where K and K’ are proportionality constants. Now if Y is the ratio of concentrations and X is the ratio of measured areas, this becomes a linear equation of the form
Y=mX+b
(13)
Table I gives a series of samples and
peak areas to which a linear equation was applied using a least squares fit. A relative standard deviation of 3.3% was found for this calibration curve. The modulation technique can be applied to any short-lived gammaproducing reaction of sufficient activity and energy, and the internal standard method allows quantitative determination whenever a suitable standard is available.
LITERATURE CITED
(1) Greenwood, R. C., Reed, J. H., Kolar,
R. D., “Scintillation Spectrometry Measurements of Capture Gamma Rays from Natural Elements,” U. S. At. Energy Comm. ARF 1193-20 (Quarterly Report), 1963. (2) Isenhour, T. L., Morrison, G. H., ANAL.CHEM.38, 162 (1966). RECEIVEDfor review August 11, 1965. Accepted December 10, 1965. Research supported by the Advanced Research Projects Agency.
Influence of Spherical Diffusion on the Alternating Current Polarographic Wave JOSEPH R. DELMASTRO and DONALD E. SMITH Department of Chemistry, Northwestern University, Evanston, 111. The recent proposal by Biegler and Laitinen suggesting that spherical diffusion can influence significantly the a.c. polarographic wave, even when the d.c. process is reversible, is subjected to detailed theoretical study. Calculations of the 0.c. polarographic wave based on the stationary sphere electrode model are presented for an electrode process influenced kinetically by charge transfer and/or diffusion. The results strongly support the reasoning of Biegler and Laitinen, indicating the need to reassess several aspects of existing a x . polarographic theory. With a reversible d.c. process, a significant contribution of spherical diffusion is indicated for systems exhibiting amalgam formation. The predicted effect is much smaller when both forms of the electroactive redox couple are soluble in the solution phase. Implications regarding the interpretation of some existing experimental results and the problem of obtaining a.c. polarographic wave equations which are rigorously applicable to the dropping mercury electrode are discussed.
W
HILE EARLIER work had considered unimportant the effect of spherical diffusion on the characteristics of the a x . polarographic wave-i.e., on the faradaic impedance-two recent papers have questioned this concept ( 7 , 14). A theoretical investigation in these laboratories (14) has indicated that curvature of the electrode surface contributes significantly when the d.c. polarographic process is subject to control by rate processes in addition to diffusion. In the analysis of experimental a.c. polarographic data which was in disagreement with theoretical
equations based on the planar diffusion model, Biegler and Laitinen ( 7 ) proposed that the disparity between theory and experiment could be due to the influence of spherical diffusion. Both studies maintain that the contribution of spherical diffusion originates in the d.c. polarographic process which is extant in the a x . polarographic experiment. Spherical diffusion influences the magnitudes of the d.c. components of the surface concentrations which, in turn, play an important role in determining the magnitude of the a x . polarographic wave (7,I.C). Interestingly, these two papers describe what might be considered different mechanisms whereby a contribution of spherical diffusion is coupled to the a x . polarographic process via the d.c. process. Biegler and Laitinen ( 7 ) propose a mode whereby spherical diffusion, by itseZj, alters the magnitudes of the d.c. components of the surface concentrations from their planar diffusion values. An important consequence of this concept is the fact that it predicts a significant effect of spherical diffusion even when the d.c. process is controlled solely by diffusion-Le., with a reversible d.c. process. Our work (14) was concerned with a mechanism in which spherical diffusion can become important when operative in combination with nonequilibrium conditions (chemical or electrochemical) in the d.c. concentration profile. Influence of spherical diffusion with pure diffusion control in the d.c. process was not indicated because careful analysis of the spherical diffusion correction was confined to systems where both forms of the electrochemical redox couple are soluble in solution and characterized by equal diffusion coefficients. This appears to be the one situation in which the effect
described by Biegler and Laitinen is negligible. The implications of Biegler and Laitinen’s work go beyond suggesting that one must recognize that the magnitude of the a x . polarographic wave may be altered by spherical diffusion. It also influences considerations regarding the time-dependence of a.c. polarographic waves and the derivation of a.c. wave equations rigorously applicable to the dropping mercury electrode. As they point out ( 7 ) , spherical diffusion can serve as the origin of a timedependent a x . polarographic waveLe., a wave dependent on mercury column height-with reversible d.c. processes. Previously, the only recognized source of time-dependence was associated with d.c. processes which were influenced by rate processes in addition to diffusion (1-3, 19, 20, 28, 32). If Biegler and Laitinen are correct, it is necessary to alter earlier concepts regarding the mechanistic implications of time-dependent a.c. waves to encompass the fact that a time-dependence does not necessarily indicate that the d.c. process is influenced by slow chemical or electrochemical steps. I n addition, a method of obtainiug a x . wave equations which incorporates effects of mercury drop growth and curvature (14) must be reconsidered because it did not include the effect pointed out by Biegler and Laitinen. It was based on the analysis of the spherical diffusion contribution for systems with both forms soluble in solution and equal diffusion coefficients. An examination of the d.c. polarographic theory of Koutecky (dS,24) and an essentially qualitative extrapolation led Biegler and Laitinen to their conclusion regarding the effect of spherical diffusion on a.c. polarographic characVOL 38, NO. 2, FEBRUARY 1966
169
teristics. While their reasoning appeared sound, the possibility remained that the effects they discussed would? prove negligible on a quantitative basis. For this reason we have performed a quantitative examination of this problem, employing the stationary sphere electrode model as a basis for calculation. While this model still represents an approximation to the dropping mercury electrode (DME) , it serves to isolate for quantitative inspection the contribution of spherical diffusion. The results of these calculations are the subject of the present work. Some important implications regarding a.c. polarography a t stationary electrodes with linear d.c. potential scan and faradaic nonlinearity are implicit also in the present work, but will be discussed elsewhere.
