13.43 through 13.54 require parameter values for the Wilson or NRTL equation for

Assuming the validity of Raoult’s law, do the following calculations for the benzene(1) /toluene (2) system: (a) Given \(x_{1}=0.33\) and \(T=100^{\circ} \mathrm{C}\), find \(y_{1}\) and P. (b) Given \(y_{1}=0.33\) and \(T=100^{\circ} \mathrm{C}\), find \(x_{1}\) and P. (c) Given \(x_{1}=0.33\) and P = 120 kPa, find \(y_{1}\) and T. (d) Given \(y_{1}=0.33\) and P = 120 kPa, find \(x_{1}\) and T. (e) Given \(T=105^{\circ} \mathrm{C}\) and P = 120 kPa, find \(x_{1}\) and \(y_{1}\). (f) For part (e), if the overall mole fraction of benzene is \(z_{1}=0.33\), what molar fraction of the two-phase system is vapor? (g) Why is Raoult’s law likely to be an excellent VLE model for this system at thestated (or computed) conditions? Text Transcription: x_1=0.33 T=100^circ C y_1 y_1=0.33 x_1 T=105^circ C z_1=0.33

Assuming Raoult’s law to be valid, prepare a Pxy diagram for a temperature of \(90^{\circ} \mathrm{C}\) and a txy diagram for a pressure of 90 kPa for one of the following systems: (a) Benzene (1)/ethylbenzene (2). (b) 1-Chlorobutane (1)/chlorobenzene (2). Text Transcription: 90^circ C

Assuming Raoult’s law to apply to the system n-pentane (1)/n-heptane (2), (a) What are the values of \(x_{1} \text { and } y_{1}\) at \(t=55^{\circ} \mathrm{C}\) and \(P=\frac{1}{2}\left(P_{1}^{\mathrm{sat}}+P_{2}^{\mathrm{sat}}\right)\)? For these conditions, plot the fraction of the system that is vapor \(\mathcal{V}\) vs. overall composition \(z_{1}\). (b) For \(t=55^{\circ} \mathrm{C}\) and \(z_{1}=0.5\), plot P, \(x_{1} \text {, and } y_{1} \text { vs. } \mathcal{V}\). Text Transcription: x_1 and y_1 t=55^circ C P=1/2(P_1^sat+P_2^sat) V z_1 z_1=0.5 x_1, and y_1 vs. V

Rework Prob. 13.3 for one of the following: (a) \(t=65^{\circ} \mathrm{C}\); (b) \(t=75^{\circ} \mathrm{C}\); (c) \(t=85^{\circ} \mathrm{C}\); (d) \(t=95^{\circ} \mathrm{C}\). Text Transcription: t=65^circ C t=75^circ C t=85^circ C t=95^circ C

Of the following binary liquid/vapor systems, which can be approximately modeled by Raoult’s law? For those that cannot, why not? Table B.1 (App. B) may be useful. (a) Benzene/toluene at 1(atm). (b) n-Hexane/n-heptane at 25 bar. (c) Hydrogen/propane at 200 K. (d) Iso-octane/n-octane at \(100^{\circ} \mathrm{C}\). (e) Water/n-decane at 1 bar. Text Transcription: 100^circ C

A single-stage liquid/vapor separation for the benzene (1) /ethylbenzene (2) system must produce phases of the following equilibrium compositions. For one of these sets, determine T and P in the separator. What additional information is needed to compute the relative amounts of liquid and vapor leaving the separator? Assume that Raoult’s law applies. (a) \(x_{1}=0.35, y_{1}=0.70\). (b) \(x_{1}=0.35, y_{1}=0.725\). (c) \(x_{1}=0.35, y_{1}=0.75\). (d) \(x_{1}=0.35, y_{1}=0.775\). Text Transcription: x_1=0.35, y_1=0.70 x_1=0.35, y_1=0.725 x_1=0.35, y_1=0.75 x_1=0.35, y_1=0.775

A mixture containing equimolar amounts of benzene (1), toluene (2), and ethylbenzene (3) is flashed to conditions T and P. For one of the following conditions, determine the equilibrium mole fractions \(\left\{x_{i}\right\} \text { and }\left\{y_{i}\right\}\) of the liquid and vapor phases formed and the molar fraction \(\mathcal{V}\) of the vapor formed. Assume that Raoult’s law applies. (a) \(T=110^{\circ} \mathrm{C}\), P = 90 kPa. (b) \(T=110^{\circ} \mathrm{C}\), P = 100 kPa. (c) \(T=110^{\circ} \mathrm{C}\), P = 110 kPa. (d) \(T=110^{\circ} \mathrm{C}\), P = 120 kPa. Text Transcription: {x_i} and {y_i} V T=110^circ C

A binary mixture of mole fraction \(z_{1}\) is flashed to conditions T and P. For one of the following, determine the equilibrium mole fractions \(x_{1} \text { and } y_{1}\) of the liquid and vapor phases formed, the molar fraction \(\mathcal{V}\) of the vapor formed, and the fractional recovery \(\mathcal{R}\) of species 1 in the vapor phase (defined as the ratio for species 1 of moles in the vapor to moles in the feed). Assume that Raoult’s law applies. (a) Acetone(1)/acetonitrile(2), \(z_{1}=0.75\), T = 340 K, P = 115 kPa. (b) Benzene(1)/ethylbenzene(2), \(z_{1}=0.50\), \(\mathrm{T}=100^{\circ} \mathrm{C}\), P = 0.75(atm). (c) Ethanol(1)/1-propanol(2), \(z_{1}=0.25\), T = 360 K, P = 0.80(atm). (d) 1-Chlorobutane(1)/chlorobenzene(2),\(z_{1}=0.50\),\(\mathrm{T}=125^{\circ} \mathrm{C}\), P = 1.75 bar. Text Transcription: z_1 x_1 and y_1 V R z_1=0.75 z_1=0.50 T=100^circ C z_1=0.25 T=125^circ C

A concentrated binary solution containing mostly species 2 (but \(x_{2} \neq 1 \text { ) }\) is in equilibrium with a vapor phase containing both species 1 and 2. The pressure of this two phase system is 1 bar; the temperature is \(25^{\circ} \mathrm{C}\). At this temperature, \(\mathcal{H}_{1}=200\) bar and \(P_{2}^{\mathrm{sat}}=0.10\) bar. Determine good estimates of \(x_{1}\) and \(y_{1}\). State and justify all assumptions. Text Transcription: (x_2 neq 1) 25^circ C H_1=200 P_2^sat=0.10 x_1 and y_1

A binary system of species 1 and 2 consists of vapor and liquid phases in equilibrium at temperature T. The overall mole fraction of species 1 in the system is \(z_{1}=0.65\). At temperature T, \(\ln \gamma_{1}=0.67 x_{2}^{2} ; \ln \gamma_{2}=0.67 x_{1}^{2} ; P_{1}^{\text {sat }}=32.27 \mathrm{kPa}\) ; and \(P_{2}^{\text {sat }}=73.14 \mathrm{kPa}\). Assuming the validity of Eq. (13.19), (a) Over what range of pressures can this system exist as two phases at the given T and \(z_{1}\)? (b) For a liquid-phase mole fraction \(x_{1}=0.75\), what is the pressure P and what molar fraction \(\mathcal{V}\) of the system is vapor? (c) Show whether or not the system exhibits an azeotrope. Text Transcription: z_1=0.65 ln gamma_1=0.67x_2^2; ln gamma_2=0.67x_1^2; P_1^sat=32.27kPa; P_2^sat=73.14 kPa z_1 x_1=0.75 V

For the system ethyl ethanoate(1)/n-heptane(2) at 343.15 K, \(\ln \gamma_{1}=0.95 x_{2}^{2} ; \ln \gamma_{2}=0.95 x_{1}^{2}\) ; \(P_{1}^{\mathrm{sat}}=79.80 \mathrm{kPa} ; \text { and } P_{2}^{\mathrm{sat}}=40.50 \mathrm{kPa}\). Assuming the validity of Eq. (13.19), (a) Make a BUBL P calculation for T = 343.15 K, \(x_{1}=0.05\). (b) Make a DEW P calculation for T = 343.15 K, \(y_{1}=0.05\). (c) What are the azeotrope composition and pressure at T = 343.15 K? Text Transcription: ln gamma_1=0.95 x_2^2 ; ln gamma_2=0.95 x_1^2 P_1^sat=79.80 kPa ; and P_2^sat=40.50 kPa x_1=0.05 y_1=0.05

A liquid mixture of cyclohexanone(1)/phenol(2) for which \(x_{1}=0.6\) is in equilibrium with its vapor at \(144^{\circ} \mathrm{C}\). Determine the equilibrium pressure P and vapor composition \(y_{1}\) from the following information: ? \(\ln \gamma_{1}=A x_{2}^{2} \quad \ln \gamma_{2}=A x_{1}^{2}\) ? \(\text { At } 144^{\circ} \mathrm{C}, P_{1}^{\text {sat }}=75.20 \mathrm{kPa} \text { and } P_{2}^{\text {sat }}=31.66 \mathrm{kPa}\) ? The system forms an azeotrope at \(144^{\circ} \mathrm{C}\) for which \(x_{1}^{\mathrm{az}}=y_{1}^{\mathrm{az}}=0.294\) Text Transcription: x_1=0.6 144^circ C y_1 ln gamma_1=A x_2^2 ln gamma_2=Ax_1^2 At 144^circ C, P_1^sat=75.20kPa and P_2^sat=31.66kPa x_1^az=y_1^az=0.294

A binary system of species 1 and 2 consists of vapor and liquid phases in equilibrium at temperature T, for which \(\ln \gamma_{1}=1.8 x_{2}^{2}, \ln \gamma_{2}=1.8 x_{1}^{2}, P_{1}^{\text {sat }}=1.24\) bar, and \(P_{2}^{\text {sat }}=40.50\) kPa. Assuming the validity of Eq. (13.19), (a) For what range of values of the overall mole fraction \(z_{1}\) can this two-phase system exist with a liquid mole fraction \(x_{1}=0.65\)? (b) What are the pressure P and vapor mole fraction \(y_{1}\) within this range? (c) What is the pressure and composition of the azeotrope at temperature T? Text Transcription: ln gamma_1=1.8 x_2^2, ln gamma_2=1.8 x_1^2, P_1^sat=1.24 P_2^sat=40.50 z_1 x_1=0.65 y_1

For the acetone(1)/methanol(2) system, a vapor mixture for which \(z_{1}=0.25\) and \(z_{2}=0.75\) is cooled to temperature T in the two-phase region and flows into a separation chamber at a pressure of 1 bar. If the composition of the liquid product is to be \(x_{1}=0.175\), what is the required value of T, and what is the value of \(y_{1}\)? For liquid mixtures of this system, to a good approximation, \(\ln \gamma_{1}=0.64 x_{2}^{2} \text { and } \ln \gamma_{2}=0.64 x_{1}^{2}\). Text Transcription: z_1=0.25 z_2=0.75 x_1=0.175 y_1 ln gamma_1=0.64 x_2^2 and ln gamma_2=0.64 x_1^2

The following is a rule of thumb: For a binary system in VLE at low pressure, the equilibrium vapor-phase mole fraction \(y_{1}\) corresponding to an equimolar liquid mixture is approximately \(y_{1}=\frac{P_{1}^{\mathrm{sat}}}{P_{1}^{\mathrm{sat}}+P_{2}^{\mathrm{sat}}}\) where \(P_{i}^{\text {sat }}\) is a pure-species vapor pressure. Clearly, this equation is valid if Raoult’s law applies. Prove that it is also valid for VLE described by Eq. (13.19), with \(\ln \gamma_{1}=A x_{2}^{2} \text { and } \ln \gamma_{2}=A x_{1}^{2}\). Text Transcription: y_1 y_1=P_1^sat/P_1^sat+P_2^sat P_i^sat ln gamma_1=A x_2^2 and ln gamma_2=A x_1^2

A process stream contains light species 1 and heavy species 2. A relatively pure liquid stream containing mostly 2 is desired, obtained by a single-stage liquid/vapor separation. Specifications of the equilibrium composition are: \(x_{1}=0.002 \text { and } y_{1}=0.950\). Use data given below to determine T (K) and P (bar) for the separator. Assume that Text Transcription: x_1=0.002 and y_1=0.950

Flash calculations are simpler for binary systems than for the general multicomponent case because the equilibrium compositions for a binary are independent of the overall composition. Show that, for a binary system in VLE, \(\begin{array}{c} x_{1}=\frac{1-K_{2}}{K_{1}-K_{2}} \quad y_{1}=\frac{K_{1}\left(1-K_{2}\right)}{K_{1}-K_{2}} \\ \mathcal{V}=\frac{z_{1}\left(K_{1}-K_{2}\right)-\left(1-K_{2}\right)}{\left(K_{1}-1\right)\left(1-K_{2}\right)} \end{array}\) Text Transcription: x_1=1-K_2/K_1-K_2 y_1=K_1(1-K_2)/K_1-K_2 V=z_1(K_1-K_2)-(1-K_2)/(K_1-1)(1-K_2)

The NIST Chemistry WebBook reports critically evaluated Henry’s constants for selected chemicals in water at \(25^{\circ} \mathrm{C}\). Henry’s constants from this source, denoted here by \(k_{H i}\) appear in the VLE equation written for the solute in the form: \(m_{i}=k_{H i} y_{i} P\) where \(m_{i}\) is the liquid-phase molality of solute species i, expressed as mol i?kg solvent. (a) Determine an algebraic relation connecting \(k_{H i} \text { to } \mathcal{H}_{i}\), Henry’s constant in Eq. (13.26). Assume that \(x_{i}\) is “small.” (b) The NIST Chemistry WebBook provides a value of 0.034 \(\mathrm{mol} \cdot \mathrm{kg}^{-1} \cdot \mathrm{bar}^{-1}\) for \(k_{\mathrm{Hi}} \text { of } \mathrm{CO}_{2} \text { in } \mathrm{H}_{2} \mathrm{O} \text { at } 25^{\circ} \mathrm{C}\). What is the implied value of \(\mathcal{H}_{i}\) in a bar? Compare this with the value given in Table 13.2, which came from a different source. Text Transcription: 25^circ C k_Hi m_i=k_Hi y_iP m_i k_Hi to H_i x_i mol cdot kg^-1 cdot bar^-1 k_Hi of CO_2 in H_2O at 25^circ C H_i

