1. Classical Statistical Mechanics (ONLY!)
The Equipartition Theorem in
Classical Statistical Mechanics (ONLY!)

2. The Equipartition Theorem is Valid in Classical Stat. Mech. ONLY!!!
“Each degree of freedom in a system of particles contributes (½)kBT to the thermal average energy of the system.”
Note:
1. This theorem is valid only if each term in the
classical energy is proportional to a momentum (p) squared or to a coordinate (q) squared.
2. The degrees of freedom are associated with translation, rotation & vibration of the system’s
molecules.

3. Outline of a Proof Follows:
In the Classical Cannonical
Ensemble, it is straighforward to
show that
The average energy of a particle per independent degree of freedom = (½)kBT.
Outline of a Proof Follows:

4. Sum of single particle energies:
Proof
System Total Energy 
Sum of single particle energies:
System Partition Function Z 
Z' , Z'' , etc. = Partition functions
for each particle.

5. System Partition Function Z  Z = Product of partition functions
Z' , Z'' , etc. of each particle
Canonical Ensemble “Recipe” for the Mean
(Thermal) Energy:
So, the Thermal Energy
per Particle is:

6.  = KEt + KEr + KEv + PEv + …. KEt = (½)mv2 = [(p2)/(2m)] KEr = (½)I2
Various contributions to the Classical Energy of each particle:
 = KEt + KEr + KEv + PEv + ….
Translational Kinetic Energy:
KEt = (½)mv2 = [(p2)/(2m)]
Rotational Kinetic Energy:
KEr = (½)I2
Vibrational Potential Energy:
PEv = (½)kx2
Assume that each degree of freedom has an energy
that is either proportional to a p2 or to a q2.

7. Proof Continued! Plus a similar sum of terms containing the (qi)2
With this assumption, the total energy  has the form:
Plus a similar sum of terms containing the (qi)2
For simplicity, focus on the p2 sum above:
For each particle, change the sum into an integral over
momentum, as below. It is a Gaussian & is tabulated.
Ki  Kinetic Energy
of particle i

8. Proof Continued! Finally, Z can be written: Ki  Kinetic Energy
of particle i
The system partition function Z is then proportional to the
product of integrals like above. Or, Z is proportional to P:
Finally, Z can be written:

9. For a Monatomic Ideal Gas: For a Diatomic Ideal Gas:
Use the Canonical Ensemble “Recipe” to get the
average energy per particle per independent degree of freedom:
Note! u  <>
For a Monatomic Ideal Gas:
For a Diatomic Ideal Gas: l
For a Polyatomic Ideal Gas
in which the molecules vibrate
with q different frequencies: