Question

Metals often form several cations with different charges. Cerium, for example, forms \(\mathrm{Ce}^{3+}\) and \(\mathrm{Ce}^{4+}\) ions, and thallium forms \(\mathrm{Tl}^{+}\) and \(\mathrm{Tl}^{3+}\) ions. Cerium and thallium ions react as follows: $$ 2 \mathrm{Ce}^{4+}(a q)+\mathrm{Tl}^{+}(a q) \longrightarrow 2 \mathrm{Ce}^{3+}(a q)+\mathrm{Tl}^{3+}(a q) $$ This reaction is very slow and is thought to occur in a single elementary step. The reaction is catalyzed by the addition of \(\mathrm{Mn}^{2+}(a q)\), according to the following mechanism: $$ \begin{aligned} \mathrm{Ce}^{4+}(a q)+\mathrm{Mn}^{2+}(a q) & \longrightarrow \mathrm{Ce}^{3+}(a q)+\mathrm{Mn}^{3+}(a q) \\ \mathrm{Ce}^{4+}(a q)+\mathrm{Mn}^{3+}(a q) & \longrightarrow \mathrm{Ce}^{3+}(a q)+\mathrm{Mn}^{4+}(a q) \\ \mathrm{Mn}^{4+}(a q)+\mathrm{Tl}^{+}(a q) & \longrightarrow \mathrm{Mn}^{2+}(a q)+\mathrm{Tl}^{3+}(a q) \end{aligned} $$ (a) Write the rate law for the uncatalyzed reaction. (b) What is unusual about the uncatalyzed reaction? Why might it be a slow reaction? (c) The rate for the catalyzed reaction is first order in \(\left[\mathrm{Ce}^{4+}\right]\) and first order in \(\left[\mathrm{Mn}^{2+}\right]\). Based on this rate law, which of the steps in the catalyzed mechanism is rate determining? (d) Use the available oxidation states of \(\mathrm{Mn}\) to comment on its special suitability to catalyze this reaction.

Short answer

The rate law for the uncatalyzed reaction is given by Rate = k[\(\mathrm{Ce}^{4+}\)]^2[\(\mathrm{Tl}^{+}\)]. This reaction is unusual because it involves a simultaneous three-ion collision, which is low in probability. The rate-determining step for the catalyzed reaction is the first step, \(\mathrm{Ce}^{4+}(a q)+\mathrm{Mn}^{2+}(a q) \longrightarrow \mathrm{Ce}^{3+}(a q)+\mathrm{Mn}^{3+}(a q)\). Manganese is particularly suited as a catalyst for this reaction because it can easily change between its oxidation states +2, +3, and +4, facilitating the reaction between cerium and thallium ions.

Step-by-step solution

a) Rate Law for the Uncatalyzed Reaction

Since the given reaction is thought to be an elementary and slow one, we can write the rate law directly from the balanced equation: $$ 2 \mathrm{Ce}^{4+}(a q)+\mathrm{Tl}^{+}(a q) \longrightarrow 2 \mathrm{Ce}^{3+}(a q)+\mathrm{Tl}^{3+}(a q) $$ The rate law for an elementary reaction is directly proportional to the concentrations of the reactants raised to their stoichiometric coefficients. In this case: Rate = k[\(\mathrm{Ce}^{4+}\)]^2[\(\mathrm{Tl}^{+}\)] where k is the rate constant.

b) Unusual Aspect of the Reaction and Why It Might be Slow

The unusual aspect of the uncatalyzed reaction is that it involves a bimolecular collision between two cerium ions and one thallium ion. It is rare for three ions to simultaneously collide with the correct orientation to undergo a reaction. The probability of such a collision happening is low, which would explain why this reaction is slow.

c) Rate-Determining Step for the Catalyzed Reaction

The rate law for the catalyzed reaction is given as being first order in \(\left[\mathrm{Ce}^{4+}\right]\) and first order in \(\left[\mathrm{Mn}^{2+}\right]\). That is: Rate = k[\(\mathrm{Ce}^{4+}\)][\(\mathrm{Mn}^{2+}\)] Considering the possible reactions in the mechanism: (i) \(\mathrm{Ce}^{4+}(a q)+\mathrm{Mn}^{2+}(a q) \longrightarrow \mathrm{Ce}^{3+}(a q)+\mathrm{Mn}^{3+}(a q)\) (ii) \(\mathrm{Ce}^{4+}(a q)+\mathrm{Mn}^{3+}(a q) \longrightarrow \mathrm{Ce}^{3+}(a q)+\mathrm{Mn}^{4+}(a q)\) (iii) \(\mathrm{Mn}^{4+}(a q)+\mathrm{Tl}^{+}(a q) \longrightarrow \mathrm{Mn}^{2+}(a q)+\mathrm{Tl}^{3+}(a q)\) The only reaction that matches the given rate law is the first reaction: \(\mathrm{Ce}^{4+}(a q)+\mathrm{Mn}^{2+}(a q) \longrightarrow \mathrm{Ce}^{3+}(a q)+\mathrm{Mn}^{3+}(a q)\) Therefore, the first step of the catalyzed mechanism is rate-determining.

