We
will show that the variation in pressure in the Earth’s
atmosphere, assumed to be at a uniform temperature, is given by ,
whereMis the molar mass of the air. We will also
show that
the number of molecules per unit volume at the heightyabove
the sea level varies as.
Consider
a vertical column of unit cross-sectional area. We set up the
equilibrium condition of an air column of width
and unit cross-sectional area at height y
from the surface of the Earth. Its weight
will be balanced by the pressure difference at its top and bottom
faces. That is
This gives a
first order differential equation for the function
,
Assuming that the air
column is in thermodynamic equilibrium and its equation of state is
described by the ideal gas equation,
where
M is the molar mass
of air. We now have a differential equation giving the variation of
air density with height y.
It is
Solution of this
differential equation is
where
is the density of air at the surface of the Earth. Under the
assumption that the temperature of the air column does not vary with
height y, we find the
following equation for pressure variation:
.
An alternative form of the
ideal gas equation is
where N
is the total number of molecules in volume V.
Or
where n
is the number of molecules per unit volume. We thus find
varies as