(a) Calculate the particle partition function for a ficti-
tious system of molecules at 80 K in which the accessi-
ble energies are 0, 2, 6, and 12 quanta, and the dege-
neracies are I, 3, 5, and 7, respectively. Each quantum
is 5.0 X 10-22 J.
(b) Which level is the most highly populated at
equilibrium?
(c) What fraction of the molecules occupies the most
populated level?

Consider a gas in equilibrium with the surface of a solid. Some of the molecules will be adsorbed onto the surface, depending upon the gas pressure. A simple statistical model accounting for this treats the surface as a two-dimensional lattice of M sites, each of which could be unoccupied or occupied by at most one of the molecules of the gas. Let the partition function of an unoccupied site be 1 and of an occupied site be q(T). Assuming the adsorbed molecules do not interact with each other, the partition function of N molecules adsorbed onto M sites is Q(N, M, T) = M(M-N)! The binomial coefficient prefactor accounts for the number of ways of distributing N molecules among M sites. By using the fact the adsorbed molecules are in equilibrium with the gas phase molecules (considered to be an ideal gas), show that the fractional coverage, θ-N/M, as a function of the gas pressure can be expressed as 1+aP' and derive an expression for α. The fractional coverage as a function of pressure (at fixed temperature) is called an adsorption isotherm, and this model gives the so-called Langmuir adsorption isothernm Hint: since the molecules on the surface and in the gas are in equilibrium, their chemical poten- tials must be equal, and chemical potentials can be evaluated from the partition function. Write the particle partition function for an ideal gas as qideal -c(T)V (using the same arguments presented in Question 10). The use of Stirling's approximation, In N! ~ NInN-N, may alsd be required