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An Introduction to Thermal Physics

Daniel V. Schroeder

Physics

1st Edition 路 356 pages

ISBN: 9780201380279

Chapter 3: Q. 3.35 (page 119)

In the text I showed that for an Einstein solid with three oscillators and three units of energy, the chemical potential is $渭 = - 系$ (where $系$ is the size of an energy unit and we treat each oscillator as a "particle"). Suppose instead that the solid has three oscillators and four units of energy. How does the chemical potential then compare to $- 系$ ? (Don't try to get an actual value for the chemical potential; just explain whether it is more or less than $- 系$ .)

Short Answer

Expert verified

$渭 < - 鈭�$

Step by step solution

01

Explanation of Solution

Given:

For $N = 3$ and $q = 3$ the chemical potential is $渭 = - 鈭�$

02

Calculation

The chemical potential formula is

$渭 = {(\frac{螖 U}{螖 N})}_{S}$

The entropy for $N = 3$ and $q = 3$ is

$S = k \ln 惟$

$S = k \ln \frac{鈭� N + q - 1}{\frac{| q - 1 | q}{鈭� 3 + 3 - 1}}$

$S = k \ln \frac{| 3 - 1 | 3}{\underset{炉}{鈭� 5}}$

$S = k \ln \frac{| 2 | 3}{\frac{| 2 |}{| 2 |}}$

$S = k \ln (10)$

The entropy must be constant throughout.

The entropy for $N = 3$ and $q = 4$ is,

$\begin{aligned} S = k \ln 鈦� 惟 \\ S = k \ln 鈦� \frac{鈭� N + q 鈭� 1}{\frac{| q 鈭� 1 | q}{鈭� 3 + 3 鈭� 1}} \\ S = k \ln 鈦� \frac{| 3 鈭� 1 | 3}{\underline{鈭� 5}} \\ S = k \ln 鈦� \frac{| 2 | 3}{\frac{| 2 |}{| 2 |}} \\ S = k \ln 鈦� (10) \end{aligned}$

Thus, the entropy increases so to reduce the entropy to its original value.

03

Conclusion

The chemical potential is lowered.

$渭 < - 鈭�$

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Most popular questions from this chapter

In Problem 2.18 you showed that the multiplicity of an Einstein solid containing N oscillators and q energy units is approximately

$惟 (N, q) 鈮� {(\frac{q + N}{q})}^{q} {(\frac{q + N}{N})}^{N}$

(a) Starting with this formula, find an expression for the entropy of an Einstein solid as a function of N and q. Explain why the factors omitted from the formula have no effect on the entropy, when N and q are large.

(b) Use the result of part (a) to calculate the temperature of an Einstein solid as a function of its energy. (The energy is $U = q 系$ , where $系$ is a constant.) Be sure to simplify your result as much as possible.

(c) Invert the relation you found in part (b) to find the energy as a function of temperature, then differentiate to find a formula for the heat capacity.

(d) Show that, in the limit $T 鈫� 鈭�$ , the heat capacity is $C = N k$ . (Hint: When x is very small, $e^{x} 鈮� 1 + x$ .) Is this the result you would expect? Explain.

(e) Make a graph (possibly using a computer) of the result of part (c). To avoid awkward numerical factors, plot $C / N k$ vs. the dimensionless variable $t = k T / 系$ , for t in the range from 0 to about 2. Discuss your prediction for the heat capacity at low temperature, comparing to the data for lead, aluminum, and diamond shown in Figure 1.14. Estimate the value of $系$ , in electron-volts, for each of those real solids.

(f) Derive a more accurate approximation for the heat capacity at high temperatures, by keeping terms through $x^{3}$ in the expansions of the exponentials and then carefully expanding the denominator and multiplying everything out. Throw away terms that will be smaller than $(系 / k T)^{2}$ in the final answer. When the smoke clears, you should find $C = N k [1 - \frac{1}{12} (系 / k T)^{2}]$ .

In Problem 1.55 you used the virial theorem to estimate the heat capacity of a star. Starting with that result, calculate the entropy of a star, first in terms of its average temperature and then in terms of its total energy. Sketch the entropy as a function of energy, and comment on the shape of the graph.

Experimental measurements of the heat capacity of aluminum at low temperatures (below about $50 K$ ) can be fit to the formula

$C_{V} = a T + b T^{3}$

where $C_{V}$ is the heat capacity of one mole of aluminum, and the constants $a$ and $b$ are approximately $a = 0.00135 J / K^{2}$ and $b = 2.48 脳 10^{- 5} J / K^{4}$ . From this data, find a formula for the entropy of a mole of aluminum as a function of temperature. Evaluate your formula at $T = 1 K$ and at $T = 10 K$ , expressing your answers both in conventional units $(J / K)$ and as unitless numbers (dividing by Boltzmann's constant).

In Problem $2.32$ you computed the entropy of an ideal monatomic gas that lives in a two-dimensional universe. Take partial derivatives with respect to $U, A$ , and N to determine the temperature, pressure, and chemical potential of this gas. (In two dimensions, pressure is defined as force per unit length.) Simplify your results as much as possible, and explain whether they make sense.

Starting with the result of Problem 2.17, find a formula for the temperature of an Einstein solid in the limit $q 鈮� N$ . Solve for the energy as a function of temperature to obtain $U = N 系 e^{- 系 / k T}$ (where $系$ is the size of an energy unit).

See all solutions

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