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Chapter 5: 5.19 (page 163)
In the previous section I derived the formula . Explain why this formula makes intuitive sense, by discussing graphs of F vs. V with different slopes.
The equilibrium condition occurs when the slope of a curve of F verses V is equal.
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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?
Because osmotic pressures can be quite large, you may wonder whether the approximation made in is valid in practice: Is really a linear function of to the required accuracy? Answer this question by discussing whether the derivative of this function changes significantly, over the relevant pressure range, in realistic examples.
The enthalpy and Gibbs free energy, as defined in this section, give special treatment to mechanical (compression-expansion) work, . Analogous quantities can be defined for other kinds of work, for instance, magnetic work." Consider the situation shown in Figure 5.7, where a long solenoid ( turns, total length ) surrounds a magnetic specimen (perhaps a paramagnetic solid). If the magnetic field inside the specimen is and its total magnetic moment is , then we define an auxilliary field (often called simply the magnetic field) by the relation
where is the "permeability of free space," . Assuming cylindrical symmetry, all vectors must point either left or right, so we can drop the symbols and agree that rightward is positive, leftward negative. From Ampere's law, one can also show that when the current in the wire is I, the field inside the solenoid is , whether or not the specimen is present.
(a) Imagine making an infinitesimal change in the current in the wire, resulting in infinitesimal changes in B, M, and . Use Faraday's law to show that the work required (from the power supply) to accomplish this change is . (Neglect the resistance of the wire.)
(b) Rewrite the result of part (a) in terms of and , then subtract off the work that would be required even if the specimen were not present. If we define W, the work done on the system, to be what's left, show that .
(c) What is the thermodynamic identity for this system? (Include magnetic work but not mechanical work or particle flow.)
(d) How would you define analogues of the enthalpy and Gibbs free energy for a magnetic system? (The Helmholtz free energy is defined in the same way as for a mechanical system.) Derive the thermodynamic identities for each of these quantities, and discuss their interpretations.
The density of ice is 917 kg/m*.
(a) Use the Clausius-Clapeyron relation to explain why the slope of the phase boundary between water and ice is negative.
(b) How much pressure would you have to put on an ice cube to make it melt at -1°C?
(c) ApprOximately how deep under a glacier would you have to be before the weight of the ice above gives the pressure you found in part (b)? (Note that the pressure can be greater at some locations, as where the glacier flows over a protruding rock.)
(d) Make a rough estimate of the pressure under the blade of an ice skate, and calculate the melting temperature of ice at this pressure. Some authors have claimed that skaters glide with very little friction because the increased pressure under the blade melts the ice to create a thin layer of water. What do you think of this explanation?
In the previous section I derived the formula . Explain why this formula makes intuitive sense, by discussing graphs of F vs. V with different slopes.
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