Q5(b). Langmuir isotherm at very low and very high pressure

Assumptions and derivation

1. Adsorption stops at a monolayer; all sites are equivalent; the chance of adsorption at a site does not depend on whether neighbouring sites are occupied (no interaction between adsorbed molecules); adsorption and desorption are in dynamic equilibrium.
2. With N sites and fractional coverage θ: rate of adsorption =kₐP N(1-θ) (gas pressure times vacant sites); rate of desorption =k_dNθ.
3. At equilibrium kₐP(1-θ)=k_dθ. With K=kₐ/k_d: θ=(KP)/(1+KP)
4. In terms of the volume adsorbed, θ=V/V_∞, the linear form is P/V=(P)/(V_∞)+(1)/(KV_∞): a plot of P/V against P is a straight line, slope 1/V_∞.

(i) Very low pressure

1. KP≪1, so 1+KP≈1: θ=KP
2. Coverage rises linearly with pressure (the initial straight part of the isotherm, like Henry's law). Most sites are empty, and each new molecule finds a vacant site.
3. For a surface-catalysed decomposition, rate =kθ=kKP: first order in the gas (PH₃ on W at low pressure).

(ii) Very high pressure

1. KP≫1, so 1+KP≈ KP: θ→1
2. The surface is saturated with a complete monolayer; more pressure adds nothing. The isotherm is flat.
3. Rate =kθ=k: zero order, independent of pressure (NH₃ on W, PH₃ on W at high pressure).
4. In between, θ≈ KPⁿ with 0<n<1, which is the Freundlich form.

Application. the Langmuir form gives the kinetics of heterogeneous catalysis (Haber process on iron, hydrogenation on nickel) and, by fitting V_∞, a first estimate of the surface area of an adsorbent.