Theoretical Treatment of Pulsed Voltammetric Stripping at the Thin Film Mercury Electrode Robert A. Osteryoung and Joseph H. Christie D e p a r t m e n t of Chemistry, C o l o r a d o State University, f o r t Collins, Colo. 8052 7
The theory of pulsed voltammetric stripping at the thin film mercury electrode is treated as a two-level approximation: as a series of single potential steps and as a series of double potential steps. The first approximation gives results that are quite similar to linear scan voltammetric stripping. The results obtained are in accord with experiment in the limit of large pulse amplitudes.
It has long been recognized that pulse polarography exhibits a high discrimination against double layer charging current, while linear scan voltammetry measures the sum of the faradaic and charging currents ( 1 ) . We, therefore, expect that pulsed stripping voltammetry would exhibit a greater sensitivity than linear scan stripping because of the greater discrimination against charging current. A further advantage of pulsed stripping analysis resides in the fact that it is essentially a repetitive technique, some of the material stripped from the electrode during the pulse being replated into the electrode in the waiting time between pulses. The same material is “seen” many times in the pulsed stripping experiment, but only once in the linear scan stripping experiment. Since the mercury thin film electrode can be rotated during the plating process, much smaller deposition times are possible with this electrode than with the hanging mercury drop electrode; the higher surface-to-volume ratio for the thin film electrode results in much higher stripping currents than for the hanging drop. The support of choice is graphite, which eliminates the problems of intermetallic compound formation attendant on the use of platinum, gold, or silver supports. All of these expected advantages of pulsed stripping from the thin film mercury electrode have been experimentally demonstrated (2). A quantitative theory is necessary for guidance in further optimization of the stripping experiment. An approximate theory, valid for films of the thickness used in our experimental study, is presented in this paper. Plating Process. We assume that the plating is carried out a t a mercury thin film electrode of thickness 1 and area A from which the metal of interest M is initially absent. We ignore the possibility of codeposition of the mercury film with the metal M. The electrode is rotated a t a rate ut (Hz) during the plating process, which is carried out a t a potential very much more negative than the halfwave potential for the couple M-n/M(Hg). The concentration of metal in the amalgam a t the end of the plating process is ck* and this is related to the total charge, q M , passed for the deposition of the metal.
tration gradients exist in the electrode a t the start of the stripping process and that the electrode is a t rest with respect to the solution during the stripping. These initial conditions correspond to those assumed by DeVries ( 3 ) and DeVries and Van Dalen ( 4 ) in their complete treatment of linear scan stripping from the thin film electrode. Boundary Conditions. Following DeVries ( 3 ) , we can write the integral equation relationships between surface concentrations and current.
n t
7
where
and E , is the plating potential and El,* is the reversible half-wave potential for the M f n / M ( H g ) couple. We assume that the electrode reaction is reversible; the surface boundary condition is given by the Nernst equation.
(5)
nF
= exp[j&E
-
E,,,)]
(6)
Eliminating the surface concentrations among Equations 2, 3, and 5 gives the integral equation for the current.
q M = nFAICR* (1) We assume that the concentration of the metal ion in solution is sufficiently near zero that it may be neglected with respect to CK*. We further assume that no concen-
This equation is similar to Equation 7 of De Vries; the major difference, other than slightly different notation, is that we have not specified the potential-time waveform that is applied. We could, in principle, define any E ( t ) , and therefore t ( t ) , and develop a solution by standard integral equation solution techniques. For a pulse waveform, with slow scan rates and a large disparity between pulse width ( 5 4 0 msec) and waiting time (20.5 sec), such straight-forward calculations are extremely lengthy, both because of the small time spacing required and because of the slow con-
( 1 ) J. B Flato,Anai Chem., 44, ( 1 1 ) . 75A (1972). (2) T. P. Copeland. J. H. Christie, R. A. Osteryoung. and R. K . Skogerboe, Anal. Chem., 45, 2171 (1973).
(3) W . T. De Vries, J . Eiectroanai Chern., 9, 448 (1965) (4) W. T. De Vries and E. Van Dalen, J. ElectroanaC Chem.. 14, 315 ( 1967).
8 W 8 8 8 8
8 8 8
8 W
8 8 8
8 nlE. E
W
,l
Figure 1 . Current functions for t h i n film
8 8
electrodes
8 8
A . Linear scan stripping. H = ( l ) = ; (2) 0.6; ( 3 ) 0.06 E . Pulsed stripping J = ( 4 ) m ( 5 ) 3.16: (6) 1.0
vergence of the kernel function in the summation term in Equation 7 . We have, therefore, adopted the technique of developing a n approximate solution valid for the conditions of pulse amplitude and film thickness shown to be optimum in our experimental study. All calculations were done in FORTRAN or FOCAL on a Digital Equipment Corporation PDP-12 computer.
APPROXIMATION I As a first approximation, we assume t h a t each pulse is a separate experiment, being carried out on a n electrode with concentration Ci+*; this experiment is equivalent to single potential step chronoamperometry. This first approximate treatment reveals the gross features of the current-potential relationship. This treatment ignores the depletion due t o preceding pulses or, equivalently, assumes that all material stripped during the pulse is replated into the electrode from the solution by the time that the next pulse is applied. Each pulse, therefore, starts from the same initial condition, but goes to a different pulse potential. Under our present assumptions, is time-independent and Equation 7 may be solved exactly by straight-forward Laplace transform techniques.
(9)
The potential dependence and the thickness dependence of the current are contained in the function G(t,>); the coefficient of this function is the Cottrell coefficient, which, when multiplied by the thickness-independent first term of G(t,J2), gives the usual equation for normal pulse voltammetry (polarography) in a semi-infinite medium. A large number of G(t,>) us. n ( E - E l ! z ) curves have been calculated for various values of J . Some of these curves for larger values of J are shown in Figure 1. Also shown for comparison in Figure 1 are several analogous d ( x , H ) curves for linear scan stripping calculated by the method of DeVries and Van Dalen ( 4 ) using their Equation 6. The proper dimensionless parameters for linear scan stripping are 352
A N A L Y T I C A L CHEMISTRY, V O L . 46, N O . 3, M A R C H 1974
8
nF RT
x = -ut
-
where u is the scan rate. Note that both sets of curves give the proper limits as 1 a, the normal pulse polarographic wave, and the linear scan peak characteristic of semiinfinite linear diffusion. Both techniques give peaked current-potential curves which shift cathodic and decrease in height as 1 becomes smaller (at constant t or v ) . Figure 2 shows the thickness dependence of the peak potential of G(t,J2). Xote that the peak potential is linearly dependent on log (J) for J < -0.1. A similar linear dependence of peak potential on log ( H ) was noted by DeVries and Van Dalen ( 4 ) for linear scan stripping. Over the same range of J , the maximum value of G(t,>) is a direct linear function of J . Figure 3 shows that G(t,,J2)/J, where ti, is the value of t a t the peak potential, is constant to within 1%for J < 0.2. Again, a similar result obtains for linear scan stripping ( 4 ) . Thus, in the limit of thin films ( J < 0.2) the function G(t,P) reduces to a function of the form G(t,P) = J.G'(t/t,)
(13)
Note that J affects G' only by way of determining the peak potential. The function of G'(t/t,,) is shown in Figure 4, together with the analogous function Z(t/t,,) for linear scan stripping calculated using Equation 22 of DeVries and Van Dalen ( 4 ) . Substituting from Equations 13, 10, and 1 into Equation 8, we obtain, for the limit of thin films.
Further substituting the value of G'(1) = 0.245, we obtain the remarkably simple result for the peak current for pulsed voltammetric stripping I , , = -0.138q,I/t (15) The peak current is predicted by this simplified treatment to be dependent only on the amount of metal plated into
I
'
i
t y
.coo
I
c
o
1.
