Description 1 Calculation Manual for Phase-Equilibrium Engineers and Investigators

Description 2 縄文研究その1　---弱肉強食生物則に対抗--- 守矢神長官による狩猟縄文は関ケ原の合戦まで永存

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Calculation Manual for Phase-Equilibrium Engineers and Investigators

Ver. 1 Sep. 1, 2024,

Ver. 2 Dec. 30, 2024

Preface

This manual includes four chapters 1-4 which were presented at MTMS’24 held in Koriyama, Japan, Aug. 27 – 29, 2024. Chapter 5 is its application which is still added. This manual, pdf version including figures, is sold at the price of 3000 JPY from TC Lines JP. A Perl program including k-parameter tables for predicting the γ_i^∞-power functions of critical ratios is sold (90,000 JPY). I hope that this manual contributes to phase-equilibrium engineers and investigators.

Satoru Kato

TC Lines JP

Professor, Emeritus, Tokyo Metropolitan University

kato@tc-lines.com (order in this e-mail address)

30 Aug. 2024, Chino, Japan

第2版のまえがき

　第五章「三定数Wohl式によるPx, xyデータの相関と熱力学解釈」を加えた。Web翻訳ソフト(Google 翻訳やMS 翻訳)を使って日本語から望みの言語に翻訳できる。Wohlパラメータデータ集、および、このページの図入り別刷り「Calculation Manual for Phase-Equilibrium Engineers and Investigators」はe-mailアドレス　kato@tc-lines.com から注文できる。

加藤　覚

茅野市豊平、2026年正月

Contents

Chapter 1. How to Classify Liquid Mixtures

Chapter 2. How to Use Critical Ratios (the critical ratio is defined in the World Patent application, PCT/ JP2012/ 074514)

Chapter 3. How to Predict Infinite Dilution Activity Coefficients

Chapter 4. How to Define Hypothetical Liquids

第五章　三定数Wohl式によるPx, xyデータの相関と熱力学解釈

Chapter 6. Applications

6.1 How to Identify Temperature Limits for Activity Coefficient Models

6.2 How to Find the Ratio of Azeotrope Strength to Molecular Disorder Strength

6.3 How to Eliminate Unreliable Px Data

6.4 How to Calculate Multicomponent VLE

6.5 How to Calculate Constant Pressure VLE

6.6 How to Estimate Molecular Disorder and Strength in Liquid Mixtures

6.7 How to Apply Entropy and Enthalpy Calculations to War Game Performance and Intermediate-step Chess Game Performance

Chapter 7 Products List

Chapter 1. How to Classify Liquid Mixtures

Introduction

The Margules equation can predict the activities of non-azeotropic binary liquid mixtures. Meanwhile, the van Laar equation covers azeotropic and two-liquid phase forming mixtures. The remaining problem is the application criteria of the Wilson equation applying to alcohol-hydrocarbon mixtures. The present manual uses the Wohl equation for establishing the criteria.

Models and Calculation Manual for Chapter 1

Non-azeotropic mixtures.

The Margules equation is used for the activity calculations of binary non-azeotropic mixtures:

lnγ₁ = x₂² [A + 2(B – A)x₁] (1-1)

lnγ₂ = x₁² [B + 2(A– B)x₂] (1-2)

A = lnγ₁^∞ (1-3)

B = lnγ₂^∞ (1-4)

Azeotropic and two-liquid phase forming mixtures.

The van Laar equation is used for the activity calculations of binary azeotropic and two-liquid phase forming mixtures:

lnγ₁ = z₂² [A + 2(BC-A)z₁] (1-5)

lnγ₂ = z₁² [B + 2(A/C-B)z₂] (1-6)

z₁ = Cx₁/(Cx₁ + x₂) (1-7)

z₂ = 1 – z₁ (1-8)

C = A/B (1-9)

Alcohol-hydrocarbon azeotropic mixtures.

The Wilson equation is used for the activity calculations of binary alcohol-hydrocarbon mixtures:

lnγ₁ = -ln(x₁ + Λ₁₂x₂) + x₂[Λ₁₂/(x₁ + Λ₁₂x₂) - Λ₂₁/(Λ₂₁x₁ + x₂)] (1-10)

lnγ₂ = -ln(x₂ + Λ₂₁x₁) – x₁[Λ₁₂/(x₁ + Λ₁₂x₂) - Λ₂₁/(Λ₂₁x₁ + x₂)] (1-11)

lnγ₁^∞ = -ln(Λ₁₂) + (1 - Λ₂₁) (1-12)

lnγ₂^∞ = -ln(Λ₂₁) – (Λ₁₂ – 1) (1-13)

Distinction criteria for the use of the Wilson equation.

Alcohol-hydrocarbon mixtures are classified into azeotropic mixtures. However, if the van Laar equation is used, correlations result in two-liquid phase forming mixtures (Prausnitz, ref. 1 page 232). Therefore, a criterion is needed for the use of the Wilson equation. The Wilson equation should be used for alcohol-hydrocarbon mixtures, if the following relationship is satisfied:

Δ_ave = (1/100)Σ|y_k,vanLaar – y_k,Wohl| > 0.01 (1-14)

The quantity, y_k,vanLaar, is calculated using the van Laar equation at k-th liquid phase mole fraction increasing with 0.01 division from 0 to 1. Furthermore, the quantity, y_k,Wohl, is calculated using the Wohl Equation at the k-th liquid phase mole fraction as follows (K. Wohl, Trans. AIChE 42 (1946) 215 - 249) :

lnγ₁ = z₂² [A + 2(BC-A)z₁] (1-15)

lnγ₂ = z₁² [B + 2(A/C-B)z₂] (1-16)

z₁ = Cx₁/(Cx₁ + x₂) (1-17)

z₂ = 1 – z₁ (1-18)

C = q₁/q₂ (1-19)

Parameters A and B in Equations (1-15) and (1-16) are determined at the minimized Δ_ave value. Meanwhile, the A and B values of the van Laar equation must represent the experimental VLE data. The Wohl equation is used in the criterion equation, because it can express the azeotropic and two-liquid phase forming behavior appearing in xy relationships, while the Px relation shows non-azeotropic behavior, as shown in Chapter 4.

Multicomponent activity coefficients.

The present manual recommends the multicomponent NRTL equation. The binary NRTL equation satisfactorily predicts the activity coefficients of non-azeotropic, azeotropic and two-liquid phase forming mixtures, except for alcohol-hydrocarbon mixtures. Determine the binary parameters of the NRTL equation, in advance, using binary Px or xy data.

Data Sources

Vapor pressure calculations

Vapor pressures are calculated using the Clapeyron equation. Details are shown in Chapter 3.

Constant temperature binary VLE and LLE data.

Constant temperature VLE and LLE data are cited from ref. 2. Furthermore, constant temperature LLE data are cited from ref. 3. The vapor-liquid equilibria are expressed as follows:

Py_i = γ_ix_ip_is (i = 1, 2) (1-20)

γ_i = γ_i^(Pa) (φ_is/φ_i^V) exp[V_i^L(P-P_a)/RT] exp[V_i⁰(P-p_is)/RT] (1-21)

Activity coefficient equations are used in Equation (1-20), assuming that pressure effects cancel out in Equation (1-21). Parameters A, B and Λ_ij are optimized using fittings to constant temperature Px data (ref. 2).

Results and Discussion for Chapter 1

The xy calculations of non-azeotropic mixtures.

In Figure 1-1, y₁ is plotted versus x₁ for the methanol (1) + water (2) binary at 323.15 [K]. The Margules equation is used for the calculation of non-azeotropic mixtures, in which Px data are correlated, in advance, for determining optimum A and B parameters. In Figure 1-1, the determined parameters are used for calculating xy relationships. Therefore, data scatterings have been removed in Figure 1-1. Figure 1-1 shows that experimental data are in good agreement with pure prediction data described in Chapter 3. The mixture properties of the predicting system are not used in the pure prediction. In Figure 1-2, y₁ is plotted versus x₁ for the water (1) + n, n-dimethylformamide (2) binary at 373.15 [K]. The Margules equation is used for the calculations of non-azeotropic mixtures. Other activity coefficient equations, including the Wilson, NRTL and UNIQUAC equations, satisfactorily correlate non-azeotropic mixture data. However, a two-liquid phase forming behavior appears, if the van Laar equation is used for this binary system (ref. 2, part 1 page XXXVI).

The xy calculations of azeotropic mixtures.

In Figure 1-3, y₁ is plotted versus x₁ for the ethanol (1) + water (2) binary at 323.15 [K]. The van Laar equation is used for the calculations of minimum azeotropic mixtures. In Figure 1-4, y₁ is plotted versus x₁ for the acetone (1) + chloroform (2) binary at 323.15 [K]. The van Laar equation is used for the calculations of maximum azeotropic mixtures. Other activity coefficient equations, including the Wilson, NRTL and UNIQUAC equations, can calculate the VLE of azeotropic mixtures.

The calculations of two-liquid phase forming mixtures.

