could not be diminished or altered by further purifications. In this way they were the first to establish the numerical value for the dissociation of water. Subsequently many other methods were developed for the determination of the same value, which will be described later, and which corroborated Kohlrausch's findings. The degree of dissociation of water is, as was already stated, very slight, i.e., the H+ and 0H~ ions have a very great tendency to unite and to form H2O. On the other hand, most electrolytes when dissolved in water dissociate into ions to a much greater extent, and some are even completely dissociated. This unique behavior of water has two noteworthy consequences : 1. If a strong electrolyte, such as KCl for example, which yields neither H+ nor 0H~ ions, is dissolved in water, then the number of H+ and OH" ions yielded by the water itself is so insignificant in comparison with that of the other ions present, that many of the properties of the given solution may be completely accounted for by the ionization of the dissolved electrolyte, as, for example, the con- ductivity and the depression of the freezing point. The dissocia- tion of the water itself is not, however, influenced by a strong electrolyte dissolved in it. 2. But, if an electrolyte which itself yields either H+ or OH" ions be present in an aqueous solution, then the dissociation of the solvent, water, is fundamentally affected. These special electro- lytes have such remarkable properties that they had been placed in a separate class long before the theoretical basis of their peculiar behavior was clearly understood. The electrolytes which give off * F. Kohlrausch und A. Heydweyller, Zeitschr. f. physikal. Chem. 14, 317 (1894). LAWS OF ELECTROLYTIC DISSOCIATION 11 H+ ions are those that had been known as "acids" and those giving off 0H~ ions as "bases," while all the other electrolytes had been designated as "salts." Thus the H+ and 0H~ ions occupy a peculiar position among the other ions by virtue of their affinity for each other being so great that they are always but very little dissociated, and secondly, because they are the ions of the commonest solvent, water. Tlie chemical equation representing the dissociation of water is: H2O ;^ H+ + OH- (1) in which H+ stands for the positively charged hydrogen-ion and 0H~ for the negatively charged liydroxyl-ion. The other conceivable manner of dissociation of water according to the equation: H2O ;:± 2H+ + 0 = is, quantitatively speaking, so negligible that it is not in any way demonstrable. Since the H2O molecule is capable of yielding H+-ions, it may be regarded as that of a dibasic acid dissociating in two successive steps: 1. H2O ;=i H+ + OH- 2. OH- ?i H+ + 0= But as a rule, as far as the present day methods are capable of detecting such changes, the dissociation process of the second step occurs to a very much smaller extent than that of the first step. Furthermore, since in the second step of the dissociation shown above in addition to H+-ions also OH~-ions are produced, water may also be considered^as a base. The acidic and basic properties of water are as matter of fact very weak. In this respect water is entirely amphoteric, or, its "reaction" is completely neutral. In terms of the law of mass action the reaction stated in equation (1) may be expressed as the following equilibrium relationship: [H+] X [0H-] [H2O] = k (2; The brackets in the above equation indicate the concentration of the chemical "species" they inclose, and this concentration is always understood to be in terms of gram-molecules or in mols per 12 HYDROGEN ION CONCENTRATION liter. Since the degree of dissociation of water is extremely small, the concentration of the undissociated H2O molecules does not show any measurable decrease, and, therefore, the concentration of the undissociated water is, within the limits of experimental error, equal to the total concentration of the water, which is, of course, a constant quantity. Then by transposing the term [H2O] to the right side in equation (2) and designating [H2O] k = kw the above equilibrium expression becomes: [H+] . [0H-] = kw (3) kw is the dissociation constant of water, and for every definite tem- perature it has a definite value. The active masses indicated in the above formulation of the mass law may be considered as being equal to their respective concentra- tions only in very dilute solutions and in the ideal gaseous state. The active mass of water indicated in (2) as [H2O] is not easily defined. According to Nernst" its value is to be assumed as one proportional to the vapor pressure of water. Thus, for instance, the values for [H2O] in pure water and in a 1 M" sugar solution are not quite identical, that of the latter being 2 per cent less. The statement that the active mass of water in different solutions is proportional to their respective vapor pressures is valid only for solutions at the same temperature. The active mass of water in solutions not at the same temperature cannot be assumed as being simply proportional to the vapor pressures. The dissociation constant of water is then a constant value for any given temperature. It varies with the temperature to a greater extent than the dissociation constants of most electrolytes. This depends on the fact that the value [H2O] varies greatly with the temperature. While it is true, as was stated above, that [H2O] is not to be assumed as being simply proportional to the vapor pres- sures at varying temperatures, still this value is doubtlessly in some, not as yet clearly understood, manner related to the vapor pressure. The wide variations of kw depends not so much upon any significant variations of the value k itself in equation (2) as upon variations in the value of the factor [H2O]. A satisfactory explanation of these relations has not as yet been derived either from thermodynamic data or those of the molecular theory. Closely 6 W. Nernst. Theoretische Chemie, 7th ed. P. 680. LAWS OF ELECTROLYTIC DISSOCIATION 13 related to the high degree of dependance of the dissociation constant upon the temperature is the relatively high heat of dissociation of water. The dissociation constant of water kw at 22°C. amounts to 1 X 10~", i.e., one ten-millionth squared. From this it follows that in pure water the H+- and OH~-ions are each present in a concentra- tion of 1 X 10~^, or, one ten-millionth gram-ions per liter (one ten-millionth normal in respect to each ion) . If this figure seems to be extremely small, it is, nevertheless, a sharply defined value. If it be recalled that one gram-molecule of any substance contains 6.2 X 10-^ molecules (Loschmidt), then in pure water there must be present 6.2 X lO'^* X 10"^ = 6.2 X lO^^ of H+- also of OH --ions per liter, or 6.2 X 10^*^ = 62 billion per cubic millimeter. One liter of water contains 1000/18 = 55.56 mols of H2O, or 55.56 X 6.2 X 10^3 = 344 X lO^^ molecules of H2O. The same liter on the other hand contains, as seen above, 6.2 X 10^® H+-ions. Therefore, out of every 555 million H2O molecules one is dissociated. 4. The properties of water and of its ions. The definitions of acids, bases and ampholytes Water occupies a most unique position among liquid compounds. Leaving aside some of its peculiarities, such as its maximal density at 4°, its expansion on freezing, its very great surface tension, etc., only those of its properties will be dealt with here which affect its behavior as a solvent for electrolytes. These special properties are best explained on the basis of its exceptionally large dielectric con- stant. The significance of this important factor may be conveyed as follows: Two electrically and oppositely charged particles of which one carries a positive charge ei and the other a negative charge e2 and separated by a distance r are attracted towards each other with the force equal to , when the two are in a vacuum (or with practically the same force in a gaseous medium). When, however, the same two particles are found in a solid or liquid medium, the force of attraction operating between them becomes smaller, and its formulation is now — =^. In this expression D represents the dielectric constant of the medium. The greater the value of D of a medium the smaller is the electrostatic attraction 14 HYDROGEN ION CONCENTRATION between any two oppositely charged particles found in ifc. Water has almost the largest dielectric constant of all substances as shown in the table 1 (the figures given*^ are mostly for 17 to 18°C.). Only polyhydric alcohols (glycerin) and perhaps the acid amides and the nitriles approach the magnitude of the dielectric constant of water, and formamide and hydrocyanic acid alone surpass it. Because of this factor all electrostatic forces of attraction are greatly diminished in water and the dissociation of electrolytes dissolved in it is powerfully enhanced. This interdependence between the dielectric constant of the solvent and the degree of TABLE I Table of dielectric constants Hydrogen-gas 1.000500 Aniline 7.32 Air 1.000576 Amyl alcohol 16.0 Hexane l.SS Diamond 16.47 Petroleum 2.0 Ammonia 22. 0 Wood, paper 2.0-7.0 Ethyl alcohol 25.4 Benzene 2. 26 Methyl alcohol 35. 4 Olive oil 3.0 Nitrobenzene 36.45 Ice (at -18°) 3.16 Acetonitrile 38.8 Bromine 3.2 Glycol 41. 2 Octyl alcohol 3.4 Glycerol 56 Urea (solid) 3.5 Acetamide (fused) 59.2 Ethyl ether 4.3 Glycollic acid nitrile 68 Chloroform 4. 95 Water 81. 7 Sodium chloride (solid) 6.12 Formamide 94 Acetic acid 6.3 Hydrocyanic acid 95 dissociation of solute electrolytes is universally applicable, and it was simultaneously discovered in 1893 by Nernst'' and J. J. Thom- son.^ The quantitative relationship between the degree of dissociation of the solute and the dielectric constant D of the solvent was worked out by P. Walden.^ He compared the behavior of solutions of a ' The figures were taken chiefly from Landolbt-Bornstein, Physikalisch- chemische Tabellen. Berlin, 1912. 7 W. Nernst, Gottinger Nachr. No. 12 (1893); Zeitschr. f. physikal. Chem. 13,531 (1894). 8 J. J. Thomson, Phil. Mag. 36, 320 (1893). 9 P. Walden, Zeitschr. f. physikal. Chem. 54, 228 (1905), also 94, 263 and 374 (1920). LAWS OF ELECTROLYTIC DISSOCIATION 15 certain salt (tetraethyl ammonium iodide) in various solvents having the same degree of dissociation. A simple relationship between D and c, the corresponding concentration, was obtained D = = constant, or, for an electrolyte possessing the same degree of dissociation in various solvents the corresponding "linear concentration^^ (i.e., the cube root of the concentration) is proportional to the dielectric constant of the solvent. It can be easily shown that the cube root of the concen- tration of the solute is inversely proportional to the average distance between any of its adjacent molecules. The ions of water have also very special properties. According to the Rutherford-Bohr hypothesis of the atomic structure the H+- ion consists of one positively charged atomic nucleus unaccom- panied by any (negative) electron. It is the smallest mass aggre- gate bearing a positive charge. The smallest negatively charged body is the electron whose mass is only 1/1800-th of a H+-ion, and an analogously small positively charged mass is unknown. In addition to being the smallest known positively charged particle, the H+-ion has the smallest atomic radius of any atomic structure. Furthermore, since it has no shell of electrons surrounding it, it can reach into closer proximity with negatively charged particles than any other positive ion. Also, since the force of attraction of opposite charges increases as the square of the distance, the H+-ion can be held more firmly by negative ions than any other mono- valent positive ion. The properties of the OH~-ion are not as easily explained on the basis of the atomic model. There is a possibility of presenting this entire question in a way which will obviate the necessity of endowing the OH~-ion with special and extreme properties, and which will permit the unique position of the H+-ion to suffice for the elucidation of the problem. This particular way may be stated as follows: Of all electrolytes in aqueous solution the acids and bases are the least dissociated. Tl:ere are a few exceptions to this rule, HCl, HNO3, H2SO4, NaOH, KOH, Ba(0II)2 and some others, which are just as strongly disso- ciated as the salts. But most free acids and bases are weak electro- lytes, while, as a general rule, the salts belong to the class of strong 16 HYDROGEN ION CONCENTRATION electrolytes, including also the salts of weak acids and bases, such as ammonium acetate and ammonium carbonate.^" In case of the acids the common cause for this behavior is explicable on the basis of the atomic model. The center of the minute H+-ion is able to attain closer proximity than any of the larger ions to the center of electrostatic force of the particles to which it is attracted. The force of attraction increasing greatly with diminishing distance, the H+-ion is held more firmly by the anions than any other cation. This conception does not appear to be capable of ready application to the bases. The OH~-ion is spatially more diffuse than the H+- ion. It cannot be in any sense asserted, for instance, that the OH~-ion is held more firmly by cations than the Cl~-ion. The NH+-ion does not bind the OH~-ion any more firmly than it does the Cl~-ion; i.e., the molecular union NH4OII occurs just as spar- ingly as NH4CI does. Not a single weak base exists in which, analogously to the weak acids, its basic nature is due to its slightly dissociable OH-groups. The countless organic compounds con- taining OH-groups are, without any exception, ^^ non basic. The N-free organic compounds which exhibit a basic character usually hold, by a double bond, an 0-atom capable of forming an oxonium ion, in a way analogous to the formation of NH4+-ions from NH2 groups. The usual explanation of the basic nature of NH3 and of every amine is as follows : The reaction NH3+ + H2O -^ NH4+ + OH" is assumed. The existence of the molecular entity NII4OH is problematic, and there is nothing compelling us to accept its existence; it is only accepted in order to justify the origin of the NH4+ ion. With at least equal justification its origin may be explained by the reaction; NH3 + H+ -^ NH4+, or, stating it in words, NH3 binds an H+-ion and thus becomes an NH4+ ion. Accordingly, a base may be defined as a molecular species, electro-neutral in itself, which on uniting with or binding an H'^-ion becomes a positive io7i. In complete analogy with the above an acid is defined as a molecular species, electro-neutral in itself, which by splitting off an H-ion becomes a negative ion. The strong alkalies do not apparently conform to this definition of a base. But this view is only superficial. The dissociation of 1" That portioa of these salts which is not hydrohjzed is strongly dissociated. '1 Except for the quartenary ammonium bases which are strong electrolytes and which behave in the same way as NaOH, as detailed below. LAWS OF ELECTROLYTIC DISSOCIATION 17 NaOH is usually written as: NaOH -^ Na+ + 0H~, and it seems indeed as if the H-ions play no part in it. But it is to be remem- bered that the symbol Na+ does not represent the actual constitu- tion of this ion. The ions are being constantly hydrated, and while the changing number of molecules of the water of hydration (see below) is not usually included in the formula for the Na+ ion, yet a sjonbol including 1 molecule of H2O as (NaH20)+ is hardly less justifiable than the sjanbol Na+. In this method of representation the sodium ion arises from sodium hydroxide by the addition of a H+ ion, and not by splitting of an 0H~ ion. NaOH + H+-^(NaH20) + Or, if it is not desirable to be restricted to the above formulation, the same result can be represented as: NaOH + H+-^Na+ + H2O That is to say: one molecule of NaOH binds or otherwise employs one H+ ion to form one Na+ ion with one molecule of H2O. In this way the strong bases can be brought within the scope of the above general definition of a base. If it be desired to include the amphoteric electrolytes in this same general scheme, then they would be defined as molecular species which are capable of both binding and yielding H-ions. If, for example, glycocoll be designated as H-R-NH2, its double dissocia- tion will be expressed as: 1. H-R-NH2 + H+^(HRNH3)+ and 2. H-R-NH2 > H+ + (RNHj)- Hitherto reaction 1 lias been usually written as H-R-NH2 + H2O -> H-R-NH3OH -^ (H-R-NH3)+ + OH- but the actuality of the molecular complex HRNH3OH is entirely questionable. Therefore, there is the possibility of correlating all aspects of acidity and alkalinity with the H+ ion without any reference to the 0H~ ion. But this is only a possibility and not an absolute necessity. Indeed, there are cases in which it is preferable and easier to explain the formation of an anion by the addition of 18 HYDROGEN ION CONCENTRATION an 0H~ ion rather than by the loss of an H+ ion, as for instance in the formation of the bicarbonate ion: CO2 + OH- -* HCO3- As will be shown later, in the discussion of the adsorbability of the ions, the H+ ion among the cations and the 0H~ ion among the anions are also unique because of their extraordinarily great adsorbability. Since the 0H~ ion occupies just as an exceptional position among the anions as the H+ ion among the cations, it is well in discussing the question of acid and base formation not to press the H+ ion to the foreground at the expense of the 0H~ ion. Briefly restated: The yielding of an H+ ion is completely equivalent to the addition of an 0H~ ion and reversely. In many cases these two processes are so nearly identical that it is impossible to decide in favor of one or the other mode of reaction. The following definitions may now be stated: an acid is a mole- cule, electrically neutral in itself ivhich yields by dissociation H+ ions (or adds to itself 0H~ ions); a base is a molecule which yields 0H~ ions (or adds to itself H+ ions); an amphoteric electrolyte or an ampholyte is a substance ivhich unites in itself both of these properties. In these definitions special emphasis is designedly laid upon the words "a, molecule electrically neutral in itself." A negatively charged particle may bind H+ ions and thereby neutralize its elec- tric charge, e.g., CH3COO- uniting with H+ to form CH3COOH. Such a molecule which unites with a H+ ion and is electrically neutrahzed is an acid ion or an anion. A molecule, which on unit- ing with a H+ ion acquires a positive charge is a base (NH3 + H+-^ NH4+). A molecule, which binds a 0H~ ion with the resulting neutralization of its charge is a basic ion or a cation (Na+ + OH- = NaOH). A molecule which on binding OH" ions acquires a negative charge is an acid (CO2 + 0H"~— ^HCOs" and SO3 + 0H-->HS04-). A molecule which already has a positive (or negative) charge which on binding an H+ ion (or 0H-) doubles its charge is a divalent cation (or anion), as illustrated by hydrazine for cations: NH2NH2 + H+-^NH2NH3+ and further NH2NH3+ + H+-^NH3NH3++, and by carbonic acid for anions: CO2 + OH" — > HCOs", HCOa" + 0H~ -^C03= + H2O (this last hydrated ion reaction can be stated as HC03--^C03= + H+). As will be seen in the chapter on ionic LAWS OF ELECTROLYTIC DISSOCIATION 19 adsorption, the above multiplicity of ways of writing reactions is not a futile exercise but quite a useful procedure. 5. The influence of dissolved substances upon the dissociation constant of water It was stated above that every substance dissolved in water has an influence upon its dissociation constant as a result of changes in the pressure of the water vapor and in the mass of the undissociated water. This influence is however very slight. The vapor pressure of a unimolar solution of a nonelectrolyte (sugar, urea, etc.) is only two per cent less than that of pure water, in solutions of binary electrolytes this effect is approximately doubled. But molar solu- TABLE 2 Alcohol Dielectric constant Vkw at 25°C. ■per cent 0 84 4.6 X 10-^ 7.4 78 4.1 24.0 67 3.4 41.8 56 2.7 64.8 43 1.5 86.6 32 0.73 92.6 29 0.47 97.4 27 0.28 99.8 26 0.30 tions are of such great concentration that they may be left out of consideration in this discussion. And even with such high concen- trations the effect upon the value of kw is within the present limits of error involved in the determination of k^. Let us next consider the conditions under which a dissolved sub- stance markedly affects the dielectric constant of water. The dielectric constant and kw diminish in value in aqueous solutions of alcohol with increasing concentrations of alcohol. But in this case also only the higher concentrations have a marked effect. The figures in table 2 were obtained by Lowenharz^- for aqueous alco- holic solutions at 25°C. " Lowenherz, Zeitschr. f. physikal. Chem. 20, 283 (1896). 20 HYDROGEN ION CONCENTRATION (By kw the value of is understood, where H2O is [H2OJ assumed to be proportional to the partial pressure of the water- vapor in an aqueous alcoholic solution.) In a unimolar alcoholic solution, which contains 4.6 per cent alcohol, the value of kw is not easily distinguishable from that of kw for pure water. For all "watery" phases in living tissues the usual value of k^ (that of pure water) is assumed. This value, however, may not be applied to such "lipoid" phases as cell membranes, nerve fibers, etc. It has been stated recently by R. Keller^^ that colloidal aqueous solutions, especially those of proteins, have a distinctly smaller dissociation constant than pure water. The reason why this highly important problem has not been accurately solved long ago lies in the technical difficulties of determining the dielectric constant in such well conducting fluids as aqueous solutions. If Keller's state- ment is confirmed, then it will have an important bearing upon the calculation of dissociation constants and of ionic equilibria. The quantitative estimation of these values will not be considered for the time being. 6. The influence of dissolved electrolytes upon the state of dis- sociation of water An electrolyte dissolved in water breaks up more or less com- pletely into its ions. If one of these ionic species is the H-ion, then the concentration of the hydrogen ions of the water is in- creased a certain number of times. But since equation (3) on page 12 must remain unclianged, it follows that the concentration of OH-ions must decrease the same number of times. If the dissolved electrolyte yields to the aqueous solution OH-ions then the exactly reverse effect occurs. An electrolyte which yields only ions other than H- or OH-ions has no effect on the dissociation of the water. H-ions are given off by acids and acid salts; OH-ions are produced by bases and basic salts; electrolytes which yield neither of these are the true neutral salts, (as NaCl, for example). It follows, therefore, that while every aqueous solution must contain H+ or OH" ions it may react either acid, neutral or alkaline. A neutral solution is characterized by the fact that it contains equal " R. Keller, KoUoid-Zeitschr. 29, 193 (1921). LAWS OF ELECTROLYTIC DISSOCIATION 21 numbers of H+ and OH" ions, and at 22°C. it is lO"'' normal. An acid solution contains more than 10~^ iV H+ ions and less than 10~^ A'' 0H~ ions, an alkaline solution represents the reverse con- dition. And yet the strongest of acid solutions is not absolutely- free of 0H~ ions, nor is the strongest alkaline solution entirely free of H+ ions. In order to define the state of acidity, neutrality or alkalinity of a fluid it is only necessary to state its [H+] or [0H~]. Since the results of such measurements are usually stated in terms of [H+], this time-established convention will be retained henceforth K ■^ St.- I - [M'J- iiO~ 2jo' dro' Fia. 1 «a7^ and the long term "hydrogen ion concentration" will be replaced by the shorter term "Hydrogen number." Therefore, the defini- tion of a Neutral reaction is : [H"*"] = 10"^ Acid reaction is : [H+] = > 10~^ Alkaline reaction is : [H+] = < 10 ~' The number 10~'^ above is in terms of normality or gram ions per liter. As stated above the [0H~] is always dependent upon the [H+] in the manner of the following relationship : ^ !i! LIBRARY 22 HYDROGEN ION CONCENTRATION If the [0H~] be graphically represented in a rectangular system of coordinates as a function of [H+] then the hyperbola shown in figure 1 is obtained. It is of interest to find out at what reaction the sum of H+ and 0H~ ions of a solution is at its minimum. An inspection of the above figure shows that it occurs at the neutral reaction, [H+] = 10~'^. Mathematically this point can be derived in the following way: Let the sought value [H+] + [0H~] be designated as u. Then u = [H+J + IOH-] Km u = [H+] + [H+] du k. d[H+] [H+]2 Setting this derivative = 0, we obtain as the minimal condition [H+] = V^ [H+] = IOH-] = Vk^ which represents the reaction of neutrality. That this value repre- sents a minimum and not a maximum is shown by the fact that its second derivative d[H+]2 [H+]2 must always be positive. 7. The hydrogen ion exponent and the pH scale The statement [H+] = 10"^ may also be expressed as log [H+] = -7, or, -log [H+] = 7. S0renseni'' replaced the term log [H+] by the symbol pH.^^ The acidic or basic property of a solution 1^ S. P. L. Sorensen, Enzymstudien II. Biochem. Zeitschr. 21, 131 (1909). 16 S0rensen's original symbol was PH"^; the present author uses ph for -log[H+], and the symbol h for [H+]. In order to avoid adding to the already confusing multiplicity of these symbols the form pH adopted by the Amer- ican Society of Biological Chemists and the symbol [H+] now in current use in this country will be adhered to in this translation to designate S0renson's exponent and the molar concentration of hydrogen ions respectively. The abbreviation "hydrion" for hydrogen ion will also be used. — Translator. LAWS OF ELECTROLYTIC DISSOCLA.TION 23 may thus be just as well defined by stating its pH as by its [H+] value. Quite properly the use of the pH symbol has become uni- versal. Its advantages are in typographic simplification and also in the fact that the methods for measuring hydrion concentration yields values only in terms of log [H+] from which [H+] must be calculated. This roundabout way is avoided in adopting the pH units for defining the reaction. It will be furthermore shown that the graphic representation of the effects of H+ ions under various conditions in terms of the logarithmic symbol possesses peculiar advantages. The graphic representation of important function yields in this way easily comprehensible symmetric curves, which would not be obtained otherwise. It is best then at once to become accustomed to employ the symbol pH. TABLE 3 pH (HT pH [H+J n.OO 1.00 X 10-° n.50 3.16 X 10-°+ » n.05 8.91 X 10-"+i(= 0.891 X lO-'') n.55 2.82 X 10-°+ 1 n.lO 7.94 X 10-°+ 1 n.60 ;2.51 X 10-°+^ n.l5 7.18 X 10-°+ 1 n.65 2.24 X 10-°+ 1 n.20 6.31 X 10-°+ 1 n.70 2.00 X 10-°+ 1 n.25 5.63 X 10-°+ 1 n.75 1.78 X 10-°+ 1 n.30 5.02 X 10-°+ 1 n.80 1.59 X 10-°+ 1 n.35 4.47 X 10-°+ 1 n.85 1.41 X 10-°+ 1 n.40 3.98 X 10-°+ 1 n.90 1.26 X 10-"+ 1 n.45 3.55 X 10-°+ 1 n.95 1.12 X 10-°+ 1 The calculation of [H+] in terms of pH may be elucidated by the following example: 1. Let [H+] = 1.00 X 10-7; then log [H+] = -7 or pH = 7 2. Let [H+1 = 2.00 X 10-^; then log [H+] = 0.301-7 pH = 6.699 Table 3 will be found of use for recalculating [H+] into terms of pH. n in this table is any integral number. The [H+] and pH of some of the more commonly used acids and bases are given below. The dissociation figures of HCl and NaOH are calculated from their conductivities (table 4). Since the dissociation constants of most acids and bases do not 24 HYDROGEN ION CONCENTRATION vary widely with the temperature, the pH values of acids may be used without great error at any temperature. With bases on the other hand the pH varies widely, while the pOH is quite independent of the temperature. The dissociation constant of water changes TABLE 4 iVHCl 0.1 A^HCI 0.01 NB.C\ 0.001 A^HCl 0.0001 iVHCl N acetic acid 0. 1 A^ acetic acid. .. 0.01 A'^ acetic acid. 0.001 N acetic acid ATNHs 0.1 NNHs 0.001 A^ NH3 0.001 N NH3 N NaOH 0.1 N NaOH 0.01 A^ NaOH 0.001 A^NaOH H+] 0.80 0.084 0.0095 9.7 X 10-" 9.8 X 10-* 4.3 X 10-3 1.36 X 10-3 4.3 X 10-* 1.36 X 10-* 1.7 X 10-12 5.4 X 10-12 1.7 X 10-11 5.4 X 10-1" 9.0 X 10-1* 8.6 X 10-1* 7.6 X 10-13 7.4 X 10-11 pH 0.10 1.076 2.022 3.013 4.009 2.366 2.866 3.366 3.866 11.77 11.27 10.77 10.27 14.05 13.07 12.12 11.13 at 18° TABLE 5 . [H+] pH [0H-] pOH In 0.01 A^HCl at ■ f 18° [ 38° 9.5 X 10-3 9.5 X 10-3 2.022 2.022 7.6 X 10-13 3.6 X 10-12 12.12 11.44 and In 0. 1 A^ NH3 at ■ f 18° [ 38° 5.4 X 10-12 2.5 X 10-11 11.27 10.60 1.7 X 10-12 1.7 X 10-12 11.77 11.77 greatly with variations of temperature. On the same basis the pOH of acids varies considerably with the temperature (table 5). 8. The numerical value of the dissociation constant of water The numerical value of kw is determined by various methods, the principles of which are as follows: LAWS OF ELECTROLYTIC DISSOCIATION 25 a. From the conductivity oj pure water. F. Kohlrausch and R. Heydeweiller^'^ determined the conductivity of absolutely pure water. Since the conductivities of the H- and OH-ions were known, the concentration of these ions could be calculated from the ob- served conductivity of pure water. At 25° the product [H+] X [0H-] = k„ = 1.1 X 10-1^ b. From the catalytic capacity oj pure water. At the suggestion of van't Hoff, J. J. A. Wijs^^ determined the velocity of saponification of methyl acetate by pure water. Tlie OH-ions function as catalyzers and the velocity of the reaction is proportional to their concentra- tion. This observed velocity was compared with that of the same saponification carried out by the use of an alkali solution of known [0H~] concentration and the [0H~] of pure water was easily calcu- lated from these data, and furthermore in pure water [H+l = [0H~]. Here it was found that at 25° [H+] X [0H-] = k^ = 1.44 X 10-'\ c. From the degree of hydrolysis oj salts. Salts of a weak acid and a strong base in aqueous solution are in part split into free acid TABLE 6 Temperature 10° 15° 25° 40° 50° kw 0.31 0.46 1.05 2.94 5. 17 X 10-1* and free base, which in turn yield to the solution by electrolytic dissociation H+ and 0H~ ions. But since the base is stronger, its dissociation predominates, and as a result there is an excess of 0H~ ions over H+ ions. In this case the dissociation constant of the water may be derived, if the dissociation constant of the weak acid is known. This point will be taken up again later. Or the [0H~] may be determined by a catalytic experiment as above in (b). Arrhenius^^ obtained from the hydrolysis of sodium acetate at 25° kw = 1.21 X 10-1* From the hydrolysis of other salts several investigators obtained similar figures. Lunden^4 found from a study of the hydrolysis of salts of a weak acid and a weak base the figures shown in table 6. 1^ F. Kohlrausch and A. Heydweiller, Zeitschr. f. physikal. Chem. 14, 317 (1894). 1^ J. J. A. Wijs, Zeitschr. f. physikal. Chem. 11, 492 and 12, 253 (1893). 18 Sv. Arrhenius, Zeitschr. f. physikal. Chem. 1, 631 (1887). ''' H. Lunden, Journ. de Chim. phys. 5, 574 (1907). 26 HYDROGEN ION CONCENTRATION d. The hydrogen ion concentration method. By means of the concentration chains, which are to be described later, the [H+] of any solution may be determined. If the [H+] of a certain NaOH solution of known [0H~] is determined by this method then kw may be calculated. WilheLm Ostwald-" was the first to call atten- TABLE 7 Temperature kw X 10'" pkw = ~ ^°S kw i .e., [H+] of pure water 16 0.63 14. 200 0.79 17 0.68 14. 165 0.82 18 0.74 14. 130 0.86 19 0.79 14. 100 0.89 20 0.86 14.065 0.93 21 0.93 14.030 0.96 22 1.01 13.995 1.00 23 1.10 13.960 1.05 24 1.19 13.925 1.09 25 1.27 13.895 1.13 26 1.38 13.860 1.17 27 1.50 13.825 1.23 28 1.62 13. 790 1.27 29 1.76 13.755 1.33 30 1.89 13.725 1.37 31 2.04 13.690 1.43 32 2.19 13. 660 1.48 33 2.35 13.630 1.53 34 2.51 13. 600 1.59 35 2.71 13.567 1.65 36 2.92 13. 535 1.71 37 3.13 13.505 1.77 38 3.35 13.475 1.83 39 3.59 13.445 1.89 40 3.80 13.420 1.95 tion to this method; W. Nernst'^ introduced an important correc- tion into the calculations of the method, and since then it has been used by various authors. Lowenherz- found: 2» Wi. Ostwald, Zeitschr. f. physikal. Chem. 11, 521 (1893). 21 W. Nernst, Zeitschr. f. physikal. Chem. 14, 155 (1894). ^2 R. Lowenherz, Zeitschr. f. physikal. Chem. 20, 283 (1896). LAWS OF ELECTROLYTIC DISSOCIATION 27 Temperature 19° 25" . 0.8 X 10-1* 1.41 X 10-" The technic of the concentration chains has been worked out recently in great detail, and it must be considered as the most accurate of the methods. The most reliable measurements were made by S. P. L. S0rensen-^ and yielded the average figure of kw = 0.73 X lO-i" = lO-i"!* at 18° Michaelis-^ using the same method obtained the figures given in table 7. They represent the results of a large number of experi- ments and of graphic interpolations. In the same way by determining the pH of NaOH solutions, whose pOH was calculated from conductivity data, Lewis, Brighton and Sebastian^s found at 25°C. kw = 1.012 X 10-1*; pk^ = 13.995 Hence for pure water at 25° we have the practically exact values [H+] = 1.00 X 10-^ or pH = 7.00. The temperature coefficient of kw may be calculated from van't Hoff's equation (called the "reaction isochore" by Nernst) which is dlnkw U dT RT2 By substituting the corresponding numerical values and by con- verting into common logarithms, the following expression is obtained for 20°C.: dlnkw 13700 dT 4.571 • (293)2 in which we replace U, the heat of the reaction H+ + 0H~ = H2O, " S. P. L. S0rensen, Biochem. Zeitschr. 21, 131 (1909). " See first edition (1914) of this book. 25 Lewis, G. N., Brighton, T. B. and Sebastian, R. L., Journ. of the Amer. Chem. Soc. 39, 2245 (1917). 28 HYDROGEN ION CONCENTRATION by the heat of neutrahzation of HCl + NaOH (13700 cal., Berthe- lot's value). This gives us the value of the coefficient as ^" = - 0.0349 dT This value calculated from the author's figures given above for the interval of 16 — 26° is = —0.0340, which is in good agreement with the theory. The smaller variations in the newer as compsftred with the older measurements is explained by a better elimination of errors es- pecially those due to the diffusion potential. In view of the fact that the very numerous studies of S0rensen were all made at the same temperature, and yield a very reliable average value, while those of the author cover a wide range of temperature, the S0rensen figure for 18° is recommended as the most credible, and the author's figures are to be used for the determination of the temperature coefficient. S0rensen's figure for 18° is lO"^''-^'*, which is an average obtained from a number of values varying between 10~^*-^^ and IQ-u.m 'pj^g author's values for 18° obtained at different times varied between IQ-^^-''^ and lO""^'*-^^ The author's average value obtained in 1913 as given in the above table (7) is 10~^*-^^ which agrees well with S0rensen's average. It will be shown later that all of the pH measurements made until the present will have to be subjected to a certain correction. The uncertainty is based on the following considerations. The [H+] of a dilute NaOH solution is measured by comparison with the [H+] of a dilute HCl solution, whose [H+] is assumed to be known. On the other hand the [OH"] of the NaOH solution is assumed to be known. Until recently there was no doubt as to the value of the [H+] of a HCl solution of definite concentration, or as to the [OH"] of a NaOH solution of definite concentration. These were calculated from the respective degrees of dissociation of the acid and base, which in turn were calculated by comparing the molar conductivities of the solutions at a definite dilution and those at infinite dilutions. As will be seen later these calculations are erroneous. It is possible that the values for pkw provisionally recommended above are really smaller (at most 10 per cent smaller) . LAWS OF ELECTROLYTIC DISSOCIATION 29 9. The general application of the mass law to the electrolytic dissociation of acids It was shown above how electrically neutral molecules like those of water can break up into two oppositely charged ions. The reverse phenomenon by means of which these two ions can combine to form electrically neutral molecules can be shown just as well. It will depend upon the external conditions whether in any given case the reaction will proceed in one or the other direction. A free H-ion and a free OH-ion coming spatially into near proximity of each other as a result of molecular motion will unite to form H2O. On the other hand the impact of a heterogeneous molecule, occur- ring in a suitable way, will split off from the H2O molecule a H- ion. In general, for every given external condition (especially that of temperature) there will be established a certain statistically con- sidered situation in which the number of ions forming is equal to that of the disappearing ions. But such are exactly the conditions for the application of the law of mass action, and the same symbols as are used for stating the reactions of non-electrolytes may be employed in this case. Thus to the "chemical equation," 1 mol alcohol + 1 mol acetic acid ;=i 1 mol ethyl acetate + 1 mol H2O, corresponds the equation: 1H+ + 1 OH- = 1 H2O And the formula of the mass law as used in [EtOH]X[AcOH] [EtOAc] X [H2O] corresponds in this case to [H+] X [0H-] = k [H2OI = k, or, [H+] X [0H-] = kw. This states the law of the electrolytic dissociation of water which has been discussed earlier. The dissociation of an acid dissolved in water will now be considered. If the acid is designated as HS and its component ions as H+ and S~, then the mass law can be stated in the form of the following equilibrium equation : [HS] ^ ^^> 30 HYDROGEN ION CONCENTRATION The constant k is a definite characteristic of every acid. It varies with the temperature although, as a rule, not very greatly. It is the dissociation constant or the affinity constant of the acid. Its magni- tude is, as Wilhehn Ostwald pointed out, the only rational measure of the strength of an acid. |"TT+1 [-Q— 1 In order to avoid misunderstandings the value |^ will be [Hb] henceforth taken as the "dissociation constant" and its reciprocal rxTOl rTT4.i ro-1 ^s the "affinity constant." The dissociation constant is l-n J Lb J a measure of the force which tends to decompose (ionically) the acid; the affinity constant is the force which tends to reunite its ions. By the terms "force" and "affinity" of a chemical reaction is understood that amount of maximal work C which may be ob- tained from the decomposition of one mol of the molecular com- pound involved, as for instance, in the case when 60 grams of acetic acid decompose into 59 grams of acetate ions + 1 gram H+ ions. The necessary condition however is that the molecular compound involved be present in one molar concentration, and that a suffi- ciently large reaction volume be chosen so that the concentration is not measurably affected by the decomposition. The relation of this amount of work C to the constant k is ex- pressed C = RT . Ink where k has its above given meaning. If the reacting molecular species are in solution in any concentration such as [AcOH], [Ace- tate"], [11+] then the maximal work W obtained from the decom- position of one mole is expressed as: W = C + RTlnX '^"°«' [Acetate-] [H+] * See following section. t The figure given is that found by Michaelis and Garmendia (Biochem. Zeitschr. 67, 431 (1914). Abbott and Bray (J. Amer. Chem. Soc. 31, 760 (1909) had previously found the value 1.95 X lO"'. It must be stated at once that in the light of some recent developments (cf. Chapter III) some of the above figures must be revised, as well as the figures in the tables of dissociation constants given on pages 59 and 72. But none of these newer corrections are large. LAWS OF ELECTROLYTIC DISSOCIATION 31 TABLE 8 Table of dissociation constants of some of the weaker acids The figures given below are compiled from Landolt-Bornstein, Physikalish- Chem. Tabellen, 4 Ed. Berlin 1912, also from some more recent data. The observers are Wi. Ostwald, Luther, Bodliinder, Rothmund and Drucker, Walker and Cormack, Lunden, A. A. Noj'^es, Walden, Madsen, H. Euler, Michaelis and Rona. Acid Arsenious Arsenic Boric Boric Boric Boric Cacodylic Carbonic* (1st stage) . . . Carbonic (2nd stage) Phosphoric (1st stage).. Phosphoric (2nd stage )t Phosphoric (3rd stage).. H2S Nitric Formic Butyric Acetic Acetic Acetic Acetic Acetic Acetic a-fructose a-glucose Glycerin Glycol Lactic j3-oxybutyric Saccharose i, 1, r Tartaric Benzoic Phenol Salicylic , Uric Temperature k 25° 6 X 10-10 25° 5 X 10-' 15° 5.5 X 10-10 25° 6.6 X 10-10 25° 6.4 X 10-10 40° 8.5 X 10-10 25° 6.4 X 10-^ 18° 3.0 X 10-7 25° 6 X 10-11 25° 1 X 10-2 8.8 X 10-8 3.6 X 10-1' 18° 9.1 X 10-8 6.0 X 10-* 2.1 X 10-* 18° 1.5 X 10-' 10° 1.83 X 10-' 18° 1.82 X 10-' 25° 1.86 X 10-6 40° 1.80 X 10-' 50° 1.74 X 10-' 100° 1.11 X 10-' 18° 6.6 X 10-13 3.6 X 10-13 7 X 10-1' < 10-iS ext remely small 25° 1.38 X 10-« 18° 3.86 X 10-' 18° 1.14 X 10-'» 25° 9.7 X 10-* 25° 6.0 X 10-' 25° 1.09 X 10-'" 20-30° 1.05 X 10-' 1.5 X io-« (See footnote at bottom of page 30.) 32 HYDROGEN ION CONCENTRATION In different acids the value of k varies within wide limits. It could be supposed that every H-atom of a molecule has the ten- dency to become dissociated into a H-ion. But this tendency depends upon the chemical configuration of the compounds, and under certain conditions it may become extremely small. Thus, for instance, sugar and also alcohol may be conceived of as acids with extraordinarily small dissociation constants, which in the case of sugar can barely be measured, and in the case of alcohol it is so small as to completely escape measurement by any means at our disposal. It may be stated, however, that wherever the methods for the determination of the constant are still sufficient, the constant has a well defined value, in spite of its small order of magnitude. The upper limit of the dissociation constants cannot be discussed at this point without additional elucidation. The study of the strong acids, such as HCl, HBr, HI, HNO3, H3PO4 (the first stage), does not yield uniform values for k, and they will be discussed later. On the other hand, the weaker and the weakest acids follow well in their dissociation the above stated laws. Table 8 gives the dissociation constants of some of the weak acids at 18°. The table also shows the effect of temperature upon k in the case of a few acids. 10. True and apparent dissociation constants of acids. Pseudo-acids In the case of some acids the free acid is not known to exist in the free state, and only its anhydride is known. Such is the case with H2CO3 which is not known in its free state, and only its an- hydride, CO2, is known. When CO2 is dissolved in water the resulting solution has an acid reaction. From this circumstance it is deduced that H+ and HCO3"" ions were formed in the solution. The mode of formation of these ions can be explained in two ways: It may be either assumed that the CO2 molecule acquires by addi- tion an 0H~ ion: CO2 + H+ + OH- =± HCO3- + H+ or, since the H+ ions play no part in the reaction, as the following: CO2 + OH- ^ HCO3- LAWS OF ELECTROLYTIC DISSOCIATION 33 ' From which follows the equilibrium reaction i [CO,] [OH-] , , fr ^ iL \ r ,^ i^h [HCQ3-] = '^ i ^ i^^^^l -^ ^^X"^^ kw I Substituting for [0H~] its value ^-^^ as derived from equation [h+] ^ i (3), page 12, we obtain ■ [CO,] X k^ , \ = Ki 1 [H+1 [HCOai or [CO,] ki = k2 (1) [H+] [HCO3-] k. This equation represents the hypothetical chemical reaction of CO2 ;=i H+ + HCO3- and k2 could be used as a true "dissociation constant." But in reality it is only an "apparent" dissociation constant, for the last chemical reaction given above does not occur. But the formation of ions by carbonic acid in solution can be accounted for in another way. The first step is assumed to be the formation of a hydrate: CO2 + H2O ^ H2CO3 and whose equilibrium state is [CO2] [H2O] [H2CO3] = k3 (2) By transposing the constant value H2O to the right side of the equation and by combining it with ks to form a new constant we obtain [CO.,] [H2CO3] = k4 (3) k4 has a very large value, i.e., there is only a small amount of H2CO3 in the presence of a great excess of CO2. The molecule H2CO3 dissociates electroly tically : H2CO3 ?=i H+ + HCO3- 34 HYDROGEN ION CONCENTRATION with the following state of equilibrium [H+] [HCO [H2CO3] = k6 (4) ^e^'- in which ks is the true dissociation constant of the real carbonic acid, H2CO3. But the value of [H2CO3] is not known, and only that of [CO2] is known. The latter is in fact the concentration of total free carbonic acid in solution, for the molecular species H2CO3 is present in extremely minute concentration as compared with that A/^i AtC?^ - of the molecular species CO2 in solution. By substituting for H2CO3 •217 /L.U in (4) its value in (3) above, we arrive at T>tc6 O^ [H+] [HCO3-] ^ ks ^ ^:{: i\\' [CO2] k4 ' L£>r4 ^tS I fc^'This expression is identical (reciprocal) with equation (1). There- O/ "*y^ ^T^ fore, ke is also only an apparent dissociation constant. fi^ C44A1^A6T^'^^^^ apparent "dissociation" constant is the only one which one , ^_< can measure. The true dissociation constant of carbonic acid, \ Z ^ < H2CO3, remains an unknown quantity. But this apparent constant 6X^ ^-pPAjZ'iiiay be employed as well as a true constant in formal calculations ^ and operations. According to Thiel and Strohecker-" the true dissociation constant of the true H2CO3 is very large, the minimum probable value being 5 X lO—*, according to Strohecker-^ it is 4.4 X 10~^, which is a much greater value than those of acetic and formic acids. This is quite plausible, since H2CO3 is in fact oxyformic acid. An idea of the relative strength of true carbonic acid may be gathered from the following simple experiment: A few drops of neutral red are added to some of a 0.1 N NaHCOs solution. The indicator shows a yellow color in the alkaline solution. Just enough of 0.1 iV HCl is added to change the color of the solution to red, thus indicating an acid reaction. On waiting a few seconds it is observed that the color changes back to yellow. Upon a further addition of HCl the color becomes red again and then slowly returns to a yellower stage. The reason for this phe- nomenon lies not in the gradual escape of gaseous CO2 but in the 26 Thiel, A. and Strohecker, R., Ber. d. Dtsch. Chem. Ges. 47, 945 (1914). "Strohecker, Zeitschr. f. Unters. d. Nahr-u. Genussm. 31, 121 (1916). LAWS OF ELECTROLYTIC DISSOCIATION 35 liberation by the HCl from the bicarbonate of the true and strong H2CO3, which then gradually decomposes into CO2 and H2O. This experiment is made possible by the relatively slight velocity of the reaction H2COS-* H2O + CO2 This case of an "apparent dissociation constant" is by no means limited to carbonic acid alone. It must be remembered that it is in reality quite impossible to distinguish in a given case a true from an apparent dissociation constant. This point is so impor- tant that it will be amplified by some examples. Glucose is an ex- traordinarily weak acid. In attempting to picture the chemical constitution of glucose and of its ions we face the possibility of repre- senting the constitution of this sugar in a number of various ways. We may regard it, according to E. Fischer's original structural formula, as an aldehyde, or, according to ToUens, as a cychc com- pound with its ring closed by means of one 0-atom. The latter may exist in two stereo-isomeric forms as the a- and /3-glucose. Recently even more isomers of the cyclic kind were found. All of these forms are actual and coexist in a state of equilibrium. Furthermore, another configuration is possible, the enolic form of Wahl and Neuberg which is derived from Fischer's aldehyde form. The formulas below show this derivation: I is the aldehyde which in II forms a hydrate and, as is shown in II, the hydrate splits off a molecule of H2O again to form the enol form shown in III. /OH H— C=0 H— C< H— C— OH OH— C— H OH— C— OH H OH— C c c c i i i I II III Aldehyde Aldehyde Enol hydrate This enolic form is common to glucose, mannose and fructose. Thus, when this enol adds a molecule of H2O and passes to the aldehyde again, the resulting sugar of this process may be either glucose, mannose or fructose, and in fact the velocity of this process 36 HYDROGEN ION CONCENTRATION is inversely proportional to the hydrion concentration.-^ Since the concentration of the sugar ions must be also inversely proportional to the [H+], it may then be surmised that the transformation of the sugar must occur through its ions. Furthermore, since, as was shown above, this transformation must occur by means of the enolic form, it follows that the ions must be in an enolic form, and that there cannot be any enolic sugar present in this process beside the ions. In other words: Sugar is a dissociable acid only in its enolic form, and in this form it is a quite strong acid. This is entirely plausible, since all enols are strong acids, while their corresponding aldehydes or ketones are not. And the sugar is a weak acid only because its only modification which possesses acid properties is present in such small amounts in comparison with its other chemical forms. Many analogous cases are met with among the indicators. Phenol- phthalein has an apparently very low dissociation constant. In the free state it has the constitution shown in I below. This form is tautomeric with the form shown in II. OH HO /^ A^ o I C .OH II The form I is colorless, while the tautomer II because of its quinonoid double bond is colored, and because of its carboxyl group strongly acidic. The latter form cannot exist in appreciable amounts together with the colorless isomer and only its ions may so occur, and the constitution of these ions may only be the one shown by formula II. Thus the colorless phenolphthalein I is considered as a very weak acid whose ions are colored red, only because in its solu- tions modification II, which is a strong acid, is present in minute concentrations. All color changes of the indicators depend upon 28 L. Michaelis and P. Roaa, Biochem. Zeitschr. 47, 447 (1912). LAWS OF ELECTROLYTIC DISSOCIATION 37 the differences between the constitution of their ions and the un- dissociated molecules, and the dissociation constants of all indicators are of the "apparent" type. Those molecular species which are not acid by virtue of their constitution, but which may be transformed by tautomeric change into acids were designated by Hantzsch^^^ as pseudo-acids. The proof for this conception can be demonstrated in a way analogous to that used above for the case of carbonic acid. When acid is added to a red alkaline solution of phenolphthalein the first change ensuing is to the red undissociated acid of form II, and then secondarily the change to the colorless form I. With phenol- phthalein, however, these steps occur so rapidly that they cannot be distinguished. But Hantzsch succeeded in demonstrating the phenomenon with suitable dyes, such as crystal violet and many others. When alkali is added to a solution of one of these dyes and the conductivity is determined at the same time, it is found that it decreases with time, showing the disappearance in the solution of a strong electrolyte. A simple experunent with a slowly changing indicator may be performed as follows. To 100 cc. of a weakly alkaline solution, such as conductivity water or a buffer solution of phosphates, is added one cc. of acid fuchsin or of water blue (1:1000). The solu- tion becomes gradually paler until the color reaches a definite intensity depending upon the pH of the solution, at which intensity it becomes stationary. At room temperature this process may take one half to one hour, while at 50° but a few minutes. Acid fuchsin and water blue are indicators in which the two steps described above are distinctly separated. The establishment of ionic equilibrium, like all other ionic reactions, must occur with great rapidity, whereas the tautomeric transformation of the colored form of an indicator into the colorless is a slower process. 11. The dissociation of bases The same holds true for the dissociation of bases as for acids, if [0H-] is substituted wherever [H+] occurs. If the molecule of a base be designated as BOH, then its dissociation occurs as BOH ^ B+ + OH- 28a See footnote, p. 97. 38 HYDROGEN ION CONCENTRATION and the dissociation constant is [B+] [0H-] k = [BOH] The principles of the apparent dissociation constants apply also in this case as well as for the acids. Here also it is necessary to differentiate between strong and weak bases. The strong bases are the caustic alkalis, such as NaOH, KOH. They are ionized to a great extent, and the exact determination of their dissociation rela- tions is fraught with the same difficulties as in the case of the strong TABLE 9 Dissociation constants of some weak bases Temperature k Ammonia 10° 18° 25° 40° 50° 100° 25° 25° 25° 25° 25° 40° 40° 40° 25° 40° 1.63 X 10-5 Ammonia 1.75 X 10-5 Ammonia 1.87 X 10-5 Ammonia 1.98 X 10-5 Ammonia 1.96 X 10-5 Ammonia 1.35 X 10-5 Ethylamine 5.6 X 10-< Diethylamine 1.26 X 10-» Triethylamine 6.4 X 10-4 Urea 1.5 X 10-" Aniline 4.6 X 10-1" Guanine 8.4 X 10-12 Caffeine 4.1 X 10-11 Creatinine 3.7 X 10-9 Pyridine 2.3 X 10-" Xanthine 4.8 X 10-" acids and will be dealt with later together with them. Also the slightly soluble bases, such as Cu(0H)2, AgOH are probably to be classed with the strong, greatly dissociated bases. The fact that they do not produce in water a marked alkaline reaction is to be attributed exclusively to the slightness of their solubility but not to any lack of dissociation. A base such as AgOH is not known in its free, pure state. But it is highly probable that the traces of Ag20 which dissolve in water go into solution entirely as AgOH, and that the latter is practically completely dissociated, so that actually almost only Ag+ and OH" ions are present in solution. LAWS OF ELECTROLYTIC DISSOCIATION 39 Of the organic bases only the quartenary ones are comparable to those above. The other organic bases, especially the important ones, such as the amino- and oxonium-bases, as well as some others, are to be regarded as pseudo-bases. It was shown earlier in this chapter how the basic character of NH3 may be explained either by the hypothesis of the formation of an extremely small amount of NH4OH and its subsequent dissociation, or, in a simpler way, by means of the addition reaction NH3 + H+ — > NH4+. The same holds true for all other weak bases. In table 9 the (apparent) dissociation constants of some weak bases are given. As it was stated before, the dissociation constants of all organic bases are ''apparent." Nothing is experimentally known of the true dissociation constant of NH4OH. But it will not be far amiss to include it among the quite strong bases in analogy with the alkali metal hydroxides. Ammonia, NH3, possesses as a base an appar- ently very small constant only because it forms in solution im- measurably small amounts of NH4OH. But the NH4OH is best considered as being almost completely dissociated. 12. The hydrogen ion concentration in pure acid solutions Thus far it has been established in a purely qualitative sense that acids increase the hydrion concentration of water and that bases decrease it. We shall now pass to the quantitative relations existing between the [H+] and the kind of acid, its dissociation constant and its concentration. Given a solution of an acid in water, then the concentration of its acid ions must be equal to that of the H+ ions. Therefore, we can substitute in equation (1) on page 29 [H+] = [S~] and obtain the expression [HS] or, [H+1 = Vk [HS] Let us assume that A moles of an acid are dissolved in one liter of solution. Part of the acid is dissociated into H-ions and acid ions, the rest of it remaining undissociated. The concentration of 40 HYDROGEN ION CONCENTRATION the unionized part is [SH] = [A] — [H+]. From which it follows that fH+]= [A] - IH+] k ■V' [H+] = l/- + kA - - For weak acids this formula is simplified in the following manner: When the dissociation constant of the acid is very small then its ionization is naturally very slight, less than one per cent, for example. It can then be said with a fair degree of accuracy that the con- centration of the undissociated acid is equal to that of the total acid, or, [HS] = [A]. Hence it follows that [H+] = Vk X [A] (la) The [H+] is, therefore, proportional to the square root of the concentration of the total acid. The analogous rule holds for the case of a weak base where [0H-] = Vk X [B] where [B] is the concentration of the total base. By substituting for [0H-] its value as ^-77:^ the relation becomes IH+] = Vk X [B] For the weakest acids whose dissociation constants are of the order of magnitude of that of water, as well as for moderately strong acids in extremely low concentrations the above formulas do not hold. For in these cases the H-ions yielded by the water cannot be ignored in relation to those of the acid. In such cases the [H+] is determined in the following way: Here also as above, be- cause of the very slight degree of dissociation [HS] = [A] and [A] Further, in accord with the law of electro-neutrality which LAWS OF ELECTROLYTIC DISSOCIATION 41 demands that the sum of the total positive ions be equal to the sum of the total negative ions: [H+] = [S-] + [0H-] (2) and thirdly, IOH-, = j^ (3) By eliminating [S~] in equation (2) by using (1), and then sub- stituting for [0H~] by the use of (3), we obtain w ^ kjA] k^ [H+] "^ [H+] or [H+] = Vk [A] + kw (4) It is to be observed from the above that the [H+] may never become less than k^, or below true neutrality, not even with an infinitely small k; and also that acids whose k differs very little from kw raise the [H+] of water to so slight an extent that solutions of sugar, for instance, are to be considered as being virtually "neutral." The acid nature of sugar becomes evident only in its ability to raise the [H+] of solutions of bases, in which case the [0H~] concentration is decreased without the sugar actually "bind- ing" the NaOH molecule. In a mixture of two weak acids, whose total concentrations are Ai and A2, the concentrations of undissociated molecules are Si and S2 and the dissociation constants are ki and k2 respectively, the [H+] may be calculated in the following way: 1. [Sr] + [S2-] = [H+] [S2-] lH+1 , Hence [H+] = VkiSi + k2S2 42 HYDROGEN ION CONCENTRATION But since the concentrations of the undissociated acids, [Si] and [S2], in the case of weak acids are virtually equal to those of the total acids, or to [Ai] and [A2] respectively, the above expression becomes for practical purposes: [H+] = VkiAi + k,A3 (5) For example if we take a solution containing one mole of acetic acid per liter (ki = 2 X 10~^) and one mole of carbonic acid (ka = 3 X 10~^), then its hydrion concentration is [H+] = V2 X 10-5 + 3 X 10-7 which is practically equal to that of a one molar acetic acid solu- tion without carbonic acid, where [H+] = -v/2 X 10-s Even the hydrion concentration of a solution of one molar CO2 + 0.1 molar acetic acid solution, [H+] = Vo.l X 2 X 10-5 + 3 X 10-' = 4.5 X 10-^ is practically the same as that of a pure 0.1 M acetic solution where [H+] = 4.48 X 10~^ This illustrates the fact that the stronger acid suppresses the dissociation of a weaker acid. Many years ago Arrhenius-^ established the law which states: When isohydric solutions (solutions of the same [H'^]) of different acids are mixed, the [H+] is not changed on mixing. This law may be easily derived from the data given above in this section. Let Ai and A2 be the concentrations of the two acids whose dissociation constants are ki and k2. The concentrations Ai and A2 are so chosen that the [H+] of the two solutions is the same. This con- dition is fulfilled when (see (la) on p. 40) ki X Ai = k2 X A2, and then of course [H+] = V ki X Ai = Vk2 X A2. Let us now mix m volumes of the first solution with n volumes of the second. Then the concentration of the first acid in the mixture is ; — Ai, m + n n and the concentration of the second is ; — A2. According m + n to (5) above the hydrion concentration of the mixture is: -V' [H+] = t/ki • — ^ Ai 4- kj m + n ' m + n ■^ Sv. Arrhenius, Zeitschr. f. physikal. Chem. 2, 284 (1888). LAWS OF ELECTROLYTIC DISSOCIATION 43 Since kiAi = k2A2, [H+] J \ m + n va -\- nl and [H+] is therefore unchanged. But this law is only valid for mixtures of the free acids alone in the absence of their salts. It is not applicable to two isohydric solutions of any composition. 13. Mixtures of weak acids and their alkali salts: Regulators or Buffers While it is true that acids and bases are capable of presenting all possible gradations of dissociation, this is not the case for their alkali salts. The latter all belong to the class of strong electrolytes to which the law of mass action is not applicable without further amplification. The same is true of the chlorides (bromides, etc.) of the weak bases. For the sake of simplifying calculations we may assume at once that the salts are completely dissociated under all conditions. This assumption is but very little removed from reality, and the results which are to be derived from it possess quite a high degree of accuracy and form the foundation of the theory of regu- lators or buffers.^*^ When to a solution of a weak acid its alkali salt is added, the dissociation of the acid becomes greatly depressed, and to a pro- gressively greater extent with increasing amounts of the salt. Thus by adding varying amounts of sodium acetate to acetic acid it is possible to obtain at will any low hydrion concentration. The same results may be obtained with a great variety of other such acid- salt mixtures. The calculation of the [H+] of such mixtures is of special importance and is met with very frequently in experimental procedures. Let us take for example a mixture of acetic acid and sodium acetate. The expression applicable to all such cases is: [S-] X [H^] [SHI ^^ Literature: Contributions from Nernst's Laboratory: Fels. Zeitschr. f. Elektrochem. 10,208 (1904) ; Salesski, ibid. 10, 204 (1904). AlsoS.P.L.S0rensen, Biochem. Zeitschr. 21, 131 (1909); Lawrence J. Henderson, Ergebn. d. Physiol. 8, 254 (1909); L. Michaelis and P. Rona, Biochem. Zeitschr. 23, 364 (1910). 44 HYDEOGEN ION CONCENTRATION from which fFT+i — ^ t^^J _ u [undissociated acid] [S~] [acid ion] In such a mixture the concentration of the undissociated acid is- virtually equal to the concentration of total free acetic acid, for the latter is very slightly dissociated. The concentration of the anions in this mixture is virtually equal to that of the sodium acetate, for the latter is assumed to be completely dissociated, and the acetate ions given off by the free acetic acid are negligible in quantity com- pared to those given off by the salt. The expression given above therefore becomes: [H+] = k X [^^^^ ^^^^'^^ ^^'^^^ (1) [Na— acetate] This expression is an approximation, since it is based on certain assumptions which are not absolutely exact. But it is quite valid for a great variety of cases with a very fair degree of accuracy. The deviations found under certain conditions, which will be dwelt upon later, are so small that they can well be used only as corrections for the above approximation. For somewhat stronger acids, such as tartaric acid, in which the dissociation of the free acid may not be disregarded, the concentra- tion of the undissociated acid is not assumed to be equal to that of the free acid [E] but is expressed as [E] — [H+]; and the concentra- tion of the anions is not equal to that of the salt, for the free acid yields also anions to the same extent as it does H-ions. Instead of the above equation (1) we obtain: ^ k([E] - [H^]) [Na-salt] + [H+] ^ ^ From this the following quadratic equation for [H+] is derived: vc ^^^ ^ _ [N.