Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 1.40.
The barometric equation was derived under the assumption that the atmosphere is stable, so that at a given height the pressure is equal to the weight of the column of air of unit cross-sectional area above that altitude. The barometric equation is
Earlier, we assumed that was constant, which allowed us to integrate the equation. In reality, the temperature decreases with increasing . If the temperature gradient passes a critical value, the density of the warmer air near the surface drops to a point where it starts to rise, resulting in convection, or the mass movement of air. Similarly, the density of the cooler air higher up is large enough that it falls towards the surface, so that a cycle is set up.
If we assume that the velocity of these air masses is high enough that little heat is lost or gained as the air moves vertically, we can use the adiabatic formula to analyze the situation. For adiabatic expansion, the pressure, volume and temperature are related by
where and are constants and . Taking differentials we get
Simplifying, we get
The average molecular mass can be found from the mass of a mole of dry air at room temperature and 1 atm pressure, which is :
so the critical temperature gradient is
If the temperature gradient reaches around 10 degrees per kilometre, we can expect convection to occur. This is known as the dry adiabatic lapse rate.