PhysicsQuestion - density

Using the ideal gas equation PV = nRT
where P and V are the respective pressure and volume of n moles of air at temperature T. R is the Universal Gas Constant (= 8.31 J/mo.K)

Wrting n = m/M, where m is the mass of air and M is its molar mass (= 0.029 kg/mol)
Hence, PV = mRT/M
P = (m/V)RT/M = dRT/M
where d = m/V, is the density of air
i.e. d = PM/RT

The variation of pressure P with altitude follows the equation:
P = (Po).exp(-Mgh/RT)
where Po is the pressure at surface, g is the acceleration due to gravity.
[You may refer to the web-page for derivation of theis formula:
http://www.math24.net/barometric-formula.html ]

Therefore, d = [Po.exp(-Mgh/RT)]M/RT
This equation gives the variation of air density with altitude.

Density at surface, (do) = (Po)M/RT
thus, d/(do) = exp(-Mgh/RT) = exp(-0.029 x 9.81 x 4000/(8.31 x 273))
d/(do) = 0.606 = 60.6%