A vessel initially contains 1 kmol of H2 and 4 kmol of N2.

Determine the change in the Gibbs function \(\Delta G^{\circ}\) at \(25^{\circ} \mathrm{C}\) in kJ/kmol, for the reaction \(\mathrm{CH}_{4}(\mathrm{~g})+2 \mathrm{O}_{2} \rightleftarrows \mathrm{CO}_{2}+2 \mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) using (a) Gibbs function of formation data. (b) enthalpy of formation and absolute entropy data.

Calculate the equilibrium constant, expressed as \(\log _{10} K\), for \(\mathrm{CO}_{2} \rightleftarrows \mathrm{CO}+\frac{1}{2} \mathrm{O}_{2}\) at (a) 500 K, (b) \(1800^{\circ} \mathrm{R}\). Compare with values from Table A-27

Calculate the equilibrium constant, expressed as \(\log _{10} K\), for the water-gas shift reaction \(\mathrm{CO}+\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \rightleftarrows \mathrm{CO}_{2}+\mathrm{H}_{2}\) at (a) 298 K, (b) 1000 K. Compare with values from Table A-27.

Calculate the equilibrium constant, expressed as \(\log _{10} K\), for \(\mathrm{H}_{2} \mathrm{O} \rightleftarrows \mathrm{H}_{2}+\frac{1}{2} \mathrm{O}_{2}\) at (a) 298 K, (b) \(3600^{\circ} \mathrm{R}\). Compare with values from Table A-27

Using data from Table A-27, determine \(\log _{10} K\) at 2500 K for (a) \(\mathrm{H}_{2} \mathrm{O} \rightleftarrows \mathrm{H}_{2}+\frac{1}{2} \mathrm{O}_{2}\). (b) \(\mathrm{H}_{2}+\frac{1}{2} \mathrm{O}_{2} \rightleftarrows \mathrm{H}_{2} \mathrm{O}\). (c) \(2 \mathrm{H}_{2} \mathrm{O} \rightleftarrows 2 \mathrm{H}_{2}+\mathrm{O}_{2}\).

In Table A-27, \(\log _{10} K\) is nearly linear in 1/T: \(\log _{10} K=C_{1}+C_{2} / T\), where \(C_{1}\) and \(C_{2}\) are constants. For selected reactions listed in the table. (a) verify this by plotting \(\log _{10} K\) versus 1/T for temperature ranging from 2100 to 2500 K. (b) evaluate \(C_{1}\) and \(C_{2}\) for any pair of adjacent table entries in the temperature range of part (a).

Determine the relationship between the ideal gas equilibrium constants \(K_{1}\) and \(K_{2}\) for the following two alternative ways of expressing the ammonia synthesis reaction: 1. \(\frac{1}{2} \mathrm{~N}_{2}+\frac{3}{2} \mathrm{H}_{2} \rightleftarrows \mathrm{NH}_{3}\) 2. \(\mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftarrows 2 \mathrm{NH}_{3}\)

Consider the reactions 1. \(\mathrm{CO}+\mathrm{H}_{2} \mathrm{O} \rightleftarrows \mathrm{H}_{2}+\mathrm{CO}_{2}\) 2. \(2 \mathrm{CO}_{2} \rightleftarrows 2 \mathrm{CO}+\mathrm{O}_{2}\) 3. \(2 \mathrm{H}_{2} \mathrm{O} \rightleftarrows 2 \mathrm{H}_{2}+\mathrm{O}_{2}\) Show that \(K_{1}=\left(K_{3} / K_{2}\right)^{1 / 2}\).

Consider the reactions 1. \(\mathrm{CO}_{2}+\mathrm{H}_{2} \rightleftarrows \mathrm{CO}+\mathrm{H}_{2} \mathrm{O}\) 2. \(\mathrm{CO}_{2} \rightleftarrows \mathrm{CO}+\frac{1}{2} \mathrm{O}_{2}\) 3. \(\mathrm{H}_{2} \mathrm{O} \rightleftarrows \mathrm{H}_{2}+\frac{1}{2} \mathrm{O}_{2}\) (a) Show that \(K_{1}=K_{2} / K_{3}\). (b) Evaluate \(\log _{10} K_{1}\) at 298 K, 1 atm using the expression from part (a), together with \(\log _{10} K\) data from Table A-27. (c) Check the value for \(\log _{10} K_{1}\) obtained in part (b) by applying Eq. 14.31 to reaction 1

Evaluate the equilibrium constant at 2000 K for \(\mathrm{CH}_{4}+\mathrm{H}_{2} \mathrm{O} \rightleftarrows 3 \mathrm{H}_{2}+\mathrm{CO}\). At 2000 K, \(\log _{10} K=7.469\) for \(\mathrm{C}+\frac{1}{2} \mathrm{O}_{2} \rightleftarrows \mathrm{CO}\), and \(\log _{10} K=-3.408\) for \(\mathrm{C}+2 \mathrm{H}_{2} \rightleftarrows \mathrm{CH}_{4}\).

For each of the following dissociation reactions, determine the equilibrium compositions: (a) One kmol of \(\mathrm{N}_{2} \mathrm{O}_{4}\) dissociates to form an equilibrium ideal gas mixture of \(\mathrm{N}_{2} \mathrm{O}_{4}\) and \(\mathrm{NO}_{2}\) at \(25^{\circ} \mathrm{C}\), 2 atm. For \(\mathrm{N}_{2} \mathrm{O}_{4} \rightleftarrows 2 \mathrm{NO}_{2}, \Delta G^{\circ}=5400 \mathrm{~kJ} / \mathrm{kmol}\) at \(25^{\circ} \mathrm{C}\). (b) One kmol of \(\mathrm{CH}_{4}\) dissociates to form an equilibrium ideal gas mixture of \(\mathrm{CH}_{4}, \mathrm{H}_{2}\), and C at 1000 K, 5 atm. For \(\mathrm{C}+2 \mathrm{H}_{2} \rightleftarrows \mathrm{CH}_{4}, \log _{10} K=1.011\) at 1000 K.

