A piston can freely move inside a horizontal cylinder closed from both ends. Initially, the piston separates the inside space of the cylinder into two equal parts each of volmek `V_0` in which an ideal gas is contained under the same pressure `p_0` and at the same temperature. What work has to be performed in order to increase isothermally the volume of one part of gas `eta` times compared to that of the other by slowly moving the piston ?

Explanation

Let the agent move the piston to the right by `x, 1 n` equilibrium position,
`p_1 S + F_("agent") = p_2 S`, or , `F_("agent") = (p_2 - p_1) S`
Work done by the agent in an infinitesmal change `dx` is
`F_("agent"). dx = (p_1 - p_2) S dx = (p_2 - p_1) d V`
By applying `pV = constant`, for the two parts,
`p_1(V_0 + Sx)= p_0 V_0` and `p_2 (V_0- Sx) = p_0 V_0`
So, `p_1 - p_2 = (p_0 V_0 2 S x)/(V_0^2 - S^2 x^2) =(2 p_0 V_0 V)/(V_0^2 - V^2)(where Sx = V)`
When the volume of the left end is `eta` times the volume of the right end
`(V_0 +V) = eta (V_0 - V)`, or `V = (eta - 1)/(eta + 1) V_0`
`A = int_0^V (p_2 - p_1) dV = int_0^V (2p_0 V_0 V)/(V_0^2 - V^2)dV = -p_0 V_0[1 n (V_0^2 - V^2)]_0^V`
=`-p_0 V_0[ 1 n(V_0^2 - V62)- 1n V_0^2]`
=`-p_0 V_0[1 n{V_0^2((eta - 1)/(eta + 1^2))V_0^2} - 1 n V_0^2]`
=`-p_0 V_0(1 n(4 eta)/((eta + 1)^2)) = p_0 V_0 1 n((eta + 1)^2)/(4 eta)`.