The variation of pressure with density in a thick gas layer is given by where C and n are constants. Noting that the pressure change across a differential fluid layer of thickness dz in the vertical z–direction is given as dP= ‒ρg dz, obtain a relation for pressure as a function of elevation z. Take the pressure and density at z = 0 to be P0 and ρ0, respectively.

Step 1:

To obtain the relation for the variation of pressure with altitude with respect to the density of the air. The change in pressure P with altitude is given by

= ----(1)

Where is density of the air

g is the acceleration due to gravity (assuming constant)

z is the vertical height (altitude)

Step 2:

Variation of the pressure with density is termed as

P = C-----(2)

Where C and n are constants

Step 3:

If the density of the air at sea level is and the pressure at sea level is P0, the equation (1) can be written as

P0 = C-----(3)

Dividing (2) and (3)

=

=

= -----(4)

Step 4:

Substituting (4) in (1)

=

separating variables, we get

= ----(5)