-saltl+k _^ l/UNa-saltl + kV ^ ^ ^^^ This is the first correction for the approximate formula. Prac- tically, it is seldom of any significance. The second much more im- portant correction has the purpose of rectifying the approximate LAWS OF ELECTROLYTIC DISSOCIATION 45 assumption of the total dissociation of the Na-salt. This point will be considered in the discussion of the strong electrolytes. Analogously, in the case of a mixture of NH3 + NH4CI the fol- lowing approximate expression holds good: [0H-] = k [NH3] [NH4CI] (3) The following method may be used for the experimental proof of equation (1) : If in a series of solutions the concentration of the sodium acetate be kept constant, at 0.1 N for example, and the concentration of the free acetic acid be varied, it will be observed that within wide limits this variation will not show any significant deviations from the postulated law. But the verification of the formula will not be so complete, if the above conditions be reversed, TABLE 10 Dilution [H+] pH 1 1.54 X 10-^ 6.813 1/2 1.25 X 10-^ 6.904 1/5 1.02 X 10-' 6.990 1/20 0.853 X 10-' 7.069 1/50 0.817 X 10-' 7.088 00 0.80 X 10-' 7.10 (extrapolated) and the concentration of the salt be varied with a constant con- centration of the free acid. In the latter case it will be found that the dissociation of the sodium acetate which had been assumed to be always complete, varies somewhat wdth its concentration. This condition becomes noticeable when a given mixture of acetic acid and its sodium salt is diluted with water. For a solution of 0.1 A'' acetic acid + 0.1 A^ sodium acetate the following figures were found by the gas chain method : [H+] pH Undiluted 2.42 X 10-5 2.16 X 10-5 4.615 Diluted X 5 4.665 Thus the effect of dilution is just barely demonstrable. In any case it is so slight that equation (1) may be regarded for ail practical purposes as quite accurate and vahd. 46 HYDROGEN ION CONCENTRATION The deviation from this approximate equation is more marked in the case of a solution of a mixture of primary and secondary sodium phosphates. If these salts be assumed to be totally dissociated, then for a mixture of their solutions it would be expected that rTX4-i 1 [primary phosphate] [H+] = k [secondary phosphate] But it is to be remembered that the secondary phosphate is the Na-salt of the weak acid — "primary phosphate." If the above expression were entirely correct, then the absolute concentration of the phosphates would be immaterial. This is true to a certain extent, but not in so strictly proportional a manner as in the case of the acetate mixtures, for the degree of dissociation of dibasic phosphate depends to a much greater extent upon the dilution. Electrometric determinations of the [H+] of a mixture of equal parts of M/15 K2HPO4 + M/15 NaH2P04 yielded the figures in table 10.31 14. The two-acids problem It is frequently desired to calculate the [H+] obtaining in a mix- ture of two weak acids and of its alkali salts. This particular con- dition was designated as the two-acids 'problem. ^^ A general solu- tion of this problem is quite difficult and is barely to be found. But since this problem has a practical bearing, as will be seen later, it will be here detailed for certain simple conditions which should suffice for the handling of any practical question which may arise. For this purpose the derivation given in the preceding section will be modified, and we shall return once again to the mixture of a weak acid and its alkali salt. Such a mixture may also be obtained by mixing a weak acid of concentration A with a strong base of concentration L, where L < A. In such mixture then the concen- tration of the salt = L and that of the free acid in excess = A — L. Then the equation (1) on page 44 may be transformed into [H+] = k ^^^ (la) Similarly a mixture of two acids and their salts may be thought of 31 L. Michaelis and A. Kriiger, Biochem. Zeitschr. 119, 307 (1921). LAWS OF ELECTROLYTIC DISSOCIATION 47 as being produced by mixing the two acids, whose dissociation con- stants are ki and k2 and whose concentrations are Ai and A2 respec- tively, with a base whose total concentration is L, where L < (Ai + A2) . Of this amount of base a part Li is bound by the first acid and a part L2 by the second, in such a manner that Li + L2 = L (lb) In accord with the mass law: [H+] = k, ^^^ (Ic) [H-] = k. ^^^ (Id) L2 It is now necessary to recollect that equations (1) as well as (la) are only approximations and may be only applied when the second fractional factor on the right side has a value lying between 0.01 and 100. In further calculations the conditions under which they are applicable will always be borne in mind. If now in (Ic) we take Li = L — L2 and eliminate L2 by means of (Id) a quad- ratic equation for [H+] is obtained which is capable of solution only in a physical sense: , = ki(Ai - L) + k2(A - L) 2L frMAi^ ^,, L) + k2(A2 - L) 2L 2 kl • k2 + —I— (Ai + A2 - L) (le) The problem under consideration has the following useful prac- tical application. In the determination of [H+] by indicator methods we add to the given solution some indicator, which is an acid, and we tacitly assume that this addition of acid does not alter the [H+] of our solution. The solution in question is as a rule of the nature of a single acid buffer, such as, for example, a mixture of NaHCOs + H2CO3. Let us carry out the calculation for a most extreme case. Let the solution in question be extremely poor in buffer value, such as ordinary water + conductivity water. Such a solution may be obtained by mixing 0.00300 A'' calcium bicarbonate (NaHCOs will do as well) and 0.00030 A^ H2CO3. In this case L 48 HYDROGEN ION CONCENTRATION = 0.003, Ai = 0.0033 and ki is given as = 3 X 10-^ We now add to this solution an indicator, m-nitrophenol (k2 = 5 X 10~^) in the concentration of A2 = 0.003 N. In reality barely one half of this amount is actually needed, although, as it happens, m-nitro- phenol possesses weak tinctorial power, and much more of this dye is required than in the case of the other indicators. If the [H+] of tlie solution before the addition of the indicator is calculated from (la), then we obtain on the basis of the given data the value [H+] = 3 X 10~^ After the addition of the indicator we obtain by the use of (le) [H+] = 5.83 X lO-^. In a similar way, assuming various values for the concentration of the indicator, A2, and retaining all the other data, the figures in table 11 are obtained. TABLE 11 A2 (mols per liter) [H+l pH Error in pH due to indicator 0.003 5.83 X 10-s 7.23 -0.29 0.0015 4.66 X 10-« 7.33 -0.20 0.00075 3.93 X 10-» 7.41 -0.11 0.00037 3.46 X 10-8 7.46 -0.06 0.00018 3.27 X 10-8 7.49 -0.03 0 3.0 X 10-8 7.52 In using m-nitrophenol one employs, for example, a 0.3 per cent, or, 0.027 molar solution. In adding 0.3 cc. to 20 cc. of conductivity water the concentration of the indicator becomes in round numbers 0.0004 N, and the pH error due to the indicator is —0.06, a value falling well within the experimental limits of error. In reality much smaller amounts of indicator suffice. In the case of all other indi- cators which have greater coloring powers and which can be used in much smaller concentrations this error is certainly of no sig- nificance. 15. The degree of dissociation and the dissociation-residue of acids The degree of dissociation of an acid in a pure aqueous solution is the term given to the ratio of the dissociated portion to the total amount of the acid. This definition will be extended so that it may also embrace the acid-salt mixtures. Restating then, by the degree of dissociation of an acid we understand in general the ratio LAWS OF ELECTROLYTIC DISSOCIATION 49 of the acid ions to the total amount of acid radicals, irrespective of the form in which they may happen to occur. The degree of disso- ciation of acetic acid in a mixture of acetic acid and its Na-salt is, therefore, represented by the ratio of the acetate anions to the total amount of acetate radicals in the free acid as well as in the Na-salt. The dissociation residue^'- is represented by the ratio of the undissociated (acetic) acid radicals to the total amount of (acetic) acid radicals present. These two conceptions will be re- peatedly utilized in our further discussions. With such applications in mind, the question arises: What is the relation of the degree of dissociation and of the dissociation-residue to the hydrion concen- tration? The degree of dissociation (as well as the residue) is con- sidered as a function of [H+], and thus, contrary to the foregoing procedure, the [H+] is treated as an independent variable. Since in the acid-salt mixtures we have found the ready means of obtain- ing any desired [H+] for any practical purpose, the assumption of [H+] being an independent variable may now appear as quite con- ceivable. According to the above definition, the degree of dissociation a may be expressed as [Si (X = [A] where S~ as previously represents the acid anions and A the total acid in whatever form present. Let us now recall the equation on page 43. rg-1 According to the definition the degree of dissociation a = ttt and [A] = [HS] + [Si Therefore, [Si a = [HS] + [Si 32 L. Michaelis, Biochem. Zeitschr. 33, 182 (1911). 50 HYDROGEN ION CONCENTRATION rrr^-i [a— 1 By substituting for [HS] its value in (1), namely, r , we obtain a k + [H+] (2) Thus this equation represents a as a function of [H+]. The dissociation constant of the acid, k, appears in this equation as a constant value characterizing the function, which will be here designated as the parameter of this function. The function a = . -. is graphically represented by the curve in figure 2 which shows a hyperbola cutting the zero ordinate at a sharp angle. 0 1 2 3 V 5 10 Kxio-'' Fig. 2. The degree of dissociation-(a)-curve of an acid whose dissociation constant = 1 X 10~'. Abscissa — [H+], ordinate-a. This representation is not especially clear, and the highly practical significance of this function rests on the fact that by means of a simple logarithmic transformation of the abscissa a very compre- hensible graphic form is obtained. Instead of the abscissa repre- senting the [H+] in our new form it stands for pH, and the curve shows a as a function of pH. pH is an antibatic function of [H+]; i.e., the pH curve falls with the rising [H+] curve. Our function may then be written as k a = k + 10-PH (2b) and is represented graphically in figure 3 on page 51. LAWS OF ELECTROLYTIC DISSOCL\TION 51 The curve is distinguished by the following characteristics: 1 . This curve can be divided into three parts : The first and third portions run asymptotically to the X-axis, the first at the height 0, the third at the height 1. 2. Between these two portions lies an almost straight-line steep portion which passes into the as^nnptotes at sharp angles of in- flection. 3. Strictly speaking this middle portion is not a straight line, but is slightly curved or S-shaped, and it has a point of inflection at the ordinate height of |. The projection of this point on the abscissa gives the negative logarithm of the dissociation constant k. If we 0.5 *» 5 6 7 8 9 10 If Fig. 3. Dissociation curve of an acid whose constant is k = 1 X 10 ~^. Abscissa-pH, ordinate-a. The scale of the ordinates is enlarged X 5 with respect to the scale of the abscissa. At the point marked x a = f, and the projection of the ordinate on the abscissa gives the value of pK (the negative logarithm of the dissociation constant). state, analogously with the previously used method, — log k = pK, then it may be said that the parameter of this function is pk, and that geometrically it means that it denotes that value on the X-axis (pH-axis) for which the ordinate = \. If similar disso- ciation curves be drawn for various values of pK (therefore for acids of various strengths) then all such curves will be parallel. They may be graphically derived from the given curve by simply moving it horizontally on its abscissa until the projection of its ordinate value of \ coincides with the value of pK of the particular acid. 4. Finally, this graphic representation has the special advantage insofar as the dissociation-residue curve is an exact mirror image of the dissociation curve. 52 HYDROGEN ION CONCENTRATION The dissociation-residue p is expressed according to the defini- tion as and hence [HS] p = [HS] + [S- [H+] = 1 - a 1 k + [H+] k 1 4- [H+] _ 10-PH '' ~ k + 10-pH i m^^ ~~^ ^ \ N 1 V 0.5 \ V \ V \ V ^ __ 5,^ _ 6 10 11 (3) (3b) Fig. 4. Dissociation-resfdwe curve of an acid under same conditions as in figure 3. It also represents the dissociation curve of a base whose dissociation constant is k = 1 X 10"''. The graphic construction is best derived from the relationship a = 1 — p, and it is at once perceived from figure 4 that the curve is a mirror image of the a-curve, and that its parameter is exactly the same. The mathematical analysis of these functions is as follows: Differentiating the function k + [H+] (1) with respect to log [H+] we obtain da d d In [H+] d log [H+] d [H+] '^ d In [H+] " d log [H+] a d [Hi LAWS OF ELECTROLYTIC DISSOCIATION 53 where log is the decimal and In is the natural logarithm, therefore, ^" = - AXi^ . I, 10 (2) d log [H+] (k + [H+]2) and differentiating once more with respect to log [H+] we obtain d at d^ a d d log [H+] d [H+] d In [H+] d log [H+]2 d [H+] d In [H+] d log [H+1 k . [H+] • (k - [H+]) (k + [H+]3 (In 10)2 (3) Letting this expression = 0 we obtain the conditions for the transformation point of the a-or log [H+]-curve, This condition obtains, as may be readily seen when [H+] = k, or pH = pK. By substituting in (1) [H+] = k, we have a = \. Thus at the transformation point, 1. pH = pK 2. a = i which was to be demonstrated. The same holds true for the residue curve which at the transfor- mation point ordinate is sjTnmetrical with the dissociation curve. The angle at which the curve intersects the ordinate in the vicin- ity of the transformation point is of interest. According to the fundamental laws of differential calculus the differential quotient represents the tangent of the unknown angle ip. Since at the point concerned [H+] = k, it follows by substitution in (2) above that d « In 10 2.303 ^ ^„„ = - — — = = 0.576 d log [H^ and arc tan ^ = almost exactly 30°. In order to obtain the correct angle by this method one must take care that in the graphic repre- sentation the ordinates are drawn to the same scale as the abscissa. If the scale of the ordinates is n times that of the abscissa then the tangent of the angle of intersection is n times greater. In the curves of figures 3 and 4, n = 5, and therefore, the tangent = 2.88 and the angle of inclination is approximately 71°. 54 HYDROGEN ION CONCENTRATION All of the above applies entirely and directly to the dissociation of the bases, and all that is needed is to substitute [0H~] for [H+] and pOH for pH. Therefore for a base, [0H-] " k 1 + -—■ 1 + k [0H-] But [0H-] = — [H+] therefore, substituting for [0H~] above, The a-curve of a base with a dissociation constant k is, therefore, identical with the p-curve of an acid, whose dissociation constant . kw is -j^, and the p-curve of a base of constant k is identical with the kw a-curve of an acid whose constant is -^. k A simple verification of the course of a dissociation curve will be found in the following experiment. p-Nitrophenol is a weak acid whose ions are yellow and whose undissociated molecule is color- less. If the same small amount of this indicator be added to a series of solutions adjusted to varying pH values by the acetate buffers, then the degree of dissociation a of the p-nitrophenol and simultaneously the intensity of its color will vary with the pH of the solution. In figure 11, p. 98, the pH is given on the abscissa and the intensity of color, or the degree of dissociation a, of the dye on the ordinate. The course of the curve corresponds exactly to the theoretical a-curve, and, if pK be determined graphically by the method given above it will be found to have the value of 7.18, while ahnost the same value, 7.28, was observed by means of the conductivity method (calculated for the same temperature, 18° from the data of Euler and Bolin and also of Lunden). LAWS OF ELECTROLYTIC DISSOCIATION 55 16. The dissociation of polybasic acids A dibasic acid may be defined as a molecule possessing two ionogenic H-atoms. The dissociation of each of the two H-ions must be considered separately. Thus oxalic acid is dibasic, but the tendency to dissociate is different in the two H-ions. The acid dissociates in two stages with each of these stages having a distinct dissociation constant. If the acid radical remaining after the removal of both H-atoms be designated as S and the whole acid molecule as SH2, then the chemical equations representing the two stages are: 1st stage: SH2 -^ SH" + H+ 2nd stage: SH" -^ S" + H+ To each of these two reactions the mass law applies in the follow- ing way: [SH-] [H^ [SH2] [Si [H+] = ki [SH-] = k In order to arrive at a clear understanding of dissociation, the conceptions of the degree of dissociation and of dissociation-residue will be discussed also in this connection. Let the concentration of undissociated acid be designated as [A], that of the primary ions as [A~] and of the secondary ions as [A=], then ai the degree of dissociation of the first stage may be defined as [A-] ai = [A] + [A-] + [Ai and ai the degree of dissociation of the second step as: "' [A] + [A-] + [Ai and the dissociation-residue, p, as A p = [A] + [Ai + [Ai 56 HYDROGEN ION CONCENTRATION The two dissociation constants are: [A-] [H+] [Ai [H+] ko = [A-] From the above equations it follows: ai [A-] ^ ^ [A-] 1 [A] IA-] a, [A=] ^ [A=] ^ 1 [A-] [A=] - = 1 + - — - + - — - p ^ [A] ^ [A] and through elimination of [A], [A-] and [A=] we finally obtain :^^ 1 ai = a2 = ^ kx ^ [H+] [H+] [H 1 + -7-^ + ki • kj ki ki 1 + 7^ + [H+] [H+] The curves shown in figures 5, 6 and 7 are graphic representations of these functions having definite values for the two parameters, ki and k2 and with the pH as the abscissa. The dissociation-residue or p-curve is almost exactly identical with that of a monobasic acid, and its deviations from the latter are so slight that they are not noticeable. The primary dissociation curve, ai, shows the most marked changes. It belongs to the same type as the p-curve of an amphoteric electrolyte, which is to be discussed later, inasmuch as it shows a maximum. Just as in the case of the ampholyte p-curve the kind of the maximum formation of the curve depends upon the 33 L. Michaelis, in: C. Oppenheimer, Handb. d. Biochem. Erganzungs- band, p. 57 (1913). 1.0 0.9 0.8 0.7 ae 0.5 0.V as o.z 0.1 oc, A ^ ^ 9 / 1 \ / \ / \ / 1 / I i / \ \ / \ 3/ \ / \ \ y ? ^,\ 1 \ 1 1 1 1 A \ \ \ \ \ r \ a, y y ^2 V a, -n -13 -12 -It -10 -9 -8 -7 -6 -S -¥ -3 -Z -1 C Fig. 6. For the case in which ki = 10"^ k2 = IQ-io 0.9 0.8 0.1 0.6 0.5 O.t 0.3 0.2. 0.1 0 cc. ^ N Y ? \ / \ / \ / \ 1 i t '1 \ la Al V 1 j \ J ° ^ "V.^ V -1 t -1 3 -1 3 -/ / -1 0 -i 1 -i ? -5 ' -6 -/ " -4 ' -J t -i. ? -1 0 Fig. 7. For the case in which ki = 10^', k. = IQ-^ Figs. 5 to 7. p — dissociation-residue curves, ai — primary dissociation curves, a^ — secondary dissociation curves. Abscissa are log [11^]. 57 58 HYDROGEN ION CONCENTRATION values of both dissociation constants of the acid; in fact it depends upon the ratio ki:k2. If ki is much greater than k2, then a wide rather flat maximum is formed in which ai is almost = 1 as in figure 5. When ki is but a little greater than ko, as in figure 6, then the maximum has a sharper angle and it does not quite reach the value 1 ; and when the values of ki and k2 are still closer together then these two characteristics are even more pronounced as shown in figure 7. The limiting case is where ki = k2, i.e., when the acid is capable of dissociation to the same extent in both stages. In this case ai is always = 0, which means that only undissociated acid molecules and divalent acid ions are formed and no univalent ions. This last case is not met with in weak electrolytes. The two dissociation constants have more or less differing values. They approach each other quite closely in the case of succinic and fumaric acids, while the two constants differ very widely in maleic and carbonic acids. Wherever the two acid (COOH) groups are spa- tially close to each other in the molecule, as in maleic acid, then the acid nature is quite pronounced and the acid is strong in the primary stage, while in its second stage only a small residue of acid is present in excess. But if the two COOH — ^groups are situated far apart, then each behaves in a manner more independent of each and exert a more nearly equal effect. The position of the maximum in the ai-curve is of particular interest. In order to evaluate the maximum it is necessary to differentiate the function ai = f ([H+]) with respect to [H+] and to set the derivative = 0. It is much easier to calculate the minimum or the inverse function which is and whose derivative is OCl_ _ J^ _ k-2 d [H+] ~ ki [H+l^ By setting this derivative = 0 we obtain the minimum for J_ or the maximum for ai ai [H"] = Vki X k2 LAWS OF ELECTROLYTIC DISSOCIATION 59 which tells us that ai is at its maximum when [H+] is equal to the geometric mean of the two dissociation constants, or when pH = pki + pka The relation of this maximal value becomes evident when it is introduced into the equation for ai ai max = 1 + 2 VE TABLE 12 Table of dissociation constants of polybasic acids Dibasic acids ki k2 ki:k2 oxalic 3.8 X 10-2 1.5 X 10-3 6.6 X 10-* 9.4 X 10-* 1.4 X 10-2 9.7 X 10-" 3.3 X 10-^ 1.3 X 10-3 1.7 X 10-2 2 X 10-« 4.9 X 10-5 2.1 X 10-« 2.7 X lO-o 3.2 X 10-6 2.6 X 10-^ 4.5 X 10-* 6 X 10-11 3.1 X 10-« 5 X 10-« 2.6-0.85 X 10-9 780 Malonic 710 Succinic 25 Fumaric 29 Maleic 5400 Tartaric 21 5 Carbonic 5500 o-Phthalic 419 Sulfurous 3400 Uric 775-2350 Tribasic acida ki k2 k3 Phosphoric Very high ca. 9 X 10-3 8.2 X 10-* 8.8 X 10-8 3.2 X 10-5 3 6 X 10-1* Citric 7 0 y 10-' VI When 2W— is very much smaller than unity, then the maximal value of ai becomes practically = 1 and at the same time the maxi- mum of the curve is broad and fiat, while in other cases a more pointed maximum is obtained, and its value is less than 1. When k2 is much smaller than ki (fig. 5) the ascending and de- scending portions of ai represent a part of the dissociation curve and of the dissociation-residue curve as well, and the projections of the two 0.5 ordinates on the abscissa give the values of pki and pk2. Both of these points are marked by circles on the curve in figure 5. 60 HYDROGEN ION CONCENTRATION Furthermore there need be no difficulty in calculating the some- what compHcated functions for poly basic acids. As long as the difference between any two consecutive dissociation constants remains < 10^, the curves may be compounded from the individual dissociation and dissociation-residue curves. The same is true of the polybasic bases, by simply substituting [0H-] for [H+]. As a result of the consecutive dissociation in stages the last ionization stages in polybasic acids can occur only in a strongly alkaline reaction. Thus for example, the concentrations of the carbonate, COs", and the tertiary phosphate, POf ions are practically negligible in the blood, in which only HCOy, HPOf , and H2P07-ions are found. Table 12 gives the dissociation constants of some dibasic and tri- basic acids (chiefly taken from Landolt-Bornstein, Physik. Chem. Tabellen. k2 for uric acid is given by A, Kanitz^^). 35 17. The dissociation of amphoteric electrolytes Amphoteric electrolytes or ampholytes are substances which behave as acids as well as bases. As acids they form salts with bases, and as bases they react with acids to form salts. The ampho- lytes may be divided into two large groups: those in which the acidic and the basic radicals are spatially apart, and those in which the two radicals are one and identical. To the latter group belong many metal hydroxides. Thus Zn(0H)2 may dissociate either as Zn++ + 20H~, or as H+ + ZnO=. The most important ampho- lytes of the first type are the amino-acids. Thus by virtue of its NH2 group glj^cocoll is a base, while by virtue of its COOH-group it is at the same time an acid. It has long been known that it reacts with HCl to form a hydrochloride and with NaOH to form 3« A. Kanitz, Zeitschr. f. physiol. Chem. 116, 96 (1921). 35 Literature: G. Bredig, Zeitschr. f. Elektrochem. 6, 33 (1899).— K. Winkel- blech, Zeitschr. f. physikal. Chem. 36, 546 (1901).— J. Walker, Zeitschr. f. physikal. Chem. 49, 82 (1904) and 51, 706 (1905).— H. Lunden, K. vetenskap. Nobelinstitut 1 (1908) ; Arkiv for Kemi, Mineralogi och Geologi (Vetenskaps- Akad., Stockholm) 2, No. 11; Zeitschr. f. physikal. Chem. 47, 476 (1906).— L. Michaelis, Biochem. Zeitschr. 24, 79 and 30, 143 (1910); 33, 182 (1911); 47, 250 (1912) ; Nernst-Festschrift, 308 (1913) ; Biochem. Zeitschr. 103, 225 and 106, 83 (1920).— S. P. L. Sorensen, Ergebn. d. Physiol. 12, 303 (1912). LAWS OF ELECTROLYTIC DISSOCIATION 61 the sodium salt. But it displays its basic nature to its full extent in a strongly acid solution and its acid nature in a strongly alkaline solution. In pure solution it behaves as a very weak electrolyte and shows a just barely demonstrable acid reaction, because of a predominance of the acidic property of the COOH-group over the basic property of the NH2-group. Assuming that the ampholyte is present in solution in concentration [A], as an acid it forms anions of [A~] concentration, and its acid dissociation constant is ka. At the same time in the same solution it also functions as a base whose dissociation constant is kb and forms cations of [A+] concentration, ka and kb must be different in value, and either the acidic or the basic character of the ampholyte predominates. Of all known ampho- lytes the difference between ka and kb is least in hemoglobin. In discussing the dissociation relations of the ampholytes it is simplest to begin with a consideration of the dissociation residue, p, which is defined as the ratio of the undissociated portion to the total concentration [A] of the ampholyte: [A] - [A+] - [A-] p = [A] On the other hand the degree of dissociation must be given separately for the cations and the anions. For the anions it is: [A- a = [A] and the degree of dissociation of the cations is: ^ [A+] or ^ [A] The mass law finds its apphcation in the following relationships: [A-] X [H+] = ka X [U] where U is the undissociated portion of the ampholyte, and further- more, [A+] X [Oil-] = kb X [U] It follows, therefore, (A-l - t. jM ,„ 62 HYDROGEN ION CONCENTRATION and [A] - [A+] - [A-] = [U] = [A] -K^- kb ^^ [H+] [0H-] From which follows [A] [U] = ka kh 1 + 7TT7^ + [H+] [0H-] and hence the sought value of p may be stated as [U] 1 [A] 1 , ka , kb [H+] [OHi substituting for [0H~] its value as j— ^ we obtain: " = — r^-T ® 1 + J^ + ^ [H+] ^ [H+] ^ k^ ^ ' In comparing the expression with that for ai in a dibasic acid as given on page 56, we find them identical, if we only substitute in the latter ka for k2 and : — for ki. As in a previous instance (see Ivb page 54) the specific replaceability of an acid constant by 7— is ka equally valid. Also in respect to the formation of the maximum the p-curve corresponds to the ai-curve. All analytic loci of the ai curve hold good in this curve, if we substitute kb for k2 and — ka for ki. In the ai-curve the width of the maximum depended upon the ki:k2 ratio, therefore in this case it is determined by the product ka X kb. Figure 8 shows the course of various p-curves in which the product ka X kb has different values, ka is assumed to be equal to kb in these curves. When ka X kb is very small, about 10"^", then the maximum elevation of the curve spreads over a very wide stretch. When this product is larger, about 10"^^, then this eleva- tion becomes smaller, the maximum point may be readily seen, and its ordinate value is around 1. With a still greater increase LAWS OF ELECTROLYTIC DISSOCIATION 63 of ka X kb the maximum rises less and less, and the elevation zone becomes narrower. Thus for ka X kb = 10~^^ = kw the maximum value is around only 1/3. However, such ampholytes and those with a still greater ka X kb product are not known to exist. The p-curves of all electrolytes having the same ka X kb value are parallel and are but horizontally shifted, as it were, on a common abscissa, as shown in figure 9. In each of such p-curves an ascend- ing and a descending arm are differentiated. The former is almost exactly the p-curve of an acid whose dissociation constant is ka, the latter is an a-curve of an acid whose constant is — i.e., a base kb whose dissociation constant is kb. This relation is practically true whenever ka X kb is very small. It follows then that the p-curve 1.3 0.5 r /f ?^ N^ r- "> s. ^ 1 io / I \ 1 \ \ / '"t h / \ \ / / / / \ \ / / 1 / } I \ j / / idI ml \ \ \ \ i 1 / n /' \ \ \ ' \ / \ \ \ [ \ 1 J I ' J r ^ x\\\ ) L 1 \ \ / I Ja ^^^i^^\ i^ V N \ i^ 1 ■is j£9fL -10 -1 0*1 Fig. 8. Dissociation-residue curves for various amphoteric electrolytes of diflferent ka X kb values. Each curve is marked with its appropriate ka X kb value. It is assumed throughout that ka = kb. of an ampholyte may be constructed by considering it in the first place as an acid with the constant ka and secondly as a base with the constant kb and then by uniting the two respective p-curves. If, however, ka X kb is larger, then this method is not exact, as may be seen from the comparison of the curves in figures 9 and 10. The deviation is greatest in the vicinity of the maximum, but the abscissa value corresponding to the maximum remains at least unchanged. The hydrion concentration corresponding to the maxunum of the dissociation-residue curve is designated as the isoelectric point of the ampholyte. This term was originally used to designate the point at which the sign of the charge of a colloid changed. For example, the negatively charged gold sol becomes positively charged 64 HYDROGEN ION CONCENTRATION on the addition of A1+++ salts; that concentration of the A1+++ salt at which this change occurred was called the isoelectric point. This conception was introduced by Hardy^*^ for proteins, when he showed that they are negatively charged in alkaline solution and positively in acid solution. According to Hardy, although it was not specifically emphasized, the isoelectric point of protein was 10 05 f \ /• \ / \ 1 \ \ \ 1 I f" 1 1 1 c. TV6 fi cU k 4 \ / \ \ \ 1 1 \ \ / / \ y^ X ^v. ■16 IqgH ■10 0 *1 Fig. 9. Dissociation-residue curves for two ampholytes in each of which kaX kb = 10~'*, but curve 1 is for the case where ka = kb = 10~^, while curve 2 (dotted) is for the case where ka = 10"^ and kb = 10~^^. I.U \ > r "^ ^■^ \ \ \ \ \ 0.5 0 -It J 9 log H -1L 1 -■} i ? *i Fig. 10. The dissociation-residue curve (pa) of an acid whose ka = 10~^ also of a base whose kb = 10~^ likewise. If curve pa is followed to its intersec- tion with pb and then pb is followed downward, a curve is obtained which is almost identical with curve 1 in figure 9. equal to the reaction of neutrality. The definition of the isoelectric point of an ampholyte as the maximum of the p-curve as given above was developed by Michaelis in 1910. ^*' The maximum of the p-curve can also be analytically defined. 36 W. B. Hardy, Journ. of Physiol. 33, 251 (1905). LAWS OF ELECTROLYTIC DISSOCL\TION 65 It is easier to calculate it as the minimum of the inverse, 1/p, func- tion as follows: 1 ka kh , , p IH+J k„ and its derivative in respect to [H+] is: p ka kb . d [H+] [H+]2 k w by setting this derivative = 0 we obtain the minimum of 1/p or the maximum of p, which is = V^ [H+] for p-max. = \r^ k^ (1) The isoelectric point I is, therefore, that [H+] which is determined by the equation = #^ ka r k^ (2) Kb It is characterized by the following properties: 1. At its isoelectric point the sum of the anions and cations of an ampholyte in a given concentration is at a minimum. 2. The concentration of the anions of an ampholyte at its iso- electric point is equal to that of the cations. These conditions may be proved as follows (according to (1) and (2), page 61): i '^ ' = IhT """^ '^ ' = 15ST By substituting for [H+] its value at the isoelectric point, -^ kw, and for [0H~] the corresponding value kb k w ^H-1 Ik . "Ik "■a kb • k. = V^: k W7 66 HYDROGEN ION CONCENTRATION we obtain [A-] = [U] . , '^^ and [A+] = [U] kb €^' Vf: or [A-] = [U] y^^ and [A+] = [U] ^^^^ Therefore [A-] = [A+] In the above derivations it is assumed that the electrically un- charged part of the ampholyte is present in one chemical modifica- tion only. But the following case is also conceivable. The ampho- lyte may exist in two forms, Ai and A2 (such as lactam and lactim forms), so that the cations have only that conJSguration which corre- sponds to Ai and the anions only that which correspond to A2. The idea would suggest itself then that in equation (1), page 61, Ui should be substituted for U, and in (2) U2 for U, where Ui is the concentration of the undissociated Ai form and U2 that of the undissociated A2 form. The author stated this idea first^^ and sub- sequently it was developed by Eckweiller, Noyes and Falk.^^ These latter authors modified the equation for the isoelectric point in the manner shown = V^ J ^,_„ , [A+] [A=] [Ai] [A2-] in which A and Ai are the two tautomeric modifications of the ampholyte. This newer interpretation is, however, not quite cor- rect. For in our experiments we determined not the equilibrium between the ampholyte ions and of any tautomeric form of the un- charged ampholyte molecule, but only the equilibrium between the ions and the undissociated ampholyte molecules in toto. Or, in other words, our experimentally determined values of ka and kb "Michaelis and Davidsohn, Biochem. Zeitschr. 30, 143 (1910); see p. 149, concerning theobromin. 38 H. Eckweiller, H. M. Noyes and K. G. Falk, Journ. of Gen. Physiol. 3, 291 (1921). LAWS OF ELECTROLYTIC DISSOCIATION 67 denote, as always, the apparent dissociation constants. Thus our original equation remains correct, and the contingent deviations from the directly determined isoelectric point and from that calcu- lated from the dissociation constants will have to be explained in some other way. Latest developments in the theory of ampholyte dissociation The most marked advance in the theory of dissociation is based upon a contribution by Bjerrum,^^^ and its important forerunner being an article by Adams. '^"^ Accordingly it is no longer correct to state the ampholyte ions (Zwitter-Ionen) are present only in extremely small amounts. For the aliphatic amino-acids it seems to hold true that in their "undissociated" state they are present almost entirely as amphoteric ions +NH3— R— C00~, while the amount of the molecular species NH2 — R — COOH must be consid- ered as being negligible. Isoelectric glycocoll is to a certain extent an inner salt between the NH2- and the COOH-groups which is completely dissociated, in analogy with ammonium acetate consisting in solution entirely of NH4+ and CHaCOO^-ions. Upon the ad- dition of acid to a solution of glycocoll, the ionization of the R — COO~-group is suppressed, and there are formed +NH3CH2COOH- ions (just as with ammonium acetate NH4+ + CH3COOH would be formed). Upon adding alkali, the ionization of the ammonium- group is suppressed, and the formation of NH2— CH2 — COO~-ions occur (analogously to the reaction of ammonium acetate + alkali -> NH3 + CH3COO-). ka, hitherto designated as the acid dissociation constant, relates to a process involving the ammonium group, and kb, hitherto desig- nated as the basic dissociation constant, relates to a process involving the COO-group. The following method of presentation^^*= appears to be the simplest : We shall consider as the undissociated form the molecular species +NH3 — CH2 — COOH, on the grounds that this ion contains the greatest number of atoms, and because all other forms of glycocoll may be conceived to arise by the process of splitting off H-ions from 38a N. Bjerrum, Zts. physik. Chem. 104, 147 (1923). 38b E. Q. Adam.s, J. Amer. Chem. Soc, 38, 1503 (1906). 38c L. Michaelis and M. Mizutani, Zts. physik. Chem. 116, 152 (1925). 68 HYDROGEN ION CONCENTRATION this molecular species. Hence in such a case we may consider an ion as the undissociated molecule, whereas in the past we have been accustomed to consider the electroneutral molecule as the undis- sociated form. This ion then dissociates, similarly to a dibasic acid, in the following way: 1st Stage: +NH3-R-COOH -^ +NH3-R-COO-+H+ 2nd Stage: +NH3-R-COO- -^ NH2-R-COO-+H+ The equilibrium constants are, for the first stage: [+NH3-R-COO-] [H+] [+NH3-R-COOH] ~ ' and for the second stage : [NH2-R-COO-] [H+] [+NH3-R-COO-] ~ ' The relation of ki and k2 to the hitherto employed (and still widely used and for most practical purposes usable) values ka and kb is: ka = k2 kw For the aromatic amino acids (such as aminobenzoic acid) this conception does not apply so well. In these the molecular species NH2 — R — COOH may actually exist in measurable concentrations along with the ampholyte ion +NH3— R — C00~. As for the am- photeric electrolytes with weaker acidogenic radicals, such as amino- phenol, on the other hand, the older conception is the more correct one. Glycocoll is a stronger acid than acetic acid and a stronger base than ammonia. Its inner salt (the ampholyte-ion) is therefore extremely little dissociated hydrolytically which corresponds to stating that there are very few molecules in the form NH2 — CH2 — COOH. It is very probable that also the proteins in the isoelectric state exist in the ampholyte-ion form. The amphoteric nature of such ions LAWS OF ELECTROLYTIC DISSOCIATION 69 manifests itself in their ability to add Il-ions (to the C00~ group) as well as to split off H-ions (from the NH3+-group). The experimentally observed fact described in the text to the effect that the product ka X kb of any of the known ampholytes has never been found to be equal to or greater than kw, has been recently shown by Bjerrum^^* to be a necessary consequence of the above developed conception. 18. The determination of the isoelectric point The isoelectric point of the ordinary ampholytes (the colloidal ampholytes will be dealt with later) may be determined by a variety of methods. 1. The first method is especially applicable to easily soluble ampholytes and is based on the following principle. When acid is added to a dilute buffer solution, i.e., to a solution of a definite [H+] but of considerable buffer value, the [H+] rises, and on the addition of a base to the same solution the [H+] falls. Let us desig- nate again that [H+] which corresponds to the isoelectric point of an ampholyte by the symbol I. Then the ampholyte in solution will behave as an acid as long as [H+] < I, and as a base when [H+] > I, In the first case, if added in suffi.cient concentration, it will raise the [H+] of the buffer solution, and in the second case it will diminish it. Only at its isoelectric point does the ampholyte remain without any effect upon the buffer, for then it behaves prac- tically as a non-electrolyte; or stated more exactly, because the very few ions yielded by it consist of equal numbers of anions and cations. The number of H-ions cannot, therefore, be increased in the buffer solution on the addition of the isoelectric ampholyte, because there are no corresponding number of negative ions to maintain the electro-neutrality, and for a similar reason the [H+] cannot be diminished. The following experiment-^^ is quoted in corroboration of this theory: A constant amount of an aqueous solution of phenylalanine was added to each of a series of buffer solutions consisting of a constant small amount of sodium acetate and increasing amounts of acetic acid. The pH of these solutions, determined electrometrically, is given in table 13. The same could be demonstrated for glycocoll. Its isoelectric " L. Michaelis, Biochera. Zeitschr. 47, 250 (1912). 70 HYDROGEN ION CONCENTRATION point was calculated at pH 6.09 and was found by the above de- scribed method to be between pH 6.3 and 5,8, 2, With difficultly soluble ampholytes the following method^" may be used which is based upon the principle that the undissociated portion of an ampholyte is less soluble than its ionized portion. An equal ample amount of an ampholyte is added to each of a series of hot acetate buffer solutions and the appearance of crystalliza- tion is observed, A suitable ampholyte for this purpose is m- aminobenzoic acid. In the following table the original pH of the TABLE 13 Phenylalanine as base. 3.75 4.10 4.27 Isoelectric point calculated at pH 4.58 pH before addition of phenylalanine pH after addition of phenylalanine Phenylalanine as acid , 4.66 5.07 5.26 5.45 5.73 4.01 4.22 4.43 4.48 4.57 4.55 4.78 4.74 4.72 4.77 Difference in pH +0.28 +0.12 +0.16 -0.01 -0.11 -0.29 -0.52 -0.73 -0.96 buffer solutions is given in the upper and the observed extent of crystallization in the lower line: pH 5.4 5.1 4.8 4.5 4.2 3.9 3.6 3.3 3.1 Crystallization: 0 0+ + + + + + + + + + + 000 It appears then that the crystallization maximum is in the close vicinity of pH 4.2, while the isoelectric point, as calculated from the dissociation constants found in the literature, is at pH 4.1. The crystallization optimum of p-aminobenzoic acid may be determined with equal definiteness. The relative sharpness of definition of such optima depends upon the value of the product ka X kb, just as the sharpness of the maximum of the dissociation-residue curve " L. Michaelis and H. Davidsohn, Biochem. Zeitschr. 30, 140 (1910), LAWS OF ELECTROLYTIC DISSOCIATION 71 (see figure 8) depends upon the same value. Tiie sharply defined minima of the aminobenzoic acids corresponds to p-aminobenzoic acid ka X kb = 3 X 10~'^ ??i-aminobenzoic acid ka X kb = 2 X 10~^^ On the other hand, it has long been known that such ampholytes as tyrosine and arsenious acid have quite poorly defined (flat) maxima. These are soluble in strong acids and bases but quite insoluble in the wide intermediate range of reaction. No sharply defined optimum in relation to the pH of solution is to be found in this case, and the small order of the corresponding values of the dissociation constants products is to be noted: Tyrosine ka X kb = 1 X IQ-^o Arsenious acid ka X kb = 6 X 10"^* Employing the above method Michaelis and Davidsohn obtained the following values: o-Aminobenzoic acid. m-Aminobenzoic acid Aspartic acid 1.21 X 10-5 1.63 X 10-5 1.5 X 10-" 2.33 X 10-12 1.22 X 10-" 1.2 X 10-12 I calc. 1.7 X 10-" 8.8 X 10-5 8.7 X 10-" I found 1.6 X 10-" 6.3 x'lO-5 9.3 x'lO-" It may he of interest to compare the results obtained by various in- vestigator's, partly for varying temperatures {table 15). The agreement among the above figures is quite good, especially when they are reduced to the same temperature. The uncertain- ties which existed formerly in regard to some of the figures were due to the fact that the influences of salt concentration had not been properly understood and corrected for. With due consideration for some of the newer principles the lack of agreement among some of the older figures will disappear. This applies to all dissociation constants, and not only to those of the ampholytes. Of all of the above ampholytes only arginine, lysine and his- tidine have (in their second stage of dissociation) larger kb values than those of ka; the isoelectric point of only these ampholytes is at an alkaline reaction. 72 HYDROGEN ION CONCENTRATION TABLE 14 Table of dissociation constants of amphoteric electrolytes at 25° The isoelectric point values in this table were calculated from the equation I = ,— ; , on the assumption that k„ = 10 ~". Arginine, 2nd stage Arginine, 1st stage Lysine, 2nd stage I/ysine, 1st stage Leucine GlycocoU Alanine Histidine, 2nd stage Histidine, 1st stage Phenylalanine Tyrosine Leucylgh^cine Alanylgh^cine Glj^cylglycine Aspartic acid. Arsenious acid o-Aminoben- zoic acid . . . p-Aminoben- zoic acid . . . m-Aminoben- zoic acid . . . kb Observer <1.1 X 10-14 1.0 X 10-1- 1.8 X 10-10 1.8 X 10-1" 1.9 X 10-i» 2.2 X 10-9 2.5 X 10-9 4.0 X 10-9 1.5 X 10-8 1.8 X 10-8 1.8 X 10-8 1.5 X lO-" 6 X 10-i« 1.06 X 10-5 1.21 X 10-5 1.6 X 10-5 2.2 X 10-12 1.0 X 10-^ 1.1 X 10-12 <1.1 X 10-^ 2.3 X 10-12 2.7 X 10-12 5.1 X 10-12 5.0 X 10-13 5.7 X 10-9 1.3 X 10-12 2.6 X 10-12 3.0 X 10-1' 2.0 X 10-11 2.0 X 10-11 1.2 X 10-12 1 X 10-1" 1.37 X 10-12 2.33 X 10-11 1.20 X 10-12 Kanitz Kanitz Kanitz Kanitz Winkelblech Winkelblech Winkelblech Kanitz Kanitz Kanitz Kanitz Euler Euler Euler Winkelblech, Lunden Wood Lunden Winkelbech, Walker Winkelbech, Walker Isoelectric point ca. 3 X 10-11 ca. 3 X lO-i" 8.8 X 10-7 8.2 X 10-7 6.1 X 10-7 6.2 X 10-8 4.4 X 10-6 3.9 X 10-s 2.2 X 10-« 3.0 X 10-« 1.1 X 10-3 2.4 X 10-5 2.8 X 10-1 7.2 X 10-3 l.l X 10-" LAWS OF ELECTROLYTIC DISSOCIATION 73 TABLE 15 Glycocoll, ka Glycocoll, kb Glycylglycocoll, ka Glycylglycocoll, kb 1.8 X 10- -10 1.2 X 10- -10 1.05 X 10- -10 2.7 X 10- -12 1.93 X 10- -12 1.7 X 10- -12 1.8 X 10- -8 3.3 X 10- -9 2 X 10- -11 0.95 X 10- -11 Temperature 25° 17.5° 18° 25° 17.5° 18° 25° 18° 25° 18° Observer* Winkelblech Michaelis and Rona Dernby Winkelblech Michaelis and Rona Dernby Euler Dernby Euler Dernby * Winkelblech, Zeitschr. f. physikal. Chem., 46, 546 (1901). Michaelis and Rona, Biochem. Zeitschr. 49, 248 (1913). Dernby, Cpt. rend, des travaux du lab. de Carlsberg. 11, 265 (1916). H. Euler, Hofmeisters Beitr. 7, 1 (1906). 19. Amphoteric ions Bredig pointed out that one of the ionization forms of an ampho- lyte of the general type NH2-R-COO must be an amphoteric ion ("Zwitter-Ion") which is positively as well as negatively charged, such as -""NHs-R-COO". This molecular species may also be conceived of as an inner salt formation arising between the COOH- and the NH2-groups. Thus, for instance, CH3COOH + NHs;:^ CH3COO- + NH4 + In the same manner we may assume that the salt arising from the internal salt-formation of the amino-acid dissociates electrolytically and produces the ion -"-NHs-R-COO". It is not possible to demonstrate the existence of this molecular species by the usual method of demonstrating ions, i.e., the migration in one direction in an electric current, for this amphoteric ion is electrically neutral. Besides, it is undoubtedly present in extremely small amounts. Furthermore, an amino-acid being the salt of so weak an acid and so weak a base always hydrolyses to a great extent, and the product of hydrolysis of the amphoteric ion is the usual form of the amino- acid. Just as the hydrolysis product of ammonium acetate is ammonia and acetic acid, or, NH4+ -f CHaCOO-;^ NH3 + CH3COOH ;v S) 0« A^ <1 -•01 74 HYDROGEN ION CONCENTRATION or, as it is customarily written CH3COO NH4 + HoO ;^ CH3COOH + NH4OH the amphoteric ion Hkewise yields by hydrolysis the ordinary amino- acid molecule: +NH3-R-C00-^ NH2-R-COOH Since one mol of amino-acid yields one mol of amphoteric ion, it follows that for a given amino-acid the concentration of the amphoteric ion must be an equivalent fraction of the concentration of the undissociated amino-acid. There is no direct method up to the present time of demonstrating concretely the existence of the amphoteric ions. Because of their very small concentration they are not capable of disturbing to a measurable extent the equilibrium among the other known dissociation products of amino-acids. 20. The hydrogen ion concentration of a pure solution of ampholyte It is apparent that a pure aqueous solution of an ampholyte must have a neutral reaction whenever ka = kb, that it must be acid in reaction when ka > kb and alkaline when ka < kb. The exact calculation of the hydrion concentration, however, is not a simple process. S0rensen^^ adopted the following method of calculation. Let A represent the total concentration of the amino-acid and A+, A~ and U the concentrations of its cations, anions and undissociated portion respectively. It then follows A+ + A- + U = A (I) A+ X H+ = k X U (II) A+ X OH- = kli X U (III) H+ X OH- = kw (IV) and furthermore, in accordance with the law of electroneutrality, A+ + H+ = A- + OH- (V) If the values of A, ka and kb be given, then the five unknown values of U, A+, A~, [H+] and [0H~] may be calculated by solving " S. P. L. Sorensen, Ergebn. d. Physiol. 12, 495 ff. (1912). LAWS OF ELECTROLYTIC DISSOCIATION 75 the five above equations. In carrjdng out the calculation for [H+], we get from (I), (II) and (III) and from (II, III, V): k a H+ OH- By substituting this value of U in (VI) we obtain; ka kb * \H+ OH-/ H+ OH- H+ - OH- in which we can further substitute kw/[H+] for [0H~]. This gives an equation of the fourth order for [H+] as given below, where x stands for [H+j X* + x3 ^^ + A^ + x^ ^ (ka - kb) - X ^ (kaA + kj - 1^ . ka • kw = 0 iVII) kb The general solution of this equation is difficult. It can have only- one real positive root, because it contains only one sign change, irrespective of the sign of the coefficient of x- (Descartes' law). The practical solution rests upon the permissibility of disregarding the first (x*) or the last member of the equation, depending upon the numerical relationships of the individual constants. Further- more equation (VII) may be factored in the following way (this may be tested by simply multiplying out) : (x2-k„jx(x2 + x^ + x.A + ^jkw-xXAXkw(^-l j=0 or Q X R - S = 0> By letting S = 0, i.e., by making ka = kb, equation (VIII) is satisfied when either Q or R = 0. When R = 0, no real positive 76 HYDROGEN ION CONCENTRATION root for X is obtained. On the other hand, when Q = 0, we obtain X = Vkw, which means that a solution of an ampholyte in ivhick ka = kb, has a neutral reaction in all concentrations. Now let us write (VIII) in the form Q = R or, S it Since R is always positive, then x- > k^, when S is positive; and x^ < kw when S is negative. But S is positive or negative when ka > kb or ka < kb respectively. In other words: lohen ka > kb the solution of the ampholyte is always acid, and when ka < kb it is always alkaline. Equation (VII) solved for A becomes (..-.,)(.. + . i^ + ... I) A = -^ '-^ ■ ^^^ ^ (IX) Now, when ka > kb, and, therefore, according to the above reasoning, also x- > kw, then both factors of the numerator are positive, consequently the denominator must also be positive, i.e., we must have X < which denotes that x is smaller than the [H+] at the isoelectric point, or, that the [H+] of the solution must lie between the neutral reaction and that of the isoelectric point. Indeed, with decreasing concentrations of the electrolyte x approaches the neutral reaction, while it approaches the isoelectric point with increasing concentra- tions. By writing equation (IX) in the form (.- V kw \ / A \ / , , ^w , kwka X + V kw ) I x2 + X \- ■ _ _v -^ / \ ^_j^ kb kb '^w'^a . — X^ kb LAWS OF ELECTROLYTIC DISSOCIATION 77 we perceive that A must increase with increasing values of x. For A = °o , X must have its maximum value, which, according to the last italicized rule above, may only be that of the isoelectric point. In other words, the more concentrated the ampholj^te solution the closer does its [H+] approach that of the isoelectric point of the ampholyte. For A = 0 we obtain, on the other hand, the other limiting value, x = Vkw, or the neutral reaction. By applying the above outlined general solution of equation (VII) and the rules developed from it, the value of x may be es- timated for any special case. By substituting this estimated value in equation (VII) together with the appropriate values for ka, kb and kw, it becomes possible to determine whether one or more terais of the expression become extremely small in respect to the other terms, and may then be disregarded. Thus approximate values for x are obtained, which may be verified by substitution in (VII), and, whenever necessary, corrected by actual trial. By this method S0rensen made the following calculations for pure aqueous solutions of glycocoll, for which ka = 1.8 X 10~^", kb = 2.7 X 10-12, kw = 10-14; Concentration of the solution: A °° 1 10-1 10-2 10-3 10-4 pH 6.088 6.089 6.096 6.155 6.413 6.782 The isoelectric point is at pH = 6.088. 21. The hydrogen ion concentration and the solubility of weak acids This problem takes us back again to the simple electrolytes. It is commonly known that free acids are less soluble than their alkali salts and that bases are less soluble than their chlorides. Since, as was shown above, the relative amounts of free acids and of their ions, as well as of salts, depends upon the hydrion concentration, it follows, therefore, that the hydrion concentration of a solution must have an effect upon the solubility of the electrolytes. Let us next define the concept of solubility. We shall choose for illustra- tive purposes the solution of a fairly insoluble acid. When water is saturated with solid benzoic acid a system is finally obtained which has the following composition: A sediment of solid benzoic acid which is not electrolytically dissociated to any measurable extent. 78 HYDROGEN ION CONCENTRATION In the supernatant solution we have molecules of undissoeiated benzoic acid, H-ions and benzoate-ions. (In addition to the H-ions there must also be present OH-ions in accordance with the laws of dissociation. But since these play no part in this consideration they may be ignored). Two interpretations may then be given to the expression "solubility of solid benzoic acid." Either it may mean the concentration of those molecules of undissoeiated benzoic acid which are present in the saturated solution, i.e., the concen- tration in the saturated supernatant solution of the molecular species of which the sediment is composed. Secondly, the term "solubility" in this case may be regarded as being denoted by the concentration of all the benzoic acid molecules regardless of the form in which they are present in the saturated solution. We shall designate the first as the "partial solubility," or X, of benzoic acid and the second as the "total solubility," or A. Furthermore, one may also speak of the "solubility of the benzoate ions." It is apparent that the benzoate ions are much more soluble in water than the free benzoic acid, which is easily demonstrated by the greater solubility of alkali benzoates. It is also occasionally stated that in a pure benzoic acid solution the state of saturation exists only for the undissoeiated acid, and that, on the other hand, the benzoic acid solution cannot be saturated in respect to the benzoate ions. But this statement is not in accord with the true concep- tion of solubility. Solubility is represented by the concentration reached by a substance in solution, when the solution is in contact with the solid phase. Thus the definition of solubility includes two contiguous phases, and solubility represents the state of equi- librium between the two phases. Without the contact with the solid phase the solubility of a substance cannot be properly defined. Under certain conditions a metastable supersaturated solution may be obtained without the presence of a solid phase. But as soon as a small amount of the solid substance is added to such a super- saturated solution, the excess of the solute separates out until the true state of saturation is reached. Therefore, it cannot be properly stated that a saturated solution of benzoic acid is not saturated in respect to the ions of the acid. For we are dealing here with the definition of the solubility of the undissoeiated acid alone of which the solid phase is composed. As for the acid anions, the amount of LAWS OF ELECTROLYTIC DISSOCIATION 79 these in solution is entirely conditioned by the mass law relationship of concentrations, in the following manner: [Acid anions] X [H+] _ jj [Undissociated acid] k is in this case the dissociation constant of the acid. Since the concentration of the undissociated acid is equal to the "partial solubility" of the acid, X, therefore, it follows that [Acid anions] = [Hm The "total solubility" of the acid, A, is therefore k • X [H+]+k ^ = ' + W] - ' ~wr or, by substituting the values given in (3) on page 52. X A = - P The total solubility. A, is, therefore, dependent upon the [H+] of the solution, while X, the partial solubility, remains constant (for a given temperature). This calculation is based upon the assumption that the undissociated salt of the acid is not present in solution. In such cases where this condition is not fulfilled a cor- rection may be applied, but we need not consider this point in this general discussion at present. These relations may be also conceived in other ways. It has been shown that certain solid salts (e.g., silver salts) at higher temperatures conduct the electric current not as metals but as electrolyte solutions to a small but noticeable extent and display material transport of ions. It must be assumed then that at least a very small part of a solid electrolyte dissociates, i.e., it consists of transportable ions."*- Thus in a saturated solution of benzoic acid *- The present day conception of a crystalline electrolyte consisting onlj' of ions arranged in a space lattice ("Raumgitter") has nothing to do with the above discussion. For the ions arranged in the space lattice are not in free motion; they are ions only insofar as they have the same intra-atomic struc- ture as the ions of a solution in Faraday's sense; but they are not ions insofar as they are not freely movable. 80 HYDROGEN ION CONCENTRATION in water both the solid phase and the liquid phase consist of undis- sociated acid, H-ions and acid anions. According to the Henr}^- Nernst law of distribution each of these three molecular species has its own distribution coefficient in respect to each phase, and the saturated solution should consequently contain these three com- ponents distributed between its two phases in concentrations lim- ited by the appropriate coefficients. But this would lead to an unequal concentration of the positive H-ions and of the negative acid ions in solution, which would be impossible because of the electro-static forces arising from such a condition. Instead of this, a condition of equilibrium is established in the system entailing equal concentrations of positive and negative ions in solution and the formation of an electric potential difference at the boundary of the liquid and solid phases (this point to be discussed later). This conception is based upon the distribution coefficient of only one of the molecular species, the undissociated acid, which determines the "partial solubility" of the benzoic acid, and it is also based upon the mass action law which regulates the concentrations of the other molecular species in solution so as to maintain equilibrium. A biologically important example of the relation of the hydrion concentration to the solubility of an acid is furnished by the case of uric acid^^ Unfortunately this problem is not yet completely solved. The investigation of this question has hitherto taken a tortuous path because of a lack of a clear conception of the inter- dependence of solubility and hydrion concentration. We shall attempt to interpret the available data in accord with the above theory. First the partial solubility, X, of uric acid must be deter- mined. For this purpose its ionization must be completely sup- pressed by means of the addition of HCl. His and Paul found a solubility in 1.0 A' HCl at 18° of one mol in 7137 Hters, or, X = 1.401 X 10~^. According to the same authors the constant of dis- sociation in the first step, ki = 1.5 X 10"" (according to Gudzent, ^3 Literature : His and Paul, Pharmazeut. Zeit. 1900. — Paul, Zeitschr. f. physiol. Chem. 31, 1 and 64 (1900).— Gudzent, Zeitschr. f. physiol. Chem. 56, 150 (1908);! 60, 25, 38; 63, 253, 455 (1909).— Bechhold and Ziegler, Biochem. Zeitschr. 20, 189 (1909); 24, 146 (1910).— Ringer, Zeitschr. f. physiol. Chem. 67, 332 (1910).— Kohler, Zeitschr. f. physiol. Chem. 70, 360; 72, 169; Zeitschr. f. klin. Med. 78, 1.— Lichtwitz, Zeitschr. f. physiol. Chem. 84, 416 (1913).— Schade andBoden, Zeitschr. f. physiol. Chem. 83, 347; 86, 416 (1913).— A. Kan- itz, Zeitschr. f. physiol. Chem. 116, 96 (1921). LAWS OF ELECTROLYTIC DISSOCIATION 81 ki = 2.3 X 10"" at 37°). Therefore, at any hydrion concentra- tion (as long as it is not so small as to permit the uric acid, which is dibasic, to begin the second dissociation stage) the solubility at 18° is: at 18° A = 1.40 X 10-^ (^^^^+1^1X^0-) Thus for example at the reaction of the blood, [H+] = 4.5 X 10~^, A = 4.77 X 10-3 which is thirty times the value of A in iV HCl. A pure saturated solution of uric acid must have at 18° a hydrion concentration according to (la) page 40. [H+] = Vl-o X 10-6 • 1.4 X 10-" = 1.45 X 10"^ The degree of dissociation of this solution is, according to (2), page 42. 1.5 X 10-6 0.094 1.5 X 10-6 + 1 45 X 10-5 Paul and His found experimentally the value of a at 18° to be 0.095, and, moreover, conversely they derived the dissociation constant from the above value of a. The verification of the theory is then afforded by the fact that X and A are calculable from each other. The solubility of uric acid calculated in the above manner did not have the hoped for significance in physiology and pathology because of the following two complications: The above considerations are valid only for the system in which uric acid is the sohd phase. The presence of the alkali salt of the weak acid did not enter into the derivation, for the alkali salts of .aknost all acids are much more soluble than the acids themselves. But in this respect uric acid forms an exception. The primary- sodium urate is quite insoluble by itself, while in the presence of NaCl it is an unusually insoluble salt. Furthermore, according to Gudzent, the primary salt e.xists in two modifications, the lactam .and the lactim forms, the solubilities of which are different. If, therefore, one of these forms is present as the solid phase, then the solubility of uric acid becomes a very complicated problem; 82 HYDROGEN ION CONCENTRATION especially since the presence of NaCl depresses enormously the solu- bility of the sodium urate. Thus we have the remarkable case in which the presence of the alkali-salt, at least in NaCl-containing solutions, diminishes, instead of increasing, the solubility of a weak acid. Recently Kanitz calculated from available data the second dissociation constant of uric acid, and he came to the conclusion that the primary sodium urate in pure solution is markedly hydro- lyzed into free uric acid and the secondary sodium salt. Such a solution contains, therefore, free uric acid which must exercise a great influence upon the solubility of the urate. On the other hand, the sodium urate tends to an abnormally great extent to form very stable supersaturated solutions, even in the presence of the urate as the solid phase. Schade considered these supersaturated solu- tions as colloidal in nature, but Kohler and Gudzent showed this assumption to be improbable by various means (ultrafiltration, compensation dialysis, conductivity). Schade and Boden showed that uric acid solutions may be easily made to form gels, which however slowly crystallized out. On the other hand, Gudzent demonstrated by means of compensation dialysis that in the blood uric acid is present in true solution. At any rate, Bechold and Ziegler found that they could dissolve large amounts of free uric acid in blood serum, and they designated this condition as the "overfilling" ("Uberfiillung") of the blood with uric acid. Evidently in this case the uric acid is present in the blood in the form of supersaturated sodium urate in solution. The whole problem demands extensive reinvestigation, for we have not yet reached an adequate solution. 22. The relation of hydrion concentration to the solubility of difficultly soluble salts The solubility of true neutral salts (NaCl, KNO3, etc.) does not depend upon the hydrion concentration which is the case for the salts of strong bases and weak acids, and reversely, for the salts of strong acids and weak bases. A biologically important instance is furnished by the solubility of calcium carbonate, which was in- vestigated by Rona and Takahashi."*^ We shall discuss this problem in a somewhat different way from these authors, and more in accord with our previous considerations. ^^ P. Rona and D. Takahashi, Biochem. Zeitschr. 