Question

The molecules of a given mass of a gas have root mean square speeds of $100m/s$ at ${27}^{\circ }C$ and $1.00$ atmospheric pressure. What will be the root mean square speeds of the molecules of the gas at ${127}^{\circ }C$ and $2.0$ atmospheric pressure?

Answer 1

Given,

$(v_1{)}_{rms}=100m/s$

$T_1=27+273=300K$

$T_2=127+273=400K$

$\because M=$ moles mass for a gas, $M$ is constant.

As we know, $v_{rms}=\sqrt{\frac{3RT}{M}}$

$\therefore v_{rms}\propto \sqrt{T}$

$\frac{(v_1{)}_{rms}}{(v_2{)}_{rms}}=\sqrt{\frac{T_1}{T_2}}$

$\frac{100}{(v_2{)}_{rms}}=\sqrt{\frac{300}{400}}$

$(v_2{)}_{rms}=\frac{100\times \sqrt{400}}{\sqrt{300}}$

$=\frac{100\times 2\times 10}{10\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}}=\frac{200\times \sqrt{3}}{3}$

$(v_2{)}_{rms}=\frac{200}{\sqrt{3}}m/s$

$\text{Final Answer}:\frac{200}{\sqrt{3}}m/s$.