In Figure 1-5, y₁ is plotted versus x₁ for the 2-butanone (1) + water (2) binary at 323.15 [K]. As shown in Figure 1-5, correlations using the van Laar equation are satisfactory at x₁ < x_1low = 0.04. However, correlations are insufficient at x_1high ( = 0.726) < x₁. The problem occurs, if other activity coefficient equations, including the NRTL and UNIQUAC equations, are used. The problem is not solved yet. In Figure 1-6, y₁ is plotted versus x₁ for the water (1) + furfural (2) system at 293.15 [K]. In Figure 1-7, y₁ is plotted versus x₁ for the cyclohexane (1) + aniline (2) system at 298.15 [K]. Figures 1-6 and 1-7 demonstrate that the van Laar equation satisfactorily calculate the xy data of asymmetric binary systems.

Distinction criteria applying to the Wohl equation.

In Figure 1-8, y₁ is plotted versus x₁ for the ethanol (1) + 2, 2, 4-trimethylpentane (2) binary at 273.15 [K]. The van Laar equation is used for the correlation of experimental Px data. However, the correlation provides a two-liquid phase forming behavior. The best fitting obtained from the Wohl equation to the xy data calculated using the van Laar equation provides Δ_ave = 0.058, satisfying the criteria, Equation (1-14). Therefore, the Wohl equation is fitted to the xy data obtained using the van Laar equation. As shown in Figure 1-8, the agreements between experimental and calculated using the Wilson equation are satisfactory. Other activity coefficient equations, including the NRTL and UNIQUAC equations, provide similar two-liquid phase forming behaviors for alcohol-hydrocarbon binary systems. Equation (1-14) should be used to alcohol-hydrocarbon mixtures, although the Equation satisfies other mixtures, such as amin + water and acetate + water binaries.

Figure 1-1. Non-azeotropic y₁ vs. x₁ for the methanol (1) + water (2) system at 323.15 [K] calculated using the Margules equation, x-axis: x₁, y-axis: y₁, (●) Dul Itskaya et al. cited from ref. 2, part 1 page 45, (●) Mc Glashan et al. cited from ref. 2, part 1 page 56, (×) Kurihara et al. cited from ref. 2, part 1c page 77, (yellow solid line) pure prediction shown in Chapter 3.

Figure 1-2. Non-azeotropic y₁ vs. x₁ for the water (1) + n, n-dimethylformamide (2) system at 373.15 [K] calculated using the Margules equation, x-axis: x₁, y-axis: y₁, (●) Doering cited from ref. 2, part 1c page 389, (yellow solid line) pure prediction shown in Chapter 3.

Figure 1-3. Minimum azeotropic y₁ vs. x₁ for the ethanol (1) + water (2) system at 323.15 [K] calculated using the van Laar equation, x-axis: x₁, y-axis: y₁, (●) Dul Itskaya et al. cited from ref. 2, part 1 page 161, (●) Udovenko and Fatkulina cited from ref. 2, part 1 page 191, (×) Chaudhry et al. cited from ref. 2, part 1a pages 117 and 118, Pomberton and Mash cited from ref. 2, part 1a page 143, Wilson et al. from ref. 2, part 1a page 155 and Kurihara et al. from ref. 2, part 1c page 198, (yellow solid line) pure prediction shown in Chapter 3.

Figure 1-4. Maximum azeotropic y₁ vs. x₁ for the acetone (1) + chloroform (2) system at 323.15 [K] calculated using the van Laar equation, x-axis: x₁, y-axis: y₁, (●) Abbott et al. cited from ref. 2, part 3b page 9, (●) Abbott and van Ness from ref. 2, part 3b page 10, (×) Abbott and van Ness from ref. 2, part 3b page 1, (×) Goral et al. cited from ref. 2, part 3b pages 17 and 18, Abbott and van Ness from ref. 2, part 3c page 108, Gorbunov from ref. 2, part 3c page 110, Mueller and Kearns from ref. 2, part 3+4 page 101, Ravincvich and Nikolaev from ref. 2, part 3+4 page 104, Roeck and Schrceder from ref. 2, part 3+4 page 113 and Schmidt from ref. 2, part 3+4 page 120, (yellow solid line) pure prediction shown in Chapter 3.

Figure 1-5. two-liquid phase forming y₁ vs. x₁ for the 2-butanone (1) + water (2) system at 323.15 [K] calculated using the van Laar equation, x-axis: x₁, y-axis: y₁, (●) experimental data by Sokolova and Morachevsky cited from ref. 2, part 1b page 208, (●) correlation of the experimental data using the van Laar equation with A = 3.44 and B = 1.82, (yellow solid line) pure prediction providing mutual solubilities x_1low = 0.04001 and x_1high = 0.7267.

Figure 1-6. Asymmetric two-liquid phase forming y₁ vs. x₁ for the water (1) + furfural (2) system at 293.15 [K], x-axis: x₁, y-axis: y₁, (●) Briggs and Comings cited from ref. 3, part 1 page 272, (yellow solid line) pure prediction

Figure 1-7. Asymmetric two-liquid phase forming y₁ vs. x₁ for the cyclohexane (1) + aniline (2) system at 298.15 [K], x-axis: x₁, y-axis: y₁, (●) Buchner and Kleyn cited from ref. 3, part 1 page 365, (yellow solid line) pure prediction.

Figure 1-8. y₁ vs. x₁ for the ethanol (1) + 2,2,4-trimethylpentane at 273.15 [K], (▲) experimental data by Kretschmer et al. cited from ref. 2, part 2a page 501 (●) correlation fitted to the experimental data using the van Laar equation, (red solid line) xy data calculated using the Wilson equation, fitted to the van Laar equation, because minimized Δ_ave (= 0.058) > 0.01 holds.

Conclusion for Chapter 1

Non-azeotropic VLE data are calculated using the Margules equation, while the van Laar equation should be used for the VLE calculations of azeotropic and two-liquid phase forming mixtures. Use the Wilson equation, if Equation (1-14) is satisfied for the alcohol-hydrocarbon mixtures.

Nomenclature for Chapter 1

A Margules parameter defined in Equation (1-3)

B Margules parameter defined in Equation (1-4)

C A/B defined in Equation (1-9)

P system pressure

P_a reference pressure

P_Ci critical pressure of component i

p_is vapor pressure of component i

q₁ molecular surface area of component 1

q₂ molecular surface area of component 2

R gas constant

T system temperature

V_i^L partial molar volume of component i

V_i⁰ molar volume of pure liquid i at T

x₁ liquid phase mole fraction of component 1

x₂ liquid phase mole fraction of component 2

x_1low, x_1high mutual solubility

y_i vapor phase mole fraction of component i

y_k k-th vapor phase mole fraction

z₁ modified mole fraction of component 1 defined in Equation (1-7)

z₂ modified mole fraction of component 2 defined in Equation (1-8)

Δ_ave average deviation determined using Equation (1-14)

γ_i activity coefficient of component i

γ_i^∞ infinite dilution activity coefficient of component i

φ_is fugacity coefficient of the saturated vapor i at p_is

φ_i^V vapor phase fugacity coefficient of component i in the mixture at P

Λ_ij Wilson parameter defined in Equations (1-10) and (1-11)

Superscript

(P) reference pressure at P

∞ infinite dilution

Chapter 2. How to Use Critical Ratios

Introduction

Using binary constant temperature VLE and LLE data, the present manual shows that infinite dilution activity coefficients are expressed using the power functions of critical ratios.

Models and Calculation Manual for Chapter 2

Using binary constant temperature VLE and LLE data, Kato showed that infinite dilution activity coefficients are expressed using critical ratios as follows (S. Kato, MTMS’24):

lnγ_i^∞ = lna_Xi + b_XilnX (i=1, 2) (2-1)

The parameter, γ_i^∞, denotes the infinite dilution activity coefficient of component i. The critical ratio, X, is defined as follows (PCT/ JP2012/ 074514; Kato et al., J. Chem Eng. Data 2011, 56, 4927-4934):

X = (p_1s + p_2s) / (P_C1 + p_2s) (2-2)

The quantities, p_1s and p_2s, denote the vapor pressures of pure component 1 and 2, respectively, while P_C1 denotes the critical pressure of component 1. It should be noted that the value of X is determined, if temperature, T, and pure substances 1 and 2 are fixed. In VLE analyses, lower boiling point substances are specified as component 1. Therefore, at the critical point of component 1, X = 1 holds. In many cases, low-boiling point VLE data cover X < 0.2. The utmost advantage of using Equation (2-1) is that it enables us to purely predict pre-power constant, a_Xi, and consonant, b_Xi. The details of the pure prediction of a_Xi and b_Xi are shown in Chapter 3.

Data Sources

Constant temperature binary VLE data are cited from ref. 2, in which Px data are correlated using the Margules and van Laar equations. Furthermore, mutual solubility data in ref. 3 are used to determine van Laar parameters.

