Past exam of the physics course of the year 4 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / Section A / A2

For this molecular escape-speed temperature estimate, $v_{\mathrm{rms}}^2=3k_{B}T/m$. Equating this to the surface escape-speed square $v_{\mathrm{esc}}^2=2G{M}_{\oplus }/R_{\oplus }$ givesThis evaluation keeps the paper's $R_{\oplus }=6.4\times 10^6\mathrm{m}$ and $m=28u$, and uses externally documented $G{M}_{\oplus }=3.9860\times 10^{14}{\mathrm{m}}^3{\mathrm{s}}^{-2}$, $u=1.660539040\times 10^{-27}\mathrm{kg}$ and $k_B=1.38064852\times 10^{-23}\mathrm{J}{\mathrm{K}}^{-1}$.^[ref][ref] The corresponding escape speed is about $11.2\mathrm{km}{\mathrm{s}}^{-1}$.

This is the requested speed-scale comparison. Atmospheric escape can occur from the high-speed tail at lower temperature; molecular dissociation and ionization also prevent treating an actual gas at this temperature as unchanged neutral nitrogen molecules.