COMMUNICATIONS TO THE EDITOR
when we observe exclusion. On the other hand, Horne, Konyushka, Plank, and others ascribe at least part of the effect to a preferential adsorption of water; some refer also to the exclusion from the smallest pores of the solute, but not of water molecules. Unfortunately, Horne, Konyushka, and Plank do not take into account the geometric effect, an effect which by its nature must always be considered. If there is in addition preferential water adsorption or ion reaction, these effects are superimposed upon the geometric effect. The geometric effect arises when there is a difference between the sizes of the solute and solvent species. The larger species is effectively more dilute near the solution-solid interface. It is more dilute because, for species touching the interface, the distance between an average point (for a sphere, the center) of the geometric shape representing the larger species and the interface is greater than the corresponding distance for the smaller species. There is therefore a compensating enrichment of the larger species in the bulk phase. This effect becomes important where the interface area is several hundred square meters per milliliter of solution. Horne statee quite correctly that ion reaction with the surface must be included in any analysis of these systems; we have either repressed reaction or calculated its amount in all of our work.’ He suggests that a correlation between our sizes and the viscosity B coefficient indicates we are really observing water structure at the interface, not ion hydrate size. We had, however, earlier shown8 that correlation with the B coefficient is about as good as the B coefficient correlation with sizes determined in other ways.8 Our sizes scatter around those of Nightingaleg and others and there is no doubt that our method, involving as it does an assumption of the size of the “average” water cluster and a knowledge of the pore size distribution, is less accurate than most other methods. These sources of error are not relevant to the discussion of the question of the reason for ion exclusion, since ion hydrates are of different sizes, and therefore the geometric argument given earlier must apply to at least some ions. One must make at least approximate calculations for each solution-solid experiment to determine whether or not the exclusion factor is important. Is there on hydrated silica gel a water layer impermeable to ions? Our sizes, similar to those of others, calculated on the assumption there is no such layer, seem to be consistent with that assumption. However, the meaning of “size” varies from one method of determination to another and our experimental error is large. Yet it is interesting to note that the ions of CsN03 “see” exactly the same pore volume of silica gel as does water
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and the ions of RbN03 “see” almost as much. Even though some of these ions are structure breakers, it is generally true that the less an ion is able to hydrate, the more easily it penetrates all of the pore volume water penetrates. It is difficult to explain this fact on the basis of a hydration layer and it is easily explainable on the basis of ion hydrate size. ~
(8) R. W. Maatman, J. Netterville, H. Hubbert, and B. Irby, J . Miss. Acad. Sci., 8,201 (1962). (9) E. R. Nightingale, Jr., J . Phys. Chem., 6 3 , 1381 (1959).
DEPARTMENT OF CHEMISTRY DORDTCOLLEQE SIOUXCENTER,IOWA51250
RUSSELLW. MAATMAN
RECEIVED OCTOBER 24, 1966
Double-Layer Perturbation without Equilibrium between Concentrations and Potential Sir: I n the application of the three recently proposed general equations1i2 for the treatment of nonsteadystate electrode processes without a priori separation of charging and faradaic currents, one needs explicit forms of the time derivatives (drldt) of the surface excesses of reactant and product. An explicit form of the dr/dt, based on equilibrium considerations, was proposed2 and applied to impedance ~ t u d i e s . ~This relationship does not take into account the nonequilibriurn between potential and the concentrations of reactant and product at the electrode for a finite exchange current iodifferent from zero. This concept of nonequilibrium is new in double-layer studies, to our knowledge, and is believed to be essential to the interpretation of nonsteady-state and periodic electrode processes. Application is made to the amalgam electrode M z + ze = M(Hg), but ideas carry over to other processes. The value of dr/dt for nonequilibrium can be expressed either directly in terms of the variables involved or by correcting for nonequilibrium the dr/dt corresponding to equilibrium. The two approaches are discussed and lead to the same final form of dF/dt. For the model used, one has for a small perturbation and a finite i o different from zero
+
d r +/dt
=
(br+/hc’+) ~ ‘ M , E (dc +/dt)Z=.o4(br+/bCs&+ ,E(dCM/’dt)z=o (br+/~E)c~+,c~,(dE/dt)(1)
(1) P. Delahay, J . Phys. Chem., 70, 2067 (1966). (2) P. Delahay, ibid., 70, 2373 (1966). (3) P. Delahay and G. G. Susbielles, ibid., 70, 3150 (1966).
