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Chapter 9: Q58E (page 407)
Show that. using equation, density of statesfollows fromlocalid="1658380849671"
The expression for density states is .
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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.)
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 .
Derivation of equation: Our model for calculatingis equation (9-26), whose denominator is the total number of particlesand whose numerator is the total energy of the system, which we here call. State with the denominator:
Insert the quantum gas density of states and an expression for the distribution. usingto distinguish the Bose-Einstein from the Fermi-Dirac. Then change variables:, and factorout of the denominator. In the integrand will be a factor
Using, a sum of two integrals results, each of Gaussian form. The integral thus becomes two terms in powers of. Repeat the process. but instead find an expression forin terms of, using
Divide your expression forby that for. both in terms of. Nowcan safely be eliminated by using the lowest-order expression forin terms of.
Exercise 67 calculates the minimum total energy in a system of spin fermions and is applicable to conduction electrons in a metal. The average particle energy is the total energy divided by the number of particles .Show that the average particle energy of a conduction electron at low temperature is . This form is convenient, being rather simple, and it can easily he put in terms of and via equation .
Not surprisingly. in a collection of oscillators, as in other thermodynamic systems, raising the temperature causes particles' energies to increase. Why shouldn鈥檛 point be reached where there are more panicles in some high energy state than in a lower energy. state? (The fundamental idea, not a formula that might arise from it. is the object.)
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