Polarographic Kinetic Currents for First-Order Preceding and Regeneration Reactions at Spherical Electrodes J. R. Delmastro and G . L. Booman Idaho Nuclear Corporation, Idaho Falls, Idaho 83401 The linear integral equation which arises in treating the first-order preceding reaction mechanism in spherical geometry for large potential-step electrochemical experiments in the limiting current region is solved by a numerical method. The exact analogy between this mechanism and the first-order fractional regeneration mechanism is demonstrated. Numerically generated data are fitted to a Chebyshev polynomial for efficient tabulation of results. Tabulated Chebyshev coefficients and special forms of analytical solutions permit rapid generation of working curves for both of the mechanisms considered, even on a small laboratory computer. The analytical solution for the first-order, one-half regeneration mechanism in spherical geometry is presented.
MANY EXPERIMENTAL large potential-step investigations of electrode processes have employed hanging drop electrodes. Nevertheless, neglect of electrode curvature characterizes most theoretical treatments of such processes. There are several reasons why existing theoretical treatments are usually restricted to the linear diffusion electrode model. Intuitively, it was thought that the spherical nature of a hanging drop electrode would not significantly influence results because kinetic studies of rapid homogeneous chemical reactions could be performed under conditions where the thickness of the reaction layer is much smaller than the diffusion layer thickness. Hence, the species participating in the chemical reaction would not “see” the curvature of the electrode at normal measurement times. Another reason that existing treatments are usually restricted to the linear diffusion model is because the analytical solutions for many spherical diffusion problems cannot be obtained. The development of numerical methods suitable for treating such problems has removed this dficulty. An electrode process involving a homogeneous chemical reaction which was treated in spherical geometry is the second-order, one-half regeneration mechanism ( I , 2). This treatment indicated that spherical diffusion can exert a significant influence in large potential-step investigations of the kinetics of coupled homogeneous chemical reactions. Because spherical diffusion can exert a significant influence at normal measurement times, it is necessary to consider numerical solution of such polarographic boundary value problems when the analytical solutions cannot be obtained. Two electrochemical mechanisms of interest for which the general analytical solution in spherical geometry cannot be obtained are Mechanism 1A involving a first-order or pseudo firstorder chemical reaction preceding the charge transfer step and Mechanism 1B involving a first-order or pseudo firstorder fractional regeneration of the depolarizer from the product of the electrode reaction. Mechanism 1A.
* 0+ n e kl
Y
R
ki
(1) G. L. Booman and D. T. Pence. ANAL.CHEM.. 37. 1366 (1969. (2) D. T. Pence, J. R. De1mastro;and G. L. Booman, ibid., 41, 737 (1969).
Mechanism 1B. O i - n e e R R
kc CY -.+
-
P
0
+ electroinactive products
(A list of definitions is given in Notation Definitions.) Theoretical treatments of these mechanisms for planar electrodes and the dropping electrode have already appeared. Koutecky presented theoretical treatments for Mechanism 1A under steady-state conditions for planar (3) and expanding planar ( 4 ) electrodes. Mechanism 1B with half (5) and total (6) regeneration of the depolarizer was also treated within the framework of the expanding plane diffusion model. Theory for the total regeneration mechanism with diffusion to planar electrode has been presented by several workers (7-9). Koutecky and Cizek solved the boundary value problems with diffusion to an expanding sphere electrode for Mechanism 1A under steady-state conditions (IO) and for the total regeneration mechanism (11). More recently Buck (12) and Guidelli and Cozzi (13) presented general theoretical treatments for these mechanisms with diffusion to planar electrodes. From the latter two theoretical treatments, it can be shown that the large potential-step response for these mechanisms in the limiting current region is exactly the same if we identify the equilibrium constant K = kl/k2 with @/CY 1) and (kl kz) with k,. Thus, each value of the equilibrium constant corresponds to a particular fraction of the depolarizer regenerated and working curves of i k / i d us. generalized time