581
Anal, Chem. 1985, 57,581-585
Potentiometric Stripping Analysis: Theory, Experimental Verification, and Generation of Stripping Polarograms Abul Hussam and J. F. Coetzee* Department of Chemistry, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
The theory for potentiometric stripping analysis is presented. Equations based on an assumed Initial parabolic concentration gradient of the metal amalgam are derived for the transient potential as a function of time and for the transition time. Results are compared for conventional electrodes and for fiber microelectrodes. I n addition, the theoretical basis for the generation of stripping (pseudo-) poiarograms by potentlometric stripping analysis is also presented. Experimental verification of m e of the theoretical relationships is provided.
Table I. Comparison of Potentiometric Stripping Analysis with Voltammetric Stripping Analysis
Potentiometric stripping analysis (PSA) is a useful and relatively new trace analytical tool (I). It differs from the much better known voltammetric stripping analysis (VSA) in that the preconcentrated analyte is stripped chemically, rather than electrochemically. The two techniques are compared in Table I. Both oxidative (1)and reductive (2) PSA, corresponding to anodic and cathodic VSA, have been developed. We have recently extended PSA to the determination of the alkali metal ions in aqueous samples by addition of appropriate cosolvents (3). Although PSA already has been widely exploited for analytical purposes, its underlying theory has not been sufficiently examined. We have recently (3)reported equations for the transition time and the transient potential in PSA; these were based on an assumed time-independentconcentrationgradient of the metal ion during stripping. A t approximately the same time, Chau et al. (4) presented a theoretical treatment based on an assumed initial linear concentration gradient of the metal in the mercury electrode and of a time-dependent concentrationgradient of the metal ion during stripping. They further assumed that the diffusion layer thickness of the metal ion remains unchanged during stripping. Experimental proof was provided for the case of deposition in a stirred solution and stripping in a quiet solution, conditions similar to those used in VSA. In this paper, we consider an initial parabolic concentration gradient of the metal in mercury, which is a more realistic model, and stripping in a stirred solution, which is the usual condition for PSA. Relationships derived for macroelectrodes and for microelectrodes (fibers) are compared. We also consider the effect of the deposition potential on the stripping time and of its application in the generation of stripping (or pseudo-) polarograms. Finally, advantages and limitations of PSA in the speciation of metal ions are cornpared with those of VSA.
where k is the formal rate constant for the deposition reaction at a potential E d , Cm+O,= 0,td) and CR(z = I&) are the surface concentrations of the metal ion and the reduced metal, A is the area of the electrode, CY is the transfer coefficient, 0 = (nF/RT)(Ed - E O ) , and all other terms have their usual meaning. For a stirred solution and assuming that the concentration of Mn+does not change appreciably during the deposition time, eq 2 applies
-
1. Deposition (Preconcentration)Step Mn+ ne xHg MtHg), electrode: thin film of mercury on glassy carbon substrate 2. Stripping Step 2.1 voltammetric stripping M(Hg), - ne Mn+ + xHg monitor i or Ai = f ( E ) 2.2 potentiometric stripping M(Hg), t (n/2)Hg2+ Mn+ + ( x + n/2)Hg monitor E = f ( t )
+ +
-
-
where D is the diffusion coefficient, 6 is the diffusion layer thickness, and CoMn+ is the bulk concentration of the metal ion. De Vries and Van Dalen (5, 6) have shown that the reduced metal concentration profile CR(X,td) in the mercury film is best approximated by CR(X,td) = COR
+ kX2
(3)
where
(4) and
(5) Here, DR is the diffusion coefficient of the metal in the mercury phase and 1 is the thickness of the mercury film. Equation 3 will become more and more accurate at increasing td and decreasing 1, and should apply to typical PSA experiments. The surface concentration of the amalgam, CR(x = l , t d ) , can then be calculated from eq 3
THEORY Concentration gradients during the two steps (1and 2.2) outlined in Table I are shown schematically in Figure 1. Since two uncoupled diffusion processes are involved, two variables (z and y) are used. Step 1. Potentiostatic Deposition. The general expression for the current resulting from the potentiostatic deposition of the metal as a function of deposition time, td, is i(td) = nFAk[CMn+(Y= 0,td) eXp(-d) - CR(X = l,td) expK1 - 4011 (1)
By use of eq 1, 2, and 6, the potentiostatic deposition current can be expressed by the relation
Therefore, at any deposition potential, COR and k can be calculated by using eq 4, 5, and 7.
