Question

The kinetics of the reaction $$ 2 \mathrm{Fe}^{3+}+\mathrm{Sn}^{2+} \rightarrow 2 \mathrm{Fe}^{2+}+\mathrm{Sn}^{4+} $$ has been studied extensively in acidic aqueous solutions. When \(\mathrm{Fe}^{2+}\) is added initially at relatively high concentrations, the rate law is $$ R=k\left[\mathrm{Fe}^{3+}\right]^{2}\left[\mathrm{Sn}^{2+}\right] /\left[\mathrm{Fe}^{2+}\right] $$ Postulate a mechanism that is consistent with this rate law. Show that it is consistent by deriving the rate law from the proposed mechanism.

Short answer

A mechanism involving a fast equilibrium and a slow step is consistent with the observed rate law.

Step-by-step solution

Understand the Reaction and Given Rate Law

The given reaction is \( 2 \mathrm{Fe}^{3+} + \mathrm{Sn}^{2+} \rightarrow 2 \mathrm{Fe}^{2+} + \mathrm{Sn}^{4+} \). The observed rate law is \( R = k[\mathrm{Fe}^{3+}]^2[\mathrm{Sn}^{2+}]/[\mathrm{Fe}^{2+}] \). This indicates that the reaction might not be a simple one-step process as it involves \( \mathrm{Fe}^{3+} \) and \( \mathrm{Sn}^{2+} \) in the numerator and \( \mathrm{Fe}^{2+} \) in the denominator which points to a possible equilibrium or intermediate in the reaction mechanism.

Propose a Possible Mechanism

Propose a plausible reaction mechanism:1. \( \mathrm{Fe}^{3+} + \mathrm{Sn}^{2+} \rightleftharpoons \mathrm{Fe}^{2+} + \mathrm{Sn}^{3+} \) (fast equilibrium)2. \( \mathrm{Fe}^{3+} + \mathrm{Sn}^{3+} \rightarrow \mathrm{Fe}^{2+} + \mathrm{Sn}^{4+} \) (slow rate-determining step)This mechanism suggests an intermediate complex referred to as \( \mathrm{Sn}^{3+} \). The first step is in fast equilibrium while the second step is the slow rate-determining step.

Write the Rate Law from the Mechanism

From the proposed mechanism, the rate of the reaction is determined by the slow step:\[ R = k' [\mathrm{Fe}^{3+}][\mathrm{Sn}^{3+}] \]Since \( \mathrm{Sn}^{3+} \) is an intermediate, we use its concentration from the fast equilibrium first step. From the equilibrium:\[ K = \frac{[\mathrm{Fe}^{2+}][\mathrm{Sn}^{3+}]}{[\mathrm{Fe}^{3+}][\mathrm{Sn}^{2+}]} \]Solving for \( [\mathrm{Sn}^{3+}] \):\[ [\mathrm{Sn}^{3+}] = K \frac{[\mathrm{Fe}^{3+}][\mathrm{Sn}^{2+}]}{[\mathrm{Fe}^{2+}]} \]

Combine to Derive the Observed Rate Law

Substitute \( [\mathrm{Sn}^{3+}] \) from the equilibrium expression into the rate law from the slow step:\[ R = k' [\mathrm{Fe}^{3+}]\left( K \frac{[\mathrm{Fe}^{3+}][\mathrm{Sn}^{2+}]}{[\mathrm{Fe}^{2+}]} \right) \]Simplify to find:\[ R = k' K \frac{[\mathrm{Fe}^{3+}]^2 [\mathrm{Sn}^{2+}]}{[\mathrm{Fe}^{2+}]} \]Let \( k = k' K \), so we have:\[ R = k \frac{[\mathrm{Fe}^{3+}]^2 [\mathrm{Sn}^{2+}]}{[\mathrm{Fe}^{2+}]} \]This derived rate law corresponds with the given rate law, confirming the proposed mechanism.

Key concepts

Rate Law

The rate law of a chemical reaction provides insight into how the concentration of reactants affects the speed of the reaction. For the given reaction, the rate law is expressed as \( R = k[\mathrm{Fe}^{3+}]^2[\mathrm{Sn}^{2+}]/[\mathrm{Fe}^{2+}] \). This mathematical form indicates that:

• The reaction rate is directly proportional to the square of the concentration of \(\mathrm{Fe}^{3+}\).
• It is also proportional to the concentration of \(\mathrm{Sn}^{2+}\).
• Interestingly, the concentration of \(\mathrm{Fe}^{2+}\) appears in the denominator, suggesting it plays a unique role in possibly stabilizing an intermediate state.

Understanding this equation helps identify which steps are important within the reaction mechanism and how each component affects the overall process.

Reaction Mechanism

A reaction mechanism is a detailed stepwise description of the transformation of reactants into products. For the given reaction, we postulate a mechanism consisting of the following two steps:

• The first step is a fast equilibrium reaction: \( \mathrm{Fe}^{3+} + \mathrm{Sn}^{2+} \rightleftharpoons \mathrm{Fe}^{2+} + \mathrm{Sn}^{3+} \). This suggests that reactant molecules briefly form products and intermediates, quickly reaching equilibrium.
• The second step is identified as the slow, rate-determining step: \( \mathrm{Fe}^{3+} + \mathrm{Sn}^{3+} \rightarrow \mathrm{Fe}^{2+} + \mathrm{Sn}^{4+} \). The formation of \(\mathrm{Sn}^{4+}\) is the critical point that dictates the overall reaction speed.

This mechanism illustrates the sequential interactions and transformations, confirming its validity by equating to the observed rate law. Understanding each part of this sequence is crucial for predicting how changes in conditions will affect the reaction.

Intermediate Complex

An intermediate complex is a transient, often unstable entity that forms during a reaction mechanism. In the proposed mechanism, \( \mathrm{Sn}^{3+} \) acts as this intermediate. It plays a vital role:

• Formed during the first equilibrium phase, it exists if the conditions favor the establishment of equilibrium.
• Involved in the slow, rate-determining step, it is consumed to form final products.
• Due to its transient nature, its concentration is low, influencing how it's represented in the rate law derived from the mechanism.

The intermediate complex \( \mathrm{Sn}^{3+} \) is often crucial in bridging initial reactants to final products. Understanding these intermediates can help chemists manipulate reactions for desired outcomes, optimizing efficiency and yield.