(a) A feed containing equimolar amounts of acetone(1) and acetonitrile(2) is throttled to pressure P and temperature T. For what pressure range (atm) will two phases (liquid and vapor) be formed for \(T=50^{\circ} \mathrm{C}\)? Assume that Raoult’s law applies. (b) A feed containing equimolar amounts of acetone(1) and acetonitrile(2) is throttled to pressure P and temperature T. For what temperature range \(\left({ }^{\circ} \mathrm{C}\right)\) will two phases (liquid and vapor) be formed for P = 0.5(atm)? Assume that Raoult’s law applies. Text Transcription: T=50^circ C (^circ C)

A binary mixture of benzene(1) and toluene(2) is flashed to 75 kPa and \(90^{\circ} \mathrm{C}\). Analysis of the effluent liquid and vapor streams from the separator yields: \(x_{1}=0.1604\) and \(y_{1}=0.2919\). An operator remarks that the product streams are “off-spec,” and you are asked to diagnose the problem. (a) Verify that the exiting streams are not in binary equilibrium. (b) Verify that an air leak into the separator could be the cause. Text Transcription: 90^circ C x_1=0.1604 y_1=0.2919

Ten (10) \(\mathrm{kmol} \cdot \mathrm{hr}^{-1}\) of hydrogen sulfide gas is burned with the stoichiometric amount of pure oxygen in a special unit. Reactants enter as gases at \(25^{\circ} \mathrm{C}\) and 1(atm). Products leave as two streams in equilibrium at \(70^{\circ} \mathrm{C}\) and 1(atm): a phase of pure liquid water, and a saturated vapor stream containing \(\mathrm{H}_{2} \mathrm{O} \text { and } \mathrm{SO}_{2}\). (a) What is the composition (mole fractions) of the product vapor stream? (b) What are the rates \(\left(\mathrm{kmol} \cdot \mathrm{hr}^{-1}\right)\) of the two product streams? Text Transcription: kmol cdot hr^-1 25^circ C 70^circ C H_2O and SO_2 (kmol cdot hr^-1)

Physiological studies show the neutral comfort level (NCL) of moist air corresponds to an absolute humidity of about 0.01 kg \(\mathrm{H}_{2} \mathrm{O}\) per kg of dry air. (a) What is the vapor-phase mole fraction of \(\mathrm{H}_{2} \mathrm{O}\) at the NCL? (b) What is the partial pressure of \(\mathrm{H}_{2} \mathrm{O}\) at the NCL? Here, and in part (c), take P = 1.01325 bar. (c) What is the dewpoint temperature \(\left({ }^{\circ} \mathrm{F}\right)\) at the NCL? Text Transcription: H_2O (^circ F)

An industrial dehumidifier accepts 50 \(\mathrm{kmol} \cdot \mathrm{hr}^{-1}\) of moist air with a dew point of \(20^{\circ} \mathrm{C}\). Conditioned air leaving the dehumidifier has a dew point temperature of \(10^{\circ} \mathrm{C}. At what rate \(\left(\mathrm{kg} \cdot \mathrm{hr}^{-1}\right)\) is liquid water removed in this steady-flow process? Assume P is constant at 1(atm). Text Transcription: kmol cdot hr^-1 20^circ C 10^circ C (kg cdot hr^-1)

Vapor/liquid-equilibrium azeotropy is impossible for binary systems rigorously described by Raoult’s law. For real systems (those with \(\gamma_{i} \neq 1\)), azeotropy is inevitable at temperatures where the \(P_{i}^{\mathrm{sat}}\) are equal. Such a temperature is called a Bancroft point. Not all binary systems exhibit such a point. With Table B.2 of App. B as a resource, identify three binary systems with Bancroft points, and determine the T and P coordinates. Ground rule: A Bancroft point must lie in the temperature ranges of validity of the Antoine equations. Text Transcription: gamma_i neq 1 P_i^sat

The following is a set of VLE data for the system methanol(1)/water(2) at 333.15 K: (a) Basing calculations on Eq. (13.24), find parameter values for the Margules equation that provide the best fit of \(G^{E} / RT\) to the data, and prepare a Pxy diagram that compares the experimental points with curves determined from the correlation. (b) Repeat (a) for the van Laar equation. (c) Repeat (a) for the Wilson equation. (d) Using Barker’s method, find parameter values for the Margules equation that provide the best fit of the \(P-x_{1}\) data. Prepare a diagram showing the residuals \(\delta P\) and \(\delta y_{1}\) plotted vs. \(x_{1}\). (e) Repeat (d) for the van Laar equation. (f) Repeat (d) for the Wilson equation. Text Transcription: G^E/RT P-x_1 delta P delta y_1 x_1

If Eq. (13.24) is valid for isothermal VLE in a binary system, show that: \(\left(\frac{d P}{d x_{1}}\right)_{x_{1}=0} \geq-P_{2}^{\mathrm{sat}}\left(\frac{d P}{d x_{1}}\right)_{x_{1}=1} \leq P_{1}^{\mathrm{sat}}\) Text Transcription: (dPdx_1)_x_1=0 geq-P_2^sat(dP/dx_1)_x_1=1 leq P_1^sat

The following is a set of VLE data for the system acetone(1)/methanol(2) at \(55^{\circ} \mathrm{C}\): (a) Basing calculations on Eq. (13.24), find parameter values for the Margules equation that provide the best fit of \(G^{E} / R T\) to the data, and prepare a Pxy diagram that compares the experimental points with curves determined from the correlation. (b) Repeat (a) for the van Laar equation. (c) Repeat (a) for the Wilson equation. (d) Using Barker’s method, find parameter values for the Margules equation that provide the best fit of the \(P-x_{1}\) data. Prepare a diagram showing the residuals \(\delta P\) and \(\delta y_{1}\) plotted vs. \(x_{1}\). (e) Repeat (d) for the van Laar equation. (f) Repeat (d) for the Wilson equation. Text Transcription: 55^\circ mathrmC G^E/RT P-x_1 delta P delta y_1 x_1

The excess Gibbs energy for binary systems consisting of liquids not too dissimilar in chemical nature is represented to a reasonable approximation by the equation: \(G^{E} / R T=A x_{1} x_{2}\) where A is a function of temperature only. For such systems, it is often observed that the ratio of the vapor pressures of the pure species is nearly constant over a considerable temperature range. Let this ratio be r, and determine the range of values of A, expressed as a function of r, for which no azeotrope can exist. Assume the vapor phase to be an ideal gas. Text Transcription: G^E/RT=A x_1x_2

