# Computation of Isobaric Vapor-Liquid Equilibrium Data for Binary and Ternary Mixtures of Methanol, Water, and Ethanoic Acid fromT, p, x, and HmE Measurements

Authors: Daming Gao; Hui Zhang; Peter Lücking; Hong Sun; Jingyu Si; Dechun Zhu; Hong Chen; Jianjun Shi
Journal: Journal of Thermodynamics
 (2012)

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## Abstract

Vapor-liquid equilibrium (VLE) data for the strongly associated ternary system methanol + water + ethanoic acid and the three constituent binary systems have been determined by the total pressure-temperature-liquid-phase composition-molar excess enthalpy of mixing of the liquid phase (p,
T,
x, HmE) for the binary systems using a novel pump ebulliometer at 101.325 kPa. The vapor-phase compositions of these binary systems had been calculated from Tpx and HmE based on the Q function of molar excess Gibbs energy through an indirect method. Moreover, the experimental T, x data are used to estimate nonrandom two-liquid (NRTL), Wilson, Margules, and van Laar model parameters, and these parameters in turn are used to calculate vapor-phase compositions. The activity coefficients of the solution were correlated with NRTL, Wilson, Margules, and van Laar models through fitting by least-squares method. The VLE data of the ternary system were well predicted from these binary interaction parameters of NRTL, Wilson, Margules, and van Laar model parameters without any additional adjustment to build the thermodynamic model of VLE for the ternary system and obtain the vapor-phase compositions and the calculated bubble points.


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## Body

## 1. Introduction

New strategies for the correlation and accurate prediction of the vapor-liquid equilibrium (VLE) data play a vital role in the distillation and separation process in chemical industry. The common technique for obtaining VLE data is by direct measurement on the system. That is to say, when the VLE is established, and phases are sampled and analyzed. Consequently, the experimental technique must be rather highly accurate to ensure meaningful results in the operation of equilibrium stills. Actually, when the vapor-phase components are sampled and analyzed, the whole compositions of components in solution and vapor have been changed. Accordingly, the behavior of the systems has been changed with the amount of compositions. Moreover, it has been long realized that the analysis of vapor-liquid composition for the infinite dilute solution is very difficult. In addition, for VLE measurements of mixtures containing a highly volatile compound, the accurate measurement of the vapor-phase composition can be difficult. This fact, coupled with the necessity for much analytical work, tends to enhance interest in exploring new methods for the determination of equilibrium data that do not involve sampling and analysis of the vapor-phase components.Several methods have been explored for the calculation of component behavior from gross solution behavior. Van Ness and coworkers have suggested the classification of these methods into two categories, direct and indirect methods [1]. The direct methods involve calculation of vapor compositions by integration of the coexistence equation, a first-order differential equation derived from the Gibbs-Duhem equation relating phase compositions at equilibrium. Hala and coworkers have given a detailed discussion of the basic direct method [2], and Van Ness and coworkers have discussed techniques for handling nonconstant temperature or pressure conditions as well as nonideal vapor phase behavior [1, 3]. Moreover, in the total pressure method one can calculate y from T, p, x measurements using an indirect method discussed by Mixon et al. [4].Of all the methods, the indirect methods involve first the measurement, by some appropriate means, the temperature and total pressure of the system, the liquid-phase compositions, and subsequent calculation of vapor-phase compositions there from. These methods usually involve ascertaining which of selected solution equations to the Gibbs-Duhem equation lead to the best fit to the experimental data, and of the determination of the parametric values producing the best fit. For example, Barker has developed a procedure based on the assumption that the excess free energy can be represented as a polynomial function of composition [5].There is a basic difference in the degree of rigor associated with the direct method and the indirect method of Barker. In the former, one makes no assumptions about the solution behavior for the nature of molecular interactions. Solution behavior is determined directly from the experimental data. The Barker method necessitates the assumption of a particular model and the estimation of its parameters, this deficiency in the method of Barker has been recognized by Tao, who has presented another indirect method in which the necessity for the a priori assumption of a particular functional form for the excess free energy has been removed. Tao’s procedure involves calculation of the activity coefficient essentially by integration of an equation resembling the coexistence equation. His procedure, though indirect, retains the rigor usually associated with the direct method [6]. The method of Tao appears specific to binary systems and does not seem to be easily generalized. However, an indirect method such as that of Barker is readily and easily generalized to ternary and higher-ordered systems, but this method retains the disadvantage of lacking rigor as compared with the direct method.This paper presents the vapor-liquid compositions calculated based on the measurements of VLE data for temperature-total pressure-liquid-phase composition-molar excess enthalpy energy of mixing of the liquid phase (T, p, x, HmE) at 101.325 kPa according to the Q function of molar excess Gibbs energy through an indirect method. We know that the reaction of methanol carboxylated with carbon monoxide is the most common and important technology for synthesis of ethanoic acid in the chemical industrial process. Modeling the thermodynamic properties and correlating and predicting the phase equilibria of a mixture involving associating components forming hydrogen bonding such as carboxylic acid remain a challenging problem, since such systems show extremely nonideal behavior and the formation of monomer, dimer, and even trimer in vapor and liquid phase. In addition, for the VLE measurements of vapor-phase components containing a highly volatile compound such as carboxylic acid, the accurate measurement of the vapor-phase composition can be difficult. Many attempts have been made to describe the vapor-liquid equilibria of carboxylic acid containing mixtures using the concept of multiscale association [7]. Arlt reported the isothermal VLE data of a new apparatus for phase equilibria in reaction mixtures containing water with ethanoic acid and propanoic acid at (333 to 363) K [8]. Xu and Chuang have developed a new correlation for the prediction of the vapor-liquid equilibrium of methyl acetate-methanol-water-ethanoic acid mixtures [9]. Sawistowski and Pilavakis explored the vapor-liquid equilibrium behavior of the quaternary system methyl acetate-methanol-water-ethanoic acid modeled by using the Margules equation in combination with Marek’s method for the association of ethanoic acid [10]. Moreover, Guan et al. investigated that the isobaric vapor-liquid equilibria for water + ethanoic acid + n-pentyl acetate, isopropyl acetate, N-methyl pyrrolidone, or N-methyl acetamide were correlated and predicted by both the nonrandom two-liquid (NRTL) and universal quasichemical activity coefficient (UNIQUAC) models used in combination with the Hayden-O’Connell (HOC) method [11–14]. In our recent work, we have concluded that the VLE data for the associating systems containing the carboxylic acid system can be correlated and predicted [15, 16]. Although the VLE data of the mixture containing the associating systems were previously reported by the different research groups [17–21], respectively, the VLE data for the associating binary system containing the carboxylic acid have been still extensively studied because of the extensive association effects occurring in them and the difficulty of properly calculating the activity coefficients [22–25]. Nominally, the system is binary but in practice, it is multicomponent. Because the carboxylic acid monomer undergoes partial dimerization and even higher polymerization. This association is attributed to the formation of hydrogen bonds and occurs in both the vapor and liquid phase. Therefore, the challenge for the VLE data of the associating systems has evoked more and more researchers to focus on new strategies for exploring them. The VLE data for methanol + water + ethanoic acid ternary system and the constituent binary systems are indispensable in distillation separation process to the product of methanol carboxylation through the correlation and prediction by the new method, while some of the isobaric VLE data for these binary and ternary systems are correlated and predicted earlier [9–25]. To provide the new correlation and prediction for some necessary basic thermodynamic data on the separation process of methanol carboxylation, therefore, it is very indispensable for these systems studied on the VLE data of the constituent binary and ternary systems using the new method. This paper reports a novel correlation and prediction for the VLE data for these systems containing the associating component that has been developed. The VLE data for methanol + water + ethanoic acid system and constituent binary systems were measured by the total pressure-temperature-liquid-phase composition-molar excess enthalpy of mixing of the liquid phase (p, T, x, HmE) for the binary systems using the novel pump-ebulliometer at 101.325 kPa. Owing to the association of ethanoic acid molecules, Marek’s method in combination with Hayden-O’Connell (HOC) model was used to deal with the associating properties of the liquid and vapor phase and the nonideality of vapor phase, respectively [26–28]. However, the nonideality of liquid phase was corrected by the calculation of its activity coefficient, which was obtained based on NRTL, Wilson, Margules, and van Laar models as the function of T, x through the nonlinear fit of the least-squares method. NRTL, Wilson, Margules, and van Laar models were applied to correlate the VLE data for the three constituent binary systems, and the model parameters together with the deviations of temperature and vapor-phase molar fractions calculated from T, p, x, HmE according to the Q function of molar excess Gibbs energy by the indirect method were obtained by the least-squares method. The VLE data of the ternary system were well predicted from these binary interaction parameters of NRTL, Wilson, Margules, and van Laar models without any additional adjustment to build the thermodynamic model of VLE for the ternary system and obtain the vapor-phase compositions and the calculated bubble points. The excess Gibbs free energies for these binary systems as the function of liquid-phase composition and activity coefficient were calculated through the activity coefficient correlation to NRTL model parameters with the experimental data.


