Voltammetric titrations involving metal complexes: Effect of kinetics and diffusion coefficients

The Science of the Total Enuironment, 60 (1987) 45-55 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

45

VOLTAMMETRIC TITRATIONS INVOLVING METAL COMPLEXES: EFFECT OF KINETICS AND DIFFUSION COEFFICIENTS*

HERMAN

P. VAN LEEUWEN

Laboratory for Physical and Colloid Chemistry, Agricultural Wageningen (The Netherlands) (Received

University, De Dreijen 6, 6703 BC

February 4th, 1986; accepted March 17th, 1986)

ABSTRACT This paper presents a theoretical consideration of the voltammetric titration of ligand with metal, under the condition that the diffusion coefficients of the complex and the free metal are unequal. The subject is closely connected with the (homogeneous) kinetic properties of the metal complex system. It is explained how, for the cases of labile, quasi-labile and non-labile complexes, an experimental titration curve must be interpreted. The known extrapolation procedures for finding an equivalence point can only be used in the extreme cases of non-labile and strong, labile complexes. In many situations the interpretation is more involved, if not impossible. It is strongly recommended that every voltammetric complex titration be accompanied by a proper characterization of the kinetic features (and, if necessary, the diffusion coefficient) of the complex. INTRODUCTION

Several advanced modes of voltammetry are suitable for heavy metal speciation in natural waters, as has been shown by several authors, notably Niirnberg et al. [l-4]. To a large extent it is Nurnberg’s strong promotional activities that have led to the present situation where voltammetric titration procedures are among the most popular in environmental metal speciation studies. In particular, stripping voltammetric techniques such as DPASV, which are extremely sensitive and technically simple, have been shown to be applicable to all kinds of samples [5]. However, voltammetric titration is unfortunately not always as simple a matter as one would infer from the light-hearted treatments in many publications in the environmental literature. For instance, in the case of metal/organic matter complexes, there are a number of complications which disturb the direct relationship between the measured voltammetric signal and the free metal (or free metal plus labile complexes) concentration in solution. In this typical example, a primary complicating factor is the difference between the mobilities, i.e. diffusion coefficients, of the free metal and the complex species. Electroanalysts have paid some (but not too much) attention to the case of simultaneously diffusing labile species with different mobilities [6, 71, and its *Dedicated

to the memory of Professor

Dr Hans Wolfgang

Niirnberg.

46

relevance to the analysis of natural systems has recently been indicated more than once [B, 91. In addition to the mobility effects, there is the possible problem of relatively low rates of the complex formation/dissociation reactions. The latter problem has been exposed by Ruiic [lo], who has shown how drastically a titration curve can be affected. In a recent paper, Cleven et al. [ll] have carefully considered the dynamics of the electrochemical reduction of a metal from a solution containing free and complexed metal, the two species having different diffusion coefficients. It was explained that a mean diffusion coefficient is operative under certain conditions involving the kinetics of the complexation/dissociation reactions. The problems of unequal mobilities and kinetics are thus linked and therefore should be treated simultaneously. This connection was one of the reasons for preparing the present paper, but there are others. In the course of a metal titration, the ratio between the concentrations of metal and ligand changes and this means that the electrochemical lability changes as well. Thus homogeneous kinetics may be important in one part of the curve but essentially immaterial in another. Likewise, a mean diffusion coefficient may be operational for only a part of a certain titration curve. This state of affairs determined the two-fold purpose of the present paper: (i) to relate the complex formation/dissociation kinetics to the operational use of a mean diffusion coefficient D; and (ii) to trace the changes of kinetics (and ensuing electrochemical labilities) and the validity of D in the course of a titration of ligand with metal. THEORY

AND DISCUSSION

In electroanalytical practice with metal complex systems, one may meet different conditions with respect to the metal-to-ligand ratio, titration sequence, etc. In the case of environmental samples, the most common practice is to titrate the ligand with metal, or to study the system at a few discrete metal-to-ligand ratios, realized by the standard addition of metal. We shall tacitly base our considerations on the case where the ligand L is titrated with the electroactive metal ion M, forming the electroinactive complex ML (charges are omitted for clarity) ka

M + L-_-ML kd

The rate constants /z, (complex formation) and kd (complex dissociation) are related to the stability constant K, via k,/kd = K. Although our treatment is valid for any combination of values of the diffusion coefficients L& and &,, it will be assumed that DMi, (z Q) < Q,, . In general, the voltammetric titration curve has a shape as outlined in Fig. 1. In comparison with the blank, we see a reduced slope in the initial part of the curve (region A), an increase in the slope in intermediate regions (B), and finally an approach to the slope of the blank at a sufficiently high excess of

metal

Fig. 1. Typical distinguished. for finding an extrapolation

added

shape of a voltammetric titration curve. Three different regimes (A, B and C) are The broken line is the reference titration (blank). Existing extrapolation procedures equivalence point include intersection with the axis (b) and intersection of double (a).