For the ethanol(1)/chloroform(2) system at \(50^{\circ} \mathrm{C}\), the activity coefficients show interior extrema with respect to composition [see Fig. 13.4(e)]. (a) Prove that the van Laar equation cannot represent such behavior. (b) The two-parameter Margules equation can represent this behavior, but only for particular ranges of the ratio \(A_{21} / A_{12}\). What are they? Text Transcription: 50^circ C A_21/A_12

VLE data for methyl tert-butyl ether(1)/dichloromethane(2) at 308.15 K are as follows: The data are well correlated by the three-parameter Margules equation [an extension of Eq. (13.39)]: \(\frac{G^{E}}{R T}=\left(A_{21} x_{1}+A_{12} x_{2}-C x_{1} x_{2}\right) x_{1} x_{2}\) Implied by this equation are the expressions: \(\ln \gamma_{1}=x_{2}^{2}\left[A_{12}+2\left(A_{21}-A_{12}-C\right) x_{1}+3 C x_{1}^{2}\right]\) \(\ln \gamma_{2}=x_{1}^{2}\left[A_{21}+2\left(A_{12}-A_{21}-C\right) x_{2}+3 C x_{2}^{2}\right]\) (a) Basing calculations on Eq. (13.24), find the values of parameters \(A_{12}, A_{21}\), and C that provide the best fit of \(G^{E} / R T\) to the data. (b) Prepare a plot of ln \(\gamma_{1}, \ln \gamma_{2}, \text { and } G^{E} /\left(x_{1} x_{2} R T\right) \text { vs. } x_{1}\) showing both the correlation and experimental values. (c) Prepare a Pxy diagram [see Fig. 13.8(a)] that compares the experimental data with the correlation determined in (a). (d) Prepare a consistency-test diagram like Fig. 13.9. (e) Using Barker’s method, find the values of parameters \(A_{12}, A_{21}\), and C that provide the best fit of the \(P-x_{1}\) data. Prepare a diagram showing the residuals \(\delta P\) and \(\delta y_{1}\) plotted vs. \(x_{1}\). Text Transcription: G^E/RT=(A_21x_1+A_12x_2-Cx_1x_2)x_1x_2 ln gamma_1=x_2^2[A_12+2(A_21-A_12-C)x_1+3Cx_1^2] ln gamma_2=x_1^2[A_21+2(A_12-A_21-C)x_2+3Cx_2^2] A_12, A_21 G^E/RT gamma_1, ln gamma_2, and G^E/(x_1x_2RT) vs. x_1 P-x_1 delta P delta y_1 x_1

Equations analogous to Eqs. (10.15) and (10.16) apply for excess properties. Because \(\ln \gamma_{i}\) is a partial property with respect to \(G^{E} / R T\), these analogous equations can be written for \(\ln \gamma_{1} \text { and } \ln \gamma_{2}\) in a binary system. (a) Write these equations, and apply them to Eq. (13.42) to show that Eqs. (13.43) and (13.44) are indeed obtained. (b) The alternative procedure is to apply Eq. (13.7). Show that by doing so Eqs. (13.43) and (13.44) are again reproduced. Text Transcription: ln gamma_i G^E/RT ln gamma_1 and ln gamma_2

The following is a set of activity-coefficient data for a binary liquid system as determined from VLE data: Inspection of these experimental values suggests that they are noisy, but the question is whether they are consistent, and therefore possibly on average correct. (a) Find experimental values for \(G^{E} / R T\) and plot them along with the experimental values of ln \(\ln \gamma_{1} \text { and } \ln \gamma_{2}\) on a single graph. (b) Develop a valid correlation for the composition dependence of \(G^{E} / R T\) and show lines on the graph of part (a) that represent this correlation for all three of the quantities plotted there. (c) Apply the consistency test described in Ex. 13.4 to these data, and draw a conclusion with respect to this test. Text Transcription: G^E/RT ln gamma_1 and ln gamma_2

Following are VLE data for the system acetonitrile(1)/benzene(2) at \(45^{\circ} \mathrm{C}\): The data are well correlated by the three-parameter Margules equation (see Prob. 13.37). (a) Basing calculations on Eq. (13.24), find the values of parameters \(A_{12}, A_{21}\), and C that provide the best fit of \9G^{E} / R T\) to the data. (b) Prepare a plot of \(\ln \gamma_{1}, \ln \gamma_{2}, \text { and } G^{E} / x_{1} x_{2} R T \text { vs. } x_{1}\) showing both the correlation and experimental values. (c) Prepare a Pxy diagram [see Fig. 13.8(a)] that compares the experimental data with the correlation determined in (a). (d) Prepare a consistency-test diagram like Fig. 13.9. (e) Using Barker’s method, find the values of parameters \(A_{12}, A_{21}\), and C that provide the best fit of the \(P-x_{1}\) data. Prepare a diagram showing the residuals \(\delta P \text { and } \delta y_{1}\) plotted vs. \(x_{1}\). Text Transcription: 45^circ C A_12, A_21 G^E/RT ln gamma_1, ln gamma_2, and G^E/x_1x_2RT vs. x_1 P-x_1 delta P and delta y_1 x_1

Rationalize the following rule of thumb, appropriate for an equimolar binary liquid mixture: \(\frac{G^{E}}{R T}(\text { equimolar }) \approx \frac{1}{8} \ln \left(\gamma_{1}^{\infty} \gamma_{2}^{\infty}\right)\) Text Transcription: G^E/RT(equimolar) approx 1/8 ln (gamma_1^infty gamma_2^infty)

Problems 13.43 through 13.54 require parameter values for the Wilson or NRTL equation for liquid-phase activity coefficients. Table 13.10 gives parameter values for both equations. Antoine equations for vapor pressure are given in Table B.2, Appendix B. For one of the binary systems listed in Table 13.10, based on Eq. (13.19) and the Wilson equation, prepare a Pxy diagram for \(\mathrm{t}=60^{\circ} \mathrm{C}\). Text Transcription: t=60^circ C

Problems 13.43 through 13.54 require parameter values for the Wilson or NRTL equation for liquid-phase activity coefficients. Table 13.10 gives parameter values for both equations. Antoine equations for vapor pressure are given in Table B.2, Appendix B. For one of the binary systems listed in Table 13.10, based on Eq. (13.19) and the Wilson equation, prepare a txy diagram for P = 101.33 kPa.

