Theory of Polarographic Kinetic Currents for Second-Order Regeneration Reactions at Spherical Electrodes II. Numerical Solution of the Integral Equations for Steady-State Behavior Joseph R. Delmastro
Idaho Nuclear Corporation, Idaho Falls, Idaho 83401 A numerical method is described for solving the nonlinear, steady-state integral equation which arises in treating the second-order, one-half regeneration mechanism for large potential-step electrochemical experiments at stationary spherical electrodes. Computational procedures employed for efficiently obtaining results used in the preparation of working curvesare discussed. These results demonstrate that for spherical diffusion the upper limit of the kinetic-to-diffusion current ratio is less than two and is highly dependent on the effective electrode sphericity. This numerical method is readily adaptable to first-order mechanisms which are difficult to solve analytically.
Nicholson and Shah (2)in theoretical treatments of stationary electrode polarography with rapid triangular potential scan. Our method of solution of the nonlinear, steady-state integral equation for the second-order, one-half regeneration mechanism under limiting current conditions at a stationary spherical electrode is analogous to Nicholson’s treatment (3) of stationary electrode polarography with rapid triangular potential scan for the mechanism involving a rapid dimerization following charge transfer O+ne=R 2R+Z
SEVERAL METHODS have been proposed for treating polarographic boundary value problems which are difficult to solve analytically. Booman and Pence ( I ) discussed various methods which have been employed to obtain approximate analytical solutions for electrode reaction mechanisms with second-order homogeneous kinetic complications. They proposed a generalized numerical method of treating such problems by solving simultaneously a set of finite-difference equations which describe the diffusion and kinetic processes for each generalized time increment. By applying this method to large potential-step electrochemical experiments at stationary spherical electrodes with the second-order, one-half regeneration mechanism, O+nezR k 2R+O+Z
The desired numerical solution of the appropriate nonlinear integral equation is obtained by converting the integralequation to one involving only dimensionless parameters, integrating by parts to remove any singularities in the kernels, replacing the integrals by their corresponding finite sums, and then solving the resulting system of nonlinear algebraic equations by Newton-Raphson iteration on a digital computer. Theoretical working curves of the ratio of the kinetic current to the diffusion controlled current vs. generalized time are obtained. The proposed method is presented in sufficient detail to show that large potential-step polarographic boundary value problems can be solved efficiently by the sequence of steps outlined above. The method is developed for large potentialstep electrochemical experiments at stationary spherical electrodes, with planar electrode behavior treated as a limiting case. Computational procedures employed for efficiently obtaining results used in the preparation of working curves for planar and spherical electrodes are discussed. Finally, the method is applied to the corresponding first-order, onehalf regeneration mechanism. Comparison of results provided by numerical solution with the analytical solution for this first-order case permits an assessment of errors inherent in this numerical method of solving integral equations and in the computational procedures which were employed.
they obtained theoretical working curves of the ratio of the kinetic current to the diffusion controlled current vs. generalized time. A list of notation is given in the Notation Definitions. Due to errors in the concentration gradients caused by limitations on the practicable number of diffusion cells, difficulties arise in calculating the large generalized time portion of the theoretical working curves. These difficulties which arise in the finite-difference method led us to develop the following alternative method for calculating this segment of the working curves. The method developed involves numerical solution of the nonlinear integral equation corresponding to a rapid secondorder chemical reaction k 2R-+O+Z
under limiting current conditions at a stationary spherical electrode. Such a solution would provide independent verification of results of the finite-difference method if a sufficient region of overlap exists between the two numerical methods. The numerical method utilized in solving the appropriate integral equation is closely related to the one employed by (1) G.L.Booman and D. T. Pence, ANAL.CHEM., 37, 1366 (1965).
