# Joule expansion - why is the change in internal energy 0 for a gas expanding into a vacuum?

I understand that no work is done on gases expanding into a vacuum as there is no external pressure the gas is expanding against, $$P_{ext}\Delta V = 0$$.

But thinking in terms of the gas particles accelerating into the vacuum due to the pressure gradient, isn't the gas particles acceleration increasing the total KE of the particles in the system.

## 2 Answers

As long as your system is thermally isolated from the environment so that the expansion is adiabatic, the internal energy must stay constant during the expansion. This follows directly from the first law of thermodynamics.

But thinking in terms of the gas particles accelerating into the vacuum due to the pressure gradient, isn't the gas particles acceleration increasing the total KE of the particles in the system.

Here you are making two mistakes. First, you mix up the macroscopic picture of the gas (pressure gradient) and the microscopic picture (molecular motion). In a dilute (ideal) gas approximation, there are no forces acting on the molecules. They simply fill whatever volume is available to them by their random motion. This explains why in an ideal gas, the velocity distribution of molecules does not change upon free expansion, hence the internal energy and temperature remain constant.

The second mistake is the automatic identification of the kinetic energy of molecules with the internal energy of the gas. This is only valid in an ideal gas. In a real gas, the internal energy consists of the kinetic energy of the molecules and the energy of their interaction. It is the total internal energy that remains constant, not just the kinetic energy. In dilute gases, the molecular interactions are typically attractive, which means that upon free expansion, the molecules slow down: the kinetic energy decreases while the interaction energy increases by the same amount. As a consequence, the temperature (not the internal energy) of dilute gases typically decreases upon free expansion. This is the cooling effect you feel when you take a can of spray and let all the gas escape at once.

Assuming an adiabatic expansion, there can be no change in internal energy. In order for the gas molecules at the interface between the evacuated and gas occupied chambers to accelerate into the vacuum, other parts of the gas behind the interface must do work to "push" the gas molecules into the evacuated space. That work results in a loss of internal energy of the portions of the gas doing the work. If it is an ideal gas, all of energy used to do the work would be kinetic. Consequently, the increase in kinetic energy of the accelerated molecules would equal the decrease in kinetic energy of the gas doing work, for an overall change in kinetic energy of zero, at least for an ideal gas.

In the case of a real gas, it has internal potential energy as well. But as @Tomas Brauner points out, real gases undergo a temperature decrease. That would, of course, be contrary to an overall increase in the internal kinetic energy of the gas.

Hope this helps.