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forces molecules tend to fall to the bottom of the vessel but intense thermal
motion prevents precipitation of the molecules, and the molecules are
distributed, so that their concentration gradually decreases with increasing
height.

We derive the variation of pressure with height
assuming that the gravitational field is uniform, the
temperature is constant and the mass of all the

molecules is the same. If the atmospheric pressure at
a height hequal to p, then at a height h  dh it is
equal to p  dp (with dh> 0, dp <0, since p

decreases with increasing h).
The pressure difference at the heights h and h + dh
is equal to

dp    gh (1.11.1)

Figure 1.11.1 If we take into account the following Mendeleev-
Clapeyron equation

m m pM
pV  RT     (1.11.2)
M V RT

we will obtain

Mg m pM
dp   RT     (1.11.3)
M V RT
When integrating equation 11.3 we obtain

dp Mg Mg
     dh  ln p    h ln C (1.11.4)
p RT RT

where С - constant of integration. Potentiating this expression, we get the
following:

Mg
p  C  ( exp ) h  . (1.11.5)
RT
The constant of integration C can be determined from the initial
condition

P(h = 0) = P , where P is the average sea level atmospheric pressure.
0
0

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