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Anal. Chem. lS87, 59,2456-2460
(9) Chloupek, J. 8.; Danes, V. 2.; Danwsovq, B. A. Collect. Czech. Chem. Commun. 1933,5 , 469,527.
(18) Howson, M. R.; House, W. A.; Pethybridge, A. D. Analyst (London)
(IO) Ferguson, A. L.; Van Lente. R.; Hitchens, K. J. Am. Chem. SOC. 1932,54, 1285.
(19) Culberson,'C-In Marine Nectrochemlstry;WhRfleld, M., Jagner, D., Eds.; Wiley: New York, 1981. (20) Whitfield, M.; Butler, R. A.; Covington, A. K. Oceanol. Acta 1985, 8 ,
1986. 111. 1215.
(11) Covlngton, A. K.; Whalley, P. D.; Davison, W. Anal. Chim. Acta 1985, 169, 221. (12) Covlngton, A. K.: Whalley, P. D.; Davison, W. Analyst (London) 1983, 108. i528. (13) Covington, A. K.; Whalley, P. D.; Davlson, W. Pure Appl. Chem. 1985,
423. (21) Bates, R. G. CRC Crit. Rev. Anal. Chem. 1981, IO, 247. (22)Dohmer, R. E.; Wegmann, D.; Morf, W. E.; Simon, W. Anal Chem. 1988, 58,2585.
(14) Davison. W.; Harbinson, T. R. Anal. Chim. Acta 1988, 187, 5 5 . (15) Ersson, E. K. Vatten W8$, 4 7 , 266. (16) Van den Berg, L. Anal. Chem. 1960,32, 828. (17) Jones, C.; Marsicano, F.;Williams, D. R. Sci. TotalEnviron.1987, 64, 211.
RECEIVED for review February 6, 1987. Accepted June 15, 1987. Financed by the Surface Water Acidification Programme of the Royal Society.
57 - . , a77 -.. .
Investigation of pH-Dependent Complex Equilibria at Low Ligand to Metal Ratio by Nonlinear Least-Squares Fit to Linear-Sweep or Cyclic Voltammetric Data Harald Gampp Institut de Chimie, UniuersitP de Neuchdtel, CH-2000 Neuchdtel, Switzerland
Metal-llgand equlllbrla can be studled voltammetrlcally provided the chemlcal equlllbrla are rapld and one of the equllibrlum specles can be reversibly reduced/oxldlzed. I n the case of strong complexatlon, the use of voltammetry Is no longer limlted by the need to work wlth an excess of Ilgand. StaMllty constants are straightforwardly dedennlned from data obtalned at low ligand to metal ratlo by uslng least-squares and slmulatlon methods. SpecHlcally, the common case of pHaependent e q u M a Is ctkumed, where the experhnmtal data are obtained from voltammetrlc/pH potentlometrlc tltratlons. Synthetlc data show the effect of stablllty constants on peak potentials and currents. As a chemlcal example, the complexation of copper( I I ) by oxalate is studied.
