The temperature of an ideal gas is increased from $27^{\circ} \mathrm{C}$ to $927^{\circ} \mathrm{C}$. The r…

The temperature of an ideal gas is increased from $27^{\circ} \mathrm{C}$ to $927^{\circ} \mathrm{C}$. The r.m.s. speed of its molecules becomes
  1. twice.
  2. half.
  3. four times.
  4. one-fourth.

Solution

$\begin{aligned} & \text { We know } \mathrm{v}_{\mathrm{rms}}=\sqrt{\frac{3 \mathrm{RT}}{\mathrm{M}}} \\ \Rightarrow \mathrm{v}_{\mathrm{rms}} & \propto \sqrt{\mathrm{T}} \\ \therefore \quad & \frac{\mathrm{v}_{\mathrm{rms}}}{\mathrm{v}_{\mathrm{rms}}}=\sqrt{\frac{\mathrm{T}_1}{\mathrm{~T}_2}}=\sqrt{\frac{300}{1200}}=\sqrt{\frac{1}{4}} \\ & \frac{\mathrm{v}_{\mathrm{rms}_1}}{\mathrm{v}_{\mathrm{rms}}}-\frac{1}{2} \\ \therefore \quad \mathrm{v}_{\mathrm{rms}_2} & =2 \cdot \mathrm{v}_{\mathrm{rms1}}\end{aligned}$

Asked in: MHT CET 2023 (09 May Shift 2)

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