I'm only stuck on question 9. I got w=at^4/12 but I didn't use many of the formulas given in the prob so I don't know if I used the right approach. Do I need to take the integral of at?

ording to aQ, = - MCd7,, where & is the specific heat of the rock, assumed to be temperature independent. If the plant operates at the Carnot limit, calculate the total amount IV of electrical energy extractable from the rock, if the temperature of the rock was initially T, - 7,, and if the plant is to be shut down when the temperature has dropped to T. = Ty. Assume that the lower reservoir temperature 7, stays constant. At the end of the calculation, give a numerical value, in kWh. for A/ = 10utkg (about 30 km'). C = 1 3 g ' K ', T, = 600 C, 7, = 110 C, 7, = 20 C. Watch the units and explain all steps! For comparison: The toval electricity produced in the world in 1976 was between I and 2 times 10"* k Why. 9. Cooling of nonmetallic solid to T = 0. We saw in Chapter 4 that the heat capacity of nonmetallic solids at sufficiently low temperatures is proportional to T', as C = aT'. Assume it were possible to cool a piece of such a solid to T = 0 by means of a reversible refrigerator that uses the solid specimen as its (varying!) low-temperature reservoir, and for which the high-temperature reservoir has a fixed temperature 7, equal to the initial temperature I, of the solid. Find an expression for the electrical energy required. 10. Itreversible expansion of a Fermi gas. Consider a gas of N noninteracting, spin | fermions of mass A, initially in a volume , at temperature z, = 0. Let the gas expand irreversibly into a vacuum, without doing work, to a final volume Vy. What is the temperature of the gas after expansion if , is sufficiently large for the classical limit to apply? Estimate the factor by which the gas should be expanded for its temperature to settle to a constant final value. Give numerical values for the final temperature in kelvin for two cases: (a) a particle mass equal to the electron mass, and N/V - 1022 cm 3, as in metals; (b) a particle mass equal to a nucicon, and N/V = 1010, as in white dwarf stars