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Statistical physics assessed problem sheet

Release date Friday 26th January 2024, covering material in section 1 – 7.

1. Consider a system with three allowed states of energies of magnitude 0, 200KB , and 300KB . Find the entropy per particle of the system if it is in thermal equilibrium with a heat bath for

(a) a heat bath at 100 K

(b) a heat bath at 1,000 K,

(c) a heat bath with a temperature close to absolute zero, and

(d) a heat bath with the temperature tending to infinity. (5 marks)

2. Consider a molecule of mass m in the Earth’s atmosphere. The molecule can be treated as a system in thermal equilibrium with the rest of the atmosphere at temperature T. Show that the probability  of  the  molecule  being  in  a  microstate  at  height  Z varies  as  p ∝ e-z/λ ,  find  an expression for λ and find its approximate value for T ≈ 290 K for an oxygen molecule. The mass of an oxygen molecule is ~5.3 × 10-26 kg. (5 marks)

3. Consider an insulated box of total volume V divided into two parts by a partition wall. On one side of the partition, which has volume V0 , there are N molecules of a dilute ideal gas. The other side is initially empty. The partition is removed, and the gas expands to fill the whole box.

(a) After the expansion, what is the probability that at a given molecule will be found in the original volume V0

(b) What is the probability that all N molecules will be found in the original volume V0 ?

(c) Using the fundamental postulate of statistical mechanics, show that Ω = Ω0 N where Ω is

the statistical weight for the system after the expansion and Ω0 is the statistical weight of the system with all molecules in the original volume V0 .

(d) Hence show that the change in Boltzmann entropy is ∆SB  = NKB ln . (5 marks)