On a sunny day, the temperature and pressure of atmosphere varying with height z from sea level as $T=T_0(1-\frac{z}{z_0}) and P=P_0{(1-\frac{z}{z_0})}^{\alpha }.$ Where $T_0$ and $z_0,$ are ${25}^{0}C$ and 25 km. The temperature pressure and density of air at sea level is $T_0={25}^{°}C,P_0=1$ atm and 1.2 $kg/m^3$ respectively. Assume that air follows ideal gas law in the given condition, i.e., $\frac{\rho T}{P}$ =constant.

Find the value of  $\alpha .$ 1 atm $={10}^{5}N/m^2$ and $g=10m/s^2$ (g is uniform)