$($a$)$ Suppose we have two gases, A and B$,$ adsorbing on the same surface, with different equilibrium

constants $K_A$ and $K_B$ :

$A(g)+SA(a)$

$B(g)+SB(a)$

Derive expressions for both the total coverage ${}_A+{}_B$ and the mole fraction of adsorbed A$,$

$x_A=\frac{{}_A}{{}_A+{}_B}.$

$($b$)$ Again assuming Langmuir$-$type behavior, for adsorption from a mixture of $N$ different gases,

obtain a simple expression for the mole fraction of gas $k$ in the adsorbed state, in terms of the

partial pressures of all the gases. $($The mole fraction of $k$ is the fraction of molecules $($or moles$)$

on the surface that are of type $k.)$