5.1 Ground Electronic Term Atoms
215
Upon utilizing Eq. (3.2.22) for the de Broglie wavelength ), U tr (T ) is
obtained as
U tr (T ) =
3
2 Nk B T .
(5.1.2)
From this result, we can immediately obtain the expression
C V ,tr (T ) ≡
∂U tr
∂T
N,V
=
3
2 Nk B
(5.1.3a)
for the translational heat capacity C V ,tr for N atoms at constant volume, or
equivalently, the expression
c V ,tr ≡
C V ,tr
N
=
3
2 k B
(5.1.3b)
for the translational heat capacity c V ,tr per atom. Similarly, we obtain the pressure
P from the defining relation as
P = k B T
∂ ln Z tr
N
∂V
N,T
=
Nk B T
V
,
or equivalently,
P V = Nk B T =
2
3 U tr
(5.1.4)
for the equation of state, which we recognize as the ideal gas law. Finally, we can
employ the expression
S tr = k B ln Z
tr
N +
U tr
T
,
(5.1.5)
to obtain an explicit expression for the entropy associated with purely translational
motion. The expression so obtained is known in thermodynamics as the Sackur–
Tetrode equation and can be obtained from Eq. (5.1.1) by employing Stirling’s
approximation for the factorial of N. Remember that Stirling’s approximation for
N ! is obtained from the Stirling approximation for the natural logarithm of N !, i.e.,
ln N! ! N ln N − N ln e = ln(N/e) N . With this approximation, we may write the
partition function Z tr
N as
Z
tr
N = Z tr (β, V ; N) =
V N
N! 3N
eV
NN 3
N
.
(5.1.6)
If we now substitute Eq. (5.1.2) for U tr and Eq. (5.1.6) for Z tr
N into Eq. (5.1.5) for
S tr , we obtain the Sackur–Tetrode equation,