The hydrogenation of ethene on copper obeys the rate law v=(k[H2]^1 / 2[C2 H4])/((1+K[C2 H4])^2)

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The hydrogenation of ethene on copper obeys the rate law...

The hydrogenation of ethene on copper obeys the rate law $$ v=\frac{k\left[\mathrm{H}_{2}\right]^{1 / 2}\left[\mathrm{C}_{2} \mathrm{H}_{4}\right]}{\left(1+K\left[\mathrm{C}_{2} \mathrm{H}_{4}\right]\right)^{2}} $$ where $k$ and $K$ are constants. Mechanistic studies show that the reaction occurs by the Langmuir-Hinshelwood mechanism. How are $k$ and $K$ related to the rate constants for the individual steps of the reaction mechanism? What can you conclude about the relative adsorption of $\mathrm{H}_{2}(\mathrm{~g})$ and $\mathrm{C}_{2} \mathrm{H}_{4}$ (g) to the copper surface from the form of the observed rate law?

The hydrogenation of ethene on copper obeys the rate law
$$
v=\frac{k\left[\mathrm{H}_{2}\right]^{1 / 2}\left[\mathrm{C}_{2} \mathrm{H}_{4}\right]}{\left(1+K\left[\mathrm{C}_{2} \mathrm{H}_{4}\right]\right)^{2}}
$$
where $k$ and $K$ are constants. Mechanistic studies show that the reaction occurs by the Langmuir-Hinshelwood mechanism. How are $k$ and $K$ related to the rate constants for the individual steps of the reaction mechanism? What can you conclude about the relative adsorption of $\mathrm{H}_{2}(\mathrm{~g})$ and $\mathrm{C}_{2} \mathrm{H}_{4}$ (g) to the copper surface from the form of the observed rate law?

Key Concepts

Dissociative Adsorption and Reaction Order

For some reactants, such as diatomic molecules, the adsorption on a metal surface may be dissociative. This process can lead to fractional reaction orders, as the effective concentration of adsorbed species scales as a fractional power of the gas-phase partial pressure. The half-order dependence on one reactant in the rate law is indicative of a dissociative adsorption process, revealing details about the molecularity of the adsorption step.

Interrelation of Rate Constants in the Mechanism

In a multi-step catalytic process, the observed rate constant (k) in the overall rate law is typically a composite of the individual rate constants for the elementary steps, including adsorption, surface reaction, and desorption. Similarly, the experimentally determined adsorption equilibrium constant (K) relates directly to the intrinsic rate constants of adsorption and desorption. Understanding these relationships helps in discerning which step controls the overall reaction rate and in predicting how changes in conditions might affect the kinetics of the reaction.

Langmuir-Hinshelwood Mechanism

This mechanism describes heterogeneous catalytic reactions in which both reactants first adsorb on a solid catalyst surface before reacting with each other. It involves the adsorption of reactant molecules, surface reactions between adsorbed species (which often include the rate?determining step), and then desorption of the product. The rate law is derived by considering these steps and applying the principles of adsorption isotherms and reaction kinetics on the surface.

Adsorption Equilibrium Constant

In heterogeneous catalysis, the adsorption equilibrium constant (often denoted as K) is defined as the ratio of the adsorption rate constant to the desorption rate constant. It reflects the affinity of a reactant for the catalyst surface. A high value of K indicates strong adsorption. This concept is crucial when deriving the rate law under the assumption that adsorption–desorption steps are in rapid pre-equilibrium relative to the slower surface reaction.

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