density of the air

Using the ideal gas equation PV = nRT
where P and V are respectively the 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 3500/(8.31 x 273))
d/(do) = 0.645 = 64.5%

mass density is proportional to number density,

so only need to find n/n_0

(n is the number density at an altitude of 3.5km
and n_0 is the number density at the surface.)

p=nkT (P is pressure)

so only need to find p/p_0

And p=p_0 e^(-mgz/kT)

This can be deduced from Boltzmann's distribution or using

the concept of equilibrium.

(p-dp)S-pS = n mg Sdz

dp/dz= -nmgdz =-pmgdz/kT

dp/p = -(mg/kT)dz

Integrate both side give

p=p_0 e^(-mgz/kT)

P.S.

p_0 = Nmg/S

which can also be derived by using the concept of equilibrium.

Nmg=R

p_0=R/S = Nmg/S