Consider the homogeneous nucleation of rain (liquid water) from water vapor at 298 K and atmospheric pressure assuming supersaturation of the water vapor such that  PH2O = 0.1 atm. The surface energy of liquid water in contact with humid (saturated) air at T = 298 K is γ_LV = 7.2 · 10⁻⁶ J/cm².
(a) Calculate ∆GV for the condensation of water vapor to liquid water (i.e., "rain formation") through the reaction H2O(g) → H2O(l) where the standard change in enthalpy, ∆H◦, and standard change in entropy, ∆S◦ are −44.0 kJ/mol and −118.9 J/(mol · K), respectively.
(b) Calculate the critical nucleus size r∗. The radius of a water molecule is about 1.5 ̊A. How many water molecules are clustered within a critical nucleus size r∗?
(c) Calculate the critical free energy change ∆G^∗.