The OstwaldâÂÂFreundlich equation governs boundaries between two phases; specifically, it relates the surface tension of the boundary to its curvature, the ambient temperature, and the vapor pressure or chemical potential in the two phases.
The OstwaldâÂÂFreundlich equation for a droplet or particle with radius is:
One consequence of this relation is that small liquid droplets (i.e., particles with a high surface curvature) exhibit a higher effective vapor pressure, since the surface is larger in comparison to the volume.
Another notable example of this relation is Ostwald ripening, in which surface tension causes small precipitates to dissolve and larger ones to grow. Ostwald ripening is thought to occur in the formation of orthoclase megacrysts in granites as a consequence of subsolidus growth. See rock microstructure for more.
In 1871, Lord Kelvin (William Thomson) obtained the following relation governing a liquid-vapor interface:
where:
In his dissertation of 1885, Robert von Helmholtz (son of the German physicist Hermann von Helmholtz) derived the OstwaldâÂÂFreundlich equation and showed that Kelvin's equation could be transformed into the OstwaldâÂÂFreundlich equation. The German physical chemist Wilhelm Ostwald derived the equation apparently independently in 1900; however, his derivation contained a minor error which the German chemist Herbert Freundlich corrected in 1909.
According to Lord Kelvin's equation of 1871,
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:
Since is the mass of one molecule of vapor or liquid, then
Hence
Thus
Since
then
Since , then . If , then . Hence
Therefore
which is the OstwaldâÂÂFreundlich equation.