49, 370 (1913). LAWS OF ELECTROLYTIC DISSOCIATION 83 The partial solubility of tlie molecular species CaCOs is, inde- pendently of the [H+], = X. Therefore, the total solubility is A = X + [Ca++] where [Ca++] is that concentration of Ca++ ions which is in equi- librium with the CaCOs concentration X. This [Ca++] depends upon [H+] in the following way: In solution there exists a state of equilibrium between calcium monocarbonate and calcium bicarbonate. or Therefore, CaCOa + H2CO3 ;:± Ca(HC03)2 CaCOa + H2CO3 5:^ Ca++ + 2 HCO3- [CaCOs] [H2CO3] ki (1) [Ca++] [HC03-]2 But in the solution, [CaCOs] = X Furthermore, H2CO3 ^ H+ + HCO3- hence Substituting in (1) the value of [H2CO3] found in (2), X X kj X [H+[ [Ca++] X [HCO3-] or, = ki tCa-]=^^X ^'^ ki [HCO3-] and X kj [H+ A = X + -— X ki [HCO3-] It is evident then that the solubility of CaCOa depends not only upon the [H+] but also upon the concentration of the bicarbonate 84 HYDROGEN ION CONCENTRATION ions present in tiie solution, regardless whether these later are de- rived from the solid phase or are added to the solution extraneously as NaHCOs. Analogously to the assumption of certain approxima- tions in the first part of this book, we may assume the concentra- tion of the bicarbonate ions as being equal to that of the total bicarbonate in solution. Furthermore it can be shown by means of a solubility determination in a strongly alkaline reaction that X is no longer analytically measurable and that practically X = 0. The above equation is, therefore, simplified to the form A = K [H^ [HCO3 The following values were found for A at 18°: [HCO3-I X 0.9» [H+l A (mol/liter) K 0.017 0.0425 0.0152 2.4 X 10-7 9.0 X 10-« 9.3 X 10-8 4.93 X 10-3 7.3 X 10-" 2.03 X 10-3 349 345 332 Average 342 Therefore, at 18' A = 342 X [HCOs- mols per liter For the conditions existing in the blood plasma, where [H+] = 3 X 10-8 and [HCO3-] = 0.02 N, a solubility of 0.022 g. Ca++ or 0.0294 g. CaO per liter is calculated. In reality one finds more CaO in the plasma, at all events more than 0.1 g., and at times even 0.17 g. CaO per liter. In the whole blood there is about 0.1 g. CaO per liter. It appears, therefore, that the plasma contains more Ca than the amount corresponding to the solubility of CaCOa, and it seems as if this were a case of a supersaturated solution. This assumption is all the more plausible, since in practice it is quite easy to prepare such supersaturated lime solutions, even in the absence of protein. Thus, when instead of solid CaCOs a CaCl2 solution is * This is a correction factor which must be introduced in order to take account of the fact that the active mass of the bicarbonate ions is somewhat Bmaller than the concentration of the calcium bicarbonate. LAWS OF ELECTROLYTIC DISSOCIATION 85 used and is rendered slightly alkaline with NaHCOs, a clear stable supersaturated lime solution is obtained. Dialysis experiments have demonstrated that the greater portion of the Ca in serum is present in a dialysable form. This excludes the possibility that the excess of calcium in the serum could be present in the form of a colloidal suspension or as a solution of a carbonate or phosphate. That portion of the calcium which in the compensation dialysis experi- ments was found to be non-diffusible is but a small fraction of the excess calcium present, and this portion only may be regarded as combined in an undissociated calcium-protein compound. Brinkman and van Dam,"*^ using another method, confirmed the solubility data of Rona and Takahashi (22 mg. Ca++ per liter, for [H+] = 3 X 10-8 and 20°), and they also found 22 mg. Ca++ ions in the ultrafiltrate of blood serum at [H+] = 3 X 10-^. In the serum itself, in those few cases in which their method could be ap- plied, they found the same amount of Ca++ ions. If Rona's com- pensation dialysate and Brinkman's ultrafiltrate be regarded as being to a certain degree the same, then all of these results may be explained only on the following assumption. The ionized calcium is present in the serum in a concentration within the limit of its actual solubility and not in a supersaturated state, and that the excess calcium is present but to a small extent as a protein complex and largely in true solution in an undissociated form of unknown nature. The following further comment is to be made upon these investi- gations. In the first place Rona's as well as Brinkman's solubility determinations apply to a temperature of 20°. Secondly they assume the [H+] in blood to be 3 X 10~^ whereas for 38° it should be taken as 4.5 X 10"^. Thirdly, it must be pointed out that the theory of activity, which is to be discussed in the next chapter, could not yet have been taken into consideration, and it is prob- able that the factor of 0.9 (see footnote on page 84) is too low. *5 R. Brinkman and E. van Dam. Kon. Akad. Wet. Amsterdam 22, 762 (1920). The method used by these authors for the determination of Ca''"''' ions consisted of establishing the amount of oxalate to be added to a solution containing calcium in order just to exceed the limit of solubility of the calcium oxalate produced. Since the solubility product [Ca"'"''] X [oxalate ~] is con- stant (0.055 millimols per liter at 20°) it is possible from the above amount of the added alkali oxalate to calculate the Ca''"'' present. All the other methods of estimating calcium determine the total calcium. 86 HYDROGEN ION CONCENTRATION These three conditions all point in the direction of the probability that the solubility of calcium as determined in these experiments is too low. It is possible that the discrepancy between the solubility of calcium in plasma as based on theory and its actual calcium content is not as great as these investigations appear to indicate. It may be seen from the above example how much more important the exact knowledge of natural constants is than hypotheses are for the clear conception of physiological facts. It is surprising that such physiologically important constants have not yet been deter- mined. The theories of formation of calculi and of the deposition of lime in the blood vessels are truly phantastic, as long as it is not established whether or not the calcium of the blood is at all in a supersaturated state. How complicated the previously discussed problem of uric acid is may be appreciated at this juncture, when it is recalled that in this case the slight solubility of the free acid is combined with the slight solubility of its Na-salt. The preceding chapter concerned itself with a slightly soluble acid whose salts are easily soluble (benzoic acid), while the present chapter deals with slightly soluble salts whose components (CO2, Ca(H0)2) are more soluble than the salt. In the case of uric acid we have a condition not yet theoreti- cally investigated, where both the free acid and its salts are but slightly soluble. 23. The relation of the hydrion concentration to the solubility of a slightly soluble ampholjrte If the partial solubility of the slightly soluble ampholyte be = X, then the total solubility is A = X + A+ + A- where A+ and A~ denote the ampholyte in its cationic and anionic forms respectively. The experimental investigations apply here insofar as the solu- bility minimum has been studied. This condition obtains when [A+] + [A""] is at a minimum, therefore, at the isoelectric point. Thus we return to the consideration fully discussed previously on page 70. Finally it must be recalled once more that all of this discussion LAWS OF ELECTROLYTIC DISSOCIATION 87 of the solubility is made on the approximate assumption that the salts of the slightly soluble acids or bases are totally dissociated. As soon as the presence of appreciable amounts of undissociated salts is taken into account, then certain corrections must be intro- duced, which will be discussed in a later chapter dealing with strong electrolytes and the ionic activity theory. All of the above deduc- tions apply only to such cases in which the concentration of the undissociated electrolytes in the solution in question remains quite small. 46 24. The hydrolysis of salts It has been assumed throughout above that all salts are completely electrolytically dissociated. But the salts have also another form of dissociation. Since a salt is produced by the general reaction Acid + base ;=i salt + H2O we must assume, in accord with the theory of equilibrium, that this reaction also takes place in the opposite direction, i.e., that the salt partially "dissociates hydrolytically" or "hydrolyses" to form acid and base. It is obviously imperative to establish the degree or extent of this hydrolysis. First case. The salt of a strong acid and a strong base. If the acid be designated as AH, the base as BOH and the salt as C, then according to our assumption we have present in solution only the ions A~, B+, H+ and 0H~. Since A~ and B+ in a strong acid and base have no affinity for the H+ and 0H~ ions, it follows that they have no effect upon the [H+] and [0H~], that is to say, in the aqueous solution the hydrion concentration is not affected by such a salt. These salts are not at all hydrolyzed, and the solution of a salt of this type is neutral in reaction. These salts are therefore called neutral salts. Second case. The salt of a strong acid and a weak base whose dissociation constant is kb. In this case the salt is hydrolyzed to a small extent with the formation of a little free acid and free base. But since the base dissociates less than the acid, the solution has a *^ Taken chiefly from N. Bjerrum. Die acidimetrische and allialimetrische Titration. Stuttgart. 1914. 88 HYDROGEN ION CONCENTRATION slightly acid reaction. Let us now study the degree of hydrolysis 7 of the salt which is defined as ■ ■ = [1!1 ^ [C] where [C] denotes the total concentration of the salt. For the general solution of the problem let us consider the co- existence of the following laws: [H+] X [0H-] = k^ (1) which is the dissociation equation of water. Since the acid produced in the hydrolysis was assumed to be a strong acid, its concentration is therefore = [H+]. But since in the hydrolysis as much base is produced as acid, it follows that [H+] = [0H-] + [BOH] (2) Furthermore the equation for the dissociation of the base is [BOH] ^^'^ ^^^ and finally, [B+] + [BOH] = [C] (4) From these four equations involving the four unknown values [H+], [0H-], [B+] and [BOH] each of the latter may be calculated, and knowing the value of [H+] that of 7 may be obtained. But these calculations are of a rather involved character. It suffices ordinarily to carry out the calculations, with some simplifying assumptions, for certain limits, as: a. Assuming that the hydrolysis is very slight (7 < 0.01, [H+] < 10-3). Then in equation (4) the value of [BOH] as a term of a sum is negligible in respect to that of [B+] and in this case [B+] = [C] (4a) and solving for [H+] by substituting in (2) the value for [OH"] from LAWS OF ELECTROLYTIC DISSOCIATION 89 (1) and the value for [BOH] from (3), and by substituting [C] for [B"^] as in (4a), we obtain: [H+]2 = [c] p^ + kw (5) kb h. Assuming a moderate hydrolysis (7 > 0.01 and [H+] > 10~^) we may disregard the value of [0H~] in equation (2) in respect to the other terms of the sum [BOH]. Thus the equation becomes [H+] = [BOH] (2a) and furthermore the value for [B+] in (4) still remains closely enough under the conditions: (4a) (7) [B+] = [C] on substituting these two values in (3) we obtain [H+]2 = [C] X k^ kb and -Vi ' x^- C] "^ kb (8) c. In the case of strong hydrolysis we may likewise assume for all purposes instead of equation (2) that [H+] = [BOH] (2a) from which we derive: [H+]» = [C] . ^ - [H+] . ^ (9) kb kb and Vi 1 k w .[C] kb y = , - 1 (10) V' + 4-^. + v 4 [C] • kb 90 HYDROGEN ION CONCENTRATION If the salt is the product of a strong base and a weak acid whose dissociation constant is ka, then the same relationships as given above are appHcable except that [OH"] is substituted for [H+] and the degree of hydrolysis is defined as T = [OH- [C] TABLE 16 Table of values for the degree of hydrolysis at room temperature k kw = = 10-" 1 N 0.1 N 0.01 N 0.001 N 00 10-7 10-6 3.3 X 10-6 10-5 1.4 X 10-5 10-* 10-2 10 -« 10 -s 10-" 10-" 10-2 1.05 X 10-* 10-" 3.2 X 10-6 3.2 X 10-* 3.2 X 10-3 0.032 10-* 10-3 10-2 3.3 X 10-* 10-6 3.2 X 10-2 1 10-8 0.032 10-10 0.095 0.62 0.27 10-12 0.095 0.27 0.92 TABLE 17 Table of pH values in solutions of hydrolytic salts pH values in the solution of a salt of a weak base and a strong acid, or pOH- values in the solution of a salt of a strong base and a weak acid. K is the dis- sociation constant of the weaker constituent. k 1 AT 0.1 iV 0.01 AT 0.001 N 00 7.0 7.0 7.0 7.0 10-2 6.0 6.5 6.9 7.0 10-* 5.0 5.5 6.0 7.0 10-6 4.0 4.5 5.0 6.0 10-8 3.0 3.5 4.0 4.5 10-10 2.0 2.5 3.0 3.6 10-12 1.02 1.6 2.2 3.0 In table 16 the values given by Bjerrum of the degree of hydrolysis were calculated for various values of the dissociation constants, the latter given in round figures. The values above the upper stair-line are obtained from equa- tion (6), the values between the stair-lines from equation (8) and LAWS OF ELECTROLYTIC DISSOCIATION 91 those below from equation (10). From the practical standpoint it appears even more important to tabulate the pH values of such solutions. These have been calculated by the author and are shown in table 17. Thus in solutions of NH4CI (kb approximately = 10~^) and of sodium acetate (ka approximately = 10~^) the following the pH values obtain: c IM 0.1 Af 0.01 M 0.001 M NH4C1 4.5 9.5 5 9 5.5 8.5 6.5 Na-acetate 7.5 25. Notes on the methods for determining dissociation constants and the isoelectric point 1. The oldest method for the determination cf the dissociation constant of a sunple electrolyte consists of measuring the conduc- tivity of its solutions in various dilutions. Wilhelm Ostwald found empirically that = k (1 - a) V where a is the degree of dissociation at the concentration -> v being that volume of solution that contains one mol of the solute, value of a is determined by the expression The = a [(1) where Ac is the molar conductance at the concentration c and A ^ denotes its limiting value at infinite dilution. Ostwald's law, as given in equation (1) is but an expression of the mass law applied to dissociation. According to definition, for instance, it is true for an acid that s- (2) 92 HYDROGEN ION CONCENTRATION where S~ is the concentration of the acid anions and S is that of the total acid. According to the mass law [S-] X [H+] IS] - [S-] If in the above expression we substitute for the concentration g- [S~] its equivalent — , where v is the volume and S~ is the absolute molar amount of the anions, and by recalling that in a pure acid solution [S~] = [H+], we obtain = k S - S- (S - S-) V (1 - a) V 2. The second method depends upon the determination of the degree of hydrolysis of the sodium salt of the weak base or the chloride of the weak acid studied. The relation of this factor to k follows obviously from the preceding chapter. The degree of hydrolysis may be estimated by two methods: a. By determining the [H+] from the catalytic action on the hydrolysis of sucrose, methyl acetate, etc. b. By the electrometric determination of the [H+]. This method has been little employed. 3. The third method depends upon the determination of [H+] in a definite mixture of the acid with its Na-salt, for which purposes various methods may be employed. The theoretical basis of this procedure is given by formula (1) on page 44. Each of these methods has its own field of applicability. But there are conditions under which either the experimental data become uncertain, or, where the equations which are but approxuna- tions applying within certain limits are no longer applicable. It is beyond the scope of this book to deal with this question in greater detail. Neither are all the possible methods exhausted with those given above. Thus, for instance, a very good method for an acid LAWS OF ELECTROLYTIC DISSOCIATION 93 of moderate strength is to measure the [H+] in a pure solution of the acid and to calculate k by applying equation (la) on page 40. The second method given above has been developed by Lunddn who determined the degree of hydrolysis of the salt of the acid studied and of a weak base of a known dissociation constant. CHAPTER II The Theory of the Quantitative Determination of Acidity AND Alkalinity summary of contents For the characterization of the acid nature of a solution it is necessary to know: (1) The titratable acidity which may he further subdivided into neutralizing capacity and equivalent capacity, (2), the hydrogen ion concentration or pH value, and (3) the buffer value. The theory of indicators and of the titration curve is developed for the elucidation of the concept of titratable acidity; the principles of the most important methods for the estimation of the pH are then briefly outlined; and finally the conception of buffer value is developed. 26. Definitions and statement of problem The acidity of a solution is defined by a number of values, and the sharp demarcation and differentiation of these values has become possible with the advent of physico-chemical methods. In the older anal3rtical chemistry these values were either not differentiated at all, or else to an insufficient extent. It is true that in previous decades excellent methods were found for the estimation of acidity and alkalinity in the form of the titration, but the reason for this is due to the fact that for strong acids and bases the various values which define acidity practically coincide. Furthermore, for certain of the weaker acids and bases some indicators had been empirically found, such as phenolphthalein and methyl orange, which enabled our prede- cessors to determine the equivalent content of these acids and bases. Thus experiment had demonstrated that the equivalent content of an acetic acid solution could be correctly determined by a titration against a standard NaOH solution with phenolphthalein, that an incorrect value was obtained when methyl orange was used as an indicator in the same titration, and that the reverse was true for the titration of ammonia against sulfuric acid. At any rate, the acidity of a solution had been defined by its content of acid equivalents. 94 DETERMINATION OF ACIDITY AND ALKALINITY 95 Therefore, according to this definition a normal hydrochloric acid and a normal acetic acid solution were of exactly the same acidity. On the other hand, it could not be entirely ignored that these two equivalent acid solutions were in some respects quite different, and a stronger acidic property had to be conceded to the hydrochloric acid than to the acetic acid. There were obviously two entirely different properties which were designated as acidity, and only one of these properties could be quantitatively estimated by means of the titra- tion. For the other property there were only qualitative approxima- tions. Thus it was known, for instance, that normal hydrochloric acid attacked and dissolved zinc better than normal acetic acid, that the former changed congo-red paper to a bluer tint, that it had a more corrosively acid taste, that it inverted cane sugar more rapidly, etc. By applying the principles of the ionic theory we may now define acidity quantitatively in three ways, which do away with the above described uncertainties. Furthermore, we shall learn of a fourth value or factor which is necessary for the complete characterization of the acid properties of a solution. One of our new definitions, "titratable acidity," corresponds some- what to the older definition. It is the number of equivalents of free acid in the solution. But as soon as we wish to define the methods by which these equivalents are determined, we find that the older conception is inadequate. For, if we were to ask a chemist of a few decades ago to what endpoint was an acid solution to be titrated he would say : until the neutral reaction was reached ; and as a means of recognizing this point he would probably suggest the change of color of phenolphthalein. We now know that such a reply contains two errors. In the first place, a solution consisting of equivalent parts of acetic acid and NaOH, or of sodium acetate, is not neutral, but, because of the ensuing hydrolysis, it is somewhat alkaline (see table, page 80). Secondly, the turning point of phenolphthalein is not at the neutral, but at an alkahne reaction. Therefore, our analyst will in the end be practically quite correct, in spite of his erroneous explanation. But he could not have explained why he should not have just as well used methyl orange in his titration. At best he would say that "methyl orange is not sensitive enough to acetic acid" without being able to explain why it was more sensitive to hydrochloric acid. In reahty the endpoint of the titration of acetic acid is not at the 96 HYDROGEN ION CONCENTRATION neutral reaction, but at the hydrion concentration corresponding to that of a solution of sodium acetate, which in turn corresponds to the turning point of phenolphthalein, but not to that of methyl orange. If we should wish to attribute a significance to the actual point of neutrality then we would have to differentiate two endpoints in our titration: first, the attainment of the neutral point and secondly the reaching of the point of equivalence. The number of equivalents of the base which were used up in reaching the neutral point may be designated as the neutralizing capacity and the number of base equivalents used to reach the point of equivalence as the equivalent capacity, or, the true titration capacity. It is only when a strong acid is titrated against a strong base that these two conceptions coincide. We must now learn to know the methods by means of which these two values are determined. 1 . The determination of neutralizing capacity For the estimation of this point an indicator is needed which shows a definite color change at the neutral reaction. Such indicators are, litmus (solution), neutral red, or phenol red. Exactly neutral water is colored with litmus, and the investigated acid solution is titrated until it shows the same blue-violet shade of color of its litmus as the standard solution. Since it is very difficult to obtain really pure water, it is more prac- ticable to employ instead of the water a mixture consisting of 6.9 volumes of M/15 Na2HP04 and 3.1 volumes of M/15 KH2PO4 solu- tions. The reaction of such a mixture is exactly neutral at 18°, according to S0rensen. The determination of the neutralizing capacity is of little practical importance. 2. The determination of the equivalent capacity For the discussion of this topic a . preliminary and more detailed explanation of the theory of indicators is required, which follows. 27. The theory of indicators The indicators are acids or bases whose shade of color is changed by the hydrion concentration of the solution. Wilhelm Ostwald' 1 W. Ostwald, Lehrb. d. allgem. Chem., 1891, p. 799 and Z. physikal. Chem 9, 579 (1892). DETERMINATION OF ACIDITY AND ALKALINITY 97 has long ago established the theory, which in its essentials is still valid, and according to which this change is explained by the assump- tion that the ions of a dye-stuff possess a color different from that of its undissociated molecule. An apparent objection to this theory arose from the work of Hantzsch' who showed indicators to be pseudo- acids and bases, and according to whom the change of color is not due to ionization but to tautomeric rearrangement. But since these two processes are inseparably related to each other, no real contra- diction can be found in these two theories. The theory of Hantzsch serves only to elucidate further the mechanism by which the color changes of indicators occur. We may even go further in the same direction. In accord with the conception of the atomic model the configuration of the atoms in the molecule is determined by electro- static forces. It is, therefore, not surprising that the appearance of a free charge, as it occurs in the process of ion formation, may recast the configuration of the atoms in the molecule. Thus we can still retain the older conception and state that an indicator is a substance which is characterized by a difference in color between its unionized and its ionized states. For practical purposes two groups of indicators are recognized: the one-color indicators which change from a colorless to a colored state, as phenolphthalein, and the two-color indicators, such as lit- mus, w^hich change from one color to another. It is more convenient in the development of the theory to begin with the one-color indica- tors. Let us choose, as an example, p-nitrophenol. This is an acid whose dissociation constant, kg = 6.5 X 10~^, or, pk = 7.19. The free acid is colorless, while its ions are yellow. Therefore at any given [H+J a part of this dye is present as the colorless free acid and another part in the fonn of yellow colored ions. The proportion of the ions to the total amount of indicator in solution is its degree of dissociation, or a. This latter value can be quite easily determined experimentally. Thus we may place the same amount of indicator into a second vessel and render it strongly alkaline with sodium hydroxide in which case the indicator will be completely dissociated and show its maximum color intensity. If now we compare colori- metrically the color intensity of the first solution with that of the second, and if this numerical relation of the concentration of the indicator be designated as the "degree of color," or, as F, then F = a. 2 H. Hantzsch, Ber. deut. chem. Ges. 46, 1537 (1913); ibid. 48, 158 (1915). 98 HYDROGEN ION CONCENTRATION It follows then that a as well as F are functions of the pH, of the type of the dissociation curve. Figure 11 shows one of such curves for p-nitrophenol, which has already been discussed on page 54. As will be readily seen this indicator is practically colorless in the pH range of 0 to 5.0. The pH at which it begins to show color depends upon its concentration. It may be stated in a general way that with concentrations of indica- tors usually employed in titrations the color becomes noticeable when the indicator is 5 per cent dissociated. In this particular case, this occurs at about pH 5.2, at most at pH 6.0. This so-called transfor- mation point of the indicator does not in any way correspond with the neutral reaction, pH 7.0. Let us now consider the significance of this pH value of 5.2 to 6.0 in a titration. Three cases must be distin- guished. First case: Titration of a strong acid with a strong base. Let us assume that we add to 10 cc. of N HCl increas- ing amounts of N NaOH. The acid had been diluted with water to a volume of 100 cc. This volume is so little altered in the process of the titration that we may- consider it as remaining constant. This assumption simplifies the calculation and introduces no marked error. In figure 12 the amount of the alkali added in cubic centimeters is shown on the abscissa and the resulting pH on the ordinate. The latter is calculated on the assumption that the acid, the alkali and the salt formed are always completely dissociated and that the amount of indicator added is so small that it does not influence the state of equilibrium between the acid and the base. In the beginning of the experiment we have 0.1 A'' HCl, therefore the pH = 1.0. After the addition of 1 cc. of the base we have a 0.09 N HCl solution, or pH = 1.05. In this manner we may con- tinue our calculation until we approach the vicinity of the neutral Fig. 11. The abscissa represent pH val- ues and the ordinates represent the color intensity of p-nitrophenol. The curve as drawn is the calculated theoretical disso- ciation-(a)-curve of the indicator, the circles are the experimentally observed values. DETEEMINATION OF ACIDITY AND ALKALINITY 99 point. Having added 9.5 cc. N NaOH we still have a 0.005 N HCl solution, pH 2.3. With 10 cc. of the base added we have neutrality or pH 7. On the further addition of the alkali free base is present in solution and the indifferent NaCl. From this point we may calcu- late the resulting pOH and therefore the pH. The lowest curve of figure 12 is obtained in this manner. The curve is characterized by its initial flat or horizontal course and then by the sudden and abrupt rise at the "end-point of the titra- tion." In this region the addition of a few drops of the alkali causes a steep increase of the pH from about 3.0 to 11.0. If we should ask what relation the transformation point of the nitrophenol (pH 5.0 13 12 11 10 9 8 7 6 5 3 pHz 1 i- > pb :no /^ ■^ — / ' V AC atr i ^tc ^ / ^ ■^ , HL / Y — '^ 13 12 11 10 9 8 7 6 5 V 3 PUz 0 123^56739 10 11 12 13 can n Na OH Fig. 12. Titration curves of 10 cc. portions of A^ HCl, acetic acid and phenol with A^ NaOH. ~ ^ — — p,-— / / I ^ /vc] r iti "f / - 5c 1UUJ_ i> 0 1 2 3 V 5 6 ccm n NaOH 7 a 9 10 H 1Z Fig. 13. Titration curves of 5 cc. and 10 cc. portions of A'^ HCl. to 6.0) bears to this curve, we shall see at once that this point falls within the rapidly ascending portion of the curve. Thus p-nitro- phenol gives us the correct end point of the titration. But it is also to be observed that the use of this particular indicator did not in any way mark the point pH 5.0 to 6.0. The same result could be obtained with an indicator whose transformation point is at pH 7.0 (litmus) or pH 8.0 (phenolphthalein). The amount of the alkali which in this titration changes the pH of the solution from 5.0 to 9.0 is less than one drop. From this we arrive at the rule: To titrate a strong acid with a strong base one may choose an indicator whose transformation point lies between pH 5.0 and pH 9.0. 100 HYDROGEN ION CONCENTRATION Second case: Titration of a weak acid with a strong base. Ten cubic centimeters of N acetic acid are diluted with water to 100 cc. and titrated with iV NaOH. Again we construct a curve which repre- sents the change in pH with increasing quantities of added base. To calculate the pH values in this case the following principles are applied : 1. The pH of the pure acetic acid solution is calculated from equa- tion (la) page 40. [H+] = Vk X [A] 2. On the addition of NaOH we have in solution a mixture of free 'acetic acid + sodium acetate. The pH of these mixtures is derived from equation (1), page 44. _ [free acetic acid] [sodium acetate] 3. As soon as an equivalent quantity of the base had been added we have a 0.1 A'' solution of pure sodium acetate. At this state the pH value is calculated from the hydrolj^sis formulae (see table, page 90). 4. When an excess of base had been added we deal with a dilute NaOH solution; assuming total dissociation the pOH and hence pH may be obtained. In this manner the middle curve in figure 12 is obtained. The end point in this titration is also characterized by a sudden rise in the pH curve. But here also the transformation point of the nitrophenol (pH 5.0 to 6.0) bears no relation to this sudden rise of the curve. This rise covers the range of pH 7.0 to 11.0. Neither nitro- phenol (pH range 5.0 to 6.0) nor litmus (pH 7.0) but phenolphthalein alone (pH 8.0 to 8.5) can be used as an indicator for reaching the true end-point of this titration. The dissociation constant of phe- nolphthalein is pk = 9.7 and hence its transformation point is at about pH 8.0 to 8.5. Among the weak acids acetic acid (k = 2 X 10~^) is a fairly strong one. If instead of acetic acid a still weaker acid, such as boric acid or phenol (pk = 10), then an entirely different curve is obtained (upper curve in figure 12). In this case the pH values are calculated just as in the case of acetic acid from the equations stated above. DETERMINATION OF ACIDITY AND ALKALINITY 101 No definite sudden rise is observed in this case. The "equivalence point" Hes at about pH 11.5. If a very weak acid of the general type of boric acid is to be titrated it can only be done in the following way: A suitable indicator, such as ?/i-nitro-benzene-azosalicylic acid (Alizarine Yellow GG), is added to a pure solution of primary sodium borate, and an equal amount of the same indicator is added to the boric acid solution; the latter is then titrated until the same color as that in the sodium borate solution is reached. Since the rise on the pH curve through the point of equivalence is very small, the color change of the indicator is very gradual, and the titration could barely be carried out without the aid of a standard for comparison. 13 12 11 10 9 ^ / 8 7 6 5 J ^^ ^^ 7^ -^ V ^ -^ •> / . . — —■ — ' 7 Q 1 Z 3 US 6 7 a 9 10 n 12 13 n 15 16 17 18 19 20 Z1 22 ccm n NaOH Fig. 14. Titration curve of a mixture consisting of 10 cc. A'' HCl + 10 cc. A'' acetic acid. The limits of error of such titrations vary with the conditions. A theory of titration error has been advanced by Bjerrum.' This theory may be roughly summarized by stating that the steeper the rise of the pH curve through the "point of equivalence" the smaller the error. In connection with the above it is interesting to observe the behav- ior on titration of a mixture consisting of a strong and of a weak acid. Let us titrate a mixture containing 10 cc. of A^" HCl and 10 cc. of N acetic acid in a volume of 100 cc. with N NaOH. The pH curve of this titration will follow the course shown in figure 14. The calculation of the pH values is carried out as follows : 1. As long as free HCl is present it suppresses almost completely ' See footnote 7. 102 HYDROGEN ION CONCENTRATION the dissociation of the acetic acid, and the pH values are calculated from the free HCl alone. 2. As soon as the HCl is neutralized the calculation is the same as in the preceding case for CH3COOH + NaOH. The curve shows a marked upward bend which corresponds to the disappearance of free HCl and which occurs at about pH 3.0. An indicator which marks off this pH value by a sharp color change is suitable for the determination of HCl in the mixture, a-dinitro- phenol, for example, is such an indicator, showing its first color change at about pH 3.0. If the titration is then continued with phenol- phthalein, the acetic acid content may also be determined in addition. There is no significant difference between a one-color and a two- color indicator. In the case of the latter any observed mixed color consists of the optical mixture of the two individual different colored components. Thus, for intance, when a strongly acid red and a strongly alkaline blue litmus solution are placed one in front of the other so that one looks through both at the source of light, an optical impression is obtained of the same violet tint as would be produced by a litmus solution of an appropriate hydrion concentration. This topic will be more fully discussed in the part on methodology. 28. The determination of the hydrion concentration, or the esti- mation of the actual acidity The determination of the hydrion concentration cannot be achieved by means of any titration method. The standard method used for this purpose is the measurement of the electromotive force developed in a hydrogen concentration chain. Schematically drawn this chain consists of two electrodes or half-elements, each being a piece of platinum saturated with hydrogen, and each of these electrodes is immersed into one of the two parts of a liquid system.^ The two parts of the liquid system are a solution of known [H+] and the unknown solution. The way in which the liquid junction between the solutions is arranged will be discussed in the part on methods. Let us at first assume that the known solution is 1 normal in respect to H-ions, [H+] = 1, or pH = 0. The electromotive force (E. M.F.) ^ An excellent schematic illustration of the gas chain will be found in W. M. Clark, The Determination of Hydrogen Ions, 2nd edition, Baltimore, 1922, Fig. 13, p. 145. DETERMINATION OF ACIDITY AND ALKALINITY 103 of such a concentration chain is according to Nernst's theory (the details of which will be discussed later) E.M.F. = RT In - C2 in which formula Ci and C2 are the concentrations of the current-pro- ducing ions in the two solutions. All other ionic species but those that can be yielded by the electrodes (i.e., in this case all ions other than H-ions) are of no significance. R is the gas constant, T the absolute temperature, and In the natural logarithm. If the E.M.F. is to be given in volts and the logarithm converted from the natural into the common system then the value of R becomes 0.000,198,3. Ci being = 1 and Ca the desired [H+] in this case, we have for any temperature T in our concentration chain the relationship: E. M. F. = -0.000,198,3 (t + 273) log [H+] or for 18°C. E. M. F. = -0.0577 X log [H+] hence , , ^, E.M.F. log H+ = ^ 0.0577 and E.M.F. pH = 0.0577 Thus the measurement of the E.M.F. gives us the pH after mere division by a constant (for a given temperature) factor. The details of this method will be given in the part on experimental technic. Another method for the determination of the hydrion concentra- tion is the so-called indicator method. The use of the indicators in this case is quite different from that in the case of titration. The indicator is simply added to the solution in question and its pH is determined by the shade of color produced. This can be done in two ways. One method which was carefully worked out by S0rensen^ consists of matching the shade of the indicator color in the unknown against a known buffer solution showing the same shade of indicator color. The buffer solution is prepared by mLxing certain known ' S. P. L. S0renseii, Biochem. Zeitschr. 21, 131 (1909). 104 HYDROGEN ION CONCENTRATION stock solutions which can be easily reproduced, and the pH of such a mixture is once for all established by standardization in a concentra- tion (gas) chain. The other method which was developed by the author'' depends upon the colorimetric estimation of the degree of color of one-color indicators. This is achieved by determining the relative amounts of indicator required to impart to the unknown solution and to an alkali solution the same shade of color. Since the shade of color depends upon the degree of color transformation and dissociation constant of the indicator (cf., page 97), the pH of the solution is obtained directly. No buffer solutions are required for this method. In its principle the method is also applicable to two- color indicators, and in this form it was developed by Bjerrum'' and shortly before the author's work by Gillespie,^ And yet the procedure with one-color indicators is simpler, and the necessary one-color indicators covering the entire pH range were first given by the author. Both indicator methods depend upon the gas chain method; the buffer method insofar as the buffer solutions must be standardized by the gas chain, and the method without buffers insofar as the determination of the dissociation constants of the indicators is best carried out by means of the gas chain method, although this can also be accomplished in other ways, such as by con- ductivity measurements. The details of the indicator methods will be fully described in the experimental part of this work. A third, and historically the oldest, method is based upon the prin- ciple that the catalytic effect of a solution upon the velocity of cer- tain chemical reactions depends upon its [H+], as in the case of the hydrolysis of cane sugar, esters etc., or upon its [0H~] as in the case of the saponification of esters. This method will also be dwelt upon in greater detail in the part on methods. 29. The resistance of solutions to change in reaction and buffer value^ The values for equivalent acid content, neutralizing capacity and the hydrion concentration do not exhaust the definition of the acidic or « L. Michaelis und A. Gyemant, Biochem. Zeitschr. 109, 166 (1920).— L. Michaelis und R. Kruger, Biochem. Zeitschr. 119, 307 (1921). '' N. Bjerrum, Die acidimetrische und alkalimetrische Titration. Stutt- gart. « L. J. Gillespie, Journ. of the Amer. Chem. Soc. 42, 742 (1920). 9 Cf. Koppel und K. Spire, Biochem. Zeitschr. 65, 409 (19141 DETERMINATION OF ACIDITY AND ALKALINITY 105 basic nature of a solution. There is still another value which deter- mines this nature and which can be best illustrated by the following example. Let us take a liter of normal acetic acid. According to data given on p. 40 its hydrion concentration is [H+] = 3.4 X 10~^. Let us also take a Hter of 3.4 X 10~^ N HCl. The two solutions have the identical [H+] and yet very different equivalent acid content. But they also differ in another respect. If we add to each one cubic centimeter of a 3.4 iV NaOH solution the resulting relative changes in the [H+] of the two solutions will be widely different. The HCl solution has become neutral in reaction, while the acetic acid has remained still acid in reaction. The acetic acid solution, in spite of having an identical [H+], presents a greater reserve of acid; it contains, as it were, latent or potential H-ions. On the other hand the HCl solution is less resistant to the addition of alkali in respect to the change of its [H+]. This resistance cannot be directly expressed in terms of the equivalent content of a solution. This can be seen from the following example. A solution of acetic acid in pure water on one hand and in a normal sodium acetate solution on the other hand have the same equivalent content of free acid. But if the same amount of NaOH solution be added to each, the [H+] of the pure acid solution will change to a much greater extent than in the acid- acetate mixture. The pure acid solution yields more readily, the acid salt mixture changes its [H+] relatively much less. This resist- ance to change in reaction is a property of the greatest significance in physiology. The acids arising in the metabolic processes (carbonic acid, lactic acid) change the reaction of the tissue fluids. For the proper functioning of the organism it is necessary that the tissue fluids have a great enough buffering effect to prevent the increase in hydrogen ion concentration to a deleterious extent. On the other hand, this resistance must permit a great enough increase in the [H+] to set in motion the automatic responses of the organism for the removal of these acids. This concept of resistance requires there- fore a more precise definition which can be arrived at in the following manner : If to a liter of the acid solution increasing amounts of A^ NaOH solution be gradually added, then for at least a very small period of this addition the change in pH will be proportional to the amount of alkali added. This applies at least to infinitely small amounts of added alkali. As the addition of alkali progresses, this proportionality 106 HYDROGEN ION CONCENTRATION disappears. Let us now designate a very small amount of added dL alkali by dL^ then t-^ becomes the direct measure of the "buffer value" of the solution. Let us now calculate this value for certain cases. We can do this graphically by making use of the titration curves shown in figs. 12 and 13 page 99). Since in these curves pH is represented by the ordinates and L by the abscissae, it follows that dL T~^ at each point is equal to the cotangent of the angle of inclina- tion of the curve. As it will be readily seen the buffer value is greater for every given amount of alkali with any acid the greater the equivalent concentration of free acid. If this concentration should approach zero then the buffer value also approaches zero. This will mean then that in very dilute acid solutions small amounts of alkali will effect very large changes in the pH. But not only the concen- tration of free acid, but also the nature of the acid itself has an effect on the buffer value. In comparing the curve for HCl and for a weak acid, such as acetic, we observe only in the curve of the latter a flat or wide point of inflection indicating the formation of the maximum of the buffer value, while the minimum is shown by a steep point of inflection. The flat point of inflection or the maximum occurs at half of the equivalent neutralization, i.e., in the presence of equal concentrations of acetic acid and of sodium acetate. • This may be demonstrated analytically in the following manner: Let the total amount of acetic acid in the solution be designated by S, the amount of added alkah by L, then S — L = the amount of free acetic acid and L = the amount of sodium acetate. Now in any mixture of acetic acid and sodium acetate we have TTT-n 1 ff^ee acid] , S — L [H+J = k -— = k INa acetate] L <-(!-) therefore, pH = — log k — I 7 — 1 J and the buffer value v d L d L L (S - L) V = ■ = — ; r- = — • • (I) d pH ^ , /S \ S log e ^ ^ d lc~ log (I - 1) DETERMINATION OF ACIDITY AND ALKALINITY 107 The maximum of the buffer value is then established when dp (S - 2 L) = 0 dL S • log e or, L.f (II) Therefore, it occurs in a mixture of equal concentrations of acetic acid and sodium acetate. The numerical relation of the buffer value j/ in a mixture of a weak acid and its salt may be obtained from equation (I) above: 1 [bound acid] ... »» X log e = - = , ^ , — r X free acid (III) rs [total acidj If for the sake of simplicity we neglect the factor log e in the above calculation permitting a shght alteration in the definition, and assume V X log e to be equal to the buffer value where v has its former value dL , . . of "TTtj ^tien equation III gives us a very simple expression for the buffer value. This holds only for a mixture of an acid and its salt, and that only within the limits within which the approximation formula [free acid] [H+] = k salt] is applicable. This range may be approximately estimated between the values pH = pk + 2 and pH = pk — 2. The reciprocal value tj may be designated as the "yielding power. "^'^ The dissociation constant of the acid does not enter into the expres- sion III. Hence it follows that : The "yielding power" and with it the buffering effect of an acid-salt mixture are independent of the nature of the acid. Thus the relatively strong acetic acid and the very weak boric acid when appropriately mixed with their respective sodium salts have the same buffering effect, except that the ma.ximum of their buffer effects is at different pH values. This maximum may be evaluated in the following way: 1" "Nachgiebegkeit" — frequently used by the author in the sense of the reciprocal of buffer value. — Translator. 108 HYDROGEN ION CONCENTRATION From equation II we see that the maximum obtains in mixtures consisting of equivalent amounts of the acid and of the salt; but in such a mixture [H+] = k, the dissociation constant of the acid. The strength of the acid therefore influences only the pH range in which the mixture of the acid and its salt acts as a good buffer. It should now be clear as to just what constitutes a poorly or a well buffered solution. The magnitude of the above defined buffer value is the sole measure for it. Thus distilled water and NaCl solutions are poorly buffered; river and sea waters are still relatively poorly buffered; much better in this respect is blood, and still better are the special buffer solutions prepared in the laboratory. The first coherent and exhaustive exposition of these relations was given by Koppel and Spiro (1. c.) Their derivation is terminologi- cally slightly different from the one given here, insofar as we have attempted to adhere closely to the derivations contained in the earlier chapters. Koppel and Spiro take also for their point of departure the conception that given two solutions of the same pH, of which one is only a strong acid and the other a buffer mixture, the first will be more susceptible to the addition of base, as far as change in pH is concerned, than the second. They designated the action of the buffer as a "moderating" one, and the buffers themselves they named "moderators." By the "yielding power" of a solution we have hitherto understood its behavior toward the addition of small amounts of either strong acid or base. But practically even of greater interest is the yielding power of a buffer solution to the addition of a small amount of the buffering acid itself. The commonest physiologically encountered buffer is the system CO2 + NaHCOs. There is a constant formation in the metabolic processes of CO2, and the obvious question arises — what is the special buffer effect of such a carbonate buffer towards the addition of free CO2? In a mixture of a free weak acid in concentra- tion A and of its alkali salt in concentration S we have the relation therefore, pH = — log k — log A + log S DETERMINATION OF ACIDITY AND ALKALINITY 109 Upon the addition of an amount dA of CO2 we have dpH 1 — — = - (log e) - dA A By making use of the definitions given above, we may designate the ratio of the decrease of CO2 to the increase in pH as the "special buffer effect" of the mixture, v', which becomes simply v' = + -r-, or, in terms of the reciprocal value, the "special buffer value," N', N' = A ■ The "special buffer effect" of a buffer mixture consisting of CO2 + NaHCOz is therefore independent of the concentration of the bicar- bonate and is inversely proportional to the concentration of the carbon dioxide. This remarkable law expresses a mode of defense of the living organism against changes in pH which might be caused by the increase of CO2. Augmentation of CO2 in the blood sets in motion the automatic processes which lead to the elimination of the CO2 present in excess. But should these automatic processes temporarily fail because of an excessive strain upon them, then the accumulation of CO2 in the blood results in a progressively decreasing change in pH. It will be shown later that in acidosis there is a smaller concentration of free CO2 in the blood than normally. The "special buffer effect" of such blood is considerably below the normal value, or in other words, the acidotic blood is more poorly buffered than normal blood. The same amount of CO2 arising in the metabolism changes the pH of the acidotic blood more than it would the pH of normal blood. The irritation produced by a given amount of CO2 in all regulatory centers controlled by the pH of the blood, is greater in acidosis than in the normal state. ^^ " Simultaneously with the publication of the second German edition of this book there appeared an article by Van Slyke (D. D. Van Slyke, Journ. Biol. Chem., 52, 525, (1922)) in which the theory of buffers was treated on the same basis but much more thoroughly. For further study of this problem this article is especially recommended. CHAPTER III The Dissociation of Strong Electrolytes summary of contents The strong electrolytes appear to deviate from the law of mass action as regards their dissociation equilibria. But this appears only because of an erroenous interpretation formerly given to conductivity measure- ments. The way in which the state of dissociation of strong electrolytes was calculated from these measurements can no longer withstand the criticism of our day. The clarification and correction of these earlier inaccuracies yielded the theory of activity. This theory is presented in a simple form in accord with the points of view developed in the litera- ture, and which is also quite sufficient for the working purposes of the physiologist. The strong electrolytes are practically always completely dissociated, and, furthermore, if weak electrolytes be present along with them in solution, they appear to raise the dissociation constants of these weak electrolytes. In physiological processes it is not the absolute dissociation constants of weak electrolytes (such as CO2) but the constants reduced to the physiological salt co7itent that are of significance. 30. The deviations from the law of mass action and their significance Hitherto all of our considerations were based on the assumption that the "strong" electrolytes are always completely dissociated in solution and that they have no other influence on the ionic equilib- rium apart from that defined by the mass law. These assumptions are particularly valid for very dilute solutions of the strong elec- trolytes. It is now our purpose to introduce certain corrections requisite for the higher concentrations. The great interest of this problem lies not only in the insignificant corrections of our earlier calculations, but also in the theoretic elucidation of the character of these deviations. These enigmatic deviations from the otherwise well accepted laws of physical chemistry are being gradually cleared 110 DISSOCIATION OF STRONG ELECTROLYTES 111 up by careful investigations.' Our presentation of this subject is but a preliminary attempt, and there is much in it that still awaits ultimate solution. The perusal of this section of our book then cannot yet yield the degree of satisfaction to be obtained from a well established chapter of a natural science. The peculiarity of the strong electrolytes first became evident in those apparent deviations from the mass law which were found in their dissociation. For the purpose of introducing our problem let us consider the determination of the dissociation constant of KCl, which by the mass law is defined as [KCl] k = [K+] [C1-] The method of estimating this constant k appears to be simple at first sight. All one needs is to determine the degree of dissociation in KCl solutions of various concentrations. If this is designated as a, then k = , X c 1 — a where c is the total concentration KCl, this being Ostwald's dilution law which has rendered such excellent service in the estimation of dissociation constants of the weak electrolytes. Of the different ways for the determination of the degree of dissociation the con- ductivity method was alwaj^s considered the best. This method is based on the following assumptions: (1) The conductivity of a solu- tion of an electrolyte is a purely additive function of the conductivi- ties of its component ions. (2) Every individual ionic species possesses within wide limits a constant molar conductivity. The latter is proportional to the motility of the ionic species (assumed to be constant), i.e., the rate of migration imparted to an ion by the action of a homogeneous electric field of unit strength. From these assumptions we derive the following expression which serves as a measure of the degree of dissociation, a, A a = — 00 * For the earlier accounts see K. Drucker, Anomalie der starken Electrolyte. Stuttgart, 1905. 112 HYDROGEN ION CONCENTRATION A is here the molar conductivity of the electrolyte at the concentra- tion c (i.e., the specific conductivity at concentration c, divided by c) and A 00 is the limiting value of this molar conductivity for infinite dilution. For such infinite dilution the dissociation may be assumed to be complete, or a = 1. If this rule be applied to a weak electrolyte a value of k is obtained which is independent of the concentration, as has been shown a very great number of times, and as it is illustrated for acetic acid, as an example, in table 18. TABLE 18 1 c A 100 X a observed calcd. for pK = 4.75 0.994 1.27 0.402 0.42 2.02 1.94 0.614 0.60 15.9 5.26 1.66 1.67 18.1 5.63 1.78 1.78 1,500. 46.6 14.7 15.0 3,010. 64.8 20.5 20.2 7,480. 95.1 30.1 30.5 15,000. 129. 40.8 40.1 TABLE 19 Molar con- centration KCl HCl * KOHt Na-acetatet c a k a k a k a k 0.001 0.973 0.043 (0.993) (0.14) (0.981) (0.06) (0.958) 0.022 0.01 0.941 0.16 0.972 0.34 0.917 0.10 0.894 0.077 0.1 0.861 0.56 0.920 1.06 0.894 0.75 0.778 0.273 1.0 0.755 2.3 0.791 3.0 0.771 2.6 0.525 0.58 3.0 0.679 4.3 0.565 7.2 0.590 2.6 0.278 0.32 * Aoo t Aoo J Am 380.5. 238.6. 78.5. Should we now attempt to apply the same method to KCl we should find that the value for k obtained will not be constant, but will vary in a definite direction. It will definitely increase with the concentrations of the salt. The same is true for all strong electro- lytes such as HCl, HBr, HI, NaOH, KOH, as well as for all salts, and even for such salts as ammonium acetate. Definite indications of this relationship are already evinced by the strongest of the weaker electrolytes, such as picric acid, tartaric acid, etc. Table 19 illus- DISSOCIATION OF STRONG ELECTROLYTES 113 trates this point for KCl, HCl, KOH and Na-acetate, calculated from the conductivity measurements of Kohlrausch and his coworkers. It has been attempted purely empirically to define the value of k in different ways, and we have an empirical expression given by van't Hoff- as well as one by Rudolphi.^ Van't Hoff's formula is Of' X c , = k = k (1 - aV and Rudolph i's (1 -a) Both are interpolation formulae that are practically usable. Arrhenius* had already attempted an explanation of these findings. He assumed that because of the contraction of the volume of water ("electrostriction") accompanying the process of solution of salts, alterations occurred in the dielectric constant, the osmotic pressure and the motility of the ions. As a consequence of this the dielectric constant and hence the dissociating capacity of the solvent upon the electrolyte are increased with the increasing concentration of the electrolyte which is being dissolved. In order to demonstrate this idea Arrhenius showed that the dissociation constant of a weak electrolyte (acetic acid) increases when a strong electrolyte (NaCl) is added to its solution. Thus the inverting capacity of acetic acid for <;ane sugar was also materially increased by the addition of salt. After allowing for that portion of the effect which he ascribed to the purely catalytic effect of the salt, he observed a still further accelera- tion of the catalysis by the salt, which he ascribed to the increased dissociation of the acetic acid effected by the presence of the salt. But even without such elaborate presentations, the foresight of theoretical chemists has long recognized that this contradiction to the mass law is only an apparent one. As Nernst puts it in his textbook: "it is highly probable, although on still unknown grounds, that neither the electric conductivity nor the osmotic pressure (lowering of the freezing point) give us an accurate scale for measuring the degree of dissociation." 2 van't Hoff, Zeitschr. f. physikal. Chem. 18, 300 (1895). » C. Rudolphi, Zeitschr. f. physikal. Chem. 17, 385 (1895). * Sv. Arrhenius, Zeitschr. f. physikal. Chem. 31, 198 (1899). 114 HYDROGEN ION CONCENTRATION A second method of estimating the degree of dissociation is by means of determining the lowering of the freezing point. From the latter one may calculate the molar concentration of the solution, and this may be related to that molar concentration which is to be expected by assuming a total lack of dissociation. If the latter concentration is c and the former c X i (i being "van't Hoff's coeffi- cient"), then with binary electrolytes i — 1 = a. It is conspicuous that values for a. obtained by this method 'do not agree exactly with those obtained by the conductivity method. But there is still another method for the estimation of a, namely, the concentration chain method. This method is appKcable to a large series of electrolytes. By means of concentration chains one can measure the concentration of H+, 0H~, Cl~ and some other ions and hence determine the degree of dissociation of many acids, such as HCl, HNO3 and of salts, such as KCl, NaCl, etc. Accurate investi- gations by this method have been made but recently^ when the technic has been considerably improved. Bjerrum^'* called attention to the fact that the values of a jdelded by the three methods did not agree. Table 20 taken from Bjerrum shows some of the evidence. The apparent degree of dissociation obtained from the freezing point method, fo, is designated as the osmotic factor; analogously f/x is the conductivity factor; and that determined by the concentration chain as the ''activity factor," fa. As is seen from table 20, at the lowest concentrations the deviations do not appreciably exceed the limits of experimental error, but at the higher concentrations they do exceed it beyond all doubt. Since these different methods give series of three different results, it is justifiable to entertain doubts as to the reUabihty of all the three, or to doubt whether these three methods of measurement yield actually degree of dissociation figures. The key to the solution of this enigma will be found in the following considerations which are based on the investigations of Bjerrum,^^ Milner^ and of Ghosh.^ The usual formula for the law of mass action is generally » It must be mentioned that O. Jahn (Zeitschr. f. physikal. Chem. 33, 545 and 35, 1 (1900)) had long since observed a lack of agreement between degree of dissociation values of strong electrolytes calculated from conductivity and concentration chain measurements. He also recognized that the con- ductivity data were misleading, while he attached full value to his calcula- tions from the concentration chain data. 6a N. Bjerrum Zts. Elektrochem. 24, 321, 1918. DISSOCIATION OF STRONG ELECTROLYTES 115 stated in terms of the concentrations of the reacting molecular or ionic species. This is justifiable only insofar as the concentration is proportional to the active mass. In reactions in gaseous systems or in reactions between electroneutral molecules in solution the concept of the active mass may be quite accurately defined. In fact, it is the value which is proportional to the osmotic partial pressure of the molecular species involved. Actually the law of mass action, as it is thermodynamically conceived, makes no mention whatever of any concentration relationships but refers to pressures.^ Since under conditions of ideal gases and of extremely dilute solutions the osmotic pressure of a molecular species is proportional to its concentration, therefore it is permissible, for electroneutral molecules in very dilute solutions, to substitute concentration relations for those of osmotic pressui-e. For higher concentrations, however, this does not hold TABLE 20 Apparent degree of dissociation of KCl Molar concentration fo f^ fa 0.001 0.01 0.1 1.0 0.985 0.969 0.932 0.854 0.979 0.941 0.861 0.755 0.943 0.882 0.762 0.558 true any longer. Only with nonelectric molecular species (e.g., sucrose) concentrations up to 1 M may be considered in this respect, with very httle error, as solutions of infinite dilution. In more con- centrated solutions the osmotic pressure is affected by the forces of attraction operating among the molecules, and because of these the pressure is no longer proportional to the concentration but increases at a lesser rate than the concentration. It has been for a while supposed that with ionized substances the "state of infinite dilution" was to be assumed up to the same concentrations as with electro- neutral molecules. The impossibihty of determining experimentally the concentration of an individual ionic species strengthened this tacit assumption. But this is just where the error is involved. If in the case of electroneutral molecules the effect of the forces of attraction becomes evident in 1 M concentrations, then in the case of ^ At one time Arrhenius spoke of it on this account as of the "pressure action law" in distinction from the "mass action law." 116 HYDROGEN ION CONCENTRATION ions with their free charges this should happen at much greater dilution. Furthermore, it cannot possibly be true that the con- ductivity of a concentrated electrolyte solution be simply an additive function composed of the sum of the conductivities of its ions, as it is in the case of very dilute solutions, for it is impossible to conceive that ions in a concentrated state could pass in the proximity of each other without producing a variety of force effects. Hence it follows that conductivity measurements made in concentrated solutions of ions cannot give us an accurate measure of the degree of dissociation; in the case of strong electrolytes at least they must simulate too low a degree of dissociation. Since this apparent degree of dissociation value is always large, i.e., very little less than 1, the above mentioned authors proposed the hypothesis to the effect that the so-called strong electrolytes are always completely dissociated. This hypothesis cannot as yet be proven as being absolutely exact. It is possibly analogous to such an acknowledged theory as, for example, the one asserting that in the solid crystallized state KCl does not exist in undissociated molecules but only as K+ and Cl~ ions definitely spatially arranged. But if instead of "totally" we should sa3^ "almost totally" or "practically totally" dissociated, then we would at all events approach the true state of the matter much closer than did the older assumptions. We shall assume then with Bjerrum that in a 1 A?" KCl solution, for example, the molecular species KCl is practically non-existent, and that each of the K+ and Cl~ ions are present in a 1 M concentration. But the active mass of the K+ or Cl~ ions in a 1 iY KCl solution is to be taken, however less than 1000 times that in a 0.001 N solution. Bjerrum was led to the establishment of his hypothesis of the total dissociation of strong electrolytes by his observation that the absorp- tion of light by solutions of colored strong electrolytes was inde- pendent of concentration and of other factors, provided there is no formation of complex ions. The chromium salts afforded convenient experimental material, because in solutions of these the formation of complex salts occurs very slowly, and it is possible for a time to work with complex-free solutions, which is not so easy with such other metal salts as those of iron. It was found that the light absorption of a gram-molecule of any chromium salt was quite independent of the concentration of its solution and of the acid anion with which it was combined. Since in all weak colored electrolytes (indicators) changes DISSOCIATION OF STRONG ELECTROLYTES 117 in the state of ionization are accompanied by changes in color, BjeiTum concluded that in the case of the chromium salts the state of dissociation did not change with dilution, in contradiction to the results apparently indicated by the molecular lowering of the freezing point. Of the possibilities of regarding chromium salts either as always undissociated or as always completely dissociated he chose the latter. It is the more probable assumption, because on the basis of our present day knowledge of the electric conductivity of salts, the former possibility is not acceptable, and because it agrees much better with the results deduced from the lowering of the freez- ing point methods. This hypothesis aids us to orient ourselves, at least in a preliminary way, concerning the concentration of the ions in solutions of strong electrolytes. The problem still before us is to obtain quantitative relations also of conductivity, osmotic pressure and the active mass. 31. The influence of ionic charge on the conductivity As long ago as 1912 Hertz^ investigated the effect of the interionic electric forces upon the conductivity of the ions, and his results indicated that with increasing concentration the conductivity of the ions must decrease, even when no change in the degree of dissocia- tion takes place. Since quantitative values which are not accurately known entered into his calculations, such as the mean free path of the ions, it must remain for future investigations to test his calculations. But in any case, quahtatively. Hertz's assertion is incontrovertible that in a mixture of positive and negative ions present in higher con- centrations the total conductivity is not simply the algebraic sum of the conductivities of the separate ionic species, as it is in the case of the most dilute solutions. Consequently it is doubtlessly also incorrect to interpret the A:A^ ratio as the "degree of dissociation." I. Chandra Ghosh^ has recently proposed a promising way of calculating the conductivity of strong electrolytes. His point of departure was to the effect that the interionic potential occurring in electrolyte solutions does not disappear (see following section). Let A denote the "autopotential" per mol of the electrostatic re- ■> J. Hertz, Ann. d. Physik (4) 37, 1 (1912). 