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Chapter 5: Q. 5.4 (page 155)
In a hydrogen fuel cell, the steps of the chemical reaction are
Calculate the voltage of the cell. What is the minimum voltage required for electrolysis of water? Explain briefly.
The minimum voltage required is We=1.23 eV
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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-?
Derive a formula, similar to equation 5.90, for the shift in the freezing temperature of a dilute solution. Assume that the solid phase is pure solvent, no solute. You should find that the shift is negative: The freezing temperature of a solution is less than that of the pure solvent. Explain in general terms why the shift should be negative.
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.
Is heat capacity (C) extensive or intensive? What about specific heat (c) ? Explain briefly.
The compression factor of a fluid is defined as the ratio PV/NkT; the deviation of this quantity from 1 is a measure of how much the fluid differs from an ideal gas. Calculate the compression factor of a Van der Waals fluid at the critical point, and note that the value is independent of a and b. (Experimental values of compression factors at the critical point are generally lower than the Van der Waals prediction, for instance, 0.227 for H22O, 0.274 for CO22, and 0.305 for He.)
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