Volume 71, Number 9 February 1067
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COMMUNICATIONS TO THE EDITOR
where r + is the surface excess of Mz+; c, the concentration; c', the bulk concentration; E , potential; 2, distance from the electrode; t, time. When io + CO, only two variables need be considered, instead of three in the writing of dr+/dt; Le., it is not possible to vary E when both cs+ and C'M are constant; likewise one cannot vary one of the cs's while keeping constant the other cs and E . However, this is not the case when io is finite. Finally, only the last term on the right-hand side of eq 1 has to be retained when io = 0 for the model used here. I n the other, intuitive approach we express dr+/dt as the sum of two components corresponding to equilibrium and nonequilibrium conditions. Thus one has for a finite io different from zero
+
similar to eq 1. These results ought to be used, rather than the forms previously ~ r i t t e n in , ~ the analysis of nonsteady-state and periodic electrode processes. Revised calculations have been completed for the electrode impedance and will soon be carried out for other relaxation processes. Acknowledgment. This work was supported by the National Science Foundation and the Office of Naval Research. We thank Dr. J. E. B. Randles, University of Birmingham, and Dr. G. C. Barker, Atomic Energy Research Establishment, Harwell, for most valuable comments. DEPARTMENT OF CHEMISTRY NEWYORKUNIVERSITY NEWYORK,NEWYORK 10003
dr+/dt = ~:br+/bcs+)c8h~(dC+/dt)Zro
P. DELAHAY K. HOLUB G. SUSBIELLES G. TESSARI
RECEIVED NOVEMBER 14, 1966
(dr+/bcsif)c8+(dciw/dt)z=o -!-
(~r+/bE)c~+,c~,(dl7,t/dt)(2) where qot is the charge-transfer overvoltage. The Barium Ion Exchange of Linde 13-X first two terms on the right-hand side of eq 2 correspond to the behavior of a reversible electrode for Sir: Recent articles102present data on alkaline earth which I?+ at equilibrium is solely a function of the two ion exchange of the synthetic zeolite Linde 13-X (NaX). independenl variables cs+ and C'M. These terms in I n ref 2 the data were presented as equilibrium data and eq 2 account for the change of r + resulting from the the free energies for the complete conversion of NaX variation of c+ from cs+ to ( c + ) ~ &and the variation of into the alkaline earth forms were computed from them. E from the equilibrium value E, to that corresponding I n both ref 1 and 2 the data indicate that a maximum of to (c+)%=Oand ( c ~ ) ~ =namely, o, E, V d , 'qd being the 74% of all the sodium ions in NaX can be replaced by diffusion overvoltage. The last term on the rightBa2+ ions. We began a study of alkaline earth exhand side of eq 2 accounts for the variation of I?+ change of NaX prior to the publication of the works when E varies from E, Vd to E, Vd 'qat. Note cited above and are in essential disagreement with porin, ,eq ~ 1 is not the same as (bI'+/ that ( b r + / b c S + ) , ~ The purpose of this communication is tions of them. and that the same remark applies to (W+/ t o report our Ba2+ ion-exchange data. ~ c ~ M ) ~ s and + , E (br+/bcsdc~+. The contents of the anhydrous unit cell of the NaX Equation 2 can be written in the same form as eq 1 and 2 and in our work are crystals used in ref 1 as follows. One has dE/dt = dq/dt, 'q = Vd ?lot, and 'qd is given by the Nernst equation. Hence one Naas [(A102)~&iO2)106 1 has for a small perturbation Nas7 [(A102)~7(Si02)105 I dr+/dt = [i:dI'+/bcs+),~, - ( R T / z F ) X and (I/(;'+) (ar+/'W c ~ + , c ~ n(dc+/dt)z=o Il Na.95[ ( A ~ O Z ) ~ ~ ( S ~I O Z ) I O ~ [(br+/bcsbf)cn+ ( R T / Z F )x We have discussed lattice structure and cation loca(I/(;'M) (br+/bE)~n+,,~,](dc~/dt)z=o tions in a previous papera3 For our present purpose it (br+/amc8tvcsM(dE/dt) (3) is sufficient to note that there are exchangeable cations located in two independent but interconnecting threeThis result is of the same form as eq 1, and correlations between the different coefficients of the time derivatives (1) L. V. C . Rees and C. J. Williams, Trans. Faraday Soc., 61, 1481 are immediate. One could also derive eq 3 directly (1965). from eq 1 without splitting dr+/dt as was done in eq (2) R. M.Barrer, L. V. C. Rees, and M. Shamsuzzoha, Inorg. Nucl. 2. Chem., 28, 629 (1966). Expressions for d h / d t and d(q zFI'+)/dt are (3) H.S. Sherry, J . Phys. Chem., 70, 1158 (1966).
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The Journal of Physical Chemistry
