JOSEPHR. DELMASTRO AND DONALD E. SMITH
2138
An Examination of Some Methods for Obtaining Approximate Solutions to the Expanding-Sphere Boundary Value Problem in
Direct Current Polarography'
by Joseph R. Delmastro28 and Donald E. Smith2b Department of Chemistry, Northwestern University, Evanston, Illinois 60601
(Received December 6 , 1086)
The present work considers whether two approximate theoretical methods, representing extensions of the Lingane-Loveridge approach, can satisfy the need for reasonably accurate assessment of the combined effects of drop growth and curvature in dc polarography while avoiding the disadvantage of extreme complexity often associated with the rigorous approach. The approximate methods are subject to detailed evaluation by comparing polarographic waves predicted by these methods with those given by rigorous solutions to the expanding-sphere problem. Mechanistic schemes considered are the reversible case, the quasi-reversible case, and systems with catalytic and preceding coupled chemical reactions. Both approximate methods are found to have considerable merit. Equations of competitive accuracy and greater simplicity relative to rigorously derived expressions are obtained. Of particular interest is a relatively simple expression for the quasi-reversible dc polarographic wave which reproduces the predictions of the highly cumbersome rigorous theory to within 1%. Evidence is obtained indicating that the assumption of mathematical separability of drop-growth and curvature effects represents an approximation of general applicability and reasonable accuracy.
Introduction Since Ilkoviit first derived the well-known equation for the diffusion-controlled limiting current in dc polarography on the basis of the expanding-plane electrode model,s a number of workers have undertaken modification of the Ilkovi6 equation to take into One of account the curvature of the mercury drop."'' the first attempts to account theoretically for both the curvature and expansion of the electrode surface was made by Lingane and Lo~eridge.~They assumed that since the Ilkovi6 theory showed that the expansion of the electrode surface into the solution can be accounted for by multiplying the equation for diffusion to a stationary plane electrode by ('/3)"', an equation accounting for curvature and expansion of the electrode surface should be obtained by an analogous operation on the diffusion-current expression for the stationarysphere electrode. Their intuitive arguments led to the equation for the polarographic limiting current (see Appendix for notation definitions)
Upon rearrangement and substitution of the appropriate expression for ro, they obtained ~ L L=
gtnFACo*D,'/'(I
+ 44.6Do"a~'"m-''') ( 2 )
While they clearly outlined the mathematical (1) This work was supported by National Science Foundation Grants GP-2670 and GP-5778. (2) (a) Present work taken in part from Ph.D. Thesis of J. R. Delmastro, Northwestern University, 1967. (b) To whom reprint
inquiries should be addressed. (3) D. IlkoviE, Collection Czech. Chen. Cornnun., 6 , 498 (1934). (4) J. J. Lingane and B. A. Loveridge, J . Am. Chem. Soc., 72, 438 (1950). (5) H. Strehlow and M. von Stackelberg, 2. Elektrochem., 54, 61 (1950). (6) T. Kambara, M. Suzuki, and I. Tachi, Bull. Chem. Soc. Japan, 23, 219 (1960). (7) T.Kambars and I. Tachi, ibid., 23, 225 (1950).
