46
2 Macroscopic Thermodynamics
does not depend upon volume, we may obtain U(T ) for an ideal gas via Eq. (2.3.8)
as
U ideal gas (T ) =
T
0
C V (T ) dT ,
with the constant of integration equal to zero, as U(0) = 0. For an ideal gas of
particles that possess only translational motion, C V is temperature-independent and
has the value C V =
3
2 Nk B , so that U ideal gas (T ) =
3
2 Nk B T .
To obtain H ideal gas (T ) from U ideal gas (T ), we have only to employ the equation
of state for the ideal gas to replace P V in Eq. (2.3.2) by Nk B T to give
H ideal gas (T ) =
T
0
C V (T ) dT + Nk B T ,
or, for an ideal gas of structureless particles, H ideal gas (T ) =
5
2 Nk B T .
We may now employ our expression for U ideal gas (T ), together with expression
(2.3.13b), to obtain the Helmholtz energy A ideal gas (T , V ) for an ideal gas of
structureless particles as
A ideal gas (T , V ) = −Nk B T
ln
V
N
+
3
2
ln
2πmk B T
h 2
+ 1
.
(2.3.16)
A similar expression for G ideal gas (T , P ) can be obtained from H ideal gas (T ) and the
ideal gas law.
Example 2.4 Thermodynamic functions for a quantum ideal gas.
Let us consider specifically a photon gas in thermal equilibrium, contained in a
cylinder that is closed at one end, sealed by a movable piston, and maintained at a
constant temperature T . According to Eq. (1.2.12), an isothermal quasistatic volume
change from V to V + V gives rise to a change, U , in the internal energy U of
U =
4σ
c
T
4 .
Because the pressure, P , of a photon gas is a function of T alone, the work done on
the photon gas during this process is given, via W = −
P (T )dV , as
W = −
4σ
3c
T
4 .
From the First Law of Thermodynamics, Eq. (2.2.4b), the energy transfer into or out
of the photon gas will then be given by
Q =
4σ
3c
T
4 V .