摘要:

DISCUSSIONS OF THE FARADAY SOCIETY No. 24, 1957 INTERACTION IN IONIC SOLUTIONS THE FARADAY SOCIETY Agents for the Society’s Publications : The Aberdeen University Press Ltd. 6 Upper Kirkgate AberdeenThe Faraday Society reserves the copyright of all Communications published in the ‘’ Discussions ” PUBLISHED . . . 1958 PFUNTED IN GREAT BRITAIN AT THE UNIVERSITY PRESS ABERDEENA GENERAL DISCUSSION O N INTERACTION IN IONIC SOLUTIONS A GENERAL DISCUSSION on Interaction in Ionic Solutions was held in the Uni- versity Laboratory of Physiology, South Parks Road, Oxford (by kind permission of the Vice-Chancellor) on the 17th, 18th and 19th September, 1957. The President, Mr. R. P. Bell, F.R.S., was in the Chair and about 250 members and visitors were present. The following overseas members and guests were welcomed by the President : Dr.T. Ackermann (Hamburg), Prof. H. Akamatu (Tokyo), Dr. J. M. Austin (Christchurch, N.Z.), Prof. and Mrs. C. J. F. Bottcher (Leiden), Mr. and Mrs. Floyd Buckley (Washington), Dr. R. L. Chooke (Houston, Texas), Dr. F. Coussemant (France), Dr. G. Dallinga (Amsterdam), Dr. M. Eigen (Gottingen), Dr. H. Eisenberg (Israel), Prof. and Mrs. H. Falkenhagen (Rostock), Prof. and Mrs. H. S. Frank (Pittsburgh), Prof. H. L. Friedman (Los Angeles), Prof. W. R. Gilkerson (South Carolina), Dr. Gertrud Glauner (Germany), Dr. Rolf Haase (Aachen), Prof. and Mrs. H. S. Harned (Con- necticut), Dr. Y . Haven (Eindhoven), Mr. J. Heemskerk (Amsterdam), Dr. Hellin (France), Prof. E. Jozefowicz (Lodz), Dr. M. Kaminsky (Marburg), Dr.and Mrs. G. Kelbg (Rostock), Dr. A. A. Krawetz (Ohio), Dr. and Mrs. P. Leprince (France), Mr. P. T. McTigue (Melbourne), Dr. R. Marcus (Cali- fornia), Dr. H. Mendel (Amsterdam), Dr. Mary L. Miller (Stamford, Corn.), Prof. L. Onsager (Yale), Mr. B. Ottar (Norway), Dr. Otto Redlich (California), Prof. R. A. Robinson (Singapore), Dr. D. H. Samuel (Israel), Dr. A. Silber- berg (Israel), Dr. M. J. Sparnaay (Eindhoven), Mr. V. R. Stimson (New England, N.S.W.), Dr. George Szasz (Zurich), Mr. G. A. J. Voetelink (Okla- homa), Prof. Dr. Kurt Zuber (Istanbul). A particularly warm welcome was accorded to Prof. Herbert S. Harned, of Yale Univcrsity, on the occasion of his delivery of the Ninth Spiers Memorial Lecture, which is printed in full in the present volume.A GENERAL DISCUSSION O N INTERACTION IN IONIC SOLUTIONS A GENERAL DISCUSSION on Interaction in Ionic Solutions was held in the Uni- versity Laboratory of Physiology, South Parks Road, Oxford (by kind permission of the Vice-Chancellor) on the 17th, 18th and 19th September, 1957.The President, Mr. R. P. Bell, F.R.S., was in the Chair and about 250 members and visitors were present. The following overseas members and guests were welcomed by the President : Dr. T. Ackermann (Hamburg), Prof. H. Akamatu (Tokyo), Dr. J. M. Austin (Christchurch, N.Z.), Prof. and Mrs. C. J. F. Bottcher (Leiden), Mr. and Mrs. Floyd Buckley (Washington), Dr. R. L. Chooke (Houston, Texas), Dr. F. Coussemant (France), Dr. G. Dallinga (Amsterdam), Dr. M. Eigen (Gottingen), Dr. H. Eisenberg (Israel), Prof. and Mrs.H. Falkenhagen (Rostock), Prof. and Mrs. H. S. Frank (Pittsburgh), Prof. H. L. Friedman (Los Angeles), Prof. W. R. Gilkerson (South Carolina), Dr. Gertrud Glauner (Germany), Dr. Rolf Haase (Aachen), Prof. and Mrs. H. S. Harned (Con- necticut), Dr. Y . Haven (Eindhoven), Mr. J. Heemskerk (Amsterdam), Dr. Hellin (France), Prof. E. Jozefowicz (Lodz), Dr. M. Kaminsky (Marburg), Dr. and Mrs. G. Kelbg (Rostock), Dr. A. A. Krawetz (Ohio), Dr. and Mrs. P. Leprince (France), Mr. P. T. McTigue (Melbourne), Dr. R. Marcus (Cali- fornia), Dr. H. Mendel (Amsterdam), Dr. Mary L. Miller (Stamford, Corn.), Prof. L. Onsager (Yale), Mr. B. Ottar (Norway), Dr. Otto Redlich (California), Prof. R. A. Robinson (Singapore), Dr. D. H. Samuel (Israel), Dr. A.Silber- berg (Israel), Dr. M. J. Sparnaay (Eindhoven), Mr. V. R. Stimson (New England, N.S.W.), Dr. George Szasz (Zurich), Mr. G. A. J. Voetelink (Okla- homa), Prof. Dr. Kurt Zuber (Istanbul). A particularly warm welcome was accorded to Prof. Herbert S. Harned, of Yale Univcrsity, on the occasion of his delivery of the Ninth Spiers Memorial Lecture, which is printed in full in the present volume.CONTENTS PAGE NINTH SPIERS MEMORIALECTURE- Some Recent Experimental Studies of Diffusion in Liquid Systems. By H. S. Harned . . 7 GENERAL INTRODUCTION. By R. P. Bell . . 17 I. GENERAL THEORY- The Statistical Interpretation of the Theory of Strong Electrolytes. Determination of General and Specific Ionic Interactions in Solution. Thermal Effects of the Interactions between Ions of Like Charge.By The Specific Interaction of Two Ions in a Strong Aqueous Electrolyte. Ion Distribution in Dilute Aqueous Solutions of Single Binary EIectro- GENERAL DrscussIoN.-Dr. M. Eigen, Prof. H. Falkenhagen and Dr. G. Kelbg, Prof. H. S. Frank, Dr. G. M. Bell and Dr. S. Levine, Dr. M. J. Sparnaay, Mr. H. E. Wrigley, Mr. H. L. Friedman, Mr. H. A. C. McKay, Prof. T. F. Young, Dr. Y . C. Wu and Dr. A. A. Krawetz, Prof. C. J. F. Bottcher, Prof. E. A. Guggenheim, Prof. C. W. Davies, Dr. R. J. P. Williams . . . 66 By H. Falkenhagen and G. Kelbg . . 20 By M. Eigen . . 25 T. F. Young, Y. C. Wu and A. A. Krawetz . . 37 By S. Levine and H. E. Wrigley . . 43 lytes. By E. A. Guggenheim . . 53 TI. INCOMPLETE DISSOCIATION- Introduction. C. W. Davies . . 83 Tonic Interaction Dissociation and Molecular Structure.By 0. Redlich and G. C. Hood . . 87 The Study of Some Ion-Pairs by Spectrophotometry. By W. D. The Dissociation Constant of Copper Sulphate in Aqueous Solution. Bale, E. W. Davies, D. B. Morgans and C. B. Monk . . 94 By W. G. Davies, R. J. Otter and J. E. Prue . . 103 Thermodynamics of Ion Association. By G. H. Nancollas. . 108 GENERAL DIscussroN.-Dr. G. A. H. Elton, Dr. R. H. Jones and Dr. D. I. Stock, Dr. J. E. Prue, Dr. F. J. C. Rossotti, Prof. C . W. Davies, Dr. C. B. Monk, Dr. M. C. R. Symons, Dr. R. H. Jones and Dr. D. I. Stock, Dr. R. J. P. Williams, Prof. H. Falkenhagen, Prof. C. W. Davies, Mr. Otter, Dr. V. S. Griffiths, Prof. E. A. Guggenheim, G. H. Nancollas, Dr. R. J. Gillespie, Dr. W. H. Lee and Dr.D. J. Miller, Dr. P. A. H. Wyatt, Dr. 0. Redlich, Dr. J. S. Rowlinson, Dr. M. Eigen, Dr. J. F. Duncan, Dr. N. Rajeswara Rao, Dr. H. Eisenberg . . 114 56 CONTENTS m. ION-SOLVENT INTERACTION- Structural Aspects of Ion-Solvent Interaction in Aqueous Solutions: A Suggested Picture of Water Structure. By H. S. Frank and Wen-Yang Wen . PAGE A New Approach to the Study of Hydration of Ions in Aqueous Solu- tions. By 0. Ya. Samoilov . Thermodynamic Properties of Aqueous Hydrochloric Acid. The Role of Ion-Solvent Interactions. By S. R. Gupta, G. J. Hills and D. J. G. Ives . A Theory of Ion-Solvent Interaction. By A. D. Buckingham . Properties of Concentrated Acid Solutions. By K. N. Bascombe and R. P. Bell . A Chemical Hydration Treatment of Concentrated Acid Solutions.Ion-Solvent Interaction and the Viscosity of Strong-Electrolyte Solu- Hydration of H+ and OH- Ions in Water from Heat Capacity By P. A. H. Wyatt . tions. By M. Kaminsky . Measurements, By T. Ackermann . Ionic Hydration and the Surface Potential of Aqueous Electrolytes. By J. E. B. Randles . X-Ray Diffraction of Aqueous Electrolyte Solutions. By C. L. van Panthaleon van Eck, H. Mendel and W. Boog . The Effect of Environmental Changes upon the Ultra-Violet Absorption Spectra of Solvated Anions. By M. Smith and M. C. R. Symons . GENERAL D1scussroN.-Dr. 0. Ya. Samoilov, Dr. J. Koefoed, Prof. D. H. Everett, Prof. D. D. Eley, Prof. H. S. Frank, Mr. L. D. Christie and Mr. T. A. Turney, Dr. S. Lengyel, Dr. R. Parsons, Dr. J. E. B. Randles, Dr. A. D. Buckingham, Dr. R. J. Gillespie, Prof. H. Falkenhagen, Dr. V. S. Griffiths, Dr. D. G. Tuck, Dr. M. Kaminsky, Dr. G. A. H. Elton, Dr. 0. Redlich, Dr. P. A. H. Wyatt, Dr. M. Eigen, Dr. Mendel, Mr. C. L. van Panthaleon van Eck and Mr. W. Boog, Dr. C . B. Monk, Dr. R. H. Jones and Dr. D. I. Stock, Prof. E. J6zefowicz, Dr. R. J. P. Williams, Dr. M. C. R. Symons . 133 141 147 151 158 1 62 171 180 194 200 206 216 Author Index . . 239

ISSN:0366-9033    

DOI:10.1039/DF9572400001

出版商:RSC    

年代:1957

数据来源: RSC

摘要:

NINTH SPIERS MEMORIAL LECTURE SOME RECENT EXPERIMENTAL STUDIES OF DIFFUSION IN LIQUID SYSTEMS* BY HERBERT S. HARNED I take this opportunity to thank the President and Council of the Faraday Society for the invitation to give the Ninth Spiers Memorial Lecture. I am very grateful to receive this honour and the privilege to present at this meeting some experimental results on diffusion in liquid systems which I believe have helped to give and will continue to give an impetus to progress in this difficult field. The subject of diffusion provides an appropriate introduction for a symposium on electrolytes because its investigation includes so many of the properties of ionic soIutions. The thermodynamics of these solutions is involved since the gradient of the chemical potential is one of the forces which cause the ions to move.The theoretical interpretation of salt diffusion and self-diffusion brings us directly to the dficult subject of the dynamics of steady-state processes which involves the mobilities and magnitudes of the ions, their partial volumes, their interactions with the solvent molecules and the influences of the viscosities of the solutions. DIFFUSION OF ELECTROLYTES AND THE SELF-DIFFUSION OF IONS When an electrolyte diffuses into a solvent, electrical neutrality requires that both ions move with the same average velocity. Under these conditions the ionic atmospheres suffer no deformation and the time of relaxation effect which is pro- portional to the difference between the average velocities of the two ions vanishes.In this process, however, the electrolyte and solvent move in opposite directions and an electrophoretic effect occurs. Further, since the chemical potential of the electrolyte varies throughout the solution, a thermodynamic term will occur in the theoretical equation for the diffusion coefficient. The simplest case of self-diffusion is that of a single ionic species at very low concentration (radioactive tracer ion) diffusing in a solution containing the non- radioactive species of the ion. The net effect of this process is the replacement of the radioactive ion by the inactive ion. Solvent displacement does not occur so that the effect of electrophoresis is negligible. On the other hand, since positive and negative ions move relative to one another, the time of relaxation effect becomes the important factor in dilute solutions.The thermodynamic term reduces to unity since the activity coefficient of a species at very low concentration in a medium of constant ionic strength does not vary sensibly with the concentration. THEORETICAL EQUATIONS FOR DIFFUSION COEFFICIENTS OF AN ELECTROLYTE IN DILUTE SOLUTION ACCORDING TO ONSAGER AND FUOsS1 The equation for computing diffusion coefficients of electrolytes in dilute solutions is (1) where 9 is the diffusion coefficient in cm2 see-1, v is the number of ions into which the electrolyte dissociates, R is the gas constant in erg mole deg.-1 and Tis the * Contribution No. 1461 from the Department of Chemistry of Yale University, 7 9 = 1000vRT(F/c) ( 1 + c- y:),8 EXPERIMENTAL STUDIES OF DIFFUSION I N LIQUID SYSTEMS absolute temperature.The term (z/c) is a concentration dependent mobility term. The second parenthesis includes the thermodynamic term which results from the requirement that one of the forces which cause diffusion is the gradient of the chemical potential.;! The concentration, c, is in moles/l. and the activity coefficient, yk, is on the molar concentration scale. Onsager and Fuoss obtained an equation for the mobility term which reduces to upon substituting the numerical values of the universal constants. Here, 21 and 22 are the valencies of the ions, v1 and YZ are the number of positive and negative ions resulting from the dissociation of one molecule of electrolyte, A10 and A20 are the limiting equivalent ionic conductances and A0 is their sum. D is the dielectric constant of the solvent and 70 is its viscosity.I is the ional concentration, zcgi2, K is the reciprocal of radius of the ionic atmosphere and a is the distance parameter of the Debye and Huckel theory. The quantity, +(~a), is the exponential integral function of the theory, values of which have been tabulated.3 At the lower limit of concentration, the thermodynamic term becomes unity and the mobility term reduces to the first member on the right of eqn. (2). As a result, eqn. (1) and (2) yield for the limiting diffusion coefficient, 90, a result originally obtained by Nernst.4 LIMITING EQUATION FOR THE SELF-DIFFUSION COEFFICIENT OF AN ION Onsager5 has obtained a limiting equation for the diffusion coefficient of an ion at very small concentrations in an electrolytic solution of otherwise constant composition.Gosting and Harned 6 and Wang 7 have arranged this equation in the following form, suitable for numerical computation. Thus, In this equation AjO is in coulombs sec cm-1 V-1, F is the Faraday in coulombs and R is in joules deg.-1 mole-1. The quantity, d(wj), depends on the number and valence types of ions present. For example, for the diffusion of an ion of type (1) present at very low concentration into a solution containing ions of types (2) and (3), c1 N 0, c2 I zz 1 = c3 I 23 I, and It is interesting to contrast the limiting numerical equations for salt diffusion Thus, for the self-diffusion of the iodide ion in potassium and self-diffusion. iodide and the sodium ion in sodium iodide, and 9 1 X lo5 = 2.045 - 0*473dc, 9~~ X lo5 = 1.334 - 0*2682/c, are obtained by substituting the numerical values of the quantities in the theoretical equations (4) and (5). The limiting theoretical equations for rubidium and cesium chlorides reduce to 9 X lo5 = 2.056 - 1-206-v'c, (8) and 9 X 105 = 2.046 - 1*1962/c, (9)H .S . HARNED 9 respectively.8 The quite considerable differences between the limiting slopes for self-diffusion and salt diffusion illustrated by these equations affords the unique opportunity of examining the electrophoretic contribution (salt diffusion) and the time of relaxation effect (self-diffusion) independently. For this purpose, very precise determinations of the diffusion coefficients in dilute solution will be required.RECENT EXPERIMENTAL METHODS The experimental values of the diffusion Coefficients which will be surveyed in this paper were almost all obtained by the conductometric method, the Gouy interference method and the capillary method for self-diffusion.9 The conductometric method is the only one which is particularly adapted for measurements in dilute solutions of electrolytes. At its best, it is capable of a very high accuracy. So far an overall accuracy of the order of f 0.2 of the diffusion coefficient at concentrations from 0.0005 to higher molarities has been obtained . * 0 The Gouy interference method has the advantage of general applicability to both electrolyte and non-electrolyte solutions. It is limited, however, to the more concentrated solution range from approximateiy 0-05 M to very concentrated solutions.The accuracy is about 0.1 x.11 The measurement of the self-diffusion coefficients of ions has not reached the precision of the most recent measurements of salt or molecule diffusion. Most of the self-diffusion coefficients have been determined by the use of diaphragm cells 12 or by the open end capillary method.13 The overall accuracy of these results is about 2 % at concentrations above 0-05 M. EXPERIMENTAL RESULTS AND THEORY FOR SALT DIFFUSION IN DILUTE SOLUTIONS Eqn. (1) and (2) have been tested for a number of salts whose diffusion co- efficients have been determined by the conductometric method at concentrations below 0.01 M.14 It was to be expected that the limiting eqn. (3) would prove to be exact although these conductometric determinations of diffusion coefficients are the only ones precise enough to prove its validity.The electrophoretic terms of the theory are small relative to the magnitude of diffusion coefficient. For example, at 0.01 M they amount to about 0-5 % of the diffusion coefficients of potassium, rubidium and caesium chlorides, 0-3 % for sodium chloride and about 1 % for the alkaline earth chlorides. Although the problem of determining effects of these magnitudes is a difficult one, our results for the majority of the salts show that the estimated values of these electrophoretic terms are of the right sign and size. Notable exceptions are calcium and lanthanum chlorides and magnesium and zinc suiphates. Harned and Hudson 14 have shown that the deviations of the bi- valent sulphates can be accounted for by assuming that the ion-pairs in these solutions migrate more rapidly than the ions.The behaviour of the diffusion coefficient of lanthanum chloride which exhibits the greatest departure from theory of the salts so far studied is illustrated by fig. 1. The concentration is expressed in moles/l. Curve (a) represents the caIculation by eqn. (1) and (2) which include both the thermodynamic term and the electro- phoretic terms. Curve (d) was obtained upon neglecting the effect of electro- phoresis. Curve (c) represents the experimental values. At 0.006 M the calcu- lated and observed results differ by 1.8 %. It is quite apparent that the inclusion of the electrophoretic effect improves the calculation considerably.The viscosity of lanthanum chloride solutions increases with concentration to the extent of 1 % at 0.006 M. if the mobilities are assumed to be proportional to yo/q, and this correction applied to the values represented by curve (a) the dashed curve (6) is obtained which is about 1 % lower than the calculated result at 0.006 M. The limiting equation of the theory is represented by curve (e).10 EXPERIMENTAL STUDIES OF DIFFUSION I N LIQUID SYSTEMS Even though the diffusion coefficient of lanthanum chloride does not appear to conform exactly with theory, our calculations indicate that the electroDhoretic 4,30 u) 0 - 1.20 X Q 1 I ! * - - '0 2 4 cx /03 6 FIG. 1 .-The diffusion coefficient of lan- thanum chloride. (a) By eqn. (1) and (2) ; (b) by eqn.(1) and (2) with viscosity factor ; (c) observed; (d) electrophoresis neglected ; (e) limiting equation. correction is of the right sign and almost of the right magnitude. A similar de- parture from theory of somewhat lesser magnitude was observed for the calcium chloride diffusion coefficient. A summary of these experiments leads to the following conclusions : (i) The Nernst limiting equation is exact. (ii) The thermodynamic term ac- counts for a large part of the change in diffusion coefficients in diIute solutions. (iii) The Onsager and Fuoss electro- phoretic correction for the mobilities is of the right sign and magnitude and is valid up to an ional concentration of 0.01 for many electrolytes. The data for calcium and lanthanum chlorides exhibit the largest departure from the theory.Bivalent sulphates conform to theory if the mobility of the ion-pairs is assumed to be greater than the mean mobility of the ions. From the preceding considerations, it appears that the diffusion coefficients of many of these strong electrolytes can be computed by eqn. (1) and (2) more accurately than they can be deter- mined experimentally. This conclusion suggested that diffusion data can be employed for determining activity co- efficients in dilute solutions. In the next section, we shall find that this pro- cedure is a promising one. ACTIVITY COEFFICIENTS IN DILUTE SOLUTIONS FROM DIFFUSION COEFFICIENTS 15 Rearrangement of eqn. (1) leads to and Fortunately, the limiting value of 2'/c* is fixed independently of the diffusion coefficient since where 9 ' ~ is the limiting theoretical slope of the logarithm of the activity co- efficient according to the Debye and Hiickel theory.By plotting 9 ' / c * against dH. S. HARNED 11 to this limiting value, the activity coefficient may be readily evaluated by graphical integration. Table 1 contains values of activity coefficients of a number of electrolytes computed from diffusion data, many of which have not been published elsewhere. Values of the distance parameter a, necessary for the computation of (&?/c), are given in the last column of the table. It is to be noticed that for silver nitrate the values at 0-005 and 0.01 M are identical in the third decimal place with those given by MacTnnes who used cells with liquid junction and transference numbers.It is indeed difficult to find another method by means of which the activity coefficients of these nitrates and perchlorates can be determined at these low concentrations at 25". The table also includes values obtained from as yet unpublished data for the diffusion coefficient of potassium chloride in 0.25 M and 0.75 M sucrose solu- tions. It is true that one must estimate the mobility term by eqn. (2). But this term depends on the limiting equivalent conductances which may be evaluated with high accuracy and a small correction for electrophoresis. In fact, for potas- sium chloride at 0.01 M, the omission of the electrophoretic correction only affects the activity coefficient by one in the third decimal place. TABLE MO MOLAR ACTIVITY COEFFICIENTS y + AT 25" FROM DIFFUSION DATA c (molesil.) salt &NO3 AgN03Mt 1 ~ ~ 0 ~ LiN03 * NaN03* CsN03" LiC104 KC104 KC11 * KCP* - D 78.54 78.54 78-54 78.54 78.54 78.54 78-54 7854 76-60 72-67 0.0005 0-974 - -9746 -9746 -975 -975 -975 -976 *974 -972 0.001 0-963 0.9646 ~9646 a964 -964 -966 *967 -963 -961 I 0.002 0-949 0.9508 -9508 -950 ~950 -9.54 -955 -949 -945 - 0.005 0.922 (.922) 0.9251 -9252 -924 -924 -932 -935 -922 -917 0.0 1 0.894 (.894) 0.8983 -8983 496 -896 -913 -915 -894 -889 4A0> 3.5 3.5 3.5 3.5 3.5 3-5 5.0 5.0 3-8 3-8 1 in 0-25 M solution of sucrose in water. * preliminary computations from the dissertation of Joseph A. Shropshire, Yale 7 MacInnes, Principles of Ekctrochemistry (Reinhold Publishing Corporation, New 2 in 0.75 M solution of sucrose in water.University, June 1957. York, 1939), p. 164. It would seem that this method should prove of considerable value for the future determination of activity coefficients. It employs one of the most accurate of measuring devices, the alternating current bridge. The value of 9'/d is known at zero concentration from exact theory. The definite integral includes the whole area under the graph and this area is a measure of the deviation of the activity coefficient from unity. Incidentally, for univalent electrolytes, the graphs have little curvature. SOME OBSERVATIONS OF SELF-DIFFUSION OF IONS AND WATER IN SALT SOLUTIONS We shall limit the discussion of self-diffusion to two illustrations. Fig. 2 shows a plot of the self-diffusion of the iodide ion in potassium iodide solutions deter- mined by Mills and Kennedy.16 The limiting expression of the theory for this system is eqn.(6) and is represented by the straight line drawn from the limiting value of the self-diffusion coefficient. The tendency of the experimental results to converge towards the theoretical limiting values is characteristic of the be- haviours of systems of this kind. However, a more exact proof of the validity of the limiting equation offers a difficult challenge to the experimenter since precise results at concentrations less than 0.01 M will be required.12 EXPERIMENTAL STUDIES OF DIFFUSION IN LIQUID SYSTEMS An illuminating illustration which shows the effect of viscosity upon self- diffusion of ions and water due to Wang 17 is shown in fig. 3. Here are the ratios dC FIG.2.-Self-diffusion of iodide ion in potassium iodide solution. dC FIG. 3.-Comparison of the self-diffusion coefficients of the sodium ion, chloride ion and water with viscosities of sodium chloride solutions.H. S . HARNED 13 of the self-diffusion coefficients of water, the sodium ion and chloride ion in sodium chloride solutions relative to their values in water. Comparison of these curves with the plot of the viscosity ratio, yo/y, also shown in the figure, leaves no doubt as to the importance of viscosity. One matter which may prove of unusual interest is that from 0 to 0.5 M sodium chloride concentration the self-diffusion of water is changed slightly if at all by salt addition. Indeed, the first addition of salts of ions with larger crystallographic radii, such as potassium iodide, appears to increase the self-diffusion of the water molecules.If these capillary-tube self-diffusion measurements are reliable, they indicate that salts with larger ions are good lubricants for water. THE DIFFUSION COEFFICIENT OF SODIUM CHLORIDE AT 25" The limit of validity of the theoretical equations (1) and (2) is illustrated by fig. 4. Here the diffusion coefficient of sodium chloride at 25" is plotted against the square root of the concentration. The results below 0.01 M were obtained by 1.60 m 5 1.55 9 1.50 I I 0 0.5 1.0 1.5 2.0 .t'c FIG. 4.-Difiusion coefficient of sodium chloride in water at 25". 0 conductometric ; optical ; diaphragm. the conductometric method, while those at higher concentrations were determined by the optical method 18 and the diaphragm cell method.19 The calculated values are shown by the graph in the upper part of the figure. In computing this graph, no correction for the viscosity was applied.It is apparent from this figure that the observed diffusion coefficients begin to deviate from the calculated ones at about 0.01 M and that they become increasingly less than those calculated as the concentration increases. Since the viscosity of the solution increases, its effect should decrease the mobilities of the ions and the diffusion coefficient. The character of the results shows that this effect is of the right sign. Robinson and Stokes20 have attempted to adopt the Hartley and Crank equation.21 for the purpose of computing diffusion coefficients of electrolytes in concentrated solutions.They introduce the viscosity correction and also the factor which rests upon the assumption that some of the ions transport a fixed number rz of solvent molecules during their migration. From a theoretical point of view, there are many uncertainties inherent in the application of the equation derived by Robinson and Stokes. For instance, the electrophoretic corrections, applicable to14 EXPERIMENTAL STUDIES OF DIFFUSION IN LIQUID SYSTEMS dilute solutions may be far from valid in the most concentrated ones. Further, it is known that the self-diffusion of water varies considerably with salt concentration, a factor neglected in calculating the solvation parameter, n. At present, we shall not consider the application of the Hartley-Crank equation to electrolytes but shall describe its use to derive a formal equation for the simpler type of system, that of two non-electrolytes, throughout their entire range of concentration.THE HARTLEY-CRANK EQUATION AND THE SYSTEM: CYCLOHEXANE + BENZENE If the diffusion of each of the components in a system of s components is ex- pressed in volume flowing in respect to a fixed plane of reference, it has been shown 22 that there will be (s - 1)2 independent coefficients of diffusion. Therefore, a system of two components may be described by a single diffusion coefficient, a result clearly demonstrated by Hartley and Crank.23 Their reasoning leads to the equation where gW is the diffusion coefficient, N is Avogadro's number and NA, NB are the mole fractions of components A and B.The thermodynamic term is expressed in terms of the mole fraction of component A and the rational activity coefficient, fA. The quantities, UA and UB, are lengths characteristic of each of the diffusing com- ponents which by analogy with the quantity, b ~ , in Stoke's equation are related to the velocities of the diffusing species. Singular values of UA and OB in non- electrolyte systems may be obtained by extrapolation of the measured diffusion coefficients. From precise measurements of the diffusion coefficients of both components in the system, diphenyl + benzene, Sandquist and Lyons24 found that the quantity 9/( 1 + c*) is a linear function of the relative viscosity (yo - $/yo. In completely miscible systems the intercepts of graphs of these quantities at NB = 1 and NA = 1 will be 9:,~ and 5!3;,~, respectively. Therefore, in dilute solutions on the mole fraction scale 9 i / ( d In NAfAld In NA)qO, A/q = sl, A + kl ATA/7]0, A, gg/(d 1n NAfAld In NA)TO,B/q = a g , B + k2 AqB/qO, B.5!3'/(d In NA fA/d In NA) N RT/N&T (1 5 ) where A ~ A = q - 7 0 , ~ and kl is a constant. Similarly, (1 6 ) (1 7) From eqn. (14), as NB approaches zero. Hence Similarly, Upon substitution of these values of CTA and CTB in the Hartley-Crank equation, we obtain as a formal expression for the variation of the diffusion coefficients over the entire concentration range, NA = 0 to NA = 1.H . S . HARNED 15 Lyons and Rodwin25 have determined the diffusion coefficients of each of the components in their dilute solutions by Gouy interferometry.From these results and graphs of eqn. (15) and (16), 9 { , ~ 9 ; ~ and the slopes kl and k2 have been evaluated. Table 2 contains the experimental values of gff and those calculated by eqn. (20). Values of the limiting diffusion coefficients and the constants, kl and k2 are included in the table. The thermodynamic data employed in this calculation were derived from the vapour pressure measurements of Scatchard, Wood and Mochel.26 TABLE 2.-oBSERVED AND CALCULATED DIFFUSION COEFFICIENTS FOR TKE SYSTEM BENZENE (A) + CYCLOHEXANE (B) AT 25" g;,* = 21-01 x 10-5; . g v x 105 = 18.76 x 10-5; kl = 5-4 x 10-5; k2 = 5.32 X 10-5 9 x 1 0 5 g v x i o 5 (obs.) (calc.) NB (obs.) (calc). p x 105 NB 0~0000 -0100 ~0501 -091 1 -1 101 .2000 -3017 -3658 21.01 20.90 20.44 19.99 19-78 19.03 18.45 18-22 21.01 0.3961 18-15 17.95 20.92 -5205 17.96 17-76 20.43 -641 8 17.99 17.98 20.0 1 -7795 18.29 18-40 19.84 -9299 18.66 18-70 19-06 -9800 18.76 18-83 18.53 1.oooO 18.76 18.76 18.09 - I The agreement of the calculated with the observed values is good over the entire range of concentration. This computation is unusually interesting since the viscosities of benzene and cyclohexane are quite different and since both the vis- cosity of the mixtures and the relative viscosity exhibit pronounced minima when the mole fraction of benzene is 0.76.The investigations described in this article were supported in part by the Atomic Energy Commission, U S.A., under Contract AT-(30-1)-1 375. 1 Onsager and Fuoss, J . Physic.Chem., 1932,36,2689. 2 The Scienti$c Papers of J. Willard Gibbs, letter to W. D. Bancroft, (Longmans, Green and Co., New York, 1906), p. 430. 3 Harned and Owen, The Physical Chemistry of Electrolytic Solutions (Reinhold Publishing Corp,, New York, 1950), p. 129. 4 Nernst, Z. physik. Chem., 1888, 2, 613. 5 Onsager, Ann. N. Y. Acad. Sci., 1945, 46, 241. 6 Gosting and Harned, J. Amer. Chem. SOC., 1951, 73, 159. 7 Wang, J. Ainer. Chem. Soc., 1952,74, 1183. 8 The limiting equation for salt diffusion is given by Harned and Owen, luc. cit., p. 179, eqn. (6-10-7). 9 For reviews of earlier methods for salt diffusion, see Longsworth, Ann. N. Y. Acad. Sci., 1945, 46, 211 ; and Harned, Chem. Rev., 1947,40,461. 10 The conductometric method is described by Harned and French, Ann.N. Y. Acad. Sci., 1945, 46, 267. Harned and Nuttall, J. Amer. Chem. SOC., 1947, 69, 736. 11 For descriptions of the theory and technique of this optical method, see Longsworth, Ann. N. Y. Acad. Sci., 1945, 46, 611. Kegeles and Gosting, J. Amer. Chem. SOC., 1947, 69, 2517. Coulson, Cox, Ogston and Philpot, Proc. Roy. SOC. A , 1948, 192, 382. Gosting and Onsager, J. Amer. Chem. SOC. 1952, 72, 66. Gosting, Hanson, Kegeles and Morris, Rev. Sci. Instr., 1949, 20, 209. Gosting and Morris, J. Amer. Chem. Soc., 1949, 71, 1998. l2 Adamson, J. Chem. Physics, 1947, 15, 76. 1 3 Anderson and Saddington, J. Chem. SOC., 1948, S381. Wang and Kennedy, J. Ainer. CJiem. SOC., 1950, 72, 2080. Wang, J. Amer. Chem. SOC., 1952, 74, 1182.16 EXPERIMENTAL STUDIES OF DIFFUSION I N LIQUID SYSTEMS 14LiC1, NaCl, Harned and Hildreth, Jr., J.Amer. Chem. SOC., 1954, 73, 650. KCl, Harned and Nuttall, J. Amer. Chem. SOC., 1947, 69, 736; 1949, 71, 1460. RbC1, Harned and Blander, J. Amer. Chem. SOC., 1953, 75, 2853. CsCl, Harned, Blander and Hildreth, Jr., J. Amer. Chem. SOC., 1954,76,4219. KNO3, Harned and Hudson, J. Amer. Chem. SOC., 1951, 73, 652. AgN03, Harned and Hildreth, Jr., J. Amer. Chem. Soc., 1951, 73, 3292. LizS04, Na2S04, Harned and Blake, Jr., J. Amer. Chem. SOC., 1951,73, 2448. Cs2SO4, Harned and Blake, Jr., J. Amer. Chem. SOC., 1951, 73, 5882. MgC12, BaC12. Harned and Polestra, J. Amer Chem. SOC., 1954, 76, 2061. CaC12, Harned and Parker, J. Amer. Chem. SOC., 1955, 77, 255. SrC12, Harned and Polestra, J. Amer. Chem. SOC., 1953,75,4168. LaCl3, Harned and Blake, Jr., J. Amer. Chem. SOC., 1951,73,4255. &Fe(CN)6, Harned and Hudson, J. Amer. Chem. Soc., 1951,73, 5083. LiC104, KC104, Harned, Parker and Blander, J. Amer. Chem. SOC., 1955, 77, 2071. MgSO4, Harned and Hudson, J. Amer. Chem. SOC., 1951, 73, 3781. ZnS04, Harned and Hudson, J. Amer. Chem. SOC., 1951, 73, 5880. 15 Harned, Proc. Nat. Acad. Sci., 1954, 40, 551. 16 Mills and Kennedy, J. Amer. Chem. SOC., 1953, 75, 5695. 17 Wang, J. Physic. Chem., 1954, 68, 686. 18 Vitagliano and Lyons, J. Amer. Chem. SOC., 1947, 78, 1549. 19 Stokes, J. Amer. Chem. SOC., 1950, 72, 2243. 20 Robinson and Stokes, Electrolyte Solutions (Butterworth Scientific Publications, 21 Hartley and Crank, Trans. Faraday SOC., 1949, 45, 801. 22 Onsager, Ann. N. Y. Acad. Sci., 1945, 46, 241. 23 Hartley and Crank, Trans. Faraday SOC., 1949,45, 801. 24 Sandquist and Lyons, J. Amer. Chem. SOC., 1954, 76, 4641. 25 Lyons and Rodwin ; Prof. Philip A. Lyons has given me the result of this cal- 26 Scatchard, Wood and Mochel, J. Physic. Chem., 1934, 43, 119. London, 1955), p. 309-322. culation prior to publication.

