What it does
Heat a gas and the energy goes into its molecules. A single atom can only store energy by moving (along x, y and z). A two-atom molecule can also turn end over end (two ways). A hot molecule can vibrate too, its atoms bouncing on the bond like masses on a spring. The law of equipartition says that, on average, every one of these independent ways (each degree of freedom, f) holds the same energy, ½kT per molecule.
This page shows each molecule moving, turning and vibrating, counts its degrees of freedom, and works out from them the internal energy U = (f/2) nRT, the molar heat capacities Cv = (f/2) R and Cp = Cv + R, and their ratio γ = 1 + 2/f. A table puts all four molecules side by side, and a mixer finds Cv and γ for a blend of a one-atom and a two-atom gas.
It serves Class 11 Physics, Kinetic Theory (law of equipartition of energy, specific heat capacity of monatomic, diatomic and polyatomic gases) and Thermodynamics (Cp − Cv = R and the adiabatic exponent γ). Start from kinetic theory to see where ½kT per direction comes from, and use γ in carnot engine and in the gas laws in gas law calculator.
How to use it
- Pick a molecule: one atom (helium, neon, argon), two atoms (nitrogen, oxygen), three in a line (carbon dioxide) or bent (water vapour).
- Tick Hot gas: count vibrations too to unfreeze the vibrations. At room temperature they hardly take part for light molecules, so the box starts unticked.
- Temperature and amount change the internal energy U. Cv, Cp and γ do not depend on them in this model.
- In Mix two gases, type the moles of a one-atom gas and of a two-atom gas.
Worked example
As the page opens, 1 mol of helium at 300 K: f = 3, so Cv = 3/2 × 8.314 = 12.47 J/(mol K), Cp = 20.79 J/(mol K), γ = 5/3 = 1.667, and U = 3/2 × 1 × 8.314 × 300 = 3742 J.
Nitrogen (rigid): f = 3 + 2 = 5, Cv = 5/2 R = 20.79, Cp = 7/2 R = 29.10, γ = 7/5 = 1.4. With the vibration counted, f = 7 and γ = 9/7 = 1.286.
Water vapour (rigid): f = 3 + 3 = 6, γ = 4/3. Carbon dioxide, hot: 3 + 2 + 4 × 2 = 13.
Mixing 1 mol of helium with 1 mol of nitrogen: Cv = (1.5 + 2.5) R / 2 = 2R, so γ = 3R / 2R = 1.5.
Check yourself
Five fresh questions: γ and Cv for a gas, the internal energy of a diatomic gas, γ of a mixture, and which gas has the largest γ. Answers are checked on the page and the working is shown. Nothing is saved.
Tips
- Count f as 3 (moving) + rotations (0, 2 or 3) + 2 × each vibration you allow.
- A line of atoms (N₂, CO₂) has only 2 rotations; a bent molecule (H₂O) has 3.
- A molecule of N atoms has 3N ways in total. Take away the moving and turning ones and the rest are vibration modes.
FAQ
Why are vibrations “frozen” at room temperature? Vibrations come in quantum steps that are big compared with kT at room temperature for light molecules, so they rarely get excited. Equipartition, a classical idea, cannot explain that; quantum physics does. As a gas gets hotter, γ slowly falls as the vibrations wake up.
Why is Cp bigger than Cv? At constant pressure the gas expands as it warms and does work pushing the surroundings back. That extra energy, R per mole per kelvin for an ideal gas, has to be supplied as heat.
Do real gases follow this? Noble gases match the one-atom values very closely, and air at room temperature has γ very close to 1.4. Heavier molecules and hot gases fall between the rigid and fully vibrating values.
Where the numbers come from
- Law of equipartition of energy: ½kT per quadratic degree of freedom; a vibration mode counts twice (kinetic and potential); 3N = translations + rotations + vibration modes; Cv = (f/2)R, Cp = Cv + R (Mayer's relation), γ = 1 + 2/f: standard kinetic theory.
- Gas constant R = 8.314462618 J/(mol K) (CODATA 2018).
- Checked against hand counts and an independent Python calculation.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.