12.3.2: Rate Laws for Interchange Mechanisms

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Interchange (I) Mechanisms take place in one concerted step where there is no intermediate, or the intermediate is elusive. The reactio nessentially occurs in once step in which bond forming and bond breaking is concerted, as shown below. The species, \(\color{green}{\ce{[Y\bond{...}ML_{n}\bond{...}X]^{\ddagger}}}\), can be defined as either a transition state or a very short-lived intermediate.

\[\begin{array}{rc}
\text{Step 1 (Concerted):} & \ce{ML_{n}X + Y <=>}\textcolor{green}{ \ce{[Y\bond{...}ML_{n}\bond{...}X]^{\ddagger}}} \ce{+ <=>[k_1][k_{-1}] ML_{n}Y + X}
\end{array} \nonumber \]

The rate law of this one-step reaction depends on whether the relative rates of the forward and reverse reactions. When the reaction is practically irreversible, then the rate depends only on the formation of product, and not its reaction to re-form product. In this case, the rate law is a simple second-order rate law.

\[\frac{d\ce{[ML_{n}Y]}}{dt}=k_1 \ce{[ML_{n}X][Y]} \nonumber \]

On the other hand, if the rate of re-formation of reactant is significant, then the rates of forward and reverse reactions must be considered in the rate law.

\[\frac{d\ce{[ML_{n}Y]}}{dt}=k_1 \ce{[ML_{n}X][Y]} - k_{-1}\ce{[ML_{n}X][X]} \nonumber \]

In the latter case where there is a dependence of rate on both [X] and [Y], an effective strategy is to employ the condition where solution concentration of both [X] and [Y] are simultaneously high so that the reaction is effectively a competition of two pseudo-first order reactions that depend primarily on the metal complex concentrations. In such a case as this, another helpful piece of information is the equilibrium constant for the reaction (\(K=\frac{k_1}{k_{-1}}\)). The rate law is written below.

\[\text{When [X] and [Y] are very high: } \frac{d\ce{[ML_{n}Y]}}{dt}=k_1 \ce{[ML_{n}Y]} - k_{-1}\ce{[ML_{n}X]} = -\frac{d\ce{[ML_{n}X]}}{dt} \nonumber \]

One distinguishing feature of an I mechanism compared to a D mechanism can be determined from such an experiment under high concentration of X and Y. In the case of a D mechanism and sufficiently high concentration of incoming ligand, [Y], the reaction demonstrations saturation kinetics to give a pseudo-first order rate law depending only on the concentration of reactant metal complex. In the case of I, the reaction rate is not a simple first order rate law because even under high [Y] and [X], there is a dependence of rate on concentration of the product complex.

Reactions that are \(I_a\) or \(I_d\) are distinguished by the relative strength of the M-X and M-Y bond in the transition state. These reactions can be distinguished by varying the Identities of X and Y to determine how the reaction rate depends on identity of X and Y. If there is a large dependence on the incoming ligand, and less so on the outgoing ligand, it could be classified as an \(I_a\) mechanisms. The opposite is also true.