DEVELOPMENT OF METHOD Theory for the Stationary Spherical Electrode. The boundary value problem for the second-order, one-half regeneration mechanism with semi-infinite spherical diffusion is
so - D o [a;: -+-at -
f aa:]
fkCR2 2
(1)
(2) R. S. Nicholson and I. Shain, ibid., 36, 706 (1964). VOL. 41, NO. 6,MAY 1969
747
t
> 0;r
=
ro: Co
=
0
(44
By introducing the transformation
$ = 2CO
+ CR
(5)
Application of boundary condition 4a to Equation 16 gives the following nonlinear integral equation for the limiting current
and assuming equality of diffusion coefficients Do
=
DR
=
D
(6)
this boundary value problem is converted to (7)
For sake of convenience in numerical solution, this integral equation can be made dimensionless by employing the substitutions V
(1 8)
kCo*u
i k ( T )= nFACo* d X k m @ ( T )
(19)
where T
Solution for the surface concentration of $ by the method of Laplace Transformation yields
=
kC,*t
The dimensionless form of Equation 17 is
where where E(x) = exp(xz)erfc(x) and erfc(x) denotes the complement of the error function. An analytical expression for the surface concentration of the electrolytically generated species R can be obtained if we assume that the second-order chemical reaction is sufficiently rapid to produce a steady-state by compensation of the rates of the chemical reaction and diffusion. In solving for the surface concentration of species R, one assumes that the chemical reaction is sufficiently rapid so that both the aCRjat and (2/r)aCRJar terms in Equation 2 can be neglected. The consequences of the latter approximation will be discussed in a later section of this paper. Under these conditions the boundary value problem for species R becomes
t = 0;r
>,
io:CR =
0
t > O ; r = r o : - ac, - --i(t) ar nFAD
Defining the diffusion controlled current (iD,J in the usual manner for a stationary spherical electrode, we obtain the expression
for calculation of theoretical working curves by solving Equation 21 for the current function @(T). Numerical solution of Equation 21 for @(T)is essentially as described by Nicholson and Shah (2). Division of the range of integration into N equally spaced subintervals of width 6 by the change of variable V = 8v and the definition m = TIS yields
(13)
(14)
1
--I
Q E [ Q d6(m - v)] dv
=
2
(24)
Equation 12 may be integrated directly to yield an expression for the flux of R. Upon application of the boundary conditions stated in Equations 13 and 14, we obtain for the surface concentration
After removal of the point of singularity in the kernel through integration by parts and replacement of the resulting RiemannStieltjes integral by its corresponding finite sum we obtain
At this point, combination of Equations 5, 11, and 15 yields for the surface concentration of species 0
m-1
748
ANALYTICAL CHEMISTRY
L
Table I. Values of 6 Employed and Corresponding Range of T
The roots of this system of nonlinear equations are evaluated by the same procedure as was outlined by Nicholson (3). The corresponding values of the current ratios are calculated from the @(m)'sby using the definition of m and Equation 23. In calculating @(m) from Equation 25, the special term [@(n7 1) - @(m)] may be neglected. It also occurs in numerical treatment of the corresponding purely diffusion controlled case and its negligiblity has been demonstrated under these conditions. The spherical electrode current function expressed by Equation 25 reduces to the corresponding planar electrode current function
6
Range of T
6
Range of T
0.25 0.50
0.25-100 0.50-200 1.OO-500 2.00-1000
4.00 8.00 16.00 32.00
4.00-2000 8.00-4000 16.00-8000 32.00-1 0240
1.oo 2.00
+
I-
in - 1
1
upon expansion of exp(xZ)erfc(x) in the power series for small values of the argument, retaining only two terms in the series expansion. 2x 1 -exp(x2)erfc(x) (27)
-
t' 7r
Equation 26 was also derived from the boundary value problem for the linear diffusion case. Because the term -1 x [@(m 2Q\; 6
-
I-
+ 1) - @(m)]
can be disregarded, linear diffusion is operative under steadystate conditions when the following inequality holds
i
+ 1)
(28)
Alternatively, this condition may be written
because n7 >> (i - 1) when numerical calculations are performed as described--.e., m >> 1 and the most recent values of @(m) receive the heaviest weighting in the summation. The inequality expressed by Equation 29 may be employed to predict the value of QT1/2 for linear diffusion to be operative within an arbitrary error limit. This relationship may be expressed in terms of more conventional parameters as V' Ft/ro’s is
+
Results for Q = 0.20 are presented in Table 11. This increase in accuracy over the linear diffusion results is probably attributable to the present definition of @(T)as a decaying function, with the current ratio being proportional to PW(T)/(l dTQTl/z). Thus, the current function at a spherical electrode does not vary over as wide a range as for the planar electrode. It can be fitted more accurately by generalized time steps than can the corresponding linear diffusion current function. Charge Ratios. The ratio of the kinetic charge to the diffusion charge can be experimentally measured more accurately than the current ratio ik/iD. Theoretical charge ratios can be obtained by integration of the electrode current. Results of the finite-difference method were integrated to yield charge ratios for the time interval between T = 0 and the time when a steady-state is established, as discussed previously (7). The charge ratios for the steady-state region, which were obtained by a simple trapezoidal integration procedure employing normalized values of the current function @, are presented in Table I11 of the first paper (7). Evaluation of Errors in the Numerical Method. To aid in an analysis of errors which occur in the numerical method which was applied to the steady-state integral equation for the second-order, one-half regeneration mechanism, the method was applied to the corresponding first-order, one-half regeneration mechanism. Results could then be compared with the analytical solution for this mechanism. Numerical solution of the steady-state integral equation for the first-order, onehalf regeneration mechanism at a spherical electrode is readily accomplished by the same sequence of steps which was applied to the second-order mechanism, except that the (2/r)dCR/drterm is retained when the steady-state approximation is made. The current function ~ ( ris) defined as
