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By how many folds the temperature of a gas would increase when the root mean square velocity of the gas molecules in a container of fixed volume is increased from $5 \times 10^4 \mathrm{~cm} / \mathrm{s}$ to $10 \times 10^4 \mathrm{~cm} / \mathrm{s}$ ?
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Four
Four
r.m.s. velocity $\mathrm{V}_{\mathrm{rms}}=\sqrt{\frac{3 R T}{M}}$
i.e., $\frac{\mathrm{V}_1}{\mathrm{~V}_2}=\sqrt{\frac{\mathrm{T}_1}{\mathrm{~T}_2}}$
$\frac{5 \times 10^4}{10 \times 10^4}=\frac{1}{2}=\sqrt{\frac{\mathrm{T}_1}{\mathrm{~T}_2}}$
$\therefore \mathrm{T}_2=4 \mathrm{T}_1$
i.e., $\frac{\mathrm{V}_1}{\mathrm{~V}_2}=\sqrt{\frac{\mathrm{T}_1}{\mathrm{~T}_2}}$
$\frac{5 \times 10^4}{10 \times 10^4}=\frac{1}{2}=\sqrt{\frac{\mathrm{T}_1}{\mathrm{~T}_2}}$
$\therefore \mathrm{T}_2=4 \mathrm{T}_1$
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