## 2. Modeling Section

There is an added complexity when working with carboxylic acids because they associate in the vapor and liquid phases. This association can be represented by assuming that the organic acid exists as monomer and dimer according to the Marek’s method [26, 27]. This fact, coupled with the necessity for much analytical work, tends to enhance interest in exploring the new strategies for correlation and prediction of the VLE data for the systems containing the associating carboxyl acid.According to the Marek’s chemical theory, there are monomer and dimer carboxylic acid molecules in both liquid and vapor phases. The equilibrium constant of vapor-phase dimerization,CEV, is calculated by the expression
(1)CEV=yD*(yM*)2p
and the equilibrium constant of liquid-phase dimerization, CEL, is calculated by the expression
(2)CEL=xD*(xM*)2.In equations mentioned above,yD* and xD* are the mole fractions of dimers of ethanoic acid molecules in both vapor and liquid phases, respectively, and CEV can be defined as the function of temperature by the expression obtained from the literature [27]
(3)CEV=exp(7290T-21.980),
where CEV  was presented in kPa−1 and T in K.When the dimers of ethanoic acid molecules mainly exist, the binary system for the methanol or water(1) + ethanoic acid (2) is nominally binary; however, it is actually ternary for the methanol or water (1) + monomer ethanoic acid (M) + dimer ethanoic acid (D) system. In this system, the mole fractions of vapor-liquid equilibrium phases are y1*, yM*, yD*, x1*, xM*, and xD*, respectively. Therefore, the VLE relations of the nonassociating and associating components are individually calculated by the expressions
(4)py1*Φ1=p1sx1*γ1,pyM*ΦM=pMsxM*γM,pyD*ΦD=pDsxD*γD.In (4), the actual mole fractions can be denoted by the apparent mole fractions of easily determined components (y1, y2, x1, and x2 multiplied by a modified coefficient. Likewise, the measured apparent vapor pressure saturated of ethanoic acid (p2s) multiplied by a modified coefficient can also denote the actual vapor pressures saturated of monomer and dimer ethanoic acid (pMs, pDs). Consequently, the apparent compositions and vapor pressures saturated substitute for the actual ones, and the VLE relation of the nonassociating component, methanol or water (1), is expressed by
(5)py1α1Φ1=p1sx1β1γ1,
where
(6)α1=2(2-y2)1-y2-1+4CEVpy2(2-y2)1+1+4CEVpy2(2-y2),(7)β1=2(2-x2)1-x2-1+4CELpx2(2-x2)1+1+4CELpx2(2-x2),
and for ethanoic acid (2), which is the associating component, its relation is expressed by
(8)py2α2Φ2=p2sθ2sx2β2γ2,
where
(9)α2=21+1+4CEVpy2(2-y2),(10)β2=21+1+4CELx2(2-x2),(11)θ2s=-1+1+4CEVp2s2CEVp2s.In (5) to (11), α1 and α2 can be regarded as modified coefficients for the deviation from ideality in vapor phase on account of association. The fugacity of coefficients Φ1 and Φ2 is not negligible, and their values were obtained through the HOC model [28]. From another point of view, θ2s can be viewed as a modified coefficient for the vapor pressure of the associating component. Moreover, β1 and β2 denote modified coefficients of the deviation from ideality in the liquid phase by reason of the existence of association, and γ1 and γ2 express the deviation from ideality in liquid phase because of other factors. Herein, in liquid phase, the modified coefficients of deviation from ideality, β1 and β2, can be incorporated to the activity coefficients, γ1 and γ2, respectively. So (5) and (8) are obtained by the expressions
(12)py1α1Φ1=p1sx1γ1,(13)py2α2Φ2=p2sθ2sx2γ2.


## 3. Experimental Section

### 3.1. Materials

Methanol (99.8 mass %) and ethanoic acid (99.8 mass %) were purchased from Sigma. The purities of the chemicals are provided by the manufacture’s specifications. Ultra sound was used to dispel the solvent air in the materials, which were dried on molecular sieve (pore diameter 30 nm from Fluka) to remove all possible traces of moisture before use, but no other treatments were applied. The densities and refractive indices at 298.15 K and normal boiling points at 101.325 kPa of the pure component were compared with the literature values of Riddick et al. [29]. The results show that the measured values are approximately in agreement with those of the literature, as presented in Table 1. The measurement method of the composition dependence of densities and refractive indices has been previously reported [30].Table 1
Physical properties of the pure compounds: densitiesρ, refractive indexes nD at 298.15 K and normal boiling points Tb.
Compound
ρ
/kg·m−3
n
D
T
b/K
expt
lit
a
expt
lit
a
expt
lit
a
Methanol
786.44
786.37
1.3264
1.3265
337.71
337.69
Water
997.01
997.05
1.3324
1.3325
373.16
373.15
Ethanoic acid
1043.90
1043.92
1.3716
1.3718
391.53
391.15
aRiddick et al. [29].