-APPROXIMATE SOLUTIONS TO
EXPANDING-SPHERE BOUNDARY VALUE
2139
PROBLEMS
time t, i. the corresponding current at a stationary steps leading to eq 1 and 2, Lingane and Loveridge were sphere electrode, and iep the current for an expandingnot explicit in stating the basic assumption underplane electrode. The assumption that drop growth lying their derixtion. Normally, one would expect and curvature perturbations do not couple then permits this to be of little consequence as the assumption one to represent (i. - i,) as the correction for drop should be apparent from the method of derivation. curvature and (iep - i,) as the correction for drop However, careful study indicates that an ambiguity growth. The same assumption suggests that addiexists in the case of the Lingane-Loveridge method in tion of these corrections t o i, should yield an expresthat two alternative assumptions appear consistent sion for the current a t an expanding sphere electrode, with their result. One may interpret the Linganeiesz. That is, one may write Loveridge derivation as equivalent to assuming that the contribution of spherical diffusion can be simply ies2 = i, (iep - ip) (is - i,) (6) represented as an alteration of the effective diffusion or coefficient which is calculable with the aid of stationary-sphere theory. On this basis Laitinen, Kivalo, ies2 = iep is i, (7) and Oldham'5 presented an equation for the diffusionRecalling the relevant expressions for the limiting curcontrolled (reversible) dc polarographic wave which rentav 16 17 allegedly accounted for drop growth and curvature. Their equation was derived from the expression based . nFACo*Do'/' on the expanding plane diffusion model by multiplying 2, = (7rt) 1/1 DO'" and DR'/?by the factors (1 44.6Do'/"t'/6rnm11/a) and (1 4 , 1 . 6 D ~ ' / ' t ' / ~ ~ respectively. ~-'/')), The sign (9) in the spherical correction factor for DR is dependent on whether the reduced form is soluble in solution (+ sign) or in the electrode phase (- sign). Their is = nFACo*Dol/' Dt'] (10) expression for the entire reversible dc polarographic wave may be written and substituting them in eq 6 yields eq 1 , so it is clear that the Lingane-Loveridge method is consistent with nFACo*Do'/P(l 44.6Do'"t'~6rn-''8) X the principle of separability of drop-growth and (1 f 44.6D~~"t~'/'rn-'/') curvature effects. (1 f 44,6D~%~/'rn-~/') The intuitive method underlying the Lingane-Love(1 44.6Do1/'t/'rn-'/')e' ridge equation obviously represents a rather uncertain theoretical base. Fortunately, an assessment of the (3) accuracy of eq 2 was forthcoming in the form of more where rigorous theoretical formulations of the expandingsphere boundary value problem. Kambara and Tachi (4) showed that eq 2 is an approximate solution to the MacGillavry-Rideal equation for diffusion to an
+
+
+ -
#
+
*
+
+
+
1
Equation 3 reduces to the Lingane-Loveridge equation for the limiting current (eq 2) when the electrode potential becomes sufficiently cathodic (e'+ 0). An alternative interpretation of the basic assumption underlying the Lingane-Loveridge derivation is that coupling of the perturbations on the mass transfer process associated with electrode growth and curvature may be neglected; i.e., the contributions of drop growth and curvature may be considered separable or additive. A more explicit statement of this principle can be given in terms of a mathematical relationship as follows. Let i, denote the current flowing at a stationary plane electrode after an electrolysis
(8) T. Kambara and I. Tachi, "Proceedings of the First International Polarographic Congress, Prague, 1951," Vol. 1, PiZrodovC deck6 Vydavatelstvi, Prague, 1951,p 126. (9) T. Kambara and I. Tachi, Bull. Chem. SOC. Japan, 25, 286 (1952). (10) H.Matsuda, ibid., 26,342 (1953). (11) J. Kouteckjl, Czech. J. Phys., 2, 50 (1953). (12) V. G. Levich, "Physicochemical Hydrodynamics," PrenticeHall, Englewood Cliffs, N. J., 1962,Chapter 10. (13) R. Subrahmanya, Can. J . Chem., 40, 289 (1962). (14) A. Kimla and F. b f e l d a , Cotleetion Czech. Chem. Commun., 28, 3206 (1963). (15) K. B. Oldham, P. Kivalo, and H. A. Laitinen, J . A m . C h m . SOC.,75, 5712 (1953). (16) F. G.Cottrell, 2.Physik. Chem., 42, 385 (1902). (17) P. Delahay, "New Instrumental Methods in Electrochemistry," Interscience Publishers Inc., New York, N. Y.,1954, pp 61-63.
