Amount of Work done by the system.

Consider a piston cylinder arrangement with gas filled in cylinder. If the piston moves from position 1 to position 2 as shown in figure with volume changes from v1 to v2, the amount of work W by the system will be,

a. _v1 ∫ ^v2 p dV
b. _v1 ∫ ^v2 p V
c. _v1 ∫ ^v2 v dp
d. none of the above

Correct Answer : a. _v1 ∫ ^v2 p dV

Explanation :

Initially at position 1, let the pressure and volume be p₁ and V₁ respectively. The piston is the only boundary which moves with gas pressure. Let the piston move out to a new final position 2. The pressure and volume will be p₂ and V₂ respectively. At any intermediate point let the pressure be p and volume be V. These macroscopic properties are significant for equilibrium state. When the piston moves as infinitesimal distance dl, and area of the piston is 'a', the force F acting on the piston F = p*a and the infinitesimal amount of work done by the gas will be,
dW = F * dl = padl = pdV
where dV= adl
Therefore, total work done W by the piston when it moves out from position 1 to position 2 will be,
W = _v1 ∫ ^v2 p dV