J . Phys. Chem. 1990, 94, 1047-1050 where T , and TA are constants. This is eq 5 of the text and corresponds to the leading portion of a 1/T perturbation expansion, treated by Zwanzig.’ The T, parameter is positive and TA usually is also, since experimental results show that T typically decreases with temperature (although exceptions are not rare, as seen in Table Ill). The constants may be related to the Boyle temperature T g ,at which the second virial coefficient B2 vanishes. Since B, is proportional to (1 - T / T ) ,we have x2-x-(TA/70) with x = TB/TO;hence
=0
(B2)
x = y2{1 + [1 - 4 ( r A / 7 0 ) ] ” 2 ] (B3) When the temperature dependence of T is weak enough to neglect, T i= T , becomes approximately equal to the Boyle temperature. Appendix C Coefficients for Temperature Dependence Table I11 tabulates the quantities T &/dT and T ar/aT. From , which occur in eqs 3 these the coefficients (uo, To) and ( T ~ TA)
1047
and 5 may be obtained by using
-T &/aT = ( U O / ~ ~ ) [ U ( T ) / U ~ ] ~ ( T (/CTl O ) )”~ and -T aT/aT =
TO( T A /
T)
(C2)
with everything evaluated at the reference temperature (20 “C). This yields uo =
T, =
+1
u7/6[u
2 au/ar]-l/6 ~
(C3)
q[u+ 1 2 ~ a ~ / a r ] / [ 1 2 ~ a ~ / a ~(]c 4p ) TO
=
T
+ Tar/aT
T* = T[-T a T / a q / [ T
+ T aT/ar]
(C5)
(C6)
Note that Table 111 retains insignificant figures in order to avoid round-off error in such calculations.
Frequency Response Analysis of Surface Reactions in Flow Systems J. R. Schrieffert and J. H. Sinfelt* Corporate Research Science Laboratories, Exxon Research and Engineering Company, Annandale. New Jersey 08801 (Received: April 28, 1989)
A frequency response analysis is presented for two examples of surface processes occurring in flow systems. The first is concerned with the adsorption and desorption of a simple gas. The second involves a two-step catalytic sequence in which an adsorption-desorption step is followed by a reaction step. Both examples illustrate the utility of frequency response data in yielding basic kinetic information beyond that obtainable with a flow system in the conventional steady-state mode of operation.
I. Introduction Investigations of the kinetics of surface-catalyzed reactions are commonly conducted under steady-state conditions in flow systems. Frequently, the rate data are obtained at low conversion levels to minimize the variation of reactant concentrations throughout the catalyst bed. The term differential reactor is often used in describing a system operated in this manner. In a typical investigation, one varies the concentration of a reactant in a stream of gas entering the catalyst bed and observes the effect on the rate of reaction. The data are then fitted to a rate equation derived for a postulated sequence of reaction steps. While studies of this kind represent an important part of any catalytic investigation, the amount of information that one can obtain on the rate constants of the various steps involved in a reaction is limited. In studies of fast chemical reactions in solution, valuable information on rate constants has been obtained with the aid of relaxation methods, in which a system in equilibrium is perturbed slightly by a rapid change in some external parameter which influences the The adjustment of the system to a new equilibrium is followed and characterized by one or more relaxation times. In one version of the method, the external parameter influencing the equilibrium is oscillated at some frequency w . The response of the system, as indicated by the concentration of a reactant or product, depends on the time required to attain the new equilibrium. If this relaxation time is large compared to the period of the external oscillation, the reaction will not sense the oscillation. No oscillations in concentration are ‘Permanent address: Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106.
