6.2 Diatomic Molecules
293
spin partition functions would also occur for any homonuclear diatomic molecule.
Moreover, as it is a manifestation of quantum mechanics, we should expect it to
play a more important role for essentially microscopic phenomena. Indeed, as we
shall see shortly, there is a profound effect on the rotational spectra of homonuclear
diatomic molecules in general, while only the bulk properties, especially the heat
capacity, of the homonuclear hydrogen isotopologues H 2 , D 2 , and T 2 are profoundly
affected by this nonseparability.
If we examine the experimental temperature dependence of the molar heat
capacity at constant volume for a series of homonuclear diatomic gases, we find
that for gaseous N 2 , O 2 , and Cl 2 , for example, C V ,rot (T ) has the value
5
2 R at all
temperatures T for which these substances are gases. For the three homonuclear
hydrogen isotopologues, C V ,rot (T ) varies with T for low temperatures but for
temperatures above about 350 K, C V ,rot (T ) for these gases has also attained the
value
5
2 R. This tells us, therefore, that for sufficiently ‘high’ temperatures, the bulk
(thermodynamic) properties of any homonuclear diatomic gas behaves as does a
heteronuclear diatomic gas, for which the factorization z rot−nuc (T ) = z rot (T )z nuc is
rigorous. As we have already specified for heteronuclear diatomic molecules that in
this context ‘high’ temperature means temperatures T such that T rot , let us
examine what happens for a homonuclear diatomic molecule when T rot .
When T rot , we may approximate sums over j by integrals over j , in which
case, the sums over j even and j odd are each given by
1
2
all j
(· · · ) ∼
1
2
∞
0
(2j + 1)e
−j (j+1)) rot /T dj ,
(6.2.96)
or, equivalently,
1
2
all j
(· · · ) ≡
T
2 rot
.
(6.2.97)
When this procedure is applied for bulk hydrogen gas, we therefore obtain the
corresponding classical limits for z ortho (T ) and z para (T ) as
z
cl
ortho (T ) =
3T
2 rot
;
z
cl
para (T ) =
T
2 rot
,
(6.2.98)
with the consequence that the classical limit of Eq. (6.2.93) will be
z
cl
rot−nuc (T ) =
3T
2 rot
+
T
2 rot
= 4
T
2 rot
.
(6.2.99)
If we recall that for a proton I a has the value
1
2 , then the factor 4 in this result simply
represents the number of nuclear spin pair-states, i.e., z nuc ≡ (2I a + 1) 2 , so that we
have
293
spin partition functions would also occur for any homonuclear diatomic molecule.
Moreover, as it is a manifestation of quantum mechanics, we should expect it to
play a more important role for essentially microscopic phenomena. Indeed, as we
shall see shortly, there is a profound effect on the rotational spectra of homonuclear
diatomic molecules in general, while only the bulk properties, especially the heat
capacity, of the homonuclear hydrogen isotopologues H 2 , D 2 , and T 2 are profoundly
affected by this nonseparability.
If we examine the experimental temperature dependence of the molar heat
capacity at constant volume for a series of homonuclear diatomic gases, we find
that for gaseous N 2 , O 2 , and Cl 2 , for example, C V ,rot (T ) has the value
5
2 R at all
temperatures T for which these substances are gases. For the three homonuclear
hydrogen isotopologues, C V ,rot (T ) varies with T for low temperatures but for
temperatures above about 350 K, C V ,rot (T ) for these gases has also attained the
value
5
2 R. This tells us, therefore, that for sufficiently ‘high’ temperatures, the bulk
(thermodynamic) properties of any homonuclear diatomic gas behaves as does a
heteronuclear diatomic gas, for which the factorization z rot−nuc (T ) = z rot (T )z nuc is
rigorous. As we have already specified for heteronuclear diatomic molecules that in
this context ‘high’ temperature means temperatures T such that T rot , let us
examine what happens for a homonuclear diatomic molecule when T rot .
When T rot , we may approximate sums over j by integrals over j , in which
case, the sums over j even and j odd are each given by
1
2
all j
(· · · ) ∼
1
2
∞
0
(2j + 1)e
−j (j+1)) rot /T dj ,
(6.2.96)
or, equivalently,
1
2
all j
(· · · ) ≡
T
2 rot
.
(6.2.97)
When this procedure is applied for bulk hydrogen gas, we therefore obtain the
corresponding classical limits for z ortho (T ) and z para (T ) as
z
cl
ortho (T ) =
3T
2 rot
;
z
cl
para (T ) =
T
2 rot
,
(6.2.98)
with the consequence that the classical limit of Eq. (6.2.93) will be
z
cl
rot−nuc (T ) =
3T
2 rot
+
T
2 rot
= 4
T
2 rot
.
(6.2.99)
If we recall that for a proton I a has the value
1
2 , then the factor 4 in this result simply
represents the number of nuclear spin pair-states, i.e., z nuc ≡ (2I a + 1) 2 , so that we
have