Voltammetric techniques are frequently used in order to study metal-ligand equilibria in solution. For the common case of a reversible electrode process and of rapid chemical equilibria, the only effect of complexation is a shift of the voltammetric wave provided the ligand is present in excess over the metal. The basis of the respective methods for determining equilibrium constants is that under these conditions the observed shift of the current-potential curve is a simple function of the concentration of the ligand in the bulk solution and of the stability constants (ref 1 and further references therein). Early applications were restricted to single equilibria (2) and later it was shown that extension to multiple equilibria involving one (3)or several ligands (4) is equally possible. Data evaluation using either graphical (3)or least-squares methods (5-7) is straightforward and a detailed error analysis has been given (8). However, it will not always be possible to study an equilibrium system in the presence of excess ligand, e.g. because the ligand may not be sufficiently soluble. A situation where it is not desirable to use the ligand in excess occurs in the case of stepwise formation of very stable complexes ML; (i = 1, 2, ...,p): For ligand to metal ratios greater than p the species ML, will be formed almost exclusively and only the pth equilibrium could be studied accordingly. Current-potential curves obtained under conditions of nearly stoichiometric or
substoichiometric ligand to metal ratio may have shapes which are different from those expected for reversible systems (1) as has been shown for polarography (9, 10) and cyclic voltammetry (CV) (11, 12). Recently, Saveant and Xu have presented a theoretical study of the effect of ligand concentration and of the stability constant on linear-sweep voltammograms for the case of a metal ion which is reversibly reduced and which forms a 1:1 complex with a ligand (13). They also showed working curves which allow one to determine the value of the stability constant from individual voltammetric curves. For the case of a metal ion forming both 1:1and 1:2 complexes Philp and co-workers reported simulated cyclic voltammograms (11,12,14) and differential pulse polarograms (15) and discussed the influence of stability constants and ligand to metal ratio f on both shapes and positions of the current-potential curves. Using a trial-and-error procedure, they were able to estimate stability constants by fitting calculated to experimental cyclic voltammograms (11,12). Thus, in the case of multiple equilibria the stability constants do affect the voltammetric curves. The magnitude of this effect depends not only on the values of the stability constants but also on the ligand to metal ratio f. Accordingly, f should be varied in a voltammetric equilibrium study. However, iff is small there is no simple relation between the current-potential curves and the stability constants and the bulk concentrations of metal and ligand (8). It is the purpose of this paper to show that equilibrium constants can be straightforwardly determined from voltammograms obtained a t low ligand to metal ratio f by using a nonlinear least-squares method in combination with a simulation program. Specifically, the common case of a pH-dependent equilibrium system will be discussed where an obvious way to change f consists in varying the pH of a given solution of metal and ligand. As a chemical example, the complexation of copper(I1) by oxalate is studied by combined cyclic voltammetric-pH titration.
EXPERIMENTAL SECTION Cyclic voltammograms were obtained with a potentiostat (MetrohmE506), a signal generator (Metrohm E612),and an x-y recorder (Hewlett-Packard7040A). A standard three-electrode
0003-2700/87/0359-2456$01.50/0 0 1987 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 59, NO. 20, OCTOBER 15, 1987
arrangement was used with a Pt-coil counter electrode, a saturated calomel (SCE) reference electrode, and a hanging mercury drop working electrode (HMDE, Metrohm). pH was measured with a combined glass electrode and a pH meter (Metrohm 632); buffers of pH 4.00 and 7.00 (Metrohm) were used for calibration. A pH-potentiometric titration of oxalic acid (1.48 mM) with NaOH (0.4 M) was carried out and the pK of the ligand was determined by using the program TITFIT (16). For the CV-pH titrations two different solutions were prepared (a) [Cu2+]= 0.182 mM, [Ox2-] = 0.191 mM; (b) [Cu2+]= 0.241 mM, [Ox2-] = 0.124 mM. In each experiment HN03 (0.02 and 0.2 M) was stepwise added to 25 mL of the respective solution in order to vary the pH between 8 and 1. After each addition of acid both the cyclic voltammogram and the pH were measured. Experimentswere carried out in a titration vessel thermostated at 298 K under Nz in aqueous solution of constant ionic strength (I = 0.50 M, KN03). Voltammograms were recorded at 0.1 V s-* scan rate with an electrode surface of 0.014 cm2. Analytical grade chemicals and doubly distilled water were used throughout.
