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Chapter 7: Q. 7.69 (page 323)

(a) As usual when solving a problem on a computer, it's best to start by putting everything in terms of dimensionless variables. So define t=T/Tcc=μ/kTc,andx=ϵ/kTc. Express the integral that defines μ, equation 7.122, in terms of these variables. You should obtain the equation

(b) According to Figure 7.33, the correct value of cwhen T=2Tcis approximately -0.8. Plug in these values and check that the equation above is approximately satisfied.

(c) Now vary μ, holding Tfixed, to find the precise value of μfor . Repeat for values of T/Tcranging from 1.2up to 3.0, in increments of 0.2. Plot a graph of μas a function of temperature.

Short Answer

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T=2Tcd

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01

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Most popular questions from this chapter

Consider a two-dimensional solid, such as a stretched drumhead or a layer of mica or graphite. Find an expression (in terms of an integral) for the thermal energy of a square chunk of this material of area , and evaluate the result approximately for very low and very high temperatures. Also, find an expression for the heat capacity, and use a computer or a calculator to plot the heat capacity as a function of temperature. Assume that the material can only vibrate perpendicular to its own plane, i.e., that there is only one "polarization."

Each atom in a chunk of copper contributes one conduction electron. Look up the density and atomic mass of copper, and calculate the Fermi energy, the Fermi temperature, the degeneracy pressure, and the contribution of the degeneracy pressure to the bulk modulus. Is room temperature sufficiently low to treat this system as a degenerate electron gas?

A white dwarf star (see Figure 7.12) is essentially a degenerate electron gas, with a bunch of nuclei mixed in to balance the charge and to provide the gravitational attraction that holds the star together. In this problem you will derive a relation between the mass and the radius of a white dwarf star, modeling the star as a uniform-density sphere. White dwarf stars tend to be extremely hot by our standards; nevertheless, it is an excellent approximation in this problem to set T=0.

(a) Use dimensional analysis to argue that the gravitational potential energy of a uniform-density sphere (mass M, radius R) must equal

Ugrav=-(constant)GM2R

where (constant) is some numerical constant. Be sure to explain the minus sign. The constant turns out to equal 3/5; you can derive it by calculating the (negative) work needed to assemble the sphere, shell by shell, from the inside out.

(b) Assuming that the star contains one proton and one neutron for each electron, and that the electrons are nonrelativistic, show that the total (kinetic) energy of the degenerate electrons equals

Ukinetic=(0.0086)h2M53memp53R2

Figure 7.12. The double star system Sirius A and B. Sirius A (greatly overexposed in the photo) is the brightest star in our night sky. Its companion, Sirius B, is hotter but very faint, indicating that it must be extremely small-a white dwarf. From the orbital motion of the pair we know that Sirius B has about the same mass as our sun. (UCO /Lick Observatory photo.)

( c) The equilibrium radius of the white dwarf is that which minimizes the total energy Ugravity+Ukinetic· Sketch the total energy as a function of R, and find a formula for the equilibrium radius in terms of the mass. As the mass increases, does the radius increase or decrease? Does this make sense?

( d) Evaluate the equilibrium radius for M=2×1030kg, the mass of the sun. Also evaluate the density. How does the density compare to that of water?

( e) Calculate the Fermi energy and the Fermi temperature, for the case considered in part (d). Discuss whether the approximation T = 0 is valid.

(f) Suppose instead that the electrons in the white dwarf star are highly relativistic. Using the result of the previous problem, show that the total kinetic energy of the electrons is now proportional to 1 / R instead of 1R2• Argue that there is no stable equilibrium radius for such a star.

(g) The transition from the nonrelativistic regime to the ultra relativistic regime occurs approximately where the average kinetic energy of an electron is equal to its rest energy, mc2Is the nonrelativistic approximation valid for a one-solar-mass white dwarf? Above what mass would you expect a white dwarf to become relativistic and hence unstable?

At the center of the sun, the temperature is approximately 107K and the concentration of electrons is approximately 1032 per cubic meter. Would it be (approximately) valid to treat these electrons as a "classical" ideal gas (using Boltzmann statistics), or as a degenerate Fermi gas (with T≈0 ), or neither?

In Section 6.5 I derived the useful relation F=-kTln(Z)between the Helmholtz free energy and the ordinary partition function. Use analogous argument to prove that ϕ=-kT×ln(Z^), where Z^ is the grand partition function and ϕis the grand free energy introduced in Problem 5.23.
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