Question

The overall equation and rate law for the gas-phase decomposition of dinitrogen pentoxide are \(2 \mathrm{~N}_{2} \mathrm{O}_{5}(g) \longrightarrow 4 \mathrm{NO}_{2}(g)+\mathrm{O}_{2}(g) \quad\) rate \(=k\left[\mathrm{~N}_{2} \mathrm{O}_{5}\right]\) Which of the following can be considered valid mechanisms for the reaction? I One-step collision II \(2 \mathrm{~N}_{2} \mathrm{O}_{5}(g) \longrightarrow 2 \mathrm{NO}_{3}(g)+2 \mathrm{NO}_{2}(g) \quad[\) slow \(]\) \(2 \mathrm{NO}_{3}(g) \longrightarrow 2 \mathrm{NO}_{2}(g)+2 \mathrm{O}(g)\) [fast] \(2 \mathrm{O}(g) \longrightarrow \mathrm{O}_{2}(g)\) [fast] III \(\mathrm{N}_{2} \mathrm{O}_{5}(g) \rightleftharpoons \mathrm{NO}_{3}(g)+\mathrm{NO}_{2}(g)\) [fast] \(\mathrm{NO}_{2}(g)+\mathrm{N}_{2} \mathrm{O}_{5}(g) \longrightarrow 3 \mathrm{NO}_{2}(g)+\mathrm{O}(g) \quad\) [slow] \(\mathrm{NO}_{3}(g)+\mathrm{O}(g) \longrightarrow \mathrm{NO}_{2}(g)+\mathrm{O}_{2}(g) \quad[\) fast \(]\) \(\mathrm{IV} 2 \mathrm{~N}_{2} \mathrm{O}_{5}(g) \rightleftharpoons 2 \mathrm{NO}_{2}(g)+\mathrm{N}_{2} \mathrm{O}_{3}(g)+3 \mathrm{O}(g) \quad[\) fast \(]\) \(\mathrm{N}_{2} \mathrm{O}_{3}(g)+\mathrm{O}(g) \longrightarrow 2 \mathrm{NO}_{2}(g)\) [slow] \(2 \mathrm{O}(g) \longrightarrow \mathrm{O}_{2}(g) \quad[\) fast \(]\) \(\mathrm{V} \quad 2 \mathrm{~N}_{2} \mathrm{O}_{5}(g) \longrightarrow \mathrm{N}_{4} \mathrm{O}_{10}(g)\) [slow] \(\mathrm{N}_{4} \mathrm{O}_{10}(g) \longrightarrow 4 \mathrm{NO}_{2}(g)+\mathrm{O}_{2}(g)\) [fast]

Short answer

Only Mechanism III is valid.

Step-by-step solution

Understand the Given Rate Law

The rate law given is: \(\text{rate} = k\left[\text{N}_2 \text{O}_5 \right]\). This implies that the reaction is first-order with respect to \(\text{N}_2 \text{O}_5 \).

Evaluate Mechanism I: One-step Collision

For a one-step mechanism to be valid, the rate law derived from the stoichiometry must match the experimental rate law. However, for a single-step, a bimolecular collision would imply a second-order rate law, which contradicts the given first-order rate law. Hence, Mechanism I is not valid.

Evaluate Mechanism II

The first step: \(2 \text{N}_2 \text{O}_5 (g) \longrightarrow 2 \text{NO}_3 (g) + 2 \text{NO}_2 (g) \ [\text{slow}] \) suggests that the rate-determining step involves \( \left[ \text{N}_2 \text{O}_5 \right]^2 \), which implies a second-order rate law. This does not match the first-order rate law. Therefore, Mechanism II is not valid.

Evaluate Mechanism III

The second step: \( \text{NO}_2 (g) + \text{N}_2 \text{O}_5 (g) \longrightarrow 3 \text{NO}_2 (g) + \text{O} (g) \ [\text{slow}] \) matches the given rate law. The rate-determining step involves one molecule of \( \text{N}_2 \text{O}_5 \), giving \( \text{rate} = k \left[\text{N}_2 \text{O}_5 \right] \). Hence, Mechanism III is valid.

