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Chapter 5: Q.5.80 (page 208)
Use the Clausius-Clapeyron relation to derive equation 5.90 directly from Raoult's law. Be sure to explain the logic carefully.
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Sulfuric acid,
The hydrogen sulfate ion, in turn, can dissociate again:
The equilibrium constants for these reactions, in aqueous solutions at 298 K, are approximately 10 and 10*, respectively. (For dissociation of acids it is usually more convenient to look up K than . By the way, the negative base-10 logarithm of K for such a reaction is called pK, in analogy to pH. So for the first reaction pK = -2, while for the second reaction pK = 1.9.)
(a) Argue that the first reaction tends so strongly to the right that we might as well consider it to have gone to completion, in any solution that could possibly be considered dilute. At what pH values would a significant fraction of the sulfuric acid not be dissociated?
(b) In industrialized regions where lots of coal is burned, the concentration of sulfate in rainwater is typically 5 x 10 mol/kg. The sulfate can take any of the chemical forms mentioned above. Show that, at this concentration, the second reaction will also have gone essentially to completion, so all the sulfate is in the form of SOg. What is the pH of this rainwater?
(c) Explain why you can neglect dissociation of water into H* and OH in answering the previous question. (d) At what pH would dissolved sulfate be equally distributed between HSO and SO2-?
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.
Seawater has a salinity of , meaning that if you boil away a kilogram of seawater, when you're finished you'll have of solids (mostly localid="1647507373105" ) left in the pot. When dissolved, sodium chloride dissociates into separate and ions.
(a) Calculate the osmotic pressure difference between seawater and fresh water. Assume for simplicity that all the dissolved salts in seawater are .
(b) If you apply a pressure difference greater than the osmotic pressure to a solution separated from pure solvent by a semipermeable membrane, you get reverse osmosis: a flow of solvent out of the solution. This process can be used to desalinate seawater. Calculate the minimum work required to desalinate one liter of seawater. Discuss some reasons why the actual work required would be greater than the minimum.
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.
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.
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