# Chemistry - Work done in expanding a gas reversibly and irreversibly

I will call your processes (1), (2), and (3). I assume (1) is reversible isothermal expansion, (2) is irreversible isothermal expansion, and (3) is unspecified.

The key idea is that the inequality $$w_\text{rev} \geq w_\text{irrev}$$ can only be applied to reversible and irreversible analogues of the same process. (1) and (2) satisfy this requirement because (1) is the reversible analogue of (2), but (1) is not the reversible analogue of (3).

Physically, (1) is like having a pile of sand on top of a container (containing an ideal gas) whose lid is a piston, and then slowly blowing away the sand. (2) is the same scenario, except you sweep off the sand in one grand gesture. But (3) is a different scenario entirely.

If we refer to the first law, $$\Delta U = q + w$$, it's clear that, given an initial and a final state, we can make $$w$$ as large as we want as long as we have a commensurate value of $$q$$. Just because processes (1), (2) and (3) all have the same initial and final states does not mean that it makes sense to apply the inequality $$w_\text{rev} \geq w_\text{irrev}$$ to any two of them.