8 I. Chandra Ghosh, Transact. Chem. Soc. London, 113, 449, 627, 707, 790 (1918); Zeitschr. f. physikal. Chem. 28, 211 (1921); see also: H. Kallmann, Zeitschr. f. physikal. Chem. 98, 433 (1921). 118 HYDROGEN ION CONCENTRATION ciprocal effect in an ionized solution of a completely dissociated electrolyte, whose molecule jdelds n ions. Then the electric con- ductance of this solutionis such as if only the fraction n X e"'^''"^^ of the mol were freely motile, while the remainder of the mol did not participate in the conduction of the current. This latter part of the mol is the one which, according to the earlier conceptions, was con- sidered as the undissociated molecule. But according to Ghosh the electrolyte is completely ionized, just as according to the modern theory of structure of crystals, a salt in its soHd cr>^stalUzed state is totally ionized. Only at this juncture a deviation from the ordinary conception of an ion slips in, as it were. From the very beginning of the theory of electrolytic dissociation the term ion stood for a charged molecule (or "half-molecule") which moves under the in- fluence of the electric current, but which was not originally split off from the uncharged molecule by the action of the current. Thus two properties had to be interrelated in order to characterize the ion: the free charge and the free motion of the particles. After it had been recognized that even in the crystal only these charged "half-molecules" and no salt-molecules proper were present, these were likewise called ions, although they lack the second charac- teristic, the free motility. Therefore, in accord with the definition of an ion, we shall either say: a dissolved electrolyte is always completely dissociated into ions, but only a fraction of it is freely motile; or: a dissolved electrolyte is always only partially dissociated into ions. It does appear, however, that the first mentioned, the newer conception of Ghosh, will become the dominant one in the future. It is even very possible that this conception is valid not only for the strong electrolytes, but for all electrolytes in general. If this is true, then a weak acid becomes under all conditions what we designated earher as a "pseudo-acid" (see page 37), which exists in two tautomeric forms, of which one is not dissociated at all and the other completely dissociated. The value a = — , when it is taken not as denoting the degree of Aco dissociation but rather in the sense of Ghosh as expressing the effect of the autopotential A, is calculated as e~^^"^^, or we can state it as A = -nRT In a DISSOCIATION OF STRONG ELECTROLYTES 119 If the electrol}i:e solution be diluted, then the mean ionic distance is altered and consequently also the autopotential changed. Therefore, at infinite dilution an amount of electric work is performed to the extent of —> where e is the electric elementary quantum (the charge on one ion), D the dielectric constant and r is the original mean ionic distance. Furthermore r = \ ^z^ (where No is the number of y 2JNo molecules per liter, and V the volume of the solution), hence ^ No e2 Ay2l7o A = and therefore, D \/v No e2 \/^ D VV ° = 2 RT In a. From the last expression the value of a may be calculated, and then one finds good agreement between observed and calculated values of a. for a large number of strong electrolytes, provided it is assumed at the same time that their dissociation is complete at each concentration used. 32. The effect of the ionic charge upon the lowering of the freezing point In two excellent papers Milner^ shows that in a mixture containing equal numbers of positive and negative ions the electrostatic forces exercise a characteristic effect upon the average spatial distribution of the ions. Milner applied a theorem proposed by Boltzman: when a large number of molecules is distributed in a defined space, then any two neighboring molecules in a given period of time are separated by varying distances for which a quite definite statistical mean can be determined. When these molecules are conceived to be ions, and, furthermore, when the numbers of positive and negative ions are equal, then we find that the distance between any two neighboring ions of the same charge, which always repel each other, 9 S. R. Milner, The Virial of a Mixture of Ions. Phil. Mag. 23, II, 551 (1912) ; The Effect of Interionic Forces on the Osmotic Pressure of Electrolytes. Ibid. 25, II, 742 (1913). 120 HYDKOGEN ION CONCENTRATION Oj9S is on the average greater than the mean distance between two oppositely charged ions which attract each other. Between two univalent ions there is a potential difference which may be expressed e- as — , where e is the charge on the ion and r is the distance. The potential between two ions of the same charge is reverse in sign to that existing between two ions oppositely charged, which is due to repulsion in the first and to attraction in the latter case. If the mean distance between any two neighboring identically charged ions were equal to that between two oppositely charged ions, then the algebraic sum of the individual potential differences would be equal to zero. But since these distances are not equal, therefore the total poten- tial is not = 0, and as a result there is an average potential dif- ference, in the sense that the attraction exceeds the repulsion. In such a solution, therefore, there is stored a certain amount of potential difference. This de- duction is used by Milner in his second paper to calculate the effect of the ionic charge upon the lowering of the freezing point of a solution. As a result of very complicated calculations he ob- tained the curves shown in figure 15. This figure shows in the dotted curve how the molar lowering of the freezing point should change with the concentration of an elec- trolyte consisting of two univalent ions, assuming the usual mass action law; while the solid line curve represents the results of Milner 's calculations. The crosses indicate the values for KCl found by freezing point determinations. With the exception of the lowest con- centrations for which the experimental technic is uncertain and the highest concentrations for which the accuracy of the calculations becomes uncertain, the observed points agree quite well with the calculated values. Bjerrum called the mutual effect of the ions in regard to their osmotic pressure "the Milner effect." It is, therefore, unnecessary to assume any measurably incomplete dissociation of KCl in order to explain the course of its lowering of the freezing point. OSO 0.05 Concentration Fig. 15 (Taken from Bjerrum) DISSOCIATION OF STRONG ELECTROLYTES 121 33. The three deviation-coefficients, £„, f^, fa, according to Bjerrum In order to calculate the relative osmotic pressure of an ion from its concentration, c (assuming the osmotic pressure of a solution which approaches in character an ideal gas to be equal to 1) the concen- tration must be multiplied by a factor fo, the osmotic coefficient, which is always less than 1. The relative conductivity is similarly obtained (assuming the conductivity when it is not affected by electrostatic forces to be equal to 1) bj^ multiplying by the factor f^, the con- ductivity coefficient. Also, the active mass is obtained by multiplying the concentration by the factor fa, the activity coefficient. In the preceding sections it has been attempted to attribute a further significance to the coefficients, f^u and fo, and now we come to the activity coefficient, which is the most significant for our purposes. We shall first outline briefly Bjerrum's attempt^'* to derive theoreti- cally fa from fo, and then the procedure for the practical determina- tion of fa. 1, According to Bjerrum's thermodynamic derivation the follow- ing relationship exists between the osmotic factor fo, determined by the freezing point method, and the activity factor fa and also the concentration c: f. + ^°-i + -^ (1) dc dc which is to be taken by all means only as an approximation formula for dilute solutions. From this, therefore, one can calculate fa from fo, at least under certain simple conditions, and in that particular case when the osmotic coefficient is given with sufficient accuracy by the freezing point method. Instead of the above complicated function Bjerrum has also worked out a simpler approximate method for the calculation of the activity coefficient in a solution of a strong electrolyte. Thus for a solution of a strong electrolyte whose molar concentration is c and which is composed of two equivalent ions (i.e., two univalent or two divalent etc. ions, as in the case of KCl, MgS04, but not Na2S04) he obtained the expression 26 3/- - log fa = — n2 Vc (la) 122 HYDROGEN ION CONCENTRATION where D is the dielectric constant of the solvent and n the valence of the ions. Therefore, for aqueous solutions it becomes - log fa = 0.32 • n2 \/c (2) This formula is derived under certain assumptions from the one above (la). But it is to be used only as an approximation for rough estimates. The factor 0.32 is in reality not quite accurately applic- able to the different electrolytes; thus for KCl it appears, according to Bjerrum, that the factor 0.25 and for HCl the factor 0.2 are more exact. Let us now calculate for an example the value of fa for a 0.1 A'^ KCl solution, from the formula — log fa = 0.25 n'-\/c then c being given as 0.1, - log fa = 0.25 V^ = 0.25 • 0.464 = 0.116 log fa = 0.884 - 1 fa = 0.766 This formula is quite serviceable for dilute solutions up to 0.1 A''. Besides, as stated above, the appropriate factor for \/c must be chosen for each kind of electrolyte. As for the practical application of this formula no more will be said for the present. 2. But the most accurate method for the determination of fa is the following : It is well established that in extremely dilute solutions fa = 1, i.e., the activity and concentration become identical. With- out this assumption the concept of activity has no meaning. Now, if we set up a concentration chain with hydrogen electrodes and two HCl solutions of which one is extremely dilute (ci) and the other of any conveniently chosen concentration Co, then according to our above outlined consideration the electromotive force developed will not be E = -— - In — F C2 but E = — - In F c; ' i a For the very dilute solution of concentration Ci we may omit the activity factor, for it is here = 1, Since the values of Ci, C2 and E are either known or measurable, fa may therefore be calculated from the DISSOCIATION OF STRONG ELECTROLYTES 123 iabove expression. All that is necessary is to choose the concentration Ci SO small that the activity factor could be assumed to be = 1. Such determinations can be made for all those ions for which rever- sible electrodes are available, especially H+, Cl~, 0H~; and elec- trotytes such as, for example, HCl, KOH, KCl, NaCl, etc. are appropriate for this purpose. Thus Noyes and Mac Innes^" employing this method found the values given in table 21 for the activity factor. TABLE 21 Molar concentration — logc f^ Noyes und Maclnnes fa for KCl c KCl LiCl HCl KOH Bjerrum 0.001 3.000 0.979 0.976 0.943 0.003 2.523 0.943 0.945 0.990 0.9S2 0.005 2.301 0.924 0.930 0.965 0.975 0.010 2.000 0.890 0.905 0.932 0.961 0.882 0.030 1.523 0.823 0.848 0.880 0.920 0.050 1.301 0.790 0.817 0.855 0.891 0.100 1.000 0.745 0.779 0.823 0.846 0.200 0.699 0.700 0.750 0.796 0.793 0.762 0.300 0.523 0.673 0.738 0.783 0.769 0.500 0.301 0.638 0.731 0.773 0.765 0.700 0.155 0.618 0.734 0.798 0.772 1.0 0.000 0.593 0.752 0.829 0.786 0.558 2.0 -0.301 — 1.040 — 3.0 -0.477 1.164 1.402 — 34. The hydration of ions and its significance for the activity Bjerrum's approximation for the calculation of the activity factor may be applied with sufficient accuracy to electrolyte solutions of concentration up to about 0.2 molar, and, therefore, quite within the limits of concentration most frequently encountered in physiological investigations. But for higher concentrations it fails so completely that, for example, for a 3 A^" solution of HCl the activity factor was experimentally found to be > 1 (see table 21), while theoretically this factor must decrease with increasing concentration. According to Bjerrum the cause for this discrepancy lies in the fact that the activity of the ions is influenced not only by the interionic electrostatic forces, " Journ. Amer. Chem. Soc. 42, 239 (1920). 124 HYDROGEN ION CONCENTRATION but also by the degree of hydration of the ions^^ which changes with the concentration. Only the essential features of this conception will be touched upon at this juncture. There are many grounds for beheving that there exists an affinity between the molecules of a dissolved substance and the solvent, and that the solvent is not merely the space in which the molecules are free to move. This affinity for the solvent becomes even more probable in the case of ions. At least two theories can be presented concerning the nature of this affi.nity. In the first place, it may be assumed that each ion forms a hydrate stoichiometrically with a definite number of water molecules analo- gously to the formation of salts with water of crystallization. It can be conceived that in concentrated solutions with an insufficient excess of water the hydration of ions does not reach its maximum extent. But ions of different degrees of hydration cannot be con- sidered equivalent from the standpoint of the mass law. It can further be conceived that only the few unhydrated ions that are present take part in the electromotive activity or in chemical reac- tions, or stated in other words, that the ions are first dehydrated when thej^ become active. It is also conceivable that a very small number of water-free ions are in a state of equilibrium with the hydrated ions. Then it would follow that the active ionic mass is proportional to the concentration of the water-free ions. As long as the water is present in a large excess the ratio of water-free to hydrated ions will remain constant, and the active masses of the ions will be proportional to their respective concentrations. On the other hand, in concentrated solutions, this ratio is disturbed in the direction of the water-free ions, and the activity of the ions will be apparently too great. Bjerrum had even attempted to calculate the degree of hydration of ions from this deviation of the observed from the calculated activity values. In reality it can be easily observed from the preceding table how at the higher concentrations the factor fa exceeds the value 1, which without these conceptions would be quite inexplicable. The second possible theory would be based on the supposition that there are no hydrated ions formed in a stoichiometric way. Instead " O. Sackur (Zeitschr. f. physiol. Chem. 70, 493 (1910)) had already made use of the theory of hydration in an attempt to explain the above discrepancy in the case of strong electrolytes. DISSOCIATION OF STRONG ELECTROLYTES 125 it may be assumed that there is a molecular attraction between every individual ion and all water molecules, which rapidly decreases with the distance from the water molecule. In this way we arrive at the idea that every ion is surrounded by a layer of closely adhering water molecules, and this in turn is surrounded by other layers of progressively less adhering water molecules. At the same time it is to be assumed that the reactivity of the ions (or their "activity") is diminished by being bound by water, and that this curtailment decreases in the presence of but little water. Attempts to determine the hydration capacity of different ions^^ by various methods based on the first conception outlined above led to contradicting results. Thus Nernst gives the following figures for infinitely dilute solutions (taken from Riesenfeld) : H(0) K(20) Ag(3o) iCd, Cu(55) Na(70) Li (150) OH(IO) ISO4, I, Br, Cl(20) N03(25) ClOsCSS) The figures in parentheses represent the numbers of bound H2O- molecules. On the other hand Bjerrum found: H(8) CI (2) K(0) Thus the uncertainty concerning such figures is very great, and each of the many methods tried yielded different values. But at all events this uncertainty would seem to speak against the idea of stoichiometrically defined hydrates and it should be interpreted as being in favor of the theory of a general attraction existing between the ions and the surrounding water-molecules. ^^ On the other hand, almost all investigators agree on the serial sequence of the degree of hydration of the ions. Thus it is always observed that the hydration of the H-ion is the least of all. With the alkali metal ions the degree of hydration decreases with increasing atomic weight in the manner. Li > K > Na > Rb > Cs while with the halogen ions, reversely, it increases with increasing atomic weight, but not to so great an extent. The obvious interpretation of this phenomenon is the following: The attraction for the water is evidently exercised by the free charge 12 Nernst, Theoret. Chem. 7. Aufl. 1913, P. 413. " K. Fajans, Naturwissenschaften 9, 729 (1921). 126 HYDROGEN ION CONCENTRATION of the ion. In the cations the free charge is located on the atomic nucleus. The greater the atomic radius the greater the distance from the unbound water-molecules and the smaller is the attraction. On the other hand, in the anions the free charge is on the periphery in the form of the valence electron, and its attraction for the water is but slightly influenced by the size of the atomic radius insofar as the greater or smaller distance of the positive nucleus weakens more or less the electric field of the electron. The attraction between the ions and the water had been conceived as an electrostatic one. A water-molecule is to be considered as being electroneutral when viewed remotely, so to speak, but more closely considered, it has electropositive as well as electronegative places, and positive ions are attracted to the negative places and conversely. Therefore, it may be readily imagined that each ion is first surrounded by a spherical layer of water-molecules and then by a second and perhaps by a third laj'er which are progressively less closely attached. The number of water-molecules which are firmly held in the inner layer depends upon spatial possibilities, much in the manner of Werner's coordination number. The hydrogen ion, in spite of its free positive charge, can, because of its small atomic radius, bind less water than any other ion. With these intimations we must end our discussion of the devia- tions in concentrated solutions from Bjerrum's theory, especially in view of the incomplete state of knowledge of the subject and of the limited scope of this book.^^^ We see then that the underlying basis of the activity theory is beset with many difficulties. Nevertheless it leads us to an impor- tant and fundamentally quite well established result. It is only "* The activity theory has been recently greatly advanced and developed in the experimental work of G. N. Lewis and his collaborators, the theoretical studies of N. Bjerrum, and particularly through the fundamental work by Debye and Huckel. These developments have been so wide in scope that they can scarcely be summarized in a supplement. For an extensive study of these the reader is referred to G. N. Lewis and M. Randall, Thermodynamics, N. Y., 1923, wherein the methods of determination of activity coefficients as well as their thermodynamic derivation are particularly well given ; also to the excel- lent summary of Debye's theories in E. Huckel, Zur Theorie der Electrolyte in Ergebn. d. exact, naturwiss., Vol. 3, Berlin, 1924, both of which books pre- suppose a good knowledge of electrophysics and of thermodynamics. DISSOCIATION OF STRONG ELECTROLYTES 127 because of this that the preceding theoretical observations were made. The result is the following. 35. The significance of the activity theory for the pH of buffers In a mixture of a free weak acid and of its sodium salt the relation (page 44) holds: = k ,f5^ (1) [Na — salt] where k is the dissociation constant of the acid, and assuming that: (1) the salt is completely dissociated, and (2) as it must be again emphasized, that the total concentration of electrolytes (i.e., ions) in the solution is sufficiently small, so that concentration and activity may be taken as being equal to each other. As was demonstrated in the last chapters, the first assumption is quite valid for all practical purposes. The basis on which the in- complete dissociation of Na-salts had been previous^ assumed has been shown to be untenable, and at least all Na-salts are much more strongly dissociated than it had been assumed; and even at that, in 0.1 N solutions for example, a 90 per cent dissociation had been assumed for these salts. And now we must provisionally extend it to 100 per cent. But the second assumption is valid only for very dilute solutions and besides for buffer solutions free of neutral salts; and these are physiologically unimportant. In fluids which are of physiological interest we find much more frequently buffer solutions of the follow- ing type: A small amount of free weak acid + a small amount of its Na-salt + a relatively large amount of neutral salt (NaCl) ; for example, in the blood: CO2 of the order of magnitude 0.002 molar NaHCOa of the order of magnitude 0.02 molar NaCl of the order of magnitude 0. 12 molar "a'^ or in sea-water : CO2 of the order of magnitude 0.0002 molar Bicarbonates of the order of magnitude 0.002 molar NaCl of the order of magnitude 0.6 molar 128 HYDROGEN ION CONCENTRATION or in the urine : Primary phosphates 1 Secondary phosphates I about 0.01 molar NaCl 0.1-0.2 molar In the first portion of this book it has been stated that NaCl has no influence on the hydrion concentration of a solution and that equation (1) above remains unaffected. But at this juncture we must correct this statement. For we can no longer let the denomina- tor in (1) stand for the concentration of the salt of the weak acid, or, what amounts to the same thing, for the concentration of the anions of this acid. What we now want is the activity of these anions. Equation (1) must now be restated as [free acid] a,rT+, = k (2) '^ ' Activity of the acid anions a[H+] is the activity of theH-ions. The true concentration of the acid anions is known from the composition of the buffer and from the chemical analysis; it is S' to retain our formerly used symbol. Then, activity of the acid anions = fa X S'. The true concentration of the H-ions is, on the other hand, entirely unknown. We have no direct method of measuring it, and probably it is of no great interest to us, for the effectiveness of the H-ions appears to depend only upon the arn+j, "the H-ion activity." At this point therefore we shall agree to accept a change of significance of our sjanbols. From this point on the symbol [H+] will represent not the concentration but the activity of H-ions, and pH its negative log- arithm. Then equation (1) becomes: [H+] = k X [free acid] ^^^ fa X [salt of this acid] where the brackets still represent the concentration as usual. What determines then the value of fa? Since the Na-salt of the weak acid is present in but a small amount in comparison with the neutral salt (NaCl), it is evident that this excess of the NaCl exercises almost the sole effect upon the activity of the buffer acid ions. The electrostatic forces to which each ion is subjected determine the activity coefficient; and since the ions of the NaCl are present in excess, they alone must determine the activity factor of the buffer acid ions. Therefore, when in a series of such solutions the amount of NaCl is kept constant, but always in excess, while that of the DISSOCIATION OF STRONG ELECTROLYTES 129 buffer acid is varied, then fa will remain for this 'particular buffer acid a constant value; and hence fa depends upon: 1. the nature, and especially the valence, of the buffer acid, but not upon its concentra- tion, as long as it remains relatively small, and 2. the kind, especially the valence, of the ions, and the concentration of the neutral salt in excess. In equation (3) above we may designate -la and then instead of (3) introduce as the corrected buffer equation the important equation ^ [Na - salt of the acid] ^ ' and the k' which is substituted for the previous k we shall designate as the reduced dissociation constant of the acid, which has a well defined value for every NaCl concentration of the solution, k' may be operated with as well as k. Since k' > k, the entire effect of the influence of the neutral salt is such as to make it appear as if ike dissociation constant of the acid were increased by it, i.e., as if the acid has become stronger. 36. The reduced constants of physiologically important acids If we should wish to apply our buffer equation developed above to physiological problems, we must know not only the k-values for the CO2, phosphoric acid and sunilar buffers, but also their k'-values reduced to the corresponding content of sodium chloride and other neutral salts. It has been experimentally confirmed that in a dilute buffer solu- tion containing a constant amount of NaCl in excess k' is a constant. Therefore it is entirely possible to render the field of experimental physiology entirely independent of the theory of activity; and all that is necessary for this purpose is to determine experimentally the k' values for every physiologically important acid and for every significant concentration of neutral salt. There are still very few exact data on this subject, and it is a pressing task for the immediate future to make available numerical data. Michaehs and Kruger^^ determined the following k' values for acetic acid. " L. Michaelis and R. Kruger, Biochem. Zeitschr. 119, 307 (1921). 130 HYDROGEN ION CONCENTRATION These values for pk' for acetic acid were determined electro- metrically in mixtures each containing 0.02 A^ acetic acid + 0.02 Na-acetate and in addition 0.5 A^ of a neutral salt: Neutral salt RbCl IK2SO4 KCl KBr NH4C1 NaCl LiCl iBaCb ^CaClj pk' of acetic acid.. . 4.587 4.587 4.569 4.569 4.512 4.482 4.452 4.382 4.268 In a mixture of 0.1 A'' Na-acetate with any concentration of acetic acid the value of pk' was found to be 4.616. In a mixture of 0.02 N acetic acid + 0.02 N Na-acetate but free of any other salt, pk' = 4.665. dv aoes aois 0.02 0.25 ooj aof 0.05 o.oe 0.07 o.os 0.09 0.1 o.iz5 0.15 Fig. 16 o.z 0.3 OM 0.5 0.6 0.7 asojf For entirely salt-free solutions of acetic acid pk = 4.733 (see table, page 31). These figures show that in the first place, pk' of acetic acid in 0.5 N NaCl solution is 4.482 instead of 4.733, as it is in a salt-free solution, and, secondly, that the effect varies with the different neutral salts. For the second dissociation constant of phosphoric acid Michaelis and Kruger obtained pk' values shown in figure 16. The abscissa represent the neutral salt content of the solution in terms of normality, the ordinates are the pk' values. The limiting pk' value for infinitely small salt content, assymptotically approached by all the curves, is 7.10. It can be seen from these experiments with phosphoric acid that the value of pk' is influenced not only by the concentration of the DISSOCIATION OF STRONG ELECTROLYTES 131 neutral salt (which alone occurs in Bjerrum's approximation equation for the activity factor), but is also greatly affected by its specific nature. Thus the influence of the alkali cations increases in the series Rb K Na Li; with anions it increases from ^ SO4 to CI; Br appears to have the same effect as CI. The most important for physiology is the reduced constant of carbonic acid, for it represents one of the most significant constants of nature. It will be dealt with in greater detail in a subsequent section. With the change of pk' accompanying variations of the salt- content, the entire dissociation curve {a- or p-curve) is simply displaced horizontally through the presence of the neutral salt. With the perception of this phenomenon the study of ionic equi- librium was greatly advanced, especially for the physiologists.^^'' 37. Another exposition on the effect of strong electroljrtes^^ It appeared necessary in the presentation of this entire chapter to follow the historical development of this subject which is still in progress. But now we may attempt to find another method of repre- sentation, which comprises the physiologically important problems just as well, and which avoids many difficult complications. The concentration of hydrogen ions is measured by means of the concentration chain. One solution whose unknown [H+] = hi is in contact with a hydrogen-platinum electrode, and its potential \i TTi = 0.058 log r , where ho denotes that hydrion concentration which Ho the solution would have if tt = 0. The second solution of hydrion concentration ha is in contact with the second hydrogen-platinum h2 electrode, and here the potential is 1:2 = log 7-. It is usually assumed that the ho values in the first and second equations are identical, and hence the electromotive force E of the chain is / hi h,\ E = TTi - 7r2 = 0.058 I log log — ) \ ho ho/ '^ Concerning recent conceptions of the reduced constants see E. Warburg (Biochem. Journ. 16, 153, 1922). 15 This section is based upon the theory of concentration chains which is detailed in a later chapter. 132 HYDROGEN ION CONCENTRATION or E = 0.05S log -^ E and h2 being known, hi is easily obtained. Criticism is invited when the question is raised as to whether it is justifiable to assume always that the two ho values are identical. If one of these solutions were aqueous and the other in anilin, it would never occur to anyone to regard the ho values as identical. But also in the case when both solutions are aqueous but the one is relatively poor and the other relatively rich in salt content, it is not correct to assume the same ho value for both. In such a case, as of a gas chain consisting of a 0.001 N HCl on the one side and 0.001 A^ HCl + 1 A^ KCl on the other, we find, after allowing for the diffusion- potential, an E.M.F. of a few (2.5) millivolts. Previously, when it had been assumed that the degree of dissociation of the HCl in the two solutions was unequal, this E.M.F. was explained by the differ- ence in the [H+] concentrations of the two solutions. But, as we learned above, this assumption rests on an erroneous interpretation, and we are no longer justified in accepting such an explanation. If we should actually suppose, as in the case above, that hi = ho exactly, then the E.M.F. of such a chain can be explained only by assuming that the two ho values are unequal. If we designate this value as ho in the electrolyte-poor solution and h'o in the electrolyte-rich solu- tion, then the E.M.F. of this chain should rather be stated as / hi h" \ E = TTi - 7r2 = 0.05S ( log }^ - log ^1 j but since hi = ho, ho' E = 0.05S • log — ho Hence, we can express this condition by saying that: The electro- lytic solution pressure of the Ho in an aqueous solution of low electro- lyte content is somewhat different from that in an aqueous solution rich in electrolytes. And yet, when we apply, as we always do, the earlier incorrect formula for the calculation of the ho value from the E.M.F. of concentration chains, we obtain a purely calculated figure which cannot represent the actual concentration of the H-ions. But it is this calculated value which we designate as the result of our DISSOCIATION OF STRONG ELECTROLYTES 133 measurements and commonly accept as the desired [H+]- or pH value. An error was then made by using an incorrect formula for the calcula- tion of the H-ion concentration, and we designate this erroneously calculated concentration as the H-ion activity. And yet this calculated value appears to be of significance in the physical sense. In many cases in which H-ions exercise any effect whatever the effect is proportional not to the true H-ion concentra- tion but apparently rather to their activity. Stated in other words: two solutions having an equal potential towards an Ho-electrode frequently exhibit an equal effectiveness of their respective H-ions. The limits within which this provisional statement is valid must be established by future investigation, now that the underlying prin- ciples are to a certain extent clarified. In any event we have derived our justification to retain for the present, as a measure of the actual acidity, the values for [H+] and Hkewise pH calculated from the E.M.F. of the concentration chain in place of the true hydrion con- centration. AN EXAMPLE FOR THE DIFFERENT METHODS OF REPRESENTATION OF THE EFFECT OF STRONG ELECTROLYTES It may be useful to clarify further the differences between the various explanations by means of an illustrative example. Let us take the concentration chain. ^^ H, 0.001 N HCl 0.001 iV acetic acid + 0.001 AT Na - acetate H2 (1) In this case all the solutions are of such great dilution that in accord with any theory whatever the HCl and Na-acetate may be assumed to be completely dissociated, and any possible effect of the ions upon the properties of the solvent may be neglected. We can, therefore, say that in the 0.001 A^ HCl, hi = 0.001 N, and for the calculation of hi in^the acetate mixture we may apply Nernst's formula (at 19°) hi E = 0.579 log — n2 " The two solutions in the actual arrangement are not in direct contact, but are connected by means of a saturated KCI solution, in order to avoid the diffusion potential which will otherwise arise. 134 HYDROGEN ION CONCENTRATION E being found to be at 19°C = 0.0997 volts, then by calculation pH = 4.722 and [H+] = 1.90 X 10"^ Now, the dissociation constant of acetic acid, as determined by the conductivity method, is 1.86 X 10~^ From the buffer equation we should expect to find in the acetate mixture [Acetic acid] [Na — acetate] which is in very good agreement with the above found value of 1.90 X 10~^. As long as the ionic content of our solutions does not exceed 0.001 N, the theory, built upon the basis of the simple buffer equation, will yield a fairly complete picture of the relation of things. Now let us take the concentration chain^^ H, 0.001 N HCl 0.1 N acetic acid + 0.1 iV Na — acetate Ho at 19°C, it is found that E = 0.0930 volts. By applying again Nernst's formula we obtain for the acetate mixture pH = 4.606 and [H+] = 2.48 X 10"^. How are we to explain then that in one equi- molar mixture of acetic acid and Na-acetate we found [H+] = 1.90 X 10~^andinthe other [H+] = 2.48 X 10~^, according as tO whether the concentration was 0.001 A^ or 0.1 Nf 1. The first older theory says: in 0.1 N solution Na-acetate is [acetic acid] not totally dissociated, therefore, in this fH+] is not = k^ , [IN a-acetate J k [acetic acid] ^ , ^ , , ^ ,. . but = — TTz ——.'Where y stands for the degree of dissocia- 7 [Na-acetate] tion of Na-acetate in 0.1 N solution. If we assume that the value of k (1.90 X 10~^) found in the first concentration chain to be 1.90 correct, then y = r-— = 0.766. But as we have now come to 2.48 believe, in reality y = 1, or very nearly so. Therefore, this inter- pretation is incorrect. 2. The activity theory says: Because of the Na-acetate content of the solution, the activity of the ions becomes different from their true concentration. The Na-acetate is completely dissociated, there- ^^ With same arrangement of chain as indicated in preceding footnote. DISSOCIATION OF STRONG ELECTROLYTES 135 fore, the concentration of acetate-ions, Cacetate = 0.1 A''. The activity of the acetate ions aacetate = fa X Cacetate, where fa, the activity factor, is < 1. Furthermore, what is measured by means of the concentration chain is not the concentration of H-ions, Ch+, but their activity, aH+, and the relation between the two is a.Tj+ = I a X Oh+ where again f'a < 1, but is not necessarily = fa. The buffer equation must therefore be restated as k [acetic acid] ^^ fa' [Na — acetate] If we let f'a = 0.766, then we obtain the desired value an^ = 2,48 X 10~^. The concentration chain measurement does not yield anything du-ectly concerning the value Ch+, so long as there is no way of determining the value of f'a. 3. The third explanation says: As a solvent, the solution rich in salt content does not behave in the same way as pure water. Con- sequently, the solution pressure of the Ho-Pt electrode towards it is not quite equal to that towards water. Hence the second of the above chains does not represent, strictly speaking, a concentration chain, for even two solutions of the same [H+], one in salt-poor water, the other containing much salt, will give rise to an E.M.F. of a few volts. This adventitious E.M.F. must be subtracted from the observed E.M.F. of the second chain before we can utihze it as a true concentration chain for the calculation of pH. The calculation carried out without this correction gives a value of pH which is in a certain respect too low, the calculated [H+] is hence too great by a definite factor. Besides this, it is possible that in a solution containing much salt the dissociation constant of acetic acid is greater than in a salt-poor solution. These two conditions bring about the result that pH calculated in the usual way has the value of only a calculation figure, which may deviate to the extent of about 0.1 from the original definition of pH. But since the correction which we would have to introduce demands the knowledge of certain data which are not yet completely available, it is for the time being more practicable to define the uncorrected figure of our calculation as the pH value. CHAPTER IV The State of Dissociation of Acids and Bases During Actual Salt Formation summary of contents Not all salts are to be included with the strong, always totally dis- sociated electrolytes. In the presence of undissociated salt molecules there arise in the above developed dissociation laws complications which are discussed below. 38. Hitherto we based our considerations on the assumption that of all electrolytes only the weak acids and bases are incompletely dissociated, while all other electrolytes are completely dissociated. Doubtlessly this assumption of complete dissociation is but one of approximation which for such electrolytes as KCl must be practically quite exact. But there are also undissociated salt molecules. We are not as yet very well informed concerning them, but nevertheless the following facts are known: In the first place it cannot be asserted in the case of solutions of organic salts in organic solvents that these salts are completely dis- sociated, which point will be taken up in greater detail in a later chapter. Secondly, we are not justified in ascribing, without further quahfication, the properties of salts of the strong or moderately weak acids and bases to the salts of very weak acids and bases. If equiva- lent amounts of acetic acid and aniline are mixed in an aqueous solution, then this solution of aniline acetate differs from a KCl solution insofar as here a far reaching hydrolysis occurs. Secondly, we cannot, in this case, as we would with a KCl solution, deny the existence of the undissociated salt, aniline acetate. Many electrolyte effects would become more comprehensible, if such "actual" salt- formation were assumed. It appears to the author that, for example, the explanation of electrolyte effect on protein solutions is not very adequate without the assumption of the formation of "actual" protein salts. Just which of the ionic species tend to form undis- sociated molecules cannot yet be accurately stated. It is only 136 DISSOCIATION OF ACIDS AND BASES 137 known with a fair degree of certainty that many of the compounds of H- and OH-ions formed with oppositely charged ions (i.e., acids and bases) are but poorly dissociated. It seems probable that those ionic species which share w^ith H- and OH-ions the property of being strongly adsorbed on charcoal are also those which tend in the direction of true salt formation. The proteins bind the ions of organic dye-stuffs, of alkaloids, etc. It is out of the question that a salt of protein and methylene blue or quinine base belong to the type of always completely dissociated electrolytes. It must be even supposed that the same applies to protein salts of calcium' and indeed to all protein salts. These intimations may suffice for the present to explain why the subject of true salt formation is brought into the discussion. In an illustrative way our problem is as follows: When a base is added to acetic acid then the ensuing ionic equilibrium in the solution has been hitherto represented on the assumption that the molecular entity, "Na-acetate," was not at all formed. Let us now progress a step forward and assume that, after all, this molecular entity is formed, and indeed, according to the equation CH3COO- + Na+ ;^ CH3COO Na in the state of equilibrium [CH3COO-] [Na+] = ka [CH3COO Na] where kg is the dissociation constant of the salt. The advent of this molecular species affects the ionic equilibrium in the solution. Of the many problems arising out of this complica- tion we shall limit ourselves to but a few. It is unessential, in this particular case of sodium acetate, that the formation of undissociated molecules is as yet scarcely considered as a practical possibility. 39. The formation of the ionic equilibrium with true salt-formation^ Let us take the following system. A very small amount of acid A is present in the concentration a in solution. The concentration of its undissociated free molecules is a, its anions A~ are in concentra- tion a~, and besides the cation H+ present in h-concentration there 1 Cf. P. Rona and Gyorgy, Biochem. Zeitschr. 56, 416 (1913). 2 L. Michaelis, Biochem, Zeitschr. 103, 225 (1920). 138 HYDROGEN ION CONCENTRATION is present another cation 1+ in concentration i+ which represents a sufficiently great excess so that the certain amount of salt formation from the interaction of the acid and 1+ does not lead to a material diminution in I+. In addition we must have for the maintenance of electroneutrality a corresponding amount of some other anions, 7 Fig. 17 4 ynoo ' /aoi /w A ^ 1 -fiS 1 6 7 pH — - 70 11 Fig. is Figs. 17 to 19. The functions p, o-, and a. of an acid whose dissociation con- stant ka = 10~', in the presence of a neutral salt of an univalent cation in the concentration 0.1 iV, with the assumption that the acid and this cation form a salt, whose dissociation constant kg has the value marked on each curve. which, however, exercise no effect upon the state of dissociation of the acid A. We shall now express the state of dissociation of this acid as a function of h. We define a- The degree of dissociation a. = -~ (1) DISSOCIATION OF ACIDS AND BASES The dissociation residue p a a The degree of salt formation <^ = 3 139 (2) (3) 0\ Fig. 20 6 '^ ' aoi ■^ 0 3t567 8 9 70 11 Fig. 21 0/^ " ^ ^^001 0.1 3 \ ( J ) t > / ' t ? i > li. Q 11 cc Fig. 22 Figs. 20 to 22. The functions p, a-, and a of an acid whose dissociation con- stant is ka = 10 ~', in the presence of a salt of an univalent cation in the concen- trations marked on the curves, assuming that the acid and this cation form a salt whose dissociation constant ka = 10 ~^. where s is the concentration of the true salt formed by the acid A and the cation I+. According to the definitions a a + a~ + 3 (4) 140 HYDROGEN ION CONCENTRATION and the mass law demands that a~ h = ka X a a-i + = kg X s From these six equations we ( derive : 1 p 1 + 1 a 1 + h i+ ka"^k3 1 (5) (6) (I) (II) (III) 1 + ^h^+l Of these functions that for p is in the simplest form. In the absence of salt-formation its usual form is 1 -t (la) i+ The value ka in (la) is substituted in (I) by ka( r" + 1), and this ■kg latter value is constant for a given concentration of added cations. This p-curve is therefore identical with that of any other acid whose i+ . dissociation constant amounts to ka (r~ + 1) without true salt forma- ks tion. The meaning of this in other words is as follows: If we vary the pH of a very dilute acetic acid solution by the addition of a very small amount of NaOH, then the p-curve has its originally derived form (page 000). But if this solution also contains an excess of a salt in constant concentration, then the resulting p-curve is such as it would be of an acid with a somewhat greater dissociation i+ constant k'a = ka (r" + 1). Graphically this is shown by a small ks horizontal displacement of the p-curve, this horizontal displacement i+ is to the extent of —log (r" + 1). This is shown in figure 17 for ks several assumed values of ks, and in figure 20 for various values of 1.+ DISSOCIATION OF ACIDS AND BASES 141 The a-curve is here no longer an exact mirror image of the p-curve. The characteristic of this curve is that a reaches the value 1 more slowly, or, practically, does not reach it at all. The cr-curve appears here for the first time. The curve of a + o- would be identical with the original a-curve, i.e., it would be a mirror image of the p-curve. The a-curve first becomes definitely marked at high pH values, that is to say, the true salt formation becomes first apparent at a certain degree of alkaUnity. This condition can be restated as follows: The H-ion and the other cation present compete for the combination with the acid. Only when there is a relative lack of H-ions does the other cation succeed in the formation of a salt. For the bases all these conditions are just as valid, after making the appropriate substitutions. 40. The behavior of ampholytes in the process of true salt- formation The relations of an ampholyte to this process are more involved. We shall derive only the p-function. We shall make the following designations: 1. a — the total concentration of the ampholyte. 2. a' and a' — the respective concentrations of the ampholyte cations and anions. 3. a — the concentration of the free undissociated ampholyte molecules. 4. Si — the concentration of the salt formed by the ampholyte as an acid with a cation. 5. Sii — the concentration of the salt formed by the ampholyte as a base with an anion. 6. ka and kb — the dissociation constants of the ampholyte as acid and base respectively. 7. ki and kn — the dissociation constants of the two ampholyte salts; ki of that with the metal-cation and kn of that with the acid- anion. Then we shall define as: = p — the dissociation residue = OL — the degree of d'.ssociation of the cations a a _ a 142 HYDROGEN ION CONCENTRATION a' "T = a' — the degree of dissociation of the anions SI ~ = ffi — the degree of dissociation of the cation salt Sll ~ = (7ii — the degree of dissociation of the anion salt The following equilibrium conditions obtain: a'.h' = ka.a (1) a*. oh' = kb.a (2) a'.i' = ki.s, (3) a".i' = kii.Sn (4) a + a' + a' + Si + Si = a (5) h'.oh' = kw. (6) a Hence for p = - a 1 p = - ^ h- ( —\ - kw \ kj^y h i"Q (7) The p-curve is formally the same as that without salt-formation, except that its parameters have a slightly different meaning. The path of the curve is such as it would be if the two dissociation con- stants of the ampholyte were increased by the presence of the electro- lyte in excess (just as in the case of the p-curve of a simple acid or base) . A further analytic study yields for the [H+] (or h) at the maximum value of p V [H+] at pn,ax = l~-k^ -^ ^i^ (8) kb ^ / i' ^ 0 + ^. This value should be compared with that for h at pmax given in (1) page 65. And for p^ax itself we find 1 Pmax — 1 + v^t-3(-a (9) DISSOCIATION OF ACIDS AND BASES 143 The curves in figure 23 below give an idea of the change in the p-curve in the presence of salts. Here we have a solution of an ampholyte with ka = 10~^ and kb = 10"^^ Its p-curve may be thought of as being the composite of the p-curves of an acid with ka = 10-^ (marked pa on the curve) and of a base with kb = TTZTi — 10~^ (marked pb on the curve), pmax corresponds to pH 5. Let us suppose that changes in pH are brought about by the addition of buffer mixtures which do not form true salts with the ampholyte. Now let us add to our solution a salt whose anion will combine with the ampholyte to form a salt with the constant ki = 0.1, and whose cation will likewise form a salt with the constant ka = 0.01. This will result in the displacement of the pa curve to p'a and of the pb- curve to p'b- The total new curve can be approximately pieced together from the curves p'a and p'b and by rounding out the angle Fig. 23 at which these two meet but leaving the maximum above the same point on the abscissa. In this way the lowest curve of the figure (drawn with the — . — . — line) is obtained. The maxima of the original and of this latter curve are indicated by small circles. We see that the maximum has been changed in a twofold way by the addition of the salt. In the first place, its ordinate value became smaller, and this is because the p'a and p'b-curves are situated closer to each other than the pa- and Pb-curves, secondly, the maximum was displaced horizontally towards the right, and this is because the values of ki and kn are different. In the case where ki = kn only the first ordinate displacement would occur. But in this case the p maximum is in the first place diminished in its absolute value, and, secondly, it is displaced to a different pH value. On superficial consideration it might be supposed that these maxima represent the two isoelectric points, and consequently that the isoelectric points were shifted by the addition of the salt. But this is not the case as we shall at once see. 144 HYDROGEN ION CONCENTRATION 41. The influence of salt-formation upon the isoelectric point Of particular importance in this coimection is the definition of the isoelectric point. Earlier in this book it could well be defined as: •=V^" I P = [H+] for p max But tliis definition is only justified in the absence of true salts of the ampholyte. The logical general definition of the isoelectric point is given as that hydrion concentration at which an equal number of positive and negative amphotyte ions are present. In order to transfer this definition analytically to this particular case it is necessary to introduce the conception of the degree of charge, X. By means of quantitative transference experiments it can be determined how much of the ampholyte is transported in a current of definite intensity, in a unit of time and in the direction o the current, while the amount migrating against the current is calculated as a negative quantity. When the ampholyte is solely in the form of cations, then its transference is maximal. In general, only a fraction of this maximal amount is transferred. It is this fraction that we shall designate as the degree of charge X. It is primarily dependent upon the hydrion concentration and also upon true salt formation by the ampholyte. It must be expressed as = a — a X may be either positive or negative. The isoelectric point can now be defined as that [H+] at which a = a , and therefore, when X = 0. The calculation of X is based upon the following relations (compare with (2) and (1), page 142). a = a • ki oh' . k a Hence ^ a • kb a • ka DISSOCIATION OF ACIDS AND BASES 145 Consequently ^ - • ' -^ — = a — a = A = p kb _ ka\ oh' h) a B}^ substituting the value of p from (7) p. 142. kb • ^^' ka kw h' , „, ^ = \ , ■ . .,. ^^ rrr (10) kb • h- / i' \ ka / i' \ In the special case, when no true salt-formation occurs, i.e., when either no extraneous ions are present, or when their affinity for the ampholyte ions is infinitely small (when i" = i^ = 0, or ki = ku = co ) expression (10) becomes kb ka ^ oh' h The isoelectric point is, therefore, that [H+] at which X = 0. Hence, from the general expression for X as well as for the special expression for Xo we derive the same value for the isoelectric point as we did originally. LP. - y^i K This means that the isoelectric point is not deflected by true salt-forma- tion, which is a very remarkable conclusion. And since it has just been demonstrated that the value of pmax is deflected, it follows that in the case of true salt-formation the values of pmax and I. P. are not identical. 42. The extension of the conception of the reduced dissociation constant The buffer equation in its last interpretation stated that in the presence of an excess of neutral salt ((4), page 129). [H-.] = k' f^'"^ ^'^^^ [salt of the acid] 146 HYDROGEN ION CONCENTRATION The difference between k' and k was interpreted in the sense of the change of activity of the acid-ions because of the presence of the neutral salt. If we should furthermore assume that the cation of the neutral salt can form a true salt with the anion of the weak acid as we have done formerly, and, as we still must assume in certain cases — then we find the following: Let us consider only the dissociation residue, p. The first approxi- mation for p was P = -^ (1) The activity theory demands that, in the presence of neutral salt^ we substitute for the constant k the constant k' reduced to the cor- responding salt concentration, and hence, P=^ (2) If we now assume that the anion of the buffer acid and the neutral salt cation (of concentration i) form a salt whose dissociation constant is constant kg, then according to (I), page 140, p = rr-T-- r (3> Equation (3) differs from (2) inasmuch as the value of k' in (2) is substituted in (3) by k' (t- +1). Since with a constant salt content the value within the parenthesis is a constant, this entire difference can be summarized as k", and equation (3) becomes P = -^ (3a> In the formal sense, nothing is changed, except that another value of k is used, i.e., the p-curve is only horizontally displaced; this is a formal procedure similar to that when k' was earlier sub- stituted for k. DISSOCIATION OF ACIDS AND BASES 147 It could be doubted if the conception of "a constant reduced to a definite salt content" would still hold, if the premise of the total dissociation of all salts were invalidated. It has been shown that such doubt is unjustifiable. Only it cannot always be distinguished in individual cases whether the apparent change of k to k' is based upon an activity effect or upon true salt-formation. As far as the physiological utilization of the "reduced constant" is concerned this is of no import; for our ionic equilibria can still be calculated if only the "reduced dissociation constants" of the acid for the corresponding neutral salt content of the solution is known, no matter how com- phcated the theoretical derivation of this constant may be. In any event, by using the properly reduced constant, the amount of un- dissociated acid molecules can be irrefutably calculated. What remains undetermined are the relative amount of the remaining acid molecules dissociated into ions and those bound into true salt molecules. All of the above considerations are valid for the univalent ions, but if a weak acid forms a salt with a divalent ion (as, for example, acetate buffer + CaCl2), then not only does the horizontal displace- ment of the p-curve ensue, but the curve simultaneously also becomes steeper. Concerning the more detailed calculation of these relations the original article' should be referred to. Special complications also arise when we assume the acid molecules in solution are asso- ciated with two or more molecular species and approach the col- loidal state. Concerning this aspect also the above mentioned original article should be consulted. 3 L. Michaelis, Biochem. Zeitschr. 106, 83 (1920). CHAPTER V Electrolytic Dissociation in Non-aqueous Solutions summary of contents The little that is known concerning dissociation in non-aqueous solutions is described, insofar as it promises to become of physiological interest. 43. Apart from a few exceptions, such as, for example, hydro- cyanic acid, the electrolytic dissociation is not nearly as great in any solvent as it is in water. The smaller the dielectric constant of a solvent the smaller is its dissociating capacity for the electrolytes dissolved in it. In the living organism there are in addition to the watery phases also lipoidal phases. These are chemical systems in which the fundamental substance is not water but substances of the nature of true fats, lecithins, and related compounds. These phases form in the organism either microscopic or else ultramicroscopic structures as colloidal solutions, or, continuous phases, such as in the medullary sheaths of the nerves. Many membranes, particu- larly, contain lipoids. Whether the lipoids form a continuous phase, or whether the lipoidal and the non-lipoidal constituents are arranged in mosaic form within the membrane structure, is not definitely estabUshed. But whether the lipoid substances are continuous, or whether they are to be conceived of as forming a "disperse" phase within an aqueous phase serving as a dispersion medium, in any event, every individual lipoid droplet is a chemical system with a lower dielectric constant than water, and which, as a solvent for electrolytes, behaves differently from water. Our knowledge concerning the dissociation of electrolytes in such organic solvents is still very full of gaps. It is a pressing need to fill these gaps, for that alone will enable us to explain the role of the lipoids, especially as a cause of bioelectric phenomena. The reason for our inadequate information hes in the difficulties inherent to the methods. The method which, in aqueous solutions, is the most useful in the study of dissociation, the measurement of conductivity, 148 DISSOCIATION IN NON-AQUEOUS SOLUTIONS 149 is very much more difficult when appUed to hpoids. This is because the conductivity of the hpoids themselves is so extraordinarily small, and electrolytes dissolved in them are so poorly dissociated that the total conductivity in all such cases remains very small. Besides, it is generally quite impossible to obtain the value of Aa, by extra- polation. The other method which led to further results was that of the concentration chains. The measurement of the electromotive force in these chains is easily carried out in aqueous solutions. These methods are all based upon the measurements of electric currents, or more specially, upon compensation of currents by oppositely directed currents. The hpoids offer such tremendous resistance to the electric current that measurable current intensity is not generally obtainable. It is chiefly electrostatic tension that can be measured in these cases. Special instruments have been devized for this purpose, such as the quadrant electrometer by Thomsen and its newer modification, the binant electrometer, by Dolezalek. These methods were successfully appUed by Cremer,^ Haber and Klemen- siewicz,^ Loeb,^ and Beutner^ to determine the degree of dissociation of electrolytes, covering, to be sure, only a part of the above de- veloped general purpose. Only a few points of support are available for attacking our problem. First, a certain regularity discovered by P. Walden^ must be mentioned. As was stated earlier (page 14), Walden found that on comparing those concentrations (c) of a given electrolyte which have the same degree of dissociation in different solvents, it appeared that the cubic roots of the concentrations were directly proportional to the dielectric constants D of the correspond- ing solvents, or, Di : D2 = a/c*! : V^Cj Let us find, for instance, that concentration c of an electrolyte in different solvents at which the degree of dissociation, a = 0.5. Now 1 Cremer, Zeitschr. f. Biol. 47, 1 (1906). 2 Haber and Klemensiewicz, Ann. d. Physik [4] 26, 927 (1908). 2 Jacques Loeb, Journ. of Gen. Physiol. 3, 667 (1921). * R. Beutner, Die Entstehung elektrischer Strome in lebenden Geweben. Stuttgart 1920. 5 P. Walden, Zeitschr. f. physik. Chem. 54, 228 (1905) ; 94, 263 and 372 (1920). 150 HYDROGEN ION CONCENTRATION according to Ostwald's dilution law, or, according to the mass law, c = k 1 - a where k is the dissociation constant of the electrolyte. By letting a = 0.5 we obtain 0.25 c = k 0.5 or c = 2 k therefore, c is proportional to k. By substituting k for c in the above equation Di : Ds = 'V'^ : \'^U where ki and k2 are the dissociation constants of the electrolytes in the two solvents. Then Walden also found that actually this relation was fairly valid for a large series of electrolytes. But the electrolytes tested by him were exclusively of the strong type (such as KCl, Nal, AgNOg, N(C2H5)4l, etc.) to which Ostwald's dilution law cannot be apphed. Thus there are still difficulties in the way of estabhshing Walden's theory. Certain further elucidation concerning the dissociation of electro- lytes in organic solvents is found in the investigations of R. Beutner (I.e.): Many of the inorganic electrolytes, such as KCl, very soluble in water, are not at all measurably soluble in oils.® A much better insight can be had by working with organic acids and bases and with their salts. Thus, if salicylic acid is dissolved in nitrobenzene, the conductivity increases perceptibly; hkewise with dimethyl- toluidine. Now, if equivalent amounts of salicylic acid and dimethyl- toluidine be dissolved in the same solvent, the resulting conductivity is appreciably greater than the sum of the individual conductivities. We may conclude from this that the salt is more strongly dissociated than the acid or the base, as is always the case in aqueous solutions. ^ The author uses the term "oil," in this connection and in subsequent chapters, to designate any oily liquid, generally forming a separate phase when in contact with an aqueous phase. — Translator. DISSOCIATION IN NON-AQUEOUS SOLUTIONS 151 But also the hydrolysis of this salt is much greater in oil than it is in water. The greater part of the acid and of the base persist along- side of each other. This can be seen from the following: When we add more base to a mixture of much acid and little base, then we should expect that the conductivity would rise in proportion to the added base, if the latter was completely utilized for salt formation. But it is observed that the conductivity increases less than expected. Consequently, it must be assumed that the base was not completely used up for the formation of salt, i.e., that the salt remains extensively hydrolyzed, even in the presence of an excess of acid. As experimental proof for this view the following observations made by Beutner may be quoted: 1 M solution of salicylic acid in nitrobenzene showed at a high temperature (not so soluble in the cold) a conductivity of 3.5 reciprocal megohms; calculated for room temperature 2 recipr, megohms 1 M solution of dimethyltoluidine in nitro- benzene 0.1 recipr. megohms 1 M solution of salicylic acid + 1 M dimethyl- toluidine in nitrobenzene 430 recipr. megohms 1/25 M solution of salicylic acid + 1/100 M dimethyltoluidine in nitrobenzene 18.5 recipr. megohms 1/25 M salicylic acid + 1/50 M dimethyl- toluidine in nitrobenzene 23.8 recipr. megohms (instead of 37, as it should have been, if all the base added were used for salt-formation) 1/25 M salicylic acid + 1/25 M dimethyltolui- dine in nitrobenzene 27. 