where the instantaneous potential, E(t), is given by
E(t) = E d . c . - AE shut
(4)
Employing step-by-step operations identical to those used by Matsuda (18) yields the system of integral equations (14)
l'
exp (Doro-2u)erf c (Do1/%--1u1/2) x
Qo(t - u)du
- DR1/2To-'pe@jX
lt
eXp(DRTo-2~)erfC(DR1/2To-'u1/2) X
THEORETICAL
Theory for the a.c. polarographic wave with the simple electrode process in which charge transfer and diffusion are the sole rate determining steps (the quasi-reversible case) will be examined. Derivation of the a x . wave equations will be outlined only briefly because the method of derivation, including the basic assumptions, has been given elsewhere (14). Two basic cases will be considered: (A) both forms of the electroactive redox couple soluble in solution and (B) the reduced form soluble in the electrode, as in amalgam formation. A list of notation definitions is given in Appendix 1. A. Derivation of A.C. Polarographic Current for Diffusion to a Stationary Spherical Electrode with Both Redox Forms Soluble in Solution. This boundary value problem has been stated and solved elsewhere ( I S , 14,21). Expressions for the surface concentrations are
-'exp (DRT, -'U)
Following the same procedure as in earlier work (14, 36) yields for Ql(t)
where p = 0,1,2, 3. * .
m
QPW
nFAE (m) (7)
p = 1 - a
(11)
P-0
x
I(&) Equations 1 and 2 are substituted into the absolute rate expression (17)
P
nFAE (x) Ql(t)
= nFACo*D,1/2
(12) requires solution of Equation 5 for p = 1 (68). The integral equation for p = 1 may be written
i(t) -=
nFAkh
(3) 170
(6)
Equations 5-11 define all current components flowing under ax. polarographic conditions with a quasi-reversible system, within the framework of the stationary sphere electrode model. Derivation of the small amplitude fundamental harmonic alternating current given by
Dol/'+o-l exp (D,T,-~u) X
D E'
Qo(t - u)du (14)
ANALYTICAL CHEMISTRY
Substituting from the relation obtained from Equation 5 for p = 0 (the integral equation for the direct current),
Q,(t - u)du +DR1/?,-'eBj
it
x
eXp(DRTo-2U)erfC(DR11?0-1U1/2)
Qo(t
The combination of Equations 17, 25, 26, 28, and 31 represents the rigorous, closed form solution for the small amplitude a.c. polarographic wave within the framework of the original assumptions (stationary sphere electrode model, both forms soluble in solution, quasireversible system, etc.). However, for any practical situation further simplification of the solution is justified because, for normal experimental conditions (14,16)
X
- u)du
(24)
and employing Equation 12 leads to the expression for the fundamental harmonic a.c. polarographic current,
I(wt)
I,,,F(t)G(w) sin (ut
=
+ e)
(26)
where
I,,,
=
X2t 2 50 (37) is approximately applicable, the d.c. process may be considered reversible, but not necessarily the a.c. process. Equation 31 then reduces to
(26)
n2F2AC,*(~D,)'/2AE
4RT cosh2
Further simplification of the foregoing equations is possible for the special cases in which the d.c. and a x . processes are influenced kinetically by diffusion alone (reversible d.c. and a x . processes). When the charge transfer rate is sufficiently rapid that the statement
(i)
so that [l - exp(a2t)erfc(at1/2)] (38)
where Thus, G(w) and
rt
erfc(Do1/Pro-1u1/2)&o(t - u)du DR"~T,-' erfc
sd
These expressions are obtained from Equation 31 by noting that Equation 37 also implies that X >> T , - ~ D , ~and /~ >> To-1DR"2, retaining only the first two terms from the binomial expansion of the term in braces in Equation 30 and, when appropriate, keeping only the first term in the asymptotic series for exp(z2)erfc(z) for large 1: (11). The expressions for G(w) and e (Equations 35 and 36) remain unchanged. Equation 38 is equivalent t o the statements
-
exp (DRT,-'u) X
Qo( t - u)du (27)
[(e + r
G(w) =
e reduce to
1
0
- Po - P R Y
(1
+
- Uo - UR)'
and e is given by Equation 17. Completion of the solution requires knowledge of the form of the function &,(t) found in F(t). This is realized by solving Equation 24 which was previously shown to yield the expression (14)
(28)
The absence of the parameter ro indicates that the functions G(w) and e are not influenced by spherical diffusion. That is, Equations 35 and 36 are the same as would be obtained from the
F(t) = 1
sd
+ Do1/2ro-1
esp(D0r2u) X
erfc(D,1'2r,-1u1/2)&~(t - u)du -
lt
DR1/27'o-1 exp ( D ~ r ~ -erf ~ uc (DR1/z ) X
r,"~"~))&,(t - u)du (40) exp(a-%) erfc(a-t1/2)
- -[ a + a- - a+
- T,-'(D,'/~ + DE'/') + r,-2D0112DR112 a+ To-2D,1/2DRl/2 exp (a+2t)erfc(a+t1/2)+ } (29) a-a,
ej~o-l(Dol/2 - DR112)2 X (1 ej)(eiDo1I2 DR'")
+
+
exp (a2t)erfc (at1l2)} (41)
Substituting Equation 29 in Equation 27, evaluating the integrals with the aid of the convolution theorem (IO),and algebraic rearrangement gives for F ( t ) :
planar diffusion model (8, 22, 28, 37). It is obvious that the influence of spherical diffusion on the a.c. polarographic wave is reflected solely in the function F ( t ) .
