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"A cylinder have a piston whose mass is insignificant and moves without friction. One mole of an ideal gas is confined into the cylinder. If we expand isotermally the gass at temperature 298K, against a constant external pressure of 0.1 atm, the gas pressure goes from 1atm to 0.1 atm. Calculate the work need for the expansion."

I have 2 answers, and I don't know why the second one is wrong:

1) Pressure is constant. W = P(ext)(V2-V1) = P(ext) nRT(1/P2-1/P1) = 0,9.1.8,31.298 = 2.23kJ

2) Gas Pressure is P, so the difference of pressure is P1=(P-0,1).

W = ∫P1.dv.

V = nRT/P

dv = -nRT/P²dP

W = ∫-(P-0.1)nRT/P² dP = 2476(ln(P1/P2) - 0.1(P2-P1)) = 2476(2.3+0.09) = 5,91kJ

Why the second is wrong, and more important, why the first one is right? What is the definition of work?

[]'s

João