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