Volume 71, Number 7 June 1067
JOSEPHR. DELMASTRO AND DONALD E. SMITH
2140
expanding sphere electrode. Rigorous treatments of the diff usion-controlled limiting current a t an expanding-sphere electrode have been presented by Kouteckf,“ Levich,12 and Matsuda.lo Their solutions can be expressed in the form of an infinite series
I n the Kouteckf and Levich treatments, only the al term contributes significantly to the instantaneous current under normal experimental conditions. A value of 39.6 is found for al in both cases, compared to the value of 44.6 associated with the corresponding term in the Lingane-Loveridge equation (eq 2). Matsuda’s treatment yielded a slightly different result with a significant a2 term compensating for a slightly smaller a1 term (a1 = 36.3), so that his theory yields essentially the same results as the Koutecky-Levich expression. The magnitude of the al term, which represents the spherical diff iision contribution for a dropping mercury electrode (dme), differs significantly (- 13%) in the rigorous solution and the approximate equation of Lingane and Loveridge. However, since this term represents a small (10-15%) contribution to the total diffusion current, the approximate Lingane-Loveridge equation conies surprisingly close to predicting the same diffusion current magnitudes as the rigorous solution. The diffusion currents predicted by eq 2 are slightly larger (-2%) than those predicted by the more exact expressions. The direction of the error is as expected since the Lingane-Loveridge method is equivalent to assuming that the effects of drop growth and curvature do not couple, while in fact the movement of the electrode surface should tend to decrease the contribution of spherical diffusion. A mathematical basis for this statement has been provided by Kouteckf and von Stackelbergls who, by appropriate manipulation of the MacGillavry-Rideal equation, identified a term manifesting coupling of the dropgrowth and curvature perturbations. Nevertheless, the rather emall difference between eq 2 and the rigorous expression provides evidence that the assumptions inherent in the Lingane-Loveridge equation are reasonably accurate as far as the polarographic limiting current is concerned. Koutecky and Ciiek have extended the method for rigorously solving the MacGillavry-Rideal equation to several more complex electrode reaction mechanisms in which chemical reactions or charge transfer compete with diffusion for rate control.1g*” These studies indicate that rigorous solution of the MacGillavryRideal equation for most mechanistic schemes will The Journal of Phyaical Chemistry
prove extremely tedious and, unlike the diffusioncontrolled case, the final results will often assume rather complex forms involving series solutions whose coefficients are defined by cumbersome recurrence relat i o n ~ . These ~~ facts apparently have discouraged most workers from applying the rigorous theory to the analysis of experimental results as attested by the almost exclusive use of the expanding plane theory. In many applications it would be advantageous to be able to estimate the magnitude of the spherical diffusion contribution with the aid of a conveniently implemented approximate theory which is accurate to within a few per cent in estimating the current magnitude (the corresponding error in the expanding plane theory is 10-20%11J8,21), The success of the Lingane-Loveridge equation in predicting the diffusion-controlled limiting current with reasonable accuracy led us to consider whether a similar correction for the effects of drop growth and curvature can be obtained for the entire dc polarographic wave with various electrode reaction mechanisms by applying either of the assumptions leading to equation 2. The present work evaluates the following two approximate methods originating in the assumptions which are consistent with the Lingane-Loveridge equation: (1) correcting the theory based on the expanding-plane electrode model for spherical diffusion by modifying the diffusion coefficients by “spherical correction factors” of the type used by Oldham, Kivalo, and Laitinen;lS (2) assuming that the effects of drop growth and curvature do not couple so that eq 7 is applicable over the entire dc polarographic wave. Either of these approximate methods would be quite useful if proven sufficiently accurate. Method 1 is simplest in both implementation and final result. However, the literature abounds in the essential theoretical equations for i,, i,, and i,, so that application of method 2 also will be quite convenient with most mechanisms, although the final results will prove somewhat more cumbersome than those of method 1.
