Molar Heat Capacity of a Gas Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 10/10/2026
The molar heat capacities of an ideal gas are calculated from its number of degrees of freedom f: Cv=(f/2)×R (at constant volume), Cp=Cv+R (at constant pressure). For a monoatomic gas (f=3, such as helium), the adiabatic index γ=Cp/Cv is exactly 5/3≈1.667; for a diatomic gas like air (f=5), γ is exactly 1.4 — the value used to calculate the speed of sound in air.
Explanation
The molar heat capacity of a gas measures the amount of heat needed to raise the temperature of one mole of that gas by one kelvin. According to the equipartition theorem, each degree of freedom of a molecule (each independent way it can store kinetic energy) contributes (1/2)×R to its molar heat capacity at constant volume. A monoatomic molecule (such as helium or argon) has only 3 degrees of freedom, all translational along the three directions of space — exactly the same ones already counted in our average kinetic energy of a gas calculator. A diatomic molecule (such as the nitrogen or oxygen in air) additionally has 2 rotational degrees of freedom around the two axes perpendicular to the bond between its two atoms (rotation around the bond axis itself does not count, since the moment of inertia around that axis is negligible), bringing its total to f=5. The heat capacity at constant pressure Cp is always greater than Cv by exactly R (Mayer's relation), because at constant pressure the gas must also perform expansion work against the external pressure in addition to raising its temperature. The ratio γ=Cp/Cv, called the adiabatic index or Laplace coefficient, is a quantity of considerable practical importance: it determines the speed of sound in a gas, and governs a gas's behavior during rapid compression or expansion (adiabatic, with no heat exchange with the surroundings), as in a heat engine or a bicycle pump that heats up as it compresses air.
Example: heat capacities of air (diatomic gas)
Inputs
Degrees of freedom: f=5 (diatomic gas, air/N₂/O₂ at room temperature).
Calculation
Cv = (5/2) × 8.314462618 ≈ 20.79 J/(mol·K). Cp = Cv + R ≈ 29.10 J/(mol·K). γ = Cp/Cv = 29.10/20.79 = 1.4 exactly.
Result
Air has an adiabatic index of exactly 1.4 — the value used in the formula for the speed of sound and in the study of adiabatic transformations in heat engines.
Frequently asked questions
Why doesn't rotation around a diatomic molecule's bond axis count as a degree of freedom?
Because the moment of inertia of a diatomic molecule around its own bond axis is negligible: almost all of the molecule's mass is concentrated very close to this axis (in the atomic nuclei themselves), unlike the two other rotation axes perpendicular to the bond, where mass is distributed at a significant distance from the axis. A rotation with no significant mass far from the axis can only store negligible energy, which explains why only 2 of the 3 possible rotation axes actually count for a diatomic molecule.
Why is Cp always greater than Cv?
Because at constant pressure, a heated gas necessarily expands (ideal gas law), meaning it must perform mechanical work against the external pressure in addition to raising its own internal temperature. At constant volume, on the other hand, the gas cannot expand and all the supplied heat goes exclusively toward raising its temperature. This extra energy needed at constant pressure, exactly R per mole per kelvin, explains why Cp always exceeds Cv by this same amount, regardless of the gas considered.
Does this formula apply exactly to real gases?
It is a very good approximation for many gases under moderate conditions, but it assumes that all the degrees of freedom considered are fully thermally "active," which is not always true in practice: a molecule's VIBRATIONAL degrees of freedom (the periodic stretching of its bond, distinct from the simple overall motion already covered by the root-mean-square speed calculator), not counted in this calculator, only become significant at high temperature, which explains why CO₂, for example, behaves like an f=6 gas only at elevated temperature, but closer to f=5 at room temperature.