<p>At such low temperatures (<span class="math-container">$\sim10\,\mathrm{K}$</span>), the rotational and vibrational degrees-of-freedom of the hydrogen molecule <a href="https://en.wikipedia.org/wiki/Molar_heat_capacity#Frozen_and_active_degrees_of_freedom" rel="nofollow noreferrer">freeze out</a> and it acts like a monatomic ideal gas with only translational degrees of freedom, so <span class="math-container">$C_v\sim (3/2)R$</span> per mole is probably a better estimate.</p>
<p>The &quot;freeze out&quot; is because rotational and vibrational states are quantized, so when <span class="math-container">$kT &lt;&lt; \hbar\omega$</span>, thermally driven transitions become almost impossible.  (Understanding the temperature dependence of the specific heat of molecular hydrogen was a <a href="https://doi.org/10.1007/s00407-009-0053-2" rel="nofollow noreferrer" title=" Astonishing Successes and Bitter Disappointment: The Specific Heat of Hydrogen in Quantum Theory., Gearhart, C.A. Arch. Hist. Exact Sci. 64, 113–202 (2010).">major challenge and then success</a> of early quantum mechanics.
) For hydrogen molecules, the fundamental vibration mode is <a href="https://arxiv.org/abs/1304.6390" rel="nofollow noreferrer" title="Fundamental Vibration of Molecular Hydrogen, G. D. Dickenson,et al., Phys. Rev. Lett. 110, 193601 – Published 8 May 2013">4161 <span class="math-container">$\mathrm{cm^{-1}}$</span> = 0.52 eV</a>, corresponding to a temperature of about 6000 K.  This is why vibrational modes can be ignored at room temperature and <span class="math-container">$C_v\sim (5/2)R$</span>.
The energy difference between the lowest energy rotational states (<a href="https://en.wikipedia.org/wiki/Spin_isomers_of_hydrogen#Allowed_rotational_energy_levels" rel="nofollow noreferrer">ortho- and para-hydrogen</a>) is <a href="https://doi.org/10.1515/zna-1979-0206" rel="nofollow noreferrer" title="Ortho and Parahydrogen in Interstellar Material, Reeves, Robert R. and Harteck, Paul., Zeitschrift für Naturforschung A, vol. 34, no. 2, 1979, pp. 163-166.">0.34 kcal/mol</a> =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 <span class="math-container">$C_v\sim (3/2)R$</span>.</p>
<p>I don't think the dust contributes much to the total heat capacity, but it is <a href="https://doi.org/10.1016/j.molap.2017.11.001" rel="nofollow noreferrer" title="H2 formation on interstellar dust grains: The viewpoints of theory, experiments, models and observations,, Valentine Wakelam et al., Molecular Astrophysics, Volume 9, 2017, Pages 1-36">likely important</a> for catalyzing transitions to reach equilibrium.</p>

CC BY-SA 4.0
2023-04-09T21:21:40.690Z
2023-04-10T03:10:01.367Z
2023-04-10T03:10:01.367Z
David Bailey
<p>Star forming <a href="https://en.wikipedia.org/wiki/Molecular_cloud" rel="nofollow noreferrer">molecular clouds</a> and especially <a href="https://en.wikipedia.org/wiki/Bok_globule" rel="nofollow noreferrer">Bok globules</a> are low temperature <span class="math-container">$\sim 10 $</span> K environments with density on the order of <span class="math-container">$10^{-18}$</span> kg/m<span class="math-container">$^3$</span>, mostly hydrogen with about 1% dust. What is their heat capacity?</p>
<p>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 <a href="https://en.wikipedia.org/wiki/Molar_heat_capacity#Rotational_modes_of_a_diatomic_molecule" rel="nofollow noreferrer">approximation for a diatomic gas</a> and assume <span class="math-container">$C_v\approx (5/2)R$</span> per mole?</p>

<interstellar-medium><heat>
What is the specific heat capacity of interstellar clouds?
Anders Sandberg
# What is the specific heat capacity of interstellar clouds?

(Asked by: username_0 on 2018-12-16)

Star forming [molecular clouds](https://en.wikipedia.org/wiki/Molecular_cloud) and especially [Bok globules](https://en.wikipedia.org/wiki/Bok_globule) 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](https://en.wikipedia.org/wiki/Molar_heat_capacity#Rotational_modes_of_a_diatomic_molecule) and assume $C\_v\approx (5/2)R$ per mole?

**Tags:** interstellar-medium, heat

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## 1 answers

### Answer by username_1 on 2023-04-09. Score: 6 | Accepted answer

At such low temperatures ($\sim10\,\mathrm{K}$), the rotational and vibrational degrees-of-freedom of the hydrogen molecule [freeze out](https://en.wikipedia.org/wiki/Molar_heat_capacity#Frozen_and_active_degrees_of_freedom) 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](https://doi.org/10.1007/s00407-009-0053-2 " Astonishing Successes and Bitter Disappointment: The Specific Heat of Hydrogen in Quantum Theory., Gearhart, C.A. Arch. Hist. Exact Sci. 64, 113–202 (2010).") of early quantum mechanics.
) For hydrogen molecules, the fundamental vibration mode is [4161 $\mathrm{cm^{-1}}$ = 0.52 eV](https://arxiv.org/abs/1304.6390 "Fundamental Vibration of Molecular Hydrogen, G. D. Dickenson,et al., Phys. Rev. Lett. 110, 193601 – Published 8 May 2013"), 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](https://en.wikipedia.org/wiki/Spin_isomers_of_hydrogen#Allowed_rotational_energy_levels)) is [0.34 kcal/mol](https://doi.org/10.1515/zna-1979-0206 "Ortho and Parahydrogen in Interstellar Material, Reeves, Robert R. and Harteck, Paul., Zeitschrift für Naturforschung A, vol. 34, no. 2, 1979, pp. 163-166.") =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](https://doi.org/10.1016/j.molap.2017.11.001 "H2 formation on interstellar dust grains: The viewpoints of theory, experiments, models and observations,, Valentine Wakelam et al., Molecular Astrophysics, Volume 9, 2017, Pages 1-36") for catalyzing transitions to reach equilibrium.


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