Barometric Formula and Boltzmann Distribution

We can glean another well-known relation, the density distribution of a gas in a homogeneous gravitational field, by using the same procedure. The Earth s atmosphere whose density decreases just about exponentially with altitude is a good example of this. Similar to what we did in the last section, we will consider the particles at a given altitude h to be molecules of a substance B(h). The basic potentials h) of the substances B(/r) only differ from each other by the above-mentioned molar potential energy w/t = M g h because they are chemically identical  

The exchange of particles between different altitudes can be described as a reaction of the following type  

When temperatures are uniform, sooner or later equilihrium is established for all the substances so that the potential becomes the same for aU altitudes, fi(h) = // for all h. Applying the mass action equation, we obtain 

RTIMg represents the range of the exponential distribution. This is the altitude at which the gas concentration has fallen to 1/e = 36.8 % compared to the value c(0) at sea level. For nitrogen as the main component of the atmosphere at 300 K, the range 

The altitude at which the concentration has fallen to V2c(0), the so-called halfheight hn, is a factor of ln2 lower, 6,300 m. This is a little more than the highest point of Mt. Kilimanjaro. The air around this peak has therefore about half the density of the air on the coast. 

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