both of which result from application of the Nernst equation to the d.c. components of the surface concentrations a t the outset of the derivation (34, 35). If the condition
e i x
0 , T + a ; C, = C,*
(48a)
> 0 , r + 0; C Ris bounded
(48b)
(49) Solution of this system of equations by the method of Laplace Transformation and neglecting the finite volume term yields for the surface concentrations
c
(DRl12e-4 - Dol12ePi)]1I12
All other terms are the same as in Case A. In the course of obtaining these results, one encounters two integrals whose values were not given in earlier work (14). The integrals and their associated values are
(59)
Substitution of Equation 58 in Equation 57, integration, and algebraic rearrangement gives
1
exp(DRr,-2u) sin w(t - u)du =
eXp(DRTo-2U) X erf c (DR1i2rOo-1U1i2)
+
I
~DR'/~T,-' eXp(DRTo-2U)
dU
(50)
and C,. is given by Equation 1. Equation 50 follows from the work of Reinmuth (31). Substituting Equations 1 and 50 into Equation 3 and proceeding as before, one obtains the system of integral equations
Do 2r,-'exp (D
-%) X
+
DR1/27'o-1eXp(DRTo-2U) x erfC(DR1/2To-1U112) 2DR1/2ro-1x
+
- U)dU}
+ wsinwt]
(56)
Substituting with the aid of the relation obtained from Equation 51 for p = 0 and employing Equation 12 leads to an expression for the fundamental harmonic alternating current which is identical in form to Equation 25. The solution differs from Case A only in the forms of PR and U R , as already stated, and in the form of F ( t ) which for this case may be written
F(t) = 1
e r f ~ ( D , ~ ~ ~ r QP--d(t , - ~ ~~ ~u)du ~)]
1
DRT,-' coswt
+ (ae-i -khep)D1/2Q0(t)+ -ai
[erfC(DR1i2To-1U1/2) - 2]&0(t-
U)dU
(51)
(57)
The significance of p , & ( t ) , etc. is the same as in Case A. The integral equation for p = 1 is solved as before,
As before, completion of the solution is effected by obtaining a closed form expression for &,@). This expression is (see Appendix 2)
eXp(DRTo-2U )
172
&p--d(t
ANALYTICAL CHEMISTRY
1
b-(b-
- b+) exp(b-2t) erfc (b-t1/2) -
1 exp(b+%)e r f ~ ( b + t ~ / ~(60) )] b+(b- - b+) Thus, Equations 17,25,26, 28, 59, 60 and the new definitions corresponding to Equations 53 and 54 comprise the rigorous solution for the case of amalgam formation, within the framework of the other original assumptions associated also with Case A. As in the previous case, the simplification associated with Equations 32-34 is applicable for normal experimental conditions so that Equations 35 and 36 may be employed for G(w) and the phase angle, e. Again, only the function F ( t ) reflects the influence of spherical diffusion, and differences in the a.c. pqlarographic behavior between Cases A and B are also manifested only in this function. As above, one may obtain simplified forms when the d.c. process is reversible, where
is applicable and F(t) reduces to
11 - exp(b2t)erfc(bt1l2)1 (62) where
Equation 62 is equivalent to the expressions
F(t) = 1
sb
+ D,1/2~o-1
exp (D , T , - ~ ~erfc ) (Dol/'+o-lul/*)X ~
(
- tu)du - D R ~ / ' + ~ - ~ L
exp (DRY,-%)[erfc(DR1'21'o-1u1/2) 21&0(t - u)du (64)
exp(b2t)erfc ( b W )
t
(65)
both of which result from application of the Nernst equation to the d.c. components of the surface concentrations a t the beginning of the derivation (15). When the a.c. process is Nernstian, Equations 42, 43, and 44 remain applicable. DlSCUSSlON
A. Relationship of Theoretical Results to Earlier A.C. Polarographic Theory. T o establish a frame of reference for further discussion of quantitative predictions of the above equations, it is helpful to examine their similarities and differences in relation to existing ax. polarographic theory. As in earlier theoretical work (19, 28, 36-38), it is found that the quasireversible a.c. polarographic wave can be expressed in terms of four functions, which we have represented here as I,,,, G(u), e, and F ( t ) [other symbols, such as F ( h t 1 / 2 )and G ( u ~ Q - ~ )have , been used previously instead of F ( t ) and G ( u ) ] . I,,, represents the well-known current amplitude expression associated with the reversible a x . polarographic wave derived on the basis of the planar diffusion model (9, 12, 18, 27, 33). The functions G(u) and F ( t ) have been interpreted as correction terms accounting for the isfluence of charge transfer kinetics on the a x . and d.c. processes, respectively (29, 86-88). The greater the deviations of G(w) and F ( t ) from
unity, the greater the deviations of the a.c. or d.c. components of the surface concentrations from their Nernstian values. The phase angle, e, depends only on the parameter, 1/2w/X, and for this reason is said to be responsive only to the status of the a.c. process with the simple quasi-reversible system. As shown above, the frequency dependent functions, I,,,, G ( w ) , and 6 are uninfluenced by spherical diffusion and the interpretation that G ( w ) and e represent a measure of the deviation of the a.c. process from Nernstian conditions appears valid. Only the function responsive to conditions in the d.c. process, F ( t ) , is altered from its previously calculated form by consideration of the effects of spherical diffusion. Both the stationary and expanding plane models yield expressions for F(t) which are equivalent to the statement (14, 19,28).
