Thermal Physics - Heat Capacity

For an ideal diatomic gas in thermal equilibrium, the ratio of the molar heat capacity at constant volume at very high temperatures to that at very low temperatures is equal to

A. 1
B. 5/3
C. 2
D. 7/3
E. 3

(GR9677 #79)

Solution:

C_V  for diatomic gas: C_V  = C_Vtrans + C_Vrot  + C_Vvib

For very low T, only translational component contributes
→ C_V(T≪) = ³/₂ Nk

For very high T, all 3 components (translational, vibrational, rotational) contribute
→ C_V(T≫) = (³/₂ + 1 + 1) R = ⁷/₂ Nk

Ratio C_V(T≪) / C_V(T≫)  = 7/3

Answer: D

Thermal Physics - Diatomic Molecule

In a gas of N diatomic molecules, two possible models for a classical description of a diatomic molecule are:



Which of the following statements about this gas is true?

A. Model I has a specific heat c_v = ³⁄₂Nk
B. Model II has a smaller specific heat than Model I
C. Model I is always correct
D. Model II is always correct
E. The choice between Models I and II depends on the temperature

(GR8677 #87)

Solution:

Equipartition of Energy: E = ½ fNkT
Specific heat: c_v = ∂E/∂t = ½ fNk

f = Degree of Freedom (DoF)

(A) FALSE
Model I is diatomic molecules in low temperature
→ 5 DoF (3 translational + 2 rotational)
→ c_v = ⁵⁄₂Nk

(B) FALSE
Model II is diatomic molecules in high temperature
→ 7 DoF (3 translational + 2 rotational + 2 vibrational)
→ c_v = ⁷⁄₂Nk

(C) and (D) are vague answers.

Answer: E