Statistical Mechanics - Physics at Oregon State University

76 CHAPTER 4. STATISTICS OF INDEPENDENT PARTICLES.
to the previous expression. We replace log(N!) by N log(N) − N only, since all
other terms vanish in the thermodynamic limit.
Back to thermodynamics.
Once the Helmholtz free energy is known, the entropy and pressure can be
calculated and we obtain again
∂F
p = −
=
∂V T,N
NkBT
(4.33)
V
∂F
S = −
= NkB log(
∂T V,N
nQ(T )
) +
n
5
(4.34)
2
Note that since n ≪ nQ(T ) the entropy is positive. This shows that we expect
differences if the density becomes comparable to the quantum concentration.
That is to be expected, because in that case there are states with occupation
numbers that are not much smaller than one anymore. Also note that this
classical formula for the entropy contains via nQ(T ). This is an example of a
classical limit where one cannot take = 0 in all results! The entropy in our
formalism is really defined quantum mechanically. It is possible to derive all of
statistical mechanics through a classical approach (using phase space, etc), but
in those cases is also introduced as a factor normalizing the partition function!
We will discuss this in a later chapter.
Check of Euler equation.
The Gibbs free energy is defined by G = F + pV and is used for processes
at constant pressure. For the ideal gas we find
G = µN (4.35)
This result is very important as we will see later on. We will show that it holds
for all systems, not only for an ideal gas, and that it puts restrictions on the
number of independent intensive variables. Of course, from thermodynamics we
know that this has to be the case, it is a consequence of the Euler equation.
Heat capacities.
Important response functions for the ideal gas are the heat capacity at constant
volume CV and at constant pressure Cp. These functions measure the
amount of heat (T ∆S) one needs to add to a system to increase the temperature
by an amount ∆T :
∂S
CV = T
∂T V,N
= 3
2 NkB
(4.36)