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Chapter 5: Q 5.82 (page 208)
Use the result of the previous problem to calculate the freezing temperature of seawater.
Therefore, the freezing temperature of seawater is
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Problem 5.64. Figure 5.32 shows the phase diagram of plagioclase feldspar, which can be considered a mixture of albite and anorthite
a) Suppose you discover a rock in which each plagioclase crystal varies in composition from center to edge, with the centers of the largest crystals composed of 70% anorthite and the outermost parts of all crystals made of essentially pure albite. Explain in some detail how this variation might arise. What was the composition of the liquid magma from which the rock formed?
(b) Suppose you discover another rock body in which the crystals near the top are albite-rich while the crystals near the bottom are anorthite-rich. Explain how this variation might arise.
Suppose you have a box of atomic hydrogen, initially at room temperature and atmospheric pressure. You then raise the temperature, keeping the volume fixed.
(a) Find an expression for the fraction of the hydrogen that is ionised as a function of temperature. (You'll have to solve a quadratic equation.) Check that your expression has the expected behaviour at very low and very high temperatures.
(b) At what temperature is exactly half of the hydrogen ionised?
(c) Would raising the initial pressure cause the temperature you found in part (b) to increase or decrease? Explain.
(d) Plot the expression you found in part (a) as a function of the dimension- less variable t = kT/I. Choose the range of t values to clearly show the interesting part of the graph.
Go through the arithmetic to verify that diamond becomes more stable than graphite at approximately 15 kbar.
In this problem you will derive approximate formulas for the shapes of the phase boundary curves in diagrams such as Figures 5.31 and 5.32, assuming that both phases behave as ideal mixtures. For definiteness, suppose that the phases are liquid and gas.
(a) Show that in an ideal mixture of A and B, the chemical potential of species A can be written where A is the chemical potential of pure A (at the same temperature and pressure) and . Derive a similar formula for the chemical potential of species B. Note that both formulas can be written for either the liquid phase or the gas phase.
(b) At any given temperature T, let x1 and xgbe the compositions of the liquid and gas phases that are in equilibrium with each other. By setting the appropriate chemical potentials equal to each other, show that x1and xg obey the equations = and where represents the change in G for the pure substance undergoing the phase change at temperature T.
(c) Over a limited range of temperatures, we can often assume that the main temperature dependence of comes from the explicit T; both are approximately constant. With this simplification, rewrite the results of part (b) entirely in terms of TA, and TB (eliminating ). Solve for x1and xgas functions of T.
(d) Plot your results for the nitrogen-oxygen system. The latent heats of the pure substances are. Compare to the experimental diagram, Figure 5.31.
(e) Show that you can account for the shape of Figure 5.32 with suitably chosen values. What are those values?
Most pasta recipes instruct you to add a teaspoon of salt to a pot
of boiling water. Does this have a significant effect on the boiling temperature?
Justify your answer with a rough numerical estimate.
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