Question

Derive a plausible mechanism for the following reaction in aqueous solution, \(\mathrm{Hg}_{2}^{2+}+\mathrm{Tl}^{3+} \longrightarrow 2 \mathrm{Hg}^{2+}+\mathrm{Tl}^{+}\) for which the observed rate law is: rate \(=k\left[\mathrm{Hg}_{2}^{2+1}\right]\) \(\left.[\mathrm{T}]^{3+}\right] /\left[\mathrm{Hg}^{2+}\right].\)

Short answer

A plausible mechanism consists of two steps: (1) Equilibrium step: \(Hg_{2}^{2+}\) \leftrightarrow 2 \(Hg^{2+}\) (2) Rate-determining step: \(Hg^{2+}\) + \(Tl^{3+}\) \longrightarrow \(Tl^{+}\) + \(Hg^{2+}\). The proposed mechanism matches with the given rate law.

Step-by-step solution

Understand and Interpret the Rate Law

The rate law is written as rate = k[\(Hg_{2}^{2+}\)][\(Tl^{3+}\)]/[\(Hg^{2+}\)]. Here, k is the rate constant and brackets refer to molar concentrations. The rates of reactions concerning each reactant are indicated in the rate law. This rate law suggests that the reaction rate increases with the concentration of both \(Hg_{2}^{2+}\) and \(Tl^{3+}\), and decreases with the concentration of \(Hg^{2+}\). The fact that the rate decreases with the concentration of \(Hg^{2+}\) points to an equilibrium step before the rate-determining step in the proposed mechanism.

Formulate a Plausible Mechanism

Considering the information derived from the rate law, a plausible reaction mechanism involves two steps where the first step is an equilibrium step and the second step is the rate-determining step. The first step can be written as \(Hg_{2}^{2+}\) \leftrightarrow 2 \(Hg^{2+}\), with the reverse reaction favoured. In this step, the \(Hg_{2}^{2+}\) ion dissociates into two \(Hg^{2+}\) ions and this is a fast equilibrium step. The second step is \(Hg^{2+}\) + \(Tl^{3+}\) \longrightarrow \(Tl^{+}\) + \(Hg^{2+}\). In this step, an \(Hg^{2+}\) ion reacts with a \(Tl^{3+}\) ion to form a \(Tl^{+}\) and \(Hg^{2+}\) ion, and this is the slow rate-determining step.

Check the Mechanism with the Rate Law

Now, the proposed mechanism should be checked with the given rate law to see if it fits. According to the rate law, the reaction is first order with respect to both \(Hg_{2}^{2+}\) and \(Tl^{3+}\) ions, and minus first order with respect to \(Hg^{2+}\) ions. This fits with our proposed mechanism as the rate-determining step involves one \(Hg^{2+}\) and one \(Tl^{3+}\) ion and the equilibrium step involves one \(Hg_{2}^{2+}\). Thus, our proposed mechanism is probable according to the given rate law.

Key concepts

Rate Law

The rate law is fundamental to the study of reaction kinetics, as it connects the reaction rate to the concentrations of reactants. It is expressed as an equation where the rate of the reaction is equal to the rate constant (\( k \)) multiplied by the concentrations of the reactants, each raised to a power representing their order in the reaction. Understanding the rate law allows us to predict how changes in concentrations affect the speed of the reaction.

For example, in the given exercise, the rate law signifies that the reaction rate increases with the concentration of \(Hg_{2}^{2+}\) and \(Tl^{3+}\), but decreases with the concentration of \(Hg^{2+}\). These relationships tell us not just how the reaction proceeds, but also give hints about the underlying mechanism, showing that the mechanism is likely to involve these species in critical steps.

Reaction Kinetics

Reaction kinetics is the branch of chemistry that deals with the study of the rates of chemical reactions and the factors that affect these rates. By examining aspects such as concentration, temperature, and catalysts, chemists can determine how fast a reaction will proceed. In the context of our discussion, kinetics helps us to explore the relationship between reaction rate and reactant concentrations, as informed by the rate law.

Understanding kinetics is essential for formulating mechanisms, as it provides insight into the steps involved in a reaction and their respective speeds. Such knowledge is not only academically interesting but also has practical applications in designing chemical processes and pharmaceuticals.

Equilibrium Step

An equilibrium step in a reaction mechanism is one where the forward and reverse reactions occur at the same rate, establishing a dynamic equilibrium. This step is often rapid and reversible, as is suggested by the rate law for the exercise given. In the proposed mechanism, the first step \(Hg_{2}^{2+} \leftrightarrow 2 Hg^{2+}\) is the equilibrium step, where \(Hg_{2}^{2+}\) dissociates into two \(Hg^{2+}\) ions and then can recombine.

When such a step is present, concentrations of the species involved rarely change dramatically during the reaction, because as soon as the products form, they rapidly revert to reactants. This balance must be considered when analyzing reaction rates and when determining the overall rate law.

Rate-Determining Step

The rate-determining step is the slowest step of a chemical reaction mechanism, and it essentially sets the pace for the entire reaction. This step will have the highest energy barrier to overcome and is therefore the most sensitive to changes in reactant concentrations. In our exercise, the rate-determining step involves the reaction of \(Hg^{2+}\) with \(Tl^{3+}\), forming \(Tl^{+}\) and another \(Hg^{2+}\) ion.

Since this step dictates the overall reaction rate, understanding and identifying the rate-determining step is crucial to optimizing conditions for a reaction and can provide valuable insights into how to control and expedite chemical processes in industrial and laboratory settings.