If y is the height above sea level show that the decrease

Question

(a) If y is the height above sea level, show that the decrease of atmospheric pressure due to a rise of dy is given by

dP/P = (-Mg/RT)dy,

where M is the molar mass of air, g is the acceleration of gravity, and T is the temperature at the height y.

(b) If the decrease of pressure in part (a) is due to an adiabatic expansion, show that

dP/P = (γ/γ-1)(dT/T).

 (c) From parts (a) and (b), using some of the numerical data of Sec. 5.7, calculate d_T/d_y in kelvin per kilometer.