A New Approach to the Solution of Electrochemical Problems Involving Diffusion

SIR : Many electrochemical experiments are carried out with conditions under which Fick’s linear laws

a ar J(r,t) = D - C(r,t)

and

a a 2 - C(r,t) = D - C(r,r) at ar2

are obeyed. Here C(r,t) is the concentration of some electro- active species at a distance r from the electrode at time t because commencement of the experiment, J(r,t) is the flux of that species and D is its diffusion coefficient. Frequently, moreover, the diffusion field is semiinfinite and conditions are initially uniform-i.e.,

C(m,t) = C(r,O) = C, a constant. (3)

To specify the experiment completely requires a further equation representing the boundary condition at the electrode surface. Formulation of this boundary condition generally demands the stipulation of both (a) the electrolysis regime-e.g., potentiostatic conditions, linearly increasing current, etc. (I)-and (b) an assumed electrochemical behaviour-e.g., complete reversibility, obedience to Volmer kinetics, etc. (2) .

The boundary condition may be a statement of how the surface concentration varies with time

C(O,t) = f ( t > (4a)

or how the surface flux varies with time

or, more generally (3), may express an interrelationship between the surface concentration and the surface flux

The standard method [for a typical example see Delahay and Berzins (41 of dealing with such electrochemical situations is to solve (often with the help of Laplace transformation) the system of Equations 1 through 4 completely, yielding an expression for the concentration as a function of distance and time-i.e.,

C(r,t> = f(r,t). ( 5 )

This expression is then utilized to provide either C(0,t) or J(0,t) which in turn [cia (a) and (b)] provides the sought rela- tionship between potential or current and time.

Three criticisms may be levelled at this standard method. First, the mathematics is unnecessarily complicated in that a bivariate function C(r,t) must be derived, whereas a much simpler univariate function C(0,t) or J(0,t) is sought. Second, because second-order partial differential equations are in-

(1) K. B. Oldham, ANAL, CHEM., 40, 1912 (1968). (2) K. J. Vetter, “Electrochemical Kinetics,” Academic Press,

(3) K. B. Oldham, ANAL. CHEM., 41, 936 (1969). (4) P. Delahay and T. Berzins, J. Amer. Chem. SOC., 75, 2486

New York & London, 1967, Chapter 2.

( 195 3).

herently difficult to solve, only very simple electrolysis regimes, (a), can be accommodated. For example, it may be necessary to assume potentiostatic conditions even when a small uncompensated resistance is known to be present, because the boundary condition which incorporates an ohmic term renders the partial differential equation intractable. Third, it is necessary to assume a priori an electrochemical behavior, (b). This is especially undesirable in experiments whose purpose is to decipher electrode kinetics.

A way of avoiding these objections would be to combine Equations 1,2 , and 3 into a single partial differential equation, which, by insertion of the r = 0 condition, might then be simplified to an ordinary differential equation before the boundary condition 4 is imposed. Recent work in this laboratory has shown that

is such an ordinary differential equation. [It has been sug- gested that “extraordinary differential equation” might be a better designation.] This equation embodies the requirements of Fick’s laws as well as Equations 3 and applies equally to all electrolysis regimes and all electrochemical behaviors. The crucial step in this approach to a voltammetric problem is to solve Equation 6 and the applicable Equation 4 simul- taneously. The formal derivation of Equation 6 and its application to practical electrochemical problems will be the subject of a forthcoming publication.

The semidifferentiation operator, d112/dx 1/2, is defined by

Its properties will be detailed elsewhere (5). Broadly speak- ing, d112/dx1’2 bears the same relation to d/dx as does d/dx to d2/dx2.

As a simple example, consider a potential-step (6, 7) experi- ment in which the electrode reaction is an irreversible reduc- tion. In this case J(0,t) is proportional to C(O,t), the constant of proportionality being the [potential-dependent] electro- chemical rate constant k . Equation 6 then becomes

which expression is a relationship between an independent variable t and its dependent variable C(0,t). We can solve this equation either by consulting a table of semiderivatives (5), or by the following technique. On semidifferentiation, Equation 8 yields

k d 1 / 2 d dt1I2 dt C(0,t) = - - C(O,r), _ _ ~ (9)

because zero is the result of semidifferentiating t - 1 1 2 with

(5) K. B. Oldham and J. Spanier, unpublished work. (6) H. Gerischer and W. Vielstich, Z . Phys. Chem. (Frankfurt 1 ,

am Main), 3, 16 (1955). (7) H. Gerischer and W. Vielstich, ibid., 4, 10 (1955).

1904 ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969

respect to t. The d1’2C(0,t)/dt1’2 may now be eliminated be- tween Equations 8 and 9 to produce the first order ordinary differential equation

C(r,r) = ‘ d k2C(0,t) kC - C(0,t) = ~ - - dt D 6;

Standard methods [see, for example, Murphy (a)] thence lead to the solution

A more powerful application of this novel approach is to a situation in which the electrode potential is not a known analytical function of time. A description of the numerical technique required in this instance will be deferred until a full presentation of the mathematical treatment can be given. (11) C(0,t) = c exp [ $1 erfc [ k $1.

