Kelvin equation

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 Kelvin equation is used for determination of pore size distribution of a porous medium using adsorption porosimetry. The equation is named in honor of William Thomson, commonly known as "Lord Kelvin".

The Kelvin equation may be written in the form

$\ln {p \over p_0}= {-2 \gamma V_m \over rRT}$

where $p$ is the actual vapour pressure, $p_0$ is the saturated vapour pressure, $\gamma$ is the surface tension, $V_m$ is the molar volume, $R$ is the universal gas constant, $r$ is the radius of the droplet, and $T$ is temperature.

Equilibrium vapor pressure depends on droplet size. If $p_0 < p$ , then liquid evaporates from the droplets.

If $p_0 > p$ , then the gas condenses onto 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 $V$ 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.

References

W. T. Thomson, Phil. Mag. 42, 448 (1871)
S. J. Gregg and K. S. W. Sing, Adsorption, Surface Area and Porosity, 2nd edition, Academic Press, New York, (1982) p.121
Adamson and Gast, Physical Chemistry of Surfaces, 6th edition, (1997) p.54

Kelvin equation

See also

References