Some Considerations Regarding a Simplified ) and the Formulation of Koutecky's F ( ~ Function Matsuda-Ayabe +(t) Function for Instantaneous D. C. Polarographic Currents Donald E. Smith,' Thomas G . McCord, and Hoying L. Hung Department of Chemistry, Northwestern University, Euanston, Ill.
60201 m
This work presents an examination of a simplified, approximate expression for the Kouteckg F ( x ) function and the Matsuda-Ayabe I)&) function which frequently arise as equivalent theoretical descriptions of dc polarographic instantaneous currents. The simplified expression i n question amounts to a generalization of the work of Meites and Israel which was confined to the irreversible polarographic wave. An evaluation of the accuracy of the simplified formulation as a function of the mechanistic and kinetic situation is given. The results indicate that it is of accuracy sufficient to permit its use in most applications where the F(x) and the $0)functions are relevant.
KOUTECK~. has shown that a higher-order transcendental function, usually designated F(x), frequently appears in the theoretical treatment of the instantaneous d c polarographic currents based o n the expanding plane approximation to the dropping mercury electrode (1-4). Employing the method of dimensionless parameters (1-4, he found that most mechanistic schemes involving kinetic influence by heterogeneous charge transfer and/or fast coupled homogeneous first-order chemical reactions give rise t o this function, with catalytic systems being a notable exception (5). Only the argument of F(x), which contains the relevant kinetic parameters, varies from one mechanism of this class t o another. The importance of F ( x ) in polarographic theory has been further enhanced by the finding that it also arises in the theoretical treatment based o n the more exact expanding sphere electrode model (6). That a single higher-order transcendental function serves as the basis for the expanding plane theoretical formulation of the polarographic wave with the broad class of mechanisms just mentioned was confirmed independently by Matsuda and Ayabe (7,8). Their approach was quite different in that they employed the method of Laplace transformation t o obtain a n integral equation which was solved by a series solution method. The ubiquitous function in the Matsuda-Ayabe treatment has been designated #({) (7). Both F ( x ) and #(E) represent infinite series solutions of the polarographic boundary value problem. F ( x ) has been expressed in the form (1-4) 1
To whom reprint inquiries should be addressed,
(1) J. KouteckS;, Collection Czech. Chem. Communs., 18, 597 ( 1953). ( 2 ) Zbid., 19, 1093 (1954). (3) Ibid., 20, 116 (1955). (4) Zbid., 21, 1056 (1956). (5) Zbid., 18, 311 (1953). (6) J. KouteckS; and J. Cizek, Collection Czecli. Chem. Communs., 21, 836 (1956). (7) H. Matsuda and Y . Ayabe, Bull. Cliem. SOC.Japan, 28, 422 (1955). (8) Zbid., 29, 134 (1956).
F(x) =
i=l
Y*Xi
where
(4)
while +({) was expressed (7)
r is the Euler G a m m a Function. It is readily shown that the treatments of Kouteckjl and Matsuda-Ayabe are equivalent and that
where
t =
v'l:- x
(7)
For convenience, the remainder of the present discussion will dwell explicitly o n the Kouteck? F ( x ) function. However, all conclusions and remarks are directly applicable to the Matsuda-Ayabe formulation, as Equations 6 and 7 attest. Because of the ubiquitous appearance of F ( x ) in theoretical polarographic wave equations, it has been the basis for calculation of numerous electrochemical and chemical rate constants from data o n instantaneous polarographic currents. Such calculations have been made reasonably convenient, despite the complexity of the F ( x ) expression, by the development of extensive tables of F ( x ) L'S. x (9). However, even greater convenience would result if one were able t o employ for F(x) a simple, approximate expression of reasonable (9) J. Weber and J. Koutecky, Collectiori Czecli. Chem. Commurls., 20, 980 (1955). VOL. 39, NO. 10, AUGUST 1967
1149
accuracy analogous to the approximate formula developed for the F(x) function which appears in theoretical expressions for average polarographic currents with the same class of mechanisms (1-4, 9, 10). It has been shown that
E1121 = E” D
- -In nF =
(1 3)
DOI-aDRu
(Standard notation is employed here-notation definitions are given a t the end of this discussion.) Employing currents calculated from Equations 9-13, Meites and Israel (16) showed 1
+ 0.886 (1; X)
x cs. log[(id - i)/i]
