Question

In this problem you will model the mixing energy of a mixture in a relatively simple way, in order to relate the existence of a solubility gap to molecular behavior. Consider a mixture of A and B molecules that is ideal in every way but one: The potential energy due to the interaction of neighboring molecules depends upon whether the molecules are like or unlike. Let n be the average number of nearest neighbors of any given molecule (perhaps 6 or 8 or 10). Let  ?₀ be the average potential energy associated with the interaction between neighboring molecules that are the same (A-A or B-B), and let  ?_AB be the potential energy associated with the interaction of a neighboring unlike pair (A-B). There are no interactions beyond the range of the nearest neighbors; the values of  ?₀ and  ?_AB are independent of the amounts of A and B; and the entropy of mixing is the same as for an ideal solution.

(d) Show that the slope of the mixing energy function is finite at both end-points, unlike the slope of the mixing entropy function.

(e) For the case  ?_AB >  ?₀ , plot a graph of the Gibbs free energy of this system vs. x at several temperatures. Discuss the implications.

(f) Find an expression for the maximum temperature at which this system has a solubility gap.

(g) Make a very rough estimate of  ?_AB - ?₀ for a liquid mixture that has a solubility gap below 100 degrees Celsius.

(h) Plot the phase diagram (T vs. x) for this system.

Answer 1