May molecules of ideal gases have an inner structure?

Question

The following question is probably very elementary: whether molecules of ideal gases may have optic properties? As far as I understand, when one discusses optic properties, one assumes that molecules of the material have some inner structure, in particular different energy levels.

The question is whether existence of such an inner structure may contradict the assumption that the gas is ideal.

Answer 1

If one defines an ideal gas as a gas made up of non-interacting entities, then the answer is yes, the gas may indeed have internal structure. The equation of state for such a gas will still be $pV = nRT$, but its energy will not be $E = (3/2) RT$ per mole. The exact form of the energy equation will depend on the internal energies of the molecules. The entropy will also depend on the inner structure.

All this is easy to see using statistical mechanics, but that's hard to show here. Suffice it to say that a verbal, but somewhat imprecise explanation is that $pV = nRT$ depends only on the translational motion of the molecules. This motion is almost always independent of the internal structure of the molecules.

Answer 2

An ideal gas can have internal degrees of freedom, and they contribute to the specific heat, but the ideal gas law is still obeyed so long as the following conditions are satisfied:

• The deBroglie wavelength of the particle at the typical thermal energy $kT$ is significantly smaller than the interparticle separation (so that the particle phase space may be approximated as classical)
• The total interaction volume of the particles (the volume of the little spheres where the particles feel a force between one another) is much smaller than the volume of the container

Then the ideal gas law $P=nT$ holds. This law only assumes that the temperature is only a function of the internal energy, not the volume, or equivalently that the internal energy is purely a function of the temperature (and the particle number) not the volume.

In the most fundamental terms, the ideal gas assumption is that the phase space can be factorized as follows:

$$ S(U,V,N) = N\log({V\over N}) + S(U) $$

This is a factorization, because entropy is a log. It says that the phase space volume is the thermodynamic limit of $V^N\over N!$ (the total possibilities for N particles to occupy volume V, ignoring the excluded volume effect) times a certain volume from the momentum allowed at a given internal energy. The $N!$ is for identical particles, and you can ignore it if N isn't changing.

The partial derivative of S with respect to V is the pressure over the temperature, and it is {N\over V} as advertised, while the partial derivative of S with respect to U is the inverse temperature, and it is only a function of U. The ideal gas law works for relativistic gasses, for photons, or for a gas of electron quasiparticles in a semiconductor, or for the pressure exerted by a solute on a semipermeable membrane in a water solution. In these cases, the energy as a function of the momentum is changed completely, but the entropy is still the logarithm of $V^N$ plus an energy dependent part.

If the particles can be excited, the specific heat will change, as the temperature gets big enough for the thermal energy to excite the quantum levels of the internal structure. For energy levels which affect visible light, the temperature where you see this structure in the specific heat is when $kT=hf$ and this is about 9,000 degrees for 1 eV photons.