0.1
1 0
1
1
nlE
i
I
.IO0
loo
.E
~
0
100
), mV
Figure 4. Thickness independent current functions for ( A ) linear scan and ( B ) pulsed stripping
*********e*****
L.--.E
.4
Lop ll~~0,Il
Figure 3. Peak value of current function divided by thickness parameter illustrating constancy for thin film region
the electrode and on the pulse width, provided that the electrode is thin enough and the pulse width is long enough that l/va;;t'< 0.2. For equally thin films or equivalently slow scan rates, the linear scan stripping peak current is given by (at 25 "C) (4) I , = -11.6n~q, (16) The peak currents for the two techniques are equal For n = 1, v = 1 V/sec, this when nvt = 1.195 X corresponds to t = 11.95 msec; both of these conditions are near the usual limits of the techniques. One may, therefore, conclude that the two techniques are essentially equivalent in their faradaic response. On the other hand, linear scan stripping exhibits a major double layer charging current component (which is also linearly dependent on Y) while pulsed stripping voltammetry discriminates almost completely against double layer charging current, except in the presence of appreciable uncompensated resistance (2, 5). Concentration Profiles. It is of interest to consider the concentration profiles of M in the electrode and M+n in the solution during the application of the pulse. Under the condition of our present assumptions, the concentration profiles are given by
*
(-b;)
+ &)}
erfc(jJ erfc-
iz(-l)'-l[ (t-') + I,_,
erfc( 1.I
t + l
e r f c ( j J 4-
(17)
+
X
I-')%(
n ( E - Ell2)= -33.37 mV. / * / O R = (c) 10; ( d ) 100 msec
l + t
t
t
Figure 5. Relative concentration profiles for M in the amalgam and M+" in the solution
t-
z(-l)j ,=I
Ill
-
-") urn
&)]
(18)
(5) T R Copeland, J H Christie, R A Osteryoung, and R K Skogerboe, Anal Chem , 45,995 (1973)
sec;
t =
( a ) 0.1; ( b ) 1.0;
These equations can be derived by substitution of Equations 8-10 into Equations 2 and 3, or more simply from the Laplace transforms of these equations. Figure 5 shows the concentration profiles calculated from Equations 17 and 18 for various times for a value of l 2 / D = ~ sec and n(E - E l / * ) = -33.37 mV (corresponding to the peak potential for t = 10 msec). Note that the profiles decay rapidly as the time increases and that the profiles are essentially flat for times 1 10 msec (52 I0.1). Figure 6 shows the surface concentration of M as a function of n(E - E1/2) for several values of J and Figure 7 shows the surface concentration of M a t the peak potential as a function of log J. It is interesting that the surface concentration of M is almost constant at the peak potential; compare Figure 3 for the peak current. The flat Profiles shown in Figure 5 lead Us to the use of Approximation I1 in which we attempt to account for the
ANALYTICAL CHEMISTRY, VOL. 46, NO. 3, MARCH 1974
9
353
1
-
lo!.
;
10.3-
.
....
e
. . . .. e
e
10"-
e
8
e
.loo Y
" ( E . E,,,) m V
10.'
Figure 6. Relative surface concentrations as a function of potential for pulsed stripping (A) /2/DRt =
e .
e
0
200
. . . 0
e
-
e
.300
-100
,200
(6) Z / D H t = 10(E
. E,,,),
0
mV
Figure 8. Fraction of charge lost for a single step as a function of step potential / * / D R = 3.75 X
sec: r = 7.5 X
sec; t , = 0.5 ( 0 ) ;5.0 ( B )
SeC
0.46
1
e
e
-6
.4
-2
0
LOg(I'ID,ll
Figure 7. Relative surface concentration at the peak potential for pulsed stripping as a function of thickness parameter
depletion of material from the electrode due to the application of the pulses. It should be noted that we have developed no dependence of pulse current on pulse rate. This deficiency is removed by the use of Approximation 11.
APPROXIMATION I1 As an improved treatment, we assume that after a pulse to potential E for time 7, the potential is returned to E, for a time t , during which the surface concentration of M+n, Co(0, t > 7 ) is equal to zero. This experiment is equivalent to double potential step chronoamperometry. During this overall process, the net charge lost from the electrode to the solution will be the integral of the current flowing between t = 0 and t = t , + 7 . One can derive an integral equation representation of this charge, analogous to the basic equation for double potential step chronocoulometry (6)
and since we assume that Co(0, t > becomes for q ( t w T )
+
T)
-$&E
, , v.
Figure 9. Fraction of charge retained after a series of steps under Approximation I I / * / O R = 3.75 X No.
1 2
3 4
sec; r = 7.5 X
sec
Scan rate, mV/sec
tar sec
1.0 2.0 2.0 5.0
0.5 0.5 1.0 1.0
= 0, this equation
The integrals remaining in Equation 21 can be expressed in terms of the function du
Substituting from Equation 13 for Co(0, t _
