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The Isothermal Atmosphere

The gravitational potential energy of a gas molecule of mass  m  at an altitude  h  above sea level is given approximately by   $\varepsilon = m g h$,  where  g = 9.81 m/s².  Here we neglect the decrease of  g  with altitude, which is a good approximation over a few dozen miles. Next we pretend that the temperature of the atmosphere does not vary with altitude, which is untrue, but perhaps relative to 0 K it is not all that silly, since the difference between the freezing (273.15 K) and boiling (373.15 K) points of water is less than 1/3 of their average. For convenience we will assume that the whole atmosphere has a temperature  T = 300 K  (a slightly warm "room temperature"). In this approximation, the probability   ${\cal P}(h)$  of finding a given molecule of mass  m  at height  h  will drop off exponentially with  h:

$${\cal P}(h) \; = \; {\cal P}(0) \; \exp \left( - {mgh \over \tau} \right)$$

Thus the density of such molecules per unit volume and the partial pressure  p_m  of that species of molecule will drop off exponentially with altitude  h:

$$p_m(h) \; = \; p_m(0) \; \exp \left( - {h \over h_0} \right)$$

where  h₀  is the altitude at which the partial pressure has dropped to  1 / e  of its value  p_m(0)  at sea level. We may call  h₀  the "mean height of the atmosphere" for that species of molecule. A quick comparison and a bit of algebra shows that

$$h_0 = {\tau \over m g}$$

For oxygen molecules (the ones we usually care about most)   $h_0 \approx 8$ km. For helium atoms   $h_0 \approx 64$ km and in fact He atoms rise to the "top" of the atmosphere and disappear into interplanetary space. This is one reason why we try not to lose any helium from superconducting magnets etc. - helium is a non-renewable resource!