91Ó°ÊÓ

The \(\mathrm{HF}(\mathrm{g})\) chemical laser is based on the reaction $$ \mathrm{H}_{2}(\mathrm{~g})+\mathrm{F}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{HF}(\mathrm{g}) $$ The mechanism for this reaction involves the elementary steps \(-\Delta_{\mathrm{r}} I I^{\circ} / \mathrm{kJ} \cdot \mathrm{mol}^{-1}\) at \(298 \mathrm{~K}\) (1) \(\mathrm{F}_{2}(\mathrm{~g})+\mathrm{M}(\mathrm{g}) \stackrel{\stackrel{k_{1}}{\rightleftarrows}}{\stackrel{k_{-1}}} 2 \mathrm{~F}(\mathrm{~g})+\mathrm{M}(\mathrm{g}) \quad+159\) (2) \(\mathrm{F}(\mathrm{g})+\mathrm{H}_{2}(\mathrm{~g}) \stackrel{\stackrel{k_{2}}{\rightleftarrows}}{\stackrel{k_{-2}}} \mathrm{HF}(\mathrm{g})+\mathrm{H}(\mathrm{g}) \quad-134\) (3) \(\mathrm{H}(\mathrm{g})+\mathrm{F}_{2}(\mathrm{~g}) \stackrel{k_{3}}{\Longrightarrow} \mathrm{HF}(\mathrm{g})+\mathrm{F}(\mathrm{g}) \quad-411\) Comment on why the reaction \(\mathrm{H}_{2}(\mathrm{~g})+\mathrm{M}(\mathrm{g}) \rightarrow 2 \mathrm{H}(\mathrm{g})+\mathrm{M}(\mathrm{g})\) is not included in the mechanism of the HF(g) laser even though it produces a reactant that could participate in step (3) of the reaction mechanism. Derive the rate law for \(d[\mathrm{HF}] / d t\) for the above mechanism assuming that the steady-state approximation can be applied to both intermediate species, \(\mathrm{F}(\mathrm{g})\) and \(\mathrm{H}(\mathrm{g})\)

Step by step solution

01

Identify the role of each reaction step

Analyzing the given elementary steps, we observe each step's contribution. Step (1) dissociates \(F_2\) to provide \(F\) radicals, step (2) uses \(F\) radicals to create \(HF\) and \(H\) radicals, and step (3) uses these \(H\) radicals to further produce \(HF\).

02

Explain absence of direct H2 dissociation

Although the dissociation of \(H_2\) to produce \(H\) radicals could theoretically contribute to step (3), it is not included likely because its rate or the energy required is not favorable compared to the generation of necessary \(H\) radicals via step (2). This energy consideration makes other pathways more realistic for the laser mechanism.

03

Apply steady-state approximation to intermediates

For the steady-state approximation, assume that the concentrations of intermediates \(F(g)\) and \(H(g)\) remain constant over time. Thus, set the rate of formation equal to rate of consumption for both intermediates to simplify the equations.

04

Derive steady-state expressions for intermediates

For \([F]\): rate of formation = \(k_1[F_2][M] + k_3[H][F_2]\) and rate of consumption = \(k_{-1}[F]^2[M] + k_2[F][H_2]\). Therefore, \(0 = k_1[F_2][M] + k_3[H][F_2] - k_{-1}[F]^2[M] - k_2[F][H_2]\). Then, solve for \([F]\).For \([H]\): rate of formation = \(k_2[F][H_2]\) and rate of consumption = \(k_3[H][F_2]\). Therefore, \(0 = k_2[F][H_2] - k_3[H][F_2]\). Then, solve for \([H]\).

05

Substitute steady-state expressions into rate law for HF

The rate of HF formation is given by \(d[HF]/dt = 2k_2[F][H_2] + k_3[H][F_2]\). Substitute the steady-state expressions for \([F]\) and \([H]\) obtained from step 4 into this rate expression.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Elementary Reactions

Elementary reactions are the simplest forms of chemical reactions that involve only a small number of molecules, usually two or three, directly participating in a single step. They are fundamental building blocks in complex reaction mechanisms.
These reactions follow a well-defined stoichiometry and occur in a single collision event, meaning that they directly correspond to changes at the molecular level.

• Reaction Order: An elementary reaction's order can be determined directly from its stoichiometry. For example, if the reaction involves one molecule of A and one molecule of B, it is second order with respect to each reactant.
• Rate Laws: For elementary reactions, the rate law can be written directly from the reaction equation. For instance, in the reaction \(A + B \rightarrow C\), the rate law can be expressed as \(rate = k[A][B]\).

In the context of the HF chemical laser mechanism, each step such as \(\text{F}_2 + \text{M} \rightleftharpoons 2 \text{F} + \text{M}\) is an elementary reaction. This step involves the dissociation of \(\text{F}_2\) into two \(\text{F}\) atoms, which is crucial for progressing through the overall reaction pathway.

Steady-State Approximation

Steady-state approximation is a method used to simplify the rate equations of complex reaction mechanisms, especially when dealing with intermediate species. It assumes that the concentration of reactive intermediates does not change significantly over the course of the reaction.
This approach allows us to set up equations where the rate of formation of an intermediate is equal to its rate of consumption, effectively making its net change zero.

• Application: This technique is ideal for reactions with fast initial steps that produce intermediates quickly. Once formed, these intermediates are rapidly consumed, maintaining a low and relatively constant concentration.
• Benefits: Using the steady-state approximation simplifies the mathematics involved, allowing chemists to focus on the concentrations of more stable, longer-lived species.

In our reaction mechanism for the HF laser, the intermediates are \(\text{F}(g)\) and \(\text{H}(g)\). Applying the steady-state approximation to these species involves equating their rates of formation and consumption, simplifying the path to finding the overall rate law for the reaction.

Reaction Mechanism

A reaction mechanism is a detailed step-by-step description of how a chemical reaction occurs at the molecular level. It reveals the sequence of elementary reactions that transform reactants into products.
Understanding the mechanism provides insight into how bonds are broken and formed and helps in predicting the behavior and rate of the reaction.

• Identification of Intermediates: Reaction mechanisms often involve transient species known as intermediates, which are not present in the initial or final product states.
• Energy Barriers: Each step in a mechanism has an associated energy barrier that must be overcome for that step to occur. The size of these barriers affects the rate of each step and the overall reaction.
• Role of Catalysts: Catalysts often play a crucial role in mechanisms by providing alternative pathways with lower activation energies.

In the case of the HF laser reaction mechanism, each step is an elementary reaction that builds up to the production of \(\text{HF}(g)\). The absence of the \(\text{H}_2\) dissociation step in our specified mechanism suggests that other paths are more energetically favorable for generating the necessary \(\text{H}\) radicals through \(\text{F}(g)\), emphasizing the selectivity and efficiency embedded in reaction mechanisms.