At what temperature does the rms speed of

(a) A nitrogen molecule

(b) A hydrogen molecule equal the escape speed from the earth’s surface?

(c) You’ll find that these temperatures are very high, so you might think that the earth’s gravity could easily contain both gases. But not all molecules move with v_rms.
There is a distribution of speeds, and a small percentage of molecules have speeds several times v_rms. Bit by bit, a gas can slowly leak out of the atmosphere as its fastest molecules escape.
A reasonable rule of thumb is that the earth’s gravity can contain a gas only if the average translational kinetic energy per molecule is less than 1% of the kinetic energy needed to escape. Use this rule to show why the earth’s atmosphere contains nitrogen but not hydrogen, even though hydrogen is the most abundant element in the universe.