# The heat Capacity of a Diatomic Gas

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```					   Chapter 15

The Heat Capacity of a Diatomic
Gas
15.1 Introduction
• Statistical thermodynamics provides deep
insight into the classical description of a
MONATOMIC ideal gas.
• In classical thermodynamics, the principle of
equipartition of energy fails to give the
observed value of the specific heat capacity for
diatomic gases.
• The explanation of the above discrepancy was
considered to be the most important challenge
in statistical theory.
15.1 The quantized linear oscillator
• A linear oscillator is a particle constrained to move
along a straight line and acted on by a restoring
force      F=-kx2
d x
F = ma= d t = -kx
m     2

• If displaced from its equilibrium position and
released, the particle oscillates with simple
harmonic motion of frequency v, given by         1 K
v
2   m

Note that the frequency depends on K and m, and is
independent of the amplitude X.
• Consider an assembly of N one-dimensional
harmonic oscillators, in which the oscillators
are loosely coupled so that the energy
exchange among them is small.

• In classical mechanics, a particle can oscillate
with any amplitude and energy.

• From quantum mechanics, the single particle
energy levels are given by
EJ = (J + ½) hv , J = 0, 1, 2, …..
• The energies are equally spaced and the
ground state has non-zero energy.
• The internal degrees of freedom include
vibrations, rotations, and electronic excitations.

• For internal degrees of freedom, Boltzmann
Statistics applies. The distinguish ability arises
from the fact that those diatomic molecules
have different translational energy.

• The states are nondegenerate, i.e. gj = 1
• The partition function of an oscillator
• Introducing the characteristic temperature θ,
where θ = hv/k

        2 3
                      
 e 1  e  e  e  ......
2T

T    T   T

                      
• The solution for the above eq. is (in class
derivation)
The distribution function for B-statistics is

2
• Note that B statistics and M – B statistics have
the same distribution function, the eq derived
in chapter 14 for internal energy is also valid
here.
U = NkT2
since

                              
1   e T              1    1 
U  Nk               Nk         
 2      
1 e T             2
eT  1 
                               
1
0
For T → 0                         
eT 1

For

thus
15.3Vibrational Modes of Diatomic
Molecules
• The most important application of the above
result is to the molecules of a diatomic gas

• From classical thermodynamics

for a reversible process!
Since

Or

At high temperatures
At low temperature limit

On has

So

approaching zero faster
than the growth of (θ/T)2 as T → 0
Therefore       Cv  0 as T  0

The total energy of a diatomic molecule is made
up of four contributions that can be separately
treated:
1. The kinetic energy associated with the
translational motion
2. The vibrational motion
3. Rotation motion (To be discussed later)

Example: 15.1 a) Calculate the fractional number
of oscillators in the three lowest quantum
states (j=0, 1, 2,) for T   4 and forT  
Sol:
J=0
• 15.2) a) For a system of localized
distinguishable oscillators, Boltzmann
statistics applies. Show that the entropy S is
given by                NJ 
S    N J ln    
J         N 

• Solution: according to Boltzmann statistics
NJ
g
n
W  N!         J
J 1 N
J
So
S  k ln w             1
ln w  ln N !  N J ln g J   ln N J
g J  1  ln 1  0
ln w  ln N !  N J ln 1   ln N J !
ln w  ln N ! ln N !
ln w   ln       J ln 1    J
ln w   ln     J ln  J
   J
ln w    J ln     J ln  J
ln w    J ln   ln  J 
         
ln w    J ln 
         

 J        
  
S     j ln     
 
 j
n
j 
S     j ln   
J 1          

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