Theory The present work evaluates approximate methods 1 (18) J. Kouteck? and M. von Stackelberg in “Progress in Polarography,” P. Zuman, Ed., with the collaboration of I. M. Kolthoff, Vol. I, Interscience Publishers Inc., New York, N. Y., 1962, Chapter 2.
(19) J. Koutecki and J. Eiiek, Collection Czech. Chem. Commun., 21, 836 (1966). (20) J. Kouteck? and J. Eiiek, {bid., 21, 1063 (1956). (21) J. and I. Smoler in “Progress in Polarography,” P. Zuman, Ed., with collaboration of I. M. Kolthoff, Vol. I, Interscience Publishers Inc., New York, N. Y., 1962, Chapter 3.
Kits
APPROXIMATE SOLUTIONS TO EXPANDING-SPHERE BOUNDARY VALUEPROBLEMS
and 2 by comparing dc polarographic waves calculated on the basis of these methods with those calculated from exact solutions of the MacGillavryRideal equation provided by Kouteckf and coworkers.11s1s-20 This is done for de polarographic waves in which: (a) diffusion is the sole rate-determining step (the reversible case); (b) charge transfer and diffusion are the rate-determining steps (the quasi-reversible case); (c) chemical reactions coupled to the charge-transfer step and diffusion are the rate-determining steps. To aid in comparison of accuracy and complexity, of mathematical expressions associated with the various theoretical electrode models, both equations and results of calculations are given for all relevant theoretical models, including equations based on the stationary plane and sphere electrode models, suitably adapted for applicability to polarography. In all cases the expression for the electrode area appropriate to the dme [A = 8.515 X 10-3(mt)z/8]is employed in calculations. Through this operation on the electrode area term, the theoretical expressions for the potentiostatic response of stationary plane and sphere electrodes are transformed to approximate theoretical models for the polarographic experiment with the dme. Calculations based on the resulting equations are then performed in the context of the polarographic experiment; i.e., parameters of importance in stationary electrode experiments such as initial (rest) potential have no relevance, while neglect of depletion effects and other assumptions inherent in most theoretical polarographic work are employed (cf. ref 18). Such utilization of the stationary-plane theory represents one of the earliest approaches to obtaining a quantitative theory for the polarographic ~ a v e , ' ~which ~'~ yields an expression in which effects of drop growth and curvature are neglected. The stationary-sphere theory applied in this manner represents an approximation to the dme which ignores contributions of drop growth. I . Difusion-Controlled Processes. For the case of ne e R, in which a simple electrode process, 0 charge transfer does not exert rate control, two basic cases will be considered: (A) both forms of the electroactive oxidation-reduction couple soluble in solution, and (B) the reduced form soluble in the electrode, as in amalgam formation. The equations for the stationary- and expanding-plane diffusion models are independent of the mode of diffusion of the reduced form, due to the symmetry of the diffusion process.ls Equations presented for the amalgam case will neglect the finite volume effect.11*22s23 It has been shown that this effect does not become important for normal polarographic conditions.22 When the equations differ
+
2141
between cases A and B, the upper sign will refer to case A (as in eq 3 ) . An expression for the polarographic current with a reversible electrode process which is based on the stationary plane model (i,)may be written15
where j is given by eq 4. The corresponding expression for diffusion to an expanding plane has the form24
For the stationary spherical electrode model, it has been shown that 23,26-27
e~(Dol'ziDR1/z)2r0-1 (1 e')(e"O1'z f OR1/')exp(aA2t)erfc(a,l'/')}
+
(14)
where
For all potentials encompassed by the polarographic wave, Kouteckf's rigorous solution of the expandingsphere problem assumes the formllp's
As stated before, eq 3 is the appropriate expression for the expanding-sphere model on the basis of approximate method 1. Method 2 yields (substituting eq 12, 13, and 14 in eq 7) ._
(22) W.H.Reinmuth, Anal. Chem., 33, 185 (1961). (23) I. Shain and K. J. Martin, J . Phya. Chem., 65,254 (1961). (24) J. Heyrovsky' and D. Ilkovi:, Collection Czech. Chem. Commun., 7, 198 (1935).
