Chapter 17, Stability and free energy Video Solutions, Thermodynamics: A complete undergraduate course

Problem 1

Consider again the two examples illustrated in Figure 3.1: a ball rolling in a bowl, and an atom emitting a photon. Do not discuss the whole universe. Rather, choose some suitable boundary to define the system under discussion, identify the constraints, and hence identify what property of the system dictates the direction of the process.

Jennifer Hudspeth

Numerade Educator

03:10

Problem 2

Consider again the apparatus described in Exercise 7.18 of Chapter 7.
(i) Calculate the initial and final total energy and entropy in the two cases.
(ii) Suggest a physical process whereby the entropy increases in case A, and explain where the energy lost by the system goes in case B.
(iii) Now suppose that after the mass is increased to $m_2$, instead of releasing the piston, it is simply moved to a given height $h$ and fixed there. Consider the total energy $U(S, h)$ and entropy $S(U, h)$. Plot graphs of $S\left(U_i, h\right)$ and $U\left(S_i, h\right)$ as a function of $h$, where $U_i, S_i$ are the total energy and entropy after the mass is increased but before the piston is moved (treat 1 mole of gas at STP for the initial conditions, and take $m_2=2 m_1$ ). Explain how these graphs relate to the answers to Exercise 7.18 of Chapter 7.

Manish Jain

Numerade Educator

02:12

Problem 3

Consider an isolated chamber divided into two parts by a membrane possessing surface tension $\sigma$ and area $A$. Let the two parts have equal temperature $T$, and let them have pressures $p_1, p_2$ and volumes $V_1, V_2$. Each part can be treated as a simple compressible system. (i) By considering a fluctuation in which a ripple appears on the membrane, changing its area without affecting $V_1$ and $V_2$, show that if the volumes are fixed then in thermal equilibrium the membrane adopts the shape of least area consistent with the fixed volumes. (ii) By considering a fluctuation in which $V_1$ grows and $V_2$ shrinks and $A$ also consequently changes, show that the equilibrium condition $\mathrm{d} S=0$ gives $\sigma=\left(p_1-p_2\right) \mathrm{d} V_1 / \mathrm{d} A$ (cf. equation (14.10)). Hence relate the pressure difference at any point on the surface to the local radius of curvature of the surface.

Narayan Hari

Numerade Educator

01:26

Problem 4

After equation (14.15) we observed that the entropy of a surface increases with its area, but in question 17.3 we argued that at maximum entropy, the area is minimized. What is going on?

David Collins

Numerade Educator

03:28

Problem 5

Adapt the argument of equations (17.9)-(17.11) to the case of a system at fixed temperature, and hence confirm the Maxwell construction illustrated in Figure 15.6.

Uma Kumari

Numerade Educator

01:45

Problem 6

Consider the example system shown in Figure 17.8. Model the two fluids as ideal gases, and suppose the rubber has a constant surface tension $\sigma$ and heat capacity $C$.
(i) Show that changes in the total entropy and energy are governed by
$$
\begin{aligned}
\mathrm{d} S= & \frac{C \mathrm{~d} T}{T}+\frac{N_1 k_{\mathrm{B}} \mathrm{d} V_1}{V_1}+\frac{C_1 \mathrm{~d} U_1}{U_1} \\
& +\frac{N_2 k_{\mathrm{B}} \mathrm{d} V_2}{V_2}+\frac{C_2 \mathrm{~d} U_2}{U_2}, \\
\mathrm{~d} U= & \mathrm{d} U_1+\mathrm{d} U_2+C \mathrm{~d} T+\sigma \mathrm{d} A,
\end{aligned}
$$
where $C_1, C_2$ are the heat capacities of the two gases, $T$ is the temperature of the rubber, $A=4 \pi r^2, V_1=(4 / 3) \pi r^3$, and $V_2=V-V_1$. Hence show that the maximum entropy occurs when
$$
\left(p_1-p_2\right) \mathrm{d} V_1=\sigma \mathrm{d} A
$$
where $p_1, p_2$ are the pressures in the fluids.
(ii) Now suppose the chamber is in thermal contact with a reservoir at fixed temperature. Show that, in an isothermal change, the free energy is governed by
$$
\mathrm{d} F=\sigma \mathrm{d} A-p_1 \mathrm{~d} V_1-p_2 \mathrm{~d} V_2,
$$
and hence obtain the equilibrium condition again.

Khoobchandra Agrawal

Numerade Educator

02:47

Problem 7

By starting from the entropy argument given in Section 17.1.3, establish the following stability conditions:
$$
\begin{array}{ll}
\left.\frac{\partial^2 U}{\partial S^2}\right|_{V, N} \geq 0, & \left.\frac{\partial^2 U}{\partial V^2}\right|_{S, N} \geq 0, \\
\left.\frac{\partial^2 F}{\partial T^2}\right|_{V, N} \leq 0, & \left.\frac{\partial^2 F}{\partial V^2}\right|_{T, N} \geq 0, \\
\left.\frac{\partial^2 H}{\partial S^2}\right|_{p, N} \geq 0, & \left.\frac{\partial^2 H}{\partial p^2}\right|_{S, N} \leq 0, \\
\left.\frac{\partial^2 G}{\partial T^2}\right|_{p, N} \leq 0, & \left.\frac{\partial^2 G}{\partial p^2}\right|_{T, N} \leq 0 .
\end{array}
$$
[See Appendix B for the method.] In summary, the thermodynamic potentials are convex functions of their extensive variables and concave functions of their intensive variables.

Narayan Hari

Numerade Educator