Consider a two-component solution of \(N_{A}\) atoms of type \(A\) and \(N_{B}\) atoms of type \(B\), which are supposed to be randomly distributed over \(N\left(=N_{A}+N_{B}\right)\) sites of a single lattice. Denoting the energies of the nearest-neighbor pairs \(A A, B B\), and \(A B\) by \(\varepsilon_{11}, \varepsilon_{22}\), and \(\varepsilon_{12}\), respectively, write down the free energy of the system in the Bragg-Williams approximation and evaluate the chemical potentials \(\mu_{A}\) and \(\mu_{B}\) of the two components. Next, show that if \(\varepsilon=\left(\varepsilon_{11}+\varepsilon_{22}-2 \varepsilon_{12}\right)<0\), that is, if the atoms of the same species display greater affinity to be neighborly, then for temperatures below a critical temperature \(T_{c}\), which is given by the expression \(q|\varepsilon| / 2 k\), the solution separates out into two phases of unequal relative concentrations.

[For a study of phase separation in an isotopic mixture of hard-sphere bosons and fermions, and for the relevance of this study to the actual behavior of \(\mathrm{He}^{3}-\mathrm{He}^{4}\) solutions, see Cohen and van Leeuwen \((1960,1961)\).]