### 3.2. Apparatus and Procedure

A new type of magnetic pump ebulliometer described in detail by Qiu et al. [31] was used to measure the boiling points with different liquid phase compositions at the 101.325 kPa. The experimental main apparatus for pump-ebulliometer is schematically shown in Figure 1. The recirculation still is entirely constructed from borosilicate glass. The main parts are a magnetic recirculating bump (F), a magnetic stirrer (G), a thermo-well (filled with a conducting oil) provided to enable good measurement of equilibrium temperature (H), a part-heating tube (resistance heating wire inserted) (K), and a tube window to observe in VLE status (I). The apparatus is an all-glass dynamic recirculation still with total volume of about 1.00 × 10−4 m3. During the run, to avoid the over heating, the still was submerged in a constant temperature bath at about 3°C below the equilibrium boiling point, which was obtained by the Nichrome wire in the part-heating tube (K) to partially heat the known mass of the material. The equilibrium pressure was measured using a Fischer digital manometer with a precision of ±0.01 kPa. The pressure measurement for the manometer had an uncertainty of ±0.07 kPa, as provided by the manufacturer’s specifications. The total uncertainty of calibration and pressure measurement is estimated to be ±0.15 kPa because of the uncertainty of the calibration and measurement errors. Since the barometric pressure changed slightly, the experimental temperatures of the systems were automatically calibrated to that at 101.325 kPa with self-adjusted pressure system. The temperature was measured using a Heraeus QuaT100 quartz thermometer with a thermosensor, with an accuracy of ±0.01 K. The calibration of the thermometers was carried out at the accredited calibration laboratory (Quality and Technique Bureau, Anhui). The total uncertainty of calibration and temperature measurement is evaluated to be ±0.085 K because of the uncertainty of the calibration, the probe’s position, and the pressure fluctuations. The equilibrium temperature, T, was measured by means of the thermosensor inserted into the thermowell (filled with conducting oil) (H). In each experiment, a known mass of the material was introduced from the injector into the still from feed inlet (B) and heated to equilibrium boiling point of the system at a fixed pressure of 101.325 kPa by an automatic pressure regulation system. The liquid mixtures of required composition were prepared gravimetrically, with the use of a Sartorius electronic analytic balance (model ER-182A) with an accuracy of ±0.0001 g. The values of mole fraction were reproducible to ±0.0001 and have uncertainty of 0.01%. The ebulliometer was charged with the mixture of desired composition, and the boiler was then heated by nichrome wire wound around the boiler. After the liquid mixture started boiling, the bubbles along with the drops of liquid slowly spurted on the thermowell one by one through the tube window observed in VLE status (I). When the VLE state was attained by adjusting the pressure to 101.325 kPa, it remained constant for 20 min to ensure the stationary state, and then the boiling temperature was measured.Figure 1
Schematic illustration for the structure of pump ebulliometer. (A) Normal distilling adaptor; (B) feed inlet; (C) coiled glass thread; (D) constant temperature introducing tube; (E) mixed vial section; (F) magnetic recirculating bump; (G) magnetic stirrer; (H) measured thermowell (filled with conducting oil); (I) tube window to observe in VLE status; (J) internal heating tube; (K) part-heating tube (resistance heating wire inserted).A flow microcalorimeter (model 2107, LKB produkter, Bromma, Sweden) was applied to measure the molar excess enthalpies,HmE, of the mixtures. The electrical-calibration apparatus and its operating procedure have been described elsewhere by Francesconi and Comelli [34]. Two automatic burets (ABU, Radiometer, Copenhagen, Denmark) were used to pump liquids through the mixing cell of the calorimeter. The performance of the calorimeter was checked by measuring the HmE of the test mixture hexane + cyclohexane system reported by Gmehling [35]. Agreement with literature is better than 0.5% over the central range of mole fraction of hexane.The uncertainties in mole fraction andHmE are estimated to be 0.0005 and 0.5%, respectively. The liquid-phase mole fraction of component i, xi, could be calculated from the known mass of the material added to the still. The vapor-phase mole fraction of component i, yi, was calculated from the experimental T, p, x, and HmE data based on the Q function (the function of molar excess Gibbs energy) by the indirect method.


## 3.1. Materials

Methanol (99.8 mass %) and ethanoic acid (99.8 mass %) were purchased from Sigma. The purities of the chemicals are provided by the manufacture’s specifications. Ultra sound was used to dispel the solvent air in the materials, which were dried on molecular sieve (pore diameter 30 nm from Fluka) to remove all possible traces of moisture before use, but no other treatments were applied. The densities and refractive indices at 298.15 K and normal boiling points at 101.325 kPa of the pure component were compared with the literature values of Riddick et al. [29]. The results show that the measured values are approximately in agreement with those of the literature, as presented in Table 1. The measurement method of the composition dependence of densities and refractive indices has been previously reported [30].Table 1
Physical properties of the pure compounds: densitiesρ, refractive indexes nD at 298.15 K and normal boiling points Tb.
Compound
ρ
/kg·m−3
n
D
T
b/K
expt
lit
a
expt
lit
a
expt
lit
a
Methanol
786.44
786.37
1.3264
1.3265
337.71
337.69
Water
997.01
997.05
1.3324
1.3325
373.16
373.15
Ethanoic acid
1043.90
1043.92
1.3716
1.3718
391.53
391.15
aRiddick et al. [29].


## 3.2. Apparatus and Procedure

A new type of magnetic pump ebulliometer described in detail by Qiu et al. [31] was used to measure the boiling points with different liquid phase compositions at the 101.325 kPa. The experimental main apparatus for pump-ebulliometer is schematically shown in Figure 1. The recirculation still is entirely constructed from borosilicate glass. The main parts are a magnetic recirculating bump (F), a magnetic stirrer (G), a thermo-well (filled with a conducting oil) provided to enable good measurement of equilibrium temperature (H), a part-heating tube (resistance heating wire inserted) (K), and a tube window to observe in VLE status (I). The apparatus is an all-glass dynamic recirculation still with total volume of about 1.00 × 10−4 m3. During the run, to avoid the over heating, the still was submerged in a constant temperature bath at about 3°C below the equilibrium boiling point, which was obtained by the Nichrome wire in the part-heating tube (K) to partially heat the known mass of the material. The equilibrium pressure was measured using a Fischer digital manometer with a precision of ±0.01 kPa. The pressure measurement for the manometer had an uncertainty of ±0.07 kPa, as provided by the manufacturer’s specifications. The total uncertainty of calibration and pressure measurement is estimated to be ±0.15 kPa because of the uncertainty of the calibration and measurement errors. Since the barometric pressure changed slightly, the experimental temperatures of the systems were automatically calibrated to that at 101.325 kPa with self-adjusted pressure system. The temperature was measured using a Heraeus QuaT100 quartz thermometer with a thermosensor, with an accuracy of ±0.01 K. The calibration of the thermometers was carried out at the accredited calibration laboratory (Quality and Technique Bureau, Anhui). The total uncertainty of calibration and temperature measurement is evaluated to be ±0.085 K because of the uncertainty of the calibration, the probe’s position, and the pressure fluctuations. The equilibrium temperature, T, was measured by means of the thermosensor inserted into the thermowell (filled with conducting oil) (H). In each experiment, a known mass of the material was introduced from the injector into the still from feed inlet (B) and heated to equilibrium boiling point of the system at a fixed pressure of 101.325 kPa by an automatic pressure regulation system. The liquid mixtures of required composition were prepared gravimetrically, with the use of a Sartorius electronic analytic balance (model ER-182A) with an accuracy of ±0.0001 g. The values of mole fraction were reproducible to ±0.0001 and have uncertainty of 0.01%. The ebulliometer was charged with the mixture of desired composition, and the boiler was then heated by nichrome wire wound around the boiler. After the liquid mixture started boiling, the bubbles along with the drops of liquid slowly spurted on the thermowell one by one through the tube window observed in VLE status (I). When the VLE state was attained by adjusting the pressure to 101.325 kPa, it remained constant for 20 min to ensure the stationary state, and then the boiling temperature was measured.Figure 1
Schematic illustration for the structure of pump ebulliometer. (A) Normal distilling adaptor; (B) feed inlet; (C) coiled glass thread; (D) constant temperature introducing tube; (E) mixed vial section; (F) magnetic recirculating bump; (G) magnetic stirrer; (H) measured thermowell (filled with conducting oil); (I) tube window to observe in VLE status; (J) internal heating tube; (K) part-heating tube (resistance heating wire inserted).A flow microcalorimeter (model 2107, LKB produkter, Bromma, Sweden) was applied to measure the molar excess enthalpies,HmE, of the mixtures. The electrical-calibration apparatus and its operating procedure have been described elsewhere by Francesconi and Comelli [34]. Two automatic burets (ABU, Radiometer, Copenhagen, Denmark) were used to pump liquids through the mixing cell of the calorimeter. The performance of the calorimeter was checked by measuring the HmE of the test mixture hexane + cyclohexane system reported by Gmehling [35]. Agreement with literature is better than 0.5% over the central range of mole fraction of hexane.The uncertainties in mole fraction andHmE are estimated to be 0.0005 and 0.5%, respectively. The liquid-phase mole fraction of component i, xi, could be calculated from the known mass of the material added to the still. The vapor-phase mole fraction of component i, yi, was calculated from the experimental T, p, x, and HmE data based on the Q function (the function of molar excess Gibbs energy) by the indirect method.


## 4. Results and Discussion

### 4.1. Calculation of Vapor-Phase Mole Fractiony for the Binary Systems from Tpx and HmE

The vapor-phase mole fractionyi of the components was calculated from
(14)yiΦip=xiγipis,
and the fugacity of coefficients  Φi of the components was obtained by the expression
(15)Φi=φ^iVφisexp[-ViL(p-pis)RT],
where φ^iVis the fugacity coefficient of component i in the vapor mixture, φis is the fugacity coefficient of component i at saturation, ViL is the molar volume of component i in the liquid phase, R is the universal gas constant, T is the experimental temperature, p is the total pressure (101.325 kPa), and pis is the vapor pressure of pure component i. The Antoine equations were applied to calculate the values of these vapor pressures. The Antoine constants Ai, Bi, Ci, and the values of Tc, pc, Vc, Zc and ω  were obtained from Shi et al. [36], as shown in Table 2. The Poynting correction factor was also included in the calculation of fugacity at the reference state. The liquid molar volumes were evaluated from the modified Rackett equation [37].Table 2
Antoine coefficients of the compounds and published parameters [36] used to calculate fugacity coefficients: critical temperature Tc, critical pressure pc, critical volume Vc, critical compression Zc, and acentric factor ω of pure compounds.
Compound
A
i
B
i
C
i
T
c/K
p
c/MPa
V
c/m3·kmol−1
Z
c
ω
Methanol
7.19736
1574.99
−34.29
512.6
8.096
0.118
0.224
0.559
Water
7.07404
1657.46
−46.13
647.3
22.048
0.056
0.229
0.344
Ethanoic acid
6.42452
1479.02
−56.34
594.4
5.786
0.171
0.200
0.454According to the Gibbs-Duhem equation, any extensive molar thermodynamic property of a given phase, such as the Gibbs and Helmholtz energies, the enthalpy, and molar volume, must satisfy the following differential relation:(16)(∂M∂T)p,xj≠idT+(∂M∂p)T,xj≠idp-∑i=1CxidM-i=0.In (16) M is a general molar property, xi the molar fraction of component i in the phase under consideration, xj≠i the pertinent set of compositions, C the number of components, and M-i the partial contribution of component i to M.When considering VLE, the molar excess Gibbs energy,GE, can be evaluated from measurable (T, p, x) data using activity coefficient relations. Replacing M by GE/RT in the following equation yields the well-known relation
(17)-HmERT2dT+VmERTdp-∑i=1Cxidlnγi=0,
where HmE and VmE are the molar excess enthalpy and volume of mixing of the liquid phase. According to the thermodynamic principles, the activity coefficients γi of the components were calculated from the expression as follows:
(18)Q=GERT=∑i=1Cxilnγi,
where Q is the function of molar excess Gibbs energy and combining (17) and (18) yields
(19)d(GERT)-∑i=1Clnγidxi=∑i=1Cxidlnγi=-HmERT2dT+VmERTdp.Application to a binary system gives(20)d(GERT)-lnγ1γ2dx1=-HmERT2dT+VmERTdp.Simultaneous solution of (18) and (20) yields
(21)lnγ1=GERT+x2[ddx1(GERT)+HmERT2dTdx1-VmERTdpdx1],lnγ2=GERT-x1[ddx1(GERT)+HmERT2dTdx1-VmERTdpdx1].Equation (18) substituted into (21) reduced it to
(22)γ1=exp[Q+(1-x1)(dQdx1+HmERT2dTdx1-VmERTdpdx1)],γ2=exp[Q-x1(dQdx1+HmERT2dTdx1-VmERTdpdx1)].For a binary system comprised of species 1 and 2 at VLE state, at constant pressure,(dp/dx1)=0, substitution into (22) reduced it to
(23)γi=exp[Q+(1-xi)[(dQdxi)+HmERT2(dTdxi)]],(i=1,2).The right-hand side,(HmE/RT2)(dT/dxi), of (23) cannot be neglected. Proper use of (23) requires the availability of heats of mixing as a function of composition and temperature. The activity coefficients γi of the components as functions of the excess Gibbs energy are as follows:
(24)γ1=exp{Q+(1-x1)[(dQdx1)+HmERT2(dTdx1)]},γ2=exp{Q-x1[(dQdx1)+HmERT2(dTdx1)]}.From (14), this equation is rearranged to obtain
(25)yi=xiγipisΦip.Because∑yi=1, for the binary system, the equation may be summed to give
(26)x1γ1φ1sp1sexp[V1L(p-p1s)/RT]φ^1Vp+x2γ2φ2sp2sexp[V2L(p-p2s)/RT]φ^2Vp=1.In (26), solved for yi by difference method. Suppose that [0,1] is subdivided into n subintervals [xk,xk+1] of equal step size h=1/n by using xk=kh for k=0,1,…,n. In k difference point, we obtain
(27)Fk=1-{x1γ1φ1sp1sexp[V1L(p-p1s)/RT]φ^1Vp}k-{x2γ2φ2sp2sexp[V2L(p-p2s)/RT]φ^2Vp}k=0.Meanwhile, (23) may be shown as follows:
(28)γi|k=exp{Qk+(1-xi|k)[Q|k+1-Q|k-12/h+HmERT2(dTdx1)|k]}.And (27) is linearized to obtain:
(29)-Fk=ΔQ|k-1(δFkδQ|k-1)+ΔQ|k(δFkδQ|k)+ΔQ|k+1(δFkδQ|k+1).The number ofn linear equation from (28) is solved for ΔQ|k by chasing method
(30)Qkj+1=Qkj+tΔQkj,
where t is relaxation factor, finally, yik (k=1~n) is obtained by difference method. The block diagram for calculation procedure was detailedly shown in Figure 2.Figure 2
Block diagram for the calculationy by T, p, x, and HmE.


### 4.2. Calculation of Vapor-Phase Mole Fractiony for the Binary Systems from Model

There are many methods concerning the correlation and prediction of VLE data. The model-free approach data treatment of vapor-liquid equilibrium is also one of the best strategies for the correlation and prediction of VLE data. Wisniak’s group has developed that the novel model-free computation techniques and limiting conditions have been applied to VLE data for azeotropic systems [42]. Moreover, Segura and coworkers reported that a model-free approach dealt with VLE data in application of ternary systems [38, 43]. For the three binary systems of this study, the activity coefficients were correlated with the NRTL [39], Wilson [40], Margules [41], and van Laar [44] equations, respectively. The interaction parameters optimized were achieved by the objective function (OF) minimized using the least-squares fitting as follows:
(31)OF=∑i=1N(xi,cal-xi,exp)2,
where N is the number of experimental points xi,cal and xi,exp are the liquid-phase mole fraction of component i calculated and experimental values from the (12) or (13) and from measured data, respectively.Because carboxylic acids are always present in an associated form, like a dimer or trimer, in both the vapor and liquid phases even at low pressures, a significant deviation in fugacity coefficient may exist using the ideal gas assumption. To illustrate the deviation from ideal behavior, Marek’s chemical theory and HOC model were applied to deal with the associating component and modify the deviation from idealities of vapor phase [26–28], respectively. The Poynting correction factor was also included in the calculation of fugacity at the reference state. The liquid molar volumes were evaluated from the modified Rackett equation [37]. However, under isobaric conditions, the most volatile component cannot exist in the liquid state, only as superheated vapor. Hence, there is no way to calculate or measure this property for the molar volumes of the pure liquids. Therefore, the correct procedure for isobaric measurements is to calculate the overall values of Aij and Aij, as adjustable parameters, and not the values of the interaction excess energy. Hence, for Wilson model, Aij and Aij were reported as adjustable parameters.The activity coefficients computed on the basis of NRTL model were used to evaluated dimensionless excess Gibbs function at 101.325 kPa for three binary systems over the overall range of composition. Liquid-phase mole fractionx1, experimental boiling point temperature  Texp, calculated bubble point temperature Tcal, vapor-phase mole fraction y1 from the model, activity coefficients γ1 and γ2 using NRTL equation correlation, fugacity coefficients φ^1V and φ^2V, and the molar excess enthalpies of mixing of the liquid phase HmE are included in Table 3. The T-x1-y1 diagrams for the methanol (1) + water (2), methanol (1) + ethanoic acid (2), and water (1) + ethanoic acid (2) three binary systems at 101.325 kPa are shown in Figures 3, 4, and 5, respectively. The plot of excess Gibbs energy function GE/RT versus liquid-phase mole fraction x1 for the three binary systems is shown in Figure 6. All the mixtures exhibit deviations from ideality with a range that may be attributed to interactions leading to the formation of various associated aggregates. Observed nonideal behavior is indicative of the magnitude of experimental activity coefficients γi, as well as of the variation of excess Gibbs energy function, GE/RT, with composition, as depicted in Figure 6. The obtained absolute maximum values of GE/RT for the methanol (1) + water (2), methanol (1) + ethanoic acid (2), and water (1) + ethanoic acid (2) three binary systems are 0.1703, 0.0180, and 0.0892, respectively. The values of excess Gibbs energy function GE/RT are positive for methanol (1) + water (2) and water (1) + ethanoic acid (2) binary systems. However, for methanol (1) + ethanoic acid (2) system, the values of those are negative in the overall range of mole fraction. GE/RT values follow the order methanol (1) + water (2) > water (1) + ethanoic acid (2) > methanol (1) + ethanoic acid (2). The absolute maximum value of GE/RT is approximate at an equimolar fraction in three binary systems. Figures 3–5 show that the comparison of the predicted vapor-phase and experimental liquid-phase compositions with those of the literature [10, 11, 32, 33]. Comparing with the values of vapor-phase and liquid-phase component from the literatures, the values of those from the paper are in good agreement with the literature, as shown in Figures 3–5. The results have demonstrated that the methods for Tpx and HmE are appropriate for representing the experimental data of the three binary systems. The optimum model interaction parameter of liquid activity coefficient and the absolute average deviations for the different models, and from Tpx and HmE are listed in Table 4. Herein, we obtained the results by the four different types of correlations for the prediction of activity coefficients in these systems, which reveal that the deviations of NRTL, Wilson, Margules, and van Laar equations are reasonably small in Table 4. For comparison, the mean deviations obtained by Gmehling and Onken [32, 33, 45, 46] are also shown in the Table 4. It can seem that two sets of deviation values are comparable. Since the superiority of one method over the others is not always obvious, practice must rely on experience and analogy. The comprehensive comparisons of four of the methods (NRTL, Wilson, van Laar, and Margules) were made in Table 4. From the data analysis, the temperature deviations between the experimental and calculated values of four different types of model are very similar in the three binary systems, and the vapor-phase mole fraction deviations between calculated values from Tpx and HmE, and from the model are very similar. Therefore, the activity coefficient models are appropriate for representing the experimental data of the three binary systems. In Table 4, the absolute average deviations dT of the difference between boiling point temperature from experiment and bubbling point temperature from calculation by NRLT model parameters for the three binary systems are 0.28°C, 0.51°C, and 0.33°C, respectively. And the absolute average deviations dy of the difference between vapor-phase mole fraction from Tpx and HmE calculation and from NRTL model calculation are 0.0072, 0.0054, and 0.0096, respectively.Table 3
VLE data for the methanol(1) + water (2), methanol (1) + ethanoic acid (2), and water (1) + ethanoic acid (2). Three binary systems at 101.325 kPa: liquid-phase mole fraction x1, experimental boiling point temperature Texp, calculated bubble point temperature Tcal, vapor-phase mole fraction y1, activity coefficients γ1 and γ2 using NRTL equation correlation, fugacity coefficients φ^1V and φ^2V, and molar excess enthalpies of mixing of the liquid phase HmE.
x
1
T
exp/K
T
cal/K
y
1
γ
1
φ
^
1
V
γ
2
φ
^
2
V
H
m
E/J·mol−1
Methanol(1) + water (2)
0.0000
373.15
373.15
0.0000
1.0000
0.9882
0.00
0.0484
366.00
365.54
0.2704
2.1436
0.9852
1.0034
0.9875
225.74
0.0539
365.05
