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Chapter 9: Q9CQ (page 403)
By considering its constituents, determine the dimensions (e.g. length, distance over lime. etc.) of the denominator in equation. Why is the result sensible?
It follows that the denominator must be dimensionless so that the average energy also has units of energy
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The electrons鈥� contribution to the heat capacity of a metal is small and goes to as . We might try to calculate it via the total internal energy, localid="1660131882505" , but it is one of those integrals impossible to do in closed form, and localid="1660131274621" is the culprit. Still, we can explain why the heat capacity should go to zero and obtain a rough value.
(a) Starting withexpressed as in equation (34), show that the slope atis.
(b) Based on part (a), the accompanying figure is a good approximation to when is small. In a normal gas, such as air, whenis raised a little, all molecules, on average, gain a little energy, proportional to . Thus, the internal energy increases linearly with , and the heat capacity, , is roughly constant. Argue on the basis of the figure that in this fermion gas, as the temperature increases from to a small value , while some particles gain energy of roughly , not all do, and the number doing so is also roughly proportional to localid="1660131824460" . What effect does this have on the heat capacity?
(c)Viewing the total energy increase as simply = (number of particles whose energy increases) (energy change per particle) and assuming the density of states is simply a constant over the entire range of particle energies, show that the heat capacity under these lowest-temperature conditions should be proportional to . (Trying to be more precise is not really worthwhile, for the proportionality constant is subject to several corrections from effects we ignore).
At high temperature, the average energy of a classical one-dimensional oscillator is , and for an atom in a monatomic ideal gas. it is . Explain the difference. using the equipartition theorem.
Somehow you have a two-dimensional solid, a sheet of atoms in a square lattice, each atom linked to its four closest neighbors by four springs oriented along the two perpendicular axes. (a) What would you expect the molar heat capacity to be at very low temperatures and at very high temperatures? (b) What quantity would determine, roughly, the line between low and high?
Show that. using equation, density of statesfollows fromlocalid="1658380849671"
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