The term (ae-i - P)D1/2&,,(t)/khe-ajis insignificant when the d.c. process is Nernstian leading to the earlier interpretation that F(t) represents a measure of the deviations of the d.c. components of the surface concentrations from their Nernstian values. Comparison of Equation 66 with Equation 27 and 57 shows that a main source of the difference between theory based on planar diffusion models and theory based on spherical diffusion is associated with the integrals in Equations 27 and 57 and their contribution to the closed form solutions, Equations 31 and 60. Only when both forms of the redox couple are soluble in solution and possess equal diffusion coefficients does the spherical diffusion theory (Equation 27) yield an expression for F ( t ) equivalent to Equation 66 (14) [Note that even when Equation 66 is applicab!., the various electrode models yield different Qo(t) functions and thus different closed-form solutions ( l 4 ) ] . Unlike the term (ae-j - P)D1/2&o(t)/khe-aj,the integrals found in Equations 27 and 57 do not become negligible when the d.c. process is reversible (c.f., Equations 40 and 64). Thus, the earlier interpretation of the F ( t ) function must be altered. I n view of present evidence, we submit that a more appropriate interpretation is that deviations of F ( t ) from unity are a measure of deviations of the magnitudes of the d.c. components of the surface concentrations from their planar diffusion, Nernstian values. This takes account of the fact that Nernstian conditions may exist, but F ( t ) may differ from unity if spherical diffusion alters significantly the magnitudes of the surface concentrations ( 7 ) . The fact that the stationary sphere electrode model leads to expressions for F ( t ) in the form of Equations 38 and 62 with reversible d.c. processes has an
interesting implication with regard to the keystone of a x . polarographic theory; the theory for the reversible wave. This work would suggest that the classical expression for the reversible a x . polarographic wave
I(&) = n2F2AC,*(wD,)1/ZAEsin wt 4RT cosh'b)
(
+
4)
(67)
which has been accepted for work with the D M E for over a decade (9, 12, 18, 27, SS), must be replaced by expressions which account for spherical diffusion. The present work indicates that better approximations to the real case would be the expression (from Equations 25, 26, 38,43, and 44)
1
(
e x p ( ~ ~ t ) e r f c ( a t ~sin / ~ ) ]ut
;>
+-
(68)
for both forms soluble in solut'ion and the expression (from Equations 25, 26, 43, 44, and 62)
I(&)
=
{
n?F2AC,*(wD,)"24E 4RT cosh2(ji)
1
1+
( + 1)
exp(b2t)erfc(bt112)] sin ut
-
for the reduced form soluble in the electrode. As will be seen, the differences between Equations 67 and 68 will often be negligible, but very significant differences exist between Equations 67 and 69. The question as to whether Equations 68 and 69 are sufficiently precise for work with the D M E is discussed below. B. Results and Discussion of Quantitative Calculations. The foregoing derivation and discussion makes apparent that a contribution of spherical diffusion is manifested in the a.c. polarographic wave equations. However, simple perusal of the equations does not permit one to conclude with any certainty that predicted effects of spherical diffusion are sufficiently large that they would be of concern in data analysis. To answer this question, extensive calculations have been performed with the aid of an IBM 709 digital computer utilizing the above equations. Some typical results are depicted in Figures 1-11. VOL 38, NO. 2, FEBRUARY 1966
173
E,-E~2("oltsl
Figure 1. Calculated ax. polarograms showing influence of spherical diffusion
Perhaps representative of the nature of the majority of results obtained in these calculations are those depicted in Figure 1. The value of kh (10-I cm. second-') employed is sufficiently large that the d.c. process is reversible-i.e., Equations 38 and 62 are applicablewhile the influence of charge transfer kinetics on the a.c. process is small, but detectable. It is seen that the predicted influence of spherical diffusion is very small for the case where both redox forms are soluble in solution, despite the fact that the diffusion coefficients differ by a factor of two. A very slight time dependence is predicted; one which would be difficultly detected experimentally. The contribution of spherical diffusion to the peak current magnitude is 1.6% and 2.7% relative to the -
Ed< E:?,
~ ~ 0 1 1I s
Figure 4. Calculated a.c. polarograms showing influence of spherical diffusion with reduced form soluble in electrode
I
x
lo-' cm. Sac.-', 7 = 25.0' c., k h = 1.00 LY = 0.50, n = 1.00, C,* = 1.00 X 10-3M, r,, = 0.0528 cm., AE = 5.0 X l o V 3volt, w = 40 7 radians set.-', D o = 5.00 c m 2 sec.-I sec.-I, DR = 1.00 X
x
Same parameters as Figure 3, except D R = 5.00 X 10-5 cmn2sec.-l
-
Calculated from planar diffusion model for any drop life Calculated from stationary sphere model for reduced form soluble in electrode; drop life = 3 seconds Calculated from stationary sphere model for reduced form soluble in electrode; drop life = 9 seconds , , , Calculated from stationary sphere model for reduced form soluble in solution; drop life = 3 seconds * ** Calculated from stationary sphere model for reduced form soluble in s o b tion; drop life = 9 seconds Polarograms depict current at end o f drop life
--..-.---
-.-.-. . .. -- - -
Edc-
Ivclts1
Figure 3. Calculated ax. polarograms showing influence of spherical diffusion with reduced form soluble in electrode
peak current magnitude is 10% and 1Sy0 relative to the planar diffusion current amplitude for t = 3 and 9 seconds, respectively. In general, all calculations performed for conditions corresponding to a reversible d.c. process yielded results similar to those shown in Figure 1. The requirement that diffusion coefficients differ suppresses considerably the influence of spherical diffusion when both forms are soluble in solution. Even when D oand D R differ by a factor of four, the correction is not overly large, as seen in Figure 2. However, it should definitely be considered experimentally significant. I n most
r
= 25' C., k h = m (reversible wave), n = 2.00, C,* = 1.00 X 10-3M, ro = 0.0528 cm., A€ = 5.00 X volt, w = 40 T radians ret.-', Do = 5.00 X c m 2 set.-', D E = 1.OO X l o -5 cm.2 set.-'
-
Calculated from planar diffusion model for any drop life Calculated from stationary sphere model; drop life = 3 seconds Calculated from stationary sphere model; drop life = 9 seconds
_..______ !o.oBo
-0.040
ow0 E ~ ~ ~ E(volts) ),
om0
00s
Figure 2. Calculated ax. polarograms showing influence of spherical diffusion with both redox forms soluble in solution
r
= 25' C.,
kh
= m (reversible wave), n =
1.00, C: = 1.00 X lo-%, io = 0.0528 cm., A€ = 5.00 X volt, o = 40 ?r radians set.-', Do = 2.50 X sec.-l, D R =
1.00 X 10-5 cm.2 set.-?
---------
...... 174
Calculated from planar diffusion model for any drop life Calculated from stationary sphere model; drop life = 3 seconds Calculated from stationary sphere model; drop life = 9 seconds ANALYTICAL CHEMISTRY
.., .,.
planar diffusion current amplitude for t = 3 and 9 seconds, respectively. On the other hand, the spherical diffusion effect predicted for the case where the reduced form is soluble in the electrode is remarkably large, contributing a substantial time dependence to the a.c. polarographic wave. Spherical diffusion also introduces a significant alteration in peak potential and a slight dependence of peak potential on drop life. In this case, the predicted contribution of spherical diffusion to the
Ede.- E>? Ivolts
I
Figure 5. Calculated a.c. polarograms showing influence of spherical diffusion with reduced form soluble in electrode Same parameters as Figure 3, except D o = 1.00 X cmS2ret.-' and n = 1.00
1
I
Edc- E>2 I volts i
Figure 6. Calculated a x . polarograms showing influence of spherical diffusion with both redox forms soluble in solution
Figure 7. Calculated a.c. polarograms showing influence of spherical diffusion with reduced form soluble in electrode
T = 2 5 ’ C., k h = 1.00 X 10-3 cm. sec.-I, CL = 0.500, n = 1.00, C,* = 1.00 X 10-3M, ro = 0.0528 cm., A€ = 5.00 X volt, w = 40 x radians sec.-I, Do = 1 .OO X 1 0-6 c m 2 set.-', sec.-l, drop life = 6 seconds DR = 2.5 X 10-6
. ..... ----
T = 25’ C., k h = 6.00 X 1 O-a cm. sec.-l, CY = 0.500, n = 1.00, C,* = 1.00 X M, r, = 0.0528 cm., A€ = 5.00 X 10-3 volt, w = 40 A radians set.-', D o = DR = 5.00 X 10-6 sec.-I, drop life = 6 seconds
. ... ..
