Chapter 5.4, Problem 58P

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 $u_0$ be the average potential energy associated with the interaction between neighboring molecules that are the same $\left(A-A\text{or}B-B\right)$, and let $u_{AB}$ be the potential energy associated with the interaction of a neighboring unlike pair $\left(A-B\right)$. There are no interactions beyond the range of the nearest neighbors; the values of $u_0$ 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.

1. Show that when the system is unmixed, the total potential energy due to all neighbor–neighbor interactions is $\frac{1}{2}Nn{u}_0$. (Hint: Be sure to count each neighboring pair only once.)
2. 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.)
3. 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 $x\left(1-x\right)$. Sketch this function vs. x, for both possible signs of $u_{AB}-u_0$.
4. Show that the slope of the mixing energy function is finite at both end–points, unlike the slope of the mixing entropy function.
5. For the case $u_{AB}>u_0$, plot a graph of the Gibb free energy of this system vs. x at several temperatures. Discuss the implications.
6. Find an expression for the maximum temperature at which this system has a solubility gap.
7. Make a very rough estimate of $u_{AB}-u_0$ for a liquid mixture that has a solubility gap below 100°C.
8. Plot the phase diagram (T vs. x) for this system.