The frequent application of Equation 8 to data o n average polarographic currents (10-14) is practical evidence of the desirability of a simplified formulation of F(x). This need is accentuated somewhat by the fact that modern analog and digital recording devices are more conveniently applied t o instantaneous current measurements so that the use of the F(x) function relative t o the F(x) function is likely t o increase. Our own search for a simplified, but accurate formulation of F ( x ) arose in the study of a c polarographic theory based o n the expanding plane electrode model, where the F(x)function also arises with a variety of mechanistic schemes. In attempts to reduce the general ac polarographic equations to certain simplified limiting laws, we found the exact series solution of F(x)to be intractable (15). Although we have not found a n explicit proposal of a simplified expression for F ( x ) in the literature, a paper by Meites and Israel (16) has implicitly provided a formulation of the F(x) function with the desired characteristics. Their work, which was concerned with irreversible polarographic waves, has received little attention and its full implications appear t o have gone undeveloped. The present work is concerned with a n examination of the implications of the paper of Meites and Israel regarding a simplified expression for F(x) and with a n evaluation of the accuracy of this approximate expression. RESULTS AND DISCUSSION
For a simple reaction scheme O + n e S R where the charge transfer rate is sufficiently slow that only the rate of the forward charge transfer step must be considered (the so-called “totally irreversible system”) and where t h e reduced form, R,is initially absent from the solution, Kouteck? showed that the polarographic reduction wave obeys the relationship (1) (9)
where
(10) J. Heyrovsky and J. Kuta, “Principles of Polarography,” Academic Press, New York, 1966. (11) A. A. Vlcek, in “Progress in Inorganic Chemistry,” F. A. Cotton, Ed., Vol. 5, Interscience, New York, 1963, Chap. 4. (12) R. Brdicka, in “Advances in Polarography,” I. S. Longmuir, Ed., Vol. 2, Pergamon, New York, 1960, p. 665. (13) R. Brdicka, Z . Elektrochem., 64, 16 (1960). (14) R. Brdicka, Collection Czech. Cliem. Commun. Suppl., 19, 541 (1954). (15) T. G. McCord and D. E. Smith, unpublished work, Northwestern University, Evanston, Ill., 1966. (16) L. Meites and Y. Israel, J . Am. Chem. Soc., 83,4903 (1961). 1 150
ANALYTICAL CHEMISTRY
is linear. The least-
square “best” equation for the straight line was shown to be log
1;I? x
=
-0.1300
+ 0.9163 log[i/(id - i)]
(14)
Equation 14 represented a very accurate fit to the theoretical line over most of the rising portion of the polarographic wave, Thus, Meites and Israel concluded that the irreversible polarographic wave would closely obey the equation
In
(F)
(15)
Although it was not indicated by Meites and Israel, one can readily show that Equations 14 and 15 are equivalent to stating that the F(x) function may be represented approximately by the relationship
As implied in the Meites-Israel paper, the accuracy of Equation 16 is quite good, except for the first l0-15X of the range of values traversed by F ( x ) [0 5 F ( x ) 5 11. This is illustrated in Table I where a comparison of F ( x ) values calculated from the exact (Equations 1-4) and the approximate (Equation 16) formulation is given for a sampling of x values. These results indicate that the use of Equation 16 in place of the exact expression for studies of totally irreversible waves will be sufficiently accurate except for the first lO-15z of the wave. The same remarks apply to polarographic waves involving the well-known mechanism k
O f n e e R - Y where a rapid (kt > IO), irreversible homogeneous chemical reaction follows a Nernstian charge transfer step (1, 3). In this case Equations 9, 11, and 12 are applicable, while the definition of x becomes (3)
Equation 16 can be applied to the analysis of these “chemically irreversible” waves with restrictions o n accuracy identical to the case of total irreversibility due to slow charge transfer. The gross inaccuracy of Equation 16 in reproducing the exact F(x)at potentials corresponding to the foot of the polarographic waves of the foregoing types of irreversible systems is not particularly serious. Data at the foot of polarographic waves are seldom subjected to quantitative treatment, and in the cases where foot-of-the wave data are of quantitative interest, the well-known limiting form of the exact F(x) expression,
