Statistical Thermodynamics
Homework II
Instructor: I-Chung Lu
1.
Gas molecules at a pressure low enough that gas can be considered ideal are in
contact with a two-dimensional surface at which some of the molecules may be
absorbed. The partition function for a single gas molecule is qgas, and the
partition function for a single molecule absorbed on the two-dimensional surface
is qad. If there are M adsorption sites on a two-dimensional surface, and N
identical molecules adsorbed, where M >> N so that the adsorbed molecules do
not interact with each other, the partition function for the adsorbed molecules is
𝑄(𝑁, 𝑀, 𝑇) =
𝑀!
(𝑞 )#
𝑁! (𝑀 − 𝑁)! !"
where the factorials arise from the number of ways of distributing N
indistinguishable molecules over M distinguishable adsorption sites (since they
are fixed on the surface of the graphite.)
(a) Use this partition function to develop an expression for the fraction of the M
#
adsorption sites (𝜃 = $) that are occupied as a function of the pressure of an
ideal gas, qgas, and qad.
(b) Discuss the difference between 𝜃 in low pressure and high pressure
conditions.
2.
Calculate the equilibrium constant of the reaction𝐻% + 𝐷% ⇌ 2𝐻𝐷 at room
temperature.
3.
(𝐻𝑖𝑛𝑡: ∆𝐸 ≈ 0)
Consider a solid consisting of N atoms. Treat each atom as bound to a fixed
center of force and behaving as independent harmonic oscillator of frequency 𝑣.
The binding energy of each atom in the ground state is 𝜀, so that the energy
states of single oscillator are (𝑛ℎ𝑣 + 𝜀) with n=1,2, 3…... For simplicity,
suppose that each atom can oscillate only in one dimension, and use the Einstein
model of the crystal, where u is the interaction energy for each molecule at its
equilibrium lattice site.
(a) Express the thermodynamic properties A, U, S, and Cv of this onedimensional crystal?
(b) Discuss the behavior of these equations at the limits of low temperature.
4.
(a) Please derive the Sackur-Tetrode equation in the following formula.
S = 3@2 𝑅 ln 𝑀 + 5@2 𝑅 ln 𝑇 − 𝑅 ln 𝑃 − 1.1517𝑅 (𝑃 𝑖𝑛 𝑏𝑎𝑟; 𝑀 𝑖𝑛 𝑔⁄𝑚𝑜𝑙 )
(b) Calculate the molar entropy of argon at 25℃ and 1 bar, and compare your
result with the experimental value of 154.8 JK-1mol-1.