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Chapter 12: Q4Q (page 507)
Explain why it is a disadvantage for some purposes that the specific heat of all materials decreases a low temperature.
The temperature increases very fast when adding to the specific heat substance.
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Consider an object containing 6 one-dimensional oscillators (this object could represent a model of 2 atoms in an Einstein solid). There are 4 quanta of vibrational energy in the object. (a) How many microstates are there, all with the same energy? (b) If you examined a collection of 48,000 objects of this kind, each containing 4 quanta of energy, about how many of these objects would you expect to find in the microstate 000004?
For a certain metal the stiffness of the interatomic bond and the mass of one atom are such that the spacing of the quantum oscillator energy levels is J. A nanoparticle of this metal consisting of atoms has a total thermal energy of role="math" localid="1657867970311" J. (a) What is the entropy of this nanoparticle? (b) The temperature of the nanoparticle is 87 K. Next we add J to the nanoparticle. By how much does the entropy increase?
Young鈥檚 modulus for copper is measured by stretching a copper wire to be about . The density of copper is about , and the mass of a mole is .Starting from a very low temperature, use these data to estimate roughly the temperature T at which we expect the specific heat for copper to approach 3 kB . Compare your estimate with the data shown on a graph in this chapter.
In Chapter 4 you determined the stiffness of the interatomic 鈥渟pring鈥� (chemical bond) between atoms in a block of lead to be 5 N/m, based on the value of Young鈥檚 modulus for lead. Since in our model each atom is connected to two springs, each half the length of the interatomic bond, the effective 鈥渋nteratomic spring stiffness鈥� for an oscillator is
4 脳 5 N/m = 20 N/m. The mass of one mole of lead is 207 g (0.207 kg). What is the energy, in joules, of one quantum of energy for an atomic oscillator in a block of lead?
Buckminsterfullerene, C60, is a large molecule consisting of 60 carbon atoms connected to form a hollow sphere. The diameter of a C60 molecule is about 7脳10-10 m. It has been hypothesized that C60 molecules might be found in clouds of interstellar dust, which often contain interesting chemical compounds. The temperature of an interstellar dust cloud may be very low; around 3 K. Suppose you are planning to try to detect the presence of C60 in such a cold dust cloud by detecting photons emitted when molecules undergo transitions from one rotational energy state to another. Approximately, what is the highest-numbered rotational level from which you would expect to observe emissions? Rotational levels are l= 0, 1, 2, 3, 鈥�
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