A sample contains mixture of helium and oxygen gas. The ratio of root mean square speed of helium and oxygen in the sample, is:

We need to find the ratio of r.m.s. speeds of helium to oxygen at the same temperature.

Recall the r.m.s. speed formula.

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

where $$R$$ is the gas constant, $$T$$ is absolute temperature, and $$M$$ is the molar mass.

Calculate the ratio.

At the same temperature, $$R$$ and $$T$$ are the same for both gases:

$$ \frac{v_{He}}{v_{O_2}} = \sqrt{\frac{M_{O_2}}{M_{He}}} $$

Molar masses: $$M_{He} = 4$$ g/mol, $$M_{O_2} = 32$$ g/mol

$$ \frac{v_{He}}{v_{O_2}} = \sqrt{\frac{32}{4}} = \sqrt{8} = 2\sqrt{2} $$

The correct answer is Option (2): $$2\sqrt{2}$$.