At what temperature will N2 molecules have sufficient velocity to escape the Ear

Question

At what temperature will N2 molecules have sufficient velocity to escape the Earth's gravitational pull? The escape velocity from the Earth is 11.2 km/ s
Show your work please, Im messing up the equation at some point. At what temperature will N2 molecules have sufficient velocity to escape the Earth's gravitational pull? The escape velocity from the Earth is 11.2 km/ s At what temperature will N2 molecules have sufficient velocity to escape the Earth's gravitational pull? The escape velocity from the Earth is 11.2 km/ s
Show your work please, Im messing up the equation at some point.

Explanation / Answer

You have to use some basic statistical physics arguments. We assume here that the gas is perfect,
for an isotropic distribution of the velocities, the average kinetic energy is given by:

Ec = 3/2 N k T = N * 1/2 *m <V^2>

where k is the boltzmann constant, N the number of molecules, m the mass of a molecule and <V^2> the rms (root mean square) velocity of a molecule.

Then : sqrt(<V^2>) = sqrt (3 k T / m) or if you want the temperature: T =m <V^2>/(3k).

using m = M/N_A where N_A is the avogadro number and using R=N_A* k = 8.31 J/mol/K

T=M<V^2>/(3R) where M=28g/mol

then using the proper units (m/s, kg ...)
T=28 * 10^-3 * (11.2 * 10^3 )^2 / (3*8.31) = 1.41 * 10^5 K
which is... quite hot, isn't it?

Morality : N2 will not leave earth even if there is a huge global warming

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