BI-IONIC POTENTIALS BY F. HELFFERICH Max-Planck-lnstitut fur physikalische Chemie Gottingen Received 30th January 1956 Assuming steady-state conditions the Nernst-Planck equations can be used to deter-mine the concentration gradients which are found to exist across not only the membrane but also such adherent liquid “ films ” as are unaffected by stirring. By use of the inter-face concentrations thus obtained the membrane potential can be calculated. In contrast to concentration cells bi-ionic cells have a strong tendency towards film control which, in certain cases cannot be overcome even by violent stirring. Therefore with counter-ions of different valences a change in stirring conditions may give rise to very large membrane potential changes and to a reversal in sign of the potential.In addition the Nernst-Planck equations permit the calculation of the concentration profiles ; it is found that the slower ion is accumulated by the membrane. The calculated potentials are in fair agreement with experimental values obtained with and without stirring. 1. INTRODUCTION An ion exchange membrane separating two electrolyte solutions maintains across itself an electric potential difference which can be measured with suitable electrodes. Potentials between solutions of the same electrolyte but of different concentrations are customarily defined as concentration potentials and those between solutions of different electrolytes but of equal equivalent concentration as bi-ionic potentials : solution a membrane solution /I AX [ containingAandB 1 BX If the concentrations at the interfaces are known the bi-ionic potential can be cal-culated from the general equation for the e.m.f.of a Voltaic cell. In the preceding paper 1 Scatchard’s quasi-thermodynamic treatment 2 has been extended to systems having concentration gradients not only in the membrane but also in the solutions. Thus concentrations at the interfaces were calculated from experimental e.m.f. data. In this paper the Nernst-Planck flux equations are used to predict membrane potentials and concentration profiles which can be compared with experimental results. 2. PREVIOUS THEORIES (1.1) z*c = ZBC$ Bi-ionic potentials have been reported by Michaelis,3 Marshall et al.,4 Meyer and Bernfeld,s Sollner et a/.,6 7 Manecke,8 and Wyllie.9 Using the Henderson equation 10 for the liquid-junction potential and in it replacing concentrations by activities Marshall 4 derived equations for the bi-ionic potential.His approach however as Wyllie 9 pointed out later while leading to the diffusion potential within the membrane ignores interfacial “ Donnan ” potentials which make an essential contribution to the membrane potential especially where counter-ions of different valences are involved. The mechanism of the bi-ionic potential was considered in detail by Sollner.6.7 He found that not only was the relative mobility of the counter-ions important but also the selectivity of the membrane material played a significant role. According to his views when the membrane is in a bi-ionic cell one ion is preferred to the same extent as when the membrane is in equilibrium with a mixed solution containing 8 84 BI-IONIC POTENTIALS both counter-ions.His equation for the membrane univalent electrolytes), reflects the two factors relative mobility (DA/EB) in, ( y B / y A ) by the membrane.* potential (restricted to uni-(2.1) and preferential adsorption Wyllie 9 combines the theories of Sollner and Marshall. He uses the Henderson equation (with activities) for the diffusion potential in the membrane adds the Donnan potentials and as Sollner corrects for the assumed preferential adsorp-tion by multiplying the mobility ratio with the activity coefficient ratio. His equations are as is (2. l) in accordance with those derived quasi-thermodynamically (except for correction terms).It should be mentioned here that the Nernst-Planck treatment though it leads to basically the same equations indicates that any preferential adsorption taking place in bi-ionic cells is completely different from that in equilibrium systems and has no effect on the membrane potential the activity coefficient ratio in (2.1) being due to the difference in the Donnan potentials ; on the other hand in Wyllie’s approach (where Donnan potentials are included) the use of activity coefficients in the Henderson equation introduces a factor which is exactly compensated by his correction for the assumed preferential adsorption. The flux equations have been integrated in a general form by Goldmann,ll Teorell,l2 and for ions of different valences by Schlogl.13 Bi-ionic potentials are only a special case of this more general solution.However this procedure requires, as does any previous approach the knowledge of the concentrations at the inter-faces. 