Problem 5.58. 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 behaviour. Consider a mixture of A and B molecules that is ideal in every way but one: The potential energy due to the interaction of neighbouring molecules depends upon whether the molecules are like or unlike. Let n be the average number of nearest neighbours of any given molecule (perhaps 6 or 8 or 10). Let n be the average potential energy associated with the interaction between neighbouring molecules that are the same (4-A or B-B), and let uAB be the potential energy associated with the interaction of a neighbouring unlike pair (4-B). There are no interactions beyond the range of the nearest neighbours; the values of 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 neighbor-neighbor interactions is . (Hint: Be sure to count each neighbouring 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.
(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 this function vs. x, for both possible signs of .
(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 , 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 for a liquid mixture that has a
solubility gap below 100°C.
(h) Use a computer to plot the phase diagram (T vs. x) for this system.
An ideal mixture of A and B molecules is given to us:
n= the average number of nearest neighbours
=average potential energy associated with the interaction between neighbouring molecules that are the same
uAB= the potential energy associated with the interaction of a neighbouring unlike pair
A) If there are N molecules and we count every interaction twice, the total potential energy due to all neighbor-neighbor interactions is given by:
B) Total potential energy when the mixture is mixed
C) Change in energy upon mixing
D) The mixing energy function's slope is:
Hence the function has end points.
E) To plot the graph, we can use a variety of Gibbs free energy values and temperatures. Allow the temperatures to be as follows:
The Gibbs free energy function is:
Deriving it with x and equating to zero,
Finding the value of x
By doing double derivation, we get
Second derivation for maximum point
Now let's set it to zero and determine the maximum temperature expression:
Using these values in temperature term
At T = 144.93 K, a liquid mixture has a solubility gap.
H) Graph for T vs x:
Figure 5.35 (left) shows the free energy curves at one particular temperature for a two-component system that has three possible solid phases (crystal structures), one of essentially pure A, one of essentially pure B, and one of intermediate composition. Draw tangent lines to determine which phases are present at which values of x. To determine qualitatively what happens at other temperatures, you can simply shift the liquid free energy curve up or down (since the entropy of the liquid is larger than that of any solid). Do so, and construct a qualitative phase diagram for this system. You should find two eutectic points. Examples of systems with this behaviour include water + ethylene glycol and tin - magnesium.
A muscle can be thought of as a fuel cell, producing work from the metabolism of glucose:
(a) Use the data at the back of this book to determine the values of and for this reaction, for one mole of glucose. Assume that the reaction takes place at room temperature and atmospheric pressure.
(b) What is maximum amount of work that a muscle can perform , for each mole of glucose consumed, assuming ideal operation?
(c) Still assuming ideal operation, how much heat is absorbed or expelled by the chemicals during the metabolism of a mole of glucose?
(d) Use the concept of entropy to explain why the heat flows in the direction it does?
(e) How would your answers to parts (a) and (b) change, if the operation of the muscle is not ideal?
The compression factor of a fluid is defined as the ratio PV/NkT; the deviation of this quantity from 1 is a measure of how much the fluid differs from an ideal gas. Calculate the compression factor of a Van der Waals fluid at the critical point, and note that the value is independent of a and b. (Experimental values of compression factors at the critical point are generally lower than the Van der Waals prediction, for instance, 0.227 for H22O, 0.274 for CO22, and 0.305 for He.)
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