0003-2700/85/0357-0581$01.50/00 1985 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 57, NO. 3, MARCH 1985 H g - F i l m D i f f u s i o n Layer
Bulk
Equation 13 can be generalized for any oxidant (ox) by simply replacing JHg2+ in eq 10 by Jo,.For values of much less than unity, the term in brackets in eq 13 reduces to (nD~t)'/'/21. Equation 13 then can be simplified to
When t is equal to the transition time, T', CR(1,7') = 0, so that X
Y
Inserting eq 4, 5, 7, and 10 for COR, k , i(td), and JHgz+,respectively, into eq 15, eq 16 is obtained
0
Hg"
Mn'
Y
Figure 1. Assumed concentration profiles during (a) potentiostatic deposition and (b) potentiometric stripping.
Step 2.2. Stripping of the Deposited Metal by an Oxidizing Agent. To calculate the change of CR as a function of the transient time, t , Fick's second law for the linear diffusion of R in the mercury film under proper initial and boundary conditions must be solved
A t t = 0, the potentiostatic control is removed and the potential of the mercury electrode is monitored as a function of time. The initial and boundary conditions are given in eq 9-12, where JHgz+ represents the flux of the stripping agent, mercury(I1) ion.
t = 0, 0 < x
12/3DR. Equation 16 then takes the simple form
(9)
The initial condition (eq 9) states that CR is a parabolic function of x. Unlike in VSA, in PSA the stripping process begins immediately after the deposition step; therefore, a parabolic gradient is a realistic one. The boundary condition in eq 10 is valid only for a stirred solution; JHgztduring stripping in a stationary solution is a time-dependent function related by the Cottrell equation JHgz+(stationary) = D1/2Hg2+CoHgi+/ (?rt)l/', making eq 8 more difficult to solve. Solving eq 8 for CR(x = 1,t) by conventional Laplace transformation, eq 13 is obtained.
This is the theoretical basis for quantitative applications of PSA. It shows that under the assumed conditions, all reduced metal is removed from the thin mercury film at the transition time, so that successive stripping transitions should be free of any cumulative effect of preceding oxidations, a distinct advantage over the bulk mercury electrode. Transition times directly proportional to concentration of analyte have also been observed in chronopotentiometry in a thin-layer cell (9), chronopotentiometry with i t1I2 (10) and chronopotentiometric stripping analysis at a thin-film mercury electrode (11). Calculation of CMn+(y,t). To find the concentration of Mn+in the diffusion layer of the solution phase during the stripping process, the diffusion equation aCM"+(Y,t)
at
d2cMWk( y , t )
= DM"+
dY2
(18)
must be solved with the following initial and boundary con-
t > 0, y = 6:
CMM"+(6,t)=
The solution of eq 18 at y = 0 is
coMul"+
ANALYTICAL CHEMISTRY, VOL. 57, NO. 3,MARCH 1985 583 A
When 6/(Dt)liz >> 3, the term in the summation part of eq 22 becomes less than 0.0001, meaning that it can be neglected cm2 s-l). With cm and D = when t > 10 ms (for 6 = this approximation, eq 23 is obtained.
Y \
If it is assumed that a steady state concentration of M"" is reached after a short time, then CM"t(0,t)can be derived from Fick's first law.
