440 SHORT COMMUNICATIONS

of an edge which corresponds to a single step reaction*. Thus

d In i/d~l ~ (n~+I)nF/3RT

the apparent Tafel slope gives the number of atoms in the critical nucleus.

(6)

Electrochemistry Research Laboratories, Department of Physical Chemistry, University of Newcastle upon Tyne, Newcastle upon Tyne, NE1 7RU (England)

R. D. Armstrong

R. D. ARMSTRONG AND J. A. HARRISON, J. Electrochem. Soc., 116 (1969) 328. 2 M. FLEISCHMANN AND H. R. THIRSK, Advan. Electrochem., 3 (1963) 200. 3 K. J. VETTER, Electrochemical Kinetics, Academic Press, New York, 1967, p. 324. 4 T. ERDEY-GRuz AND M. VOLMER, Z. Physik. Chem., 157 (1931) 165. 5 J. ZELDOVITCH, J. Exp. Theoret. Phys. (U.S.S.R,), 12 (1942) 525. 6 A. BEWICK, M. FLEISCHMANN AND H. R. THIRSK, Trans~ Faraday Soe., 58 (1962) 2200.

Received July 24th 1970

J. Electroanal. Chem., 29 (1971) 439-440

Logarithmic analysis of two overlapping d.c. polarographic waves III. Very close waves with equal slopes

In the previous paper t the analysis of two overlapping d.c. polarographic waves for reversible or totally irreversible electrode processes has been proposed. In the case of very close d.c. polarographic waves with equal slopes it was suggested that from the values of A T =exp (aEt) and A~ = exp (aE2), which can be determined ex- perimentally, A 1 = exp (aE~) and A2 = exp (aE~) are obtained. E 1 and E 2 are potentials at the intersection of asymptotes (E ~ + ~ ) and (E ~ - o 0 ) of the composite loga- rithmic curve with the abscissa, and a=~nF/RT (or nF/RT in a reversible case). A t and Az are calculated from. A* and A* values using the following equations"

A* = (mA t + A2)/(m+ 1) (1) and

A* = (m+ 1)Ai Az/(mA2 + A1) (2)

where m = gal/ia2 is the diffusion current ratio. In the second step the half-wave poten- tials are evaluated from the logarithms of A1 and A 2. The analysis of very close over- lapping d.c. polarographic waves is quite complex for practical use, and it would be interesting to evaluate an explicit relation for easier and more precise calculations. The asymptotes mentioned above are defined by the following equations :

x* = (mxt + x2)/(m + 1) (3) and

x* = (m+ 1)XlX2/(mx2 +xl) (4)

where Xl*=A l*exp ( - aE) , x '~=A*exp(-aE) , x t = A t e x p ( - a E ) and x z = A 2

* The opinion 6 that the rate of nucleation can be constant and independent of potential is incorrect. In the limit n c = 1 in which case nucleation can no longer be said to occur.

Y. Electroanal. Chem., 29 (1971) 440-442

SHORT COMMUNICATIONS 441

exp ( - aE). For the inflection point of the composite logarithmic curve the following relation can be written:

[ x, + xl . ] (5) Xi = I I + x i x2 /x ,~ j i = ~ 1 ~2/i

from which • , ~ _ xax2 = ( x l x 2 ) - x i (6)

It can be seen that:

X 1 -[- X 2 -~- X~ At- X 1 X 2 / X ~ -~- X~ At- (X:~/X~) ½ (7)

Combining eqns. (6) and (7) the following equation can be obtained :

" ~ i ¢v2! --

and analogously :

x~ + (x"; x ~ ) ~ = * * * ~ Ix, + (x,/~.) ],,. Generally we can write:

x,,. = ½Ix* + (x*/x*)~] _+ [¼{x* + (x*/x*)~}'- (xT ~*)~]+ (8)

The sign before the square root will be (+) for xl and ( - ) for x2. Using eqn. (8) which is valid at the potential of the inflection point we can determine x~ and x2 directly as

X.

l 50 mV I

POTENTIAL

Fig. 1. Logarithmic analysis of two very close overlapping d.c. polarographic waves with equal slopes.

J. Electroanal. Chem., 29 (1971) 440-442

442 SHORT COMMUNICATIONS

the straight lines (parallel with x* and x* asymptotes) which are defined by the solu- tions of eqn. (8). The diffusion current ratio, m, can be evaluated from the following relation :

m = ( x T - - x 2 ) / ( x 1 - - x ~ ) ~- X 1 ( x ~ - - x 2 ) / x 2 (x 1 -- x~) (9)

which is valid at the potential of the inflection point. One example of such an analysis is presented in Fig. 1. The values ofxi, x* and

x* at the inflection point follow from Fig. 1 as"

x * = 4 , x * = l and Xi=(X*X*)k-=2 Introducing x* and x~ values in eqn. (8) the values of xl and x2 become"

xl = 3 + x / 7 and Xz = 3 - x / 7

from which

m = (1 + ~/7)/(x/7- 1) ~ 2.2

From the total limiting current i a the particular diffusion currents of the each wave alone can be calculated as: idl = mia/(m+ 1) ~ 0.687 i a and ia2 = ia/(m+ 1) ~ 0.313 i a. Thus eqns. (8) and (9) can be applied for an easy and sufficiently accurate separation and the analysis of very close overlapping d.c. polarographic waves with equal slopes. The method proposed was applied for the analysis of the polarogram 2 given by the system U 0 2 ( C 0 3 ) 3 , UO2(H202)(CO3) 2, 1 m NaeCO 3.

Acknowledgement The author is grateful to Mrs. Vera Zuti6, whose experimental results initiated

this work.

Center for Marine Research, Institute "Rudjer Bo~kovid", Zagreb, Croatia (Yugoslavia)

IVICA RU2Id

1 I. RU~I8 AND M. BRANICA, a r. Electroanal. Chem., 22 (1969) 243. 2 V. ~UTI6, private communication.

Received 3rd August 1970

J. Electroanal. Chem., 29 (1971) 440-442