for Mechanism 1A can also be applied to Mechanism 1B. Such working curves can be generated by numerical solution of the boundary value problem by the method developed earlier (14, 15). Although values of the ratio of the kinetic current to the diffusion current can be generated by numerical solution of polarographic boundary value problems, efficient presentation of results for the preparation of working curves becomes difficult when the current ratio i k / i d is a function of two or more parameters. This is the case for all reaction mechanisms which involve coupled homogeneous reactions in spherical geometry. For example, the current ratio for the secondorder, one-half regeneration mechanism in spherical geometry
+
(3) J. Koutecky and R. Bridicka, Collect. Czech. Chem. Commun., 12, 337 (1947). (4) J. Koutecky, ibid., 18, 597 (1953). (5) J. Koutecky, R. Brdicka, and V . Hanus, ibid., 18, 611 (1953). (6) J. Koutecky, ibid., 18, 311 (1953). (7) Ibid., p 183. ( 8 ) P. Delahay and G. Stiehl, J. Amer. Chem. SOC.,74, 3500 (1952). (9) S . Miller, ibid., 74, 4130 (1952). (10) J. Koutecky and J. Cizek, Collect. Czech. Chem. Commun., 21, 836 (1956). (11) Ibid., p 1063. (12) R. P. Buck, J . Electroanal. Chem., 5 , 295 (1963). (13) R. Guidelli and D. Cozzi, ibid., 14, 245 (1967). (14) R. S. Nicholson and I. Shain, ANAL.CHEM., 36, 706 (1964). (15) J . R. Delmastro, ibid., 41, 747 (1969). VOL. 41, NO. 11, SEPTEMBER 1969
1409
is a function of the generalized time parameter and the effective electrode sphericity. Hence, current ratios were presented at logarithmically-spaced values of generalized time for selected values of effective electrode sphericity (1, 2). An alternative to the direct tabulation of values is to present the coefficients obtained by fitting the values to a suitable polynomial approximation. Such coefficients could then be used for rapid calculation of working curves, even on a small general purpose digital computer. For electrode reaction mechanisms which can be solved analytically, coefficients obtained by fitting results to a polynomial approximation would still be quite useful because the analytical expressions frequently involve higher order transcendental functions and different expressions for various values of the kinetic parameters, thus making calculations extremely difficult on a small general purpose digital computer. Therefore, polynomial fitting of both numerical and analytical results represents a possible solution of the data presentation problem for the kinetic region. Simplified forms of analytical expressions may be employed for calculation of the steady-state region of working curves. The numerical method developed elsewhere (15) is applied to the integral equation for the limiting current with the preceding reaction mechanism (Mechanism 1A) at spherical electrodes. The exact analogy between this mechanism and the first-order, fractional regeneration mechanism is illustrated by comparison of the expressions for the concentration of the oxidized form of the redox couple at the surface of a spherical electrode. Computation procedures employed for efficiently obtaining numerical results are discussed. Results predicted by the analytical solution (12) for the preceding reaction mechanism with linear diffusion are used to evaluate a Chebyshev polynomial fitting procedure. This fitting procedure is then applied to numerical results obtained for Mechanism 1A in spherical geometry. Chebyshev coefficients are presented for selected values of the effective electrode sphericity and logarithmically-spaced values of (kl k2)t in the kinetic region of working curves of i& us. (kl k2)t. These coefficients permit rapid calculation of this segment of the working curve for any value of the chemical equilibrium constant K. Because of the exact analogy between the preceding reaction mechanism and the first-order, fractional regeneration mechanism, these tabulated coefficients also permit rapid calculation of the kinetic segment of the working curve for the regeneration mechanism with any fraction cv/p of the depolarizer regenerated. Equations are presented which can be employed for calculation of the short-time and steady-state regions of working curves on a small general purpose digital computer. Even though the general analytical solution for Mechanism 1A or 1B in spherical geometry cannot be obtained, an analytical solution is possible for the first-order, one-half regeneration mechanism (a/P = l / 2 ) , corresponding to the special case of the preceding reaction mechanism with an equilibrium constant of unity if we identify k , with (kl k2). The analytical solution derived for this special case permits verification of results of numerical solution for the kinetically-significant case of Mechanism 1A with K = 1 and evaluation of the precision of fit of the polynomial approximation employed in the tabulated results.