Results and Discussion for Chapter 2

In Figure 2-1, γ₁^∞ and γ₂^∞ are plotted versus X for the benzene (1) + heptane (2) binary at 20 [K] < T – 273.15 < 155 [K]. The Margules equation is used for the non-azeotropic mixtures. Figure 2-1 shows that γ₁^∞ and γ₂^∞ data are represented using power functions. The power functions are almost passing through the origin, lnγ_i^∞ = 0 at lnX = 0, which characterizes binary hydrocarbon mixtures. In Figure 2-2, γ₁^∞ is plotted versus 1/T for the benzene (1) + heptane (2) binary at 20 [K] < T – 273.15 < 155 [K]. Figure 2-2 demonstrates that data scatterings are the same between Figures 2-1 and 2-2. The utmost disadvantage of γ_i^∞ = ae^b/T- type function is that partial molar excess enthalpy and entropy loses temperature effects. In Figure 2-3. γ₁^∞ and γ₂^∞ are plotted versus X for the benzene (1) + cyclohexane (2) binary at 8 [K] < T – 273.15 < 150 [K]. Figure 2-3 shows that correlation lines are passing through the origin.

In Figures 2-4 and 2-5, γ₁^∞ and γ₂^∞ are plotted versus X for the methanol (1) + water (2) and ethanol (1) + water (2) systems, respectively. Figures 2-4 and 2-5 demonstrate that data scatterings are not small for alcohol + water mixtures. In Figures 2-6 and 2-7, γ₁^∞ and γ₂^∞ are plotted versus X for the acrylonitrile (1) + water (2) and acetaldehyde (1) + water (2) systems, respectively. Figures 2-6 and 2-7 show that straight lines are obtained at γ_i^∞ >> 1. In Figures 2-8, 2-9 and 2-10, γ₁^∞ and γ₂^∞ are plotted versus X for the water (1) + n, n-dimethylformamide (2), water (1) + n, n-dimethylacetamide (2) and acetone (1) + chloroform (2) systems, respectively. Figures 2-8, 2-9 and 2-10 show that straight lines are obtained at γ_i^∞ ≦ 1. Figures 2-11 and 2-12 demonstrate that two-liquid phase forming mixtures and minimum azeotropic mixtures provide straight lines, if lnγ_i^∞ is plotted versus lnX.

Figure 2-1. γ_i^∞ vs. X for the benzene (1) + heptane (2) binary at 20 [K] < T – 273.15 < 155 [K] calculated using the Margules equation, x-axis: X, y-axis: γ₁^∞ and γ₂^∞, (●) γ₁^∞, (●) γ₂^∞, data are cited from ref. 2, part 6b, pages 120 – 157, 6c 464 – 472 and 6e 574 – 587.

Figure 2-2. γ₁^∞ vs. 1/T for the benzene (1) + heptane (2) binary at 20 [K] < T – 273.15 < 155 [K] calculated using the Margules equation, x-axis: X, y-axis: γ₁^∞, (●) γ₁^∞, data are cited from ref. 2, part 6b, pages 120 – 157, 6c 464 – 472 and 6e 574 – 587.

Figure 2-3. γ_i^∞ vs. X for the benzene (1) + cyclohexane (2) binary at 8 [K] < T – 273.15 < 150 [K] calculated using the van Laar equation, x-axis: X, y-axis: γ₁^∞ and γ₂^∞, (●) γ₁^∞, (●) γ₂^∞, data are cited from ref. 2, part 6b, pages 204 – 239, 6c 215 – 231 and 6d 250 – 272.

Figure 2-4. γ_i^∞ vs. X for the methanol (1) + water (2) binary at -10 [K] < T – 273.15 < 115 [K] calculated using the Margules equation, x-axis: X, y-axis: γ₁^∞ and γ₂^∞, (●) γ₁^∞, (●) γ₂^∞, data are cited from ref. 2, part 1, pages 38 – 73, 1a 49 – 57, 1b 29 – 33 and 1c 57 – 99.

Figure 2-5. γ_i^∞ vs. X for the ethanol (1) + water (2) binary at 10 [K] < T – 273.15 < 130 [K] calculated using the van Laar equation, x-axis: X, y-axis: γ₁^∞ and γ₂^∞, (●) γ₁^∞, (●) γ₂^∞, data are cited from ref. 2, part 1 pages 157 – 196, 1a 117 – 155, 1b 83 – 108 and 1c 181 – 253.

Figure 2-6. Two-liquid phase forming γ_i^∞ vs. X for the acrylonitrile (1) + water (2) binary at 0 [K] < T – 273.15 < 100 [K] calculated using the van Laar equation, x-axis: X, y-axis: γ₁^∞ and γ₂^∞, (●) γ₁^∞, (●) γ₂^∞, data are cited from ref. 2, part 1 pages 38 – 73, 1a 49 – 57, 1b 29 – 33 and 1c 57 – 99.

Figure 1-7. γ_i^∞ vs. X for the acetaldehyde (1) + water (2) binary at 0 [K] < T – 273.15 < 100 [K] calculated using the Margules equation, x-axis: X, y-axis: γ₁^∞ and γ₂^∞, (●) γ₁^∞, (●) γ₂^∞, data are cited from ref. 2, part 1 pages 83 – 94, 1a 78 – 81, 1b 38 – 40 and 1c 134.

Figure 2-8. γ_i^∞ vs. X for the water (1) + n, n-dimethylformamide (2) binary at 30 [K] < T – 273.15 < 100 [K] calculated using the Margules equation, x-axis: X, y-axis: γ₁^∞ and γ₂^∞, (●) γ₁^∞, (●) γ₂^∞, data are cited from ref. 2, part 1c pages 389 – 401.

Figure 2-9. γ_i^∞ vs. X for the water (1) + n, n-dimethylacetamide (2) binary at 20 [K] < T – 273.15 < 80 [K] calculated using the Margules equation, x-axis: X, y-axis: γ₁^∞ and γ₂^∞, (●) γ₁^∞, (●) γ₂^∞, data are cited from ref. 2, part 1a pages 319 – 322.

Figure 2-10. Maximum azeotropic γ_i^∞ vs. X for the acetone (1) + chloroform (2) binary at 0 [K] < T – 273.15 < 55 [K] calculated using the van Laar equation, x-axis: X, y-axis: γ₁^∞ and γ₂^∞, (●) γ₁^∞, (●) γ₂^∞, data are cited from ref. 2, part 3b pages 9 – 24, 3c 108 – 113 and 3+4 87 – 125.

Figure 2-11. Two-liquid phase forming γ_i^∞ vs. X for the 2-butanone (１) + water (2) binary at 0 [K] < T – 273.15 < 120 [K] calculated using the van Laar equation, x-axis: X, y-axis: γ₁^∞ and γ₂^∞, (●) γ₁^∞, (●) γ₂^∞, data are cited from ref. 2, part 1 pages 359, 1a 271, 1b 206 – 210 and ref. 3, part 1 page 217.

Figure 2-12. Minimum azeotropic γ_i^∞ vs. X for the ethanol (１) + 2,2,4 -trimethylpentane (2) binary at 0 [K] < T – 273.15 < 76 [K] calculated using the van Laar equation, x-axis: X, y-axis: γ₁^∞ and γ₂^∞, (●) γ₁^∞, (●) γ₂^∞, data are cited from ref. 2, part 2a pages 501 – 504, 2c 467 and 2h 501.

Conclusion for Chapter 2

The infinite dilution activity coefficients are expressed with power functions of the critical ratios.

Chapter 3 How to Predict Infinite Dilution Activity Coefficients

Introduction

The present manual provides a pure prediction method for infinite dilution activity coefficients of binary systems, in which (i) vapor pressures are calculated using the Clapeyron equation, (ii) experimental infinite dilution activity coefficient data are correlated using the power functions of critical ratios, (iii) partial molar excess entropy and enthalpy are calculated from the power functions, (iv) the exponents of the power function are calculated using the linear approximation of partial molar excess enthalpy, (v) the universal line for exponents is established, (vi) the universal line for pre-power constants is presented and finally (vii) a criterion judging the data quality of infinite dilution activity coefficients is presented.

Models and Calculation Manual for Chapter 3

Relationships between temperature and the infinite dilution activity coefficients

The infinite dilution activity coefficient of component i in a binary system is related with the critical ratio, X, as follows (S. Kato, MTMS’24):

lnγ_i^∞ = ln a_Xi + b_Xi ln X (i= 1, 2) 　 (3-1)

The critical ratio, X, is defined as follows:

X = (p_1s + p_2s) / (P_C1+p_2s) 　 (3-2)

The constants a_Xi and b_Xi are system dependent. It was shown that Equation (3-1) holds at temperatures covering T < T_Ci using 2700 binary VLE and LLE systems (S. Kato, MTM’S24).

Partial molar excess quantity definition

The partial molar excess enthalpy of component i is defined as follows:

h_i^E∞ = R[∂lnγ_i^∞/∂(1/T)] (3-3)

Using Equations (3-1) to (3-3), b_Xi is related with h_i^E∞as follows:

b_Xi ln X = θh_i^E∞ / RT 　　　　(3-4)

The temperature parameter, θ, is defined as follows:

θ = T ln X / [∂ln X /∂(1/T)]　 　 (3-5)

It should be noted that parameters, θ and lnX, are temperature dependent. Furthermore, they are pure substance paramters. Meanwhile, partial molar excess entropy is defined as follows:

lnγ_i^∞ = h_i^E∞ / RT – s_i^E∞/R (3-6)

Using Equations (3-1) to (3-6), the ratio of entropy to enthalpy is expressed as follows:

(s_i^E∞/R)/( h_i^E∞/RT) = -θ ln a_Xi / (b_Xi ln X) + ( 1 – θ)　 (3-7)

Equations (3-4) and (3-7) are starting equations for the γ_i^∞-pure prediction. They include two parameters, a_Xi and b_Xi, to be purely predicted.

