Imagine that particles interact in pairs. If the interaction energy of two particles is 2ε, then half of that (ε) can be associated with each particle. Let εA, εB, εX represent the interaction energy per particle for pairs of As, Bs, and unlike particles, respectively. (For attractive interactions, these energies are negative.) There are N f particles of type B and N(1 − f ) of type A.
(a) Show that the change in potential energy (equal and opposite to the change in thermal energy) is given by
(b) In the common case where attraction between like particles is stronger than that between unlike particles, 2εX − εA − εB > 0, so the answer to part (a) is positive. Potential energy rises and thermal energy falls, and the answer to (a) represents energy that must be added to the system to keep its temperature constant. As can be seen, ∆G = −T ∆Sm − T ∆Si must be concave upward everywhere to avoid a solubility gap. What restriction does this place on the factor 2εX − εA − εB for any value of f?
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(c) Suppose that εA = εB = −0.25 eV and εX = −0.20 eV. At what temperature would A and B be miscible for all f ?