Problems 13.43 through 13.54 require parameter values for the Wilson or NRTL equation for liquid-phase activity coefficients. Table 13.10 gives parameter values for both equations. Antoine equations for vapor pressure are given in Table B.2, Appendix B. For one of the binary systems listed in Table 13.10, based on Eq. (13.19) and the NRTL equation, prepare a Pxy diagram for \(\mathrm{t}=60^{\circ} \mathrm{C}\). Text Transcription: t=60^circ C

Problems 13.43 through 13.54 require parameter values for the Wilson or NRTL equation for liquid-phase activity coefficients. Table 13.10 gives parameter values for both equations. Antoine equations for vapor pressure are given in Table B.2, Appendix B. For one of the binary systems listed in Table 13.10, based on Eq. (13.19) and the NRTL equation, prepare a txy diagram for P = 101.33 kPa.

Problems 13.43 through 13.54 require parameter values for the Wilson or NRTL equation for liquid-phase activity coefficients. Table 13.10 gives parameter values for both equations. Antoine equations for vapor pressure are given in Table B.2, Appendix B. For one of the binary systems listed in Table 13.10, based on Eq. (13.19) and the Wilson equation, make the following calculations: (a) BUBL P : \(t=60^{\circ} \mathrm{C}, x_{1}=0.3\). (b) DEW P : \(t=60^{\circ} \mathrm{C}, y_{1}=0.3\). (c) P, T ? flash: \(t=60^{\circ} \mathrm{C}, P=\frac{1}{2}\left(P_{\text {bubble }}+P_{\text {dew }}\right), z_{1}=0.3\). (d) If an azeotrope exists at \(t=60^{\circ} \mathrm{C}, \text { find } P^{\text {az }} \text { and } x_{1}^{\mathrm{az}}=y_{1}^{\mathrm{az}}\). Text Transcription: t=60^circ C, x_1=0.3 t=60^circ C, y_1=0.3 t=60^circ C, P=1/2(P_bubble+P_dew), z_1=0.3 t=60^circ C, find P^az and x_1^az=y_1^az

Problems 13.43 through 13.54 require parameter values for the Wilson or NRTL equation for liquid-phase activity coefficients. Table 13.10 gives parameter values for both equations. Antoine equations for vapor pressure are given in Table B.2, Appendix B. Work Prob. 13.47 for the NRTL equation.

Problems 13.43 through 13.54 require parameter values for the Wilson or NRTL equation for liquid-phase activity coefficients. Table 13.10 gives parameter values for both equations. Antoine equations for vapor pressure are given in Table B.2, Appendix B. For one of the binary systems listed in Table 13.10, based on Eq. (13.19) and the Wilson equation, make the following calculations: (a) BUBL T : P = 101.33 kPa, \(x_{1}=0.3\). (b) DEW T : P = 101.33 kPa, \(y_{1}=0.3\). (c) P, T ? flash : P = 101.33 kPa, \(T=\frac{1}{2}\left(T_{\text {bubble }}+T_{\text {dew }}\right), z_{1}=0.3\). (d) If an azeotrope exists at P = 101.33 kPa, find \(T^{\mathrm{az}} \text { and } x_{1}^{\mathrm{az}}=y_{1}^{\mathrm{az}}\). Text Transcription: x_1=0.3 y_1=0.3 P=1/2(T_bubble+T_dew), z_1=0.3 T^az and x_1^az=y_1^az

Problems 13.43 through 13.54 require parameter values for the Wilson or NRTL equation for liquid-phase activity coefficients. Table 13.10 gives parameter values for both equations. Antoine equations for vapor pressure are given in Table B.2, Appendix B. Work Prob. 13.49 for the NRTL equation.

Problems 13.43 through 13.54 require parameter values for the Wilson or NRTL equation for liquid-phase activity coefficients. Table 13.10 gives parameter values for both equations. Antoine equations for vapor pressure are given in Table B.2, Appendix B. For the acetone(1)/methanol(2)/water(3) system, based on Eq. (13.19) and the Wilson equation, make the following calculations: (a) BUBL P : \(t=65^{\circ} \mathrm{C}, x_{1}=0.3, x_{2}=0.4\). (b) DEW P : \(t=65^{\circ} C, y_{1}=0.3, y_{2}=0.4\). (c) P, T ? flash : \(t=65^{\circ} \mathrm{C}, P=\frac{1}{2}\left(P_{\text {bubble }}+P_{\text {dew }}\right), z_{1}=0.3, z_{2}=0.4\). Text Transcription: t=65^circ C, x_1=0.3, x_2=0.4 t=65^circ C, y_1=0.3, y_2=0.4 t=65^circ C, P=1/2(P_bubble+P_dew), z_1=0.3, z_2=0.4

Problems 13.43 through 13.54 require parameter values for the Wilson or NRTL equation for liquid-phase activity coefficients. Table 13.10 gives parameter values for both equations. Antoine equations for vapor pressure are given in Table B.2, Appendix B. Work Prob. 13.51 for the NRTL equation.

Problems 13.43 through 13.54 require parameter values for the Wilson or NRTL equation for liquid-phase activity coefficients. Table 13.10 gives parameter values for both equations. Antoine equations for vapor pressure are given in Table B.2, Appendix B. For the acetone(1)/methanol(2)/water(3) system, based on Eq. (13.19) and the Wilson equation, make the following calculations: (a) \(B U B L T: P=101.33 \mathrm{kPa}, x_{1}=0.3, x_{2}=0.4\). (b) \(D E W T: P=101.33 \mathrm{kPa}, y_{1}=0.3, y_{2}=0.4\). (c) P, T ? flash : \(P=101.33 \mathrm{kPa}, T=\frac{1}{2}\left(T_{\text {bubble }}+T_{\text {dew }}\right), z_{1}=0.3, z_{2}=0.2\). Text Transcription: BUBLT: P=101.33 kPa, x_1=0.3, x_2=0.4 DEW T: P=101.33 kPa, y_1=0.3, y_2=0.4 P=101.33 kPa, T=1/2(T_bubble+T_dew), z_1=0.3, z_2=0.2

Problems 13.43 through 13.54 require parameter values for the Wilson or NRTL equation for liquid-phase activity coefficients. Table 13.10 gives parameter values for both equations. Antoine equations for vapor pressure are given in Table B.2, Appendix B. Work Prob. 13.53 for the NRTL equation.

The following expressions have been reported for the activity coefficients of species 1 and 2 in a binary liquid mixture at given T and P: \(\gamma_{1}=x_{2}^{2}\left(0.273+0.096 x_{1}\right) \quad \ln \gamma_{2}=x_{1}^{2}\left(0.273-0.096 x_{2}\right)\) (a) Determine the implied expression for \(G^{E} / R T\). (b) Generate expressions for ln \(\gamma_{1} \text { and } \ln \gamma\) from the result of (a). (c) Compare the results of (b) with the reported expressions for ln \(\gamma_{1} \text { and } \ln \gamma\). Discuss any discrepancy. Can the reported expressions possibly be correct? Text Transcription: gamma_1=x_2^2(0.273+0.096x_1) ln gamma_2=x_1^2(0.273-0.096x_2) G^E/RT gamma_1 and ln gamma

Possible correlating equations for ln \(\gamma_{1}\) in a binary liquid system are given here. For one of these cases, determine by integration of the Gibbs/Duhem equation [Eq. (13.11)] the corresponding equation for ln \(\gamma_{2}\). What is the corresponding equation for \(G^{E} / R T\)? Note that by its definition, \(\gamma_{i}=1 \text { for } x_{i}=1\). (a) \(\ln \gamma_{1}=A x_{2}^{2}\) ; (b) \(\ln \gamma_{1}=x_{2}^{2}\left(A+B x_{2}\right)\) ; (c) ln \(\gamma_{1}=x_{2}^{2}\left(A+B x_{2}+C x_{2}^{2}\right)\) ; Text Transcription: gamma_1 gamma_2 G^E/RT gamma_i=1 for x_i=1 ln gamma_1=Ax_2^2 ln gamma_1=x_2^2(A+Bx_2) gamma_1=x_2^2(A+Bx_2+Cx_2^2)

A storage tank contains a heavy organic liquid. Chemical analysis shows the liquid to contain 600 ppm (molar basis) of water. It is proposed to reduce the water concentration to 50 ppm by boiling the contents of the tank at constant atmospheric pressure. Because the water is lighter than the organic, the vapor will be rich in water; continuous removal of the vapor serves to reduce the water content of the system. Estimate the percentage loss of organic (molar basis) in the boil-off process. Comment on the reasonableness of the proposal. Suggestion: Designate the system water(1)/organic(2) and do unsteady-state molar balances for water and for water + organic. State all assumptions. Data: \(T_{n_{2}}\) = normal boiling point of organic = \(130^{\circ} \mathrm{C}\). \(\gamma_{1}^{\infty}\) = 5.8 for water in the liquid phase at \(130^{\circ} \mathrm{C}\). Text Transcription: T_n_2 130^circ C gamma_1^infty

Binary VLE data are commonly measured at constant T or at constant P. Isothermal data are much preferred for determination of a correlation for \(G^{E}\) for the liquid phase. Why? Text Transcription: G^E

Table 13.10 gives values of parameters for the Wilson equation for the acetone(1)/ methanol(2) system. Estimate values of \(\ln \gamma_{1}^{\infty} \text { and } \ln \gamma_{2}^{\infty} \text { at } 50^{\circ} \mathrm{C}\). Compare with the values suggested by Fig. 13.4(b). Repeat the exercise with the NRTL equation. Text Transcription: ln gamma_1^infinity and ln gamma_2^infinity at 50^circ C

For a binary system derive the expression for \(H^{E}\) implied by the Wilson equation for \(G^{E} / R T\). Show that the implied excess heat capacity \(C_{P}^{E}\) is necessarily positive. Recall that the Wilson parameters depend on T, in accord with Eq. (13.53). Text Transcription: H^E G^E/RT C_P^E

A single \(P-x_{1}-y_{1}\) data point is available for a binary system at \(25^{\circ} \mathrm{C}\). Estimate from the data: (a) The total pressure and vapor-phase composition at \(25^{\circ} \mathrm{C}\) for an equimolar liquid mixture. (b) Whether azeotropy is likely at \(25^{\circ} \mathrm{C}\). Data: \(\text { At } 25^{\circ} \mathrm{C}, P_{1}^{\text {sat }}=183.4 \text { and } P_{2}^{\text {sat }}=96.7 \mathrm{kPa}\) \(\text { For } x_{1}=0.253, y_{1}=0.456 \text { and } P=139.1 \mathrm{kPa}\) Text Transcription: P-x_1-y_1 25^circ C At 25^circ C, P_1^sat=183.4 and P_2^sat=96.7kPa For x_1=0.253, y_1=0.456 and P=139.1 kPa

A single \(P-x_{1}\) data point is available for a binary system at \(35^{\circ} \mathrm{C}\). Estimate from the data: (a) The corresponding value of \(y_{1}\). (b) The total pressure at \(35^{\circ} \mathrm{C}\) for an equimolar liquid mixture. (c) Whether azeotropy is likely at \(35^{\circ} \mathrm{C}\). Data: \(\text { At } 25^{\circ} \mathrm{C}, P_{1}^{\text {sat }}=120.2 \text { and } P_{2}^{\text {sat }}=73.9 \mathrm{kPa}\) \(\text { For } x_{1}=0.389, P=108.6 \mathrm{kPa}\) Text Transcription: P-x_1 35^circ C y_1 At 25^circ C, P_1^sat=120.2 and P_2^sat=73.9kPa For x_1=0.389, P=108.6 kPa

The excess Gibbs energy for the system chloroform(1)/ethanol(2) at \(55^{\circ} \mathrm{C}\) is well represented by the Margules equation, \(G^{E} / R T=\left(1.42 x_{1}+0.59 x_{2}\right) x_{1} x_{2}\). The vapor pressures of chloroform and ethanol at \(55^{\circ} \mathrm{C}\) are \(P_{1}^{\text {sat }}=82.37 \text { and } P_{2}^{\text {sat }}=37.31 \mathrm{kPa}\). (a) Assuming the validity of Eq. (13.19), make BUBL P calculations at \(55^{\circ} \mathrm{C}\) for liquid-phase mole fractions of 0.25, 0.50, and 0.75. (b) For comparison, repeat the calculations using Eqs. (13.13) and (13.14) with virial coefficients: \(B_{11}=-963, B_{22}=-1523, \text { and } B_{12}=52 \mathrm{cm}^{3} \cdot \mathrm{mol}^{-1}\). Text Transcription: 55^circ C G^E/RT=(1.42x_1+0.59x_2)x_1x_2 P_1^sat =82.37 and P_2^sat=37.31kPa B_11=-963, B_22=-1523, and B_12=52 cm^3 cdot mol^-1

Find expressions for \(\hat{\phi}_{1} \text { and } \hat{\phi}_{2}\) for a binary gas mixture described by Eq. (3.38). The mixing rule for B is given by Eq. (10.62). The mixing rule for C is given by the general equation: \(C=\sum_{i} \sum_{j} \sum_{k} y_{i} y_{j} y_{k} C_{i j k}\) where Cs with the same subscripts, regardless of order, are equal. For a binary mixture, this becomes: \(C=y_{1}^{3} C_{111}+3 y_{1}^{2} y_{2} C_{112}+3 y_{1} y_{2}^{2} C_{122}+y_{2}^{3} C_{222}\) Text Transcription: hat phi_1 and hat phi_2 C=sum_i sum_j sum_k y_i y_j y_k C_ijk C=y_1^3 C_111+3 y_1^2 y_2 C_112+3y_1 y_2^2 C_122+y_2^3 C_222

A system formed of methane(1) and a light oil(2) at 200 K and 30 bar consists of a vapor phase containing 95 mol-% methane and a liquid phase containing oil and dissolved methane. The fugacity of the methane is given by Henry’s law, and at the temperature of interest Henry’s constant is \(\mathcal{H}_{1}=200\) bar. Stating any assumptions, estimate the equilibrium mole fraction of methane in the liquid phase. The second virial coefficient of pure methane at 200 K is \(-105 \mathrm{cm}^{3} \cdot \mathrm{mol}^{-1}\). Text Transcription: H_1=200 -105 cm^3 cdot mol^-1

