Question

These data were collected for the reaction between hydrogen and nitric oxide at \(700^{\circ} \mathrm{C}\) : \(2 \mathrm{H}_{2}(g)+2 \mathrm{NO}(g) \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(g)+\mathrm{N}_{2}(g)\) $$ \begin{array}{cllc} \text { Experiment } & {\left[\mathrm{H}_{2}\right]} & {[\mathrm{NO}]} & \text { Initial Rate }(\mathrm{M} / \mathrm{s}) \\ \hline 1 & 0.010 & 0.025 & 2.4 \times 10^{-6} \\ 2 & 0.0050 & 0.025 & 1.2 \times 10^{-6} \\ 3 & 0.010 & 0.0125 & 0.60 \times 10^{-6} \end{array} $$ (a) Determine the order of the reaction. (b) Calculate the rate constant. (c) Suggest a plausible mechanism that is consistent with the rate law. (Hint: Assume the oxygen atom is the intermediate.)

Short answer

The reaction order is 2 as it is first order in both \(H_2\) and \(NO\). The rate constant \(k\) is 9.6 M^{-1} s^{-1}. Based on this rate law, a plausible mechanism is: (1) \(H_2 + NO \longrightarrow H_2O + N\), (2) \(N + NO \longrightarrow N_2 + O\), (3) \(H_2 + O \longrightarrow H_2O\), with the first step being the rate-determining step.

Step-by-step solution

Determine the Order of the Reaction

The order of the reaction can be determined by comparing the rate of reaction between different experiments. Comparing experiments 1 and 2, when the concentration of \(H_2\) is halved, the rate is also halved, indicating a first order dependence on \(H_2\). Comparing experiments 1 and 3, when the concentration of \(NO\) is halved, the rate is also halved, indicating a first order dependence on \(NO\). Thus, the reaction is second order: first order in \(H_2\) and first order in \(NO\). It can be written as: \(rate = k[H_2][NO]\).

Calculate the Rate Constant

The rate constant \(k\) can be calculated by rearranging the rate equation and inserting values from any of the experiments. For this scenario, data from experiment 1 will be used: \(k = rate / ([H_2][NO]) = 2.4 * 10^{-6} M/s / (0.010 M * 0.025 M) = 9.6 M^{-1} s^{-1}\).

Suggest a Plausible Mechanism

A plausible mechanism that is consistent with the rate law involves the formation of an intermediate species. As hinted in the question, assume that the Oxygen atom forms the intermediate. The proposed mechanism can be as follows: (1) \(H_2 + NO \longrightarrow H_2O + N\), (2) \(N + NO \longrightarrow N_2 + O\), (3) \(H_2 + O \longrightarrow H_2O\). The rate-determining step, which is the slowest step, should be the step that matches the rate law determined earlier. In this case, step 1 leads to a rate law that matches the determined rate law: \(rate = k [H_2][NO]\), thus it this can be considered as the rate-determining step.

Key concepts

Reaction Order

The reaction order indicates how the concentration of reactants affects the rate of the reaction. In this specific problem, we determine the reaction order by observing how changes in concentration influence the rate.

Comparing experiments identical apart from the concentration of one substance helps us deduce the order. In this exercise, when the concentration of hydrogen \([H_2]\) was halved (from 0.010 to 0.005 M), the rate also halved (from 2.4 to 1.2 x 10^{-6} M/s), indicating that the reaction is first order with respect to hydrogen. Similarly, halving the concentration of nitric oxide [NO] (from 0.025 to 0.0125 M) halved the rate again, indicating a first order dependence on NO. Thus, the reaction is confirmed to be first order with respect to both H₂ and NO.

Consequently, the overall reaction order is the sum of the orders for each reactant, which is 1 + 1 = 2 (second order reaction). Understanding reaction order is crucial as it informs us about the rate at which chemical reactions occur, and assists in determining the rate law.

Rate Law

The rate law is an equation that relates the rate of a reaction to the concentrations of its reactants. From the provided data and observations, we've determined that our rate law is:

rate = k[H_2][NO]

In this equation:

• 'rate' is the speed at which the reactants turn into products.
• 'k' is the rate constant, which remains constant at a fixed temperature and signifies how fast the reaction is.
• '[H_2]' and '[NO]' represent the concentrations of the reactants.

The form of this rate law emphasizes its dependence on both hydrogen and nitric oxide, which play pivotal roles in determining how quickly the reaction proceeds.

When writing the rate law, it's important to consider experimental data for each reactant's impact, instead of solely relying on balanced chemical equations.

Reaction Mechanism

The reaction mechanism provides a step-by-step description of the pathway taken by molecules to convert reactants into products. This encompasses all the individual stages leading to the final product. A mechanism that's consistent with the rate law provides deeper insights into the reactivity and sequence of events occurring on a molecular level.

In this case, the suggested mechanism aligns well with the experimentally determined rate law:

• First step: \( H_2 + NO \to H_2O + N \)
• Second step: \( N + NO \to N_2 + O \)
• Third step: \( H_2 + O \to H_2O \)

The slowest step in a reaction mechanism is the rate-determining step, which dictates the speed of the overall reaction. Here, the first step \( H_2 + NO \to H_2O + N \) is considered rate-determining because its rate law matches \( rate = k[H_2][NO] \).

Understanding mechanisms helps predict reaction behavior and assists in the design of reactions and chemical processes.

Rate Constant

The rate constant \( k \), is a fundamental part of the rate law equation and has significant importance in chemical kinetics. It quantifies the intrinsic speed of a reaction at a given temperature. In simpler terms, the rate constant is what ties together rate, concentrations, and reaction order in the rate law.

To determine the rate constant for this reaction, we utilize the data from Experiment 1:

k = \frac{Rate}{[H_2][NO]} = \frac{2.4 \times 10^{-6} M/s}{0.010 M \times 0.025 M} = 9.6 M^{-1} s^{-1}

- The units of \( k \) depend on the overall order of the reaction. For second-order reactions, as in this case, the units are typically \( M^{-1} s^{-1} \).

The rate constant is sensitive to temperature changes; thus, it varies with temperature but not with concentration. Calculating \( k \) accurately is crucial as it helps predict how fast a reaction proceeds under specific conditions.