The linear approximation of exponents b_X1 and b_X2

The present investigation uses the following linear approximation:

b_Xi = c₁ h_i^E∞/RT + c₂θ+ c₃ ln X + c₄ (3-8)

The parameters c₁ to c₄ are constants. Each term in the right side of Equation (3-8) is the consisting quantity of b_Xi shown in Equation (3-4). Eliminating h_i^E∞/RT from Equations. (3-4) and (3-8), b_Xi is given as follows:

b_Xi,pre=(c₂θ+c₃ ln X+c₄)/(1-c₁ ln X/θ) 　(i = 1, 2) 　　　　(3-9)

In the pure prediction process, first, the T, θ and lnX values of a γ₁^∞-predicting binary system are determined. Using data fitting, it is possible to regress the c₁ to c₄ constants to the experimental b_X1, h₁^E∞/RT, θ and lnX values of the γ₁^∞-predicting binary system. Constants c₁ to c₄ for the second component are similarly determined using experimental b_X2, h₂^E∞/RT, θ and lnX values. Moreover, the c- parameters in Equation (3-8) must satisfy the following requirement;

∂b_Xi/∂T = 0 (3-10)

Meanwhile, the parameter c₁ is a linear function of 1/(h_i^E∞/RT). Furthermore, constants, c₂ to c₄, linearly increase with increasing h_i^E∞/RT as follows:

c₁ = k₁₁/(h_i^E∞/RT)+k₂₁ 　　 (3-11)

c₂ = k₂₁h_i^E∞/RT+k₂₂ (3-12)

c₃ = k₃₁h_i^E∞/RT+k₃₂ (3-13)

c₄ = k₄₁h_i^E∞/RT+k₄₂ (3-14)

Parameters k_ij denote constants. In addition, k-values are determined using the following quadratic function, if binaries are chosen form the 2700 systems, in which their θ values are close to that value of γ_i^∞-predicting system.

k_ij = l (lnX)² + m lnX + n (i = 1, 2, 3, 4, j=1, 2) (3-15)

The parameters l, m and n are constants. For the use of pure prediction, the k_ij values must be tabulated, in advance, in a k-value parameter table for b_X1. The k-value parameter table, supplied from the TC Lines JP, includes 2700 binary systems, in which each system has k_ij values determined using parameter fittings to b_X1 experimental values. Empirically, experimental b_Xi values are related with prediction values, b_Xi,pre, as follows:

b_xi,pre = αb_Xi + β (i= 1, 2) (3-16)

Parameters, α and β, are universal constants. Their values are empirically determined using experimental b_Xi values and predicted b_Xi,pre values. It should be stressed that pure prediction is possible, because the constants are universal. Two components must have the same universal constants.

The pure prediction of b_X1 and b_X2

The pure prediction of b_X1 consists of following steps: (i) fix the system temperature, T, and a binary system forming a γ_i^∞-predicting system, (ii) calculate θ using Equation (3-5) and lnX using Equation (3-2). (iii) determine constants l, m and n in Equation (3-15) using data regression to k_ij obtained from the k-value parameter table for b_X1. Binaries are arbitrarily chosen from the table including 2700 systems, if they have θ values close to that of the γ_i^∞-predicting system, (iv) assume a b_X1 value, (v) determine h₁^E∞/RT value using Equation (3-4), (vi) determine c₁ to c₄ values using Equations (3-11) to (3-14), (vii) calculate b_X1,pre value using Equation (3-9), (viii) calculate b_X1 using Equation (3-16), finally, (ix) repeat the trial-and-error process between (iv) and (viii) steps until the same value of b_X1 is obtained. The same procedure is applied to determine b_X2 value using a k-value parameter table for b_X2, separately determined. The calculation process is pure prediction, because the experimental values of b_X1 and b_X2 are not used in the calculation procedures.

The linear approximation of pre-power constants a_X1 and a_X2

Following the linear approximation used for the b_Xi exponent, the present investigation assumes the pre-power constant, a_Xi, as follows:

lna_Xi /b_Xi= d₁ [(s_i^E∞/R)/( h_i^E∞/RT)]+ d₂θ+ d₃lnX + d₄ (i=1, 2) (3-17)

The parameters d₁, d₂ d₃ and d₄ denote constants. Each term in the right side of Equation (17) is the consisting quantity of a_Xi shown in Equation (3-7). Eliminating T s_i^E∞/ h_i^E∞ in Equations (3-7) and (3-17), a_Xi is given as follows:

lna_Xi,pre /b_Xi = [d₁(1-θ) + d₂θ+ d₃lnX + d₄]/(1+d₁θ/lnX) (i = 1, 2) (3-18)

Similarly to b_Xi cases, parameters d₁ to d₄ are linear function of T s_i^E∞/h_i^E∞ as follows:

d₁ = k₁₁/( T s_i^E∞/h_i^E∞)+k₂₁ 　　　　　 (3-19)

d₂ = k₂₁ T s_i^E∞/h_i^E∞ +k₂₂ (3-20)

d₃ = k₃₁ T s_i^E∞/h_i^E∞+k₃₂ (3-21)

d₄ = k₄₁ T s_i^E∞/h_i^E∞+k₄₂ (3-22)

The same relationships, Equations (3-15) and (3-16), are applied to this case.

The pure prediction of a_X1 and a_X2

The same procedure determining b_X1 and b_X2 are applied to the cases of lna_X1/b_X1 and lna_X2/b_X2. It should be noted that b_X1 and b_X2 are predetermined without using a_X1 and a_X2. The Marquardt method is used for the optimization of parameters c₁ to c₄ and d₁ to d₄. Calculations are performed using programing codes written in Perl, an interpreter language.

Data Sources

Vapor pressures

Vapor pressures are calculated using the Clapeyron equation as follows:

p_is = P_Ciexp(C_Clap1T_Clap³ + C_Clap2T_Clap2 + C_Clap3T_Clap) (3-23)

T_Clap = 1 – 1/T_ri (3-24)

Clapeyron temperature, T_Clap, is advantageous, because it allows to define hypothetical liquids (S. Kato, AIChE J. 51 (2005) 3275-3285). Vapor pressure data are cited from ref. 5 which uses the Wagner equation covering temperatures T < T_Ci. Furthermore, the Antoine equations are cited from ref. 2 and ref. 3.

Constant temperature binary VLE and LLE data

Constant temperature VLE and LLE data are cited from ref. 2. Furthermore, constant temperature LLE data are cited from ref. 3. They include a total number of 2700 binary systems. Each binary system includes three or more datasets at different temperatures. Infinite dilution activity coefficients are calculated as follows: the Margules equation is used for non-azeotropic mixtures, because Margules parameters are reported in ref. 2. The van Laar equation is used for azeotropic and two-liquid phase forming mixtures with the parameters reported in ref. 2. For LLE data, van Laar parameters are determined using LLE data reported in ref. 3. Meanwhile, the Wilson equation is used for alcohol-hydrocarbon mixtures along with the distinction criteria identified in Chapter 1, in which the Wohl equation is used.

Results and Discussion for Chapter 3

Vapor pressures calculated using the Clapeyron equation

In Figure 3-1, P_ri is plotted vs. T_Clap for ethanol at temperatures 20 [K] < T – 273.15 < 120 [K]. Figure 3-1 shows that the Clapeyron equation satisfactorily correlates vapor pressures calculated using the Wagner equation. Furthermore, Figure 3-1 demonstrates that reduced pressure almost linearly increases with increasing Clapeyron temperature, passing through the origin.

Infinite dilution activity coefficients correlated using the power function of critical ratio

In Figure 3-2a, γ₁^∞ and γ₂^∞ are plotted vs. X for the benzene (1) + heptane (2) binary. In Figure 3-2a, bold solid lines represent power function correlations. Figure 3-2a demonstrates that power function correlations are satisfactory. In Figure 3-2b, γ₁^∞ and γ₂^∞ are plotted vs. 1/T for the benzene (1) + heptane (2) binary. Figures 3-2a and 3-2b show that correlation errors are the same. Furthermore, Figure 3-2a includes dotted lines, representing power functions satisfying ideal solution behavior at the critical temperature of benzene. Figure 3-2a shows that, for hydrocarbon mixtures, power function correlations satisfying ideal solution approximation at the critical temperature of lower-boiling point substance are acceptable.

The values of partial molar excess enthalpy and entropy at infinite dilution

It is partial molar excess quantities that provide pure prediction methods for infinite dilution activity coefficients. In Figure 3-3a, h₁^E∞/R and h₂^E∞/R are plotted vs. X for the ethanol (1) + water (2) binary at temperatures 10 [K] < T - 273.15 < 200 [K]. Figure 3-3a shows that partial molar excess enthalpies calculated using Equations (1), (2) and (3) vary with increasing temperature. The constants, a_Xi and b_Xi, are correlated using γ_i^∞ values calculated from the van Laar equation, because the ethanol + water binary is classified into an azeotropic mixture. In Figure 3-3b, s₁^E∞/R and s₂^E∞/R are plotted vs. X for the ethanol (1) + water (2) binary at temperatures 10 [K] < T - 273.15 < 200 [K]. Figure 3-3b shows that partial molar excess entropies calculated using Equations (3-1), (3-2), (3-3) and (3-6) vary with increasing temperature. It should be noted that enthalpies and entropies are constant, if γ_i^∞= a exp(b/T)-type temperature effects are used.