+
+
bC0 dt t = 0; r
2 ro: C y + CO = Cy* + Co*
(3a)
c-o--_co*_- -K CY
CY*
t > 0 ; r=ro:
bC0 i(t) D O T = - nFA
bCY = O Dybr
where
K = kijkz
(5)
This boundary value problem is readily solved for the surface concentrations by the method of LapIace Transformation after assuming equality of diffusion coefficients for species Y and 0. D y = Do= D
(6)
/.t
r .
lik
1
- Dil2ro-l E(D1/2ro-1u1/z)] du
(8)
where
E(X)
k
=
ki .f k2
=
exp(X2)erfc(X)
(9)
(10)
and erfc denotes the complement of the error function. For ( n D t ) 2/ro 51.0, can be calculated from the analytical steadystate solution-Equation 30-by employing the above mentioned series approximations for E(X). In Equation 30, E{ [K (1 Q) Q]i"l/z/ approaches zero for large values of the argument. Thus, the infinite-time or stationary-state limit of the steady-state solution is
+
+
+
This result may also be obtained for the infinite-time limit of Equations 24 and 28. Equation 46 predicts that stationarystate limiting values of the current ratio can vary from essentially unity to K/(K 1). A limiting value of significantly less than unity is due to the convergent nature of the spherical diffusion field for the oxidized form of the depolarizer. Working curves for the fractional regeneration mechanism may be generated by the same general procedure which was discussed in the previous paragraph for Mechanism 1A. The working curve for the regeneration mechanism with a given fraction a//3of the depolarizer regenerated and fixed Q is most easily calculated by determining which value of K in Mechanism 1A it corresponds to. This result is calculated from K = p/a - 1. The working curve for this value of K is then generated by the procedure discussed above. Finally, current l)/K to take into account the ratios are multiplied by ( K definition of in terms of Co*for Mechanism 1B instead of the total concentration (Co* Cy*)for Mechanism 1A. Thus, to generate the working curve for the one-half regeneration mechanism (a/P = 1/2), one would calculate the working curve for K = 1 from the Chebyshev coefficients, analytical solution for linear Wusion, and the steady-state spherical
+
+
+
T
P
0.051 0 1 2 0.062 0 1 2 0.083 0 1 2 0.100 0 1 2
0.130 0 1 2 0.160 0 1 2 0.210 0 1 2 0.260 0 1 2 0.320 0 1 2 3 0.410 0 1 2 3 0.510 0 1 2 3 0.620 0 1 2 3 0.830 0 1 2 3
1.000 0 1 2 3
1.300 0 1 2 3 0 1.600 1 2 3 2.100 0 1 2 3 2.600 0 1 2 3
Table I. Sets of Coefficients for Discrete Values of T and Q Chebyshev C:oefficients a p Chebyshev Coefficients a p 1 Q = 0.01 Q = 0.05 Q = 0.50 Q = 0.00 Q = 0.00 Q = 0.01 Q = 0.05 Q = 0.50 T W = 0.34, W = 0.25 W = 0.12 W = 0.05 P W = 0.34 W = 0.25 W = 0.12 W = 0.05 1.0117 1.0115 1.0129 1.1278 1.0082 3.200 0 1.1797 1.2100 1 ,2424 0.5008 0.5010 0.4957 0.5010 0.5000 1 0.5130 0.5067 0.4929 -0.0041 -0.0049 -0.0056 -0.0051 2 -0.0898 -0.1050 -0.1209 -0.0634 0.0046 0.0078 3 -0.0133 -0.0062 1.0152 1.0135 1.0135 1.0100 0 4 0 0.0006 0 0.5010 0.5008 0.5000 0.5010 1.2538 1.1291 -0.0050 -0.0059 -0.0067 -0.0060 1.1836 1.2169 4.100 0 0,4953 1 0.5155 0.5080 0.4914 1.0194 1.0171 1.0168 1.0132 2 -0.0917 -0.1088 -0.1268 -0.0641 0.5010 0.5010 0.5008 0.5000 3 -0.0160 -0.0076 0 .0050 0.0092 -0.0066 -0.0077 -0.0088 -0.0077 4 0.0010 0.0005 0 0 1.0194 1.0227 1.0158 1.0201 0 0 5 0.0005 0 0.5007 0.5009 0.5001 0.5010 1.2592 0 1.1825 1.2183 1.1276 5.100 -0.0079 -0.0092 -0.0105 -0.0090 1 0.4906 0.5177 0.5094 0.4952 1 ,0237 1 ,0203 1 .0251 1 .0284 2 -0.0911 -0.1099 -0.1298 -0.0633 0.5007 0,5001 0.5009 0.5010 0.0101 0,0050 3 -0.0184 -0.0092 -0.0102 -0.0117 -0.0134 -0.0112 0 4 0 0.0013 0.0007 0 0 5 0.0007 0 1.0277 1.0340 1.0247 1.0300 0.5011 0.5006 0.5009 0.5002 1.2158 1.2604 1.1238 1.1780 6.200 0 -0.0124 -0.0141 -0.0162 -0.0132 0.4902 0.4954 1 0.5109 0.5199 2 -0.0887 -0.1089 -0.1306 -0,0615 1.0339 1.0430 1.0318 1 ,0379 0.0049 3 -0.0208 -0.0108 0.0105 0.5008 0.5003 0.5011 0.5005 0 4 -0.0006 0.0014 O.OOO9 -0.0159 -0.0181 -0.0206 -0.0163 0 5 0.0008 0 0 1 ,0396 1 ,0386 1.0515 1.0455 1,2557 1.1140 0 1 ,2062 1.1659 8.300 0.5004 0.5008 0.5012 0.5004 0.5136 1 0.4961 0.5232 0.4908 -0.0193 -0.0219 -0.0249 -0.0192 2 -0.0820 -0.1041 -0.1286 -0.0567 1.0459 1.0613 1.0542 1.0463 0.0041 3 -0.0243 -0.0138 0.0099 0.5003 0.5007 0.5013 0.5006 0.0013 0 4 -0.0014 0.0014 -0.0232 -0.0263 -0.0298 -0.0224 0 0 5 0.0010 0.0007 0 0 -0.0006 -0.0005 1.1552 1.1966 