5 recipr. megohms (instead of 74) A quantitative utilization of these figures in the calculation of dissociation constants can not unfortunately be carried out, for Ostwald's dilution law, i.e., the mass law, is not vahd for electrolyte solutions in oils on grounds which are as yet to be worked out. This is probably due to molecular associations or aggregates, which electroneutral molecules tend so strongly to form. A second difference is in the fact that in oily solutions there are no acids or bases which are as strongly dissociated as the salts: there are no "strong" acids and bases. Here we have only weak electrolytes — namely all salts, and extremely weak electrolytes — namely, all acids and bases. 152 HYDROGEN ION CONCENTRATION The definition of acid and alkaline reaction would be the same as in aqueous solution; at the neutral reaction [H+] = [0H~], but methods for the determination of the reaction are still lacking. A gas chain with an oily Hquid has not yet been investigated. Nor is the dissociation constant of water (in oil) known [H+] [0H-] _ H2O (where [H2O] represents again a value proportional to the vapor pressure). Doubtlessly it depends to a great extent upon the nature of the oil and especially upon its dielectric constant. In our present state of knowledge it is useless to attempt to apply any indicator method to oil phases. All that can be done at present is to disclose this great gap. The principal difference between the two classes of electrolytes, 1-acids and bases, 2-salts, appears to be still evident even in solvents with small dielectric constants. In this connection also H- and OH-ions exhibit their specific pecuharity. This is all the more remarkable, since the other peculiarity of the H- and OH-ions, i.e., their abnormally high conductivity, is not in evidence in these solvents. H- and OH-ions, for example, have no greater motility in organic water-free solvents than alkali or halogen-ions. The abnormally high conductivity of H- and OH-ions was explained by Arrhenius in the following way: When a moving H-ion comes in contact with a H20-molecule, it combines with the OH-group of the latter which immediately liberates its H-group as an H+-ion. In this way in the process of conduction the portion of the path cor- responding to the thickness of the H20-molecule is saved, and an apparent impression of an abnormally high motility of the H-ion results. The same holds for the OH-ion. In general it appears that in every solvent those ions which the solvent itself is capable of yielding possess an abnormally high conductivity. This theory, is similar to the one once proposed by Grotthus concerning general electrolytic conduction, and which had been dominant before the Arrhenius theory of dissociation appeared. PART II The Ions, Particularly the Hydrogen Ions, as Sources OF Electric Potential Differences INTRODUCTION 44. All solutions of electrolytes contain equal amounts of posi- tively and negatively charged ions. Deviations from this proposition are possible only to so slight an extent that it is beyond the scope of analytical chemical methods to detect the inequalities between the two ionic species. But, nevertheless, because of the large amounts of free electricity which arise in such cases, this inequality is demon- strable by physical methods, and it is the cause of a number of electric phenomena. It is not our purpose to treat of these as extensively as of ionic equilibria. But since the methods for measur- ing hydrion concentration are based upon these phenomena they must be discussed here insofar as our understanding of these methods requires it. Furthermore, our explanations in general will be made in the direction in which they are to have a probable physiological bearing. Among the potential differences brought about by the unequal partition of ions, several groups may be differentiated; such as: 1. electrode potentials, or those which are located on the surface of a metal in contact with a solution; 2. diffusion -potentials, or those which ensue at the contact of two different electrolyte solutions or of two solutions of unequal concentration; 3. phase boundary potentials which are generally located on the boundary surface between two phases, in such a manner that one phase shows a potential towards the other phase; 4. membrane potentials, which are based upon the impermeabihty of certain membranes for certain ions; 5. adsorption potentials which also develop on a phase boundary surface, but in such a way that the boundary layer of a phase exhibits a potential difference towards the rest of the same phase. 155 CHAPTER VI The Electrode Potentials summary of contents Nernsfs theory of metal electrode potentials is developed and applied to concentration chains, of which the hydrogen gas chains are the most important for physiological work. The application of Nernsfs theory to chains with various metal electrodes is briefly explained and the electromotive series is described. 45. The single potential of an electrode A metal immersed in a solution generally shows a difference of potential towards the solution. This long familiar fact has been theoretically worked out after the establishment of the ionic theory and developed by Nernst. It has long been recognized that this potential difference between a metal and a solution depends, first, upon the nature of the metal, secondly, upon the composition of the solution, and thirdly, to a small extent, upon the temperature. The composition of the solution refers to the solvent as well as to the solute electrolyte. As the solvent, water alone will first be con- sidered. As far as the electrolyte is concerned, only the concentra- tion of that ionic species is of importance which is capable of yielding ions of the metal involved. All other ionic species which may be present in the solution are immaterial as far as the value of the potential difference is concerned. Thus the potential of a silver electrode towards a 0.001 N AgNOa solution is the same as that towards a 0.001 N solution of AgClOs, or of any silver salt, i.e., if a galvanic chain were to be made of these two electrodes then the resulting E.M.F. = 0. But if one silver electrode is immersed in a 0.01 N AgNOs solution on one side and another silver electrode is immersed in a 0.001 A^ AgNOs solution on the other side, a current will be generated by this chain. Whether or not there be present in these solutions another salt, 0.01 N KNO3 for example, is immate- rial, insofar as this salt does not by chemical reaction alter the con- 156 ELECTRODE POTENTIALS 157 centration of the dissolved Ag+-ions. On the other hand, the addition of NaCl would precipitate almost all of the Ag+ as AgCl, and only insofar as the precipitation is concerned has NaCl an effect upon the potential. Therefore, at the very beginning of our theoretical derivation of Nernst's electrode potential we shall state the experimental fact that the potential difference of a metal toward a solution depends only upon the concentration of the ions of the same metal present in the solution. This can be formulated as: ^ = f (c) where tt is the potential and c is the concentration of the ionic species concerned. Now it is our purpose to define the function f (c) in greater detail. To this end let us utihze the principle of thermo- dynamics : When a chemical system passes reversihly from one state into another, the maximal work which can he gained from such change of state is independent of the manner in which this change occurs as long as it remains reversible. Now then, when a metal electrode is used as one pole of a galvanic element, experience shows that the process occurring at this electrode, on closing the circuit, is in many cases reversible. According to the conditions and according to the direction of the resulting current, the metal will be either dissolved or deposited. Thus, if the current is allowed to flow until one gram ion of Ag+ is dissolved, and then by means of another source of current an oppositely directed current of the same ampere-hour value as the first is sent through the same system, then exactly the same amount of silver will be deposited as had been previously dissolved. Hence the process is reversible. The initial and final states of this system (if the process is not re- versed) differ, inasmuch as at the end there is one gram-ion of Ag+ more in the solution than in the beginning. This change in the system can be brought about in another reversible way, different from the electrolytic process, i.e., by removing water from the solution until it is correspondingly concentrated and then restoring the volume by the addition of a solution of the same higher concentration. The removal of the water must needs be performed in a reversible manner, such as by means of an osmotic piston impermeable to Ag-ions. With such a piston the solution is compressed, as it were^ until its concentration reaches the desired value. 158 HYDROGEN ION CONCENTRATION In the one case we can calculate the electrical work and in the other case the osmotic work performed. The electrical work is equal to the product of the potential difference ir and the transported amount of electricity, or 96,450 coulombs, wliich is attached to one gram- ion of silver. This amount of 96,450 coulombs is designated as 1 F (faraday). The electric work W is, therefore, x F. If the potential were increased to t + d tt, then the work would be in- creased by the amount of d W = F d r. The osmotic work performed in the above cited case consists entirely of the work of compression; the restoration to the original volume does not demand nor ^deld any work, for no change of con- centration is involved. This osmotic pressure depends upon the osmotic pressure against which it is exerted, which means that it depends upon the concentration of the Ag-ions already present in the solution. Therefore it can not be defined without going into further explanation. But this much can be said: if the osmotic process consists of a dilution, and if the osmotic pressure is at first p and then p — d tt, then the work to be obtained in the latter case is less by v-dp than in the first case. Here v stands for the volume con- taining one gram-ion in solution. When, on the other hand, the osmotic process consists of compression (concentration), then the sign is to be reversed and taken as negative. Thus the osmotic work must be equal to the electrical work, which can be expressed in terms of differentials as F d TT = — V dp RT Since p v = RT and v = — , hence P Fd7r= dp P _ _ RT dp F * p and integrating: RT , 1 ^ IT = -—- la — h Constant F p ELECTRODE POTENTIALS 159 RT This integration constant can be replaced by -;^ X In P and our r equation becomes RT P TT = — X In - (1) F p When p = P, then w = 0. Hence P represents that osmotic pressure of the ionic species involved against which the electrode has 0 potential. Herein lies the physical interpretation or signifi- cance of the integration constant. Since in very dilute solutions the osmotic pressure is proportional to the concentration, the above equation can also be stated as TT = —- X In - (2) 1" c where c is the ionic concentration in the given solution and C that concentration against which the electrode potential would be = 0. The meaning of the osmotic pressure of an ionic species must be here further dealt with in somewhat greater detail. For extremely dilute solutions there is no difficulty in this respect, for here con- centration and pressure are proportional to each other. But in higher concentrations this is no longer strictly true, and in such cases equation (1) above must be used, in terms of pressures. Form- erly no doubt was entertained in accepting as the osmotic pressure that value which was obtained from freezing point determinations. Thus, for example, if it was found that the lowering of the freezing point of a 1 A'' KCl solution was 1.8 times as great as that of a 1 iV cane-sugar solution, it was concluded that the "osmotic pressure" of the KCl solution was 1.8 times as great. Furthermore, it was assumed that, because of incomplete dissociation, there were 1.8 times (instead of twice) as many individual molecules (i.e., ions) in the KCl solution as in the sugar solution. Now we assume that there are actually twice as many molecules in the KCl solution (see page 116), but that the electrostatic forces have an effect upon the lowering of the freezing point. We have also shown that these forces also exercise another influence upon the active 77iass of the ions, and it is this active mass that we must obviously take to rep- resent the "pressures" p and P above. Thus, when in following the old convention, we speak of "osmotic pressure," it must be 160 HYDROGEN ION CONCENTRATION realized that we must no longer take it in the sense of that "osmotic pressure" which had been derived from freezing point determinations, but that it would be much better to substitute the term "active mass" for that of "osmotic pressure." The active mass is obtained from the concentration by multiplying the latter by the factor fa, the factor being so chosen that the other- wise approximate equation (2) should correspond exactly to the measured E.M.F. Properly speaking, this brings us into a vicious circle. For, an equation such as (2) is set up, it is observed that it is not quite correct, some of its values are altered by means of empiric factors, and then, naturally, it is found that the formula is entirely correct. While there is a grain of truth in this objection, it is not entirely justifiable, for we have shown above, in a quantitative way, the causes for the necessity of these corrections, and, in a few suitable cases, it was also quantitatively possible to arrive at fairly accurate results on the basis of certain verified assumptions. Furthermore, we have seen that even the uncorrected equation is evidently entirely vahd, as a limiting law, for high dilutions. It is beyond doubt that in the near future the activity factor will be derived from other than electromotive force data. Then the reason for the above vicious circle objection will disappear. Considering the newness of the conception of the activity factor, it appeared only proper to point out these difficulties and to forestall this all too easily raised objection. If instead of the activity theory we choose the theoiy given on page 131, we could then state even more clearly, the following: The concentration C of the current yielding ions against which the metal electrode has a potential of 0, varies with the amount and the nature of all the ions present in solution (not only the current pro- ducing ions). Therefore, when the concentration of the current- yielding ions is greatly altered, without maintaining the total electro- lyte content of the solution at a constant level by a corresponding change of the non-current-yielding ions, then not only is the value of RT C c changed, but also that of C. Hence, in the equation tt = —^ In — the value C cannot be strictly considered as being a constant. 46. The concentration chain There is no adequate method for the measurement of any given smgle potential of an electrode, but two electrodes are put together ELECTRODE POTENTIALS 161 into a chain, its E.M.F. is measured and taken to represent the difference between the single potentials. Chains may be made up of two different metals, or of electrodes of the same metal in contact with solutions of different concentrations, these latter being the concentration-chains. In accord with the main purpose of this book, we shall concern ourselves chiefly with the theory of such concentra- tion chains. Let us take two silver electrodes immersed in solutions of AgNOa whose concentration on the left side is Ci and on the right side q,%. We shall assume that at the place of contact of the two solutions there exists no difference of potentials, a condition frequently re- ahzed and with which we shall later deal separately. We have then in this case RT , C RT , C Ci ■ F C2 and the EMF RT, C C2 Jii = iri — TTz = -=r In — • — F Ci C E = — — In — F ci Hence the constant C disappears from the equation. Such chains are easily reaHzed, and, therefore, we must first discuss the numerical evaluation of the factor R. Since it is desired to obtain E in volts, it is necessary to calculate the gas constant R in terms of electrostatic units. Note: When the valence of the current-yielding ions is n, then the equation correspondingly becomes E = — — In — nF ci 47. The numerical evaluation of Nemst's formula in volts The value R is defined according to the gas laws as p V po Vo R = T 273.09 162 HYDROGEN ION CONCENTRATION where p is the pressure of an ideal gas, v is the volume which con- tains one mol of this gas at this pressure and at the absolute tem- perature T; Po and Vo are the corresponding values for 0°C., and 273.09 is the absolute temperature value of 0°C. Taking Berthelot's figure, at 0°C, at 1 atmosphere of pressure, Vo = 22,412 cc; Po = 1 atmosphere = 76 cm. mercury, which, at 0°C = 980,665 X 76 X 13,595 = 1,013,280 dynes per sq. cm. Therefore R = 83,157,720 ergs. 10^ ergs = 1 absolute joule; 1 absolute joule = 0.99,966 inter- national joules. Hence R = 8.313 international joules or volt coulombs. 1 faraday, F = 96, 450 coulombs. Now by substituting these numerical values in our equation we obtain for the E.M.F. of a concentration chain: 8.313 . (273 + t) , ci , ii< = 7:tt: in ~ volts 96 540 Co Dividing the natural logarithm by 0.4343 we change to the Briggsian logarithms (base 10) and obtain E = 0.000,1985 • (273 + t) log - C2 = 6 • log — C2 TABLE 22 t B t e t e t 9 t °c. e t e °c. "C. °c. "C. 0 0.0542 16 0.0573 22 0.0585 28 0.0597 34 0.0609 40 0.0621 5 0.0552 17 0.0575 23 0.0587 29 0.0599 35 0.0611 45 0.0631 10 0.0561 18 0.0577 24 0.0589 30 0.0601 36 0.0613 50 0.0641 12 0.0566 19 0.0579 25 0.0591 31 0.0603 37 0.0615 60 0.0661 14 0.0569 20 0.0581 26 0.0593 32 0.0605 38 0.0617 70 0.0680 15 0.0571 21 0.0583 27 0.0595 33 0.0607 39 0.0619 80 0.0700 Table 22 gives the values of d (RT) for various temperatures. By applying these values and the proper experimental conditions, Nernst's equation for concentration chains can be verified with great accuracy. The experimental data of former years frequently did not exceed in accuracy the difference of 2 to 3 millivolts between the ELECTRODE POTENTIALS 163 observed and calculated values. This was because of the imperfect understanding of the dissociation of strong electrolytes and because of the incomplete abohtion of the diffusion potential at the contact of the two solutions. It seems that the most appropriate arrange- ment for the demonstration of the absolute exactness of Nernst's equation for concentration chains is the following : Pt HCl (ci) saturated HCl (Co) Pt H2 + 1.0N KCl I KCl solution + 1.0 iV KCl II H2 Here the concentrations Ci and C2 of the HCl must be relatively- small in respect to the KCl concentration (1.0 N), i.e., they should be between 0.0001 and 0.05 N. The excess of KCl accomplishes the following purposes: (1) The degree of dissociation of the HCl (or better, the activity factor of the H+-concentration) is sufficiently near equaUty in both solutions so that the ratio of the HCl concentra- tions is actually equal to that of their H+-activities, and (2) the diffusion potential remains infinitely small. Such chains give E.M.F. values which rarely show a maximum deviation from the calculated figures of 0.0005 volt, and the average of repeated measure- ments usually agrees perfectly with the theory. Thus this verifica- tion of the theory constitutes the convenient method of checking the correctness of the experimental technic for any worker in this field. 48. Reversible electrodes for concentration chains The condition sine qua non, for Nernst's equation is the reversibility of the electrodes. This means that, if the circuit of the chain is closed for a definite period, and then if the exact amount of current produced is sent through the chain in the reversed direction, the original state of the chain must be reestabhshed exactly. Each of the two electrodes must be capable of functioning reversibly. This is alwaj^s the case when a metal is immersed into a solution of its salt and (the circuit being open) no chemical reaction can occur. Thus Zn in dilute H2SO4 is not a reversible electrode, since it goes into solution spontaneously, but Zn in ZnS04 is such an electrode. Zn in CUSO4 is not reversible, for the spontaneous reaction occurs, Zn + CUSO4 -^ ZnS04 + Cu. Hg in water is not a reversible elec- 164 HYDROGEN ION CONCENTRATION trode, since, depending upon the amounts of impurities in the water, especially of O2 and CO2, varying traces of Hg will go into solution. On the other hand, Hg in a saturated HgCl solution is a reversible electrode, in spite of the fact that the solubility of calomel depends on the total Cl~ content of the solution. For its dissociation occurs in the following manner: Hg CI ^ Hg+ + Cl- hence [Hg CI] [Hg+] [C1-] In a saturated calomel solution the concentration of HgCl mole- cules is constant, hence, [Hg+] X [Cli = a constant Therefore the concentration of the Hg-ions in solution is inversely proportional to that of the Cl-ions. Therefore, two Hg-electrodes immersed in two solutions of different concentrations of KCl, both saturated with calomel, form a reversible concentration chain for Hg-ions. Since the ratio of the Hg-ions is reciprocally equal to that of the Cl-ions, such a chain may also be used as a reversible chain for Cl-ions. These are designated as electrodes of the second order, or, reversible electrodes for anions. These electrodes may therefore be utilized for the determination of Cl-ion concentrations. An alloy of a noble with a common metal behaves, in the electro- motive sense, as if it consisted of the common metal only. The same is true of gas-electrodes. Platinum (as well as gold, palladium, iridium) absorbs H2-gas from an atmosphere of hydrogen, the absorption equilibrium being determined by the partial pressure of the hydrogen. Such an electrode behaves as if it consisted only of a metal-like, conductive hydrogen, hence it is a reversible electrode for H+-ions. 49. The theory of gas chains We saw above that when a concentration chain is made up of Pt — H2 electrodes and of two solutions of a different [H+], (hi and h2) then its E.M.F. is: lii = — — In — ELECTRODE POTENTIALS 165 hi being known, then h2 can be found by determining E. The appHcation of this principle in practice will be discussed in detail when the methods of determination of [H+] are described later. But at present it is always assumed that the Ho-pressure at both electrodes is the same. When such is not the case, then the theory of gas chain requires certain amplifications. In such a case an E.M.F. is developed, even when the [H+] on both sides is the same. The original equation for concentration chains ^ RT, c, RT, c, may assume the simphfied form of £ = ■?•"- F C2 only when the constant C has the same value in both terms on the right side of the equation. But this is not true when the gas pressure varies, i.e., the potential of a gas electrode also depends upon the gas pressure about it. Our problem now is to measure the E.M.F. of a gas chain in which the [H+] is the same throughout, but in which the pressure of hydrogen gas at the two electrodes is different. The platinum electrodes are in equilibrium with two hydrogen atmos- pheres whose respective partial pressures are pi and P2: the solution in which they are immersed is the same on both sides and of the same [H+] designated by h. On closing the circuit we observe that on the side of the higher partial pressure H2-gas goes into solution as H+-ions and on the other side gaseous hydrogen is formed. If we permit the current to pass long enough until 1 mol of H2-gas dis- appears on the left and as much is formed on the right side, then the change of the state of the system is the same as would occur when 1 mol of H2-gas passes from a vessel in which it was under a pressure of pi into another whose pressure is p2. And the work performed is hence equal to the work yielded by a gas, when 1 mol of it expands from a pressure pi to a lesser pressure p2, or to the work W = RT In — . The electric work resulting in the same final state P2 of the system is equal to the product of the electromotive force E by the number of transported coulombs. Since 1 mol of hydrogen 166 HYDROGEN ION CONCENTRATION amounts to 2 gram-ions of H+, the number of coulombs transported in this case will amount to 2 F. Therefore, Pi P2 and or 2 F . E = RT In 2F p2 0.0001983 , pi E = — — log — volts 2 F ^ p2 Table 23 gives the values of E of a chain in which the ratio of the two partial pressures is 1 :n. TABLE 23 E in millivolts l:n At 18° At 37° 1 0 0 1.05 0.6 0.6 1.1 1.2 1.2 1.2 2.3 2.4 1.5 5.1 5.4 2 8.7 9.2 10 28.9 30.8 100 57.7 61.5 By "pressure of H2-gas" we always mean the partial pressure. Thus, for instance, in an atmosphere of hydrogen saturated with water vapor by H2-pressure we mean only the partial pressure of hydrogen and not that of the water vapor. These "pressure-chains" are of interest in the practice of pH determinations, for the pressure of the atmosphere of hydrogen in the electrode vessels depends in the first place upon the barometric pressure, and, secondly, because of its continuous saturation with water vapor, also upon the temperature. Gases which form reversible electrodes on platinum surfaces for their own ionic species are primarily hydrogen and chlorine. Oxygen ELECTRODE POTENTIALS 167 is also electromotively active, and at high temperatures (over 300°) and by employing a solid electrolyte (glass) instead of an electrolyte solution, such 02-electrode behaves, as Haber^ found, exactly as a OH~-ion jdelding electrode would lead us to expect. In other words, a chain consisiting of a H2- and of an 02-electrodes, both immersed into one and the same Hquid of any composition, the so-called oxy- hydrogen gas chain, has that E.M.F. which can be calculated from the chemical affinity of the reaction of the explosive mixture of oxygen and hydrogen. At lower temperatures, however, the usual oxy-hydrogen gas electrode shows a poorly adjusted potential which is about 0.3 volt too low. Otherwise one could determine 0H~- concentrations with an O2 electrode, just as we determine the H+- concent rat ions with an H2-electrode. But in reahty this is not pos- sible. Even though Barendrecht- claims the opposite, he disagrees with all other investigators. As a cause of this irregular behavior of the 02-electrode it is assumed that in the presence of oxygen the platinum is covered by a thin layer of an oxide; hydrogen, on the other hand, reduces this layer and, therefore, the H2-electrode be- haves correctly from the standpoint of theory. 50. The electromotive series The potential of a metal against a solution is determined by the nature of the electrode metal as well as by the concentration of the corresponding ions in the solution. Up to this point we have assumed the nature of the electrode metal as being constant and we have discussed only the effect of the concentration. Now we shall briefly deal with the effect of the nature of the metal. The various electro- lytic solution tensions of the metals could be characterized by giving that concentration of ions against which the metal shows zero potential. But since the absolute value of single potentials cannot be very accurately determined, it is better to select another way. That potential difference which exists between a metal and a normal solution of its ions is arbitrarily assumed to be standard. Among such standards that of any one metal may be arbitrarily set equal to 0. At the suggestion made by Nernst we designate the potential 1 F. Haber, Ann. d. Physik [4]. 26, 927 (1908); Idem., Thermodynamics of Technical Gas Reactions. London 1908. 2 H. P. Barendrecht, Akad. Wetensch. Amsterdam 27, 1113, 1236, 1406; 28, 23 (1919). 168 HYDROGEN ION CONCENTRATION difference between a hydrogen electrode (Pt — H2) under one atmos- phere pressure of hydrogen-gas and a solution normal with respect to the H-ion as equal to zero. In this way it was possible to arrive at the electromotive series of the metals and some non-metals shown in table 24, which could be used just as hydrogen in the form of a platinum electrode covered with gas.^ With the help of Nernst's equation and these values one can calculate the potential between a metal and a solution containing ions of this metal in any concentration. Conversely, the E.M.F. of the concentration chain being known, the unknown concentration TABLE 24 Elect •olytic soluV '.on tension at 25°C. Li+ - 3.305 re++ - 0.43 Pt + + + 0.863 Rb + - 3.205 T1 + - 0.32 Au + + 1.5 K+ - 3.203 - 2.993 Co + + Ni + + - 0.29 - 0.22 Na + HSO4- + 2.6 Ba+ + - 2.7 Pb + + - 0.12 SO4- + 1.9 Sr + + - 2.767 H + ± 0.00 0- + 0.43* Ca + + - 2.55 As + + + + 0.292 OH- + 0.88t Mg + + - 1.55 Cu + + + 0.34 ci- + 1.35 A1 + + + - 1.276 Bi + + + + 0.391 Br- + 1.08 Mn + + - 1.075 Hg + + 0.86 I- + 0.54 Zn + + - 0.766 Ag + + 0.8 F- + 1.9 Cd + + - 0.40 Pd + + + 0.789 S= - 0.55 Sn + + - 0.1 CH3COO- + 2.5 * t These two values are in reference to a solution 1 A'' with respect to 0H~- ions. They are frequently given in tables in reference to a 1 A' H"*" solution, in which case they are +1.23 and +1.68 respectively. of the metal ions in the solution can be calculated. This is partic- ularly important for the cases where complex ions are formed and of the relatively insoluble metal salts. Thus, for example, the solubility of AgCl can be estimated by determining the concentration of the Ag+-ions in the solution in the chain Ag AgNos 1 molar AgCl, sat. solution Ag 3 Some of these values were not determined in the direct experimental way, but were indirectly calculated. ELECTRODE POTENTIALS 169 When KCN is added to a solution of a silver salt, a complex AgCN ion is formed and the Ag+-ions disappear from the solution except for an extremely minute residue. This residue can also be estimated by means of the concentration chain. 50a. Oxidation-Reduction Potentials and the Quinhydrone Elec- trode. A. Theory. In recent years, since the appearance of the last German edition of this book oxidation-reduction potentials have assumed considerable importance, a brief outline of the underlying theory will be given here as an addendum only. By the term oxidation we understand either: 1. The addition of oxygen (or of an OH group), or the loss of hydro- gen, or, 2. The gain of a positive charge, or the loss of a negative charge. These four processes are obviously quite different in nature, but their consequences may be identical, so that frequently it cannot be discriminated by which one of these four possible ways the end result of oxidation has been achieved. Thus, for example, in the reaction between metallic sodium and water, Na + HoO -* NaOH + § H2 -^ Na+ + OH" + § H2 the oxidation of the sodium may be interpreted either in terms of an addition of oxygen (OH group) to the sodium, or, as an addition of a positive charge to the metallic sodium at the expense of the H+ ions of the dissociated HoO molecules. Reduction is exactly the reverse process to that of oxidation and needs no further elucidation. There are some types of oxidation and reduction which in practice can be made to occur with any reasonable velocity only by means of an exchange of electric charges. If we have in solution simultaneously a substance and its oxidation product, but no other substances which could act either as oxidizing of reducing agents for the original pair of substances, nothing will happen. For example, in a solution containing Fe++ and Fe+++ salts the following reaction is conceivable: Fe++ + Fe+++^ Fe+++ + Fe++ This reaction, however, will have no discernable consequences, since as much of Fe"*^ disappears as is formed, and the sa"me applies to the Fe+++. 170 HYDROGEN ION CONCENTRATION But if into the same solution we immerse such a chemically indiffer- ent electrode as one of platinum or of gold and make it a part of a closed circuit, the Fe++ ions may receive an additional charge (be oxidized) from the current, while the Fe+++ may give up one of its charges (be reduced) to the current. According as to whether the reaction Fe++ + 0 ^ Fe +++ depending chiefly upon the concentration ratio, tends to occur pre- dominatingly in the one or in the other direction, an increase of the oxidation or of the reduction product will result. The chemical force with which the above oxidation-reduction reaction tends to occur in the one and in the other direction depends upon the specific chemical nature of the process involved and, secondly, upon the concentrations. There is a definite concentration ratio at which the forces tending towards oxidation and towards reduction are mutually compensating. If we designate this par- ticular concentration ratio by Co, then it can be stated that oxidation fFe+++l will predominate when \^ ^^, > Co while the reduction process [i^ e++J [Fe+++] will prevail when -^ _^^ < Co ^ [Fe++] Let us now take a galvanic chain: Pt [Ye+++] = ci [Fe++] = c'l [Fe+++] = C2 [Fe++] = c'2 Upon closing the chain with a metallic conductor between the two electrodes, an electric current will generally be produced in this cir- cuit, let us say from left to right within the chain, hence from right to left in the external circuit. We shall also assume that the volumes of the two solutions are sufficiently great, so that no appreciable change in their concentrations will occur on the passage of a small current. When 1 faraday (96500 coulombs) had flown through any cross section of this circuit, the electric work A = Fx will result, where t is the E.M.F. The same effect which had been produced by the current, could result from an osmotic transport in which one mol of Fe+++ in concentration Ci is isothermally and reversibly com- pressed to the concentration C2, which would entail the work, A = RTln - , and at the same time 1 mol of Fe"^ is correspondingly diluted Ci ELECTRODE POTENTIALS 171 from c'2 to c'l the work amounting to RTln —j-. But since the electric C2 work of the current and the osmotic work are maximal, they must be equal to each other, hence Fa- = RTln - + RTln ^ Ci c 2 or, ^,^^ RT, C2c'i IT = E.M.F. = -— - In — - F cic'2 when -T- = Co, then c'2 E.M.F. = ^ In ^^ + ^ In Co = ^ In ^^ + K (1) F ci F F ci where K is a constant characterizing the given chemical reaction. Therefore equation (1) represents also the potential of a single Fe+++ — Fe++ electrode, in which equation an additive constant is left undetermined. B. The Quinhydrone Electrode — An Application. When an incompletely dissociated electrolyte participates in such a reaction as outlined above, some special aspects arise which may be developed into a very useful practical application, as in the case of the quinonehydroquinone electrode. Hydroquinone, C6O2H6, is a very weak dibasic acid. Quinone, C6O2H4, on the other hand, is practically a non-electrolyte. When- ever a mixture of both of those substances is placed in contact with a platinum electrode in a closed circuit the following process takes place : C602Hr ;^ C6O2H4 + 20 Secondary hydroquinone ion Quinone In analogy with equation (1) above, the potential of such a single electrode is RT o' E.f,n|+K (2) where c' is the concentration of the quinone and c is the concentra- tion of the secondary hydroquinone ions. The value 2 appears in 172 HYDROGEN ION CONCENTRATION the divisor of the right hand term because the secondary hydro- quinone ion is bivalent, and, therefore, the transport of F coulombs is equivalent to the osmotic transport of ^ mol. The ratio of the concentrations of the secondary hydroquinone ions and of the total hydroquinone is regulated by the mass law in the following manner: [CeOsHe] where [C6H2O6] means the concentration of the undissociated hydro- quinone molecules. When the reaction is not too alkaline, pH < about 8.5, so that the primary as well as the secondary ions of the hydroquinone form but an insignificant portion of the total hydroquinone, the concentration of the undissociated hydroquinone is practically equal to that of the total h3^droquinone. Hence the last equation may be written [C.O,H.-, = ypp where [C6H2O6] means the total concentration. By substituting (3) into equation (2), we obtain E = — In + K 2F k [CeOoHe] which may be further simplified by substituting Ki for K, so as to include k, thus ^ RT , [CBO2H4] [H+] ^ = T ^" [CeO.He] + '"' When in a series of such different electrodes the ratio [Hydroquinone] . , ^, , 1 ,, rTT_Li • • 1 — rr— : ^ — is kept constant, = K2, and only the [H+j is varied, iQumone] then E. = 5T ,„ Ell + K. - 51 ,0 IH^I + K. F K2 F This means that the potential depends upon the pH, exactly as it does with hydrogen-gas electrodes. By setting up a chain consisting , , 1 • 1 • , , 1 . • [Quinone] . , , of two such electrodes m which the ratio ,^^ ; -■ "i ^^ ^"^ same, HydroqumoneJ ELECTRODE POTENTIALS 173 but the pPI values are different, one of these being known and the other unknown, the latter may be determined as in the ordinary Ho- gas chain. For pH values > 8.5 the above described limitation renders the method unusable. In practice the mixture of quinone-hydroquinone in a constant proportion is replaced by the addition to the solution whose pH is to be measured of a small amount of quinhydrone. This is a molecular addition compound consisting of 1 mol hydroquinone + 1 mol quinone, and which in solution splits into its components. For a more detailed account of the theory of oxidation-reduction potentials and of the use of the quinhydrone electrode the reader is referred to the original literature."* «E. Biilman and collaborators, Ann. de China. 15, 109 (1921); Ibid. 16, 321 (1921). Trans. Faraday Soc. 19, Part 3, (1924). Jour. Chem. Soc. 125, 1954: (1924). S. P. L. S0rensen, M. S0rensen and K. Linderstr0m-Lang. Compt. rend, trav. lab. Carlsberg, 14, No. 14 (1921); K. Linderstr0m-Lang. Ibid. 16, No. 3 (1925). W. M. Clark and collaborators. U. S. Public Health Reports, 38-39„ (1923-4). CHAPTER VII Diffusion Potentials (Potential Differences at Liquid Junctions) SUMMARY of CONTENTS Potential differences arise at the junction of two aqueous solutions which differ in kind or in the concentration of the dissolved electrolytes. These can he explained and calculated on the basis of Nernsfs diffusion theory. The calculation is simple in a few special cases, but in general it is difficult. Since these potentials are of little significance in the realm of physiology, and since they can, in experimental technic, be readily avoided, their calculation may well be omitted in this work. The most suitable method of preventing diffusion potentials is through the use of a salt bridge consisting of a saturated KCl solution. 51. When a solution has in different places a var5dng content of electrolytes a potential diJfference arises between the places of different composition. The difference in composition may either be one of concentration of the same electrolyte or one of the kind of electrolyte involved; or it may consist of both of these variations at once. But the solvent is necessarily the same throughout the solution, and, as a rule, it is water. Cases in which this last condition is not fulfilled will be dealt with in another chapter. Nernst furnished an explana- tion of these potentials on the basis of the ionic theory in the follow- ing manner. Let us assume that we have in the left part of a horizon- tal tube an HCl solution of the concentration Ci and in the right hand part an HCl solution whose concentration is C2. At the junction of these two solutions diffusion occurs, and then a concentration gradient is formed. The relative incline of such a gradient depends upon the difference of the concentrations and also upon the rate of diffusion (or diffusion coefficient) of the diffusing substance. The more rapidly the substance diffuses the smaller the gradient. If two different substances are present in solution, then each develops its own gradient independent of the other. Since HCl in solution is really a mixture of H- and Cl-ions, the same would apply in this 174 DIFFUSION POTENTIALS 175 case, if it were not for the hindering effect of the electrostatic attrac- tion operating between these oppositely charged ions. The more rapidly diffusing H-ions would establish a flatter gradient, the Cl-ions a steeper gradient. But the electrostatic attractions prevent this. The H-ions are retarded by the Cl-ions, the Cl-ions are accelerated by the H-ions, and an average diffusion gradient results. It is true that the H-ions are slightly in advance, but only insofar as the potential difference developed here permits it. Consequently the more dilute solution becomes positively charged with respect to the more concentrated one. The amount of such predominating H-ions is extremely minute and is beyond chemical measurement, and it is only recognized through the potential difference which develops under these conditions. The values of this potential difference can be, according to Nernst,^ calculated in the following way. Let us recall the two ends of the tube the left end of which contains HCl in concentration ci, and the right in concentration C2. Now these two ends are connected by an external metal conductor. The immersed electrodes in this case are assumed to have, for the sake of simplicity, equal and opposite potential differences with respect to the solutions and thus counter- balance each other. Then the E.M.F. of this chain is simply equal to the diffusion potential r. The circuit is now kept closed until F coulombs have passed through the transverse section of the tube. Then the electric work x X F has been accomphshed in a reversible manner. 2 The effect of this work is that a certain amount of H-ions migrated from the stronger into the weaker solution and a certain amount of Cl-ions migrated in the opposite direction; in fact a total of 1 gram-ions had been transported. If the migration-velocity of the H-ions = u, and that of the Cl-ions = v, then of 1 mol of ions the u . v fractions — -; — of H-ions and — ; — of Cl-ions have migrated. The u + V u H- V osmotic work to be obtained when 1 mol of H-ions passes from a solution of concentration Ci into one of concentration C2 is = Ci u u Ci RT In - : hence of — ; — moles H-ions the work is — ; — RT In — , and C2 u -f- V u + V C2 ^ W. Nernst, Zeitschr. f. physikal. Chem. 2, 611 (1888); and especially in Ibid. 4, 129 (1889). ^ On the assumption that the period of time consumed in this experiment is so short that the displacement of substances in the solution through mere diffusion can be neglected. 176 HYDROGEN ION CONCENTRATION correspondingly for the Cl-ions it is = osmotic work derived would then be V Ci — ^ RT In -. U + V C2 U — V Ci —-— RT In -. U + V C2 The total Since the amount of the maximal work to be derived depends only upon the initial and final states of the system, regardless of the way or method of procedure, as long as it functions reversibly, it follows that tt F = u — V _ Ci --— RT In -. U -f V C2 By transposing F to the right side and bj^ making the numerical transformations shown on page 162, we obtain: TT = ^ ~ ^ X 0.0001983 T log - volts U + V C2 This formula agrees quite well with experimental findings. Thus the difference between the migration velocities of the cation and anion is very important in determining the value of tt. Table 25, taken from Kohlrausch, gives the migration velocities of ions at 18 °C. TABLE 25 u= v = H 318 33.44 43.55 64.67 67.5 68. 64. 46. 46. 46. 51. 55. 45. 54.02 OH 174. Li F 46.6 Na CI 65.44 K Br 67.0 Rb I 66.5 Cs CNS 56.6 NH4 Acetate 35. i Zn Propionate 35. i Cu Butyrate 31. A Cd NO3 61.78 i Ca CIO3 64. ^Ba ... IO3 48. iMg Aff TABLE 26 HCl + 0.0378 volts LiCl - 0.0187 volts NaOH - 0.0346 volts NaCl - 0.0116 volts KOH - 0.0265 volts KCl - 0.0004 volts NH4CI - 0.0007 volts NH4NO3 + 0.0011 volts DIFFUSION POTENTIALS 177 It can readity be seen from this table that large values for u — v can only result, when one of the ions is either a H-ion or an OH-ion. Hence, the diffusion potential of two solutions of a binarv^ electrohie whose concentrations are in the ratio of 1:10 have been calculated for 18°C. to be as shown in table 26. This method of calculation is appHcable only when two solutions of differing concentrations of one and the same electrolyte are in contact. For all other cases, the calculation of the diffusion potential is much more comphcated. The presentation of equations generally applicable to any solutions containing any mixtures of electrolytes is a difficult and a still unsolved problem. For the case where two solutions are in contact, one of M^hich contains one electrolyte in concentration Ci (and the velocities of its cations and anions are Ui and Vi) , and the other con- tains a different electrolyte in concentration Co (whose ions have the velocities of U2 and V2), the following general equation for tt has been derived by Planck^: ■K = 0.0001983 T log r where ^ is denned by the following theoretical equation : log log f f C2U2 — CiUi Cl r C2 — Ci C2V2 — f ViCi , C2 , , C; — f Ci log h log f Cl When the concentrations Ci and C2 are equal to each other, then the calculation is simphfied considerably. Then, as can be readily demonstrated, the value .t is easily calculated, and, for two solutions of two different binary univalent electrolytes of the same concen- tration Planck's formula becomes TT = 0.0001983 T log Hl±Z? U2 + Vl where ui and Vi are the velocities of the cation and anion, respectively of the one electrolyte, and, correspondingly, U2 and V2 for the other electrolyte. Thus from this formula we obtain, for example, the 3 Planck, Wiedemanns Annalen 40, 561 (1890). 178 HYDROGEN ION CONCENTRATION figures in table 27 (for tt) for solutions of equal concentrations at 18°C. When more than one electrolyte are present in each of the two solutions the following substitutions must be made in the above formulas : UiCi substituted by u/ci' + Ui'd" + .... U2C2 substituted by U2'c2' + U2"c2" + . . . . ViCi substituted by Vi'ci' + Vi"c2" + . . . . V2C2 substituted by V2'c2' + V2"c2" + . . . . where the subscripts 1 and 2 denote, as before, the two solutions in contact, while the superscripts (', ", etc.) refer to the different kinds of ions. Planck's equations (the first, general one, as well as the second, special one) apply to the case where all electrolytes consist of two TABLE 27 HCl, NaCl 0.0315 volts NaCl, KCl - 0.0044 volts HCl, KCl 0.0268 volts NH4NO3, KCl 0.0006 volts NaOH, NaCl - 0.0173 volts KOH, NaCl - 0.0127 volts univalent ions. When all the ions present are n-valent then the potential amounts to -th of its value as shown by the equation. n But in cases where ions of different valences occur the calculation assumes great difficulties which have not yet been solved. The condition on which Planck based his calculations is that the two solutions which are in contact should not be mechanically mixed and that the above mentioned temporarily stationary concentration gradient had been established. P. Henderson^ making a somewhat different assumption concern- ing the connecting boundary between the two solutions, deduced for certain conditions an equation which has the advantage over Planck's of not being theoretically transcendental. Henderson assumed that the connecting boundary is not the result of spontaneous diffu- sion of the two solutions in contact, but that it ensues through a * P. Henderson, Zeitschr. f. physikal. Chem. 59, 118 (1907); 63, 325 (1908). DIFFUSION POTENTIALS 179 continuous mechanical process of mixing between the two solutions in such a way that the composition of solution I gradually passes through the intermediate mixture zone into the composition of the solution II. The assumption for the temporary constancy of such a potential is that the two solutions have at least one ion species in common. The Henderson equation is (Uj - Vj) - (Uji - Vjj) V{ + Vj' ^ = 0.0001983 • T . _ , , ,/,, — r^^n-TTT. i«g (Uj' + V/) - (u„' + V„') U„' + Vj/ in which Ui = UiCi + U2C2 + U3C3.... Vi = viCi + V2C2 + V3C3 Ui' = UiWiCi + U2W2C2 + U3W3C3. . . . Vi' = ViWiCi + V2W2C2 + V3W3C3 where c = the concentration (for a negative ion, c). u = the mobiUty of a cation and v of an anion, w = the valence (w for an anion). The subscripts I, II refer to the two solutions, the subscripts 1, 2, 3, etc., to the different ion species. This equation was further modified by Gumming. ^ Planck's and Henderson's equations do not yield the same values, except under certain conditions, such as, for instance, when each of the two solutions contains but one salt, in the same concentration, and both having one ion species in common. It is to be expected, therefore, that the diffusion potential should in general represent a somewhat fluctuating value depending on whether Planck's or Henderson's conception of the contact zone is assumed. Indeed, Chanoz''' has shown that the diffusion potential changes with time. Thus when a chain is arranged of the type Electrode 11 Solution I | Solution II | Solution I || Electrode A B we should expect that its E.M.F. would be equal to zero, for here not only the electrode potentials but also the diffusion potentials * Gumming, Transact. Faraday Soc. 8, 86 (1912). 6 Chanoz, Ann. de I'univ. Lyon, Nouv. S6r. 1, 18 (1906), cited from P. Henderson. 180 HYDROGEN ION CONCENTRATION at A and B disappear. But if the zone of contact at A were old and the one at B freshly renewed, then a potential difference of up to six millivolts will develop, which will gradually diminish on standing. This fluctuation of the potential with time has also been found by a number of investigators.'' Recently an ingenious apparatus has been devised by Lamb and Larson^ by means of which they succeeded in maintaining a constantly self renewing fresh junction of the solutions, and they obtained potential values of remarkable reproducibility. Without some such arrangement it can scarcely be even expected to attain the sharply defined boundary surfaces which were assumed by Planck. And since the time effect must be such that the potential gradually diminishes, in practice one frequently finds values for potentials which are smaller than those calculated according to Planck's equation. Recently Fales and Vosburgh (1. c.) have even stated that between a saturated (4.1 N) KCl solution and 0.1 to 1.0 N HCl-solution there arises no measurable diffusion potential at all. Under these circumstances, it must be said that the value of a diffusion potential is not as sharply defined as it should uncondi- tionally be required in concentration chain measurements for anal}d:i- cal purposes, for which it is frequently employed. For this reason it is less important in the practical use of concentration chains to calculate exactly the diffusion potential than it is to arrange the chain in such a way that the diffusion potential should become ex- tremely small and neghgible. The method of achieving this purpose will be described as follows : All these diffusion potentials originate in the differences among ion mobilities. Between electrolytes whose individual ions have equal mobilities no diffusion potential develops. The electrolytes in which the difference between the values of u and v is the smallest are KCl (64.6 and 65.5) and a few others such as NH4NO3 (64 and 61.7). Solutions of different concentrations of KCl or those of NH4 NO3 have therefore extremely small diffusion potentials. Thus ^ A. Weyl, Messung von Diffusionspotentialen konzentrierter Chlorid- losungen. Dissert. 1905. Bjerrum, Zeitschr. f. Elektrochem. 17, 58, 389 (1911). G. N. Lewis and F. F. Rupert, Journ. of the x\mer. Chem. Soc. 33, 299 (1911); A. C. Gumming and E. Gilchrist, Transact. Faraday Soc. 9, 174 (1913) ; H. A. Fales and W. C. Vosburgh, Journ. of the Amer. Chem. Soc. 40, 1291 (1918). * A. B. Lamb and A. T. Larson, Journ. of the Amer. Chem. Soc. 42, 229 (1920). DIFFUSION POTENTIALS 181 between any two solutions of KCl there arises no potential of any practical importance, as these calculated figures show: volts Ci :C2 = 1 : 10 0 . 0004 1 : 100 0 . 0008 1 : 1000 0. 0012 And even these values must surely be much too high for contact boundaries that are not very sharply defined. Hence the diffusion potential arising between any electrolyte solution and a KCl solution is always smaller than the one with a solution of any other salt in which the mobilities of the cations and anions differ very markedly. Consequently it should be possible to diminish greatly the diffusion potential between any two solutions by interposing between them a KCl solution. This effect of KCl increases with the increasing concentration of its solution, since then the conduction is due more and more to the K- and Cl-ions. In most cases then it suffices to interpose a saturated (4J N) KCl solution in order to practically abolish the diffusion potential. A wide use is made of this principle in practical concentration chain measurements. In most cases then the diffusion potential may be neglected when a KCl solution is used as a liquid bridge. And yet in veiy acid (pH < 3) and in very alkahne solutions the aboKtion of this potential is not quite complete. In such cases Bjerrum's^ empirically derived extrapolation method is apphed. The E.M.F. of the chain is measured once with the interposition of a 1.75 M KCl solution and then it is measured again with a 3.5 M KCl solution. If a difference between these measurements is found, this difference is to be taken into account in order to obtain the approximately true value of the potential difference of the electrodes. For example volts p.d. with 1.75 M KCl 0.2048 p.d. with 3.5 M KCl 0.2068 Then the true value is 0.2088 This extrapolation must amount to but a few millivolts, if its claim to certainty is to be assumed. Of all physiological fluids such an extrapolation is taken into consideration only in the gastric juice 9 N. Bjerrum, Zeitschr. f. physikal, Chem. 53, 428 (1905). 182 HYDROGEN ION CONCENTRATION because of its high acidity. It must scarcely ever exceed 1 to 2 niiHivolts. The question is of greater importance in deahng with concentration chains in which strong acids and bases are involved, as for example in the determination of the dissociation constant of water or in the standardization of the normal hydrogen electrode. Here the re- quired extrapolation values are frequently so great that pure acid solutions are never used, but instead we employ acid solutions contain- ing KCl, in order to decrease the diffusion potential. But since this affects the activity of the H-ions, which can still be only approxi- mately calculated, therefore, a practical limitation is hereby imposed upon the attainable accuracy of all pH determinations. It may be said that such measurements are inaccurate to the extent of at least 0.5 milHvolts, which corresponds to a probable error in pH of about 0.01. 52. The physiological significance of diffusion potentials It is evident that electrode potentials do not occur in the living organisms, for there are no free metals present. But the problem arises as to whether or not the potential differences observed in the organism could be explained as being diffusion potentials. But it is out of question that diffusion potentials have any considerable share in these tissue potentials. For the potentials that have been observed between injured and uninjured parts are of the order of magnitude of 0.02 to 0.05 volts and even above. Such high diffusion potentials can only develop under extreme conditions such as never occur in the living organism. In the almost always approximately neutral and NaCl-containing tissue fluids not even a diffusion potential of 0.01 volt can ever arise as will be readily seen from the discussion given above. CHAPTER VIII Potentials at Phase Boundaries summary of contents At the boundary of two phases there generally arises a potential difference which depends upon the specific distribution of a dissolved electrolyte between the phases. If we assume that every ionic species has a characteristic distribution coefficient, then the coefficients of the cations and of the anions of an electrolyte will not be equal. Hence it follows, in the first place, that the distribution law does not hold for the whole electrolyte when it does hold for its individual ion components, and secondly, that a potential difference arises at the phase boundary and that this potential bears a thermodynamically calculable relation to the electrolytic solution pressures of the various ionic species in each phase. The potential difference can be calculated from the concentra- tions of any ionic species common to the two phases, and the E.M.F. of a chain involving two such boundaries may vary, according to the conditions, from zero to a certain maximal value. The latter is com- parable to that of an ordinary concentration chain with metal electrodes. With respect to their electromotive behavior the different ions as well as the solvents are arranged in a serial sequence. These phase boundary potentials appear to be of great physiological importance, for it is prob- able that many of the potential differences found in living organisms are of this nature. 53. The origin and calculation of phase-boundary potentials Properly speaking, electrode potentials are phase boundary poten- tials as well. But here we shall consider as such only those potentials which arise without the participation of metals at phase boundaries. Both phases in the case to be considered shall be solutions — not, as in diffusion potentials, miscible liquids which can become homo- geneous through diffusion, but rather immiscible fluids separated by a sharply defined boundary at the place of contact. These phases can exist side by side in complete equilibrium, whereas the diffusion potential is characterized by a lack of equihbrium. 183 184 HYDROGEN ION CONCENTRATION It will be useful to recall clearly the following points. A current producing chain can only be represented by a system which, when considered as a whole, is not in a state of chemical equilibrium. The E.M.F. of such a chain is the algebraic sum of all the single potential differences present between the poles of the chain. Such a single potential is always found between two contiguous laj^ers. These two contiguous layers may coexist in chemical equihbrium (metal electrode potentials, phase boundary potentials, and see below for Donnan's membrane potentials), i.e., the electric force is com- pensated by an opposing force. Or they may not coexist in a state of equihbrium (diffusion potentials). Two contiguous layers, con- sidered as an isolated chemical system, retain their potential differ- ence as long as they remain in chemical equilibrium (e.g., a single silver electrode immersed into a AgNOa solution) ; and they gradually lose their potential difference when the latter depends only upon the lack of equilibrium (e.g., the boundary layers between a concentrated and a dilute HCl solution which are in contact; when these two solutions become homogeneous through diffusion, the potential difference disappears, without any electric current having been generated) . Two true aqueous solutions never form distinct phases in relation to each other. For this purpose there are required two solutions having immiscible or poorly miscible solvents, e.g., water and oil. As a rule in the following discussion we shall assume an aqueous phase in contact with an ''oil" phase as it was designated by Beutner. A better way of stating this would be: two phases of different dielectric constants. The first fundamental principles for a rational examination of boundary potentials are to be found in a paper by Nernst^ who also in this field must be regarded as the founder of the theoiy; and then also in the equally important investigations of Haber.- The signifi- cance of these investigations of Nernst and of Haber for physiologi- cal processes had not been recognized for a long time by physiologists, although these authors were fully aware of the far reaching physio- 1 W. Nernst, Zeitschr. f. physikal. Chem. 9, 140 (1892); Riesenfeld, tTber elektrolytische Erscheinungen und elektromotorische Krafte an der Grenze zweier Losungsmittel. Diss. Gottingen 1901 ; Nernst and Riesenfeld, Ann. d. Physik. [4] 8, 600 (1902). 2 F. Haber, Ann. d. Physik. [4] 26, 927 (1908); Haber and Klemensiewicz, Zeitschr. f. physikal. Chem. 67, 385 (1909). POTENTIALS AT PHASE BOUNDARIES 185 logical import of their ideas and pointed them out.^ R. Beutner/ a former student of Haber's and under the encouragement of Jacques Loeb, has recently given us a substantial development of this prob- lem, which we shall discuss next. But we must also give full credit to the theories of Nernst and Haber. While there are no fundamental differences between them, they are both indispensable. We shall now show thermodynamically that, in general, between two solutions consisting of immiscible solvents a potential difference must arise, even though they be in a state of chemical eciuilibrium with respect to each other. Let us, for example, shake up portions of water and of guaiacol or of amyl alcohol with some HCl, so that the electrolyte is distributed in the two solvents and is consequently in equiUbrium. The concentration of HCl in the solutions is different. Now we shall transfer the mixture of these two solutions into a U-tube, in which they will sepa- rate with a sharply defined boundary between them. If we now dip some metal electrodes into the upper ends of the solutions and connect them by a metal conductor, we shall obtain an arrange- ment of a galvanic chain (fig. 24). We shall choose for our electrodes hydrogen-platinum electrodes which, in contact with a solution con- taining H-ion, have a well defined potential cal- culable from Nernst 's equation. No electric current can originate in this chain, since it is a system in a state of complete equilibrium. Any current should bring about a change in the concentration of the electrolyte, but since we have here a state of equihbrium, no spon- taneously occurring process can be produced which would result in a disturbance of the equihbrium. This is an important part of the second law of thermodynamics. Consequently the E.M.F. of this chain must be equal to zero. On the other hand, we can also estimate the potentials at the electrodes. At electrode I the potential is Fig. 24 TTi = RT In Si ki ^ Cf. L. Michaelis, Dynamik der Oberflachen. Dresden 1909. * R. Beutner, Die Entstehung elektrischer Strome in lebenden Geweben. Stuttjrart 1920. 