Calculated from planar diffusion model Calculated from stationary sphere model using approximate form of F(f) (Equation 66) Colculated from stationary sphere model without approximation
systems of this type encountered in experimental studies, differences in diffusion coefficients will be much smaller rendering the spherical correction insignificant. On the other hand, the solubility of the reduced form in the electrode produces marked deviations of the predicted alternating current magnitude from planar diffusion theory regardless of the relative magnitudes of D o and DR. Figures 3-5 illustrate the magnitude of the spherical diffusion contribution to the reversible a x . polarographic wave for systems of this type. The situation changes when k h is sufficiently small that the d.c. process is influenced by charge transfer kinetics (kh < cm. second-I). The factor (ae-3 - p)D1”Qo(t)/khe-a3becomes significant in the expression for F ( t ) and a substantial spherical diffusion contribution is noted regardless of whether the reduced form is soluble in the solution or electrode. Even with D o = D E and the reduced form soluble in solution, a significant spherical diffusion effect is predicted as noted previously (14). Figures 6 - 11 illustrate results of calculations for such small kA values. In Figures 6 and 7 we illustrate for reference the a s . polarogram calculated from the stationary plane electrode model (14) and the a x . polarogram calculated on the basis of Equation 66 for F(t). I n the latter, a spherical diffusion correction is still introduced through the term Qo(t) (cf. Equations 29 and 58), but neglect of the terms associated with the integral in Equations 27 and 57 renders this calculation approximate. Figure 6 illustrates that when both redox forms are soluble in solution the spherical diffusion contribution will be significant, but the approximate form of
-.......
Calculated from planar diffusion model Calculated from stationary sphere model using approximate form of F ( f ) (Equation 66) Calculated from stationary sphere model without approximation
F ( t ) (Equation 66) will often be sufficiently accurate. Once again the situation differs when the reduced form is soluble in the electrode. The spherical correction and the term associated with the integrals are both quite significant as shown in Figure 7 . One interesting prediction of equations presented here regarding a x . polarographic behavior with a d.c. process influenced by charge transfer kinetics is related to the cross-over point. This effect is predicted for observations of a.c. polarograms as a function of
mercury column height or drop life on the basis of planar diffusion (3, 13, 28) and the overly simplified spherical diffusion model which assumes applicability of Equation 66 (14). At the d.c. potential where ae-3
-p
=
0
(70)
these theoretical models predict that alternating current will be independent of column height, and that the direction (sign) of the variation of alternating current with variation of column height will change (13). Obviously, this can
Edc- E:/* I volts i
Figure 8. Calculated a x . polarograms showing influence of spherical diffusion with reduced form soluble in electrode T = 25‘ C., kh = 8.00 X 10-acm. sec.-l, a = 0.500, n = 1.00, Co* = 1.00 X M, ro = 0.0528 cm., A€ = 5.00 X 10-3 volt, w = 40 x radians sec.-I, Do = DR = 5.00 X 10-8 sec.-1
-
Calculated from stationary sphere model;
drop life =
9 seconds Calculated from stationary sphere model; 3 seconds
drop life =
VOL 38, NO. 2, FEBRUARY 1966
175
I
/
CROSS-OVER POINT
r.
1
_----0140
-0070
OOOO
0.070
I
E~.-E>* i VOIIS)
-0140
Figure 9. Calculated a.c. polarograms showing influence of spherical diffusion with reduced form soluble in electrode Parameters same as Figure 8, except sec. -l
kh
= 4.00
be true only when Equation 66 is applicable so that the integrals in Equations 27 and 57 represent perturbations on this behavior. As has already become apparent, the contribution of these additional terms is small when both redox forms are soluble in solution so that earlier concepts regarding the cross-over point are altered only slightly. For example, when D R = 2 0 , the cross-over point calculated on the basis of the more rigorous expression given here differs by only 4 mv. from that corresponding to Equation 70. On the other hand, the expressions derived here for the case of amalgam formation predict that the crossover point will be shifted significantly from the planar diffusion value (Equation 70), or eliminated completely because of the spherical diffusion effect. The fate of the cross-over point in this situation is very sensitive t o the value of k h . These effects are illustrated in Figures 8-10. An additional alteration on the planar diffusion predictions is that no unique value of d.c. potential is associated with the cross-over point for all values of column height. Instead, the position of the cross-over point depends on the particular values of column height (drop-life) associated with any pair of a.c. polarograms in question. This effect is illustrated in Figure 11. The same phenomena is predicted when both redox forms are soluble in solution if D o# D R ,but is essentially insignificant for any normal combination of Do and DR. The nature of the results shown in Figures 1-11 indicates that recent theory assessing (14) the spherical diffusion correction may be considered reasonably sound when both redox forms are soluble in solution (unless Do and DR differ considerably), but it is significantly inaccurate for systems exhibiting amalgam formation. 176
ANALYTICAL CHEMISTRY
x
OOOO
-0070 Edc,- E>z
1
0070
f volts)
Figure 10. Calculated a.c. polarograms showing influence of spherical diffusion with reduced form soluble in electrode
cm.
