# Heat capacity of gas at zero K

Sunfire
Hello All,

I have been trying to find an established expression cp = cp(T) for a gas asymptotically close to zero K, as well as showing how does the function grow/behave in the low-temperature regions.

This can be an idealized mono/diatomic gas

Thank you

Dickfore
At low enough temperatures (such that the thermal De Broglie wavelength becomes comparable to the inter-particle distance), the gas eventually becomes degenerate, and quantum effects become important. Then, the type of the statistics obeyed by the particles plays a crucial role. Namely, if the particles are bosons, they tend to condense in the ground state, and there is a jump in heat capacity associated with this condensation. If the particles are fermions, then the heat capacity behaves linearly with temperature.

These conclusions are valid for ideal gases. When interaction between the particles is taken into account, these effects change slightly. For bosons, the interaction causes some of the particles to get excited from the ground state. For fermions, if the interaction is attractive, we may have a formation of Cooper pairs that condense to a superfluid state.

Sunfire
I am looking for a (hypothetical) gas that remains gas very close to T=0... and for the expression c_p(T) for such a gas. Because this concept still has to be reminiscent of a gas, I imagine a mono- or diatomic gas would fit the bill.

A Fermi gas would be applicable to an electron gas, and Bose-gas would still be comprised by elementary particles that obey Bose statistics. These would not work for me

Dickfore
But, in reality, nothing remains a gas at T = 0. It would either solidify in a crystal, or as in the case of helium, remain liquid at atmospheric pressure. For helium, there are two isotopes, 4He, which behaves as a Bose liquid, and 3He, which behaves as a Fermi liquid. The quantum effects I was referring to are quite prominent in this substance and were measured experimentally even before a theory had been put forward to explain them.

Sunfire
I actually need the c_p(T) for non-condensing gas. Again, this can be an idealized gas, not a real one; and T may not be precisely zero, but asymptotically close to it.

I need the mathematical form of c_p(T), especially between [0+epsilon, 300K], epsilon --> 0
The only reference I have so far is "Introduction to Statistical Mechanics and Thermodynamics" by Keith Stowe. On page 305 he plots a graph c_p(T) for a hypothetical diatomic gas, which spans from infinity to ~10,000K (vibrational mode) then to about 10K (rotational mode), then the low-temp region (translational mode), going towards the absolute zero when c_p --> 0