Catalyst equation help
A catalyst consisting of palladium on an α-Al2O3 support, Pd/α-Al2O3, has been used for the oxidation of CO at room temperature:

Equation 1
CO(g) + ½O2(g) = CO2(g)

Under certain conditions, the oxidation reaction was found to involve competitive adsorption of the reactants with CO being non-dissociatively adsorbed and oxygen undergoing dual-site adsorption:

Equation 2
ka
CO(g) + * ↔ CO(ad)
Kd

Equation 3
ka’
O2(g) + 2* ↔ O(ad) + O(ad)
kd’

The rate-limiting step is then a bimolecular reaction between CO(ad) and O(ad):

Equation 4
CO(ad) + O(ad) → CO (ad)

Carbon dioxide can be assume to be weakly adsorbed and, as a consequence, desorbs as quickly as it is formed.

(i) For the competitive adsorption of CO show that the following expression can be derived by equating the rates of adsorption and desorption of CO:

Equation 5
θCO = bCOpCO(1 – θCO – θO)

Where θCO and θO are the fractional surface coverages of CO and O, respectively. The quantity bCO (= ka/kd) is the adsorption coefficient for CO and pCO is the partial pressure of CO. (Hint In your working you should represent the total number of adsorption sites by N and provide expressions for both the rate of adsorption and the rate of desorption of CO.)

(ii) Equation 5 can be used in a more detailed analysis of the mechanism to derive the following two expressions:

Equation 6
θCO = (bCOpCO) / (1 + bCOpCO + (bO2pO2)1/2)

and

Equation 7
θO = θCO((bO2pO2)1/2 / bCOpCO)

where b02(= ka’ / kd’) and pO2 are, respectively, the adsorption coefficient and partial pressure of O2.

Given Equations 6 and 7, in conjunction with the information about the rate-limiting step at the start of this question, show that the theoretical rate equation takes the form:

Equation 8
r = (kθbCOpCO(bO2pO2)1/2) / {1 + bCOpCO + (bO2pO2)1/2}^2

(iii) The experimental rate equation for the CO oxidation reaction, under conditions for which the mechanism given in part (ii) is valid, takes the form:

Equation 9
r = (kR(pO2)1/2) / pCO

How can this result be rationalised in terms of the theoretical rate equation (Equation 8) that has been derived for the mechanism?