0022-3654/90/2094-1047$02.50/0
then observed. At the other extreme, the relaxation time may be so small that the concentration will oscillate in phase with the external oscillation. When the relaxation time is comparable to the period of the external oscillation, the concentration oscillates at the imposed frequency but is out of phase with it. The phase lag provides a measure of the relaxation time, which in turn contains information on rate constants. Frequency response studies of this type have been utilized very little for kinetic investigations in surface catalysis. An application to chemisorption kinetics has been reported by Naphtali and P ~ l i n s k i . ~In , ~their investigation the pressure variation induced by a sinusoidal variation of the volume of a closed adsorption system was measured. More recently, similar types of investigations have been reported by others.s-8 Adsorption and desorption rate constants have been reported for the chemisorption of ethylene on zinc oxide6 and of hydrogen on a rhodium catalyst.8 The application of frequency response methods to surface reactions in flow systems is a natural extension of these investigations. We consider such an application here. The essence of the method is very simple. The concentration of a reactant in a fluid (1) Eigen, M. Discuss. Faraday SOC.1954, 17, 194. (2) Eigen, M.; de Maeyer, L. In Technique of Organic Chemistry. Vol VIII - Part I I . Rates and Mechanisms of Reactions, 2nd ed.; Weissberger, A., Ed.; Interscience Publishers: London, 1963; pp 895-1054. (3) Naphtali, L. M.; Polinski, L. M. J . Phys. Chem. 1963, 67, 369. (4) Polinski, L. M.; Naphtali, L. M. Adu. Caral. 1969, 19, 241. (5) Yasuda, Y. J. Phys. Chem. 1976,80, 1867. ( 6 ) Yasuda, Y. J . Phys. Chem. 1976,80, 1880. (7) Goodwin, J. G., Jr.; Sayari, A.; Lester, J. E.; Marcellin, G. Am. Chem. SOC.,Diu. Per. Chem., Prepr. 1984, 29, 690. (8) Marcellin, G.;Lester, J. E.; Mitchell, S. F. J . Catal. 1986, 102, 240.
0 1990 American Chemical Society
Schrieffer and Sinfelt
1048 The Journal of Physical Chemistry, Vol. 94, No. 3, 1990
stream prior to entering a vessel containing a bed of adsorbent or catalyst particles is caused to oscillate with frequency w about an average value. At a given distance through the bed, the concentration of a product species formed via a surface reaction also oscillates about some average value with the same frequency w . As a result of finite reaction times, there will in general be a phase shift between the oscillating parts of the product and reactant concentrations. If the amplitude of the imposed oscillation in the initial reactant concentration is small compared to the concentration, the system can be treated as a linear one. The approach is similar to that employed previously by others for investigating diffusional processes in flow systems9-I2 and the transfer of matter between a moving fluid phase and a stationary solid phase in a chromatographic c01umn.l~ It is also closely related to reported work on the use of frequency response for analyzing the dynamic behavior of flow reactor system^.'^-^^ In the current paper we present a frequency response analysis for two examples of surface processes occurring in flow systems. The first is limited to the adsorption and desorption of a simple gas. The second involves a two-step catalytic sequence, in which an adsorption-desorption step is followed by a reaction step. For simplicity, we assume that only one species is adsorbed on the surface in both examples. We show how basic kinetic information beyond that obtainable from conventional steady-state flow reactor studies can be derived from frequency response data. We assume that diffusional effects do not have a significant influence on the frequency response but do suggest a simple way to approximate the effect of surface diffusion in the analysis. The influence of pore diffusion in porous particles and of axial diffusion between particles in the flow of gas through a bed of such particles has been considered by others? If such effects are important in a given situation, they can be incorporated in the analysis presented here. 11. Adsorption-Desorption Process We consider the adsorption from a gas stream of a species such as C O onto the surface of a solid adsorbent packed in a tubular vessel of unit cross-sectional area. The gas stream, which comprises a mixture of the adsorbing species and a nonadsorbing carrier gas, flows through the vessel with a velocity u. Before the gas stream enters the vessel, the concentration of the species to be adsorbed is caused to oscillate with a frequency w about some average value. Within the vessel, the adsorption-desorption process can be represented as kl2
A + S C B
(1)
k21
where A is the species to be adsorbed from the gas phase, S is a bare surface site, and B is the adsorbed species. The parameters k 1 2and kzl are rate constants for adsorption and desorption. If the symbol Cj (where j is either A, S, or B) is used to represent the concentration of a given species, we can write the following equations for the time derivatives of CA, C,, and Cs at a given distance z through the vessel a c A / a t = -vacA/az - k&Cs
ac,/at
= -acs/at = k&Cs
+ k2,C~
(2)
- kZlcB
(3)
