While reading Free-expansion, I got this:

Real gases experience a temperature change during free expansion. For an ideal gas, the temperature doesn't change, [...]

I know that temperature doesn't change for free-expansion of ideal gas since internal energy is a function of temperature and since, $\Delta U= 0\;,$ temperature remains same.

But the later statement? Why does the temperature change for real gas? Does the internal energy of real gas not depend on temperature? Can anyone please explain this fact?


The internal energy of a real gas depends on both temperature and pressure. So, if U remains constant and pressure changes, the temperature must change. In the ideal gas limit of very low pressures, the pressure dependence of real gases weakens and approaches zero.


The reason that the temperature changes in an expansion/compression of a real gas at constant internal energy can be quite intuitively understood.

Suppose that the interactions between the gas molecules are repulsive. If we compress the gas while fixing its energy, the average intermolecular distance decreases, leading to the increase in the potential energy. Accordingly, the kinetic energy, as well as the temperature, has to decrease (energy conservation).

Similar lines of reasoning apply for other cases, i.e., expansion or compression of a gas with repulsive or attractive interactions.


Experiment which Joule tried to fulfil is termodinamically unstable process. Meaning that is goes through states in which pressure fields are not propagate fast enough, and often there is no pressure or temperature at all.

If you have a gas, which is not standing in "termodinamical balanced state", imaging you have every atom having random velocity. So you have no state of temperature. You could have something looking like temperature after finging some median velocity, but velocities is all you stand with. There is no temperature.

The same problem goes when you have all atoms moving towards same direction, technically you would obtain some "temperature" like median but that is wrong - termodinamically you can only look into bodies with immovable center of mass.

What this wiki article is absolutely right is that $dS=dQ/T$ formula is not working.

Here goes some more info about why Joule didn't find temperature change. https://en.wikipedia.org/wiki/Joule_expansion#Real-gas_effect But that article calls it "adiabatic".

Also you can do free expansion and many other fun experiments numerically with real gases with LAMMPS and many other packages containing data for real gases.

Also precious commentary from @CuriousOne notes that we could have "mind experiment" (concerning only first and final states of our mechanical system), but we could also note that ideal gas shows in that unstable processes that its model is not "complete". In a sense that it does not have potential, but its molecules are clearly somehow "organized". For real gas, there is "close" potential and "far" potential, which could have non-mechanical nature, so everything could happen there.

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    $\begingroup$ By free expansion they mean that the gas is not performing any work. The volume changes, but that doesn't change the velocities for an ideal gas because there are no inter-molecular forces. For a real gas the larger average distance between molecules changes the internal energy distribution, which does change the temperature. $\endgroup$ – CuriousOne Jan 23 '16 at 4:12
  • $\begingroup$ So I guess I just remove that part discussing "adiabatic". $\endgroup$ – sanaris Jan 23 '16 at 4:26
  • $\begingroup$ "and often there is no pressure or temperature at all"? $\endgroup$ – dryairship Mar 23 '18 at 14:50

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