Determine the extent to which dissociation occurs in the following cases: One lbmol of \(\mathrm{H}_{2} \mathrm{O}\) dissociates to form an equilibrium mixture of \(\mathrm{H}_{2} \mathrm{O}, \mathrm{H}_{2}\), and \(\mathrm{O}_{2}\) at \(4740^{\circ} \mathrm{F}\), 1.25 atm. One lbmol of \(\mathrm{CO}_{2}\) dissociates to form an equilibrium mixture of \(\mathrm{CO}_{2}, \mathrm{CO}\), and \(\mathrm{O}_{2}\) at the same temperature and pressure.

One lbmol of carbon reacts with 2 lbmol of oxygen \(\left(\mathrm{O}_{2}\right)\) to form an equilibrium mixture of \(\mathrm{CO}_{2}\), CO, and \(\mathrm{O}_{2}\) at \(4940^{\circ} \mathrm{F}\), 1 atm. Determine the equilibrium composition.

The following exercises involve oxides of nitrogen: (a) One kmol of \(\mathrm{N}_{2} \mathrm{O}_{4}\) dissociates at \(25^{\circ} \mathrm{C}\), 1 atm to form an equilibrium ideal gas mixture of \(\mathrm{N}_{2} \mathrm{O}_{4}\) and \(\mathrm{NO}_{2}\) in which the amount of \(\mathrm{N}_{2} \mathrm{O}_{4}\) present is 0.8154 kmol. Determine the amount of \(\mathrm{N}_{2} \mathrm{O}_{4}\) that would be present in an equilibrium mixture at \(25^{\circ} \mathrm{C}\), 0.5 atm. (b) A gaseous mixture consisting of 1 kmol of NO, 10 kmol of \(\mathrm{O}_{2}\), and 40 kmol of \(\mathrm{N}_{2}\) reacts to form an equilibrium ideal gas mixture of \(\mathrm{NO}_{2}\), NO, and \(\mathrm{O}_{2}\) at 500 K, 0.1 atm. Determine the composition of the equilibrium mixture. For \(\mathrm{NO}+\frac{1}{2} \mathrm{O}_{2} \rightleftarrows \mathrm{NO}_{2}\), K = 120 at 500 K. (c) An equimolar mixture of \(\mathrm{O}_{2}\) and \(\mathrm{N}_{2}\) reacts to form an equilibrium ideal gas mixture of \(\mathrm{O}_{2}, \mathrm{N}_{2}\), and NO. Plot the mole fraction of NO in the equilibrium mixture versus equilibrium temperature ranging from 1200 to 2000 K. Why are oxides of nitrogen of concern?

One kmol of \(\mathrm{CO}_{2}\) dissociates to form an equilibrium ideal gas mixture of \(\mathrm{CO}_{2}\), CO, and \(\mathrm{O}_{2}\) at temperature T and pressure p. (a) For T = 3000 K, plot the amount of CO present, in kmol, versus pressure for \(1 \leq p \leq 10\) atm. (b) For p = 1 atm, plot the amount of CO present, in kmol, versus temperature for \(2000 \leq T \leq 3500 \mathrm{\ K}\).

One lbmol of \(\mathrm{H}_{2} \mathrm{O}\) dissociates to form an equilibrium ideal gas mixture of \(\mathrm{H}_{2} \mathrm{O}, \mathrm{H}_{2}\), and \(\mathrm{O}_{2}\) at temperature T and pressure p. (a) For \T=5400^{\circ} \mathrm{R}(\), plot the amount of \(\mathrm{H}_{2}\) present, in lbmol, versus pressure ranging from 1 to 10 atm. (b) For p = 1 atm, plot the amount of \(\mathrm{H}_{2}\) present, in lbmol, versus temperature ranging from 3600 to \(6300^{\circ} \mathrm{R}\).

One lbmol of \(\mathrm{H}_{2} \mathrm{O}\) together with x lbmol of \(\mathrm{N}_{2}\) (inert) forms an equilibrium mixture at \(5400^{\circ} \mathrm{R}\), 1 atm consisting of \(\mathrm{H}_{2} \mathrm{O}, \mathrm{H}_{2}, \mathrm{O}_{2}\), and \(\mathrm{N}_{2}\). Plot the amount of \(\mathrm{H}_{2}\) present in the equilibrium mixture, in lbmol, versus x ranging from 0 to 2.

An equimolar mixture of CO and \(\mathrm{O}_{2}\) reacts to form an equilibrium mixture of \(\mathrm{CO}_{2}\), CO, and \(\mathrm{O}_{2}\) at 3000 K. Determine the effect of pressure on the composition of the equilibrium mixture. Will lowering the pressure while keeping the temperature fixed increase or decrease the amount of \(\mathrm{CO}_{2}\) present? Explain.

An equimolar mixture of CO and \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) reacts to form an equilibrium mixture of \(\mathrm{CO}_{2}\), CO, \(\mathrm{H}_{2} \mathrm{O}\), and \(\mathrm{H}_{2}\) at \(1727^{\circ} \mathrm{C}\), 1 atm. (a) Will lowering the temperature increase or decrease the amount of \(\mathrm{H}_{2}\) present? Explain. (b) Will decreasing the pressure while keeping the temperature constant increase or decrease the amount of \(\mathrm{H}_{2}\) present? Explain.

Determine the temperature, in K, at which 9% of diatomic hydrogen \(\left(\mathrm{H}_{2}\right)\) dissociates into monatomic hydrogen (H) at a pressure of 10 atm. For a greater percentage of \(\mathrm{H}_{2}\) at the same pressure, would the temperature be higher or lower? Explain.

Two kmol of \(\mathrm{CO}_{2}\) dissociate to form an equilibrium mixture of \(\mathrm{CO}_{2}\), CO, and \(\mathrm{O}_{2}\) in which 1.8 kmol of \(\mathrm{CO}_{2}\) is present. Plot the temperature of the equilibrium mixture, in K, versus the pressure p for \(0.5 \leq p \leq 10\) atm.

One kmol of \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) dissociates to form an equilibrium mixture of \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g}), \mathrm{H}_{2}\), and \(\mathrm{O}_{2}\) in which the amount of water vapor present is 0.95 kmol. Plot the temperature of the equilibrium mixture, in K, versus the pressure p for \(1 \leq p \leq10\) atm.

A vessel initially containing 1 kmol of \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) and x kmol of \(\mathrm{N}_{2}\) forms an equilibrium mixture at 1 atm consisting of \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\), \(\mathrm{H}_{2}\), \(\mathrm{O}_{2}\), and \(\mathrm{N}_{2}\) in which 0.5 kmol of \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) is present. Plot x versus the temperature T for \(3000 \leq T \leq 3600 \mathrm{\ K}\).

A vessel initially containing 2 lbmol of \(\mathrm{N}_{2}\) and 1 lbmol of \(\mathrm{O}_{2}\) forms an equilibrium mixture at 1 atm consisting of \(\mathrm{N}_{2}\), \(\mathrm{O}_{2}\), and NO. Plot the amount of NO formed versus temperature T for \(3600 \leq T \leq 6300^{\circ} \mathrm{R}\).

A vessel initially containing 1 kmol of CO and 4.76 kmol of dry air forms an equilibrium mixture of \(\mathrm{CO}_{2}\), \(\mathrm{CO}, \mathrm{O}_{2}\), and \(\mathrm{N}_{2}\) at 3000 K, 1 atm. Determine the equilibrium composition.

A vessel initially containing 1 kmol of \(\mathrm{O}_{2}\), 2 kmol of \(\mathrm{N}_{2}\), and 1 kmol of Ar forms an equilibrium mixture of \(\mathrm{O}_{2}, \mathrm{\ N}_{2}\), NO, and Ar at \(2727^{\circ} \mathrm{C}\), 1 atm. Determine the equilibrium composition.

One kmol of CO and 0.5 kmol of \(\mathrm{O}_{2}\) react to form a mixture at temperature T and pressure p consisting of \(\mathrm{CO}_{2}\), CO, and \(\mathrm{O}_{2}\). If 0.35 kmol of CO is present in an equilibrium mixture when the pressure is 1 atm, determine the amount of CO present in an equilibrium mixture at the same temperature if the pressure were 10 atm.

A vessel initially contains 1 kmol of \(\mathrm{H}_{2}\) and 4 kmol of \(\mathrm{N}_{2}\). An equilibrium mixture of \(\mathrm{H}_{2}\), H, and \(\mathrm{N}_{2}\) forms at 3000 K, 1 atm. Determine the equilibrium composition. If the pressure were increased while keeping the temperature fixed, would the amount of monatomic hydrogen in the equilibrium mixture increase or decrease? Explain.

Dry air enters a heat exchanger. An equilibrium mixture of \(\mathrm{N}_{2}\), \(\mathrm{O}_{2}\), and NO exits at \(3882^{\circ} \mathrm{F}\), 1 atm. Determine the mole fraction of NO in the exiting mixture. Will the amount of NO increase or decrease as temperature decreases at fixed pressure? Explain.

A gaseous mixture with a molar analysis of 20% \(\mathrm{CO}_{2}\), 40% CO, and 40% \(\mathrm{O}_{2}\) enters a heat exchanger and is heated at constant pressure. An equilibrium mixture of \(\mathrm{CO}_{2}\), CO, and \(\mathrm{O}_{2}\) exits at 3000 K, 1.5 bar. Determine the molar analysis of the exiting mixture.

An ideal gas mixture with the molar analysis 30% CO, 10% \(\mathrm{CO}_{2}\), 40% \(\mathrm{H}_{2} \mathrm{O}\), 20% inert gas enters a reactor operating at steady state. An equilibrium mixture of CO, \(\mathrm{CO}_{2}, \ \mathrm{H}_{2} \mathrm{O}, \ \mathrm{H}_{2}\), and the inert gas exits at 1 atm. (a) If the equilibrium mixture exits at 1200 K, determine on a molar basis the ratio of the \(\mathrm{H}_{2}\) in the equilibrium mixture to the \(\mathrm{H}_{2} \mathrm{O}\) in the entering mixture. (b) If the mole fraction of CO present in the equilibrium mixture is 7.5%, determine the temperature of the equilibrium mixture, in K.

A mixture of 1 kmol CO and 0.5 kmol \(\mathrm{O}_{2}\) in a closed vessel, initially at 1 atm and 300 K, reacts to form an equilibrium mixture of \(\mathrm{CO}_{2}\), CO, and \(\mathrm{O}_{2}\) at 2500 K. Determine the final pressure, in atm.

Methane burns with 90% of theoretical air to form an equilibrium mixture of \(\mathrm{CO}_{2}\), CO, \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g}), \mathrm{H}_{2}\), and \(\mathrm{N}_{2}\) at 1000 K, 1 atm. Determine the composition of the equilibrium mixture, per kmol of mixture.

Octane \(\left(\mathrm{C}_{8} \mathrm{H}_{18}\right)\) burns with air to form an equilibrium mixture of \(\mathrm{CO}_{2}, \ \mathrm{H}_{2}\), CO, \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\), and \(\mathrm{N}_{2}\) at 1700 K, 1 atm. Determine the composition of the products, in kmol per kmol of fuel, for an equivalence ratio of 1.2.

Acetylene gas \(\left(\mathrm{C}_{2} \mathrm{H}_{2}\right)\) at \(25^{\circ} \mathrm{C}\), 1 atm enters a reactor operating at steady state and burns with 40% excess air entering at \(25^{\circ} \mathrm{C}\), 1 atm, 80% relative humidity. An equilibrium mixture of \(\mathrm{CO}_{2}, \mathrm{H}_{2} \mathrm{O}, \mathrm{O}_{2}\), NO, and \(\mathrm{N}_{2}\) exits at 2200 K, 0.9 atm. Determine, per kmol of \(\mathrm{C}_{2} \mathrm{H}_{2}\) entering, the composition of the exiting mixture.

Carbon dioxide gas at \(25^{\circ} \mathrm{C}\), 5.1 atm enters a heat exchanger operating at steady state. An equilibrium mixture of \(\mathrm{CO}_{2}\), CO, and \(2527^{\circ} \mathrm{C}\) exits at \(\mathrm{O}_{2}\), 5 atm. Determine, per kmol of \(\mathrm{CO}_{2}\) entering, (a) the composition of the exiting mixture. (b) the heat transfer to the gas stream, in kJ. Neglect kinetic and potential energy effects.

