The Kelvin equation describes the change in vapour pressure due to a curved liquid–vapor interface, such as the surface of a droplet. The vapor pressure at a convex curved surface is higher than that at a flat surface. 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.
Formulation
The original form of the Kelvin equation, published in 1871, is:
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.
This may be written in the following form, known as the Ostwald–Freundlich equation:
where is the actual vapour pressure, is the saturated vapour pressure when the surface is flat, is the liquid/vapor surface tension, is the molar volume of the liquid, is the universal gas constant, is the radius of the droplet, and is temperature.
Equilibrium vapor pressure depends on droplet size. If the curvature is convex, is positive, then If the curvature is concave, is negative, then
As increases, decreases towards , and the droplets grow into bulk liquid.
If the vapour is cooled, then decreases, but so does . This means increases as the liquid is cooled. and may be treated as approximately fixed, which means that the critical radius must also decrease. The further a vapour is supercooled, the smaller the critical radius becomes. Ultimately it can become as small as a few molecules, and the liquid undergoes homogeneous nucleation and growth.
The change in vapor pressure can be attributed to changes in the Laplace pressure. When the Laplace pressure rises in a droplet, the droplet tends to evaporate more easily.
When applying the Kelvin equation, two cases must be distinguished: A drop of liquid in its own vapor will result in a convex liquid surface, and a bubble of vapor in a liquid will result in a concave liquid surface.
History
The form of the Kelvin equation here is not the form in which it appeared in Lord Kelvin's article of 1871. 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. In 2020, researchers found that the equation was accurate down to the 1nm scale.
Derivation using the Gibbs free energy
The formal definition of the Gibbs free energy for a parcel of volume , pressure and temperature is given by:
where is the internal energy and is the entropy. The differential form of the Gibbs free energy can be given as
where is the chemical potential and is the number of moles. Suppose we have a substance which contains no impurities. Let's consider the formation of a single drop of with radius containing molecules from its pure vapor. The change in the Gibbs free energy due to this process is
where and are the Gibbs energies of the drop and vapor respectively. Suppose we have molecules in the vapor phase initially. After the formation of the drop, this number decreases to , where
Let and represent the Gibbs free energy of a molecule in the vapor and liquid phase respectively. The change in the Gibbs free energy is then:
where is the Gibbs free energy associated with an interface with radius of curvature and surface tension . The equation can be rearranged to give
Let and be the volume occupied by one molecule in the liquid phase and vapor phase respectively. 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 is the Boltzmann constant. Thus, the change in the Gibbs free energy for one molecule is
where is the saturated vapor pressure of over a flat surface and is the actual vapor pressure over the liquid. 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 is
The maximum value occurs when the derivative equals zero. The radius corresponding to this value is: