Use a kinetic argument to derive the BET isotherm for multilayer adsorption. Hint: let θi denote the surface area covered by i layers of surface molecules, where i = 0, . . . , n (n denotes the largest number of adsorbed surface layers, and 0 corresponds to the bare surface). At equilibrium, θ0 must be constant, and therefore the rate of evaporation from the first layer must be equal to the rate of condensation onto the bare surface. If k−1 and k1 denote the rates of evaporation and condensation, one can write k−1θ1 = k1Pθ0,
where P is the pressure of the system. Similarly, at equilibrium, θ1 must be constant, and k1Pθ0 + k−2θ2 = k2Pθ1 + k−1θ1. These two expressions can be combined to yield k−2θ2 = k2Pθ1. Similar arguments can be applied to successive layers to arrive at a recursive relation. You may use the fact that _∞
i=1 xi = x/(1−x) and _∞ i=1 ixi = x/(1−x)2.