Given the same room temperature of 300 K for helium and argon, we are tasked with finding the ratio of their average kinetic energies per molecule.

Formula for Kinetic Energy per Molecule:

$ \mathrm{K.E.} = \frac{\mathrm{f}}{2} \mathrm{kT} $

In this equation:

$\mathrm{f}$ is the degrees of freedom, which is 3 for both helium (He) and argon (Ar) since they are monatomic gases.

$k$ is the Boltzmann constant.

$T$ is the temperature in Kelvin.

Kinetic Energy Comparison:

For both helium and argon, since $\mathrm{f} = 3$, the expression simplifies:

$ \frac{\mathrm{K} \cdot \mathrm{E}_{\mathrm{He}}}{\mathrm{K} \cdot \mathrm{E}_{\mathrm{Ar}}} = \frac{1}{1} $

Thus, the average kinetic energy per molecule for both gases is the same at the same temperature, leading to a ratio of:

$ 1:1 $

This result implies that despite their differences in molar masses (helium being 4 g/mol and argon 40 g/mol), the average kinetic energy per molecule remains equal at the same temperature.