Saturated water vapor at \(15 \mathrm{\ lbf} / \mathrm{in} .^{2}\) enters a heat exchanger operating at steady state. An equilibrium mixture of \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g}), \mathrm{H}_{2}\), and \(\mathrm{O}_{2}\) exits at \(4040^{\circ} \mathrm{F}\), 1 atm. Determine, per kmol of steam entering, (a) the composition of the exiting mixture. (b) the heat transfer to the flowing stream, in Btu. Neglect kinetic and potential energy effects.

Carbon at \(25^{\circ} \mathrm{C}\), 1 atm enters a reactor operating at steady state and burns with oxygen entering at \(127^{\circ} \mathrm{C}\), 1 atm. The entering streams have equal molar flow rates. An equilibrium mixture of \(\mathrm{CO}_{2}\), CO, and \(\mathrm{O}_{2}\) exits at \(2727^{\circ} \mathrm{C}\), 1 atm. Determine, per kmol of carbon, (a) the composition of the exiting mixture. (b) the heat transfer between the reactor and its surroundings, in kJ. Neglect kinetic and potential energy effects.

An equimolar mixture of carbon monoxide and water vapor at \(200^{\circ} \mathrm{F}\), 1 atm enters a reactor operating at steady state. An equilibrium mixture of \(\mathrm{CO}_{2}\), CO, \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\), and \(\mathrm{H}_{2}\) exits at \(2040^{\circ} \mathrm{F}\), 1 atm. Determine the heat transfer between the reactor and its surroundings, in Btu per lbmol of CO entering. Neglect kinetic and potential energy effects.

Carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) and oxygen \(\left(\mathrm{O}_{2}\right)\) in a 1:2 molar ratio enter a reactor operating at steady state in separate streams at 1 atm, \(127^{\circ} \mathrm{C}\) and 1 atm, \(277^{\circ} \mathrm{C}\), respectively. An equilibrium mixture of \(\mathrm{CO}_{2}\), CO, and \(\mathrm{O}_{2}\) exits at 1 atm. If the mole fraction of CO in the exiting mixture is 0.1, determine the rate of heat transfer from the reactor, in kJ per kmol of \(\mathrm{CO}_{2}\) entering. Ignore kinetic and potential energy effects.

Methane gas at \(25^{\circ} \mathrm{C}\), 1 atm enters a reactor operating at steady state and burns with 80% of theoretical air entering at \(227^{\circ} \mathrm{C}\), 1 atm. An equilibrium mixture of \(\mathrm{CO}_{2}\), CO, \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\), \(\mathrm{H}_{2}\), and \(\mathrm{N}_{2}\) exits at \(1427^{\circ} \mathrm{C}\), 1 atm. Determine, per kmol of methane entering, (a) the composition of the exiting mixture. (b) the heat transfer between the reactor and its surroundings, in kJ. Neglect kinetic and potential energy effects.

Gaseous propane \(\left(\mathrm{C}_{3} \mathrm{H}_{8}\right)\) at \(25^{\circ} \mathrm{C}\), 1 atm enters a reactor operating at steady state and burns with 80% of theoretical air entering separately at \(25^{\circ} \mathrm{C}\), 1 atm. An equilibrium mixture of \(\mathrm{CO}_{2}\), CO, \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g}), \mathrm{H}_{2}\), and \(\mathrm{N}_{2}\) exits at \(1227^{\circ} \mathrm{C}\), 1 atm. Determine the heat transfer between the reactor and its surroundings, in kJ per kmol of propane entering. Neglect kinetic and potential energy effects.

Gaseous propane \(\left(\mathrm{C}_{3} \mathrm{H}_{8}\right)\) at \(77^{\circ} \mathrm{F}\), 1 atm enters a reactor operating at steady state and burns with the theoretical amount of air entering separately at \(240^{\circ} \mathrm{F}\), 1 atm. An equilibrium mixture of \(\mathrm{CO}_{2}\), CO, \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \mathrm{O}_{2}\), and \(\mathrm{N}_{2}\) exits at \(3140^{\circ} \mathrm{F}\), 1 atm. Determine the heat transfer between the reactor and its surroundings, in Btu per lbmol of propane entering. Neglect kinetic and potential energy effects.

One kmol of \(\mathrm{CO}_{2}\) in a piston–cylinder assembly, initially at temperature T and 1 atm, is heated at constant pressure until a final state is attained consisting of an equilibrium mixture of \(\mathrm{CO}_{2}\), CO, and \(\mathrm{O}_{2}\) in which the amount of \(\mathrm{CO}_{2}\) present is 0.422 kmol. Determine the heat transfer and the work, each in kJ, if T is (a) 298 K, (b) 400 K.

Hydrogen gas \(\left(\mathrm{H}_{2}\right)\) at \(25^{\circ} \mathrm{C}\), 1 atm enters an insulated reactor operating at steady state and reacts with 250% excess oxygen entering at \(227^{\circ} \mathrm{C}\), 1 atm. The products of combustion exit at 1 atm. Determine the temperature of the products, in K, if (a) combustion is complete. (b) an equilibrium mixture of \(\mathrm{H}_{2} \mathrm{O}, \mathrm{H}_{2}\), and \(\mathrm{O}_{2}\) exits. Kinetic and potential energy effects are negligible.

For each case of Problem 14.45, determine the rate of entropy production, in kJ/K per kmol of \(\mathrm{H}_{2}\) entering. What can be concluded about the possibility of achieving complete combustion?

Hydrogen \(\left(\mathrm{H}_{2}\right)\) at \(25^{\circ} \mathrm{C}\), 1 atm enters an insulated reactor operating at steady state and reacts with 100% of theoretical air entering at \(25^{\circ} \mathrm{C}\), 1 atm. The products of combustion exit at temperature T and 1 atm. Determine T, in K, if (a) combustion is complete. (b) an equilibrium mixture of \(\mathrm{H}_{2} \mathrm{O}, \mathrm{H}_{2}, \mathrm{O}_{2}\), and \(\mathrm{N}_{2}\) exits.

