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.
Wilmer Olivares
Grupo de Quimica Tfdrica Departamento de Quimica Facultad de Ciencias Universidad de Los Andes Merida, Venezuela
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) a,Ar
(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
2968
The Journal of Physical Chemistry, Vol. 89, No. 13, 1985
a cell consisting of the bulk and dispersed metal electrodes immersed in the same solution of an indifferent electrolyte. As in this case no transfer of Mez+ particles occurs across the solid/ electrolyte interface, it is not clear why the dispersion of the bulk metal should be involved in the imaginary process used for the calculation of the potential difference.
Comments Central Research Institute for Chemistry of the Hungarian Academy of Sciences Budapest, H-1525, Hungary
G . HorPnyi
Received: July 18, 1984; In Final Form: March 25, 1985
