Suppose the pressure `p` and the density `rho` of air are related as `p//rho^n = const` regardless of height (`n` is a constant here). Find the corresponding temperature gradient.

Explanation

We have, `(dp)/(dh) = - rho g` (Sec 2.13)…(1)
But, from `p = C rho^n` (where `C` is, a const) `(dp)/(d rho) = Cn rho^(n - 1)` ….(2)
We have from gas low `p = rho (R )/(M) T`, so using (2)
`C rho^n = rho ( R)/(M).T`, or `T = (M)/(R) C rho^(n - 1)`
Thus, `(dT)/(d rho) = (M)/(R) .C (n - 1) rho^(n - 2)` ...(3)
But, `(d T)/(dh) = (dT)/(d rho).(d rho)/(dp).(dp)/(dh)`
So, `(dT)/(dh) = (M)/(R) C(n - 1) rho^(n -1) (1)/(C n rho^(n -1))(- rho g)= (- Mg (n -1))/(n R)`.