Question

You have been given a box filled with two gases - oxygen, and an unknown gas. It is seen that at any temperature of the mixture, the rms speed of the unknown gas is always four times that of oxygen. Can you identify this gas?

• A. Hydrogen
• B. Helium
• C. Carbon dioxide
• D. Argon

Answer 1

The correct option is A

Hydrogen

From a kinetic form of the ideal gas equation, we can write -

$PV=\frac{2}{3}N\left(\frac{1}{2}m{v}_{rms}^2\right)=Nk\left(\frac{2\times kineticenergy}{3k}\right)=NkT,$

from which we obtain -

$v_{rms}=\sqrt{\frac{3kT}{m}}$

$=\sqrt{\frac{3\left(N_{A}k\right)T}{\left(N_{A}m\right)}}$

$=\sqrt{\frac{3RT}{M}}$.

$\therefore$ For two different gases with molar masses $M_1$ and $M_2$

(rearranging equation (1) -)

$\begin{array}{c}\frac{v_{rms,1}}{v_{rms,2}}=\sqrt{\frac{M_2}{M_1}}\end{array}$

$\Rightarrow M_2=M_1\times \frac{v_{rms,1}^2}{v_{rms,2}^2}$.

For oxygen, M=32gms; thus the molar mass of the unknown gas,

(given $v_{rms,1}:v_{rms,2}=1:4$), will be -

$M_2=32\times \frac{1^2}{4^2}g$

=2 g.

An Avogadro number of which gas's molecules has a mass of 2g? Hydrogen $\left(H_2\right)$!