The concentrations are expressed as molecules or sites per unit volume of gas in the vessel. Equation 2 is simply the equation of continuity for species A with appropriate terms for its disappearance by adsorption and reappearance by desorption. Equation 3 for the time derivative of C, assumes that B is immobile on the surface. For some imposed time dependence of CAat the inlet ~~
~
(9) Deisler, P F., Jr.; Wilhelm, R. H. Ind. Eng. Chem. 1953, 45, 1219. (IO) McHenry, K. W., Jr.; Wilhelm, R. H . AIChE J . 1957, 3(1), 83 ( 1 1 ) Wehner, A.; Wilhelm, R. H. Chem. Eng. Sci. 1958, 6, 89. (12) Kramers, H.: Alberda, G.Chem. Eng. Sci. 1953, 2, 173. (13) Rosen, J B.; Winsche, W. E. J . Chem. Phys. 1950, 18, 1587. (14) Leder, F.; Butt, J. B. AIChE J. 1966, 12(6), 1057. (15) Douglas, J. M. Ind. Eng. Chem., Process Design Deu. 1967,6(1), 43. (16) Crider, J. E.; Foss, A. S.AIChE J . 1968, 14(1), 77
to the vessel, Le., a given CA(O,t),eq 2 and 3 can be solved to give the concentration c A ( l , t ) at the exit ( z = 1 ) of the vessel. We assume that only cA(0,t) and c A ( l , t ) can be observed experimentally. If the system is initially at steady state, so that the time derivatives of CA,C,, and Cs in (2) and (3) are equal to zero, the steady-state concentrations CA,C,, and Cs are independent of z and
k l 2 C ~ C s= kZlCB
(4)
Suppose that oscillations are imposed on the concentration CA at the inlet to the vessel, such that At any distance z through the vessel, the concentrations CA,C,, and Cs will also oscillate. Consequently, we can write the equation Cj(z,t) =
C.J + c.(z)eiwz/Wwr J
(6)
where the subscript j refers to species A, S, or B. In general, the amplitudes cj(z) of the oscillations are complex quantities with phase shifts. Since we will be concerned with small oscillations and therefore linear equations, we can work with complex concentrations, taking the real part in the end. If there were no adsorbent in the vessel, the species S and B would not exist and eq 6 would simply represent a traveling concentration wave for the species A. The amplitude cA would then be real everywhere and independent of z. Since the amplitudes are assumed to be small, terms involving the product of cA and cs may be neglected when eq 6 for each species is substituted into (2) and (3). In making the substitution, we note that conservation of surface sites requires that cs = -c,, and we obtain the equations C,
=
Ll$A/(L
- iw)
acA/az = -KCA
(7)
(8)
which contain the new parameters
The solution of eq 8 is CA(z) = CA(o)e-'' (12) The amplitude ~ ~ ( 1where ) . 1 is the value of z at the exit of the vessel, would be determined as a function of w in a typical experiment. From its absolute value and its phase shift relative to cA(0), one would obtain the real and imaginaryparts of_the parameter K as a function of w . The parameters k 1 2and k for the value of cAemployed in the experiment are then determined from eq 11. If one repeated the 4etermination of cA(I) as a funstion of-w for various values of CA,a differentjet of values of k 1 2and k would be obtained for each value of CA. The true rate constants k12and kzl are then derived from eq 10. The value of Cs for each value of CAis then obtainable from eq 9, and, with the aid of eq 4, one can also obtain the corresponding value of C,. The sum of Cs and gives the total concentration L of surface sites capable of participating in the adsorption-desorption process. Obviously, none of these parameters can be obtained if the oscillations in CA(0) are omitted in the experiments. In such a steady-state situation, the effluent gas stream is identical with the inlet stream, and no information is available. As indicated in the Introduction, diffusional effects are not included in the analysis. In some circumstances they could influence the frequency response. For example, in the simple adsorption-desorption example considered here, it has been assumed that a molecule adsorbed on a given site remains at that site until it desorbs. In general, the molecule would diffuse a mean distance x over the surface before desorbing, where x can be approximated by (Drr)l/zwith D being the diffusion coefficient and 7r the mean residence time on the surface. In order for surface diffusion to
cB
Frequency Response Analysis of Surface Reactions
The Journhl of Physical Chemistry, Vol. 94, No. 3, 1990 1049
have a significant influence on the frequency response, the diffusion length must be on the scale of the wave length A = 2 m / w of the concentration modulation. We might reasonably assume that x I0.1X. If a is the hopping length for diffusion and ?h is the hopping time, we can approximate D by the quantity a2/?h. Therefore, we have the requirement that ?h/?, 5 1 0 0 ( ~ / A ) ~ . Assuming a = IO-* cm and X = 1 cm, we see that ?h/T, I 1 0 - l ~ in order for surface diffusion to have a measurable influence on the frequency response. If one ignores any difference in frequency factors for desorption and surface diffusion, this implies that the activation energy for surface diffusion must be lower than that for desorption by 19 kcal/mol for the detection of a diffusional influence at room temperature. In the adsorption-desorption example which we have just considered, the net effect of surface diffusion can be described approximately by replacing the desorption rate constant k21by an effective rate constant k'21 = kZ1 + where K is equal to w/u.
