Q$1.$ The reduction of metal oxides with carbon monoxide produces a product layer comprising

of porous metal $($M$)$ surrounding an unreacted core of the metal oxide $(M{O}_2)$ :

$M{O}_2(s)+2CO(g)M(s)+2C{O}_2(g)$

The reduction reaction is first order with respect to carbon monoxide. Two samples of metal

oxide are tested under the same conditions for $45$ minutes. Sample $1$ has a particle diameter of

$5mm$ and $36\%$ conversion. Sample $2$ has a particle diameter of $2mm$ and $72\%$ conversion.

a$)$ Prove that the rate controlling mechanism at the conditions above is the surface

reaction.

b$)$ The experiments in part a$)$ were conducted in a pure $CO$ atmosphere at a pressure of

$125$kPa and a temperature of $700K.$ The atomic weight of the metal is $90\frac{g}{m}ol$ and the

density of the oxide is $6\mathrm{.}67\frac{g}{c}{m}^3.$ Using the data from sample $1,$ determine the surface

reaction rate constant $k_s^{\text{'}\text{'}}($in $\frac{m}{s}).$

c$)$ What value of effective diffusivity $($in $\frac{m^2}{s})$ would increase the time to reach $36\%$

conversion for a $5mm$ particle by $50\%$ compared with the rate being solely determined

by the surface reaction?