J . Phys. Chem. 1985, 89, 2966-2967
2966
and in any "double substrate" run with persulfate, the rate is always intermediate between the separate runs (instead of being their sum). This is discussed a t the bottom of p 1180 of ref 2 in terms of the simultaneous oxidation of Et,SO and i-PrOH. This effect is another unusual and intriguing aspect of oxidations by peroxydisulfate. (6) If we assume that Berhman's data are correct, the rate equations should be rate = k l P k2PS0,where Sois the initial sulfoxide concentration in a given run. This does not mean that the second term is second order but merely that the overall first-order rate constant, k , is a linear function of So at higher concentrations. In a recent study of the persulfate oxidation of sulfonamides: however, the reaction was truly second order at all Sovalues, since the integrated second-order plots were linear.
+
Registry No. S2OS2-,15092-81-6.
The Institute for Theoretical Studies POB 13690 El Paso, Texas 79912
L. S. Levitt
Received: November 2, 1984; In Final Form: April 5, 1985
TABLE I: Reduced Potential at the Axis of a Cylindrical Microcapillary, &, for a Reduced Surface Potential 6, = 2 dn Ka
0.01 0.05 0.1 0.2 0.5 1 2 3
5 7
numerical 1,9999 1.9977 1.9909 1.9647 1.8050 1.4278 0.7652 0.3637 0.0667 0.0390
Oldham
Olivares
et al. 1.9999 1.9977 1.9910 1.9647 1.8054 1.4278 0.766 0.367 0.071 0.012
al. 1.9999 1.9977 1.9910 1.9647 1.804 1.415 0.723 0.316 0.048 0.007
Sir: The linearized Poisson-Boltzmann equation for the inner region of a cylindrical charged microcapillary immersed in an aqueous electrolyte solution, valid for small reduced surface potentials 4a< l , was solved by Rice and Whitehead.' For larger potentials Levine et aL2 gave an approximate analytical solution which has several regions of validity. In a previous work3 we extended the range of application of the simple linear solution of Rice and Whitehead for reduced potentials 0 C 4, C 6 and reduced micropore radius 1 < KU < 1000, using a variational approach for solving the nonlinear e q ~ a t i o n .The ~ variational trial function was a simple parametrization of the linear solution
where Io(x) is the zeroth modified Bessel function at the reduced distance x = Kr and p is the only variational parameter, obtained by minimization of the free energy functional
Is( z)+ cosh 4 - 1
J [ d ] = - 2 ? r ~ l i ( xdx 1 d 4
Levine
Martinov
1.9999 1.9977 1.9908 1.9640 1.801 1.409
1.9999 1.9977 1.9910 1.9647
$a2
12(A
+ B4:)
(4)
where A =1
+
(KU)2/3
(5)
and
B = O.OI(1 - 5(KU)3/2)
(5)
for 0 C KU C 1 and C 6. The correct values for the potential at the axis in the variational approximation are given in Table I and give very good agreement with the numerical data and other existing approximations for small values of KU and 4,. We also calculated the values corresponding to Oldham's approximation using our own iteration routine and obtained values slightly different from those reported by Sigal and Ginsburg. These values are in better agreement with the numerical solution of Sigal as shown in Table I. For application in systems such as zeolites, where a i= 10 A, we fall in the region KU < 1 for a 0.1 M 1-1 electrolyte at 25 O C . We thus would like to compare the existing approximations with special attention to this range of KU values. In the region of values of KU < 1, the potential satisfies the condition +(x) > 1 for all values of x inside the cylinder, particularly for surface potentials > 2. This falls in the so-called region I11 of Levine's solution, which actually has a simple expression for the potential
