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ANALYTICAL CHEMISTRY, VOL. 51, NO. 9, AUGUST 1979
Pseudopolarograms: Applied Potential-Anodic Stripping Peak Current Relationships Mark S. Shuman" and John L. Cromer Department of Environmental Sciences and Engineering, School of Public Health, University of North Carolina, Chapel Hill, North Carolina
27514
data extremely well. In general, diffusion layer theory though approximate, is useful for stirred solutions (12). With the relationships given below, overall reversibility can be evaluated and any change in reversibility with changes in solution composition can be observed. The utility of this analysis is diagnostic-if stripping voltammetry is reversible, the treatment of pseudopolarogram half-wave potentials analogous to polarographic half-wave potentials may be justified.
Applied potential-anodic stripping peak current relationships for reversible and nonreversible electrochemical reactions were generated using diffusion layer theory. Theoretical correlations between pseudopolarogram half-wave potentials and pre-electrolysistime, heterogeneous rate constants and electrode radius were obtained and relationships between applied potential and log [ ( i , i ) / j ] for both reversible and nonreversible reactions were calculated. The anodic stripping voltammetry of TI(1) agreed with the theory for reversible reactions.
-
THEORY Derivation is based on the typical ASV experiment in which the solution is stirred and the mercury electrode has an area A and volume V. If the electrode is a mercury drop, the volume is determined by its radius, r. I t is assumed that reduced species R forms a soluble amalgam and that for the electrode reaction
Anodic stripping voltammetry (ASV) and other voltammetric techniques are used to analyze trace metals and identify trace metal species in natural waters. Applications were reviewed recently by Davison and Whitfield ( I ) . In some procedures, a shift in ASV peak potential with changes in solution composition is observed and interpreted in terms of a change in composition of the major metal species reduced during the ASV accumulation step (2-5). There is no theoretical basis for this interpretation and reasoning is by analogy with polarography, although ASV is not strictly analogous to polarography. Stationary electrodes, stirred solutions, and longer electrolysis times which create much higher amalgam concentrations are used with ASV; ASV currents result from metal amalgam oxidation; and the high amalgam concentrations formed in ASV can lead to solid phase or intermetallic compound formation. Gardiner ( 6 ) found no consistent peak potential shifts when he investigated formation of inorganic and organic complexes of cadmium with ASV and suggested t h a t interpretation was not straightforward. "Pseudopolarograms", t h a t is, stripping peak current plotted as a function of applied pre-electrolysis potential, were used by Nurnberg et al. ( 7 ) to identify two P b carbanato complexes in seawater and by Branica et al. ( 4 9 ) as a general procedure for evaluating complexation of cadmium in marine samples. The potential a t half the maximum current, analogous t o the half-wave potential of polarography, was assumed to reflect the overall activation energy of the electrode reaction and used to indicate complexation. To date, no general theory of pseudopolarograms has appeared in the literature and it is not clear whether experimental evidence for complexation and calculations of formation constants using pseudopolarograms are valid. An approximate expression for pseudopolarogram El,2 as a function of pre-electrolysis time was presented by Zirino and Kounaves ( I O ) for a simple reversible charge transfer reaction, but their approach did not result in a general equation for pseudopolarograms nor did they consider other relationships. T h e theory of pseudopolarograms for reversible and nonreversible electrode reactions as developed below is based on Nernst diffusion layer theory. T h e development for the reversible case is very similar to that of Lee (11)for a rotating mercury electrode. Lee showed theory fit his experimental 0003-2700/79/03S 1-1546$01 OO/O
O+neGR
(1)
