The Equipartition Theorem in
Classical Statistical Mechanics (ONLY!)
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.
• 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:
Proof
System Total Energy 
Sum of single particle energies:
   ' ' ' ' ' '
System Partition Function Z 
Z  e
 / kT
 e
 ( '  ''  ''' ) / kT
 Z ' Z ' ' Z ' ' '
Z' , Z'' , etc. = Partition functions
for each particle.
System Partition Function Z 
Z  e
 / kT
 e
 ( '  ''  ''' ) / kT
 Z ' Z ' ' Z ' ' '
Z = Product of partition functions
Z' , Z'' , etc. of each particle
Canonical Ensemble “Recipe” for the Mean
(Thermal) Energy:
2   ln Z 
U  NkT 

 T V
So, the Thermal Energy
U
 ln Z
   
per Particle is:
N

Z   e / kT   e( ' '' ''') / kT  Z ' Z ' ' Z ' ' '
  ln Z 
U  NkT 2 


T

V
  
U
 ln Z

N

• 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.
Proof Continued!
• With this assumption, the total energy  has the form:
2
2
2
  b1 p1  b2 p2   b f p f
Plus a similar sum of terms containing the (qi)2
• For simplicity, focus on the p2 sum above:
Z   e / kT   e( '  ''  ''' ) / kT  Z ' Z ' ' Z ' ' '
• For each particle, change the sum into an integral over
momentum, as below. It is a Gaussian & is tabulated.


0
e
 bi pi2
dpi  
1 / 2
y i   1 / 2 pi


0
e
bi yi2
dyi  
1 / 2
Ki
Ki  Kinetic Energy
of particle i
Proof Continued!
Z   e / kT   e( ' '' ''') / kT  Z ' Z ' ' Z ' ' '


e
0
 bi pi2
dpi  
1 / 2


0
e
bi yi2
dyi   1/ 2 K i
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:
y i   1 / 2 pi

P e
0
 b1 p12
dp1


0
e
 b2 p22

dp2  e
0
 b f p 2f
dp f
• Finally, Z can be written:
Z   1/ 2 K1   1/ 2 K2  1/ 2 K f    f / 2 K1K2 K f
Z   1/ 2 K1   1/ 2 K2  1/ 2 K f    f / 2 K1K2 K f
• Use the Canonical Ensemble “Recipe” to get the
average energy per particle per independent degree of freedom:
  
 ln P
f
f

 kT

2 2
• 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:
Note! u  <>
3
u  RT
2
5
u  RT
2
u  (3  q) RT