Chapter 9: Chemical Reaction Equilibria (Solution)
Problem 9.1: Water vapor decomposes according to the following reaction:
H_2O\rightarrow H_2+1/2O_2Derive expressions for the mole fraction of each species in terms of the extent of reaction assuming that the system contained n0 moles of water vapor initially.
Problem 9.2: The following reaction occurs in a mixture consisting of 2 mol methane, 1 mol water, 1 mol carbon monoxide, and 4 mol hydrogen initially.
CH_4+H_2O\rightarrow CO+3H_2Deduce expression relating the mole fractions of various species to the extent of reaction.
Problem 9.3: A system consisting of 2 mol methane and 3 mol water is undergoing the following reaction
CH_4+H_2O\rightarrow CO+3H_2 CH_4+2H_2O\rightarrow CO_2+4H_2Derive expressions for mole fractions in terms of the extent of reactions.
Problem 9.4: The following gas-phase reactions occur in a mixture initially containing 3 mol ethylene and 2 mol oxygen.
C_2H_4+1/2O_2\rightarrow\left(CH_2\right)_2O C_2H_4+3O_2\rightarrow2CO_2+2H_2ODerive expressions for mole fractions in terms of the extent of reactions.
Problem 9.5: Calculate the equilibrium constant at 298 K of the reaction N2 + 3H2 → 2NH3, given that the free energy of formation of ammonia at 298 K is –16,500 J/mol.
Problem 9.6: Calculate the standard free energy change at 298 K in the gas-phase alkylation of isobutane with ethylene to form neohexane.
C_4H_{10}(g)+C_2H_4(g)\rightarrow C_6H_{14}(g)The free energies of formation at 298 K are –21,000 J/mol, 68,460 J/mol, and –9,950 J/mol for isobutane, ethylene, and neohexane respectively.
Problem 9.7: Calculate the equilibrium constant at 673 K and 1 bar for the reaction
N_2(g)+3H_2(g)\rightarrow2NH_3(g)assuming that the heat of the reaction remains constant in the temperature range involved. Take the standard heat of formation and standard free energy of formation of ammonia at 298 K to be – 46,110 J/mol and –16,450 J/mol respectively.
Problem 9.8: Is the following reaction promising at 600 K?
NaOH(s)+CO(g)\rightarrow HCOONa(s)The free energy of formation, the heat of formation, and the specific heat of the components are given below:
| \triangle G_{f,298}^0 (J/mol) | \triangle H_{f,298}^0 (J/mol) | CP (J/mol K) | |
| CO (g) | –1.37×105 | –1.11×105 | 26.64 + 7.7125×10–3T |
| HCOONa (s) | –6.17×105 | –6.51×105 | 76.86 |
| NaOH (s) | –3.79×105 | –4.27×105 | 52.50 |
Problem 9.9: Methanol is produced by the following reaction:
CO(g)+2H_2(g)\rightarrow CH_3OH(g)The standard heat of formation of CO (g) and CH3OH (g) at 298 K are –110,500 J/mol and –200,700 J/mol respectively. The standard free energies of formation are –137,200 J/mol and – 162,000 J/mol respectively. (a) Calculate the standard free energy change and determine whether the reaction is feasible at 298 K. (b) Determine the equilibrium constant at 400 K assuming that the heat of the reaction is constant. (c) Derive an expression for the standard free energy of reaction as a function of temperature if the specific heats of the components are:
C_P=3.376R+0.557\times10^{-3}RT-0.031\times10^5RT^{-2}\;for\;CO C_P=3.249R+0.422\times10^{-3}RT+0.083\times10^5RT^{-2}\;for\;H_2 C_P=2.211R+12.216\times10^{-3}RT-3.450\times10^{-6}RT^2\;for\;CH_3OH(d) Use the equation obtained in part (c) to calculate the equilibrium constant at 400 K and compare it with the result in part (b).
Problem 9.10: Calculate the equilibrium constant at 298 K for the reaction
C_2H_4(g)+H_2O(g)\rightarrow C_2H_5OH(g)given the following data:
| S_{298}^0 (J/mol K) | H_{298}^0 (J/mol K) | |
| C2H4 (g) | 220.95 | 48986 |
| H2O (g) | 189.12 | -241997 |
| C2H5OH (g) | 278.00 | -238941 |
Problem 9.11: The standard free energy change for the reaction
C_4H_8(g)\rightarrow C_4H_6(g)+H_2(g)is given by the relation
\triangle G_T^0=1.03665\times10^5-20.9759T\ln\left(T\right)+12.9372TWhere \triangle G_T^0 is in J/mol and T is in K. (a) Over what range of temperature is the reaction promising from a thermodynamic viewpoint? (b) For the reaction of pure butene at 800 K, calculate the equilibrium conversion for operation at 1 bar and 5 bar. (c) Repeat part (b) if the feed consists of 50 % (mol) butene and the rest inerts.
Problem 9.12: Calculate the equilibrium constant for the vapor-phase hydration of ethylene to ethanol at 600 K
The following data are available:
| \triangle H_{f,298}^0\times10^{-3} (J/mol) | \triangle G_{f,298}^0\times10^{-3} (J/mol) | CP (J/mol K) | |
| Ethylene | 52.61 | 68.46 | 11.886 + 120.12×10–3T –36.649×10–6T2 |
| Water | –241.818 | –228.57 | 30.475 + 9.652×10–3T + 1.189×10–6T2 |
| Ethanol | –235.1 | –168.49 | 29.358 + 166.9×10–3T –50.09×10–6T2 |
Problem 9.13: The equilibrium constant at 420 K for the vapor-phase hydration of ethylene to ethanol according to the reaction
C_2H_4+H_2O\rightarrow C_2H_5OHis 6.8×10–2 and the standard heat of reaction at 298 K is –45.95×103 J. The specific heat data are as follows:
| CP (J/mol K) | |
| Ethylene | 11.886 + 120.12×10–3T –36.649×10–6T2 |
| Water | 30.475 + 9.652×10–3T + 1.189×10–6T2 |
| Ethanol | 29.358 + 166.9×10–3T –50.09×10–6T2 |
Formulate general relationships for estimating the equilibrium constant and standard free energy change as functions of temperature.
Problem 9.14: For the vapor-phase hydration of ethylene to ethanol according to
C_2H_4+H_2O\rightarrow C_2H_5OHthe equilibrium constants were measured at temperatures 420 K and 600 K. They are 6.8×10–2 and 1.9×10–3 respectively. The specific heat (J/mol K) data are:
| CP (J/mol K) | |
| Ethylene | 11.886 + 120.12×10–3T –36.649×10–6T2 |
| Water | 30.475 + 9.652×10–3T + 1.189×10–6T2 |
| Ethanol | 29.358 + 166.9×10–3T –50.09×10–6T2 |
Develop general expressions for the equilibrium constant and standard free energy change as functions of temperature.
Problem 9.15: The water–gas shift reaction
CO(g)+H_2O(g)\rightarrow CO_2(g)+H_2(g)takes place at 373 K. The equilibrium constant KP for this reaction at 537 K = 9.8×10–4. The heats of formation at 298 K are: CO = –110,525 J/mol, CO2 = –393,509 J/mol, H2O = –241,818 J/mol. Calculate the equilibrium constant at 1000 K.
Problem 9.16: Calculate the fraction of pure ethane that would dehydrogenate at 750 K and 5 atm, if the following reaction goes to equilibrium.
C_2H_6(g)\leftrightarrow C_2H_4(g)+H_2(g)∆G0 for the reaction at 750 K is 42.576 kJ. Assume ideal behavior.
Problem 9.17: Ethanol can be prepared by the following vapor-phase reaction from ethylene:
C_2H_4(g)+H_2O\leftrightarrow C_2H_5OH(g)The value of ∆G0 for the above reaction at 1 bar and 398 K is 5040 J. Calculate the conversion obtained if an isothermal reactor operating at 398 K and 2 bar is fed with a mixture containing 50 mol percent ethylene and 50 mol percent steam. Assume that equilibrium is reached at the exit of the reactor and the gases behave ideally.
Problem 9.18: A gaseous mixture containing 30 % CO, 50 % H2, and the rest inert gas is sent to a reaction chamber for methanol synthesis. The following reaction occurs at 635 K and 310 bar.
CO(g)+2H_2(g)\rightarrow CH_3OH(g)Assuming that the gas mixture behaves as an ideal solution calculate the percent conversion of CO given that Kf = 5×10–5 and Kϕ = 0.35.
Problem 9.19: Estimate the maximum conversion of ethylene to alcohol by vapor phase hydration at 523 K and 34 bar.
C_2H_4(g)+H_2O(g)\rightarrow C_2H_5OH(g)The equilibrium constant varies with temperature as
\ln\left(K\right)=\frac{4760}T-1.558\ln\left(T\right)+2.22\times10^{-3}T-0.29\times10^{-6}T^2-5.56The steam–ethylene ratio in the initial mixture is 5.0. The fugacity coefficients for ethylene, ethanol, and water vapor are 0.98, 0.84, and 0.91.
Problem 9.20: Ethanol is manufactured by the vapor-phase hydration of ethylene to ethanol according to the reaction,
C_2H_4(g)+H_2O(g)\rightarrow C_2H_5OH(g)Starting with a gas mixture containing 25 % ethylene and 75 % steam, determine the composition of the products if the reaction was carried out at 400 K and 1 bar. The standard free energy of reaction at 400 K is 4548.3 J.
Problem 9.21: What would be the equilibrium yield of ethanol at 1 bar and 373 K in the following reaction?
C_2H_4(g)+H_2O(g)\rightarrow C_2H_5OH(g)The reactant stream consists of an equimolar mixture of steam and ethylene. The standard free energy change may be taken as \triangle G_{373}^0 = 1264 J/mol.
Problem 9.22: Calculate the equilibrium percentage conversion of nitrogen to ammonia at 700 K and 300 bar, if the gas enters the converter with a composition of 75 % (mol) hydrogen and 25 % (mol) nitrogen. For the reaction
1/2N_2+3/2H_2\rightarrow NH_3equilibrium constant may be taken as K = 9.1×10–3. Assume that Kϕ= 0.72.
Problem 9.23: The gases from the pyrites burner of a contact sulphuric acid plant have the following composition: SO2 = 7.80 %, O2 = 10.80 % and N2 = 81.40 %. This is then passed into a converter where the SO2 is converted to SO3. The temperature and pressure in the converter are 775 K and 1 bar. The equilibrium constant for the reaction
SO_2+1/2O_2\rightarrow SO_3may be taken as K = 85. Calculate the composition of gases leaving the converter.
Problem 9.24: One mol carbon at 298 K reacts with 2 mol oxygen at 298 K to form an equilibrium mixture of CO2, CO, and O2 at 3000 K and 1 bar. If the equilibrium constant K = 0.328, determine the equilibrium composition.
Problem 9.25: One mol carbon at 298 K and 1 bar reacts with 1 mol oxygen at 298 K and 1 bar to form an equilibrium mixture of CO2, CO, and O2 at 3000 K and 1 bar in a steady flow process. Determine the equilibrium composition and heat transfer for this process if the equilibrium constant K = 0.328. The standard heat of formation are 393.509 kJ/mol for CO2, 110.525 kJ/mol for CO. The mean heat capacity of products = 45 J/mol K.
Problem 9.26: Pure N2O4 at a low temperature is diluted with air and heated to 298 K and 1 bar. The following reaction occurs
N_2O_4(g)\rightarrow2NO_2(g)If the mole fraction of N2O4 in the N2O4–air mixture before dissociation begins is 0.2, calculate the extent of decomposition and mole fraction of NO2 and N2O4 present at equilibrium. The standard free energy change for the reaction at 298 K = 4644.7 J/mol.
Problem 9.27: Methanol is manufactured according to the reaction
CO(g)+2H_2(g)\rightarrow CH_3OH(g)The reaction is carried out at 400 K and 1 bar. The standard heat of reaction at this condition is – 9.4538×104 J and the equilibrium constant is 1.52. Analysis of the equilibrium vapor product from the reactor shows 40 % hydrogen. An equilibrium gas mixture can be treated as an ideal gas. (a) Determine the concentrations of CO and CH3OH in the product. (b) If the reaction occurred at 500 K and 1 bar starting with the same feed as in part (a) would you expect the concentration of hydrogen in the equilibrium mixture to be greater or less than 40 % mole? Why?
Problem 9.28: Determine the maximum percentage of ethane that may get dehydrogenated to ethylene at 750 K and 5 bar according to the reaction
C_2H_6(g)\rightarrow C_2H_4(g)+H_2(g)The standard free energy of reaction at 750 K is 4.2593×104 J.
Problem 9.29: Hydrogen cyanide can be produced by the gas-phase nitrogenation of acetylene according to the reaction
N_2(g)+C_2H_2(g)\rightarrow2HCN(g)The feed to the reactor consists of an equimolar mixture of acetylene and nitrogen. The temperature of the reaction is 575 K. At this temperature, the standard free energy of the reaction is 3.0181×104 J. Determine the percentage of cyanide in the reaction mixture if (a) The pressure is 1 bar (b) The pressure is 200 bar. The fugacity coefficients for HCN, C2H2, and N2 may be taken as 0.607, 0.942, and 1.07 respectively.
Problem 9.30: For the synthesis of ammonia according to the reaction
1/2N_2+3/2H_2\rightarrow NH_3a mixture consisting of 0.5 mol N2, and 1.5 mol H2 is send to the reactor. The equilibrium mixture behaves as an ideal gas. Show that the extent of reaction ε is given by
\varepsilon=1-\left(1+1.299K_P\right)^{-1/2}Problem 9.31: For the reaction
SO_2(g)+1/2O_2(g)\rightarrow SO_3(g)in equilibrium at 775 K what pressure is required for a 90 percent conversion of SO2 if the initial mixture is equimolar in the reactants. Assume ideal gases. Take the free energy of the reaction at 775 K to be –2.8626×104 J.
Problem 9.32: 1-butene is dehydrogenated to 1,3-butadiene according to the reaction
C_4H_8(g)\rightarrow C_4H_6(g)+H_2(g)Determine the extent of reaction at equilibrium at 900 K and 1 bar with (a) 1 mol butene as the reactant (b) a reactant mixture consisting of 1 mol butene and 10 mol steam. The following free energy functions and heat of formation data are available:
| \left(G_{900}^0-H_{298}^0\right)/T (J/K) | H_{f,298}^0 (J) | |
| C4H6 | −336.41 | 1.1024×105 |
| C4H8 | −368.56 | −1.256×102 |
| H2 | −145.54 | —— |
Problem 9.33: An experimental investigation of the effect of temperature on the reaction
A(g)+B(g)\rightarrow C(g)gave the following equilibrium compositions at 373 K and 473 K. The pressure was maintained at 1 bar. At 373 K, yA = 0.414, yB = 0.414 and yC = 0.172; At 473 K, yA = 0.179, yB = 0.179 and yC = 0.642. What will be the equilibrium composition at 423 K and 10 bar if equimolar quantities of the reactants are used?
Problem 9.34: Determine the ranges of temperature and pressure for which the equilibrium conversion is at least 10 % in the following reaction:
CO(g)+2H_2(g)\rightarrow CH_3OH(g)Assume that stoichiometric quantities of reactants are used. The standard free energy of formation of methanol and CO are respectively –1.626×105 J/mol and –1.374×105 J/mol at 298 K. The standard heat of formation at 298 K are –2.013 ×105 J/mol and –1.106×105 J/mol. The heat of the reaction may be assumed to remain constant.
Problem 9.35: The equilibrium constant for the oxidation of SO2 to SO3 according to the reaction
SO_2(g)+1/2O_2(g)\rightarrow SO_3(g)was found to be related to temperature as
-R\ln K=-\frac{22630}T+5.281\ln T-0.959\times10^{-2}T+0.28\times10^{-5}T^2-7.68where T is in K and K is in (bar)–1/2. A feed mixture containing 12 % SO2, 9 % O2, and 79 % N2 is reacted at 749 K and 1 bar. Calculate the fractional conversion of SO2.
Problem 9.36: Ethanol is produced by vapor-phase hydration of ethylene:
C_2H_4(g)+H_2O(g)\rightarrow C_2H_5OH(g)(a) Calculate the equilibrium constant at 298 K using the following data:
| S_{298}^0 (J/mol K) | H_{298}^0 (J/mol) | |
| C2H4 (g) | 220.85 | 4.9×104 |
| H2O (g) | 189.12 | -2.42×105 |
| C2H5OH (g) | 278.00 | −2.39×105 |
(b) If the standard free energy for the reaction can be expressed as
\triangle G^0=-4.0486\times10^4+26.92T\ln T-37.72T-6.65\times10^{-3}T^2where ∆G0 is in J/mol ethanol. Discuss the feasibility of the reaction for the temperature range of 298 K to 733 K. (c) Calculate the equilibrium conversion of ethylene at 450 K at a pressure of 1 bar and 5 bar when the feed contains equimolar quantities of ethylene and water. (d) Repeat part (c) when the feed contains 100 % excess water. (e) Calculate the equilibrium conversion at 450 K and 1 bar, if the feed consists of 1 mol ethylene, 2 mol water, and 1 mol inert gas.
Problem 9.37: Acetic acid is esterified in the liquid phase with ethanol at 373 K and 1 bar to produce ethyl acetate and water according to the reaction
CH_3COOH(l)+C_2H_5OH(l)\rightarrow CH_3COOC_2H_5(l)+H_2O(l)The feed consists of 1 mol each of acetic acid and ethanol, estimate the mole fraction of ethyl acetate in the reacting mixture at equilibrium. The standard heat of formation and standard free energy of formation at 298 K are given below:
| CH3COOH (l) | C2H5OH (l) | CH3COOC2H5 (l) | H2O (l) | |
| \triangle H_f^0 (J) | −484500 | −277690 | −463250 | −285830 |
| \triangle G_f^0 (J) | −389900 | −1174780 | −318218 | −237130 |
Assume that the heat of the reaction is independent of temperature and the liquid mixture behaves as an ideal solution.
Problem 9.38: The esterification of ethanol with acetic acid occurs in an aqueous solution as follows:
C_2H_5OH(aq)+CH_3COOH(aq)\rightarrow CH_3COOC_2H_5(aq)+H_2O(l)The free energies of the formation of acetic acid, ethanol, and ethyl acetate in a hypothetical 1 molal solution at 298 K are –3.9645×105 J, –1.8053×105 J, and –3.3296×105 J respectively. The free energy of the formation of water at 298 K is –2.3735×105 J. What is the equilibrium constant? Starting with a dilute equimolar mixture of ethanol and acetic acid, calculate the extent of the reaction and the molalities of ethyl acetate and acetic acid in the equilibrium solution. Assume dilute solution behavior.
Problem 9.39: Carbon dioxide is reduced by graphite according to the equation
C(s)+CO_2(g)\rightarrow2CO(g)Assuming that equilibrium is attained at 1000 K and 1 bar, calculate the degree of completion of reduction of CO2. The following data are available:
| \left(G_{1000}^0-H_0^0\right)/T (J/mol K) | H_{f,0}^0 (J) | |
| CO | −204.57 | −1.1389×105 |
| CO2 | −226.54 | −3.9343×102 |
Problem 9.40: Carbon dioxide is reduced by graphite according to the equation
C(s)+CO_2(g)\rightarrow2CO(g)Calculate the effect of pressure on the degree of completion of pure CO2 at 1000 K assuming total pressures of 1, 2, and 3 bar. The gas mixture may be treated as ideal gas and an equilibrium constant value of K = 1.778 may be assumed.
