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Chapter 5: 5.13 (page 159)
Use a Maxwell relation from the previous problem and the third law of thermodynamics to prove that the thermal expansion coefficient (defined in Problem 1.7) must be zero at T=0.
Coefficient of Expansion becomes Zero at T=0.
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In Problems 3.30 and 3.31 you calculated the entropies of diamond and graphite at 500 K. Use these values to predict the slope of the graphite- diamond phase boundary at 500 K, and compare to Figure 5.17. Why is the slope almost constant at still higher temperatures? Why is the slope zero at T = 0?
Osmotic pressure measurements can be used to determine the molecular weights of large molecules such as proteins. For a solution of large molecules to qualify as "dilute," its molar concentration must be very low and hence the osmotic pressure can be too small to measure accurately. For this reason, the usual procedure is to measure the osmotic pressure at a variety of concentrations, then extrapolate the results to the limit of zero concentration. Here are some data for the protein hemoglobin dissolved in water at :
| Concentration (grams/liter) | (cm) |
| 5.6 | 2.0 |
| 16.6 | 6.5 |
| 32.5 | 12.8 |
| 43.4 | 17.6 |
| 54.0 | 22.6 |
The quantity is the equilibrium difference in fluid level between the solution and the pure solvent,. From these measurements, determine the approximate molecular weight of hemoglobin (in grams per mole).
An experimental arrangement for measuring osmotic pressure. Solvent flows across the membrane from left to right until the difference in fluid level,, is just enough to supply the osmotic pressure.
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 S fixed, each is taken with V fixed, and N is always held fixed. From the thermodynamic identity (for U ) 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 H, F, and G ). Hold N fixed in all the partial derivatives; other Maxwell relations can be derived by considering partial derivatives with respect to N, 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.
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°C to 40°C. Notice that the vapour pressure approximately doubles for every 10° increase in temperature.
(b) Suppose that the temperature on a certain summer day is 30° C. What is the dew point if the relative humidity is 90%? What if the relative humidity is 40%?
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.
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