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 h, and the mean molecular weight of air is M.

Explanation

Suppose the pressure at height h is p and that at `h+dh is p+dp. Then `
`dp=-rhog dh. …(i)`
Now considering any small volume `DeltaV` if air of mass
`Deltam, `
`pDeltaV=nRt=(Deltam)/(M)RT`
`p=(Deltam)/(DeltaV) (RT)/(M)=(rhoRT)/(M)`
`or, =rho=(M)/(RT)p.`
Putting in (i),
`dp=-(M)/(RT)pg dh`ltbr.`or, `int_(p_(0))^(p)(dp)/(p)=int_(0)^(h)-(M)/(RT)g dh`
`or, In(p)/(p_(0))=-(Mgh)/(RT)`
where p_(0) is the pressure at h=0.`
Thus, `p=p_(0) e^(-(Mgh)/(RT)).`