the flux of oxidized species 0 is equal to the flux of R a t the electrode surface. The Nernst diffusion layer theory assumes that there is a stagnant layer of thickness 6 through which transport is solely by diffusion and that the concentration gradient across this layer is linear so that
where CO* is the bulk concentration of 0, Coo is the concentration at the electrode surface and x distance is measured from the electrode surface. T h e current is given by
i = nFADo
co*
-
coo
6
where Do is the diffusion coefficient of 0. A limiting current il is defined for the case where Coo = 0, a t large negative overpotentials
il = nFADo
CQ*
-
6
(4)
For a reversible reaction, the Nernst equation holds, so that at any potential E
where Eco is t,he potential at unit concentration of 0 and R. In the case of a nonreversible electrode reaction, the current is defined as
where h, is the standard heterogeneous rate constant for
C
1979 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 51, NO. 9, AUGUST 1979
Reaction 1. The bulk concentration of R in the mercury after t seconds of accumulation is given by R 1 f R = i(t) d t (7) ~
nFV
o
by 3 / r corresponding to a hanging mercury drop of radius r. The relationship between q , E , and t is
1
q = nFV Co* exp
A flux equation can be written that applies to both reversible and nonreversible cases
where J0 and JR are the diffusion layer thicknesses for 0 and R, respectively, and DR is the diffusion coefficient of R. Reversible Electrode Reaction. Equation 8 can be rearranged to give
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U'hen the potential is very negative, it can be shown that Equation 15 reduces to
q = nFA Co*
Do -
60
(16)
t = qI
where q1 is the limiting quantity of electricity. Equations 15 and 16 are combined, the result is
When
where p = D R b 0 / 6 ~ D o . Substitution into Equation 3 and use of the definition of il in Equation 4 yields
i = k,
-
k,
3Dot
J' i(t) d t
__
(10)
rbo
I-)]
1
where
kl =
11
11 -
1+ (P/O)
a n d h2 =
-
11
P
nFV C O * [ ~+ ( ~ / @ ) l
Differentiation with respect to time, separation of variables, and integration yields In i ( t ) = -k,t const. (11)
-
1 (17)
An expression with the E l , 2of the pseudopolarogram is obtained from Equation 17:
+
Evaluation of the constant a t t = 0 gives In k l so that
i(t) = h, e
(12)
k2t
Integration over time gives the amount of electricity passed during preelectrolysis,
q = ( k , / h , ) (1 - e-kgt)
(13)
a measure of the quantity for 0 electrodeposited in the mercury drop. The stripping current, i,, is measured from the experiment and is related to q by 1,
(18) Nonreversible Electrode Reaction. From Equations 8 and 9 the expression for the concentration of R at the electrode is
CR
(14)
= Kq
where A is a constant whose magnitude depends on experimental parameters such as electrode volume and potential sweep rate. The equation could be expressed in terms of z, but this has no advantage and introduces another empirical constant; therefore, in the remaining discussion the equations are in terms of q . Also, the ratio of electrode area to volume where it appears in the following equations has been replaced
[
-anF nFAk,Co* exp RT ( E - E , " ) ]
- nFV
fo
iso
t
idt+--nFADop
(19)
Substitution of Coo and CR3 into Equation 6 gives Equation 20. In a manner similar to the reversible case, Equation 20 can be differentiated with respect to time, rearranged, and integrated twice to obtain 4 , see Equation 21. I t can be shown that at large negative potentials, this also reduces to Equation 16.
3h,
r
J t idt exp[
( 1 - a)nF
RT
( E - Ec")
1
(20)
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ANALYTICAL CHEMISTRY, VOL. 51, NO. 9, AUGUST 1979
As with the reversible case, an expression for (SI - q ) / q can be obtained and is given in Equation 22. An expression with Eli2of the nonreversible pseudopolarogram can be written as in Equation 23. Separation of variables is not possible for this equation or for Equation 18 and - Ec" must be calculated numerically. Limiting Cases. I t can be shown that equations for nonreversible charge transfer approach reversible charge transfer equations in the limit of large h,. In addition, for any k,, at sufficiently long electrolysis times the bulk concentration of R in the electrode builds up to a degree that the reverse reaction rate is significant. In the limit of long times, both Equations 18 and 23 reduce to
!
!
