I do not know how to begin answering the following problem. 10.8 Suppose we have

Question

I do not know how to begin answering the following problem.

10.8 Suppose we have a gas in thermal equilibrium at temperateT. Each molecule of the gas has mass m. (a) What isthe ratio of the number of molecule at the Earth's surface to thenumber at height h (with potential energy mgh)?(b) What is the ratio of the density of the gas at heighth to the density at the surface? (c) Would you expect thissimple model to give an adequate description of the Earth'satmosphere?

Thanks in advance for your help.

Explanation / Answer

This is an application of the boltzmann distribution formula n ~ e -Energy/kT On earth surface, the number is proportional to nsurface ~ e-E/kT     At a height h above, the molecules have extra gravitational PE =mgh nh ~ e -(E + mgh)/kT So the ration of numbers are nh/nsurface = e -mgh/kT part b: since density is number per unit volume, the ratio is densityh /densitysurface = e-mgh/kT part c: no, because the molecules are not in perfect thermal equilibriumand there are other factors such as wind, high altitude thermalinversion, etc. hope this helps!

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