Q4(c). Pressure dependence of the order of a surface-catalysed decomposition

Plausible mechanism

1. The reactant A is adsorbed quickly and reversibly on a vacant surface site S, and the adsorbed molecule decomposes slowly; the products are weakly held and leave at once: 4pt] A(g) + S ⇌A-S (fast, equilibrium) A-S →products + S (slow)
2. Langmuir assumptions: one layer (monolayer); all sites equivalent, one molecule on each; no interaction between adsorbed molecules; adsorption and desorption in dynamic equilibrium.
3. With θ the fraction of sites covered, adsorption needs a vacant site and desorption an occupied one: kₐ p (1-θ)=k_d θ. With K=kₐ/k_d: θ=(Kp)/(1+Kp) (Langmuir isotherm)
4. Only adsorbed molecules react, so the rate is proportional to θ: v=-(dp)/(dt)=kθ=(kKp)/(1+Kp)

How the order changes with pressure

• Low pressure, Kp≪1: θ≈ Kp, so v=kKp: first order. The surface is nearly bare and the number of adsorbed molecules is proportional to the pressure. A weakly adsorbed gas (small K) stays in this region, for example HI on platinum.
• High pressure, Kp≫1: θ→1, so v=k: zero order. Every site is occupied and more gas cannot add molecules to the surface. A strongly adsorbed gas reaches this region at low pressures, for example HI on gold. PH₃ on tungsten shows both limits: first order at low pressure and zero order at high pressure.
• Intermediate pressure: the order is n=(dln v)/(dln p)=(1)/(1+Kp), which falls smoothly from 1 to 0; at Kp=1 it is ½. Over a limited range the rate fits v=k'pⁿ with 0<n<1, the fractional order that corresponds to a Freundlich isotherm, θ=c₁p^(1/c₂).
• Tests: 1/v=1/k+1/kK·1/p, so 1/v against 1/p is a straight line (intercept 1/k, slope 1/kK). The half-life is constant at low pressure (2/kK) but proportional to the starting pressure at high pressure (p₀/2k).
• Temperature: at low pressure the measured rate constant is kK, so the apparent activation energy is the true one plus Δ_(ads)H (negative), and is lower; at high pressure it is the true value.

Product inhibition: if a product P is strongly adsorbed on the same sites, θ_A=(K_Ap_A)/(1+K_Ap_A+K_Pp_P) and, for K_Pp_P≫1+K_Ap_A, v≈(kK_Ap_A)/(K_Pp_P): first order in A and order -1 in the product. The decomposition of NH₃ on platinum at 1000°C, slowed by strongly adsorbed H₂, follows -dp_(NH₃)dt=k_cp_(NH₃)p_(H₂).

Application. the same coverage argument with two adsorbed reactants, v=kθ_Aθ_B (Langmuir–Hinshelwood mechanism), describes the oxidation of CO to CO₂ on platinum, one of the reactions in automobile catalytic converters.