Evaluate Mechanism IV

The first step: \(2 \text{N}_2 \text{O}_5 (g) \rightleftharpoons 2 \text{NO}_2 (g) + \text{N}_2 \text{O}_3 (g) + 3 \text{O} (g) \ [\text{fast}] \), followed by the slow step involving \( \text{N}_2 \text{O}_3 \), would imply a rate law involving more than one molecule of \( \text{N}_2 \text{O}_5 \), which would not match the first-order rate law. Therefore, Mechanism IV is not valid.

Evaluate Mechanism V

The first step: \(2 \text{N}_2 \text{O}_5 (g) \longrightarrow \text{N}_4 \text{O}_10 (g) \ [\text{slow}] \), involves two molecules of \( \text{N}_2 \text{O}_5 \), implying a rate law of the form \( \text{rate} = k \left[\text{N}_2 \text{O}_5 \right]^2 \). This does not match the given first-order rate law. Thus, Mechanism V is not valid.

Key concepts

reaction mechanisms

Reaction mechanisms detail the step-by-step sequence through which reactants are transformed into products. For the decomposition of dinitrogen pentoxide (N2O5), the overall reaction is:
2 N2O5(g) → 4 NO2(g) + O2(g).
A mechanism must align with the observed rate law. A reaction can occur in a single step or multiple steps. These steps can be classified as 'fast' or 'slow' where the slow step is often the rate-determining step, influencing the overall reaction rate. In our case, a valid mechanism must produce the given rate law: rate = k[ N2 O5].

• Mechanism I, a one-step collision, assumes a single-step reaction. This would result in a second-order rate law (rate = k[ N2 O5] 2), which does not match our first-order rate law.
• Mechanism II suggests multiple steps, with a slow initial step involving two N2 O5 molecules. This again implies a second-order rate law, making it invalid.
• Mechanism III proposes a fast equilibrium followed by a slow step. This slow step matches the observed rate, making this mechanism valid.
• Mechanisms IV and V both imply rate laws that do not align with the given first-order rate law, thus they are invalid.

rate law

Rate law expresses the relationship between the rate of a chemical reaction and the concentration of its reactants. For a reaction: aA + bB → products, the rate law could be written as: rate = k[A]^m [B]^n. Here, 'k' is the rate constant, and 'm' and 'n' are the reaction orders with respect to each reactant, determined experimentally.
For the decomposition of N2O5, the given rate law is rate = k[ N2 O5]. This signifies the reaction is first-order with respect to N2O5, meaning the rate depends linearly on the concentration of N2O5.
Let’s evaluate:

• Mechanism I suggests a one-step collision, implying a bimolecular process, leading to a second-order rate law which contradicts the given first-order law.
• Mechanism II, with a slow step involving 2 N2 O5 molecules, suggests a second-order rate law, disqualifying it.
• Mechanism III has a slow step involving one N2 O5 molecule, correctly producing a first-order rate law, thus it fits our requirement.
• Mechanism IV and V also do not match the first-order rate law because they involve steps suggesting higher-order rate laws.

kinetics

Kinetics studies the speed of chemical reactions and the factors affecting this speed. For the decomposition of N2O5, the kinetics is governed by the rate law:
rate = k[ N2 O5].

• The observed first-order kinetics implies that the reaction rate depends directly on the concentration of N2O5.
• Total reaction time is influenced by the slowest step in the reaction mechanism, the rate-determining step.

Mechanisms:

• In Mechanism I, a single-step bimolecular collision does not align with the kinetic data, as it suggests a second-order process.
• Mechanism II’s initial slow step also predicts a second-order rate, conflicting with our first-order observation.
• Mechanism III accurately reflects the kinetic data: the slow step dictates a first-order dependence on N2O5, validating its position as the correct mechanism.
• Mechanisms IV and V also fail to align with the kinetic data due to suggested higher-order dependence on N2O5.

To conclude, understanding the rate-determining step in kinetics is essential for establishing accurate rate laws and validating reaction mechanisms.