Thermal Physics - Maxwell-Boltzmann Statistics

A large isolated of N weakly interacting particles is in thermal equilibrium. Each particle has only 3 possible non degenerate states of energies 0, ε, and 3ε. When the system is at an absolute temperature T ≫ ε/k, where k is Boltzmann’s constant, the average energy of each particle is

A. 0
B. ε
C. 4ε /3
D. 2ε
E. 3ε

(GR8677 #67)

Solution:

In thermal equilibrium:
→ Each particle has roughly equal probability of being in any of the three states.
→ The average energy of each particle is total energy/3 = (0+ε+3ε)/3 = 4ε/3

Answer: C

Note:

Complete Calculation:

T ≫ ε/k → 1 ≫ ε/kT 
ε/kT → 0
e^-ε/kT → e⁰  = 1

Partition function: Z = ∑_i e^-ε_i/kT
i = 3 → Z = 3

Probability: P_i = e^-ε_i/kT/Z
→ P₁ = P₂  = P₃  = 1/3

Energy: E = ∑_i E_iP_i
→ E = E₁P₁ + E₂P₂ + E₃P₃ = (0+ε+3ε)/3 = 4ε/3