Anal. Chem. 1985. 57.585-591
PSA is free from errors caused by iR drop and by the presence of oxidizable impurities in solution. In conclusion, the theory presented in this paper applies to stripping in a stirred solution. For the case of stripping in a stationary solution, when the flux of the stripping agent is a function of time, it may still be possible to solve the diffusion equations (15). It also would be interesting to study the kinetics of metal ion complexation during stripping into solutions containing ligands; the advantages of using flow cells for such purposes have been pointed out by Jagner et al. (16). The theory for such processes can be worked out by including the proper kinetic terms in the diffusion equations. Finally the use of microelectrodes (graphite or noble metal fibers) offers special advantages for PSA, as shown by eq 28 and also by other considerations; we shall report elsewhere (12) experimental proof that these expectations can be realized.
ACKNOWLEDGMENT We thank S. G. Weber of this department for valuable discussions.
585
LITERATURE CITED Jagner, D.; Aren, K. Anal. Chlm. Acta 1978, 100, 375. Chrlstensen, J. K.; Kryger, L.; Mortensen, J.; Rasmussen, J. Anal. Chlm. Acta 1980, 121, 71. Coetzee, J. F.; Hussam, A.; Petrick, T. R. Anal. Chem. 1983, 55, 120. Chau, T. C.; Li, D. Y.; Wu, Y. L. Talanta 1982, 29, 1083. De Vries, W. T.; Van D a h , E. J . flectroanal. Chem. 1984, 8 , 366. De Vries, W. T.; Van D a h , E. J . flectroanal. Chem. 1987, 14, 315. Zirino. A.; Kounaves, S. P. Anal. Chem. 1977, 48, 56. Brown, S. D.; Kowalskl, B. R. Anal. Chem. 1979, 51, 2133. Christensen, C. R.; Anson, F. C. Anal. Chem. 1963, 35, 205. Hurwitz, H.; Gierst, L. J . flectroanal. Chem. 1961, 2 , 128. Perone, S. P.; Brumfield, A. J . flectroanal. Chem. 1967, 13, 124. Coetzee, J. F.; Hussam, A,, unpublished results, University of Pittsburgh, 1984. Buffle, J. J . flectroanal. Chem. W81, 125, 273. Mortensen, J.; Britz, D. Anal. Chlm. Acta 1981, 131, 159. Hussam, A. Ph.D. Thesis, University of Pittsburgh, 1982. Anderson, L.; Jagner, D.; Josefson, M. Anal. Chem. 1982, 5 4 , 1371.
RECEIVED for review August 2,1984. Accepted November 16, 1984. This work was supported by the National Science Foundation under Grant No. CHE-8106778.
Pulse Polarography: Effects of Electrode Sphericity on the Current-Potential Curves in Normal Pulse Polarography, Reverse Pulse Polarography, and Differential Pulse Polarography Jesus Galvez
Laboratory of Physical Chemistry, Faculty of Science, Murcia 239169, Spain
Equations whlch take Into account the spherlclty of the DME for reverslble electrode processes In normal pulse polarography (NPP), reverse pulse polarography (RPP), and dlff erentlal pulse polarography (DPP) have been derlved In a rlgorous way. We have consldered the cases where the reduction product dlssblves both in the electrolyte solutlon and In the electrode. I n DPP the effect exerted by the spherlclty of the electrode Is much more marked for systems Involving amalgam formation than for reductions to a solutlon-soluble product. I n RPP thls influence Is very slgnlficant for both types of processes, although It acts In an opposlte way for amalgam-forming systems than for those systems where the reduced form Is soluble In the solution. Flnally, in NPP the current-potentlal curves show only a small dependence on whether there Is amalgam formation or not.
Table I. Notation and Definitions
A@) m
distance from the center of the electrode electrode radius at time t time-dependent electrode area rate of flow of mercury
r
Euler gamma function
r r0
9-
51
Y K, 3,
e T
f(T)
g(7)
M
z
(3rn/4~d)'/~ (12Di/7$2)1/2t1/6
(DA/DB)'/' exp(nF(Ei- E o ) / R T ) (1 f KJ/(1+ YKi) 7 p l 7/11 (=0.7868)
tflh, + t?
