Thermodynamics

JEE Main level, single correct, medium

171 of 190

Question 171 of 190

An ideal monatomic gas with pressure PP, volume VV and temperature TT is expanded isothermally to a volume 2V2 V and a final pressure PiP_i. If the same gas is expanded adiabatically to a volume 2 V2 \mathrm{~V}, the final pressure is PaP_a. The ratio PaPi\frac{P_a}{P_i} is

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Answer: (D) 2−2/32^{-2 / 3}

For isothermal process :
PV=Pi2VP=2Pi\begin{aligned} & P V=P_i 2 V \\ & P=2 P_i \end{aligned}
For adiabatic process
PVγ=Pa(2 V)γ\mathrm{PV}^\gamma=\mathrm{P}_{\mathrm{a}}(2 \mathrm{~V})^\gamma
(∵(\because for monatomic gas γ=5/3)\gamma=5 / 3)
[From (i) ]
or, 2PiV53=Pa(2 V)53\quad 2 \mathrm{P}_{\mathrm{i}} \mathrm{V}^{\frac{5}{3}}=P_a(2 \mathrm{~V})^{\frac{5}{3}}
⇒PaPi=2253⇒PaPi=2−23\begin{aligned} & \Rightarrow \frac{P_a}{P_i}=\frac{2}{2^{\frac{5}{3}}} \\ & \Rightarrow \frac{P_a}{P_i}=2^{\frac{-2}{3}} \end{aligned}

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