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:
ΔVmix = 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).
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.
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.
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
b. Two-parameter Margules equation
11.26. Fit the specified model to the methanol(1) + benzene(2) system T-x-y data at 760 mmHg by minimizing the sum of squares of the pressure residual. Plot the resultant fit together with the original data for both phases (Hudson, J.W., Van Winkle, M. 1969. J. Chem. Eng. Data 14:310), using
a. One-parameter Margules equation
b. Two-parameter Margules equation
11.27. VLE data for the system carbon tetrachloride(1) and 1,2-dichloroethane(2) are given below at 760 mmHg, as taken from the literature.23
a. Fit the data to the one-parameter Margules equation.
b. Fit the data to the two-parameter Margules equation.
c. Plot the P–x–y diagram at 80°C, based on one of the fits from (a) or (b).
11.28. When only one component of a binary mixture is volatile, the pressure over the mixture is determined entirely by the volatile component. The activity coefficient for the volatile component can be determined using modified Raoult’s law and an activity coefficient model can be fitted. The model will satisfy the Gibbs-Duhem equation and thus an activity coefficient prediction can be made for the nonvolatile component. Consider a solution of sucrose and water. The sucrose is nonvolatile. The bubble pressures of water (1) + sucrose (2) solutions are tabulated below at three temperatures.
a. Fit the one-parameter Margules equation to the water data at the temperature(s) specified by your instructor. Report the values of A12.
b. Prepare a table of γ1 values and plot of the experimental and fitted/predicted ln γ1 versus x1 for water and sucrose over the range of experimental compositions for the temperature(s) specified by your instructor.
c. Prepare a table of values and on the same plot as (b) add a curve for ln γ2* for the temperature(s) specified by your instructor.
d. Prepare a table of values and a plot of osmotic pressure (in MPa) for the solution versus C2 (g/L) at 25°C. The density at 25°C can be estimated using ρ(g/mL) = 0.99721 + 0.3725w2 + 0.16638w22 where w2 is wt. fraction sucrose. Include a curve of the osmotic pressure expected for an ideal solution.
e. Calculate the osmotic pressure (MPa) using the activity of water modeled with the one-parameter Margules equation at 25°C fitted in part (a). Add it to the plot in part (d).
f. Calculate the second and third osmotic virial coefficients (for concentration units of g/L) at 25°C by fitting the calculations from part (d). Add the modeled osmotic pressure to the plot from (d).
g. From the temperature dependence of the one-parameter Margules parameter fitted in (a), show that the parameter may be represented with f(1/T(K)). Then provide a model for the excess enthalpy and the parameter value(s) that represent the experimental data.
11.29. Red blood cells have a concentration of hemoglobin (Mw ~ 68000) at 0.3 M. The osmotic pressure a body temperature (37°C) is 0.83 MPa. Water can permeate the cells walls, but not hemoglobin.24
a. Using only the second osmotic coefficient, determine the coefficient value (L/g), and determine the activity of water at the conditions given above.
b. Calculate the ideal solution osmotic pressure at the conditions given above.
c. Suppose we were to transfer red blood cells in a laboratory solution at 37°C (blood banks need to do this). We want the external glucose solution to match the red blood cell’s internal osmotic pressure to avoid swelling or shrinking of the cells. If glucose has an osmotic pressure of 2 MPa at 0.7 M and 37°C, what glucose concentration (g/L) would match the internal osmotic pressure to keep the blood cells stable? What is molality of the resulting glucose solution? Comparing molalities, what can you infer about the solution non-idealities of the glucose solution compared to the hemoglobin solution?
11.30. Osmotic pressure of bovine serum albumin (BSA) has been measured at 298.15 K and various pH values by Vilker, V.L., Colton, C.K., Smith, K.A. 1981. J. Colloid Int. Sci. 79:548, as summarized in the table below. The investigators report the BSA molecular weight in their sample as 69,000.
a. Regressing all data, determine the second and third osmotic virial coefficients for pH 7.4.
b. Regressing all data, determine the second and third osmotic virial coefficients at pH 4.5.
11.31. Boric acid is a common supplement to make ophthalmic solutions isotonic. It is entirely undissociated at normal ophthalmic conditions.
a. Estimate the concentration (wt%) of boric acid to prepare a solution that is isotonic with human blood.
b. Estimate the concentration (wt%) of boric acid that should be added to a 0.025wt% solution of Claritin to make it isotonic. The molecular formula of Claritin is listed at ChemSpider.com as C22H23ClN2O2.CopycopyHighlighthighlightAdd NotenoteGet Linklink
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