Statistical Mechanics - Physics at Oregon State University

4.3. GAS OF POLY-ATOMIC MOLECULES. 77
Cp = T
∂S
=
∂T p,N
5
2 NkB
(4.37)
where we have used
∂S ∂S ∂S ∂V
=
+
(4.38)
∂T p,N ∂T V,N ∂V T,N ∂T p,N
with
∂S
NkB
∂V = T,N V and
∂V
NkB V
∂T = p,N p = T .
It is obvious that we need Cp CV , since at constant pressure we have to do
work against the outside world. This is an example of an inequality in thermal
physics which relates two functions and holds in general for all systems. This
relation was derived in thermodynamics based on equilibrium principles, and
our current microscopic theory is in agreement with those general observations.
Ratio of heat capacities.
The ratio of the heat capacities is often abbreviated by γ and is used in a
number of formulas. For the ideal gas γ = Cp 5 = CV 3 . From an experimental point
of view this is a useful ratio, since it can be measured directly. From equation
(4.38) we see that
γ = 1 +
∂S
∂V
∂V T,N ∂T p,N
∂S
−1 ∂T V,N
∂S ∂V ∂T
γ = 1 +
∂V T,N ∂T p,N ∂S
∂T ∂V
γ = 1 −
∂V ∂T
S,N
p,N
V,N
(4.39)
(4.40)
(4.41)
In any system where the ideal gas equation of state is pV = NkBT obeyed we
find
∂T
= (1 − γ)
∂V S,N
T
(4.42)
V
or in regions where γ is constant we see that for adiabatic processes T ∝ V 1−γ ,
which is equivalent to say that (again using the ideal gas law) for adiabatic
processes pV γ is constant.
4.3 Gas of poly-atomic molecules.
Internal motion is independent.