Methane at \(77^{\circ} \mathrm{F}\), 1 atm enters an insulated reactor operating at steady state and burns with 90% of theoretical air entering separately at \(77^{\circ} \mathrm{F}\), 1 atm. The products exit at 1 atm as an equilibrium mixture of \(\mathrm{CO}_{2}\), CO, \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \mathrm{H}_{2}\) and \(\mathrm{N}_{2}\). Determine the temperature of the exiting products, in \({ }^{\circ} \mathrm{R}\). Kinetic and potential energy effects are negligible.

Carbon monoxide at \(77^{\circ} \mathrm{F}\), 1 atm enters an insulated reactor operating at steady state and burns with air entering at \(77^{\circ} \mathrm{F}\), 1 atm. The products exit at 1 atm as an equilibrium mixture of \(\mathrm{CO}_{2}\), CO, \(\mathrm{O}_{2}\), and \(\mathrm{N}_{2}\). Determine the temperature of the equilibrium mixture, in \({ }^{\circ} \mathrm{R}\), if the combustion occurs with (a) 80% of theoretical air. (b) 100% of theoretical air. Kinetic and potential energy effects are negligible.

For each case of Problem 14.49, determine the rate of exergy destruction, in kJ per kmol of CO entering the reactor. Let \(T_{0}=537^{\circ} \mathrm{R}\).

Carbon monoxide at \(25^{\circ} \mathrm{C}\), 1 atm enters an insulated reactor operating at steady state and burns with excess oxygen \(\left(\mathrm{O}_{2}\right)\) entering at \(25^{\circ} \mathrm{C}\), 1 atm. The products exit at 2950 K, 1 atm as an equilibrium mixture of \(\mathrm{CO}_{2}\), CO, and \(\mathrm{O}_{2}\). Determine the percent excess oxygen. Kinetic and potential energy effects are negligible.

A gaseous mixture of carbon monoxide and the theoretical amount of air at \(260^{\circ} \mathrm{F}\), 1.5 atm enters an insulated reactor operating at steady state. An equilibrium mixture of \(\mathrm{CO}_{2}\), CO, \(\mathrm{O}_{2}\) and \(\mathrm{N}_{2}\) exits at 1.5 atm. Determine the temperature of the exiting mixture, in \({ }^{\circ} \mathrm{R}\). Kinetic and potential energy effects are negligible.

Methane at \(25^{\circ} \mathrm{C}\), 1 atm enters an insulated reactor operating at steady state and burns with oxygen entering at \(127^{\circ} \mathrm{C}\), 1 atm. An equilibrium mixture of \(\mathrm{CO}_{2}\), CO, \(\mathrm{O}_{2}\), and \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) exits at 3250 K, 1 atm. Determine the rate at which oxygen enters the reactor, in kmol per kmol of methane. Kinetic and potential energy effects are negligible.

Methane at \(77^{\circ} \mathrm{F}\), 1 atm enters an insulated reactor operating at steady state and burns with the theoretical amount of air entering at \(77^{\circ} \mathrm{F}\), 1 atm. An equilibrium mixture of \(\mathrm{CO}_{2}\), CO, \(\mathrm{O}_{2}, \mathrm{H}_{2} \mathrm{O}(\mathrm{g})\), and \(\mathrm{N}_{2}\) exits at 1 atm. (a) Determine the temperature of the exiting products, in \({ }^{\circ} \mathrm{R}\). (b) Determine the rate of exergy destruction, in Btu per lbmol of methane entering, for \(T_{0}=537^{\circ} \mathrm{R}\). Kinetic and potential energy effects are negligible.

Methane gas at \(25^{\circ} \mathrm{C}\), 1 atm enters an insulated reactor operating at steady state, where it burns with x times the theoretical amount of air entering at \(25^{\circ} \mathrm{C}\), 1 atm. An equilibrium mixture of \(\mathrm{CO}_{2}\), CO, \(\mathrm{O}_{2}, \mathrm{H}_{2} \mathrm{O}\), and \(\mathrm{N}_{2}\) exits at 1 atm. For selected values of x ranging from 1 to 4, determine the temperature of the exiting equilibrium mixture, in K. Kinetic and potential energy effects are negligible.

A mixture consisting of 1 kmol of carbon monoxide (CO), 0.5 kmol of oxygen \(\left(\mathrm{O}_{2}\right)\), and 1.88 kmol of nitrogen \(\left(\mathrm{N}_{2}\right)\), initially at \(227^{\circ} \mathrm{C}\), 1 atm, reacts in a closed, rigid, insulated vessel, forming an equilibrium mixture of \(\mathrm{CO}_{2}\), CO, \(\mathrm{O}_{2}\), and \(\mathrm{N}_{2}\). Determine the final equilibrium pressure, in atm.

A mixture consisting of 1 kmol of CO and the theoretical amount of air, initially at \(60^{\circ} \mathrm{C}\), 1 atm, reacts in a closed, rigid, insulated vessel to form an equilibrium mixture. An analysis of the products shows that there are 0.808 kmol of \(\mathrm{CO}_{2}\), 0.192 kmol of CO, and 0.096 kmol of \(\mathrm{O}_{2}\) present. The temperature of the final mixture is measured as \(2465^{\circ} \mathrm{C}\). Check the consistency of these data.

Estimate the enthalpy of reaction at 2000 K, in kJ/kmol, for \(\mathrm{CO}_{2} \rightleftarrows \mathrm{CO}+\frac{1}{2} \mathrm{O}_{2}\) using the van’t Hoff equation and equilibrium constant data. Compare with the value obtained for the enthalpy of reaction using enthalpy data.

Estimate the enthalpy of reaction at 2000 K, in kJ/kmol, for \(\mathrm{H}_{2} \mathrm{O} \rightleftarrows \mathrm{H}_{2}+\frac{1}{2} \mathrm{O}_{2}\), using the van’t Hoff equation and equilibrium constant data. Compare with the value obtained for the enthalpy of reaction using enthalpy data.

Estimate the equilibrium constant at 2800 K for \(\mathrm{CO}_{2} \rightleftarrows \mathrm{CO}+\frac{1}{2} \mathrm{O}_{2}\) using the equilibrium constant at 2000 K from Table A-27, together with the van’t Hoff equation and enthalpy data. Compare with the value for the equilibrium constant obtained from Table A-27.

