Question

Assume that the temperature remains essentially constant in the upper part of the atmosphere. Obtain an expression for the variation in pressure in the upper atmosphere with height, the mean molecular weight of air is ` M`.

Answer 1

Suppuose the pressure at height `h` is `p` and the at `h+dh` is `p+dp`.Then
`dp=-rhog dh." "...(i)`
Now considering any small volume `DeltaV` of air mass, `Deltam`,
`pDeltaV=nRT=(Deltam)/(M)RT`
or, `p=(Deltam)/(DeltaV) (RT)/(M)=(rhoRT)/(M)`
or, `rho=(M)/(RT)`
Putting in (i)
`dp=-(M)/(RT)pgdh`
or,`overset(p)underset(p_(0))int(dp)/(P)=overset(h)underset(0)int-(M)/(RT)gdh`
or, In`(P)/(p_(0))=-(Mgh)/(RT)`
Where `P_(0)` is the pressure at `h=0`
Thus,`P=P_(0)e^((-Mgh)/(RT))`