364.88
0.3009
2.1147
0.9849
1.0042
0.9875
248.74
0.0607
364.40
364.12
0.3213
2.0799
0.9846
1.0053
0.9874
276.88
0.0696
363.50
363.19
0.3492
2.0358
0.9843
1.0069
0.9874
312.53
0.0806
362.25
362.13
0.3866
1.9835
0.9839
1.0092
0.9873
354.36
0.0911
361.35
361.21
0.4126
1.9360
0.9836
1.0117
0.9872
392.94
0.1056
360.05
360.05
0.4492
1.8739
0.9832
1.0155
0.9871
443.59
0.1197
359.05
359.04
0.4762
1.8172
0.9829
1.0197
0.9870
489.29
0.1354
358.60
358.02
0.4885
1.7584
0.9826
1.0248
0.9869
538.32
0.1436
358.00
357.53
0.5047
1.7293
0.9824
1.0277
0.9869
562.20
0.1586
356.95
356.70
0.5323
1.6788
0.9822
1.0334
0.9868
602.83
0.1815
355.55
355.56
0.5677
1.6083
0.9819
1.0428
0.9867
659.52
0.2092
354.35
354.35
0.5969
1.5325
0.9815
1.0555
0.9865
719.43
0.2202
354.10
353.91
0.6031
1.5049
0.9814
1.0609
0.9865
741.25
0.2380
353.25
353.24
0.6237
1.4633
0.9812
1.0701
0.9864
772.62
0.2648
352.05
352.32
0.6519
1.4066
0.9809
1.0847
0.9862
812.77
0.2918
351.35
351.48
0.6681
1.3561
0.9807
1.1005
0.9861
846.82
0.3166
350.30
350.76
0.6923
1.3148
0.9805
1.1157
0.9860
870.15
0.3442
349.05
350.02
0.7197
1.2740
0.9803
1.1336
0.9859
889.06
0.4015
347.95
348.61
0.7431
1.2038
0.9800
1.1730
0.9856
909.80
0.4282
347.90
348.00
0.7443
1.1767
0.9798
1.1924
0.9854
911.28
0.4644
347.20
347.21
0.7613
1.1449
0.9796
1.2194
0.9853
903.04
0.5240
346.20
345.98
0.7860
1.1026
0.9793
1.2656
0.9850
869.55
0.6091
344.80
344.33
0.8231
1.0595
0.9789
1.3338
0.9845
783.83
0.6899
343.31
342.85
0.8633
1.0325
0.9785
1.3995
0.9840
666.92
0.8213
341.20
340.56
0.9221
1.0086
0.9779
1.5038
0.9832
419.31
1.0000
338.10
337.66
1.0000
1.0000
0.9771
0.00
Methanol(1) + ethanol acid (2)
0.0000
390.15
391.04
0.0000
1.0000
0.9703
0.00
0.0359
387.32
386.23
0.1574
0.9388
0.9765
0.9999
0.9685
77.28
0.0410
386.21
385.56
0.1664
0.9393
0.9768
0.9999
0.9682
86.70
0.0648
383.89
382.68
0.2719
0.9419
0.9779
0.9997
0.9666
134.05
0.0741
382.73
381.90
0.3133
0.9428
0.9782
0.9996
0.9660
149.55
0.0859
381.20
380.60
0.3333
0.9441
0.9787
0.9995
0.9652
171.14
0.1040
380.03
379.06
0.3830
0.9460
0.9791
0.9992
0.9639
205.37
0.1280
378.46
377.35
0.4602
0.9485
0.9797
0.9988
0.9622
245.43
0.1629
375.62
374.70
0.5213
0.9521
0.9802
0.9981
0.9595
299.80
0.1919
373.21
372.43
0.5873
0.9551
0.9805
0.9973
0.9572
341.32
0.2510
369.31
368.44
0.6495
0.9609
0.9808
0.9954
0.9528
411.44
0.3391
364.35
363.56
0.7538
0.9692
0.9809
0.9916
0.9461
490.73
0.4401
358.54
357.71
0.8232
0.9776
0.9805
0.9857
0.9388
536.56
0.5829
351.77
350.75
0.8999
0.9874
0.9797
0.9748
0.9292
526.43
0.7050
346.47
346.39
0.9389
0.9936
0.9790
0.9633
0.9217
449.37
0.7509
344.90
344.06
0.9533
0.9955
0.9787
0.9583
0.9189
401.45
0.8109
342.81
342.35
0.9693
0.9974
0.9783
0.9515
0.9155
330.61
0.8711
341.60
340.75
0.9766
0.9988
0.9779
0.9442
0.9122
241.41
0.9161
339.65
340.00
0.9859
0.9995
0.9776
0.9384
0.9097
163.78
1.0000
338.15
337.66
1.0000
1.0000
0.9771
0.00
Water(1) + ethanoic acid (2)
0.0000
390.15
391.04
0.0000
1.0000
0.9703
0.00
0.0665
387.22
387.67
0.1140
1.4161
0.9899
1.0025
0.9693
245.31
0.0749
387.00
387.31
0.1288
1.4046
0.9898
1.0031
0.9692
273.17
0.0855
386.48
386.88
0.1603
1.3903
0.9897
1.0040
0.9691
307.50
0.0995
385.86
386.34
0.1938
1.3722
0.9896
1.0054
0.9689
350.32
0.1507
384.16
384.60
0.2802
1.3117
0.9892
1.0121
0.9683
489.63
0.1722
383.44
383.96
0.3155
1.2888
0.9891
1.0156
0.9681
540.35
0.1926
382.99
383.39
0.3375
1.2684
0.9890
1.0193
0.9679
585.57
0.2168
382.39
382.77
0.3676
1.2458
0.9890
1.0242
0.9677
632.34
0.2502
381.53
381.98
0.4101
1.2171
0.9889
1.0316
0.9674
689.92
0.2936
380.94
381.06
0.4407
1.1838
0.9887
1.0425
0.9670
750.61
0.3303
380.20
380.36
0.4815
1.1589
0.9887
1.0528
0.9667
788.98
0.3756
379.40
379.57
0.5253
1.1318
0.9886
1.0666
0.9663
821.39
0.4465
378.27
378.48
0.5890
1.0965
0.9885
1.0908
0.9658
841.59
0.4946
377.61
377.82
0.6274
1.0768
0.9885
1.1087
0.9654
836.35
0.5615
376.80
376.99
0.6777
1.0543
0.9884
1.1355
0.9649
802.60
0.6036
376.24
376.51
0.7137
1.0427
0.9884
1.1533
0.9645
767.63
0.6517
375.59
376.00
0.7546
1.0315
0.9884
1.1746
0.9642
714.78
0.7067
375.11
375.46
0.7849
1.0213
0.9884
1.1998
0.9637
639.01
0.7045
375.11
375.48
0.7848
1.0217
0.9884
1.1988
0.9638
642.76
0.7507
374.71
375.05
0.8173
1.0148
0.9883
1.2207
0.9634
568.24
0.8008
374.31
374.62
0.8501
1.0091
0.9883
1.2451
0.9630
475.20
0.8723
373.80
374.05
0.9011
1.0035
0.9883
1.2809
0.9624
323.52
0.9116
373.54
373.76
0.9315
1.0016
0.9883
1.3009
0.9620
230.13
0.9546
373.24
373.45
0.9630
1.0004
0.9883
1.3231
0.9616
121.95
1.0000
373.15
373.15
1.0000
1.0000
0.9882
0.00Table 4
Correlation parameters for activity coefficients, average deviation for studied systems and vapor-phase composition mean absolute deviation from the literatures.
Equation
Parameters or deviations
Methanol(1) + water (2)
Methanol(1) + ethanoic acid (2)
Water(1) + ethanoic acid (2)
(
g
12−g11)/J·mol−1
−436.50
−85.50
−352.42
(g21−g22)/J·mol−1
1159.55
0.98
715.43
NRTLa
α
12
0.241
0.30
0.23
dT/K
0.28
0.51
0.33
dy
0.0072
0.0054
0.0096
Wilsona
Λ
12/J·mol−1
389.37
50.46
644.71
Λ
21/J·mol−1
368.45
95.58
−87.59
dT/K
0.43
0.47
0.64
dy
0.0036
0.0059
0.0053
Margulesb
A
12
0.95
0.12
0.29
A
21
0.65
−0.46
0.57
dT/K
0.37
0.32
0.56
dy
0.0049 (0.0057)c
0.0075 (0.0234)c
0.0086 (0.0123)c
van Laarb
A
12
0.86
0.24
0.32
A
21
0.56
0.03
0.55
dT/K
0.27
0.58
0.47
dy
0.0069
0.0078
0.0098
aWilson’s interaction parameters (J·mol−1), NRTL’s interaction parameters (J·mol−1). bMargules and van Laar interaction parameters (dimensionless). dT=∑|Texp-Tcal|/N; N: number of data points; Tcal: calculated bubble point from model, K; Texp: experimental boiling point temperature, K. dy=∑|ycal-ymod|/N; N: number of data points; ycal: calculated vapor-phase mole fraction fromTpx; ymol: calculated vapor-phase mole fraction from model. cThe values in parentheses from the literatures [38–41].Figure 3
T
-
x
1-y1 diagram for methanol (1) + water (2) at 101.325 kPa: □, vapor-phase mole fraction y1 from Tpx and HmE; ■, vapor-phase mole fraction y1 from the literature [32, 33]; °, liquid-phase experimental temperature; ●, liquid-phase mole fraction x1 from the literature [32, 33]; —, NRTL correlation temperature.Figure 4
T-x1-y1 diagram for methanol (1) + ethanoic acid (2) at 101.325 kPa: □, vapor-phase mole fraction y1 from Tpx and HmE; ■, vapor-phase mole fraction y1 from literature [10]; –-, dot line, vapor-phase mole fraction y1 with Block diagram for the calculation y by T, p, x, and HmE neglected from our previous work [15]; °, liquid-phase experimental temperature; ●, liquid-phase experimental temperature from literature [10]; -·-·-, dash dot, liquid-phase experimental temperature from our previous work [15]; —, NRTL correlation temperature.Figure 5
T-x1-y1 diagram for water (1) + ethanoic acid (2) at 101.325 kPa: □, vapor-phase mole fraction y1 from Tpx and HmE; ■, vapor-phase mole fraction y1 from literature [11]; ●, liquid-phase experimental temperature from the literature; °, liquid-phase experimental temperature [11]; —, NRTL correlation temperature.Figure 6
Excess Gibbs energy functions (GE/RT) versus liquid-phase mole fraction of component 1 (x1) diagram: °, methanol (1) + water (2); ●, water (1) + ethanoic acid (2); □, methanol (1) + ethanoic acid (2).


