Q.How is rms (root mean square) speed related to molar mass of the gas? Why is there no hydrogen in the Earth's atmosphere?

Since v_rms is proportional to 1/square root of M, lighter gases move faster; hydrogen's rms speed is high enough to be comparable to Earth's escape velocity, so hydrogen molecules have escaped Earth's atmosphere over geological time.

The root mean square (rms) speed of the molecules of an ideal gas at absolute temperature T is

v_rms = square root of (3RT/M)

where R is the universal gas constant and M is the molar mass of the gas.

This shows that v_rms is inversely proportional to the square root of the molar mass:

v_rms proportional to 1/square root of M

So, at the same temperature, gases with smaller molar mass (lighter molecules) have a higher rms speed than gases with larger molar mass (heavier molecules). For example, hydrogen (M about 2 g/mol) has a much higher rms speed than nitrogen (M about 28 g/mol) or oxygen (M about 32 g/mol) at the same temperature.

Why there is no hydrogen in Earth's atmosphere:

Earth's escape velocity is about 11.2 km/s. For a gas molecule to permanently escape Earth's gravitational pull (especially from the upper atmosphere, where collisions are rare), a significant fraction of its molecules must be moving at speeds close to or exceeding this escape velocity.

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