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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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The methods of this section can also be applied to reactions in which one set of solids converts to another. A geologically important example is the transformation of albite into jadeite + quartz:
Use the data at the back of this book to determine the temperatures and pressures under which a combination of jadeite and quartz is more stable than albite. Sketch the phase diagram of this system. For simplicity, neglect the temperature and pressure dependence of both S and V.
Ordinarily, the partial pressure of water vapour in the air is less than the equilibrium vapour pressure at the ambient temperature; this is why a cup of water will spontaneously evaporate. The ratio of the partial pressure of water vapour to the equilibrium vapour pressure is called the relative humidity. When the relative humidity is 100%, so that water vapour in the atmosphere would be in diffusive equilibrium with a cup of liquid water, we say that the air is saturated. The dew point is the temperature at which the relative humidity would be 100%, for a given partial pressure of water vapour.
(a) Use the vapour pressure equation (Problem 5.35) and the data in Figure 5.11 to plot a graph of the vapour pressure of water from 0掳C to 40掳C. Notice that the vapour pressure approximately doubles for every 10掳 increase in temperature.
(b) Suppose that the temperature on a certain summer day is 30掳 C. What is the dew point if the relative humidity is 90%? What if the relative humidity is 40%?
Effect of altitude on boiling water.
(a) Use the result of the previous problem and the data in Figure 5.11 to plot a graph of the vapor pressure of water between and . How well can you match the data at the two endpoints?
(b) Reading the graph backward, estimate the boiling temperature of water at each of the locations for which you determined the pressure in Problem 1.16. Explain why it takes longer to cook noodles when you're camping in the mountains.
(c) Show that the dependence of boiling temperature on altitude is very nearly (though not exactly) a linear function, and calculate the slope in degrees Celsius per thousand feet (or in degrees Celsius per kilometer).
Use the data at the back of this book to verify the values of and quoted above for the lead-acid reaction 5.13.
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?
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