WOLFRAM|DEMONSTRATIONS PROJECT

Low-Temperature Heat Capacity of Hydrogen Molecules

​
temperature (K)
200
ortho
H
2
para
H
2
3:1 mixture
equilibrium
H
2
HD
Hydrogen
H
2
is the lowest boiling molecular species, remaining a gas down to 20K. At and above room temperature,
T≳300K
, the rotational degree of freedom is fully excited; thus the rotational contribution to heat capacity approaches its equipartition value,
R
per mole. Owing to the exceptionally small moment of inertia of
H
2
, rotation becomes inactive at temperatures below about 50K. However, the heat capacity behaves anomalously as the temperature is lowered. This anomaly was first explained by Dennison in 1927. Since
H
2
is a homonuclear molecule, only half of its rotational states are accessible. In the singlet nuclear-spin state, known as parahydrogen (p-
H
2
) only even-
J
rotational states are accessible; in the triplet nuclear-spin state, known as orthohydrogen (o-
H
2
) only odd-
J
rotational states are accessible. The molecular partition functions representing the rotational and nuclear spin degrees of freedom are given by
q
ortho
=3
Σ
Jodd
(2J+1)
-J(J+1)Θ/T
e
,
q
para
=
Σ
Jeven
(2J+1)
-J(J+1)Θ/T
e
.
The rotational energies are given by
E
J
=J(J+1)
2
ℏ
/2I
with
(2J+1)
-fold degeneracies. It is convenient to define the rotational characteristic function
Θ=
2
ℏ
/2Ik
, equal to 87.57 for
H
2
and 65.70 for HD. The factors 1 and 3 represent the degeneracies of the para and ortho nuclear spin states, respectively.
The rotational contribution to heat capacity per mole can be calculated using
rot
C
(T)=
∂
∂T
R
2
T
∂logq
∂T
. This can be plotted for o-
H
2
, p-
H
2
and a 3:1 mixture which exists in hydrogen gas at room temperature. The two forms do not interconvert unless a catalyst, such as activated charcoal or platinum is present, so the 3:1 ratio will persist as the temperature is lowered. In the presence of a catalyst, the partition function can be represented by its equilibrium value
q
equi
=
q
ortho
+
q
para
, with the sum running over both even and odd
J
. This will be reflected in a heat capacity
rot
C
equi
(T)
that reaches a maximum in excess of
2R
around
T≈40K
. Para
H
2
in the
J=0
state, with a purity around 99.7%, can be obtained by cooling the equilibrium mixture down to 20K. (There also exist elaborate procedures for obtaining pure o-
H
2
.)
The isotopomer HD is a heteronuclear diatomic molecule, with the nuclear spin-molecular rotational partition function given by
q
HD
=6
∞
∑
J=0,1…
(2J+1)
-J(J+1)
Θ
HD
/T
e
,
Θ
HD
=65.70
K.
The nuclear-spin degeneracy equals
(2
I
P
+1)(2
I
D
+1)
, where the spins
I
of the proton and deuteron are
1
2
and 1, respectively.
You can select any combination of five heat-capacity curves over a temperature range. These can be identified using the tooltip.