5.1 Ground Electronic Term Atoms
217
we obtain S(T , V ; N) as
S(T , V ; N) = k B
ln
z N
N!
+ βNu(T , V )
.
Upon employing the Stirling approximation to ln N ! and the expression for u(T , V )
obtained in Example 4.1 of Chap. 4, we see that S(T , V ; N) is given by
S(T , V ; N) = Nk B
ln
V
NN 3
+ ln
1 − e −η
η
+
7
2
−
η
e η − 1
,
which we may also write in the form
S(T , V ; N) = Nk B ln
V e
7
2
NN 3
+ Nk B
ln
1 − e −η
η
−
η
e η − 1
.
That this expression for the entropy of an ideal gas of structureless classical particles
predicts that the entropy can become negative for sufficiently large values of the
parameter η, see also Problem 10, indicates that there are problems with this
classical result in the presence of a sufficiently strong gravitational field.
To obtain expressions for the translational Gibbs energy and chemical potential,
we also use this approximation for Z tr
N , together with the defining relation for G in
terms of A and P V . We thus obtain
G tr (T , V ; N) = A tr + P V = −k B T ln Z
tr
N + Nk B T
= −Nk B T ln
eV
NN 3 + Nk B T ln e
= −Nk B T ln
V
NN 3
.
(5.1.8)
As V = Nk B T /P for an ideal gas, we may also write the final result as a function
of temperature and pressure as
G tr (T , P ; N) = − Nk B T ln
k B T
P P 3
.
(5.1.9)
Because the chemical potential μ for a pure substance is merely the Gibbs energy
per particle, we see that the translational chemical potential is given by
μ tr (T , P ) =
G tr (T , P ; N)
N
= − k B T ln
k B T
P P 3 .
(5.1.10)