ISSN:0366-9033    

DOI:10.1039/DF9572400007

出版商:RSC    

年代:1957

数据来源: RSC

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GENERAL LNTRODUCTION BY R. P. BELL The subject of electrolyte solutions occurs frequently in the Transactions of‘ the Faraday Society from 1903 onwards, and an interesting account of these early discussions is given by C . W. Davies in his article on “ Solutions ” in the memorial volume The First Fiftv Years, which marked the fiftieth anniversary of the Society in 1953. After a rather inconclusive discussion in 1919 on “ The Present Position of the Theory of Ionization ”, the meeting held in Oxford in 1927 on “ The Theory of Strong Electrolytes ” represents a landmark in the history of the subject. It was a most successful meeting, at which the visitors from overseas included Bjerrum, Bronsted, Fajans, Harned, Hiickel, Onsager, Scatchard and Ulich. Since 1927 the Society has held no discussions on this general topic, though there have been two on colloidal electrolytes, and it is therefore natural to take the 1927 meeting as a starting point for introducing the present one.In 1927 the classical paper of Debye and Hiickel was only four years old, and Onsager’s treatment of conductivity had only just appeared. Most of the discussion, therefore, centred round the theory of interionic attraction and its application to very dilute solutions, and the success of this essentially physical attack on the problem carried with it a tendency to discount any specific chemical explanations. It is a tribute to the pioneer work of Debye and Hiickel that their equations form the basis of all the papers on ion-ion interaction presented to the present meeting.There have, of course, been extensions of their original work, and more powerful methods have been devised for tackling the statistical problems involved, as exemplified by the paper of Falkenhagen and Kelbg to the present meeting. Similarly, Guggenheim discusses critically the best way of combining the Debye-Huckel expressions with the concept of ion-pairing introduced by Bjerrum. The latter may be regarded as a way of circumventing the difficult part of the calculation, but the model is still an electrostatic one based on Coulomb’s law. If two ions approach close to one another it is questionable whether the macroscopic dielectric constant of the solvent can still be used, and this problem is considered in the paper by Levine and Wrigley : fortunately it appears that the correction is not a very large one, at least for the example chosen. In 1927 the emphasis was on the complete dissociation of salts and strong acids, and a good deal of time was spent in demolishing spurious degrees of dis- sociation arrived at by ignoring interionic forces.Since that time the idea of incomplete dissociation has been gradually creeping back into the subject, and the second section of our discussion is devoted to this subject. A similar alter- nation of viewpoint can be traced in the theory of non-electrolyte solutions. Some forty years ago incorrect assumptions about the laws of intermolecular force led Dolezalek and others to assume compound formation and association in almost every type of solution, for example in mixtures of liquid nitrogen and argon.A better understanding of intermolecular forces showed later that these assumptions were in many cases incorrect, and it became fashionable to ignore chemical explanations of non-ideal behaviour. More recently, however, the pendulum has swung back in the other direction, partly because we have a better understanding of special types of attraction such as hydrogen-bonding and donor- acceptor interaction, and partly because it is realized that when dealing with short- range forces the choice between a chemical and a physical formulation may often be only a matter of convenience : for example, one of the most successful ways of treating the thermodynamics of mixtures of non-polar molecules is the “ quasi- chemical approximation ”.1718 GENERAL INTRODUCTION The problem is a more difficult one for electrolyte solutions, since the Coulombic forces are long-range in nature; it is, of course, this circumstance which leads to the special features of the interionic attraction theory. For ions of high charges or solvents of low dielectric constant the Bjerrum picture of ion-pairs has been widely used, but, as shown clearly in Guggenheim’s paper, the degree of association arrived at depends considerably on the choice of parameters. It is possible that additional attractive forces of a short-range nature come into play when two ions approach closely, but this is difficult to ascertain: for example, there has been much argument as to whether the hydration shells of ions are displaced when they come into contact.However, many authors have used experimental data to derive association constants without reference to the nature of the association forces, and this approach has been adopted particularly by C. W. Davies, who is giving the introductory paper to our second section. For many salts, such as calcium or magnesium salts of dicarboxylic acids, there can be no doubt about the physical reality of the association constants derived, but for others, such as the bivalent sulphates, the position is much less clear. The usual procedure in treat- ing thermodynamic data or conductivities is to ascribe to association all deviations from a standard expression, and, as emphasized by Redlich and Hood, there are obvious dangers in this procedure, which we shall no doubt discuss.In this situation it is desirable to investigate several different properties of the solution, particularly ones not affected by environment. Absorption spectra have been widely used here, and the paper by Davies, Otter and Prue compares the association constants derived for copper sulphate from its ultra-violet absorption and from other sources. (It is somewhat ironical that copper sulphate should be treated in this way, since the fact that copper sulphate solutions obey Beer’s law accurately in the visible part of the spectrum was part of the evidence originally quoted by Bjerrum and others in support of the complete dissociation of strong electrolytes.) In this work, as in the paper by Monk and his collaborators, the interpretation of the optical measurements is still dependent upon the expression assumed for activity coefficients, and it is important to discuss under what con- ditions this difficulty can be avoided.In many respects the use of Raman spectra or of nuclear magnetic resonance had advantages over absorption spectra, though they are at present limited to concentrated solutions. The paper by Redlich and Hood discusses these types of measurement, and shows gratifying agreement for a number of strong acids. The third section of our discussion deals with ion-solvent interaction or solvation. The 1927 Discussion touched hardly at all on this subject, though it is interesting to note that the second Discussion ever held by the Faraday Society (in 1907) was on “ Hydration in Solution ”, and it is chastening to see that many of the points debated fifty years ago are still uncertain today.Like incomplete dissociation, the problem of ion-solvent interaction has been returning gradually into prominence during the last twenty or thirty years, and it occupies more than half the papers in the present discussion. Its temporary eclipse was probably due to the great success of the Debye-Huckel theory, in which the solvent enters only as a medium of known dielectric constant. However, this approach is adequate only for the concentration dependence of properties of very dilute solutions, and solvation must be taken explicitly into account as soon as it affects an appreciable proportion of the solvent molecules : this was first pointed out by Bjerrumin 1919, and the same idea has recently been revived in the treatment of concentrated solutions given by Stokes and Robinson and by Glueckauf.More- over, ion-solvent interaction is of primary importance in determining the individual molar properties of salts in solution, for example entropies, specific heats and molar volumes, all of which are considered in this section. Early attempts to represent ion-solvent interaction used a physical picture of a continuous medium, notably Born’s treatment of the free energy of solution, and various modifications which attempted to take into account electrostrictionR. P . BELL 19 or dielectric saturation. However, this model is unsatisfactory, and there is no doubt that it is necessary to take into account the molecular structure of the solvent.Thus in the paper by Buckingham, which represents the most " physical " approach in this section, the inner shell of water molecules are treated individually, and the remainder of the solvent as a continuous dielectric. At the other extreme lies the representation of ion-solvent interaction as a chemical process with a stoichiometric solvation number, and this approach is used in the paper of Wyatt and of Bascombe and Bell on the properties of concentrated acid solutions. The concept of stiochiometric hydration is obviously only a crude approximation, and the paper by Samoilov stresses the fact that the nature of the interaction may vary greatly in different cases: he pays particular attention to the time factors involved in the movement of water molecules near an ion.Further, there is no doubt that any complete treatment must take into account the ordered structure of water and the effect of ions on it. Water itself presents a difficult structural problem even in the absence of ions, and it is not surprising that this method of approach is at present qualitative rather than quantitative. Frank has been the pioneer in this field, and his paper with Wen-Yang Wen describes recent developments. The same concepts of ordering and disordering by ions are used in various ways by Gupta, Hills and Ives, by Ackermann, and by Kaminsky in accounting for a wide range of properties. In this discussion we have a number of papers on the use of new experimental methods for investigating ionic solutions. As already mentioned, the use of Raman spectra by Young and of nuclear magnetic resonance by Redlich and others provides the most reliable information about degrees of association in concentrated solutions. The well-established methods of X-ray diffraction have been applied in a novel way to salt solutions by van Eck and his collaborators, with conclusions which differ in some respects from those derived from other methods. Randles shows in his paper how a study of the surface properties of electrolyte solutions can give information about ionic hydration, and Eigen has used measurements of rate processes in ionic solutions to distinguish between different kinds of inter- action. Finally, Smith and Symons have studied the ultra-violet absorption of anions as a function of environment and give an interpretation in terms of ion- solvent interaction: this type of measurement is also important when using ab- sorption spectra for studying ionic association It is to be hoped that much more work will be done on ionic solutions by less conventional methods such as those just mentioned, since it seems likely that major advances in our understanding of these solutions can only come from radically new experimental information.

ISSN:0366-9033    

DOI:10.1039/DF9572400017

出版商:RSC    

年代:1957

数据来源: RSC

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I. GENERAL THEORY THE STATISTICAL INTERPRETATION OF THE THEORY OF STRONG ELECTROLYTES BY H. FALRENHAGEN AND G. KELBG Institut fur Theoretische Physik, Universitat, Rostock, Germany Received 1st July, 1957 If we follow the development that the theory of strong electrolytes has taken in the last few years, the dominant factor of this work was, first, to derive the Debye-Hiickel limiting law by means of strict statistical methods and to state the assumptions and approximations contained in it. Then, great efforts were made to take the actual volume of the ions into consideration by introducing an additional repelling potential in order to extend the theory to higher concentration ranges. Whereas the Debye-Huckel theory furnishes the limiting law very simply and elegantly, the more exact statistical methods of Kirkwood, Mayer, Bogoljubow and others require an extraordinarily great amount of calculation.Therefore it is certainly useful to con- sider whether it is not possible to derive the relation between the potentials of the average forces and the average electric potentials from the fundamental statistical equations with less amount of mathematics, in a way which is not much less strict. Thus in this paper we shall point out that in a short way using the Kirkwood equations of the second kind we may obtain the desired connection, and that furthermore it is possible to extend the theory of strong electrolytes by introducing short-range forces. The considered system niay be represented by a canonical ensemble, the poten- tial energy UN of which consists additively of the potentials UQ between pairs i and j of particles. If a particle is separated we may write N i = 1 and we obtain the following relation between the partition functions nN and nN-l in the configuration range N Zj is equal to the quotient of the configuration integrals.By means of the cor- relation functions gN ( n ~ = nNgN) we come to the following development : where hj = exp (- BUG) - 1 are defined in the usual way. all the co-ordinates with the exception of rj and r, yields An integration over 1 fik.fiegkes drk dre + . . ] . (4) The higher distribution functions in this still exact development may now be ap- proximated. Jf we use the potentials of the average forces F?,, w k e s , wkers, etc., the following superposition principle may be usedH .FALKENHAGEN AND G . KELBG 21 That means that the average force on an ion s is equal to the sum of the average forces which we get if the ion k or the ion e, etc., were alone present. This is certainly valid only at very high dilutions and represents the most drastic approx- imation. On employing (5), we may sum up the series in (4) and obtain the be- ginning of an e-function of the argument 1 nk J j j k gks drk , 4 may now be found from the condition for I ri - r, I 3 co from eqn. (4) and the resulting equation is k Ts = 19, -- k T C ?!k Ih,c{gks - 1 >dr'k. (6) If we take into consideration a repelling potential of rigid spheres additional to the Coulomb term, we may divide the integral in (6) into two parts, at first in- tegrating over an area about rj which does not contain the sphere 1 Y ) - rk I -: n.In this first integralAk may be linearized k Cj <'k This approximation is the better, the smaller - is. We find I DkTiz 1 4n 3 3 y = 4 n a 3 -a3 n If we add and subtract Rks drk , then, in the first integral, we find the average Jm d",3 3 electric potential &(rst>) at the place ri in the neighbourhood of a s-ion at the place r , . Thus we get ]{gh - 1 ) drk. (9) ej ck - W, == uj, Cj \/rs ~ RT nk 1 { 1 - D 1 rj - rk 1 4.1 -a3 3 Since the third term is of higher order in n than the second, in first approximation, we find the statement q, = Ujs 4- ej FS (r,, rj) employed by Debye-Hucltel. A further discussion of the eqn. (9) has not been given up to now. Moreover, some remarks may be made about methods which are less known.Worthy of emphasis are the papers of Bogoljubow, Glaubermann and Juchnowski.1 Strict investigations of the Debye-Huckel approximation were conducted by means of the Bogoljubow-Yvon-Born-Green equation. The authors develop the par- tition functions formally according to the powers of V/N. If we employ the re- duced Coulomb potential which is regular everywhere e. e. Dr uii = -{I - exp(- a r ) ) , for the radial partition functions we get the solutions g i i = l - - - 1 eicj (v) - exp (- DkT p2 - 9 2 Y * The same result was obtained by G. Kelbg in an investigation of eqn. (6).22 STATISTICAL INTERPRETATION OF STRONG ELECTROLYTES Here are and x is the reciprocal of the average radius of the ionic atmosphere. For the free energy there results the simple expression Furthermore the theory of Moller is worth mentioning.2 The Born-Green equations are solved for a Coulomb potential plus a repelling potential of ab- solutely hard spheres.The calculations are very complicated ; the results are given in fig. 1 and 2. Moreover, the question about the change of the ionic 0 3 5 051 0 4 7 0 4 3 0-39 0 - 3 5 0 4 1 I atmosphere is cleared up. In fig. 3 the electric charge density is shown. If we write down the results of Hiickel and Krafft 3 and Eigen and Wicke 4 too, we see that Huckel and Kt-afft are in contradiction to the theory of Moller. On sur- veying these papers we may say that good results have been obtained as regards establishing the Debye-Hiickel statement, but that still much work is to be done concerning a theory for higher concentration ranges.The situation with regard to the theory of irreversible processes looks some- what worse than the strict molecular theory which is still in the stage of develop- ment. But by means of general statements we succeeded in extending the Debye-Hiickel-Onsager-Falkenhagen theory by introducing an ion diameter and by employing certain new boundary conditions. Here we have to mention the papersH. FALKENHAGEN AND G. KELBG 23 of Kaneko,fi Falkenhagen and Kelbg,6 Pitts 7 and Fuoss and Onsager.8 For various substances the theory is valid up to 0.1, sometimes even up to 0.5 mole/l., in 1 : 1-electrolytes. d 2 c FIG. 2.-The osmotic coefficients 1 - g of aqueous solutions of 1 : 1 electrolytes and dependence on the square root of the concentration in mole/l.t = 0" C ; + LiC104, A NaBr, 0 LiCI, OKCI. Huckel- Kraff t \--- FIG. 3.-Qualitative course of charge density in the neighbourhood of a central ion Moreover, Kelbg6 introduced the ionic radius into the theory of the Wien effect. It was possible to fix the A-coefficient by means of a perturbation cab culation. A comparison of this theory with the experiment was performed by Fucks and Tesch 9 (see fig. 4).24 The effect of the radius on the viscosity of strong electrolytes was examined by Falkenhagen, Kelbg and Pitts.7 Evidence is given that the ion radius alone is not able to reproduce the characteristic variations of the viscosity coefficient. STATISTICAL INTERPRETATION OF STRONG ELECTROLYTES P- 12- I I- 10 - 9 - 8- 7- b 5- 4- 3- 2- I 0 .......Wien - Fucks and Tesch --Wilson -.- Falkonka9o~ and Kalb9 E 4% I00 It0 rho Ibo FIG. 4.-Conductivity increase AA in dependence on the field power E in a MgS04 solution. C = 5 x 10-3 mole/l. Comparison with the measurements of Wien, the theory of Wilson and the theory of Falkenhagen-Kelbg. MgS04, c = 5 x 10-3 mole/] A -A&O AA = 100 ~ AE- 0 T = 293.2 77 =. 1.005 x 10-2 poise D = 80.3 Other influences are of greater importance, e.g. structure change effects of the solvent, hydrolysis, polarization effects. The theory of the A-coefficient by Falkenhagen is here compared with the recent experiments by Kaminsky.10 The following tables show a good agreement with the theory. TABLE 1 TABLE 2 A theor. KI 18 0.0045 0.0048 Li2SO4 15.1 0.0 1 60 0.01 60 0.0 1 64 A expt. 0.0045 - ^^ .- 0.0049 - ^ ^ _ ^ 20 - _ 0.0165 - -- _- to C A expt. A theor. I 0 c - ^ - _ - 20 0.0041 0.0050 30 0.0052 0.0051 35 0.0052 0.0052 40 0.0054 0.0054 25 0.U161 0.01 66 30 0.0 1 69 0.0168 42.5 0.0173 0.0 172 1 Bogoljubow, ProbEeme der dynamischen Theorie in der statistischen Physik (Moskau, 1946). Glaubermann and Juchnowski, J. Expt. Physics, 1952, 22, 562, 572 ; 1953, 25, 580. 2 Moller, Diss. (Rostock, 1956). 3 Huckel and Krafft, 2. physik. Chern., 1955, 5, 305. 4 Wicke and Eigen, Naturwiss, 1951, 38, 453 ; 1952, 39, 545 ; 2. Elektrochem., 1952, 5 Kaneko, J. Chern. Soc., 1935, 56, 793, 1320; 1937, 58, 985. 6 Kelbg, Diss. (Rostock, 1954). 7 Falkenhagen and Kelbg, 2. Elektrochem., 1952, 56, 834. 9 Fucks and Tesch, 2. Physik., 1957, 148, 53. 10 Kaminsky, 2. physik. Chem., 1956, 8, 173. 56, 551 ; 1953, 57, 319 ; 2. Naturforsch., 1953, 8a, 161. Falkenhagen and Kelbg, 2. Elektrochem., 1954, 58, Pitts, Proc. Roy. Soc. A , 653. 1953, 217, 43. 8 Fuoss and Onsager, Proc. Nat. Acad. Sci., 1955, 41, 274.

ISSN:0366-9033    

DOI:10.1039/DF9572400020

出版商:RSC    

年代:1957

数据来源: RSC

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DETERMINATION OF GENERAL AND SPECIFIC IONIC INTERACTIONS IN SOLUTION BY M. EIGEN Max-Planck-Institut fur physikalische Chemie, Gottingen, Germany Received 17th September, 1957 The determination of “ general ” and “ specific ” ionic interactions from static thermo- dynamic data involves in general a certain amount of arbitrariness. By using a dynamical method to record the relaxation spectrum of the system it is possible to separate the different kinds of ionic interaction. Some applications of the method to 2 : 2-electrolytes in aqueous solution are described. The results are compared with those of recent statis- tical and thermodynamic studies. The application of thermodynamics to a concrete system is independent of special assumptions about its microscopic structure. Any physical interpretation of thermodynamic data, however, requires a statistical picture of the system and depends on our knowledge of the molecular interaction forces which generally express themselves only summarily in the observable quantities.With respect to electrolyte solutions most of our knowledge about the structure and interaction effects has been deduced from analyses of thermodynamic data such as activity coefficients, etc. However, the present state of electrolyte theory allows an unequivocal distinction between different interaction effects only in limiting cases : (i) ‘‘ strong ” electrolytes (e.g. alkali halides) at Iow concentrations : Coulombic forces determine the “ non-ideal ” behaviour of the solution, leading to the well-known &terms of Debye and Huckel ; (ii) “ weak ” electrolytes (such as ammonia or acetic acid in aqueous solution) : chemical interactions, considered using the law of mass action, are de- cisive for their behaviour-at least at concentrations C > K,, where Kc is the dissociation constant.Generally for electrolytes at higher concentrations several factors complicate the interpretation of thermodynamic data and lead to arbitrary definitions of “ weak ” and “ strong ”. At close interionic distances the Coulombic forces between two or more ions are strongly influenced by the special structure of the solvent and by the ion-solvent interaction (solvation). A superposition of the electrostatic poten- tials using the Coulombic term z,z,e$Dr with the macroscopic dielectric constant D and a “ cut off” at a hard-sphere radius ro does not represent the actual “ poten- tial of the average force ”.In addition, at very close distances the non-electro- static forces become important and are superposed on the electrostatic ones, which here correspond to some kind of “ association ”. These uncertainties (rather than difficulties in the exact statistical treatment which could in principle be over- come 1) restrict the quantitative interpretation of experimental data. In this situation it would be of great value to have an experinzental method allowing a direct and separate determination of the different interaction effects taken into account in the thermodynamic functions. The requirements for such a method seem to be met by one which hitherto has been scarcely applied to the elcctrolyte problem, viz., relaxation spectrometry.In the present paper the prin- ciple, applications and limitations of this method will be described. To characterize the different types of interaction we shall use the terms “ general ” and “ specific ”. A general ionic interaction is that described by the 2526 SPECIFIC IONIC INTERACTIONS Coulombic term with the macroscopic dielectric constant. This holds for interionic distances r > r$ + r& where rg includes several layers of solvent molecules (hydration shells) and where the potential is a monotonic function of distance. For Y < YO” + the electrostatic interaction energy cannot be represented by such a function, as one cannot use the macroscopic dielectric constant.There are several favoured positions depending on the number of solvation layers between the ions. In addition to these “ specific ” electrostatic effects, chemical interactions of the ions may occur. A dissociation process therefore proceeds stepwise. METHOD PRINCIPLE The relaxation technique and especially its application to problems of re- action kinetics in electrolyte solutions has already been described elsewhere.2 If a single step equilibrium (e.g. A+ + B- + AB) is disturbed by changing an ex- ternal parameter, the new equilibrium state will be reached within a certain time, the “ relaxation time ” (for definition cf. eqn. (4)). If, however, the equilibrium consists of several steps, for instance in the consecutive reaction system : A+ + B- * (AB)1 + (AB)2 + .. . =+ (AB),, (1) or in a system like the ionic atmosphere with its multiple configurations, the time dependence is determined by a set of time constants-a “ spectrum ” of relaxation times-which may be either discrete or a continuous distribution about a given time. Specific interactions show a discrete spectrum due to discrete reaction steps, whereas the general interaction shows a continuum corresponding to the multiplicity of similar states in the ionic atmosphere. One is the analysis of relaxation times to obtain information about reaction kinetics. The other is the identification of different equilibria by separating the corresponding parts of a thermodynamic function on the time axis. It is this application that we wish to take up here. Let us consider the ionic interaction due to several influences which all con- tribute to the thermodynamic functions of the system. Any static change in such a function f (e.g.volume, heat content, etc.) caused by changing an external para- meter 7 (e.g. temperature, pressure, etc.) may be written as There are two main applications of the relaxation technique. where the ni represent the concentrations of the diiferent stages of interaction (nA+, nABI, nABz . . .). Information about the individual parts of this sum can only be obtained by theoretical considerations, involving the above-mentioned difficulties. If, however, we study the dynamical function dfldv-for instance for a periodic change in 7-the single terms of the sum become “marked” by a frequency factor, due to the fact that the equilibria of the different steps will not be reached at the same time.This allows an experimental separation of the single terms, assuming that the relaxation times in the spectrum differ sufficiently. The dynamical function [dfld9]dyn then has to be written : with where the change ( 3 f l 3 ~ ) ~ , . . . . ni may be assumed instantaneous and certain ex- ternal conditions have to be maintained. $ i ( w i ) is a complex function containing frequency terms of the form, 1/(1 + wZri2) or WTi/(l + w%i2) and the ni are inM. EIGEN 27 this case transformed concentration parameters (normal variables). There are observable quantities which correspond directly to the real or imaginary parts of such functions, for instance the velocity and absorption of sound.Examples for sound absorption spectra are given in fig. 2. From this a separation of the single terms (&/3P)i of the compressibility is easily possible. THEORETICAL TREATMENT A general thermodynamic theory of relaxation phenomena has been developed by Meixner.3 The kinetic theory of relaxation is a “ physical specification ” of the more general thermodynamic treatment. For analytical problems-as in the present application or for a study of the mechanisms of fast reactions-it is of advantage to use this theory, as it contains more information on the physical mechanisms involved.* In the following a short review will be given. With respect to the electrolyte problem three cases are of special interest : (i) the single- step mechanism due to a specific interaction, (ii) the spectrum resulting from a series of specific steps, (iii) the relaxation of the ionic atmosphere and its influence on the specific interaction (for more details, see ref.(2), (6)). (i) The one-step relaxation mechanism is characterized mathematically by a linear differential equation in the variable SC, the deviation from the equilibrium concentration ( 6 ~ = c - c> : dC C - c dt T Y (4) - =- ___ - d(6C) - 8C - dt r or for C = const. : ki k2 (Example : A+ + B- + AB ; 8CA = SC, =- SCAB ; 1 / ~ = k2 + kl(C7, +c)). One can observe the relaxation process directly or indirectly by dispersion or absorption measurements. As an example we consider the expression for the sound absorption due to the relaxation of a binary dissociation equilibrium in the pressure wave.For electrolytes in aqueous solution the dependence of the relaxation time on the thermodynamic state (constant s, v or T, v and so on, cf. ref. (3), (2)) as well as the influence of the temperature wave may be neglected (Cp M CV). Then we obtain, where Ap is the excess-absorption per wavelength A, defined by the exponential decrease of the energy in a progressive plane sound wave with increasing distance d : E = EO exp - (@/A). The frequency factor ~ / ( 1 + COW) shows a maximum at o = 11. (OJ = 2m, v = frequency), reaching the value 1/2. This behaviour (shown in fig. 1) represents a “ monochromatic ” relaxation process. The absolute value of the absorption is determined by AK/K (< l), the ratio of that part of the compressibility due to the relaxing system, to the total compressibility of the solu- tion (here adiabatic = isotherm).AK may be calculated from the degree of dis- sociation a and the difference of partial molar volumes A V = V, + VB - VAB : (6) 1 3v da ~r(1 - a) (An2 AK =-- - (-) - = Cop--- 210 3~ dp 2 - a R T ‘ (CO = molar concentration of the electrolyte.) * A kinetic theory of single relaxation steps due to first-order transformations has been developed by Kronig.4 For a generalization of this theory for more complicated chemical systems (n partners with any possible one- or multiple-step reactions between them), cf. ref. (2). A treatment of a first-order consecutive reaction system is also given by Bauer.528 SPECIFIC IONIC INTERACTIONS Any influence of general interactions is neglected here.Eqn. (6) is therefore valid only for very weak electrolytes. Such one-step equilibria are found in protolytic systems. Relaxation technique has been used to determine the overall reaction rate of the very fast protolytic reactions.2~9 (ii) Normally the specific interaction cannot be treated as a one-step mechan- ism. The recombination of oppositely charged ions is hindered by hydration layers, which have to be removed stepwise. In the general case the specific interaction may be characterized by a consecutive reaction system of the form given in (1). This system has a spectrum of relaxation times, in which the number of time constants corresponds to the number of independent steps. On the other hand, a single relaxation time does not necessarily correspond to a single step in the reaction scheme.Similarly, as for the normal frequencies in a system of coupled oscillators (which are not identical with the resonance frequencies of the single groups), the relaxation times have to be calculated from the whole system of (linearized) rate equations. Mathematically this is a problem of '' principal axis transformation ", in which one has to find a new set of concentration variables (normal variables yi) instead of the given variables xi (= SC,, SCB, SC,,,. . . .). The new variables have to fulfil the condition : ji = - Yi/Ti, (7) where ~i is the corresponding relaxation time (9 = dy/dt). The transformations for a number of systems have been carried out and are discussed in detail else- where.2.6 Let us consider the results for the following simple example : kl2 k23 A+ + B- + AB' + AB with k12, k21, > k23, k32 I k21 I1 k32 111 (For detailed calculations and a discussion in connection with the sound ab- sorption spectra of 2 : 2-electrolytes, see ref.(6).) The linearized rate equations are : (9) x1 = - ki2X1 + k21x2, i 2 = k;2x1 - (k2l f k23b2 $- k32X3, i 3 = k23x2 - k32x3, with SC, = SC, = x1 I and kj2 = kl2(G + G), where the c are the equilibrium concentrations. SC*B = x3 The transformation to a new system : ii = - yi/Ti may be written in the vectorial forms : --+ --t y = Mx, (10) and + --t x = M-ly, where the matrices M and M-1 are given by 1 1 1 withThe The M. EIGEN 29 relaxation times result as eigenvalues of the functional equation : corresponding parts of the thermodynamic functions can be calculated from the transformations.pressi bility : In this way we obtain for the relaxing parts of the com- . . (Av3)2; av3 = av,,,. RT The principal result of these calculations is that there are two absorption maxima due to two independent steps, which do not necessarily correspond to the single steps in the reaction scheme. (Here 7 1 = CO corresponds to the condition of constant overall concentration of the electrolyte, which reduces the number of independent variables.) Under the special assumption k;2, k21 > k23, k32, the first step of the reaction is identical with the relaxation effect due to 7 3 . For CAB -+ 0 (7 = ,8 = ot) it be- comes identical with the binary one-step-equilibrium treated above. The other effect ( T ~ ) , however, corresponds to the final equilibrium between the state AB and the two states A+, B- and AB‘.Here we can find first order as well as second order processes, depending on the concentrations in the different states. The relaxation time 72 is almost independent of concentration, except in a small transition range at concentration C, for which ki2 k21. (Small concentrations C, : kj2 < k21,~2 = l/k32 ; high concentrations C, : k;2 > k21,q = I/(k23$k32)). If Cat becomes very small compared to CA and C,, we have again the binary one- step-equilibrium. Then A K ~ becomes equal to Co ’(’ - - ’) - v2)2, /3 being identical with cc in eqn. (6). In this case 7 2 is given by l/(k32 + k23k;2/k21) and AV2 by It is easily possible to extend the theory to more complicated cases.The interpretation of the results, however, then becomes much more difficult also. (iii) The general ionic interaction has to be considered in any case at higher ionic concentrations. If the equilibrium in the ionic atmosphere is disturbed, the resulting relaxation process includes displacements of all ions in the atmo- sphere. Such a process cannot be represented by a discrete-step mechanism. According to Debye and Falkenhagen the relaxation of the ionic distribution (which is represented by a distribution function F) is described by the differential equation : 2-/3 RT AK-11 -t AK1-m. 3t = 117(1 P A + 2-1 PB (div grad F - K$F> (16) or in the dimensionless form for the centrosymmetric case :30 SPECIFIC IONIC INTERACTIONS + Here the pi are frictional coefficients of the ions in water and KO = ri2; - Zi2,lj-J is the well-known " reciprocal radius of the ionic atmosphere " of the Debye- Hiickel theory.The dimensionless parameters are s = KOT and ti@ with being the so-called relaxation time of the ionic atmosphere. The definition of this quantity differs materially from the relaxation time introduced by eqn. (4), the solution of which (with ?? = const.) yields an exponential decay : [6C] = [W]O exp (- t/T). (18) In comparison with this the solution of (17) under similar conditions (disappear- ance of a small perturbation 6F) is This solution does not correspond to a " monochromatic " relaxation process, but to a continuous distribution caused by the multiplicity of similar configura- tions in the ionic atmosphere.This becomes still more evident if we write the solution in the form (t > O!) : 1 cos (sA)* fi p(A) = - - + - sin (s~*)] ; A* = (?)+, [I 2/2h* 2 (20) which can be obtained by an inverse Laplace transformation of (19). (For comparison, a discrete spectrum would be represented by a sum of expon- ential functions with different 71.) The sound absorption coeficient due to the relaxation of the ionic atmosphere has been calculated by Hall.7 The frequency term of the absorption coefficient p has the form : which may be compared to that of the single step relaxation mechanism ( ~ - / ( l + COW)). Both terms are graphically represented in fig. 1. For the ionic atmosphere the maximum occurs at we 4, the maximum value of f(d) being about 0-3.For comparison, the maximum of ~ ( w T ) occurs at WT = I, where the absolute value is 0-5. Furthermore, the absorption curve due tof(w8) is broad- ened compared to that of ~ ( w T ) . This shape is to be expected for a superposition of steps with a continuous distribution of relaxation times and can be distinguished from that of a " monochromatic " relaxation mechanism. In addition to the relaxation of the ionic atmosphere itself we have to consider also the influence of the general interaction on the specific relaxation effects. A general treatment has to start from a system of differential equations including the system of rate equations (e.g. (9)) and the equation (16) in which ki2 and kZ1 as t Somewhat different results have been obtained by Leontovich.8M. EIGEN 31 well as KO are time dependent.(The rate constants kiz and k21 are functions of the time dependent function F(qt) of the ionic distribution, as the mutual inter- action of the ions depends also on the average potential of the ionic atmosphere. The quantity KO in eqn. (16) includes the ionic concentration, which is time de- pendent due to the reactions.) Such a general treatment has to be applied if the relaxation times are all of the same order of magnitude, for instance in diffusion-controlled recombinations, the rates of which are comparable to that of any (also diffusion control1ed)rearrange- ment in the ionic atmosphere. Protolytic reactions in aqueous solution are such diffusion-controlled processes, but in most of these cases studied hitherto, the ionic concentrations are small enough to justify neglecting the general interaction and to permit an exact determination of the " chemical " relaxation time (examples : N H 3 , CH3COOH, self-dissociation of H2O).9 FIG.1.-Frequency term of the absorption per wave length. ~ ( w T ) according to a " monochromatic " relaxation effect (single step in " specific " interaction). f(w8) accord- ing to a continuous distribution of relaxation times (continuous spectrum of the " general " interactions in the ionic atmosphere). In many cases, however, the specific reactions proceed much slower than any change in the ionic atmosphere. Then one can treat the relaxation of the general and specific interaction separately. The equilibrium of the ionic atmosphere is established within a time in which no essential change of the ionic concentrations occurs. On the other hand the influence of the general on the specific interaction is due only to the equilibrium in the ionic atmosphere, and this may be described by introducing (time independent) " activity coefficients f " * into the rate equations.Under this supposition we have to consider the following modifications in the above treated example (eqn. (8)) : in eqn. (9), ki2 is given by The general form of the transformations is not changed. In calculating AK, we have to consider the influence of activity coefficients on the equilibrium concentrations * The introduction of the thermodynamic activity coefficients into rate equations is only justified in the special case of a slow rate-determining step.Also in this case the activity coefficient is not necessarily identical with the experimental value as usually tabu- Iated in literature. It is only due to the general interaction of the ions (i.e. for distances > r$ + r2, whereas experimental values usually contain also the specific interactions represented by a " degree of dissociation ".32 SPECIFIC IONIC INTERACTIONS and their dependence on pressure (concentration dependence of the partial molar volumes). Then we obtain : with k;2 according to (22). (The compressibility part of a one-step equilibrium has the same form as AK~.) A Vc is the difference of partial molar volumes at the concentration CO, con- taining concentration dependent terms (cf. eqn. (14)). The relaxation time is also in- fluenced by the activity coefficients and their concentration dependence according to ki2 (eqn.(22)). The above treatment is only justified if the three time lags (8,72,73) differ sufficiently (i.e. at least by a factor of 10, in the order : 8 < 73 < 72). The examples discussed below fulfil this condition. At present an unequivocal distinction of the different effects from relaxation spectra is only possible in these cases. (The special form of eqn. (23) has been chosen in order to illustrate the simple limiting cases as shown above ; (for evaluations, cf. ref. (6)).) APPLICATIONS In principle we can use any type of relaxation technique to study general and specific ionic effects (e.g. high electric field, temperature or pressure relaxation methods %9).The high electric field technique has been used primarily for the study of reaction kinetics,g whereas the most interesting results with respect to the problems considered in this paper have been obtained by the sound absorption technique. Some results, obtained by Tamm, Kurtze and co-workers 10 in TABLE 1 .-SOUND ABSORPTION OF ELECTROLYTES IN AQUEOUS SOLUTIONS v = 104 - 3 x io*c/sec; co = 10-3 - 10-1 M type examples 1 : 1 c' strong ") NaCl, KCl 1 : 1 (" weak") NH3 (cf. ref. (2), (9)) 2: 1 MgC12 1 : 2 Li2SO4, Na2S04, Na2C03 3 : 1 AlC13 3 : 1 La(N03)3 (cf. ref. (10)) 2 : 2 sulphates, thiosulphates, chromates 1:3 Na3P04 3 : 2 Ads04) 3 relaxation effects remarks not detectable " monochromatic " one step not detectable rise at high fre- quencies, relaxation time< 10-9 sec small effect at high frequencies broad maximum due to superposition of several effects spectra with 2 discrete maxima nearly completely " dissociated '' relaxation due to the known dis- sociation equilibrium nearly completely '' dissociated " probably specific interaction (not yet identified).(H2SO4 shows a dissociation * typical effect of one-step nearly completely " dissociated " specific interaction, several stages with similar relaxation times specific interaction (discussion below) specific interaction (inter- }at ~ % ~ ~ ~ ~ ~ c i e s pretation complicated by hydrolysis effects) * The one-step mechanism is typical for protolytic systems (NH3, CH3COOH, H2S04, H20, etc.) according to the special mechanism of proton migration (penetration through hydration she1ls)Z.M .EIGEN 33 Gottingen, are represented in fig. 2. (For a more detailed discussion, cf. Eigen, Kurtze, Tamm.lls 6 ) The characteristic features of these results are summarized in table 1. The most interesting behaviour is shown by the 2 : 2-electrolytes : our dis- cussion will be confined to this group. As has been shown elsewhere,ll both absorption maxima which generally are found for this group are due to specific interactions between the divalent anion and cation. This has been concluded from measurements of the absolute values of relaxation time and absorption FIG. 2.-Sound absorption spectra of electrolytes in aqueous solution (at 20°C) ac- cording to measurements of Tamm, Kurtze and Kaiser (relative to water). (Most curves represent average values for 10-2 to 10-1 M solutions, for which no essential concentration dependence of Ap/Co has been observed. coefficient, its concentration and temperature dependence and from a study of the influence of other electrolytes (including acids and bases).(The possibility of solvent-solvent, ion-solvent or intramolecular relaxation effects in the frequency range considered above can be excluded without uncertainty.) The simplest possible mechanism allowing an explanation of all hitherto found results may be represented by the following scheme : A++ + B-- + AB' + AB" + AB"' I I1 I11 IV The first equilibrium (1-11) corresponds to diffusion-controlled steps of interaction of the completely hydrated ions (general interaction). A relaxation effect due to this interaction should be expected at frequencies higher than lo9 c/sec (for con- centrations of about 0.1 M, at which most of the measurements have been carried out).This interaction would be present in all electrolyte solutions, especially in those which do not show any detectable absorption below 3 x lOBc/sec. B34 SPECIFIC IONIC INTERACTIONS Therefore the relaxation effects found in 2 : 2-electrolytes should be connected with further consecutive reaction steps in the above scheme (11-III, III-IV). That these discrete steps correspond to physical reality is shown by the fact that the maxima-at least that at lower frequencies-have the shape of “ monochromatic ” relaxation curves; otherwise we should expect a continuum due to a continuous probability function of the distances of separation.There exists, of course, the possibility of more complicated mechanisms in which not all relaxation steps are detectable. A theoretical treatment of the first absorption maximum (10s cjsec) assuming a one-step dissociation equilibrium has been given by Bies.12 This theory holds only for small concentrations ; it cannot explain the behaviour for concentrations above 10-2 M, where the relaxation time as well as the absolute value of Ap/Co becomes almost independent of concentration.* (A slight decrease is again ob- served at CO > 10-1 M for both maxima.) As shown in the preceding paragraph, the existence of further relaxation effects cannot be neglected in an exact theoretical treatment. The example treated above seems to describe the experimental facts in a more extended concentration range. At low concentrations (k;2/k12 < 1) we have a transition to an apparent (one-step) binary dissociation equilibrium (cf.ref. (1 l), (12)) for the mechanism represented by A K ~ (cf. eqn. (14)), whereas at higher con- centrations the influence of the first-order steps predominates. The experiments show clearly that the first maximum is due to an effect strongly influenced by the individual properties of the cation. The relaxation time varies with the radius of the cation by orders of magnitude (e.g. Be2+, Mg2+, Mn2+ with maximum frequencies of 103, 105 and 3 x 106c/sec),t whereas it is nearly independent of the anion (e.g. SO$-, S20,2-, CrOi-). The rate-determining step for 72 has an activation energy of about 6-8 kcal/mole (MgS04).The second maximum is almost independent of the radius of the cation as is to be expected for an interaction of partly hydrated ions. The quantitative interpretation of the data, however, requires a more detailed knowledge of the quantities involved in (23), such as the “real” activity co- efficients, partial molar volumes and their concentration dependence as well as the absolute value of the equilibrium constants. (Some of these quantities may be determined by other methods.) In addition, the detailed reaction mechanism of the transitions 11-111 and 111-IV seems to be of more complicated nature. There is some evidence, obtained by a study of the influence of pH, that intermediate hydrolysis stages may occur in the decomposition or formation of the hydration shells.The stationary concentration of free protons or hydroxyl ions, however, must be very small, but there may exist a complex of the form MgOH+ . . . HSOZ (A mixture of MgC12 and NaOH in solution, which separately do not show de- tectable excess absorption, has a strong absorption at 105-106 cJsec, presumably due to formation of MgOH+.) CONCLUSIONS We may summarize the results as follows. The sound absorption measure- ments demonstrate the existence of specific ionic interaction effects for 2: 2- electrolytes. These probably become effective when the ions approach to a dis- tance less than three water layers. They should be mainly of electrostatic nature, but the interaction potential cannot be described by the Coulombic term with a macroscopic dielectric constant, neither with the constant of pure water nor * Such a constancy can not be explained by the decrease of the activity coefficients with increasing concentration.If one assumes a dissociation equilibrium characterized by a degree of dissociation oc , then the “ real ” activity coefficient of the ions is given by (fexp/ cc ), where fexs is the experimental value usually tabulated in literature. (Axp/ oc ) generally shows a mmimum and rise at higher concentrations. f The rise for the absorption of &SO4 with decreasing frequencies seems to indicate a maximum at about 103 c/sec.M. EIGEN 35 with an effective one changing monotonically with decreasing distance. The specific nature of the effects is demonstrated by the existence of the discrete spectrum, in which the relaxation times differ by several orders of magnitude and strongly depend on individual properties of the cation.To estimate the total amount of electrolyte in specific interaction one may assume an overall equilibrium constant between 10-2 and lO-3mole/l (attributed to fi in eqn. (23)). More accurate values for the constants of the individual steps should be obtainable. There is an interesting correlation between the results obtained in this paper and the concentration dependence of activity coefficients (as shown for some examples in fig. 3). Electrolytes (e.g. MgClz) which do not show detectable 015 ---- 1.0 [m/es//ifer] 1.5 1/2 FIG. 3.-Mean activity coefficients of electrolytes in aqueous solution at 25" C. sound absorption in the frequency range considered above are characterized by a minimum and subsequent strong rise of the activity coefficients at higher con- centrations.On the other hand, the minimum and rise do not occur for those electrolytes which are distinguished by strong sound absorption effects (e.g. the divalent sulphates). In the statistical treatment of the general interaction effects (cf. ref. (1)) a large " distance of closest approach " is responsible for minimum and rise in the concentration dependence of activity coefficients. The activity coefficientsf* which are due only to the general interactions of the hydrated ions (large distances) would therefore generally show a minimum and rise which vary from case to case much less than is actually found.13 The disappearance of the minimum then has to be attributed to a specific interaction which may be described summarily by introducing a " degree of dissociation " a, the experimental value f being then af*.Especially for 2: 2-electrolytes the activity coefficient can be described theoretically in either of two ways, in one of which one assumes only36 SPECIFIC IONIC INTERACTIONS general interactions with a small " closest distance ", say 4 A ; the other way consists in assuming a larger distance, say 6-8 A with additional specific inter- actions.13 From the thermodynamic or statistical point of view one cannot distinguish between the two modes of description, whereas the experiments de- scribed above favour the second one. Finally we should mention some difficulties and limitations of the relaxation method.An experimental separation of the different effects is only possible if the relaxation times differ sufficiently, otherwise we obtain again a sum of all effects present, and the same difficulties arise as in static methods. On the other hand, if the relaxation times differ widely the measurements have to be extended over a wide frequency range, which involves some experimental difficulties. (The half-width of relaxation absorption curves is large compared with that dge to a resonance effect.) Furthermore, the error limits of present measurements are still relatively high and the resolution has to be improved in order to permit the detection of small effects (which may be present also in some 1 : 1- and 2 : 1- electrolytes) or to allow the recognition of small deviations of a given curvs type.Hiedemann and Spence14 showed that absorption curves due to continuous dis- tributions of relaxation times around a given value do not differ essentially, even if the distributions are described by mathematically rather different functions. As " chemical " relaxation processes appear to be rather specific this restriction should be less effective than in usual relaxation spectrometry (e.g. dipole relaxation effects, etc.). In addition to the experimental difficulties we have to consider the theo- retical ones. For spectra with more than one or two maxima the quantita- tive interpretation of data is complicated by the fact that in the spectra occur only the normal variables, which are a superposition of those of the real steps, separately still unknown in most cases (e.g. the volume change due to a single step). On the other hand, the method can give information on the existence or non- existence of specific effects which generally cannot be detected directly by static thermodynamic methods. Therefore it is superior to these methods, in so far as it gives additional information, such as do optical spectroscopic methods also.15 Apart from an improvement of the experiments as described above, a great number of electrolytes still have not been investigated in detail. The author is greatly indebted to Prof. K. Tamm and Dr. L. de Maeyer for valuable discussions and especially to Mr. G. Schwarz, who did some of the calculations. 1 Mayer, J. Chem. Physics, 1950, 18, 1926. Kirkwood and Poirier, J. Physic. Chem., 2 Eigen, (a) Furaday Soc. Discussions, 1954, 17, 194. (b) Chemische Relaxation 3 Meixner, KolloidZ., 1953, 134, 3. 5 Bauer, J. Chem. Physics, 1953, 21, 1888. 1954, 58, 591. (Steinkopf, Darmstadt, 1958). Kronig, Physik. Z., 1938, 39, 823. Eigen and Tamm, 2. Elektrochem., 1957, in press. Hall, J. Acoust. SOC. Amer., 1952, 24, 704. * Leontovich, Expt. Theor. Physic. U.S.S.R., 1938, 8,40. 9 Eigen, de Maeyer and Schoen, 2. Elektrochem., 1955,59,483, 986. 10 Tamm and Kurtze, Acustica, 1953,3,33. Tamm, Kurtze and Kaiser, Acustica, 1954,4, 380. Liebermam, Physic. Rev., 1949, 76, 1520. Wilson and Leonard, J. Acoust. SOC. Amer., 1954,26,223. Tamm, in Encyclopedia of Physics (ed. S. Flugge, Springer Verlag, 1957), in press. 11 Eigen, Kurtze and Tamm, 2. Elektrochem., 1953, 57, 103. 12 Bies, J. Chem. Physics, 1955, 23, 428. 13 Wicke and Eigen, 2. Elektrochem., 1953, 57, 319 ; J. Physic. Chem., 1954, 58, 702. 14 Hiedemann and Spence, 2. Physik., 1951, 133, 109. 15Bale, Davies, Morgans and Monk, this Discussion. Redlich and Hood, this Discussion. Young, Monograph on the Symposium of the Theoretical Division of the h e r . Electrochem. Soc. (Washington, 1957), to be published. Manes, J. Chem. Physics, 1953, 21, 1791.

ISSN:0366-9033    

DOI:10.1039/DF9572400025

出版商:RSC    

年代:1957

数据来源: RSC

摘要:

THERMAL EFFECTS OF THE INTERACTIONS BETWEEN IONS OF LIKE CHARGE BY T. F. YOUNG, Y . C. Wu AND A. A. KRAWETZ * The George Herbert Jones Chemical Laboratory, University of Chicago, Chicago, Illinois Received 5th July, 1957 The heat absorbed during the mixing of binary solutions of pairs of univalent electro- lytes with common ions has been measured. The heat effects are somewhat smaller than those observed when electrolytes without common ions are mixed but are by no means negligible. When both of two cations are small, or when both are large, heat is generally absorbed ; when one is large and the other small heat is usually evolved. The molal heats of mixing when one molal solutions are mixed are approximately quadratic functions of the solute mole fraction, x3. The relative partial molal enthalpies are therefore approximately linear functions of x3.Studies of the thermodynamic properties of mixtures of strong electrolytes have been summarized in the treatise written by Harned and Owen.1 The be- haviour of relatively concentrated solutions led to the formulation of the Harned rule,2 which may be restated in terms of the partial molal free energy thus : the partial molal excess free energy of an electrolyte is nearly a linear function of the concentration of the second electrolyte when the total concentration is constant. The partial molal volume of an electrolyte in ternary aqueous solutions con- taining two electrolytes of the 1 : 1-charge type frequently varies in a similar manner. Wirth 3 determined the partial molal volumes of hydrochloric acid and sodium chloride in a series of aqueous solutions in which the sum of the two concentra- tions was constant.He found that the partial molal volume of each of the two solute constituents is a linear function of the concentration of one of them. Wirth and Collier4 published precise density data showing that the same rule is obeyed by aqueous solutions of perchloric acid and sodium perchlorate. It was shown by Young and Smith 5 that the partial molal volumes of perchloric acid and sodium perchlorate can be calculated from the measured densities of the respective binary solutions if the following empirical principle is employed- when two solutions of the same molality are mixed the volume of the ternary solution produced is equal to the sum of the volumes of the binary solutions.Aqueous solutions of perchloric acid and sodium perchlorate obey this principle almost perfectly at 25" C . Solutions of hydrochloric acid and sodium chloride do not obey the principle quite so closely. There is a significant increase in volume when equi-molal solutions of these electrolytes are mixed. To discuss the behaviour of ternary solutions the following symbols will be needed : n2 will represent the number of moles of the first solute, i.e., the number of moles of the second constituent of the solution, and n3 will denote the number of moles of the other solute. The symbols, x2 (= n2/(n2 + n3)) and x3 will repre- sent the respective " solute mole fractions ". It will be convenient normally to write equations for the mixing of equi-molal solutions so as to involve one mole of solute (n2 + n3 = 1).The increase in volume which occurs during each such * present address : Wright Air Development Center, Wright-Patterson Air Force Base, Dayton, Ohio. 3738 THERMAL EFFECTS mixing process will be denoted by the symbol A,K The symbol /3 is defined by the equation : (1) It was shown that /3 for hydrochloric acid and sodium chloride mixtures is constant for each value of rnz , the molality of the ternary solution resulting from a mixing process. In other words, A,V is a quadratic function of x3. It was also found that /3 is proportional to rnz . Thcse two conditions are sufficient to insure that the partial molal volume will be a linear function of x3. The similar variation of the partial molal volumes of the two solutes in mixtures of perchloric acid and sodium perchlorate is a particular example of the consequences of these two conditions.The coefficient /3 of that pair of electrolytes is zero or negligibly small throughout the composition range investigated. Hence both of the " sufficient " conditions exist. A, V = / 3 ( ~ 2 ~ 3 ) = / 3 ( ~ 3 - ~ 3 2 ) . 500 - - L i C l 400 0 -100 I \ I \ I \ 4. \ \ \ -200 - I I I I 0.00 0.25 0.50 0 - 7 5 I . Solute mole froction, X, o f N a C I 0 FIG. 1.-The relative partial molal enthalpies of LiCl and NaCl in ternary aqueous solutions of the two electrolytes. The dashed lines were calculated from the heats of dilution of the two electrolytes in simple binary solutions without the utilization of the heat of mixing.The solid lines represent calculations based upon all of the information available. The differences, due to the heats of mixing, are relatively large for this pair of electrolytes. The total molality rn is unity and the temperature is 25" C. The heat absorbed in an isothermal mixing is analogous to the volume increase which occurs in the same mixing operation. When no heat of mixing is observed the calculation of the relative partial molal enthalpies requires no data other than the heats of dilution of the respective binary solutions. When an appreciable heat effect is observed, however, a complete calculation requires information concerning both heats of dilution and heats of mixing. The relative partial molal heat content of lithium chloride in a mixture con- taining both lithium chloride and sodium chloride is represented by the upperT .F . YOUNG, Y . C. W U AND A . A . KRAWETZ 39 solid line of fig. 1. The other solid line represents r, the relative partial molal heat content of sodium chloride, in the same series of solutions. Note that of lithium chloride decreases as its solute mole fraction decreases whereas of sodium chloride increases as the solute mole fraction of that salt decreases. The limiting values approached as the respective solute mole fractions approach zero are approximately equal.6 The dashed lines of fig. 1 represent approximate calculations of the relative partial molal enthalpies. They differ from the exact calculations in that the heats of mixing were ignored, i.e., treated as if they were zero.The approximate values show the same general trends as the exact values. The success of the approximate calculation is especially impressive for sodium chloride because of the large variation of from about - 200 cal mole-1 to + 200 cal mole-*. The differences between the exact calculations and the approximate ones are larger than they would be for a typical series of ternary solutions because the heats of mixing for this pair of cations is relatively large for two electrolytes with a common anion. The similarity of fig. 1 to the plots of partial molal volumes and of logarithms of activity coefficients suggests that much about the interactions between ions can be learned from a study of the heats of mixing of salts. The number of electrolytes whose activity coefficients can be determined from electromotive force methods is unfortunately limited by the small number of satisfactory reversible electrodes available.Nearly all pairs of electrolytes, however, can be studied in the calori- meter. The work described here is a study of the heats of mixing of pairs of solutions of univalent electrolytes with common anions and of a few pairs with common cations. THERMODYNAMICS OF THE PLAN OF MEASUREMENT The objective of the calorimetric work was the determination of the heat absorbed when a solution of solute MX is mixed with a solution of NX of the same molality, m'. The heat absorbed will be denoted by A,H. The mixing process is then denoted by eqn. (2) : [nzMX(m') + *H20] + [n3NX(m') + *H20] -+ [n2MX -l- n3NX + *HzO].(2) The brackets [ ] denote that the substances represented by the enclosed symbols are in a single solution. The asterisk denotes simply that the solution contains water in the amount necessary to conform to the molality, m'. Since the ternary solution is to contain one mole of solute, n2 + n3 is one mole. The calorimeter was equipped with a " pipette " containing a volume approx- imately one-ninth of the total volume of liquid in the calorimeter. It was necess- ary to formulate a stepwise programme for the mixing of successive quantities of one of the solutions, e.g. the solution of NX, with the ternary solution of MX and NX produced in an earlier mixing experiment. A solution containing 0-45 to 0-5 mole of NX in a total of one mole of MX and NX was produced in about six steps.In the first step the pipette was filled with a solution of NX which was then mixed with eight or nine times its volume of a solution of MX. This step is repre- sented by the thermodynamic equation : h, a MX + *H2Ol + ((1 - n2, a) NX + *HDI -+ [n2, a MX 4- (1 - nZ,,)NX + *H20] ; AH,. ( 3 4 Here the subscript 2 refers to the solute MX and the subscript a to the first step of the mixing programme. Hence n2,a denotes the number of moles of MX (per mole of solute) in the first step. The heat absorbed per mole of solute is AHa. In the second step a portion of the ternary solution produced in the first was discarded. The remainder was then mixed with a second quantity of the solution40 THERMAL EFFECTS of NX. Eqn. (3b) represents the process “ normalized ” to one mole of solute in the second ternary solution : Ju [nz, MX + (1 - n2, NX + *H20] + [(l - Ju) NX + *€I201 -+ [Jun2, MX + (1 - JUn2,J NX + *H201; AHb.(3b) The sum o f AH, and J ~ A H ~ is A,H for the production of the ternary soIution in which the solute mole fraction of NX (regarded as the third constituent of the solution) is produced from the binary solutions. That sum is A,H of eqn. (2) for x3 = 1 - Jun2, In the third step a fraction Jb of the ternary solution produced in the second step was mixed with a third quantity of the solution of NX, and the heat absorbed, AH, was determined. The mole fraction of NX in the third ternary solution is x3 = 1 - JuJbn2,u and A,H for the production of that solution from the original binary solutions is After a fourth mixing step A,H = AH, + JbaHb 3- JJbAH,.A,H = AHd f JcAHc $- JbJ,AHb + JuJbJcAHu, (4) ( 5 ) and X3 = 1 - JuJbJcn2, a . For the complete determination of a curve a fifth and a sixth step were necessary; x 3 had then risen nearly to 0.5. Another series of mixings was then made for the other half of the curve, the range between x 3 = 1.00, x3 = 0-5. To make possible the investigation of more systems only a few points, instead of ten or twelve, were determined for some of the curves. These few points indicate the general shapes of the curve with sufficient accuracy to show how the heat effects depend upon composition and the nature of the ionic species. EXPERIMENTAL The design of the calorimeter was essentially that described by Young and Smith,s except that stirring was done by propellers of a new and more efficient type, to be de- scribed elsewhere.The water was distilled twice, the second time from an alkaline permanganate solution in a block-tin still, heated by steam. Solutions were prepared from more concentrated stock solutions which had been prepared from known weights of water and salts of C.P. or “analytical” grade. Each stock solution was analyzed for chloride gravimetrically, and then diluted with a weighed quantity of water to the desired molality . RESULTS COMMON ANIONS Observed values of AmH are represented by two series of circles in fig. 2. The filled circles represent measurements made by M. Smith, the open circles new measurements made in the course of this research. Two of the open circles are omitted because they are essentially identical with the two extreme points on the curve for LiCl + NaCl.The curves shown were calculated from empirical equations of the type : AmH = (U - 6 ~ 3 ) ~ 2 ~ 3 . (6) It is not certain that values of b are significant in some cases. The curves may be sym- metrical with respect to x3 = 0 5 . More work will have to be done before that question can be answered. The lines drawn are the best representations of the measurements which have been made at this time, however. More important now than the deviations from the quadratic relationship of eqn. (2) is the fact that all of the curves do conform reasonably well to the quadratic relationship. COMMON CATIONS A few measurements of A,& have been made for pairs of electrolytes with the same positive ion.The results are in table 1.T. F. YOUNG, Y . C . WU AND A . A . KRAWETZ 41 TABLE HEATS OF MIXING, 4,,,€€ (CAL MOLE-1) FOR PAIRS OF ELECTROLYTES WITH COMMON CATIONS. 4,H HAS BEEN ESTIMATED FOR x3 = 0.5 FROM FIRST-STEP MIXINGS ONLY. THE TEMPERATURE IS 25" C AND THE TOTAL MOLALITY IS ONE MOLE/KG OF WATER electrolytes AmH KC1 + KBr 0.80 NaCl + NaBr 0.79 electrolytes AmH LiCl + LiBr 0.81 KC1 + KNO3 0.34 HCI t NaC I 00 0.2 0.4 0.6 0 8 1.0 Solute m o l t fraclion X3 FIG. 2.-The heat absorbed, AmH of eqn. (2) in cal/moIe of solute, against x3, the solute mole fraction of the second electrolyte of the pair that is shown adjacent to each curve. Observations made by two independent observers are distinguished by 0 and 0. / \ \ / \q c I t CIC I/ -50.I I I I DISCUSSION From the point of view of thermodynamics the most significant generalization to be noted in these results is that A,,H for each pair of electrolytes is approxi- mately a quadratic function of x. This fact is more readily observed in fig. 3 in which d(A,H)/dx, is plotted. A perfect quadratic should be represented in fig. 3 by a straight line passing through the point: A,H = 0 and x = 0.5. Each line is approximately straight and does pass only a little below or above that point. The quadratic relationship coupled with the observation that /3 is approximately proportional to m for values of m less than unity is the explanation for the essentially straight solid lines of fig. 1 which were calculated from the measure- ments reported here.42 THERMAL EFFECTS The heat effect for the mixing of sodium chloride with another solute is largest for hydrochloric acid, smaller for lithium chloride and is negative for salts of the larger ions (larger crystal radii).For salts other than sodium chloride the trends are not so regular. Some uniformity is apparent, however. If salts of two of the three smaller cations are mixed heat is absorbed. Likewise, if two salts of the three larger cations are mixed heat is absorbed. If, however, the chloride H + 0 . 2 5 0 . 7 5 0 . 2 5 0.75 0 . 2 5 0 - 7 5 0 - 2 5 0 . 7 5 0 . 2 5 0.75 Solute moic froction, X3 FIG. 3 . T h e derivative d(AmH)/dx3 against x3. of one of the smaller cations is mixed with the chloride of one of the larger cations heat is evolved. Moreover, the larger heat effects are observed only when at least one of the cations is small. That the division of the cations into these two groups may be significant is indicated by the viscosity B coefficients7-9 and the partial molal volumes 10.11 of the ions. We are indebted to Dr. M. Smith for the measurement of several values of A,,,H which have not been published elsewhere. This work was supported in part by the Office of Naval Research. 1 Harned and Owen, The Physical Chemistry of Electrolytic Solutions (Reinhold, 2 Robinson and Stokes, Electrolyte Solutions (Butterworths, London, 1959, 427. 3 Wirth, J. Amer. Chem. SOC., 1940, 62, 1 128. 4 Wirth and Collier, Jr., J. Amer. Chem. SOC., 1950, 72, 5292. 5 Young and Smith, J. Physic. Chem., 1954,58, 716. 6 Bronsted, J. Amer. Chem. Soc., 1923,45,2898. 7 Laurence and Wolfenden, J. Chem. Soc., 1934, 1144. 8 Jones and StaufTer, J . Arnev. Chem. Soc., 1940, 62, 336. 9 Gurney, Ionic Processes in Solution (McGraw-Hill, New York, 1953), 163. 10Bernal and Fowler, J. Chem. Physics, 1933, 1, 515. 11 Stokes and Robinson, Trans. Furaduy SOC., 1957, 53, 301. New York, 1943, 1950).

ISSN:0366-9033    

DOI:10.1039/DF9572400037

出版商:RSC    

年代:1957

数据来源: RSC

摘要:

THE SPECIFIC INTERACTION OF TWO IONS IN A STRONG AQUEOUS ELECTROLYTE BY S. LEVINE AND H. E. WRIGLEY Dept. of Mathematics, University of Manchester Received 1Ofh July, 1957 A method of calculating the interaction of two univalent ions in water at small sepa- rations is developed. The region of water surrounding a typical ion is divided into two parts, a co-ordination or hydration shell and the rest of the water. The water beyond the co-ordination shells is treated as a continuous dielectric medium. Assuming a given configuration of molecules in contact with the ions, the induced dipole moment in each hydration shell is obtained by introducing the reaction field. The interaction energy is determined by first evaluating the corresponding energy of the two ions with their co- ordination shells in V ~ C U O and then adding on the work of bringing up the dielectric.The case of non-overlapping co-ordination shells is considered and also that of smaller sepa- rations at which the overlapping is not yet sufficient to necessitate the removal of any molecule from these shells. Deviations from the usual Coulomb energy with the macroscopic value of the dielectric constant are obtained at small ionic separations. The correction to this classical energy takes the form c4 c6 c7 @+@+@+. . .9 where R is the separation and the C’s are constants. Numerical computations have been carried out for potassium fluoride, on the assumption that the K and F ions have the same radius and polarizability. The correction is found to be positive for both like- charged and oppositely-charged ion-pairs and is about 8 % at a separation of 6-8 A. The free energy of interaction of two ions at a distance apart in an aqueous medium may be written as (1) where e and e’ are their charges, E: is the dielectric constant of the pure water and W(R) is a correction term which becomes significant at small separations.I t is the purpose of this paper to develop a method of determining W(R). The region of water surrounding an ion is divided into two parts, a “ co-ordination ” or hydration shell and the rest of the water, which is treated as a continuous dielectric medium. (For brevity, an ion and its co-ordination shell will be called an ion complex.) This approach is preferable to one in which the whole of the water is regarded as a continuum with variable dielectric properties, since the validity of macroscopic theory within the ion complex is doubtful.Since an accurate determination of W(R) presents considerable difficulties, particularly at very small X, in the present paper we shall choose a simplified model which, however, illus- trates the main features of our general approach. In particular, dielectric satura- tion and electrostriction are neglected in the medium surrounding the ion complexes and the values of R are such that either the hydration shells do not overlap or overlapping is possible without the removal of a co-ordination molecule from its shell. V(R) = ee’/& + W(R), POTENTIAL DISTRIBUTION DUE TO ION COMPLEXES For a uni-univalent electrolyte there are three cases: case I, two positive ions, case 11, two oppositely charged ions, case 111, two negative ions.It is 4344 SPECIFIC INTERACTION OF TWO IONS sufficient to examine cases I and IT. For simplicity, it is assumed that in all cases the two ions have the same radius and polarizability and that the configuration of water molecules in the two hydration shells can differ only in the sense in which the dipoles are pointing. The water dipoles of an ion complex are assumed to point either away from a positive ion or towards a negative ion when R is infinite ; this simplifies the mathematics. It is quite possible to extend the cal- culations to more precise models and orientations of the molecules in the co- ordination shells,l-4 but this will not be attempted here. Let 01 be the centre of the ion carry- ing the (positive) charge e and 0 2 that of the other ion with charge e in case I and - e in case IT, where e is the electronic charge.The distance 0 1 0 2 = R and P is any point in the vicinity such that 01P = r l , 02P = r2 and 81 and 02 are the angles between 0 1 0 2 and 01P and 02P respectively (fig. 1). If P lies in the continuous medium the electrostatic potential $0 at P satisfies Laplace's equation and may be expanded in spherical harmonics, namely, P + e ' _ c FIG. 1. retaining only three terms. Throughout this paper, we use the convention that if alternative plus and minus signs appear in an equation, the upper sign refers to case I and the lower sign to case 11. In case I the normal derivative of $0 at the median plane between the two ions is zero and in case 11, $0 = 0 on the plane.For a specified configuration of the water molecules in the hydration shells, $0 will depend on the azimuthal angle about the line 0102. In a more elaborate treatment the energy of interaction would be determined for each configuration of the co-ordination molecules and then the appropriate average over all such con- figurations can be interpreted as W(R). However, it is expected that the depend- ence on the azimuthal angle will be small and so as a first approximation this will be neglected. By making use of the expansion, valid for I rl/R I < 1 5 (Po (COS 62) = 1, P1 (cos 02) = cos &), the following convenient form for 40 at the boundary of the ion complex 01, which is assumed to be a sphere of radius a, is obtained :S .LEVINE AND H. E. WRIGLEY 45 The potential #i inside the co-ordination shell of ion 01, for values of r1 greater than the distance of any charge inside the shell from the origin 01 can be similarly expressed as +i = Bo + - + B1r1 + - cos 61 + B2r12 + - P2 (cos 8,) + . . . . (5) rl ( :2) :3) We need to determine the seven constants in the formulae (4) and (5). EVALUATION OF p AND 4 The constants p and q represent the dipole and quadrupole moments of the charge distribution in the ion complex 01. Suppose that there are four water molecules in each hydration shell forming a tetrahedral configuration with the dipoles pointing towards a negative ion and away from a positive ion. Then at R = CO both dipole and quadrupole moments vanish, i.e.p = q = 0. How- ever, at small R the reaction field6 that one ion complex and the surrounding dielectric medium produce inside the other ion complex will induce dipole and quadrupole moments. Inside the ion complex 0 1 , the potential from which this reaction field is derived is (6) +R = ~~r~ cos el + ~ ~ r ~ 2 ~ ~ (cos 8,) + . . . . If co-ordinate axes x, y, z with centre at 0 1 and the z axis along 0 1 0 2 are intro- duced, then (6) reads +R = BIZ + 92(2z2 - x2 - y2). (7) The dipole moment p is determined as follows. Let the centre 01 of a given water dipole in the ion complex have Cartesian co-ordinates x, y, x and spherical polar co-ordinates r, + rw, 81, $, where ri and rw are the radii of the ion and of a water molecule respectively. Then x = (Is + I.,) sin 81 cos 4, y = (ri + rw) sin 61 sin 4, z = (ri+ rw) cos 61.(8) At R = co the dipole lies on the radial line pointing away from and thus 81 and $ define its direction, where 81 is the angle between the dipole and the line 0 1 0 2 . When the dipole is subjected to the reaction field at finite R, the angle 4 is unaltered since 3$~/34 = 0 but 81 becomes 81 -- /3 say, where it may be assumed that p is small. Thus if pw is the dipole vector its components are (9) where pw is the dipole moment of a water molecule. As a first approximation the interaction between adjacent water molecules inside a hydration shell may be neglected compared with that between the ion and a hydration molecule. Then, making use of (7)-(9), the energy associated with the field acting on the given water dipole is readily calculated to be pw sin (81 - cos $, pw sin (8, - p) sin 4, pw cos (6, - p>, + B2~.w(ri + rw)(2 cos 61 cos (01 - /3) - sin 61 sin (81 - /$I, (10) where the first term on the right represents the interaction of the dipole with the ion.The angle p is now determined by requiring that w be a minimum. The condition dw/dp = 0 reads46 SPECIFIC INTERACTION OF TWO IONS The dipole moment in the direction 0 1 0 2 produced by this change in orientation of the water dipole is given by p,(cos (81-/3)-cos el> = pw sin 81 (tanp + . . .) + pw cos 191(--+ tan2p + . . .). (12) The expression (11) for tan P is substituted into (12) and the resulting form for (12) is developed as a power series in B1 and B2. The sum of (12) over the four molecules of the ion complex is obtained as follows.We may suppose that the four water molecules of 0 1 are situated at alternate corners of a cube (fig. 2). FIG. 2. It is now convenient to introduce co-ordinate axes XI, y’, z’ which are attached to the ion complex such that the point 01 is the origin and the cube edges are parallel to the axes. Further let I, rn, n be the direction cosines of the line 0 1 0 2 relative to these axes. If ,E denotes summation over the four molecules then it is readily seen that Zsin281 = 4 - Z c o s 2 8 1 = 8/3 since z cos2 el = 4t(z + rn + n)2 + (I - rn - n)2 + ( - I - m + n)2 + (- I + nz - 4 2 1 = 4/3. Similarly, C sin 81 sin 281 = 2.Z (cos 81 - cos3 81) = 0. It follows that those terms in (12) which are linear in B1 and B2 contribute to the induced dipole moment in the direction 0 1 0 2 an amount Other such linear terms are due to the polarizabilities 01, and ai of a water molecule and ion respectively.The z-component of the induced dipole of the water molecule at x, y , z is Summing over the four molecules and noting that Zcos 81 = 0, we obtain the contribution - 4cc,B1. The corresponding induced dipole moment of the ion is - ~$1. Adding up the three terms, the induced dipole moment due to the homogeneous part of the reaction field is - (010 + 4a, + ai)B1 =- aB1 where 01 may be described as the polarizability of the ion complex. We may expect B1 > 0 in case I and B1 < 0 in case 11. It is noteworthy that cc is independent of the orientation of the tetrahedral configuration of the ion complex.The sum over the four molecules of the higher terms in (12) depend in general on the orientation of the tetrahedral arrangement, but this will be ignored. A convenient orientation is that for which the cube in fig. 2, has its edges parallel - *BlpwC(ri + rw121eI =- EOBl, say. - cc,(B1 + 2B2z) =- ccc,(B1 + 2B2(ri + rw) cos e,>.S . LEVINE AND H . E. WRIGLEY 47 to the x , y , z axes. (This means that in fig. 2 the cube is rotated until the z' axis coincides with the line 0 1 0 2 . ) Then it may be verified that the contribution to the induced dipole moment from the quadratic terms in B1 and B2 on the right- hand side of (12) is simply - ' ~ ' ~ w [ ( ~ i + rw)s/e2]B1B2 =- aB1B2, say. Thus the induced dipole moment is p =- CiBl - 0B1B2 + . .., (1 3) neglecting higher terms. The determination of q is considerably more difficult and we shall only estimate its value. As with (T, the dependence of q on the orientation of the tetrahedral arrangement will be ignorcd, and it is sufficient to retain the first term on the right of (6). Suppose that the orientation of the ion complex is such that the axes x, y, z, which we denote here by X I , x2, x3, are the principal axes associated with the tensor ec say, which defines the quadrupole moment and, in addition, let ell = 132. Then making use of the usual formula for the potential due to a quadrupole,sS 7 it may be shown that q = e33 - ell. (14) For simplicity we follow Bernal and Fowler 1 and assume that each water molecule consists of two charges & e at a distance d = p,/e apart, with the negative charge - c at the centre of the molecule.A particular orientation of the ion complex 0 1 which satisfies the required conditions on eQ is that already introduced above to evaluate G. Then the contribution to q from the rotation of the water dipoles (assumed to be rigid), when subjected to the field ] B1 I is where ,f3 is obtained from (1 1) by putting B2 = 0 and sin 81 = 4 2 / 3 . This result still applies if the cube is rotated about 0 1 0 2 , since the potential distribution re- mained independent of the azimuthal angle +. There will be a contribution to q from the induced charge distributions in the water molecules and ion. To determine this the quadrupole moments produced by a homogeneous field are required for a water molecule and an ion. Since these do not seem to be known we shall make a rough estimate of this polariza- bility term with the aid of the following simple model.It is assumed that inside a water molecule of the ion complex 0 1 , a negative charge - r],e, where 7, is of order one, is displaced a distance s from its centre in the direction 0 1 0 2 , leaving behind a charge Twe. Then s > 0 in case I and s < 0 in case 11. The contribution to e33 is i 2~wes(ri + rw) cos w - Twes2, where o is the angle between 0 1 0 2 and the radial line from the point 01 to the centre of the molecule. Since Zr cos w = 0, summation over the four molecules yields the contribution to 4. The corresponding contribution from the ion is written as - 4ui2B12/qe. The other components of the tensor eij are unaffected by this polarizability effect.Thus, adding together the three terms above 4 = X W , where The second term on the right of (16) is independent of the orientation of the tetra- hedral configuration provided the water molecules are assumed to be isotropic. For simplicity, we shall choose yw = ~i = 1. Since both terms in (16) are probably48 SPECIFIC INTERACTION OF TWO IONS rough estimates and are of the same order of magnitude, the value of x may be in considerable error. Furthermore the form (2) for #O implies that in case I1 x is replaced by - x for a negative ion. Fortunately in the present calculations, x is found to be so small that it may be neglected as a first approximation. APPLICATION OF BOUNDARY CONDITIONS The five remaining constants in the expressions for #o and #i are obtained by applying the boundary conditions at ~1 = a, namely, On equating the coefficients of cos 81 and P2 (cos 81) in the usual manner we have the four relations : (21) The remaining relation which determines Bo is not required.It is convenient to solve these equations by a method of successive approximations as follows. Eqn. (18) and (19), yield A1 and B1 in terms of A2, B2 and B1B2, namely, ~ X B I 2ea 6Ala 3A2 12A2a 2B2a--=~ &-&--- a4 [ cR3 R4 a4 *TI a where, introducing t = a/R and u = a/a3, Good approximations sides of (22) and (23) c = 1- u - (1 + 2U)/€, D = 1 - u + (1/2€)(1 -t 2u) ct3, (26) to A1 and B1 are given by the first term on the right-hand respectively. Substituting these values for A1 and B1 into (20) and (21), we can solve for A2 and B2 and so obtain (27) (28) A2 =& 2 e 3 [ ( 1 -;)(I - 3 5 3 ) ”s 45 z2] X’t , 3 EF where x’ = eX/a6, a dimensionless quantity, and G = 1 - - 1 F 2t(2 + 4).andS. LEVINE AND H. E. WRIGLEY 49 These values for B1, A2 and B2 can then be inserted into the right-hand sides of (22) and (23) and this yields better approximations for A1 and B1. Clearly this iteration process can be repeated to yield any degree of accuracy but this was not found necessary. The terms in the square brackets in (27) and (28) which are proportional to X I were found to be of the order of 10-6 and 10-5 respectively and therefore have been neglected. EVALUATION OF INTERACTION ENERGY The energy V(R) may be obtained by carrying out the following steps.(i) It is imagined that a single ion complex with its water molecules in the configuration corresponding to intinite R is situated in vacuu. The internal energy of interaction of this complex is not required. The dielectric medium is now brought up from infinity and surrounds the complex, under the condition that the charge distribution inside the complex is held fixed. Then the work of introducing the dielectric is 6 9 7 where V is the volume of the dielectric, Eo is the electric field in the vacuum before it is filled by the dielectric and E is the corresponding field in the dielectric medium after this operation. Since the dipole and quadrupole moments of the (undis- torted) charge distribution in the ion complex are zero and we are ignoring the higher multipoles, Eo .E = e 2 / ~ 4 , where r is the radial distance from the centre of the ion complex. Thus (30) reads The corresponding energy of the two ion complexes at separation R is now evaluated in three stages as follows. (ii) Consider the two ion complexes at infinite separation in vacuo with their charge distributions in the state corresponding to this position. This distribution must now be changed to that actually prevailing at the given finite separation R. The energy expended in rotating a water molecule of the complex 01, through an (ri + rw)2 Expanding the expression (1 1) for substituting into (32), the sum of (32) -- - 'we [+ tan2 p +. . .I. (ri + rwI2 tan p into a power series in B1 and B2 and then (32) over the four molecules of one complex is + e (33) p w h f r w ) 4 B 2 2 t....Only the coefficient NO is independent of the orientation of the ion complex and we have again chosen the particular orientation used previously. The work required to polarize the water molecules and ions is next found. If X is the electric field on a water molecule in complex 0 1 , due to its ion, the work of polarizing this water molecule is &@&(x - V#R) (x - V#R) - *awx2, where X is the magnitude of the vector X. If x , y , z denotes the position of the centre of the molecule relative to 0 1 as centre, then X has components and the components of V { ! R can be determined from (7). Summing over the four molecules and adding on the corresponding work of polarizing the ion itself, we obtain (e/(ri + rwI3>(x9 Y , 23 W," = 2aw{B12 + 2B22(ri + ~ w ) 2 > + 4CtiB12.(34)50 SPECIFIC INTERACTION OF TWO IONS The work of displacing the charge inside the two ion complexes is then 2w1 = 2(W1' + W,") = aB12 + yB22 + . . ., where y = 8cc,(ri + r d 2 + (2pw/c)(ri + r d 4 . It is observed that the leading term in this expression is the work of polarizing the ion complexes in a homogeneous field of intensity I B1 I. The resulting charge distribution is held fixed for the remaining two operations. (iii) The two ion complexes are moved in vacuu from infinite distance apart to separation R and the work required is evaluated. This is (35) e2 2ep 2p2 2eq R R2 R3 2 W 2 = & - & - & - & R j + . . . . (iv) The dielectric is again introduced to fill the vacuum surrounding the two complexes, stationed at distance R apart.The expression (30) yields the work of bringing up the dielectric. Here E =- V$O and if YO is the potential function f + . ., (37) then Eo =- VYo. Since both #O and Yo satisfy Laplace's equation, this work may be expressed as applying Green's theorem. Here S is the surface bounding the volume V of the dielectric and 3/3n denotes differentiation along the normal to S drawn outwards from V. It is convenient to replace V by the infinite half-volume bounded by the median plane and the sphere r1 = a and take twice the value obtained for the energy (38). On the median plane 3$0/3n = 3Yo/3n = 0 in case 1: and $0 = YO = 0 in case I1 and so there is no contribution. The integral over the surface at infinity (e.g.an infinite hemisphere) also vanishes. Either surface integral in (38) may be used. To calculate the first integral over the sphere rl = a we substitute the expansion (4) for #*, evaluated at rl = a and the corresponding expansion Then, neglecting the terms in q since these are very small, 2 +'j(*,R2+;;z ea A1 f- 2Ala *-) 2 R3 R4 ea2 3Ala2 R4 + ~ d ~ ~ ~ ) ( + ~ = t ~ ) + . A2 . .I. (40) R5 +3 f-f- '( E R ~S . LEVINE AND H . E. WRIGLEY 51 After some algebraic manipulation, the energy of interaction can finally be expressed as w?) = 2Wl + w2 + w3 - WO) f 1 + b3t3 -1 b5t5 3- b6t6 -I- . . . , (41) ER 1 where b3, 05 and b6 are slowly varying functions of t = a/R, given by 3F b6 ='F -[ 1 (1 -+) ( 1 + 5u - g) - 2 3 3 0 C 9 whereH=- 1 -- 4 1 +- f- 0 "( r) - ( iE) 2 0 6 ' Here the dimensionless quantities y' = y/a5 and (T' = oela6 have been introduced, We have ignored higher powers of t since these would also require a consideration of higher terms in the expansions (2) and (5).The values of b3, b5 and b6 are shown in the table for a KF solution. The K and F ions are assumed to have the same radius ri = 1.33 A and the same polar- izability tci = 0.93 X 10-24 cm3 (the mean of their actual polarizabilities). Choos- ing Y, = 1-38& 01, = 1.68 x 1 0 - 2 4 ~ ~ 3 and p = 1*84D, it is found that cc = 15.3 x 10-24 cm3. An upper estimate of a is given by a = ri + 2r, = 4-09 A, when u = a/a3 = 0.223. If a tetrahedral structure is assumed for the water surrounding an ion, the radius of the sphere touching the nearest molecules which are not in contact with the ion is 3.09 A and this gives a lower estimate of a when u = 0.518. The distance R = 6.18 A is approximately equal to the closest distance of approach of the two ions without causing the disturbance of the tetra- hedral structure inside each shell.The effective dielectric constant of the ion complex is readily seen to be 2 = 1 + 474(4na3/3) = 1 + 3u. Thus 7 =1-66 for the upper estimate and 2-56 for the lower estimate of a. The fist value of C seems too small since it is expected that when dielectric saturation of the co- ordination molecules is complete 2 = d, where n is the refractive index of the ion complex. Of course, our value of a can only be approximate since a crude model has been chosen for the co-ordination shell. We have computed b3, b5 and b6 for the two values of u and for t = 3 (contact of the ion complexes) 6 and +. It is seen that the correction to the Coulomb energy is positive in both cases I and 11, i.e.the correction behaves as an additional repulsion. It is instructive to consider the case where u = 0 when we have two point charges at the centres of spherical cavities. The corresponding values of b3, b5 and bg are given in the table and again there is repulsion. Now there is a well-known theorem in electrostatics that if the dielectric constant of the medium in an electric field is reduced but the source of the field is unchanged then the energy is increased. Hence the removal of the water molecules from the co- ordination shells to produce the spherical cavities must lead to an increase in energy and this may be regarded as the origin of the positive correction. The coefficient b3 becomes negative for u > 1 (when E' > 4) and this shows that in order to obtain a negative correction the value of the polarizability a must be appreciably greater than that estimated here.TABLE 1 ul h) case I I1 0 (a’ = y’ = 0) I 0.223 I1 (a = 4.09A) I I1 (a = 3.09A) 0,518 65 b6 b3t3 4- b5t5 f bgt6 b3 5 a 3 t t 3 t 0.583 0.527 0.501 0.744 0.673 0,656 -0.373 - 0.345 - 0.332 0.090 0.032 0.008 a419 *458 ,481 -582 -635 -650 -29 1 -311 -322 -075 so30 -008 -615 -534 -497 ~750 *677 660 -a827 - ‘779 - -756 a087 *U31 so08 0350 a438 -469 ,585 -639 ,655 *680 -729 .73& -073 -030 -008 -697 -551 ‘487 a775 a697 -679 - 2.007 - 1.915 - 1.870 *080 -029 -008 -288 -382 -437 a600 ~656 *673 1-713 1.791 1.833 -081 so30 so08 When a = 3.09& x’ = 1.31 x 10-3, U’ = 0-385, 7’ = 0.515. When a = 4-09 A, x’ = 2.44 x 10-4, U’ = 0.072, 7’ = 0.127. 1 Bernal and Fowler, J . Chem. Physics, 1933, 1, 515. 2 Eley and Evans, Trans. Furuduy SOC., 1938, 34, 1093 ; Everett and Coulson, Trans. Furuduy SOC., 1940,36,633 ; Verwey, Rec. trav. chim., 1941,60,887 ; 1942, 61, 127. 3Lennard-Jones and Pople, Proc. Roy. SOC. A, 1951, 205, 155. Pople, Proc. Roy. SOC. A, 1951, 205, 163. Duncan and Pople, Trans. Furuduy SOC., 1953, 49, 217. 4 Campbell, J. Chern. Physics, 1952, 20, 141 1. 5 Hobson, Spherical and Ellipsoidul Harmonics (Cambridge University Press, 193 l), 6 Boettcher, Theory of Electric PoZurization (Elsevier, Amsterdam, 1952). 7 Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941). p. 140.

ISSN:0366-9033    

DOI:10.1039/DF9572400043

出版商:RSC    

年代:1957

数据来源: RSC

摘要:

ION DISTRIBUTION IN DILUTE AQUEOUS SOLUTIONS OF SINGLE BINARY ELECTROLYTES BY E. A. GUGGENHEIM Dept. of Chemistry, The University, Reading Received 17th July, 1957 Appraisal of the Debye-Hiickel theory should distinguish between the model (complete ionization, rigid spherical ions) and the approximation (ze$ < kT). For typical 2 : 2 salts in water the model is valid, the approximation is not. A combination of Miiller’s more accurate solution with a modified form of Bjerrum’s conventional definition of association leads to values of osmotic coefficients and activity coefficients effectively in- dependent of any arbitrary assumptions or conventions. These calculated values are moreover in good agreement with experimental measurements of freezing points on six sulphates of bivalent metals having effective mean diameters between 3-5A and 4-0k There is also agreement with e.m.f.measurements on ZnS04 and CdS04 with mean dia- meters in the same range. So-called association constants, in contrast to the osmotic coefficients and activity coefficients, have values which can vary by 20 % according to the precise form of the arbitrary conventions from which they are computed. 1. INTRODUCTION The long-standing question whether typical strong electrolytes are completely ionized in dilute aqueous solution has received answers depending partly on ex- periment and partly on prejudice, fashion and especially on the interpretation of the words “ completely ionized ”. In the interest of brevity and simplicity I shall confine this discussion to dilute aqueous solutions of single binary electrolytes, in fact 1 : 1 and 2 : 2 electrolytes. It is generally agreed that 1 : 1 salts, apart from a few exceptions such as some thallous salts, are almost completely ionized.On the other hand convincing evidence has accumulated that many 2 : 2 salts, notably magnesium oxalate and the zinc salts of dicarboxylic acids are far from completely ionized, but the more extreme view that sulphates of bivalent metals are also incompletely ionized is at least questionable. Some recent precision measurements of freezing points of dilute solutions of these salts have prompted a thorough reappraisal of the theory. 2. NOTATION The following symbols will be used : C number of ions of each sign per unit volume, e fundamental charge, rt z charge number of ion, T absolute temperature, k Boltzmann’s constant, E permittivity of medium (water), L Avogadro’s constant, po density of solvent (water), m molality, I ionic strength, # electrical potential, r distance between two ions, a diameter of ions, s = z2e2/ekT, a length ; in water at 0°C 4~122 = 27.74 A, K = (8nCs)t, a reciprocal length, 5354 ION DISTRIBUTION zc = l / ~ r , a pure number, y = sly, a pure number, b = s/a a pure number, 4 = ze#/kT, a pure number, a0 y activity coefficient, #J osmotic coefficient, cc coefficient in Debye-Huckel formulae, d 6 fraction of ions associated, K-1 association constant of molality scale.length defined by K ~ O = Ij(mo1e kg-l)), distance used to define conventional “ association ”, 3. ION DISTRIBUTION As already mentioned I shall confine this discussion to aqueous solutions of single binary electrolytes, in fact 1 : 1 and 2: 2 electrolytes.The only model with which I am concerned is the simple one chosen by Debye and Hiicke1,l namely, complete ionization to rigid spherically symmetrical ions in a continuous medium with the macroscopic permittivity of the solvent water. The question whether the two kinds of ions are of equal size is unimportant ; only the closest distance of approach of ions of opposite sign is significant, that is the mean of the diameters of the cations and anions. For this model Debye and Hiickel derived the so-called Poisson-Boltzmann differential equation, which in our notation can be written dZ$[du2 = 24-4 sinh 4, (1) with the boundary conditions $ + O as U + O , (2) ud$/du = b when u = 1 / K a .(3) As long as 4 Q 1, formula (1) may be replaced by the approximation d2#/du2 = u-44, (4) which has the relevant solution ( 5 ) where A is a constant determined by the boundary condition (3). If b is sufficiently small the condition 4 < 1 is satisfied for all values of r > a and the well-known approximate solution of Debye and Hiickel is valid. When b is not sufficiently small, the condition 4 < 1 is not satisfied for all values of r > a but is of course satisfied for sufficiently large values of r. The solution of the form (5) is then valid for small values of u (large r ) and the solution for larger values of zi (smaller r ) can be obtained, as described by Miiller,2 by numerical integration with adjust- ment of the constant A to fit the boundary condition (3).The numerical integra- tions have been done for A = 2, l , 0.5, 0.25. If we denote by 40 the value of when r = a, then 40 - b is the part of 4 0 due to the interaction between the chosen ion and the remaining ions. In other words it is zelkT times the average potential at the place occupied by a chosen ion due to the remaining ions. This quantity is shown in fig. 1 plotted (curves labelled M) against (&+d for b = 7.0, 3*5,2-8,2-5 and compared with the Debye-Hiickel approximation (curves labelled D) Included in the plot are some points calculated by Miiller. These points lie nearly but not quite on the same curve as mine. Whether Muller’s arithmetic or mine is the more accurate I do not know.In any case the discrepancy is small enough to be unimportant. It is immediately clear from this plot that the ap- proximation of Debye and Hiickel is useless when b = 7 but useful, if not accurate, when b < 2.8. This means that when z =f 1 the approximation is useful for values of a as small as 2-5 A, but when z = f 2 the approximation is useless for 4 = Au exp (- l/u)? 4 0 - b = + K S / ( t + Ka). (6)E. A . GUGGENHEIM 55 values of a less than about lOA. Since most ions may be expected to have dia- meters about 4 A it is evident that the Debye-Huckel approximation should be good for 1 : 1 electrolytes but useless for 2: 2 electrolytes. This conclusion is of course not new. It was reached by Bjerrum 3 in 1926 by a different train of reasoning.2.0 1.0 13-90 0 Let us now consider the distribution of ions of either sign in the neighbourhood of a chosen positive ion. The average excess number of negative ions at distances between r and r + dr over the number if there were complete randomness is d a 2 2b 4.rrC(e+ - l ) v 2 dr = - (ed - 1) The average excess number of positive ions at distances between r and Y + dr over the number if there were complete randomness has the negative value (8) K2a2 26 4nC(ed - l)r2 dr = - (e-4 - 1) The average excess number of negative ions over positive ions at distances between r and r + dr has the positive value K2a2 b 4nC(d - e-4)b)y2 dr = 8nC sinh + r2 dr = - sinh56 ION DISTRIBUTION Since the whole solution surrounding the chosen positive ion contains just one more negative ion than positive ions, the integral of (9) must be unity.Thus These quantities have been evaluated and are shown in fig. 2-5 for four cases. The relation (10) provides a check on the arithmetic. It was found to be satisfied to within about 1 %. Fig. 2 and 3 relate to 1 : 1 electrolytes with z = f l , b = 1.75, a N 4 A. The respective values of KU are 0.521 and 0.283 corresponding at 0°C to nz = 0.164 mole kg-1 and m = 0.049 mole kg-1. The broken curves with a maximum shown by an arrow represent the Debye-Hiickel approximation. We 0.10 0 . 0 5 e '9-1 2 bw' 0 - 0 . 0 5 0 1 5 10 15 20 f/Q FIG. 2. see that even at the high molality 0.164 the error introduced by this approximation is small and is confined to a strikingly small region. This small region makes a correspondingly small contribution to the interaction energy between a chosen ion and the rest of the solution.Hence follows the complete success, not only of the model but also of the formulae of Debye and Huckel for 1 : 1 electrolytes in water. The respective values of ~a are 0.1 65 and 0.078 corresponding to rn = 0.00410 mole kg-1 and m = 0-00092 mole kg-1. We are at once struck by the contrast with fig. 2 and 3. The maximum of the Debye-Huckel approximation predicted at the place indicated by a downward arrow and often referred to as the mean thickness of the ionic atmosphere has disappeared and its place is taken by a very flat in- flection. This behaviour is not novel. It was predicted both by Bjerrum 3 and by Fuoss: but, so far as I know, scale diagrams for practically important examples have not hitherto been produced. It is clear from these plots that for 2 : 2 electro- lytes the Debye-Huckel approximation is quite useless even for values of m as Fig.4 and 5 relate to 2 : 2 electrolytes with z = f 2, b = 7.0, a N 4 A.E. A . GUGGENHEIM 57 I t 1 1 0 1 5 10 IS 20 r/o FIG. 3. 5 10 15 2 0 f/Q FIG. 4.58 ION DISTRIBUTION low as 10-3 mole kg-1. The statement that the Debye-Huckel approximation will still be valid in the limit rn -+ 0 may satisfy a mathematician, but it is prac- tically useless since we shall find that it becomes accurate only at values of rn below 10-5 mole kg-1, at which dilutions precise measurements are impossible. Study of the general shape of the curves in fig. 4 and 5 leads naturally to Bjerrum’s3 method of approach.The essential feature of this approach is to choose a suitable length d and then use quite different techniques for dealing with the two ranges r > d and r < d. Bjerrum actually chose d = 4s and called this distance q. If this method of treatment is correct and accurate, then the values finally obtained for the osmotic coefficient and related quantities should be independent of the choice of d over some non-zero range. Bjerrum was aware 0.1 s 0.10 0 . 0 5 0 0 212 b=7 U=3-96A 1 I I I 0 1 5 10 IS 20 r / a FIG. 5. of this and in fact verified for the example z = f 1, a = 1 A that the difference in the values obtained for the activity coefficients were not greatly affected by choosing d = 4s instead of d = 3s. I propose to study this point more deeply and more quantitatively.Having chosen a suitable value for d, Bjerrum used the Debye-Hiickel ap- proximation in the range r > d substituting d for the distance of closest approach since this is by definition the smallest allowed value of r in this range. From a study of fig. 1 we have already reached the conclusion that the Debye-Hiickel approximation require b < 2.8 or thereabouts. This means that we should choose d > 10 when z = f 2. For the region r < d, Bjerrum used the approximation of treating each pair of ions whose distance apart is less than d as an electrically neutral molecule. This may be a useful approximation for d < 8 A or there- abouts but it seems rather unreasonable with Bjerrum’s choice of d = 3s = 14A when z =f 2.This approximation is moreover unnecessary. We can treat these ion pairs accurately by introducing a correction analogous to the second virial coefficient of a gas. The only restriction on the validity of this correction is that d must be sufficiently small for penetration of the range r < d by a thirdE . A . GUGGENHEIM 59 ion to be negligible. A rough estimate indicates that we may safely use values of d up to about 14A. I would like to call the range Y > d the Debye-Huckel range or DH range and to call the range Y < d the Bjerrum range or B range. Bjerrum used the names “ free ions ” for r > d and “ associated ions ” for Y < d. Since these names are widely used, I shall continue to use them. But this terminology has unfor- tunately been misunderstood and misinterpreted.In particular La Mer 5 has said “ Bjerrum’s treatment consequently resolves itself into a more or less arbi- trary reduction of the concentration of free ions ”. There is in fact nothing arbi- trary except the particular choice of d made by Bjerrum. I cannot too strongly emphasize that the choice of d is largely immaterial and I shall show that when z =Jr 2 the values obtained for osmotic coefficients are practically unchanged when d is varied from 10 A to 14A. This range is shown in fig. 4 and 5 by the double-headed arrow. 4. OSMOTIC COEFFICIENT I shall now enumerate the formulae used for the calculations omitting deriva- tions. The standard Debye-Hiickel formulae may be written as a Q = - l+y--- 21n(1+ y ) } . Y3 3 { 1 f Y and for aqueous solutions at 0” C we have a = 1.1235 kg$ molef and a0 = 3.085 A.Correspondingly Bjerrum’s formulae become in our notation (14) I ’ = (1 - S)I, K- 1 = 477pos3QL = 161-6 Q kg mole-1 for water at 0” C, (1 7) Values of this integral were recalculated by use of tables of Ei(y) and are given in table 1. Bjerrum’s 3 calculated values for b = 5 and b = 10 seem to be slightly wrong. These errors have been taken over by Harned b Q(b) b Q(b) and Owen.6 2 O~Oo(i0 6 1.0408 Finally my formula for the 0s- 2.5 0.1880 7 1.4168 TABLE ~.-VALWS OF ~ ( 6 ) = 1: e~y-4 dy 1.996 2.95 1 4626 2.8 0-2746 8 9 3 03253 04413 10 3.5 motic coefficients 4 is + = (1 - S)(l - *22KI’~~(I’*d/~O)) 0.5495 11 7.656 0.771 1 12 13.41 08968 4- 3 m + PI, (19) 4 p-1 = 3Q~3/d3. (20) 5.5 5 This differs from Bjerrum’s formula 5.6 0-9240 by the term p.As already mentioned this term is analogous to a term closely related to the second virial coefficient of a gas and the formula for /3 is derived in the appendix. In the calculations had values in the range 0.01 to 0.03.60 ION DISTRIBUTION All the calculations of osmotic coefficients related to 2 : 2 electrolytes, so that z = 2, and to aqueous solutions at 0°C. Cal- culations were made for b = 7 and b = 8 corresponding to a = 3.96A and 3-47A respectively. For both values of a at each molality the calculation was They are summarized in table 2. B 0 0 g 2 8 2 8 0 0 0 0 tE. A . GUGGENHEIM 61 repcated for three values of d approximately 14A, 11.2A and low. Whereas this variation in d implies a variation of about 20 % in K the values obtained for the osmotic coefficients are changed by only 3 % or less. This is as should be if these calculations are, as I believe, physically meaningful. Inclusion of the small term p significantly improves the agreement between sld = 2.0 and sld = 2.5 but has much less effect on the agreement between s/d = 2-5 and s/d = 2.8.We can now verify that for 2 : 2 electrolytes with ions of diameter about 4A the Debye-Huckel approximation can become valid only at such extreme dilutions as to be practically useless. The condition for the Bjerrum short-range effect to be unimportant compared with the Debye-Huckel long-range effect is K-lm < C C Z ~ ~ , or nz) <( tcz3K. With tc N 1 kg) mole-), z = 2 and K” 0.004 mole kg-1, this requires nz* < 0-03 mole) kg-), say rn+ < 3 x 10-3 mole3 kg-j or nz < 10-5 mole kg-1.5. COMPARISON WITH EXPERIMENT It is interesting to compare our calculated values of the osmotic coefficient with values obtained experimentally from measurements of freezing points. Such measurements have been made recently at Reading by Brown and Prue 7 on the I ?I I I I + cu a Zn A Mg o Ca x co o Ni A + 3r I O * ~ C t t 1 t : I I I I I 1 0 0 . 0 0 5 0.010 0.015 0*020 0 . 0 2 5 m FIG. 6. sulphates of copper, zinc, magnesium, calcium, cobalt and nickel. The com- parison is shown in fig. 6. The only adjustable parameter in these curves is the effective diameter a since, as already emphasized, the curves are unaffected by varying d between lOA and 14A. I think this comparison shows beyond any reasonable doubt that the colligative properties of these sulphate solutions can be quantitatively explained by the model of Debye and Huckel, namely, complete ionization to rigid spherical ions in a continuum with the macroscopic permittivity of the solvent water.The only auxiliary assumption is that for all these sulphates the closest distance of approach between cation and anion lies between 3-5A and 4 A, which is what one would expect 7 from distances in crystalline hydrates.62 ION DISTRIBUTION 6. MEASUREMENTS OF E.M.F. Activity coefficients can of course also be calculated from the same premises, but comparison with experiment is unconvincing because of the adjustable extra- polation to m = 0. Nevertheless I have constructed fig.7 to show the comparison between calculated values of y and experimental values based on e.m.f. measure- ments of the cell Zn 1 Zn, Hg I ZnS04 (as) I PbS04 I Pb, Hg I Pb at 25°C. The curves relate to b = 7 and b = 8 corresponding at 25°C to N = 4.08 A and a = 3.57 A respectively. By using the same values of b rather than the same values of a as at O'C, it is possible to transcribe calculations from 1.0 0 - 9 0.8 0 . 7 0.6 0 . 5 0 - 4 Y 0.3 0 -2 0.1 0 I I I I - + I I I 1 1 I 0 0.005 O*OIO 0.01 5 0.020 0 0 2 5 m FIG. 7. 0°C to 25°C by a mere shift in the molality scale. We have I/mole kg-1 = K2ao2 with a0 = 3-085A at 0°C and with a0 = 3.048A at 25°C. The comparison with experiment depends critically on the value chosen for E", the standard e.m.f. of the cell.For the measurements of Cowperthwaite and La Mer8 I have accepted their value E" = 410436 mV. For the measurements of Bray 9 I have used the value E" = 41 1-05 mV to bring them into concordance with the other measurements. The difference of 0.2 mV can be explained by a small difference in the physical state of the Pb/PbS04 electrode. The agreement between the calculated and experimental values is all that could be desired, but it is only fair to mention that by choice of an entirely different value for E" one could force an agreement with the Debye-Huckel approximation which we have good reason to believe to be physically meaningless in this application. Thus in fig. 7 a shift of 10 mV in the value of E" will bring the experimental points on to the Debye-Hiickel curve labelled D-H.This is in fact how Bray interpreted his own measurements.E . A . GUGGENHEIM 63 La Mer and Parks10 have made e.m.f. measurements of the analogous cell with cadmium replacing zinc. With a suitably chosen value of E" the plot superposes so completely on that for ZnS04, that it did not seem worth while reproducing it. 7. CONCLUSION I have used Miiller's method of numerical integration to achieve a quantitative analysis and a slight improvement of Bjerrum's method of approach. It is diffi- cult to make a detailed comparison of this approach with that of Gronwall11 which uses series of odd powers of b with coefficients which are functions of Ka. The leading term in b is the Debye-Huckel approximation and the coefficients of higher powers of b are complicated functions of K a ; those of b3 and of b5 have been tabulated. The speed of convergence of these series is determined by the vaIue of b rather than that of Ka.For example, we have when K a = 0.1 : 1 - 4 = - 1 x 0.1 x 0.8662 b + 0.0616 (4r + 0.0501 ($jr + . . ., (21) 6 1 0.1 2 1.1 - l n y = - x -b + 0.1513 and when Ka = 0.2, 1 - 4 = - 6 1 x 0.2 x 0.7588 b -+ 0.1020 (A)3 + 0-0074(~)' + . . ., (23) - lny = - 1 x - b 0.2 4- 0.3075 ( $ ) 3 + 0.1541 (i)5 + . . ., 2 1.2 and when K a = 0.4 : 1 .- 4 = - 6 1 x 0.4 x 0.5986 b + 0.0744 ($)3- 0*1167(;)l+ . . ., (25) - lny = - 1 x 0.4 - b + 0.4152 (3, + 0.0481 (3 + . . . . 2 1.4 In each formula the first term on the right is the Debye-Huckel approximation while the second and third terms are due to Gronwall.It is clear that for the values of b between 7 and 8, with which we are concerned, the series are con- verging rather slowly and it seems unlikely that higher terms are negligible. Nevertheless the $ values for b = 7.5 calculated from the first three terms of Gronwall's series have been included in fig. 6 as the broken curve. Whether this curve is a better or worse approximation than ours is a matter of opinion, but in either case it strengthens the evidence against incomplete ionization. I prefer Bjerrum's approach partly because it clearly shows up the relative con- tributions of the D-H range and the B range and partly because the mathematics involved are so much more elementary, It is more interesting to compare our approach with that of Davies 12 who describes the behaviour of single binary electrolytes in terms of a " dissociation constant " K and a standard formula for the activity coefficient of the dissociated part of the electrolyte lny' =- azz - I ' ) - 0.2019.(1 + I'* I have three comments to make on this. First, if by " incomplete dissociation " is meant anything stronger than what we know must happen when two rigid spherical ions approach within a distance q = +s, then in the case of the sulphates64 ION DISTRIBUTION of bivalent metals the evidence is against anything of the kind. Secondly, if by " incomplete dissociation " is meant nothing more than what Bjerrum called " association " then the value of the constant K for 2 : 2 electrolytes must depend significantly on the choice of the distance d by which association is defined.In other words, if d is not specified there is for 2 : 2 electrolytes an indeterminacy of about 20 % in the value of K. Thirdly, the smallest value of d for which the Debye-Huckel approximation is at all valid is about 1OA and consequently when z =i 2 Davies' formula for y' does not correspond to any physical reality. I suggest that for 2 : 2 electrolytes it would be an improvement to replace the formula (27) by I '* lny'=-4a -- (1 + I'* which corresponds roughly to d~ 9 A. The values so obtained for K-1 would describe the extent of association to within a distance of about 9A. If there is any kind of association stronger than that due to electrostatic attraction between rigid ions this will be revealed by a value of K-1 significantly greater than 200 mole-1 kg.A preliminary abstract of this paper was presented to the Theoretical Division of the American Electrochemical Society at a meeting in Washington in May 1957. 1 Debye and Huckel, Physik. Z., 1923,24, 185. 2 Miiller, Physik. Z., 1927, 28, 324. 3 Bjerrum, Kgl. Danske Vid. Selsk. Math-fys. Me&., 1926,7, (9). 4 FUOSS, Trans. Faraday SOC., 1934,30, 967. 5 La Mer, Trans. Amer. Electrochem. SOC., 1927, 61, 631. 6 Harned and Owen, The Physical Chemistry of Electrolytic Solutions (Reinhold, 7 Brown and Prue, Proc. Roy. SOC. A, 1955,232, 320. 8 Cowperthwaite and La Mer, J. Amer. Chem. SOC., 1931,53,4333. 9 Bray, J. Amer. Chem. SOC., 1927, 49, 2372. 10 La Mer and Parks, J. Amer. Chem. SOC., 1931, 53, 2040. 11 Gronwall, Proc. Nut. Accd.Sci., 1927, 13, 198. Gronwall, La Mer and Sandved, 12 Davies, J . Chem. SOC., 1938, 2093. 1950), p. 123. Physik. Z., 1928, 29, 358. APPENDIX The following symbols will be used : N Nl N2 = N - N1 NO number of solvent molecules, VO molecular volume of solvent, number of ions of each sign, number of free ions of each sign, number of ion pairs, d c1 = 1 exp (- E/kT)47irzdr = 4rs3Q, 0 F P1 P2 PO free energy, chemical potential of a free ion, positive or negative, chemical potential of an ion pair, chemical potential of solvent.E . A. GUGGENHEIM 65 Initially we ignore the electrostatic interactions between free ions. The non-trivial 2(N1 In N1 - N I } 4- N2 In N2 - N2 - 2{N1 In (NOVO - &Nu)} - N2 In (~Novo), (Al) where the factors 2 take account of the two kinds of ions, positive and negative. The corresponding terms in p1IRT are tcrms in FIRTare then to a first approximation and those in p2/RT are The condition for equilibrium between free ions and associated pairs P2 = 2P1 leads to Now introducing activity coefficients of the free ions to take account of the long-range D-H effect we replace (A5) by which at high dilutions reduces to This is Bjerrum’s formula for association equilibrium equivalent to (1 6). For the relevant terms in - po/RT we obtain from (Al) . N2 which by virtue of (A7) is to a first approximation equivalent to Now inserting the D-H correction for the free ions we replace (A9) by where $D-H denotes the Debye-Huckel expression for the osmotic coefficient 4. (A10) it follows that From = (1 - ~ $ D - H + PS(l + ul.) = (I - S)#JD-H + @(I + d3/3@3). (A1 1) Formula (A1 1) is equivalent to formula (19) and differs from Bjerrum’s approximation by the presence of the factor (1 + u/x). The above argument is admittedly far from rigorous. In particular I am dissatisfied with the replacement of (A6) by (A7). Nevertheless I believe that the inclusion of the term u/a in (All) is a small but significant improvement, the more so the larger the value chosen for d. C

ISSN:0366-9033    

DOI:10.1039/DF9572400053

出版商:RSC    

年代:1957

数据来源: RSC

摘要:

