# Chemical Potetnial.

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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 $$N=4\pi n(R)exp(-MgR/\tau)\int_{R}^{\infty}drr^2exp(MgR^2/(r\tau)$$ 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
$$\tau log(n(R)/n_Q)=\tau log(n(r)/n_Q)+Mg(r-R)$$
where $$n_Q=(M\tau /2\pi\hbar^2)^\frac{3}{2}$$ and N/V=n where V is the volume of the concentration, now i get that:
$$N=V*n(R)*exp(-Mg(r-R)/\tau)$$
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?

Last edited:

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First of all, remember that the density will vary with distance from the Earth's surface. This means you can't just multiply by a volume to get particle number; you'll have to integrate it.

Gold Member
how exactly?
I mean:
n(r)/V=n(R)*exp(-Mg(r-R)/(k_B*T)

how to procceed from here?
I mean N=integral(n(r)/V)dV
where dV=r^2sin(theta)drd(theta)d(phi).
how to evalute V i mean i can see ad hoc what it needs to be from what i need to show, but it doesnt make much sense to me at least, i mean from what i see V should be an exponenetial without any factor attach to it in order to make its units of volume.