Chapter 3

Electromagnetic radiation in an evacuated vessel of volume \(V\) at equilibrium with the walls at temperature \(T\) (blackbody radiation) behaves like a gas of photons having internal energy \(U=a V T^{4}\) and pressure \(P=1 / 3 a T^{4}\), where \(a\) is Stefan's constant. (a) Plot the closed curve in the \(P-V\) plane for a Carnot cycle using blackbody radiation. (b) Derive explicitly the efficiency of a Carnot engine which uses blackbody radiation as its working substance.

A Carnot engine uses a paramagnetic substance as its working substance. The equation of state is \(M=n D H / T\), where \(M\) is the magnetization, \(H\) is the magnetic field, \(n\) is the number of moles, \(D\) is a constant determined by the type of substance, and \(T\) is the temperature. (a) Show that the internal energy \(U\), and therefore the heat capacity \(C_{M}\), can only depend on the temperature and not the magnetization. Let us assume that \(C_{M}=C=\) constant. (b) Sketch a typical Carnot cycle in the \(M-H\) plane. (c) Compute the total heat absorbed and the total work done by the Carnot engine. (d) Compute the efficiency of the Carnot engine.

A heat engine uses blackbody radiation as its operating substance. The equation of state for blackbody radiation is \(P=1 / 3 a T^{4}\) and the internal energy is \(U=a V T^{4}\), where \(a=7.566 \times 10^{-16}\) \(\mathrm{J} /\left(\mathrm{m}^{3} \mathrm{~K}^{4}\right)\) is Stefan's constant, \(P\) is pressure, \(T\) is temperature, and \(V\) is volume. The engine cycle consists of three steps. Process \(1 \rightarrow 2\) is an expansion at constant pressure \(P_{1}=P_{2} .\) Process \(2 \rightarrow 3\) is a decrease in pressure from \(P_{2}\) to \(P_{3}\) at constant volume \(V_{2}=V_{3}\). Process \(3 \rightarrow 1\) is an adiabatic contraction from volume \(V_{3}\) to \(V_{1}\). Assume that \(P_{1}=3.375 P_{3}, T_{1}=2000 \mathrm{~K}\), and \(V_{1}=10^{-3} \mathrm{~m}^{3}\). (a) Express \(V_{2}\) in terms of \(V_{1}\) and \(T_{1}=T_{2}\) in terms of \(T_{3}\) (b) Compute the work done during each part of the cycle. (c) Compute the heat absorbed during each part of the cycle. (d) What is the efficiency of this heat engine (get a number)? (e) What is the efficiency of a Carnot engine operating between the highest and lowest temperatures.

Experimentally one finds that for a rubber band $$ \begin{aligned} &\left(\frac{\partial J}{\partial L}\right)_{\mathrm{T}, M}=\frac{a T}{L_{0}}\left[1+2\left(\frac{L_{0}}{L}\right)^{3}\right] \quad \text { and } \\\ &\left(\frac{\partial J}{\partial T}\right)_{L, M}=\frac{a L}{L_{0}}\left[1-\left(\frac{L_{0}}{L}\right)^{3}\right] \end{aligned} $$ where \(J\) is the tension, \(a=1.0 \times 10^{3} \mathrm{dyn} / \mathrm{K}\), and \(L_{0}=0.5 \mathrm{~m}\) is the length of the band when no tension is applied. The mass \(M\) of the rubber band is held fixed. (a) Compute \((\partial L / \partial T)_{J, M}\) and discuss its physical meaning. (b) Find the equation of state and show that \(\mathrm{d} J\) is an exact differential. (c) Assume that the heat capacity at constant length is \(C_{L}=1.0 \mathrm{~J} / \mathrm{K}\). Find the work necessary to stretch the band reversibly and adiabatically to a length of \(1 \mathrm{~m}\). Assume that when no tension is applied, the temperature of the band is \(T=290 \mathrm{~K}\). What is the change in temperature?

Blackbody radiation in a box of volume \(V\) and at temperature \(T\) has internal energy \(U=a V T^{4}\) and pressure \(P=1 / 3 a T^{4}\), where \(a\) is the Stefan- Boltzmann constant. (a) What is the fundamental equation for blackbody radiation (the entropy)? (b) Compute the chemical potential.

For a low-density gas the virial expansion can be terminated at first order in the density and the equation of state is \( P=\frac{N k_{\mathrm{B}} T}{V}\left[1+\frac{N}{V} B_{2}(T)\right] $$ where \)B_{2}(T)\( is the second virial coefficient. The heat capacity will have corrections to its ideal gas value. We can write it in the form $$ C_{V, N}=\frac{3}{2} N k_{\mathrm{B}}-\frac{N^{2} k_{\mathrm{B}}}{V} F(T) $$ (a) Find the form that \)F(T)\( must have in order for the two equations to be thermodynamically consistent. (b) Find \)C_{P, N}$. (c) Find the entropy and internal energy.

