5.4 Atomic Gas Thermodynamic Functions
233
As both A/T and ln Z are independent of T and V for the electronic and nuclear
degrees of freedom, both U and P have no contributions arising from them. Thus,
U and P are determined solely by the translational degrees of freedom for the atoms
as
U =
3
2 Nk B T ,
P = Nk B T /V .
Similarly, as the heat capacities at constant volume and pressure, C V and C P , are
given by the temperature partial derivatives of U and H with the volume (and
number of atoms) held fixed, they too will be determined only by the translational
contributions, namely, C V =
3
2 Nk B and C P =
5
2 Nk B , so that γ ≡ C P /C V has the
value 5/3. Finally, from the total differential (5.4.4) for dA, we may also obtain an
expression for the chemical potential μ as
μ(T , V ) =
∂A
∂N
T ,V
= μ tr (T , V ) − k B T ln(ω
e
0 ω
n
0 )
= −k B T ln
V ω e
0 ω n
0
NN 3 (T )
.
(5.4.9)
Moreover, as the chemical potential for a pure substance is the Gibbs energy per
particle (here atom), we have G(T , V ; N) = Nμ(T , V ).
5.4.2 Influence of Excited Electronic States
Some atoms have excited electronic states that are thermally accessible at normal
temperatures, meaning temperatures lying roughly in the temperature range of
250 K to 1000 K. Many of the elements in Groups 4A through 7A of the periodic
table possess low-lying excited atomic states: of special note are C, O, and F in
the second row, Si, S, and Cl in the third row, as well as many of the transition and
lanthanide metals and their ions. We shall see in Chap. 8, for example, that such lowlying atomic electronic states play an important role in understanding the magnetic
properties of a number of europium and samarium salts.
We shall write the electronic partition function for such an atom as
z el (T ) =
∞
i=0
ω
e
i e
−βββ e
i ,
(5.4.10)
with e
i = e
i − e
0 . To obtain an expression for U el (T , N), we shall employ the
fundamental statistical mechanical definition of U to obtain
233
As both A/T and ln Z are independent of T and V for the electronic and nuclear
degrees of freedom, both U and P have no contributions arising from them. Thus,
U and P are determined solely by the translational degrees of freedom for the atoms
as
U =
3
2 Nk B T ,
P = Nk B T /V .
Similarly, as the heat capacities at constant volume and pressure, C V and C P , are
given by the temperature partial derivatives of U and H with the volume (and
number of atoms) held fixed, they too will be determined only by the translational
contributions, namely, C V =
3
2 Nk B and C P =
5
2 Nk B , so that γ ≡ C P /C V has the
value 5/3. Finally, from the total differential (5.4.4) for dA, we may also obtain an
expression for the chemical potential μ as
μ(T , V ) =
∂A
∂N
T ,V
= μ tr (T , V ) − k B T ln(ω
e
0 ω
n
0 )
= −k B T ln
V ω e
0 ω n
0
NN 3 (T )
.
(5.4.9)
Moreover, as the chemical potential for a pure substance is the Gibbs energy per
particle (here atom), we have G(T , V ; N) = Nμ(T , V ).
5.4.2 Influence of Excited Electronic States
Some atoms have excited electronic states that are thermally accessible at normal
temperatures, meaning temperatures lying roughly in the temperature range of
250 K to 1000 K. Many of the elements in Groups 4A through 7A of the periodic
table possess low-lying excited atomic states: of special note are C, O, and F in
the second row, Si, S, and Cl in the third row, as well as many of the transition and
lanthanide metals and their ions. We shall see in Chap. 8, for example, that such lowlying atomic electronic states play an important role in understanding the magnetic
properties of a number of europium and samarium salts.
We shall write the electronic partition function for such an atom as
z el (T ) =
∞
i=0
ω
e
i e
−βββ e
i ,
(5.4.10)
with e
i = e
i − e
0 . To obtain an expression for U el (T , N), we shall employ the
fundamental statistical mechanical definition of U to obtain