How pressure of saturated water vapor depends on temperature (Clausius-Clapeyron law)

The exact Clausius-Clapeyron equation is presented in formula (4) in the Russian version of "Main Findings". Here we do not aim to derive it, but will discuss its nature. Molecules of the liquid collide with each other chaotically. Their velocities change. After collision, the molecule can either increase or decrease its velocity and kinetic energy. Therefore, there is always a certain portion of "highly energetic" molecules in the liquid, which kinetic energy E is significantly higher than the average for all molecules. This share is proportional to exp(−E/kT), where T is temperature in degrees Kelving, k = 1.38 × 10⁻²³ J/degree Kelvin/molecule.

In order to overcome the intermolecular forces in the liquid and move free to the gas phase (evaporate), the water molecule must have energy E in excess of a certain value set by the molecular properties of water. For water this value is E_H2O = 7.3 × 10⁻²⁰ J/molecule, so E_H2O/k ~ 5300 K. The exponent for the share of "highly energetic" molecules is thus written as exp[−(5300 K)/T]. For small temperature changes around T₀ ~ 300 K this exponent corresponds to doubling of the number of molecules with E > E_H2O per each ten degrees of temperature increase.

For this reason the saturated (i.e. maximum) concentration of water vapor above the liquid water surface behaves in the same manner. This is illustrated in the model. The equilibrium state is reached when the number of molecules leaving the liquid phase per unit time is equal to the number of molecules returning to liquid from the gas phase per unit time. When temperature drops, the number of "highly energetic" molecules decreases, the number of molecules leaving the liquid phase per unit time decreases as well, while the number of molecules returning to the liquid phase from gas is initially practically unaffected by temperature. In the result, the saturated concentration of water vapor decreases. The temperature dependence of saturated water vapor concentration underlies the biotic pump physics.

Partial pressure of water vapor cannot exceed the saturated one (red line), the latter drops twofold for each ten degrees of temperature decrease. In the static atmosphere water vapor partial pressure must drop twofold for each 9 km of height increment (blue line). When the red line finds itself below the blue line, as is the case at the observed lapse rate of 6.5 K/km, the atmosphere cannot be static. Water vapor condenses and disappears from the gas phase. There appears a dynamic air flow from areas with less intense condensation to areas where condensation is more intense.