Chemical Potential and Atmospheric Molecule Distribution at Constant Temperature

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In summary, the question is asking how to calculate the number of particles in the atmosphere, and the answer is that you need to integrate it.
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MathematicalPhysicist

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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.
 
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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.
 
  • #3
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 doesn't 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.
 

What is chemical potential?

Chemical potential is a thermodynamic concept that measures the potential energy of a substance in a given environment. It is defined as the change in internal energy of a system when one mole of a substance is added, while keeping all other conditions constant.

How does chemical potential affect atmospheric molecule distribution at constant temperature?

At constant temperature, the chemical potential of a substance determines its distribution in the atmosphere. Molecules with higher chemical potential tend to be more abundant, as they have a higher potential to move from areas of higher concentration to areas of lower concentration.

What factors influence chemical potential?

Chemical potential is influenced by temperature, pressure, and concentration. At a constant temperature and pressure, the chemical potential will depend on the relative concentrations of the substances involved.

What is the relationship between chemical potential and diffusion?

Diffusion is the movement of particles from an area of higher concentration to an area of lower concentration. The rate of diffusion is directly proportional to the difference in chemical potential between the two regions; the greater the difference, the faster the diffusion will occur.

How is chemical potential related to the equilibrium state of a system?

In an equilibrium state, the chemical potential of a substance is the same in all regions of the system. This means that there is no net movement of particles between regions, as the chemical potential is equalized. Chemical potential plays a crucial role in determining the equilibrium state of a system.

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