MHT CET · Physics · Thermodynamics
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
- A \(2 \sqrt{2} \mathrm{~T}\)
- B \(4 \mathrm{~T}\)
- C \(\sqrt{2} \mathrm{~T}\)
- D \(2 \mathrm{~T}\)
Answer & Solution
Correct Answer
(C) \(\sqrt{2} \mathrm{~T}\)
Step-by-step Solution
Detailed explanation
Given, \(\mathrm{PV}^{3 / 2}=\) constant or \(\mathrm{TV}^{1 / 2}=\) constant
\( \begin{aligned} & \therefore \mathrm{T}_1 \mathrm{~V}_1^{1 / 2}=\mathrm{T}_2 \mathrm{~V}_2^{1 / 2} \\ & \frac{\mathrm{T}_2}{\mathrm{~T}_1}=\left(\frac{\mathrm{V}_1}{\mathrm{~V}_2}\right)^{1 / 2}=\sqrt{2} \\ & \mathrm{~T}_2=\sqrt{2} \mathrm{~T}_1=\sqrt{2} \mathrm{~T} \end{aligned} \)
\( \begin{aligned} & \therefore \mathrm{T}_1 \mathrm{~V}_1^{1 / 2}=\mathrm{T}_2 \mathrm{~V}_2^{1 / 2} \\ & \frac{\mathrm{T}_2}{\mathrm{~T}_1}=\left(\frac{\mathrm{V}_1}{\mathrm{~V}_2}\right)^{1 / 2}=\sqrt{2} \\ & \mathrm{~T}_2=\sqrt{2} \mathrm{~T}_1=\sqrt{2} \mathrm{~T} \end{aligned} \)
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