Molecular Gases: Rotational and Vibrational Degrees of Freedom

A monatomic gas stores energy only in translation. A molecule can also rotate and vibrate, and each internal motion is a set of quantized levels with its own energy scale. The heat capacity of a molecular gas therefore depends on temperature: a mode contributes its equipartition share only when the temperature is high enough to excite it, and stays frozen below. The observed heat capacity of hydrogen climbs a staircase from $\frac{3}{2}R$ at low temperature to $\frac{5}{2}R$ once rotation is active and toward $\frac{7}{2}R$ once vibration turns on. This lesson computes each step from the partition function, giving the quantitative account behind the qualitative staircase.

Factorization of the molecular partition function

The energy of a diatomic molecule separates, to a good approximation, into independent contributions,

the separation resting on the Born–Oppenheimer picture — electrons move fast enough to define a potential for the slow nuclear motion, and the bond vibration is fast compared with the molecular rotation. Because the energy is a sum, the Boltzmann factor is a product and the single-molecule partition function factorizes,

The $N$-molecule partition function keeps the translational indistinguishability factor, $Z=z^N/N!$, so $\ln Z=N\ln z-\ln N!$ and each factor of $z$ contributes additively to the free energy, the energy, and the heat capacity. The heat capacity is thus a sum of independent pieces,

each computed from its own partition function. Translation contributes the monatomic $C_V^{\mathrm{tr}}=\frac{3}{2}N{k}_B$; the electronic factor is a constant $g_0$ for temperatures far below the first electronic excitation, contributing nothing to $C_V$. Rotation and vibration are the temperature-dependent pieces.

The rigid rotor

A diatomic molecule rotating about an axis perpendicular to its bond is a rigid rotor with moment of inertia $I=\mu r_0^2$, where $\mu$ is the reduced mass and $r_0$ the bond length. Its quantized energy levels are

with degeneracy $2J+1$ for the $2J+1$ orientations of the angular momentum. The rotational temperature ${\theta }_{\mathrm{rot}}$ sets the scale; for hydrogen it is about $85\text{K}$, and for heavier molecules it is smaller. The rotational partition function is

Two limits are needed. At high temperature, $T\gg {\theta }_{\mathrm{rot}}$, the levels are closely spaced on the scale $k_{B}T$ and the sum becomes an integral. Substituting $u=J(J+1)$, $\mathbf{d}u=(2J+1)\mathbf{d}J$,

Then $\ln z_{\mathrm{rot}}=\ln T+\text{const}$, so $U_{\mathrm{rot}}=N{k}_{B}T$ and

the two rotational degrees of freedom each contributing $\frac{1}{2}{k}_{B}T$, exactly the equipartition value. At low temperature, $T\ll {\theta }_{\mathrm{rot}}$, only the first two terms survive,

and the heat capacity falls to zero exponentially, $C_V^{\mathrm{rot}}\sim N{k}_B(2{\theta }_{\mathrm{rot}}/T{)}^2{e}^{-2{\theta }_{\mathrm{rot}}/T}$. Rotation is frozen out below ${\theta }_{\mathrm{rot}}$, and its full $N{k}_B$ is available only above it.

The vibrating bond

The bond stretches as a one-dimensional harmonic oscillator of angular frequency $\omega$, with levels ${\epsilon }_n=\hslash \omega (n+\frac{1}{2})$ and a vibrational temperature ${\theta }_{\mathrm{vib}}=\hslash \omega /k_B$, of order $6000\text{K}$ for hydrogen. The partition function is a geometric series,

The mean vibrational energy and heat capacity follow from $U=-\partial \ln Z/\partial \beta$,

The high-temperature limit is $C_V^{\mathrm{vib}}\to N{k}_B$, the full equipartition value for a mode with both a kinetic and a potential quadratic term ($\frac{1}{2}{k}_{B}T$ each). The low-temperature limit is $C_V^{\mathrm{vib}}\to N{k}_B({\theta }_{\mathrm{vib}}/T{)}^2{e}^{-{\theta }_{\mathrm{vib}}/T}\to 0$, frozen below ${\theta }_{\mathrm{vib}}$. Because ${\theta }_{\mathrm{vib}}\gg {\theta }_{\mathrm{rot}}$ for typical molecules, the two modes switch on at well-separated temperatures.

