Question

The molecules of a given mass of a gas have root mean square speed of $100m{s}^{-1}$ at $27{}^{\circ }C$ and 1 atmospheric pressure. What will be the root mean square speeds of the molecules of the gas at $127{}^{\circ }C$ and 2 atmospheric pressure?

• A. $\frac{150}{\sqrt{3}}m/s$
• B. $\frac{125}{\sqrt{3}}m/s$
• C. $\frac{200}{\sqrt{3}}m/s$
• D. $100\sqrt{3}m/s$

Answer 1

The correct option is C

$\frac{200}{\sqrt{3}}m/s$

We know that for a given mass of gas

$v_{rms}=\sqrt{\frac{3RT}{M}}$

where $R$ is a gas constant , $T$ is temperature in kelvin and $M$ is molar mass of the gas. Clearly for a given gas, $v_{rms}\propto \sqrt{T}$ as $R$, $M$ are constants.

$\frac{{\left(v_{rms}\right)}_1}{{\left(v_{rms}\right)}_2}=\sqrt{\frac{T_1}{T_2}}$
Given $(v_{rms}{)}_1=100m/s$

$T_1=27+273=300K$

$T_2=127+273=400K$

$\frac{100}{(v_{rms}{)}_2}=\sqrt{\frac{300}{400}}=\frac{\sqrt{3}}{2}$

$(v_{rms}{)}_2=\frac{2\times 100}{\sqrt{3}}=\frac{200}{\sqrt{3}}m/s$