(region C). This general shape is by no means characteristic for a specific type of complex, e.g. a relatively weak complex. As we shall see, there can be a number of different chemical/electrochemical circumstances that apparently lead to the same type of titration curve. These different circumstances will be considered for the different stages A, B and C.

metal

Region A, large excess of ligand In the region of low metal-to-ligand ratios, the ligand concentration is approximately constant and it is allowed to adopt a quasi-monomolecular reaction with Kc,

=

K’ =

k;/kd =

cMJcM

(2)

where ki = kacL. A further distinction between the possible types of situation is made by considering the chemical kinetic properties of the system in relation to the dynamic nature of the electrochemical experiment. For a metal complex system, containing different metal species with unequal diffusion coefficients, the ensuing criteria for electrochemical lability have recently been formulated [Ill. On the basis of a comparison between the chemical reaction rate parameter kd(&/k~)1’2 and the dynamic diffusion flux parameter (DML/t)1’2, these criteria were formulated as

k, (DM/DML)1’2k~“‘tl”

< 1 non-labile

(3a)

z 1 quasi-labile

(3b)

s 1 labile

(3c)

48

The derivation of Eqn (3) is based on the flux equation for semi-infinite linear diffusional transport to the surface of an electrode. For other experimental conditions, e.g. in the case of a DPASV pre-electrolysis with stirring, we arrive at analogous (but approximative) expressions. These are essentially generalizations of the well-known Davison criteria [12]: (4

6

DgK’(k,

+ k’,)-“2

1 non-labile

z 1 quasi-labile

i $ 1 labile

(4a)

(4b) (4c)

where, in the present case of unequal diffusion coefficients, the diffusion layer thickness 6 is a function of hi,. The functionality is of the type L&, with c( normally between l/3 and l/2, depending on the convective conditions [13]. Equations (3) and (4) show that metal complexes are only labile for certain ranges of values of (i) the rate constants for association/dissociation of the complex, (ii) the diffusion coefficients of the metal complex and the free metal, and (iii) the time scale of the dynamic experiment (or, equivalently, convective parameters such as stirring rate or rotation speed (not electrolysis duration) in stripping voltammetric experiments). 1. Non-labile complexes This situation is trivial. The voltammetric response is directly related to the concentration of free metal in solution. However, the distinction between this simple case and, for example, the case of a labile complex with low diffusion coefficient (to be discussed below), is not possible from a titration curve containing the current as the voltammetric response. In order to make such a distinction by electrochemical means, one has to turn to the potential characteristics, either determined from one stripping voltammogram or taken from a composed pseudopolarogram [14]. In the case of a non-labile species, the halfwave potential or peak potential will be the same as for the reference case of purely free metal, whereas in the case of the labile complex there will be a shift towards more negative potentials. In stripping voltammetric experiments this distinction is not always possible, since ligand saturation may obscure the stripping voltammogram [15]. This complication persists down to low metal-toligand ratios because the metal is preconcentrated by an appreciable factor (typically of the order of 100). Variation of the duration of the electrolysis step or medium change after the accumulation may solve the problem. 2. Quasi-labile complexes In the regime of intermediate parameter values where Eqn (3b) or (4b) applies, the dynamic response is (partly) governed by the dissociation/association kinetics of the complex system. For a proper speciation analysis under these conditions, it will be necessary to consider the complete time-scale dependence of the signal. This will be a cumbersome task in the most general case, where we have both appreciable amounts of free metal ions (plus labile

49

complexes, if any) and a kinetic contribution of such a magnitude that a concentration gradient for the quasi-labile complex has to be taken into account [1618]. For the present case of unequal diffusion coefficients, the rigorous solution for the dynamic voltammetric response is not available. Only for the rather approximative situation where the reaction layer theory applies is an analytical solution - given by Koutecky [6] a long time ago ~ at hand. It is important to note that in the course of a titration the lability of a given complex is not fixed. The ligand concentration, while still being in ex.cess over the total metal concentration, decreases upon addition of metal. This means that h’,and K’ decrease as well and, according to criteria (3) and (4) the lability ilzcreases. Thus, on going from region A to region B in a titration curve, the slope may increase for purely kinetic reasons. The practical impa.ct of the lability change is not very important for the initial region A, since the extent of change is small there. However, in region B the effect may take more drastic proportions, as we shall see below. 3. Labile complexes Electrochemically labile complexes are very common in environmental samples [5, 91. Voltammetric speciation on the basis of analysis of currents is only possible by the grace of the difference between the diffusion coefficients of the labile complex and the free metal. The simultaneous diffusion of ML and M towards the electrode surface gives rise to a diffusion layer with a thickness that is intermediate between those for pure ML and pure M. It has been explained that a mean diffusion coefficient D will be operative [6, 7, 111. This l5 is related to the relative concentrations of M and ML, provided. that the exchange of M between the complexed and the free state is fast on the timescale of the experiment. The latter condition may be expressed as kr,t, h,,t 9

1

(5)

and it is important to note the distinction between conditions (3~) and (4~) relating to electrochemical lability. With (5) satisfied, D can be given as

and, for example, for the case of semi-infinite response obeys the general proportionality current

rx D”“cT*

linear diffusion,

the dynamic

(7)

where c++= c$ + c&,. The physical significance of Eqn (6) is simply that any individual M, which diffuses towards the electrode surface, spends a fraction c; /CT* of the time as free M and the complementary fraction c&/c; of the time as the complex ML. This is only achieved if the number of conversions of M into ML and vice versa is much higher than unity, so that condition (5) is obeyed. In passing, we may note that condition (5) could be formulated equally well in terms of the reaction

50

-I

1

,c--___---

I/

-E

Fig. 2. Polarographic characteristics for labile metal complexes with diffusion coefficient 4, unequal to h. Situation of excess ligand. I: Reference case of pure M. II: Labile complex ML; C*m/Cti

=

20;

&.I&,

complex; c&/c;

complex = 0.1. III. Labile = 20; D,, irrelevant.

ML; c$Jc&

= 20; D,,/D%

--t 0. IV: Non-labile

layer thicknesses for the association and dissociation reactions, the criterion being that both layers should be thin compared with the electrochemical diffusion layer. The significance of a mean diffusion coefficient D is well illustrated by Fig. 2, where some normal polarographic characteristics for labile complexes are given. The shift of the wave along the potential axis is a function of both c&/c~ and &,lD, . This point will not be considered any further here, except that we may note that, with decreasing DML/DM, IX,,, shifts in the positive direction, thus (partly) compensating for the primary effect of the complexation. The decrease of the limiting current in cases II and III in Fig. 2 is solely due to the inequality of &,, and &. The difference between curves II and IV illustrates the contribution from the complex, the total’current being substantially higher than the sum of the separate currents for free metal and complex. This important characteristic, directly following from Eqn (6), is elucidated more clearly by curve III, in comparison with curve IV. For the limiting case of &, -+ 0, under experimental conditions where the dynamic response obeys a squareroot dependence on D (Eqn (7)), the two different cases correspond with (i) curve III (labile): current cc D”‘cG -+ D~c&~~~c~“~,derived from Eqns (6) and (7); and (ii) curve IV (non-labile):

current

CCDgc;.

Thus a complex that is itself immobile can contribute greatly to the current, thereby drastically modifying the measured signal. Physically, this is explained by the fact that inside the diffusion layer the complex produces free metal ions because it easily dissociates so as to adjust its concentration to the local free metal concentration. The different possibilities for the nature of the initial part of the titration curve are summarized in Table 1. Not all of the possibilities are equally likely

51 TABLE

I

Nature of initial slope of a voltammetric

titration

curve for different types of complex

Zero slope

Non-labile,

Intermediate

Non-labile, weak (c& # O)“Sb Quasi-labilea Labile with hL < h

Reference

slope

slope (approximately)

aAny L&L. bin the bulk of the solution

strong complex (c& + 0)

Labile with 4,

z h

there is assumed to be equilibrium.