### 4.3. Correlation and Prediction of VLE of Ternary System

The binary interaction parameters of the NRTL, Wilson, Margules, and van Laar model given in Table4 were used to correlate and predict the VLE data of the ternary system. VLE data for methanol (1) + water (2) + ethanoic acid (3) at 101.325 kPa included liquid-phase mole fraction x1, x2, and x3, experimental boiling point temperature Texp, calculated bubble point temperature Tcal, calculated vapor-phase mole fraction y1, y2, and y3, activity coefficients γ1, γ2, and γ3, the average deviation in the bubble temperatures of the ternary system using NRTL equation correlation listed in Table 5. The absolute average and maximum deviation between the boiling point from experimental data and the bubble point from NRTL model calculation are 0.77°C and 1.94°C, respectively. The average and maximum deviations using Wilson, Margules, and van Laar equation individually are 0.79°C, 1.90°C, 0.85°C, 1.89°C, and 0.96°C, 1.91°C. Diagram of VLE for the ternary system methanol (1) + water (2) + ethanoic acid (3) at 101.325 kPa is shown in Figure 7.Table 5
VLE data for the methanol(1) + water (2) + ethanoic acid (3) Ternary system at 101.325 kPa: liquid-phase mole fraction x1, x2, and x3, experimental boiling point temperature Texp, calculated bubble point temperature Tcal, vapor-phase mole fraction y1, y2, and y3, activity coefficients γ1, γ2, and γ3 using NRTL equation correlation.
x
1
x
2
x
3
T
exp/K
T
cal/K
y
1
y
2
y
3
γ
1
γ
2
γ
3
0.7919
0.2080
0.0001
341.08
340.94
0.9108
0.0870
0.0022
1.0152
1.5155
0.9163
0.7502
0.1967
0.0531
342.79
342.12
0.9024
0.0882
0.0094
1.0156
1.5068
0.9294
0.6923
0.1816
0.1261
344.60
343.90
0.8886
0.0872
0.0242
1.0147
1.4953
0.9444
0.6623
0.1841
0.1536
345.45
344.77
0.8783
0.0910
0.0307
1.0152
1.4838
0.9493
0.5917
0.1645
0.2438
348.36
347.27
0.8561
0.0895
0.0544
1.0114
1.4719
0.9632
0.4619
0.4554
0.0827
349.12
348.30
0.7566
0.2242
0.0191
1.1037
1.2749
0.9612
0.4122
0.3196
0.2682
350.38
352.17
0.7339
0.1929
0.0732
1.0461
1.3352
0.9808
0.5333
0.1483
0.3184
350.67
349.55
0.8338
0.0881
0.0781
1.0072
1.4623
0.9722
0.4263
0.4203
0.1534
351.53
350.19
0.7359
0.2254
0.0387
1.0874
1.2853
0.9763
0.4946
0.1375
0.3679
351.93
351.19
0.8167
0.0869
0.0964
1.0040
1.4558
0.9772
0.3166
0.4774
0.2060
352.84
354.29
0.6481
0.2893
0.0627
1.1178
1.2310
1.0116
0.3959
0.3903
0.2138
353.19
351.89
0.7162
0.2258
0.0580
1.0733
1.2939
0.9847
0.2537
0.5812
0.1651
354.90
355.93
0.5880
0.3565
0.0555
1.1972
1.1675
1.0536
0.2148
0.6454
0.1398
356.16
357.08
0.5479
0.4013
0.0508
1.2669
1.1308
1.0932
0.1750
0.4020
0.4230
363.36
363.66
0.4673
0.3513
0.1814
1.0664
1.2364
1.0246
0.0585
0.9055
0.0360
364.18
364.83
0.2965
0.6797
0.0238
1.9547
1.0150
1.5148
0.1563
0.3589
0.4848
365.42
365.67
0.4367
0.3420
0.2213
1.0459
1.2516
1.0186
0.0478
0.9228
0.0294
366.09
365.86
0.2627
0.7164
0.0209
2.0514
1.0104
1.5759
0.1413
0.3246
0.5341
366.99
367.35
0.4109
0.3321
0.2570
1.0318
1.2630
1.0145
0.0404
0.9348
0.0248
367.10
366.66
0.2359
0.7454
0.0187
2.1259
1.0077
1.6231
0.0350
0.9435
0.0215
367.41
367.30
0.2142
0.7689
0.0169
2.1843
1.0059
1.6602
0.1290
0.2963
0.5747
368.56
368.80
0.3888
0.3218
0.2895
1.0214
1.2720
1.0116
0.0241
0.9592
0.0167
368.82
368.78
0.1626
0.8230
0.0144
2.2986
1.0032
1.7357
0.0203
0.9657
0.0140
369.63
369.35
0.1425
0.8449
0.0126
2.3510
1.0023
1.7687
0.0175
0.9704
0.0121
370.14
369.79
0.1266
0.8622
0.0112
2.3906
1.0017
1.7938
0.0154
0.9740
0.0106
370.54
370.13
0.1141
0.8759
0.0100
2.4220
1.0013
1.8136
0.0884
0.2313
0.6803
371.87
373.49
0.3017
0.3004
0.3979
1.0004
1.2864
1.0068
0.0779
0.2038
0.7183
373.44
375.08
0.2771
0.2814
0.4415
0.9933
1.2933
1.0050
0.0690
0.1805
0.7505
375.20
376.50
0.2548
0.2629
0.4822
0.9878
1.2987
1.0037
0.0625
0.1635
0.7740
375.65
377.59
0.2376
0.2480
0.5144
0.9841
1.3022
1.0030
0.0413
0.1802
0.7785
379.37
379.33
0.1656
0.2871
0.5473
0.9859
1.2885
1.0044
0.0359
0.1567
0.8074
379.87
380.59
0.1486
0.2617
0.5897
0.9812
1.2942
1.0032
0.0314
0.1369
0.8317
381.03
381.69
0.1338
0.2380
0.6282
0.9775
1.2986
1.0023
0.0280
0.1222
0.8498
381.64
382.55
0.1220
0.2192
0.6588
0.9749
1.3016
1.0018
Deviations
d
T
=
0.77Figure 7
Diagram of VLE for the ternary system methanol(1) + water (2) + ethanoic acid (3) at 101.325 kPa: °, liquid-phase mole fraction; ●, vapor-phase mole fraction.