Equation 11 is a well-known result (9, IO), and its derivation here is noteworthy only in the novelty of its appearance other than as a special case of the expression

KEITH B. OLDHAM

Science Center North American Rockwell Corp. Thousand Oaks, Calif. 91360

RECEIVED for review June 5, 1969. Accepted September 5, 1969.

(8) G. M. Murphy, “Ordinary Differential Equations and Their Solutions,” D. van Nostrand Co., Inc., Princeton, N. J. (1960).

(9) M. Smutek, Chem. Listy, 45,241 (1951). (10) M. Smutek, Collect. Czech. Chem. Commun., 18, 171 (1953).

Additional Comments on the Precision Standardization of Ceric Sulfate Solutions

SIR: In recent work Schlitt and Simpson ( I ) failed to ob- serve the directional-dependent titration error we reported (2 ) for the standardization of ceric sulfate against arsenious oxide in the presence of osmium tetroxide-the Gleu (3) procedure- and which we attributed to incomplete reoxidation of the osmium catalyst when ceric was the titrating agent. Schlitt and Simpson did find entirely similar behavior but only in the presence of nitrate ion. Furthermore, both investigations agreed in that the reverse titration with As(II1) eliminated all errors. This led Schlitt and Simpson to suggest that our ceric reagent, which was obtained from the G . F. Smith Chemical Co. supposedly as a pure solution of 0.1N ceric sulfate in 1N sulfuric acid, was probably a solution of ceric ammonium ni- trate in sulfuric acid.

Fortunately a portion of the ceric solution from our earlier work was still on hand, and tests did indeed show the presence of substantial amounts of both nitrate and ammonium ions. Quantitative analyses were carried out by standard procedures of this Laboratory: ammonia by distillation after the addi- tion of excess sodium hydroxide and nitrate by difference after reduction with ferrous and silver(1) catalyst to ammonia fol- lowed by the same distillation (4, 5). The results were 0.0569M and 0.0678M, respectively, for ammonium and ni- trate ions, and the ceric concentration was 0.0982M. These values, while far from stoichiometric agreement with (NH& Ce(NO&, amply confirm the suspicion of Schlitt and Simpson.

The presence of nitrate and ammonium impurities would certainly not be expected on the basis of information pub-

(1) R. C. Schlitt and K. Simpson, ANAL. CHEM., 41, 1722 (1969). (2) A. J. Zielen, ibid., 40, 139 (1968). (3) K. Gleu, 2. Anal. Chem., 95, 305 (1933). (4) J. E. Varner, W. A. Bulen, S. Vanecko, and R. C. Burrel,

(5) J. M. Pappenhagen, ibid., 30,282 (1958). ANAL. CHEM., 25, 1528 (1953).

lished by Smith (6) on the preparation of his reagents. The sulfato-ceric acid solution is, in fact, specifically recommended for direct use by the consumer because dissolving ceric bisul- fate in dilute sulfuric acid is a troublesome procedure. Also this was not a case of a single bottle of contaminated reagent. Another type and batch of G. F. Smith sulfato-ceric acid so- lution (0.5N ceric in 2N sulfuric acid) was on hand, and this solution gave a much stronger “brown ring” nitrate test than the 0.1Nceric solution.

The critical experiment and confirmation of the Schlitt and Simpson results remained to be done-namely, if the use of a nitrate-free ceric reagent would eliminate the directional-de- pendent error reported in our earlier work. Accordingly a solution of 0.1N ceric sulfate in 1N sulfuric acid was prepared as suggested by Schlitt and Simpson using G. F. Smith crystal- line Ce(HS04)4 as the starting material. The ceric bisulfate and hot solution were equilibrated for 6 hours on a hot plate- magnetic stirrer; and the solution was then cooled, filtered, stored in a brown glass bottle, and allowed to age for 2 days before use. This ceric solution gave a negative “brown ring” nitrate test.

Inasmuch as directional dependence of the titration was the only variable to be tested, potentiometric titrations on 2-meq samples were performed in both directions with a single 0.067N arsenious oxide solution and a constant 0.76 pmole of osmium tetroxide per titration. Other than the ceric reagent, the equipment and procedures used were all identical to the earlier work (2) . This simple comparison of the two solutions was selected in order to obtain the highest possible precision in the titrations without undue concern over the absolute ac- curacy of the standardization. The results are summarized in

( 6 ) G. F. Smith, “Cerate Oxidimetry,” The G. Frederick Smith Chemical Co., Columbus, Ohio, 1942, pp 1 - 1 1 ; ibid., 2nd ed., 1964, pp 33--36.

ANALYTICAL CHEMISTRY, VOL. 41, NO. 13, NOVEMBER 1969 1905