Volume 71, Number 7 June 1967
2142
JOSEPH R. DELMASTRO AND DONALD E. SMITH
?I. Electrode Processes Kinetically Controlled by Charge Transfer and Diffusion. As in the reversible case, one must consider whether the reduced form is soluble in the solution (case A) or electrode phase (case B) in deriving theory for diffusion to a stationaryor expanding-spherical electrode. The same sign convention will be employed as with the reversible case. Theory for diffusion to a stationary planar electrode yields for this case28--30 &
= nFACo*
(E
)"ihe-ai exp(X2t) erfc(xt'/a) (18)
+X + f 0;") + XI2 F 4[To-2D~/'DR"' +
2b* = ro-l(Dol/' f OR'/')
{ [ro-'(Dt/'
khT0-l
(&"'e -
* D~'/'e@j)]}'''
(25)
The rigorous solution of the expanding-sphere boundary value problem for the quasi-reversiblecase has been provided by Kouteckf and &iek.lg They showed that
where A =
@e kh
-4 + e8i)
p = 1 - a D =D ~ D R ~
(19) (204
The expanding-plane electrode model produces the expressions1 la2
where
x = 7/'2Xt'" 7
GO, G&,
(22)
and F ( x ) is defined and tabulated by Kouteckf and c ~ - w o r k e r s . ~ ~Recently, J~ the appropriate relations for the stationary-sphere model were presented for both cases A and B. It was found that2'J4
and F ( x ) are defined in Kouteckf and Ciiek's article, where G+(x) and G-(x) are represented as GA(x) and G B ( x ) ,respectively, in the Kouteckf-Ciiek paper.lg x is given by eq 22. Approximate method 1 gives the relationship
where I-
(25) I. Shain and D. S. Polcyn, J. Phys. Chem., 65, 1649 (1961). (26) W. G.Stevens and I. Shain, Anal. Chem., 38, 865 (1966). (27) J. R. Delmastro and D. E. Smith, ibid., 38, 169 (1966). (28) M. Smutek, Collection Czech. Chem. Commun., 18, 171 (1953). (29) T. Kambara and I. Tachi, Bull. Chem. SOC.Japan, 25, 136 (1962). (30) P.Delahay, J. Am. Chem. Soc., 75, 1430 (1953). (31) J. Koutecki, Collection Czech. Chem. Commun., 18, 597 (1953). (32) H. Matsuda and Y. Ayabe, Bull. Chem. SOC.Japan, 28, 422 (1955). (33) J. Weber and J. Kouteok:, Collection Czech. Chem. Commun.,20, 980 (1965).
The Journal of Phy&
Chemistw
APPROXIMATE SOLUTIONS TO EXPANDINGSPHERE BOUNDARY VALUEPROBLEMS
2, e&j
(1 f 44.6D2%"8m-"a)
}
kd
ah
-
*
exp(ah2t) erfc(a+t'/')
*
DR1/a)
*
r o - ~ ( ~ o ' / ~D ~ ' ' ~ )
ro-2D;/lDR'12 a&
nFACo*Do'/'kc'/' 1 el
+
(33)
)
Kouteckf and Cifek solved the boundary value problem for the simple catalytic mechanism with diffusion to an expanding sphere.20 They obtained
-(b* - T ~ - ~ ( D ~*' / ' a& - b*
) exp(b*Zt) erfc(bht'/')
T~-~D~'/~DR'/~
b*
=
+
This is obtained by replacing Dol/' by Do'/'(l 44.6Do''*t''8m-''a) and DR'" by DR'/'(l 44*6D2'v''6m - '/a) in eq 21. Method 2 yields (substituting eq 18, 21, and 23 in
1
.