The heat-capacity staircase

Adding the pieces gives the temperature dependence of the molar heat capacity, expressed per mole with $N{k}_B=R$:

• Below ${\theta }_{\mathrm{rot}}$: only translation is active, $C_V=\frac{3}{2}R$.
• Between ${\theta }_{\mathrm{rot}}$ and ${\theta }_{\mathrm{vib}}$: translation and rotation, $C_V=\frac{3}{2}R+R=\frac{5}{2}R$.
• Above ${\theta }_{\mathrm{vib}}$: all three, $C_V=\frac{5}{2}R+R=\frac{7}{2}R$.

The heat capacity rises in two steps, each smoothed over about a decade of temperature around the relevant $\theta$. For hydrogen the lower step sits near $85\text{K}$ and the upper near $6000\text{K}$; in practice the molecule dissociates before the vibrational plateau is fully reached, so the $\frac{7}{2}R$ value is approached rather than cleanly attained. The staircase is the direct, measurable signature of quantized internal levels: a classical molecule with continuous rotation and vibration would show the flat $\frac{7}{2}R$ at all temperatures, and its absence was one of the early failures of classical statistical mechanics that the equipartition lesson flagged.

The symmetry number and ortho/para hydrogen

For a homonuclear molecule the two nuclei are identical, and a rotation by $\pi$ maps the molecule onto itself. Counting all rotational orientations then double-counts every physical configuration, and the high-temperature partition function must be divided by the symmetry number $\sigma$,

The symmetry number changes the additive constant of the entropy but not the heat capacity, which depends only on $\ln T$. Its microscopic origin is the exchange symmetry of identical nuclei, and for hydrogen this origin has an observable consequence. The two protons are spin-$\frac{1}{2}$ fermions, so the total molecular wavefunction must be antisymmetric under their exchange. Exchange combines a spatial part, whose parity under nuclear interchange is $(-1{)}^J$, with a nuclear-spin part. The symmetric nuclear-spin states (total nuclear spin $1$, three states, ortho hydrogen) must pair with odd $J$; the antisymmetric nuclear-spin state (total spin $0$, one state, para hydrogen) pairs with even $J$.

At high temperature the two species populate odd and even $J$ in the $3:1$ ratio of their nuclear-spin weights, and the effective symmetry number $\sigma =2$ emerges as the average over that mixture. Because interconversion between ortho and para is slow — it requires flipping a nuclear spin, weakly coupled to the rotational motion — cooled hydrogen behaves as a nonequilibrium mixture of two gases with distinct rotational partition functions, and its low-temperature heat capacity depends on the ortho/para ratio. This is a direct thermodynamic fingerprint of nuclear-spin statistics, and it was historically decisive evidence that the proton has spin $\frac{1}{2}$.

Summary

• The molecular partition function factorizes, $z=z_{\mathrm{tr}}{z}_{\mathrm{rot}}{z}_{\mathrm{vib}}{z}_{\mathrm{el}}$, so $C_V$ is a sum of independent mode contributions.
• The rigid rotor has ${\epsilon }_J=k_B{\theta }_{\mathrm{rot}}J(J+1)$ with degeneracy $2J+1$; above ${\theta }_{\mathrm{rot}}$, $z_{\mathrm{rot}}\to T/{\theta }_{\mathrm{rot}}$ and $C_V^{\mathrm{rot}}\to N{k}_B$, frozen below.
• The harmonic bond has ${\theta }_{\mathrm{vib}}=\hslash \omega /k_B$; $C_V^{\mathrm{vib}}\to N{k}_B$ above ${\theta }_{\mathrm{vib}}$ and freezes exponentially below.
• With ${\theta }_{\mathrm{rot}}\ll {\theta }_{\mathrm{vib}}$, the molar heat capacity climbs a staircase $\frac{3}{2}R\to \frac{5}{2}R\to \frac{7}{2}R$; for hydrogen the steps sit near $85\text{K}$ and $6000\text{K}$.
• Homonuclear molecules carry a symmetry number $\sigma =2$; for hydrogen the underlying nuclear-spin statistics split the gas into ortho ($3$, odd $J$) and para ($1$, even $J$) species, observable in the low-temperature heat capacity.