# 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 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 Kelvin equation may be written in the form

$\ln {\frac {p}{p_{\rm {sat}}}}={\frac {2\gamma V_{\text{m}}}{rRT}},$ where $p$ is the actual vapour pressure, $p_{\rm {sat}}$ is the saturated vapour pressure when the surface is flat, $\gamma$ is the liquid/vapor surface tension, $V_{\text{m}}$ is the molar volume of the liquid, $R$ is the universal gas constant, $r$ is the radius of the droplet, and $T$ is temperature.

Equilibrium vapor pressure depends on droplet size.

• If the curvature is convex, $r$ is positive, then $p>p_{\rm {sat}}$ • If the curvature is concave, $r$ is negative, then $p As $r$ increases, $p$ decreases, and the droplets grow into bulk liquid.

If we now cool the vapour, then $T$ decreases, but so does $p_{\rm {sat}}$ . This means $p/p_{\rm {sat}}$ increases as the liquid is cooled. We can treat $\gamma$ and $V_{\text{m}}$ as approximately fixed, which means that the critical radius $r$ must also decrease. The further a vapour is supercooled, the smaller the critical radius becomes. Ultimately it gets as small as a few molecules, and the liquid undergoes homogeneous nucleation and growth. A system containing a pure homogeneous vapour and liquid in equilibrium. In a thought experiment, a non-wetting tube is inserted into the liquid, causing the liquid in the tube to move downwards. The vapour pressure above the curved interface is then higher than that for the planar interface. This picture provides a simple conceptual basis for the Kelvin equation.

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.

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 $p$ is replaced by the solubility of the solid (or a second liquid) at the given radius, $r$ , and $p_{0}$ by the solubility at a plane surface. Hence small particles (like small droplets) are more soluble than larger ones.