4-20. Consider a gas in equilibrium with the surface of a solid. Some of the molecules of the gas will be adsorbed onto the surface, and the number adsorbed will be a function of the pressure of the gas. A simple statistical mechanical model for this system is to picture the solid surface to be a two-dimensional lattice of M sites. Each of these sites can be either un- occupied, or occupied by at most one of the molecules of the gas. Let the partition function of an unoccupied site be 1 and that of an occupied site be q(T). (We do not need to know q(T) here.) Assuming that molecules adsorbed onto the lattice sites do not interact with each other, the partition function of N molecules adsorbed onto M sites is then M! Q(N, M, T) = N! (M -- N)! [q(T)]" The binomial coefficient accounts for the number of ways of distributing the N molecules over the M sites. By using the fact the adsorbed molecules are in equilibrium with the gas phase molecules (considered to be an ideal gas), derive an expression for the fractional coverage, 0= N/M, as a function of the pressure of the gas. Such an expression, that is, (p), is called an adsorption isotherm, and this model gives the so-called Langmuir adsorption isotherm.

4-20. Consider a gas in equilibrium with the surface of a solid. Some of the molecules of the gas will be adsorbed onto the surface, and the number adsorbed will be a function of the pressure of the gas. A simple statistical mechanical model for this system is to picture the solid surface to be a two-dimensional lattice of M sites. Each of these sites can be either un- occupied, or occupied by at most one of the molecules of the gas. Let the partition function of an unoccupied site be 1 and that of an occupied site be q(T). (We do not need to know q(T) here.) Assuming that molecules adsorbed onto the lattice sites do not interact with each other, the partition function of N molecules adsorbed onto M sites is then M! Q(N, M, T) = N! (M -- N)! [q(T)]" The binomial coefficient accounts for the number of ways of distributing the N molecules over the M sites. By using the fact the adsorbed molecules are in equilibrium with the gas phase molecules (considered to be an ideal gas), derive an expression for the fractional coverage, 0= N/M, as a function of the pressure of the gas. Such an expression, that is, (p), is called an adsorption isotherm, and this model gives the so-called Langmuir adsorption isotherm.