The relation obeyed by a perfect gas during an adiabatic process is $\mathrm{PV}^{3 / 2}=$ constant. The…

The relation obeyed by a perfect gas during an adiabatic process is $\mathrm{PV}^{3 / 2}=$ constant. The initial temperature of the gas is ' $\mathrm{T}$ '. when the gas is compressed to half of its Initial volume, the final temperature of the gas is
  1. $2 \sqrt{2} \mathrm{~T}$
  2. $4 \mathrm{~T}$
  3. $\sqrt{2} \mathrm{~T}$
  4. $2 \mathrm{~T}$

Solution

Given, $\mathrm{PV}^{3 / 2}=$ constant or $\mathrm{TV}^{1 / 2}=$ constant $\begin{aligned} & \therefore \mathrm{T}_1 \mathrm{~V}_1^{1 / 2}=\mathrm{T}_2 \mathrm{~V}_2^{1 / 2} \\ & \frac{\mathrm{T}_2}{\mathrm{~T}_1}=\left(\frac{\mathrm{V}_1}{\mathrm{~V}_2}\right)^{1 / 2}=\sqrt{2} \\ & \mathrm{~T}_2=\sqrt{2} \mathrm{~T}_1=\sqrt{2} \mathrm{~T} \end{aligned}$

Asked in: MHT CET 2021 (20 Sep Shift 1)

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