to occur in practice, but they all may occur in principle. The implications for the correct evaluation of an equivalence point will be discussed later. Here we want to emphasize that for a distinction to be made between the different cases with intermediate slope, independent means will be necessary. In an electrochemical experiment there are tools for doing this. The distinction between labile and non-labile is made on the basis of the potential characteristics, i.e. a non-labile complex gives no shift. The detection of quasi-lability is most directly done by varying the time-scale of the experiment, i.e. pulse duration in pulse polarography or rotation speed/stirring rate in stripping voltammetry [12, 171. Regions B and C, around and beyond the equivalence point With the condition of excess ligand no longer being fulfilled, we have to abandon the quasi-monomolecular scheme for the complexation reaction. Consequently, the lability criteria (as formulated by Eqns (3) and (4)) are not applicable. As already stated, during the course of a titration, the lability of a complex increases. Apart from being interesting in itself, this feature is helpful in the sense that a system that has been found to be labile in the initial part of the titration and which does not show special complications will certainly be labile in regions B and C. This observation is especially useful for finding the correct equivalence point, an item to which we shall return for each of the different cases to be discussed. The case of a complex that is non-labile in any stage of the titration is of course trivial. The voltammetric response always corresponds with the free metal concentration in solution, irrespective of whether the complexation reaction is at equilibrium or not. An increasing slope in the titration curve reflects the reduction of the extent of complexation of M, until eventually no more complexation takes place and the slope of the blank is reached. Figure 3 shows the typical shape of the titration curve for a non-labile complex. The initial slope of such a curve is zero, unless the complex is weak or there is no equilibrium in solution. However, neither of these conditions is very likely to hold for non-labile complexes. In any situation, the equivalence point is found as the intersection of the excess metal asymptote with the CT*axis, because eventually all of the ligand is converted to the non-labile ML.

52

Fig. 3. Shape of a voltammetric titration curve in the case of the formation of non-labile complexes. The solid line refers to a strong complex, the dotted line to a relatively weak complex. The broken line is the reference case (blank).

It almost goes without saying that in the case of quasi-labile complexes, with all the participating species at comparable activity levels, there is no explicit expression for the voltammetric response. There are so many independent variables (in principle, seven: lz, , lz,, DML,DM, c&, c& , t)that even numerical approaches to the problem would be cumbersome. Qualitatively, but with extra care, one could use the arguments as given before for region A/case (ii). We have already indicated that, generally, the lability of complexes increases in the course of the titration with metal. In the case of a system that gives quasi-labile species in the initial stage, it may thus be anticipated that they could very well be labile beyond the equivalence point. This means that extrapolation of the initial portion (Fig. 4a) will lead to erroneous results. The excess metal region of the curve, from where the second extrapolation is carried out, involves a contribution from the complex. This contribution cannot be taken into account by using the initial part of the curve because it is of a kinetic nature, not solely reflecting the diffusion of complex. The intersection with the axis (Fig. 4b) will be an underestimate of the real value, and the same (or the opposite, depending on the parameter values) is true for the intersection with some extrapolation of the initial curve (Fig. 4a). Hence, for the purpose of finding an equivalence point, such a type of titration curve must be disregarded. The experimental recognition of this case is of course quite essential and the most adequate tool is the variation of the (effective) time-scale (or the stirring speed in a stripping experiment). The case of a complex that is labile right from the beginning of the titration would seem to be relatively simple. This is, indeed, basically true, but there is an essential change in going from region A to regions B and C. Let us consider the numerical example, as represented by Fig. 5. The values for the different

53

Fig. 4. Shape of a voltammetric titration curve in the case of a complex which is quasi-labile in the initial stage. The broken line is the reference case. The intersections a and b are found by extrapolation. Further explanation is given in the text.

Fig. 5. Shape of a voltammetric titration curve in the case of the formation of a labile complex with K = lo’, c: = 10m5M and L&,/L& = 0.02. Curve 1 is calculated with D”‘cf and curve 2 with *. the dotted parts indicate the regions where validity is lost. For comparison, the D,$c;, + D,Ii2cM> curve for K --t co, ct = 10m5M and L&,,/L&, = 0.02 is included (-.-.-.). The broken line is the reference case (blank).

54

Fig. 6. Schematized concentration profiles for a voltammetric region C (beyond the equivalence point), L& < 4.

titration experiment; labile complex,

parameters K, CL and hL/DM are typically met in natural samples, e.g. for metal/humic acid complexes [B]. First of all, it is clear that initially the titration curve is linear. In the case of not-too-strong complexes, the slope does not only depend on hL, but also on K, CL* and &. This is connected with the fact that the (nearly constant) ratio between c& and c& is not small enough to render the final term in the right-hand side of Eqn (6) negligible. Hence, extrapolation of the initial part as a basic step in finding the equivalence point is not applicable in this case. But, before considering this aspect, let us examine the remaining part of the curve. If one continues to calculate the curve on the basis of the averaged diffusion coefficient D, one winds up with anomalous results beyond the equivalence point. For example, there are_ combinations of K, DMLand & (by themselves not unreasonable) for which Eqn (6) generates slopes in region C higher than the reference value. The reason for this can be found in the foregoing treatment. The lability of the complex ML increases on going from region A to region C, because k’, decreases. However, this also means that in the course of a titration, especially beyond the equivalence point where the ligand concentration drops strongly, the fulfillment of condition (5) gets lost. This point is easily understood in terms of reaction scheme (1): with cL low, the association rate will be low and it is no longer true that any M is frequently converted into ML while diffusing towards the electrode surface. The concentration profiles shown in Fig. 6 illustrate this in terms of the equilibrium between M, L and ML. In the diffusion layer set up by M, the equilibrium condition requires only a minor adaptation of cML. Under these circumstances, the complex ML (labile as it is) is left to develop its own diffusion layer. Thus we have two distinct diffusion layers with their own thicknesses 6, and hhIL,determined by the respective diffusion coefficients. For high c&/c&, this situation approaches the case of independent diffusion of the species ML and M. Thus the current becomes proportional to w * + D$Lc&] instead of to D” ‘ c,*. According to this explanation, the slope ]D, CM of the titration curve asymptotically approaches the reference value (see Fig. 5).

