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Chapter 9: Q4CQ (page 402)
A scientifically untrained but curious friend asks, "When I walk into a room, is there a chance that all the air will be on the other side?" How do you answer this question?
One present only one microstate, mathematically it is one of infinity.
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When a star has nearly bumped up its intimal fuel, it may become a white dwarf. It is crushed under its own enormous gravitational forces to the point at which the exclusion principle for the electrons becomes a factor. A smaller size would decrease the gravitational potential energy, but assuming the electrons to be packed into the lowest energy states consistent with the exclusion principle, "squeezing" the potential well necessarily increases the energies of all the electrons (by shortening their wavelengths). If gravitation and the electron exclusion principle are the only factors, there is minimum total energy and corresponding equilibrium radius.
(a) Treat the electrons in a white dwarf as a quantum gas. The minimum energy allowed by the exclusion principle (see Exercise 67) is
Note that as the volume Vis decreased, the energy does increase. For a neutral star. the number of electrons, N, equals the number of protons. If protons account for half of the white dwarf's mass M (neutrons accounting for the other half). Show that the minimum electron energy may be written
Where, R is the star's radius?
(b) The gravitational potential energy of a sphere of mass Mand radius Ris given by
Taking both factors into account, show that the minimum total energy occurs when
(c) Evaluate this radius for a star whose mass is equal to that of our Sun 2x1030kg.
(d) White dwarfs are comparable to the size of Earth. Does the value in part (c) agree?
What is special about a metastable Stale, and why is it so useful ina laser? Why wouldn't a non-metastable state at the same energy work?
This problem investigates what fraction of the available charge must be transferred from one conductor to another to produce a typical contact potential. (a) As a rough approximation treat the conductors as 10 cm square plates apart-a parallel-plate capacitors so that , where . How much charge must be transferred from one plate to the other to produce a potential difference of ?(b) Approximately what fraction would this be of the total number of conduction electrons in a piece of copper. which has one conduction electron per atom?
A particle subject to a planet's gravitational pull has a total mechanical energy given by , whereis the particle's mass.M the planet's mass, and Gthe gravitational constant. It may escape if its energy is zero that is, if its positive KE is equal in magnitude to the negative PE holding if to the surface. Suppose the particle is a gas molecule in an atmosphere.
(a) Temperatures in Earth's atmosphere may reach . Referring to the values obtained in Exercise 45 and given that and . should Earth be able to "hold on" to hydrogen ? 10 nitrogens ? (Note: An upper limit on the number of molecules in Earth's atmosphere is about ).
(b) The moon's mass is times Earth's. its radius 0.26 times Earth's, and its surface temperatures rise to . Should it be able to hold on to these gases?
We claim that the famous exponential decrease of probability with energy is natural, the vastly most probable and disordered state given the constraints on total energy and number of particles. It should be a state of maximum entropy ! The proof involves mathematical techniques beyond the scope of the text, but finding support is good exercise and not difficult. Consider a system ofoscillators sharing a total energy of just . In the symbols of Section 9.3. and .
What do your finding suggests?
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