186 HYDROGEN ION CONCENTRATION where Ci is the concentration of hydrogen ions in the water, and ki is that concentration of H-ions in the same solvent against which the potential would be zero. At electrode II the potential is Co TTo = RT In — with the corresponding meaning for C2 and k2. The difference between these two electrode potentials is, therefore, (I) This potential difference can become equal to zero when 2L - hi C2 ka But since — represents a constant value, therefore, the ratio - must also be constant, i.e., when any acid is distributed in any con- centration between an aqueous and an oil phase with the establish- ment of an equilibrium, the ratio of the H-ion concentrations in the two phases will remain constant. It can be equally well shown that when chlorine electrodes are taken instead of hydrogen electrodes the ratio of Cl-ion concentrations in the two phases will remain the same whatever chloride in whatever concentration be involved. Briefly stated : it can be shown that any given ionic species must be distributed between an aqueous and an oil phase in accord with its unalterable distribution coefficient. But this is not possible, as can be demonstrated both practically and theoretically. The theoretic proof of this impossibihty can be given in the sense of Nernst's theory as follows: If we first assume that every ion-species, as well as any other molecular species, has a definite distribution coefficient, then it should certainly be different for the various ion-species. Therefore, in the case of HCl, the H-ions will have a different dis- tribution coefficient from that of the Cl-ions, and the two phases must have different distributions with respect to H- and Cl-ions. But for electrostatic reasons this is impossible. Neither can the true distribution correspond to the specific coefficients of the CI- or H-ions, but it must He somewhere in between these. And, hkewise, this true distribution must be different in an HCl solution from POTENTIALS AT PHASE BOUNDARIES 187 that in an aniline chloride solution. Hence it is not possible that the distribution of Cl-ions should always be the same regardless of the kind or of the amount of chloride present, and consequently the ratio — cannot be constant under all conditions. C2 Practically, this point was demonstrated by R. Beutner as follows : when water and an oil phase, such as guaiacol, were shaken on the one hand with NaCl, and on the other hand with aniline chloride, it was found that the aqueous phase took up more CI when NaCl was used than when aniline chloride was used. Evidently that which apphes to CI as such need not by all means apply to Cl~-ions, and the same was found when the conductivity of the guaiacol solu- tions in the two cases was determined. In the case of the aniline chloride the Cl-ion content of the guaiacol was much higher than in the case of NaCl. Hence the distribution of Cl-ions, or, in general of any ion species, between water and guaiacol is not in all cases the same. Since the ratio Ci/c2 in equation (I) represents a constant value, and ki/k2 being a constant, then tti — 7r2 must always be other than zero. But since the E.M.F. of the whole chain must be zero, it follows that a third potential difference tts must be present, and it must amount to tti — W2. The algebraic sum of these three values must be equal to zero. This potential xs resides in the boundary surface of the water and oil phases. The potential difference at the boundary of two phases which contain an ion in common in the concentrations Ci and C2 is TT RT Here k2 is that concentration of this ion in phase II against which a metal electrode of the same kind would produce zero potential, and ki is the concentration of the same ion in phase I against which a metal electrode of the same kind would likewise 3deld zero poten- tial. Or, stated otherwise, ki and k2 are the electrolytic solution tensions of the ionic species concerned in phases I and II respectively. Thus these two values, k2 and ki, are quite problematical quanti- ties which frequently have no physical meaning. For, if we should wish to use, for example ammonium acetate as the electrolyte, we find that we have available neither an "ammonium-metal" nor an 188 HYDROGEN ION CONCENTRATION acetate-ion yielding gas to use for our electrodes. And so k2 and ki, are in reality nothing but integration constants for purposes of calculation. In fact, k2/ki is but a single constant to which we attribute the same significance. Hence we can state that ^&^ TT = RT In — + K (1) C2 And yet this constant K does possess important meaning, its value relates specifically to the ion-species, being a definite charac- teristic value of every ion-species. If a current-yielding chain be arranged to contain two such phase boundaries with a potential at each, then the total potential E of the system will be E = ,r - TTi = RT In — + K - ( RT In ^' + K' ) C2 y C 2 J Only in that case, where all the phases have only one ion-species in common will the individual values of K be identical and disappear and Ci, C2, c/, 0,% will then signify only the concentrations of the common ion. But if the various phases involved contain more than one ion- species in common the equation can just as well be appHed to each of these ions. But since in all combinations of phases in which water is included H-ions are present, it would follow for all such chains rTT+] TT = RT In —-^ + Constant IH'^J2 where the constant depends upon the nature of the solvent only. This result may be easily deduced from Beutner's general equation (even though Beutner himself did not come to this conclusion). Nevertheless it is still very difficult to determine experimentally the hydrion concentration in non-aqueous phases. So much for the thermodynamic explanation of phase boundary potentials. It will be worth while to add another method of repre- sentation, that of Nernst, which is the older one in point of time. It follows from Nernst's conception of the distribution of the salt in the two solvents. The final state attained corresponds, as was shown above, neither to the true distribution equiUbrium of the POTENTIALS AT PHASE BOUNDARIES 189 cations nor to that of the anions. If KCl is used, the oil phase will contain either too much K+and too little Cl~, or perhaps the reverse. There will be a tendency in the system to approach the true equi- librium for each of the ionic species. Since because of this tendency a separation of the ions is required, a difference of potential will develop. And, furthermore, the separation will be accompanied by the formation of an electric double layer of ions whose potential will be such that the separating force will be counterbalanced by the attraction between the two layers of ions. The calculation of this effect can be carried out in some such manner as the following.^ Suppose that equilibrium having been reached the concentrations of the electrolyte in the two solutions are Ci and C2. Now the cations tend to distribute themselves in the ratio of lio: where a is the true distribution coefficient of the cation in question, and the anions likewise tend to distribute themselves in the ratio of \:^. This tendency is opposed by the electrostatic forces which develop because of this unequal distribution of ions. When one mol of an ionic species migrates from solution 1 to solu- tion 2 (in order to attain true equihbrium), then an electric current is produced yielding work amounting to ttF. If the same replace- ment or transposition of ions occurred mechanically (osmotically) C]Q: then the amount of work RT In — would be produced. We can C2 thmk of this transition as occurring in two stages. First one mol of cations is transferred from one medium in which its concentra- tion is Ci into a medium of another kind where the concentration is Cia. This process does not require any work. Now we dilute the latter solution from concentration cia to C2 which requires the Cior amount of work RT In — . The actual stable distribution which C2 ensues must be so defined that RT , ci a ■R = — In F C2 By the same reasoning we arrive at the second condition, namely, RT , ci /3 TT = — — — In F C2 • As given by Michaelis; cf. footnote 3. 190 HYDROGEN ION CONCENTRATION and hence RT , ci • a RT , ci . /5 TT = -—- in = — In F Co F cj Therefore, whenever a = ^, then tt = 0. By determining analytically (by conductivity measurements or similar methods) Ci and C2, we derive the following two equations de- fining the three quantities, t, a and /S: RT ci ' a TT = -— - In F C2 Ci • a C2 „ C2 ^ / = y/ a • p or C2 ci |8 ci If we compare the former equation RT , ci RT , k2 TT = — In — — In -- F Co F ki with the present equation which we will now write in the form of RT , ci , RT , we see at once that k2 — = ot ki and that likewise for the second ion-species it is ki Thus we see that the distribution coefficient of an ion-species bears a definite, and a thermodynamically calculable, relationship to the above defined values k2 and ki. This principle can be restated as follows: The oil-solubility (in terms relative to water-solubility) of an ion- species bears a thermodynamically deducible relationship to its elec- trolytic solution tension in the oil. A truly amazing conclusion! The relation of solubihty to electrolytic solution tension can also be derived in the following way: POTENTIALS AT PHASE BOUNDARIES 191 Let US take a special case in which the two ion species are distrib- uted between the phases I and II according to their true distribu- tion coefficients. (This is only possible when both the anion and cation of the electrolyte used have the same distribution coefficient a). Then each of the ions is in concentration ac in phase I and in concentration c in phase II. Now, if we dip reversible metal elec- trodes into the two solutions, we obtain as the electrode potential Ca • C on the left, tti = RT In—, and on the right 7r2 = RT In- , where A ij A and B denote the electrolytic solution tensions on the left and on the right side respectively. Since in this case a boundary poten- tial cannot develop, and hence the chain cannot yield a current, it follows that TTi — 7r2 = 0, i.e. : RT In — = RT In - A B or a _ J. A ~ B Stated in words this means: Every ion-species tends to so dis- tribute itself between two phases that the ratio of its concentrations (or better, its active masses) in these phases should be the same as that of the electrolytic solution tensions of similar electrodes in the same phases. The attainment of such an equihbrium, as was said above, is only then possible, when the distribution coefficients of the anion and cation are equal. In general, however, this kind of distribution is prevented by the resulting potential difference. Then from the joint action of the two partial forces, the tendency towards distribution and the electrostatic force, a kind of equilibrium results, which is different from the one to be expected on the basis of the true distribution coefficients. 54. Phase boundary chains The same happens with a single phase potential as with a single electrode potential: it does not generate any electric current. Only a combination of two different metal electrode potentials or of two 192 HYDROGEN ION CONCENTRATION different boundary potentials produces under certain conditions (not always) a curtent-jdelding chain. One such arrangement would be: + Na-oleate, aqueous solu- tion II Na-oleate in oil (oil layer rich in salt) III IV NaCl in oil NaCl, (oil laj^er aqueous poor in salt) solution TTl 7r2 The lack of equilibrium between two contiguous layers will be found in this system at 2. But this is not a phase boundary and it can only be the seat of an ordinary diffusion potential. As was stated above (page 176), such a potential for the common salts may amount at most to a few millivolts, while this chain produces several centivolts. We have here two phase boundaries, both behaving as metal electrodes; and the difference between their potentials constitutes the E.M.F. of this metal-free chain, which could be closed by means of a saturated KCl solution and thus rendered free of diffusion potentials as well as metal-free. In order to obtain large E.M.F. values it is important to choose an inorganic salt and an organic salt both of which are easily soluble in the oU phase. This is comparable to a metal-containing chain in which metals as far apart as possible in the electromotive series are chosen as electrodes. The calculation of the E.M.F. of a chain of this type is begun by finding that ion-species which is common to all the phases. In this case it is the Na-ion. This chain can be designated as "a reversible chain with respect to Na+." If now we designate the concentra- tions of Na+ in the four vessels I, II, III and IV of the chain as Ci, cii. Cm and Civ respectively, we shall have at boundary sur- face 1: XI = RT In — + K and at the boundary 3: ■"■2 = RT In ai -IV ^m + K POTENTIALS AT PHASE BOUNDARIES 193 hence the total E.M.F. of the chain is E = TTi - TTo = RT • In "II "III "IV = RT In — "IV on ^11 The potential at the zone of contact 2 is of the character of a diffusion potential, and in most cases may be disregarded as being negligible, especially if it is remembered that the great mobility of the H- and OH-ions, which in aqueous solutions is the important source of all diffusion potentials, is non-existent in oily solvents. R. Beutner (1. c.) verified this equation experimentally in the following way: Guaiacol, whose specific conductance is 0.1 reciprocal megohms (r.m.), is shaken up with 0.1 M aqueous solution of NaCl. After the distribution equilibrium is established the conductivity rises to 1.2 r.m., an increase of 1.1 r.m. Then another portion of the guaiacol is shaken up with a 0.1 M aqueous solution of dimethyl- anihne hydrochloride. The conductivity rises to 59.0 r.m., an increase of 58.9 r.m. Since the conductances of the various ion- species are not very different, we may take the ionic concentration as being proportional to the conductance. The conductance of the pure guaiacol must be, of course, subtracted. Then the E.M.F. of this chain was determined: Calomel 0.1 N NaCl NaCl in dimethyl- 0.1 iV dimethyl- Calomel electrode in water guaiacol aniline— HCl in guaiacol aniline— HCl in water electrode "U "in + "IV The common ion is Cl~, and we obtain Cjii 58.9 = 1; "IV ^u 1.1 hence E = 0.0001983- T- log 58.9 1.1 = 0.100 volt The observed value was 0.091 volts. The figures in table 28 were obtained in a similar way for guaiacol 194 HYDROGEN ION CONCENTRATION chains, in which other electrolytes instead of dimethylaniUne- HCl were employed (the NaCl side being left unchanged). If it is kept in mind that the conductance is but an approximate measure of the ionic concentration, it becomes evident that the calculated and observed values are in very significant agreement. Table 29 shows the results of a few more of Beutner's experiments with p-cresol. These relationships represent perhaps some of the most important recent advances in physiological physics, for here we have succeeded for the first tune in obtaining electromotive TABLE 28 Conductance of guaiacol after shaking r.m. E.M.F. in volts Calculated Found 0 1 M Dimethylaniline HCl 59.0 10.3 1.9 0.3 -0.100 -0.056 -0.012 +0.043 -0.091 0. 1 M Aniline HCl -0.059 0.1 M KCl -0.011 0.1 MMgCla +0.045 TABLE 29 After shaking with Conducts ance of the oil phase Calculated E.M.F. (in volts) for the chain: Observed E.M.F. KCl 8 r.m. 13.2 19.3 6.7 80 630 1300 -KCI/KNO3+ 0.013 -KC1/KSCN+ 0.022 -KCl/NaCl+ 0.005 -NaCl/Na-Salic.+ 0.063 -NaCl/Na-01eate+ 0.117 +KCl/Dim-Tol-ClH-0.128 0.009 KNO3 0.025 KSCN 0.009 NaCl 0.081 Na-salicvlate 0.120 Na-oleate Dimethyl toluidine-HCl . . -0.119 forces of the order of magnitude of those observed physiologically, without the participation of free metals. It is also possible to arrange chains with phases which have no ion in common, as, for example, the following: Na2S04 in water Oil KCl in water The oil contains Na2S04 in its left boundary layer in the corre- sponding distribution equilibrium, and KCl, in a similar way, in Na2S04 NaCl Oil Oil in water in water POTENTIALS AT PHASE BOUNDARIES 195 its right boundary layer. The E.M.F. of such a chain can be cal- culated more easily by imagining an aqueous NaCl solution inter- posed in the following manner: KCl in water By the method outlined above we can calculate the values (tti — Wi) and (tts — TTa). Since the total E.M.F. is not affected by the interposition of the NaCl solution, it follows that Tra = X3. Hence the total E.M.F. is where each of the terms is calculable. 55. The direction of the current in phase boundary chains In Beutner's chains, in which an oil phase is the middle member, the direction of the current may vary. It depends upon whether cation or anion is considered the common ion species. Thus in the arrangement : Water + NaCl Oil + salicylic acid Water + NaCl (ci > C2) Ci C2 the electrolytes distribute themselves in the vicinity of the boundary surfaces, as soon as the phases are brought into contact. A trace of NaCl passes from the water into the oO, and a small amount of sahcylic acid passes from the oil into the aqueous phase. In the oil phase, the mixture NaCl + salicyUc acid is partly changed into Na-sahcylate + HCl. In the aqueous phase this rearrangement does not occur, for here the weaker salicyhc acid does not displace the much stronger hydrochloric acid from its salts. In the oil, on the other hand, as it was pointed out on page 151, the difference between weak and strong acids is not nearly so pronounced, and the above change may and does occur. Hence, as a result of electro- lytic dissociation, we find in the oil phase the following ions: Sali- cylate", Na+, and traces of H+. The composition of the oil phase is not homogeneous, for each of its boundaiy surfaces attains a 196 HYDROGEN ION CONCENTRATION state of equilibrium with respect to its adjacent aqueous phase. Actually, the ionic distribution may be represented as: Water Na+ (ci) CI- (ci) Salicylate H+ in traces Oil Salicylate" Na+ Na+ (c'l) (c'2) Water Na+ (C2) CI- (C2) Salicylate" H+ in traces If we neglect the ions present in traces only the arrangement becomes: Water Oil Water Na+ (ci) Salicylate- Na+ (C2) CI- (ci) Na+ Na+ CI" (ca) (c'l) (c'2) Hence, the ion common to all three phases is the Na-ion only. We can, therefore, trace the formation of the potential differences at the phase boundaries to the tendency of Na-ions present in the oil, in a state of electrostatically evoked equilibrium, to approach the true equilibrium, such as would ensue in the absence of electro- static opposition, by repassage into the aqueous phase. The free Na-ions upon reaching the water phase render it positive with re- spect to the oil, and that side of the water phase is the more positive which is adjacent to the part of the oil phase containing the higher concentration of Na-ions. If, however, the common ion is a nega- tive ion, as in the chain: Water Oil Water Na+ (c) Aniline-ions Na+ (C2) CI- (C2) CI- (c'l) CI- (c'2) CI- (Co) then the same considerations apply to the Cl-ion, and the side which was the more positive in the former chain becomes in this chain the more negative. This is the manner in which the direction of the current is determined in these chains. 56. The concentration effect When two aqueous solutions containing an electrolyte in different concentrations, Ci and C2, are separated by an oil phase, then, according to our experience, the E.M.F., of such a chain may He POTENTIALS AT PHASE BOUNDARIES 197 within two definite limiting values. The one limiting value is that E.M.F. which would be jdelded by an ordinary concentration chain containing a metal for the common ion species involved, Ci • or = RT In — , with the sign depending on the positive or negative C2 charge of the ion. The other limiting value of the E.M.F. is zero. Between these two fall all the observed values. The necessary condition for the maximal concentration effect is that the concentration of the cormnon ion-species in the entire mxiss of the oil phase should be constant, although the concentration of this common ion in the two adjacent aqueous phases is different. For when in the equation given on page 193 ''i ^iii E = RT In -!- X -^ Cm becomes = Cjv, and we actually obtain E = RT In - C2 K this condition is incompletely or not at all fulfilled, as when the concentration of the common ion is weaker on the boundary of the oil phase with the weaker aqueous solution than it is on the boundary of the oil with the stronger solution, or when, in other words, Ci < Cjy and Cu < Cm, then E < RT In — . C2 Let us see just when this condition for the maximal concentration effect is fulfilled. 1. The first example of the maximal concentration effect is the so-called glass-chain, which has been first used by Cremer.^ This author was the first to conceive the idea of imitating a physiological membrane by means of an extremely thin walled glass sheath, which can be easily blown out at the end of a test tube. In this way glass bulbs are obtained whose walls are a few hundredths of a millimeter in thickness, and which show at room temperatures, or better at 40-50"", a distinct^ demonstrable electric conductivity. In accord with aheady estabhshed ideas concerning the conductivity of glass, Cremer had to assume that he was dealing with true elec- 6 M. Cremer, Zeitschr. f. Biol. 47, 1 (1906). 198 HYDROGEN ION CONCENTRATION troljd^ic conductivity in which ionic transport was involved. For E. Warburg^ had abeady shown that at 200°C the conductivity of glass was of electrolytic nature, and undoubtedly Cremer was justified in interpreting the conductivity of much thinner glass walls as being of electrolytic nature. Cremer made the striking observation, which was not appreciated until much later, that an aqueous acid solution inside of his thin glass bulb showed a poten- tial difference of about one volt towards an alkaline solution outside of the bulb (the value observed was 0.55 volts, to which a correction for the diffusion potential had to be added). Cremer recognized the glass wall as the source of this electromotive force, and he also quite correctly recognized the great significance of the acidity of the solution in determining the value of this E.M.F. But his interpretation is somewhat different from that given later by Haber. Haber and Klemensiewicz^ were the first to advance the theory which is still held today, and they made exact meas- urements and extensive application of the glass chain. When two solutions of unequal hydrion concentration are separated by an extremely thin glass wall, a potential difference arises between the two solutions which is of the same mag- nitude as that developed in a hydrogen gas chain as described on page 161. The arrangement of the glass chain is as shown in figure 25. The solution of hydrion concentration h2 is placed in a large beaker the solution of hydrion concentration hi is in a very thin walled glass bulb which is immersed into the first solution. The current is led away on one side by means of a calomel electrode and on the other by a platinum wire (instead of the latter it would be better to use a calomel electrode also, in order that the two electrode poten- tials compensate each other). E is an electrometer, best of all a binant electrometer. If in a series of experiments the inner solu- tion be kept constant and the outer solution varied, we find for a ^ Cited by Georg Meyer, Wiedemanns Ann. 40, 244 (1890). « F. Haber and Klemensiewicz, Ann. d. Physik [4] 26, 927 (1908) Fig. 25. Glass-chain POTENTIALS AT PHASE BOUNDARIES 199 tenfold change in h2 a difference of 58 millivolts. All other ionic species which may be present in the solutions are without effect, as was recently stated by Freundlich and Rona.^ The glass chain may be used as well as the gas chain for pH measurements, and especially for electrometric titration for which Haber gives beautiful examples. The glass chain represents a chain reversible for H-ions only, in distinction from Beutner's oil-chains which, as we saw above, can be arranged to be reversible for any given anion or cation. The only possible explanation, in Haber's sense, is that the H-ion constitutes the common ion for the two aqueous phases and for the glass phase, and that the concentration of H-ions in the glass is constant and is independent of the adjacent TABLE 30 C2 E.M.F. Cl Observed Calculated maximal effect* M M volts volts i tV 0.021 0.040 tV 1 0.025 0.040 A U^O 0.034 0.040 ^55 TsVo 0.041 0.040 * Ci:c2 ratio in all of these cases is 1:5. effect is TT = 0.058 X log 5 = 0.0406 volts. Hence the calculated maximal media. Haber assumes that the glass contains traces of water of imbibition (swelHng), and that this factor determines the H-ions in the glass. 2. As we saw above, Beutner succeeded in obtaining more or less complete concentration effects by employing oily liquids as the middle phase. The first used by him were chains with salicyhc aldehyde as the oil phase: KCl in water Cl Salicylic aldehyde KCl in water Cs with these he found the E.M.F. values given in table 30. ' H. Freundlich and P. Rona, Sitzungsber. d. Preuss. Akad. d. Wissensch. (1920), 397. 200 HYDROGEN ION CONCENTRATION It is a striking result that at the lower salt concentrations the effect obtained approaches the calculated maximal effect, while at the higher concentrations the observed values are increasingly- low. The obtained values were temporarily completely reversible; and in a series of experiments when ci was kept constant while C2 was varied in both directions, Beutner always obtained the same E.M.F. values for the same values of C2. As the necessary condition for the maximal effect we specified above that the concentration of the common ion should be the same at the two sides of the oil phase, while it is unequal in the two aque- ous phases. Let us now see how this condition is fulfilled in this particular chain. Salicylic aldehyde is decidedly acid in character, partly because of its phenolic OH-group and partly because it always contains traces of salicylic acid. In contact with water ahnost none of these acid groups pass into the aqueous phase, since the distribution coefficient with respect to water is here extremely small. But a small amount of KCl passes from the contiguous aqueous phase into the oil phase. But, as we saw in the chapter on dissociation in non-aqueous solvents, hydrochloric acid in oil does not behave as a strong but as a weak acid, and hence we have in the oil phase the double decomposition: KCl + salicylic acid ?=i K-salicylate + HCl a reaction which could not have occurred in aqueous solution. In this way the KCl equilibrium is disturbed, the sahcyfic acid attract- ing, to a certain extent, more K-ions into its solution. If in the oil phase sahcyfic acid should be a stronger acid than HCl, then the former will not remain in the free state, but it will attract enough potassium to transform all of the acid in the oil into the K-salt, independently of the KCl concentration in the aqueous phase. In this case, the K+-concentration will be the same throughout the oil phase. But should the saficylic acid in the oil solution not behave as a stronger acid than HCl, but rather dissociate to the same extent as HCl, even then the same result will ensue, with greater or lesser completeness, depending upon the conditions. When the KCl concentration in the water phase is very small, then, after the establishment of equilibrium, the K+ will be present in the oil almost entirely as K-salicylate. But when the KCl con- centration in the water is very great, then it will also be greater in POTENTIALS AT PHASE BOUNDARIES 201 the oil, and the potassium will be present in the oil not only as the sahcylate but also, to a relatively marked degree, as the chloride. In the latter case the total amount of K+ in the oil phase can no longer remain independent of the HCl concentration in the water phase. This is why it happens that the concentration effect ap- proaches its maximal value the more closely the more dilute the aqueous KCl solutions, which was experimentally confirmed, as shown above. When NaCl was used instead of KCl the E.M.F. values shown in table 31 were found. The concentration effect is smaller in this case, but the effect of the absolute salt concentration is still quite obvious. TABLE 31 C) cs E.M.F. in volts Observed Calculated 1 TO 1 BO 1 ^50 TTSBO 0.024 0.026 0.030 0.040 0.040 0.040 Another oil-chain in which the E.M.F. values closely approached the maximal effect was the following: Water dimethyl aniline — HCl Ci Salicylic aldehyde Water dimethylaniline — HCl C2 (Ci > C2) Cl C2 E.M.F. Observed Calculated molar 0.1 0.02 malar 0.02 0.004 volts 0.042 0.037 volts 0.040 0.040 In this case also the acidic nature of the oily solvent is responsi- ble for the constant concentration of cations (i.e., dimethylanihne- ions) within it, in spite of the differences between Ci and C2. 202 HYDROGEN ION CONCENTRATION On the other hand in the chain Water Salicylic Water Na-salicylate aldehyde Na-salicylate Ci = 0.1 molar C2 = 0.5 molar the observed E.M.F. value was only 0.005 volts, instead of the calculated maximal value of 0,040 volts; thus we have here an approach to the other hmiting value of R.M.F. = 0. This is due to the fact that, in this case, the anion, the salicylate-ion, is so much more soluble in the oil phase than the cation, that it predominates in the distribution. But being an anion it cannot be fixed, as a cation would be, by the acidic oil. Hence the necessary conditions for the maximal effect are lacking; on the side where the aqueous phase contains more sahcylate-ions, the oil also contains more of them; or, the concentration of saHcylate-ions in the oil phase is not invariable. The theoretical considerations which determine whether and how far the E.M.F. approaches this or that limit may be stated as follows: The salt tends to distribute itself between the phases. If this dis- tribution is affected by the simultaneously occurring separation of oppositely charged ions, then the difference of potential arises. But if no or equal electrostatic forces interfere on both sides of the oil phase, then no E.M.F. is developed. The last example, with Na-salicylate, presents the simplest case. If the above theory is correct, then it should be experimentally de- monstrable that in this case the common ion can distribute itself according to a definite distribution coefficient and unhindered by other forces. In other words, it must be shown that in those cases where the E.M.F. = 0 the common distribution law for electrolyte is valid and that the ratio of its concentrations in the two phases is invariably independent of the absolute amount present. This demonstration was given by Beutner for the chain arranged with salicylic aldehyde and sodium salicylate: 12.5 cc. of salicylic aldehyde were shaken with 50 cc. of a 0.1 molar aqueous solution of sodium salicylate until the distribution equilibrium was estab- lished. The conductance of the oil phase which was originally 0.7 r.m. is now 2.56 r.m. On using a 0.5 molar solution of sodium salicylate instead of the 0.1 molar, the conductance is 14.46 r.m. These two observed conductances are in the ratio of 1:5.6, while the ratio of the concentrations of the two aqueous solutions is 1:5, POTENTIALS AT PHASE BOUNDARIES 203 Since the conductances are here closely proportional to the concen- trations, it is hence demonstrated that distribution law is appHc- able to the partition of sodium salicylate between an oil and an aqueous phase. Consequenth^ no concentration effect can arise. On the other hand, when dimethylaniline-HCl was used in such a chain the relations were as follows: conductance Salicylic aldehyde shaken with 1 molar aqueous solution of dimethylaniline-HCl 78 r.m. Salicylic aldehyde shaken with 1/50 molar aqueous solu- tion of dimethylaniline-HCl 19. 9 r.m. In this case, while the aqueous concentrations were in the ratio of 1 :50, the ratio of the ionic concentrations in the oil phase was 1 :2.5, i.e., the distribution law did not apply here. The ionic content of the oil phase remained constant and unaffected by the concen- tration in the external aqueous phase. The same conditions are true for Haber's glass chain. Here the common ion is the H-ion, whose presence in the glass is probably due to the dissociation of the traces of contained water. But since only the pure water, and not the dissolved electrolytes, penetrates into the glass, the H+-concentration of the glass remains constant and unaffected by that of the aqueous phase. The case of the usual concentration chain may also be interpreted in the same sense. Being given two phases, viz., metallic silver and an aqueous solution of AgNOs, it may be said that the silver contains freely motile electrons (since it possesses the metallic property of conductivity), and hence it may be said to contain "Ag-ions." The metal represents, as it were, a phase consisting of a silver salt, whose cation is Ag"'"and whose anion is its free electron. The aqueous solution, on the other hand, contains only Ag- and NOs-ions, the electrons do not penetrate into the water phase, they are "not water-soluble." Therefore the concentration of "Ag- ions" in the metal is independent of the Ag+-concent ration in the solution and is invariable. Hence a metal must yield a maximal concentration effect, when it is arranged in the order of an oil chain. The E.M.F. of the chain AgNOa Cx Metallic I AgNOa Ag I C2 204 HYDROGEN ION CONCENTRATION when its ends are led off by means of two similar metal electrodes (such as mercury-calomel-electrodes in 0.1 N KCl) may be easily calculated. All electrode potentials are removed, with the excep- tion of those on the two sides of the silver plate. These two sides, Ci according to Nernst, have in fact the potential difference of RT In—, C2 which means that this chain is in reality one form of arrangement of the usual (Nernst's) concentration chain. It can be imagined that the metal silver strip is split into two plates which are con- nected by a wire, and one of which is in contact only with the left and the other only with the right solution. The whole of this ar- rangement represents the usual concentration chain into whose circuit are introduced two calomel electrodes, the potentials at which are mutually aboHshed. It is hence permissible to neglect these two calomel electrodes, and then the above arrangement be- comes a typical concentration chain for Ag-ions. This demonstrates that Nernst's concentration chain ma}^ be conceived of as a special case of the phase boundary chain. The following is the simplest case in which the concentration effect = 0. Two aqueous solutions of different concentrations of the same electrolyte are separated by an oil. The oil is chemically completely inert, is neither acid nor basic in character and does not enter into any chemical reaction with the electrolyte. In such a chain the E.M.F. is zero, in spite of the concentration differences, as in the chain. Water Indifferent oil Water + KCl + KCl + KCl (Ci) (c'l) (c'2) (C2) of ions. [K+] = ci [K+] = c'l [K+] = c'2 [K+] = = C2 [C1-] = ci [CI -] = c'l [C1-] = c'2 [C1-] = = C2 The essential point involved in this case is that K+ concentration is everywhere in the chain equal to the Cl~ concentration. In this chain neither K+ nor CI" may be regarded as the common ion. In the case of the K+ we should have as was shown on page 193. E = RT In ^' X ^^ C2 Cl POTENTIALS AT PHASE BOUNDARIES 205 In the case of the Cl~ ions we should have, because of the nega- tive charge of the ion: E = - RT In ^^ X ^^ C2 Cl Both equations must be correct, and this can onl}^ be possible when E = 0 (or, what amounts to the same thing, when the frac- tion following In is equal to 1). Only whenever it happens that the concentration of the K-ions in any one of the four parts of the chain is different from that of the Cl-ions, can the E.M.F. be other than zero. Such is the case when a second electrolyte is present, or when the oil phase itself is of an acid or basic character. 57. The ionic series The chains considered above contained at both ends solutions of the same electrolyte in different concentrations, and insofar they are analogous to metal concentration chains. But if we employ different electrolytes at the two ends, then we obtain an analogy to chains containing different metals. As the simplest possible case we shall choose that in which two different electrolytes, but possess- ing an ion in common, are used on the two sides in the same con- centration, as, for example, Water NaCl or, Water NaCl Oil Oil Water KCl Water NaSCN In the first case the common ion is the anion, in the second the cation. Here the E.M.F. of the chains depends upon the difference in the solubihty of the two electrolytes in the oil phase. If we as- sume that the Cl~ is more soluble than the Na+ ion, then the oil should become negatively charged with respect to the NaCl solu- tion. But if it should happen that on the other side of the oil the difference between the oil-solubility of Cl~ and that of K+ is smaller, then the KCl solution will become less positively charged with re- spect to the oil. Hence the KCl solution becomes the negative pole of this chain. Hence, if in a chain of this order, we should al- 206 HYDROGEN ION CONCENTRATION ways use NaCl at the left end var5dng the salt at the right end, the latter will be negative whenever the difference between the oil- solubiUties of its two ions is less than that between the oil-solubiHties of the two ions of the NaCl solution; and it will be positive when this difference is greater. If we should vary only the cation on the right side solution, the right end will become the more negative the more oil-soluble is the cation in comparison with the Na+ ion. If only the anion is varied, then the right end will become the more positive with the increasing oil-solubiKty of this anion in comparison with the Cl~ ion. In Beutner's experiments shown in table 32 0.1 N NaCl solution was used uniformly on the left side (of the oil) and on the right 0.1 iV solutions of the salts indicated in the first column of TABLE 32 Salt (a) NaCl (b) Nal (c) NaSCN (d) Dimethylaniline-HCl (e) KCl (f) CaCU Acidic oil phase Basic oi Salicylalde- hyde containing salicylic acid Nitrobenzene + i Mol o-Nitroben- zoic acid Nitrobenzene + |Mol Dimethyl- aniline I II III millivolts millivolts millivolts 0 0 0 0 + 3 +68 + 15 + 16 +92 -110 -210 - 6 - 24 - 44 0 + 40 + 2 - 3 o-Toluidine IV millivolts 0 + 133 + 140 - 25 0 + 10 the table. In experiments a, b, c the anion was varied. It will be seen that the positive effect increases in the series Cl~, I~, SCN~. We must therefore conclude that this also represents the sequence of the increasing oil-solubility of the ions. In fact, the ionic series remains unaltered regardless of the oil used for the middle phase of the chain. It is true that the quantitative differences between the ions may not remain the same, but the sequence in the series is unchanged These ionic series, which we encounter here for the first time and to which we shall return later more extensively, are as follows: Decreasing negative effect, — decreasing oil solubility Cations K Na Ba Ca Mg Increasing positive effect, — increasing oil-solubility Anions SOi CI Br I SCN POTENTIALS AT PHASE BOUNDARIES 207 The nature of the oil in the middle phase is not quite as immaterial as might appear from this tabular summary, for the differences between the individual ions within the series are accentuated to various degrees depending upon the oil. Thus, if the oil is of an acid character (either by itself or in virtue of an added oil-soluble organic acid), then the differences between the cations are strongly marked, while those between the anions are shght. In a basic oil (or with an added organic base) this effect is reversed, as would be expected from the stated theory. But apart from this general rule specific effects of the nature of the oil manifest themselves. Thus in a chain with NaCl at one end and Na2S04 at the other we should expect from the above rule the Na2S04 side to be negative. Such is the case with o-, m-, p- cresol, phenol, guaiacol, acetophenone, acetoacetic ester (all of which jdeld an E.M.F. of 20-30 millivolts); aniline, toluidine (which yield even higher E.M.F. values). On the other hand, benzaldehyde and salicylaldehyde give E.M.F. values of nearly zero, while with cin- namic aldehyde, anisaldehyde and benzophenone even a positive charge of the Na2S04 side was found. Evidently the above rules are obeyed b}- all of those oils which are definitely acid or basic in character in virtue of NH4 or OH-groups; while with oils of a chemically indifferent character the relationships are not so un- equivocal, which is quite comprehensible in view of our earlier explanations. 58. The electromotive series of the oils We have hitherto considered chains in which the asymmetry of arrangement lay in the difference between the two aqueous phases. But chains can be had whose aqueous phases are of the same com- position and whose oil phase is asymmetrical, the latter consisting of two oils, as, for instance, I II III Aqueous phase | Oil I Oil II I Aqueous phase Such chains, with the two aqueous phases being identical, jdeld an E.M.F. obviously caused by the potential difference between the single potentials I and III. As for example, in the chain Calomel electrode in 0.1 N KCl in water Cresol Benzaldehyde Calomel electrode in 0.1 N KCl in water 208 HYDROGEN ION CONCENTRATION Some of Beutner's measurements are shown in table 33. Thus we have an "electromotive series" for oily Hquids as well as for the metals. But also the chain Aqueous salt solution Nitrobenzene Nitrobenzene + picric acid Aqueous salt solution studied long ago by Cremer^" may also be included in this group of chains. For in this case the salt distribution on the two sides is unequal; the dissolved picric acid effecting a higher concentration of cations, which, in addition, is but httle dependent upon the con- centration of the salt in the adjacent aqueous phase. TABLE 33 Benzaldehyde Benzaldehyde Benzaldehyde Benzaldehyde Benzaldehyde Benzaldehyde Benzaldehyde Benzaldehyde Benzaldehyde with Cresol with Phenol with Guaiacol with Toluidine with Benzyl alcohol... with Amyl alcohol. . . . with Acetoacetic ester with Acetophenone — with Dimethylaniline. V0LT3 -0.13 +0.13 +0.08 +0.05 +0.04 +0.02 -0.05 -0.07 -0.09 to -0.13 The nitrobenzene dissolved on the right side in the picric acid imparts to the latter a new character. The addition of an organic acid to an oil renders the oil ''more positive" in the electromotive series. In pure oils, however, it has not been found as a rule that they are more positive according to the strength of their acidity. Furthermore, it is not merely a question of the amount of acid in the oil, but also of the degree of dissociation of this acid. The dielectric constant of the oil phase is also of great importance. Of two oils containing the same amount of acid the one with the smaller dielectric constant is the more negative. Thus: Nitrobenzene saturated with nitrobenzoic acid Positive Without added benzene + an equal volume of Negative benzene i» M. Cremer, Zeitschr. f. Biol. 47, 1 (1906). POTENTIALS AT PHASE BOUNDARIES 209 The benzene, diminishing the dielectric constant of the oil phase decreases the dissociation of the dissolved acid and thus renders the oil phase more negative. ^^ The correlation of the chemical con- stitution of an oil with its position in the electromotive series still needs further investigation. 59. Physiological applications of the theory of boundary potentials It appears appropriate at this juncture to examine the relation of the above observations to the electromotive forces found in living tissues. In fact, these currents produced without metals, observed by the physiologists several decades ago, served as a stimulus for the above described studies which were undertaken to elucidate biological processes. The fundamental physiological observations date back to DuBois-Rej^mond^- which stated briefly were as fol- lows: Whenever a living tissue (muscle, nerve, leaf or fruit) was injured (cut, crushed, etc.) at any place, and an injured and an uninjured place were connected by means of unpolarizable electrodes an electric current of 0.02-0.1 volts was produced, the injured spot being negative with respect to the uninjured spot. Also whenever the uninjured surfaces of any two related organs (tendon and muscle, root and leaf or stem) were connected in the same manner, a cur- rent was obtained (the tendon being negative towards the muscle, the root negative towards the leaf). These observations provoked the hope of explaining the nature of vital processes on the basis of electric phenomena. Physical science of that day had not yet reached an adequate interpretation of those observations; for it was then still impossible to reproduce so great potential differences without the use of metals. A "molecular theory" attempted by DuBois-Rejinond was of a purely descriptive character. Soon afterwards L. Hermann^^ discovered that injured parts of the same tissue showed no potential differences towards each other, and hence " This finding will be recalled later in the section dealing with Coehn'a Rule. ^'' E. Du Bois-Reymond, Untersuchungen iiber tierische Elektrizitat. Berlin 1848, 1849, 1860, C. Reimer; Gesammelte Abhandlungen zur Muskel und Nervenphysik. Leipzig 1875; see also Helmholtz, AUgemeine Monatsschr. f. Literatur u. Kunst. Kiel, April 1852. " L. Hermann, Weitere Untersuchungen zur Physiologic der Muskeln und Nerven. Berlin 1867. 210 HYDROGEN ION CONCENTRATION he ascribed the observed current to the destructive (death) proc- esses at the place of injury, a theory which subsequently was ex- panded into the so-called "alteration theory." When, later, dif- fusion potentials became better known, it was attempted to utilize them for the explanation of these phenomena. ^^ But diffusion potentials of corresponding order of magnitude can arise only as an exception in the diffusion of free mineral acids toward neutral solu- tions, and besides, they demand conditions never met with in the living organism. Even when the presence of such acids as lactic acid is taken into account, the maximal diffusion potentials could amount to no more than a few millivolts, whereas currents of injury usually amount to several centivolts. Wilhelm Ostwald^^ gave a material stimulus to the reinvestigation of this problem. In the course of an investigation of the electrical properties of precipitation membranes he expressed the conjecture that the currents observed in muscles and nerves, and even in the electric fishes, have their origin in the properties of similar membranes. He ascribed to these membranes the peculiarity of being permeable either only to anions or only to cations. The diffusion of an electrolyte through such a membrane would therefore result in a separation of the ions and hence in a potential difference. This explanation has the great distinction of being the first to introduce the phase boundary as the place of origin of an electromotive force. And yet the concep- tion of ionic permeability did not prove to be capable of adequate and quantitative investigation. To be sure, the physiologists attempted to extend this theory by trying to find the ions to which living membranes are impermeable. Bernstein^*^ ascribed this property to the K-ions. Hober" could not confirm this. He thought to have shown that the permeability of the cell membrane to the different ions varied with the effect of different ions upon the membrane. He assumed that the colloidal state of the mem- brane changed under the influence of the various ions, and that along with it the permeability varied. This assumption appeared all the more plausible, as the ionic series, so frequently encountered i« Oker-Blom, Pfliigers Arch. f. d. ges. Physiol. 84, 191 (1901). '» Wi. Ostwald, Zeitschr. f. physikal. Chem. 6, 71 (1890). '« Bernstein, Pflligers Arch. f. d. ges. Physiol. 92, 521 (1902). ^' Hober, Physikalische Chemie der Zelle und Gewebe. Leipzig 1911, p. 487. POTENTIALS AT PHASE BOUNDARIES 211 in colloidal chemistr\'', were shown also to exercise their influence upon the electric properties of membranes. Hober's studies fur- nish valuable experimental data, and the important question at present is whether these phenomena can be best explained in terms of Hober's theory of perm.eability or by the theory of forces at the phase boundary. Cremer^^ correlated the variable permeabihty to ions in Ostwald's sense with the ionic mobilities, and regarded these potentials as being analogous to diffusion potentials. He identified the more diffusible ion as that ion which has a great mobility in the medium into which it penetrates. It was also Cremer who first succeeded in arranging experimentally a two-phase chain without the use of metals, such as: Aqueous solution Nitrobenzene Nitrobenzene + picric acid Aqueous solution In order to explain the observed potential differences in terms of Cremer 's theory it. would be necessary to assume incredibly great differences in the mobilities of the several ions involved. The phase boundary theories discussed in the preceding sections represent a development the way to which had been paved by Nernst's^^ electromotive theoiy of metal electrodes. The experi- mental and theoretical investigations along this path were carried out in a most thorough manner by Haber,"^" as well as previously by Cremer, although with not quite the same interpretation. These investigations were followed by those of Beutner referred to above and which represent a distinct step in advance, inasnmch as they can be dnectly apphed to the interpretation of bioelectric phenom- ena. The place of origin of the electromotive forces is, accord- ing to the latter author's conclusions, at the boundary surface lying between aqueous phases and the lipoidal membranes of living cells and tissues. And, indeed, all such electric phenomena become easily exphcable by merely assuming that such membranes repre- '»M. Cremer, Zeitschr. f. Biol. 47, 1 (1906); Nagels Handb. d. Physiol. 4, 868 (1909). »9 W. Nernst, Zeitschr. f. physikal. Chem. 9, 140 (1S92); Nernst and Riesen- feld, Ann. d. Physik. 8, 600 (1902). -°F. Haber, Ann. d. Physik. 25, 917 (1908); Haber and Klemensiewicz, Zeitschr. f. physik. Chem. 67, 385 (1909). 212 HYDROGEN ION CONCENTRATION sent non-aqueous oil-phases containing oil-soluble organic acids. In order to prove Beutner's theory it is necessary to show that on varying the kind and the concentration of the electrolytes in the aqueous phase the resulting changes in the E.M.F. are quahtatively and quantitatively the same as those obtained in artificially repro- duced oil-chains containing an acid (or acidified) oil-phase. The electromotive forces observed in physiological systems must possess the following properties in order to fulfill the conditions of Beutner's theory: 1. A potential difference also must be observed between two uninjured places when they are connected by means of solutions of an electrolyte of different concentrations. This is shown in the following experiments: (a) With differing concentrations of the same electrocute. The E.M.F. between two uninjured parts of an apple were measured, with the following solutions being used as connecting liquids: *»-*• El Tf ET » J Calculated maxirnal E. M. F. observed concentration effect M/10 NaCl and M/10 NaCl. . 0 M/IO NaCl and M/50 NaCl... 0.029-0.024 volts ] M/50 NaCl and il//250 NaCl.. . 0.042-0.036 volts !> 0.040 volts M/250 NaCl and M/1250 NaCl.. . 0.041-0.038 volts J showing that the more dilute solution is always positive. The agreement with the results obtained from artificial oil-chains is so complete that even with the higher salt concentrations, the concentration effect is well below the theoretical maximal effect obtainable. With the more dilute solutions, the theoretical maxi- mal effect is reached. With NaBr the concentration effect was more pronounced. Calculated maximal NaBrAT/S and A^/40 , 0.040 volts 1 AV40 and A''/200 0.041 volts \ 0.040 volts iV/200 and A^/IOOO 0.042 volts J The concentration effect proved to be reversible whenever no permanent injury was inflicted upon the apple peel by the stronger salt solutions. POTENTIALS AT PHASE BOUNDARIES 213 With other materials the effect produced was less and not so regular, for example. rri .1 „. Ccdctdaled maximal Turtle carapace: g^gct KCl N/IO and N/50 0.010 volts ] N/50 andA^/250 0.015 volts f 0.040 volts Ar/250 andiV/1250 0.021 volts J Human finger nails: NaCl 10 N/8 and A^/8 0.017 volts ] N/8 and A^/80 0.023 volts I , N/80 and iV/800 0.028 volts f "•"^' ^ ^"""^ iV/800 and iV/8000 0.030 volts J In muscles a much smaller effect was obtained: KCI N/8 and N/40 0.002 volts ] N/40 and iV/200 0.006 volts [ 0.040 volts iV/200 and AT/IOOO 0.008 volts J (6) An E.M.F. must also be produced when two uninjured places are connected by means of solutions of the same concentration but of different electrolytes. The influence of the ions in the negative direction must follow in the same series sequence as in the "oil- chains." This point had already been demonstrated in Hober's studies, which, however, could not at the time be interpreted in the sense of the later experiments. It was shown that when the con- necting fluid was varied at an uninjured place, the negative effect of the anions was expressed in the series: S0=4 CI- Br- I- NO3"" SCN- and of the cations: K+ Na+ Ba++ Cu++ Mg++ Furthermore, for a closer conception of the phenomenon it was necessary to find out whether the oil-layer represented by the apple peel is a single oil phase, such as salt solution | oil salt solution which would correspond to salt solution apple peel Free fluid of apple pulp 214 HYDROGEN ION CONCENTRATION or, whether different oil phases were present in layers, as salt solution | oil I oil II | salt solution such as, for example. Salt solution Nitrobenzene + picric acid Nitrobenzene without or with less picric acid Salt solution corresponding to, in the case of the apple, Salt solution Outer layer of apple peel Inner layer of apple peel Free fluid of apple pulp a- The following of Beutner's experiments answered this question. Two places on the apple were con- nected by means of iV/50 KCl solution (fig. 26). In the case of the uninjured apple (I) the E.M.F. was zero, of course. With a shght injury (II), a current of 0.041 volt was produced. With further injury (III) the effect remained unchanged. When the injury or hollowing out was carried on until but 0.25 cm. of pulp remained (IV), the E.M.F. sank to 0.020 volt. And, finally, with the peel alone, (V) an E.M.F. of but 0.01 volt remained. On the other hand, when the concentration of the connecting fluid at the uninjured places was varied, the concentration effect appeared exactly the same as in the entirely uninjured apple. The interpretation of this observation is only pos- sible when it is assumed that the apple peel does not Fig. 26. (Taken represent a single oil-phase. Only the outermost from Beutner) layer of the peel may represent such a single oil phase, and hence, when placed between two salt solu- tions of the same concentration, it produced no current. Experi- ments II and IV above can only be explained by assuming a varying composition of the peel in its various layers, and that the outer laj^ers contain in their oil phases an organic acid, wliich is absent or is present in a lesser concentration in the inner layers. b+ POTENTIALS AT PHASE BOUNDARIES 215 60. Potentials at precipitation membranes The theoiy of cell membrane potentials is based upon the analogy existing between an "oil' '-phase and a cell membrane. But there is still an older model for cell membranes, which reproduced very well indeed its properties of permeabihty, namely M. Traube's precipitation membranes. Traube's membranes were adopted by Pfeffer in his well known cells for the qualitative and quantitative study of osmotic pressure and semipermeabiHty. The fact that these copper ferrocyanide membranes showed the same impermea- bility for many dissolved substances as was assumed in the explana- tion of shrinking and swelling phenomena in cells placed into non- isotonic solutions, gave a great stimulus to the entire theory of semipermeabiHty. And without this we could scarcely ever have arrived at the idea of a membrane which is permeable for cations but not for anions. The questions arise naturally whether such precipitation membranes can also be the seat of potential differ- ences, whether these are analogous to the cell membrane potentials, and whether Beutner's theory is apphcable to them, or, whether in this case the older idea of a specific permeability for definite ions is effective instead. Disregarding the older investigations studied such chains. - Secfrvmeter Fig. 27 of Bruning,^* Beutner His arrangement was the following (fig. 27): A ilf/40 warm solution of potassium ferrocyanide in 10 per cent gelatin solution was poured into a glass tube open at both ends, permitted to gel, and immersed in a beaker containing M/20 CuS04 solution. At the zone of contact a brown precipitation membrane of copper ferrocyanide was immediately formed, barely perceptible at first, but increasing in thickness with time. The upper end of the gelatin layer and the outer copper solution were led off by means of calomel electrodes. An E.M.F. of 0.10-0.12 volts was observed, the CuS04 solution being positive. Then the composition of the outer solution was varied, keeping its osmotic pressure constant throughout by the addition of sugar in order not to destroy the 21 Briining, Pflugers Arch. f. d. ges. Physiol. 117, 409 (1907). 216 HYDROGEN ION CONCENTRATION membrane mechanically through the osmotic differences between the inner and the outer fluids. When the concentration of the CUSO4 only was varied, no definite constant change in the E.M.F. was observed. But when some KCl was added to a constant con- centration of the CUSO4 the resulting E.M.F. values were observed to be definitely and irreversibly dependent upon the concentration of the added KCl, as shown by the average figures given in table 34. But these are the exact values for a KCl concentration chain (0.040 volts). The same results were obtained with NaCl, HCl, NH4CI, when the concentrations were varied, and on comparing chains containing different chlorides of the same concentration differences were again TABLE 34 Difference 5*0 mol. CUSO4 + 1 ^Q mol. CUSO4 + uJo mol. CuSO^ 4- Yo^oo j\ mol. CUSO4 + h* 1 JO mol KCl mol KCl mol KCl mol KCl 0.020 volts 0.061 volts 0.098 volts 0.12 volts * An E.M.F. of =» was theoretically to be expected in this case. But small amounts of K-salts (K2SO4) diffuse through the membrane into the solution (see text below), hence the finite, but uncertain and irregular values for the E.M.F. observed. found. In brief, — this form of chain is in every respect analogous to the chain. Salt solution I Salicylic aldehyde + salicylic acid Salt solution II and it represents, for any cation, a reversible chain. The anions were shown to be without any effect, just as a chain whose middle phase consists of an acid oil. The analogy with such oil chains is so complete that it is permissible to apply the same theory to both. This theory may be stated as follows: The formation of the cupric ferrocyanide occurs according to the equation: 2 CUSO4 + K4Fe(CN)6 = 2 K2SO4 + Cu2Fe(CN)6 POTENTIALS AT PHASE BOUNDARIES 217 Thus there is also formed K2SO4, and it is present on both sides of the precipitation membrane. Although the membrane is con- sidered impermeable for salts, yet it is certainly not so in the abso- lute sense. The fact that the membrane becomes thicker in the course of time shows that the salts do penetrate to a certain extent. As far as the potassium ferrocyanide solution is concerned the dif- fusion of the K2SO4 is immaterial. In the CUSO4 solution, on the other hand, it is only then unimportant, when it already contains a previously added large amount of a potassium salt, otherwise it is just the trace of K2SO4 which, on diffusing through, affects the E.M.F. of the chain appreciably. Thus our chain may be schema- tized as: Solution of K"*" ions Ci Membrane Solution of K"*" ions C2 In order to explain the concentration effect it must be assumed that the membrane is a phase which also contains K-ions, but in an almost constant concentration, just as in the case of the acid- containing oil phase. We may assume for this purpose that the membrane does not consist of entirely pure Cu2Fe(CN)6. These amorphous substances "carry along with them on precipitation a small portion of the soluble salts," or, there is formed some sort of combination of Cu2Fe(CN)6 with K2SO4. It can also be shown analytical!}' that the Cu2Fe(CN)e precipitate always contains alkali metal salts. It is uncertain whether, in this case, adsorption, solid solution or chemical combination is involved. But the very fact that the membrane holds K-salts in a firm way is sufficient in itself for our purposes. In the case of the acid-containing oil it was the affinity of the acid for the cations (or bases) which produced the almost constant concentration of cations in the portions of the oil phase in contact with the aqueous electrolyte solutions. In this case of precipitation membranes it is an affinity of the amorphous membrane substance for the salt which produces a similar effect. It is therefore possible to explain membrane potentials on the same basis as "oil-chain-potentials." Whether this explanation is the only correct one the future will show, but it does possess the advan- tage in that it can be easily correlated with other conceptions in a readily demonstrable way. We shall see in the next chapter that 218 HYDROGEN ION CONCENTRATION membranes msiy give rise to electric phenomena bj^ an entirely different mechanism. 61. Polarization phenomena at phase boundaries All of the above considerations of phase boundary potentials relate to the conditions where chemical equihbrium prevails, at least at the contiguous boundary layers. These potential differences are comparable to the E.M.F. of an open galvanic chain without a current. When such a galvanic chain is closed, the resulting current produces a greater or lesser change, according to the con- ditions, in the concentrations of the solutions, and along with it a change in the E.M.F. which is designated as polarization. This polarization may result either from the current produced by the E.M.F. of the chain itself or from a current sent out by an external E.M.F. through the chain. In the same way polarization, and hence a change in the phase boundary potential, may occur in phase boundary chains either when they are short circuited for a longer period of tune, or when an external current is sent through them. Nernst and Riesenfeld" furnished the theory and the discussion of these polarization phenomena: The arrangement, from left to right is as follows: (1) water saturated with phenol, (2) phenol saturated with water, (3) water saturated with phenol again; an electrolyte is distributed among these phases in a state of equilibrium, and, furthermore, for the sake of simphcity it is assumed that the phase boundary potential is zero. An electric current is sent through this system from left to right for a period of time, until one faraday (96,500 coulombs) has flowed through every cross section. The dissolved electrol5^e consists of two univalent ions whose relative speeds are u for the cation and v for the anion. In general the value of u in water will not be the same as in phenol, hence let us designate it as Ui in water and U2 in phenol; and hkewise we shall differentiate Vi from V2. Let us consider any given cross section of the path of the current in the water. We find that the fraction of cations concerned in the Ui transport of one faraday is — y — - = ni. This fraction is known as . . . . Vi the transference number of the cation. Likewise the fraction , Ui 4- Vi " Nernst and Riesenfeld, Ann. d. Physik [4] 8, 600 (1902). POTENTIALS AT PHASE BOUNDARIES 219 = mi of the anions is involved. Hence ni + mi = 1. On the U2 other hand, in the phenol the distribution is — — — = n2 for the V2 cations, and for the anions — ; = m2, and again, n2 + m2 = 1, ' U2 + V2 Let us now consider that cross section of the left hand water phase which is in direct contact with the phenol phase. Through this cross section the cations are moving from left to right, and the .anions from right to left. In this cross section ni cations are moving inward from the left and n^ cations outward to the right. If ni > D2 then ni — n2 cation will accumulate in this cross section. On the -other hand, in the same cross section m2 anions are moving from the right and mi anions to the left. When ni > n2, then mi must be < m2, and hence m2 — mi anions accumulate in this cross section. Altogether Ui — n2 cations and m2 — mi anions accumulate. But since m2 = 1 — n2 and mi = 1 — ni, therefore, m2 — mi = ni — n2, i.e., the total electrolyte accumulated in this cross section amounts to 2 (ni — n2). The same consideration may be applied to the bound- ary layer of the phenol phase, in which we shall find a correspond- ing diminution of the electrolyte in the cross section. The same process occurs in the two boundary layers on the right hand side. Thus we find that the current produces an increase in electrolyte concentration in the water and a decrease in the phenol phase in both boundaries. Because of the spontaneous diffusion imme- diately arising between portions of solutions of unequal concentra- tion, the changes of concentration at the boundary surfaces actually observed do not quite reach the expected values. When, reversely, ni < n2, we obtain an increase of concentration in the phenol and a decrease in the water layer. If the phase boundary potential was equal to zero at the beginning, it can no longer remain 0 after the passage of the current, because of the above changes in concentra- tion at the boundaries. Thus a potential difference arises as a polarization phenomenon at the boundary surfaces. The two oppositely directed potentials of the two boundaries constitute the ►electromotive force of the polarization, which, in turn, must diminish the electric current, just as in the polarization of an electrolyte solution between two platinum electrodes. CHAPTER IX Membrane Potentials summary of contents When two aqueous electrolyte solutions are separated by a membrane which is impermeable for one of the ionic species present, a potential difference arises between the two solutions, and also an unequal distri- bution of the ions, including even the freely diffusible ions, ensues. This effect diminishes with increasing concentrations of the electrolytes. The relation of these potentials to the phase boundary potentials described in the preceding chapter is discussed. 