The parameter p is a function of both the reduced potential and the radius, which to within a few percent is given by the expression p2 = ex~(a4;) + P(4a/Ka)2
0.753 0.362 0.067 0.011
satisfactorily fitted the numerical data, our approximation was unsatisfactory. Unfortunately, these authors used eq 3 with the wrong value of and also applied it outside its range of validity, since for KU < 1, the parameter p has a completely different behavior that can be calculated instead by the relation p = l +
Comments on the Calculatlon of the Potential Inside a Charged Microcapillary
et
(7)
(3)
for 1 < KU C 100 and 0 < C 6. The fitting constants CY and /3 have the values 0.0412 and 0.0698, respectively. This last constant was erroneously transcripted as 0.698 in our original paper.3 In an attempt to compare our approximation with an exact numerical solution of the nonlinear Poisson-Boltzmann equation, Sigal and Ginsburg5 used eq 1 and 3 to calculate the potential at the axis of the microcapillary, do. The results obtained were also compared to the power series suggested by Oldham et a1.6 Sigal and Ginsburg concluded that, while Oldham's solution (1)C.L. Rice and R. Whitehead, J . Phys. Chem., 69,4017 (1965). (2) S . Levine, J. R. Marriott, G. Neale, and N . Epstein, J . Colfoid Interface Sci., 52, 136 (1975). (3) W. Olivares, T. Croxton, and D. A. McQuarrie, J . Phys. Chem., 84, 867 (1980). (4)W. Olivares and D. A. McQuarrie, Biophys. J., 15, 143 (1975). (5)V. L.Sigal and Yu. Ye. Ginsburg, J. Phys. Chem., 85, 3730 (1981).
where
40
In (16Bo/(~a)')
(8)
and Bo =
(4 + e q K a ) 2 ) ' / 2 - 2 (4
+ e"(Ka)2)'/2 + 2
(9)
The values obtained for 4o by this analytical expression are essentially identical with those obtained by Oldham's iteration and compare very well to our approximation and to the numerical data, as can be seen in Table I. More recently, Martynovl gave analytical expressions valid for KU C 0.3 and $a C 5 and KU > 2 and 4, C 5 . Both of these expressions fare very well to the numerical solution, as shown in Table I. The region between 0.3 C K RC 1 is not covered by either of Martynov's expressions.
(6) I. E. Oldham, F. J. Young, and J. F. Osterle, J . Colloid Sci., 18, 328
(7)G.A. Martynov and S. M. Avdeev, Kolliodn. Zh., 44,702 (1982).
(1963).
0022-3654/85/2089-2966$01 S O /O
0 1985 American Chemical Society
J . Phys. Chem. 1985,89, 2967-2968 W e thus have shown that for Ka < 1 and 0, < 4, which corresponds to R C 30 A and { potentials less than 100 mV for a 0.01 M 1-1 electrolyte at 25 OC, the existing analytical approximations are numerically equivalent. Contrary to that stated by Sigal and Ginsburg, the variational approach not only gives adequate agreement with the theory of Levine et al. for the electrokinetic quantities3 but also is quite satisfactory in predicting the potential itself for small microcapillaries.
