# Kelvin equation

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The Kelvin equation describes the change in vapour pressure due to a curved liquid/vapor interface (meniscus) with radius $r$ (for example, in a capillary or over a droplet). The vapor pressure of a curved surface is higher than that of a flat and non-curved 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 Kelvin equation may be written in the form

$\ln {p \over p_0}= {2 \gamma V_{\rm{m}} \over rRT}$

where $p$ is the actual vapour pressure, $p_0$ is the saturated vapour pressure, $\gamma$ is the surface tension, $V_{\rm{m}}$ 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 $p>p_0$, then liquid evaporates from the droplets
• If $p , then the gas condenses on to the droplets increasing their volumes

As $r$ increases, $p$ decreases and the droplets grow into bulk liquid.

If we now cool the vapour, then $T$ decreases, but so does $p_0$. This means $p/p_0$ increases as the liquid is cooled. We can treat $\gamma$ and $V_{\rm{m}}$ 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.

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 positively curved liquid surface or a bubble of vapor in a liquid, will result in a negatively curved liquid surface.

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.[1]

## References

1. ^ Robert von Helmholtz (1886) "Untersuchungen über Dämpfe und Nebel, besonders über solche von Lösungen" (Investigations of vapors and mists, especially of such things from solutions), Annalen der Physik, 263 (4) : 508-543. On pages 523-525, Robert von Helmholtz converts Kelvin's equation to the form that appears here (which is actually the Ostwald-Freundlich equation).

## Further reading

• Sir William Thomson (1871) "On the equilibrium of vapour at a curved surface of liquid," Philosophical Magazine, series 4, 42 (282) : 448-452.
• S. J. Gregg and K. S. W. Sing, Adsorption, Surface Area and Porosity, 2nd edition, Academic Press, New York, (1982) p.121
• Arthur W. Adamson and Alice P. Gast, Physical Chemistry of Surfaces, 6th edition, Wiley-Blackwell (1997) p.54
• Butt, Hans-Jürgen, Karlheinz Graf, and Michael Kappl. "The Kelvin Equation." Physics and Chemistry of Interfaces. Weinheim: Wiley-VCH, 2006. 16-19. Print.