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.