1.2488 1.1049 10.000 0 1.0543 1 ,0573 1.0665 1.0751 1 0.5254 0.5157 0.4919 0.4968 0.5009 0.5014 0.5005 0.5001 2 -0.0760 -0.0991 -0.1254 -0.0522 -0.0286 -0.0324 -0.0367 -0.0266 3 -0.0266 -0.0160 0.0088 0.0034 0 0 -0.oo09 -0.0007 0.0016 0 4 -0.0022 0.0012 0 5 0.0011 0.0008 0 1.0685 1.0791 1.0893 1 ,0624 0.0005 6 0 0 0 0.5016 0.5003 0.4999 0.5013 -0.0342 -0.0387 -0.0438 -0,0307 1.1367 1.1790 1.2342 1.0881 13.000 0 -0.0013 -0.0008 0.0005 0 1 0.5285 0.5189 0.4944 0.4980 1.0798 1.0919 1.1035 1 ,0703 2 -0,0654 -0.0897 -0.1185 -0.0439 0.5018 0.5018 0.4997 0.5001 3 -0,0297 -0,0194 0.0063 0.0021 -0.0399 -0.0451 -0.0510 -0.0346 4 -0.0037 o.Ooo5 0.0019 0 0.0008 -0.0018 -0.0011 0.0006 5 0 0.0010 0 0.0010 6 0 0 0 0.0007 1.1132 1.1275 1,0987 1 ,0827 0.4992 0.5022 0.4995 0.5028 1.1621 1.1197 1.2187 1.0715 16.000 0 -0.0493 -0.0558 -0.0630 -0.0408 1 0.5308 0.5214 0.4969 0.4991 -0.0028 -0.0015 0.0012 0.0012 2 -0,0554 -0.0805 -0.1109 -0.0356 3 -0,0318 -0,0221 0,0036 0.0010 1.1116 1.1280 1.1442 1.0907 4 -0.0052 0.0021 0 0 0.4988 0.5037 0.5026 0.4990 5 0.0007 0.0010 0 0 -0.0558 -0.0632 -0.0713 -0.0449 6 0.0008 0 0 0 -0.0037 -0.0019 0.0017 0.0016 1.1687 1.1304 1.1495 1.1018 21.000 0 1.0946 1.1364 1.1935 1.0450 0.4981 0.5052 0.5033 0.4982 1 0.5334 0,5246 0,5007 0,5004 -0.0651 -0.0740 -0.0836 -0.0504 2 -0.0406 -0.0662 -0.0983 -0.0224 0.0026 0.0022 -0.0053 -0.0026 3 -0.0339 -0,0252 0 0 4 -0.0076 -0.0020 0.0021 0 1.1882 1.1449 1.1664 1.1100 5 0 0.0010 0 0 0.4971 0.5067 0.5039 0.4976 6 0.0008 0 0 0 -0.0723 -0.0826 -0.0934 - 0.0545 -0.0068 -0.0032 -0.0036 0,0028 26.000 0 1.0732 1.1137 1.1699 1 ,0206 1 0.5268 0.5350 0.5038 0.5009 1.1619 1.1867 1.2123 1,1191 2 -0.0278 -0.0534 - 0.0864 -0.0102 0.4956 0.5090 0.5049 0.4968 3 -0.0348 -0.0272 -0.0036 -0.0008 -0.0808 -0.0929 -0.1056 -0.0591 4 -0.0097 -0.0036 0 0.0051 0.0019 -0.0091 -0.0042 0.0035 5 -0.0006 0.0007 0 0 1.1726 1.2001 1.2290 1.1245 6 0.0008 0 0 0 0.5057 0.4942 0.5110 0.4962 0.0005 7 0 0 0 -0.0862 -0.0998 -0.1141 - 0.0617 -0.0111 -0.0052 0,0064 0.0041 (Continued on page 1416)
VOL. 41, NO. 11, SEPTEMBER 1969
0
1415
T
P
32.000
0
1 2 3 4 5
6 7 41.000
0
1 2 3 4 5
6 7 51.000
0
1 2 3 4 5
6 7
Table 1. Continued Chebyshev Coefficients a p Q = 0.00 Q = 0.01 Q = 0.05 Q W = 0.34 W = 0.25 W = 0.12 W 1.0510 0.5360 -0.0146 -0.0350
-0,0118 -0.0016 0.0006 0.0005 1.0233 0.5365 0.0020 -0.0342 -0.0142 -0.0030 0
0.0006 0.9979 0.5361 0.0171 -0.0324 -0.0162 -0.0044 0
0.0005
1.0898 0.5284 -0.0398 -0.0285 -0.0053
1,1441 0.5066 -0.0732 -0.0066 0.0015
0
0 0 0
0.0005 0
1,0592 0.5295 -0.0223 -0.0290 -0.0074
1.1100 0.5095
-0.0554 -0.0095 0.0008 0 0 0
0 0 0
1.0306 0.5297 -0.0061 -0.0284 -0.0093 -0.0012
1.0771 0.5113 -0.0382 -0.0113 0 0 0 0
0 0
= =
0.50 0.05
0.9938 0.5010
0.0033 -0.oo09 0
0 0 0
special analytical solution permits verification of results of numerical solution and evaluation of the precision of fit of the tabulated Chebyshev coefficients for this kinetically-significant case. Because derivation of this analytical solution involves a non-routine application of the Laplace Transformation method and cumbersome algebraic expressions, it is presented elsewhere (22). The analytical solution for the first-order, one-half regeneration mechanism may be written in three different forms, depending on whether the effective electrode sphericity Q is less than, equal to, or greater than one-half. Q
< 0.5:
0.9581 0.4999 0.0211 0 0 0 0 0
0.9237 0.4977 0.0382 0.0023 0 0 0 0
diffusion expression. The resulting current ratios would then be multiplied by (K l)/K =2 to convert them to results for the one-half regeneration mechanism. For the regeneration mechanism, generalized time is defined by T = k,t. Current ratios for the fractional regeneration mechanism will vary from unity for small values of T to limiting values dependent on the fraction regenerated a//3and the effective electrode sphericity Q. Stationary-state limiting values of the current ratio for the first-order, fractional regeneration mechanism may be calculated from the expression
+
Q = 0.5:
[(T
Q
This result may be obtained from Equation 46 by replacing K by @/a - 1) after multiplying by (1 K)/K to account for definition of i D , sin terms of Co*for Mechanism 1B instead of Equation 23. Thus, the stationary-state limiting value for the first-order, one-half regeneration mechanism at a planar electrode is two, but can be significantly less than two at spherical electrodes because of the divergent nature of the spherical diffusion field for the reduced form of the depolarizer. Similar behavior was observed in calculations for the second-order, one-half regeneration mechanism (2).
