12.3.4: Preassociation Complexes

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Most octahedral complexes react through either an associative or dissociative interchange mechanism ($I_a$ or $I_d$). Although the rate laws should be different for these two cases, it is difficult to distinguish between them. The difficulty lies in the seemingly conflicting observations from experiments performed under limiting conditions of the incoming ligand. For example, in the case of the reaction of hexaaquochromium(III) ( $\ce{[Cr(H2O)6]}$ ) with ammonia ( $ \ce{NH3}$ ), the rate laws determined at high and low concentrations of the incoming ligand give different apparent rate laws. Under most conditions, the rate law appears to indicate a dissociative mechanism (ie rate is independent of the $\ce{[NH3]}$). But, at very high $\ce{[NH3]}$, the rate law appears to indicate an associative mechanism (ie the rate depends on $\ce{[NH3]}$).

This information might seem contradictory. However, it can be explained by the formation of a transient ion pair, usually called an encounter complex, in a step prerequisite to the rate-determining step(s). The Eigen-Wilkins Mechanism is based on this idea.

The Eigen-Wilkins mechanism

The Eigen-Wilkins mechanism is also a rate law, and it governs the reactions of octahedral metal complexes. This mechanism does not define the rate-limiting step; rather, it defines the existence of a pre-equilibrium step (ie an initial step that is not rate-determining) that results in formation of an encounter complex. The encounter complex is a short-lived ion pair of the metal complex and the incoming ligand; it is an intermediate formed through Coulomb interactions. For the conversion of a generic metal complex, where X is the leaving group and Y is the entering group, the overall reaction, pre-equilibrium step, and formation of products from the encounter complex are shown below:

\[\begin{array}{rcc}
& \text{CHEMICAL EQUATION} & \text{EQUILIBRIUM EXPRESSION} \\
\hline \text{Formation of Encounter Complex} & \ce{ML5X + Y<=>[{k_1}][{k_{-1}}] (ML5X*Y)} &\ce{$K_E = \ce{\frac{[(ML5X*Y)]}{[ML5X][Y]}}$} \\
\text{Formation of Products:} & \ce{(ML5X*Y) ->[{k_2}] ML5Y + X} &\ce{$K_2 = \ce{\frac{[ML5Y][X]}{[(ML5X*Y)]}}$}\\
\hline \text{Overall Reaction:} & \ce{ML5X + Y <=> ML5Y + X} & \ce{$K = \ce{\frac{[ML5Y][X]}{[ML5X][Y]}}$} \\
\end{array} \nonumber \]

Fast pre-association

One possibility (that is most common) is that $k_1$ and $k{-1}$ are much larger than $k_2$. In this case, the encounter complex forms quickly, and once it forms, it also can quickly fall apart to re-form the reactant complex. When $k_2$ is relatively small then reaction of the encounter complex to form the product is the rate-determining step, and sometimes the concentration of encounter complex can be experimentally determined. However, it is rare that this is the case. A second, more common scenario iis that $k_2$ is also relatively large. Assuming that the formation of the encounter complex is still a fast step, then the rate still depends on the reaction of that encounter complex to form product. In either case, one path for deriving the rate laws is given below. The rate of formation of the product is defined by the second step:

\[\text{Rate}=\frac{d\ce{[ML5Y]}}{dt}=k_2 \ce{[(ML5X*Y)]} \nonumber \]

Although the encounter complex may exist in high enough concentration to be detected, that is rare and technically difficult. Rather, the concentration of the encounter complex can be determined by manipulation of the equilibrium constant expression and some helpful assumptions. The equilibrium constant for formation of the encounter complex is:

\[K_E = \ce{\frac{[(ML5X*Y)]}{[ML5X][Y]}} \nonumber \]

This can be rearranged to solve for concentration of the encounter complex.

\[\ce{[(ML5X*Y)]} = K_E\ce{[ML5X][Y]} \label{encounter} \]

...and we can re-write the rate experssion to remove the encounter complex from the expression. But by doing so, we remove the problem of the unmeasurable encounter complex being in the expression, but add the problem of the reactant metal complex being part of the expression:

\[\text{Rate}=k_2 K_E\ce{[ML5X][Y]} \nonumber \]

The problem is that we don't precisely know what the value of $\ce{[ML5X]}$. We only know the total amount we initially added. But, we assume that $k_1$ is large, and so the initial concentration of the metal complex may change immediately to produce some of the encounter complex. As long as $k_2$ is relatively small, we can assume that the initial total concentration of metal complex at the start of reaction consists primarily of the reactant complex plus the encounter complex:

\[\ce{[M]_{total}}=\ce{ [ML5X] +[(ML5X*Y)]} \nonumber \]

We can substitute XXX into XXX to remove the encounter complex from the expression, then rearrange to solve for the reactant metal ion concentration, $\ce{[ML5X]}$:

\[\begin{array}{rl}
\ce{[M]_{total}}&=\ce{ [ML5X]} +K_E \ce{[ML5X][Y]} \\
&=\ce{ [ML5X]}(1 +K_E \ce{[Y]}) \\
\ce{[ML5X]} & = \frac{\ce{[M]_{total}}}{1 +K_E \ce{[Y]}}
\end{array} \nonumber \]

Now this allows us to do one more substitution into the rate expression to get the rate into terms of constants and solution concentrations that are known:

\[\text{Rate}=k_2 K_E\frac{\ce{[M]_{total}[Y]}}{1 +K_E \ce{[Y]}} \nonumber \]

This rate expression is complex, but can be simplified under extremely high or low concentrations of the incoming ligand, [Y].

High [Y]: If the recation proceeds at high concentration of Y, then the denominator, $1 +K_E \ce{[Y]}$ becomes approximately $K_E \ce{[Y]}$. The expression is simplified to a first-order rate law under high [Y]:

\[\text{Rate}=k_2 K_E\frac{\ce{[M]_{total}[Y]}}{K_E \ce{[Y]}}=k_2 \ce{[M]_{total}} \nonumber \]

This would allow determination of $k_2$ because the rate could be directly observed by varying the concentration of $\ce{[M]_{total}}$ at high [Y].

Low [Y]: If the reaction is run at very low [Y], then the denominator, $1 +K_E \ce{[Y]}$ becomes approximately 1. We can let the observed rate constant, $k_{obs}=k_2K_E$ to get a simplified second-order rate law under very low [Y]:

\[\text{Rate}=k_2 K_E\frac{\ce{[M]_{total}[Y]}}{K_E \ce{[Y]}}=\frac{d\ce{[ML5Y]}}{dt}=k_{obs} \ce{[M]_{total}}[Y] \nonumber \]

The observed rate constant can be measured experimentally, and $K_E$ can be determined theoretically ($k_{obs}=k_2K_E$)