Entropy-enthalpy compensation rule

The pure prediction of infinite dilution activity coefficients uses partial molar excess entropy-enthalpy compensation rule. In Figure 3-4, s₁^E∞/R and s₂^E∞/R are plotted vs. h₁^E∞/RT and h₂^E∞/RT at temperatures 40 [K] < T - 273.15 < 120 [K]. Figure 3-4 shows that partial molar excess entropy-enthalpy compensation rule almost holds for the hexane (1) + heptane (2) binary, because correlation lines are passing through the origin. However, the methanol (1) + water (2) binary is remote from the compensation rule, because correlation lines deviate from the origin. Of the 2700 binaries compiled in ref. 2 and ref. 3, a very limited number of binaries satisfies the partial molar excess entropy-enthalpy compensation rule. Figure 3-4 demonstrates that partial molar excess entropy-enthalpy compensation rule cannot be a universal relationship.

The preparation of k-value parameter table for b_X1, b_X2, a_X1 and a_X2

To demonstrate the preparation procedure of the k-value parameter table for b_X1, the ethanol (1) + water (2) binary is selected as a γ_i^∞-predicting binary system. First, system temperature is fixed at 274.15 [K]. The values of θ from Equation (3-5) and lnX from Equation (3-2) are 0.3798 and -7.943, respectively. Next, using data regression, c₁ to c₄ constants are determined from Equation (3-8), in which b_X1 = 0.0726, h₁^E∞/RT = -1.519, θ = 0.3798 and lnX = -7.934 of the γ_i^∞-predicting binary system are used. Determined constants are listed in Table 3-1. In the calculation, the experimental value of b_X1 from Equation (3-1) is determined from lnγ₁^∞ vs. lnX plot using experimental data cited from ref. 2. The van Laar equation is used in this case, because the ethanol + water system is classified into an azeotropic mixture at 274.15 K. The value of h₁^E∞/RT is calculated using Equation (3-3) or Equation (3-4). Next, a δ-closest binary is selected using the following filter, δ:

|(θ_k – θ_pre)/ θ_pre| < δ (3-25)

|(lnX_k - lnX_pre)/lnX_pre| < δ (3-26)

The suffix k denotes k-th binary selected from the 2700 a_Xi, b_Xi-registered binary systems. The suffix pre denotes γ_i^∞-predicting binary, the ethanol (1) + water (2) binary in this case. In the case of ethanol (1) + water (2) binary, the δ-closest binary was the ethanol (1) + 1-propanol (2) system at the smallest δ (= 0.013). That means two binaries, γ_i^∞-predicting binary and δ-closest binary, are selected satisfying δ ≦ 0.013. In the next step, c₁ to c₄ constants of the δ-closest binary are determined using b_X1, h₁^E∞ values of the δ-closest binary system. These values are included in Table 3-1 with the determined c₁ to c₄ values. Finally, using Equations (3-11) to (3-14), k_ij values are determined by solving simultaneous two linear equations including c_i (i = 1, 4) values and h_i^E∞ values of the γ_i^∞-predicting binary and δ-closest binary. Determined k_ij values of the ethanol (1) + water (2) binary are listed in Table 3-1. The procedures are repeated at temperatures covering 1 [K] ≦ T – 273.15 ≦ T_max, in which T_max denotes the maximum temperature providing the existence of two binaries, γ_i^∞-predicting binary and δ-closest binary. The value of T_max was 45 + 273.15 [K] for the ethanol (1) + water (2) binary. In this temperature range, b_X1,pre is constant. Preparation procedures for b_X2, a_X1 and a_X2 are the same with that of b_X1.

The determination of universal lines

In Figure 3-5, k₁₁ is plotted versus lnX at 274.15 [K] using 75 binary systems satisfying |(θ_k – θ_pre)/ θ_pre| < 0.05 at θ_pre= 0.3798 for the ethanol (1) + water (2) γ_i^∞-predicting binary. Figure 3-5 shows that data are very scattering. The other k_ij values show similar scattering. However, the following modifications on k₂₁ and k₂₂ greatly improve the convergence of the universal lines, that is, Equation (3-16):

k₂₁ = (θ/lnX - k₁₂ – k₃₁lnX - k₄₁)/θ (3-27)

k₂₂ = -(k₁₁ + k₃₂lnX + k₄₂)/θ (3-28)

Equation (3-27) is obtained from the equality in the coefficients of h_i^E∞, θ/lnX = k₁₂ + k₂₁θ + k₃₁lnX + k₄₁. Furthermore, Equation (3-28) arises from the intercepts, k₁₁ + k₂₂θ + k₃₂lnX + k₄₂ = 0. In Table 3-2, to establish the universal lines, 25 binaries are selected, because they have a large number of VLE and LLE datasets. In Figure 3-6, using binaries shown in Table 3-2, b_X1pre is plotted versus experimental b_X1. The following converged universal line is established:

b_Xipre = b_Xi (i=1, 2) (3-29)

The same converged line is obtained between b_X2pre and b_X2 as shown in Figure 3-6. In Figures 3-7, experimental b_X1 and b_X2 are directly compared with predicted values using the universal line, Equation (3-29). The pure prediction is satisfactory, because prediction errors are less than 1 % at |b_Xi| > 0.2.

A similar calculation process is applied to lna_Xi/b_Xi. In Figure 3-8, the following universal line for lna_Xi/b_Xi is shown:

lna_Xipre/b_Xi = (lna_Xi/b_Xi) (i=1, 2) (3-30)

The values of k_ij are scattering again in this case. However, the following modifications on k₂₁ and k₂₂ greatly improve the universal line:

k₂₁ = - (lnX/θ+ k₁₂ + k₃₁lnX + k₄₁)/θ (3-31)

k₂₂ = (1-θ)lnX/θ- k₁₁ – k₃₂lnX – k₄₂)/θ (3-32)

Equation (3-31) is obtained from the equality in the coefficients of Ts_i^E∞/h_i^E∞, -lnX/θ= k₁₂ + k₂₁θ + k₃₁lnX + k₄₁. Furthermore, Equation (3-32) arises from the intercepts, k₁₁ + k₂₂θ + k₃₂lnX + k₄₂ = (1 -θ)lnX/θ. In Figure 3-9, experimental a_X1 and a_X2 are directly compared with the values predicted using the universal line, Equation (3-30), and the modifications, Equations (3-31) and (3-32). The pure prediction is satisfactory, because prediction errors are less than 1 %.

The establishment of VLE and LLE data quality test

Using the pure prediction procedures of a_Xi and b_Xi (i= 1, 2), γ_i^∞ is calculated from Equation (3-1). In Figure 3-10, experimental γ_i^∞ is directly compared with predicted for the 25 binaries shown in Table 3-2 at 274.15 [K]. The agreement between the two is encouraging. Pure predictions often provide data quality judgement criteria. Therefore, the γ_i^∞-pure prediction procedure is applied to 100 binaries selected from the 2700 binary systems including 25 binaries in Table 3-2. A criterion, ±30 %, was obtained, because experimental　γ_i^∞ values are identical with predicted γ_i^∞ values within the criterion which satisfies the statistical ±95 % data reliability limit. Finally, the present investigation proposes the following criteria for the VLE and LLE data quality judgement:

|(γ_i^∞_exp - γ_i^∞_pre)/ γ_i^∞_exp| < 0.3 (i=1,2) (3-33)

Suffix exp denotes experimental data. Suffix pre denotes predicted data. Applying Equation (3-33) to the 25 binary systems in Table 3-2, the tetrachloromethane + benzene binary provided the nonreliable values of both γ₁^∞ and γ₂^∞.

Figure 3-1. The vapor pressures of ethanol at 10 [K] < T – 273.15< 120 [K], x-axis: T_Clap (=1-1/T_ri), y-axis: P_ri (=p_is/P_Ci), (●) calculated from the Wagner equation using parameters cited from ref. 4, (−) correlated using the Clapeyron equation.

Figure 3-2a. γ_i^∞ vs. X for the benzene (1) + heptane (2) binary, x-axis: X, y-axis: γ₁^∞ and γ₂^∞, (●) benzene, (■) heptane, (−) power function correlations, (. . .) power function correlations with ideal solutions at X = 1, γ₁^∞ and γ₂^∞are calculated using the Margules equation fitted to the constant temperature VLE data cited from ref. 2, part 6b 120-157, part 6c 464-472 and part 6e 574-587.

Figure 3-2b. γ_i^∞ vs. 1/T for the benzene (1) + heptane (2) binary, x-axis: 1/T, y-axis: γ₁^∞ and γ₂^∞, (●) benzene, (▲) heptane, (−) exponential function correlations, γ₁^∞ and γ₂^∞ are calculated using the Margules equation fitted to the constant temperature VLE data cited from ref. 2, part 6b 120-157, part 6c 464-472 and part 6e 574-587.

