Question

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

• A. 64P
• B. 32P
• C. P/64
• D. 16P

Answer 1

Answer: (C)

First, isothermal expansion
$P V=P^{\prime}(2 V)$
(For isothermal process, $P V=$ constant)
$P^{\prime}=\frac{P}{2}$
Then, adiabatic expansion
$P^{\prime}(2 V)^{\gamma}=P_{f}(16 V)^{\gamma}$
(For adiabatic process, $P V^{\gamma}=$ constant)
$\frac{P}{2}(2 V)^{5 / 3}=P_{f}(16 V)^{5 / 3} $
$P_{f}=\frac{P}{2}\left(\frac{2 V}{16 V}\right)^{5 / 3}=\frac{P}{2}\left(\frac{1}{8}\right)^{5 / 3}=\frac{P}{2}\left(\frac{1}{2^{3}}\right)^{5 / 3}$
$=\frac{P}{2}\left(\frac{1}{2^{5}}\right)=\frac{P}{64}$