GENERAL DISCUSSION Dr. M. Eigen (Gottingen) said: Prof. Falkenhagen mentioned the work of Fuchs and Tesch concerning the Wien effect of aqueous 2: 2-electrolytes. As found by means of sound absorption measurements (cf. fig. 2 in my paper) these electrolytes show-even at high concentrations-at least two relaxation effects (T - 10-6 and 10-9 sec) due to specific interactions of the ions. If the impulse duration in Fuch’s work is smaller than 10-6 sec the measured field effect refers only to such interaction effects, the relaxation time of which is essentially shorter than 10-6 sec, i.e. the “ weaker ” specific and the general ionic interaction. Prof. H. Falkcnbgen and Dr. G. Kelbg (Rosrock) said : The results in our paper using eqn. (6) are nearly independent of the following models.(i) We took into consideration a repulsive potential corresponding to absolutely hard spheres. (ii) We used the reduced Coulomb potential (see (10)). Both models give the characteristic functional dependence on the concentrations for the activity co- efficients and the peak of the curves. One has to be very careful in deducing the molecular laws of forces from the thermodynamical data. In reply to Dr. Eigen, the time of relaxation in the case used by Fucks and Tesch is 6 x 10-9 sec. The impulses in their method are of magnitude 7.4 x 10-7 sec. That means that the dispersion effect can be neglected. A comparison of our theory with Wilson’s theory has not been made. The comparison of the A-coefficients at higher concentrations seems to be necessary. This is a task of the future.Prof. H. S. Frank (Pittsburgh) said: In connection with the discussion of the statistical interpretation of strong electrolytes I should like to call attention to a point of view which has recently been expressed 1 that the concentration range in which the ion cloud can conform to the general requirements of the Debye-Hiickel theory is less extended than has generally been supposed. Everyone knows that there must be some limit to this range, and few people would claim that at 1 mole/l., say, in a 1 : 1-electrolyte in water, the theory was valid. Fuoss and Onsager have recently2 taken 0.01 M as the concentration above which it cannot be depended upon. On the basis of considerations similar to theirs, Thompson and I prefer to set the limit at about 0.001 My and lower for higher valence types, or in solvents of lower dielectric constant.The reasons for the failure of the theory arise ultimately from what may be called the two-handed nature of the Debye-Hiickel Poisson-Boltzmann equation, and the need, if the theory is to be applicable, for the right hand, so to speak, to know what the left hand is doing. It is well known that C$ in the Poisson equation, v2$, = - 4-rrp/D, is a proper (average) electrostatic potential whereas zj = e+ in the (linearized) Boltzmann expression mi = mi0 1 - -- is properly a potential of average force. These two potentials can be equated with each other only if the averaging processes involved in the two cases are compatible, and there have been many discussions of the requirements for this in terms of fluctuations.It seems, however, that a sharper point can be put on this dis- cussion than has been done in the past if it is carried out by considering the actual numbers of ions which are required to produce the principal effects. The classical Debye-Hiickel potential around a central ion is ( :$) Zie eKa e-Kr D l + K a r +(r) =- - - 2.e eKa e-Kr and this corresponds to a distributed charge density p(r) = - -!- ~ 2 - - . 4~ 1 + ~ a r 1 Frank and Thompson, Electrochem. SOC. Symposium, May, 1957. 2 J. Physic. Chcm., 1957, 61, 668. 66GENERAL DISCUSSION This places in a spherical shell of radius r and thickness dr a total charge 67 This charge makes, at the site of the central ion, a contribution P(r)dr to the potential, arising from the ion cloud, to which the central ion is subjected.This C(v)dr - ZieK eKa comes out to be P(r)dr = - - - - - e-Krd(Kr). The whole influence the Dr of the cloud on the central ion is thus standard result. But also, the fraction f(R) of the effect on the central ion arising from the spherical region lying within Y = R is Z i e K D(1 + Ka? a function of K and therefore of the concentration c. Again, the average total number N(R) of + and - ions lying within Y = R can be computed, advantage being taken of the fact that, in this approximation, for a solute of symmetrical charge type, the excess of ions of one sign just equals the defect of ions of the opposite sign. For such a salt, therefore, N(R) = - n(R3 - a3) -. When R and a are in A, this comes out N(R) = 0*00505c(R3 - a3).It is thus easy to compute that for a 1 : 1-solute in water at 25" C at c = 0.01 moIe/l., using a = 4A, the first 50 % of the influence on the central ion comes from the region lying within the radius R = 25A, and that this region contains, on average, 0.80 ions of both signs. The next 25 % of the effect comes from the region lying between R == 25 and R = 46.2 A, and this region contains, on average, 4.2 ions of both signs. We are suggesting that the theory is invalid in 0.01 M solutions because the number, 0-8 ion (one ion present 80 % of the time), producing the " inner " 50 % of the effect must be subject to such large fluctuations that the potential of average force and the average electrostatic potential energy can no longer be equated to each other.The structural description of this failure is that the equations are making impossible demands upon the ions-for example, that 0.8 average ion distribute itself, in a sphere of radius 25A, according to a continuous radius law which requires spherical symmetry for its formulation. It is, moreover, required that the central ion, together with this 0.8 " inner " ion, present a sphericalIy symmetric aspect to the rest of the solution, from which comes the remaining 50 % of the influence which the central ion feels. When there is no place in the solution where the required spherical symmetry exists it does not appear that the situation can be saved by averaging over configurations. At extreme dilutions these difficulties do not arise. Thus at c = 10-8 mole/l.the first 25 % of the effect on the central ion comes from 33.9 ions lying within R= 8760 A, the next 25 % from 440 ions lying between R = 8760 and R = 21,100 A, and the third 25 % from 3316 ions lying between R =21,100 and R = 42,190 A. These numbers are all large enough to permit the averaging operations to be carried out without violation of the spherical symmetry requirements. It therefore appears that at c = 10-8 mole/l. the theory is good and that at c = 10-2 mole/l. it is not. At 10-3 mole/l. the situation still seems to be bad. Considerations of this kind, however, do not permit a straightforward estimate of the concentration at which the theory actually fails. For this we turn to a somewhat intuitive interpretation of the comparison betwee I 1 /K, the character- istic length of the Debye-Huckel theory, and a " lattice length " 2, defined, for a symmetrical charge-type solute, as I= (i:)' ____ , the average distance at which an ion has an oppositely charged nearest neighbour.A different lattice might be assumed, but this would not affect the argument. 4 2 Nc 3 100068 GENERAL DISCUSSION Going to the limiting law approximation, we note that Pi = - ZieK/D can be interpreted as meaning that the central ion “ sees its image ” at a distance 1 / ~ . Now l / ~ in a 1 : 1-electrolyte in water at room temperature is m 3c-* A. I on the other hand, is m 9.4c-AA. Curves of 1 / ~ and 1 against c intersect at c = co = 0.001 14 mole/l., both lengths there being about 90A. At higher dilutions l / ~ > and the central ion “ sees its image ” farther away than its nearest-neighbour ions are, in fact, located.This is clearly because of partial cancellation of the effects of + and - ions. For c > CO, on the other hand, 1 / ~ > I, and the central ion purports to see its image nearer at hand than the average distance of oppositely charged ions. In the same terms as above, this corresponds to a “negative can- cellation ” of the effects of + and - ions. While this formulation would not, standing alone, justify with any rigour the rejection of the theory for c > CO, the circumstance that co does, in fact, lie within the concentration range in which the fluctuation difficulties have become acute leads us to make this postulate : that for c < co the Debye-Huckel formulation is valid; for c 3 co it is not.Of interest in this connection is an empirical finding related to activity co- efficients, namely, that for almost all 1 : 1 salts in water, when logf’ is plotted against d, the curve becomes accurately linear at about c = 0.001 mole/l. and continues linearly over a 20- to 100-fold range of concentration. This seems to provide some experimental support for the notion that at or near c = co some change does in fact take place in the physical nature of the ion cloud. Prof. H. Falkenhagen and Dr. G. Kelbg (Rostock) (communicated) : Prof, Frank speaks about the limit of validity of the theory of strong electrolytes. Con- siderations of this should be made from two aspects. As in every physical theory, a certain model forms the basis of the theory of strong electrolytes, too; it ap- proaches reality only roughly.The character of approximation of the model still depends on the electrolyte under consideration. Furthermore a strict solution is not possible within the limits of the model, because of the complexity of the basic equations. Approximations must be carried out. It is with these physical- mathematical simplifications that Prof. Frank’s remark deals. But his con- siderations are not connected with Kirkwood’s strict statistical equations nor with those given in our paper. He goes back to the results of the Debye-Hiickel theory and draws the conclusion that the limit of the theory is at about 0.001 mole/l. If we consider the special model of a gas comprised of hard spheres with point charges at their centre which is embedded in a continuous medium of the dielectric, this limit is, to our mind, uncertain.Lastly it depends on the demands which we make on the accuracy of the calculations within the limits of the model used. We feel more inclined to say the following. If we look at the eqn. (5), (7) and (9) in our paper, then assuming eqn. ( 5 ) the character of the approximations is the better the smaller I ejek ]/DkT“ is and the smaller (4n/3, n, a3 is in comparison to 1. (n = number of ions in 1 cm3.) If we draw the theoretical curves, say for osmotic coefficients, perhaps even up to 1 mole/l. and observe the position of the experimental points, no one will overlook the approximations introduced and will try to draw absolutely certain conclusions in the range cf higher concentrations for other theoretical considerations.Strictly speaking it seems to be impossible, too, to draw absolutely certain conclusions from the theory even at 1/1000 mole/l. because of the inadequacy of the models. If we keep this in mind, the use of the statistical theory of strong electrolytes only below a concentration range of 1/1000 mole/l. seems to be a little exaggerated with regard to a comparison with the experi- mental results. We mention that in the following papers the concentration ranges of the calculations are the following. Glaubermann and Juchnowski : 1 0 < c < I (see the equations by Bogoljubow). Poirier : 2 0 < c < 0.5. Haga : 3 0 < c < 0.1 (c mole/l.). 2 J. Chem. Physics, 1950, 18, 1426. - 1 loc.cit. 3 J . Physic. SOC. Japan, 1953, 8, 714.GENERAL DISCUSSION 69 What was said for the thermodynamical theory is valid, too, in the theory of irreversible processes. In particular, equations for the equivalent conductivity of strong electrolytes have been developed which, on extrapolation to higher concentration ranges, say, 0.3-0.5, or even 1 mole/l. for 1-1 electrolytes, give good agreement with the experiment. At least these formulae will keep their validity as long as no better ones have been derived, if they are used with caution for higher concentration ranges as extrapolated or half-empirical formulas. Dr. G. M. Bell and Dr. S . Levine (Manchester University) said: Recently Hiickel and Krafft 1 (H.K.), making use of the earlier work of Kirkwood,2 have objected to the theories of Eigen and Wicke 3 (E.W.) and Falkenhagen and Kelbg 4 (F.K.) on the grounds that they are not strictly derived from statistical mechanics.In their present paper, however, F.K. quote some results of Moller * which seem to agree with those of E.W. but contradict those of H.K. We wish to discuss here some aspects of these conflicting theories. H.K. assume the validity of eqn. (29) in Kirkwood's paper. However, in the footnote on p. 773 of his paper, Kirkwood states that his formula is an approxim- ation and it is desirable to investigate the nature of this approximation. Consider a general electrolyte containing rn ion types and having a bulk density nio of ions of type i, charge ei. The modified Poisson-Boltzmann equation satisfied by the mean electrostatic potential $ at a general point q say, in the vicinity of an ion k located at qk say, may be expressed as s = Y = qj - q k I between ions j and k whereas Aj is a function of the distance qj - q 1 between i andj.Since uj will differ significantly from zero only for 1 Huckel and Krafft, Z. physik. Chem., 1955, 3, 135. 2 Kirkwood, J. Chem. Physics, 1934,2, 767. 3 Wicke and Eigen, 2. Elektrochem., 1952, 56, 551 ; J. Physic. Chem., 1954, 58, 703. 4 Falkenhagen and Kelbg, Ann. Physik, 1952, 11, 60. * Enderby, Thesis (London, 1952) also applied the Born-Green equation to the theory of strong electrolytes.70 GENERAL DISCUSSION where, for brevity V = V&(R). We have expanded up to quadratic terms in the Taylor's series since the linear terms vanish.Kirkwood's formula (2) is obtained by substituting the first term only on the right-hand side of (4) into (3) and the work of H.K. is based on this approximation. The second term in (4) is due to the variation of exp (- P Y) over the small region in which is effectively different from zero. R For simplicity, consider a 1 : 1-type of electrolyte and assume that the hydration shells of the positive and negative ions have the same diameter a and are im- penetrable. This means that the two integrals on the right-hand side of (4) may be equated to - 47~a3/3 and - 47ra5/5 respectively. Furthermore, if we neglect fluctuation terms and quantities of order (nF)2, then we may put Y = ej$ in (4) and so on introducing the volume correction, eqn. (1) becomes where q0 = n 2 O = n, el =- e2 = e and K is the familiar Debye-Hiickel para- meter defined by ~2 = 877nPe2/~.The second term in the square brackets, which was not considered by H.K. is by no means negligible unless R > ~ - 1 and K a < 1. In these conditions we are dealing with a very dilute electrolyte and may expect the volume correction to be small. Thus the significance of the considerations of H.K. is obscure. In Kirkwood's approach, the solvent is treated as a continuum, whereas in the theories of E.W. and F.K. the conception of numbers of available sites is introduced, i.e. the molecules of the solvent are considered to form a quasi-crystalline lattice. The equation of E.W. and F.K. which corresponds to (5) reads 8 m e V 2 # = - sinh Pe$/[1 + 2nv (cosh /3e# - l)], E where v is the volume of a hydrated ion.E.W. identify v with the excluded volume which is in fact 8u. This assumption (which has already been criticized by Robinson and Stokes I), would for small n make the correction term in (6) equivalent to the first correction term in the square brackets in (8). However, unless further con- siderations are introduced there is no reason to replace actual ionic volumes by excluded volumes in a lattice theory. An attempt to justify the distribution function leading to the eqn. (6) has been made by Schlogl2 who applied the methods of statistical mechanics (see also Dutta3). Rather earlier, Grimley and Mott4 (G.M.) arrived at the eqn. (6) 1 Robinson and Stokes, Electrolyte Solutions (London, Butterworths, 1955), p.80. 2 Schlogl, 2. physik. Chem., 1954, 202, 379. 3 Dutta, Proc. Nat. Inst. India, 1953, 19, 183. 4 Grimley and Mott, Furuduy SOC. Discussions, 1947 1 , 3 . Grimley, Proc. Roy. Soc.A, 1950, 201,40.GENERAL DISCUSSION 71 by using a restricted (or semi-) grand partition function. Bell and Levine 1 (B.L.) have applied the restricted grand partition function to electric double layers in colloidal systems to obtain the distribution law 3 l n g 3Ing" ---- 3 ni - Pi+, (7) where g is the number of arrangements in unit volume available to the hydrated ions and water molecules. We are supposing that at a given point where the mean potential is #, there are ni ions of type i (i = 1, . . ., m) and no water mole- cules which are not attached to any ion, per unit volume.The superscript' indicates that the quantity concerned is taken at a reference point where # = 0. The fluctuation term has been omitted on the right-hand side of (7). We suggest that the above authors have not used the most reasonable form of g. Let all ions be hydrated and let the partial volume of an ion complex of type i be v = hive, while that of a solvent molecule is vo. Then k In g is essentially an entropy of mixing of molecules of different sizes and there seems no reason why existing work on this subject should not be used. The zeroth-order ap- proximation is that of Flory,2 which Glueckauf 3 has more recently applied to hydrated electrolytes and we can introduce their formula immediately. If yj = vinj is the volume fraction of component i, where i = 0, 1, .. ., m then this approxima- tion yields (8) Better approximations have been obtained by various authors4 and the case investigated which seems closest to the mixture of ion complexes and water molecules is that of a tetrahedral 4-unit molecule and monomer on a face-centred cubic lattice model. It is found that Flory's formula is considerably better than the ideal entropy formula, which corresponds to the form assumed by Schlogl. If we suppose that hi and vo are uniform throughout the electrolyte, then no is given as a function of the ni by the condition of constant volume, namely m In g =- 2 ni In yi. i = O j - 1 (9) On substituting (8) into (7) and making use of (9), we derive the distribution law (10) where yio is the value of n(i = 0, 1, . . ., m) at the reference point.This can be put as Yi = YiOOO/YOO)hi exp (- ei#fkT), m ni = np [( 1 - vo 2 hjnj)/( 1 - vo 2 hjnj" j - 1 j = 1 L j = 1 where the second form applies at small electrolyte concentrations. In a 1 : 1-type of electrolyte, in which both ion types have the partial volume v = hvo, the modified Poisson-Boltmann equation for small concentrations becomes 4n 8nn A2# = - - (nl - n2)e = - [l - 2h50n (cosh pe$ - 1)] sinh Pe$. (12) E E 1 Bell and Levine, Trans. Faraday Sac., 1957,53, 143. 2 Flory, J. Chem. Physics, 1942, 10, 51. 3 Glueckauf, Trans. FaraAay Soc., 1955,51, 1235. 4 Guggenheim, Mixtures (Oxford, Clarendon Press, 1952).72 GENERAL DISCUSSION The distribution function chosen by Schlogl reads and this yields a distribution law which has the same form as (10) provided volume fractions are replaced by mole fractions.For small electrolyte concentrations this leads to the relations The form of g used by G.M. is given by eqn. (4.5) in the paper by B.L. It is as- sumed here that a permanent quasi-crystalline lattice is available to the hydrated ions, which all have the same size and indeed, that the unattached water molecules can also be grouped into water complexes of the same size as those of the ions. By using eqn. (ELL., 4.5) we can readily show that for small concentrations, where we have put hi = h(Z = 1, . . ., m). In the particular case of a 1 : 1-type with h = hl = h2 it is seen from (ll), (14) and (15) that the factors to be com- pared for the different distributions are h2 (Flory), 2h - 1 (Schlogl) and h (G.M.). Thus we have agreement between the formulae of Flory and Kirkwood at small concentrations if h = 8, which is not unreasonable.The formula of G.M. seems quite incorrect and that of Schlogl is also unsatisfactory but not to the same degree. The basis of the lattice theories of the type described above has been criticized by H.K. who do not think it proper to separate out entropy and electrical effects in this way. Fuoss 1 has also raised objections on the grounds that at the higher concentrations where the volume effect becomes significant the continuum model for the ionic atmosphere is not valid; the fluctuation terms can no longer be ignored. Now the use made by B.L. of the grand partition function to derive (7) implies that the ionic atmosphere can be divided by a family of equipotential surfaces such that the region between two neighbouring surfaces, $, t,h + A$ where ] eiA$1 < kT for each type ej contains a statistically significant number of ions.This assumption may be reasonable in the electric double layers of colloidal particles provided they are not too small, but its application to ordinary electro- lytes is much more questionable. Although a term corresponding to the first correction term in (5) is derived by the lattice theories, there is nothing to cor- respond to the other correction term in (5), which is due to the variation of the electrical potential over the field of the short-range forces. The use of Kirkwood's formula (2) is a sound, although doubtless difficult, method of dealing with the volume effects and thus it seems desirable to extend equations of the type (5) to include terms of order (nj")2.The objection of Fuoss could possibly be met by including fluctuation terms, although this has never been achieved. Such an analysis should throw light on the merits of the lattice theories which at present are inadequate. Dr. Kelbg (Rostock) (communicated): In reply to Dr. Bell and Dr. Levine I have some objections to your eqn. (5) based on the following arguments. The intermolecular potential is a non-analytical function in the range of 0 < k < co provided that the model is of hard spheres. Therefore we have to suppose that the mean potentials V k j are not analytical and consequently in (3) a development in a Taylor series is not allowed.Consideration shows that the mean potcntial 1 Fuoss, J. Amer. Chem. SOC., 1957, 79, 3301. Fuoss and Onsager, J. Physic. Chem., 1957, 61, 668.GENERAL DISCUSSION 73 is partially continuous. In the first approximation there exist two distinct differ- ential equations in the two domains n < k < 2a and 2a < k < 00. Therefore I think that we have to be careful in drawing further conclusions from eqn. (5). Dr. M. J. Sparnaay (Eindhoven) said: Although the approach given by Dr. Levine and Mr. Wrigley looks promising, an alternative method, based on statis- tical thermodynamics, may not be excluded. Consider the system, formed by two ions, whose centres are a distance R apart (R varying between 10 and 25& and water molecules filling up the space around and between the ions.The inter- action force K between the ions may be written : where the symbols have their usual meaning. The first term on the right represents mainly Coulomb interaction. Concerning the second term the following may be said : if the distance between two ions with unlike sign is increased, the move- ments of the water molecules between them become more restricted, i.e. the entropy will decrease. Thus the second term represents a repulsion. If the ions have like signs, it follows from a similar reasoning, that the force contribution due to entropy represents an attraction. Preliminary calculations indicated that the entropy effects amount to a few per cent of the Coulomb interaction, but owing to the complicated character of the calculations, this cannot be said with certainty.Dr. S. Levine and Mr. H. E. Wrigley (Manchester University) (partly com- municated) : We would like to discuss in more detail a number of points in our paper. (i) It would appear from the remarks of Dr. Spaarnay that our method is not based on statistical thermodynamics. However, we believe that our results do represent a first approximation to the free energy of interaction of two ions. It is first imagined that the free energy has been determined for a given configura- tion of all the water molecules inside the co-ordination shells at a given separation of the ions. This means that we have averaged over all positions of the molecules outside the ion complexes but have kept the molecules inside fixed in a specified configuration.The reaction field will obviously depend on the given arrangement of the co-ordination molecules. If, now, each hydrated ion is treated as a pseudo- molecular ion, six external degrees of freedom, three of translation and three of rotation, can be assigned to it. Let W and W’ represent the Eulerian angles de- fining the orientations of the two ion complexes. By averaging over all the in- ternal degrees of freedom of the two complexes, the free energy of our system E(R, W, W’) say, is determined for specified R, W and w’. Then the free energy of interaction of the two ions is defined by where E(co) is the limiting value of E(R, 0, w’) when R 3 00. This definition of V(R) can be extended to include a possible variation in the number of co- ordination molecules, but such an extension is rather premature at the present stage.In our theory the averaging over all molecules outside the co-ordination shells is achieved (although not exactly) by replacing these by a continuous dielectric medium. With regard to the internal degrees of freedom of the ion complexes, we may expect their main contributions to V(R) to be due to an entropy change. Making use of the work of Eley and Evans and of Everett and Coulson (ref. (2)), we have estimated the effect of the reaction field on the entropy associated with these internal degrees of freedom and find that this is very small. Finally, the fact that the polarizability a of the ion complex is independent of its orientation (a consequence of the symmetry in the tetrahedral structure), implies that, to a first approximation E(X, W, Z ) is independent of W and W’.74 GENERAL DISCUSSION (ii) It is possible to simplify the expression for 2W3 by replacing (#o),, = by (y!~&,-~ which is obtained from (9, in the integrand in (40).We now require BO which is given by the condition of continuity in (17). This yields Furthermore the series (41) may be rearranged so that b3, b5 and b6 are replaced by much simpler coefficients which are independent of t. These are ( t ) a n d F o = l + - 2 3 E * where Do = 1 + - - u 1 - - 2 E (iii) An alternative method of obtaining the free energy of interaction is to introduce the Maxwell stress tensor T = (~/877)(2EE - EzI), where E is the mag- nitude of the electric field E and I is the unit tensor, and then to evaluate the force between the ions.This force is given by the integral, taken over any closed sur- face in the continuous dielectric medium which encloses one ion only, of the scalar T . n . k where n is the unit normal vector to the surface and k is the unit vector parallel to the line 0 1 0 2 . It is most convenient to choose the sphere rl = a (or r2 = a) for this surface. The force method, however, is more laborious than the energy method described in our paper. (iv)The interaction energy of two point charges e and e’ at the centres of spherical cavities of radius a is simply e Bo - - 1 - - , to which V(R) reduces on puttingp = q = 0. This result can also be obtained by integrating the familiar electrostatic energy density over the total volume of the system. To avoid the difficulty of the infinite self-energy of a point charge, it is necessary to assign to each charge a radius b, which we allow to tend to zero in the appropriate manner.By making use of Green’s theorem and of the boundary conditions (17), it is a simple matter to prove that this interaction energy is { :( :>> where we integrate over the sphere of radius b and y& is defined by (5) (with p = q = O ) . (v) The theory of ionic interaction proposed in our paper is only in its initial stages and further investigations are necessary. Perhaps the most important extension is to smaller separations of oppositely charged ions, when one or more co-ordination molecules may be dislodged from their shells. Preliminary in- vestigations indicate, as to be expected, that this is a much more difficult problem than that considered in the present paper.It is also desirable to introduce more elaborate models of the water molecule and to study the effect of varying the number of co-ordination molecules. Some calculations have already been made with the co-ordination number six and the results indicate that there are no radical changes in the interaction energy V(R). Dr. H. L. Friedman (University of S. California) (partly communicated) : It is interesting to compare the results of Young, Wu and Krawetz with the special case of Harned’s rule provided by the Bronsted specific ion interaction theory.1 For 1 Harned and Owen, The Physical Chemistry of Electrolytic Solutions (Reinhold, New York, 1950).GENERAL DISCUSSION 75 this purpose it is convenient to define relative thermodynamic functions as follows, using the Gibbs free energy as an example.for a Raoult's law solution of the same concentration and amount 1 solution - G [ Gr ~ [for the real] The Raoult's law solution is defined by the equations Pl = Pl* - RTMl Cvsm,, p, = ps9 + VsRTln ms, S where p refers to the pure solvent, 8 refers to the solute in the standard state, and M1 is kg/mole of pure solvent. Then the relative function for a quantity of solution containing a kg of solvent is Gr/nlM1 e Gr = G;/M1 + zmsGi where the partial molal relative free energy of the solvent is related to the osmotic coefficient g by S G[ = RTMl(1 - g) 2 m,, S and the partial molar relative free energy of a solute species is related to the rational mean ionic activity coefficient by G,' = V, R T In ys&.These equations permit the easy calculation of Gr from published data for g and yh and it is for this reason that the definition of Raoult's law in terms of a molality concentration scale has been used. The relative enthalpy, entropy, and volume are defined in a similar way. The usefulness of these relative functions is analogous to that of excess functions. They are not the same, however, since the reference solution for excess functions is the ideal solution. For the special case of mixtures of 1 : l-electrolytes Harned's rule, G; = G(m3 = 0) + / % 3 ~ , G; = G',(m2 = 0) + 8323722, gives rise to the following equations in relative functions : G' - (m2/mt)Gr(m3 = 0) - (m3/mt)Gr(m2 = 0) = m2m3(823 + 832)/2 = AGLix, GT(m3 = 0) - G[(mz = 0) = Mit$(/!I23 - &)/2.In these equations Gr, G'(m3 = 01, G'(m2 = O), p23, p32, Gf(m2 = 0) and Gi(m3 = 0) all apply at the same value of m2 + m3 = mi, which of course is the same throughout the series of mixtures to which Harned's rule applies. Q(m3 = 0) means Gr for a solution of m3 = 0, m2 = mi. AGLk is the change in relative free energy when two end solutions of the series are combined to form an amount of mixture containing a kg of solvent. Evidently appropriate differ- entiation yields similar equations for Hr, S' and Vr, and the corresponding mixing quantities. For instance, AfGix = m*m3(/353 + I352)Pr where and AH&, = AH,, as defined by Young, Wu and Krawetz.76 GENERAL DISCUSSION Now it has been shown (cf.Harned and Owen 1) that for solutions described by the Bronsted specific ion interaction theory it is true that p23 -t p32 == 0, and on this basis a non-zero value of this sum arises from effects excluded by the theory. It is presumed that such effects are mainly differences in the interaction of like charged ions. The Bronsted theory is not, however, limited to free energy effects and analogous statements can be made about the sums ,352 -k ,&, On this basis one would expect that p23 + ,832, being a result of the interaction of two different like-charged ions in a mixture, would be independent of the nature of the common ion of opposite charge. This has been observed for ,& + for the C1- + Br- interaction in the presence of the common ions Li, Na, and K by Young, Wu and Krawetz.Mr. H. A. C. McKay (Harwell), (communicated): Young, Wu and Krawetz’s measurements prompt calculations of the molal free energies and hence entropies of mixing for the systems they have considered. The necessary formulae may be derived as follows; they do not seem to have been obtained before. We shall treat 1 : 1-electrolytes only, the extension to other valency types being readily made. Consider the mixing of m 2 moles of a solution of solute 2 with m3 moles of 3 to yield a mixed solution containing m = m 2 + m3 moles, all the solutions being at the same molality m. Then the excess Gibbs free energy of the solutions before mixing is given by /j;3 + p;2, etc- G$ + GT = 2RT(m2 In 7; 4- m3 In y; + m - m2$5 - m$:), where the y’s are activity coefficients, the 4’s are osmotic coefficients, and super- script ’ denotes a quantity referring to one of the pure solutions.The excess free energy after mixing is given by GE = 2 R T ( m 2 In y2 + nz3 In y3+ m - m$). Hence the excess free energy absorbed on mixing is given by A,GE = GE - (G?+ G f ) = 2 R T C m 2 In ( 7 2 / 7 3 + m 3 In ( ~ 3 / ~ 3 0 ) - m+ + nz2$; + m3$30i. If the Harned rule holds for both solutes : In b‘2!?’;> = - a23m3, In (73/1/03) = - a32m2, where the E’S are constant at constant m, then from the Gibbs-Duheni relation at constant rn, we have m3 0 = - J (m23 + a32)m3d1723 = - *(a23 f a32)m$, and similarly with the suffixes 2 and 3 reversed. tained into our expression for A,GE, we have simply Inserting the relations so ob- A,GE =- (a23 + m32)m2m3RT.This is a quadratic expression for A,GE of the same form as that found experi- mentally by Young, Wu and Krawetz for A,H (= A,HE, since the ideal heatGENERAL DISCUSSION 77 of mixing is zero). We may now compare the calculated values of 4,,kGE with the experimental values of AmH, for instance at the maximum in Young, Wu and Krawetz’s curves, where rn2 = 1123 = 0.5, and we may also calculate the excess mold entropy of mixing, AmSE = (AmH - A,GE>/T. The results are as follows : 1 system HCl + LiCl HCI + NaCl HC1 + KC1 HCI + CsCl LiCl + NaCl LiCl + KC1 LiCl + CsCl NaCl + KCI NaCl + CsCl KC1 + CsCl A m H cal /mole 13 32 - 3.5 - 34 21 - 16 - 49 - 9.5 - 9 1.7 A,GE cal /mole 2.7 8.2 3.7 3.1 - 15 - 8 - 27 - 3.7 - 9 1.0 AmSE cal /mole deg.0.03 0.08 - 0.02 - 0.06 0.06 - 0.03 - 0.07 - 0.02 0.00 0.002 There is a general parallelism in the trends of A,H, 4,GE and A,SE, e.g. all three quantities are largest for the HC1 + NaCl sysem and smallest (most negative) for LiCl + CsCl. The cc’s are not always true constants, nor have they always been determined down to 1 m, so the calculation of A,GE is subject to some uncertainty. In some cases the data exist for a more refined calculation. It certainly seems probable that the slight variability of the a’s would give rise to divergences of A,GE from the quadratic form, comparable with those observed experimentally for A,,H. It may be suggested that the measurements of A,H should be extended to higher concentrations, where we sometimes have more information about the behaviour of 3/2 and 7 3 than we have at 1 m.Prof. T. F. Young, Dr. Y. C. Wu and Dr. A. A. Krawetz (University of Chicago) (conzinunicated) : Mr. McKay’s remarks refer to the fundamental reason for the beginning of this work. It was the free energy function (the excess free energy) in which we were originally interested. Only a few systems can be studied directly, however, by techniques now available. The study of volumes and heats was undertaken with the hope that they would yield some suggestions concerning relationships to be expected among the changes in excess free energy which accompany the mixing of electrolytes. Of course, knowledge of the volume and thermal relationships is important in itself regardless of whether that hope is realized. The calculations made by Mr.McKay are very interesting-indeed to us they are exciting. We are grateful to him for his contribution. His tabulation shows that A,CE, A,H and TA,P have in general the same algebraic sign and are of the same order of magnitude. (There is one exception.) When the three quantities are positive TAJE is a little smaller than AmH (within about 30%) and A,GE is therefore the relatively small difference between them. When the quantities are negative TAmSE and AmGE are each approximately half of A&?. Dr. A. A. Krawetz (University OJ Chicago) (partly communicated) : Since our paper was written the investigation has been extended to include the mixing of pairs of electrolytes having no common ion. A useful relationship between the heats of mixing of electrolytes without common ions and those with common 1 see McKay, Trans.Faraday SOC., 1955,51,903, and references given there for or-values.78 GENERAL DISCUSSION ions may be illustrated with the following four salts arranged at the corners of a square : LiCl - 16.06 KCI \ / \+ 0.17 '\ / 4- 0-81 - 31*75/ The heat of mixing of the cations Li+ and K+ in the presence of Cl- (- 16-06 cal/mole) was obtained from the mixing in a calorimeter of the two electrolytes at the top of the square ; the heat of mixing of the same cations in the presence of Br- (- 17.06 cal/mole) was obtained from the mixing of the two electrolytes at the bottom of the square. The heats of mixing of the two anions were determined as indicated in the presence of Li+ and K+, respectively.They are relatively small. The sum of these four values, denoted by CO is, in this example, - 31-51 cal/mole of electrolyte. The mixing of two electrolytes situated diagonally across the square from each other we shall call a cross mixing. In a cross mixing there is no common ion. The mixing of 3 mole of LiCl and -$ mole of KBr in one molal solutions at 25" C is accompanied by the absorption of 0.17 cal. The cross mixing of the other pair of electrolytes causes the absorption of - 31.75 cal/mole of electrolyte. The sum of the two denoted by 2 X is - 31.58 cal/mole. An interesting empirical relation exists between these two sums, namely, Z X =co. In the three sets of electrolytes which have been investigated, the difference between the sums is about 0.5 % or less of the sum.Here it is 0-O7 cal/mole. Prof. C. J. F. BGttcher (Leiden) said : In the verbal introduction to the paper of Young, Wu and Krawetz, attention was drawn to an empirical relation between the six AH values, related to the mixing at 25" C of each pair, chosen from four solutions, each consisting of 0.5 kg HzO and 0-5 mole of the salts MX, MY, NX and NY respectively. The sum CO of the four heats of mixing, related to the combinations with common ions, appears to be practically equal to the sum CX of the two heats of mixing, related to the combinations without common ions. For a theoretical derivation of a relation between Co and CX we introduce the notation, represented by fig. 1. We consider three different ways for the formation of a solution, containing 0.5 mole of each of the four salts and 2 kg HzO.The three different ways and the corresponding AH terms are represented in the diagram of fig. 2. The term AFyiom in (M+ N), used in fig. 2 is defined as the heat of mixing of a solution, contaming 0.25 mole MX, 0.25 mole NX, 0.5 kg HzO and a solution, containing 0.25 mole MY, 0.25 mole NY, 0.5 kg H20. In a similar way the term The mixture of MX and NY is identical to the mixture of NX and MY. This explains the zero AH value of the second step of the third route. The diagram of fig. 2 leads to the following relations : AHations in (x + y ) is introduced. Affd -I- AHd2 = Affanions in M f AHanions in N -I- 2AHcations in (x+ Y), A H d + AHd2 = AHcations in x + AHcations in Y -I- 2AHanions in (M + N). (1) (2)GENERAL DISCUSSION 79 The quantities CO and 2 x are defined by : CO = AHanions in M + AHanions in N + AHcations in x + AHcations in y, (3) CX = A H d $ AH& (4) / \ ,' // Hd2 I \ I / I / Ifdl / FIG.1 .-The six AH terms. AHanions in M j--T-EOnsinn-+ 0.5 mole MY 0.5 kg ~~0 Oa5 NX 0.5 kg H20 AHcations in x + AHcations in Y -+ 0.5 mole NY 1 0-5 kn H?O 0.5 mole M% 0.5 mole MY 0-5 mole NX 0-5 mole NY 1 kg H20 1 kg H20 0.5 mole MX 0.5 mole NX i I 0-5 mole MX 1 2AHcations in (x + Y) c 0.5 mole MX 0.5 mole MY 0.5 mole NX 0.5 mole NY 2 kg H20 4 I FIG. 2.-Three different ways for the formation of a solution, containing 0-5 mole of each of the four salts and 2 kg H20. The corresponding AH values are indicated in the figure. From (0, (3) and (4) it follows that Combination of 12), (3) and (4) yields C x - 2'0 == 2AHcations in (x + y)- AHcations in x - AHanions in y - C x - CO 2aHanions in (M + N) - AHanions in M - AHnnions in w ( 5 ) (6)80 GENERAL DISCUSSION The experimental data of Young, Wu and Krawctz show that the heat of mixing of two cations is almost independent of the nature of the anions.In other words, AHcationsinx is almost equal to AHcatiominy. It may be expected that AH=tionsin(x+y) has a value in between the latter two. Thus the right-hand side of eqn. (5) will only slightly differ from zero, which explains the empirical relation, EX = 20. Prof. T. F. Young, Dr. Y. C. Wu and Dr. A. A. Krawetz (University of Chicago) (communicured): Prof. Bottcher’s analysis of the problem, so far as it goes, is quite correct.His argument closely parallels the intuitive reasoning which led us in our search for an equation such as the cross-square relationship. The two developments proceed, of course, in opposite directions, his to explain the relation- ship and to discover its consequences, ours to derive the equation as a consequence of various plausible assumptions concerning conditions which might be sufficient for a relationship. Prof. Bottcher has correctly stated a sufficient condition for the validity of the cross-square relationship. It is not true, however, that heats of mixing of two cations, e.g. Na+ and Kf are always the same in the presence of different anions e.g. Cl- and NO3-, even when the cross-square relation is valid. (The difference in this case is 25 %.) The results are in accord, however, with the postulate that the heat of mixing in the presence of an equimolal mixture of the anions C1- and NO3- is the arithmetic mean of the two respective heats of mixing in the presence of each of the anions singly.This is a sufficient and necessary condition for the validity of the cross-square relationship. Prof. Bottcher’s straightforward demonstration makes that clear. Let us recapitulate briefly. (i) The heats of interaction between large ions (positive or negative) are small, as usually assumed, but the heats of interaction between small cations and other cations (large or small) are often of considerable magnitude and must not be ignored in thermodynamic treatments of solutions, (ii) These heats of interaction between cations are, as a rough approximation, independent of the nature of the anions present-but they do vary considerably from anion to anion.(iii) The heat of mixing of two cations in the presence of two anions is the weighted arithmetic mean of the heats of mixing of the same cations in the presence of the respective single anions. To this third generalization there will doubtless be exceptions, especially for solutions of very high concentration and when chemical changes are involved such as the formation of weak acids, etc. This empirical relation is, however, sur- prisingly accurate for the few relatively simple cases already investigated. It and the cross-square relationship are equivalent statements of the same fact. Prof. Bottcher’s analysis leads to another result which will be useful in dealing with mixtures of electrolytes.When the molal heat of mixing of two cations in the presence of an equimolal mixture of two anions differs from the mean of the mixings in the presence of the two single anions by an amount 6, there will be an exactly equal difference 6 for the corresponding mixing of the anions in the presence of mixed and single cations. The cross-square equation must then fail by the amount, 26. Mr. H. A. C. McKay (Harwell) said: It is doubtful whether it is ever valid to use the Debye-Hiickel equation (1) In y = - xzZZ*/( 1 + I b / n o ) (notation as in Guggenheim’s paper), to interpret the empirically determined a-value as the distance of closest approach. The reason for this is the existence of a variety of phenomena which add an approximately linear term to the equation: (2) In y = - az2I%/(l + I*U/U()) + AI.It seems indisputable that such a term must arise from ionic hydration, if from no other cause, and that h may in some cases be as large as 0.5.GENERAL DISCUSSION 81 As the concentration is reduced, there is then no region in which the effect of AI becomes negligible in comparison with that of I%z/ao, so we cannot take eqn. (1) as a limiting form of (2). We can easily see this if we expand the Debye-Huckel term : In y =- ~ ~ 2 1 ) + (az2alao + A)I + higher terms. One consequence is that if we attempt to fit the data to an equation of the form of (I), the quantity we interpret as a may really be a + Aao/az2. It is better, of course, to fit the data to an equation of the form of (2).If we assume that h is exactly constant, we can calculate a best fit, and obtain separate values for a and A. However, these may still not be the true values, because X may in fact be a somewhat variable quantity. Prof. E. A. Guggenheim (Reading) (communicated) : I entirely agree with all Mr. McKay’s remarks. If the whole calculation were repeated with an assumed reasonable value of A, I am confident that the final conclusion would remain un- affected, namely there is no positive evidence of any interaction between the ions of the bivalent metals and the sulphate ions other than what would be expected from the model of rigid spheres. Prof. C. W. Davies (Aberystwyrh) said: Prof. Guggenheim’s final paragraph might give the impression that the K values originally derived 1 for 2 : 2-electrolytcs were greatly in error through the use of eqn.(27) in place of eqn. (28). This is not so, as the values were derived from conductivity measurements, at or below 171 - 0.001, using the limiting equations. The error involved in this treatment cannot be accurately assessed at present, but it is a feature of the conductivity method that errors due to the omission from the limiting Onsager and Debye- Huclcel equations of terms of higher order than 1 4 partly cancel out in the evaluation of K. Prof. E. A. Guggenheim (Reading) (communicated) : Some cancellation of errors is possible in Prof. Davies’ calculations on conductivities, but this is not borne out by the detailed calculations on MgS04 communicated by Dr.Prue and Mr. Otter. I repeat that, quite apart from errors of calculation, the value of K for electrolytes as strong as MgS04 has no precise physical meaning unless the value of d (in my notation) is both specified and used as in my formula (15) for the activity coefficient of the free ions. Dr. R. J. P. Williams (Oxford) said: In some recent work we have been seeking spectroscopic evidence for association of two kinds. The first is that referred to by Guggenheim and which is treated under the general heading of ion-pair forma- tion by Bjerrum and others. In this type of association it is assumed that the law of force operating is that between charges at a large distance. Now if ions ap- proach closely, say more closely than 5-10 A, this law of force is no longer applic- able because of the saturation of the dielectric and the introduction of exchange forces. When association at such close distances occurs we shall describe the association as complex formation and it will be convenient to think of the anion and cation as being adjacent to one another. Fortunately there are cases where the distinction between the types of association can be made unambiguously. Charged complexes of cobaltic and chromic ions can be prepared in solution and these ions exchange their ligand molecules but slowly. If we consider these com- plex ions as if they were hydrated simple i0ns-e.g. [CO(NH3)6I3+ is a model for a hydrated trivalent ion, [Co(NH;)5C1]2+ is a model for a hydrated divalent ion- we have models for all hydrated cations and anions between + 3 and - 3. The ion of opposite charge can be chosen at will. To take an example, Posey and Taube 2 studied the association of [Co(NH3)5H20]3+ with sulphate. The total distance between the centre of the anion and the cation is greater than 5.0A in 1 Davies, Trans. Faraday SOC., 1927, 23, 351. 2 Posey and Taube, J. Amer. Chem. SOC., 1953, 75, 1463. Money and Davies, Trans. Faraday Soc., 1932, 28, 609.82 GENERAL DISCUSSION this case. The association could be followed by the change in the ultra-violet absorption spectrum but no change was observed in the visible. The bands in the visible are d-d transitions and their energies depend only upon the molecules which are nearest neighbours to the cation. The d-d bands give us direct evi- dence about complex formation. Now we have observed that the d-d band of cupric sulphate solutions is not in the same position as that of cupric perchlorate and that the d-d band of cobaltous thiosulphate solutions is not in the same position as in cobaltous perchlorate solutions. We conclude that in both cupric sulphate solutions and cobaltous thiosulphate solutions there is a measurable degree of complex formation even at 0.05 M. It is hard to make accurate measure- ments at lower concentrations as the extinction coefficients are too low. It does appear, however, that the association observed is of similar strength to that found from other types of measurement. In his opening remarks the President pointed out that Bjerrum used the fact that cupric sulphate absorption in the visible followed Beer’s law as a proof that cupric sulphate was not associated. There is an alter- native explanation ; the degree of association of cupric sulphate is independent of concentration. This implies that the stoichiometric association constant must decrease with ionic strength increase. This is the case.

