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

- adsorption of A,
- surface reaction between the adsorbed A and adjacent site to produce adsorbed M and adsorbed N, and
- 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 à A.s

Rate of adsorption of A = r_{aA} = k_{aA}C_{A}q_{V}

Where q_{V} is the fraction of unoccupied sites surface.

Rate of desorption of A is = r_{dA} = k_{dA}q_{A}

At adsorption equilibrium, r_{aA} = r_{dA}

i.e., k_{aA}C_{A}q_{V} = k_{dA}q_{A}

q_{A} = K_{A}C_{A}q_{V} à
1

where K_{A} = k_{aA} / k_{dA}

Where q_{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 à M.s + N.s

-r_{A} = k q_{A}q_{V} à
2

Desorption of M and N:

M.s à M + s

N.s à N + s

Rate of desorption of M:

r_{dM} = k_{dM} q_{M}

rate of adsorption of M:

r_{aM} = k_{aM} C_{M}q_{V}

at desorption equilibrium,

r_{dM} = r_{aM}

i.e., k_{dM} q_{M} = k_{aM} C_{M}q_{V}

q_{M} = K_{M} C_{M}q_{V} à
3

where K_{M} = k_{aM}/k_{dM}

similarly,

q_{N} = K_{N} C_{N}q_{V} à
4

q_{V} can be written in terms of known quantities by the definition of q_{V} = 1 - (q_{A} + q_{M} + q_{N})

i.e., q_{V} = 1 - q_{A} - q_{M} - q_{N} = 1 - K_{A}C_{A}q_{V} - K_{M}C_{M}q_{V} - K_{N}C_{N}q_{V}

q_{V} + q_{V}(K_{A}C_{A} + K_{M}C_{M} + K_{N}C_{N}) = 1

q_{V} = 1 / (1 + K_{A}C_{A} + K_{M}C_{M} + K_{N}C_{N}) à
5

substituting for q_{A} and q_{V} in equation 2, we get

-r_{A} = k q_{A}q_{V }= k K_{A}C_{A}q_{V}^{2} = k K_{A}C_{A} / (1 + K_{A}C_{A} + K_{M}C_{M} + K_{N}C_{N})^{2}

The overall rate equation is

**-r _{A} = k K_{A}C_{A} / (1 + K_{A}C_{A} + K_{M}C_{M} + K_{N}C_{N})^{2}**