### 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:

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:

A + s → A.s

Rate of adsorption of A = raA = kaACAθV

Where θV is the fraction of unoccupied sites surface.

Rate of desorption of A is = rdA = kdAθA

At adsorption equilibrium, raA = rdA

i.e., kaACAθV = kdAθA

θA = KACAθV → 1

where KA = kaA / kdA

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

A.s + s → M.s + N.s

-rA = k θAθV 2

Desorption of M and N:

M.s → M + s

N.s → N + s

Rate of desorption of M:

rdM = kdM θM

raM = kaM CMθV

at desorption equilibrium,

rdM = raM

i.e., kdM θM = kaM CMθV

θM = KM CMθV → 3

where KM = kaM/kdM

similarly,

θN = KN CNθV → 4

θV can be written in terms of known quantities by the definition of θV = 1 - (θA + θM + θN)

i.e., θV = 1 - θA - θM - θN = 1 - KACAθV - KMCMθV - KNCNθV

θV + θV(KACA + KMCM + KNCN) = 1

θV = 1 / (1 + KACA + KMCM + KNCN) → 5

substituting for θA and θV in equation 2, we get

-rA = k θAθV = k KACAθV2 = k KACA / (1 + KACA + KMCM + KNCN)2

The overall rate equation is

-rA = k KACA / (1 + KACA + KMCM + KNCN)2