Reference no: EM132291624

Assignment - Problem Set

1. Derive Stirling's approximation. That is, show that in the limit of large N,

lnN! ∼ N ln N - N

Hint: You will need to approximate a particular sum by an integral. Argue that the difference between adjacent N values is extremely small compared to N:

ΔN/N << 1.

Thus, ΔN = 1 → dN, i.e. ΔN is effectively infinitesimal.

2. The canonical partition function Q for an N-particle ideal gas can by related to the partition function of a single gas particle q as

Q = q^N/N!

The canonical partition function for a mixture of two gases with N_A particles of type A and N_B particles of type B is

Q = Q_AQ_B = (q_A^N_A/N_A!)(q_B^N_B/N_B!)

Show that the average energy of the ideal gas mixture is the sum of the average energies for each gas component separately.

3. It can be shown that the canonical partition function of an N-particle monatomic ideal gas confined to a container of volume V at temperature T is given by

Q(N, V, T) = 1/N!((2πmk_BT)/h²)^(3/2)N V^N

Use this partition function to derive an expression for the average energy and the constant volume heat capacity of the monatomic ideal gas. Note that in classical thermodynamics these quantities were simply given. Your calculations show that these quantities have their origins in the specific form of the partition function for the ideal gas.

4. Calculate the probability that at 300 K a given particle in a 2D square box with a length of 1 cm is found in

a) The quantum state (n_x, n_y) = (3, 1).

b) The energy level ε₃.