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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=渭/kTc,andx=系/kTc. Express the integral that defines 渭, 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 渭, holding Tfixed, to find the precise value of 渭for . Repeat for values of T/Tcranging from 1.2up to 3.0, in increments of 0.2. Plot a graph of 渭as a function of temperature.

Short Answer

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T=2Tcd

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01

d

d

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Most popular questions from this chapter

It's not obvious from Figure 7.19 how the Planck spectrum changes as a function of temperature. To examine the temperature dependence, make a quantitative plot of the functionu(系) for T = 3000 K and T = 6000 K (both on the same graph). Label the horizontal axis in electron-volts.

For a system of particles at room temperature, how large must 系-渭be before the Fermi-Dirac, Bose-Einstein, and Boltzmann distributions agree within 1%? Is this condition ever violated for the gases in our atmosphere? Explain.

The planet Venus is different from the earth in several respects. First, it is only 70% as far from the sun. Second, its thick clouds reflect 77%of all incident sunlight. Finally, its atmosphere is much more opaque to infrared light.

(a) Calculate the solar constant at the location of Venus, and estimate what the average surface temperature of Venus would be if it had no atmosphere and did not reflect any sunlight.

(b) Estimate the surface temperature again, taking the reflectivity of the clouds into account.

(c) The opaqueness of Venus's atmosphere at infrared wavelengths is roughly 70times that of earth's atmosphere. You can therefore model the atmosphere of Venus as 70successive "blankets" of the type considered in the text, with each blanket at a different equilibrium temperature. Use this model to estimate the surface temperature of Venus. (Hint: The temperature of the top layer is what you found in part (b). The next layer down is warmer by a factor of 21/4. The next layer down is warmer by a smaller factor. Keep working your way down until you see the pattern.)

For a brief time in the early universe, the temperature was hot enough to produce large numbers of electron-positron pairs. These pairs then constituted a third type of "background radiation," in addition to the photons and neutrinos (see Figure 7.21). Like neutrinos, electrons and positrons are fermions. Unlike neutrinos, electrons and positrons are known to be massive (ea.ch with the same mass), and each has two independent polarization states. During the time period of interest, the densities of electrons and positrons were approximately equal, so it is a good approximation to set the chemical potentials equal to zero as in Figure 7.21. When the temperature was greater than the electron mass times c2k, the universe was filled with three types of radiation: electrons and positrons (solid arrows); neutrinos (dashed); and photons (wavy). Bathed in this radiation were a few protons and neutrons, roughly one for every billion radiation particles. the previous problem. Recall from special relativity that the energy of a massive particle is 系=(pc)2+mc22.

(a) Show that the energy density of electrons and positrons at temperature Tis given by

u(T)=鈭�0鈭�x2x2+mc2/kT2ex2+mc2/kT2+1dx;whereu(T)=鈭�0鈭�x2x2+mc2/kT2ex2+mc2/kT2+1dx

(b) Show that u(T)goes to zero when kT鈮�mc2, and explain why this is a

reasonable result.

( c) Evaluate u(T)in the limit kT鈮�mc2, and compare to the result of the

the previous problem for the neutrino radiation.

(d) Use a computer to calculate and plot u(T)at intermediate temperatures.

(e) Use the method of Problem 7.46, part (d), to show that the free energy

density of the electron-positron radiation is

FV=-16蟺(kT)4(hc)3f(T);wheref(T)=鈭�0鈭�x2ln1+e-x2+mc2/kT2dx

Evaluate f(T)in both limits, and use a computer to calculate and plot f(T)at intermediate

temperatures.

(f) Write the entropy of the electron-positron radiation in terms of the functions

uTand f(T). Evaluate the entropy explicitly in the high-T limit.

Change variables in equation 7.83 to 位=hc/系 and thus derive a formula for the photon spectrum as a function of wavelength. Plot this spectrum, and find a numerical formula for the wavelength where the spectrum peaks, in terms of hc/kT. Explain why the peak does not occur at hc/(2.82kT).
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