De,
111. Two-step Catalytic Reaction In this example we extend the analysis in section I1 to include an irreversible reaction step subsequent to the adsorption-desorption step. Thus, we have the sequence k
ka
A+S&B-C+S kll
where C is a gas-phase product species and k23 is the rate constant for its formation from the adsorbed species B. The time derivatives of the concentrations of the various species in (13) are given by the equations
+ k2lCB acB/ar= -acs/at = kl2CACS - (kzl + k23)~B acc/at = -uacc/az + k23C~ acA/az- k&&s
a c A / a t = -0
(14) (15) (16)
As in the previous example, the time derivatives in eq 14-16 are all equal to zero at steady state, but the concentrations Cj are now functions of z. Thus, we have the relations -U a C A / a Z -
k l 2 c ~ c s+ k 2 l c ~= 0
k]2cACs - (k21 + k23)CB = 0 -U aCc/az + k23CB = o
(17)
(18) (19)
On substituting eq 6 into eq 14-16, and noting eq 17-19, and the relation cs = -cB, one finds CB
=
where the parameters K'
k12cA/(k
+ k23 - iw)
(20)
acA/az = -K'cA
(21)
ac,/az =
(22)
K'
~K'cA
and y are given by
= ( k 1 2 / u ) [ ( w 2+ k23k
+ k232) - i w k ] / [ ( k+ k23)' + w21 (23)
y = (k232
and k12and and 22 are
+i
+
~ k ~ ~ ) w2) / ( k ~ ~ ~(24)
k are given by eq 9 and 10.
The solutions of eq 21
CA(z) = CA(o)e-"'
(25)
cC(z) = -ycA(z)
(26)
One determines c ~ ( z and ) cc(z) at the exit of the vessel, i.e., at z = I, for various values of w. From the absolute values of cA(I) and +(I) and their phase shifts relative to cA(0), one obtains the real and imaginary paris 0-f the parameters K' and y as functions of w . The parameters k, k I 2 ,and k23 are then obtained from eq 23 and 24. _The parameter k23 is a trye rate constant, but the parameters k and k 1 2are functions of CA(l).If the dependence of ~ ~ ( and 1 ) cc(l) on w is determined fer various values of cA(I), which are readilysbtaincd by varying CA(0),one finds a different set of values of k and k 1 2for each value of cA(l).As in the previous example, the true rate constants k 1 2and k z l are then
obtained from eq 10, and the quantities CS(l)and cB(l) correare obtainable from eq 9 and 18. sponding to each value of cA(l) The sum of Cs(1) and cB(l) then gives the total concentration L of surface sites of interest for the reaction. Steady-state flow reactor studies do not provide information on all of these parameters. At steady state, the rate r of the two-step catalytic reaction considered here is given by the equation r = k23bLCA/( 1 + bCA)
(27)
where b is given by = k12/(k21 + k23)
(28)
By determining the rate as a function of cAat low conversions to approach the condition of a differential reactor, one can determine the quantities k23L and b. However, this amount of kinetic information is clearly less than that obtainable from frequency response studies.
