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Overall Rate Equation for a Heterogeneous Reaction

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Develop the overall rate equation for the gas-phase heterogeneous reaction A →
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 →
A.s

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

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

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

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

i.e., k_{aA}C_{A}θ_{V} = k_{dA}θ_{A}

θ_{A} = K_{A}C_{A}θ_{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_{A}θ_{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_{M}θ_{V}

at desorption equilibrium,

r_{dM} = r_{aM}

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

θ_{M} = K_{M} C_{M}θ_{V} →
3
where K_{M} = k_{aM}/k_{dM}

similarly,

θ_{N} = K_{N} C_{N}θ_{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_{A}C_{A}θ_{V} - K_{M}C_{M}θ_{V} - K_{N}C_{N}θ_{V}

θ_{V} + θ_{V}(K_{A}C_{A} + K_{M}C_{M} + K_{N}C_{N}) = 1
θ_{V} = 1 / (1 + K_{A}C_{A} + K_{M}C_{M} + K_{N}C_{N}) →
5
substituting for θ_{A} and θ_{V} in equation 2, we get

-r_{A} = k θ_{A}θ_{V }= k K_{A}C_{A}θ_{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}

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Last Modified on: 04-Feb-2022

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