+ 3) Zo(')(T/2) + (T + 2) Zl(O)(T/2)]- 0.5n (1 + 0.5
(49)
> 0.5:
+
~ ~ 8 p ' )5 ' (1 +8Q2 16Q4)"ZoYT/2)] + < T (4Q2 y )+ E1 [ ( ~4Q2) -d 1T ] (1
64Qa
Although the general analytical solution for Mechanism 1A or 1B in spherical geometry could not be obtained, an analytical solution by the method of Laplace Transformation was derived for the one-half regeneration mechanism. This, solution corresponds to the special case of the preceding reaction mechanism with an equilibrium constant of unity. This 1416
ANALYTICAL CHEMISTRY
n=i
4Q2-
(12" + 16Q2
'1
(1
+ Qd?rT)-'
(50)
(22) J. R. Delmastro and G. L. Booman, AEC Report IN-1274, Idaho Falls, Jdaho, 1969.
In these equations Zpcn)(X) denotes the nth derivative of the modified Bessel function, and Equation 10 defines E(X). Working curves for discrete values of Q may be calculated from these equations. The term involving a summation of higher derivatives of the modified Bessel function of order zero fails to converge when Q approaches one-half. This does not represent a serious difficulty because Equation 49 holds for Q = 0.5. Working curves were calculated by employing polynomial approximations of Z o ( o ) ( x ) , e-xZo(o)(X), zl(0)(X), e-xzl(o)(X) and the asymptotic expansion of ZP(0)(X) for large order P (23). Higher order modified Bessel functions are required for calculation of derivatives of ZdX) by accumulation of Zp(O)(X) with appropriate binomial coefficients (23). Recurrence relations cannot be used to calculate ZdX), Z3,(X). . . successively if ZdX) and h(X) are known because there is a rapid build-up of error when P exceeds X (24). Therefore, all the higher order modified Bessel functions are themselves calculated from the two highest order modified Bessel functions by recurring backwards (24)
L(x)
= ZP+1
+
2P ZP(X) X
0
+I
T
Figure 1. Ratio of kinetic to diffusion current VS. generalized time for first-order, one-half regeneration mechanism Effective electrode sphericity factor Q of 0.00 in curve A , 0.01 in B, 0.05 in C, and 0.50 in D
-’
Equations 48 and 50 may be employed for calculation of working curves for Q < 0.333 and Q > 0.75, respectively, with fewer than fifty terms in the summation. Current ratio results obtained from Equations 48-50 are in excellent agreement with the analytical steady-state solution for spherical diffusion for values of T greater than about 10, irrespective of the effective electrode sphericity. Figure 1 illustrates working curves for the first-order, one-half regeneration mechanism for the generalized time range 0.051 5 T 5 51.0 calculated from Equations 48-49 for the four Q values employed earlier-0.00, 0.01, 0.05, and 0.50. Results depicted as the Q = 0.00 curve were calculated by using a Q value of because a rapid loss of significant figures occurs when smaller Q values are used with Equation 48. Agreement between these results and those calculated from the equation presented earlier for linear diffusion ( I ) is within 0.2 % for the largest value of T employed. Figure 1 depicts the influence of spherical diffusion on the electrode response. These effects were discussed qualitatively in the previous section of this paper. After multiplying the current ratios by K/(K 1) and redefining T as (kl k2)f,results depicted in Figure 1 may be applied to the preceding reaction mechanism with an equilibrium constant of unity. Errors in the numerical method of solving the integral equation for the preceding reaction mechanism were evaluated by comparing results obtained by numerical solution with those provided by the analytical solution corresponding to the K = 1 case. Numerical results for the 200th time point calculated for K = 1 agreed within 0.25 % with results provided by Equations 48-50 for all values of Q over the entire T range 0.05151.0. The precision of the Chebyshev series fit obtained from numerical results was evaluated by recurrence to obtain the current ratio for K = 1 and comparison with results provided by the analytical solution. The Chebyshev coefficients tabulated in Table I provide current ratios which are within 0.3zagreement with results depicted in Figure 1. For the
+
I
+
(23) M. Abramowitz and L. A. Stegun, Eds., “Handbook of Mathematical Functions,” National Bureau of Standards, U. S.Government Printing Office, Washington, D. C., 1964, Chapter 9. (24) E. T. Goodwin and Staff, Mathematics Division, National
Physical Laboratories, “Modern Computing Methods,” Phiiosophical Library, Inc., New York, N. Y.,1961, Chapter 13.
