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.

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]

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.