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The question is from kittel's book, thermal physics:

If n concentaration of moleclues at the surface of earth, M the mass of a molecule and g and gravitational acceleration at the surface, show that at constant temprature the total number of molecules in the atmosphere is [tex]N=4\pi n(R)exp(-MgR/\tau)\int_{R}^{\infty}drr^2exp(MgR^2/(r\tau)[/tex] where tau is the tempratue divided by boltzman's constant, and r is measured from the centre of the Earth and R is the radius of the earth.

my attempt at solution:

Now obviously this is a question of chemical potenital, i.e

[tex]\tau log(n(R)/n_Q)=\tau log(n(r)/n_Q)+Mg(r-R)[/tex]

where [tex]n_Q=(M\tau /2\pi\hbar^2)^\frac{3}{2}[/tex] and N/V=n where V is the volume of the concentration, now i get that:

[tex]N=V*n(R)*exp(-Mg(r-R)/\tau)[/tex]

but I'm not sure how to calculate V the volume here, any suggestions?

obviously if i solve this then i will show the identity but how?

thanks in advance.

If n concentaration of moleclues at the surface of earth, M the mass of a molecule and g and gravitational acceleration at the surface, show that at constant temprature the total number of molecules in the atmosphere is [tex]N=4\pi n(R)exp(-MgR/\tau)\int_{R}^{\infty}drr^2exp(MgR^2/(r\tau)[/tex] where tau is the tempratue divided by boltzman's constant, and r is measured from the centre of the Earth and R is the radius of the earth.

my attempt at solution:

Now obviously this is a question of chemical potenital, i.e

[tex]\tau log(n(R)/n_Q)=\tau log(n(r)/n_Q)+Mg(r-R)[/tex]

where [tex]n_Q=(M\tau /2\pi\hbar^2)^\frac{3}{2}[/tex] and N/V=n where V is the volume of the concentration, now i get that:

[tex]N=V*n(R)*exp(-Mg(r-R)/\tau)[/tex]

but I'm not sure how to calculate V the volume here, any suggestions?

obviously if i solve this then i will show the identity but how?

thanks in advance.

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