Figure 3-3a. h_i^E∞/R vs. X for the ethanol (1) + water (2) binary at 10 [K] < T - 273.15 < 200 [K], x-axis: X, y-axis: h₁^E∞/R and h₂^E∞/R [K]. (●) ethanol, (▲) water, Equations (3-1), (3-2) and (3-3) are used with a_X1 = 7.21, b_X1 = 0.0726, a_X2 = 2.25, b_X2 = -0.0248, γ₁^∞ and γ₂^∞ data are calculated using the van Laar equation fitted to the constant temperature VLE data cited from ref. 2, part 1 150-196, part 1a 116-155, part 1b 83-111 and part 1c 176-253.

Figure 3-3b. s_i^E∞/R vs. X for the ethanol (1) + water (2) binary at 10 [K] < T - 273.15 < 200 [K], x-axis: X, y-axis: s₁^E∞/R and s₂^E∞/R. (●) ethanol, (▲) water, Equations (3-1), (3-2), (3-3) and (3-6) are used with a_X1 = 7.21, b_X1 = 0.0726, a_X2 = 2.25, b_X2 = -0.0248, γ₁^∞ and γ₂^∞ data are calculated using the van Laar equation fitted to the constant temperature VLE data cited from ref. 2, part 1 150-196, part 1a 116-155, part 1b 83-111 and part 1c 176-253.

Figure 3-4. s_i^E∞/R vs. h_i^E∞/RT at 40 [K] < T - 273.15 < 120 [K], x-axis: h₁^E∞/RT and h₂^E∞/RT, y-axis: s₁^E∞/R and s₂^E∞/R, (♦) s₁^E∞/R vs. h₁E^∞/RT for the hexane (1) + heptane (2) binary, (■) s₂^E∞/R vs. h₂^E∞/RT for the hexane (1) + heptane (2) binary, (▲) s₁^E∞/R vs h₁^E∞/RT for the methanol (1) + water (2) binary and (×) s₂^E∞/R vs h₂^E∞/RT for the methanol (1) + water (2) binary, γ₁^∞ and γ₂^∞ data are calculated using the Margules equation cited from ref. 2.

Table 3-1. The values of k_ij for the ethanol (1) + water (2) system at 274.15 [K].

	γi^∞-predicting binary system	δ-closest binary system
	ethanol (1) + water (2)	ethanol (1) + 1-propanol (2)
θ	0.3798	0.3751
lnX	-7.943	-8.024
bX1	0.0726	0.1143
h1^E∞/RT	-1.519	-2.445
c1	0.70488	0.70488
c2	2.92333	2.9233
c3	0.01008	0.01008
c4	0.82153	0.82152
k11	0.0744
k21	-0.7225
k31	0.0323
k41	-0.2646
k12	0.7353
k22	1.1570
k32	0.0891
k42	0.1744

Figure 3-5. lnX vs. k₁₁ for the binaries satisfying |(θ_k – θ_pre)/ θ_pre| < 0.05 with θ_pre = 0.3798 for the ethanol (1) + water (2) binary at T = 274.15 [K], x-axis: lnX, y-axis: k₁₁, (◆) k₁₁, (−) correlation using Equation (3-15).

Table 3-2. 25 binary systems used for the establishment of the universal lines.

Nt	NP	NT	first comp. (1) + second comp. (2)
148	70	78	ethanol + water
107	34	73	water + acetic acid
93	43	50	methanol + water
74	52	22	benzene + cyclohexane
74	40	34	benzene + ethanol
68	34	34	acetone + water
66	37	29	methanol + benzene
64	48	16	tetrachloromethane + benzene
62	31	31	benzene + heptane
59	40	19	acetone + chloroform
58	30	28	acetone + methanol
55	17	38	2-propanol + water
48	23	25	1-propanol + water
45	36	9	1,4-dioxane + water
44	10	34	formic acid + water
42	28	14	acetone + benzene
41	17	24	heptane + toluene
41	26	15	benzene + 1-propanol
40	38	2	methanol + cyclohexane
39	18	21	water + 1-butanol
38	19	19	tert-butanol + water
36	25	11	ethanol + heptane
35	17	18	ethyl acetate + ethanol
35	19	16	methyl tert-butyl ether + methanol
35	16	19	2-butanone + watr
Nt: total number of VLE and LLE datasets in ref. 2 and 3 NP: number of constant pressure datasets
NT: number of constant temperature datasets

Figure 3-6. The universal line for b_X1 and b_X2 obtained using 25 binaries listed in Table 2, x-axis: b_X1 and b_X2, y-axis: b_X1pre and b_X2pre, (◆) b_X1pre calculated using Equations (3-9) and (3-11) to (3-15), (−) the universal line for b_Xi: b_Xipre = b_Xi, (●) b_X2pre.

Figure 3-7. Comparison between experimental b_Xi and predicted b_Xipre calculated using the universal line, Equation (3-29), for the 25 binaries included in Table 2, x-axis: experimental |b_Xi|, y-axis: predicted |b_Xipre|, (●) |b_X1| vs. |b_X1pre|, (●) |b_X2| vs. |b_X2pre| and (yellow solid line) diagonal line.

Figure 3-8. The universal line for lna_X1/b_X1 and lna_X2/b_X2 obtained using 25 binaries listed in Table 3-2, x-axis: lna_X1/b_X1 and lna_X2/b_X2, y-axis: lna_X1pre/b_X1 and lna_X2pre/b_X2, (◆) lna_X1pre/b_X1 calculated using Equations (3-18) to (3-22), (3-31) and (3-32), (−) the universal line for lna_Xi/b_Xi: lna_Xipre/b_Xi = lna_Xi/b_Xi, (●) lna_X2pre/b_X2.

Figure 3-9. Comparison between experimental a_Xi and predicted a_Xipre calculated using the universal line, Equation (3-30), for the 25 binaries included in Table 3-2, x-axis: experimental a_Xi, y-axis: predicted a_Xipre, (●) a_X1 vs. a_X1pre, (●) a_X2 vs. a_X2pre and (yellow solid line) diagonal line.

Figure 3-10. Establishment of VLE and LLE data quality judgement criteria calculated using the 25 binary systems shown in Table 3-2 at 274.15 [K]. x-axis: experimental γ_i^∞, y-axis γ_i^∞ predicted using pure prediction method for a_Xi and b_Xi, (●) γ₁^∞, (●) γ₂^∞, (yellow solid line) diagonal line, (blue solid line) +30% reliability line, (purple solid line) -30% reliability line.

Conclusion for Chapter

The present manual provides a pure prediction method for γ_i^∞ values. Furthermore, a criterion for data quality test, Equation (3-33), has been established. The pure prediction needs temperature and pure substance data including p_1s and p_2s. It is stressed that the pure prediction method satisfactorily predict the xy relationships of symmetric non-azeotropic mixtures as shown in Figure 1-1, asymmetric non-azeotropic mixtures as shown in Figure 1-2, minimum azeotropic mixtures as shown in Figure 1-3, maximum azeotropic mixtures as shown in Figure 1-4, asymmetric and non-azeotropic tow-liquid phase forming mixtures as shown in Figure 1-7, two-liquid phase forming mixtures as shown in Figures 1-5 and 1-6.

Nomenclaature for Chapter 3

a_Xi pre-power function constant for component i defined in Eq. (3-1)

b_Xi exponent of power function for component i defined in Eq. (3-1)

C_Clap1, C_Clap2, C_Clap3 constants defined in the Clapeyron Eq. (3-12)

c₁, c₂, c₃, c₄ constants defined in Eq. (3-8)

d₁, d₂, d₃, d₄ constants defined in Eq. (3-10)

h_i^E∞ partial molar excess enthalpy of component i at infinite dilution

P_Ci critical pressure of component i

P_C1 critical pressure of component 1

p_is vapor pressure of component i

P_ri reduced pressure of component i

p_1s vapor pressure of component 1

p_2s vapor pressure of component 2

R gas constant

s_i^E∞ partial molar excess entropy of component i at infinite dilution

T system temperature

T_Ci critical temperature of component i

T_Clap Clapeyron temperature defined in Eq. (3-13)

T_ri reduced temperature (= T/T_Ci) of component i

X critical ratio defined in Eq. (3-2)

γ_i^∞ infinite dilution activity coefficient of component i

θ temperature parameter defined in Eq. (3-5)

Chapter 4　How to Define Hypothetical Liquids

Introduction

Liquids and vapor-liquid equilibria (VLE) exist at the temperature T_C1 ≦T. The Equation-of-states approaches have been commonly used for the analyses of high-pressure VLE. However, a very limited number of investigations tried to apply thermodynamics to binary VLE analyses at temperatures, T_C1 ≦T < T_C2, because hypothetical liquids are needed above the critical temperature of lower boiling-point substances. The reduced temperature of the component i, ln(p_is/P_Ci), is proportional to the Clapeyron temperature factor, 1-1/T_ri, at the temperature range, 0.7T_Ci < T < T_Ci. Therefore, it is natural to extend the proportionality to the temperature range T_Ci < T, which defines the stable vapor pressure values of the hypothetical liquids of pure substance i. It is rational to apply the Wohl activity coefficient equation to the high-pressure VLE analyses, because the Wohl equation most rigidly satisfy the Gibbs-Duhem equation (S. Kato, Fluid Phase Equilib. 297 (2010) 192 - 199).

