The root mean square
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 $\frac{x}{\sqrt{3}} \mathrm{~ms}^{-1}$. The value of $x$ will be

Solution:

$\mathrm{v}_{\mathrm{rms}}=\sqrt{\frac{3 \mathrm{RT}}{\mathrm{M}}}$

$\mathrm{v}_{\mathrm{rms}} \propto \sqrt{\mathrm{T}}$

$\frac{\left(\mathrm{v}_{\mathrm{rms}}\right)_{2}}{\left(\mathrm{v}_{\mathrm{rms}}\right)_{1}}=\sqrt{\frac{\mathrm{T}_{2}}{\mathrm{~T}_{1}}}$

$=\sqrt{\frac{400}{300}}$

$=\frac{2}{\sqrt{3}}$

$\left(\mathrm{v}_{\mathrm{rms}}\right)_{2}=\frac{2}{\sqrt{3}}\left(\mathrm{v}_{\mathrm{rms}}\right)_{1}$

$=\frac{2}{\sqrt{3}} \times 200$

$\left(\mathrm{v}_{\mathrm{ms}}\right)_{2}=\frac{400}{\sqrt{3}} \mathrm{~m} / \mathrm{s}$

Ans. 400