# Kelvin Equation

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## Kelvin equation

[′kel·vən i‚kwā·zhən]## Kelvin Equation

an equation that characterizes the change in the vapor pressure of a liquid or the solubility of solids that results from curvature of the interface of adjacent phases (the solid-liquid or liquid-vapor contact surface). For example, above spherical drops of a liquid the saturated vapor pressure *P* is greater than the pressure *p*_{0} above a flat surface at the same temperature *T.* Correspondingly, the solubility *c* of a solid with a convex surface is greater than the solubility *c*_{o} of flat surfaces of the same substance.

The Kelvin equation was derived by W. Thomson (Lord Kelvin) in 1871 on the basis of the equality of chemical potentials in adjacent phases that are in a state of thermodynamic equilibrium. It has the form *p*/*p*_{0} = *c* /*c*_{0} = exp (2 *σν/rRT*), where *r* is the radius of average curvature of the phase interface, *σ* is the interphase surface tersion, *ν* is the molar volume of the liquid or solid whose vapor pressure *ρ* or solubility *c* is a component of the equation, and *R* is the gas constant. For spherical particles *r* is equal in absolute magnitude to their radius.

According to the Kelvin equation, a decrease or increase in the vapor pressure and solubility depends on the sign of curvature of the surface of the substance under consideration: an increase corresponds to a convex surface (*r* > 0), and a decrease corresponds to a concave surface (*r* < 0). Thus, in contrast to the cases considered above, the vapor pressure in a bubble or above the surface of a concave meniscus in a capillary is lowered (*p < po*). Since the values of *p* and *c* are different for particles of different dimensions and for surface areas that have depressions and protrusions, the Kelvin equation defines the direction of the transport of the substance (from greater to lesser values of *p* and *c*) in the process of the system’s transition to a state of thermodynamic equilibrium. In particular, this leads to a situation in which large drops or particles grow as a result of the evaporation or solution of smaller drops or particles, and uneven surfaces are smoothed out by solution of the protrusions and filling of the depressions. Noticeable differences in pressure and solubility are found only for sufficiently small *r.* Therefore, the Kelvin equation is used most widely to characterize the state of small entities (particles of colloidal systems and nucleations of the new phase) and in the study of capillary phenomena.

N. V. CHURAEV