+
+
+
+
1 r / 3 7 ~ ' / 5 4 4 r 3 / 8 1 ... 1 - 5r2/48 + 7PO(Y 1 ) 6 A / ( l l r ( l + YKd)
1/(1
...
+ rK2) - 1/(1 + YK,)
other definitions are conventional Pulse polarography (NPP, RPP, DPP) has become one of the most powerful electroanalytic techniques both for chemical analysis and for the study of electrode processes (1-13). Regarding the theory of the current-potential curves several approaches have been developed, although in all of them, the expanding plane electrode model (EP) was normally adopted for the DME. Only Los and co-workers (14, 15) and more recently Galvez and co-workers (16-22) have also considered the more rigorous expanding sphere electrode model (ES),but just for some types of limiting currents in NPP. The aim of 0003-2700/85/0357-0585$0 1.50/0
the present paper is, therefore, to extend the theory of the current-potential curves in pulse polarography with the following assumptions (a) The treatment adopted is based in obtaining the solution of the system of differential equations which describe the boundary value problem for the ES model separately for both pulses. Under these conditions the solution to the first system of equations becomes the initial condition for the second one (20,11, 23). (b) In order to obtain these solutions we have applied the dimensionless parameter method previously described (16). 0 1985 American Chemical Society
588
ANALYTICAL
CHEMISTRY, VOL. 57, NO. 3, MARCH 1985
This allows then solving the problem through concentration profiles for any value of r and t so that the use of linearized profiles (23) can be avoided. (c) The case where the reduction product dissolves in the electrode will also be considered because this situation has not been reported in the literature to our knowledge. (d) The equations obtained will be applied to the different modes of pulse polarography mentioned above, e.g., NPP, RPP, and DPP, so that the influence exerted by the sphericity of the electrode on the current-potential curves will be discussed.
YCA*
bo = 1 + YK1
THEORY Notation and definitions are given in Table I. If we consider the reversible charge transfer reaction A
+ ne- ~ iB:
and the product B is soluble in the electrolyte solution (amalgamation reaction will be discussed later), the expanding sphere boundary value problem is given by BAcA
t
= BBcB =
+ K1
1
1 + YKl
01 =
o
and being the \E's the functions defined by Kouteckg (25). In addition, from eq 4, 6, and 8 the current, il(t), is given by
with 7 e = -p1/7 11
(1)
(=0.7868)
(15)
For t > tl we may write t = tl + t'and so, eq 1 remains valid if in eq 5 we substitute t by t '. In turn, the boundary value problem is given now by t' = 0 CA(r,t) = CA(SA,SA)
> 0, r = Po:
C d r , t ) = CB(SB,SB)
(16)
t ' > 0 , r = ro (17) where Bi is the operator (24) with
1
nF
In the following we shall suppose that the potential is set on a constant value El from t = 0 (beginningof the drop growth) to t = t l , and at t = tl it is stepped up from Elto the other constant value Ez.Under these conditions the solution of eq 1-4 during the period 0 < t < tl can be readily obtained (25) and so, if we only consider the firstiorder spherical correction we find
CA(SA,SA)= P O ( S A ) + SAP~(SA)
CB(SB,SB)= $O(SB) + SB&(SB)
(6)
K2 = exp[ =(E2 - Eo) In addition, it is held
i=AorB where Civalues in eq 16 and 20 are defined by eq 6. The solution of eq 1 (with t ' instead of t ) and eq 16-18 can be obtained proceeding as follows: With the variables
where
si =
r - ro
s; =
(
2(Dit')112
212Dit ) I 2 ( r - ro) ti
=
2(Dit ')l/' TO
7 = -
eq 1 become
(8) where
nF K1 = ex,[ =(E1 - E o )
I
t'
t,
+ t'
ANALYTICAL CHEMISTRY, VOL. 57, NO. 3, MARCH 1985
587
defined by eq 33 and Al-A4, the following expression for the current is obtained:
