Past exam of the chemistry A course of the University of Cambridge / 2023 / Paper 2 / A6 / d

Normalizing the geometric distribution requiresThere are $M{\theta }_j$ sites containing $j$ atoms each. Therefore the expected total population isConsequentlyThe quantity counted is the total number of atoms in every layer. The number of nonempty columns is instead $M(1-{\theta }_0)=Mx$.

Dividing the total adsorbed population by the number of underlying adsorption sites givesThis multilayer uptake can exceed one because it is a mean column height. The occupied-site fraction remains $1-{\theta }_0=bp<1$. If each layer has the same binding energy and microscopic states assumed in the single-layer calculation, the corresponding thermodynamic expression for $b$ is also the one obtained there; the kinetic ratio is the definition supplied by this multilayer model.

The model allows unlimited column height without weakening the binding or reducing the upward rate as a column grows. As $bp\to 1^{-}$, successive heights become nearly equally probable and the mean height diverges. At $bp=1$, the putative height weights cannot even be normalized; for $bp>1$ they increase with height and no finite stationary uptake exists.

This is a condensation-like failure of the unbounded-layer idealization, rather than a monolayer reaching its capacity. Finite gas supply, finite geometrical capacity, altered layer binding and interactions can modify or limit the actual uptake. The divergent expression is an equilibrium result only below its convergence threshold.