Kelvin equation

The Kelvin equation describes the change in vapour pressure due to a curved liquid–vapor interface, such as the surface of a droplet.

The Kelvin equation is dependent upon thermodynamic principles and does not allude to special properties of materials.

It is also used for determination of pore size distribution of a porous medium using adsorption porosimetry.

The equation is named in honor of William Thomson, also known as Lord Kelvin.

Equilibrium vapor pressure depends on droplet size.

may be treated as approximately fixed, which means that the critical radius

Ultimately it can become as small as a few molecules, and the liquid undergoes homogeneous nucleation and growth.

The derivation of the form that appears in this article from Kelvin's original equation was presented by Robert von Helmholtz (son of German physicist Hermann von Helmholtz) in his dissertation of 1885.

[3] The formal definition of the Gibbs free energy for a parcel of volume

The change in the Gibbs free energy due to this process is where

represent the Gibbs free energy of a molecule in the vapor and liquid phase respectively.

is the Gibbs free energy associated with an interface with radius of curvature

If the drop is considered to be spherical, then The number of molecules in the drop is then given by The change in Gibbs energy is then The differential form of the Gibbs free energy of one molecule at constant temperature and constant number of molecules can be given by: If we assume that

then The vapor phase is also assumed to behave like an ideal gas, so where

Solving the integral, we have The change in the Gibbs free energy following the formation of the drop is then The derivative of this equation with respect to

The radius corresponding to this value is: Rearranging this equation gives the Ostwald–Freundlich form of the Kelvin equation: An equation similar to that of Kelvin can be derived for the solubility of small particles or droplets in a liquid, by means of the connection between vapour pressure and solubility, thus the Kelvin equation also applies to solids, to slightly soluble liquids, and their solutions if the partial pressure

The equation would then be given by: These results led to the problem of how new phases can ever arise from old ones.

For example, if a container filled with water vapour at slightly below the saturation pressure is suddenly cooled, perhaps by adiabatic expansion, as in a cloud chamber, the vapour may become supersaturated with respect to liquid water.

A reasonable molecular model of condensation would seem to be that two or three molecules of water vapour come together to form a tiny droplet, and that this nucleus of condensation then grows by accretion, as additional vapour molecules happen to hit it.

The Kelvin equation, however, indicates that a tiny droplet like this nucleus, being only a few ångströms in diameter, would have a vapour pressure many times that of the bulk liquid.

Such nuclei should immediately re-evaporate, and the emergence of a new phase at the equilibrium pressure, or even moderately above it should be impossible.

Hence, the over-saturation must be several times higher than the normal saturation value for spontaneous nucleation to occur.

The chance of a fluctuation is e−ΔS/k, where ΔS is the deviation of the entropy from the equilibrium value.

[4] It is unlikely, however, that new phases often arise by this fluctuation mechanism and the resultant spontaneous nucleation.

It is more likely that tiny dust particles act as nuclei in supersaturated vapours or solutions.

In the cloud chamber, it is the clusters of ions caused by a passing high-energy particle that acts as nucleation centers.

Actually, vapours seem to be much less finicky than solutions about the sort of nuclei required.

For a sessile drop residing on a solid surface, the Kelvin equation is modified near the contact line, due to intermolecular interactions between the liquid drop and the solid surface.

is the disjoining pressure that accounts for the intermolecular interactions between the sessile drop and the solid and

This implies that a new phase can spontaneously grow on a solid surface, even under saturation conditions.