Grupo de Quimica Tfdrica Departamento de Quimica Facultad de Ciencias Universidad de Los Andes Merida, Venezuela
Wilmer Olivares
2967
The real process occurring in the cell in the course of the transformation of a differential number of moles of metal is the increase (or decrease) of the radius of metal particles forming the electrode. This process occurs via the electrodeposition of dissolved metal ions present in the solution phase. If the number of the metal particles is high, the increase in the radius will be very small. Actually this requirement must be fulfilled in order to assure the unchanged state of the system, specifically, the practical constancy of the particle dimensions. The increase of the surface area for one particle is AAi i= 8 w A r
(4)
if Ar is very small. If there are N’ particles, the total increase of the surface area will be h n a l d A. McQuarrie*
Department of Chemistry University of California Davis, California 9561 6
AA
i=
N‘AA, = 8wArN’
(5)
At the same time, in the case of the introduction of 1 mol of metal
Received: March 25. 1985
N’Ar4rza = V ,
-
It follows from eq 5 and 6 that in the limiting case ( N 0)
Comments on the Electrochemlcal Behavlor of Small Metal Particles Sir: Questions connected with the electrochemical behavior of small metal particles have recently been raised in the literature. In a recent paper by Plieth’” approximate equations were formulated for the relationship between the particle size and reversible redox potential (notation used in ref l a for the potential of the metal immersed in the solution of its own ions), the potential of zero charge, the surface potential, and the work function. Although equations proposed in ref l a are derived on the basis of thermodynamic considerations, some doubts can be raised concerning their general validity. In ref la the cell potential of the cell MeblMeZ+IMed(denoted by AtD) is correlated with the free energy of the dispersion process (AGD): AGD AtD = -ZF
(1)
It is assumed that the cell reaction is the transference of 1 mol of bulk metal into the dispersed form. For the free energy of the dispersion process the relationship
is derived, where V , is the molar volume of the metal, y is the surface tension, and r is the radius of the spherical particles. According to ref l a the quantity AGD calculated on the basis of eq 2 is the free surface energy of 1 mol of metal dispersed into particles of radius r . In order to obtain a cell potential, however, one should be concerned with a difference in electrochemical potential between two states. Plieth should have considered the differential change in free energy with number of moles. Thus, he should have calculated the free energy change with number of moles to disperse bulk metal onto the surface of particles of radius r. In fact, the free surface energy of 1 mol of metal dispersed into particles of radius r (AGD’) should be given by the following relationship: (3)
Thus, AGD differs from AGD’ and it is very important to make a distinction between them. (1) (a) W. J. Plieth, J . Phys. Chem., 86, 3166 (1982); (b) W. J. Plieth in ’Electrochemie der Metalle”, Decheme-Monographien Bd. 93., Verlag Chemie, Weinheim, 1983, p 151.
0022-3654/85/2089-2967$01.50/0
2 VM AA = r
-
(6)
Ar
a,
(7)
The work done in the cell is
where ys is equal to surface excess energy per unit area if Cr,k, = 0. It follows from this derivation that AGD which Plieth derives has nothing to do with the surface excess energy of the dispersed system. The process he should have considered is the dispersion of 1 mol of the bulk metal onto the surface of a system consisting of an infinite number of spheres of radius r, and AGD given by eq 8 is the work connected with this process. In contrast to this in ref 1a AGD is considered as the free energy of dispersion of 1 mol of bulk metal into N = vM/(4?./3) particles of radius r (see for instance p 3169 in ref la). In conclusion, it can be stated that there is no difference between eq 2 (derived in ref la) and eq 8, but the physical meaning of AGD as a differential change in free energy may be clarified on the basis of the derivation given here. Another important problem is connected with the use of the term “surface tension” for solids, as in this case it is very difficult to define the surface tension in terms of mechanical properties. (For a detailed discussion of the problem see ref 2.) According to ref l a the shift of the potential of zero charge with the dispersion of the bulk metal is equal to the shift of the reversible potential, namely €pzc,d
AGD 2yvM - €pzc,b = A t D = --zF = - r
(9)
This equation is obtained on the basis of a calculation of the energy required to disperse 1 mol of metal forming a sphere of radius R into particles of radius r assuming that the surface charge density is constant during the dispersion process. (i) Again the process considered by Plieth, namely, dispersion of 1 mol of metal into particles of radius r, would not result in AGD (see eq 3 ) . In the process considered for the derivation of eq 9 instead of AGD the expression 3 VMy(1/ r - 1/ R ) would give the change in the surface energy. (ii) In ref l a considerations leading to eq 9 involve that the number of particles (N)is determined by the relation N = R3/?. However, it follows from our more rigorous derivation that there is no correlation between the number of particles (N? and r. (iii) In the case of the study of the tpzcshift we should consider (2) J. J. Bikerman, Top. Curr. Chem., 77,1 (1978).
0 1985 American Chemical Society
