 Solution

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.