Parameters same as Figure 8, except kh = 2.00 sec.-1
X 10-3
cm.
tive spherical correction a t all potentials Upon examination of the foregoing with the relative magnitude of the equations and figures for conditions correction increasing as the d.c. potencorresponding to a reversible d.c. proctial becomes more negative. ess, one notes some interesting differBecause the source of the spherical ences in manifestations of the influence diffusion contribution to the a.c. wave of spherical diffusion on the a.c. wave lies in the d.c. polarographic process, compared to what is observed in the these differences a t first may appear theory for the reversible d.c. polarorather striking. However, their ragraphic wave (23, 24). For example, tionale lies in the fact that the d.c. wave when the reduced form is soluble in is responsive to the influence of sphersolution the spherical diffusion contribuical diffusion on the $uses of the electrotion to the reversible d.c. polarographic active species a t the electrode surface, wave is always positive (23, 2 4 , while while the a x . wave is responsive to the its contribution to the a.c. polarographic wave may be positive (when Do > DR), influence of spherical diffusion on the magnitude of the d.c. components of the negative (Do < DR) or zero (Do = DR). electroactive species a t the electrode When the reduced form is soluble in the surface (7). An examination of the exelectrode, the spherical diffusion conpressions for the sums of the surface tribution to the d.c. polarographic wave concentrations readily confirms the is negative a t anodic potentials, positive latter concept with regard to the ax. at cathodic potentials, and zero a t some wave. Assuming Nernstian behavior, intermediate d.c. potential. For the Equations 1 and 2 lead to the expression same case the a x . wave exhibits a posi-
I
I
I
-0140
-0070
OOOO
0070
I
Eds;ETb lvolts I
Figure 11. Calculated a.c. polarograms showing influence of spherical diffusion with reduced form soluble in electrode Parameters same os Figure 8, except kh = 1.00 X cm. sec.-l and calculated sec.-l, OR = 1.00 X 1 0 - 6 from stationary sphere model; drop life = 6 seconds
......
for both redox forms soluble in solution. Equations 1 and 50 yield
(Equation 74 is equivalent to Equation 12 of Reference 7) derived from planar diffusion theory, but a more accurate expression for metal ion-metal amalgam systems would be
implied by Biegler and Laitinen when amalgam formation is involved. It should be recognized that the very small spherical diffusion correction predicted when both redox forms are soluble in solution indicates that this method of obtaining Do will prove more successful and appealing for systems of this type. The dependence of the slope of the impedance plots on d.c. potential observed by Biegler and Laitinen is also a t least qualitatively consistent with spherical diffusion theory-i.e., with Equation 75. The fact that the relative contribution of spherical diffusion is predicted to increase as the potential becomes more negative may explain the observation that experimental impedance slopes yield values smaller at positive potentials and larger a t negative potentials than values calculated from planar diffusion theory by assuming the value a t the observed half-wave potential is in agreement with theory ( 7 ) . Failure to obtain the expected results in other quantitative experimental studies on metal ion-metal amalgam syatelns (2, e), including measurements on current components arising from faradaic nonlinearity (4, 6, 29), may be attiibutable to the influence of spherical diffusion. The absence of a cross-over point in studies of the time dependence of the a x . polarogram of zinc ion in 1.11 KCl (2) [ k h -y 4 X cm. second-' (30, 40) I appears qualitatively consistent with the effects depicted in Figures 8-10. At the same time, excellent agreement between planar diffusion theory and a.c. polarographic experiments for systems with both redox forms soluble in solution (19, 38,39, 41) is not surprising in view of the facts that the spherical diffusion contribution with such systems is predicted to be either small with reversible d.c. processes or significant and operative primarily through the term (ae-2 - p) D112Qo(t)/khe-QJ when the d.c. process is quasireversible (c.f., e.g., reference 19). It is not surprising that phase angle data on a zinc system (40) gave the expected values of k h , since the phase angle is predicted to be independent of spherical diffusion, regardless of whether the reduced form dissolves in the solution or electrode. In this regard it is interesting to note that earlier work also indicated that the phase angle would be uninfluenced by drop growth. Thus, if one wishes to employ a.c. polarographic conditions-i.e., net flow of direct current-in the study of the faradaic impedance and retain the simplicity
The fact that the slope of the impedance plot with spherical diffusion depends on DR and time, as well as Do, makes this method somewhat less appealing than
associated with the use of equations uncomplicated by terms accounting for the contributions of drop growth and spherical diffusion, the phase angle appears to
to the a.c. polarogram through the concentration magnitudes.
Jo
for the reduced form soluble in the electrode, The corresponding relation obtained for planar diffusion (either stationary or expanding plane) is
A comparison of Equations 71 and 72 with Equation 73 shows that the terms in braces in the former contain the spherical diffusion correction on the sum of the surface concentrations. Deviations of the terms in braces from unity quantitatively represent, in relative units, the alteration of the sum of the surface concentrations due to spherical diffusion. Referring to Equations 40 and 64, one observes that these expressions for F ( t ) equal the terms in braces in Equations 71 and 72, respectively. One concludes that when the d.c. process is reversible, the relative contribution of spherical diffusion to the magnitude of the a x . polarographic wave can be equated with the relative change in the sum of the interface concentrations due to spherical diffusion. The situation becomes more complicated when the d.c. process is influenced by charge transfer kinetics. In this case, the mode whereby spherical diffusion influences the ax. wave may be considered twofold. Spherical diffusion alters not only the magnitudes of the surface concentrations, but it also influences the magnitude of the deviation from electrochemical equilibrium. Thus, in the theoretical relations, the spherical diffusion effect appears in an additional term ( a e - j - p)D1/2Q,(t)/khe-aj. Differences between the manner whereby spherical diffusion contributes to the a.c. and d.c. polarographic waves also become more complicated, but they ultimately relate to the fact that spherical diffusion contributes to the d.c. polarogram through the fluxes and
