Assume that the temperature remains essentially constant in the upper part of the atmosphere. Obtain an expression for the variation in pressure with height in the upper atmosphere. The mean molecular weight of air is M. Assume a variable PO for the pressure at a height h = 0, where h = 0 depends on the choice of origin (could very well be at the beginning of the stratosphere, for example).
P=P0e−MghRT
Suppose the pressure at height h is P and that at (h+dh) is (P+dP). Then -
dP=−ρgdh
Now considering any small volume ΔV of air of mass Δm,
PΔV=nRΔT=ΔmMRΔT
⇒P=[ΔmΔV×RTM]=ρRTM
⇒ρ=MRTP.
Substituting -
dP=−MRTPgdh
⇒P∫P0(dPP)=h∫0(−MRTg)dh
⇒ln(PP0)=−(MghRT),
where P0 is the pressure at h = 0
Thus, P=P0e−(MghRT),