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Chapter 5: Q. 5.21 (page 163)
Is heat capacity (C) extensive or intensive? What about specific heat (c) ? Explain briefly.
Heat capacity is an extensive property
Specific heat is an intensive property.
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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掳C.
(h) Use a computer to plot the phase diagram (T vs. x) for this system.
How can diamond ever be more stable than graphite, when it has
less entropy? Explain how at high pressures the conversion of graphite to diamond
can increase the total entropy of the carbon plus its environment.
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%?
A mixture of one part nitrogen and three parts hydrogen is heated, in the presence of a suitable catalyst, to a temperature of 500掳 C. What fraction of the nitrogen (atom for atom) is converted to ammonia, if the final total pressure is 400 atm? Pretend for simplicity that the gases behave ideally despite the very high pressure. The equilibrium constant at 500掳 C is 6.9 x 10-5. (Hint: You'l have to solve a quadratic equation.)
Use the result of the previous problem to calculate the freezing temperature of seawater.
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