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Chapter 3: Q. 3.9 (page 95)
In solid carbon monoxide, each CO molecule has two possible orientations: CO or OC. Assuming that these orientations are completely random (not quite true but close), calculate the residual entropy of a mole of carbon monoxide.
The residual entropy for a carbon monoxide is calculated to be.
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In Problem 2.18 you showed that the multiplicity of an Einstein solid containing N oscillators and q energy units is approximately
(a) Starting with this formula, find an expression for the entropy of an Einstein solid as a function of N and q. Explain why the factors omitted from the formula have no effect on the entropy, when N and q are large.
(b) Use the result of part (a) to calculate the temperature of an Einstein solid as a function of its energy. (The energy is , where is a constant.) Be sure to simplify your result as much as possible.
(c) Invert the relation you found in part (b) to find the energy as a function of temperature, then differentiate to find a formula for the heat capacity.
(d) Show that, in the limit , the heat capacity is . (Hint: When x is very small, .) Is this the result you would expect? Explain.
(e) Make a graph (possibly using a computer) of the result of part (c). To avoid awkward numerical factors, plot vs. the dimensionless variable , for t in the range from 0 to about 2. Discuss your prediction for the heat capacity at low temperature, comparing to the data for lead, aluminum, and diamond shown in Figure 1.14. Estimate the value of , in electron-volts, for each of those real solids.
(f) Derive a more accurate approximation for the heat capacity at high temperatures, by keeping terms through in the expansions of the exponentials and then carefully expanding the denominator and multiplying everything out. Throw away terms that will be smaller than in the final answer. When the smoke clears, you should find .
Figure 3.3 shows graphs of entropy vs. energy for two objects, A and B. Both graphs are on the same scale. The energies of these two objects initially have the values indicated; the objects are then brought into thermal contact with each other. Explain what happens subsequently and why, without using the word "temperature."
Consider an Einstein solid for which both N and q are much greater than . Think of each oscillator as a separate "particle."
(a) Show that the chemical potential is
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(b) Discuss this result in the limits and , concentrating on the question of how much increases when another particle carrying no energy is added to the system. Does the formula make intuitive sense?
A cylinder contains one liter of air at room temperature ( ) and atmospheric pressure . At one end of the cylinder is a massless piston, whose surface area is . Suppose that you push the piston in very suddenly, exerting a force of . The piston moves only one millimeter, before it is stopped by an immovable barrier of some sort.
(a) How much work have you done on this system?
(b) How much heat has been added to the gas?
(c) Assuming that all the energy added goes into the gas (not the piston or cylinder walls), by how much does the internal energy of the gas increase?
(d) Use the thermodynamic identity to calculate the change in the entropy of the gas (once it has again reached equilibrium).
Consider an ideal two-state electronic paramagnet such as DPPH, with . In the experiment described above, the magnetic field strength was and the minimum temperature was . Calculate the energy, magnetization, and entropy of this system, expressing each quantity as a fraction of its maximum possible value. What would the experimenters have had to do to attain of the maximum possible magnetization?
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