# Restrained expansion of a piston/cylinder

Suppose an ideal gas in a piston cylinder has some initial pressure $p_1$, volume $V_1$, and temperature $T_1$. A pin, which holds the piston in place, is suddenly removed, and the gas quickly expands. Once equilibrium is reached, the final pressure of the gas is $p_2$.

I am looking at a textbook solution which claims that the work done during this process is $$W = \int p\,dV = p_2\left(V_2-V_1\right),$$ which implies that the pressure is constant ($p=p_2$) throughout the whole process, as soon as the pin is released. Is this really the case, and are there not irreversibilities in this process that must be considered?

• My first thought is that $p$ is the pressure of the environment acting against the cylinder... As in you are doing work against the atmosphere (which has a constant $p$). – Julien Mar 31 '14 at 18:17
• $p = p(V,T)$ is the pressure of the gas inside the cylinder. If the piston has area $A$ and mass $M$, and if the external pressure is $p_0$, then $p_2 = p_0 + Mg/A$. – Doubt Mar 31 '14 at 18:27
• That wasn't made clear in your question, perhaps you should reformulate it. – Julien Mar 31 '14 at 18:28