# Why do the specific heat constants vary with temperature for semi-perfect gases?

We were taught that the difference between the perfect and semi-perfect lies in that the specific heat constants for perfect gases are constant, while for semi-perfect gases they are functions of temperature and temperature only. However, both perfect and semi-perfect gases are a subset of ideal gases for which the main assumption is that the intermolecular forces are negligible. For that reason changes in potential energy associated with these forces are negligible as well.

If that is the case, what changes in a semi-perfect gas with temperature so that the same heat input does not equal the same temperature raise? After all, from my understanding of temperature it is directly proportional to average kinetic energy of the gas molecules, and hence should vary linearly with heat input.

## 1 Answer

Poly-atomic molecules have different modes of vibration or rotation, each equivalent to a quantum harmonic oscillator or quantum rotor. The modes are quantized. For vibrations at natural frequency $\omega$ the energy levels are uniformly spaced : $E_n=(n+\frac12)\hbar \omega$. For a rigid rotor of moment of inertia $I$ the energy levels are quadratically spaced : $E_l=l(l+1)\frac{\hbar^2}{2 I}$.

Each mode has a different minimum amount of energy. When the gas is in thermal equilibrium each mode holds $\frac12kT$ of energy. So a vibration mode will not be excited in a significant fraction of molecules until $\frac12kT \ge \frac12\hbar \omega$.

So different modes of vibration or rotation become active as the temperature is increased, and the capacity of each mode increases at a different rate depending on the energy spacing. This leads to the stepped relation between specific heat and temperature as seen in hydrogen :