Solved thermodynamics exercise

Stability of thermal equilibrium: concavity of the entropy and positive heat capacity

Exercise 44 · Lesson 8 — Direction of Evolution, Equilibrium and Stability

stability
concavity
entropy maximum
heat capacity
thermal equilibrium
fundamental relation

Statement

A body is described by a fundamental relation $S (U, V, N)$ . In Questions 1 to 3, its volume and number of particles remain fixed; we simply write its entropy as $S (U)$ , and define its heat capacity at constant volume as $C_{V} = (\partial U / \partial T)_{V, N}$ .

A rigid, isolated enclosure contains two identical portions of this body, with the same volume and the same number of particles, separated by a fixed, impermeable, diathermal wall. The total energy $2 U_{0}$ is fixed. Let $U_{1}$ be the energy of the first portion; the second therefore has $2 U_{0} - U_{1}$ .

Write the total entropy $S_{tot} (U_{1})$ of the enclosure. Show that the uniform distribution $U_{1} = U_{0}$ makes $S_{tot}$ stationary, and interpret this condition.
Write the expansion of $S_{tot} (U_{0} + ε)$ to second order in $ε$ . Under what condition on $S^{''} (U_{0})$ is the uniform distribution a local maximum of the entropy? What would happen otherwise?
Show that $(\frac{\partial ^{2} S}{\partial U ^{2}})_{V, N} = - \frac{1}{T ^{2} C _{V}} .$ Deduce that the stability of thermal equilibrium requires $C_{V} > 0$ . Interpret physically: how would an exchange of energy between the two portions evolve if $C_{V}$ were negative?
Now consider the monatomic ideal gas, at fixed $N$ : $S (U, V) = N k_{B} [\frac{3}{2} ln \frac{U}{N} + ln \frac{V}{N}] + N s_{0},$ where $s_{0}$ is a constant. Calculate its temperature and its heat capacity $C_{V}$ . Using the two-variable concavity criterion from the lesson, check that $S$ is concave in $(U, V)$ .
Finally, consider the model $S (U, V) = A U^{2} / V$ , with $A > 0$ and $U > 0$ . Check that this relation is extensive. Calculate the temperature, then the heat capacity $C_{V}$ in terms of $V$ and $T$ . Can this model describe a stable homogeneous substance? Conclude using the concavity criterion.

Hint

For Question 3, differentiate the relation

(\partial S / \partial U)_{V, N} = 1/ T

with respect to

U

, using the chain rule:

T

depends on

U

, and

(\partial T / \partial U)_{V, N} = 1/ C_{V}

. For Question 4, the two-variable criterion reads

S_{U U} \leq 0

S_{V V} \leq 0

and

S_{U U} S_{V V} - S_{U V}^{2} \geq 0

Detailed solution

Solution

Question 1. The two portions are described by the same function

S

, and entropy is additive:

S_{tot} (U_{1}) = S (U_{1}) + S (2 U_{0} - U_{1}) .

Differentiate with respect to $U_{1}$ . The derivative of $S (2 U_{0} - U_{1})$ is $- S^{'} (2 U_{0} - U_{1})$ , hence

\frac{d S _{tot}}{d U _{1}} = S^{'} (U_{1}) - S^{'} (2 U_{0} - U_{1}) .

For $U_{1} = U_{0}$ , the two portions have the same energy and the derivative is zero: the uniform distribution is stationary. Since $S^{'} (U) = 1/ T$ , the general stationarity condition $S^{'} (U_{1}) = S^{'} (2 U_{0} - U_{1})$ expresses the equality of the temperatures of the two portions: it is the condition for thermal equilibrium.

Question 2. Expand each term to second order about

U_{0}

, using Taylor's formula

f (U_{0} + h) = f (U_{0}) + f^{'} (U_{0}) h + \frac{1}{2} f^{''} (U_{0}) h^{2} + o (h^{2})

S (U_{0} + ε) = S (U_{0}) + S^{'} (U_{0}) ε + \frac{1}{2} S^{''} (U_{0}) ε^{2} + o (ε^{2}), S (U_{0} - ε) = S (U_{0}) - S^{'} (U_{0}) ε + \frac{1}{2} S^{''} (U_{0}) ε^{2} + o (ε^{2}) .

Adding them, the first-order terms cancel:

S_{tot} (U_{0} + ε) = 2 S (U_{0}) + S^{''} (U_{0}) ε^{2} + o (ε^{2}) .

