Two absolutely critical properties of gas are its temperature and pressure. These determine the liquifying conditions of that gas. Joule and Thompson devised this experiment that shows how the temperature varies with the pressure.

They simply let a gas expand from a high pressure environment to a low pressure environment in an adiabatic container. This is particular clever, since it imposes the constraint of constant enthalpy.

(net work = wnet)wnet = p1V1 – p2V2 (work done on gas)
ΔU = U2 – U1 = q + wnet = wnet
\ U2 – U1 = p1V1 – p2V2
Since H = U + pV
\ H2 = H1
\ constant enthalpy

They simply let a gas expand from a high pressure environment to a low pressure environment in an adiabatic container. This is particular clever, since it imposes the constraint of constant enthalpy.

The Joule-Thompson coefficient is

If m is positive then the gas cools on expansion, and if its negative it heats on expansion. The temperature at which the sign changes is called the inversion temperature. Since the gas cools on expansion if m is positive or heats on expansion if it iss negative, we can assume that if m = 0, then the temperature will remain unchanged on expansion.

If you look at a high gas pressure cylinder and watch  it release the gas, you’ll notice it cools (assuming m is positive), sometimes sufficiently so as to condense water vapour on the cylinder.