Thermal Physics - Einstein solid

One feature common to both the Debye theory and the Einstein theory of the specific heat of a crystal composed of N identical atoms is that the

1. Average energy of each atom is 3kT
2. Vibrational energy of the crystal is equivalent to the energy of 3N independent harmonic oscillators
3. Crystal is assumed to be continuous for all elastic waves
4. Speed of the longitudinal elastic waves is less than the speed of the transverse elastic waves
5. Upper cutoff frequency of the elastic waves is the same

(GR8677 #51)

Solution:

Einstein solid:

• Thermal properties of crystal
• Highly idealized model focusing only the vibrational modes of crystal
• Assumption: Each of N atoms is free to vibrate around its equilibrium position in any of 3 coordinate directions by a harmonic force with a natural frequency ω₀.

Einstein solid equipartition energy:

• Number of oscillator N = N₀,
• 3-D Harmonic Oscillator, f = 6
• U = (6/2) N₀kT = 3N₀kT

Answer: B

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

Thermal Physics - Equipartition Law

A three-dimensional harmonic oscillator is in thermal equilibrium with a temperature reservoir at temperature T. The average total energy of oscillator is

A. ½kT
B. kT
C. ³⁄₂kT
D. 3kT
E. 6kT

(GR0177 #05)

Solution:

Equipartition of Energy: E = ½ fkT

f = Degree of Freedom (DoF)

3D harmonic oscillator has 6 DoF = 3 components of momentum (kinetic energy) and 3 components of position (potential energy)
→ E = ⁶⁄₂ kT = 3kT

Answer: D

Thermal Physics - Equipartition Law

A gaseous mixture of O₂ (molecular mass 32 u) and N₂ (molecular mass 28 u) is maintained at constant temperature. What is the ratio v_rms(N₂)/v_rms(O₂) of the root-mean-square speeds of the molecules?

A. 7/8
B. √(7/8)
C. √(8/7)
D. (8/7)²
E. ln (8/7)

(GR0177 #48)

Solution:

Kinetic energy, E = ¹/₂ mv²

Equipartition of energy, E = ³/₂ kT 

¹/₂ mv² = ³/₂ kT

v² = 3kT/m

v_rms = √(3kT/m)

v_rms ∝ √(1/m)

v_rmsN2 / v_rmsO2 

= √(m_O2 / m_N2) 

= √(32u/28u)

= √(8/7)

Answer: C