A monoatomic gas at a pressure $$p,$$ having a volume $$V$$ expands isothermally to a volume $$2 V$$ and then adiabatically to a volume $$16 V.$$  The final pressure of the gas is (take $$\gamma = \frac{5}{3}$$ )

Solution :
For isothermal expansion process
$$\eqalign{ & pV = p' \times 2V\,\,\left[ {\therefore V' = 2V} \right] \cr & p' = \frac{p}{2} \cr} $$
For adiabatic expansion, $$p{V^\gamma } = {\text{constant}}$$
$$\eqalign{ & \Rightarrow p'{V^\gamma } = p''{{V''}^\gamma } \cr & \Rightarrow \frac{p}{2}{\left( {2V} \right)^{\frac{5}{3}}} = p''{\left( {16V} \right)^{\frac{5}{3}}} \cr & \Rightarrow p'' = \frac{p}{2}{\left[ {\frac{{2V}}{{16V}}} \right]^{\frac{5}{3}}} = \frac{p}{2}{\left( {\frac{1}{8}} \right)^{\frac{5}{3}}} \cr & = \frac{p}{2}\left( {\frac{1}{{32}}} \right) \cr & = \frac{p}{{64}} \cr} $$