Chemical potential homogeneous nucleation of water-vapor condensation

8 Chemical potential homogeneous nucleation of water-vapor condensation 

Exercise 2.8. Show that when a plane surface of a liquid is in equilibrium with its vapor, the chemical potentials in the liquid and vapor phases are the same. 

Solution. If a liquid is in equilibrium with its vapor, molecules may evaporate and condense without the temperature or pressure of the system changing. Hence, from Eq. (2.35), dg = 0. But, if g does not change when a molecule passes from the liquid to the vapor phase (or vice versa), the Gibbs free energy per molecule (i.e., the chemical potential) must be the same in the liquid phase as in the vapor phase. The pressure exerted by the vapor under these equilibrium conditions is called the saturation vapor pressure it depends only on the substance being considered and its temperature. 

We will now derive an expression for the difference in the chemical potentials of a vapor and its liquid at an arbitrary partial pressure e and temperature T. Applying Eq. (2.40) to one molecule of the vapor, we obtain 

In the region just around / = r, a droplet can grow or evaporate infinitesimally without any change in the energy of the system. We can obtain an expression for r in terms of e by setting d E) dR = 0 at R = r. Hence, from Eq. (2.54) 

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