d) Oxidation States of Manganese in the Catalytic Reaction

Manganese's special suitability to catalyze this reaction is due to its range of oxidation states, which are +2, +3, and +4. The reactions in the catalytic cycle involve Mn in all its possible oxidation states: 1. Mn²⁺ is oxidized to Mn³⁺ in step (i), then 2. Mn³⁺ is oxidized to Mn⁴⁺ in step (ii), and finally, 3. Mn⁴⁺ is reduced back to Mn²⁺ in step (iii). Manganese can easily change between these oxidation states to facilitate the reaction between cerium and thallium ions, thus acting as an effective catalyst.

Key concepts

Rate Law

In reaction kinetics, the rate law is an expression that relates the rate of a chemical reaction to the concentration of its reactants. For an elementary reaction, the rate law can be directly derived from the stoichiometry of the reaction. In the given uncatalyzed reaction between cerium and thallium ions, the balanced equation is \(2 \mathrm{Ce}^{4+}(a q)+\mathrm{Tl}^{+}(a q) \longrightarrow 2 \mathrm{Ce}^{3+}(a q)+\mathrm{Tl}^{3+}(a q)\).

This implies that the rate of the reaction can be expressed as: Rate = \(k[\mathrm{Ce}^{4+}]^2[\mathrm{Tl}^{+}]\), where \(k\) is the rate constant. This expression demonstrates that the reaction rate is directly proportional to the concentration of \(\mathrm{Ce}^{4+}\) squared and the concentration of \(\mathrm{Tl}^{+}\).

Understanding the rate law helps predict how changes in concentration affect the speed of the reaction. A higher concentration of reactants generally increases the reaction rate, making it crucial in optimizing and controlling chemical processes.

Catalysis

Catalysis is a process in which the rate of a chemical reaction is increased by the presence of a catalyst. The remarkable ability of a catalyst is that it is not consumed by the reaction and can be used repeatedly. In the reaction of cerium and thallium ions, the adding of \(\mathrm{Mn}^{2+}\) enhances the rate, providing an example of catalysis. The catalyst essentially provides an alternative reaction pathway with a lower activation energy, allowing the reaction to proceed more rapidly.

In this mechanism, the catalysis process involves multiple steps:

• First, \(\mathrm{Ce}^{4+}(a q)\) and \(\mathrm{Mn}^{2+}(a q)\) react to form \(\mathrm{Ce}^{3+}(a q)\) and \(\mathrm{Mn}^{3+}(a q)\).
• Then, \(\mathrm{Ce}^{4+}(a q)\) and \(\mathrm{Mn}^{3+}(a q)\) convert into \(\mathrm{Ce}^{3+}(a q)\) and \(\mathrm{Mn}^{4+}(a q)\).
• Finally, \(\mathrm{Mn}^{4+}(a q)\) reacts with \(\mathrm{Tl}^{+}(a q)\) to regenerate \(\mathrm{Mn}^{2+}(a q)\) and produce \(\mathrm{Tl}^{3+}(a q)\).

The first step of the catalyzed mechanism, which involves \(\mathrm{Ce}^{4+}(a q)\) and \(\mathrm{Mn}^{2+}(a q)\), is the rate-determining step because it aligns with the rate law: Rate = \(k[\mathrm{Ce}^{4+}][\mathrm{Mn}^{2+}]\). Catalysis allows reactions that might otherwise proceed at a prohibitive slow rate to occur at a useful speed.

Oxidation States

The concept of oxidation states is vital in understanding redox (reduction-oxidation) reactions. An oxidation state indicates the degree of oxidation of an atom in a chemical compound. Manganese is a particularly good catalyst for the reaction between cerium and thallium ions because of its ability to easily oscillate between different oxidation states.

Manganese can exist in several oxidation states, crucially +2, +3, and +4, as seen in the catalytic cycle:

• Initially, \(\mathrm{Mn}^{2+}\) is oxidized to \(\mathrm{Mn}^{3+}\).
• This is followed by further oxidation to \(\mathrm{Mn}^{4+}\).
• Finally, \(\mathrm{Mn}^{4+}\) is reduced back to \(\mathrm{Mn}^{2+}\).

This ability to readily switch oxidation states facilitates electron transfer processes in the reaction, making manganese an effective catalyst. It reduces the energy barrier by providing alternative pathways for electron transfer, demonstrating the importance and suitability of manganese in catalytic redox cycles.