We know that for an ideal gas, there is a one-to-one
correspondence between the temperature and the kinetic energy of the
gas particles. However, that does not mean that there is a one-to-one
correspondence between kinetic energy and heat energy. (In this
context, heat energy refers to whatever is measured by a heat
capacity experiment.)
To illustrate this point, let’s consider a sample of pure monatomic
nonrelativistic nondegenerate ideal gas in a cylinder of horizontal
radius r and vertical height h at temperature T. The pressure
measured at the bottom of the cylinder is P. Each particle in the gas
has mass m. We wish to know the heat capacity per particle at
constant volume, i.e. Cv/N.
At this point you may already have in mind an answer, a simple answer,
a well-known answer, independent of r, h, m, P, T, and N. But
wait, there’s more to the story: The point of this exercise is that h
is not small. In particular, mgh is not small compared to kT, where g
is the acceleration of gravity. For simplicity, you are encouraged to
start by considering the limit where h goes to infinity, in which
case the exact value of h no longer matters. Gravity holds virtually
all the gas near the bottom of the cylinder, on the scale of a few
kT/mg.
Later, if you want to come back and work the problem a second time,
with a large but finite h, that’s worth doing. Also if you want to
generalize to a polyatomic gas, that’s also worth doing.
You will discover that a distinctly nontrival contribution to the
heat capacity comes from the potential energy of the ideal gas. When
you heat it up, the gas column expands, lifting its center of mass,
doing work against gravity. (Of course, as always, there will be a
contribution from the kinetic energy.)
So, we conclude that in general, heat energy is not just kinetic
energy.