The present manual clarifies that high-pressure binary Px and xy VLE data can be satisfactorily correlated using the three parameter Wohl equation and hypothetical liquids.

Models and Calculation Manual for Chapter 4

The activity coefficient model

Using activity coefficients, the phase equilibria are described as follows (ref. 5, R. C. Reid, J. M. Prausnitz, B. E. Poling, The Properties of Gases and Liquid 1987 Chap. 8 MacGraw-Hill):

Py_i = γ_ix_ip_is (i = 1, 2) (4-1)

γ_i = γ_i^(Pa) (φ_is/φ_i^V) exp[V_i^L(P-P_a)/RT] exp[V_i⁰(P-p_is)/RT] (4-2)

Even at the high-pressure VLE, the present investigation assumes that the pressure effects on fugacity coefficients cancel out. Therefore, using the Wohl equation (K. Wohl, Trans. AIChE 42 (1946) 215 - 249), the activity coefficients are given as follows:

lnγ₁ = θ₂² [A + 2(BC-A)θ₁] (4-3)

lnγ₂ = θ₁² [B + 2(A/C-B)θ₂] (4-4)

θ₁ = Cx₁/(Cx₁ + x₂) (4-5)

θ₂ = 1 – θ₁ (4-6)

The parameters A, B and C are system dependent. Their values are constant at 0 < x₁ < 1. The Wohl equation originally uses molar volume ratios as the values of C.

Model parameter determination

The system pressure is calculated using Equations (4-1), (4-3) and (4-4) as follows:

P = γ₁x₁p_1s + γ₂x₂p_2s (4-7)

The parameter values of A, B and C are determined using the following objective function:

OF_P=(1/n)Σ|(P_k,exp – P)/P_k,exp| (4-8)

The index k varies from 1 to n. Using constant temperature Px data, A, B and C values minimizing the objective function, OF_P, are determined. Meanwhile, the vapor phase mole fraction of component 1 is given as follows:

y₁ = γ₁x₁p_1s/P (4-9)

A different set of A, B and C is determined using the following objective function:

OF_y = (1/n) | (y_1k,exp – y₁)/y_1k,exp | (4-10)

The Marquardt method is used for the determination of parameters A, B and C. Calculations are performed using programs written in Perl, an interpreter language.

Hypothetical liquid definition

Following the Clapeyron plots proposed by Kato (S. Kato, AIChE J. 51 (2005) 3275-3285), the vapor pressure of the hypothetical liquid is calculated as follows:

ln(p_is/P_Ci) = h(1-1/T_ri) (4-11)

The constant, h, is determined from the linear plot, ln(p_is/P_Ci) vs. (1-1/T_ri), passing through the origin. Using the vapor pressure data covering 0.7T_Ci < T < T_Ci, the proportionality line is extended to T > T_ci, which defines vapor pressure values of the hypothetical liquids.

Data Sources

Vapor pressures

Vapor pressure values were calculated using the Wagner equation. Wagner parameters are compiled in the literature⁵

High-pressure VLE data

High-pressure VLE data are cited from Knapp’s datasets⁴ compiled in Dechema Chemistry Data Series.

Results and Discussion for Chapter 4

The vapor pressure of hypothetical liquid

In Figure 4-1, ln(p_is/P_Ci) is plotted vs. 1-1/T_ri for four substances at temperatures covering 0.7T_Ci < T < T_Ci. Figure 4-1 demonstrates that vapor pressure values can be satisfactorily correlated with straight lines passing through the origin. Therefore, the straight lines are extended above critical to determine h values of hypothetical liquids. In Table 4-1, h values determined from Eq. (4-11) are listed. It should be noted that the h values of hydrocarbons are close to six.

The correlation of high-pressure Px and xy data

In Figure 4-2a, experimental Px and Py data are plotted vs. x₁ and y₁, respectively, for the helium-4 + nitrogen binary at 121.74 K. Figure 4-2a includes a correlated line determined by minimizing OF_P values. The minimized value, that is, the average relative deviations defined in Equation (4-12) is 0.28 % for the Px correlation.

(ARD)_P= (1/n)Σ|(P_k,exp – P)/P_k,exp|_min (4-12)

In Figure 4-2b, experimental y₁ data are plotted vs. x₁ for the helium-4 + nitrogen binary at 121.74 K. Figure 4-2b includes a correlated line determined by minimizing OF_y values. The minimized value, that is, the average relative deviations defined in Equation (4-13) is 0.27 % for the xy correlation.

(ARD )_y= (1/n)Σ|(y_1k,exp – y₁)/y_1k,exp|_min (4-13)

Figure 4-2b shows a phase-splitting trend appearing in the liquid phase, because y₁ has the maximum value. The three parameter Wohl equation provides better correlation results than two parameter equations including the Margules, van Laar, Wilson, NRTL and UNIQUAC equations. Figures 4-2a and 4-2b show that the hypothetical liquid correlations using the three parameter Wohl equations are satisfactory.

In Figure 4-3, the values of A are plotted vs. critical ratios, X, defined as follows:

X = (p_1s + p_2s)/(P_C1 + p_2s) (4-14)

As shown in Equations (4-3) and (4-4), A and B are respectively related with infinite dilution activity coefficients as follows:

A = lnγ₁^∞ (4-15)

B = lnγ₂^∞ (4-16)

Using low and near critical temperature VLE and LLE data, Kato (S. Kato, MTMS24) demonstrated that infinite dilution activity coefficients are expressed using the power function of X at X ＜ 1. Similarly to low-boiling point VLE, Figure 4-3 demonstrates that the hypothetical liquids and Px data provide the power function of X. Moreover, Figure 4-3 shows that A values determined from xy data converge to a smooth line. Therefore, hypothetical liquids and the Wohl equation are advantages, because i) the satisfactory correlations of Px and xy data over critical are obtained, further, ii) A-parameter prediction using the line shown in Figure 4-3 is possible. The A-parameter increases with increasing X. However, the increasing trends appearing in helium-4 hypothetical liquids do not appear in the other hypothetical liquids.

In Figure 4-4, A/B is plotted vs. C for the helium-4 + component 2 system including nitrogen and carbon monoxide as a component 2 at 77 K < T < 122 K. Figure 4-4 shows that C = A/B holds for the xy data. As shown in Equations (4-3) and (4-4), the Wohl equation is identical with the van Laar equation, if C = A/B holds. Using low-pressure VLE and LLE data, it is shown in Chapter 1 that the van Laar equation represents azeotropic or two-liquid phase forming behavior. Therefore, Figure 4-4 demonstrates that the xy data reflect azeotropic or two-liquid forming behavior represented by the van Laar equation. If C = 1 holds, the Wohl equation is identical with the Margules equation. However, the parameter C widely varies from -4 to 4 as shown in Figure 4-4.

In Figure 4-5, experimental Px and Py data are plotted vs. x₁ and y₁, respectively, for the hydrogen + methane binary at 144.26 K. Figure 4-5 shows that correlations are satisfactory. In Figure 4-6, A values are plotted vs. X for the hydrogen + component 2 system including nitrogen and methane as a component 2 at 77 K < T < 174 K. Figure 4-6 shows that the increasing trends of A values are very clear. In Figure 4-7, A/B is plotted vs. C for the hydrogen + component 2 including nitrogen and methane at 77 K < T < 174 K. Figure 4-7 demonstrates that the van Laar equation represents the Px data, because C = A/B holds. Meanwhile, the xy data reflect different factors, because A/B values deviate from the line representing C = A/B.

In Figure 4-8, experimental Px and Py data are plotted vs. x₁ and y₁, respectively, for the carbon dioxide + propane binary at 277.59 K. Figure 4-8 shows that Px and Py correlations fairly well represent the beak behavior appearing near x₁ = 1. Weighing objective functions may improve the correlations. In Figure 4-9a, A is plotted vs. X for the carbon dioxide + propane binary at 244. K < T < 345 K. Figure 4-9a demonstrates that, at X = 1, the A values are continuous at the critical point of lower-boiling point substance, carbon dioxide, which means that the two assumptions, Equation (4-2) and hypothetical liquids, are rational and convenient. In Figure 4-9b, B is plotted vs. X for the carbon dioxide + propane binary at 244. K < T < 345 K. Figure 4-9b shows that the decreases in B (= lnγ₂^∞) at X > 1 are striking, which means that the hypothetical liquid component, carbon dioxide, controls the total pressure satisfying P ≒ γ₁^∞x₁p_1s. In Figure 4-9c, C is plotted vs. X for the carbon dioxide + propane binary at 244. K < T < 345 K. Figure 4-9c shows that B and C have similar trends. In Figure 4-10, A/B is plotted vs. C for the carbon dioxide + propane binary at 244. K < T < 345 K. Figure 4-10 shows that C = A/B holds for the Px data at the limited range, C > 0.4, where xy data are not found in the literature. Both Px and xy data deviate from the C = A/B line at C < 0.4.

In Figure 4-11, A is plotted vs. X for the 25 lower-boiling point substances using Px data. Table 4-2 lists the 25 binaries. Figure 4-11 shows that, at X > 1, A decreases with increasing X. It is important to note that, at X = 1, the hypothetical liquids behave as ideal solutions, because the values of A (= lnγ₁^∞) are close to zero.

