Look up the average temperature of the Earth’s surface and the escape velocity a

Question

Look up the average temperature of the Earth’s surface and the escape velocity at Earth’s surface and report them at the start of your problem. Use this information to find the rms velocity of molecular hydrogen (H2) and nitrogen (N2) in the Earth’s atmosphere. Comment on how this information (and other things you’ve learned about the velocity of molecules in a gas) would impact the relative amounts of hydrogen and nitrogen in the Earth’s atmosphere. [Note: the Universe as a whole contains 74% hydrogen and 0.1% nitrogen by mass.]

Explanation / Answer

here,

From stefan Boltz man law Average temperatureof Earth, T = 279 K

Escape velocity is calcualted by using,
V =sqrt(2G*M/r)

for Earth
ve = 11200 m/s or 11.2 km/s

From StefanBoltz man Law, rms speed of molecules,
Vr = sqrt(3*K*T/m)
Where,
K = boltzman constant
m = mass of object
T is temperature

Part A:
mass of nitrogen, N2 = 2 * 2.325*10^-26 kg

vr = sqrt(3*K*T/m)
vr = sqrt(3 * 1.381 * 10^-23 * 279 / (2 * 2.325*10^-26 ))
Vr = 498.578 m/s

Part b:
mass of hydrogen, m = 2 * 1 = 2 * 1.674 * 10^-27 kg

vr = sqrt(3*K*T/m)
vr = sqrt(3 * 1.381 * 10^-23 * 279 / (2 * 1.674 * 10^-27 ))
Vr = 1858.09 m/s

They both not able to escape from atmosphere

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