PHYSICAL CHEMISTRY IN BRIEF

CHAP. 13: PHYSICAL CHEMISTRY OF SURFACES [CONTENTS] 443

subscriptr. The same quantities for a flat surface are denoted using the superscript^.

13.1.7.1 Kelvin’s equation

applies

∆μ=μr−μ^ =RTln

fr

f^

=

∫p (^) +2γ/r
p^
Vm(l)dp=
2 γ Vm(l)
r

, (13.17)

wherefris the fugacity of a substance in a droplet of the radiusr,f^ =f(T) is the fugacity
of a substance in the saturated state with a non-curved surface,γis the surface tension, and
Vmis the molar volume of the liquid.

Note:If a system contains both a liquid in the form of droplets and a common liquid (e.g.

in a beaker), the fugacity of the liquid in droplets is higher than that of vapour or of the

equilibrium liquid over the non-curved surface. The droplets will thus gradually evaporate

until they vanish completely.

The behaviour between a liquid and a solid phase may be described in a similar way (if
we assume a spherical shape of the crystals). In this case, however, we have to employ the
interfacial tensionγs` and the chemical potential. An analogy to Kelvin’s equation is the
relation

∆μ=μr−μ^ =RTln

fr

f^

=

2 γs`Vm(s)

r

, (13.18)

whereμris the chemical potential of a solid substance in a crystal of the radiusr,μ^ is the
chemical potential of a solid in an infinitely large crystal,γs`is the interfacial tension between
the solution and the solid phase, andVm(s)is the molar volume of the solid.

Note: In an exact description, the non-spherical shape of the crystal and the different

surface tensions on its different planes have to be taken into account.

13.1.8 Temperature dependence of surface tension

The surface tension of a liquid that is in equilibrium with its vapour usually decreases with
temperature and is zero at the critical point.