Q.At what temperature will the RMS speed of nitrogen ($N_2$, molar mass $28\ \text{g/mol}$) molecules equal $500\ \text{m/s}$?

The root-mean-square speed, $v_{rms} = \sqrt{\overline{v^2}} = \sqrt{3RT/M} = \sqrt{3k_BT/m}$, is the particular molecular-speed average that connects most directly to a gas's pressure and energy (since kinetic energy depends on $v^2$, not $v$). It increases with the square root of absolute temperature and decreases with the square root of molar mass, so light gases (hydrogen, helium) have far higher RMS speeds than heavy gases (carbon dioxide, oxygen) at the same temperature. The full Maxwell-Boltzmann distribution shows molecules actually possess a spread of speeds, with the most probable speed $v_p$, mean speed $\bar{v}$, and RMS speed $v_{rms}$ always ordered $v_p < \bar{v} < v_{rms}$, the whole distribution shifting to higher speeds and broadening as temperature rises.