Table I, Comparison of Approximate and Exact Formulations of F ( x ) can be employed. Equation 18 represents the use of only the first term in the exact series solution of F(x), a reasonably accurate approach for x 5 0.1. The two mechanistic schemes considered above are characterized by the fact that F ( x ) is the only potential-dependent term in the theoretical solution, so that F ( x ) must traverse all values from zero to unity over the rising portion of the polarographic wave. However, it is important to note that this situation is somewhat unique, actually corresponding t o the “worst-case” as far as the accuracy of waves calculated with the aid of Equation 16 is concerned. There are many circumstances in which a potential-dependent quantity will appear in the pre-F(x) term of the theoretical solution (see Equations 19 and 23 below). The main result of this will be that F ( x ) will not traverse all values between zero and unity over the potential range of the polarographic wave, but will tend t o assume only larger values where the accuracy of Equation 16 is more favorable. In addition, the variation of F ( x ) values may not be monotonic from small t o large values with increasing polarographic currents, as in the irreversible systems mentioned above. In appropriate cases, F(x) will traverse a minimum o r vary in a monotonic fashion from large to small values as the calculated polarographic currents increase. Thus, the region of maximum error introduced by the use of Equation 16 will not, in general, correspond t o the foot of the polarographic wave. Illustration of the foregoing statements is provided in Tables 11-IV which depict some typical results of calculations we have performed t o assess the practical accuracy of Equation 16 under various kinetic situations. The tables compare polarographic currents calculated with the aid of the exact and approximate formulations of F ( x ) as a function of dc potential. Values of F ( x ) are also given t o illustrate the magnitudes this function assumes. Table I1 gives results for a system involving rate control by charge transfer and diffusion. The charge transfer rate parameters (ks = 10-4 cm sec-I, a = 0.5) are such that both the forward and reverse charge transfer steps are significant (17), thus differing from the totally irreversible case discussed above where only the forward charge transfer rate is important. Polarographic waves of the type depicted in Table I1 are often designated as quasi-reversible (7). The relevant theoretical equation for such a system may be expressed F(X)
Exact
Approximate
Error relative to exact F ( x )
0.0175 0.0345 0,0828 0.1552 0.2749 0.5051 0.6881 0.8248 0,9269 0,9630 0.9814 0.9925
0.0143 0.0299 0.0773 0.1514 0.2755 0.5081 0.6876 0.8242 0.9272 0.9645 0,9830 0,9937
-18.2 -13.3 -6.64 -2.45 +0.22 +O. 59 -0.07 -0.07 +0.03 $0.16 +O. 16 $0.12
F(x) X
0.0200 0.0400 0.100 0.200 0,400 1 .OO 2.00 4.00 10.00
20.00 40.00 100.0
Table 11. Comparison of D C Polarographic Currents Calculated from Exact and Approximate Formulations of F ( x ) for a System with Rate Control by Diffusion and Heterogeneous Charge Transfer Ede -
(volts)
+o. 100 +O. 060
$ 0 , 020
-0.020 -0,060 -0.100 -0.140 -0.180 -0.220 -0,260 -0.300
Exact (+A)
Approximate
idc
idc
(+A)
A ( %)
(exact)
0.1554 0.4136 0.9808 2.137 4.279 7.639 11.63 14.89 16.91 17.91 18,38
0.1566 0.4134 0.9615 2.095 4.276 7.700 11.65 14.90 16.90 17.93 18.41
$0.77 -0.04 -1.97 -1.97 -0.07
0.4174 0.2512 0.1670 0.1665 0,2500 0.4149 0.6218 0.7931 0.9003 0.9533 0.9783
p = 1 - a
Table 111. Comparison of D C Polarographic Currents Calculated from Exact and Approximate Formulations of F ( x ) for a System with Rate Control by Diffusion and a Homogeneous Chemical Reaction Preceding the Charge Transfer Step
+
The potential-dependent quantity in the pre-F(x) term, (1 e’)-l, results in a substantial difference in the range of values of F ( x ) associated with the polarographic wave, relative t o the
(17) P. Delahay, “New Instrumental Methods in Electrochemistry,” Interscience, New York, 1954, p. 75.
- Ei/zr
(volts)
Exact idc (FA)
$0.100 +O ,060 +o ,020 -0.020 -0.060 -0.100 -0.140 -0.180
0.03479 0.1635 0.7434 2.946 7.783 11.82 13.24 13.58
Ed0
(22)
+0.17 +0.07 -0.06 $0.11 +0.16
Parameter values: 12 = 1.00, CO*= 5.00 X 10-3M, D o = D R = 1.00 X 10-5 cmz sec-1, k , = 1.00 X cm sec-I, cy = 0.500, t = 6.00 sec, A = 0.0350 cmz, T = 298” K. A = error in approximate calculation relative to exact calculation. Exact id0 = dc polarographic current calculated from exact F ( x ) formula. Approximate idc = dc polarographic current calculated from approximate F ( x ) formula.
were
.-
$0.80
F(x)
Approxirnate ido
(FA)
0.03479 0.1636 0.7445 2.949 7.773 11.79 13.21 13.56
A ( %)
F(x) (exact)
0.00 $0.06 +0.14 +0.10 -0.13 -0.25 -0.23 -0.15
0.9994 0.9972 0.9874 0.9488 0.8554 0.7702 0.7380 0.7302
Parameter values: n = 1.00, Co* = 5.00 X 10-3M, D o = D R = 1.00 X cm2 sec-’, k , = 5.00 sec-l, kb = 50.0 sec-l, K = 0.100,t = 6.00 sec, A = 0.0350 cm2, T = 298“ K. Notation same as Table 11.
