An ideal gas with pressure $\mathrm{P}$, volume $\mathrm{V}$ and temperature $\mathrm{T}$ is expanded…

An ideal gas with pressure $\mathrm{P}$, volume $\mathrm{V}$ and temperature $\mathrm{T}$ is expanded isothermally to a volume $2 \mathrm{~V}$ and a final pressure $\mathrm{P}_{\mathrm{i}}$. The same gas is expanded adiabatically to a volume $2 \mathrm{~V}$, the final pressure is $\mathrm{P}_{\mathrm{a}}$. In terms of the ratio of the two specific heats for the gas ' $\gamma$ ', the ratio $\frac{\mathrm{P}_i}{\mathrm{P}_{\mathrm{a}}}$ is
  1. $2^{\gamma+1}$
  2. $2^{\gamma-1}$
  3. $2^{1-\gamma}$
  4. $2^\gamma$

Solution

For isothermal expansion we have $\begin{aligned} & \mathrm{P}_1 \mathrm{~V}_1=\mathrm{P}_2 \mathrm{~V}_2 \\ & \therefore \mathrm{P}_2=\mathrm{P}_1 \frac{\mathrm{V}_1}{\mathrm{~V}_2}=\mathrm{P}_1 \times \frac{1}{2}=\frac{\mathrm{P}}{2} \\ & \therefore \mathrm{P}_{\mathrm{i}}=\frac{\mathrm{P}}{2} \end{aligned}$ For adiabatic process: $\begin{aligned} & \mathrm{P}_1 \mathrm{~V}_1^\gamma=\mathrm{P}_2 \mathrm{~V}_2^\gamma \\ & \therefore \mathrm{P}_2=\mathrm{P}_1\left(\frac{\mathrm{V}_1}{\mathrm{~V}_2}\right)^\gamma=\mathrm{P}_1\left(\frac{1}{2}\right)^\gamma=\frac{\mathrm{P}}{2^\gamma} \\ & \therefore \mathrm{P}_{\mathrm{a}}=\frac{\mathrm{P}}{2^\gamma} \\ & \therefore \frac{\mathrm{P}_{\mathrm{i}}}{\mathrm{P}_{\mathrm{a}}}=\frac{2^\gamma}{2}=2^{\gamma-1} \end{aligned}$ :

Asked in: MHT CET 2021 (23 Sep Shift 1)

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