A monoatomic gas of presuure 'P' having volume 'V' expands isothermally to a volume '2V' and then…

A monoatomic gas of presuure 'P' having volume 'V' expands isothermally to a volume '2V' and then adiabatically to a volume '16V'. The final pressure of the gas is (ratio of specific heats $=\frac{5}{3}$)
  1. $\frac{P}{16}$
  2. $\mathrm{P}$
  3. $\frac{\mathrm{P}}{32}$
  4. $\frac{\mathrm{P}}{64}$

Solution

$\gamma=\frac{5}{3}$ Case I: $\quad P_{1} V_{1}=P_{2} V_{2}$ $\mathrm{PV}=\mathrm{P}_{2} \times 2 \mathrm{~V}$ $\therefore \quad P_{2}=\frac{P}{2}$ Case II : $\quad P_{2} V_{2}^{\gamma}=P_{3} V_{3}^{\gamma}$ $\begin{array}{l} \left(\frac{P}{2}\right)(2 V)^{\gamma}=P_{3}(16 V)^{\gamma} \\ P_{3}=\frac{P}{2} \frac{(2 V)^{\gamma}}{(16 V)^{\gamma}}=\frac{P}{2}\left(\frac{1}{8}\right)^{\gamma} \\ =\frac{P}{2}\left(\frac{1}{2^{3}}\right)^{5 / 3}=\frac{P}{2}\left(\frac{1}{2}\right)^{5} \\ =\frac{P}{2 \times 32}=\frac{P}{64} \end{array}$

Asked in: MHT CET 2020 (16 Oct Shift 1)

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