# Why heat capacity of an ideal gas varies with temperature? Why can't it remain the constant?

1. Why heat capacity of an ideal gas varies with temperature? Why can't it remain the constant?

2. Is there any molecular reason why the heat capacity of a gas cannot remain constant with temperature?

3. How do we understand heat capacity of a gas at the molecular level?

• The heat capacity of helium is very constant as long as it is a gas.
– user137289
Dec 28 '19 at 19:20

In a purely classic world, heat capacity of an ideal gas would be independent on temperature. The reason can be traced back to the fact that the Hamiltonian of a classical perfect gas is purely kinetic (depends only on the momenta, not on positions) and quadratic. The well known classical statistical mechanics equipartition theorem guarantees that the average energy associated with each quadratic term in the Hamiltonian is $$\frac12 k_BT$$. So, for instance a monoatomic gas of $$N$$ atoms would have an internal energy $$U=\frac32 N k_B T$$. A gas of diatomic molecules modeled by $$N$$ rigid rods would be $$U=\frac52 k_B T$$ (three quadratic terms corresponding to the momentum of the center of mass of each "molecule", two quadratic terms for rotations around the center of mass). The resulting constant volume heat capacities would be $$\frac32 N k_B$$ in the first case and $$\frac52 N k_B$$ in the second, at any temperature. I.e. hat capacity is directly proportional to the number of quadratic terms in the Hamiltonian
However, classical mechanics and classical statistical mechanics are just approximations of a more fundamental quantum theory. It turns out that the classical equipartition theorem breaks down when temperature decreases at the point typical thermal energy $$k_BT$$ becomes comparable with the spacing between quantum energy levels. Blow that threshold, internal energy decreases with temperature faster than linearly. The resulting effect on heat capacity is sometimes referred to as freezing of degrees of freedom because in a classical descritpion it would be equivalent to an effective reduction of the number of quadratic terms in the Hamiltonian.
Rotational energy level separations of typical molecules are in the range of energis of microwaves, i.e. below $$1$$ meV. This ensures that at room temperature rotational degrees of freedom behave classically and heat capacity of a diatomic molecule is quite close to the classical value $$\frac52 N k_B$$. However, by decreasing temperature on can observe a transition to the value $$\frac32 N k_B$$.