C. Implications Concerning Experimental Studies. Perusal of the above equations and discussion should make apparent the fact t h a t the theoretical results obtained here justify the suggestion by Biegler and Laitinen (7) that the disparities between planar diffusion theory and experimental results they observed with metal ion-metal amalgam systems may be due to the effect of spherical diffusion. Indeed, essentially all aspects of their qualitative theoretical considerations are supported here. The nature of the apparent deviations of the a.c. polarographic data of Biegler and Laitinen from planar diffusion theory appears to be at least qualitatively in agreement with the calculated effect of spherical diffusion. The predicted enhancement of alternating current by spherical diffusion with amalgam formation seems to be in agreement with the consistently high values of diffusion coefficient (compared to rigorously-calculated d.c. polarographic values) calculated by Biegler and Laitinen with the aid of planar diffusion theory. The present work suggests that the method of calculating Do from slopes of impedance plots with reversible d.c. processes should not be based on the expression ( 7 ) coshz(j/2) s = 4RT n2FZC,*(2D,)l l 2
(74)
VOL 38, NO. 2, FEBRUARY 1966
177
be the experimental parameter of choice. However, it should be noted that this advantage does not apply to all mechanistic schemes (20). D. Concerning the Problem of Obtaining A.C. Polarographic Wave Equations Rigorously Applicable to the DME. The combination of present and earlier (7, 14, 68) studies indicates that neglect of drop growth and geometry in a.c. polarographic theory is frequently unjustified for work with the DRIE, suggesting that efforts to obtain equations rigorously accounting for these effects are not without merit. Some aspects of the problem of obtaining such equations have been discussed ( 1 4 ) . However, in view of the present work, a few additional remarks seem appropriate. It is likely that the desired equations can be obtained through rigorous solution of the hIacGillavry-Rideal differential equation (26) with the aid of the method of Koutecky (23, 94). Although this would appear to be the approach presenting the least pitfalls, the course of solution will likely prove tedious and cumbersome. Perhaps for this reason no attempts at this mode of solution have been reported. I t is possible that some less involved theoretical approaches can be employed which will yield only slightly less accurate results than obtainable through the AIacGillavry-Rideal equation. The method previously used to obtain a x . polarographic nave equations for the quahi-reversible system which were presumed applicable to the DME is one such example (I.$). I t was assumed in that derivation that Equation 66 was applicable to F ( t ) . The present work has shown that Equation 66 is reasonably accurate only if both forms of the redox couple are soluble in the solution phase and if their diffusion coefficients do not differ excessively. For this situation, the expression presented previously for the a x . polarographic wave which allegedly accounted for drop growth and geometry effects (Equations 33 and 34 of reference 14) is not rendered suspect by the present work. However, the corresponding expression for the case of amalgam formation (Equations 33 and 35 of reference 14) must be considered inaccurate. A second possible alternative to the solution of the MacGillavry-Rideal equation may be envisioned by noting that one can provide some rather convincing theoretical evidence that the perturbations on the mass transfer process presented by drop growth and spherical diffusion are nearly separable. I n other words, it appears that addition of the correction for drop growth calculated from the expanding plane electrode model and the correction for spherical diffusion calculated on the basis of the stationary sphere electrode 178
ANALYTICAL CHEMISTRY
model yields to a high degree of accuracy the combined contribution of drop growth and spherical diffusion obtained from the expanding sphere model-i.e., the MacGillavry-Rideal equation. The rather small difference between the Lingane-Loveridge equation (65) and Koutecky's exact expression (25, 64) represents one piece of evidence in favor of the near separability of the drop growth and geometry corrections in d.c. polarography. Further evidence can be obtained which is applicable to both d.c. and a.c. polarography (15). A complete discussion of this matter is beyond the scope of the present work and will be given elsewhere. However, if this concept is accurate, some of the expressions derived here may be sufficiently accurate in their present form for work with the DME. We refer specifically to expressions applicable to the situation where the d.c. polarographic wave is reversible-i.e., Equations 38 and 62. In this case, no influence of drop growth is predicted on the basis Of the expanding plane (28) so that, if the drop growth and geometry contributions are separable as defined above. onlv the wherical correction manifested in the stationary sphere calculation need be considered. These concepts are presently being subjected to more detailed theoretical study and experimental test. Z
ing current relative to applied alternating potential = angular frequency =time = auxiliary variable of integration = Laplace Transform variable = apparent heterogeneous rate constant for charge transfer a t Eo = charge transfer coefficient = distance from planar electrode surface = distance from the center of spherical electrode = spherical electrode radius = slope of plot of resistive or capacitive component of the faradaic impedance versus
w
t u S kh
CY
2
r
r, S
w-1/2
APPENDIX 2
Derivation of D.C. Polarographic Current with DifIusionto a Stationary Spherical Electrode and Reduced Form Soluble in the Elec&ode, Q o ( t ) is the solution to the integral equation (Equation 51 for p = 0)
"
APPENDIX 1
D'l2Qo (t)
-
e-aj
- e-aj
kh
s,'[&-
Do1/2r.-1exp(Dor,-2u) X
1
erfc(D,112ro-1u1/2) Qo(t - u)du
-
Notation Dehitions
A Di Ci
Ci*
= = = =
electrode area diffusion coefficient of species i concentration of species i initial concentration of species
i Ci7-,o = surface concentration of species i Eo = standard redox potential in European convention E(t) = instantaneous value of applied potential AE = amplitude of applied alternab ing potential Ed,o, = d.c. component of potential E;/2 = reversible polarographic halfwave potential (planar diffusion)
F
Faraday's constant R ideal gas constant T absolute temperature number of electrons transferred in the heterogeneous charge transfer step i(t) = total faradaic current Z ( d ) = fundamental harmonic faradaic alternating current e = phase angle of fundamental harmonic faradaic alternat= = = =
exp(DRr,-2u) erfc(DRl'2ro-1u1/2) +
1
2D~"~r,-' exp(Daro-2u) du (Al) which is expressed in Laplace transform space as
Algebraic rearrangement yields
where b+ and b- are defined by Equation 59. Expansion of the right side of Equation A3 by the method of partial fractions, inverse transformation, and algebraic rearrangement yields Equation 58. For the special case where X2t
2 50
(444)
Equation 58 reduces to the expression associated with the reversible wave, Equation 65.