+
1
Solution of this system of equations by the same computation procedure which was utilized for the linear diffusion case yielded results which are in good agreement with results obtained by the finite-difference method (7) for all except the three values of Q-0.20, 0.50, and 1.00. It is apparent that the semi-empirical spherical correction factor which was introduced begins to fail for large degrees of electrode sphericity. The resulting current ratios ik/iD,,are then 1 - 3 z too low. This error is not serious because the resulting working curves still have nearly the correct shape. Hence, current ratios can be normalized to correct infinite-time limiting values provided by finite-difference solution for the time-independent stationary state by the procedure discussed elsewhere (7). The corresponding integral equation stationary-state limiting values of ik/iD,,are obtained from numerical solution of Equation 32 for large values of m6(= T ) or from the resulting infinitetime limit of Equation 3 2
with
(34) Equation 3 4 may be written as
where ~ ( 7is) the solution of the system of linear equations m-1 which illustrates that the stationary-state current ratios will be two or less, depending on the degree of electrode sphericity Q. Stationary-state limiting values of ik/iD,s for various values of Q were obtained by Newton-Raphson iteration (8) of Equation 33 and substitution of the resulting value of $(T+ m ) into Equation 35. These limiting values are presented in Table I of the first paper (7). Normalizing factors were calculated from the ratio of the finite-difference stationary-state limiting values to the integral equation stationary-state limiting values of ik/iD,,. The linearly spaced, normalized current ratio results obtained after numerical solution of Equation 3 2 for various values of Q were interpolated at selected logarithmically-spaced time intervals. These results are presented in Table I1 of the first paper. Spherical diffusion does not significantly alter the rate at which a steady-state is achieved. In general spherical diffusion current ratios obtained by numerical solution of Equation 3 2 are more accurate than linear diffusion values. The per cent difference in current ratio corresponding to the final generalized time point calculated with a given value of 6 and the same time point calculated with the next larger value of 6 is smaller than for the linear diffusion case and the error decreases as Q increases. (8) J. B. Scarborough, “Numerical Mathematical Analysis,” Johns Hopkins Press, Baltimore, Md., 1958, p 192.
with
ma
=
r
=
klt
(39) The ratio of the kinetic current to the purely diffusion controlled current at a stationary spherical electrode is calculated from x(r) by the expression
The analytical solution for this mechanism under steady-state conditions is readily obtained by the method of Laplace Transformation. The more exact expression for the steadyVOL. 41. NO. 6. MAY 1969
* 751
state current ratio (obtained by retaining the (2/r)dCR/dr term in Equation 31) may be written r
Comparison of results of numerical solution for the firstorder, one-half regeneration mechanism with those calculated from the analytical solution indicates good agreement for representative values of Ql. Current ratios corresponding to the 200th time point calculated by using values of 6 greater than unity are within 0.20% of results predicted by the analytical solution for Ql = 0.01. The numerical results become more accurate as more generalized time points are calculated with a given value of 6. As Ql increases, numerically calculated current ratios approach the correct values more rapidly. For Ql = 0.20, the current ratio corresponding to the 200th time point differs from the analytical solution by less than 0.05 %. Except for the error due to the approximate nature of the spherical correction factor which was introduced into the equations for the second-order, one-half regeneration mechanism, errors in numerical solution of the first- and second-order regeneration mechanisms are probably quite comparable. Thus, errors in numerically calculated current ratios for the second-order, one-half regeneration mechanism in the steadystate region may be as large as 0.5z with linear diffusion. Typical errors for current ratios corresponding to time points beyond the 200th are probably less than 0 . 2 z . Numerically calculated current ratios for spherical diffusion, obtained from the steady-state integral equation, are probably accurate to at least 0 . 2 z after normalizing, and become more accurate as Q increases. Stationary-State Current Ratios. Numerical calculations indicate that the current ratio ik,liD,s for the first order, one-half regeneration mechanism can approach a limiting value of less than two for large values of generalized time 7 . Stationary state limiting values of iA/iD,smay be obtained from the infinite-time limit of Equation 37.
