Volume work, absolute and relative pressure confusion

[kandelabr]

hello.

i can't figure out where I'm wrong. this is the problem:

we have a cylinder, closed with a piston. the absolute pressure inside the cylinder is p₀, atmospheric pressure is p_atm. the air inside expands isothermally to some (specific) volume v_end.
i derived the equation for isothermal work directly from this one: dW = F(s)ds.

here it is (sorry for handwriting... uses less time (will learn TeX one day... )):

the final equation seems just fine, but you've probably noticed the big fat asterisk. here's why:
p₀ is absolute pressure and it exerts F(s₀) on the piston. but there is also p_atm, which does negative work equal to p_atm * A * (s_end - s₀). this fact hasn't been taken into account anywhere.

where am i wrong (the thermodynamics probably isn't wrong, right? :) )

 

[Doc Al]

What you are calculating is the work done by the gas on the piston, not the net work done on the piston. The gas is not subjected to atmospheric pressure--it's contained in the cylinder.

 

[kandelabr]

oh, i get it.
but if now i want to know how much work i can get from this piston, i need to include atmospheric pressure.