62. The membrane as the cause of a potential The phase boundary potentials which we have just considered are closely related to certain potential differences which we shall desig- nate as membrane potentials. As was stated in a preceding chapter (see page 210), Wilhelm Ost- wald suggested the idea that the impermeabihty of membranes for any ionic species may give rise to potential differences. This theory was first fully developed by Donnan^ whose investigations have proved to be exceedingly fruitful. Donnan's derivation will be first shown for the simplest possible case. Let us take a solution containing only a Na-salt of an acid, the anions of this acid as well as its undissociated molecules being unable to penetrate the membrane, and let R~ represent this anion. Let us take a second solution con- taining only NaCl, and interpose between the solutions a membrane shown by a vertical fine in the diagram: Na+ R- Na+ ci- (1) (2) 1 F. G. Donnan, Zeitsehr. f. Elektrochemie, 17, 572 (1911). For the most recent account in English see F. G. Donnan. The Theory of Membrane Equi- libria. Chemical Reviews, 1, 73 (1924). 220 MEMBRANE POTENTIALS 221 This initial state is soon altered through diffusion, and, considered quahtatively, results in the following redistribution of ions: Na+ Na+ R- ci- CI- (1) (2) Since electroneutrality must be maintained in the solution on both sides of the membrane, it follows that, after the diffusion equilibrium had been estabhshed, [Na+]i = [R-] + [Q-]: and [Na+Ja = [Cl-]2, and that neither [Na+]i = [Na+Ja nor [Cl-]i = [Cl-Ja. This equilib- rium is characterized by the fact that the maximal work which could be gained from the reversible and isothermic transport of a very small amount of Na-ions in one direction must be equal to the work expended in the reversible transport of the Cl-ions in the same direc- tion. In other words, the algebraic sum of the work, 5 A, which can be obtained from the simultaneous transport of a very small amount, 6 n, of Na-ions and of the same amount of Cl-ions must be equal to zero. The work which can be gained from the transport of 5 n moles of Na+, involving the change of concentration from [Na+]i to [Na+ja is = 5 n X RT In tzt^- The same condition being true for the [iNa+Ji Cl-ions, after the estabhshment of equilibrium we should have 6n X RT In |^^ + 5n X RT In p|l^ = O [Na+]i [Clii Hence it follows that [Na+]2 X [Cl-]2 = [Na+]i X [CI"], Certain difficulties which Donnan encountered in regard to the possible presence of undissociated molecules may be considered today as having been overcome. These difficulties were due to the inexact figures concerning the degree of dissociation of strong electrolytes obtained from calculations based on conductivity data. Therefore, we need not at present dwell at any greater length upon these difficulties. Since in solution (2) we must have [Na+Ja = [Cl-ja, but in solution (1) [Na+]i cannot be = [Cl~]i, it can be stated that [Na+], X [Clii = [Na+]? = [Cli^ 222 HYDROGEN ION CONCENTRATION For the sake of preserving the continuity of argument developed in the preceding chapter let us choose an interpretation of this phenomenon somewhat different from Donnan's. Let us imagine two electrolyte solutions separated by a membrane which is impermeable to one of the ionic species present. Let, for instance, the membrane be one of collodion and let one of the solu- tions be a KCl solution and the other of the chloride of a cation which cation cannot diffuse through the membrane. A process of diffusion ensues leading to a certain state of equilibrium. But the above non-penetrating cation cannot take part in this diffusion. If the system is initially Colloid-chloride (dissociated into colloid-cation+ + Cl~) Membrane KCl (dissociated into K+ + C1-) diffusion will at once occur through the membrane. ^ K-ions will diffuse from right to left, Cl-ions will diffuse, according to the con- ditions, from right to left, or reversely. But there is no possibility that the concentrations of K+ become the same in both solutioQS, nor could that happen for the Cl-ions. For, if this were the case, then the solution on the left would contain, because of the presence of the colloid-cations, more cations than anions, which state would violate the law of electroneutrality. It is more likely that the colloid- cations will hold back a part of the Cl-ions. For in the state of equilibrium the total sum of cations in the solution on the left (col- loid cations + K+) must be equal to its content of anions (Cl~), and, similarly, in the solution on the right side, the concentration of the K-ions must equal that of its Cl-ions. The inequahty of the con- centrations of the oppositely charged ions can be but immeasurably small, and it effects a potential difference on both sides of the mem- brane. In the first place then, we find an anal3i:ically demonstrable difference between the Cl-ion concentrations in the solutions, and, secondly, we also find a potential difference. These two phenomena are definitely related to each other, and this relation can be calculated. Thus, if after the establishment of equilibrium, we should immerse in each of the two solutions, a reversible Cl~ electrode (i.e., a gas ^ The term "colloid" in the above scheme does not in this case signify that the substance is not in molecularly dispersed solution, but it only signifies that it does not diffuse through the membrane. MEMBRANE POTENTIALS 223 Pt-Cl2 electrode, or a Hg-HgCl electrode which is reversible for C1-) and unite the electrodes by a metal conductor, we shall find that no current can flow through this system, since it is in equiUbrium. But the electrode potentials are not equal to each other. Thus, if in the final state of equilibrium the Cl~ concentration on the left is ci and on the right is C2, then the potential difference between the two electrodes is equal to RT In -. In order that the entire system (or ^ C2 this chain) be entirely devoid of an electric current flowing through it, we must assume that necessarily a potential amounting to — RT In ~ be present at the membrane, C2 If we should now assume that in addition to KCl also HCl is pres- ent, then a different state of equihbrium will ensue. This equilibrium can also be evaluated. If we designate again the concentrations of CI- in the left and right solutions Ci and C2 respectively (which are now different, of course, from the Ci and C2 used in the preceding example), the potential difference will be — RT In -. But we can use hydrogen electrodes now, and even then the chain will not yield any electric current. But this may only be true if the membrane potential is equal to RT In 77, where hi and h2 represent the [H+] of the two solu- 112 tions. The same conditions will prevail in the presence of anj^ of the common ions along with our non diffusible ion. And thus by determining, after equilibrium has been attained in the system, the concentrations of any of its component ions, for instance that of the H-ions, which are always present, and which are easily determined, we can calculate from it the following: 1. The membrane potential. 2. The distribution of any of the ionic species between the two solutions. When all the ions present are univalent, and letting the ratio of the H+-concentrations, of that in the left to that in the right solu- tion be 7 : 1, then 1. The potential = RT In 7. 2. The ratio of the concentrations of any of the positive diffusible ions, that in the left to that in the right solution, is likewise = 7, and for the negative ions it is = — . 224 HYDROGEN ION CONCENTRATION The simplest case to be had is that in which but one diffusible electrolyte is present, such as Colloid base HCl HCl (1) (2) According to the principle developed above the right to left ratio of the H+-concentrations, must be equal to the left to right Cl~-concentration ratio, thus: [H+]2 [Cl-ji [H+]i [Cl-]2 But since on the right side [H+]2 = [Cr]2, it follows that [H+[! = [Cii? = [H+]i X [Ciix (1) Let us now see how we can define more closely the distribution of ions in the simplest possible case. Let us take this system: the solution inside of a collodion thimble is that of a chloride of a non- diffusible, but molecularly dispersed, base in concentration a. Also, the H+ and Cl"~ ions are present in this inner solution in the concentrations hi and ch. In the outer solution we have HCl in the concentration h2 = CI2, but no "colloid" is present. The elec- trolytes are assumed to be completely dissociated. The values h2 = CI2 are known while hi and ch are taken as the unknown values. The following relations exist : ^& and also a + hi = cli hi = CI2 hi _ CI2 h2 cli a + hi = cli (1) hi . cli = ch^ (2) From (1) and (2) the following quadratic equation for ch is obtained : cU* - a cli - cl2« = 0 ff cli = ^ + Vt + ch^ (3) MEMBRANE POTENTIALS 225 That only the positive root is significant can be seen by setting a = 0, when cli = CI2, and not ch = — CI2. By substitution in (1) we obtain Vf hi = - ^ + t/'^. + cU" (4) Thus we see that cli > hi, i.e., the positively charged colloid ion simul- taneously attracts the negative Cl-ions and repels the positive H-ions outward through the membrane. If we assume that in addition to HCl, NaCl is also present in the solution, then in the inner solution the ions are in the concentrations a, hi, cli, ni (where Ui is the Na+-concentration) ; and in the outer solution h2, CI2, n2. And now we have the following relationships. ,^J. ^ ' T 1 ' ~ ^t' \ From the law of electroneutrality (II) no + ho = CI2 J From Donnan's law of ionic distribution We shall now solve these equations for hi, so that hi appears as a function only of a and other values carrying the subscript 2. This is done by solving (III) for hi, ehminating cli by means of (I) and ni by means of (IV) which results in cU • hi hi = - (III) hi : h2 = CI2 : ch (IV) hi : h2 = ni : na a + — • n2 + hi h2 This in turn yields a quadratic equation for hi a • h2 //a • haX^ , „ 2 CI2 iV 2 CI2/ And hence, hi hi 2 cl (5) From the last equation and (III) we can at once derive the ratio cli/cl2, and, similarly by using (IV), also ni/n2. Furthermore, we can now express the total ionic distribution and the membrane poten- 226 HYDROGEN ION CONCENTRATION tial as a function of two variables, a ("colloid" ion concentration) and CI2 (which is in general the concentration in the outer solution of any ion having a charge opposite to that of the "colloid "-ion). Let us now consider the relation of the ratio hi/h2 to a. From equation (5) we see that when a = 0 (i.e., in the absence of a "col- loid" ion), hi/h2 = 1, and that therefore the potential = RT In 1 = 0; and that the greater the value of a the greater the difference between hi/h2 and 1, and thus the greater the resulting potential. Now for the relation of hi/h2 to the value of CI2. It is such that the ratio hi/h2 increases, without forming a maximum or minimum, with increasing values of CI2. In fact, the greatest value for hi/h2 = 1 is when CI2 = 00 , and its lowest value is 0 when CI2 = 0 . For, if we- write (5) in the form y = - x-fv/x^ + 1 and assume that x is very large (i.e. that CI2 is very small) in respect to 1, then we can express the term \/x2 + 1, by developing it in a binomial series, as 11 ,1 11 xH — — , and hence, y= — + — 2x 8x3 ' ' -^ 2x 8x3 For X = 00 (i.e., for CI2 = 0), y (i.e., hi/h2) becomes = Q. It is quite easy, therefore, in a serial experiment, keeping the value' of a constant, to increase the outer concentration of the Cl-ion, CI2, by simply adding to the whole system HCl or any chloride. Hence it follows that : 1. The value of the ratio hi/h2 approaches 1 (the potential approaches 0) with the addition of HCl. 2. The same effect is obtained on the addition of any chloride, e.g.;. NaCl. This leads to the following physiologically important conclusion: All membrane potentials are decreased by the addition of neutral salts; and in colloidal solutionis rich in electrolytes only very small membrane- potentials are developed. It will be interesting to see just what potentials are to be expected under certain assumptions. If the concentration of the molecularly dispersed non-diffusible cation is taken as a = 0.01 N, then by apply- ing equation (5) (page 225) we shall find in the outer solution the- MEMBRANE POTENTIALS 227 different values for CI2 shown in table 35. When it is assumed that a = 0.1 N, the values shown in table 36 will be obtained. When the "colloid" is of the nature of an anion, then all of the above derivations may be directly applied by simply substituting oh for h and n for cl. Since ohi/oho = h2/hi equation (5) above becomes hi 2 n2 V + -^1 + 2 m TABLE 35 ch (normal) h,/h. TT millivolts 1 0.995 0.1 0.1 0.95 1.3 0.01 0.62 12.2 0.001 0.10 58 0.0001 0.01 116 0.00001 ca 0.001 ca.l74 TABLE 36 C12 ^hi/h^ TT millivolts 1 0.9.3 1.3 0.1 0.62 12.2 0.01 0.10 58 0.001 0.01 116 0.0001 0.001 174 0.00001 0.0001 232 and the two tables shown above retain their vaUdity in this case by merely changing hi/h2 to h2/hi and CI2 to n2. Let us now attempt to derive from the above consideration such useful physiological appHcations as may be possible at present. We shall take the case of a cell whose content represents a 0.01-0.1 molar solution of protein, which, insofar as it is ionized, we shall consider as being moleculary dispersed. The protein, at the pH of the tissue fluids, must be assumed to be in the form of anions, insofar as it is ionized in general. The cell is surrounded by a membrane imper- meable to protein and is bathed in a protein-free NaCl solution, approximately 0.1 A''. In such a case we should expect a membrane 228 HYDROGEN ION CONCENTRATION potential of between 1.3 and 12.2 millivolts. The latter value would represent the highest imaginable. However, such ideal conditions as: a 0.1 molar protein concentration; a complete molecular disper- sion and ionization of the protein solution; absence of protein in the surrounding fluid, can barely be found in the living organism. There- fore it is quite questionable whether Donnan's theory of ionic dis- tribution and of formation of potentials can, in this form, play an important role in the living organism. On the other hand, for cer- tain experimental conditions in colloidal chemistry the theory is of the greatest importance, especially in the following respects. It is not unconditionally requisite that the colloidal and the col- loid-free solutions be separated by an actual membrane. Procter^ was the first to dem.onstrate that a piece of solid gelatin, immersed in an aqueous solution and permitted to assume a definite state of swelling, behaves in exactly the same manner. A solution is enclosed within the meshes of the gel, and the gel itself represents, according to the pH and because of its amphoteric character, a non-diffusible acid or base. Consequently, when a piece of gelatin swells in a dilute HCl solution, at equilibrium the H+-concentration must be greater in the outer solution than within the gel, and this difference becomes less upon the addition of NaCl. This observation was verified experim.entally by Jacques Loeb,* and we shall return to this topic in our chapter on colloids. 63. The relation of membrane potentials to phase boundary potentials If it appears that the biological significance of these potentials is not ver3^ great for the living cell, it is yet possible that membrane potentials originating from another source may be of importance for the living organism. The investigations of many physiologists have led to the idea that the membranes of various cells show a specifically selective permeability for different ions. R. Hober, especially, advanced the thsor^y that cell membranes possess a general difference in permeabiHty for cations and anions. Ever since the fundamental ' H. R. Procter, Journ. of the Chem. Soc. 105, 313 (1914) ; H. R. Procter and J. A. Wilson, Journ. of the Chem. Soc. 109, 307 (1916). 4 Jacques Loeb, Journ. of Gen. Physiol. 3, 667; 3, 691; 3, 827; 4, 33; 4, 97 (1921); also J. Loeb. Proteins and the Theory of Colloidal Behavior. 2nd ed. New York, 1924. MEMBRANE POTENTIALS 229 researches of Pfeffer and of de Vries on plant cells and of Hamburger on animal cells it has been assumed that most of the water soluble substances, such as particularly sugar and NaCl, cannot penetrate the cell membrane. Those substances which do penetrate (urea, narcotics) are, according to Overton and Hans Meyer, lipoid-soluble. In later years this hypothesis has not been sustained. Many inves- tigations have since demonstrated that anions diffuse freely through the membrane of blood corpuscles, and that only the cations are non- diifusible. This question cannot be very well discussed in greater detail at this juncture. But, if this theory'- is at all correct, then all conditions and prerequisites are present for the formation of mem- brane potentials. Only the non-diffusible ion is in this case not the protein-ion, but the common alkali metal cation. Since such cations are present in the tissue fluids in much greater molar concentrations than the proteins, it becomes quite possible that they give rise to considerable membrane potentials. The only difficulty lies in the fact that thus far it has not been found possible to construct a satis- factory model for the study of such processes, or to reproduce a membrane which would be impermeable to cations alone, in the same sense in which phase boundary potentials have been so successfully reproduced artificially. This leads us to the following conjecture: If we recall Haber's glass chains or Beutner's ''oil chains," it wiU be remembered that they are considered as chains "reversible only for cations or only for anions." We could restate this by saying that the oil-membrane is "permeable" only for cations (or only for anions). This impermeability is not by any means of the same Idnd as that of the collodion membrane for protein. In the latter case the imperme- abiUty is explained on the basis of a spatial disparity between the pores of the membrane and the diffusing molecules, while in the foraier case it is based upon the limited solubility of the ions of the aqueous solution in the medium of the "oil". In the former case the membrane is represented as a system of very fine canals through which all the (non-colloidal) molecules, of the solvent as well as of the solutes, circulate freely, the solvent in the pores being the same (water) as in the two solutions. In the latter case the membrane is conceived of as a foreign phase interposed between the two solutions and representing a solvent totally different from those of the solu- tions on either side of it. This brings us back to the first statement of this section in which 230 HYDROGEN ION CONCENTRATION the close relationship of phase boundary potentials and membrane potentials was suggested. A layer of an oil interposed between two aqueous phases behaves, under certain conditions, hke a membrane impermeable to certain ions, and hence it should be possible to regard the ionic distribution at equiHbrium in both ways. It seems, how- ever, that the interpretation in terms of phase boundary potentials is the more comprehensive of the two. Donnan's interpretation should be Umited to those cases in which the impermeabihty is actually determined by a lack of diffusibiUty (of a molecule) ; where the ulti- mate cause of the potential Ues in the presence of a colloid in the aqueous solution; where the membrane may be imagined as a system of fine canals through which all substances (except the colloid) circu- late freely: and where, especially, the solvent circulating in these canals is the same as in the adjacent solutions. In such a case Donnan's interpretation is quite irreplaceable, and as such it has assumed an important place in the theory of colloidal systems. CHAPTER X Adsorption Potentials and Electrokinetic Phenomena summary of contents Adsorption is defined as that phenomenon in which a substance dis- solved in one phase becomes concentrated at the boundary surface of an adjacent phase, irrespective of the forces causing this concentration. If such adsorption involves dissolved electrolytes, then the unequal ad- sorption of positive and negative ions may give rise to a potential differ- ence at the boundary surface. By studying the adsorption of electrolytes on charcoal one may obtain an idea of the adsorbability of the different ions. That of the H- and OH-ions appears to be especially great. The adsorbing surfaces may be grouped, according to their capacity to bind H- or OH-ions, as acidoids, basoids or ampholytoids. This classification corresponds to that derived on the basis of chemical constitution for the water-soluble acids, bases and ampholytes. In the process of adsorption all cations compete with the H-ions, while all anions compete with the OH-ions. Because of an opposite exchange of ionogenic radicals between the adsorbent and the dissolved substances exchange-adsorptions arise, which may be considered almost entirely in terms of purely chemical phenomena. The adsorption of ions is the cause of the electric charge on the boundary surface. These potential differences become manifest in the phenomena of endosmosis or cataphoresis, whose history and theory will be reviewed. The tangential displacements of the double layers against each other are chiefiy involved. The electrically charged or ionic layer moving in water carries along with it water particles. According to whether the water or the solid particles are fixed by mechanical means, cataphoresis or endosmosis ensues. The effect of the dissolved electro- lytes upon the magnitude and direction of the endosmosis or the cata- phoresis permits one to predict the magnitude and direction of the adsorption potentials, and the results correspond to those to be expected from the determination of the actual adsorption of ions. In this con- nection we find again thai the adsorbents may be grouped into: always negative (acidoids), always positive (basoids) and those with a change- 231 232 HYDROGEN ION CONCENTRATION able sign of charge {ampholytoids) . Among the acidoids there are some whose sign of charge can be reversed, if not by H-ions, then by trivalent cations. In the electroendosmosis of water, the changes in the concen- trations of dissolved electrolytes on both sides of the diaphragm are interrelated, especially the changes in H-ion concentrations. Another manifestation of adsorption potentials are the hydrodynamic potentials. The adsorption potentials are electromotively inactive. When they are led off to electric measuring instruments, in the same way as the previously described phase boundary potentials, then only such phase boundary potentials are manifested. The adsorption potential is only a stage in that potential difference whose totality represents the phase boundary potential. It is best, therefore, to represent the phase boundary potential not as an unstable potential difference, but rather as a gradual change of potential within a very thin but measurable layer. Finally, it will be attempted to correlate Coehn's law with these phe- nomena, and the relation of boundary potentials to colloid chemistry will be discussed. 64. Definition of adsorption The concept of adsorption potentials as developed below includes a whole series of electric phenomena which are of great importance in biology and are of no less interest to the pure physicist. The prob- lems of this field are as yet even further removed from their ultimate solution than those presented above. It is possible that the method of presentation chosen will contribute towards further elucidation, insofar as it is unified in character and at the same time points out the existing gaps in our knowledge. Therefore, the whole method of presentation employed may be regarded as an attempt to fuse to- gether facts and theories of varied origin into a single coherent structure. The future is to show of what developments such a structure is capable. The point of departure will be the phenomonon of adsorption, especially the adsorption of ions. For this purpose it will be necessary to define adsorption in the sense in which we shall use this term. Different authors have given it different interpretations. For our purposes we shall define it as that process by which a substance accumulates at a boundary surface of two contiguous phases in a concentration higher than it exists in the interior of these two phases. The question of the forces ADSORPTION POTENTIALS AND ELECTROKINETIC PHENOMENA 233 causing this accumulation is deliberately left open, i.e., it is im- material for the present whether it is caused by physical or by the so-called chemical forces. At the time when Rutherford's atomic model is constantly furnishing greater support for the common basis of the physical and chemical properties of matter, it appears quite futile to attempt to differentiate between mechanical forces of ad- hesion and chemical attractions. We shall therefore use the term adsorption not in any sense of distinction from chemical union, and we shall take it to include cases which theoretical chemists would recognize as chemical reactions, as well as such cases which, to an investigator of the older school, would not convey the least sug- gestion of chemical affinity, such as, for example, the adsorption by charcoal of an alcohol dissolved in water. The important feature of our definition is that this binding process occurs at the boundary surface of two phases, and not in the interiors of these phases. This definition of adsorption leads to a peculiar consequence. According to Gibbs' principle, surface-active substances dissolved in w^ater must concentrate in the surface layer of the water, even though no special "adsorbent" is present, and even when the water is bound only by a gaseous space. In this case, according to our definition, the gaseous space would be the "adsorbent." While it is true that this designation is a more or less borrowed one, it cannot, nevertheless, be neglected. Thus let us take for an example two phases, A and B, and dissolved in phase A is substance C, which is either entu'ely insoluble in B, or has reached in B its equilibrium of distribution. Then the composition of the boundary layer will be determined by the effects of the following molecular interrelationships: 1. those of the molecules A - A, 2. A - B; 3. A - C; 4. B -B;5. B -C;and 6. C — C, But when phase B (a gas) is practically without mass in comparison with A, then the forces A — B, B — B, and B — C need not be considered. In spite of this, and because of the interference of the remaining forces, there occurs a concentration of substance C in the surface layer of phase A. One could offer the definition: — in this case the adsorbent is lacking; hence an adsorbent is a substance which changes such a state of distribution. But since the amount of C accumulated in the free surface layer cannot be experimentally determined, practically it is preferable to choose for our point of departure that state in which the dissolved substance C is uniformly distributed throughout in phase A, and to designate every accmnu- 234 HYDROGEN ION CONCENTRATION lation in the surface layer as adsorption. If it is desired to differen- tiate such an adsorption as described above from the usual mode, it could be designated as ''apparent adsorption.'' In these terms the empty space becomes also an adsorbent. And thus this definition facilitates the formal explanation of the phenomena of adsorption. 65. The adsorption of electrolytes by charcoal The problem reduced to its simplest form relates to the adsorption of the individual ionic species. But since it is impossible to obtain a solution of an individual ionic species, and since we can only have a TABLE 37 SOi ci- CNS- OH- The adsorption of ions from 0.1 A" solutions by charcoal Na+ 0 0 0 6 47 6 6 7 13 14 21 27 38 59 29 36 55 63 39 K+ NHi+ |Ca++ iMe + + — iZn++ _ AA1 + + + iCu + + H+ With 0.05 A^ solutions Na+.... NH4+ .. 3 H+. 0 0 16 47 70 11 11 35 73 68 solution of one electrolyte, i.e., a mixture of a negatively charged and a positively charged species of ions, so that actually we can only investigate experimentally the adsorption of the entire electrolyte molecule, and any statement concerning the adsorption of a single ionic species, which completely escapes chemical analysis, can be made only by inference, and with greater or lesser certainty. Thus we find, for instance, in studying the adsorption of different sodium salts (NaCl, NaBr, Nal, etc.) certain quantitative differences. Now if we study the calcium salts of the same anions, we find other ab- solute values, but the relative differences in the series of anions are ADSORPTION POTENTIALS AND ELECTROKINETIC PHENOMENA 235 found to be the same as before. Hence we must conclude that the adsorption properties of an electrolyte are additive functions of those of its anions and cations. But also the nature of the adsorbent as well as of the solvent, has an influence upon the process of adsorp- tion. Therefore, in an experimental study, we must vary the adsorb- ent, the cation and the anion. For the adsorbent we shall choose for the present charcoal,^ which is the prototype of the so-called chemically indifferent adsorbing agents, and we shall vars^ the anions and the cations. After some preliminary experiments with Lachs, the author in collaboration with Rona^ has carried out a systematic series of in- vestigations and has obtained the following results. Of the neutral salts all, with the exception of the sulfates of the alkali metals, were adsorbed to an analytically demonstrable extent. In all cases of true neutral salts equal amounts of cation and anion were adsorbed, no measurable amount of hydrolytic splitting having occurred in the adsorption of true neutral salts. The adsorption of a salt appeared as an additive property of its ions, thus, for instance, all sulfocyanates were more strongly adsorbed than the corresponding chlorides. Consequently the anions and cations can be arranged in series ac- cording to their adsorbabilities. In such series we can also include the H- and OH-ions, by simply comparing the adsorption of HCl with that of other chlorides and bj^ comparing the adsorption of NaOH with that of other Na-salts of equivalent concentrations. These series are Cations: Na"*" I Organic dye bases K+ Ca++ Mg++ Al^++ Cu++ H+ NH4+J Ag+ Anions: SO? CI- Br- I- SCN- OH- Organic dye acids ' Blood charcoal will always be understood. Retort-, sugar- and other charcoals do not show the same uniform behavior. 2 Lachs and Michaelis, KoU.-Z. 9, 275 (1911) ; Zeitschr. f. Elektrochemie 17, 1 and 917 (1911); Rona and Michaelis, Bioch. Zeitschr. 94, 240; 97, 85; 103, 19 (1920) ; Michaelis and Rona, Bioch. Zeitschr. 94, 225; 97, 57; 102, 268 (1920) : KoUoid-Zeitschr. 25, 225 (1919). 236 HYDROGEN ION CONCENTRATION The most striking aspect of these series is the relation to the elec- tromotive tension series, the nobler the metal the more easily are its ions adsorbed. Also the valence has a certain influence. The higher oxidation forms of the ions (Fe+++, Hg++) are exceptionally strongly adsorbable, as well as the salts of certain organic bases and acids, such as the dyes, quinine, etc. Among the halogen anions the dis- charge tension increases with increasing atomic weight. No demon- strable difference in the adsorbabihties of the different alkali metal cations was observed. This fact was also confirmed by Hartleben.^ The relative adsorption of the salts, as it was also shown for non- electrolytes, increases with the decreasing concentration of their solutions. Table 37, taken from the cited article by Rona and Michaelis, shows the results obtained by using 0.1 equivalent normal solutions of the different salts and by employing 15 g. of the same land of charcoal (Merck's blood charcoal) per 100 cc. of solution. The figures represent the per cent of the original salt content re- moved from solution by adsorption."* 66. The exceptional position of the H- and OH-ions At this juncture also our attention is especially attracted to the H- and OH-ions; they assume an exceptional position and belong to the most strongly adsorbable ions. Only the ions of the "noblest" metals, a few polyvalent (Fe+++) and some organic ions rival the H-ions, and only the most easily adsorbed acid dye ions compare with the OH-ions. In this connection an analogy to the chemical combinations among ions suggests itself. In aqueous solutions most ions combine among themselves but to a very slight extent, i.e., most salts are "strong" highly dissociated electrolytes. Only among the acids and bases are there countless weak electrolytes. This means that the H- and OH-ions bind other ions more firmly than all other ions. In an exactly similar way, the H-ion is more firmly bound by charcoal than the other cations and OH-ion more firmly bound than the other anions. To be sure, it is impossible to effect the 3 Hans Hartleben, Bioch. Zeitschr. 115, 46 (1921). * The figures in the original article were recalculated in the following man- ner. From the given data of the rough analyses a correction for the water content of the charcoal (30 per cent) was applied. Furthermore, all concen- trations were recalculated into terms of normal equivalents, for some of the original data were given in terms of molarity. ADSORPTION POTENTIALS AND ELECTROKINETIC PHENOMENA 237 adsorption of H-ions alone by charcoal, the entire acid molecule (HCl, HNO3, etc.), being adsorbed. The adsorption of the H-ions cannot proceed to their specific adsorption equihbrium, for they are held back in solution by the much less adsorbable Cl-ions (or the corresponding anions of the acid). On the other hand, more Cl-ions are adsorbed than would correspond to their specific adsorption equilibrium since some are carried along by the better adsorbable H-ions. The actual adsorption equihbrium represents an average somewhere between the specific equilibria of the two ions (i.e., the H-ion and the anion). It is of ob\dous interest to determine the adsorption of the H- and OH-ions alone. In order to achieve this end the experiment must be so arranged that the establishment of equilib- rium be hindered as little as possible by opposing forces. This can be approximately accomplished by studjdng the adsorption of HCl in presence of a large excess of KCl. The reason for this is the following : Let there be given a chemical system consisting of a dilute aqueous solution of a non-electrolyte (e.g., acetone) and of a solid adsorbent, such as charcoal. The adsorption equilibrium in this system is attained. Further, let the extension of the phases be so great, that the addition to or the removal from the solution of one mol of acetone would cause no appreciable change in concentration. Under these conditions the equilibrium attained is characterized by the fact that no work is required to transfer a relatively very small amount, e.g., 1 mol, of acetone from the charcoal surface into the solution. But if we assume that no state of equilibrium has been attained, and that the concentration of acetone is smaller in the solution than it is on the charcoal surface, then there should be present a tendenc}^ to reach equilibrium. If we could render this process reversible, we should be able to gain a certain amount of work. In order to calculate the amount of this work let us imagine the process of attaining equihbrium as occurring in two stages. In the first we shall compress the solution, by means of an osmotic piston, until it has that concentration which wou'd be in equilibrium with the charcoal surface, thus changing the concentration from c to Cq. To obtain this result we must expend an amount of work n RT In — , c where n is the number of mols of acetone in the solution. Now let 1 mol of acetone pass from the charcoal surface into the solution. 238 HYDROGEN ION CONCENTRATION Since this will occur in the state of equilibrium, no work will be per- formed. Finally, we shall dilute our solution, again by means of the osmotic piston, to its original concentration, whereby the amount of work, (n + 1) RT In Co, will be gained, since there are now n + 1 mols of acetone in the solution. The algebraic sum of the work ex- pended and gained gives the amount of work gained, RT In — . Thus the amount of work depends upon the ratio of the given concentration to the concentration at equiUbrium. When an electrolyte, such as HCl, is adsorbed, a similar process takes place. The easily adsorbable H-ion must carry along with it the less easily adsorbable Cl-ion, so that more of Cl~ is adsorbed than would correspond to the true Cl~ adsorption equiUbrium. To ac- complish this a certain amount of work must be apphed, which depends upon the ratio of the given Cl~ concentration of the solu- tion to that Cl~ concentration which would correspond to the specific Cl~ adsorption equiUbrium. When a definite amount of Cl~ had been carried along upon the charcoal surface, then the value of the ratio of the original Cl~ concentration to the final concentration approaches the closer to 1 the greater the Cl~ concentration. There- fore, the H-ions can cany along the necessary Cl-ions out of a solution rich in Cl~ almost without an expenditure of energy, and the true adsorption equilibrium of H-ions is thus established. If this theory is correct, the adsorption of HCl should be increased through the addition of KCl. This has been experunentally confirmed.^ This result was not one to be expected as a matter of course. For hitherto it had always been observed that the presence of another substance decreased the adsorption of the first substance ("adsorption- displacement")^ while in this instance the very opposite was the case. By constantly increasing the concentration of the added KCl in a 0.01 A'' HCl solution a limiting value for the adsorption of the HCl is reached which represents the true adsorption equilibrium of the H-ions. The theory was confirmed by finding that the adsorption of HNO3 when accompanied by an increasing concentration of KNO3 led to the same limiting value for H+-adsorption. In the same s Rona and Michaelis, Bioch. Zeitschr. 97, 85 (1919). « Michaelis and Rona, Bioch. Zeitschr. 15, 196 (1908) ; 16, 489 (1909) ; Masius, tJber die Adsorption in Gemischen. Diss. Leipzig 1908; Freundlich and Mas- ius, Gedenkboek van Bemmelen 1910. ADSORPTION POTENTIALS AND ELECTROKINETIC PHENOMENA 239 way it was possible to estimate the true adsorption equilibrium of OH-ions by the addition of NaCl to a NaOH solution. The striking result was in the finding that the adsorhahilily of H-ions and the OH-ions is practically the same. The full value of the experiment in demonstrating quantitatively the above relations is obtained only, when it is possible to assume that the K-ions are so poorly adsorbed in comparison with the H-ions, that they do not increase the adsorption of the H-ions, in spite of their greater concentration. This assumption is practically quite justifiable, for the very slight degree of adsorption of K-ions in com- parison with H-ions is easily demonstrable by separate adsorption tests with pure HCl and KCl solutions. The above does not hold true for acids with strongly adsorbable organic anions. Thus, for in- stance, saHcylic acid is adsorbed to a much greater extent than HCl,^ apparently because its organic anion is more strongly ad- sorbed than its H-ion. In this way the following figures were obtained. The limiting value of the adsorption of HCl, HNO3, H2SO4, KOH and NaOH, in the presence of an excess of the corresponding salts, from a 0.01 A^ solution by one per cent by weight of the same kind of charcoal (Merck's blood charcoal) was 40-45 per cent. (From a 0.01 N NaCl solution, under the same conditions, the limiting values were 2 to 5 per cent). 67. Charcoal as an insoluble ampholyte: acidoid, basoid, ampholy- toid, saloid An acid has been defined above (see p. 18) as a molecular species which, while maintaining its negative electric charge, binds OH-ions, and a base as one binding H-ions. This definition could also be ex- tended to substances which are not molecularly dispersed. Thus we could say that charcoal is an absolutely insoluble ampholyte, not molec- ularly dispersed, which binds H- and OH-ions to the same extent. Its isoelectric point is, therefore, in the vicinity of the true neutral reaction. In the section on electroendosmosis it will be shown how closely this theory agrees with the observations made upon the elec- tric charge on charcoal. An attempt to apply the concepts of acid, base and ampholyte to an absolutely insoluble substance may, however, lead to a certain ^ Unpublished results. 240 HYDROGEN ION CONCENTRATION confusion. For this concept of acid, base, etc., is bound up with the property of electric conductance in aqueous solutions. Tliis property, to be sure, does not entirely disappear in our non-mole cularty dis- persed substances, and we shall shortly recognize its share in con- ductance phenomena under the form of electrophoresis. But in order to avoid any possible confusion it would be better to restrict the use of the terms "acid, base, ampholyte, salt" for the water-soluble, true electrolytes, and the non-molecularty dispersed substances which, because of their capacity to adsorb OH-ions, are analogous to acids will be designated as acidoids. The following definitions are given : An acidoid is a non-molecularly dispersed substance which adsorbs OH-ions (or dissociates off H-ions) while maintaining a negative charge (or adsorbs H-ions on neutralization of its negative charge). A basoid is analogously defined, and charcoal is a good example of an ampJiolytoid. Between a true acid (or base, or ampholyte) and a coarsely dispersed acidoid (or basoid, or ampholytoid) the colloidal substances form a multitude of intermediate stages. In adsorbing a dissolved base an acidoid forms a salt-like surface compound which we shall designate as a saloid. Charcoal, being an ampholytoid, forms saloids, with acids as well as with bases. In addition, behaving like a true ampholyte, charcoal forms with KCl a double saloid, just as an aixiino acid, at least in its sohd state, forms a double salt compound with KCl. Also, quite analogously, the tendency of charcoal to combine with HCl or NaOH is much greater than that of combining with KCl, just as in the case of an amino acid. The author is aware of the fact that the consideration of the so- called "chemically unreactive" substances and of ampholytes under a common heading will not meet with universal acceptance, and that the phenomenon of adsorption will be regarded for a long time to come in the light of a process quite distinct from those of ordinary chemical reactions. In the age of the atomic model and of the re- duction of chemical affinity and of the forces of cohesion to a common basis of electrostatic forces, it would appear quite appropriate to do away with such hmitations. Henceforth, then, in spealdng of ad- sorption processes, it should be clearly understood that they do not represent a fundamental contrast to chemical union. Since we can only speak of adsorption when the reaction is hmited to the boundary sur- ADSOEPTION POTENTIALS AND ELECTROKINETIC PHENOMENA 241 face of two adjacent phases, the isolation of the resulting compound in a pure state is not possible, and, therefore, its stoichometric relations cannot as yet be established. At this juncture we must return to the extended definition of adsorption given on page 234. When the substance of the non- aqueous phase does not display a marked molecular attraction either for, H- or OH-ions, then it behaves towards a solution con- taining H+- or OH~-ions exactly as a mass-free space, and in such a case the accumulation of the H- or OH-ions at the surface must follow simply in accord with Gibbs' principle. It is not possible to predict whether the H- or OH-ions will show more surface-activity. But experience tells us that numerous non-aqueous phases which are chemically neither acids nor bases assume a negative charge when in contact with pure water. This is a partial aspect of Coehn's law (see below, page 288). The common basis of all these phenomena renders it probable that the non-aqueous phase does not participate in the H+- OH "-distribution, and that this distribution occurs as if the aqueous phase were in contact with an empty space. The negative charge on the non-aqueous phase in respect to the aqueous phase leads one to postulate that the OH-ions are more strongly sur- face-active than the H-ions, that, therefore, the OH-ions accumulate at the surface of the water phase, and that consequently the interior of the water phase becomes positive in respect to its own surface layer and hence also in respect to the adjacent phase. If we are to apply here our extended definition of adsorption, we must suppose that such chemically indifferent substances, as ethyl ester, benzonitrile, oils, hj'drocarbons, etc., have a greater tendency to adsorb OH-ions than H-ions, and are, therefore, to be considered acidoids. But even substances not chemically indifferent are more frequently negatively charged towards water rather than positively. Thus Freundlich and Gyemant^ have found that anihne is negatively charged towards water or towards water saturated with anihne. It could perhaps be expected that aniline, being a base, must adsorb H-ions (C6H5NH2 -f- H+ = C6H0NH3+), and that at all events it should adsorb H-ions much better than OH-ions. But this is quite erroneous. Those few aniline molecules which have combined Avith H-ions have become water-soluble aniUne-ions, and we could scarcely expect any signifi- * H. Freundlich and A. Gyemant, Zeitschr. f. physikal. Chem. 100, 182 (1922). 242 HYDROGEN ION CONCENTRATION cant amount of such would remain in the boundary layer. Other- wise aniline behaves as a chemically indifferent substance, and there is nothing in the way of OH-ions accumulating in the surface layer in accord with Gibbs' theory. It could then be stated that anihne not dissolved in water is of the character of a very weak acidoid. No matter what nomenclature is adopted, instances will be met with which are not adequately provided for. Such cases are disposed of by our "extended" definitions. The most exact of the sciences, mathematics, furnishes ample illustrations of this. Thus the defini- tion: "a — b is the sum of the terms + a and — b" is an instructive example. For, according to the original definition of a sum, a — b is no sum at all, but its exact opposite, a difference. Likewise, we shall have to accept the idea, that aniline may either be a base or an acidoid, according to the conditions and to the point of view. Whether or not the acidoid- or basoid-nature of a substance varies as its true chemical acidic or basic character, depends upon the relative intensity of the acidic or basic character. The very weak base aniline is thus an acidoid, while the much stronger base A1(0H)3 is a basoid in respect to pure water. Hence only substances with a very pronounced basic character can behave as basoids, and that is why basoids occur relatively much more rarely than acidoids. And most of these are basoids only towards a neutral or acid solution, for most of them chemically are ampholytoids possessing an iso- electric point, as, for example, AI2O3, Fe203, ZnO, and other similar substances. 68. The equivalent adsorption of ions of a salt by charcoal; the neutralization effect of charcoal The property of the H-ions of being more strongly adsorbed by charcoal than the other cations leads to the following consequence. A solution containing any number of ionic species and whose reaction is acid never becomes more acid in reaction on being shaken with charcoal. On the contrary it becomes less acid. An alkaUne solution behaves in a reverse manner under these conditions, becoming less alkaline,^ Thus, as a matter of rule, any solution on being treated with charcoal approaches neutrality, and the reaction of a neutral solution never becomes markedly acid or alkaline on treatment with charcoal. 9 W. Loffler and K. Spiro, Helvet. chim. Acta 2, 417 (1918) and 2, 533 (1919). ADSORPTION POTENTIALS AND ELECTROKINETIC PHENOMENA 243 Exactly in the same way the reaction of any aqueous solution may be made to approach neutrality by the addition of a true ampholyte whose isoelectric point lies in the vicinity of neutrality. Hence it follows that in an aqueous solution of a neutral salt, or even of a salt of an organic dye, equivalent amounts of anions and cations are ad-, sorbed, i.e., the salt molecule is adsorbed as a whole. Michaelis and Rona^'' designated this phenomenon as the equivalent ionic adsorption of salts, and established the fact that no hydrolytic cleavage of salts takes place in the process of adsorption on charcoal. The second observation stated above, to the effect that an acid or alkaline solu- tion approaches neutrality on treatment with charcoal could have been deduced from these observations of Michaelis and Rona, but was not recognized by them at the time. It was found experimen- tally by Loffler and Spiro,^ and this phenomenon can be designated as the neutralization effect of charcoal.^°'' 69. Other adsorbents Charcoal possesses the pecuharity of not itself yielding any ions to a solution. As soon as an adsorbent contains ionogenic radicals, it must be taken into account that the adsorbent itself may show a >« Michaelis and Rona, Bioch. Zeitschr. 94, 225; 97, 57; 102, 268 (1920). 1"" It developed later that the above described amphylotoid nature of char- coal is not a general property but is rather peculiar to certain varieties or preparations of charcoal. It seems to have little to do with the mineral im- purities, but to depend much more upon the organic material from which the charcoal is prepared, and above all upon the temperature at which the carbon- ization is carried out. It now appears that with rising temperature of charring all charcoals tend to reach a single definite limiting state. This fact was first brought out by Miller and Bartell (Jour. Amer. Chem. Soc, 44, 1866 (1922)) with highly heated (activated) ash free charcoal from sugar. This charcoal is not of ampholytoid character. For it adsorbs HCl but to a slight extent, does not adsorb NaOH at all, while neutral salts are adsorbed by it in such a manner that a trace of acid is adsorbed, leaving a slightlj' alkaline solution, the adsorp- tion being thus a hydrolytic one. On the other hand blood charcoal which had not been activated by thorough heating will yield all the adsorption phenomena described in this section (I . Ogawa, Biochem. Zeitschr., 172, 249 (1926jj. No recognizable differences were observed in the elementary analyses of the different charcoal prepara- tions before and after activation, making it probable that the changes occur only in the ultimate crystal structure of the substance, in spite of the fact that the x-ray diagrams obtained by the method of Debye and Scherrer show no differences between the two states. 244 HYDROGEN ION CONCENTRATION tendency to form ions or to exchange ions with the solution about it. The simplest case is that of an adsorbent of the type of an insoluble acid (silicic acid, mastic-resin acid), or of an insoluble base (metal oxides). It was found that in such cases the adsorption proceeds along the lines of ordinary chemical laws, and here it would be quite futile to attempt to demonstrate any contrast between chemical reactions and adsorptions. Thus, for example, ferric oxide (colloidal ferric hydroxide) adsorbs picric acid and eosin, because ferric picrate and ferric eosinate are insoluble. Likewise, silicic acid adsorbs methylene blue, because methylene blue-siHcate is insoluble. To be sure, the experimental study of this state is beset with unexpected difficulties. For it is almost impossible to obtain such "insoluble acids or bases" free of ions adsorbed on their surfaces with which they form insoluble salts. Thus there is no siHcic acid preparation free of Ca++,^^ and there is no colloidal iron preparation free of Cl~ (or of some other anion, depending upon the method of preparation) with which it forms an insoluble basic salt. For this reason those iono- genic adsorbents which may be conceived of as insoluble "salts" are more convenient for experimental work. These are represented by such substances as kaohn (silicates of alkahne earths) and the so- called ferric oxide (basic ferric chloride). Such adsorbents behave entirely according to the well known laws of chemical reactions and react with the electrolytes in aqueous solution by means of an ex- change of ions. A solution of methylene blue chloride reacts with kaohn according to the following scheme : Alkaline earth silicate + methylene blue chloride :^ Methylene blue silicate + alkaline earth chloride Since methylene blue silicate is insoluble, and the alkaline earth chloride is soluble, the kaohn exchanges its alkaline earth cation (especially the ever present Ca++) for the methylene blue cation, while the Cl-ion of the methylene blue remains in solution, not as HCl, however, but as CaCl2. An exchange of ions between kaolin and NaCl cannot be demonstrated by analytical methods, although doubtlessly according to general chemical laws such an exchange can take place. Not a trace of eosin or picric acid is adsorbed by kaohn, for an exchange of ions in these instances would not result in the formation of insoluble salts. " Michaelis and Rona, Bioch. Zeitschr. 97, 57 (1919), cf. p. 71. ADSORPTION POTENTIALS AND ELECTROKINETIC PHENOMENA 245 It would be of great interest to study the adsorption capacity of an indifferent substance, such as cellulose (filter paper, cotton wool), but unfortunately great experimental difficulties arise. It is quite impossible to obtain a cellulose entirely free of ash. Even after the most careful extraction with hydrochloric and hydrofluoric acids it always contains some silica and lime. In studying the adsorption of dyes on cellulose, Michaelis and Rona^- found invariably adsorp- tion by exchange of ions. Thus after the adsorption of methylene blue chloride the cation was found to have been adsorbed while the anion (CI") was found in solution as a neutral salt (CaCl2). Simi- larly, in the adsorption of NII4- eosin, the eosin anion was adsorbed, while at least 70 per cent of the NH4- ion remained in solution as a neutral salt. Since "adsorption by exchange" exercises so great an effect it has been practically impossible to ascertain the extent to which the cellulose itself participates in the adsorption process. It must be, in any case to an extremely minute extent. The abihty of the technical forms of cellulose (cotton, paper etc.) to take dyes must depend largely upon an exchange of ions with its ash con- stituents, and the latter form quite possibly an integral part of the cellulose molecule. The properties of cellulose are very different from those of charcoal, in which, to be sure, there is also no lack of these impurities, but in which the much greater adsorbing capacity of the charcoal itseK renders any effect due to the impurities neghgible. It is not the purpose of this book to exhaust the problem of ionic adsorption. The discussion offered above should only serve as an introduction to the subject of adsorption potentials, and it should suffice for this purpose. 70. The formation of a charge upon phase boundary surfaces due to adsorption of ions In the Ught of the explanations given in the preceding section we can ascribe to each kind of ion its own specific adsorbabihty. But since the electrostatic forces prevent the adsorption of the two ions of an electrolyte to measurably different extents, what really happens is the estabhshment of an adsorption equilibrium which lies somewhere between the hypothetic equihbria of the positively and >2 Michaelis and Rona, Bioch. Zeitschr. 103, 19 (1920). 246 HYDROGEN ION CONCENTRATION of the negatively charged ions. But at the same time a potential difference is formed at the phase boundary surface. Still the more adsorbable of the two ions is adsorbed to a greater extent than the other ion, this difference being demonstrable, if not by analysis, then by some other means. Charcoal in contact with dilute HCl solution adsorbs somewhat more H-ions than Cl-ions. While the difference cannot be demonstrated by analysis, it becomes evident from the fact that the charcoal surface has become positively charged towards the solution. The adsorption of the H-ions in excess pro- ceeds to such an extent that the potential due to the separation of the ions just compensates the repulsion tendency of the ions themselves. In this way can we explain the long known fact that a potential difference arises at the boundary surface of any two different phases in general. As the means of studying and recognizing this formation of electric charges we have at our disposal two apparently very different, and intrinsically quite correlated, methods: electric en- dosmosis (electroendosmosis) and electric cataphoresis (electro- phoresis) . In entering now upon the discussion of the so-called electr akinetic (Helmholtz) or electroosmotic (von Smoluchowski) 'phenomena, we shall for the time being apparently lose our connecting thi-ead with the preceding chapter, but eventually we shall reestablish the connection. 71. The earlier history of electroendosmosis In 1808 Reuss^^ made the following observation: If two tubes filled with water are partly immersed in wet clay, and a galvanic current is then sent through them by means of metal electrodes, the water rises to a higher level in the tube on the cathode side and falls to a lower level on the anode side. This was the discovery of electroendosmosis. At the same time particles of the clay become loose and travel towards the anode, as shown by the clouding of the water in the tube. This was the discovery of electrophoresis. Both phenomena are, as it were, the expression of the positive charge of the water with respect to the clay. If the clay is thought of as a sohd filter or diaphragm for the water, then only the movement of '^ F. Reuss, M6m. de la Soc. imp. des Naturalistes de Moscou. 2, 327 (1S09). ADSORPTION POTENTIALS AND ELECTROKINETIC PHENOMENA 247 the positively charged water towards the cathode would be possible. But the clay particles which are ireely movable in the water also exhibit their negative charge by their motion towards the anode. The meaning of the endosmosis was not clear, and for a time Uttle attention was paid to it. It was again described in 1816 for sand filters by Porret,^'* and then also by Becquerel, Armstrong, and Daniell. The phenomenon of electrophoresis was further described in tissue fibers by Faraday, ^^ in starch grains by DuBois Reymond,!'' and in carmin and starch by Heidenhain and Jiirgensen.^^ The first more exact investigations on electroendosmosis were carried out by Wiedemann, ^^ and the first theoretical treatment of this subject approaching the views held toda}^ was furnished by Quincke. ^^ This modern theoretical conception resulted from the observation that water does not always move in the direction of the current but at times moves against the current. In this way Quincke showed that the original idea to the effect that water "was carried along by the current" was not correct. He then proposed the theory of the electric double layer wliich has since proved to be so fruitful. Quincke assumed that at the boundary of fluids with sohds two layers of free electricity, one positive and one negative are found at a very small but finite distance from each other. The one forms a coating adhering fast to the soHd, while the other is in the fluid medium. The latter laj^er can be parallelly displaced along the former by means of a tangentially directed current of electric forces. If the solid be a particle suspended in the fluid, then this displacement is manifested in the motion of the particle under the influence of an electric field. If the solid is a porous sohd in the nature of a diaphragm blocking the path of the water, then, in an electric field, the water will migrate through the pores of the diaphragm in the opposite direction. The quantitiative relationships of this theory of these phenomena were developed by Helmholtz.-*' >^ R. Porret, Gilb. Ann. 66, 272 (1816). 15 M. Faraday, Exp. Res. Nr. 1562, 1838. 16 E. Du Bois-Reymond, Berl. Ber. 1860, p. 895. 1^ E. Heidenhain u. Jurgensen, Arch. f. Anat. u. Physiol. 1860, p. 573. 18 Wiedemann, Ann. d. Phys. u. Chem. 87, 321 (1852). '3 Quincke, Ann. d. Phys. u. Chem. 113, 513 (1861). 20 H. V. Helmholtz, Ann. d. Phys. u. Chem. 7, 337 (1879) ; Ges. Abhandlungen I, p. 855 (1882). 248 HYDROGEN ION CONCENTRATION 72. Helmholtz's theory of electroendosmosis-^ Let us imagine that a solid diaphragm with capillary pores is inserted into a tube filled vv'ith aqueous fluid. Now, on sending an electric current through this tube, we observe that the water is set into motion, and that after a short time a flow of water with a con- stant velocity will ensue, provided care is taken to prevent hydro- static difference of pressure due to the migration of the water. Let us now consider this stationary condition of our system. To begin with, it is characterized by the fact that the motion produced by the electric current is rendered uniform by the friction. Now the electric force is proportional to the electric density of the double layer a, and to the electric field intensity, H, to which the double layer is exposed. In terms of absolute units the electric force is simply equal to cH. This value remains constant throughout the entire process. If no friction were present it would produce a uniforml}^ accelerated motion of the water. But the friction, checking the motion, changes its rate from a uniformly accelerated to a uniform one. The force of friction is proportional to the specific friction coefficient of water, 7? to the velocity v, and inversely proportional to the distance between 21 The theory is not expounded here in Helmholtz's original terms and scope, for that would require of the reader an extensive knowledge of theoretical physics. It is presented in its simplified form, as given by Perrin. Helm- holtz's own presentation has the advantage of greater strength, especially in regard to the absolute value of the proportionality factors, which is not so convincing in Perrin's interpretation. But since the assumption on which Helmholtz's calculations are based have perhaps not been strictly justified (of. section 78), and since the experimental data available thus far is still insufficient to confirm the theory unqualifiedly, the mode of interpretation given here will have to suffice, in a preliminary way, for physiologists. The difficulty in the way of experimental quantitative confirmation of the theory is the following : Adsorption potentials cannot yet be recognized in any other way except through electrokinetic phenomena (electrophoresis, electro- endosmosis, hydrodynamic currents and currents of falling particles). Thus, for example, in Helmholtz's theory the wall potential is calculated from the figures given by electroendosmosis. The theory could only be confirmed by comparing the value of the potential obtained from electrokinetic data with that obtained by using some other method which is entirely independent of electrokinetic technic. Such a method is still unavailable. And thus the quantitative aspect of the theory is still uncertain, while in its qualitative aspect is of great service. ADSORPTION POTENTIALS AND ELECTROKINETIC PHENOMENA 249 the layer set in motion and the wall. Hence, in our stationary con- dition we have d where d is the distance between the moving electric layer and the resting laA^er, hence it is the thickness of the double layer. If, on the other hand, the volume >f> of the fluid displaced per unit of time is measured, then the velocity v can be expressed in the following terms, on the assumption that the diaphragm is rep- resented by a single capillary whose average cross section has the radius r: TT r^ V = ^5 from which it follows that U Tl •*s a 03 03 bl) n> a O m >> W crt & _c -4J o C ID O " •2 o ^.5 ^-^ I I 03 :3 c o3 I 4) TO o > ^ to •=! o3 to « !- « O -^ © Tl o3 M 0) %t Tl c3 03 o o 43 &43 ■*-> H 43 -fj o ^ ki -1-= ^ to 03 O Oh > CO Si M CQ o 45 O !».2 -♦J 00 li a) m .£; ® -♦J CO O (N -* 00 1-! r-I lO (N 00 00 1> LO lO oo 00 t^ 00 ^H CO CO CD (M 05 uo (» CM 00 lO ■<*< 00 O "H (M ,-1 (M rfiC0O ^H Tfi CO 1— I 2 -* CO CO (M C3 O I I I + I I + I + + + + .o <1> s t. trl _ ^ "9 ts O ^ ce «^ -=< ^ tM W _ -^ *^ " ^OOO 03 cj 03 eSX .S o :-! o o o o o o o o o o o o o o o .280^ Q 00000 00000 u?n i < ^^ ^^ '^H ^^ '^^ ^^ '*^ LO 10 i-H T-H ^H ca o >— I c^ 000000^0 00000000 00000000 00000000 o a g a o o o3 o S3 O S ADSORPTION POTENTIALS AND ELECTROKINETIC PHENOMENA 263 a *3 I t-i O o3 OQ 13 « ii bC O 02 a 05 >> bC- o3 O O 0) BO • > a a • r3 o 2 ^ -T 03 a O o oi 00 CO Tti t>. CD t-- t—i lO O b- CO 00 l-l O 1-H cv| ^ fl a C3 03 o3 03 ;-< (H ;-i -n Xi X! 1—1 (—1 a a a CI a> 0) o o O is: o o W _ -s- «? - OOO o U o3 o3 o3 ^ ^^ f^ ^^ ^^ C-1 ^^ u O dJ 1— C T-I I— C T-H ^ 1— I ^88S8m8 ^ S o (6 d 4) •T cS bO c3 0 a> o 0) P. (14 03 += '^ 2 "5 tf >- 9 '=' O o o o a « © g -a CO -a ^ -^ 03 ^ Si ^ 03 '2 ^ 03 ^ a -^-- ^ 03 !