Question

The root mean square speed of a gas molecule is $300\text{m/s}$. What will be the root mean square speed of the molecules if the atomic mass is doubled and absolute temperature is halved?

• A. $300\text{m/s}$
• B. $150\text{m/s}$
• C. $600\text{m/s}$
• D. $175\text{m/s}$

Answer 1

The correct option is B $150\text{m/s}$

Given:
Initial root mean square speed, $v_{rms}=300\text{m/s}$
To find:
Final root mean square speed when the atomic mass is doubled and absolute temperature is halved, $v_{rms}^{{}^{\prime }}=?$

We know that,
$v_{rms}=\sqrt{\frac{3RT}{M}}$, where $M$ is molar mass and $T$ is absolute temperature.
$\Rightarrow 300=\sqrt{\frac{3RT}{M}}$ .......(1)
Now, if molar mass is doubled and absolute temperature is halved, then
$v_{rms}^{\prime }=\sqrt{\frac{3RT/2}{2M}}=\sqrt{\frac{3RT}{4M}}=\frac{1}{2}\sqrt{\frac{3RT}{M}}$
$\Rightarrow v_{rms}^{\prime }=\frac{300}{2}$ [ from (1) ]
$\Rightarrow v_{rms}^{\prime }=150\text{m/s}$