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 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.

The original form of the Kelvin equation, published in 1871, is: $${\displaystyle p(r_{1},r_{2})=P-{\frac {\gamma \,\rho _{\rm {vapor}}}{(\rho _{\rm {liquid}}-\rho _{\rm {vapor}})}}\left({\frac {1}{r_{1}}}+{\frac {1}{r_{2}}}\right),}$$ where:

This may be written in the following form, known as the Ostwald–Freundlich equation: $${\displaystyle \ln {\frac {p}{p_{\rm {sat}}}}={\frac {2\gamma V_{\text{m}}}{rRT}},}$$ where ${\displaystyle p}$ is the actual vapour pressure, ${\displaystyle p_{\rm {sat}}}$ is the saturated vapour pressure when the surface is flat, ${\displaystyle \gamma }$ is the liquid/vapor surface tension, ${\displaystyle V_{\text{m}}}$ is the molar volume of the liquid, ${\displaystyle R}$ is the universal gas constant, ${\displaystyle r}$ is the radius of the droplet, and ${\displaystyle T}$ is temperature.

Equilibrium vapor pressure depends on droplet size.

As ${\displaystyle r}$ increases, ${\displaystyle p}$ decreases towards ${\displaystyle p_{sat}}$, and the droplets grow into bulk liquid.

If the vapour is cooled, then ${\displaystyle T}$ decreases, but so does ${\displaystyle p_{\rm {sat}}}$. This means ${\displaystyle p/p_{\rm {sat}}}$ increases as the liquid is cooled. ${\displaystyle \gamma }$ and ${\displaystyle V_{\text{m}}}$ may be treated as approximately fixed, which means that the critical radius ${\displaystyle r}$ 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.