Overall Stability Constants

There is another effect introduced in the experimental data above also the use of a set of ligands rather than a single ligand in binding the metal in many examples and it is this that we shall examine more fully next. Because metal ions typically provide more than one coordination site, they can bind more than one donor group, or in most cases (except where one polydentate ligand fully satisfies the coordination sphere demands of a metal ion) more than one ligand. Thus, we need to extend the equations for stability constants developed above for attachment of one ligand to account for attachment of a set of n ligands. This process occurs sequentially, not all at once. This is because ligand replacement is the result of molecular encounters, with the complex unit required to make contact with an incoming ligand with sufficient velocity and with an appropriate direction of approach so as to permit a ligand exchange to occur. The probability of all of a set of L replacing all of a set of coordinated water molecules at once (in a concerted reaction) is far too low to be considered a viable pathway. Hence after ML a series of other complexes ML2 ML3 ... ML form in a sequence of steps. Although we refer to the stability constants for each of these processes as stepwise stability constants, there is no sudden 'step' from all of

Figure 5.7 : Speciation diagrams resulting from metal-ligand titrations, with the concentration profiles of the various ML species throughout the process of successive addition of the ligand identified. At left is the pH-dependent formation of vanadyl complexes with the acetylacetonate anion; at right is the chloride ion concentration-dependent formation of chloromercury (II) species.

one species to all of another occurring in solution. Rather, and depending on the reaction conditions and relative sizes of the stepwise stability constants for each species, more than one complex will exist in solution at any one time, although one may be dominant. We can see this experimentally from analysis of accurate metal-ligand titrations, where formation curves resulting from data analysis track the concentrations of sequentially formed species. Two examples appear in Figure 5.7.

We can represent the sequential substitution steps for formation of MLx by a series of equilibria (5.18):

For each step, a stepwise stability constant can be written of the form (5.19):

The overall reaction occurring through combination of the above steps can be written as (5.20):

for which we can define an overall stability constant (5.21)

This overall stability constant simply defines the formation of the overall or ‘final’ complex, where all of one ligand (here, water) can be considered replaced by another; it does not infer anything about the mechanism of the process. Actual examples have appeared earlier in Equations (5.9) and (5.10).

It is a straightforward task to show that the overall stability constant can be represented in terms of the component stepwise stability constants by Equations (5.22) and (5.23):

From examinations of experimental results, is has been observed that, for sequential ligand substitution without change in coordination number, there is a constant size order for the sequential stability constants, shown in Equation 5.24.

K1>K2>K3 ··· >Kx (5.24)

Thus, as more and more of one type of ligand are introduced, the gain in stability in each step keeps diminishing. This occurs irrespective of whether neutral or anionic ligands are involved, as illustrated in Table 5.4.

This trend is explained to a modest level of satisfaction by a simple statistical argument. Consider a partly-substituted metal complex containing both L and OH₂ ligands. If one reaches in and plucks out 'unseen' and at random any ligand from the coordination sphere, the probability of removing L (rather than OH₂) from ML is greater than the probability of removing L from ML(m-1), simply because there is one more to choose from in the former case. At the same time, if you reach in and pluck out a ligand at random and replace it by a new L ligand (irrespective of whether you first pluck off an L or an OH₂), in some cases you will advance the number of L ligands (when you replace an OH2 by an L), and in others you will make no change by inadvertently replacing one L by another L. From a statistical standpoint, the probability of adding L to ML(m-2) is greater than the probability of adding L to ML(m-1), because there is one extra OH₂ site to choose from in the former complex. Let's represent this in a 'real' case (5.25) below.

Six water substitution sites are available in [M(OH)₆], with five water substitution sites available in [M(NH₃)(OH₂)s], so it is more probable for addition of ammonia to occur in [M(OH)6] than in [M(NH₃)(OH₂)s]. For the reverse step of removal of ammonia, [M(NH₃)₂(OH₂)₄] is more likely to lose one, as two are available versus only one in [M(NH₃) (OH₂) s]. So there is a greater driving force for adding an NH3 in the first step, and a greater driving force for removing an NH₃ in the second step. This means the concentration [M(NH₃) (OH₂) s] versus [M(OH₂)₆] will be relatively greater than that of [M(NH3)₂(OH₂)₄] versus [M(NH₃) (OH₂) s], and hence K, will be larger than K2. Thisstatistical argument can be made for each sequential substitution. Indeed, it is possible to use the above modelling to calculate the ratio of successive stability constants, and these statistical predictions match at least modestly to experimental values. Clearly, it is not the sole contributor to sequential stability (for example, the relative sizes of the two competing ligands might be expected to have some role to play when building up a set of ligands around a single central metal), but it plays a significant role for at least simple ligands.

Stability constants for Ni^2+/NH₃ and Al^3+/F follow the predicted trend smoothly, regardless of one system using a neutral ligand and the other an anionic ligand. For Cu²⁺/NH₃, although the appropriate trend is followed, the far larger than usual drop from log K4 to log Ks (Table 5.2) is evidence for the operation of the Jahn-Teller effect, which imposes elongation along one axis that does not favour strong complexation in the fifth and sixth axial positions.

There are also conditions that can lead to an exception to this general rule of size order of stability constants. For example, experimental log Km determined for [Cd (OH₂)₆]²⁺ reacting with Br follow the usual trend from K, to K3, but then there is a rise to K, and there are no further values measurable (Table 5.4); clearly something different is happening here. As it happens, we know from physical and structural studies that the complex resulting from introduction of four Br ions is a tetrahedral species [CdBr.], whereas the immediately prior species with three Brions coordinated is an octahedral species [CdBr₃(OH)₃] Thus, it is clear that in the fourth substitution step, there is a change from six-coordinate octahedral to four-coordinate tetrahedral coordination. Yet sequential addition of Br still occurs, so why should it matter so much? The answer to this lies in another quite different effect, associated with reaction entropy, which can be envisaged as reflecting the change in the level of disorder in the system upon reaction. Let's see this from the participating molecule aspect, introduced earlier. The change in the K4 step can be written as (5.26):

versus the usual situation met where there is no change in coordination number, as for K3 (5.27):

The difference is the sudden release of three water molecules when the coordination number falls from six to four. This means there is an increase in 'disorder' in this first case (or a favourable entropy change) and not in the other, influencing K4 but not K3 and other prior steps. While this entropy-based effect has been illustrated for simple ligands through a reaction where change in coordination number is the key step, in effect any mechanism that leads to an unprecedented change in disorder (or a 'burst' of released molecules) will produce an enhanced stability constant.

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مواضيع ذات صلة

Lability and Inertness in Octahedral Complexes

Kinetic Rate Constants

Thermodynamic and Kinetic Stability

Undergoing Change– Kinetic Stability

Factors Influencing Stability of Metal Complexes

Bedded Down– The rmodynamic Stability

Bedded Down– The rmodynamic Stability

Host–Guest Molecular Assemblies

Encapsulation Compounds

Compounds of Polydentate Ligands

Sophisticated Shapes

Isomer Preferences