Zep
To our knowledge, an expression for the current with diffusion to a stationary-sphere electrode and the simple catalytic mechanism has not been presented. However, the following expression is readily obtained by application of the method of the ~~~i~~~ transformation40
(29)
*
=
2143
-
.=k+t
7 nFACo*Do'/' [l 1+ej
+ 39.6D0'~'t'/"m-'/'] X
50.31D0'/Y'/"m-'/~[[8(k,t) - 0.7868$(kct)]]
{$(kct)
(35)
III. Electrode Processes Controlled by Di$usim and Chemical Reaction. 1. The Simple Catalytic Mechanism with Nernstian Conditions. The approximate methods of obtaining theory for dc polarography applicable a t the dme will be examined for the simple catalytic mechanism35represented by
8(kct) is defined in the original manuscript.20 Method 1 yields the expression
id =
+ e3
For kct > 9, eq 36 simplifies to
.
=
Zeal
R+Z+O where the oxidizing agent Z is electroinactive in the potential range of the 0-R oxidation-reduction couple and is present in sufficient excess to produce pseudofirst-order conditions. Theory presented for this mechanism assumes DO = DR. The appropriate theory for the catalytic mechanism with diffusion to a stationary planar electrode has been presented by a number of workers.as-as They found that
+ 44.6Do'/*t'~8m-'~') $(kCt)
7 nFACo*Do'/'(l (1
(36)
O + n e z R
+ kl/' erf[(k.t)'~']}
Gt
nFACo*Do'/'(l
+ 44."o'/'t''"m-'/')kc'/' 1
+ e'
(37)
The expression for the simple catalytic mechanism given by method 2, obtained by substituting eq 31, 32, and 34 in eq 7, takes the form
tz
=
,/&
nFACo*Do'/' X 1 e3
+
[$(?cot) For k,t
+ 44.57D0'/'t'/"m-'/']
(38)
>9
(31)
Kouteckf derived the following expression for the polarographic current with diffusion to an expanding plane electrode39
$(kct) is defined and tabulated in Kouteckjr's article.se For kct > 9, eq 31 and 32 reduce to a6-ag
(34) I. Shain, IC. J. Martin, and J. W. Ross, J . Phys. C h m . , 65, 259 (1961). (36) P.Delahay, "New Instrumental Methods in Electrochemistry," Interscience Publishers Inc., New York, N. Y., 1964, pp 100-104. (36) P. Delahay and G . Stiehl, J . Am. Chem. SOC.,74, 3600 (1952). (37) 8. Miller, ibid., 74, 4130 (1952). (38) J. Kouteckj., Collection Czech. Chem. Cmmzm., 18, 183 (1953). (39) J. KouteckS;, ibid., 18, 311 (1953). (40) J. R. Delmastro and D. E. Smith, unpublished work.
Volume 71, Number 7 June 1067
JOSEPHR. DELMASTRO AND DONALD E. SMITH
2144
2. Limiting Current in a System Involving a Preceding Chemical Reaction. Methods 1 and 2 will be evaluated for the electrode reaction mechanism ki
A
Y z O + n e - R
+
+ K)k'/'
=
dL
-
n F A C o * D o1l /+' (K F) X
where x k is given by eq 44. F(xJ and HC(xk)are defined in the original article.1e Approximate method 1 leads to the expression
(1
while method 2 (substituting eq 40, 43, and 45 in eq 7) gives
(40)
+
=
kl
+ k2
K = kl/kz
(41)
F(xk) is defined in the original article.a1 The corresponding equation for diffusion to a stationary-sphere electrode may be obtained by routine application of the method of the Laplace transformation.m The more exact steady-state approximation was employed i.e., neglecting only the b/dt term, but retaining the (2/r)d/& term44in Fick's second law for diffusion to a stationary-sphere electrode. The result may be expressed
exp(a2t) erfc(at"')
+ Do'/'~o-~ I (45)
where a is defined in eq 46. The Journal of Phyaical Chemistry
[K(k'/'
+
+ Doi/'ro-l) exp(a2t) erfc(ati/') + Doi/'r~-l]k'/'K exp(K2kt) erfc(Kk'/Y'/')]
(42)
The expression for the limiting current based on the expanding-plane model also was presented by Kouteckjr who showeda1
+
K(k'/' Dol/zro-l) X Doi/'ro-l) Doi"ro-l K(k'/'
where
k
+ 44.6D0"'t'/'m~~")F(x~) (48)
X
exp(K2kt) erfc(Kk'/'t'/')
+
(46)
3st
where first-order or pseudo-first-order conditions prevail with respect to the preceding chemical reaction. Theory presented will be restricted to expressions for the limiting current under conditions where a steady state is achieved; i.e., (ICl k2)t > 10." This is done since the rigorous calculation for the expanding-sphere electrode applies only under these conditions.l8 For the sake of simplicity, equality of diffusion coefficients (Do = DU = DR)is also assumed. For the conditions listed above, the expression based on the stationary-plane electrode model takes the
[K(k'/'
+ Do'/zro-l)+ Dol/lro-l
Kouteckjr and Gifek's equation for diffusion to an expanding sphere has the forml8
i,
kr
i, = nFACo*Do'/'(l
a = K(k'/'
(49)
Results and Discussion To evaluate approximate methods 1 and 2, numerous dc polarographic waves predicted by these methods have been calculated and compared with waves calculated on the basis of exact solutions for the expanding-sphere model. A CDC 3400 digital computer was utilized to facilitate this work. Some typical results are depicted in Figures 1-5. For reference purposes, these figures include polarographic waves predicted by the stationary-plane, stationary-sphere, and expanding-plane models, along with those based on method 1, method 2, and the exact solution. The stationary-electrode equations were converted to expressions relevant to the polarographic experiment as indicated at the beginning of the theoretical section. The difference in polarographic waves calculated on the basis of the stationary- and expanding-plane models indicates the magnitude of the drop-growth contribution. The magnitude of the spherical-diffusion effect can be ascertained either from differences in polaro(41) R. BrdiEka, V. Hanus, and J. Kouteck? in "Progress in Polarography,': P. Zuman, Ed., with collaboration of I. M. Kolthoff, Vol. I, Interscience Publishers Inc., New York, N. Y., 1962, Chapter 7. (42) R. P. Buck, J . Electroanal. Chem., 5 , 295 (1963). (43) J. Kouteck? and R. Brdicke, Collection Czech. Chem. Commun., 12,337 (1947).
(44) J. &ek and J. Kouteckf, ibid., 28,2808 (1963).
APPROXIMATE SOLUTIONS TO EXPANDING-SPHERE BOUNDARY VALUEPROBLEMS
-0.220
-0.160
E~~-E~(I~VOIIS)
Figure 1. Calculated dc polarograms for a reversible system with both oxidation-reduction forms soluble in solution. Plots show instantaneous currents a t end of drop life for Co* = 1.00 X M ; Do = 5.00 X lo+ cme sec-1; D R = 1.00 X lo-$ cmz sec-l; drop life = 6.00 sec; T = 25.0'; n = 2; TO = 0.0528 cmz,(at t = 6 sec); m = 1.39 mg sec-I: , calculated from rigorous expanding-sphere theory; - - -, calculated from methods 1 and 2; - - -,calculated from expanding-plane electrode model; I-, calculated from stationary-sphere electrode model; . . calculated from stationary-plane electrode model.
-
-
2145
-0.100 -0.040 ~~,~.-~~(i:lvoitsi
r0.020
Figure 3. Calculated do polarograms for irreversible system. Plots show instantaneous currents a t end of drop life for Co* = 1.00 X 10-8 M ; DO = D R = 5.00 X cmz sec-l; drop life = 5.00 sec; T = 25.0'; n = 1; TO = 0.0528 cml (at t = 5 sec); ern see-'; m = 1.67 mg sec-1; k h = 1.00 X a = 0.800. Notation same as Figure 1.
