Behavior of Perfect Gas and Kinetic Theory - Degrees of Freedom

Degrees of Freedom ($f$) is defined as the total number of independent coordinates or independent ways in which a system can possess energy.

For a system containing $N$ particles with $k$ independent relations/constraints between them, the degrees of freedom is given by $f = 3N - k$.

Monoatomic gases (e.g., $He$, $Ne$, $Ar$) have only 3 translational degrees of freedom ($f = 3$).

Diatomic gases (e.g., $O_2$, $H_2$, $CO$) at moderate temperatures have 3 translational and 2 rotational degrees of freedom, giving $f = 5$. At very high temperatures, 2 additional vibrational degrees of freedom are added.

Polyatomic non-linear molecules (e.g., $H_2O$, $NH_3$) have 3 translational and 3 rotational degrees of freedom, totaling $f = 6$.

The Law of Equipartition of Energy states that for any system in thermal equilibrium, the total energy is equally distributed among its various degrees of freedom, and each degree of freedom contributes an average energy of $\frac{1}{2} k_B T$.

The internal energy ($U$) of $n$ moles of an ideal gas is related to the degrees of freedom by the expression $U = \frac{f}{2} nRT$.