The formal definition of the Gibbs free energy for a parcel of volume , pressure  and temperature  is given by:
where  is the internal energy and  is the entropy. The differential form of the Gibbs free energy can be given as
where  is the chemical potential and  is the number of moles. Suppose we have a substance  which contains no impurities. Let's consider the formation of a single drop of  with radius  containing  molecules from its pure vapor. The change in the Gibbs free energy due to this process is
where  and  are the Gibbs energies of the drop and vapor respectively. Suppose we have  molecules in the vapor phase initially. After the formation of the drop, this number decreases to , where
Let  and  represent the Gibbs free energy of a molecule in the vapor and liquid phase respectively. The change in the Gibbs free energy is then:
where  is the Gibbs free energy associated with an interface with radius of curvature  and surface tension . The equation can be rearranged to give
Let  and  be the volume occupied by one molecule in the liquid phase and vapor phase respectively. If the drop is considered to be spherical, then
The number of molecules in the drop is then given by
The change in Gibbs energy is then
The differential form of the Gibbs free energy of one molecule at constant temperature and constant number of molecules can be given by:
If we assume that  then
The vapor phase is also assumed to behave like an ideal gas, so
where  is the Boltzmann constant. Thus, the change in the Gibbs free energy for one molecule is
where  is the saturated vapor pressure of  over a flat surface and  is the actual vapor pressure over the liquid. Solving the integral, we have
The change in the Gibbs free energy following the formation of the drop is then
The derivative of this equation with respect to  is
The maximum value occurs when the derivative equals zero. The radius corresponding to this value is:
Rearranging this equation gives the Ostwald–Freundlich form of the Kelvin equation: