14Regular Solutions
In this chapter, we learn about thermodynamics of a binary mixture. If you remember such words as a critical temperature and a miscibility gap, you should have learned in physical chemistry the thermodynamics of mixing, at least phenomenologically. Here we apply canonical‐ensemble formulation to a system of an A–B binary mixture in a liquid state. Examples include a water–acetone mixture and a water–pentanol mixture. Unlike a mixture of gas molecules, we cannot add or remove a molecule without changing the volume, that is, the mixture is incompressible. We neglect a small‐volume change upon mixing due to a packing change.
First, we consider a simple mixture of A molecules and B molecules that have an equal size. Subsequently, we consider a binary mixture of different sizes. For both mixtures, we adopt a mean‐field approximation for the interaction between the molecules. This model is called a Hildebrand regular solution.
14.1 Binary Mixture of Equal‐Size Molecules
14.1.1 Free Energy of Mixing
Suppose we mix NA molecules of A and NB molecules of B. Here, NA is not the Avogadro's number. The mixture fills a box consisting of N sites (N = NA + NB), each being occupied by either an A molecule or a B molecule (see Table 14.1). Since the two molecules have an equal size, the mole fraction xj = Nj/N (j = A, B) is identical to the volume fraction.
Table 14.1 Numbers of molecules and mole fractions in an A–B mixture.
| Molecule | Number of molecules | Mole fraction ... |