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 u_{0} 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 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.

A) Show that when the system is unmixed, the toal potential
energy due to all neighbor-neighbor interactions is ½
Nu_{0}. (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}>u_{0}, 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

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...

Can you please check my answers and tell me if they are
correct? thanks
8. Write the ionic equation for dissolution and the solubility
product (Ksp) expression for each of the following slightly soluble
ionic compounds:
(a) PbCl2
PbCl2(s) --> Pb2+(aq) +
2Cl-(aq)
Ksp
= [Pb2+][Cl-]2
(b) Ag2S
Ag2S(s) --> 2Ag+(aq)
+
S2-(aq) Ksp
=[Ag+]2[S2-]
(c) Sr3(PO4)2 Sr3(PO4)2(s)
--> 3Sr2+(aq) +
2PO43-(aq) Ksp
=[Sr2+]3[PO43-]2
(d) SrSO4 SrSO4(s)
--> Sr2+(aq) +
SO42-(aq) Ksp
=[Sr2+][SO42-]
14. Assuming that no equilibria other than dissolution are
involved,...

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