3. FLUX EQUATION TREATMENT In bi-ionic cells concentration gradients exist across both the membrane and adherent Nernst diffusion “ films ” which are unaffected by stirring.14 The flux equation treatment consists in calculating by use of the steady-state condition the concentrations at the interfaces and from these the diffusion potentials in films and membrane and the Donnan potentials at the interfaces and hence the membrane potential. (a) SIMPLIFYING ASSUMPTIONS treated (i) (ii) (iii) (iv) (4 (Vi) (vii) (viii) Only cation exchange membranes are discussed.Anion exchangers can be in the same way. Furthermore it is assumed that: The concentrations of the solutions are so small compared with the con-centration of fixed ions in the membrane that the concentration of anions in the membrane may be neglected (zx < C, C,). Transfer of solvent through the membrane may be neglected. The ratio of the diffusion constants of the counter-ions is independent of mole fraction (EA/OB = const.). The membrane/solution interfaces are sharp. Changes and gradients of swelling pressure may be neglected ; this implies a constant concentration of fixed ions in the membrane. The motion of the solution near the interface may be represented by a Nernst diffusion “ film ”. At the interfaces equilibrium is preserved between adjacent infinitesimal thin layers in membrane and film (i.e.no interfacial resistance to diffusion.) Only the steady state is considered. _ -* Heteroporosity is stressed as another important factor. Tn the forthcoming dis-cussion we account for this by using diffusion constants h experimentally determined by methods (self-diffusion or conductivity measurements) in which heteroporosity plays the same role as in bi-ionic cells F. HELFFERICH 85 Most of these assumptions have been made previously by other authors who discussed their validity in detail. If membranes with high water-permeability are used a considerable error may be introduced by (ii) despite the fact that in bi-ionic cells such as (1.1) the activity of the solvent is practically equal in both solutions .I5 The self-diffusion constants of ions in ion-exchange resins are known to depend on resin composition; however the ratio DJDB of the two diffusion constants in the same resin containing both counter-ions A and B is found to be reasonably constant and independent of mole fraction cA/(cA + cB),16 as assumed in (iii).Assumption (v) is certainly an oversimplification if counter-ions of different valence are involved. The error is larger with concentration profiles than with membrane potentials. (6) CALCULATION OF THE CONCENTRATIONS AT THE INTERFACES In contrast to concentration cells with dilute solutions where only a very small diffusion can occur we are concerned in bi-ionic cells with a continuous exchange of the counter-ions across the membrane i.e. with large ionic fluxes and corres-ponding large concentration gradients which may be located in both membrane and adherent films.In the steady state the fluxes $i of all ionic species i are constant throughout membrane and films i.e. the value of each flux is constant through all planes parallel to the interfaces no matter whether the planes are located in film or membrane i.e., According to. assumption (i) the membrane is impermeable to anions ; therefore - -4 i = 6 (3.1) $x = 0. (3 .a Electroneutrality requires that c Zi$i = 0. i (3.3) At the interfaces equilibrium is assumed : = KiCi/C (3.4) (C f CAZA + CBzB ; the partition coefficients K i are defined by (3.4) ; note that for zA + zB the ~i depend on C even in the ideal case.) By use of the Nernst-Planck flux equations (3.5) the concentrations at the interfaces can be calculated from (3.1-4).However a straightforward application of (3.5) leads to a complicated expression and is there-fore not a profitable line of approach. Instead we define integral diffusion con-stants DAB by 1 1 (C' and C" = concentrations at the left- and right-hand boundary respectively, of the considered layer (membrane or film) of thickness 1.) Using (3.6) instead of (3.5) we find for the concentrations at the interfaces a and b C," == (C - C:ZB)/Z 86 BI-IONIC POTENTIALS In the calculation the additional assumption has been made that CAZA + CBZB = C = const. (3.8) and thereby that Cx = const. Whereas in the membrane this follows from the assumptions (i) and (v) it is not strictly correct for the films because due to & = 0, the electric patential gradient acting on the anions has to be compensated for by a corresponding concentration gradient (see also footnote to 5 3c).If A and B are isotopes of the same cation (DA = DB = D) (3.7) reduces to This expression is exact as far as the film concept holds and it reveals more clearly than (3.7) the effect of the parameters D/E C / c and d/8 on the distribution of the concentration