larger values of Q, agreement is within 0.2 %. In all cases, the results calculated from the tabulated Chebyshev coefficients are slightly larger than current ratios provided by the analytical solution corresponding to K = 1. Only for small values of K will errors in the current ratios calculated from the tabulated coefficients exceed 0.5 %. CONCLUSIONS
The numerical method which has been applied to the solution of the integral equation for the preceding reaction mechanism in spherical geometry provides current ratio results which are accurate to within 0.3% for all values of the equilibrium constant. The combination of analytical solutions for the short- and long-time regions and the tabulated Chebyshev coefficients can be employed to efficiently generate working curves for the preceding reaction mechanism with any value of K or for the first-order, fractional regeneration mechanism with any fraction of the depolarizer regenerated and with any of four discrete values of electrode sphericity. These working curves may be generated on a small general purpose digital computer by using the series approximations for E ( X )and F(X) included herein. Results of the present study illustrate that fitting of data to a Chebyshev series to be used in the calculation of working curves represents a solution to the data presentation problem which arises when the current ratios are a function of more than one parameter. The numerical method developed and the Chebyshev fitting procedure employed are readily adaptable to other electrochemical mechanisms and should be applicable to other electrochemical techniques. The Chebyshev fitting procedure developed here does not preclude the existence of other fittings procedures, such as a Chebyshev series expansion in two variables, for efficient tabulation of results to be used in the preparation of working curves as required for analyzing results of experimental investigations of electrode processes at spherical electrodes. a
ap
A
NOTATION DEFINITIONS = An arbitrary constant which affects the rate of convergence of the series in Equations A l , A2, A5, and A8 = Chebyshev coefficient for polynomial of order P = Electrode area VOL. 41, NO. 11, SEPTEMBER 1969
0
1417
a/@
ct
=
= =
Cr* (Cf),-,o = Dt
=
6
=
e
=
E(Z)
=
Fraction of the depolarizer regenerated in Mechanism 1B Concentration of species i Initial concentration of species i Concentration of species i at the surface of a stationary spherical electrode Diffusion coefficient of species i Length of generalized time subinterval Maximum possible error in a given series approximation Exponential error function complement. Defined by
erf(Z)
5szm
= Error function.
eVt2dt
s,"
Defined by -
e-l'dt
erfc(2) = Error function complement. erfc(Z) = 1 erf(z) f(X) = An arbitrary function F(Z)
= Dawson's integral.
Defined by e-"
-
f,"
et2dt
d-7 in Equations 45, A10, and A13 Electrode current = Electrode current in presence of kinetic complication iD,P = Electrode current at a stationary planar electrode controlled completely by diffusion iD,* = Electrode current at a stationary spherical electrode controlled completely by diffusion Z p ( n ) ( X ) = nth derivative of the modified Bessel function of order P K = Equilibrium constant for the preceding reaction mechanism. Defined as kl/k2 k = Sum of forward and reverse chemical rate constants for the preceding reaction mechanism kl, k2 = Homogeneous rate constants for the first-order preceding reaction in Mechanism 1A kc = Homogeneous rate constant for the first-order regeneration reaction in Mechanism 1B m = Serial number of the generalized time subinterval in numerical calculations (m = 0, M>. Also used as an index (m = 2n - 1) in Equations Al, A2, A5, A8, and A10 = Number of electrons transferred in the heteron geneous charge-transfer step = Dimensionless variable of integration V = Oxidized form of the redox couple 0 = A constant which reflects the effective electrode Q sphericity for the preceding reaction or fractional regeneration mechanism r = Spherical distance coordinate = Spherical electrode radius in centimeters ro = Reduced form of the redox couple R t = Time in seconds = Dimensionless time parameter for the preceding T reaction or fractional regeneration mechanism Tmax = Maximum value of the dimensionless time parameter calculated with a given value of 6 T p ( X ) = Chebyshev polynomial of degree P in X u = Variable of integration = Dimensionless variable of integration v = Empirically selected weighting factor for fitting W the K,'s to a Chebyshev polynomial = Dimensionless current function for the first-order, X preceding reaction mechanism i i(t) in
1418
=
=
ANALYTICAL CHEMISTRY
Y
Electroinactive species
=
= Complex argument.
2
2= x
+ iy
APPENDIX EVALUATION OF E(2) =
5
Jzm
Z
e-"dt AND F(Z) = e-z*
e'ldt
FOR REAL AND COMPLEX ARGUMENTS Real Arguments. The exponential error function complement with real argument, denoted herein by E(X), may be evaluated from several different series expansions or polynomial approximations presented elsewhere (21). One method of calculating this function is to employ the asymptotic expansion for large values of the argument ( X > 3.2) and the product of the exponential and the error function complement for small values of the argument. This method cannot be used for evaluation of E ( X ) on a small general purpose laboratory computer because of the slow convergence of the absolutely convergent series for erf(X) in the range 2 < X < 3.2 or to the large number of significant figures required from polynomial approximations for the error function. A convenient method for evaluating E ( X ) can be derived from the work of Miller and Gordon (25). They presented an extensive discussion of general methods for evaluating many Fourier series and functions such as F ( X ) which arise in the solution of problems in heat conduction and electrochemical diffusion. Miller and Gordon (25) presented the following approximation which is extremely useful for evaluating functions which are defined in terms of integrals in which the integrand is of a particular form.
I-
m
"XZ
In this approximation a is an arbitrary constant which affects the rate of convergence of the resulting power series and rn = 2n - 1. Exponential error function complement, Dawson's integral, error function, and Fresnel integrals are all functions of the form required for application of this approximation. Upon evaluation of the integral on the right-hand side of Equation Al, the contribution of the lower limit to the sum is usually a summable series which can be evaluated from tables presented by Miller and Gordon (25). Because of the cumbersome notation employed in their work, it may often be convenient to use other tables of summable series (26) for evaluating this term. Application of approximation A1 to the exponential error function complement yields
For the sake of convenience in calculation we define a as (25) W. L. Miller and A. R. Gordon, J. Phys. Chem., 35, 2785
(1931).
(26) A. D. Wheelon, "Tables of Summable Series and Integrals Involving Bessel Functions," Holden-Day, Inc., San Francisco, Calif., 1968.
a = 0.1/x2
(A3)
which permits evaluation of E(X) for all values of the argument greater than unity with an error e(X) of less than 2 X lo-* with only seven terms in the evanescent series in Equation A2. For all values of the argument less than unity, E(X) is accurate in two parts in the sixth significant figure. If a smaller value is selected for a in numerical evaluation, E ( X ) may be calculated more accurately for arguments of less than unity, but significantly more terms are required in the series. Equation A2 with a defined by Equation A3 is particularly useful for calculations on a small computer because the series terms do not involve transcendental functions and convergence is extremely rapid. For fixed a defined by Equation A3, the exponential terms are constants which may be stored in the program. For large X only the sum contributes significantly to E(X). The first term on the right-hand side of Equation A2 may be set equal to zero for X greater than two. This term must be set equal to zero for X>> 7rdG because it then increases without bound. If the argument lies in the range - 1 < X < 1 and E(X) is required to an accuracy greater than two parts in the sixth significant figure, it may be evaluated conveniently from the product of the exponential and the error function complement, after calculation of the error function from its absolutely convergent series expansion or polynomial approximations (21). For negative arguments E(X) can also be evaluated from the relationship
E ( - x ) = 2 ex2 - E ( X )
(A4)
Equation A2 was programmed in FOCAL language on a PDP-S/S computer and E(X) evaluated from this expression for X greater than unity. For arguments in the range - 1 < X < 1, E ( X )was evaluated by using the absolutely convergent series for erf(X), taking the complement, and multiplying this result by the exponential. This program for calculation of E(X) is accurate to the 23-bit mantissa used in the FOCAL floating point calculations for all values of X. Equation A1 may be used to obtain the following approximation for the error function
If Equation A3 is employed for a, A5 permits evaluation of erf(X) with only seven terms in the evanescent series for all values of the argument with an accuracy comparable to that obtained in calculation of E(X). Dawson's integral, which is frequently denoted by F(X), may be evaluated from several different series approximations presented by Miller and Gordon (25). The absolutely convergent series for F(X) may be written
Equation A2 may be obtained by application of approximation A1 .
Alternatively, Equation A8 may be obtained from A2 by employing Equation 45, which relates E(iX) to F(X). If the arbitrary constant a is taken as defined by Equation A3, this series approximation permits evaluation of F ( X ) for all values .of X to the same accuracy as was obtained with Equation A2 for E(X). For larger values of X, only the summation term contributes significantly to F(X). The magnitude of the first term on the right-hand side of Equation AS is less than 10-*1 for an argument of five. Equation A8 was programmed in FOCAL language on a PDP-SfS computer and F(X) evaluated from this expression for arguments greater than unity. For arguments less than unity F ( X ) was calculated from Equation A6. For all values of the argument, this FOCAL program for calculation of Dawson's integral is accurate to the 23-bit mantissa used in floating point calculations. Complex Arguments. The exponential error function complement with complex argument 2 = x iy arises for certain ranges of parameters in the solution of some diffusion problems in electrode kinetics for transient perturbation techniques such as the galvanostatic single (27) and double (28) pulse methods, the coulostatic impulse relaxation method (29, 30), and the large potential-step method (31). Computer calculation of E(Z) for use in the analysis of coulostatic and galvanostatic data has recently been performed by several workers (32, 33). Methods used by these workers require a larger computer for calculations using the absolutely convergent and asymptotic series (32) or an interpolation procedure (33) which employs tabulated values of E ( X ) . Hence, these methods cannot be used for data analysis on a small laboratory computer. A convenient method for evaluating E(Z) is now available because approximation A1 is valid with complex argument 2 as well as with real argument X. Therefore, Equation A2 can be employed for calculation of E(Z). Equation A2 may be written in terms of real variables after defining a as
+
E(x
These two series approximations cannot be used for evaluation of F(X) for all values of X on a small laboratory computer because of the slow convergence of Equation A6 for arguments in the range 2 < X < 3.5. An evanescent series analogous to
+ iy) =
+ cos (2by) cos (2xy) + sin (2by) sin (2xy)l + cosh (26x) + cos (2by) exp (x' - y*)[e-zbzsin (2xy) + cos (2by) sin (2xy) sin (2by) cos (2xy)l + cosh (2bx) + cos (2by)
exp ( x 2 - y2) [e-2b2cos (2xy)
i
For large values of the argument (X > 3.9, the asymptotic expansion may be employed
(-49)
a = 0.1122
(27) T. Berzins and P. Delahay, J. Amer. Chem. SOC.,17, 6448 (1955). (28) H. Matsuda, S. Oka, and P. Delahay, ibid., 81, 5077 (1959). 34, 1272 (1962). (29) W. H. Reinmuth, ANAL.CHEM., (30) P. Delahay, J. Phys. Chem., 66, 2204 (1962). (31) R. Guidelli, J. Electroanul. Chem., 18, 5 (1968). (32) R. F. Martin and D. G. Davis, Abstracts, Southwest Regional Meeting of the American Chemical Society, Austin, Texas, Dec. 1968, No. 8. (33) D. J. Kooijman, J . Electround. Chem., 19, 445 (1968). VOL. 41, NO. 11, SEPTEMBER 1969
0
1419
series terms do not involve transcendental functions and convergence is extremely rapid. A useful property of E(2) is that this function of the conjugate of 2 is equal to the conjugate of E(2).