and inserting these expressions in eq 22 we obtain for ui&s‘,J and 6!j(s b) the following system of recurrent differential equations:
4
C dru’pj-l(~’.J + -(i r+p=i 3 6”iJ
where f(7)=
I
1
1 7 + -7 + -7’ + 814 + ... 3 54 -T3
+ 3j - ~ ) U ~ ~ - ~ ( S / A(25) )
+ 2Sb6‘ij - 2(i +2j)aij = - k+l+l=i c
CkIYlj(Sb)
- -
+
As expected, if Ez El, 2 0 (see eq 35) and iz(t)= il(t). In addition, the corresponding expression for the current in the expanding plane electrode model is readily obtained by making [A = 0 in eq 36
where ck
= 2(-1)k(S{)k;ck = 0 for i = 0
dr = 4(-lNr
+ 2)(~:)‘+’/3; d, = 0 for j
=0
(27)
In turn, the boundary conditions (16)-(18) now are s ’ ~+ m:
00,o
uij =
= CA(SA,SA);~O,O = CB(SB,SB)
aij = 0 unless i = j
=0
(28)
s: = 0 y’aij(0) = K26ij(0)
(29)
yi+la’ij(O) = -6’;JO)
(30)
while condition (20) becomes lim ~ z - t ~ i
~ ~ , , ( S ’ A ) = ~ A ( S~ AA , ) ; ~ ~ , ~ ( s ’ B ) = ~ B t ( ~s (31) B ), i,j(s‘B = 0 unless i = j = 0
ai,j(s’A =
Note that the functions uoj(s5)and 6,j(sb) are the solutions for the expansing plane electrode model and that they must be known before deriving the solution for the ES model. Thus, we have: Expanding Plane Electrode. If i = 0 eq 25 and 26 become d’oj 2s’,doj - 4j~Toj= -(8/3)~2~’0j-l- 40’ - 1)uoj-l
+
6”oj
+
- 4jSOj = - ( 8 / 3 ) ~ b 6 ’ ~-~40‘ - ~- 1)60j-l (32)
These functions can be obtained by combining eq 32 and 28-31 and following the same derivational pattern previously described (16). Thus, we find
~ o , ~ ( s ’ A=) CA(SA,SA) - MZCA*@O(SA)
60,o(sb)= CB(SB,SB)+ Mzc~*@&b)
(33)
where
(34)
In turn, the first functions for j > 0 are given in the Appendix. Expanding Sphere Electrode. If we consider only the first-order spherical correction, the set of equations (25 and 26) must be solved for i = 1. In the Appendix we show the procedure to follow in order to obtain the corresponding functions ulJs6,) and &J(s’B).Finally, combining eq 18, 21, and 24 and taking into account the expressions for u&L)
Note that in this case the values of the current computed with eq 39 do not depend on whether there is amalgam formation or not. Amalgam Formation. In this case we must rewrite the boundary value problem as follows: eq 2 is now t = 0, r > 0: C A = C*A, C B = 0
t > 0, r --*
CA = C*A, r
00:
-+
-m,
CB = 0
the minus sign in eq 4 and 18 is changed to plus. The procedure of derivation to follow by using these new conditions differs slightly from that adopted previously (see Appendix) although, we find that the expression for the current is also given by eq 36 if in this equation ai and Mare defined as Qi
1 f Ki =1 7Ki
+
(i = 1, 2)
where the upper sign relates to a solution-soluble product and the lower sign to the amalgamation reaction (this rule will be maintained for the next sections).
RESULTS AND DISCUSSION From eq 36 and 40 we may easily obtain the expressions for the different modes of pulse polarography by introducing the experimental conditions corresponding to the mode employed. Thus, we have the following. Normal Pulse Polarography. In NPP, the initial potential El is maintained at an initial value at which no current flows. The scan of the potential Ez is then made to the cathodic direction. Hence, K1 m and introducing this condition in eq 36 and 40 we obtain
-
(41) In turn, the limiting current can be obtained by inserting the condition E2 -a (Le., K , 0) in eq 41
-
-
588
ANALYTICAL CHEMISTRY, VOL. 57, NO. 3, MARCH 1985
;
k 0.4
4
-5
-6
+
-7
4
-8
i
1
I
0.2
i
I -O
-I0 0.I
-0,1
0
-0.2
( E - E " ) /v Figure 1. NPP current-potential curves computed from eq 41 and 42 for y = 0.7, tA= 0.15, t = 1 s,and T = 0.05: (1) without amalgam formation; (2) reduction product dissolved in the electrode.
