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A monatornic 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 $(\,Take \,\gamma = 5/3)$
$64 P$
$32 P$
$\frac{P}{{64}}$
$16P$
Solution
First, isothermal expansion
$PV = P'\left( {2V} \right);\,\,P' = \frac{P}{2}$
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}}$