It can be concluded that in the case of not-too-strong, labile complexes, the evaluation of the equivalence point is not only complicated by defining the ultimate slope beyond the equivalence point. Even less achievable is the quantitative meaning of the initial part of the curve, leaving the analyst with an

TABLE 2 Evaluation _

of the equivalence

point from a voltammetric

metal complex titration

Nature of the complex _

Procedure

Non-labile

Extrapolation

Quasi-labile

None (existing extrapolation yield erroneous results)

Labile: strong (Kc: $ L&/hL)

Intersection of double extrapolation (a in Fig. 1)

Labile: intermediate

Interpretation (extrapolation

and weak

to CT*axis (b in Fig. 1) procedures

all

only possible if DjnL is known procedures fail)

unknown overestimate of the contribution from the complex and hence with an overestimate of the equivalence point. The basic remedy for the problem is to (separately) find the values of DMLand L& so that the real nature of the curve can be established and a proper analysis set forth. Let us summarize our findings by considering the different possibilities for determining a correct equivalence point from a voltammetric titration curve (Table 2). The two extreme cases are clear and simple: with a non-labile complex one has to extrapolate to the $-axis, while with a strong, labile complex one has to take the intersection point found by double extrapolation. For the different cases in between these two, the procedure becomes more complicated, if not hopeless. Anyway, it should be clear that every vol’tammetric metal titration must bepreceded by a characterization of the kineticproperties of the complex involved. REFERENCES 1 2 3 4 5 6 I 8 9 10 11 12 13 14 15 16 17 18

H.W. Niirnberg, P. Valenta, L. Mart, B. Raspor and L. Sipos, Z. Anal. Chem., 282 (1976) 357. H.W. Nurnberg, Electrochim. Acta, 22 (1977) 935. H.W. Niirnberg and B. Raspor, Environ. Technol. Lett., 2 (1981) 457. H.W. Niirnberg and P. Valenta, in C.S. Wong, E. Boyle, K.W. Bruland, D. Burton and E.D. Goldberg (Eds), Trace Metals in Sea Water, Plenum, New York, 1983, pp. 671697. C.J.M. Kramer and J.C. Duinker (Eds), Complexation of Trace Metals in Natural Waters, Nijhoff/Junk Publ., The Hague, 1984. J. Koutecky, Collect. Czech. Chem. Commun., 19 (1954) 857. D.R. Crow, Polarography of Metal Complexes, Academic Press, London, 1969. R.F.M.J. Cleven, Ph.D. Thesis, Agricultural University, Wageningen, The Netherlands, 1984. J. Buffle, in H.W. Nurnberg, F. Pellerin and T.S. West (Eds), Report of Analytical Chemistry of IUPAC, in press. I. RuiiC, Anal. Chim. Acta, 140 (1982) 99, 331; also ref. 5, pp. 131-147. R.F.M.J. Cleven, H.G. de Jong and H.P. van Leeuwen, J. Electroanal. Chem., 202 (1986) 57. W. Davison, J. Electroanal. Chem., 87 (1978) 395. V.G. Levich, Physicochemical Hydrodynamics, Prentice Hall, Englewood Cliffs, NJ, 1962. M. Branica, D.H. Novak and S. Bubic, Croat. Chem. Acta, 49 (1977) 231. J. Buffle, J. Electroanal. Chem., 125 (1981) 273. A.A.A.M. Brinkman and J.M. Los, J. Electroanal. Chem., 7 (1964) 171; 14 (1967) 269. H.P. van Leeuwen, J. Electroanal. Chem., 99 (1979) 93. J. Galvez, C. Serna, A. Molina and H.P. van Leeuwen, J. Electroanal. Chem., 167 (1984) 15.