E(x
and rn=2n-1
(AW
This expression permits evaluation of the real and imaginary parts of E(2) for all values of the argument x iy with an error of less than two parts in the sixth significant figure with seven terms in the evanescent series in Equation A10. For large x or y only the sums contribute significantly to The first two terms on the right-hand side of Equation A10 may be set equal to zero for x greater than two or the absolute value of y greater than about ten. Equation A10 is suitable for calculations on a small laboratory computer because the
+
&(a.
- iy) = E(x + i y )
(‘413)
Equation A1 3 may be derived from relationships presented elsewhere (21). This symmetry relation is useful because when the exponential error function complement with complex argument occurs in electrochemical theory, the complex arguments always appear in conjugate pairs. Useful approximations for the error function and Dawson’s integral with complex argument may be obtained from Equations AS and A8, respectively. Equations A5 and A8 may be written in terms of real variables after substitution of Equation A9. The resulting expressions permit evaluation of erf(2) and F ( 2 ) with complex argument, even on a small laboratory computer. RECEIVED for review March 3, 1969. Accepted June 13, 1969. Work supported by the United States Department of the Interior, Office of Saline Water.
Rapid and Specific Extraction of Aluminum From Complex Mixtures with N-Benzoyl-N.Phenylhydroxylamine Spectrophotometric Determination with 8-Quinolinol Robert Villarreal, John R. Krsul and Spence A. Barker Argonne National Laboratory, Idaho Division,Idaho Falls, Idaho 83401 A highly selective method for the separation and determination of aluminum has been developed and applied to uranium-based fuels, stainless steel, and various other materials. The procedure is based on the extraction of aluminum with N-benzoyl-N-phenylhydroxylamine (BPHA) into benzene from an ammonium carbonate solution containing several masking agents. Aluminum is back-extracted into 0.20M HCI, complexed with 8-quinolino1, and the colored complex extracted into chloroform and measured at 390 nm. Of 50 metallic elements tested none gave serious interference. Milligram quantities of oxalate, tartrate, phosphate, sulfate, cyanide, and other common anions do not interfere; citrate, fluoride, and EDTA do interfere. Beer’s law is obeyed from 0-50 pg AI in 10 ml of chloroform; the optimum range i s 5-30 rg AI. At the 0.01% level in uranium alloy and the 0.05% level in stainless steel, the relative precision of the method is *2% at the 2~ level. The specificity of the method makes it applicable to many types of samples.
INGENERAL, spectrophotometric determinations of aluminum are based on measuring the color formed by aluminum with various lake-forming reagents or 8-quinolinol. The nonselectivity of these chromogenic reagents for aluminum makes several separations necessary to remove interfering ions. Direct masking of interfering ions aids in reducing the number of preliminary separations required to determine aluminum with unspecific chromogenic reagents. Lake-forming chromogens require exacting conditions and, like 8-quinolinol, are subject to interfering ions even after addition of masking 1420
ANALYTICAL CHEMISTRY
agents. Ion-exchange separation of aluminum is slow and after separation, lengthy treatment of samples is required to establish the optimum conditions for a spectrophotometric procedure. Several methods for the determination of aluminum have been summarized by Sandell (I). Reactor fuels used in Experimental Breeder Reactor I1 (EBR-11) are uranium-metal alloys which may contain all the high-yield fission products along with common metallic impurities. To study the effect of trace impurities on reactor fuel characteristics, the determination of low-level aluminum in various fuels was necessary. Several methods were tried for the determination of microgram quantities of aluminum in fuel containing uranium alloying metals, fission products, and common impurities. The often erratic results were probably due to (a) incomplete separation of interfering ions, (b) interference of unknown ions, (c) aluminum contamination in different reagents, and (d) lengthy chemical procedures. Although several procedures have been reported for the determination of aluminum in specific alloys (2-8), no satis(1) E. B. Sandell, “Colorimetric Determination of Traces of Metals,” 3rd ed, Interscience, New York, N. Y., 1959, p 219-253. (2) W. Sprain and C. V. Banks, Anal. Chim. Acta, 6 , 363 (1952). (3) A. Claassen, L. Bastings, and J. Visser, ibid., 10, 373 (1954). (4) F. J. Miner, R. P. Degrazio, C. R. Forrey and T. C. Jones, ibid., 22, 214 (1960). ( 5 ) D. W. Margerurn, W. Sprain and C. V. Banks, ANAL.CHEM., 25, 249 (1953). (6) R. J. Hynek and L. J. Wrangell, ibid., 28, 1521 (1956). (7) H. B. Evans and Hiroshi Hashitani, ibid., 36, 2032 (1964). (8) R. T. Oliver and E. P. Cox, ibid., 41, lOlR (1969).
