In 1871, Lord Kelvin (William Thomson) obtained the following relation governing a liquid-vapor interface:[1]
where:
- = vapor pressure at a curved interface of radius
- = vapor pressure at flat interface () =
- = surface tension
- = density of vapor
- = density of liquid
- , = radii of curvature along the principal sections of the curved interface.
In his dissertation of 1885, Robert von Helmholtz (son of the German physicist Hermann von Helmholtz) derived the Ostwald–Freundlich equation and showed that Kelvin's equation could be transformed into the Ostwald–Freundlich equation.[2][3] The German physical chemist Wilhelm Ostwald derived the equation apparently independently in 1900;[4] however, his derivation contained a minor error which the German chemist Herbert Freundlich corrected in 1909.[5]
According to Lord Kelvin's equation of 1871,[6][7]
If the particle is assumed to be spherical, then ; hence,
Note: Kelvin defined the surface tension as the work that was performed per unit area by the interface rather than on the interface; hence his term containing has a minus sign. In what follows, the surface tension will be defined so that the term containing has a plus sign.
Since , then ; hence,
Assuming that the vapor obeys the ideal gas law, then
where:
- = mass of a volume of vapor
- = molecular weight of vapor
- = number of moles of vapor in volume of vapor
- = Avogadro constant
- = ideal gas constant =
Since is the mass of one molecule of vapor or liquid, then
- volume of one molecule .
Hence
- where .
Thus
Since
then
Since , then . If , then . Hence
Therefore
which is the Ostwald–Freundlich equation.