-'.O
t - E t ( m V 4/s S C E )
For an electrode whose radius is greater than the diffusion layer thickness, 6, eq 23 describes CM"t(O,t),while for a planar microelectrode (radius -5 pm), or at a high stirring rate, a constant concentration gradient of Mn+ will be established within a few milliseconds after stripping begins; eq 24 then describes CM"+(O,t) adequately. This time-independent concentration gradient is a unique property of microelectrodes for which simpler mathematical expressions therefore can be used. The applications of microelectrodes in PSA will be described elsewhere (12). Shape of the Transient Potential-Time Curve. The Nernst equation during the stripping process can be written in the form
Figure 2. Plot of eq 2 6 for deaerated aqueous solution containing 2 X M Cd(C104)2 5 X lo-, M HgCI, 1 X lo-' M NaCIO,,
+
+
adjusted to pH 2.1 with HCIO,; E, = -1.0 V vs. SCE. Lines a through e apply to deposition times of 60, 120, 240, 480 and 960 s, respectively, and are offset by 20 mV for clarity. Table 11. Potentiometric Stripping Analysis of Cadmium: Theoretical vs. Experimental Correlationd run no
td, s
1 2
60 120 240
3 4
480
5
960
T,
s
1.41 3.20 6.60 13.50 26.80
-E,12a (mV vs. SCE)
-E,jzb (mV vs. SCE)
slope,CmV
637.5 641.5 646.0 652.0 660.0
641.0 644.0 648.0 654.0 660.0
26.0 28.0 26.0 26.0 28.5
From the experimental response at t = 7/2. *From the plot of Slope of the plot of footnote b. dConditions: 2 X M Cd(C104)2+ 5 X lo-, M HgC12 .C 1 X lo-' M NaClO,, pH 2.1 by adding HClO,, deaerated; Ed = -1.0 V a
Substituting eq 14 and 23 into eq 25, and assuming that Ed > 12/3DR and D M n t = DHg2+,eq 26 is obtained
log [(l - 9)/@/2]vs. -E, at 6 = 1/2. VR.
posited amalgam is carried out by a constant oxidant flux or by a constant current in a stirred solution, the shape of the transient potential-time curve will be the same.
where q5 = t/T. At t = 712, eq 27 applies
For a planar microelectrode (graphite or noble metal fiber), the transient potential E, can be obtained from eq 14,24, and 25, assuming that C"Hgz+>> coM"+.
E , = Eo - RT In nF
($)- nF In
SCE.
(1 - 4)
(28)
Therefore,for a microelectrode,the initial drop in the transient potential will occur faster than that for a macroelectrode. The analogous expression for the transient potential in chronopotentiometric stripping at a thin mercury film is given by eq 29
which differs from eq 26 only in the second term (the potentiometric part) and which is the same in the time-dependent part. This shows that whether oxidation of the de-
EXPERIMENTAL SECTION Experiments were carried out as described before (3),except that data were collected with an Aminco DASAR system,using sampling times of 20,50, or 100 ms, depending on the transition time, and using a time constant of 50 ms. Potentials were measured against the Ag' (acetonitrile)/Ag reference electrode (3) when dimethyl sulfoxide was the solvent, and against the saturated calomel electrode (SCE) when water was the solvent; the potential of the Ag+/Ag electrode is +0.30 V vs. SCE. RESULTS AND DISCUSSION Transient Potential-Time Relationship. Figure 2 provides experimental proof of eq 26 for five different deposition times for cadmium in water. In Table 11, further proof is provided by comparing E,lz values derived from such plots to values read directly from the original E , vs. t responses; agreement is reasonable. The data also show that the transition time is less sensitive to the nature of the mercury film than transient potentials are. Figure 3 provides a test of eq 27. The slope of the plot is 30.5 mV, in reasonable agreement with the theoretical value of 29.6 mV. From the intercept at T = 1, E" can be calculated from eq 27 if D and 1 are known. Equation 26 also predicts a dependence of transient potentials on the dimensionless parameter Dr/12;this dependence is illustrated in Figure 4.
584
ANALYTICAL CHEMISTRY, VOL. 57,NO. 3, MARCH 1985
1
0.60
0.70
L
0.00
- E d ( V e SCE)
1.0' 630
1
,'
I
1
1
I
1
650
640
(mv
U
2.00
-Ed(Vfi
3.00
Aqt/Aq)
-I
I
660 070
SCE)
v_l
L
2.60
074
0 7 0 260
- E d ( V c SCE)
Flgure 3. Plot of eq 27 for the same system as in Figure 2 and Table 11. Error bars represent estimated experimental uncertainty of f2 mV.
