Consider a particle of mass m in a uniform gravitational field. Its potential energy is E(h) = mgh,where h is the height above sea level and g is the earth's gravitational constant, g = 9:8 m/s2.
(i) According to the Boltzmann distribution, what is the probability p(h) of finding the particle at
height h, relative to the probability p(0) of finding it at sea level (h = 0), at temperature T?
[In truth, p(h) is a probability density, since h is a continuous variable. A more proper way to word
this question would be in terms of the probability p(h)h of finding the particle in a small range of height between h and h+h. But since (p(h)h)=(p(0)h) = p(h)=p(0), the answer would be the same.]
(ii) Now imagine that there are N such masses, all identical and non-interacting. On average, how many of these objects reside at height h, relative to the number at sea level? Express your answer in terms of relative densities, (h)=(0), where (h) denotes the average number of particles per unit volume at height h.
(iii) Many people (including me) are susceptible to altitude sickness due to oxygen depletion at elevations greater than 10,000 feet above sea level. Using your result from part (ii), estimate the density of molecular oxygen at this elevation, relative to that at sea level. In performing this calculation, assume that the earth's atmosphere has a temperature of 300 K throughout.