# Adiabatic free expansion (Joule expansion) needing a small explanation

My professor tries to demonstrate that adiabatic free expansion is an irreversible process:

1. for and adiabatic $$Q=0$$;

2. because the gas is expanding through vacuum W=0;

therefore $$\Delta U=Q-W=0$$

now I can put back the gas doing work on it: $$W=p\Delta V$$,

because $$\Delta V$$ is negative $$W$$ is negative too, so $$\Delta U=-W$$ we have an increment of the internal energy.

$$U=U(T)$$, $$\Delta U=\frac32nR\Delta T$$ so if the ΔU is positive $$\Delta T$$ is positive too,

now arrive the thing that I haven't understood, my professor said:

a) We can extract $$Q=W$$

b) $$Q\rightarrow W$$ isn't possible for the second law of thermodynamics, therefore, a free adiabatic expansion is not reversible.

I understand the second law of thermodynamics, I don't understand the passage from point a) to point b).

• It's an irreversible process because in a free expansion the gas thermodynamic parameter pressure isn't defined. Commented Feb 9, 2021 at 11:32

Your professor is saying that step b returns the gas to its original state that existed before the expansion and reversible recompression. So, overall, you have carried out a cycle (on the system), the net result of which is to take heat from a single source and produce an equal amount of work. (Incidentally, in the reversible expansion, you are supposed to be using a constant temperature reservoir, such that the internal energy doesn't change during the expansion, and the change in U is zero throughout the expansion.).

According to the second law, with a cyclic process, receiving heat from a single source at a constant temperature and converting it to work is not allowed.

• "According to the second law, with a cyclic process, receiving heat from a single source at a constant temperature and converting it to work is not allowed". But doesn't the irreversible expansion followed by a reversible isothermal compress constitute a cycle? And isn't the proof that the expansion is irreversible is that entropy of the surroundings increases by $Q/T$ in the cycle? Commented Feb 9, 2021 at 13:48