This particular boundary value problem has been previously solved by Galvez and Serna (16)and their solution is identical with eq 42. In any case, it is interesting to show that both in NPP and in dc polarography (26), the limiting current does not depend on whether the process involves amalgam formation or not. An example of this is shown in Figure 1. The characteristics of the current potential curves are similar to those previously described in dc polarography (26) and for this reason further discussion is not given here. Reverse Pulse Polarography. In RPP (6),or scan-reversa1 pulse polarography (4),the initial potential El is set on the cathodic diffusion current plateau and the potential E z is scanned anodically. This is a powerful technique to characterize electrode processes (6, 27), although the expression for the current has only been derived for the E P model (1,6). However, this model is not sufficient to explain the behavior of the current potential curves obtained experimentally and so, Osteryoung and Kirowa-Eisner reported (6): "it appears to be the case that amalgam-formingsystems and systems for which both the oxidized and reduced forms are soluble in the solution behave differently" (see also Figure 3 in ref 6). However, we show in this section that this different behavior can be explained taking into account the ES model. Thus, inserting the condition El -a (Le., K1 0) in eq 36 and 40 we obtain -+
-+
(43) where
b-
I
.-z
3
i
L
2
Figure 2. RPP current-potential curves computed from eq 46 for 7 = 0.005: (1) without amalgam formation, lA = 0.15, y = 1.1; (2) amalgam formation, EA = 0.15, y = 0.8; (3) EP model, [A = 0.
with 7p,/ll = 0.718. From eq 43 and 44 the relationship i 2 ( t ) / i d c is given by
-idt) -
-1-
idc
(46)
-
The anodic limiting current, iRp, is obtained by introducing the condition Kz a in eq 46
(47) Note that the corresponding expression for the EP model is only a particular case of eq 47 and so, for FA = 0 we have
-5% = ($)
1/2
idc
f(7)
-1
This equation is equivalent to that previously derived for Oldham and Parry (1)for this electrode model, and it is readily shown that the values of -iRp/idc computed from both equations differ about l %or less even for t' 5 0 . 2 . Equations 46 and 47 predict that the behavior of the processes involving amalgam-formation must be different from those where the product dissolves in the electrolyte solution. This is shown in Figure 2 where we have plotted current potential curves for y = 0.8 (amalgam formation), y = 1.1 (without amalgamation reaction), SA = 0.15, and 7 = 0.005. For comparison we have also included the corresponding curve for EA = 0. This different behavior is also shown in Figure 3 in which we have plotted -iRP/idc vs. F ( T )(being F ( T )the right-hand side of eq 48 for several values of 7 . These plots are practically linear and from eq 47 and 48 we find that their slope, a, is given by
Accordingly, if there is amalgam formation we have a > 1 (unless y > 10.4), while for processes where the product is
ANALYTICAL CHEMISTRY, VOL. 57,
NO. 3,
MARCH 1985
589
Table 111. Deviations of the Peak Current Values as a Function of [A and yo 28 €A
Y
5% (1)
% (11)
0.10
0.7 1.0 1.4
2.5 3.3 3.8
7.6 6.8 6.3
0.15
0.7 1.4
3.2 4.3 5.1
10.5 9.5 8.8
0.7 1.0 1.4
3.9 5.4 6.4
13.3 12.0
15
1.0
.-4 \ .-&
0.20 le
11.1
Deviations between the values of the peak current obtained with the EP and ES models (eq 55): AE = 10 mV, n = 2,7 = 0.03; (I) reduction product dissolves in the electrolyte solution; (11) amalgam-formingsvstems.
5
as a difference between iz at t = tl + t'and il at t = tl. Hence, combining eq 36 and 14 we find 5
10
15
28
F(r) Figure 3. RPP dependence of -iRp/idc (eq 47) on F(T). Other conditions are given in Flgure 2.
Table 11. Deviations of the Values of of [A and yo
-jRP/jdo as
a Function
T*=1-T
€A
Y
?& (1)
% (11)
0.10
0.7 1.4
11.3 7.9 5.7
-8.1 -5.7 -4.0
0.15
0.7 1.0 1.4
17.2 11.8 8.5
-11.4 -8.0 -5.6
0.20
0.7 1.0 1.4
23.3 15.7 11.2
-14.2 -10.1 -7.1
1.0
a Deviations between the values of -iRP/idc obtained with the EP and ES models (eq 51): (I) reduction product dissolves in the electrolyte solution; (11) amalgam-forming systems.
soluble in the solution a must be