tion mechanism is valid. Stationary-state limiting values of ik/iD,sfor the second-order, one-half regeneration mechanism calculated from Equations 33 and 35 differ from those calculated elsewhere (7) by finite-difference methods solely because of our inability to exactly take into account the contribution of the (Z/r)dCR/drterm in obtaining the steady-state solution. This causes values of the current ratio in the steady-state region to be several per cent too low when the degree of electrode sphericity is large, thus requiring normalization of results to correct stationary-state limiting values by the procedure discussed earlier. CONCLUSIONS
The numerical method which has been applied to the solution of the steady-state integral equation for the second-order, one-half regeneration mechanism provides results which are in good agreement with current ratios obtained by the finitedifference method in the region where the two methods overlap. Thus, it provides verification of the finite-difference method developed elsewhere (7). The steady-state region of working curves can be utilized to provide an independent check on the value of the diffusion coefficient calculated from the short-time asymptote. In addition, inclusion of the steadystate region in working curves permits the analysis of kinetic data to be extended over a larger time range, thus increasing the precision in evaluation of second-order rate constants. This numerical method of solving integral equations is readily adapted to other electrode reaction mechanisms. For example, it may be applied to the solution of first-order, polarographic boundary value problems in spherical geometry which cannot be solved analytically. Application of this method to large potential-step electrochemical experiments at spherical electrodes for first-order, homogeneous chemical kinetic mechanisms of interest will be presented elsewhere (9). NOTATION DEFINITIONS
with (43)
Substitution of Equation 42 into 43 yields (44) k
+
In Equation 41, E[(1 2Q1)7"2] approaches zero for large values of the argument. Thus, an expression identical to Equation 44 is obtained as the infinite-time limit of the analytical solution. Equation 44 predicts that the stationary-state will vary from essentially two for very limiting value of ik/iD,s small values of Q I to unity for extremely large values of A limiting value of significantly less than two for electrodes exhibiting a moderate to high degree of sphericity is due to the divergent nature of the spherical diffusion field of species R. The fact that the correct expression for the stationary-state limiting values of ik/iD,scan be obtained from Equation 37 indicates that the method employed in obtaining the corresponding expression for the second-order, one-half regenera-
el.
752
ANALYTICAL CHEMISTRY
ki
m n 0 @
Electrode area. Concentration of species i. Concentration of species i at the surface of a stationary spherical electrode. Initial concentration of species i. Diffusion coefficient of species i. Length of the generalized time subinterval. Error function complement. Exponential error function complement. exp(xz)erfc(x) Electrode current. Electrode current in presence of kinetic complication. Electrode current at a stationary spherical electrode controlled completely by diffusion. Homogeneous rate constant for the second-order, one-half regeneration reaction. Homogeneous rate constant for the first-order, onehalf regeneration reaction. Serial number of the generalized time subinterval. m = 0,N. Number of electrons transferred in the heterogeneous charge-transfer step. Oxidized form of the redox couple. Dimensionless current function for the secondorder, one-half regeneration mechanism. Linear combination of concentrations which obeys the pure diffusion equation.
(9) J. R. Delmastro and G. L. Booman, 1969, unpublished work.
Q QI r r0 R t T 7
A constant, defined by Equation 22, reflecting the effective electrode sphericity for the second-order, one-half regeneration mechanism. A constant, defined by Equation 39, reflecting the effective electrode sphericity for the first-order, one-half regeneration mechanism. Spherical distance coordinate. Spherical electrode radius in centimeters. Reduced form of the redox couple. Time in seconds. Dimensionless time parameter for the second-order, one-half regeneration mechanism. Dimensionless time parameter for the first-order, one-half regeneration mechanism.
U
v,v X
z
Variable of integration. Dimensionless variables of integration. Dimensionless current function for the first-order, one-half regeneration mechanism. Electroinactive species. ACKNOWLEDGMENT
Helpful discussions of the subject matter with G. L. Booman are gratefully acknowledged. RECEIVED for review October 31, 1968. Accepted February 7, 1969. Work supported by the United States Department of the Interior, Office of Saline Water.
Electrochemistry of 8,8'-Diquinolyldisulfide John J. Donahue and John W. Olver Department of Chemistry, University of Massachusetts, Amherst, Mass. The electrochemistry of 8,8'-diquinolyldisulfide, RSSR, and mercuric thiooxinate, (RS)2Hg, has been investigated in methanolic-sodium acetate and aqueous 1.OM H2SO4, using conventional polarography, cyclic voltammetry, and controlled potential electrolysis. The data indicate that RSSR is not directly reduced at the DME but is adsorbed and then undergoes a fast precursor chemical reaction to form mercuric thiooxinate, (RS)?Hg, which is electroactive. Controlled potential electrolysis shows that the (RS),Hg formed undergoes a 2e- reduction to give 8-mercaptoquinoline, RSH, as the major product while the oxidation of RSH gives (RS)2Hg. Although the chemical and adsorption equilibria involved in the reduction of RSSR and (RS)Y Hg are complex, the coulometric generation of RSH from either starting material is feasible and should facilitate the use of 8-mercaptoquinoline as an analytical reagent.
8-MERCAPTOQUINOLINE, the sulfur analog O f 8-hydroxyquinoline, was first synthesized by Edinger in 1908 ( I ) , however, specific application of this compound as an analytical reagent was not investigated until 1944 when Taylor (2) reported that 8-mercaptoquinoline was not adaptable to general analytical use because it was readily oxidized by atmospheric oxygen to 8,8-diquinolyldisulfide(2). The work of Bankovskii et al. (3-5) and Freiser and Fernando (6) clearly shows that 8-mercaptoquinoline, RSH, has useful properties as an analytical reagent. In particular, Freiser (6) has shown that certain complexes of 8-mercaptoquinoline are more stable than the corresponding 8-hydroxyquinolates ; however, reagent instability has posed a severe drawback in the cases cited.
(1) A. Edinger, Chem. Ber., 41,937 (1908). (2) J. R. Taylor, VirginiaJ. Sci., 3,289 (1944). (3) V. I. Kuznetsov, Y.A. Bankovskii, and A. F. Levinsh, J. Anal. Chem. (U.S.S.R.),13,299 (1958). (4) Y. A. Bankovskii, A. F. Levinsh, and Z. E. Liepinya, J. Anal. Cltem. (U.S.S.R.),15, l(1960). (.5 .) Y . A. Bankovskii and A. F. Levinsh. J. Anal. Chem. (U.S.S.R.). 17, 725 (1962). (6) 35.1424 . , A. Corsini. 0.Fernando. and H. Freiser. ANAL.CHEM.. (1963). I
_
We have examined the electrochemical behavior of RSSR by conventional polarography, cyclic voltammetry, and controlled potential electrolysis in order to develop a method for the coulometric generation of 8-mercaptoquinoline, thereby avoiding reagent instability problems. EXPERIMENTAL
Apparatus. Polarograms were obtained using an Indiana Instruments and Chemical Corporation Controlled Potential and Derivative Voltammeter (Model ORNL 1988A) and recorded on a Sargent Model SR recorder. All electrochemical experiments were carried out at 25 i 1 "C. Solutions were deaerated with prepurified tank nitrogen which was presaturated with solvent. The saturated calomel electrode was used as a reference electrode in all cases and the dropping mercury electrode had the following characteristics at open circuit in a solution of 1.OM H2S04, m = 2.04 mg sec-', t = 3.69 sec. Cyclic polarograms at low voltage scan rates were obtained using the Heath Model EUW-401 polarography system (Heath Co., Benton Harbor, Mich.), a conventional threeelectrode cell and a Moseley Autograf Model 135 X-Y recorder. At high sweep rates (>0.2 volt second-') the triangular sweep generator of Weir and Enke (7) was used in conjunction with the Heath Model EUW-19A operational amplifier system. A Tektronix 502A Dual Beam Oscilloscope with attached Tektronix Oscilloscope Camera, Model C-12, was used to record the current-voltage curves. A Jaissle Potentiostat Model lOOOT (Jaissle ElektronikLabor, West Germany) was used to maintain the desired potential values during the controlled potential electrolyses. A 1.0-pf capacitor was inserted in the potentiostatic feedback loop to filter 60-cycle ac noise picked up by the unshielded cell. Current-time curves were recorded using a Sargent Model SR Recorder and integrated electronically using a resistance-capacitance integrator (8). Applied potentiostatic potential and the integrator readout potential were measured using a Honeywell potentiometer Model 2730.
(7) W. D. Weir and C. G . Enke, Rev. Sei. Instr.,35,833 (1964). ( 8 ) H. C. Jones, W. D. Shults, and J. M. Dale, ANAL.CHEM., 37, 680 (1965). VOL. 41, NO. 6, MAY 1969
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