The Gibbs Equation versus the Kelvin and the Gibbs-Thomson Equations to Describe Nucleation and Equilibrium of Nano-Materials

The Kelvin equation, the Gibbs equation and the Gibbs-Thomson equation are compared. It is shown that the Kelvin equation (on equilibrium vapor pressure above nano-droplets) can be derived if the inner pressure due to the curvature (from the Laplace equation) is substituted incorrectly into the external pressure term of the Gibbs equation. Thus, the Kelvin equation is excluded in its present form. The Gibbs-Thomson equation (on so-called equilibrium melting point of a nano-crystal) is an analog of the Kelvin equation, and thus it is also excluded in its present form. The contradiction between the critical nucleus size (from the Gibbs equation) and the so-called equilibrium melting point of nano-crystals (from the Gibbs-Thomson equation) is explained. The contradiction is resolved if the Gibbs equation is applied to study both nucleation and equilibrium of nano-crystals. Thus, the difference in the behavior of nano-systems compared to macro-systems is due to their high specific surface area (Gibbs) and not to the high curvature of their interface (Kelvin). Modified versions of the Kelvin equation and the Gibbs-Thomson equation are derived from the Gibbs equation for phases with a general shape and for a spherical phase.