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Chapter 5: Q 5.25 (page 171)
In working high-pressure geochemistry problems it is usually more
convenient to express volumes in units of kJ/kbar. Work out the conversion factor
between this unit and m3
The conversion factor is
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The formula for Cp-Cv derived in the previous problem can also be derived starting with the definitions of these quantities in terms of U and H. Do so. Most of the derivation is very similar, but at one point you need to use the relation .
The partial-derivative relations derived in Problems 1.46,3.33, and 5.12, plus a bit more partial-derivative trickery, can be used to derive a completely general relation between and.
(a) With the heat capacity expressions from Problem 3.33 in mind, first considerto be a function of andExpand in terms of the partial derivatives and . Note that one of these derivatives is related to
(b) To bring in , considerlocalid="1648430264419" to be a function ofand P and expand dV in terms of partial derivatives in a similar way. Plug this expression for dV into the result of part (a), then set and note that you have derived a nontrivial expression for . This derivative is related to , so you now have a formula for the difference
(c) Write the remaining partial derivatives in terms of measurable quantities using a Maxwell relation and the result of Problem 1.46. Your final result should be
(d) Check that this formula gives the correct value of for an ideal gas.
(e) Use this formula to argue that cannot be less than .
(f) Use the data in Problem 1.46 to evaluatefor water and for mercury at room temperature. By what percentage do the two heat capacities differ?
(g) Figure 1.14 shows measured values of for three elemental solids, compared to predicted values of . It turns out that a graph of vs.T for a solid has same general appearance as a graph of heat capacity. Use this fact to explain why and agree at low temperatures but diverge in the way they do at higher temperatures.
Use the result of the previous problem to estimate the equilibrium constant of the reactionat 500° C, using only the room- temperature data at the back of this book. Compare your result to the actual value of K at 500° C quoted in the text.
Let the system be one mole of argon gas at room temperature and atmospheric pressure. Compute the total energy (kinetic only, neglecting atomic rest energies), entropy, enthalpy, Helmholtz free energy, and Gibbs free energy. Express all answers in SI units.
A muscle can be thought of as a fuel cell, producing work from the metabolism of glucose:
(a) Use the data at the back of this book to determine the values of and for this reaction, for one mole of glucose. Assume that the reaction takes place at room temperature and atmospheric pressure.
(b) What is maximum amount of work that a muscle can perform , for each mole of glucose consumed, assuming ideal operation?
(c) Still assuming ideal operation, how much heat is absorbed or expelled by the chemicals during the metabolism of a mole of glucose?
(d) Use the concept of entropy to explain why the heat flows in the direction it does?
(e) How would your answers to parts (a) and (b) change, if the operation of the muscle is not ideal?
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