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Chapter 9: Q11CQ (page 403)

The Fermi energy in a quantum gas depends inversely on the volume, Basing your answer on Simple Chapter 5 type quantum mechanics (not such quaint notions as squeezing classical particles of finite volume into a container too small). Explain why.

Short Answer

Expert verified

The description of the quantum gas as particles contained in an infinite well whose allowable energies correspond to those of a free particle leads in the dependency of the Fermi energy on volume in a quantum gas:

En=蟺2h22mL2n2

The allowed energies depend on the dimension Lof the 3-D well.

Step by step solution

01

Fermi energy

When referring to a quantum system of non-interacting fermions at absolute zero temperature, the term "Fermi energy" in quantum mechanics often refers to the energy difference between the highest and lowest occupied single-particle states.

02

Inverse Relationship between Energy and Wavelength

The Fermi energy can be considered to be inversely related to volume because the particles can be looked at as waves bounded by the walls of the container.

Because the particles can be viewed as waves bounded by the container's walls, the Fermi energy is inversely proportional to volume. The waves that are bounded inside the container get smaller as the container gets smaller. When the wavelength of something shrinks, its energy rises. As a result of the inverse relationship between energy and wavelength, it can help explain why Fermi energy is inversely related to volume.

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Most popular questions from this chapter

By considering its constituents, determine the dimensions (e.g. length, distance over lime. etc.) of the denominator in equation(9鈭�26). Why is the result sensible?

A particle subject to a planet's gravitational pull has a total mechanical energy given by Emechanical=12mv2-GMmr, whereis the particle's mass.M the planet's mass, and Gthe gravitational constant6.67脳10-11N路m3/kg2. 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 1000K. Referring to the values obtained in Exercise 45 and given that REarth=6.37脳106mand MEarth=5.98脳1024kg. should Earth be able to "hold on" to hydrogen (1g/mol)? 10 nitrogens (28g/mol)? (Note: An upper limit on the number of molecules in Earth's atmosphere is about 10-18).

(b) The moon's mass is 0.0123times Earth's. its radius 0.26 times Earth's, and its surface temperatures rise to 370K. Should it be able to hold on to these gases?

A "cold" subject,T1=300K, is briefly put in contact with s "hut" object,T2=400K, and60Jof heat flows frum the hot object io the cold use. The objects are then spiralled. their temperatures having changed negligibly due ko their large sizes. (a) What are the changes in entropy of each object and the system as a whole?

(b) Knowing only this these objects are in contact and at the given temperatures, what is the ratio of the probabilities of their being found in the second (final) state for that of their being found in the first (initial) state? What dies chis result suggest?

Consider a system of one-dimensional spinless particles in a box (see Section 5.5) somehow exchanging energy. Through steps similar to those giving equation (9-27). show that

D(E)=m1/2L鈩�蟺21E1/2

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
Uclocimns=310(3蟺2h3me32V)23N53

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

Uelectrons=9h280me(3蟺2M5mp5)131R2

Where, R is the star's radius?

(b) The gravitational potential energy of a sphere of mass Mand radius Ris given by

Ugray=-35GM2R

Taking both factors into account, show that the minimum total energy occurs when

R=3h28G(3蟺2me3mp5M)13

(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?
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