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

Let ${\epsilon }_s$ be the energy of one adsorbed atom, using the same energy reference as the gas-phase value ${\epsilon }_0$. For equivalent, noninteracting adsorption sites, choosing the occupied positions givesEvery configuration has energy $N_1{\epsilon }_s$, so the canonical lattice-gas partition function isThere is no further $1/N_1!$ factor: the combinatorial choice already counts indistinguishable adsorbates correctly. The Helmholtz free energy, using Stirling's approximation for a macroscopic number of occupied and empty sites, isHence the adsorbed-phase chemical potential at fixed $T,M$ isThe logarithm comes from configurational entropy: adding an atom is less favorable when vacancies are scarce. In the gas, the stipulated expression can be writtenwhere $\Lambda$ is the thermal de Broglie wavelength. Equality of chemical potentials at chemical equilibrium givesThereforeThe binding energy $E_b={\epsilon }_0-{\epsilon }_s$ is positive for favorable binding. The units of $b$ are inverse pressure, making $bp$ dimensionless. Choosing the adsorbed energy as zero simply sets ${\epsilon }_s=0$ and leaves the same physics; it must not remove the relative binding energy. This is the Langmuir isotherm, with low-pressure linear uptake and high-pressure monolayer saturation.

At chemical equilibrium, adsorption and desorption balance:Canceling $M$ and rearranging givesThus $K_{\mathrm{ads}}=b$ when the kinetic and thermodynamic descriptions refer to the same ideal surface and energy convention. In particular,The ratio fixes the adsorption isotherm, while the individual rate constants also determine relaxation toward equilibrium. Here $k_a$ has units of inverse pressure per time and $k_d$ of inverse time, so $K_{\mathrm{ads}}$ is dimensional. This equivalent-site arrival/departure argument is the simple case of the Langmuir adsorption model.^[ref]

For the uniform-binding multilayer adsorption model, ${\theta }_j$ describes sites of exactly height $j$, not the fraction whose height is at least $j$. At equilibrium the upward and downward fluxes on each adjacent pair of heights balance:For completeness, stationarity alone gives a height-independent probability current; the reflecting boundary at height zero sets that current to zero. This justifies pairwise balance for a normalizable equilibrium distribution.

Writing $x=K_{\mathrm{ads}}p$ gives the recurrence ${\theta }_j=x{\theta }_{j-1}$. Successive substitution then yieldsAn equilibrium geometric distribution of unbounded heights requires $0\le x<1$. The first-layer-only Langmuir adsorption model instead truncates the allowed heights at one, which changes the normalization and removes this convergence restriction.

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.