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

Thermal Physics - Maxwell-Boltzmann Statistics

Consider a system of N non-interacting particles confined in a volume V at a temperature such that the particles obey classical Boltzmann statistics. If the temperature is lowered to the point at which quantum effects become important, the pressure of the gas may differ depending on whether the particles are fermions or bosons. Let P_F be the pressure exerted by the particles if they are fermions. P_B be the pressure if they are bosons and, and P_C be the pressure the particles would exert if quantum effects are ignored. Which of the following is true?

A.
B.
C.
D.
E.

(GR8677 #88)

Solution:

Ideal gas:

Classical (Maxwell-Boltzmann Statistics):




For Bosons (photon):



→

For Fermion:
= Fermi Energy,

For the same temperature:  

Answer: B

Thermal Physics - Maxwell-Boltzmann Statistics

In a Maxwell-Boltzmann system with two states of energies ε and 2ε respectively, and a degeneracy of 2 for each state, the partition function is

A. e ^{−ε / kT}
B. 2e^{−2ε / kT}
C. 2e^{−3ε / kT}
D. e^{−ε / kT} + e^{−2ε / kT}
E. 2(e^{−ε / kT}+ e^{−2ε / kT})

(GR0177 #49)

Solution:

Canonical partition function:  Z = Σ_i e^−βE_i
β = 1/kT

For Maxwell-Boltzmann System:  Z = Σ_i g_i e^−βE_i
g_i = degeneracy

Thus, the partition function is
Z = 2e^−βε + 2e^−2βε = 2(e^{−ε / kT} + e^{−2ε / kT})

Answer: E

Thermal Physics - Maxwell-Boltzmann Statistics

An ensemble of system is in thermal equilibrium with a reservoir for which kT = 0.025 eV. State A has an energy that is 0.1 eV above that of state B. If it is assumed the systems obey Maxwell-Boltzmann statistics and that the degeneracies of the two states are the same, then the ratio of the number of the systems in state A to the number in state B is

A. e⁺⁴
B. e^+0.25
C. 1
D. e^-0.25
E. e^-4

(GR0177 #77)

Solution:

Maxwell-Boltzmann distribution:





Answer: E