$PV = P'\left( {2V} \right);\,\,P' = \frac{P}{2}$

Then, adiabatic expansion

$P'{\left( {2V} \right)^\gamma } = {P_f}{\left( {16V} \right)^\gamma }$

                              $\left( {For\,adiabatic\,process,\,P{V^\gamma } = constant} \right)$

$\frac{P}{2}{\left( {2V} \right)^{5/3}} = {P_f}{\left( {16V} \right)^{5/3}}$

${P_f} = \frac{P}{2}{\left( {\frac{{2V}}{{16V}}} \right)^{5/3}} = \frac{P}{2}{\left( {\frac{1}{8}} \right)^{5/3}} = \frac{P}{2}{\left( {\frac{1}{{{2^3}}}} \right)^{5/3}}$

$ = \frac{P}{2}\left( {\frac{1}{{{2^5}}}} \right) = \frac{P}{{64}}$