J. Electroanal. Chem., 97 (1979) 283--286 283 © Elsevier Sequoia S.A., Lausanne - -Pr in ted in The Netherlands

Short communication

DETERMINATION OF THE HETEROGENEITY FACTOR IN THE FRUMKIN ADSORPTION ISOTHERM FROM THE WIDTH OF CAPACITY PEAKS *

A. SADKOWSKI

Institute of Physical Chemistry, Polish Academy of Sciences, 01-224 Warsaw, Kasprzaka 44/52 (Poland)

(Received 21st June 1978; in revised form January 1979)

Recently Bewick and Thomas [1] proposed an approximate formula for the half-width AE0.s of the potentiodynamic peak (the width of the peak in the half of its height) as a function of the heterogeneity factor g, assuming the validity of the Frumkin isotherm. It was claimed that the formula gives the values of heterogeneity factor g with good accuracy from measured values of AEo.5 when g approaches the limiting value g = 4 (according to the sign convention adopted in ref. 1) corresponding to the phase transition in surface phase.

The initial equations were formulated in the cited paper, but the derivation w ' s not taken to the final, exact solution. Instead, it was simplified giving the approx- imate formula AE0.s = f(g). The same derivation, consequently performed, gives the exact formula and, as will be shown below, the approximation proposed in [1], because of its oversimplification, does not approximate properly this exact dependence.

This problem was solved exactly already in 1963 by Conway et al. [2] and that exact solution applies to the even more general case of the width of the capacitance peak at its height corresponding to the capacity value being any fraction of the maximum capacity, i.e. for C = 7Cmax where 0 < 7 ~ 1.

Below, we shall reformulate the derivation by Conway et al. using, for the sake of clarity and generality of final formula, reduced (dimensionless) quanti- ties.

We shall use here the sign convention for heterogeneity factor according to Gileadi and Conway [2] rather than that used in [1]. We consider the former more appropriate according to eqn. (1) describing the chemical potential of adsorbed atoms obeying the Frumkin isotherm:

0 Pad = Pad + R T ln(0/(1 -- 0)) + g R T O (1)

The last term in (1) is the correction, which should be added to the two first terms corresponding to regular surface solution [3] to allow for intrinsic hetero- geneity or interactions in the adsorbed layer.

The derivation below is valid for any electrode reaction of the type: AZoiv + x e- ~ BZd~ ' (z = an integer number) under the simplifying assumption of absence of transport control in the electrolyte solution.

* Part of Research Project No. 03.10.

284

The Frumkin adsorption isotherm for this type of reaction may be written as:

E = E s t + ( R T / z F ) ln~(1 -- 0)10} --gO (2)

Est is the standard potential [4] of an electrochemical adsorption reaction. By defining a dimensionless potential variable: e = z F ( E --Est ) / R T the iso-

therm (2) may be rewritten in dimensionless form:

e = ln((1 - - O)/O} --gO (3)

The dimensionless adsorption capacity may be defined as:

C = --d0/de (4)

From (3) and (4) it follows:

= [g + (O/(1 -- O}-']-' (5)

In potentiodynamic experiments performed reversibly, the current--potential dependence replicates the capacity--potential dependence according to:

I = d Q / d t = (dQ/dE) (dE/dt) = Cs (6)

where s is the potential sweep rate, C the adsorption capacity, directly related to the dimensionless capacity C by:

C = (C/q °) ( R T / z F ) (7)

and q0 the charge equivalent of an adsorbed monolayer. Consequently, further analysis need be concerned only with the C = C(e)

dependence. Maximum value of capacity follows from (5) and equals:

Cmax -- (g + 4)-' (8)

The width of the curve C vs. e at the height corresponding to C = 7Cmax can be found from eqns. (3) and (5) and the condition:

= 7/ (g + 4) (9)

o s o b

A,~o. 5

0.60

a 04O

0.20

I - 4 O0 - 3 . 8 0 - 3 . 6 0 - 3 . 4 0 3 .20 9

Fig. 1. Ae0.s versus g (in dimensionless units) according to eqn. (13) of this paper (curve a), and according to the approximation proposed in ref. 1 (curve b).

285

From (5) and (9) it follows:

g + (0/(1 -- 0)} -1 = (g + 4)/7 (10)

and solution of this equation gives:

01.2 = 0.5 + 0 . 5 V ~ - - 4a (11)

where a = "),/(g + 4 - - ",/g).

Allowance for (3) gives the final formula:

1 / (g + 4) (1 --"),) Ae~ = 2 In x/g + 4 -- 7g + x/(g + 4) (1 -- y) + (12) gl /

In a particular case with ~/= 0.5, formula (12) simplifies to the following exact formula for the half-width of the capacitance peak:

1 Ae0.s = 2 In ~[g + 6 + ~/(g + 8) (g + 4)] +g{(g + 4)(g + 8)) in (13)

The latter formulae may be transformed into dimensioned form by substi- tuting:

A E = ( R T / z F ) A e (14)

For g = 0 (Langmuir isotherm) formulae (13) and (14) give:

AEo.s = ( 2 R T / z F ) ln(3 + 2x/2) ~ 9 1 / z mV (T = 298 K) (15)

When g ~< --4 (g ~> 4 in Bewick and Thomas's notation), AE~ = 0 according to the interpretation of this case as a phase transition. This corresponds to an abrupt change of 0 with infinitesimal change of E (step function) and to a Dirac ti function on the C vs. e dependence.

The function described by eqn. (13) cannot be approximated by the formula proposed in [1], which in our notation may be formulated as:

Ae0.s = 0.5[(g + 4) Ls + g + 4] (16)

Both eqns. (13) and (16) give Ae0.s = 0 at the point g = --4, but even in the immediate vicinity of this point, the difference between them becomes apprecia- ble. Speaking more precisely, the difference between the exact and this approxi- mate functions is a small quantity of the same order as the increment in g. The necessary condition for a good approximation is that in the interval containing the given point, the difference between the approximate and approximating functions is a small quantity of higher order than the increment of the argument. This is equivalent to equality of values of the functions and also equality of their successive derivatives (at least one) at that point. This condition is fulfilled by the power series approximation of functions continous with their derivatives in the interval containing the given point. The impossibility of approximating function (13) by a power series, or formula (16) in the interval containing point g = --4 is caused by the discontinuity of this function in the interval containing g = --4. Both (13) and (16) are determined only f o r g ~> --4 and the right-hand derivative of (13) at g = --4 equals zero, whereas the same derivative of (16) differs from zero.

2 8 6

R E F E R E N C E S

1 A. Bewick and B. Thomas, J. Electroanal. Chem., 85 (1977) 329. 2 E. Gileadi and B.E. Conway, m B.E. Conway and J.O'M. Bockris (Eds.), Modern Aspects of Electro-

chemistry, Vol. 3. Butterworths, London, 1964, Ch. 5. 3 K. Denbigh, The Prinmples of Chemical Equilibrium, Univ. Press, Cambridge, 1968, 2nd ed., 436 pp. 4 B.E. Conway, H. Angerstem-Kozlowska and H.P. Dhar, Electroehim. Acta, 19 (1974) 455.