Köhler theory

Köhler theory describes the process in which water vapor condenses and forms liquid cloud drops, and is based on equilibrium thermodynamics. It combines the Kelvin effect, which describes the change in saturation vapor pressure due to a curved surface, and Raoult's Law, which relates the saturation vapor pressure to the solute.[1] It is an important process in the field of cloud physics. It was initially published in 1936 by Hilding Köhler, Professor of Meteorology in the Uppsala University.

Kohler curves showing how the critical diameter and supersaturation are dependent upon the amount of solute. It's assumed here that the solute is a perfect sphere of sodium chloride.

Köhler equation:

${\displaystyle \ln \left({\frac {p_{w}(D_{p})}{p^{0}}}\right)={\frac {4M_{w}\sigma _{w}}{RT\rho _{w}D_{p}}}-{\frac {6n_{s}M_{w}}{\pi \rho _{w}D_{p}^{3}}}}$

where ${\displaystyle p_{w}}$ is the droplet water vapor pressure, ${\displaystyle p^{0}}$ is the corresponding saturation vapor pressure over a flat surface, ${\displaystyle \sigma _{w}}$ is the droplet surface tension, ${\displaystyle \rho _{w}}$ is the density of pure water, ${\displaystyle n_{s}}$ is the moles of solute, ${\displaystyle M_{w}}$ is the molecular weight of water, and ${\displaystyle D_{p}}$ is the cloud drop diameter.