Use Eq. (13.13) to reduce one of the following isothermal data sets, and compare the result with that obtained by application of Eq. (13.19). Recall that reduction means developing a numerical expression for \(G^{E} /RT\) as a function of composition. (a) Methyl Ethyl Ketone(1)/toluene(2) at \(50^{\circ} \mathrm{C}\): Table 13.1. (b) Acetone(1)/methanol(2) at \(55^{\circ} \mathrm{C}\): Prob. 13.34. (c) Methyl tert-butyl ether(1)/dichloromethane(2) at \(35^{\circ} \mathrm{C}\): Prob. 13.37. (d) Acetonitrile(1)/benzene(2) at \(45^{\circ} \mathrm{C}\): Prob. 13.40. Second-virial-coefficient data are as follows: Text Transcription: G^E/RT 50^circ C 55^circ C 35^circ C 45^circ C

For one of the following substances, determine \(P^{\mathrm{sat}} / \mathrm{bar}\) from the Redlich/Kwong equation at two temperatures: \(T=T_{n}\) (the normal boiling point), and \(T=0.85 T_{c}\). For the second temperature, compare your result with a value from the literature (e.g., Perry’s Chemical Engineers’ Handbook). Discuss your results. (a) Acetylene; (b) Argon; (c) Benzene; (d) n-Butane; (e) Carbon monoxide; (f) n-Decane; (g) Ethylene; (h) n-Heptane; (i) Methane; (j) Nitrogen Text Transcription: P^sat/bar T=T_n T=0.85T_c

Departures from Raoult’s law are primarily from liquid-phase nonidealities \(\left(\gamma_{i} \neq 1\right)\). But vapor-phase nonidealities \(\left(\hat{\phi}_{i} \neq 1\right)\) also contribute. Consider the special case where the liquid phase is an ideal solution, and the vapor phase a nonideal gas mixture described by Eq. (3.36). Show that departures from Raoult’s law at constant temperature are likely to be negative. State clearly any assumptions and approximations. Text Transcription: (gamma_i neq 1) (hat phi_i neq 1)

Determine a numerical value for the acentric factor \(\omega\) implied by: (a) The van der Waals equation; (b) The Redlich/Kwong equation. Text Transcription: omega

The relative volatility \(\alpha_{12}\) is commonly used in applications involving binary VLE. In particular (see Ex. 13.1), it serves as a basis for assessing the possibility of binary azeotropy. (a) Develop an expression for\(\alpha_{12}\) based on Eqs. (13.13) and (13.14). (b) Specialize the expression to the composition limits \(x_{1}=y_{1}=0 \text { and } x_{1}=y_{1}=1\). Compare with the result obtained from modified Raoult’s law, Eq. (13.19). The difference between the results reflects the effects of vapor-phase nonidealities. (c) Further specialize the results of part (b) to the case where the vapor phase is an ideal solution of real gases. Text Transcription: alpha_12 x_1=y_1=0 and x_1=y_1=1

Although isothermal VLE data are preferred for extraction of activity coefficients, a large body of good isobaric data exists in the literature. For a binary isobaric \(T-x_{1}-y_{1}\) data set, one can extract point values of \(\gamma_{i}\) via Eq. (13.13): \(\gamma_{i}\left(x, T_{k}\right)=\frac{y_{i} \Phi_{i}\left(T_{k}, P, y\right) P}{x_{i} P_{i}^{\text {sat }}\left(T_{k}\right)}\) Here, the variable list for \(\gamma_{i}\) recognizes a primary dependence on x and T; pressure dependence is normally negligible. The notation \(T_{k}\) emphasizes that temperature varies with data points across the composition range, and the calculated activity coefficients are at different temperatures. However, the usual goal of VLE data reduction and correlation is to develop an appropriate expression for \(G^{E} / R T\) at a single temperature T. A procedure is needed to correct each activity coefficient to such a T chosen near the average for the data set. If a correlation for \(H^{E}(x)\) is available at or near this T, show that the values of \(\gamma_{i}\) corrected to T can be estimated by the expression: \(\gamma_{i}(x, T)=\gamma_{i}\left(x, T_{k}\right) \exp \left[\frac{-\bar{H}_{i}^{E}}{R T}\left(\frac{T}{T_{k}}-1\right)\right]\) Text Transcription: T-x_1-y_1 gamma_i gamma_i(x, T_{k)=y_i Phi_i(T_k, P, y)P/x_iP_i^sat (T_k) T_k G^E/RT H^E(x) gamma_i(x, T)=gamma_i(x, T_k) exp [-bar H_i^ERT(T/T_k-1)]

What are the relative contributions of the various terms in the gamma/phi expression for VLE? One way to address the question is through calculation of the activity coefficients for a single binary VLE data point via Eq. (13.19): \(\gamma_{i}=\underbrace{\frac{y_{i} P}{x_{i} P_{i}^{\mathrm{sat}}}}_{(\mathrm{A})} \cdot \underbrace{\frac{\hat{\phi}_{i}}{\phi_{i}^{\mathrm{sat}}}}_{\text {(B) }} \cdot \underbrace{\frac{f_{i}^{\mathrm{sat}}}{f_{i}}}_{(\mathrm{C})}\) Term (A) is the value that would follow from modified Raoult’s law; term (B) accounts for vapor-phase nonidealities; term (C) is the Poynting factor [see Eq. (10.44)]. Use the following single-point data for the butanenitrile(1)/benzene(2) system at 318.15 K to evaluate all terms for i = 1 and i = 2. Discuss the results. VLE data: \(P=0.20941 \text { bar, } x_{1}=0.4819, y_{1}=0.1813\). Ancillary data: \(P_{1}^{\mathrm{sat}}=0.07287 \text { and } P_{2}^{\mathrm{sat}}=0.29871 \mathrm{bar}\) \(B_{11}=-7993, B_{22}=-1247, B_{12}=-2089 \mathrm{cm}^{3} \cdot \mathrm{mol}^{-1}\) \(V_{1}^{l}=90, V_{2}^{l}=92 \mathrm{~cm}^{3} \cdot \mathrm{mol}^{-1}\) Text Transcription: gamma_i=underbrace y_iP/x_iP_i^sat_(A) cdot underbrace hat phi_i/phi_i^sat_ (B) cdot underbrace f_i^sat/f_i_(C) P=0.20941 bar, x_1=0.4819, y_1=0.1813 P_1^sat=0.07287 and P_2^sat=0.29871 bar B_11=-7993, B_22=-1247, B_12=-2089 cm^3 cdot mol^-1 V_1^l=90, V_2^l=92 cm^3 cdot mol^-1

Generate \(P-x_{1}-y_{1}\) diagrams at \(100^{\circ} \mathrm{C}\) for one of the systems identified below. Base activity coefficients on the Wilson equation, Eqs. (13.45) to (13.47). Use two procedures: (i) modified Raoult’s law, Eq. (13.19), and (ii) the gamma/phi approach, Eq. (13.13), with \(\Phi_{i}\) given by Eq. (13.14). Plot the results for both procedures on the same graph. Compare and discuss them. Data sources: For \(P_{i}^{\mathrm{sat}}\) use Table B.2. For vapor-phase nonidealities, use material from Chap. 3; assume that the vapor phase is an (approximately) ideal solution. Estimated parameters for the Wilson equation are given for each system. (a) Benzene(1)/carbon tetrachloride(2): \(\Lambda_{12}=1.0372, \Lambda_{21}=0.8637\) (b) Benzene(1)/cyclohexane(2): \(\Lambda_{12}=1.0773, \Lambda_{21}=0.7100\) (c) Benzene(1)/n-heptane(2): \(\Lambda_{12}=1.2908, \Lambda_{21}=0.5011\) (d) Benzene(1)/n-hexane(2): \(\Lambda_{12}=1.3684, \Lambda_{21}=0.4530\) (e) Carbon tetrachloride(1)/cyclohexane(2): \(\Lambda_{12}=1.1619, \Lambda_{21}=0.7757\) (f) Carbon tetrachloride(1)/n-heptane(2): \(\Lambda_{12}=1.5410, \Lambda_{21}=0.5197\) (g) Carbon tetrachloride(1)/n-hexane(2): \(\Lambda_{12}=1.2839, \Lambda_{21}=0.6011\) (h) Cyclohexane(1)/n-heptane(2): \(\Lambda_{12}=1.2996, \Lambda_{21}=0.7046\) (i) Cyclohexane(1)/n-hexane(2): \(\Lambda_{12}=1.4187, \Lambda_{21}=0.5901\) Text Transcription: P-x_1-y_1 100^circ C Phi_i P_i^sat Lambda_12=1.0372, Lambda_21=0.8637 Lambda_12=1.0773, Lambda_21=0.7100 Lambda_12=1.2908, Lambda_21=0.5011 Lambda_12=1.3684, Lambda_21=0.4530 Lambda_12=1.1619, Lambda_21=0.7757 Lambda_12=1.5410, Lambda_21=0.5197 Lambda_12=1.2839, Lambda_21=0.6011 Lambda_12=1.2996, Lambda_21=0.7046 Lambda_12=1.4187, Lambda_21=0.5901

Humidity, relating to the quantity of moisture in atmospheric air, is accurately given by equations derived from the ideal-gas law and Raoult’s law for \(\mathrm{H}_{2} \mathrm{O}\). (a) The absolute humidity h is defined as the mass of water vapor in a unit mass of dry air. Show that it is given by: \(h=\frac{\mathscr{M}_{\mathrm{H}_{2} \mathrm{O}}}{\mathscr{M}_{\mathrm{air}}} \frac{p_{\mathrm{H}_{2} \mathrm{O}}}{P-p_{\mathrm{H}_{2} \mathrm{O}}}\) where \(\mathscr{M}\) represents a molar mass and \(p_{\mathrm{H}_{2} \mathrm{O}}\) is the partial pressure of the water vapor, i.e., \(p_{\mathrm{H}_{2} \mathrm{O}}=y_{\mathrm{H}_{2} \mathrm{O}} P\). (b) The saturation humidity \(h^{\mathrm{sat}}\) is defined as the value of h when air is in equilibrium with a large body of pure water. Show that it is given by: \(h^{\mathrm{sat}}=\frac{\mathscr{M}_{\mathrm{H}_{2} \mathrm{O}}}{\mathscr{M}_{\mathrm{air}}} \frac{p_{\mathrm{H}_{2} \mathrm{O}}^{\mathrm{sat}}}{P-p_{\mathrm{H}_{2} \mathrm{O}}^{\mathrm{sat}}}\) where \(p_{H_{2} O}^{\mathrm{sat}}\) is the vapor pressure of water at the ambient temperature. (c) The percentage humidity is defined as the ratio of h to its saturation value, expressed as a percentage. On the other hand, the relative humidity is defined as the ratio of the partial pressure of water vapor in air to its vapor pressure, expressed as a percentage. What is the relation between these two quantities? Text Transcription: H_2O h=M_H_2O/M_air p_H_2O/P-p_H_2O M p_H_2O p_H_2O=y_H_2O P h^sat h^sat=M_H_2O/M_air p_H_2O^sat/P-p_H_2O^sat p_H_2O^sat

Consider the following model for \(G^{E} / R T\) of a binary mixture: \(\frac{G^{E}}{x_{1} x_{2} R T}=\left(x_{1} A_{21}^{k}+x_{2} A_{12}^{k}\right)^{1 / k}\) This equation in fact represents a family of two-parameter expressions for \(G^{E} / R T\); specification of k leaves \(A_{12} \text { and } A_{21}\) as the free parameters. (a) Find general expressions for \(\ln \gamma_{1} \text { and } \ln \gamma_{2}\), for any k. (b) Show that \(\ln \gamma_{1}^{\infty}=A_{12} \text { and } \ln \gamma_{2}^{\infty}=A_{21}\), for any k. (c) Specialize the model to the cases where k equals \(-\infty,-1,0,+1, \text { and }+\infty\). Two of the cases should generate familiar results. What are they? Text Transcription: G^E/RT G^E/x_1x_2RT=(x_1A_21^k+x_2A_12^k)^1/k A_12 and A_21 ln gamma_1 and ln gamma_2 ln gamma_1^infty=A_12 and ln gamma_2^infty=A_21 -infty,-1,0,+1, and +infty

Prove: An equilibrium liquid/vapor system described by Raoult’s law cannot exhibit an azeotrope.

Do all four parts of Prob. 13.7, and compare the results. The required temperatures and pressures vary significantly. Discuss possible processing implications of the various temperature and pressure levels.

Do all four parts of Prob. 13.9, and compare the results. Discuss any trends that appear.

Air, even more than carbon dioxide, is inexpensive and nontoxic. Why is it not the gas of choice for making soda water and (cheap) champagne effervescent? Table 13.2 may provide useful data.

Helium-laced gases are used as breathing media for deep-sea divers. Why? Table 13.2 may provide useful data.

If a system exhibits VLE, at least one of the K-values must be greater than 1.0 and at least one must be less than 1.0. Offer a proof of this observation.

An unusual type of low-pressure VLE behavior is that of double azeotropy, in which the dew and bubble curves are S-shaped, thus yielding at different compositions both a minimum-pressure and a maximum-pressure azeotrope. Assuming that Eq. (13.57) applies, determine under what circumstances double azeotropy is likely to occur.

A breathalyzer measures volume-% ethanol in gases exhaled from the lungs. Calibration relates it to volume-% ethanol in the bloodstream. Use VLE concepts to develop an approximate relation between the two quantities. Numerous assumptions are required; state and justify them where possible.

Work Prob. 13.69 for one of the following: (a) The Soave/Redlich/Kwong equation; (b) the Peng/Robinson equation.