>>^ 1-^ bfl o3 05 05 05 "S to «5 o 05 £ 05 s a 03 ^ 05 a 45 05 _ 05 ^ . el 05 3 .2 tn . ._ 05 O 43 O rd 0) •- +^ -a -£3 +3 05 'oo a 05 05 .§.s 05 05 05 fl o o3 t-. a -s 45 05 O s -^ 45 n 45 * +j 05 J2 05 a 05 -a d 45 a 45 13 fl "3 45 -|J m 3 a a 45 .J bi 45 a X OQ 05 45 -tJ OQ O 264 HYDROGEN ION CONCENTRATION dissociates off H-ions, which, in turn, go to form the external stratum of the double layer, while the negative^ charged ions form the inner layer. A gelatin ion which had combined with an H-ion (or, starting from the electroneutral hydrated form, which had spht off an OH-ion) can no longer be designated as "a gelatin membrane with a covering layer of H-ions," just as an H-ion bound to an electroneutral mole- cule (H+ + NH3 = NH4+) is no longer designated as an H-ion. Thus it may appear somewhat forced, in adhering to Perrin's con- ception, to state that the charge on a wall or membrane depends only upon a covering layer of H- or OH-ions. But in reahty there is no contradiction in this; it is merely a purely formal difference of ex- pression. If we should decide to accept the NH4+-ion as an NH3 molecule overlaid by an H+-ion, then Perrin's interpretation becomes also acceptable for gelatin or any other charged wall. But this mode of expression is barely suitable. In order to state the basic idea underlying Perrin's rule in a form adequate for our present day theory, we may say that: A wall becomes electrically charged, either by binding (adsorbing) H-ions while acquiring the charge; this is equivalent to saying that it dissociates off OH-ions, which then form an ionic layer at a finite distance from the wall; or by binding (adsorbing) OH-ions; which is equivalent to stating that it dissociates off H-ions which aggregate into a layer at a finite distance from the wall. By combining this conception with the definitions given above in section 67, this law assumes the following form: A solid wall is always positively charged with respect to an aqueous solution, if the substance of the wall is an acidoid. It is always nega- tively charged, if it is a basoid. A wall may be either positively or negatively charged, depending upon the hydrion concentration of the solution, if its substance is an atnpholytoid. Just as in the case of a true ampholyte, zero charge on an ampholytoid is present only at its isoelectric point, the charge varying directly with the hydrion concen- tration. The isoelectric point of the wall, depending upon the chemical nature of its substance, may lie at an acid, neutral or alkaline reaction, and it is represented by the middle point of a more or less wide isoelectric zone. In the case of the dissolved ampholyte, the width of this zone de- pends upon the value of the product ka X kb. For the present no corresponding value for an ampholytoid can be defined. ADSORPTION POTENTIALS AND ELECTROKINETIC PHENOMENA 265 In general, the acidoid theory still shows a total lack of any values which would determine or define the relative strength of the acidoid or basoid properties in the sense in which the dissociation constant defines the strength of the various true acids and bases. In this latter case such a scale of measurement was made available by the possibilitj' of applying the mass-law. This possibihty is lacking in the case of acidoids and basoids. And yet it appears to be a quite soluble problem for the future, to find the means of extending the definition of the acidic and basic character of substances so as to in- clude cases bordering upon the state of molecular dispersion. All that is valid for sohd walls is entirely applicable to any par- ticles suspended in water and to colloidal solutions. And thus the electric charge phenomena of colloidal solutions are rendered com- pletely analogous to the dissociation phenomena of electrolytes in true solution. The laws of dissociation hence appear as a limiting case for the condition where the dissolved electrolyte approaches molecular dispersion. On the other hand, the electric charge phe- nomena in a colloidal solution appear as a limiting case for an elec- trolyte solution in which the dissolved electrolyte become more coarsely dispersed and finally acquires the character of a solid 'Svall." 78. A consideration of the other ions The above simple approximate method of explanation was made possible, as in the case of theory of electrolytic dissociation, by the auxihary assumption which ascribed an exceptional position to the H- and OH-ions. Just as in that case we assumed that the salts of acids and bases are always completely dissociated, or, in effect, non- existent, so in this case we assume provisionally that a wall alwaj^'s adsorbs only H- and OH-ions, but no Na- or Cl-ions, for instance. The extremely shght adsorbability of NaCl on charcoal (see page 234) in contrast to NaOH or HCl, indicates that such an assumption must be justifiable in reaching an approximated general law. In order to complete the scope of this law it is again necessary to drop these sim- plifying assumptions and to reconsider it in the fight of the adsor- ability of the other ionic species. It is to be expected that the in- fluence of any ionic species is the more pronounced, the greater its adsorbabihty, and hence the more effectively it can compete with the H- or OH-ions. Accordingly, the common univalent ions, which are poorly adsorbed, have also the least influence upon the electric 266 HYDROGEN ION CONCENTRATION charge on a solid wall. And yet in higher concentrations of univalent ions this effect is markedly present. This is shown in the following experiment by Perrin. The figures give the electroendosmosis velocity together with the sign of the charge. Carborundum 1/500 A^ KOH -105 Carborundum 1/500 A^ KOH + 0.1 A^ NaBr -24 CrCU 1/150 N HCl +100 CrCla 1/150 A^ HCl + 0.1 A^ KBr +35 In describing adsorption it was stated (page 237) that the adsorp- tion of HCl or of NaOH is augmented by the addition of a neutral salt. This was explained in the sense that the forces opposing the isolated adsorption of the H- (or 0H-) ions were thereby diminished. Another explanation of the same phenomenon is suggested, as can be seen from these experiments, in the diminution by the presence of the neutral salt of the potential which develops in the adsorption of HCl or KOH. An analytic formuhzation of the regularity of this phenomenon has not yet been attempted. According to the above the decrease of potential by univalent ions could be explained without regard for their adsorbability. But they doubtlessly do exhibit an adsorbability, be it ever so slight, and they do enter into competition to a small extent with the H- and OH-ions for adsorption on boundary surfaces. This competing effect is more striking in the case of the much more strongly adsorbable polyvalent ions, as shown again by the following of Perrin's figures : (a) AI2O3 1/1000 A^ HNO3 +100 AI2O3 1/1000 A^ HCl +100 AI2O3 1/1000 A^ H2SO4 +15 (b) CrCl3 1/1000 N HNO3 +88 CrCU 1/1000 A^ HNO3 + 1/1000 A^ MgS04 +23 CrCU 1/1000 A^ HNO3 + 1/100 AT CdS04 +4 (c) CrCU Very dilute HCl +75 CrCls KH2PO4 of the same pH +7 (d) CrCU in faintly alkaline water —46 CrCla Same + 1/1000 M K3Fe(CN)6 -46 ADSORPTION POTENTIALS AND ELECTROKINETIC PHENOMENA 267 (e) CrCls in faintly acidified water +59 CrCls Same + 1/1000 M K3Fe(CN)6 +2 CrCls Same + 1/50 M K3Fe(CN)6 -20 (f) CrCU in 1/1000 N HCl +86 CrCls Same + 1/2000 M K4Fe(CN)6 +1.5 These experiments show the effect of various polyvalent ions. As seen in (d), as long as a wall remains negatively charged, even polyvalent negative ions are without effect. But the same ions exercise, even in low concentrations, a strongly diminishing effect upon the potential of a positively charged wall, as in (e) and (f); while in greater concentration they can even reverse the sign of the charge, as in (e). This activity rises markedly with the valence. Bivalent ions are more effective than univalent, and trivalent ions have a greater effect than the bivalent. The effect of the valence of the anions is also noticeable in experiment (a), where the anion is added not in the form of a neutral salt, but as a constituent of the acid itself, for HCl and HNO3 show the same activity, while H2SO4 greatly reduces the potential. The cations behave in an exactly corresponding manner: they exert an influence only upon a negatively charged wall. For example : CrCls in faintly acidified water +43 CrCU Same + 1/1000 M MgCU +43 CrCls 1/1000 A^KOH -76 CrCls Same + 1/1000 M MgCU -10 AI2O3 1/500 N NaOH -85 AI2O3 Same + 1/500 A^ Ca(N03) 2 -18 AI2O3 in faintly acidified water +41 AI2O3 Same + 1/500 AT Ca(N03)2 +41.5 Mn203 in faintly alkaline water —40 Mn203 Same + 1/500 A" Ba(N03)2 +18 Carborundum 1/500 A KOH -60 Carborundum Same + 1/1000 A La (NO3) 3 -0.7 The example with the Mn203 shows that the bivalent Ba-ion can even reverse the charge. The same could be demonstrated to an even greater extent by A1+++ or La+++, but these are so insoluble in 268 HYDROGEN ION CONCENTRATION alkaline solutions that it is impossible to obtain the concentrations of these ions necessary for the reversal of the charge. We shall later discuss in greater detail the profound influence of the trivalent ions upon the sign of the charge, under other experimental conditions. It follows from the investigations of other authors that those dia- phragms which do not become positive even in an extremely acid reaction fall into two groups: 1. those which are positively charged by trivalent cations (A1+++, La+++), and 2. those the sign of whose charge cannot even be reversed by such trivalent cations. From the table given on pages 262-3, taken from Gyemant's study, it is seen thai while agar and collodion are not positively charged by A1+++, the charge of kaohn is reversed bj^ this cation. In regard to collodion this is a confirmation of Jacques Loeb's findings.^^ The reversal of the charge on kaohn appears to be easily explicable. Its surface adsorbs A1+++ by exchange for an H-atom of its silicic acid and forms, to a certain extent, a basic sihcate, i.e., an H-atom of the siHcic acid (of the kaoHn) is replaced by an A1+++ ion, which then still has two free valences left, or, because of its dissociated state, this amounts to two free positive charges. In the case of agar or collodion this is not possible, for these substances do not possess a replaceable H-atom, and their negative charge can not be due to the splitting off of H-ions, but rather to the adsorption of H-ions. (This is possibly a case of "apparent adsorption" cf. page 234). Heesch,^^ working in Hober's laboratory, found similar^ that agar membranes could not become positively charged, even by La+++, while he could demonstrate a reversal of the sign of the charge on parchment by La+++ or by other strongly adsorbable organic cations (histone, clupein, Rhodamin S). The otherwise irreversible sign of charge of such substances as col- lodion can still be reversed by A1+++ when these diaphragms are covered by an adsorption la^^er of an amphoteric colloid. Thus J. Loeb^i showed that collodion impregnated with a solution of gelatin or albumin can have the sign of its charge reversed by H+ or A1+++, as would be the case for these two colloids taken alone. Investigations on electrophoresis show it to be, as was to be ex- pected, an exact reversal of electroendosmosis. The same material which, when used as a diaphragm, propels water towards the cathode, when it is suspended in water, migrates towards the anode. The phenomenon of cataphoresis leads so directly over into the theory^of " Heesch, Pfliigers Arch. f. d. ges. Physiol. 190, 210 (1920). ADSORPTION POTENTIALS AND ELECTROKINETIC PHENOMENA 269 colloids that we shall discuss its contributions in greater detail in the following volume on colloids. For the present we shall only point out some instructive examples of the reversal of the sign of the charge. R. Hober^^ estabhshed the fact that the ordinarily nega- tively charged blood corpuscles become easily positively charged under the influence of trivalent ions. But since their charge is also reversed by H-ions, they are, therefore, of ampholytoid nature.^^ Putter^'^ found that bacteria in colloid-free aqueous suspension are always negatively charged, even in a most acid medium; and yet their charge is reversed by A1+++. But in the presence of peptone bacteria become also positively charged by means of high H+- concentrations. 79. Summary of the theory of formation of the electric charge on walls or diaphragms The electric charging of a wall towards a solution depends upon the differences of adsorbability of the various ions. The charge of the adsorbed ion is imparted to the wall. The serial sequence of adsorb- abihty of the ions is the same as can be found with charcoal by chemical analysis. Since the H- and OH-ions are particularly strongly adsorbable, the formation of the charge can be approximately ex- plained, in the absence of polj^alent or organic ions, in terms of the difference between the adsorbabilities of the H- and OH-ions alone. On closer consideration the other ions must also be taken into account. The serial sequence of the anions is probably the same for all ad- sorbents, as for example (arranged in order of increasing adsorb- abihty) : CI- I- SCN- OH- and similarly it is also the same for the cations, namely in the order: Na+ Ca + + A1 + + + H + But the absolute extent to which anions on the one hand and cations on the other are adsorbed by any adsorbent is quite different 3^ R. Hober, Pfliigers Arch. f. d. ges. Physiol. 101, 627 (1904) ; 102, 196 (1904). " L. Michaelis and Takahashi, Bioch. Zeitschr. 29, 439 (1910); Calvin B. Coulter, Journ. of Gen. Physiol. 3, 309 (1921). " E. Putter, Zeitschr. f. Immunitatsforsch. u. exp. Therap., 32, 538 (1921). 270 HYDROGEN ION CONCENTRATION for the different substances. Thus charcoal adsorbs H- and OH-ion approximately equally well, and if we write down the two ionic series in such a way that equally adsorbable ions are placed one directly under the other, we obtain the following schematic serial arrange- ment (for charcoal) CI I SCN OH Na Ca Al H For AI2O3, which adsorbs H-ions" more strongly than OH-ions'^ the arrangement will probably be CI I SCN OH Na Ca Al H For gelatin which adsorbs OH-ions^^ more strongly than H-ions:*° CI I SCN OH Na Ca Al H For an ampholytoid adsorbent with even stronger acidic properties: CI I SCN OH Na Ca Al H For an adsorbent which remains negatively charged even in an extremely acid reaction (collodion) : CI I SCN OH Na Ca Al H i.e., the cations are displaced so far to the left in the series that their adsorbability may well be regarded as being equal to zero. There is no fundamental difference between a sohd wall, the disperse par- ticle of a colloidal solution and a single molecule in true solution. The process of adsorption for all these cases is a completely equiva- lent one, after it had become established that an NHs-molecule, a glycocoU molecule, a single casein molecule, a molecule superficially located on a colloidal casein aggregate, a molecule on the surface of charcoal, each of these merely binds an H-ion. " AI2O3 + 6H+ = 2A1 + + + + 3 H2O. 38 AI2O3 + 6 OH- = 2 (AIO3) - + 3 H2O. "HRNH2 + H+ = (HRNH3)+. " HRNH2 + OH- = (RNH2)- + H2O. ADSORPTION POTENTIALS AND ELECTROKINETIC PHENOMENA 271 The absolute amount of adsorbed H- or OH-ions depends upon their concentration. That hydrion concentration at which H- and OH-ions are equally well adsorbed, in the absence of other strongly adsorbable ions, is designated as the isoelectric point of the adsorbent. If it is at some finite [H+] then we are deahng with an ampholyte or ampholjioid. If it is at [H+] = ^, then the substance is an acid or an acidoid, and, if it is at [0H~] = o= , the substance is a base or a basoid. As soon as, in addition to the H- and OH-ions, still other ions par- ticipate in the adsorption (i.e., poorly adsorbable ions in high con- centrations, or strongly adsorbable ions in any concentration) the relations become more or less different. The presence of well ad- sorbable cations had the same effect as an increase in the [H+], while the presence of well adsorbable anions, analogously, as an increase in the [0H+]. The reduction of these laws to the Hmiting case, when H- and OH-ions only are present as actively participating ions, is surely but an arbitrary or optional method. Theoretically one could start with anj^ other pair of ions. But this representation has the advantage that the ideal limiting case — absence of other active ions — can be readily realized, which would not be possible for any other single pair of ions. This arbitrariness is no greater than that involved in ascrib- ing to the acids and bases alone of all the electrolytes an exceptional position. 80. Changes in concentration, especially of H-ions, in electro- endosmosis In comparing Helmholtz's theory of endosmosis with the modern conceptions developed later, certain incongruities become apparent. Helmholtz calculated from the potential difference between the porous diaphragm and the water, which he assumed as given, the amount of water transported by a given external electrical force. His internal friction factor enters into his equation only from the intrinsic properties of the water. Hence it appears as if any elect ro- Ij^tes dissolved in the water, which scarcely affect its viscositj^, can influence the process of endosmosis only insofar as they ma}'- alter the potential of the diaphragm. But if we accept our above developed view that the current of the external forces acts primarily upon the movable ionic layer, and that the latter only carries the water along, 272 HYDROGEN ION CONCENTRATION then we must consider that the amount of water carried along by an ion depends upon the so-called hydration of the ionic species involved (see page 123). Hence we must also consider that the amount of water transported in endosmosis at a given wall potential depends at least as much upon the water binding capacity of the ionic species passing along the wall as upon the internal friction of the water. Furthermore, Helmholtz's theory regards the water as a whole, and it does not admit the possibihty that the various con- stituents of the solution, the water and the individual ionic species in solution, may be differently displaced in the process of endos- mosis. The superiority of the newer conception could be proven if we succeded in demonstrating local changes in concentration of the dissolved ions through endosmosis. Although these changes in con- centration had been previously described, ^^ the more exact and con- sistent study of this aspect of the problem was furnished in the im- portant contributions by Bethe.^^ The fundamental observation made by Bethe was the following. During the process of electroen- dosmosis through a collodion, parchment or gelatin membrane, or through a diaphragm of powdered material, the originally neutral solution on the anode side of the diaphragm became acidified while that at the cathode side it became alkahne (or the reverse of what happens in the course of ordinary electrolysis of a neutral solution between two platinum electrodes). A change in the concentrations of the other ions was also observed, but by far the greatest and the most easily demonstrable was the change in the concentrations of the H- and OH-ions. This is but a natural consequence of the present day theorj^ of endosmosis. Bethe and Toropoff developed this theory in the following way: When, for example, the diaphragm wall is negatively charged, the am'ons are held in the pores in an adhering layer, and to these is due the negative charge on the wall. Above all others are the OH-ions so held, then the polyvalent or the strongly adsorbable organic anions, while the common univalent anions are bound but to a very slight degree. Now, if the capillary spaces be only sufficiently small, then, under certain conditions, almost all OH-ions, and possibly almost all m; ni>n and equation (VI) becomes B = - aKA - (ni - n) H^O + (ni - n - b) KOH If the solution is neutral to begin with then n and b must be small with respect to Ui, because of the slight concentration of H- and OH-ions, and the KOH term in (VI) wiU be positive, i.e., the solution win be alkaline on the anode side. Similarly, under the same conditions, the diaphragm boundary on the cathode side must become acid. These predictions based on the theory correspond, for most of the neutral salts, with the ex- perimental findings of Bethe and Toropoff. The higher the concentration of the dissolved electrolytes, and, it may be added, the larger the capillary spaces, the less hkely it is that the adsorbable ionic species will be completely bound on the walls of the pores; and, consequently, the smaller will be the changes in concentration produced by the passage of the current, and, as was pointed out above, the endosmotic transport of water resulting from concentration changes will be diminished. If the sign of the charge on the diaphragm is irreversible, and the diaphragm is of an acidoid character (collodion), the walls of the pores adsorb only OH-ions and other anions, but no cations. Ac- cording to the adsorbability of the anions, other than 0H~, which may be present, they are more or less bound by the wall and par- ticipate correspondingly more or less in the conductance of the current within the pores. The greater the adsorbability of the ions, other things being equal, the greater will be the change in hydrion concentration brought about on both sides of the diaphragm on the passage of the current. In fact, on comparing the effect of K- or Na-salts of the different anions in displacing the originally neutral reaction of a solution, Bethe and Toropoff found the following serial sequence, which was to be expected : Citrate- > I" > Br" > SOf > CI" > NO3- When the anion was kept constant and the cations varied, no such sharply defined series could be obtained which would be independent 276 HYDROGEN ION CONCENTRATION of the anion used, at least not with collodion menbranes. Parchment membranes showed a distinct decreasing effect in the series: NH4>Li>K>Na>Mg>Ba>Ca This effect still needs further study. Ampholj'toid diaphragms showed, as they did also in respect to the endosmosis of water, a reversal of the effect when a certain well defined H+-concentration was exceeded. This [H+] represents in regard to changes in concentration a point of indifference. The ex- periments were carried out by Bethe and Toropoff especially on chromated gelatin, in which the disturbing swelling phenomena do not occur, as they do with ordinary gelatin. This point of indiffer- ence of concentration displacement was, as expected, different for different salts, and for 0.01 M solutions at the following pH values: pH NaaSO^ 3.6 Na2C204 4.0 NaCl, KCl 4.3 BaCl. 4.8 C0(NH3)6C1 7.0 The important question arises as to the relation of this point of indifference to the isoelectric point. If we assume that the pH of the solution is equal to the isoelectric point of the diaphragm, then the wall of the diaphragm will not adsorb any ions, the potential of the wall is zero, and the current is conducted equally well within the pores as in the free solution. But when the water-combining power and also the mobiUty of the anions are equal to those of the cations in solution (or better, as in the case where the various cations are in the concentrations ki, ko . . . with the mobilities Ui, U2 . . . , and the anions are in the concentrations ai, 0,2 .. . with the mobilities Vi, V2 . . .; and when Uiki + U2k2 = Vi ai + V2a2 . . .), no water will be transported. Otherwise, even at the isoelectric point, some, even if slight, displacement of water will occur. The point of indifference for the motility of the water is not quite equal to the isoelectric point, the latter corresponding more closely to the point of in- difference for concentration change. In fact, Bethe and Toropoff demonstrated that the indifference points for water mobihty and for concentration change do not always coincide. A total coincidence of these two points is possible only when the dissolved electrolyte com- ADSORPTION POTENTIALS AND ELECTROKINETIC PHENOMENA 277 bines these two properties: (1) neither of its ions must have a marked degree of adsorbabiHty as compared to H- and OH-ions, (2) the mobihties of its two ions must be equal to each other. Of all the electrolytes KCl fulfills these conditions best, other binary univalent electrolytes, such as NaCl, only to a certain extent, but NaoSOi, for example, not at all. All of the authors mentioned before considered the indifference point of water mobihty as the isoelectric point of the diaphragm, but according to Bethe and Toropoff this is not quite correct. 81. History of the theory of adsorption potentials The theory of adsorption potentials, as it is developed in this section, is not to be found in this form in the literature. The fact that ions may become adsorbed is quite obvious and was stated by a number of authors, without there being any evident historic sequence on this point. The present day theory dates back chiefly to two sources, and may be thought of as the result of the blending of the two, as it were. One is the earlier description by the present author^* and the other was that given by Freundlich.^* Beginning with the first, Michaelis, on the basis of a theoretical research of Nernst's,^^ conceived a colloidal particle suspended in water, for example of a resin acid (mastic), as of a "binary electrode," which, in distinction to a simple electrode, tends to send into solution not one ion species only, but H-ions and acid anions simultaneously. From the different solution tensions of these two kinds of ions taken in conjunction with the differing concentrations in which these two ions are present in a state of true solution in the aqueous phase, the author derived the potential of the double layer at the interface. Should there be present in the solution any species of ion which is capable of forming a salt of this acid, an ionic exchange occurs in the double layer which alters its potential. Already at that time the author explained this ionic exchange by stating that it is identical with ionic adsorption. In a theoretical derivation of the ionic equilibrium the author"**^ " L. Michaelis, Physik. Chemie der KoUoide, in Richter-Koranyis Hand- buch: Physikalische Chemie und Medizin. Leipzig 1908. L. Michaelis, Dynamik der Oberfliichen. Dresden 1909. P. 49 ff. ** H. Freundlich, Kapillarchemie. Leipzig 1909. p. 213 ff. « W. Nernst, Zeitschr. f. physikal. Chem. 9, 137 (1892). " L. Michaelis, Zeitschr. f. Elektrochemie 14, 353 (1908). 278 HYDROGEN ION CONCENTRATION made the error of correlating the solution tensions of the ions with their mobiUty, which error was pointed out by F. Haber.'*'' The solution tension bears no relation to the mobiHty of the ion. Freundlich's conception is based on the idea that the ions, as well as the ordinary molecules, may be adsorbed on boundary surfaces, and that the adsorbabilities of the anion and of the cation of an elec- trolyte are unequal. He also ascribed a particularly high degree of adsorbabiHty to the H- and OH-ions. He derived the adsorption potential from the different adsorption tendencies of the anions and cations. To the adsorbents, according to their chemical (acidic or basic) character, he ascribed the tendency to dissociate off H- or OH-ions respectively, which ions would then lead to the formation of the double layer. At this juncture Freundlich fell into the same error as Michaelis did, in correlating the spHtting off of these ions with their high mobiUties. But Freundlich showed at the same time that the adsorption potential need not be identical with the potential which the same adsorbent (when it is a metal, for example) would have towards the same solution, when it is used as a pole of a galvanic chain. This point must be recognized as being of very great im- portance, and this phenomenon will become more comprehensible in section 85 on "The SpHtting of Potentials." In general there does not appear to be any great difference between FreundUch's conception and that of the author. However, one difference between the two does appear in the following respect: Freundlich takes pain to maintain as far as possible a Hne of demarca- tion between chemical reaction and adsorption on boundary surfaces. He regards the adsorbent chiefi}^ as a means of increasing the surface of the solution, and he does not attribute any particular value to the chemical nature of the adsorbent. This idea appears more distinctly in his chapters on the ordinary adsorption of non-electrolytes than it does in the above cited pages dealing with adsorption potentials. In this latter chapter Freundlich cannot help attributing a special importance to the acidic or basic nature of the adsorbent. And yet it does not appear from this chapter that he thereby relinquishes his sharp line of demarcation between adsorption and chemical reaction. The present author, on the contrary, took pains at the outset to re- move the contrast between adsorption and chemical reaction, at least in the case of chemically reactive (acidic or basic) adsorbents. " F. Haber, Ann. d. Physik. [4], 927 (1908) ; s. p. 948. ADSORPTION POTENTIALS AND ELECTROKINETIC PHENOMENA 279 The danger of delimiting chemical from physical attractions is obvious from the very nature of these processes, and a serviceable theory is quite evidently one which does not demand any such demarcation. It is under the influence of these ideas that this at- tempt is made to present a unified theory of adsorption potentials. 82. Hydrodynamic potentials It has been previously stated (page 246) that in the study of ad- sorption potentials both methods of electroendosmosis and electro- phoresis can be used. To these we must add two other methods, that of the hydrodynamic current and that of the current of falling particles. When water is forced through a porous membrane or through a capillary tube, there arises a potential difference between the two sides of the membrane or between the two ends of the capillary tubes. This phenomenon is the reverse of electroendosmosis. For in the latter water is forced through a diaphragm by the appHcation of a potential difference between its two ends, while in the former the potential difference is produced on forcing water through the dia- phragm. If the two ends of the capillary are led off by means of metal electrodes, a current is obtained as long as the flow of water through the capillary continues. Hence such electric currents are designated as hydrodynamic currents, and the potentials which cause it are called hydrodynamic potentials, '^^ This phenomenon was discovered by Quincke. ^^ A porous clay plate was cemented between the bevelled ends of two glass tubes, or a diaphragm of some powdered substance was tamped into a glass tube. Through such a tube water was forced and platinum electrodes were inserted in the hquid on both sides of the diaphragm. As long as the water was being forced through the dia- phragm an electric current manifested itself whose electromotive force was proportional to the hydrostatic pressure. The E.M.F. was independent of the structure and thickness of the diaphragm, which is in complete analogy with the phenomena involved in elec- *^ The German terms "Stromungsstrome" for "hydrodynamic currents" and "Stromungspotentiale" for "hydrodynamic potentials" are perhaps more descriptive than the English equivalents. Transi. " G. Quincke, Pogg. Ann. d. Physik. 107, 1 (1859); 110, 38 (1860). 280 HYDROGEN ION CONCENTRATION troendosmosis. The same results were obtained by Zollner/" Edlund, 51 Haga,52 ^nd Clark." The theory of hydrodynamic currents is similar to that of electroen- dosmosis, — it is the reverse of the latter. The movable ionic layer of the double layer is being sheared by the mechanical water current in the capillary. Hence the further end of the capillary becomes enriched in the ions of which this laj^er is made up. This does not occur, to be sure, to an analytically demonstrable extent, but sufficiently to give rise to a potential difference between the two ends of the tube. Figure 30 is an attempt to visuahze this process. The Hquid flows through the capillary A A' into the larger vessel BB'. As long as the liquid remained stationary a double layer was being formed in the manner indicated along the wall A. When the liquid is set in motion, it tears along with it the movable layer adjacent to it, while the layer adhering to the wall remains so, as indicated along wall A'. In this way the upper end becomes negatively and the lower end positively charged. The value E of the potential differ- ence between the ends of the tube is according to Helmholtz, and including Pellat and Perrin correction for the A 1+ 1+ 1+ !+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ 1+ neg. I +1 +1 +1 + +1 +1 +1 +1 +1 + 1 + +1 + + A \ Fig. 30 dielectric constant (see page 249), as follows: E = f ■ P-D 4 TT • 17 X (where 77 is the viscosity of the fluid, X its specific electric conduct- ance, P the hydrostatic excess of pressure between the ends of the tube, D the dielectric constant, f the potential of the double la^-er on the capillary wall — or the "adsorption potential"). The E.M.F. of the current of flow is therefore directly proportional to the hydro- static pressure and to the adsorption potential. It is inversely proportional to the viscosity of the liquid, for the more viscous the 60 F. Zollner, Pogg. Ann. d. Physik. 148, 640 (1873). " E. Edlund, Pogg. Ann. d. Physik. 156, 251 (1875) ; Wied. Ann. 1, 184 (1877). *2 H. Haga, Wied'. Ann. 2, 326 (1877) : 5, 287 (1878). " J. W. Clark, Wied. Ann. 2, 335 (1877). ADSORPTION POTENTIALS AND ELECTROKINETIC PHENOMENA 281 liquid the more slowly it moves at the same pressure and conse- quently the fewer ions it can tear along with it. The E.M.F. is also inversely proportional to the electric conductance, for the better conductor the liquid is the less of a potential difference is allowed by it to develop between the two ends of the tube/^ Furthermore, E is directly proportional to the dielectric constant, for the greater the value of the latter, the easier it is to separate the ions from each other. The proportionality factor — is introduced in order to reduce 47r the value to electrostatic units. This equation can be used to calcu- late the potential f on the wall, after having measured the electro- motive force E of the hydrodynamic current. The proof for the correctness of the absolute value of so calculated potentials suffers from the same difficulties as in the case of endosmosis. In this case, however, it depends only on the correctness of the factor — ' other- 47r wise the relative correctness of the equation may be accepted as having been demonstrated. Thus the hydrodynamic current method represents a means for the study and calculation of adsorption potentials, and also for the study of the influence of the chemical character of the wall and of the hquid upon these potentials. These studies were made chiefly in glass capillaries, and glass should have behaved, in respect to its chemical properties, quite similarly to other siUcates such as clay and kaolin. It is obviously of interest to find out whether the results obtained by the use of these methods agree with the results obtained by endosmosis and cataphoresis methods with clay and kaohn. Of the not very numerous studies in this field^^ only two are par- ticularly adequate in answering this question. These are the studies by Kruyt^'' and by FreundUch and Rona." In general, it may be " It is to be noted that in the equation for electroendosmosis (page 249) the conductivity did not enter directly. It entered only indirectly insofar as the conductance of the liquid in the pores of the diaphragm affected the fall of the potential per centimeter of the externally applied tension, or insofar ab it affected the intensity of the electric field. » A. T. Cameron and E. Oettinger, Phil. Mag. [6] 18, 586 (1909). A. Grum- bacher, Ann. d. chim. et de phys. [8] 24, 433 (1911) ; L. Riety, Ann. d. chim. et de phys. [8] 30, 1 (1913). " H. R. Kruyt, Kolloid-Zeitschr. 22, 81 (1918). ^' H. FreundUch and P. Rona, Sitzungsber. d. preuss. Akad. d. Wiss. 1920, p. 397. 282 HYDROGEN ION CONCENTRATION stated that the theoretical expectations were to a great extent confirmed. First it appeared that glass becomes in most cases negatively- charged towards aqueous solution. By means of high hydrogen ion concentrations of the solution the current potential can be reduced to zero, but the sign of the charge could not be reversed in this way. Bivalent cations (Ba++) aboHsh the current potential in lower con- TABLE 42 KCl concentra- In millivolts BaCb concen- tration E r AlCb concen- tration E tion E f 0 + ca. 350 — 0 + ca. 350 — 0 + ca. 350 50 102 19.9 10 139 10 0.5 + 52 100 57 22.4 25 79 14.2 1 - 42 250 23 23.0 50 44 16.0 2 - 122 100 25 18.1 3 - 129 500 13 23.3 200 9 12.7 4 - 100 1000 4.0 15.4 1000 1.1 7.6 10 - 52 40000 No reversal of sign of charge 100 500 6 - 1.4 HCI concentration E r p-ChJoraniline chloride concentration E r 0 + ca. 350 — 0 + ca . 350 25 77 22.1 31 114 13.9 50 43 24.8 100 22 25.4 62 65 19.6 250 7.7 22.1 124 26 16.6 500 3.1 17.5 310 12.2 14.5 1000 1.2 13.6 500 4.9 11.8 50000 No reversal of charge sign of 1000 1.8 9.7 centrations than univalent cations. Trivalent cations (A1+++) reverse the sign of the current potential even in the lowest concentrations, while at higher concentrations the potential is reduced to zero again. This was shown by the following experiment by Kruyt: Various hquids were forced through a glass capillary and jaelded the values for hydrodynamic current potential shown in table 42, at a water pressure of 1 cm. of mercury. The concentrations are given in micromols (1 /x-mol = 0.000,001 mol); a + sign before the value of ADSORPTION POTENTIALS AND ELECTROKINETIC PHENOMENA 283 E denotes that the lower end of the capillary is positive in respect to the upper end. The potential f of the wall of the capillary was calculated from the corrected Helmholtz equation. It is of particular interest to observe that notable hydrodynamic potentials were obtained only in very dilute solutions of electroly1:e. While in 10~^ N KCl solutions of 20-40 milhvolts were obtained, no measurable potentials were observed in 10~^ to 10"^ N solutions. We have already stated on various occasions that the presence of an excess of neutral salts diminishes potentials (diffusion-, interphase boundary-, membrane-potentials), and the same was the case in electroendosmosis experiments, where an excess of neutral salt caused a diminution in the amount of transported water. But in all of these cases an excess of salt of an entirely different order of magni- tude was involved. Hydrodynamic potentials may, in general, be properly observed only in electrolyte solutions of concentrations of the order of magnitude of 10"'' to 10~'* normal. The cause for this is the fact that the current potential depends upon the conductivity of the solution. By increasing the concentration of the salt, in the first place the adsorption potential f is greatly diminished; secondly, the current potential E produced by a given adsorption potential and a given pressure is very profoundly depressed. It must be recalled that in the equation given above (page 280) the conductivity X is in the denominator of the fraction. By calculating from the current potential E the potential of the wall f towards the solution, with the help of the corrected Helmholtz equation, the values given in the above table were obtained by Kruyt. It is notable that the values of the f potential do not vary as much with the electrolyte concentration as those of the E potential. This corresponds to the fact that the phenomena of endosmosis and cataphoresis, which show a greater paralleHsm with the f potential, do not depend to so great an extent upon the electrolyte concentra- tion as hydrodynamic potentials do (since the specific conductivity does not appear in the equation for endosmosis) . It will be stated only in a corollary way that in the course of sohd particles falling through a layer of water a potential difference develops between the upper and the lower ends of the water column. This is the potential of falling particles. The phenomenon is the re- verse of the hydrodynamic potential, and the two are entirely analogous. 284 HYDROGEN ION CONCENTRATION 83. The influence of electrokinetic potentials upon the boundary surface tension For the sake of completeness we must mention here a mechanical manifestation of electrokinetic potentials, a more comprehensive discussion of which will be reserved for the chapter on colloids in a later volume of this work. According to the observations of Lipp- mann, in conjunction with the discovery of the capillary electrometer and with the Helmholtz theory relating to it, the double layer at a phase boundary exercises an influence upon the surface tension. Every individual stratum of the double layer, because it consists of similarly charged particles, tends to expand or to to extend itself, thus working in opposition to the ordinary surface tension. Con- sequently the surface tension at an interphase boundary must depend upon the potential difference. Now, when one of the phases is soUd or at least much more viscous than the other, this expansion can occur only in that ionic layer which is in the more fluid or less viscous phase. This tendency to expand on the part of the layer at the surface of the liquid is antagonistic to the ordinary surface tension. The total tension is therefore dependent upon the absolute value of the electric density of the double layer. It is at a maximum when this density = 0; this is the case when the potential of the double layer is also = 0, or hence, at the isoelectric point. The theory was extended by Bredig^* to colloidal solutions, and Michaelis^^ showed, for colloidal solutions of protein substances, that, in fact, the iso- electric point (determined by cataphoresis experiments) is identical with the maximum of the surface tension (as determined by estima- tions of flocculation optima). This aspect will be further discussed in the following volume. 84. A summary of electrokinetic phenomena In conclusion it may be useful to present a systematic summaiy of all electrokinetic phenomena. Let us imagine a soUd wall and a fluid movable in reference to this wall. If an external potential is apphed in a direction tangential to the wall, a displacement of the fluid along the wall will occur. When the wall is not mechanically displaceable (a capillary or a diaphragm), the fluid is displaced along *' Bredig, Anorgan.Fermente. Leipzig 1901. *' Summarized by L. Michaelis, Nernst-Festschrift. Halle 1913. ADSORPTION POTENTIALS AND ELECTROKINETIC PHENOMENA 285 the wall: this is electroendodmosis. When the sohd wall is displace- able (suspended particles) but the fluid is not displaceable (either because it is in an enclosed chamber, or, w^hen in a U-tube, because hydrostatic differences in pressure can not be maintained), then the solid wall is moved along the fluid: — this is electrophoresis. Conversely, when the solid wall is mechanically displaced along the fluid, a potential is formed in the direction of this displacement. If the wall is fixed in place (a capillary or a diaphragm), and the fluid flows along the wall, then a hydrodynamic current is formed. The fluid being stationary and the sohd wall movable (falling particles), a potential of falling particles arises. 85. The splitting of phase boundary potentials The relation of phase boundary potentials to adsorption potentials In comparing the chapters on phase boundary potentials (pages 183 ff.) and adsorption potentials it appears rather remarkable that the potential difference occurring at an interface maj^ be considered in two entirely distinct ways. Taking, for example, the phase bound- ary" glass-water and considering the potential as a phase boundary potential, then the latter depends exclusively upon the [H+] in the solution. But if the potential is considered as an adsorption poten- tial, then it depends upon all the ions, and to an especially great extent upon any trivalent cation present in the solution, which would be of no significance for the phase boundary potential. Thus the potential difference at the interface glass-aqueous solution, for example, does not appear to be rigidly defined, but may differ accord- ing to the nature of its manifestation which is taken into account. We regarded it as a phase boundary potential when it mani- fested itself in the form of an electromotive force in a Haber glass- chain. We regard it as an adsorption potential when it manifests itself in the form of electrokinetic phenomena. This discord w^as for a long time a source of an ambiguity, which however is lately being slowly dispelled, through the explanations which were developed by Freundlich. For a better understanding of this problem we shall present the following considerations offered by Haber.^° Let us take the chain shown in figure 31. A silver electrode on the »» F. Haber, Ann. d. Physik. [4], 927 (1918), cf. p. 949. R. Beutner, Dissert., Karlsruhe (1908). 286 HYDROGEN ION CONCENTRATION right is in contact with a solution of AgNOs which is saturated with AgCl. Into the same solution on the left side there is immersed another silver electrode, but not du-ectly, being covered with a layer of fused and solidified AgCl. This system is in a state of chemical equilibrium. When the two electrodes are connected by a metal wire there will be no flow of current."' This system, therefore, con- tains three interfaces, a, b, and c. Each of these may be regarded as the source of a potential difference; c is the source of an ordinary metal electrode potential. In order to render this chain currentless the potential at c must be equal to that of a + b. Hence the intro- duction of the layer of solid AgCl has not altered the potential differ- ence between the silver electrode and the solution. This can be summarized by the following generally applicable statement: When between two phases A and C, which are in a stdte of chemical equilibrium^ a third phase B is introduced, which is also in a state of equilibrium with the first two phases, the potential difference between A and B is not altered thereby. From this it follows that the total potential from A to C is decomposed into two stages through the introduction of the J^^/ flgNO, ^ third phase. 5 5 are positively charged, and those whose constant < 5 negatively charged; see, for example, table 43. The same rule appUes to the sign of the electric charge resulting from the friction of two solid substances. For it is generally ap- «» A. Coehn, Wiedem. Ann. 64, 217 (1898). " A. Coehn and U. Raydt, Ann. d. Phys. 30, 777 (1909). ADSORPTION POTENTIALS AND ELECTROKINETIC PHENOMENA 289 parent that solid wall-potentials form the connecting link with the oldest known electric phenomenon, viz., friction electricity, which is still the most obscure of all electric phenomena. It is still unknown whether in these cases the free electricity is represented by ions or by free electrons, although for substances in a non-metaUic state we must consider the participation of free electrons as highly improbable. No contradiction can be found in Coehn rule to the laws governing the better known relationships. On the other hand to accept it as a universally apphcable law would be far fetched. For it was shown, for instance, that the sign of the charge of water against glass is reversed by the presence of a trace of an aluminum TABLE 43 Water Glycerin Alcohol Acetic acid Aniline Propionic acid. Ethyl butyrate Amyl acetate. . Chloroform. . . . Ethyl ether. . . . Butyric acid. . . Benzene Turpentine oil. Sign of charge against glass + + + + + -(I) + salt, which could certainly have no detectable effect upon the dielec- tric constant of the water. There is also no occasion to speak of the superposition of the established ionic laws upon an universally valid Coehn's rule as of a fundamental principle. Undoubtedly all these phenomena may be explained in a coordinated way. Our present day theories extend only as far as our knowledge of the ionization of the phases permits. For all other cases Coehn's rule may be accepted as being empirically correct to a certain extent. Recently R. Keller" made use of Coehn's rule for the explanation of biologically im- portant processes, especially of histologic staining. It is possible *' R. Keller, Elektrohistologische Untersuchungen an Pflanzen und Tieren. Prag 1920. 290 HYDROGEN ION CONCENTRATION that he, in spite of considerable errors, has also brought forward some useful suggestions, which, however still await experimental verification. The following interpretation of Coehn's rule appears to the author the most plausible. When one of the phases is of a distinctly iono- genic character (silicic acid, ferric oxide), then Coehn's rule does not applj^ at all; the laws covering this case were stated before. For here the dielectric constant plays a subordinate part, and, according to the nature and the concentration of the dissolved electrolytes, the water may be either positively or negatively charged with respect to the other phase. When the non-aqueous phase possesses a great adsorption capacity (charcoal), Coehn's rule does not hold again; for the charge on charcoal may be either positive or negative, again depending upon the electrolytes in the contiguous aqueous phase. On the other hand, the rule does seem to be valid in those cases where at least one of the substances involved is of a chemically indifferent nature, and where, at the same time, no great adsorption capacity, such as that of charcoal, is present, and where, therefore, the adsorp- tion of ions at the interphase boundary is of the "apparent" type (see page 234) . In these latter cases the following might be a general interpretation of Coehn's rule, which is offered with reservations only as a provisional working hypothesis. TJ ,cause of the charge on the interface hes chiefly in the traces of wat»„i .: ^ of the two ions com- posing water the OH-ion is the more si active one. Since water ionizes to a greater extent in a medium possessing the higher dielectric constant, there occurs at the surface of such a medium an accumulation of OH-ions, and hence this surface appears to be nega- tively charged with respect to its own medium. 87. The significance of adsorption potentials in physiology and colloidal chemistry The first question which occurs to us is whether adsorption po- tentials bear any relation to the electric potentials and currents which we observe with the help of electrical measuring instruments in the Hving organism. This question had to be apparently answered in the negative. For it was shown above that the adsorption poten- tials manifest themselves only through electrokinetic phenomena (electroendosmosis, etc.) and that they can not be led off by means of metalHc electrodes. As soon as we attempt to lead off any potential ADSORPTION POTENTIALS AND ELECTROKINETIC PHENOMENA 291 present on a wall to a measuring instrument we always obtain only the phase boundary potential, and not the adsorption potential, and these two, as was previously shown, are not identical. The only case in which an adsorption potential yields a direct electric mani- festation which is detectable by electric measuring instruments is that of the hydrodynamic potential (or the potential of falling particles). But as we have shown before, the conditions prevailing in the living organism are most unfavorable for the generation of hydrodynamic potentials. But, on the other hand, the adsorption potential manifestations that are undetectable by electric measuring instruments are quite clearly present in colloidal solutions. Only adsorption potentials can exercice an influence upon the state of a colloidal solution. For this state is determined by the tensions existing at the boundary surfaces of the phases. It is on these tensions that changes in the form and size of the particles, coagulation and peptization depend. The most important single agent affecting these tensions is the electric charge. But since all changes in the colloidal state depend upon the recip- rocal action of adjacent boundary surfaces, therefore only those double layers can be of significance which are located on the boundary surfaces, on the walls of the dispersed particles and in the adjacent to them la3''ers of the H" rsion medium. This introduces us to another mode of ar '- of adsorption potentials other than their electrokinetic maniieit^tions, which, moreover, prepares us for the theory of colloidal behavior. But all of these processes are really of much greater importance for the interpretation of other significant physiological problems than would first appear from the above discussion. If we consider the capillary-electrical phenomena in diaphragms with very narrow pores, such as can be observed in collodion or gelatin membranes, especially the polarization phenomena described by Bethe, then the physiological apphcability of these processes to such problems as currents of rest, action currents, and of the stimulation of muscles and nerves becomes so evidently manifold that we can hardly attempt to discuss them at greater length in this volume. Bethe'^*^ had made an important beginning in this direction in which the exceptional position of the H-ions is again emphasized. There also the question narrows itself down to the alternative as to whether to base the «8 A. Bethe, Pflugers Arch. f. d. ges. Physiol. 163, 147 (1916). 292 HYDROGEN ION CONCENTRATION interpretation of bioelectric currents, ionic permeability, stimulation, and other equally important biological processes upon the activity of forces at interfaces between an aqueous and a non-aqueous phase (page 191) or upon the capillary electric phenomena in narrow-pored gelatinous diaphragms. The limits of the designated subject of this book were somewhat exceeded in the last chapters, though it was not attempted to present a complete description of the bordering problems. However, it is just in this impossibility of sharp delimitation of the subject and in its constantly growing sphere of influence in other fields that its funda- mental importance is manifested. Nor does the future seem far dis- tant when the small portion of physical chemistry, here presented, doubtlessly improved and changed in many respects, will become the ultimate basis of that rational physiology which we expect to see developed in the next few decades. AUTHOR INDEX Abbott, 30 Adams, 67 Armstrong, 247 Arrhenius, 7, 25, 42, 113, 115 Barendrecht, 167 Bartell, 243 Bechhold, 80, 82 Becquerel, 247 Bernstein, 210 Bethe, 259, 272 ff., 291 Beutner, 149ff., 184ff., 211ff., 285 Biilmann, 173 Bjerrum, 67, 69, 87, 90, 101, 104, 116, 120£f., 126, 180, 181 Boden, 80, 82 Bodlander, 31 Bolin, 54 Boltzmann, 119 Bray, 30 Bredig, 60, 284 Brighton, 27 Brinkman, 85 Briining, 215 Burton, 253 Cameron, 281 Chanoz, 179 Clark, W. M., 102, 173 Clark, J. W., 280 Coehn, 288ff. Cormack, 31 Coulter, 269 Cremer, 149, 197-8, 208, 211 Cumming, 179, 180 Daniell, 247 Davidsohn, 66, 70 Debye, 126 Dernby, 73 Donnan, 220ff. Drucker, 31, 111 DuBois Raymond, 209, 247 Eckweyller, 66 Edlund, 280 Ellis, 251 Euler, 31, 54, 72, 73 Fajans, 125 Fales, 180 Falk, 66 Faraday, 247 114, Fels, 43 Fischer, E., 35 Freundlich, 199, 238, 241, 277-8, 281, 285, 287-8 Garmendia, 30 Ghosh, 114, 117-8 Gibbs, 233, 241 Gilchrist, 180 Gillespie, 104 Glixelli, 259 Grineff, 260 Grotthus, 149 Grumbacher, 281 Gudzent, 80, 82 Gyemant, 104, 241, 25^-61, 268, 287 Gyorgy, 137 Haber, 149, 167, 184-5, 198, 203, 211, 278, 285 Haga, 280 Hamburger, 229 Hantzsch, 37, 97 Hardy, 64 Hartleben, 236 Heesch, 268 Heidenhain, E., 247 293 294 AUTHOR INDEX Helmholtz, 209, 247, 248ff., 271 Henderson, L. J., 43 Henderson, P., 178-9 Hermann, 209 Hertz, 117 Heydweyller, 10, 25 His, 80, 81 Hittorf, 272 Hober, 210-11, 269 Htickel, 126 Jahn, 114 Jiirgensen, 247 Kallmann, 117 Kanitz, 60, 72, 80, 82 Keller, 20, 288 Klemensiewicz, 149, 184, 198, 211 Kohler, 80, 82 Kohlrausch, 10, 25, 177 Koppel, 104, 108 Kruger, R., 129 Kriiger, A., 46 Kruyt, 281-2 Lachs, 235 Lamb, 180, 250 Larson, 180 Lewis, G. N., 27, 126, 180 Lichtwitz, 80 Linderstr0m-Lang, 173 Lippmann, 284 Loeb, J., 149, 185, 228, 260, 268 Loffler, 242-3 Lowenhertz, 19, 26 Lunden, 25, 31, 60, 72, 93 Luther, 31 Maclnnes, 123 Madsen, 31 Masius, 238 Meyer, G., 198 Meyer, H., 229 Michaelis, 27, 30, 31, 36, 43, 46, 49, 56, 60, 64, 66, 67, 69, 70, 73, 104, 129, 137, 147, 185, 189, 235, 238, 242, 244-5, 260, 269, 277, 284 Miller, 243 Milner, 114, 119-20 Mizutani, 67 Moers, 9 Nernst, 7, 9, 12, 14, 26, 103, 113, 125, 156ff., 175, 184-5, 211, 218, 255, 277 Noyes, A. A., 31, 123 Noyes, H. M., 66 Oettinger, 281 Ogawa, 243 Oker-Blom, 210 Ostwald, Wilh., 26, 30, 31, 91, 96, 210 Overton, 229 Paul, 80, 81 Pellat, 249 Perrin, 248-9, 255ff., 261, 264, 266 Pfeffer, 215, 229 Planck, 177-80 Porret, 247 Procter, 228 Putter, 269 Quincke, 247, 279 Randall, 126 Raydt, 288 Reimer, 209 Reuss, 246 Reynolds, 252 Riesenfeld, 125, 184, 211, 218 Riety, 281 Ringer, 80 Rona, 31, 36, 43, 73, 82, 85, 137, 199, 235, 238, 242, 244^5, 281 Rothmund, 31 Rudolphi, 113 Rupert, 180 Sackur, 124 Salesski, 43 Schade, 80, 82 Sebastian, 27 Smoluchowski, 250£f., 287ff. AUTHOR INDEX 295 Sprensen, 22, 27, 28, 43, 60, 74, 96, 103, 173 Spiro, 104, 108, 242-3 Strohecker, 34 Takahashi, 82, 85, 269 Thiel, 34 Thomson, J. J., 14 ToUens, 35 Toropoff, 259, 272 Traube, 215 van Dam, 85 Van Slyke, 109 van't Hoff, 7, 113, 114 Vosburgh, 180 Walden, 14, 31, 149-50 Walker, 31, 60, 72 Warburg, E., 131, 198 Weyl, 180 Wiedemann, 247 Wijs, 25 Wilson, J. A., 228 Winkelblech, 60, 72, 73 Wood, 72 Ziegler, 80, 82 Zollner, 280 SUBJECT INDEX Acid, definition, 11, 16, 18 Acid-amides, 14 Acid-fuchsin, 37 Acidity, 94^109; determination, Chap- ter II ; titratable, 95 Acidoid, 240, 261 Acids, apparent dissociation of, 32; degree of dissociation of, 48; dissociation of, 29-37; dissociation constants, tables, 31, 59; mixture of two — , 46-48; polybasic, disso- ciation of, 55-60; pseudo — , 32, 118; pure, pH in solutions of, 37; solu- bility, 77, 82. Active masses, 6, 160 Activity, llOff., 132-3; coeflScient, 121ff.; factor, 114, 160; of H-ions, 128; theory, 87, llOff., 126, 127, 134-5, 160 Adsorbents other than charcoal, 243 Adsorption, Chapter X; apparent, 234, 268; definition, 231-2; equiva- lent — , of H- and OH-ions, 242; by exchange of ions, 244; of elec- trolytes, 234, of H- and OH-ions, 236, 242; of other ions, 265; of ions and charge formation, 245 Adsorption potentials, Chapter X; history of theory of, 277; relation to phase boundary potentials, 285; significance in physiology, 290 Affinity constants, 5, 30 Alcohol, 19 Alkalinity, determination of. Chapter II Ammonia, 16 Ammonium bases, 16 Ampholytes, 17; definition, 18; dis- sociation of, 60ff., 141; isoelectric point, 65-7; solubility, 86; table of dissociation constants and isoelec- tric points, 72-3 Ampholytoid, 240 Amphoteric electrolytes, see Ampho- lytes Atomic model, 8, 9, 15 Autopotential, 117 Bases, definition, 11, 16, 18; dissocia- tion of, 37-39; table of dissociation constants, 38 Basoid, 240 Blood (plasma), buffers in, 127; cal- cium ions in, 85; solubility of cal- cium carbonate in, 84; uric acid in, 80 Buffers, 43-49; 104^9; activity and pH of, 127-9; in blood, 127; in sea- water, 127; in urine, 128 Calcium ions in blood, 85 Calcium carbonate in blood, 84 Carbonic acid, dissociation constant of, 34^5 Cataphoresis, see Electrophoresis Charcoal, adsorption of electrolytes, 234; adsorption of H- and OH-ions, 236; as an ampholyte, 239, 243; neutralizing effect of, 242 Coehn's rule, 241, 288-90 Colloids, 265, 290 Concentration chains, 102, 160 Concentration effect, 196 Conductivity, coefficient, 121ff.; fac- tor, 114; of H- and OH-ions, 152; in oil phases, 150; influence of ionic charge on, 117; molar, 112, of water, 24 Crystallization optimum, 70 297 298 SUBJECT INDEX Current of falling particles, 279 Currents, hydrodynamic, 279 Degree of dissociation, 9, 14r-15; of acids, 48-54; definition, 48 Diaphragms, as acidoids, 261; iso- electric point of, 259; potential on, 259; sign of charge of, 261, 269 Dielectric constants, 13-15, 288; of water, 14; tables of, 14, 289 Diffusion potentials. Chapter VII; abolition of, 180 Dilution, effect of, on pH of buffers, 45 Dissociation, electrolytic, laws of, Chapter I; in non-aqueous solu- tions. Chapter V Dissociation of: acids, 29-37; am- pholytes, 60-69; bases, 37-9; poly- basic acids, 55-60; strong electro- lytes, Chapter III; water, 9-13, 19-22 Dissociation constants, 5; determina- tion of, 91-3; reduced, 129, 145; of water, 19-22, 24-28 Dissociation constants, tables of: acids, 31; ampholytes, 72-3; bases, 38; polybasic acids, 59; of water, effect of temperature on, 26 Dissociation residue, definition, 49; of acids, 48-54; 139; of ampholytes, 61, 141-3 Distribution coefficient, 186 Double layer, electric, 247, 280 Electric double layer, 247, 280 Electrode, gas, 164; quinhydrone, 171; reversible, 163 Electrode potentials. Chapter VI Electroendosmose, 246, 254, 284; changes of [H+] in, 271; Helm- holtz's theory, 248; Perrin's experi- ments, 258 Electrokinetic phenomena, Chapter X, summary of, 284 Electrolytes, strong, dissociation of, Chapter III; effect of pH, 131 Electrolytic dissociation, Chapter I Electromotive series of ions, 167-9; table, 168; of oils, 207 Electrons, 8 Electrophoresis, 250, 254, 284 Electrostriction, 113 Equivalent capacity of acids, 96 Exchange of ions in adsorption, 244 Falling particles, current and poten- tials of, 279, 284 Formamide, 14 Freezing point depression, 11&-123 Gas chains, theory of, 164-7 Gas electrode, 164 Gibbs principle, 241 Glass chain, 197 Glucose, 35-6 Glycerin, 14 Glycocoll, 17 Henderson's formula for diffusion potentials, 179 Hydrocyanic acid, 14 Hydrodynamic currents and poten- tials, 279, 285 Hydrogen atom, 9 Hydrogen ion, negative, 9 Hydrogen ion concentration, 21; determination of, 102-4; in acids, 39, 46, 77, 82; in ampholytes, 74, 86 Hydrogen ion exponent, 22-4 Hydrogen number, 21 Hydrolysis of salts, 87-91; table, 90 Indicators, 36-7; theory of, 9&-102 Ions, migration velocity of, 174-6; table, 175; series of, 205, 213, 235, 275 Isohydric solutions, 42 Isoelectric point of ampholytes, 65- 73, 76-7, 91-3, 143-5; determination of, 69-73, 91-3; table of, 72-3 Isoelectric point of diaphragms, 259- 261 SUBJECT INDEX 299 Liquid junctions, see Diffusion poten- tials Lithium hydride, 9 Mass action law, 3-7; deviations from, 110-7 Membrane potentials, Chapter IX; relation of to phase boundary po- tentials, 228-31 Migration velocity of ions, 174-6; table, 175 Milner effect, 120 Moderators, 108 Neutral salts, definition, 87 Neutralizing capacity, 96 Nernst's formula for concentration potentials, 103, 161; evaluation of, 161 Nitriles, 14 Oil chains, 192 Oil phases, dissociation in, 150 Ostwald's law, 91 Osmotic coefficient, 121ff. Osmotic factor, 114 Osmotic pressure of ions, 159 Oxidation-reduction potentials, 169- 73 Permeability, 210 pH scale, 22-4 Phase boundary chains, 191-5 Phase boundary potentials. Chapter VIII; and adsorption potentials, 285; calculation of, 183; origin of, 183; physiological applications of, 209; splitting of, 285 Phenolphthalein, 36-7 Phosphates, activity of, 130; buffers, 46 Planck's formula for diffusion poten- tials, 177 Polarization at phase boundaries, 218 Potentials, adsorption, Chapter X; diffusion. Chapter VII; electrode, Chapter VI; of falling particles, 279-83; hydrodynamic, 279-83; membrane. Chapter IX; oxidation- reduction, 169-73; phase boundary. Chapter \T:II; at precipitation membranes, 215 Pseudo-acids, 32, 118 Precipitation membranes, 215 Quinhydrone electrode, 177 Reaction, definition, 21 Reduced constants of acids, 129-31, 145-7 Regulators, see Buffers Reversible electrodes, 163 Saloid, 240 Salts, definition, 11, 18 Salt-formation, Chapter IV Saturation, definition, 78 Solubility, of acids, 77-86; of am- pholytes, 86-7; of Ca-carbonate, 82; of uric acid, 80; partial and total, 78 Surface tension at interfaces, 284 Two acids, problem of, 46-8 Titratable acidity, 95 Titration capacity, 96 Titration curves, 99 Transport number, 218 Uric acid, 80 Water, dielectric constant, 14; disso- ciation of, 9-28; dissociation con- stant of, 12-3, 24-8; ions of, 13; properties of, 13-9 "Yielding power," 107 "Zwitter-Ion," 73 Sans Tache Sans Tache IN THE "elder days of art" each artist or craftsman enjoyed the privilege of independent creation. He carried through a process of manufacture from beginning to end. 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