Explanation

(c) As finally the piston is in equilibrium, both the gases must be at same pressure $P_{f}$. It is given that displacement of piston be in ideal state $x$ and if $A$ is the area of cross-section of the piston. Hence the final volumes of the left and right part finally can be given by figure as
$V_{L} = \frac{V_{0}}{2} + Ax$ and
$V_{R} = \frac{V_{0}}{2} - Ax$
As it is given that the container walls and the piston are adiabatic in left side and the gas undergoes adiabatic expansion and on the right side the gas undergoes adiabatic compression. Thus we have for initial and final state of gas on left side
$P_{1}\left( \frac{V_{0}}{2} \right)^{\gamma} = P_{f}\left( \frac{V_{0}}{2} + Ax \right)^{\gamma}$ ...(i)
Similarly for gas in right side, we have
$P_{2}\left( \frac{V_{0}}{2} \right)^{\gamma} = P_{f}\left( \frac{V_{0}}{2} - Ax \right)^{\gamma}$ …(ii)
From eq. (i) and (ii)
$\frac{P_{1}}{P_{2}} = \frac{\left( \frac{V_{0}}{2} + Ax \right)^{\gamma}}{\left( \frac{V_{0}}{2} - Ax \right)^{\gamma}} \Rightarrow Ax = \frac{V_{0}}{2}\left\lbrack \frac{P_{1}^{1/\gamma} - P_{2}^{1/\gamma}}{P_{1}^{1/\gamma} + P_{2}^{1/\gamma}} \right\rbrack$
Now from equation (i) $P_{f} = \frac{P_{1}\left( \frac{V_{0}}{2} \right)^{\gamma}}{\left\lbrack \frac{V_{0}}{2} + Ax \right\rbrack^{\gamma}}$