gradients. For DCd/b?8 > 1 we have C,” = Cb = C and Ci = C,” = 0 ; the concentration gradients are located completely within the membrane (ideal membrane control). For DCd/EC8 < 1 we have C = Ci = Cg = Cg = C/2 ; the concentration gradients are located completely within the films (ideal film control).The mechanisms correspond to what is known as “ par-ticle kinetics ” and “ film kinetics ” respectively in ion-exchange kinetics. For the evaluation of (3.7) approximate values of the DAB have to be estimated. This is done in the following way. We neglect gradients of activity coefficients and assume constant Di. d8/dx can now be eliminated from (3.5) by use of (3.3) and (3.8) giving As $x = 0 there are only two fluxes $A and +B. (3.10) The second term in (3.10) vanishes when the activity coefficients are independent of the mole fraction CA/(CA + CB) and thereby of the space coordinate x. Under this additional assumption (3.10) is readily integrated (again using (3.8)) for zA =+ z, an F. HELFFERICH 87 The limiting case ( 3 . 1 2 ~ ~ ) shows that the diffusion rate is essentially governed by the diffusion constant of the r a w ion.This rule which applies to any diffusion of two ionic species interrelated by the electroneutrality condition is inherent in the NernSt-Planck equations the electric term contains as a factor the concentration Cj of the species; the potential gradient produced by the diffusion process will therefore have a large effect on the species present in high concentrations but only a small effect on the species present in low concentration. In equilibrium with dilute solutions the membrane shows a pronounced pre-ference for the counter-ion of higher valence. This is evident from the equilibrium condition (aA/ZdZB = (aB/ZB)ZA (3.13) Hence in systenis with Z < z, if a concentration gradient builds up in the film situated on the cc side the membrane accumulates the ion B and the integral diffusion constant D A B approaches DA.On the other hand the concentration profile within the membrane remains relatively insensitive to concentration changes in the film on the p side. The evaluation of (3.7) requires furthermore the knowledge of the film thickness 6 and the partition coefficients Ki. The latter can be calculated directly from ion-exchange equilibria. 6 may be determined from one e.m.f. measurement-for a given stirring rate and cell geometry-and used for the prediction of all other e.m.f. values. In computing the concentrations at the interfaces (3.9) is used to make a first rough estimate which will often reveal that the system is practically membrane-controlled or film-controlled.In either case the membrane potential can be calculated directly from (3.17) or (3.1 8) respectively. If neither applies the first estimate-for counter-ions of different valences in combination with (3.13)-can be used to obtain approximate integral diffusion constants DAB from (3.12) or (3.11). Substituting these values in (3.17) gives a second approximation for the concentrations at the interfaces. It must be emphasized that this procedure leads to only a rough approximation for the concentrations at the interfaces. In most cases however the error thereby introduced into the calculation of the mem-brane potential does not exceed a few millivolts. (c) CALCULATION OF THE MEMBRANE POTENTIAL The membrane potential is the sum of the Donnan potentials at the interfaces and the diffusion potentials in membrane and films.The Donnan potentials are given by (3.14) (negative for side cc positive for side p). The diffusion potential can be obtained from the Henderson equation (without activity coefficients!) or if the necessary data are available from the Nernst-Planck equations. Using ( 3 . 9 (3.3) and (3.8) we obtain (3.15) (3.16 88 BI-IONIC POTENTIALS Except for the second term (3.16) is equivalent to the Henderson equation without activity coefficients. This becomes evident when C = CAZA + CBZB is substituted. Henderson's assumptions being granted (as is the case here since C = const.) the activity coefficients appear only in the correction term which makes a cohtribu-tion only if gradients of activity coefficients exist.Without giving a general equation for the membrane potential-which is readily obtained from (3-14) and (3.16) by substituting appropriate values for the interface concentrations-we would stress the two limiting cases. For ideal membrane control the diffusion potentials in the films vanish and the boundary conditions for the diffusion potential in the membrane become C~Z = C, and C~Z = 0. After substituting in (3.16) and adding the Donnan potentials we find the membrane potential The first and third terms arise from the diffusion potential and the others (including that containing the " preferential adsorption factor ") from the Donnan potentials. If we retain C = const. which particularly in this case is a rather forced assumption, the sum of the Donnan potentials vanishes also and we are left with For idealfilm control the diffusion potentials in the membranes vanish.which differs from the liquid-junction potential without membrane only in that the terms Dxzx under the logarithm are missing.