The root mean square speed of molecules of a given mass
Question:

The root mean square speed of molecules of a given mass of a gas at $27^{\circ} \mathrm{C}$ and 1 atmosphere pressure is $200 \mathrm{~ms}^{-1}$. The root mean square speed of molecules of the gas at $127^{\circ} \mathrm{C}$ and 2 atmosphere pressure is $\frac{\mathrm{x}}{\sqrt{3}} \mathrm{~ms}^{-1}$. The value of $\mathrm{x}$ will be

Solution:

$(400)$

$V_{\text {rms }} \sqrt{\frac{3 \mathrm{RT}_{1}}{\mathrm{M}_{0}}}$

$200=\sqrt{\frac{3 R \times 300}{M_{0}}}$

Also, $\frac{x}{\sqrt{3}}=\sqrt{\frac{3 R \times 400}{M_{0}}}$

$(1) \div(2)$

$\frac{200}{x / \sqrt{3}}=\sqrt{\frac{300}{400}}=\sqrt{\frac{3}{4}}$

$\Rightarrow x=400 \mathrm{~m} / \mathrm{s}$