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Chapter 9: Q32E (page 404)
Using the relationship between temperature and and given in (9-16) and that between and in (9-6), obtain equation (9-17) from (9- 12). The first sum given In Exercise 30 will be useful.
The expression for is
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The exact probabilities of equation (9-9) rest on the claim that the number of ways of addingdistinct non-negative integer to give a total of is . One way to prove it involves the following trick. It represents two ways that distinct integers can add to and, respectively. In this special case.
The X's represent the total of the integers, -each row has . The represent "dividers" between the distinct integers of which there will of course be each row has . The first row says that is (three before the divider between it and ), is (no between its left divider with and its right divider with ), ) is . through are , is , and and are . The second row says that is . is , is , and all other are . Further rows could account for all possible ways that the integers can add to . Argue that properly applied, the binomial coefficient (discussed in Appendix ) can be invoked to give the correct total number of ways for any and .
For a roomtall, by roughly what percent does the probability of an air molecule being found at the ceiling differ from that of an equal speed molecule being found at the floor? Ignore any variation in temperature from floor to ceiling.
Discusses the energy balance in a white dwarf. The tendency to contract due to gravitational attraction is balanced by a kind of incompressibility of the electrons due to the exclusion principle.
(a) Matter contains protons and neutrons, which are also fanions. Why do the electrons become a hindrance to compression before the protons and neutrons do?
(b) Stars several times our Sun's mass has sufficient gravitational potential energy to collapse further than a white dwarf; they can force essentially all their matter to become neutrons (formed when electrons and protons combine). When they cool off, an energy balance is reached like that in the white dwarf but with the neutrons filling the role of the incompressible fermions. The result is a neutron star. Repeat the process of Exercise 89. but assume a body consisting solely of neutrons. Show that the equilibrium radius is given by
(c) Show that the radius of a neutron star whose mass is twice that of our Sun is only about .
What information would you need to specify the macro-state of the air in a room? What information would you need to specify the microstate?
According to Wien's law, the wavelengthat which the thermal emission of electromagnetic energy from a body of temperatureis maximum obeys.Show that this law follows from equation (9-47). To do this. Useto expressin terms ofrather than f, then obtain an expression that, when solved, would yield the wavelength at which this function is maximum. The transcendental equation cannot be solved exactly, so it is enough to show that solves it to a reasonable degree of precision.
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