What is the specific heat capacity of interstellar clouds?

Question

Star forming molecular clouds and especially Bok globules are low temperature $\sim 10 $ K environments with density on the order of $10^{-18}$ kg/m$^3$, mostly hydrogen with about 1% dust. What is their heat capacity?

I have tried to find good data for low-temperature heat capacities for hydrogen (which would be nice to have anyway), but most experiments also use dense hydrogen rather than the by-Earth-standard very low-pressure molecular gas of the clouds. Are there any decent estimates of how much a given increase in energy increases the cloud temperature? Or should one just use the ideal gas approximation for a diatomic gas and assume $C_v\approx (5/2)R$ per mole?

Answer 1

At such low temperatures ($\sim10\,\mathrm{K}$), the rotational and vibrational degrees-of-freedom of the hydrogen molecule freeze out and it acts like a monatomic ideal gas with only translational degrees of freedom, so $C_v\sim (3/2)R$ per mole is probably a better estimate.

The "freeze out" is because rotational and vibrational states are quantized, so when $kT << \hbar\omega$, thermally driven transitions become almost impossible. (Understanding the temperature dependence of the specific heat of molecular hydrogen was a major challenge and then success of early quantum mechanics. ) For hydrogen molecules, the fundamental vibration mode is 4161 $\mathrm{cm^{-1}}$ = 0.52 eV, corresponding to a temperature of about 6000 K. This is why vibrational modes can be ignored at room temperature and $C_v\sim (5/2)R$. The energy difference between the lowest energy rotational states (ortho- and para-hydrogen) is 0.34 kcal/mol =14.7 meV, corresponds to a temperature of 171 K, however, so in equilibrium at 10 K the rotational modes are also mostly inaccessible and the hydrogen gas will be all parahydrogen and $C_v\sim (3/2)R$.

I don't think the dust contributes much to the total heat capacity, but it is likely important for catalyzing transitions to reach equilibrium.