Estimate the equilibrium constant at 2800 K for the reaction \(\mathrm{H}_{2} \mathrm{O} \rightleftarrows \mathrm{H}_{2}+\frac{1}{2} \mathrm{O}_{2}\) using the equilibrium constant at 2500 K from Table A-27, together with the van’t Hoff equation and enthalpy data. Compare with the value for the equilibrium constant obtained from Table A-27

At \(25^{\circ} \mathrm{C}\), \(\log _{10} K=8.9\) for \(\mathrm{C}+2 \mathrm{H}_{2} \rightleftarrows \mathrm{CH}_{4}\). Assuming that the enthalpy of reaction does not vary much with temperature, estimate the value of \(\log _{10} K\) at \(500^{\circ} \mathrm{C}\).

If the ionization-equilibrium constants for \(\mathrm{Cs} \rightleftarrows \mathrm{Cs}^{+}+\mathrm{e}^{-} 1600\) and 2000 K are K = 0.78 and K = 15.63, respectively, estimate the enthalpy of ionization, in kJ/kmol, at 1800 K using the van’t Hoff equation.

An equilibrium mixture at 2000 K, 1 atm consists of Cs, \(\mathrm{Cs}^{+}\), and \(\mathrm{e}^{-}\). Based on 1 kmol of Cs present initially, determine the percent ionization of cesium. At 2000 K, the ionization-equilibrium constant for \(\mathrm{Cs} \rightleftarrows \mathrm{Cs}^{+}+\mathrm{e}^{-}\) is K = 15.63.

An equilibrium mixture at \(18,000^{\circ} \mathrm{R}\) and pressure p consists of Ar, \(\mathrm{Ar}^{+}\), and \(\mathrm{e}^{-}\). Based on 1 lbmol of neutral argon present initially, plot the percent ionization of argon versus pressure for \(0.01 \leq p \leq 0.05\) atm. At \(18,000^{\circ} \mathrm{R}\), the ionization-equilibrium constant for \(\mathrm{Ar} \rightleftarrows \mathrm{Ar}^{+}+\mathrm{e}^{-}\) is \(K=4.2 \times 10^{-4}\).

At 2000 K and pressure p, 1 kmol of Na ionizes to form an equilibrium mixture of Na, \(\mathrm{Na}^{+}\), and \(\mathrm{e}^{-}\) in which the amount of Na present is x kmol. Plot the pressure, in atm, versus x for \(0.2 \leq x \leq 0.3\) kmol. At 2000 K, the ionization equilibrium constant for \(\mathrm{Na} \rightleftarrows \mathrm{Na}^{+}+\mathrm{e}^{-}\) is K = 0.668.

At 12,000 K and 6 atm, 1 kmol of N ionizes to form an equilibrium mixture of N, \(\mathrm{N}^{+}\), and \(\mathrm{e}^{-}\) in which the amount of N present is 0.95 kmol. Determine the ionization equilibrium constant at this temperature for \(\mathrm{N} \rightleftarrows \mathrm{N}^{+}+\mathrm{e}^{-}\).

Carbon dioxide \(\left(\mathrm{CO}_{2}\right)\), oxygen \(\left(\mathrm{O}_{2}\right)\), and nitrogen \(\left(\mathrm{N}_{2}\right)\) enter a reactor operating at steady state with equal molar flow rates. An equilibrium mixture of \(\mathrm{CO}_{2}, \mathrm{O}_{2}, \mathrm{\ N}_{2}\), CO, and NO exits at 3000 K, 5 atm. Determine the molar analysis of the equilibrium mixture.

An equimolar mixture of carbon monoxide and water vapor enters a heat exchanger operating at steady state. An equilibrium mixture of CO, \(\mathrm{CO}_{2}, \mathrm{O}_{2}, \mathrm{H}_{2} \mathrm{O}(\mathrm{g})\), and \(\mathrm{H}_{2}\) exits at \(2227^{\circ} \mathrm{C}\), 1 atm. Determine the molar analysis of the exiting equilibrium mixture.

A closed vessel initially contains a gaseous mixture consisting of 3 lbmol of \(\mathrm{CO}_{2}\), 6 lbmol of CO, and 1 lbmol of \(\mathrm{H}_{2}\). An equilibrium mixture at \(4220^{\circ} \mathrm{F}\), 1 atm is formed containing \(\mathrm{CO}_{2}\), CO, \(\mathrm{H}_{2} \mathrm{O}, \mathrm{H}_{2}\), and \(\mathrm{O}_{2}\). Determine the composition of the equilibrium mixture.

Butane \(\left(\mathrm{C}_{4} \mathrm{H}_{10}\right)\) burns with 100% excess air to form an equilibrium mixture at 1400 K, 20 atm consisting of \(\mathrm{CO}_{2}, \mathrm{O}_{2},\ \mathrm{H}_{2} \mathrm{O}(\mathrm{g}), \mathrm{N}_{2}\), NO, and \(\mathrm{NO}_{2}\). Determine the balanced reaction equation. For \(\mathrm{N}_{2}+2 \mathrm{O}_{2} \rightleftarrows 2 \mathrm{NO}_{2}\) at 1400 K, \(K=8.4 \times10^{-10}\).

One lbmol of \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) dissociates to form an equilibrium mixture at \(5000^{\circ} \mathrm{R}\), 1 atm consisting of \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g}), \mathrm{H}_{2}, \mathrm{O}_{2}\), and OH. Determine the equilibrium composition.

Steam enters a heat exchanger operating at steady state. An equilibrium mixture of \(\mathrm{H}_{2} \mathrm{O}, \mathrm{H}_{2}, \mathrm{O}_{2}\), H, and OH exits at temperature T, 1 atm. Determine the molar analysis of the exiting equilibrium mixture for (a) T = 2800 K. (b) T = 3000 K.

For a two-phase liquid–vapor mixture of water at \(100^{\circ} \mathrm{C}\), use tabulated property data to show that the specific Gibbs functions of the saturated liquid and saturated vapor are equal. Repeat for a two-phase liquid–vapor mixture of Refrigerant 134a at \(20^{\circ} \mathrm{C}\).

