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Q.

An ideal monatomic gas with pressure P, volume V and temperature T is expanded isothermally to a volume 2V and a final pressure $P_{i}$ . If the same gas is expanded adiabatically to a volume 2 V, the final pressure is $P_{a}$ . The ratio $\frac{P_{a}}{P_{i}}$ is

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a

$2^{- 1 / 3}$

b

$2^{1 / 3}$

c

$2^{2 / 3}$

d

$2^{- 2 / 3}$

answer is D.

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Detailed Solution

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$\begin{array}{l} \begin{array}{l} For isothermal process: \\ {P V=P}_{i} 2 V \\ \Rightarrow {P=2 P}_{i} \\ For adiabatic process \\ P V^{\frac{5}{3}} = P_{a} {(2 V)}^{\frac{5}{3}} \\ \Rightarrow ({2 P}_{i}) V^{\frac{5}{3}} = P_{a} {(2 V)}^{\frac{5}{3}} \\ S o, \frac{P_{a}}{P_{i}} = \frac{2}{2^{\frac{5}{3}}} \\ \Rightarrow \frac{P_{a}}{P_{i}} = 2^{\frac{- 2}{3}} \end{array} \end{array}$

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