Overall Rate Equation for a Heterogeneous Reaction

Develop the overall rate equation for the gas-phase heterogeneous reaction A → M + N, considering the following steps:

1. adsorption of A,
2. surface reaction between the adsorbed A and adjacent site to produce adsorbed M and adsorbed N, and
3. desorption of M and N

Assume step (ii) is controlling.

Calculations:

If s is taken as the active catalyst site,

The steps taking place are:

Adsorption of A:

A + s → A.s

Rate of adsorption of A = r_aA = k_aAC_Aθ_V

Where θ_V is the fraction of unoccupied sites surface.

Rate of desorption of A is = r_dA = k_dAθ_A

At adsorption equilibrium, r_aA = r_dA

i.e., k_aAC_Aθ_V = k_dAθ_A

θ_A = K_AC_Aθ_V → 1

where K_A = k_aA / k_dA

Where θ_A is the fraction of the surface covered by adsorbed species of A.

Surface reaction between adsorbed A (i.e. A.s), and adjacent site to produce M and adsorbed N:

A.s + s → M.s + N.s

-r_A = k θ_Aθ_V → 2

Desorption of M and N:

M.s → M + s

N.s → N + s

Rate of desorption of M:

r_dM = k_dM θ_M

rate of adsorption of M:

r_aM = k_aM C_Mθ_V

at desorption equilibrium,

r_dM = r_aM

i.e., k_dM θ_M = k_aM C_Mθ_V

θ_M = K_M C_Mθ_V → 3

where K_M = k_aM/k_dM

similarly,

θ_N = K_N C_Nθ_V → 4

θ_V can be written in terms of known quantities by the definition of θ_V = 1 - (θ_A + θ_M + θ_N)

i.e., θ_V = 1 - θ_A - θ_M - θ_N = 1 - K_AC_Aθ_V - K_MC_Mθ_V - K_NC_Nθ_V

θ_V + θ_V(K_AC_A + K_MC_M + K_NC_N) = 1

θ_V = 1 / (1 + K_AC_A + K_MC_M + K_NC_N) → 5

substituting for θ_A and θ_V in equation 2, we get

-r_A = k θ_Aθ_V = k K_AC_Aθ_V² = k K_AC_A / (1 + K_AC_A + K_MC_M + K_NC_N)²

The overall rate equation is

-r_A = k K_AC_A / (1 + K_AC_A + K_MC_M + K_NC_N)²