Degrees of Freedom

So far, a gas molecule has been treated as a structureless point mass, free to move only through simple translation in three-dimensional space -- along the $x$, $y$ and $z$ directions. The number of independent coordinates needed to completely specify a molecule's state of motion is called its number of DEGREES OF FREEDOM. For a single point-mass molecule, exactly three independent velocity components are needed, so it has $f=3$ degrees of freedom, all of them translational.

A real molecule made of two or more atoms bonded together, however, can also ROTATE about axes through its own centre of mass, and this rotational motion adds further, independent degrees of freedom beyond the three translational ones:

• Monatomic gas (e.g. helium, argon, neon -- a single atom per molecule): only the $3$ translational degrees of freedom exist. A single point-like atom has no meaningful internal structure to rotate about, so $f=3$ in total.

• Diatomic gas (e.g. oxygen $O_2$, nitrogen $N_2$, hydrogen $H_2$ -- two atoms joined, roughly, by a rigid rod-like bond): besides the $3$ translational degrees of freedom, the molecule can rotate about TWO independent axes perpendicular to the bond axis (rotation about the bond axis itself contributes essentially no energy, since the atoms are treated as point masses lying ON that axis, giving negligible moment of inertia about it). So a rigid diatomic molecule has $2$ rotational degrees of freedom in addition to the $3$ translational ones, giving $f=5$ in total, at moderate temperatures where the bond's own vibration can be ignored.

• Non-linear polyatomic gas (e.g. water vapour $H_2O$, ammonia $NH_3$ -- three or more atoms NOT arranged in a single straight line): such a molecule can rotate about all THREE mutually perpendicular axes through its centre of mass (unlike a diatomic molecule, none of these three rotation axes gives negligible moment of inertia, since the atoms are no longer confined to a single line). So a non-linear polyatomic molecule has $3$ rotational degrees of freedom in addition to the $3$ translational ones, giving $f=6$ in total. …