Q.A vessel contains $0.5\ \text{mol}$ of helium (monatomic) gas at $27\,^\circ\text{C}$. Find the total internal energy of the gas, using $U = \tfrac{3}{2}nRT$.

A monoatomic gas molecule has exactly 3 degrees of freedom -- all translational.

Degrees of freedom are the independent ways a molecule can possess energy (independent coordinates needed to specify its position/configuration completely). A monoatomic gas molecule (e.g. He, Ne, Ar) is modelled as a single point particle with no internal structure to rotate or vibrate at ordinary temperatures. It can move independently along the x, y and z directions, giving it 3 translational degrees of freedom and no rotational or vibrational degrees of freedom.

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In thermal equilibrium, every independent quadratic term of energy a molecule possesses (each degree of freedom — translational, rotational, or vibrational) gets, on average, the same share of energy: (1/2) kB T per molecule.

Statement: According to the law of equipartition of energy, for a system in thermal equilibrium at absolute temperature T, the total average energy is equally distributed among all the degrees of freedom, and each degree of freedom associated with a quadratic term in the energy expression (e.g. (1/2)mv_x^2 for motion along x, or (1/2)I omega^2 for rotation about an axis) contributes on average an energy of (1/2) kB T to the total energy of the system, where kB is Boltzmann's constant.

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A diatomic gas molecule has 3 translational + 2 rotational degrees of freedom = 5 degrees of freedom at ordinary (moderate) temperatures.

A degree of freedom is an independent way in which a molecule can possess energy.

For a diatomic molecule (like O2, N2), modeled as two point masses connected by a rigid rod:

Translational degrees of freedom: The molecule as a whole can move along the x, y, and z directions — this gives 3 translational degrees of freedom (same as for a monatomic gas).

Rotational degrees of freedom: The molecule can rotate about two axes perpendicular to the line joining the two atoms (rotation about the bond axis itself is not counted, since the moment of inertia about that axis is negligible for point-like atoms). This gives 2 rotational degrees of freedom.

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