Develop the overall rate equation for the catalytic reaction A + B à C. The effect of diffusive mass transfer may be neglected. The following steps may be considered:

- Adsorption of A and B
- Surface reaction of adsorbed A and adsorbed B

For the cases

- The product C is not adsorbed
- The product C is adsorbed

Assume step (b) is the rate controlling step.

Calculations:

**(i) Product C is not adsorbed**:

Adsorption of A and B:

A + s à A.s

B + s à B.s

Where s is the vacant site.

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

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

At equilibrium r_{aA} = r_{dA}

Therefore,

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}

Similarly, for B

q_{B}* *= K_{B} C_{B}q_{V} à 2

q_{V} = fraction of unoccupied sites = 1 - q_{A} - q_{B}

substituting for q_{A} and q_{B} from equn.1 and 2,

q_{V} = 1 - K_{A} C_{A}q_{V} - K_{B} C_{B}q_{V}

i.e.,

q_{V} = 1 / (1 + K_{A} C_{A} + K_{B} C_{B})

Surface reaction:

A.s + B.s à C + 2s

-r_{A} = k q_{A}q_{B} à 3

i.e., the overall rate equation is,

-r_{A} = k K_{A}K_{B }C_{A}C_{B} / (1 + K_{A} C_{A} + K_{B} C_{B})^{2}

(ii) Product C is adsorbable:

*Adsorption of A and B:*

A + s à A.s

B + s à B.s

Reaction between adsorbed A and adsorbed B:

A.s + B.s à C.s + s

Desorption of C:

C.s à C + s

For the above steps and some similarity to the part (i)

q_{A}* *= K_{A} C_{A}q_{V}

q_{B}* *= K_{B} C_{B}q_{V }

-r_{A} = k q_{A}q_{B }à 4

and for the desorption of C:

r_{dC} = k_{dC} q_{C}

adsorption of C:

r_{aC} = k_{aC} C_{C}q_{V}

For equilibrium desorption of C,

r_{dC} = r_{aC}

Therefore,

k_{dC} q_{C} = k_{aC} C_{C}q_{V}

q_{C} = K_{C} C_{C}q_{V}

where K_{C} = k_{aC}/k_{dC}

Here, q_{V} = 1 - q_{A} - q_{B} - q_{C}

Therefore,

q_{V} = 1 / (1 + K_{A}C_{A} + K_{B}C_{B} + K_{C}C_{C })

and q_{A} = K_{A}C_{A} / (1 + K_{A}C_{A} + K_{B}C_{B} + K_{C}C_{C })

q_{B} = K_{B}C_{B} / (1 + K_{A}C_{A} + K_{B}C_{B} + K_{C}C_{C })

substituting for q_{A} and q_{B} in equn.4,

-r_{A} = k K_{A}K_{B }C_{A}C_{B} / (1 + K_{A} C_{A} + K_{B} C_{B} + K_{C} C_{C} )^{2}