Question

Derive an expression for adiabatic lapse rate.

Answer 1

Hydrostatic equation

$$dp=-g\rho dz$$

First law of thermodynamics for adiabatic process

$$mc_V dT - \frac{Vdp}{\gamma}=0$$

Where

$$\gamma=\frac{c_p}{c_V}$$

Combining the Hydrostatic equation with first law of thermodynamics for adiabatic process, we have

$$c_p dT - \frac{m}{V}dp=0$$ $$c_p dT + \frac{m}{V}g\rho dz=0$$ $$\frac{dT}{dz}=-\frac{g}{c_p}$$

This an expression for adiabatic lapse rate

$$\frac{dT}{dz}=-\frac{g}{c_p}$$