Consider a surface-catalyzed bimolecular reaction between...

Consider a surface-catalyzed bimolecular reaction between molecules A and B that has a rate law of the form $$ v=k_{3} \theta_{\mathrm{A}} \theta_{\mathrm{B}} $$ where $\theta_{\mathrm{A}}$ is the fraction of surface sites occupied by reactant $\mathrm{A}$ and $\theta_{\mathrm{B}}$ is the fraction of surface sites occupied by reactant B. A mechanism consistent with this reaction is as follows: $\mathrm{B}(\mathrm{g})+\mathrm{S}(\mathrm{s}) \stackrel{\stackrel{k_{a}^{\mathrm{B}}}{\Longleftrightarrow}}{\stackrel{k_{\mathrm{d}}^{\mathrm{B}}}} \mathrm{B}-\mathrm{S}(\mathrm{s}) \quad$ (fast equilibrium) $\mathrm{A}-\mathrm{S}(\mathrm{s})+\mathrm{B}-\mathrm{S}(\mathrm{s}) \stackrel{k_{3}}{\Longrightarrow}$ products Take $K_{\mathrm{A}}$ and $K_{\mathrm{B}}$ to be the equilibrium constants for Equations 1 and 2 , respectively. Derive expressions for $\theta_{\mathrm{A}}$ and $\theta_{\mathrm{B}}$ in terms of $[\mathrm{A}],[\mathrm{B}], K_{\mathrm{A}}$, and $K_{\mathrm{B}} .$ Use your results to show that the rate law can be written as $$ v=\frac{k_{3} K_{\mathrm{A}} K_{\mathrm{B}}[\mathrm{A}][\mathrm{B}]}{\left(1+K_{\mathrm{A}}[\mathrm{A}]+K_{\mathrm{B}}[\mathrm{B}]\right)^{2}} $$

Consider a surface-catalyzed bimolecular reaction between molecules A and B that has a rate law of the form
$$
v=k_{3} \theta_{\mathrm{A}} \theta_{\mathrm{B}}
$$
where $\theta_{\mathrm{A}}$ is the fraction of surface sites occupied by reactant $\mathrm{A}$ and $\theta_{\mathrm{B}}$ is the fraction of surface sites occupied by reactant B. A mechanism consistent with this reaction is as follows:
$\mathrm{B}(\mathrm{g})+\mathrm{S}(\mathrm{s}) \stackrel{\stackrel{k_{a}^{\mathrm{B}}}{\Longleftrightarrow}}{\stackrel{k_{\mathrm{d}}^{\mathrm{B}}}} \mathrm{B}-\mathrm{S}(\mathrm{s}) \quad$ (fast equilibrium)
$\mathrm{A}-\mathrm{S}(\mathrm{s})+\mathrm{B}-\mathrm{S}(\mathrm{s}) \stackrel{k_{3}}{\Longrightarrow}$ products
Take $K_{\mathrm{A}}$ and $K_{\mathrm{B}}$ to be the equilibrium constants for Equations 1 and 2 , respectively. Derive expressions for $\theta_{\mathrm{A}}$ and $\theta_{\mathrm{B}}$ in terms of $[\mathrm{A}],[\mathrm{B}], K_{\mathrm{A}}$, and $K_{\mathrm{B}} .$ Use your results to show that the rate law can be written as
$$
v=\frac{k_{3} K_{\mathrm{A}} K_{\mathrm{B}}[\mathrm{A}][\mathrm{B}]}{\left(1+K_{\mathrm{A}}[\mathrm{A}]+K_{\mathrm{B}}[\mathrm{B}]\right)^{2}}
$$

Key Concepts

Site Balance

Site balance is the concept that the sum of all the fractional coverages of the different species on a catalyst surface must equal one. This principle ensures that every catalytic site is either occupied by one of the reactants or remains vacant. It is fundamental in deriving the expressions for surface coverage from the adsorption equilibria, thus aiding in the development of accurate kinetic models for surface-catalyzed reactions.

Equilibrium Constants

Equilibrium constants quantify the position of equilibrium in adsorption/desorption processes on surfaces. They relate the concentration of the adsorbed species to the concentration of the species in the gas or liquid phase, providing a key link between the measurable concentration and the surface coverage. These constants are essential for expressing the coverage in terms of concentration in the derivation of the overall rate law.

Langmuir Adsorption Isotherm

This is a model that describes how molecules adsorb onto a surface, establishing a relationship between the concentration (or pressure) of the adsorbate in the surrounding phase and the fraction of surface sites occupied. It is based on the assumption that adsorption occurs on a fixed number of identical sites with no interactions between adsorbed molecules, and it forms the basis for deriving expressions for surface coverages in terms of the equilibrium constants and the concentrations.

Surface Reaction Rate Laws

Surface reaction rate laws describe the kinetics of reactions that occur on catalyst surfaces, taking into account the fractional coverage of reactants. In a bimolecular surface reaction, the rate is often proportional to the product of the coverages of the two reactive species. Understanding how these rates depend on the adsorbate concentrations and the equilibrium constants is crucial for linking molecular interactions on the surface to macroscopic observable rates.

Mechanistic Derivation

Mechanistic derivation involves breaking down a complex catalytic reaction into a series of elementary steps, often using equilibrium assumptions for fast steps and rate-determining approximations for slow ones. By identifying these steps, one can derive expressions for the fractions of occupied active sites (coverage) and ultimately substitute these into the rate law to obtain a relationship that includes all relevant concentrations and equilibrium constants.

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