Degrees Of Freedom

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An ideal monatomic gas such as helium consists of molecules that behave as point masses without rotational inertia; how many degrees of freedom does each molecule possess?

RankGuru · Accepted Answer

Degrees of freedom count the independent ways a molecule can store energy. A monatomic gas molecule, such as helium or argon, is treated as a structureless point mass that can only move along three perpendicular spatial directions, giving exactly three translational degrees of freedom. It has no rotational degrees of freedom because a point mass has negligible moment of inertia, and no vibrational modes because there is no interatomic bond. Hence the answer is 3. The value 5 belongs to a rigid diatomic molecule, which adds two rotational degrees of freedom. The value 6 would apply to a non-linear polyatomic molecule with three translational and three rotational modes. The value 2 underestimates by omitting one translational direction. This follows the NCERT law of equipartition, where each degree of freedom contributes $\frac{1}{2}k_B T$ of energy. As a consistency check, the internal energy of one mole of monatomic gas is $\frac{3}{2}RT$, exactly matching three degrees of freedom, confirming the result.

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