J. Chem. SOC., Faraday Trans. 1, 1987,83, 1719-1723 Determination of Stability Constants from Linear-scan or Cyclic-voltammetric Data using a Non-linear Least-squares Method Harald Gampp Institut de Chirnie de r Universitk, Avenue de Bellevaux 51, CH-2000 Neuchcitel, Switzerland Stability constants of binary complexes can be determined using linear-scan or cyclic-voltammetric data. In contrast to a recently applied graphical method, numerical treatment based on a non-linear least-squares refinement allows one to make optimal use of the information contained in the data set. This is illustrated by re-evaluating recent literature data. In a recent paper by Killal it was shown that linear-scan or cyclic-voltammetric data can be used to determine stability constants. This was demonstrated for metal ions M forming a series of binary complexes ML, ( i = 1, .., p ) with a given ligand L. The stability constants were obtained using a procedure based on the classical approach by DeFord and H ~ m e : ~ ? ~ for each type of complex ML,, the measured data are transformed into a polynomial of degree p - i in ligand concentration cL and plotted against cL. The respective stability constant is obtained by linear extrapolation to zero concentration. Clearly, this method was, despite its inherent disadvantage^,^^^ optimal at a time where no computing facilities were available. These disadvantages are as follows. (i) The data are transformed non-linearly and thus obtain different weights, which is neglected in the graphical evaluation. (ii) For complexes ML, with i < p - 1, the plots are not linear and, accordingly, the extrapolation to zero concentration cannot be made unambiguously.(iii) No unbiased estimates of the confidence limits of the calculated parameters are obtained. (iv) Errors in the stability constants are accumulated since in the determination of K,, the previous constants K,, ..., are used. (v) The data evaluation is rather time-consuming. Some attempts have been made to overcome these drawbacks, either by using linear regression analysis4v5 or by solving sets of simultaneous equations.6 However, these methods are not generally applicable (vide infra), and accordingly have never become popular among chemist^.^ It is the purpose of this note to demonstrate that a non-linear least-squares procedure allows a straightforward data analysis that does not suffer from any of the abovemen- tioned drawbacks.The re-evaluation of some recent literature data1 will show that in contrast to the graphical method, the full information contained in the data set is obtained by the rigorous evaluation. Theoretical Background A complex ML, is generally reduced at a more negative potential than the solvated metal ion. If both the complexation equilibria and the electron transfer at the electrode are fast, a cyclic voltammogram of an equilibrium mixture containing species ML, ( i = 0, I , . . . , p ) shows a single wave with a shape typical for a reversible (or Nernstian) system.8 Hence, the reduction occurs at the level of the free metal ion (i.e. the species which is the easiest to reduce) and the only effect of complexation is a shift of the wave.17191720 Stability Constants from Voltammetry In the following we consider a reversible process at a mercury working electrode, M n + + n e e M o , where the reduced metal is not complexed by L. Assuming constant activity coefficients, the Nernst equation can be rearranged into eqn (l).398 Thus the measured shift of the half-wave potential, AE,, can be described as a function of the concentration of free ligand, [L], and of the stability constants& (& = [MLi]/[M][LIi); all charges omitted, i.e., M stands for Mn+): Eqn (1) can be used as a fitting function and the parameters pi can be determined by any non-linear least-squares method. In the present case a simple Newton-Gauss procedureg-10 was chosen.In the general case, where the total concentration of ligand (c,) is not in large excess over that of metal (cM), the program uses the Newton-Raphson algorithmll in order to calculate the concentration of free ligand ([L]). If cL 9 cM, [L] in eqn (1) can be replaced by cL and eqn (1) can be transformed into a polynomial form : exp AEL- -1 = &(c& ( 1;) Accordingly, the unknown parameters pi can be calculated non-iteratively by linear regression anal~sis.