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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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Suppose you start with a liquid mixture of 60% nitrogen and 40% oxygen. Describe what happens as the temperature of this mixture increases. Be sure to give the temperatures and compositions at which boiling begins and ends.
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?
Suppose you have a box of atomic hydrogen, initially at room temperature and atmospheric pressure. You then raise the temperature, keeping the volume fixed.
(a) Find an expression for the fraction of the hydrogen that is ionised as a function of temperature. (You'll have to solve a quadratic equation.) Check that your expression has the expected behaviour at very low and very high temperatures.
(b) At what temperature is exactly half of the hydrogen ionised?
(c) Would raising the initial pressure cause the temperature you found in part (b) to increase or decrease? Explain.
(d) Plot the expression you found in part (a) as a function of the dimension- less variable t = kT/I. Choose the range of t values to clearly show the interesting part of the graph.
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.
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?
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