0 5 t
-c
-02
I
-c3
E-E;,VOLTS
Figure 1. Theoretical pseudopolarograms for nonreversible reactions,
k, = 10-5-10-3
nF In (1
-
ex.[%
exp[ %(El,, - E,") -018-
I t is not possible to further simplify this equation; however, the last term of Equation 24 is a constant with the value -0.005972/n volts, so that for typical hanging mercury drop electrodes and electrolysis times greater than about 10 s (values of the argument 3Do/rdo greater than about 6) .Eliz - Eco is a linear function of In t with a slope of -RT/nF and an intercept of (RTInF)In (2rdo/3Do) - 0.005972/n volts. For electrolysis times less than this, calculations using Equation 18 show that El,*- Ec" approaches zero as t approaches zero. In the limit of very small k,, Equation 22 becomes
-014
-002
and
h(l1
Flgure 2. (E,,,
Therefore for a totally irreversible reaction, the pseudopolarogram El/* is a function of k , and is independent of electrolysis time. A plot of E - Ec" vs. In ( q l - q ) / q is linear with a slope RTlanF and an intercept of ( R T / a n F ) In k$olDo.
THEORETICAL CORRELATIONS Pseudopolarograms for three values of h, are presented in Figure 1. In common with classical polarography, E l I 2shifts cathodically with decreasing k , and the current-potential curve changes shape from a quasi-reversible to a totally irreversible reaction. Figure 2 shows Eli*as a function of pre-electrolysis
- Eco) vs.
In t f o r values of k , = 10-6-10-2
time, t , for a two-electron process and typical values of DO, r, and ?j0. For the reversible case (except very small In t values), for large k , or long times (when amalgam concentrations are substantial), El/* is a linear function of In t whereas for small k , values (totally irreversible) El/*is independent of time. Figure 3 shows that El,*- Ec" shifts positive with increasing rate of reaction but is independent of k , a t large k , where the reaction approaches the reversible case. Also evident in this figure is the effect of pre-electrolysis time on the E l l 2 - log k , relationship. Longer times result in more negative El/*and
ANALYTICAL CHEMISTRY, VOL. 51, NO. 9, AUGUST 1979
50
f l y
1549
-06
.' -oa 0"
lO.300
-
I8
4 4 t
- 2 00
-100
103
000
(3;G
log
1 1 2 00
\
,771
Figure 6. ( E l i z - E c o ) vs. log ( 9 1 - 9 ) / 9 for reversible reactions,
Figure 3. ( E l I P- E c o ) vs. log k, and t = 10-10000 s
S D o t / f ? i o 5 0.1
Or
VI
'
- 121
w
I
4c-5 - 480 -
0 01
05
r,cm
Figure 4, ( E I I P- E c o ) vs. r f o r k, = 10'5-10-2
-520k
Flgure 7. Comparison of experiment with theory. ( 0 )t = 60 s, r = 0.06 cm; (0)t = 420 s, r = 0.05 cm. Dashed line is 59 mV/decade and solid line is calculated for theory, 3D,t/r?io
2
3
oc
Figure 5. (E,,, - E c o ) vs. log (9, - 9 ) / q f o r nonreversible reactions, k = 10-6-10-2
shift the onset of reversible behavior (ElI2independent of k,) to lower k , values. The dependence of ElI2- Eco on the drop radius of the hanging mercury drop electrode (HMDE) is shown in Figure 4 a t various values of k,. In general, El12shifts positive with increasing radius except for k , < where it is independent of t h e radius. For very small values of the argument 3Dot/rho (large r or small t ) and a reversible reaction, calculations show t h a t Eljz approaches Ec". Data analysis analogous t o polarography can be made by plotting preelectrolysis potential vs. log (41 - q ) / q t o test for reversibility and to obtain transfer coefficients. Figure 5 shows this data analysis for a two-electron process, k , = lo4 to lo-* and CY = 0.5. T h e curve for k , = lo4 is a straight line as it would be for polarography. However, as k , increases, the line becomes curvilinear, and is still curvilinear at k , = 10-'where t h e reaction is nearly reversible. Figure 6 shows this type of data analysis for reversible reactions with the result that the relationship is not strictly linear except for small values of
= 55.1
3Dot/rho (