Using the Clapeyron equation, solve the following problems from Chap. 11: (a) 11.32, (b) 11.33, (c) 11.34, (d) 11.35, (e) 11.40.

A closed system at \(20^{\circ} \mathrm{C}\), 1 bar consists of a pure liquid water phase in equilibrium with a vapor phase composed of water vapor and dry air. Determine the departure, in percent, of the partial pressure of the water vapor from the saturation pressure of pure water at \(20^{\circ} \mathrm{C}\).

Derive an expression for estimating the pressure at which graphite and diamond exist in equilibrium at \(25^{\circ} \mathrm{C}\) in terms of the specific volume, specific Gibbs function, and isothermal compressibility of each phase at \(25^{\circ} \mathrm{C}\), 1 atm. Discuss.

An isolated system has two phases, denoted by A and B, each of which consists of the same two substances, denoted by 1 and 2. Show that necessary conditions for equilibrium are 1. the temperature of each phase is the same, \(T_{\mathrm{A}}=T_{\mathrm{B}}\). 2. the pressure of each phase is the same, \(p_{\mathrm{A}}=p_{\mathrm{B}}\). 3. the chemical potential of each component has the same value in each phase, \(\mu_{1}^{\mathrm{A}}=\mu_{1}^{\mathrm{B}}, \mu_{2}^{\mathrm{A}}=\mu_{2}^{\mathrm{B}}\).

An isolated system has two phases, denoted by A and B, each of which consists of the same two substances, denoted by 1 and 2. The phases are separated by a freely moving thin wall permeable only by substance 2. Determine the necessary conditions for equilibrium.

Referring to Problem 14.79, let each phase be a binary mixture of argon and helium and the wall be permeable only to argon. If the phases initially are at the conditions tabulated below, determine the final equilibrium temperature, pressure, and composition in the two phases.

Figure P14.81 shows an ideal gas mixture at temperature T and pressure p containing substance k, separated from a gas phase of pure k at temperature T and pressure \(p^{\prime}\) by a semipermeable membrane that allows only k to pass through. Assuming the ideal gas model also applies to the pure gas phase, determine the relationship between p and \(p^{\prime}\) for there to be no net transfer of k through the membrane.

What is the maximum number of homogeneous phases that can exist at equilibrium for a system involving (a) one component? (b) two components? (c) three components?

Determine the number of degrees of freedom for systems composed of (a) water vapor and dry air. (b) liquid water, water vapor, and dry air. (c) ice, water vapor, and dry air. (d) \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\) at \(20^{\circ} \mathrm{C}\), 1 atm. (e) a liquid phase and a vapor phase, each of which contains ammonia and water. (f) liquid acetone and a vapor phase of acetone and \(\mathrm{N}_{2}\).

Develop the phase rule for chemically reacting systems.

Apply the result of Problem 14.84 to determine the number of degrees of freedom for the gas phase reaction: \(\mathrm{CH}_{4}+\mathrm{H}_{2} \mathrm{O} \rightleftarrows \mathrm{CO}+3 \mathrm{H}_{2}\)

For a gas–liquid system in equilibrium at temperature T and pressure p, Raoult’s law models the relation between the partial pressure of substance i in the gas phase, \(p_{i}\), and the mole fraction of substance i in the liquid phase, \(y_{i}\), as follows: \(p_{i}=y_{i} p_{\mathrm{sat}, i}(T)\) where \(p_{\text {sat }, i}(T)\) is the saturation pressure of pure i at temperature T. The gas phase is assumed to form an ideal gas mixture; thus, \(p_{i}=x_{i} p\) where \(x_{i}\) is the mole fraction of i in the gas phase. Apply Raoult’s law to the following cases, which are representative of conditions that might be encountered in ammonia–water absorption systems (Sec. 10.5): (a) Consider a two-phase, liquid–vapor ammonia–water system in equilibrium at \(20^{\circ} \mathrm{C}\). The mole fraction of ammonia in the liquid phase is 80%. Determine the pressure, in bar, and the mole fraction of ammonia in the vapor phase. (b) Determine the mole fractions of ammonia in the liquid and vapor phases of a two-phase ammonia–water system in equilibrium at \(40^{\circ} \mathrm{C}\), 12 bar.

Spark-ignition engine exhaust gases contain several air pollutants including the oxides of nitrogen, NO and \(\mathrm{NO}_{2}\), collectively known as \(\mathrm{NO}_{x}\). Additionally, the exhaust gases may contain carbon monoxide (CO) and unburned or partially burned hydrocarbons (HC). The pollutant amounts actually present depend on engine design and operating conditions, and they typically differ significantly from values calculated on the basis of chemical equilibrium. Discuss both the reasons for these discrepancies and possible mechanisms by which such pollutants are formed in an actual engine. In a memorandum, summarize your findings and conclusions.

The Federal Clean Air Act and succeeding Clean Air Act Amendments target the oxides of nitrogen NO and \(\mathrm{NO}_{2}\), collectively known as \(\mathrm{NO}_{x}\), as significant air pollutants. \(\mathrm{NO}_{x}\) is formed in combustion via three primary mechanisms: thermal \(\mathrm{NO}_{x}\) formation, prompt \(\mathrm{NO}_{x}\) formation, and fuel \(\mathrm{NO}_{x}\) formation. Discuss these formation mechanisms, including a discussion of thermal \(\mathrm{NO}_{x}\) formation by the Zeldovich mechanism. What is the role of \(\mathrm{NO}_{x}\) in the formation of ozone? What are some \(\mathrm{NO}_{x}\) reduction strategies? Write a report including at least three references.

Using appropriate software, develop plots giving the variation with equivalence ratio of the equilibrium products of octane–air mixtures at 30 atm and selected temperatures ranging from 1700 to 2800 K. Consider equivalence ratios in the interval from 0.2 to 1.4 and equilibrium products including, but not necessarily limited to, \(\mathrm{CO}_{2}\), CO, \(\mathrm{H}_{2} \mathrm{O}\), \(\mathrm{O}_{2}\), O, \(\mathrm{H}_{2}\), \(\mathrm{N}_{2}\), NO, OH. Under what conditions is the formation of nitric oxide (NO) and carbon monoxide (CO) most significant? Write a report including at least three references.

