Thermodynamics: work done on the atmosphere by an expanding gas

[LCSphysicist]

Homework Statement

A metal container, of volume V and with diathermal walls, contains n
moles of an ideal gas at high pressure. The gas is allowed to leak out
slowly from the container through a small valve to the atmosphere
at a pressure P0. The process occurs isothermally at the temperature
of the surroundings. Show that the work done by the gas against the
surrounding atmosphere is
W = P0(nv0 −V)
where v0 is the molar volume of the gas at atmospheric pressure and
temperature.

Relevant Equations

\n

V = V(n,P)
PV = nc
dV = (-cn/P²)dP + (c/P)dn
dW = (cn/P)dP - (c)dn​

This was how i tried to attack the problem, but it will end with log and probably it is wrong, any tips to fix it?

 

[Chestermiller]

The escaping gas is pushing back the surrounding atmosphere. What is the pressure of the gas remaining in the vessel after all the gas that was going to escape has escaped? What is the final total volume of all the gas that was originally in the vessel? What volume of gas still remains in the vessel in the end. What is the volume of the escaped gas in the end?

 

[LCSphysicist]

The escaping gas is pushing back the surrounding atmosphere. What is the pressure of the gas remaining in the vessel after all the gas that was going to escape has escaped? What is the final total volume of all the gas that was originally in the vessel? What volume of gas still remains in the vessel in the end. What is the volume of the escaped gas in the end?

I think i misinterpreted the question, sorry. It is easier than i thought, it is just an isobaric expansion at p0 XD