Compute the entropy, enthalpy, Helmholtz free energy, and Gibbs free energy of a paramagnetic substance and write them explicitly in terms of their natural variables when possible. Assume that the mechanical equation of state is \(m=(D H / T)\) and that the molar heat capacity at constant magnetization is \(c_{\mathrm{m}}=c\), where \(m\) is the molar magnetization, \(H\) is the magnetic field, \(D\) is a constant, \(c\) is a constant, and \(T\) is the temperature.

Show that \(T \mathrm{~d} s=c_{x}(\partial T / \partial Y)_{x} \mathrm{~d} Y+c_{Y}(\partial T / \partial x)_{Y} \mathrm{~d} x\), where \(x=X / n\) is the amount of extensive variable, \(X\), per mole, \(c_{x}\) is the heat capacity per mole at constant \(x\), and \(c_{Y}\) is the heat capacity per mole at constant \(Y\).

Compute the heat capacity at constant magnetic field \(C_{H, n}\), the susceptibilities \(X_{T}, \mathrm{n}\) and \(X_{S, n}\) ' and the thermal expansivity \(\alpha_{H, n}\) for a magnetic system, given that the mechanical equation of state is \(M=\mathrm{n} D H / T\) and the heat capacity is \(C_{M, n}=\pi c_{\prime}\) where \(M\) is the magnetization, \(H\) is the magnetic field, \(\mathrm{n}\) is the number of moles, \(D\) is a constant, \(c\) is the molar heat capacity, and \(T\) is the temperature.

A material is found to have a thermal expansivity \(\alpha_{P}=R / P v+a / R T^{2} v\) and an isothermal compressibility \(K_{T}=1 / v(T f(P)+(b / P))\) where \(v=V / n\) is the molar volume. (a) Find \(f(P)\). (b) Find the equation of state. (c) Under what conditions is this material mechanically stable?

A boy blows a soap bubble of radius \(R\) which floats in the air a few moments before breaking. What is the difference in pressure between the air inside the bubble and the air outside the bubble when (a) \(R=1 \mathrm{~cm}\) and (b) \(R=1 \mathrm{~mm}\) ? The surface tension of the soap solution is \(\sigma=25\) \(\mathrm{dyn} / \mathrm{cm}\). (Note that soap bubbles have two surfaces.)

A stochastic process, involving three fluctuating quantities, \(x_{1}, x_{2}\), and \(x_{3}\), has a probability distribution $$ P\left(x_{1}, x_{2}, x_{3}\right)=C \exp \left[-\frac{1}{2}\left(2 x_{1}^{2}+2 x_{1} x_{2}+4 x_{2}^{2}+2 x_{1} x_{3}+2 x_{2} x_{3}+2 x_{3}^{2}\right)\right] $$ where \(C\) is the normalization constant. (a) Write probability distribution in the form \(P\left(x_{1}, x_{2}, x_{3}\right)=C \exp \left(-1 / 2 x^{T} \cdot g+x\right)\), where \(g\) is a \(3 \times 3\) symmetric matrix, \(x\) is a column matrix with matrix elements \(x_{i}, i=1,2,3\), and \(x^{T}\) is its transpose. Obtain the matrix \(\boldsymbol{g}\) and its inverse \(g^{-1}\). (b) Find the eigenvalues \(\lambda_{i}(i=1,2,3)\) and orthonormal eigenvectors of \(\boldsymbol{g}\) and obtain the \(3 \times 3\) orthogonal matrix \(\boldsymbol{O}\) that diagonalizes the matrix \(\boldsymbol{g}\) (get numbers for all of them). Using this orthogonal matrix, we can write \(x^{\mathrm{T}} \cdot g \cdot x=x^{\mathrm{T}} \cdot \boldsymbol{O}^{\mathrm{T}} \cdot \boldsymbol{O} \cdot g \cdot \boldsymbol{O}^{\mathrm{T}} \cdot \boldsymbol{O} \cdot \boldsymbol{x}=\boldsymbol{a}^{\mathrm{T}} \cdot \bar{\Lambda} \cdot \boldsymbol{a}=\sum_{i=1}^{3} \lambda_{i} a_{i}^{2}\) where \(\boldsymbol{O} \cdot g \cdot \boldsymbol{O}^{\mathrm{T}}=\bar{\Lambda}\) is a \(3 \mathrm{x}\) 3 diagonal matrix with matrix elements \((\bar{A})_{i, j}=\lambda_{i} \delta_{i, j}\) and \(\boldsymbol{O} \cdot \boldsymbol{x}=\boldsymbol{a}\) is a column matrix with elements, \(a_{i}(i=1,2,3)\). (c) Compute the normalization constant, C. (d) Compute the moments \(\left(x_{i}\right)(i=1,2,3),\left\langle x_{i} x_{j}\right\rangle(i=1,2,3, j=1,2,3)\left(x_{1}^{2} x_{2} x_{3}\right)\) and \(\left\langle x_{1} x_{2}^{2} x_{3}\right\rangle+\) (Note that Exercises \(\mathrm{A.7}\) and \(\mathrm{A} .8\) might be helpful.)

A monatomic fluid in equilibrium is contained in a large insulated box of volume \(V\). The fluid is divided (conceptually) into \(m\) cells, each of which has an average number of particles \(N_{0}\), where \(N_{0}\) is large (neglect coupling between cells). Compute the variance in fluctuations of internal energy per particle \(u=U / N,\left\langle\left(\Delta u_{i}\right)^{2}\right\rangle\), in the ith cell. (Hint: Use temperature \(T\) and volume per particle \(v=V / N\) as independent variables.)

A van der Waals gas can be cooled by free expansion. Since no work is done and no heat is added during free expansion, the internal energy remains constant. An infinitesimal change in volume \(\mathrm{d} V\) causes an infinitesimal temperature change in \(\mathrm{d} T\), where $$ \mathrm{d} T=\left(\frac{\partial T}{\partial V}\right)_{U, \mathrm{n}} \mathrm{d} V $$ (a) Compute the Joule coefficient \({ }^{(\partial T / \partial V)}_{U, \mathrm{H}}\) for a van der Waals gas (note that the heat capacity \(C_{V, n}\) is independent of volume and use \(C_{V, n}=3 / 2 \mathrm{n} R\) ) (b) Compute the change in temperature of one mole of oxygen \(\left(\mathrm{O}_{2}\right)\) and one mole of carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) if they each expand from an initial volume \(V_{\mathrm{i}}=10^{-3} \mathrm{~m}^{3}\) at temperature \(T_{\mathrm{i}}=300 \mathrm{~K}\) to a final volume \(V_{\mathrm{f}}=\infty\). (For \(\mathrm{O}_{2}\) the van der Waals constant is \(a=0.1382 \mathrm{~Pa} \mathrm{~m}^{6} / \mathrm{mol}^{2}\) and for \(\mathrm{CO}_{2}\) it is \(a=0.3658 \mathrm{~Pa} \mathrm{~m}^{6} / \mathrm{mol}^{2}\).)

Two containers, each of volume \(V\), contain ideal gas held at temperature \(T\) and pressure \(P\). The gas in chamber 1 consists of \(N_{1, a}\) molecules of type \(a\) and \(N_{1, b}\) molecules of type \(b\). The gas in chamber 2 consists of \(N_{2, a}\) molecules of type \(a\) and \(N_{2, b}\) molecules of type \(b\). Assume that \(N_{1, a}+N_{1, b}=\) \(N_{2, a}+N_{2, b}\). The gases are allowed to mix so the final temperature is \(T\) and the final pressure is \(P\). (a) Compute the entropy of mixing. (b) What is the entropy of mixing if \(N_{1, \mathrm{a}}=N_{2, \mathrm{a}}\) and \(N_{1, \mathrm{~b}}=N_{2, \mathrm{~b}}\). (c) What is the entropy of mixing if \(N_{1, \mathrm{a}}=N_{2, \mathrm{~b}}\) and \(N_{1, \mathrm{~b}}=N_{2, \mathrm{a}}=0\). Discuss your results for (b) and (c).

An insulated box with fixed total volume \(V\) is partitioned into \(m\) insulated compartments, each containing an ideal gas of a different molecular species. Assume that each compartment has the same pressure but a different number of moles, a different temperature, and ( \(\left.\left.P, \mathrm{n}_{i}, T_{\mathrm{i}}, V_{\mathrm{i}}\right)_{-}\right)\)a different volume. (The thermodynamic variables for the ith compartment are If all partitions are suddenly removed and the system is allowed to reach equilibrium: (a) Find the final temperature and pressure, and the entropy of mixing. (Assume that the particles are monatomic.) (b) For the special case of \(m=2\) and parameter \(\mathrm{n}_{1}=1 \mathrm{~mol}, T_{1}=300 \mathrm{~K}, V_{1}=1 \mathrm{l}, \mathrm{n}_{2}=3 \mathrm{~mol}_{1}\) and \(V_{2}=2\) 1, obtain numerical values for all parameters in part (a).

A tiny sack made of membrane permeable to water but not \(\mathrm{NaCl}\) (sodium chloride) is filled with a \(1 \%\) solution (by weight) of \(\mathrm{NaCl}\) and water and is immersed in an open beaker of pure water at \(38{ }^{\circ} \mathrm{C}\) at a depth of \(1 \mathrm{ft}\). (a) What osmotic pressure is experienced by the sack? (b) What is the total pressure of the solution in the sack (neglect surface tension)? Assume that the sack is small enough that the pressure of the surrounding water can be assumed constant. (An example of such a sack is a human blood cell.)

A biological molecule of unknown mass can be prepared in pure powdered form. If 15 \(\mathrm{g}\) of this powder is added to a container with \(1 \mathrm{~L}\) of water at \(T=300 \mathrm{~K}\), which is initially at atmospheric pressure, the pressure inside the container increases to \(P=1.3 \mathrm{~atm}\). (a) What is the molecular weight of the biological molecules? (b) What is the mass of each molecule expressed in atomic units?

A solution of particles A and B has a Gibbs free energy $$ \begin{aligned} G\left(P, T, \mathrm{n}_{\mathrm{A}}, \mathrm{n}_{\mathrm{B}}\right)=& \mathrm{n}_{\mathrm{A}} g_{\mathrm{A}}(P, T)+\mathrm{n}_{\mathrm{B}} g_{\mathrm{B}}(P, T)+\frac{1}{2} \lambda_{\mathrm{AA}} \frac{\mathrm{n}_{\mathrm{A}}^{2}}{\mathrm{n}}+\frac{1}{2} \lambda_{\mathrm{BB}} \frac{\mathrm{n}_{\mathrm{B}}^{2}}{\mathrm{n}} \\ &+\lambda_{\mathrm{AB}} \frac{\mathrm{n}_{\mathrm{A}} \mathrm{n}_{\mathrm{B}}}{\mathrm{n}}+\mathrm{n}_{\mathrm{A}} R T \ln x_{\mathrm{A}}+\mathrm{n}_{\mathrm{B}} R T \ln x_{\mathrm{B}} \end{aligned} $$ Initially, the solution has \(\mathrm{n}_{\mathrm{A}}\) moles of A and \(\mathrm{n}_{\mathrm{B}}\) moles of B. (a) If an amount \(\Delta \mathrm{n}_{\mathrm{B} \text {, of } \mathrm{B} \text { is added }}\) keeping the pressure and temperature fixed, what is the change in the chemical potential of A? (b) For the case \(\lambda_{\mathrm{AA}}=\lambda_{\mathrm{BB}}=\lambda_{\mathrm{AB} \text {, }}\) does the chemical potential of A increase or decrease?

Consider the reaction $$ 2 \mathrm{NH}_{3}=\mathrm{N}_{2}+3 \mathrm{H}_{2} $$ which occurs in the gas phase. Start initially with \(2 \mathrm{~mol}\) of \(\mathrm{NH}_{3}\) and 0 mol each of \(\mathrm{H}_{2}\) and \(\mathrm{N}_{2}\). Assume that the reaction occurs at temperature \(T\) and pressure \(P\). Use ideal gas equations for the chemical potential. (a) Compute and plot the Gibbs free energy, \(G(T, P\), (5), as a function of the degree of reaction, \(\xi\), for (i) \(P=1\) atm and \(T=298 \mathrm{~K}\) and (ii) \(P=1 \mathrm{~atm}\) and \(T\). \(=894 \mathrm{~K}\). (b) Compute and plot the affinity, \(A(T, P, \xi)\), as a function of the degree of reaction, \(\xi\), for (i) \(P=1 \mathrm{~atm}\) and \(T=298 \mathrm{~K}\) and (ii) \(P=1\) atm and \(T=894 \mathrm{~K}\). (c) What is the degree of reaction, \(\xi\), at chemical equilibrium for \(P=1 a t m\) and temperature \(T=894\) K? How many moles of \(\mathrm{NH}_{3}, \mathrm{H}_{2}\), and \(\mathrm{N}_{2}\) are present at equilibrium? (d) If initially the volume is \(V_{0}\), what is the volume at equilibrium for \(P=\) \(1 \mathrm{~atm}\) and \(T=894 \mathrm{~K}\) ? (e) What is the heat of reaction for \(P=1 \mathrm{~atm}\) and \(T=894 \mathrm{~K}\) ?