MHT CET · Physics · Kinetic Theory of Gases
To what temperature should the hydrogen at \(327^{\circ} \mathrm{C}\) be cooled at constant pressure, so that the root mean square velocity of its molecules becomes half of its previous value?
- A \(-123^{\circ} \mathrm{C}\)
- B \(123^{\circ} \mathrm{C}\)
- C \(-100^{\circ} \mathrm{C}\)
- D \(0^{\circ} \mathrm{C}\)
Answer & Solution
Correct Answer
(A) \(-123^{\circ} \mathrm{C}\)
Step-by-step Solution
Detailed explanation
\(v_{\mathrm{ms}} \propto \sqrt{\frac{3 R T}{M}}\)
\(
\begin{array}{ll}
\Rightarrow & T \propto v_{\mathrm{rms}}^{2} \\
\Rightarrow & \frac{T_{2}}{T_{1}}=\left[\frac{v_{2}}{v_{1}}\right]^{2}=\frac{1}{4} \\
\Rightarrow & T_{2}=\frac{T_{1}}{4}=\frac{(273+327)}{4} \\
& =150 \mathrm{~K}=-123^{\circ} \mathrm{C}
\end{array}
\)
\(
\begin{array}{ll}
\Rightarrow & T \propto v_{\mathrm{rms}}^{2} \\
\Rightarrow & \frac{T_{2}}{T_{1}}=\left[\frac{v_{2}}{v_{1}}\right]^{2}=\frac{1}{4} \\
\Rightarrow & T_{2}=\frac{T_{1}}{4}=\frac{(273+327)}{4} \\
& =150 \mathrm{~K}=-123^{\circ} \mathrm{C}
\end{array}
\)
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