We saw earlier that the lapse rate in the troposphere depends on cp and g. The gravitational acceleration g is a function of latitude on all the giant planets, while cp is dominated by the heat capacity of both helium (which is monatomic and thus has only the three translational degrees of freedom leading to Cp = 2.5R Jmol—1 K— where Cp = Cv + R) and molecular hydrogen, which is a diatomic molecule and thus also has rotational degrees of freedom. For Jupiter and Saturn the contribution of other molecules to mean heat capacity is negligible. However, for Uranus and Neptune the heat capacity of methane and water vapor are also important considerations below their respective condensation levels.
The rotation of linear molecules may be modeled by the motion of a rigid rotator. From quantum mechanics (e.g., Rae, 1985) the allowed energy levels of such a rotator are
where l is an integer; I is the moment of inertia; and H is Planck's constant divided by 2a\ Equation (4.15) may conveniently be re-expressed as Et = l(l + 1)fcgOR, where kB is the Boltzmann constant, and or = h2/2kBI is known as the rotational temperature. These rotational energy levels are degenerate with a degeneracy factor gl = (2l + 1), and thus the rotational partition function is
and the contribution of rotational energy to the molar internal energy Urot is
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