The amount of sulfur dioxide \(\left(\mathrm{SO}_{2}\right)\) present in off gases from industrial processes can be reduced by oxidizing the \(\mathrm{SO}_{2}\) to \(\mathrm{SO}_{3}\) at an elevated temperature in a catalytic reactor. The \(\mathrm{SO}_{3}\) can be reacted in turn with water to form sulfuric acid that has economic value. For an off gas at 1 atm having the molar analysis of 12% \(\mathrm{SO}_{2}\), 8% \(\mathrm{O}_{2}\), 80% \(\mathrm{N}_{2}\), estimate the range of temperatures at which a substantial conversion of \(\mathrm{SO}_{2}\) to \(\mathrm{SO}_{3}\) might be realized. Report your findings in a PowerPoint presentation suitable for your class. Additionally, in an accompanying memorandum, discuss your modeling assumptions and provide sample calculations.

A gaseous mixture of hydrogen \(\left(\mathrm{H}_{2}\right)\) and carbon monoxide (CO) enters a catalytic reactor and a gaseous mixture of methanol \(\left(\mathrm{CH}_{3} \mathrm{OH}\right)\), hydrogen, and carbon monoxide exits. At the preliminary process design stage, plausible estimates are required of the inlet hydrogen mole fraction, \(y_{\mathrm{H}_{2}}\), the temperature of the exiting mixture, \(T_{\mathrm{e}}\), and the pressure of the exiting mixture, pe, subject to the following four constraints: (1) \(0.5 \leq y_{\mathrm{H}_{2}} \leq 0.75\), (2) \(300 \leq T_{\mathrm{e}} \leq 400 \mathrm{\ K}\), (3) \(1 \leq p_{\mathrm{e}} \leq 10\) atm, and (4) the exiting mixture contains at least 75% methanol on a molar basis. In a memorandum, provide your estimates together with a discussion of the modeling employed and sample calculations.

When systems in thermal, mechanical, and chemical equilibrium are perturbed, changes within the systems can occur, leading to a new equilibrium state. The effects of perturbing the system considered in developing Eqs. 14.32 and 14.33 can be determined by study of these equations. For example, at fixed pressure and temperature it can be concluded that an increase in the amount of the inert component E would lead to increases in \(n_{\mathrm{C}}\) and \(n_{\mathrm{D}}\) when \(\Delta v=\left(\nu_{\mathrm{C}}+\nu_{\mathrm{D}}-\nu_{\mathrm{A}}-\nu_{\mathrm{B}}\right)\) is positive, to decreases in \(n_{\mathrm{C}}\) and \(n_{\mathrm{D}}\) when \(\Delta v\) is negative, and no change when \(\Delta v=0\). (a) For a system consisting of \(\mathrm{NH}_{3}, \mathrm{\ N}_{2}\), and \(\mathrm{H}_{2}\) at fixed pressure and temperature, subject to the reaction \(2 \mathrm{NH}_{3}(\mathrm{~g}) \rightleftarrows \mathrm{N}_{2}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g})\) investigate the effects, in turn, of additions in the amounts present of \(\mathrm{NH}_{3}, \mathrm{~N}_{2}\), and \(\mathrm{H}_{2}\). (b) For the general case of Eqs. 14.32 and 14.33, investigate the effects, in turn, of additions of A, B, C, and D. Present your findings, together with the modeling assumptions used, in a PowerPoint presentation suitable for your class.

With reference to the equilibrium constant data of Table A-27: (a) For each of the tabulated reactions plot \(\log _{10} K\) versus 1/T and determine the slope of the line of best fit. What is the thermodynamic significance of the slope? Check your conclusion about the slope using data from the JANAF tables. (b) A textbook states that the magnitude of the equilibrium constant often signals the importance of a reaction, and offers this rule of thumb: When \(K<10^{-3}\), the extent of the reaction is usually not significant, whereas when \(K<10^{3}\) the reaction generally proceeds closely to equilibrium. Confirm or deny this rule. Present your findings and conclusions in a report including at least three references.

(a) For an equilibrium ideal gas mixture of \(\mathrm{N}_{2}, \mathrm{H}_{2}, \mathrm{NH}_{3}\), evaluate the equilibrium constant from an expression you derive from the van’t Hoff equation that requires only standard state enthalpy of formation and Gibbs function of formation data together with suitable analytical expressions in terms of temperature for the ideal gas specific heats of \(\mathrm{N}_{2}, \mathrm{H}_{2}, \mathrm{NH}_{3}\). (b) For the synthesis of ammonia by \(\frac{1}{2} \mathrm{~N}_{2}+\frac{3}{2} \mathrm{H}_{2} \rightarrow \mathrm{NH}_{3}\) provide a recommendation for the ranges of temperature and pressure for which the mole fraction of ammonia in the mixture is at least 0.5. Write a report including your derivation, recommendations for the ranges of temperature and pressure, sample calculations, and at least three references.

U.S. Patent 5,298,233 describes a means for converting industrial wastes to carbon dioxide and water vapor. Hydrogen- and carbon-containing feed, such as organic or inorganic sludge, low-grade fuel oil, or municipal garbage, is introduced into a molten bath consisting of two immiscible molten metal phases. The carbon and hydrogen of the feed are converted, respectively, to dissolved carbon and dissolved hydrogen. The dissolved carbon is oxidized in the first molten metal phase to carbon dioxide, which is released from the bath. The dissolved hydrogen migrates to the second molten metal phase, where it is oxidized to form water vapor, which is also released from the bath. Critically evaluate this technology for waste disposal. Is the technology promising commercially? Compare with alternative waste management practices such as pyrolysis and incineration. Write a report including at least three references.

Figure P14.10D gives a table of data for a lithium bromide–water absorption refrigeration cycle together with the sketch of a property diagram showing the cycle. The property diagram plots the vapor pressure versus the lithium bromide concentration. Apply the phase rule to verify that the numbered states are fixed by the property values provided. What does the crystallization line on the equilibrium diagram represent, and what is its significance for absorption cycle operation? Locate the numbered states on an enthalpy-concentration diagram for lithium bromide– water solutions obtained from the literature. Finally, develop a sketch of the equipment schematic for this refrigeration cycle. Present your findings in a report including at least three references.