A Lattice Model Describes Mixtures

In Chapter 14 we considered pure liquids or solids composed of a single chemical species. Here we consider solutions, mixtures of more than one component. (A solution is a mixture that is homogeneous.) The fundamental result that we derive in this chapter, = ° + kT In yx, is a relationship between the chemical potential p and the concentration x of one of the components in the solution. This relationship will help address questions in Chapters 16, and 25 to 30 When does one chemical species dissolve in another When is it insoluble How do solutes lower the freezing point of a liquid, elevate the boiling temperature, and cause osmotic pressure What forces drive molecules to partition differently into different phases We continue with the lattice model because it gives simple insights and because it gives the foundation for treatments of polymers, colloids, and biomolecules. 

We use the (T, V, N) ensemble, rather than (T, p,N), because it allows us to work with the simplest possible lattice model that captures the principles of solution theory. The appropriate extremum principle is based on the Helmholtz free energy, F = U - TS, where 5 is the entropy of mixing and U accounts for the interaction energies between the lattice particles. 

Suppose there are Na molecules of species A, and Nb molecules of species B. Particles of A and B are the same size—each occupies one lattice site—and 

The multiplicity of states is the number of spatial arrangements of the molecules  

The translational entropy of the mixed system can be computed by using the Boltzmann Equation (6.1), S = kin IF, and Stirling s approximation. Equation (4.26)  

---