IV. Concluding Remarks The usefulness of frequency response methods in characterizing the dynamic behavior of linear systems has been appreciated for a long time by engineers concerned with servomechanisms and problems of automatic control.I7J8 While the methods have also been extended to other problems such as the characterization of diffusional phenomena in the flow of fluids through packed beds of solid particle^,^-'^ they have not yet received much attention in basic studies of the kinetics and mechanisms of surface reactions. In the present paper the authors have attempted to provide a couple of simple illustrations of how frequency response analysis could be used in this area to yield kinetic information not obtainable from a conventional investigation with a steady-state flow reactor. Equations have been derived to show the relationship between the amplitude of the concentration wave for a component in the effluent stream from the adsorber or reactor and the corresponding amplitude for the reactant in the inlet stream. In the application of the equations, one would obtain experimental data on the absolute values of the amplitudes and their differences in phase. With increasing distance through the adsorber or reactor, the amplitude of the concentration wave for a given component decreases, and at sufficiently long distances the wave is completely 03. An obvious damped out, Le., the amplitude cj(z) 0 as z experimental consideration is the choice of conditions (w,u,z) which will make it possible to have amplitudes at the exit of the adsorber or reactor which are large enough to be determined with sufficient precision. Obviously, the choice of satisfactory conditions depends on the rate constants characteristic of the system under investigation. Since the rate constants are the unknowns to be determined, some exploratory experimentation will in general be required to establish a satisfactory range of conditions. For many catalytic reactions of practical importance, a frequency w in the range of 0.1-10 rad/s would appear to be useful in frequency response investigations. However, for adsorption-desorption processes occurring at rates which are much higher than those typical of catalytic reactions, the required frequencies could be orders of magnitude higher than 10 rad/s. The experimental aspects of frequency response studies have not been discussed in this paper. Descriptions of methods used in generating sinusoidal concentration waves have been given by In general, they involve the addition of the component of interest, at a periodically varying rate, to the main fluid stream. Equipment for producing such an oscillating flow rate has been described in detail elsewhere.IF2l
- -
(17) Cheng, D. K. Analysis of Linear Systems; Addison-Wesley: Reading, MA, 1959. (18) Coughanowr, D. R.; Koppel, L. B. Process Systems Analysis and Control; McGraw-Hill: New York, 1965. (19) Deisler, P. F., Jr. Ph.D. dissertation, Department of Chemical Engineering, Princeton University, Princeton, NJ, 1952. (20) McHenry, K. W., Jr. Ph.D. dissertation, Department of Chemical Engineering, Princeton University, Princeton, NJ, 1957. (21) Leder, F. Ph.D. dissertation, Yale University, New Haven, CT, 1965.
J . Phis. Chem. 1990, 94, 1050-1055
1050
In our treatment of the frequency response method for the study of surface reactions in flow systems, we have chosen very simple examples to illustrate the principles as clearly as possible. These simple examples focus very sharply on the kinds of additional
kinetic information which can be obtained when conventional flow-reactor investigations are modified to include the frequency response feature. The approach described can be adapted readily to other problems of interest in catalysis.
Reorganization Free Energy for Eiectron Transfers at Liquid-Liquid and Dielectric Semiconductor-Liquid Interfaces R. A. Marcus Noyes Laboratory of Chemical Physics, California Institute of Technology,? Pasadena, California 91 125 (Received: July 7, 1989)
The reorganization free energy is calculated for a reaction (i) between two reactants, each in its own dielectric medium, separated by an interface, and (ii) between a reactant and some semiconductors. An expression is also given for the rate constant of an electron-transfer reaction at an interface between reactants in two immiscible phases. Under certain conditions it is shown that the reorganization energy for the twdmmiscible-liquid system is the sum of the electrochemical reorganization energies of the two reactants, each in its own respective solvent. The reorganization energy for a semiconductor-liquid system can differ considerably from the corresponding metal-liquid value, even a factor of 2.
Introduction Some time ago I derived an expression for the reorganizational free energy X in electron-transfer reactions occurring in homogeneous s o l ~ t i o n s l -and ~ a t metal-solution interface^.^-^ Since that time there have also been studies on electron transfers at liquid-liquid interfacesG9 and at semiconductor-liquid interfaces.ID-l2 In the present paper we obtain expressions for the reorganization free energy at such interfaces, using the same approximations as those used earlier.3 In the case of a metal-liquid surface, whose r e s u l t ~ j -are ~ given for comparison (eq 1 l), the detailed electronic structure propertiesl3J4of the metal surface are neglected and throughout a local dielectric response is used for the 1 i q ~ i d . l ~ Earlier, using a charging path to produce a system with a nonequilibrium dielectric polarization, we obtained a classical statistical mechanical expression for the free energy of a system having longitudinal polarization fluctuations.16 This result was then expressed in terms of the free energies of certain hypothetical equilibrium systems16 and proved convenient for deriving expressions for reorganization free energy,3 as well as for obtaining other properties, such as spectral shifts in polar media for simple and less simple (e.g., ellipsoidal) solute shapesi7 The principal assumptions used were (1) linearity of the response of the medium to a change in electric field, (2) a static treatment of the lowfrequency motions, and (3) instantaneous response of the electronic polarization in the system to a change in electric field. We also comment on the applicability of the relation to systems with linear but nonlocal dielectric response. Theory
We consider a nonequilibrium system having some charge distribution, denoted by p l , in an environment that would be in equilibrium with a different charge distribution po. Expressed in terms of equilibrium free energies, the free energy of formation GInon- GICof this nonequilibrium system from a similar system, but one that is in thermal equilibrium, is given by eq 15 of ref 16: Glnon
- G 1e =
~
~
~
-G e . 1-0=
0
~
(1)
where GICis the free energy of the equilibrium system with charge distribution p l , is that of an equilibrium system with a 'Contribution No. 8006.
