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Chapter 5: Q 5.63 (page 195)
Everything in this section assumes that the total pressure of the system is fixed. How would you expect the nitrogen-oxygen phase diagram to change if you increase or decrease the pressure? Justify your answer.
The phase area will get smaller.
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For a magnetic system held at constant TT and HH (see Problem 5.17 ), the quantity that is minimized is the magnetic analogue of the Gibbs free energy, which obeys the thermodynamic identity
Phase diagrams for two magnetic systems are shown in Figure 5.14 ; the vertical axis on each of these figures is μ0Hμ0H (a) Derive an analogue of the Clausius-Clapeyron relation for the slope of a phase boundary in the HH - TT plane. Write your equation in terms of the difference in entropy between the two phases. (b) Discuss the application of your equation to the ferromagnet phase diagram in Figure 5.14. (c) In a type-I superconductor, surface currents flow in such a way as to completely cancel the magnetic field (B, not H)(B, not H) inside. Assuming that MM is negligible when the material is in its normal (non-superconducting) state, discuss the application of your equation to the superconductor phase diagram in Figure 5.14.5.14. Which phase has the greater entropy? What happens to the difference in entropy between the phases at each end of the phase boundary?
By subtracting from localid="1648229964064" ,or,one can obtain four new thermodynamic potentials. Of the four, the most useful is the grand free energy (or grand potential),
(a) Derive the thermodynamic identity for , and the related formulas for the partial derivatives ofwith respect to, and
(b) Prove that, for a system in thermal and diffusive equilibrium (with a reservoir that can supply both energy and particles), tends to decrease.
(c) Prove that
(d) As a simple application, let the system be a single proton, which can be "occupied" either by a single electron (making a hydrogen atom, with energy ) or by none (with energy zero). Neglect the excited states of the atom and the two spin states of the electron, so that both the occupied and unoccupied states of the proton have zero entropy. Suppose that this proton is in the atmosphere of the sun, a reservoir with a temperature of and an electron concentration of about per cubic meter. Calculate for both the occupied and unoccupied states, to determine which is more stable under these conditions. To compute the chemical potential of the electrons, treat them as an ideal gas. At about what temperature would the occupied and unoccupied states be equally stable, for this value of the electron concentration? (As in Problem 5.20, the prediction for such a small system is only a probabilistic one.)
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).
Consider a completely miscible two-component system whose overall composition is x, at a temperature where liquid and gas phases coexist. The composition of the gas phase at this temperature is and the composition of the liquid phase is . Prove the lever rule, which says that the proportion of liquid to gas is . Interpret this rule graphically on a phase diagram.
Functions encountered in physics are generally well enough behaved that their mixed partial derivatives do not depend on which derivative is taken first. Therefore, for instance,
where each is taken with fixed, each is taken with fixed, and is always held fixed. From the thermodynamic identity (for) you can evaluate the partial derivatives in parentheses to obtain
a nontrivial identity called a Maxwell relation. Go through the derivation of this relation step by step. Then derive an analogous Maxwell relation from each of the other three thermodynamic identities discussed in the text (for ). Hold fixed in all the partial derivatives; other Maxwell relations can be derived by considering partial derivatives with respect to , but after you've done four of them the novelty begins to wear off. For applications of these Maxwell relations, see the next four problems.
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