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  μ₀be the average potential energy associated with the interaction between neighboring molecules that are the same (A-A or B-B), and let  μ_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  μ₀ and  μ_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 total potential energy due to all the neighbor-neighbor interactions is (1/2)Nnμ₀. (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 parts (a) and (b) to obtain the change in energy upon mixing. Simplify the result as much as possible; you should obtain an expression proportional to x(1-x). Sketch the function vs. x for both possible signs of μ_AB -  μ₀.

(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  μ_AB >  μ₀ , 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  μ_AB - μ₀ for a liquid mixture that has a solubility gap below 100 degrees Celsius.

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

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 μ₀be the average potential energy associated with the interaction between neighboring molecules that are the same (A-A or B-B), and let μ_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 μ₀ and μ_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 total potential energy due to all the neighbor-neighbor interactions is (1/2)Nnμ₀. (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 parts (a) and (b) to obtain the change in energy upon mixing. Simplify the result as much as possible; you should obtain an expression proportional to x(1-x). Sketch the function vs. x for both possible signs of μ_AB - μ₀.

(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 μ_AB > μ₀ , 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 μ_AB - μ₀ for a liquid mixture that has a solubility gap below 100 degrees Celsius.

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

Accepted Answer

As per rule, allowed to answer first three subparts and post the remaining in the next submission.

e the average potential energy associated with the interaction between neighboring molecules that are the same (A-A or B-B), and let μAB be the potential energy associated with the interaction of a neighboring unlike pair (A-B)