5.3 Electronic and Nuclear Spin States
229
characterizes the thermal accessibility of that excited electronic state to the atom.
We mean by this that in an equilibrium ensemble of a gas of such atoms, collisions
for which the relative kinetic energy of the colliding atoms is at least equal to the
energy difference e
i may excite one of the atoms into the excited electronic state,
leaving both atoms travelling more slowly. An appropriate definition of what we
shall call a characteristic temperature, denoted as T e
i , is given by
T
e
i ≡
e
i
k B
.
(5.3.3)
We may employ this definition to rewrite Eq. (5.3.2a) in terms of characteristic
temperatures as
z el (T ) = e
−ββ e
0
ω
e
0 + ω
e
1 e
−T e
1 /T
+ ω
e
2 e
−T e
2 /T
+ · · ·
.
(5.3.4)
Thus, if T e
i T , so that T e
i /T 1, all terms other than ω e
0 may be neglected;
in such a case, z el (T ) can be represented very well by
z el (T ) ω
e
0 e
−ββ e
0 .
(5.3.5)
If we are further able to choose e
0 = 0 for our ground electronic energy, then
z el (T ) = ω e
0 is also independent of temperature. A temperature-independent
partition function makes no contribution to the thermodynamic internal energy U
or to the heat capacities C V and C P (because they are obtained from temperature
derivatives of the partition function), but it does contribute to the thermodynamic
entropy S and to the thermodynamic Helmholtz and Gibbs energies A and G,
respectively. Note, however, that such additive contributions to S, A, and G will
cancel out when S, A, and G are determined, so that it is normally not relevant
for chemical processes or thermodynamic changes.
Example 5.2 Electronic partition function for the Si atom.
We have seen that the ground term of the Si atom is a 3 P term, and that it is split
by spin–orbit coupling into three components, called levels, identified as 3 P 2 , 3 P 1 ,
and 3 P 0 . We also know (see Problem 8) that the 3 P 0 level (which is nondegenerate,
i.e., consists of a single atomic state) lies lower in energy than the other two levels,
by 77.12 cm −1 for the 3 P 1 level (with three atomic states) and by 223.16 cm −1 for
the 3 P 2 level (with five atomic states). In addition, there is a 1 D 2 level (with five
atomic states) lying 6298.85 cm −1 above the ground level and a (nondegenerate)
1 S 0 level lying 15,394.4 cm −1 above the ground level.
Let us utilize Eq. (5.3.4) to obtain an expression for z el (T ) for the Si atom:
we first determine the characteristic electronic temperatures for the various atomic
levels of Si as T e
1 = e
1 /k B = 110.0 K, T e
2 = 321.31 K, T e
3 = 9063 K, and
T e
4 = 22,150 K, so that z el (T ) becomes
z el (T ) = 1 + 3e
−110.0/T
+ 5e
−321.3/T
+ 5e
−9063/T
+ e
−22150/T
+ · · · ,
229
characterizes the thermal accessibility of that excited electronic state to the atom.
We mean by this that in an equilibrium ensemble of a gas of such atoms, collisions
for which the relative kinetic energy of the colliding atoms is at least equal to the
energy difference e
i may excite one of the atoms into the excited electronic state,
leaving both atoms travelling more slowly. An appropriate definition of what we
shall call a characteristic temperature, denoted as T e
i , is given by
T
e
i ≡
e
i
k B
.
(5.3.3)
We may employ this definition to rewrite Eq. (5.3.2a) in terms of characteristic
temperatures as
z el (T ) = e
−ββ e
0
ω
e
0 + ω
e
1 e
−T e
1 /T
+ ω
e
2 e
−T e
2 /T
+ · · ·
.
(5.3.4)
Thus, if T e
i T , so that T e
i /T 1, all terms other than ω e
0 may be neglected;
in such a case, z el (T ) can be represented very well by
z el (T ) ω
e
0 e
−ββ e
0 .
(5.3.5)
If we are further able to choose e
0 = 0 for our ground electronic energy, then
z el (T ) = ω e
0 is also independent of temperature. A temperature-independent
partition function makes no contribution to the thermodynamic internal energy U
or to the heat capacities C V and C P (because they are obtained from temperature
derivatives of the partition function), but it does contribute to the thermodynamic
entropy S and to the thermodynamic Helmholtz and Gibbs energies A and G,
respectively. Note, however, that such additive contributions to S, A, and G will
cancel out when S, A, and G are determined, so that it is normally not relevant
for chemical processes or thermodynamic changes.
Example 5.2 Electronic partition function for the Si atom.
We have seen that the ground term of the Si atom is a 3 P term, and that it is split
by spin–orbit coupling into three components, called levels, identified as 3 P 2 , 3 P 1 ,
and 3 P 0 . We also know (see Problem 8) that the 3 P 0 level (which is nondegenerate,
i.e., consists of a single atomic state) lies lower in energy than the other two levels,
by 77.12 cm −1 for the 3 P 1 level (with three atomic states) and by 223.16 cm −1 for
the 3 P 2 level (with five atomic states). In addition, there is a 1 D 2 level (with five
atomic states) lying 6298.85 cm −1 above the ground level and a (nondegenerate)
1 S 0 level lying 15,394.4 cm −1 above the ground level.
Let us utilize Eq. (5.3.4) to obtain an expression for z el (T ) for the Si atom:
we first determine the characteristic electronic temperatures for the various atomic
levels of Si as T e
1 = e
1 /k B = 110.0 K, T e
2 = 321.31 K, T e
3 = 9063 K, and
T e
4 = 22,150 K, so that z el (T ) becomes
z el (T ) = 1 + 3e
−110.0/T
+ 5e
−321.3/T
+ 5e
−9063/T
+ e
−22150/T
+ · · · ,