--a
e ,
Ed.c; E$e ( in V O l b 1
Ed%- E22 (in VOllS)
Figure 2. Calculated dc polarograms for a reversible system with the reduced form soluble in the electrode. Plots show instantaneous currents a t end of drop life. Conditions and notation same as Figure 1.
graphic waves predicted by the stationary-plane and stationary-sphere models or from differences in the expanding-plane and expanding-sphere predictions. The latter differences are of particular interest in light of the fact that the expanding-plane theory is most frequently applied to the analysis of dc polarographic
Figure 4. Calculated dc polarograms for catalytic system with Nernstian conditions. Plots show instantaneous currents a t end of drop life for Co* = 1.00 X IO-* M ; Do = D R = 5.00 X 10-8 cmz sec-1; drop life = 5.00 sec; T = 25.0'; n = 1; ro = 0.0528 cm (at t = 5 sec); m = 1.67 mg sec-l; k, = 0.20 sec-l; -- , calculated from rigorous expanding-sphere theory; - - -, calculated ., calculated from method 1; from method 2; -, calculated from expanding-plane model; , calculated from stationary-sphere model; -,calculated from stationary-plane model.
-. -. .. a
e .
-- --
-
a
data. In Figures 1-3, the waves predicted by methods 1 and 2 are represented by a single curve. In these cases, differences between the predictions of these two methods are so small ( 10).*0 It can be seen that in this case method 1 does not provide a reasonably accurate assessment of the contribution of spherical diffusion because introduction of “spherical correction factors” 44.6D1’2t1’‘m-1’3) fails to consider of the form (1 that the chemical reaction reduces the contribution of spherical diffusion, because of an associated reduction in the thickness of the diffusion layer. Method 2 yields significantly better results because the influence of the chemical reaction on drop growth and geometry effects is explicitly incorporated in the expressions for iepand i, which are employed in the application of this approximate method. For the case of homogeneous chemical reactions preceding charge transfer, calculations were restricted to the limiting-current region for reasons stated above. For convenience, the results are depicted in a plot of &/id us. log K d ( k l k&, where ik is the kinetic limiting current predicted by a particular diffusion model (eq 40, 43, 45, 47, 48, or 49), and i d is the diffusioncontrolled limiting current for the same diffusion model (eq 3, 12, 13, 14, 16, or 17 with e ) = 0). Calcu-
+
+
(45) P. Delahay, “New Instrumental Methods in Electrochemistry,” Interscience Publishers Inc., New York, N. Y., 1954,Chapter 4.
APPROXIMATE SOLUTIONS TO EXPANDING-SPHERE BOUNDARY VALUEPROBLEMS
1.0
-
Ik Id
-0.10
0.10
0.30
0.50
0.70
0.90
I
log A#*
Figure 6. Calculated ratio of kinetic limiting current with preceding coupled chemical reaction (ik) and diffusion-controlled limiting current ( i d ) us. log ~ ( k l LJW. Plots show ratios of instantaneous currents a t end of drop C y * = 1.00 X 10-3 M ; DO = DY = D R = life for Co* 5.00 X cm2 sec-I; drop life = 6.00 sec; T = 25.0’; n = 1 ; yo = 0.0328 cm (at t = 6 sec); m = 1.39 mg sec-l; , calculated from rigorous expanding-sphere theory and method 2; - - - -, calculated from method 1 and expanding-plane model; - . -, calculated from * * ., calculated from stationary-sphere model; stationary-plane model.
+
+
+
.
lations were restricted to values of the rate parameters where the steady-state approximation is valid, Le., for (kl kz)t 2 10. The results are shown in Figure 6. The predictions of method 1 are found to superimpose on those for the expanding-plane model. Values of &/id given by method 2 are in excellent agreement (
