The environmental lapse rate dT/dz characterizes the local variation of temperature with elevation in the earths atmosphere.

Question:

The environmental lapse rate dT/dz characterizes the local variation of temperature with elevation in the earth’s atmosphere. Atmospheric pressure varies with elevation according to the hydrostatic formula,
where M is molar mass, ρ is molar density, and g is the local acceleration of gravity. Assume that the atmosphere is an ideal gas. with T related to P by the polytropic
formula:
TP ^{(1−δ) ⁄ δ} = constant
Develop an expression for the environmental lapse rate in relation to M, g, R, and δ.