DATA REDUCTION Simulation of Cyclic Voltammograms. Calculations were made by using the explicit method of Feldberg (I 7) as outlined in ref 1. For a given set of parameters which define the electrochemicalexperiment (initialand switching potential, potential step size) and the chemical system (stability constants, total concentrations of metal and ligand, pH), the program calculates the (adimensional)current J/ for each value of the (adimensional) potential E. The following assumptions are made: (1) The free metal ion is reversibly reduced at a planar microelectrode; other electroactive species do not occur. (2) The chemical processes are sufficiently fast that equilibrium is always established. (3) All diffusion coefficients are equal. (4) The pH of the solution does not change during the potential scan, not even near the electrode surface. (5) With the exception of the proton, activitites are replaced by concentrations. A similar simulation program has been described by Philp and co-workers (11) and more details can be found in their article. Determination of Stability Constants from CV-pH Titrations. A nonlinear least-squares program was written which requires as input the experimental peak potentials, pH values, and total concentrations of metal and ligand and the conditions of the electrochemical experiment. The peak potentials are converted into adimensional potentials tp (I,13). The simulation program described above is called as a subroutine which for a given set of estimated stability constants returns the corresponding calculated peak potentials tP,& The stability constants are refined by applying the Newton-Gauss method using numerical derivatives (18). Calculations were done on a Hewlett-Packard HP9835 desk computer with 128K memory, equipped with a plotter HP 7225A. RESULTS AND DISCUSSION Theoretical Considerations. The effect of stability constants on cyclic or linear-sweep voltammograms obtained at different pH values for a given mixture of metal and ligand will first be discussed for the following system:
Mn+ + L H+ + L
*
Mn+ + ne-
MLn+ HL+
* Mo
(1) (2)
(3) It can be described by three thermodynamic parameters, pm (=[MLn+]/ ( [Mn+][L])), KHHL(= [L] [H+]/ [HL+]), and E O . Figure 1 shows the dependence of the (cathodic) peak po-
2457
I
/----:
lot
G
1
7
5
3
7
PH
Flgure 1. pH dependence of cathodic peak currents (upper part) and potentlals (lower part) for the system described by eq 1-3 with pKH, = 4 and log PMLvaried from 3 to 9 (cM0= c t = 0.2 mM).
tentials and peak currents for an equimolar mixture of metal and ligand. In the case of weak complexation the peak characteristics at low pH are as expected for a reversible system, i.e. qP= 1.1and tp = 0.446 (I). Clearly, under these conditions most of the ligand is protonated and complexation does not occur. Increasing pH results in a decrease of fiP and in an increase of For pH 25 both peak potential and peak current remain almost constant which is due to the fact that at high pH protonation of the ligand becomes negligible and that the concentration of complex ML is practically independent of PH. The decrease of J/p clearly indicates that the shape of the corresponding voltammograms is different from that usually observed for a reversible system ( I , 13). The minimum value of qPis 0.384 in perfect accordance with the theoretical value (13). As can be seen from Figure 1 the effect of pH on the peak current is relatively small, in particular when very stable complexes are formed. Thus, the determination of stability constants based on measurements of the peak current only does not seem to be a good approach. The effect of pH on the peak potentials is relatively large and, moreover, strongly depends on &L (see Figure 1). The limiting potentials at high pH, E p ~ i m ,are as predicted by the theory (13) Sp,lim
1.16 + In
(PMLCM')
(4)
It should be noted that the shapes of the curves shown in Figure 1do not depend on KHmand that a variation of this parameter shifts all the curves by the same amount parallel to the pH axis. Thus, from measurements of peak potentials as a function of pH it will be possible to determine as well as K"L. Clearly, in the case of this simple equilibrium system the stability constants can be obtained without using a leastsquares procedure, &L can be calculated from tpJim(eq 4), and KHm can be determined by using working curves as given in Figure 1. For more complicated equilibrium systems the Ep-pH curves depend on the total concentrations of metal and ligand and on the set of ligand and complex stability constants. Ac-
om
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ANALYTICAL CHEMISTRY, VOL. 59, NO. 20, OCTOBER 15, 1987
61
1
5
3
7
PH
Figure 2. pH dependence of the cathodic peak potentlals for the = 4 (ope! system described by eq 1-3 and 5 with pKHH = 4, log symbols) and 5 (full symbols): log,8 = 8 (A), 9 (O),and 10 (0)(c, = c: = 0.2 mM).