-look - 80
I
I
I
2.60
2 76
2 04
-Ed('/%
Aq+/Aq)
Figure 5. Stripping polarograms generated by potentiometric stripping analysis: (a) Cd in water, with 1 X M Cd(N03), + 1 X M HgCI, 1 X 10-1M KCI, and t, = 120 s;(b) Na in dimethyl sulfoxide, with 4 X M HgCI, 4- 1 X lo-' M Et,NCIO,, M NaCl 4- 1 X and t , = 60 s; (c)plot of eq 30 for data of curve a; (d) plot of eq 30 for data of curve b.
+
Surface. A potential complication in PSA is that the concentration of the stripped metal ion a t the electrode surface may become sufficiently high to induce precipitation of compounds of the metal ion on the electrode surface, thereby interfering with the stripping process. This is a complication, not only in PSA but in all stripping techniques, and it has been treated in detail by Buffle (13). The maximum value of CM"+in eq 23 is given when t r , so that
-
(CMn+)max
I
I
0.2
0.4
1
I
0.8
0.6
1.0
'+
Figure 4. Theoretlcal dependence of transient potential-time curves on values of Dr//' (eq 26). Curves 1 through 5 apply to values of 2380,595, 264, 148,and 95,respectively.
Generation of Stripping Polarograms. Equation 16 can be rewritten for the special case of a reversible deposition reaction a t a thin-film electrode in the following form:
RT RT Ed = Eo + - In (81/Dtd) + - In nF nF
- T')/T')
((7
(30)
In Figure 5, curves a and b are plots of the normalized response, ./', vs. the deposition potential for cadmium in water and sodium in dimethyl sulfoxide. Such plots are "stripping polarograms". It should be possible to use such plots to study the speciation of metal ions at concentrations too low to permit the direct use of voltammetry, in the same way as analogous plots based on voltammetric stripping have been used (8). For such purposes, PSA has the advantage over VSA that the stripping polarograms can be generated more quickly. Plots c and d (reciprocal slopes = 29 f 2 and 62 f 2 mV) show that eq 30 applies. It is noteworthy that no significant corrosion of sodium amalgam occurred in dimethyl sulfoxide during the duration of the experiments. Maximum Surface Concentration of the Metal Ion and Formation of Interfering Precipitates on the Electrode
= (4/n)CoH g 2 t ( D r / 6%) '1'
(31)
By use of the same numbers as Buffle (td = 300 s, CoM2+= 1X M, CoHg2+= 5 X lo4 M, D = 1 X cm2 s-l, and 6=3X cm), eq 17 and 31 predict a ( C M 2 t ) m a value of 4.6 X low6M, which is 9 times larger than the value calculated by Buffle. The discrepancy is caused by Buffle's assumption of a time-independent concentration gradient of M2+during the stripping process, which is not strictly valid for stripping at a macroelectrode. From eq 31, the maximum concentration of an anion, Am-, that may be present without causing precipitation of M,A, can be calculated if the solubility product of MmA, is known. Equations 17 and 31 also show that such precipitation problems can be minimized by using lower concentrations of the stripping agent and by decreasing the diffusion layer thickness by increasing the stirring rate (or by using a microelectrode). Examples of complications caused by precipitation on the electrode surface in the stripping analysis of natural water samples have been discussed by Buffle. Another example is the precipitation of (highly insoluble) alkali and alkaline-earth metal hydroxides in aprotic solvents caused by reaction of residual water with the highly reactive amalgams (3). Other Considerations. Operationally, potentiometric stripping analysis differs from chronopotentiometric stripping analysis (CPSA) only in that stripping is carried out with a constant stripping agent flux in PSA but with a constant current in CPSA. Like CPSA (and VSA), PSA is not free from perturbations caused by double layer charging (14),although this problem is less important at microelectrodes. However,
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
