Assuming that the rotational energy of diatomic molecules is independent of their translational energy, the contribution to heat capacity at constant volume, Cv = dU/dT, may be simply added to the translational heat capacity at constant volume Cv = 3R/2. Using this expression for the partition function we can calculate

0 100 200 300

Figure 4.2. Variation of molar heat capacity at constant volume of molecular hydrogen with temperature. Solid line is the expected curve from using the expression for the partition function of Equation (4.16). The dotted line is the heat capacity calculated from the revised partition function of Equation (4.18). This is the curve measured in the presence of a catalyst. For cases where ortho:para equilibration is so slow that the ortho:para hydrogen ratio may be considered to be frozen at its high-temperature 3:1 ratio, the dashed line is obtained from Equation (4.19), in good agreement with measurement.

0 100 200 300

Figure 4.2. Variation of molar heat capacity at constant volume of molecular hydrogen with temperature. Solid line is the expected curve from using the expression for the partition function of Equation (4.16). The dotted line is the heat capacity calculated from the revised partition function of Equation (4.18). This is the curve measured in the presence of a catalyst. For cases where ortho:para equilibration is so slow that the ortho:para hydrogen ratio may be considered to be frozen at its high-temperature 3:1 ratio, the dashed line is obtained from Equation (4.19), in good agreement with measurement.

the rotational heat capacity as a function of temperature for H2 (where OR = 85 K) shown in Figure 4.2 (solid line). As can be seen the rotational contribution to molar heat capacity tends to R (JmoF1 K_1) for temperatures well above the rotational temperature and tends to zero for T ^ OR. Although the predicted heat capacity using this formula agrees well with the measured rotational heat capacity for linear molecules such as CO and NO (adjusting the rotation temperature appropriately), something strange is found to happen for molecular hydrogen. If hydrogen at room temperature is cooled fairly rapidly and its heat capacity measured as a function of temperature, the dashed curve of Figure 4.2 is obtained, which is clearly rather different from that expected. Even more puzzling is that if this experiment is instead done very slowly, or in the presence of a catalyst such as activated charcoal, the dotted curve is obtained. What can be going on?

Although the theory outlined above is satisfactory for heteronuclear diatomic molecules such as CO and NO, it is not applicable for homonuclear diatomic molecules, such as H2, where the two nuclei are identical. The nuclei of the H2 molecule are protons, which are fermions and must thus be described by antisymmetric wave-

functions. Thus, the wavefunction of the molecule must change if the two protons are interchanged. The wavefunction may in fact be separated into the product of a "rotation part" and a "spin part". The rotation part describes the rotation of the two nuclei round each other, and it is found that the states / = 0,2,4,... have even exchange parity, while the states / = 1,3,... have odd exchange parity. Hence, the even, symmetric rotation states must have odd, antisymmetric spins with total spin S = 0 in order for the total wavefunction to be antisymmetric, and likewise the odd, asymmetric rotational states must have even, parallel spins with total spin S = 1. Hydrogen molecules with spins antiparallel (S = 0,/ = 0,2,4,...) are known as para-hydrogen, while hydrogen molecules with spins parallel (S = 1,/ = 1,3,...) are known as ortho-hydrogen. The S = 0 state can be shown to be a singlet state, while the S = 1 state is a triplet state.

Because there is such a fundamental difference between ortho-hydrogen and para-hydrogen, a simple summing over the rotational states in Equation (4.16) is incorrect. Instead, assuming that the hydrogen is in thermal equilibrium, the partition function is actually given by

Zrot = £(2/ + 1) exp[-/(/ + 1)or/T] + 3 £(21 + 1) exp[-/(/ + 1)or/T]. (4.18)

4ven /odd

The heat capacity derived from this partition function accurately fits the dotted curve of Figure 4.2, where hydrogen is slowly cooled in the presence of a catalyst to ensure equilibrium. What can be said about the remaining curve, however? It turns out that conversion between ortho-hydrogen and para-hydrogen via collision processes is actually quite difficult since to change / by ± 1 requires that the total spin is also changed, which is not easy unless there is a third body, such as a surface, to take away the spin. Hence, ortho-hydrogen and para-hydrogen behave almost as different gases and may be treated quasi-separately. At high temperatures, the sums in Equation (4.18) are both equal, and thus high-temperature (T ^ O), thermally equilibrated hydrogen has an ortho:para ratio of 3:1, or equivalently a para-H2 fraction f = 0.25. Room temperature is "high" in this case. The heat capacity of hydrogen that is cooled rapidly enough such that negligible ortho-para conversion occurs will have a heat capacity given by

where the ortho part is obtained from the rigid rotor partition function (Equation 4.16) summed only over odd /, and the para part is obtained also from Equation (4.16) by summing only over even /, and this then accurately models the dashed curve in Figure 4.2. In the giant planet atmospheres, air upwelling from the deep interior will have the deep ortho:para ratio of 3:1 and this ratio will slowly change as orthopara conversion proceeds at a rate governed mainly by the availability of aerosol surfaces to exchange spin angular momentum with (Fouchet et a/., 2003; Massie and Hunten, 1982). Hence, measurement of the ortho:para ratio in the giant planet atmospheres provides information on the rate of vertical upwelling and on the presence of catalytic aerosol surfaces. Observations of this fraction will be discussed later. An additional effect of so-called "lagged" ortho-para conversion is that the latent heat release can act to stabilize the vertical profile of temperature and thus inhibit convection.

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