VOL. 39, NO. 10, AUGUST 1967
1 151
Table IV. Comparison of D C Polarographic Currents Calculated from Exact and Approximate Formulations of F(x) for a System with Rate Control by Diffusion, Heterogeneous Charge Transfer and a Homogeneous Chemical Reaction Preceding the Charge Transfer Step ApproxExact imate F( X) (volts) idc (PA) i,ic (PA) A (73 (exact) $0,080 0.06444 0.06436 -0.12 0.8520 $0,040 0.2515 0.2511 -0.16 0.7116 0.000 0.8180 0.8225 +0.55 0.5224 +0.80 0.3750 -0,040 2.122 2.139 -0.080 4.383 4.414 +0.71 0.3474 -0.120 7.270 7.327 +0.78 0.4268 -0.160 9.915 9.964 $0.49 0.5391 -0.200 11.72 11.74 +0.17 0.6265 -0.240 12.72 12.72 0.00 0.6778 -0.280 13.23 13.21 -0.15 0,7042 -0,320 13.47 13.45 -0.15 0.7169 Parameter values: 11 = 1.00, Co* = 5.00 X 10-3M, DO= Dn = D, = 1.00 X 10-5 cm2 sec-l, k , = 1.00 X cm sec-', cy = 0.500, k , = 5.00 sec-1, k b = 50.0 sec-I, K = 0.100, t = 6.00 sec, A = 0.0350 cm2, T = 298" K. Edc
- Emr
irreversible case (where ei = 0 at all potentials). The result is that the error introduced by the use of Equation 16 does not exceed about 2.4% and is much smaller over most of the wave for the case considered in Table 11. The situation improves markedly as k , increases-e.g., when k , = 10-3 cm sec-I and a = 0.5, the maximum error introduced by the approximate calculation is about 0.2%. Tables I11 and IV illustrate the accuracy of Equation 16 when applied to systems with a coupled homogeneous chemical reaction preceding the charge transfer step, kf
Y e0 kb
+ ne S R
Using the Koutecki or Matsuda-Ayabe method with the ka)t > lo], one can readily steady-state assumption [ ( k f show that the theoretical expression for the polarographic wave which accounts for kinetic influence of both the preceding chemical reaction and the charge transfer step may be written ( 4 , 8)
+
where
6 = l +
k,e-"3 D1l2kTll2(1
+ K)
K = kf/kb
(27)
k r = k , f ko
(28)
In the special case where charge transfer is very rapid so that Nernstian conditions prevail-i.e., when kse-"3/ D 2kT'i (1 k ) >> 1, the definition of x reduces t o the more familiar expression
The results given in Table 111obey the predictions of Equations 23 and 29, while Equations 23-28 must be employed for the case considered in Table IV. These results for two decidedly different situations involving kinetic influence of a preceding chemical reaction also show that the accuracy of polarographic waves calculated from Equation 16 frequently can be excellent, exceeding expectations based o n considering only totally irreversible systems. One concludes from the foregoing results that use of the simplified expression for F(x)given by Equation 16 should prove both convenient and accurate for the kinetic analysis of polarographic instantaneous currents. The accuracy of this approximate formulation will vary, depending o n the mechanism, the magnitude of the associated rate parameters and the position (potential) along the polarographic wave. The error introduced normally will not exceed normal experimental error, except at the foot of totally irreversible waves which represent the "worst case." Finally, it is obvious that application of Equation 16 in combination with Equations 6 and 7 will yield a n approximate, simplified form of the Matsuda-Ayabe +([) function which will be reasonably accurate over a much wider range of E values than the approximate expression once suggested by Matsuda (18). NOTATION DEFINITIONS
electrode area diffusion coefficient of species i = activity coefficient of species i Co* = initial concentration of the oxidized form of the electroactive redox couple = standard redox potential in European convention Eo Edc = applied dc potential El,; = reversible half-wave potential based on planar diffusion model F = Faraday's constant R = ideal gas constant T = absolute temperature = number of electrons transferred in the heterogeneous n charge transfer step t = time in seconds = apparent heterogeneous rate constant for charge k, transfer at Eo CY = charge transfer coefficient k,, kb = forward and reverse first-order chemical rate constants for chemical reaction preceding charge transfer K = equilibrium constant for preceding chemical reaction k = chemical rate constant for irreversible first-order decomposition following charge transfer = instantaneous dc polarographic current i(t) = instantaneous d c polarographic current at end of i drop life = limiting instantaneous dc polarographic current a t id end of drop life
A
Di fi
= =
RECEIVED for review March 14, 1967. Accepted May 11, 1967. Work supported by National Science Foundation Grant GP-5778. One of the authors (T.G.McC) is a n N I H Graduate Fellow.
+
1 152
ANALYTICAL CHEMISTRY
(18) H. Matsuda, Z. E/ektroc/?ern.,62, 977 (1958).
