The root mean square (rms) speed of a gas is given by the formula :

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

where :

• $$k$$ is the Boltzmann constant
• $$T$$ is the temperature in Kelvin
• $$m$$ is the mass of one molecule of the gas

Given that the atomic mass of helium is 4 amu and that of argon is 40 amu, we can find the rms speed ratio by comparing the rms speeds of helium and argon, as temperature $$T$$ and Boltzmann constant $$k$$ are the same for both gases :

$$\frac{v_{rms} \left( \text{helium} \right)}{v_{rms} \left( \text{argon} \right)} = \sqrt{\frac{m_{\text{argon}}}{m_{\text{helium}}}}$$

Substituting the masses :

$$m_{\text{helium}} = 4 \text{ amu}$$

$$m_{\text{argon}} = 40 \text{ amu}$$

we get :

$$\frac{v_{rms} \left( \text{helium} \right)}{v_{rms} \left( \text{argon} \right)} = \sqrt{\frac{40}{4}} = \sqrt{10} \approx 3.16$$

Thus, the ratio of the rms speeds of helium to argon is approximately 3.16.

The correct answer is :

Option D : 3.16