Q.A sample of gas consists of $\mu_1$ moles of monoatomic molecules, $\mu_2$ moles of diatomic molecules and $\mu_3$ moles of linear triatomic molecules. The gas is kept at high temperature. What is the total number of degrees of freedom?

Definition. The degree of freedom of a system is the minimum number of independent coordinates needed to completely specify its position and configuration in space. For $N$ free point particles, the maximum possible total is $f=3N$ (three spatial coordinates per particle); if the system carries $q$ independent constraints (such as atoms within a molecule being held at fixed separations from one another), the count is reduced to $f=3N-q$.

Counted by molecule type. A monatomic molecule (a structureless point mass, e.g. helium, neon, argon) has only translational freedom: $f=3$. A diatomic molecule (e.g. hydrogen, nitrogen, oxygen) additionally has two rotational degrees of freedom (about the two axes perpendicular to the bond; spin about the bond axis itself carries negligible moment of inertia and doesn't count), giving $f=5$ at normal temperature; at very high temperature its two atoms also start to vibrate, contributing two more degrees of freedom (one kinetic, one potential energy of the stretched bond), raising the total to $f=7$. A linear triatomic molecule (e.g. carbon dioxide, O=C=O) behaves rotationally just like a diatomic molecule (only two rotational axes, since all three atoms sit on one line), so it too has $f=5$ at normal temperature and $f=7$ at high temperature. A non-linear (bent) triatomic molecule (e.g. water, H-O-H) has three genuine rotational degrees of freedom, since no …