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6 Molecular Systems
z
cl
rot−nuc (T ) = (2I a + 1)
2 T
2 rot
= z nuc z
cl
rot (T ) ,
(6.2.100)
which is the product of the rotational and nuclear spin partition functions, provided
that the rotational partition function is T /(2 rot ) for a homonuclear diatomic
molecule rather than T // rot as for a heteronuclear diatomic molecule. This
expression is typically valid to better than 1% for rot /T ≤ 0.05.
Let us now examine the percentage of H 2 molecules that belong to the para
modification as a function of temperature. We begin with the expression for the
combined partition function,
z rot−nuc (T ) = 3
j =odd
(2j + 1)e
−j (j+1)) rot /T
+
j =even
(2j + 1)e
−j (j+1)) rot /T ,
(6.2.101)
with the first and second terms corresponding, respectively, to the contributions from
the ortho and para modifications. The percentage of H 2 molecules that belong to the
para modification will therefore be given by the ratio
%pH 2 =
z para (T )
z rot−nuc
× 100%
=
j even
(2j + 1)e
−j (j+1)) rot /T
3
j odd
(2j + 1)e
−j (j+1)) rot /T
+
j even
(2j + 1)e
−j (j+1)) rot /T
× 100% .
(6.2.102)
We shall begin by examining the limiting behaviour of this ratio for T = 0 and
T → ∞. For very low temperatures, only the terms from the lowest rotational
state in each sum will survive, so that the ortho sum reduces to the term for
j = 1, i.e., 9e −2 rot /T , while the para sum reduces to the j = 0 term, i.e., 1:
the numerator and denominator for our expression thus both become 1 at T = 0,
and the H 2 is 100% para-H 2 . For the high-temperature limit, we can replace the
discrete sums over j by integrals, with the consequence that z para (T ) = T /(2 rot )
and z ortho (T ) = 3T /(2 rot ), giving z rot−nuc (T ) = T // rot , so that H 2 is only 25%
pH 2 . Thus, for temperatures T such that T rot , we obtain what is called ‘normal
hydrogen’, which is simply the high-temperature equilibrium mixture of pH 2 and
oH 2 (i.e.,
3
4 ortho plus
1
4 para). The temperature dependence of the percentage pH 2
in equilibrium hydrogen is displayed in Fig. 6.11.
We see from Fig. 6.11 that at thermal equilibrium, molecular hydrogen consists
entirely of pH 2 at T = 0 K, with the percentage pH 2 dropping fairly rapidly
with increasing temperature, attaining a 1:1 ratio of pH 2 to oH 2 at about 80 K,
and dropping to the 1:3 nuclear spin degeneracy ratio by about T = 200 K, and
remaining at the 25% pH 2 mixture ratio thereafter. This high-temperature equilibrium mixture is referred to as normal hydrogen, and designated as nH 2 . Because
pH 2 and oH 2 have their origin in the coupling of the nuclear spins I a =
1
2 of their
6 Molecular Systems
z
cl
rot−nuc (T ) = (2I a + 1)
2 T
2 rot
= z nuc z
cl
rot (T ) ,
(6.2.100)
which is the product of the rotational and nuclear spin partition functions, provided
that the rotational partition function is T /(2 rot ) for a homonuclear diatomic
molecule rather than T // rot as for a heteronuclear diatomic molecule. This
expression is typically valid to better than 1% for rot /T ≤ 0.05.
Let us now examine the percentage of H 2 molecules that belong to the para
modification as a function of temperature. We begin with the expression for the
combined partition function,
z rot−nuc (T ) = 3
j =odd
(2j + 1)e
−j (j+1)) rot /T
+
j =even
(2j + 1)e
−j (j+1)) rot /T ,
(6.2.101)
with the first and second terms corresponding, respectively, to the contributions from
the ortho and para modifications. The percentage of H 2 molecules that belong to the
para modification will therefore be given by the ratio
%pH 2 =
z para (T )
z rot−nuc
× 100%
=
j even
(2j + 1)e
−j (j+1)) rot /T
3
j odd
(2j + 1)e
−j (j+1)) rot /T
+
j even
(2j + 1)e
−j (j+1)) rot /T
× 100% .
(6.2.102)
We shall begin by examining the limiting behaviour of this ratio for T = 0 and
T → ∞. For very low temperatures, only the terms from the lowest rotational
state in each sum will survive, so that the ortho sum reduces to the term for
j = 1, i.e., 9e −2 rot /T , while the para sum reduces to the j = 0 term, i.e., 1:
the numerator and denominator for our expression thus both become 1 at T = 0,
and the H 2 is 100% para-H 2 . For the high-temperature limit, we can replace the
discrete sums over j by integrals, with the consequence that z para (T ) = T /(2 rot )
and z ortho (T ) = 3T /(2 rot ), giving z rot−nuc (T ) = T // rot , so that H 2 is only 25%
pH 2 . Thus, for temperatures T such that T rot , we obtain what is called ‘normal
hydrogen’, which is simply the high-temperature equilibrium mixture of pH 2 and
oH 2 (i.e.,
3
4 ortho plus
1
4 para). The temperature dependence of the percentage pH 2
in equilibrium hydrogen is displayed in Fig. 6.11.
We see from Fig. 6.11 that at thermal equilibrium, molecular hydrogen consists
entirely of pH 2 at T = 0 K, with the percentage pH 2 dropping fairly rapidly
with increasing temperature, attaining a 1:1 ratio of pH 2 to oH 2 at about 80 K,
and dropping to the 1:3 nuclear spin degeneracy ratio by about T = 200 K, and
remaining at the 25% pH 2 mixture ratio thereafter. This high-temperature equilibrium mixture is referred to as normal hydrogen, and designated as nH 2 . Because
pH 2 and oH 2 have their origin in the coupling of the nuclear spins I a =
1
2 of their