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Chapter 9: Q59E (page 407)
Density of states (9-39) does not depend on , the total number of particles in the system; neither does the density of states in equation (9-27). Why not?
The density states do not depend on number of particles in the system.
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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 .
The diagram shows two systems that may exchange both thermal and mechanical energy via a movable, heat-conducting partition. Because both Eand Vmay change. We consider the entropy of each system to be a function of both:. Considering the exchange of thermal energy only, we argued in Section 9.2 that was reasonable to defineas. In the more general case, is also defined as something.
a) Why should pressure come into play, and to what mightbe equated.
b) Given this relationship, show that(Remember the first law of thermodynamics.)
In a large system of distinguishable harmonic oscillators, how high does the temperature have to be for the probability of occupying the ground state to be less than?
Show that. using equation, density of statesfollows fromlocalid="1658380849671"
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