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An Introduction to Thermal Physics

Daniel V. Schroeder

Physics

1st Edition 路 356 pages

ISBN: 9780201380279

Chapter 7: Q. 7.58 (page 313)

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.

Short Answer

Expert verified

Hence, the Debye Temperature from experimental Data (graph) $T_{D} = 359.9176 K$ .

Step by step solution

01

Step 1. Given information

We have, $the speed of sound in copper is 3560 m / s$

$The Debye temperature is T_{D} = \frac{h C_{s}}{2 k_{B}} 路 {(\frac{6 N}{蟺 V})}^{\frac{1}{3}}$ .

02

Step 2. Putting the value of h , V , N ,CS ,kB.

$V = 7.11 {cm}^{3} / mol$

$C_{s} = 3560 m / s = 3560 脳 100 cm / s$

$N = 6.022 脳 10^{23} atoms / mol$

$k_{B} = 1.381 脳 10^{- 23} J / K$

$h = 6.626 脳 10^{- 34} J 路 s$

substituting all the above value in Debye Temperature we get

$T_{D} = \frac{6.626 脳 10^{- 34} J 路 s 脳 3560 脳 100 cm / s}{(2 脳 1.381 脳 10^{- 23} J / K)} {[\frac{6 脳 6.022 脳 10^{23} / mol}{蟺脳 7.11 {cm}^{3} / mol}]}^{\frac{1}{3}}$

$鈭� T_{D} = 465.3381 K$

03

Step 3. Calculating the experimental Debye Temperature.

Take any two points on experimental data calculate the slope [Approximately] Approximately take two points

Approximately take two points

$A (0, 0.75) B (18, 1.5)$

$\frac{螖 (\frac{C_{V}}{T})}{螖 (T^{2})} = \frac{1.5 - 0.75}{18 - 0}$

$= 0.0417$

04

Step 4. Finding the slope of the graph.

$Slope: \frac{螖 (\frac{C_{V}}{T})}{螖 (T^{2})} = \frac{12 蟺^{4} N k_{B}}{5 T_{D}^{3}}$

$\frac{12 蟺^{4} N k_{B}}{5 T_{D}^{3}} = 0.0417 mJ / K^{4}$

then , the Debye Temperature

$T_{D} = {[\frac{12 蟺^{4} N k_{B}}{5 脳 0.0417 脳 10^{- 3} J / K^{4}}]}^{\frac{1}{3}}$

$= {(\frac{12 蟺^{4} 脳 6.022 脳 10^{23} 脳 1.381 脳 10^{- 23}}{5 脳 0.0417 脳 10^{- 3}})}^{\frac{1}{3}}$

$= 359.9176 K$

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Most popular questions from this chapter

In Problem $7.28$ you found the density of states and the chemical potential for a two-dimensional Fermi gas. Calculate the heat capacity of this gas in the limit role="math" localid="1650099524353" $k T 鈮� 蔚_{F}$ 路 Also show that the heat capacity has the expected behavior when $k T 鈮� 蔚_{F}$ . Sketch the heat capacity as a function of temperature.

Consider a gas of noninteracting spin-0 bosons at high temperatures, when $T 鈮� T_{c}$ . (Note that 鈥渉igh鈥� in this sense can still mean below 1 K.)

Show that, in this limit, the Bose-Einstein function can be written approximately as
${\overset{炉}{n}}_{B E} = e^{鈭� (鈭� 鈭� 渭) / k T} [1 + e^{鈭� (鈭� 鈭� 渭) / k T} + 鈰�]$ .
Keeping only the terms shown above, plug this result into equation 7.122 to derive the first quantum correction to the chemical potential for gas of bosons.
Use the properties of the grand free energy (Problems 5.23 and 7.7) to show that the pressure of any system is given by In $P = (k T / V)$ , where $Z$ is the grand partition function. Argue that, for gas of noninteracting particles, In $Z$ can be computed as the sum over all modes (or single-particle states) of In $Z_{i}$ , where $Z_{i}$ ; is the grand partition function for the $i th$ mode.
Continuing with the result of part (c), write the sum over modes as an integral over energy, using the density of states. Evaluate this integral explicitly for gas of noninteracting bosons in the high-temperature limit, using the result of part (b) for the chemical potential and expanding the logarithm as appropriate. When the smoke clears, you should find
$p = \frac{N k T}{V} (1 鈭� \frac{N v_{Q}}{4 \sqrt{2} V})$ ,
again neglecting higher-order terms. Thus, quantum statistics results in a lowering of the pressure of a boson gas, as one might expect.
Write the result of part (d) in the form of the virial expansion introduced in Problem 1.17, and read off the second virial coefficient, $B (T)$ . Plot the predicted $B (T)$ for a hypothetical gas of noninteracting helium-4 atoms.
Repeat this entire problem for gas of spin-1/2 fermions. (Very few modifications are necessary.) Discuss the results, and plot the predicted virial coefficient for a hypothetical gas of noninteracting helium-3 atoms.

(a) Estimate (roughly) the total power radiated by your body, neglecting any energy that is returned by your clothes and environment. (Whatever the color of your skin, its emissivity at infrared wavelengths is quite close to $1$ ; almost any nonmetal is a near-perfect blackbody at these wavelengths.)

(b) Compare the total energy radiated by your body in one day (expressed in kilocalories) to the energy in the food you cat. Why is there such a large discrepancy?

(c) The sun has a mass of $2 脳 10^{30} kg$ and radiates energy at a rate of $3.9 脳 10^{26} watts$ . Which puts out more power per units mass-the sun or your body?

The Sommerfeld expansion is an expansion in powers of $\frac{k T}{蔚_{F}}$ , which is assumed to be small. In this section I kept all terms through order ${(\frac{k T}{蔚_{F}})}^{2}$ , omitting higher-order terms. Show at each relevant step that the term proportional to localid="1650117451748" $T^{3}$ is zero, so that the next nonvanishing terms in the expansions forlocalid="1650117470867" $渭$ and localid="1650117476821" $U$ are proportional to localid="1650117458596" $T^{4}$ . (If you enjoy such things, you might try evaluating the localid="1650117464980" $T^{4}$ terms, possibly with the aid of a computer algebra program.)

Figure 7.37 shows the heat capacity of a Bose gas as a function of temperature. In this problem you will calculate the shape of this unusual graph.

(a) Write down an expression for the total energy of a gas of $N$ bosons confined to a volume $V$ , in terms of an integral (analogous to equation 7.122).

(b) For $T < T_{c}$ you can set $渭 = 0$ . Evaluate the integral numerically in this case, then differentiate the result with respect to $T$ to obtain the heat capacity. Compare to Figure 7.37.

(c) Explain why the heat capacity must approach $\frac{3}{2} N k$ in the high- $T$ limit.

(d) For $T > T_{c}$ you can evaluate the integral using the values of $渭$ calculated in Problem 7.69. Do this to obtain the energy as a function of temperature, then numerically differentiate the result to obtain the heat capacity. Plot the heat capacity, and check that your graph agrees with Figure 7.37.

Figure 7.37. Heat capacity of an ideal Bose gas in a three-dimensional box.

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