Heterogeneous catalysis and surface reaction models

Coverage and the rate basis belong to the mechanism

In a stipulated competitive mean-field Langmuir–Hinshelwood model, both A and B adsorb on equivalent sites. With dimensionless weights w_A=K_Aa_A and w_B=K_Ba_B, vacant fraction is 1/(1+w_A+w_B), and coverages are θ_A=w_A/(1+w_A+w_B) and θ_B=w_B/(1+w_A+w_B). Fast adsorption is treated as quasi-equilibrated; the slower forward reaction has supplied total-site turnover r_site=k_LHθ_Aθ_B.

A different Eley–Rideal model uses one adsorbed reactant and a directly arriving reservoir reactant, with its separately stipulated rate normalization. The same numeric coefficient in two models does not prove a measured performance advantage or identify a real mechanism. Adjacent-site correlations, multisite binding and heterogeneous sites can invalidate a simple product-of-coverages law.

More feed can block a needed partner

For a forward-only illustrative pair reaction, set k_LH=4.00 s⁻¹ on the total-site basis and weights w_A=1.00,w_B=2.00. Coverages are 0.250 and 0.500, so turnover is 0.500 product events per total site per second. With accessible site amount 2.00 × 10⁻⁵ mol, the model product rate is 1.00 × 10⁻⁵ mol s⁻¹.

A blocking product weight w_P adds to the denominator. At w_P=4.00, θ_A=0.125, θ_B=0.250 and forward turnover is 0.125 s⁻¹. This example stipulates negligible reverse chemistry; site blocking is not the same mechanism as a reverse reaction. With w_B fixed at 2 and no product weight, turnover 4w_Aw_B/(1+w_A+w_B)² peaks at w_A=3 rather than increasing without limit.

The surface can see a different concentration from the bulk

For a nonporous isothermal first-order surface model, external film flux is k_m(c_bulk−c_surface) and intrinsic surface flux is k_s c_surface. Here both coefficients have units m s⁻¹ and use the same external area. At no accumulation, equate these fluxes rather than insert the bulk concentration into the intrinsic expression automatically.

The result is c_surface=k_m c_bulk/(k_m+k_s) and k_overall=(1/k_m+1/k_s)⁻¹. If film transport is slow, increasing intrinsic surface speed can have a limited effect on the observed flux. A mass-based coefficient requires conversion with the relevant specific area; heat and internal-pore effects would require additional balances.

An internal diffusion model introduces another convention

For a uniform spherical isothermal first-order pellet, define φ=R_p√(k_v/D_e), with pellet radius R_p, intrinsic volume-normalized coefficient k_v in s⁻¹ and effective diffusivity D_e in m² s⁻¹. The effectiveness is η_eff=3(φ cothφ−1)/φ² relative to the entire pellet at its outer-surface concentration. Its limit approaches 1 as φ approaches 0.

For this specific model, η_eff≤1. Nonisothermal or other kinetic models need different conclusions. The surface concentration is not necessarily the bulk concentration, and an observed apparent coefficient is not the intrinsic k_v. A source convention based on R_p/3 would give a parameter three times smaller; its formula must be converted consistently rather than combined with this radius convention.

Worked examples

Try it yourself

1. A turnover of 0.500 s⁻¹ acts on 2.00 × 10⁻⁵ mol of accessible sites. What amount rate follows on that basis?

   Show the answer and reasoning

   1.00 × 10⁻⁵ mol s⁻¹.

   Multiply turnover by its defined site amount.

2. Is competitive product blocking the same assumption as a reverse chemical event?

   Show the answer and reasoning

   No.

   Adsorption occupancy and reaction direction are separate parts of the mechanism.

3. Can an m s⁻¹ film coefficient be combined directly with an unrelated mass-normalized catalyst coefficient?

   Show the answer and reasoning

   No.

   Their area and amount bases must be converted consistently.

4. Does η_eff≤1 hold for every nonisothermal catalytic system?

   Show the answer and reasoning

   No.

   That bound here belongs to the specified uniform isothermal first-order pellet.

What to carry forward

• Define sites, coverage and event normalization.
• A compatible rate law does not uniquely identify a mechanism.
• Transport can separate bulk and surface conditions.
• Effectiveness depends on geometry, coefficient basis and model assumptions.