Homogeneous nucleation during a phase transition

The nucleation and growth of liquid water drops from water vapor is a very important process in nature and also in industry. If we assume that the water obeys the ideal gas law, the difference in the Gibbs energies to form a liquid water drop in its vapor phase by the phase change process can be written as 

Since Pv P°, the higher the super-saturation ratio, the higher the AG value. The number of moles, n = m/Mw where m is the mass of the droplet, Mw is the molecular mass (m = AnPpJS), r is the radius of the droplet and pL is the density of the liquid. The value of n is very small because n represents the number of moles of vapor that condense into a tiny nucleus (liquid droplet) at the vapor pressure, Pv. Then, we can write 

In order to find the total free energy difference to form the drop, AGT, we also need to consider the surface free energy effects  

Equation (356) is identical to the Kelvin equation (Equation (347)) for saturation vapor pressures (Pv = P°) at r = rc. The P v parameter is the critical vapor pressure, which corresponds to the vapor pressure when the drop radius, r equals rc, which is also the radius of curvature for a spherical drop having the critical size. We should note that Equation (356) is valid when two phases coexist in equilibrium. 

Equation (356) maybe applied to the equilibrium solubility of a solid in a liquid. In this case the ratio (P%IP°W) is replaced by the ratio ( / ), where a° is the activity of the dissolved solute in equilibrium with a flat surface, and a] is the corresponding quantity for a spherical surface. Then we may write 

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