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`.

Explanation

Suppose the pressure at height `h` is `p` and that at `h + dh` is `p + d p`. Then
`dp = - rhog d h`
`pV = nRT = (m)/(M)RT`
`p = rho (RT)/(M) rArr rho = (pM)/(RT)`
`dp = -(pM)/(RT)gdh`
`underset(p_(0))overset(p)(int)(dp)/(p) = -(Mg)/(RT) underset(0)overset(h)(int) dh`
`"ln" (p)/(p_(0)) = -(Mgh)/(RT)`
`p = p_(0) e^(-(Mgh)/(RT))`