Question

Define the term adiabatic lapse rate. obtain an expression for adiabatic lapse rate for the earth atmosphere.

Answer 1

Lapse rate shows how much changes the temperature with height

$\Gamma = - \frac{dT}{dz}$

Even in condition of thermal equilibrium, there won't be equal temperature in different layer of gas due to Boltzmann distribution. If the the gas (e.g. in atmosphere) moves in a way that $PV^\gamma =$ const, this processes is called adiabatic. In adiabatic atmosphere adiabatic lapse rate can be obtained.

From the first law of thermodynamics: $dU = \delta Q - PdV$

Since the processes are adiabatic, $\delta Q =0.$ Also, $dU = m c_V dT$ .

From the equation of adiabatic process taking the differential: $V^\gamma dP + P \gamma V^{\gamma-1} dV =0 => PdV = -\frac{V^\gamma dP}{\gamma V^{\gamma-1}}$

$pdV = -\frac{VdP}{\gamma}$

Putting all result into the first law of thermodynamics, we get

$m c_V dT = \frac{VdP}{\gamma}$

Assuming an atmosphere in hydrostatic equilibrium $dP = -\rho g dz = -\frac{m g dz}{V}$ and taking into account $\gamma = c_p/c_V$ we have

$dT = -\frac{V/m * \rho g dz}{\gamma c_V}=-\frac{ g dz}{ c_P}$

$\Gamma = -\frac{dT}{dz}=\frac{g}{c_P}$