ACKNOWLEDGMENT
The authors thank Northwestern University Computing Center for generous donation of computer time. LITERATURE CITED
( 1 ) Aylward, G. H., Hayes, J. W., J.
tional Mathematics in Engineering,’’ McGraw-Hill, New York, 1944. (11) Delahay, P., “New Instrumental Methods in Electrochemistry,” p. 74, Interscience, New York, 1954. (12) Ibid., p. 170. (13) Delahay, P., Mamantov, G., J . Am. Chem. SOC.76, 5323 (1954). (14) Delmastro, J. R., Smith, D. E., J . Electroanal. Chem. 9, 192 (1965). (15) Delmastro, J. R., Smith, D. E.,
Northwestern University, Evanston, Ill., unpublished data, 1965. (16) Gerischer, H., 2. Physik. Chem. Lei zig 198, 286 (1951). (17) asstone, S., Laidler, K. J., Eyring, H., “The Theory of Rate Processes,” pp. 575-7, McGraw-Hill, New York, 1941. (18) Grahame, D. C., J . Electrochem. SOC.
81
mer, H. H., Australian J . Chem. 17,
99, 370C (1952). (19) Hung, H. L., Smith, D. E., ANAL. CHEM.36, 922 (1964). (20) Hung, H. L., Smith, D. E., J . Electroanal. Chem. 10, in press. (21) Hurwitz, H. D., Ibid., 7, 368 (1964). (22) Kambara. T.. 2. Phuszk. Chem. ‘ (Prankfurt) 5, 52’(1955). (23) Koutecky, J., Czechoslovak J . Physics 2, 50 (1953). (24) +tecky, J.? Stackelberg, M. von,
in Progress in Polarogra hy,” P. Zuman and I. M. Kolthoff. e&., Vol. 1. Chap. 2, Interscience, New York, 1962: (25) Lingane, J. J., Loveridge, B. A., J . Am. Chem. SOC.72,438 (1950).
(26) MacGillavry, D., Rideal, E. K., Rec. Trav. Chim. 56, 1013 (1937). (27) Matsuda, H., 2.Elektrochem. 61, 489 (1967). (28) Ibid., 62, 977 (1958). (29) Paynter, J., Ph.D. thesis, Columbia University, New York, 1964. (30) Randles. J. E. B.. Somerton. K. W.. ‘ Trans. Faraday Soc.‘ 48, 951 (1952). ’ (31) Reinmuth, W. H., ANAL.CHEM.33, 185 (1961). (32) Senda, M., Kagaku KO Ryoiki, “Zokan 50,” 15 (1952). (33) Senda. M.. Tachi. I.. Bull. Chem. SOC. ‘ Japan 28,632 (1955). ’ (34) Shain, I., Martin, K. J., J . Phys. Chem. 65, 254 (1961). (35) Shain, I., Polcyn, D. S., J . Phys. Chem. 65, 1649 (1961). (36) Smith, D. E., ANAL.CHEM.35, 602 (1963). (37) Ibid., 36, 962 (1964). (38) Smith, D. E., Ph.D. thesis, Columbia University, New York, 1961. (39) Smith, D. E., Reinmuth, W. H., ANAL.CHEM.33, 482 (1961). (40) Tamamushi, R., Tanaka, N., 2. Physik. Chem. (Frankfurt)21,89 (1959). (41) Underkofler, W. L., Shain, I., ANAL. CHEM.37,218 (1965). , - - - - I -
RECEIVEDfor review August 17, 1965. Accepted November 24, 1965. Work supported by the National Science Foundation.
Cyclic Voltammetry of Some Iron Porphyrin Complexes DONALD G. DAVIS and DANIEL J. ORLERON Deparfmenf o f Chemisfry, louisiana Sfafe University in New Orleans, New Orleans, l a . Iron porphyrin complexes with pyridine and cyanide ion, individually and in mixtures, have been investigated by cyclic voltammetry at a stationary platinum electrode. It was found that the reduction of various iron(ll1) species proceeds reversibly to iron(l1) and that the iron(l1) is reversibly re-oxidized. Various complexes were identified and their Ep’s reported. The stability constants of several complexes have been measured. The greatest contribution of cyclic voltammetry is that ligand exchange reactions occurring after electron transfer can be characterized and their rates measured. The rate for the substitution of pyridine for cyanide ion in cyanopyridine hemochrome was found to be about 0.5 sec.-l, while the rate for the same substitute in cyanochemochrome was estimated as being 20 sec.-1
I
protoporphyrin chloride (hemine) is easily prepared from hemoglobin, and serves as a prototype for a wide variety of iron porphyrin complexes of great biological importance. The porphyrin ring is a planar aromatic system which contains four nitrogen atoms in such positions RON(III)
that they easily coordinate with many metal ions. The X Y plane coordination positions of the metal ion are thereby filled. Metal ions capable of sixfold coordination, such as iron(III), have two remaining coordination positions available above and below the plane of the porphyrin ring on the Z axis (4). When these positions on the Z axis are occupied by nitrogen-containing, rbonding ligands, such as pyridine, cyanide, imidazole, etc., complexes designated as hemichromes result. Hemichromes are characterized by relatively high solubility in aqueous solutions and a red color, as opposed to usual greenish black of hemin solutions. Assuming pyridine to be the ligand, the resulting hemichrome might properly be designated bispyridyl ferriprotoporphyrin (4). However, the simpler name pyridine hemichrome will be used here. The corresponding ferrous complex would be pyridine hemochrome. The chemistry of hemichromes and hemochromes in aqueous solution’ was studied extensively several years ago (11) by the techniques of spectrophotometry and zero current potentiometry. These results and others similar to them have been summarized by Clark (I) and by Falk (4). Recently some
701 22 polarographic and chronopotentiometric studies of iron porphyrin complexes have been completed in this laboratory (2), and the dropping mercury electrode polarography of iron(II1) protoporphyrin in aqueous potassium hydroxide, reported by others ( 5 ) . Studies involving the electron transfer reactions of metalloporphyrins are of interest since such reactions are important steps in the transfer of energy by the cytochromes and other respiratory enzymes in a variety of biological systems. The participation of the Z position ligands in these rapid electron transfer mechanisms has been proposed (12). Cyclic voltammetry was found to be an excellent method to investigate such systems. Equilibrium constants for the formation of various complexes can be measured, but polarography with a rotating platinum electrode is probably an easier technique (2). Of greater interest is the possibility of gaining information about the rate of electron transfer for the reduction of hemichromes and about ligand substitution reactions following the electron transfer reaction. This work is primarily concerned with the latter. The theoretical aspects of stationary VOL. 38, NO. 2, FEBRUARY 1966
179
