Problem $1.$ Langmuir Isotherm for Non$-$dissociative and Dissociative Adsorption Processes $(30$ points$)$

The Langmuir isotherm is a fundamental importance in heterogeneous catalysis since it provides the

starting basis for solid$-$catalyzed reactions involving gases and liquids. According to Langmuir theory, if A

denotes a specie in the gas bulk, and s$*$ is an active site on an adjacent catalytically active solid surface,

then the adsorption of $A$ on the active site is described by the following equilibrium reaction where As$*$

denotes the chemisorbed specie.

$A+s^{**}=A{s}^{**}$

$($a$)$ If $k_a$ is the adsorption rate coefficient and $k_d$ is the desorption rate coefficient, derive the

equation that shows the expression for the concentration of adsorbed $A$ at equilibrium is

defined by

$C_{A{S}^{**}}=\frac{C_t{K}_A{C}_A}{1+K_A{C}_A}$

where $K_A$ is the adsorption equilibrium constant and $C_T$ is the total concertation of sites $C_T=C_{S^{**}}+C_{As}^{**}.$

What is the expression for the fractional coverage $$ of the occupied site?

$($b$)$ Suppose that a diatomic species, such as $H_2$ or $O_2,$ undergoes dissociative adsorption on two

vacant sites $2s^{**}$ to form an adsorbed specie $2As**.$ If this diatomic specie is called $A_2,$ then the

surface adsorption process would be described by the following surface equilibrium equation

$A_2+2s^{**}=2A{s}^{**}$

Using an approach similar to the one used for part a$,$ show that the expression for the

concentration of adsorbed $A$ at equilibrium is now defined by the following expression

$C_{AS**}=\frac{C_t\sqrt[\phantom{2}]{K_A{C}_A}}{1+\sqrt[\phantom{2}]{K_A{C}_A}}$

What is the expression for the fractional coverage $$ of the occupied site for this case?

$($c$)$ Suppose that data on the fractional coverage $$ was available at various values of $C_A.$ Show how

one would linearize the two above derived expressions for the fractional coverage $$ to identify

the adsorption equilibrium constant $K_A.$