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 alike or different. Let n be the average number of nearest neighbors of any given molecule (perhaps 6 or 8 or 10). Let u₀ be the average potential energy associated with the interaction between neighboring molecules that are the same (A-A or B-B), and let u_AB be the potential energy associated with the interaction of a neighboring unlike pair (A-B). There are no interactions beyond the range of nearest neighbors; the value of u₀ and u_AB are independent of the amounts of A and B; and the entropy of mixing is the same as for an ideal solution.

A) Show that when the system is unmixed, the toal potential energy due to all neighbor-neighbor interactions is ½ Nu₀. (Hint: Be sure to count each neighboring pair only once)

B) Find a formula for the total potential energy when the system is mixed, in terms of x, the fraction of B. (Assume that the mixing is totally random)

C) Subtract the results of part a and b to obtain the change in energy upon mixing. Simplify the results as much as possible; you should obtain an expression proportional to x(1-x). Sketch this function vs. x for both possible signs of u_AB-u_o.

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 u_AB>u₀, plot a graph of the Gibbs free energy of this system vs x at several temperatures. Discuss

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

G) Make a rough estimate of u_AB-u_o for a liquid mixture that has a solubility gap below 100 °C.

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

Answer 1