The simplest binary phase equilibrium equation to keep in mind is the bubble pressure,
Through this equation, it is very easy to compute the implications of non-ideality and assess qualitatively whether process complications like azeotropes or LLE are likely. A simple equation to guide your assessment is the MAB estimate of A12 in the Margules one-parameter model.
When considering distillation applications you must first check that αLH > 1 at top and bottom:
We also developed Henry’s law,
We showed how to relate Henry’s law to the Lewis-Randall rule used for modified Raoult’s law and how to predict the solubilities of supercritical gases in liquid solvents with the SCVP+ model.
11.15. Practice Problems P11.1. Ninov et al. (J. Chem. Eng. Data, 40:199, 1995) have shown that the system diethylamine(1) + chloroform(2) forms an azeotrope at 1 bar, 341.55 K and x1 = 0.4475. Is this a maximum boiling or minimum boiling azeotrope? Determine the bubble temperature and vapor composition at x1 = 0.80 and 1 bar. (ANS. 331 K, 0.97) P11.2. Derive the expression for the activity coefficient of the Redlich-Kister expansion.
11.16. Homework Problems 11.1. The volume change on mixing for the liquid methyl formate(1) + liquid ethanol(2) system at 298.15 K may be approximately represented by J. Polack, Lu, B.C.-Y. 1972. J. Chem Thermodynamics, 4:469:
∆Vmix = 0.8x1x2 cm3/mol
a. Using this correlation, and the data V1 = 67.28 cm3/mol, V2 = 58.68 cm3/mol, determine the molar volume of mixtures at x1 = 0, 0.2, 0.4, 0.6, 0.8, 1.0 in cm3/mol. b. Analytically differentiate the above expression and show that
and plot these partial molar excess volumes as a function of x1. 11.2. In vapor-liquid equilibria the relative volatility αij is defined by Eqn. 10.32.
a. Provide a simple proof that the relative volatility is independent of liquid and vapor composition if a system follows Raoult’s law.
b. In approximation to a distillation calculation for a nonideal system, calculate the relative volatility α12 and α21 as a function of composition for the n- pentane(1) + acetone(2) system at 1 bar using experimental data in problem 11.11. c. In approximation to a distillation calculation for a non-ideal system, calculate the relative volatility α12 and α21 as a function of composition for the data provided in problem 10.2. d. Provide conclusions from your analysis.
11.3. After fitting the two-parameter Margules equation to the data below, generate a P-x-y diagram at 78.15°C.
11.4. A stream containing equimolar methanol(1) + benzene(2) at 350 K and 1500 mmHg is to be adiabatically flashed to atmospheric pressure. The two-parameter Margules model is to be applied with A12 = 1.85, A21 = 1.64. Express all flash equations in terms of Ki values and Ki values in terms of Modified Raoult’s law.
a. List all the unknown variables that need to be determined to solve for the outlet. b. List all the equations that apply to determine the unknown variables.
11.5. In the system A + B, activity coefficients can be expressed by the one-parameter Margules equation with A = 0.5. The vapor pressures of A and B at 80°C are PAsat = 900 mmHg, PBsat = 600 mmHg. Is there an azeotrope in this system at 80°C, and if so, what is the azeotrope pressure and composition? 11.6. The system acetone(1) + methanol(2) is well represented by the one-parameter Margules equation using A = 0.605 at 50°C.
a. What is the bubble pressure for an equimolar mixture at 30°C? b. What is the dew pressure for an equimolar mixture at 30°C? c. What is the bubble temperature for an equimolar mixture at 760 mmhg? d. What is the dew temperature for an equimolar mixture at 760 mmhg?
11.7. The excess Gibbs energy for a liquid mixture of n-hexane(1) + benzene(2) at 30°C is represented by GE = 1089 x1x2 J/mol.
a. What is the bubble pressure for an equimolar mixture at 30°C? b. What is the dew pressure for an equimolar mixture at 30°C? c. What is the bubble temperature for an equimolar mixture at 760 mmHg? d. What is the dew temperature for an equimolar mixture at 760 mmHg?
11.8. The liquid phase activity coefficients of the ethanol(1) + toluene(2) system at 55°C are given by the two-parameter Margules equation, where A12 = 1.869 and A21 = 1.654.
a. Show that the pure saturation fugacity coefficient is approximately 1 for both components. b. Calculate the fugacity for each component in the liquid mixture at x1 = 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0. Summarize your results in a table. Plot the fugacities for both components versus x1. Label your curves. For each curve, indicate the regions that may be approximated by Henry’s law and the ideal solution model. c. Using the results of part (b), estimate the total pressure above the liquid mixture at 55°C when a vapor phase coexists. Assume the gas phase is ideal for this calculation. Also estimate the vapor composition. d. Comment on the validity of the ideal gas assumption used in part (c).
11.9. a. The acetone(1) + chloroform(2) system can be represented by the Margules two-parameter equation using A12 = –1.149, A21 = –0.862 at 35.17°C. Use bubble-pressure calculations to generate a P-x-y and y-x diagram and compare it with the selected values from the measurements of Zawidzki, Z. Phys. Chem., 35, 129(1900). b. Compare the data to the predictions of the MAB model.
11.10. a. Fit the Margules two-parameter equation to the methanol(1) + benzene(2) system T-x-y data below at 90°C (Jost, W., Roek, H, Schroeder, W., Sieg, L., Wagner, H.G. 1957. Z. Phys. Chem. 10:133) by fitting to x1=0.549. Plot the resultant fit together with the original data for both phases. b. Compare the data with the predictions of the MAB model.
11.11. a. Fit the Margules two-parameter equation to the n-pentane(1) + acetone(2) system P-x-y data below at 1 bar (Lo et al. 1962. J. Chem. Eng. Data 7:32) by fitting to x1=0.503. Plot the resultant fit together with the original data for both phases. b. Compare the data with the predictions of the MAB model.
11.12. For a particular binary system, data are available: T = 45°C P = 37 kPa x1 = 0.398 y1 = 0.428
In addition, and . From these data, a. Fit the one-parameter Margules equation b. Fit the two-parameter Margules equation
11.13. The compositions of coexisting phases of ethanol(1) + toluene(2) at 55°C are x1 = 0.7186, and y1 = 0.7431 at P = 307.81 mmHg, as reported by Kretschmer and Wiebe, J. Amer. Chem. Soc., 71, 1793(1949). Estimate the bubble pressure at 55°C and x1 = 0.1, using
a. The one-parameter Margules equation b. The two-parameter Margules equation
11.14. A vapor/liquid experiment for the carbon disulfide(1) + chloroform(2) system has provided the following data at 298 K: , , x1 = 0.2, y1 = 0.363, P = 34.98 kPa. Estimate the dew pressure at 298 K and y1 = 0.6, using
a. The one-parameter Margules equation b. The two-parameter Margules equation
11.15. The (1) + (2) system forms an azeotrope at x1 = 0.75 and 80°C. At 80°C, ,
. The liquid phase can be modeled by the one-parameter Margules equation. a. Estimate the activity coefficient of component 1 at x1 = 0.75 and 80°C. [Hint: The relative volatility (given in problem 11.2) is unity at the azeotropic condition.] b. Qualitatively sketch the P-x-y and T-x-y diagrams that you expect.
11.16. Ethanol(1) + benzene(2) form an azeotropic mixture. Compare the specified model to the experimental data of Brown and Smith cited in problem 10.2.
a. Prepare a y-x and P-x-y diagram for the system at 45°C assuming the MAB model. b. Prepare a y-x and P-x-y diagram for the system at 45°C assuming the one- parameter Margules model and using the experimental pressure at xE = 0.415 to estimate A12. c. Prepare a y-x and P-x-y diagram for the system at 45°C assuming the two- parameter model and using the experimental pressure at xE = 0.415 to estimate A12 and A21.
11.17. The acetone + chloroform system exhibits an azeotrope at 64.7°C, 760 mmHg, and 20 wt% acetone.
a. Use the MAB model to predict the T-x-y diagram at 1 bar. b. Use the Margules one-parameter model to estimate the T-x-y diagram at 1 bar.
11.18. For the Margules two-parameter model estimate the total pressure and composition of the vapor in equilibrium with a 20 mol% ethanol(1) solution in water(2) at 78.15°C using data at 78.15°C:
11.19. Using the data from problem 11.18, fit the two-parameter Margules equation, and then generate a P-x-y diagram at 78.15°C. 11.20. A liquid mixture of 50 mol% chloroform(1) and 50% 1,4-dioxane(2) at 0.1013 MPa is metered into a flash drum through a valve. The mixture flashes into two phases inside the drum where the pressure and temperature are maintained at 24.95 kPa and 50°C. The compositions of the exiting phases are x1 = 0.36 and y1 = 0.62.
Your supervisor asks you to adjust the flash drum pressure so that the liquid phase is x1 = 0.4 at 50°C. He doesn’t provide any VLE data, and you are standing in the middle of the plant with only a calculator and pencil and paper, so you must estimate the new flash drum pressure. Fortunately, your supervisor has a phenomenal recall for numbers and tells you that the vapor pressures for the pure components at 50°C are
and . What is your best estimate of the pressure adjustment that is necessary without using any additional information?
11.21. Suppose a vessel contains an equimolar mixture of chloroform(1) and triethylamine(2) at 25°C. The following data are available at 25°C:
a. If the pressure in the vessel is 90 mmHg, is the mixture a liquid, a vapor, or both liquid and vapor? Justify your answer. b. Provide your best estimate of the volume of the vessel under these conditions. State your assumptions.
11.22. Ethanol(1) + benzene(2) form azeotropic mixtures. a. From the limited data below at 45°C, it is desired to estimate the constant A for the one-term Margules equation, GE/RT = Ax1x2. Use all of the experimental data to give your best estimate.
b. From your value, what are the bubble pressure and vapor compositions for a mixture with x1 = 0.8?
11.23. An equimolar ternary mixture of acetone, n-butane, and ammonia at 1 MPa is to be flashed. List the known variables, unknown variables, and constraining equations to solve each of the cases below. Assume MAB solution thermodynamics and write the flash equations in terms of K-ratios, with the equations for calculating K-ratios written separately. (Hint: Remember to include the activity coefficients and how to calculate them.
a. Bubble temperature b. Dew temperature b. Flash temperature at 25mol% vapor b. Raised to midway between the bubble and dew temperatures, then adiabatically flashed.
11.24. Fit the data from problem 11.11 to the following model by regression over all points, and compare with the experimental data on the same plot, using:
a. One-parameter Margules equation b. Two-parameter Margules equation
11.25. Fit the specified model to the methanol(1) + benzene(2) system P-x-y data at 90°C by minimizing the sum of squares of the pressure residual. Plot the resultant fit together with the original data for both phases (data are in problem 11.10), using
a. One-parameter Margules equation