Show that the variation of atmospheric pressure with altitude is given by p=pe^-

Question

Show that the variation of atmospheric pressure with altitude is given by p=pe^-(alpha)y, where alpha=pg/pP, P is atmospheric pressure at some reference level y=0, andp is teh atmospheric pressure over an infinitesimal change in altitude so that the density is approximately uniform over the infinitesimal change can be expressed from equation 14.4 as dP=-pgdy. also assume the density of air is proportional to the pressure, which, as we will see in chapter 20, is equivalent to assuming the temperature of the air is the same at all altitudes.

Explanation / Answer

we have,let dP be the infinitesimal change in pressure at an altitude y when dy is the infinitesimal change in altitude at an altitude y. is the density at altitude y.

dP=-gdy    (g is acceleration dur to gravity)

=PM/RT            

( is density at altitude y, P is pressure,Mis molecular mass,R is universal gas constant,T is temperature)

=Pk where k=M/RT=/p ..(1)

dP=-kPgdy

dP/P=-kgdy

integrating both sides within limits y varies from 0 to y and P varies from p(at reference y=0) to P at altitude y.

ln(P/p)=-kgy

P=pe-kgy=pe-gy/p ...from(1)

P=pe-y    where   =g/p

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