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Directions

This simulation builds the molecular partition function, q, of a diatomic ideal gas one contribution at a time. It shows how translational, rotational, vibrational, and electronic motion each add to q, and how the heat capacity (C_v, C_p), entropy (S), and q itself change with temperature as each mode "turns on" near its own characteristic temperature — for example, the rotational and vibrational steps in C_v(T) for H₂.

how to use this simulation

• Choose a molecule from the dropdown menu or select custom, which opens the molecular parameters, or open "molecular parameters (editable)" to build a custom molecule from its own θ_rot, θ_vib, mass, symmetry number, and electronic degeneracies. The molecular parameters can be changed using the up/down arrows on the side of the input boxes or by dragging the numbers to the left or right to decrease or increase the numbers.
• Use the checkboxes to turn translational, rotational, vibrational, and electronic contributions on and off, and watch how q, C_v, S, and C_p change in the three plots.
• Drag the temperature slider, or click and drag directly on any plot, to move the temperature; the table below the plots updates with each mode's contribution at that temperature.
• Use the temperature axis and units buttons to switch between a log or linear temperature scale and between dimensionless (R) and J mol⁻¹ K⁻¹ units.
• The dashed vertical lines mark each mode's characteristic temperature; the dotted horizontal lines mark the classical equipartition limits those heat capacities approach.
• For H₂, the "rotational treatment" menu also offers the equilibrium and frozen (normal) ortho/para mixtures, which is what real H₂ heat-capacity data follow.

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Details

The total partition function factors into independent contributions, so ln q, U, C_v, and S all add over the enabled modes:

translational — free particle in a box of volume V = kT/P:

rotational (rigid rotor):

For a heteronuclear molecule (σ = 1) the sum runs over all J. For a homonuclear molecule (σ = 2), the symmetry of the nuclear spins allows only every other J level, so the sum runs over even J only (odd J only for O₂, with energies measured from J = 1). Skipping half the levels is what produces the factor 1/σ in the high-T limit, but unlike dividing the full sum by σ, it keeps q_rot ≥ 1 at low temperature, so ln q_rot and S_rot never go negative.

As T ≫ θ_rot, C_v,rot/R → 1. On the way there, C_v,rot/R briefly overshoots: to about 1.1 for a heteronuclear molecule (all J), and to about 1.47 near T ≈ 2θ_rot for an even-J homonuclear molecule, because the first allowed excitation (J = 0 → 2) is a larger jump. The odd-J sum used for O₂ has almost no overshoot.

vibrational (harmonic oscillator, energy measured from the ground vibrational state):

electronic:

A degenerate ground state (g₀ > 1) contributes S_elec/R = ln g₀ even as T → 0. A low-lying excited state (NO, θ_el = 174 K) produces a Schottky bump in C_v.

Pressure & C_p: C_p = C_v + R for an ideal gas. Entropies use the chosen standard-state pressure and exclude nuclear-spin entropy.

H₂ spin isomers: ortho-H₂ (odd J, nuclear spin weight 3) and para-H₂ (even J, weight 1). "equilibrium" mixes the two populations at every temperature; "normal" H₂ freezes the room-temperature 3:1 ortho:para ratio, which is what most measured H₂ heat-capacity data actually follow.

Model limits: this is a rigid-rotor, harmonic-oscillator, ideal-gas model — no dissociation, anharmonicity, centrifugal distortion, or vibration–rotation coupling. Curves above roughly 2000–3000 K (H₂ up to about 3000 K) are a thought experiment, not real gas behavior. At the other extreme, the translational partition function treats the gas classically, which requires many translational states per molecule (q_trans/N ≫ 1, that is, Λ³ ≪ V/N). For light molecules at very low temperature (H₂ below about 3 K at 1 bar) q_trans/N falls below 1: the thermal wavelength is then larger than the space available to each molecule, quantum (Bose–Einstein or Fermi–Dirac) statistics are needed, and the Sackur–Tetrode entropy can even become negative. That region is shaded gray on the plots; real gases have condensed long before it is reached. The H₂ equilibrium and normal ortho/para options divide out the nuclear-spin degeneracy (the factor 1/4), so their ln q_rot and S_rot can be negative at low temperature; that is a bookkeeping convention, not an error.

Symbols

• qmolecular partition function (and q_trans, q_rot, q_vib, q_elec for each mode)
• Tabsolute temperature (K)
• θ_rotcharacteristic rotational temperature, θ_rot = h²/(8π²Ik) (K)
• Imoment of inertia of the molecule, I = μr_e², where μ is the reduced mass and r_e is the equilibrium bond length
• θ_vibcharacteristic vibrational temperature, θ_vib = hν/k (K)
• νvibrational frequency of the bond (s⁻¹)
• θ_elenergy of the first excited electronic state, divided by k (K)
• σsymmetry number (2 for a homonuclear diatomic, 1 for heteronuclear)
• Jrotational quantum number
• g₀, g₁degeneracy of the ground and first excited electronic states
• k, h, RBoltzmann constant, Planck constant, gas constant
• Λthermal de Broglie wavelength
• mmass of one molecule
• P, V, Npressure, volume, number of molecules
• Uinternal energy
• C_v, C_pheat capacity at constant volume, at constant pressure
• Sentropy
• γheat-capacity ratio, C_p/C_v

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About

This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.