AS of Mixing Flory Theory

All live quantities (AE, d5, AH, AG, and dA, or in integral forms, AE, AS, AH, AG, and AA) are state functions they do not depend on the path. Among the five, Eq. (4.4) is the most important in the study of macromolecules. It is usually put in integrated form  

In the above description, the state is specified by P, V, and T because historically thermodynamics starts with the observation of change in the properties of gases. A gas is described or specified by P, V, and T only. If we deal with solutions (a system of two or more components), we have to introduce one more independent variable, n, the number of moles of a component, which we discuss in later sections. 

We begin our study of macromolecular thermodynamics by discussing these three basic quantities AS, AH, and AG. Our interpretation of the change in a polymeric system is based on the change in the values of AS, AH, and AG. A change in the properties of the polymer always occurs whenever there is a change in the surroundings (environment). In later sections we discuss two more thermodynamic quantities v, the partial specific volume, and p, the chemical potential. Both are related to the behavior of polymer solutions. 

The Flory model for AS of mixing a polymer chain with solvents has been influential in polymer chemistry for several decades. The model assumes the validity of the lattice theory to describe the change in the molecular configuration of the polymer in the presence of a solvent, just as it describes the patterns of the crystal stracture of molecules. The central point is the filling of lattice sites in a three-dimensional space by polymer segments and solvent molecules that is, how many ways can we fill up the lattice sites  

Consider a binary solution consisting of two types of molecules, the polymer chain and the solvent, and neglect the interaction potential existing between polymer segments. Here the meaning of segment is slighly different from that 

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