Figure 4-1. vapor pressures calculated from the Wagner equation between 0.7 < T_ri < 1, x-axis: 1-1/T_ri, y-axis: ln(p_is/P_Ci), (●) helium-4, (●) hydrogen, (●) nitrogen, (●) carbon dioxide, (----) linear correlations passing through the origin.

Table 4-1. The values of h for seven substances.

Substance	h
Helium-4	3.39
Hydrogen	4.35
Nitrogen	5.60
Carbon Dioxide	6.56
Methane	5.48
Ethane	5.92
Propane	6.20

Figure 4-2a. Pxy relationships for the helium-4 + nitrogen binary at 121.74 K; x-axis: x₁ and y₁, y-axis: P [bar], (●) experimental Px data, (−) correlated Px data with A=2.24, B=-12.3 and C=-0.318, (●) experimental Py data, (−) correlated Py data with A=1.32, B=-18.2 and C=0.221; data are cited from ref. 4, page 199.

Figure 4-2b. xy relationships for the helium-4 + nitrogen binary at 121.74 K; x-axis: x₁, y-axis: y₁, (●) experimental data cited from ref. 4, (−) correlated line with A=1.32, B=-18.2 and C=0.221.

Figure 4-3. Relationships between X and A calculated for the helium-4 + nitrogen binary at 77 K < T < 122 K, x-axis: X, y-axis: A, (●) determined from Px data, (---) a power function correlating A values from Px data, (●) determined from xy data.

Figure 4-4. Relationships between C and A/B calculated for the helium-4 + component 2 including nitrogen and carbon monoxide at 77 K < T < 122 K, x-axis: C, y-axis: A/B, (●) determined from Px data, (---) correlation of A/B values determined from the Px data, (●) determined from xy data, (---) correlation of A/B values determined from the xy data; data are cited from ref. 4 in pages 199 – 204.

Figure 4-5. Pxy relationships for the hydrogen + methane binary at 144.26 K; x-axis: x₁ and y₁, y-axis: P [bar], (●) experimental Px data, (−) correlated Px data with A=0.780, B=-2.5e7 and C=0.00011, (●) experimental Py data, (−) correlated Py data with A=0.316, B=-15.8 and C=0.461; data are cited from ref. 4 in page 221.

Figure 4-6. Relationships between X and A calculated for the hydrogen + component 2 including nitrogen and methane at 77 K < T < 174 K, x-axis: X, y-axis: A, (●) determined from Px data for the hydrogen + nitrogen binary, (●) determined from Px data for the hydrogen + methane binary; data are cited from ref. 4 in pages 210 – 227.

Figure 4-7. Relationships between C and A/B calculated for the hydrogen + component 2 including nitrogen and methane at 77 K < T < 174 K, x-axis: X, y-axis: A/B, (●) determined from Px data, (---) correlation of A/B values determined from the Px data, (●) determined from xy data, (---) correlation of A/B values determined from the xy data; data are cited from ref. 4 in pages 210 – 227.

Figure 4-8. Pxy relationships for the carbon dioxide + propane binary at 277.59 K; x-axis: x₁ and y₁, y-axis: P [bar], (●) experimental Px data, (−) correlated Px data with A=0.491 B=0.693 and C=0.708, (●) experimental Py data, (−) correlated Py data with A=0.367, B=-5.79 and C=0.280; data are cited from ref. 4 in page 589.

Figure 4-9a. Relationships between X and A calculated for the carbon dioxide + propane binary at 244. K < T < 345 K, x-axis: X, y-axis: A, (●) determined from Px data, (---) a power function correlating A values from Px data, (●) determined from xy data.

Figure 4-9b. Relationships between X and B calculated for the carbon dioxide + propane binary at 244. K < T < 345 K, x-axis: X, y-axis: B, (●) determined from Px data, (●) determined from xy data.

Figure 4-9c. Relationships between X and C calculated for the carbon dioxide + propane binary at 244 K < T < 345 K, x-axis: X, y-axis: C, (●) determined from Px data, (●) determined from xy data.

Figure 4-10. Relationships between C and A/B calculated for the carbon dioxide + propane including nitrogen and methane at 244 K < T < 345 K, x-axis: C, y-axis: A/B, (●) determined from Px data, (●) determined from xy data, (---) correlation of A/B values determined from xy data; data are cited from ref. 4 in page 589.

Figure 4-11. Relationships between X and A calculated for the lower-boiling point component 1 + component 2 binary, x-axis: X, y-axis: A, (●) determined from Px data, data are cited from ref. 4; component 1 includes 25 substances excluding helium-4 and hydrogen.

Table 4-2. Binary systems used in Figure 4-11. Data are cited from ref. 4.

Component 1	component 2
N2	CO, Ar,, CO2, methane to heptane, benzene
CO	methane to propane
Ar	O2, methane
Methane	C2H2, propylene, CO2, propane to heptane, benzene to m-xylene,
	C6H12, m-cresol
CF4	fluoroform
C2H2	ethane to heptane, acetylene, CO, CO2
CO2	ethane, nitrousoxide, difluoromethane, propylene, propane to decane, dichlorofluoromethane, isobutene, 1-butene, 2-methylbutane,
	diethylether, methylacetate, methanol, C6H6, C6H12, H2O, toluene,
Ethane	propylene, isobutene, diethylether, acetone, methylacetate, methanol,
	C6H6, C6H12
Acetylene	propylene
Chlorotrifluoromethane	dichlorodifluoromethane
Carbonylsulfide	propane
Propylene	propane, 1-butene, isobutene, 2-methylbutane, ethanol, C6H6
Chloropentafluoroethane	chlorodifluoromethane
Chlorodifluoromethane	dichlorodifluoromethane
NH3	H2O
1-Butene	1,3-butadiene, butane
1,3-butadiene	butane
Pentane	C6H6
2,2-Dimethylbutane	1-pentanol
2,3-Dimethylbutane	1-pentanol
2-Methylpentane	1-pentanol
Methanol	H2O
Hexane	C6H6, isopropylalcohol, 1-pentanol
Perfluorobenzene	C6H6
C6H12	C6H6
C6H6	heptane

Conclusion for Chapter 4

It is natural and convenient to use hypothetical liquids and an approximation, Equation (4-2), because high-pressure binary Px and xy VLE data can be satisfactorily correlated using the three parameter Wohl equation and hypothetical liquids.

Nomenclature for Chapter 4

A, B, C Wohl parameters defined in Eqs. (4-3) and (4-4)

(ARD)_P average relative deviation of P defined in Eq. (4-12)

h constant defined in Eq. (4-11)

OF_P objective function defined in Eq. (4-7)

OF_y objective function defined in Eq. (4-10)

n number of data points

P system pressure

P_a reference pressure

P_Ci critical pressure of component i

P_k,exp k th experimental total pressure data

p_is vapor pressure of component i

R gas constant

T system temperature

T_Ci critical temperature of component i

T_C1 critical temperature of component 1

T_C2 critical temperature of component 2

T_ri reduces temperature of component i

V_i^L partial molar volume of component i

V_i⁰ molar volume of pure liquid i at T

X critical ratio defined in Eq. (4-13)

x₁ liquid phase mole fraction of component 1

x₂ liquid phase mole fraction of component 2

y_i vapor phase mole fraction of component i

y₁ vapor phase mole fraction of component 1

y_1k,ezp k th experimental vapor phase mole fraction of component 1

γ_i activity coefficient of component i in the liquid phase

γ₂ activity coefficient of component 2 in the liquid phase

γ_i^(Pa) acti vity coefficient of component i in the liquid phase at P = P_a

φ_is fugacity coefficient of the saturated vapor I at p_is

φ_i^V vapor phase fugacity coefficient of component I in the mixture at P

θ₁ parameter defined in Eq. (4-5)

θ₂ parameter defined in Eq. (4-6)

Superscript

(P) reference pressure at P

∞ infinite dilution

Subscript

min minimization

Literature cited

1) Prausnitz J. M., Molecular Thermodynamics of Fluid-Phase Equilibria; Prentice-Hall: Upper Saddle River, NJ, 1960.

2) J. Gmehling, U. Onken, Vapor–Liquid Equilibrium Data Collection, DECHEMA Chemistry Data Series, Vol. I, parts 1 to 8a, DECHEMA: Frankfurt, 1977 – 2001, including Landolt Bornstein (1960, 1972, 1975) and other data collections shown in part 1, pages XLIV and XLV.

3) J.M. Sorensen, W. Arlt, Liquid–Liquid Equilibrium Data Collection, Binary Systems, DECHEMA Chemistry Data Series, Vol. V, parts 1 to 4, DECHEMA: Frankfurt, 1979, including data collections shown in part 1, page XX.

4) Knapp, H.; Doring, R.; Oellrich, L.; Plocker, U.; Prausnitz, J. M., Vapor-Liquid Equilibria for Mixtures of Low Boiling Substances. DECHEMA Chemistry Data Series, Vol. VI; DECHEMA: New York, 1982.

5) R.C. Reid, J.M. Prausnitz, B.E. Poling, The Properties of Gases and Liquid 1987 MacGraw-Hill.,