~-~ Obviously, the experimental data are non-linearly transformed and appropriate weighting factors have to be introduced in order to obtain meaningful results. Since the optimal selection of weights is not always easy to make and since eqn (2) is not generally applicable, the non-linear fit using eqn (1) is to be preferred.Results and Discussion Three of the data sets measured by Killal have been evaluated by using a non-linear least-squares pr0gram~3~~ based on the Newton-Gauss method. In each case two different models (i.e., with and without pl) were fitted. The results of the numerical analysis together with the respective literature values are compiled in table 1. For the Cd2+-oxalate system the excellent fit of the experimental data by the calculated curve can be seen in fig. 1. The standard error of fit increases from 1.5 to 2.6 mV when is omitted from the model. Subjecting the corresponding variances to the F-testg shows that including p1 leads to a statistically significant improvement of the fit at a 95 % level (at a 99% level /I1 has to be rejected, however). Quite in contrast, is not necessary in order to describe the experimental data obtained from the other two equilibrium systems.Although the stability constant calculated for the 1 : 1 complex between Cd2+ and 1,3-diaminopropane seems to agree with the reported values (table l), the F-test clearly shows that it is a coincidence, thus including p1 does not improve the quality of fit. The same is true for the Cu2+-oxalate system, where the experimental data are well explained by considering only p2, as illustrated in fig. 2. In this case not even statistical criteria are necessary since the program finds that the best fit is obtained with a physically meaningless, negative value of B1. The fact that in the last two systems D1 could not be determined is by no means due to a poor mathematical procedure.Using the reported stability constants,' one easily calculates that at the lowest ligand concentration (0.01 mol dm-3) in the Cu2+-oxalate system ML, is already formed to > 93%, ML to < 7% (its concentration decreases rapidly with increasing ligand concentration), and that uncomplexed metal is never present in significant amounts. Clearly, under these circumstances p1 cannot be determ- ined. The Cd2+-1 ,3-diaminopropane system has been studied under similar conditions,l where M is not present to > 0.02% and where ML is formed to > 20% only at the two lowest ligand concentrations. Obviously, in these equilibrium systems reliable values forH. Gampp 1721 Table 1.Stability constants and standard errors obtained by nonlinear least-squares fit to cyclic vol tamme tric da taa system no. of parameters SE/mVb log & log P 2 Cd2+-oxalate C d e Cd2+-1 ,3-di- C aminopropane c d f Cu2+-oxalate C d e 3 1.5 2.42 & 0.10 2 2.6 - 2.69 2.73 f 0.03 3 1 .o 4.95 f 0.48 2 1.1 - 5.477 4.5k0.2 2 1.2 -Q 1 1.6 - 6.00 5.53+ 1 3.86 k 0.14 4.24 k0.06 4.04 4.1 f O . 1 7.58 f 0.04 7.63 & 0.02 7.59 7.2f0.5 9.15 k0.02 9.1 1 k0.02 9.13 9.54f0.5 5.1 0 f 0.04 4.95 0.07 5.16 5.1 8.30 & 0.05 8.25 f 0.05 8.3 1 8.0 - - - - a Data from ref. (1) ( I = 1.0 mol dm-3, 25 "C). * Overall standard error of fit. This work (non-linear least-squares fit). From ref. (1) (graphical method). A. E. Martell and R. M. Smith, Critical Stability Constants (Plenum Press, New York, 1982), vol.5. f A. E. Martell and R. M. Smith, Critical Stability Constants (Plenum Press, New York, 1975), vol. 2; constants refer to 1 = 0.1 mol dm-3 and 25 "C. Best fit obtained with a negative value for PI. 1oc > E 2 6C 1 cl 20 0 0.1 0.2 0.3 0.4 [oxalate]/mol dm-3 Fig. 1. Dependence of the shift of the half-wave potential AEL on the concentration of ligand in the Cd2+-oxalate system. [Experimental values from ref. (1) are represented as open squares of height 3 times the standard error of fit; calculated values obtained from a non-linear least-squares fit of three parameters are represented as a solid line.]1722 200 > E U --- I;i' 150 100 Stability Constants from Voltammetry 0.0 0.1 0.2 [ oxalate]/mol dmd3 0.3 Fig. 2. Dependence of the shift of the half-wave potential AEL on the concentration of ligand in the Cu2+-oxalate system.(Symbols are as in fig. 1; only a single parameter, &, was fitted.) p1 can only be obtained under experimental conditions where the ligand is not in large excess over the metal. Since the concentration of ligand then no longer equals its total concentration but depends on the unknown stability constants [eqn (2)], and thus the graphical method cannot be used (uide supra). Nevertheless, one might argue that the good agreement between the parameters obtained by the graphical method and by the least-squares procedure (see table 1) at least in the present case does not really disfavour the former method. Therefore, the graphical evaluation was repeated for the two oxalate equilibriq.Inspection of the plots for the Cd2+ system clearly reveals that an unbiased extrapolation does not lead to log p1 > 2.4 or to log p2 > 3.5 [cf. fig. 1 in ref. (I)]. The respective plots are such that zero or even negative p values could equally well be deduced, hence neither B1 nor Bz can be determined by this method. Only for p3 is a reliable value obtained (5.2 < log p3 < 5.3). Similarly, in the Cu2+-oxalate system the graphical method predicts a physically meaningless, negative value for pl, whereas p2 can be determined (9.0 < log p2 < 9.2). Thus, in the abovementioned equilibrium systems the graphical evaluation is strictly limited to the determination of the maximum number of coordinated ligands, p , and the respective formation constant pp.By using a non-linear least-squares method the data evaluation is considerably improved and the full information contained in the experimental data is readily obtained. Especially helpful is the fact that unbiased statistical criteria can be applied in order to decide whether a certain complex and the respective stability constant are significant or not. This is of decisive importance where unknown equilibrium systems are to be studied. To conclude, cyclic voltammetry is indeed well suited for the determination of stability constants. A least-squares procedure which is easily implemented even on inexpensive microcomputers allows one to perform a safe, complete and straightforward data analysis in a short time. Clearly, other electrochemical techniques like linear-scan voltammetry or polarography which equally allow one to determine El values can be used as well.Accordingly, these electrochemical measurements should be considered as an alternative to the commonly used pH-potentiometric or spectrophotometricH. Gampp 1723 titrations since they can be used for studying equilibria in strongly acidic or basic solution (where pH-potentiometry cannot be applied12) or for systems which do not show well developed spectral characteristics (e.g. complexes of the d 1 O metals). This work was supported by the Swiss National Science Foundation. References 1 H. M. Killa, J. Chem. Soc., Faraday Trans. I , 1985, 81, 2659. 2 D. D. DeFord and D. N. Hume, J. Am. Chem. Soc., 1951,73, 5321. 3 H. L. Schlafer, Komplexbildung in Losung (Springer, Berlin, 1961), p. 205. 4 P. Kivalo and J. Rastas, Suomen Kemi., 1957, B 30, 128; J. Rastas and P. Kivalo, Suomen Kemi., 1957, 5 L. N. Klatt and R. L. Rouseff, Anal. Chem., 1970,42, 1234. 6 J. G. Frost, M. B. Lawson and W. G. McPherson, Inorg. Chem., 1976, 15, 940. 7 F. R. Hartley, C. Burgess and R. Alcock, Solution Equilibria (Ellis Horwood, Chichester, 1980), 8 A. J. Bard and L. R. Faulkner, Electrochemical Methods (Wiley, New York, 1980). 9 P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, B30, 143. pp. 165. 1 969). 10 H. Gampp, M. Maeder and A. D. Zuberbuhler, Talanta, 1980, 27, 1037. 11 H. Margenau and G. M. Murphy, The Mathematics of Physics and Chemistry (Van Nostrand, 12 H. Gampp, D. Haspra, M. Maeder and A. D. Zuberbuhler, Znorg. Chem., 1984, 23, 3724. Princeton, 1968), pp. 492. Paper 6/1408; Received 15th July, 1986