Lesson 2 · 35 min

Entropy and heat capacity from molecular levels

N₂ and CO weigh the same, yet CO has 6 J K⁻¹ mol⁻¹ more entropy. Partition functions split the entropy into translation, rotation and vibration, reproduce the tabulated standard entropies from spectroscopic constants, and show heat capacity rising as vibration wakes up.

In this lesson

Use the partition function to say how many states a molecule can reach at a given temperature
Split the molar entropy of a gas into translational, rotational, vibrational and electronic parts, and compute a standard entropy from molecular constants
Use the characteristic temperatures θ_rot and θ_vib to explain why the heat capacity of a diatomic gas is about 7/2 R at room temperature and climbs towards 9/2 R when hot

Open a table of thermodynamic data: N₂ and CO both have a molar mass of 28 and almost the same bond length, yet their standard entropies are 191.61 and 197.66 J K⁻¹ mol⁻¹ — CO is 6.05 higher. Cl₂ is higher still, at 223.08. These numbers first came from calorimetry, heating a sample up from near absolute zero and integrating Cp/T; today the tables are computed from spectra instead, and the two agree. Why should molecules of the same size have different entropies? The answer lies in the energy levels a molecule can use.

Predict first

N₂ and CO have the same mass and almost the same bond length, and both rotate freely at room temperature. The difference: the two ends of N₂ are identical, so after half a turn it cannot be told from how it started. Which has the larger entropy, and by roughly how much?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

They are equal CO, by about R ln 2 ≈ 6 J K⁻¹ mol⁻¹ N₂, by about 6 J K⁻¹ mol⁻¹ CO, by about 60 J K⁻¹ mol⁻¹

The partition function: how many states are in reach

By the Boltzmann distribution, a state of energy ε is occupied with relative probability $e^{-\varepsilon/kT}$ . Add this factor over all states and you have the partition function q: roughly the number of states a molecule can actually reach at temperature T. When the levels are spaced much wider than kT, q ≈ 1 and only the lowest state is used; when they are much closer, q is large. A molecule’s energy splits into translational, rotational, vibrational and electronic parts, so q splits into a product and the entropy into a sum.

q = \sum_i g_i\, e^{-\varepsilon_i/kT}, \qquad S = S_\text{trans} + S_\text{rot} + S_\text{vib} + S_\text{el}

For each kind of motion

S = R\ln q + U/T

(for translation, the indistinguishability of the molecules also enters, giving the Sackur–Tetrode equation).

$q$: partition function
$g_i$: degeneracy of level i
$\varepsilon_i$: energy of level i
$U$: molar internal energy of that motion

The three kinds of motion have wildly different level spacings, best compared through characteristic temperatures: θ_rot = hcB/k and θ_vib = hcν̃/k. Translational levels are so close that at room temperature each CO molecule has about 6 × 10⁶ translational states in reach. CO’s θ_rot is only 2.8 K; room temperature is far above it and q_rot ≈ T/(σθ_rot) ≈ 108. Its θ_vib is 3084 K, though, so almost every molecule sits in v = 0, q_vib ≈ 1, and vibration adds almost nothing to the entropy. Cl₂ is heavy and “soft”, with θ_vib of only 798 K: about 7 % of its molecules are already vibrationally excited at room temperature.

Worked example

The standard entropy of CO from molecular constants

From the mass of CO, its rotational constant B₀ = 1.9225 cm⁻¹ and its vibrational fundamental of 2143.3 cm⁻¹, compute the molar entropy at 298.15 K and 1 bar.

Heat capacity: motions waking up one by one

Far above a motion’s characteristic temperature, the motion is fully “awake” and equipartition gives ½R per squared term: 3/2 R for translation, R for rotation and R for vibration (kinetic plus potential energy). Classical physics therefore predicts Cv = 7/2 R and Cp = 9/2 R for a diatomic gas — but at room temperature Cp is only 7/2 R. This puzzled nineteenth-century physicists; quantisation gives the answer: room temperature is far below θ_vib, and vibration is still “frozen”. As the temperature rises, vibration’s share climbs along the Einstein function to R and Cp rises towards 9/2 R — the model below lies almost exactly on the measured curve from the JANAF tables.

Partition functions · checked against the JANAF tables

Entropy and heat capacity from molecular levels

Temperature 298 K

Where the molar entropy comes from (1 bar)

translation 162.0
rotation 58.6
vibration 2.2
electronic 0.000
total 222.9 J K⁻¹ mol⁻¹

Molar heat capacity at constant pressure, Cp/R

Model (partition functions) 4.068
JANAF table (Shomate fit) 4.082

At 298.15 K: computed 222.87, CODATA table 223.081 J K⁻¹ mol⁻¹ (difference −0.21).

θ_rot = hcB/k = 0.35 K, θ_vib = hcν̃/k = 798 K

States available per molecule: translation 2.37 × 10⁷, rotation 424.6, vibration 1.074.

Molecules in v = 1 or above: 6.880 %.

Drag the temperature from 50 K to 3000 K and watch Cp/R climb from 7/2 towards 9/2. Cl₂, with its low θ_vib, is already clearly above 7/2 at room temperature; N₂ and CO only stir above several hundred kelvin. The electronic ground state of O₂ is triply degenerate, which adds R ln 3 = 9.13 J K⁻¹ mol⁻¹ to its entropy.

Try the calculation

The vibrational fundamental of Cl₂ is 554.4 cm⁻¹ (hc/k = 1.4388 cm K). In the harmonic approximation, what percentage of the molecules are in v ≥ 1 at 298.15 K?

Concept check

At room temperature, which gas gets the largest contribution to Cp from vibration?

HCl N₂ CO Cl₂

Key takeaways

The partition function q = Σ g e^(−ε/kT) is roughly the number of states in reach at temperature T; for independent motions q multiplies and the entropy adds.
At room temperature about 3/4 of CO’s entropy is translational and 1/4 rotational, with almost nothing from vibration; the 197.65 computed from molecular constants matches the table’s 197.66 J K⁻¹ mol⁻¹.
The symmetry number σ = 2 of a homonuclear molecule lowers its entropy by R ln 2; the triply degenerate ground state of O₂ raises it by R ln 3.
A diatomic gas has Cp ≈ 7/2 R for T ≪ θ_vib, rising towards 9/2 R as vibration wakes; Cl₂, with θ_vib of only 798 K, is already well above 7/2 R at room temperature.

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Data and references

J. D. Cox, D. D. Wagman and V. A. Medvedev, CODATA Key Values for Thermodynamics, Hemisphere, 1989 (standard entropies, via the NIST Chemistry WebBook).
M. W. Chase, Jr., NIST-JANAF Thermochemical Tables, 4th ed., J. Phys. Chem. Ref. Data Monograph 9 (1998) (Shomate heat-capacity fits, via the NIST Chemistry WebBook).
K. P. Huber and G. Herzberg, Constants of Diatomic Molecules, Van Nostrand Reinhold, 1979 (via the NIST Chemistry WebBook).
D. A. McQuarrie and J. D. Simon, Physical Chemistry: A Molecular Approach, University Science Books, 1997 (partition functions and statistical thermodynamics).