Problem 9.41: Calculate the decomposition pressure of limestone at 1000 K given that
\triangle H_{1000}^0=1.7553\times10^5J;\;\triangle S_{1000}^0=150.3\frac J{mol\;K}Problem 9.42: Ammonium chloride decomposes upon heating to yield a gas mixture of ammonia and hydrochloric acid. At what temperature does ammonium chloride exert a decomposition pressure of 1 bar? The standard heat of formation and the standard free energy of formation are as follows:
| H_{f,0}^0 (J) | G_{f,0}^0 (J) | |
| NH4Cl (s) | −314430 | −202870 |
| NH3 (g) | −46110 | −16450 |
| HCl (g) | −92307 | −95299 |
Problem 9.43: The following decomposition reaction occurs at 373 K in the liquid phase.
A\rightarrow B+CThe equilibrium constant based on the pure liquid standard state is 2. The vapor pressures are PA = 5 bar, PB = 20 bar, and PC = 2 bar. Assume that all vapors are ideal, liquid B is immiscible with the A–C liquid mixture and the A–C mixture is ideal. Calculate the equilibrium pressure and the composition of the liquid and vapor phases.
Problem 9.44: The equilibrium constant for the following reaction is found to be 2.
A(l)\rightarrow B(l)+C(l)The vapor pressures are PA = 5 bar, PB = 20 bar, and PC = 2 bar. A and C form the ideal solution and B is immiscible with either A and C or their mixtures. The system consisted of pure A initially. Find the pressure below which only a gas phase exists.
Problem 9.45: Mixtures of CO and CO2 are to be processed at temperatures between 900 K and 1000 K. Determine the conditions under which solid carbon might deposit according to the reaction
CO_2(g)+C(s)\rightarrow2CO(g)The equilibrium constants for this reaction are 0.178 at 900 K and 1.58 at 1000 K. (Hint: The activity of solid carbon is less than unity if carbon is not present in the system.)
Problem 9.46: Acetylene is catalytically hydrogenated to ethylene at 1500 K and 1 bar. Starting with an equimolar mixture of acetylene and hydrogen what will be the mole fractions at equilibrium? Assume ideal gases.
C_2H_2\rightarrow2C+H_2;\;\;K=5.2 2C+2H_2\rightarrow C_2H_4;\;\;K=0.1923Problem 9.47: What would be the equilibrium conversion of ethyl alcohol to butadiene at 700 K and 1 bar given the following reactions?
C_2H_5OH\rightarrow C_2H_4+H_2O;\;\triangle G^0=-45427\frac J{mol} C_2H_5OH\rightarrow CH_3CHO+H_2;\;\triangle G^0=-15114\frac J{mol} C_2H_4+CH_3CHO\rightarrow C_4H_6+H_2O;\;\triangle G^0=-5778\frac J{mol}Problem 9.48: The feed to a reactor consists of an equimolar mixture of A and B. Determine the equilibrium composition of the mixture if the following gas-phase reaction occurs at 1000 K and 1 bar.
A+B\rightarrow C+D;\;\;K=0.4 A+2B\rightarrow C+E;\;\;K=0.5067Problem 9.49: The following reactions occur at 1500 K and 10 bar.
A+B\rightarrow C+D;\;\;K=2.67 A+C\rightarrow2E;\;\;K=3.20The initial mixture consists of 2 mol A and 1 mol B, determine the composition at equilibrium assuming ideal gas behavior.
Problem 9.50: Determine the number of degrees of freedom in a gaseous system consisting of NH3, NO2, NO, H2O, O2, and N2.
Problem 9.51: Determine the number of degrees of freedom in a gaseous system consisting of H2O, HCl, O2, and Cl2.