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## Homework Statement

Consider a simplified model of cloud formation. Hot air in contact with the earth’s surface contains water vapor. This air rises convectively till the water vapor content reaches its saturation pressure. When this happens, the water vapor starts condensing and droplets are formed. We shall estimate the height at which this happens. We assume that the atmosphere consists of the diatomic gases oxygen and nitrogen in the mass proportion 21:79 respectively. We further assume that the atmosphere is an ideal gas, g the acceleration due to gravity is constant and air processes are adiabatic. Under these assumptions one can show that the pressure is given by

$$p=p_0\left(\frac{T_0-\Gamma z}{T_0}\right)^{\alpha}$$

Here ##p_0## and ##T_0## is the pressure and temperature respectively at sea level ##(z = 0)##, ##\Gamma## is the lapse rate (magnitude of the change in temperature ##T## with height ##z## above the earth’s surface, i.e. ##\Gamma > 0##).

(a) Obtain an expression for the lapse rate ##\Gamma## in terms of ##\gamma, R, g## and ##m_a##. Here ##\gamma## is the ratio of specific heat at constant pressure to specific heat at constant volume; ##R##, the gas constant; and ##m_a##, the relevant molar mass.

(There are few more parts to this problem, they seem to be related to (a), I will post them when I solve a) ).

## Homework Equations

## The Attempt at a Solution

Consider a layer at height ##z##, cross section ##A## and thickness ##dz##. Let ##p## and ##p+dp## be the pressure on two sides as shown in figure. The mass of the air in this layer is ##A\rho \,dz## where ##\rho## is the density of air at height ##z##. Equating the forces, I get

$$\frac{dp}{dz}=-\rho g$$

Now I am not sure what to do after this. From the ideal gas, I have ##\rho=(pm_a)/(RT)## but how do I introduce ##\gamma##?

Is my approach even correct?

Any help is appreciated. Thanks!