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An Introduction to Thermal Physics

Daniel V. Schroeder

Physics

1st Edition · 356 pages

ISBN: 9780201380279

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 $t = T / T_{c}$ $c = μ / k T_{c}, and x = ϵ / k T_{c}$ . 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 $c$ when $T = 2 T_{c}$ is approximately $- 0.8$ . Plug in these values and check that the equation above is approximately satisfied.
(c) Now vary $μ$ , holding $T$ fixed, to find the precise value of $μ$ for . Repeat for values of $T / T_{c}$ ranging from $1.2$ up to $3.0$ , in increments of $0.2$ . Plot a graph of $μ$ as a function of temperature.

Short Answer

Expert verified

$T = 2 T_{c}$ d

Step by step solution

01

d

d

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Most popular questions from this chapter

The speed of sound in copper is $3560 m / s$ . Use this value to calculate its theoretical Debye temperature. Then determine the experimental Debye temperature from Figure 7.28, and compare.

In the text I claimed that the universe was filled with ionised gas until its temperature cooled to about 3000 K. To see why, assume that the universe contains only photons and hydrogen atoms, with a constant ratio of 10⁹ photons per hydrogen atom. Calculate and plot the fraction of atoms that were ionised as a function of temperature, for temperatures between 0 and 6000 K. How does the result change if the ratio of photons to atoms is 10⁸ or 10¹⁰? (Hint: Write everything in terms of dimensionless variables such as t = kT/I, where I is the ionisation energy of hydrogen.)

For a system of fermions at room temperature, compute the probability of a single-particle state being occupied if its energy is

(a) $1 e V$ less than $μ$

(b) $0.01 e V$ less than $μ$

(c) equal to $μ$

(d) $0.01 e V$ greater than $μ$

(e) $1 e V$ greater than $μ$

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 $70$ times that of earth's atmosphere. You can therefore model the atmosphere of Venus as $70$ successive "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 $2^{1 / 4}$ . The next layer down is warmer by a smaller factor. Keep working your way down until you see the pattern.)

Sometimes it is useful to know the free energy of a photon gas.

(a) Calculate the (Helmholtz) free energy directly from the definition

(Express the answer in terms of T' and V.)

(b) Check the formula $S = - (\partial F / \partial T)_{V}$ for this system.

(c) Differentiate F with respect to V to obtain the pressure of a photon gas. Check that your result agrees with that of the previous problem.

(d) A more interesting way to calculate F is to apply the formula $F = - k T \ln Z$ separately to each mode (that is, each effective oscillator), then sum over all modes. Carry out this calculation, to obtain

$F = 8 π V \frac{(k T)^{4}}{(h c)^{3}} \int_{0}^{\infty} x^{2} \ln (1 - e^{- x}) d x$

Integrate by parts, and check that your answer agrees with part (a).

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