Exercise $5\mathrm{.}14$: Competitive adsorption and reaction rate

Consider the following heterogeneous reaction in which A and B associatively

adsorb on a catalyst surface and undergo reaction to product $P,$ which subse$-$

quently desorbs.

$A+x{}_{k_{-1}}^{k_1}A-x$

$B+x{}_{k_2}^{k_{-2}}B-x$

$A-x+B-x={?}^{k_3}P-x+x$

$P+x{}_{k_{-4}}^{k_4}P-x$

At $373K$ the equilibrium constants for adsorption are

$K_1=190,000,K_2=580,000,K_4=75,000,c\frac{m^3}{m}ol$

The heats of adsorption are

$H_1=-20,000,H_2=-30,000,H_4=-12,000,ca\frac{l}{m}ol$

The rate constant for $r_3$ is

$k_3=7\mathrm{.}6{10}^{28}$exp$(-\frac{15000}{T}),\frac{(gca{t}^2)}{(s)(c{m}^3)(mol)}$

where $T$ is in Kelvin.

$($a$)$ Develop a Langmuir$-$Hinshelwood rate expression $($i$.$e$.,$ the rate is ex$-$

pressed in terms of gas$-$phase components$)$ for the surface reaction

when you assume the surface reaction is the slow step and the adsorp$-$

tion and desorption processes are at equilibrium.

$($b$)$ Determine the magnitude of rate of the surface reaction $(r_3)$ at $373K$ if

the gas contains a $50$:$50$ mixture of A and B at a pressure of $1\mathrm{.}0$atm.

What is the value of the rate if a $50$:$10$:$40$ mixture of $A$:$B$:$P$ is present

instead? $C_m=0\mathrm{.}0008mo\frac{l}{g}$ cat.