Abstract: The canonical partition functions of polycondensation of A_a type were constructed from two viewpoints by the principle of statistical mechanics, and the explicit expressions of the equilibrium free energy and the law of mass action were obtained. Meanwhile, two new methods were used to derive the number fraction distribution function. Below the critical point of the sol-gel transition, the equilibrium free energy of sol and gel phases were given on the basis of the intrinsic invariant property of the number faction distribution function. Furthermore, the sol-gel phase transition is shown to be a kind of geometric phase transition rather than thermodynamic phase transition.

Key words: Equilibrium free energy, Law of mass action, Sol-gel phase transition