If $S^{''} (U_{0}) < 0$ , any energy transfer $ε \neq = 0$ decreases the total entropy: the uniform distribution is a local maximum, and hence a stable equilibrium. If, on the contrary, $S^{''} (U_{0}) > 0$ , a small energy transfer in either direction increases the entropy of the isolated system. Such a transfer can then occur spontaneously: the uniform distribution, although stationary, is unstable.

Question 3. By definition,

(\partial S / \partial U)_{V, N} = 1/ T

, where

T

is a function of

U

at fixed

V

and

N

. Differentiate this relation with respect to

U

. The derivative of

1/ T

with respect to

T

- 1/ T^{2}

, so, by the chain rule,

(\frac{\partial ^{2} S}{\partial U ^{2}})_{V, N} = - \frac{1}{T ^{2}} (\frac{\partial T}{\partial U})_{V, N} .

Now $(\partial T / \partial U)_{V, N} = 1/ (\partial U / \partial T)_{V, N} = 1/ C_{V}$ , hence

(\frac{\partial ^{2} S}{\partial U ^{2}})_{V, N} = - \frac{1}{T ^{2} C _{V}} .

Since $T^{2} > 0$ , the stability condition $S^{''} < 0$ from Question 2 is equivalent to $C_{V} > 0$ .

Interpretation: suppose

C_{V} < 0

, and that a small amount of energy passes from portion 2 to portion 1. Portion 1, having received energy, would cool down; portion 2, having lost energy, would warm up. Since heat flows spontaneously from hot to cold, portion 1 would then receive even more energy: the difference would grow instead of dying out. With

C_{V} > 0

, the opposite happens: the portion that receives energy warms up, gives heat to the other, and the difference dies out. Question 4. Differentiate

S

with respect to

U

at fixed

V

\frac{1}{T} = (\frac{\partial S}{\partial U})_{V} = \frac{3 N k _{B}}{2 U} ⟹ U = \frac{3}{2} N k_{B} T, C_{V} = (\frac{\partial U}{\partial T})_{V} = \frac{3}{2} N k_{B} > 0.

Calculate the second derivatives:

S_{U U} = \frac{\partial}{\partial U} (\frac{3 N k _{B}}{2 U}) = - \frac{3 N k _{B}}{2 U ^{2}} < 0, S_{V} = \frac{N k _{B}}{V}, S_{V V} = - \frac{N k _{B}}{V ^{2}} < 0,

and $S_{U V} = 0$ , since $S_{U}$ does not depend on $V$ . The determinant is

S_{U U} S_{V V} - S_{U V}^{2} = (- \frac{3 N k _{B}}{2 U ^{2}}) (- \frac{N k _{B}}{V ^{2}}) - 0 = \frac{3 N ^{2} k _{B}^{2}}{2 U ^{2} V ^{2}} > 0.

The three conditions of the criterion are satisfied everywhere: the entropy of the ideal gas is concave in $(U, V)$ , and its equilibrium is stable.

Question 5. Multiply

U

and

V

λ > 0

S (λ U, λV) = A \frac{λ ^{2} U ^{2}}{λV} = λ A \frac{U ^{2}}{V} = λ S (U, V) .

The relation is therefore extensive. Its temperature satisfies

\frac{1}{T} = (\frac{\partial S}{\partial U})_{V} = \frac{2 A U}{V} > 0 ⟹ U = \frac{V}{2 A T} .

At fixed $V$ , $U$ is proportional to $1/ T$ , whose derivative is $- 1/ T^{2}$ :

C_{V} = (\frac{\partial U}{\partial T})_{V} = - \frac{V}{2 A T ^{2}} < 0.

The heat capacity is negative: according to Question 3, this model cannot describe a stable homogeneous substance. This can be checked directly with the concavity criterion: $S_{U U} = 2 A / V > 0$ , which already violates the first condition $S_{U U} \leq 0$ . A body described by this relation would not remain homogeneous: the slightest energy fluctuation between two of its portions would grow, as described in Question 3.

Extensivity alone is therefore not enough to make a function

S (U, V, N)

an acceptable fundamental relation: stability additionally requires it to be concave. Negative heat capacities do nevertheless occur in nature, for example in stars, but only for non-additive systems dominated by gravitation, to which this reasoning does not apply.