0022-3654/90/2094- 1050$02.50/0
hypothetical charge distribution p1 - po, and Gl_Oe,op is the corresponding quantity when only an electronic response of the medium or media to the charge occurs. (In the last case, any dielectric constant would be replaced by the optical dielectric constant.) All quantities in eq 1 are calculated at a fixed position of the reactant(s). While statistical mechanical expressions can be introduced in (1) Marcus, R. A. J . Chem. Phys. 1956, 24, 966. (2) Marcus, R. A. Discuss. Faraday SOC.1960, No. 29, 21. (3) Marcus, R. A. J . Chem. Phys. 1965,43,679. (4) Marcus, R. A. ONR Tech. Rep. 1957, No. 12. Reprinted in: Special Topics in Electrochemistry; Rock, D. A., Ed.; Elsevier: New York, 1977; p 181. (5) Marcus, R. A. Can. J . Chem. 1959, 37, 155. (6) (a) Kharkats, Yu. I. Sou. Electrochem. (Engl. Transl.) 1976,12, 1257. The results in ref 6a have been utilized in: (b) Kharkats, Yu.I.; Volkov, A. G. Electroanal. Chem. 1985, 184, 435; (c) Kuznetsov, A. M.; Kharkats, Yu. I. In ref 9, p 11. (7) Geblewicz, G.; Schiffrin, D. J. J . Electroanal. Chem. 1988, 244, 27. (8) (a) Samec, Z.; Maracek, V.; Weber, J.; Homolka, D. J . Electroanal. Chem. 1981, 126, 105. (b) Samec, Z . Ibid. 1979,99, 107. (9) Articles in: The Interface Structure and Electrochemical Processes at the Boundary Between Two Immiscible Liquids; Kazarinov, V. E., Ed.; Springer-Verlag: New York, 1987. (10) Gerischer, H. Z . Phys. Chem. (Munich) 1960,26,233; 1%1,27,40. Dogonadze, R. R. In Reactions of Molecules at Electrodes; Hush, N. S . , Ed.; Wiley: New York, 1971; p 135. Mehl, W. Ibid., p 305. (1 1) E.g.: Morrison, S. R. Electrochemistry at Semiconductor and Oxidized Metal Electrodes; Plenum: New York, 1980, and numerous references cited therein. The first application of the electron-transfer theory of ref 1 to electron transfers at semiconductor electrodes was made by Dewald (Dewald, J. F. In Semiconducfors; Hannay, N. B., Ed.; Reinhold: New York. 1959; p 727). (12) See also: Inkson, J. C. J . Phys. C 1972, 5, 2599. Muramatsu, A,; Hanke, W. In Topics in Current Physics. Structure and Dynamics ofSurfaces II; Sommers, W., von Blanckenhagen, P., Eds.; Springer-Verlag: New York, 1987; Vol. 43, p 347. (13) E.g.: (a) Lang. N. D.; Kohn, W. Phys. Rev. B 1973, 7, 3541. (b) Appelbaum, J. A.; Hamann, D. R. Phys. Rev. B 1972,6, 1122. (c) Efrima, S. S u r , Sci. 1981, 107, 337. References 13a-13c employed a density functional formalism. (d) Applications of the theory, and other references, are given in: Jay-Gerin, J.-P.; Karouni, J. Solid State Commun. 1983, 48, 69. (e) Reference 15, p 403. (f) Theophilou, A. K.; Modinos, A. Ibid. 1972, 6, 601. (14) E.g.: Heinrichs, J. Solid Stare Commun.1982, 44, 893, 897. (15) Nonlocal dielectric response is discussed by Vorotyntsev: (Vorotyntsev, M. A. In The Chemical Physics of Solvation; Dogonadze, R. R., Kalmln, E., Kornyshev, A. A., Ulstrup, J., Eds.; Elsevier: New York, 1985; Part C, p 401. This series of volumes (Parts A to C) contains many helpful reviews and discussions of ion-solvent and electrode-solvent interactions. (16) Marcus, R. A. J. Chem. Phys. 1963, 39, 1734. (17) Marcus, R. A. J . Chem. Phys. 1965, 43, 1261.
0 1990 American Chemical Society
