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An Introduction to Thermal Physics

Daniel V. Schroeder

Physics

1st Edition · 356 pages

ISBN: 9780201380279

Chapter 3: Q. 3.27 (page 112)

What partial-derivative relation can you derive from the thermodynamic identity by considering a process that takes place at constant entropy? Does the resulting equation agree with what you already knew? Explain.

Short Answer

Expert verified

The relation that can be derived from the thermodynamic identity is $P = - {(\frac{d U}{d V})}_{N, S}$ .

The derived equation agrees with the first law of thermodynamics.

Step by step solution

01

Given Information

The thermodynamic identity is:

$d U = T d S - P d V$

The process is taking place at constant entropy.

Hence, $d S = 0$

02

Calculation

Since the process is taking place at constant entropy, the thermodynamic identity can be written as:

$d U = T d S - P d V d U = - P d V P = - {(\frac{d U}{d V})}_{N, S}$

This is the derived relation.

From the relation, it can be seen that the volume change is fully responsible for the energy shift. As a result, there is no heat transfer into or out of the system. Heat flows until equilibrium is reached, resulting in a rise in entropy. As a result, the equation follows the first law of thermodynamics.

03

Final answer

The derived relation is $P = - {(\frac{d U}{d V})}_{N, S}$ . It agrees with the first law of thermodynamics.

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Most popular questions from this chapter

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

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 $b = 2.48 \times 10^{- 5} J / K^{4}$ . 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 Section 2.5 I quoted a theorem on the multiplicity of any system with only quadratic degrees of freedom: In the high-temperature limit where the number of units of energy is much larger than the number of degrees of freedom, the multiplicity of any such system is proportional to $U^{N f / 2}$ , where $N f$ is the total number of degrees of freedom. Find an expression for the energy of such a system in terms of its temperature, and comment on the result. How can you tell that this formula for $Ω$ cannot be valid when the total energy is very small?

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

$Ω (N, q) \approx {(\frac{q + N}{q})}^{q} {(\frac{q + N}{N})}^{N}$

(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 $U = q ϵ$ , where $ϵ$ 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 $T \to \infty$ , the heat capacity is $C = N k$ . (Hint: When x is very small, $e^{x} \approx 1 + x$ .) 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 $t = k T / ϵ$ , 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 $ϵ$ , 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 $(ϵ / k T)^{2}$ in the final answer. When the smoke clears, you should find $C = N k [1 - \frac{1}{12} (ϵ / k T)^{2}]$ .

Use the thermodynamic identity to derive the heat capacity formula

$C_{V} = T {(\frac{\partial S}{\partial T})}_{V}$

which is occasionally more convenient than the more familiar expression in terms of $U$ . Then derive a similar formula for $C_{P}$ , by first writing $d H$ in terms of $d S$ and $d P$ .

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