Question

In constructing the phase diagram from the free energy graphs in Figure 5.30, I assumed that both the liquid and the gas are ideal mixtures. Suppose instead that the liquid has a substantial positive mixing energy, so that its free energy curve, while still concave-up, is much flatter. In this case a portion of the curve may still lie above the gas's free energy curve at T_A. Draw a qualitatively accurate phase diagram for such a system, showing how you obtained the phase diagram from the free energy graphs. Show that there is a particular composition at which this gas mixture will condense with no change in composition. This special composition is called an azeotrope.

Short answer

The entropy of a gas increases as the temperature rises; because the gas has more degrees of freedom, it has more entropy.

As a result of the negative sign of entropy, the gas curve falls. Lowering the temperature causes the curve to rise until it coincides with the liquid curve at one point, forming an azeotrope combination.

Step-by-step solution

Given information

In constructing the phase diagram from the free energy graphs in Figure 5.30, I assumed that both the liquid and the gas are ideal mixtures. Suppose instead that the liquid has a substantial positive mixing energy, so that its free energy curve, while still concave-up, is much flatter. In this case a portion of the curve may still lie above the gas's free energy curve at T_A.

Explanation

Consider the following curve, which shows the free energy of the gas and liquid at temperature TA. We can see that the gas curve is more concave than the liquid curve, indicating that the two curves intersect at two points, indicating that the liquid and gas are stable in two different composition ranges.

Explanation

Draw a tangent on a graph between x and T (the phase diagram) at the two intersection locations as indicated in the accompanying figure; this tangent intersects with the gas and liquid curves. Then draw perpendicular lines from the four intersection points on a graph between x and T (the phase diagram).

Explantion

The Gibbs free energy is given by:

$G=U+PV-TS$

At constant volume and entropy, the change in Gibbs free energy is as follows:

$dG=dU+VdP-SdT$

By increasing the temperature, we get

$\frac{\partial G}{\partial T}=-S$

The entropy of a gas increases as the temperature rises; because the gas has more degrees of freedom, it has more entropy.

As a result of the negative sign of entropy, the gas curve falls. Lowering the temperature causes the curve to rise until it coincides with the liquid curve at one point, forming an azeotrope combination.