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A monoatomic gas at a pressure P, having a volume V expands isothermally to a volume 2V and then adiabatically to a volume 16V. The final pressure of the gas is: (take $\gamma = \frac{5}{3}$)
A
64P
B
32P
C
$\frac{P}{64}$
D
16P
Detailed Solution
Isothermal: $P_1V_1 = P_2V_2 \Rightarrow PV = P_2(2V) \Rightarrow P_2 = \frac{P}{2}$
Adiabatic: $P_2V_2^\gamma = P_3V_3^\gamma \Rightarrow \frac{P}{2}(2V)^\gamma = P_3(16V)^\gamma$
$P_3 = \frac{P}{2}\left(\frac{1}{8}\right)^{5/3} = \frac{P}{64}$
Adiabatic: $P_2V_2^\gamma = P_3V_3^\gamma \Rightarrow \frac{P}{2}(2V)^\gamma = P_3(16V)^\gamma$
$P_3 = \frac{P}{2}\left(\frac{1}{8}\right)^{5/3} = \frac{P}{64}$