cordingly, the use of working curves for determining the thermodynamic parameters will be rather inconvenient and a least-squares fitting procedure is to be preferred. Nevertheless, the calculated curves show some general trends which will be useful in designing an actual CV-pH titration experiment. For a system described by eq 1-3 and 5 some calculated t,-pH curves are given in Figure 2 ( P ~ L= [MLz"+]/ ([M"+][LI2)). Due to the additional formation of a 1:2 comMn+ 2L e ML,"+ (5)
+
plex the limiting peak potentials increase. .$, depends a both on Bm and on pmL; the effect of j3m, however, becomes less important as increases. Whereas ,9mL mainly influences Spa, in addition has a marked effect on the curve shape. Increaslng values of pm result in a shift of the inflection point of the [,-pH curves to lower pH values. Again, the effect of ,Bm becomes smaller with increasing &ILL. However, the influence of @ML can be enhanced by decreasing the ligand to metal ratio and/or the total concentration of metal, i.e. under conditions that disfavor the formation of the 1:2 complex. This leads to a point of practical interest, namely, the optimum choice of the ligand to metal ratio f. Iff is too large, only omLcan be determined and, in addition, precipitation of the ligand or of a complex may occur. On the other hand, iff is too small, only pm can be determined (in the extreme case, f N 0, even this is no longer possible). Moreover, decreasing f leaves increasing amounts of uncomplexed metal which might create additional complications due to the formation of hydroxo complexes (vide infra). Fortunately, in our case of pH-dependent equilibria the choice of a reasonable f is not very critical: By addition of base to a given mixture of metal and protonated ligand the actual value off changes from 0 (pH C C pKHHL)to c L 0 / c ~ O(PH >> PK~HL). Curves calculated with the parameters used in Figure 2 for 0.5 C cLo/cMoC 4 show that the influence of pm has its maximum around pH 3, whereas pMLL has its main effect a t pH >5. (Clearly, these pH regions are determined by the pK of the ligand.) For cLo N cyo a variation of p m or P ~ produces L effects of similar magnitude in the respective pH regions. Thus, a reasonable experimental strategy will be to start from a stoichiometric mixture of metal and ligand and to measure voltammograms between pH 2 and 7. Complexation of Copper(I1) by Oxalate. The Cu(I1)oxalate system has been studied by many workers both by potentiometric (19) and by voltammetric methods (4, 11,20). Although this is a relatively simple equilibrium system where only the complexes Cu(0x) and C U ( O X ) ~are ~ -formed, considerably differing stability constants have been reported. For the same conditions (I= 0.1 M, 298 K) values range from 4.8 to 6.7 for log pMLand from 9.2 to 10.5 for log &LL (19,21).
E vs. SCE ( m V )
Flgure 3. Effect of pH on cyclic voltammograms for the copper-
(11)-oxalate system. ccuo= 0.241 mM, cmo= 0.124 mM, pH varied from 4.85 (a) to 1.43 (b) (I = 0.5 M, KNO,; 298 K).
i 3.
11
7
5
3
PH Flgure 4. pH dependence of the cathodic peak potentials for the copper(I1)-oxalate system: open (f = c o ~ / c =c 0.51) ~ and full circles (f = 1.05) of radius equal to the standard error of fit; calculated curves, -.
In view of these large discrepancies between the published stability constants, a reinvestigation of the Cu(I1)-oxalate system by an alternative method appeared to be called for. CV-pH titrations were made for ligand to metal ratios, f , of 1.05 and 0.51 (see Experimental Section for details). The total concentrations of metal and ligand wdre chosen sufficiently small in order to prevent precipitation of cupric oxalate (solubility 0.158 mM at 298 K (22)). Figure 3 shows the dependence of the cyclic voltammograms on pH for f = 0.51. Decreasing the pH leads to an increase of cathodic and anodic peak currents and to a shift of the cathodic peak toward more positive potentials. The voltammograms at high pH are rather broad and look like superpositions of two waves (curve a, Figure 3). With decreasing pH the voltammograms become more narrow, and at pH