ISSN:0366-9033    

DOI:10.1039/DF9572400066

出版商:RSC    

年代:1957

数据来源: RSC

摘要:

11. INCQMPLETE DISSOCIATION INTRODUCTORY PAPER BY CECIL W. DAW Edward Davies Chemical Laboratories, Aberystwyth, Wales Received 17th September, 1957 It is interesting to compare the situation in this field now with what it was when the Faraday Society last discussed Electrolytes in Oxford 30 years ago.1 At that meeting Debye and Onsager presented papers on the new interionic attraction theory, which established the view that complete dissociation was the rule for simple electrolytes in dilute aqueous solution. A number of apparent exceptions from this rule were mentioned by Onsager and others, and it was emphasized that the dissociation constants used in these cases need be considered as no more than a mathematical device to supplement the basic equations for long-range electrostatic interaction.The occurrence of short-range effects out- side the scope of the simple Debye model was not considered at the meeting, nor has any fundamental change of outlook won general acceptance in the inter- vening period, but a considerable amount of experimental information on a wide variety of salts has accumulated, and the present would seem to be a suitable time to bring this under review. A feature of the Bjerrum ion-association hypothesis is that it provides for strong electrolytes the same formal treatment as is applied to a weak electrolyte, and the dissociation constant (for a binary electrolyte) : K =f:azrn/(l - N), where f* is the mean activity coefficient of the free ions, and a is the fractional number of free ions, can be used without distinction to describe the behaviour, in dilute solution, of any electrolyte. To the theoretical chemist this has the drawback that quite different effects are given common treatment, with the corresponding advantage that it enables wide comparisons to be made on a basis free of any preconceptions.For the practical chemist it has the advantage of providing a simple method of predicting behaviour in electrolyte mixtures, at any rate within the range of concentration in which a common activity equation is approximately valid, and of studying abnormal salt effects in reaction kinetics and other diverse fields. Bjerrum's paper2 was limited to the interpretation of activity data, and before reviewing results one will need to be satisfied (i) that thermodynamic properties and conductivities (and, if available, other independent criteria) give the same measure of ion-pair formation and (ii) that the K value ascribed to an electrolyte does in fact express its behaviour over the whole concentration range open to test.By now about fifty electrolytes have been studied by at least two independent methods and, with one or two doubtful exceptions, the same K value has always been obtained within the (sometimes large) uncertainty of the methods. Manganous oxalate provides an example.3 Nineteen measurements of the conductance at 18" C, from high dilutions (m = 1 x 10-4, A = 74) to saturation (rn = 0.0037, A = 20-3), give calculated K's varying irregularly within the limits 1.3 - 1-4 x 10-4; and the sparingly soluble barium oxalate shows about ten times as great a solubility increase in manganous chloride as in potassium chloride of the same ionic strength, leading (if the anomaly is attributed to un- dissociated manganous oxalate) to a K value of 1.1 x 10-4.The data for lan- thanum ferricyanide4 show the apparent precision with which K values can be evaluated. Over a concentration range in which the conductance falls from 90 % 8384 INTRODUCTION of A0 to 56 % of the calculated dissociation constants at each of three temper- atures remain constant to within 0.5 %. With the results now available it should be possible to see whether or not Bjerrum’s treatment is adequate to explain the experimental figures in terms of crystallographic radii and, if anomalies appear, to find some indication of whether it is minor modifications of the electrostatic ion-pair model that are required or whether additional short-range forces must be brought into consideration.For an ion-pair with a mean ionic diameter of, say, 4.5& Bjerrum’s theory would predict no association of univalent ions in water, but K values decreasing, with increasing charge-product, from 0.14 for a 1 : 2-valent electrolyte to 3.2 x 10-5 for a 2 : 4-valent salt. The experimental data show qualitative agreement with this trend, especially in those cases to which an uncomplicated electrostatic theory can be applied with most confidence. A detailed examination, however, is not so favourable to the theory. Table 1 shows a selection of the experimental K values, and of the mean ionic diameters (a values) in A derived from them in ac- cordance with Bjerrum’s equation.Clearly the equation fails for the thallous salts and for the lead and cadmium chloride ion-pairs, which give impossibly small a values, and at the other extreme it fails for magnesium nitrate and the large number of other 2 : 1-valent salts in which ion-pairs cannot be detected. TABLE 1 .-MEAN IONIC DIAMETERS IN WATER (BJERRUM) ion-pair K x 102 a ion-pair K x 102 a TlCl ~ 1 ~ 0 ~ NaNO3 KNO, 30 47 380 160 2-6 1.1 52 15 12 39 23 20 13 6-8 2.7 1.0 1.4 3.3 2.6 1.8 1.4 6-7 4-8 4.3 6.4 5.5 5.3 4.4 8.4 5.6 0-5 0.49 047 0.0013 0.0002 0.021 0.04 0.14 0.0 I 69 0~0010 0.0181 0.0 179 4.5 4.4 4.3 1.7 1.4 4-0 4.2 5.6 5.7 4.0 7.2 7.2 The next step should be to consider the simplifying assumptions in the model used.These are that the ions are spherically symmetrical and unpolarizable, and that the solvent can be regarded as a continuous medium of uniform dielectric constant. All these may be questioned, especially the assumption that the di- electric constant effective for ions in contact, or nearly so, is that of pure water. Our test has introduced the additional assumption that the distance of closest approach of two ions in water will be given by the sum of their crystallographic radii, and this may also be greatly in error.5 Taken together these uncertainties allow for so much latitude that the electrostatic ion-pair concept may apply in many cases where Bjerrum’s equation leads to improbable a values. For instance, thallous hydroxide is weak, and its calculated a value (0.83 A) is far smaller than the sum of the crystal radii (2.84A); yet the Raman spectrum shows no evidence of covalent linking.6 When we examine other thallous salts we find that, without exhibiting the highly specific values expected for covalent compounds, they are all, including the perchlorate, uniformly weak.The only explanation that would seem to fit the facts is that the thallous ion is unsymmetrical, and behaves in ion-pair encounters as if its radius were much less than its crystal radius.C. W. DAVIES 85 Examples where a very drastic revision of the theory are necessary are fur- nished by the alkali-metal sulphates and the alkaline-earth nitrates of table l ; table 2 provides other examples. In these series of related compounds the order of a values is the reverse of that of the cation radii, and ion-pair formation has been regarded as a competition between anions and water molecules for places adjacent to a cation.Table 2 shows that this effect is absent for the carboxylate TABLE 2.-DISSOCIATION CONSTANTS IN WATER AT 25°C OH- NO; acetate- lactate- aminoacetate- s 2 0 3 2 - oxalate2- malonate2- ferricyanide 3- P3093- 10; 0.0026 0.19 0.2 0.043 0.0085 0.015 0.00037 0.00 15 0.0016 0-00049 - 0.050 0.5 0.13 0.2 0.036 0.042 0.010 0~0010 0.0032 04014 0.00034 0.1 1 0-15 0.4 0.1 1 0.12 0.0092 0.0029 0.00 14 - - 0.23 0.12 0.08 0 4 0.20 0.18 0.0047 0.0047 0.019 0.00 1 3 0.00045 ions. An adequate theory of ion-solvation should not only explain these facts but should throw general light on experimental dissociation constants, and we can hope for guidance here from section III of this Discussion.When an ion is introduced into water it will interfere with the semi-crystalline structure of the water molecules in its neighbourhood, and will also tend to impose a new order by the re-orientation of adjacent water molecules. The ion-pair is not considered in the discussions of Frank7 and Gurney 5 , but the same considerations will apply; an ion-pair dipole will form the centre of a zone of affected water molecules, and the process : ion-pair-solvent complex + cation solvate + anion solvate, may be accompanied by substantial free energy and entropy changes which are addi- tional to those calculable from Coulomb's law, and may markedly affect K values. Moreover the solvent structure, and the associated energy changes, in the zone of influence of an ion-pair are likely to be very sensitive to small changes in the ionic separation distance, so that in cases where stable ion-pairs are formed these may have configurations and diameters that are more sharply defined than in the original ion-pair concept.Table 3 gives the pK values for some further salts of bivalent cations. The sulphates are almost identical, with pK's of the right magnitude, and can clearly be considered electrostatic ion-pairs. The lower rows of the table present an TABLE 3.-- log K FOR SOME SALTS OF BIVALENT CATIONS Cu Ni Zn Co Mn Cd Mg Ca sulphate 2.30 2.40 2.27 2.47 2.28 2.29 2.16 2.28 thiosulphate - 2.06 2-40 2.05 1-95 3.92 1.84 1.98 succinate 3.40 2.39 - 243 2.25 2-82 2.05 2-0 malonate 5.75 4 1 3.68 3.7 3-33 3-29 2.83 2.60 oxalate - 5 3 4-89 4.7 3.91 3.8 3.43 3.0 aminoacetate 8.62 6.18 5.52 5.53 3.44 - 2-07 1.38 entirely different picture, with an enormous spread of pK values, and the transi- tional metals arrange themselves in the order previously reported for chelate com- pounds with a variety of ligands.8 Here a new sort of short-range interaction is the dominating influence.It is also clear that a rigid distinction between electro- static ion-pairs and complexes is hard to maintain : with the succinates the influence86 INTRODUCTION of chelation is almost lost but still seems to be apparent in the copper salt: cadmium thiosulphate, with its tendency to form complex anions, falls out of line; and even with the calcium ion the influence of chelation seems quite clear in the or-amino and a-hydroxy salts (table 2).This review of the exploratory work so far carried out in this field is necessarily brief and very incomplete. Hardly anything has been said about the methods used and their inherent uncertainties: an error of 20 % or more in the values quoted would not affect the conclusions that have been drawn. But efforts are now being directed to improving experimental accuracy, and one important result is to make available more reliable data for the temperature dependence of K values. Nancollas in his paper discusses quite a considerable body of thermo- dynamic data. Another result of improved accuracy will be to make the com- parison of different criteria of ion-pair formation more critical, and Prue and his co-workers are concerned with this vital point. The spectrophotometric method they use to study ion-pairing is relatively new-the supposed absence of such effects was once widely quoted as proof of complete dissociation-and Bale, Davies, Morgans and Monk give further applications of the method, which is clearly not as simple and free from interfering effects as is sometimes thought. All these papers provide material for valuable discussion. We also have a further interesting paper on strong acids from Redlich and his co-workers, who restrict the use of a mass-action treatment to chemical changes. 1 Trans. Faraday SOC., 1927, 23, 334-542. 2 Bjerrum, Kgl. danske vidensk. Selsk., 1926, 7 , no. 9. 3 Money and Davies, Trans. Faraday SOC., 1932, 28, 609; J. Chem. SOC., 1938, 2098. 4 Davies and James, Proc. Roy. SOC. A , 1948, 195, 1 16. 5 Gurney, Ionic Processes in Solution (McGraw-Hill, New York, 1953). 6 George, Rolfe and Woodward, Trans. Faraday Soc., 1953,49, 375. 7 Frank and Evans, J. Chem. Physics, 1945, 13, 507. 8 Mellor and Maley, Nature, 1948, 161, 436. Irving and Williams, Nature, 1948. 162, 746.

ISSN:0366-9033    

DOI:10.1039/DF9572400083

出版商:RSC    

年代:1957

数据来源: RSC