* (d) CONCENTRATION PROFILES IN THE MEMBRANE Integrating (3.10) from x = 0 at the interface a to a variable value of x within the membrane (assuming constant Di) the concentration CA is obtained as a function of the space coordinate x. The resulting equation cannot be solved with respect to CA for z -+ z,. As in this case the evaluation is somewhat tedious the discussion below is for simplicity restricted to 2 = zB and to ideal membrane control. Activity coefficients enter (3.10) only in form of the expression [d In (Tp/e)ldx].The second term in (3.10) disappears when K is inde-To begin with we integrate (3.10) -zB yA/yB-= %A K is equivalent to the ion-exchange equilibrium constant defined by K G (CB/aB)"B (aAFJzA. pendent of mole fraction and thereby of x. under this additional assumption and obtain -= Cx/d. for DA = * The assumption C = const. is as (3.5) shows compatible with Dx = 0. Actually the diffusion potentials build up anion concentration films. A more detailed calculation which is omitted here shows that C, DB DB. (3.19) # = 0 only for gradients in both is far from being constant ; for the conditions Z = Z = z and K j independent of total concentration it is found that Cg = 2cE/[(&/D.$z + z x ) h -t- 11 and ci = 2c:/[(DAD,)(' +zx)lz~ f 11.For the potential however we obtain an expression identical with (3.18) except for a slight difference in the correction term. Thus the assumption C = const. is seen to introduce practically no error into the calculation of the membrane potential even for ideal film control where it is farthest from representing the actual conditions F . HELFFERICH 89 Eqn. (3.19) is plotted in fig. 2 for different values of D A / D B ; it is seen that the membrane accumulates the slower ion. Qualitatively this result follows from the rule that the ion present in smaller concentration governs the diffusion rate (3.12~). Near the interface a the diffusion _ _ FIG. 1 .-Characteristic con-centration profile for steady-state counter-diffusion of two ions.A and By with z < 2, and BB < 5 (schematically). The system is not completely membrane controlled hence the membrane accumulates the ion of higher valence. Due to DB < DAY the smaller CA/CB is the steeper is the profile. - _ solution d f i l m membrane f i l m solutron 3c FIG. 2.-Concentration pro-file of the ion A within the membrane for steady-state counter-diffusion with ideal membrane control Z = z,, and d In K/dx = 0. (a) D A I D B = 1, (b) = 2, (c) = 5, (4 = 15, - -constant of the slower ion B is effective (because of FB < zA) ; near the interface b this is true of the faster ion A. At the same time the steady-state condition requires that +i - const. Hence the concentration profile must be steeper near a than near b 90 BI-IONIC POTENTIALS Now we consider the effect of the quantity d In K/dx in (3.10) equating for simplicity D = D,.As the second term in (3.10) contains both FA and z,, it becomes equal to zero at both interfaces ; hence the slope of the profile is the same at each interface. If d1nKldx is a monotonic function as is usual S-shaped profiles are obtained which are steeper in the middle of the membrane than at the boundaries for d In K/dx > 0 and flatter for d In K/d? < 0 (fig. 3a). The profile develops a bulge when d In K/d% has a maximum or minimum (fig. 3b) ; which ion then predominates depends on whether the former or the latter is the case and not on whether In K is positive or negative so that the preference shown by the membrane in equilibrium systems is irrelevant.- -0 ) (b) FIG. 3.-Steady-state concentration profile of ion A for ideal membrane control and D = 5 schematically. -(a) d In K/dxB > 0 (b) d In K/dSi- has a minimum. Selectivity is a membrane property that can come into play only at the inter-faces. Thus if fixed boundary conditions are given at the interfaces-as is the case when ideal membrane control is assumed-the selectivity has no bearing on the concentration profiles. However the more the system changes over to film control the more the effect of the selectivity on the concentrations at the interfaces increases. In the limiting case of ideal film control Sollner’s assumption (pre-ferential adsorption the same as in equilibrium with a mixture of both solutions) is valid but now the membrane potential is given by quite a different expression which contains neither the preferential adsorption factor nor the intra-membrane mobility ratio.4. DISCUSSION AND EXPERIMENTAL EVIDENCE (a) THE FILM HYPOTHESIS The approach outlined above differs from any previous theory in that it can account quantitatively for partial or complete film control. The film hypothesis is therefore examined first. A rough estimate with eqn. (3.9) shows that deviations from ideal membrane control are to be expected for DCd/z& < 50 and that complete film control will occur for DCd/De8 < 0.1. Taking a characteristic ion-exchange membrane with C (ion-exchange capacity) w 1 N D / D = 5 and d w 0.1 cm we find that main-tenance of complete membrane control requires the reduction of film thickness 6 to less than 10-3 cm for solutions of 10-1 N 10-5 cm for 10-3 N and 10-7 cm for 10-5 N whereas complete film control must be anticipated for 8 > 0.5 cm, 5 x 10-3 cm and 5 x lO-5cmY respectively.Stirring by usual methods only F. HELFFERICH 91 reduces the films to about 10-3 cm and a reduction to below 10-4 cm is difficult to achieve Thus bi-ionic cells with dilute solutions are likely to be partially or completely film-controlled. In cells with zA = z the second term in (3.17) is equal to zero and the membrane potentials for both ideal membrane control and film control are seen to be fairly independent of concentration. Actually their values can be very similar when D,/D w DA/DB (compare fig. 8a and b in the preceding paper 1).This may account for the fact that with the exception of Sollner 697 and Manecke,8 the significance of film control escaped the notice of previous investigators. For Z =l= z, however, due to the second term in (3.17) the membrane-controlled potential is linearly dependent on log C where the potential changes by (z - zB)/zAzB X 59 mV for an increase in C by a factor of 10 whereas the film-controlled potential is again practically independent of concentration. It is therefore evident that both potentials may differ from each other considerably in absolute value and even in sign. In the preceding paper,l bi-ionic potentials obtained with HCl + CaC12 and NaCl + CaC12 were reported (fig. 8c-d and table 2). It is seen that violent stirring can maintain membrane control in most cases but not in all.In agreement with (3.9) and (3.7) deviations are more pronounced (i) with single membranes than with packs of four (ii) with the more permeable phenosulphonic than with the Amber-plex membranes (iii) with more dilute solutions than with more concentrated and (iv) in the system HC1 + CaC12 more than in the system NaCl + CaCli (in the former DAB/DAB is relatively small due to D A B + &+ according to (3.12~2)). Nearly all the experimental results reported in that paper with or without stirring on either or both sides agree to within & 5 mV with the calculated values using 6 = 5 x 10-5 cm with stirring and 6 m 5 x 10-2 cm without stirring." The film hypothesis is thus seen to account for the extreme sensitivity towards stirring of bi-ionic cells containing counter-ions of different valences and to explain those experimental potential values which disagree with previous theories.It has been pointed out that in bi-ionic systems with ideal membrane control the concentration profiles of the counter-ions are governed by the relative mobility of the ions and to a certain extent by the dependence of the equilibrium constant on mole fraction whereas the selectivity shown by the membrane in equilibrium systems (i.e. the absolute value of the equilibrium constant) is irrelevant. Ex-perimental evidence on this point has been obtained. Concentration profiles were measured directly by analysis of a multiple membrane pack between 0.1 m solution of NaCl and HC1 and of NaCl and KCI.In the system NaCl + HCl the equilibrium constant is very close to unity and independent of mole fraction. The profile shows that the slower ion Na+ is ac-cumulated to an extent corresponding roughly to a mobility ratio &/ENa= 6 (fig. 5). The ratio of the specific conductances of H+-resin and Naf-resin is 6.1. In the system NaCl + KCl the equilibrium constant decreases with increasing mole fraction of K+ the latter being preferred to a considerable degree. The profile shows the characteristic S-shape predicted by the theory for d In K/d%B < 0, the initial slopes at the phase boundaries corresponding to a mobility ratio The results are thus in excellent agreement with the predictions from the flux equation treatment and they rule out the hypothesis that in bi-ionic systems the counter-ions are present within the membrane in the same ratio as if the membrane were equilibrated with a solution prepared by mixing equal volumes of the two solutions of the bi-ionic system since this hypothesis would require absence of any accumulation in the system NaCl + HCl and a definite accumulation of K+ over Na+ in the system NaCl + KC1.Full details will be published elsewhere. - -&/ENa N 1.2 (fig. 4). * This is the distance for linear diffusion in the holes of the cell 92 B I - ION I C POT E N TI A L S (b) NON-STEADY STATE CONDITIONS The approach thus far has required the steady state. But since the integration leading to (3.16) is carried out over one variable (CJ only the potential difference between the two boundaries of the layer under consideration is determined solely by the boundary conditions and is independent of the concentration profile within the layer.Therefore the potential reaches the steady state value as soon as the concentrations at the interfaces assume the values corresponding to the steady state; usually this occurs long before the profile within the layer has become stationary (compare the results in the preceding paper ; 1 also the glass electrode seems to behave in a similar manner). For this same reason a relative abundance of one species within the membrane has no effect on the membrane potential. KCI x= 0 d FIG. 4. (C) GENERAL CONCLUSIONS Bi-ionic cells such as (1-1) represent an excellent tool for the study of ion-exchange kinetics they permit steady-state measurements under the simplest geometrical conditions (one-dimensional diffusion).The limiting cases (3.12) apply also to " particle kinetics " in granular ion exchange resins. Thus it becomes evident that the assumption of an " effective diffusion constant " DAB independent of composition (i.e. mole fraction) used in previous approaches,l7 cannot be upheld when %A is considerably different from 5,. From a modified eqn. (3.9) a rough but simple prediction can be made for ion exchange with granular resins as to whether " film kinetics " or " particle kinetics " is to be anticipated. 5. EXTENSIONS The treatment can be readily extended to (i) systems with different film thicknesses on either side of the membrane, (ii) systems with different total concentrations in both solutions (Ca + CP) an (iii) (iv) F .HELFFERICH 93 systems in which one or both solutions contain both counter-ions diffusion through the membrane of an electrolyte (as in concentration cells) or a non-electrolyte. / (Cf CB" =I= 0). Interfacial resistance to diffusion can be accounted for by introducing inter-The experiments however facial permeability parameters gi as defined by Scott.18 provide no evidence in favour of this phenomenon in ion exchange. NoCl x-0 d FIG. 5 . The method does not lend itself to systems with more than two permeating species i.e. with more than two counter-ions or with concentrations which permit anionic fluxes. In such cases either Wyllie's approach must be used or Schlogl's 13 which is more accurate but more time-consuming.Both however are restricted to ideal membrane control and neglect gradients of activity coefficients. The experiments on which this work is based were initiated at the Massachusetts Institute of Technology under the auspices of Professor G. Scatchard. The author is indebted to him and to Dr. R. Schlogl for advice and many helpful discussions. SYMBOLS a activity X mole fraction, C concentration (moles/I.) z ionic valence (negative for anions), d membrane thickness y activity coefficient, D diffusion constant 6 film thickness, E electric potential difference K partition coefficient, K ion-exchange equilibrium constant 0 electric potential, r (differential) transference number 4 ionic flux. x space coordinate normal to interfaces, Subscripts A and B refer to cations X to the anion i to any ionic species.Superscripts C( and p refer to the left and right solution a and b to the interfaces on the left and right side of the membrane. Bars denote quantities within the membrane 94 DIFFERENTIAL RATES OF PERMEATION 1 Scatchard and Helfferich this discussion p. 70. 2 Scatchard J. Amer. Chem. SOC. 1953 75 2883. 3 Michaelis Kolloid-Z. 1933 62 2. 4 Marshall and Krinbill J. Amer. Chem. SOC. 1942 64 1814 and later publications. 5 Meyer and Bernfeld Helv. chim. Acta 1945 28 962. 6 Sollner J. Physic. Chem. 1949 53 121 1 and 1226, 7 Sollner Dray Grim and Neihof Ion Transport Across Membranes (Acad. Press, N.Y. 1954) p. 144. Dray and Sollner Biochim. Biophys. Acta 1955 18 341. 8 Manecke 2. Elektrochern. 1951 55 672. 9 Wyllie J. Physic. Chem. 1954 58 67. 10 Henderson 2. physik. Chem. 1907 59 11 8. 11 Goldmann J. Gen. Pliysiol. 1943 27 37. 12 Teorell Z. Elektrochem. 1951 55 460. 13 Schlogl Z. physik. Chem. 1954 1 305. 14 The film concept was used in ion exchange reactions first by Boyd Adamson and 15 This effect is discussed by Schlogl Z. physik. Chem. 1955,3 73 and this Discussion. 16 See for instance Soldano and Boyd J. Amer. Chem. Soc. 1953 75 6107, 17 See for instance Kressman and Kitchener Faraday SOC. Discussioizs 1949 7 90. 18 Scott Tung and Drickamer J. Chem. Physics 1951 19 1075. Myers J. Amer. chem. Soc. 1947 69 2836. Reichenberg J. Amer. Chem. SOC. 1953 75 589