## 4.1. Calculation of Vapor-Phase Mole Fractiony for the Binary Systems from Tpx and HmE

The vapor-phase mole fractionyi of the components was calculated from
(14)yiΦip=xiγipis,
and the fugacity of coefficients  Φi of the components was obtained by the expression
(15)Φi=φ^iVφisexp[-ViL(p-pis)RT],
where φ^iVis the fugacity coefficient of component i in the vapor mixture, φis is the fugacity coefficient of component i at saturation, ViL is the molar volume of component i in the liquid phase, R is the universal gas constant, T is the experimental temperature, p is the total pressure (101.325 kPa), and pis is the vapor pressure of pure component i. The Antoine equations were applied to calculate the values of these vapor pressures. The Antoine constants Ai, Bi, Ci, and the values of Tc, pc, Vc, Zc and ω  were obtained from Shi et al. [36], as shown in Table 2. The Poynting correction factor was also included in the calculation of fugacity at the reference state. The liquid molar volumes were evaluated from the modified Rackett equation [37].Table 2
Antoine coefficients of the compounds and published parameters [36] used to calculate fugacity coefficients: critical temperature Tc, critical pressure pc, critical volume Vc, critical compression Zc, and acentric factor ω of pure compounds.
Compound
A
i
B
i
C
i
T
c/K
p
c/MPa
V
c/m3·kmol−1
Z
c
ω
Methanol
7.19736
1574.99
−34.29
512.6
8.096
0.118
0.224
0.559
Water
7.07404
1657.46
−46.13
647.3
22.048
0.056
0.229
0.344
Ethanoic acid
6.42452
1479.02
−56.34
594.4
5.786
0.171
0.200
0.454According to the Gibbs-Duhem equation, any extensive molar thermodynamic property of a given phase, such as the Gibbs and Helmholtz energies, the enthalpy, and molar volume, must satisfy the following differential relation:(16)(∂M∂T)p,xj≠idT+(∂M∂p)T,xj≠idp-∑i=1CxidM-i=0.In (16) M is a general molar property, xi the molar fraction of component i in the phase under consideration, xj≠i the pertinent set of compositions, C the number of components, and M-i the partial contribution of component i to M.When considering VLE, the molar excess Gibbs energy,GE, can be evaluated from measurable (T, p, x) data using activity coefficient relations. Replacing M by GE/RT in the following equation yields the well-known relation
(17)-HmERT2dT+VmERTdp-∑i=1Cxidlnγi=0,
where HmE and VmE are the molar excess enthalpy and volume of mixing of the liquid phase. According to the thermodynamic principles, the activity coefficients γi of the components were calculated from the expression as follows:
(18)Q=GERT=∑i=1Cxilnγi,
where Q is the function of molar excess Gibbs energy and combining (17) and (18) yields
(19)d(GERT)-∑i=1Clnγidxi=∑i=1Cxidlnγi=-HmERT2dT+VmERTdp.Application to a binary system gives(20)d(GERT)-lnγ1γ2dx1=-HmERT2dT+VmERTdp.Simultaneous solution of (18) and (20) yields
(21)lnγ1=GERT+x2[ddx1(GERT)+HmERT2dTdx1-VmERTdpdx1],lnγ2=GERT-x1[ddx1(GERT)+HmERT2dTdx1-VmERTdpdx1].Equation (18) substituted into (21) reduced it to
(22)γ1=exp[Q+(1-x1)(dQdx1+HmERT2dTdx1-VmERTdpdx1)],γ2=exp[Q-x1(dQdx1+HmERT2dTdx1-VmERTdpdx1)].For a binary system comprised of species 1 and 2 at VLE state, at constant pressure,(dp/dx1)=0, substitution into (22) reduced it to
(23)γi=exp[Q+(1-xi)[(dQdxi)+HmERT2(dTdxi)]],(i=1,2).The right-hand side,(HmE/RT2)(dT/dxi), of (23) cannot be neglected. Proper use of (23) requires the availability of heats of mixing as a function of composition and temperature. The activity coefficients γi of the components as functions of the excess Gibbs energy are as follows:
(24)γ1=exp{Q+(1-x1)[(dQdx1)+HmERT2(dTdx1)]},γ2=exp{Q-x1[(dQdx1)+HmERT2(dTdx1)]}.From (14), this equation is rearranged to obtain
(25)yi=xiγipisΦip.Because∑yi=1, for the binary system, the equation may be summed to give
(26)x1γ1φ1sp1sexp[V1L(p-p1s)/RT]φ^1Vp+x2γ2φ2sp2sexp[V2L(p-p2s)/RT]φ^2Vp=1.In (26), solved for yi by difference method. Suppose that [0,1] is subdivided into n subintervals [xk,xk+1] of equal step size h=1/n by using xk=kh for k=0,1,…,n. In k difference point, we obtain
(27)Fk=1-{x1γ1φ1sp1sexp[V1L(p-p1s)/RT]φ^1Vp}k-{x2γ2φ2sp2sexp[V2L(p-p2s)/RT]φ^2Vp}k=0.Meanwhile, (23) may be shown as follows:
(28)γi|k=exp{Qk+(1-xi|k)[Q|k+1-Q|k-12/h+HmERT2(dTdx1)|k]}.And (27) is linearized to obtain:
(29)-Fk=ΔQ|k-1(δFkδQ|k-1)+ΔQ|k(δFkδQ|k)+ΔQ|k+1(δFkδQ|k+1).The number ofn linear equation from (28) is solved for ΔQ|k by chasing method
(30)Qkj+1=Qkj+tΔQkj,
where t is relaxation factor, finally, yik (k=1~n) is obtained by difference method. The block diagram for calculation procedure was detailedly shown in Figure 2.Figure 2
Block diagram for the calculationy by T, p, x, and HmE.


## 4.2. Calculation of Vapor-Phase Mole Fractiony for the Binary Systems from Model

There are many methods concerning the correlation and prediction of VLE data. The model-free approach data treatment of vapor-liquid equilibrium is also one of the best strategies for the correlation and prediction of VLE data. Wisniak’s group has developed that the novel model-free computation techniques and limiting conditions have been applied to VLE data for azeotropic systems [42]. Moreover, Segura and coworkers reported that a model-free approach dealt with VLE data in application of ternary systems [38, 43]. For the three binary systems of this study, the activity coefficients were correlated with the NRTL [39], Wilson [40], Margules [41], and van Laar [44] equations, respectively. The interaction parameters optimized were achieved by the objective function (OF) minimized using the least-squares fitting as follows:
(31)OF=∑i=1N(xi,cal-xi,exp)2,
where N is the number of experimental points xi,cal and xi,exp are the liquid-phase mole fraction of component i calculated and experimental values from the (12) or (13) and from measured data, respectively.Because carboxylic acids are always present in an associated form, like a dimer or trimer, in both the vapor and liquid phases even at low pressures, a significant deviation in fugacity coefficient may exist using the ideal gas assumption. To illustrate the deviation from ideal behavior, Marek’s chemical theory and HOC model were applied to deal with the associating component and modify the deviation from idealities of vapor phase [26–28], respectively. The Poynting correction factor was also included in the calculation of fugacity at the reference state. The liquid molar volumes were evaluated from the modified Rackett equation [37]. However, under isobaric conditions, the most volatile component cannot exist in the liquid state, only as superheated vapor. Hence, there is no way to calculate or measure this property for the molar volumes of the pure liquids. Therefore, the correct procedure for isobaric measurements is to calculate the overall values of Aij and Aij, as adjustable parameters, and not the values of the interaction excess energy. Hence, for Wilson model, Aij and Aij were reported as adjustable parameters.The activity coefficients computed on the basis of NRTL model were used to evaluated dimensionless excess Gibbs function at 101.325 kPa for three binary systems over the overall range of composition. Liquid-phase mole fractionx1, experimental boiling point temperature  Texp, calculated bubble point temperature Tcal, vapor-phase mole fraction y1 from the model, activity coefficients γ1 and γ2 using NRTL equation correlation, fugacity coefficients φ^1V and φ^2V, and the molar excess enthalpies of mixing of the liquid phase HmE are included in Table 3. The T-x1-y1 diagrams for the methanol (1) + water (2), methanol (1) + ethanoic acid (2), and water (1) + ethanoic acid (2) three binary systems at 101.325 kPa are shown in Figures 3, 4, and 5, respectively. The plot of excess Gibbs energy function GE/RT versus liquid-phase mole fraction x1 for the three binary systems is shown in Figure 6. All the mixtures exhibit deviations from ideality with a range that may be attributed to interactions leading to the formation of various associated aggregates. Observed nonideal behavior is indicative of the magnitude of experimental activity coefficients γi, as well as of the variation of excess Gibbs energy function, GE/RT, with composition, as depicted in Figure 6. The obtained absolute maximum values of GE/RT for the methanol (1) + water (2), methanol (1) + ethanoic acid (2), and water (1) + ethanoic acid (2) three binary systems are 0.1703, 0.0180, and 0.0892, respectively. The values of excess Gibbs energy function GE/RT are positive for methanol (1) + water (2) and water (1) + ethanoic acid (2) binary systems. However, for methanol (1) + ethanoic acid (2) system, the values of those are negative in the overall range of mole fraction. GE/RT values follow the order methanol (1) + water (2) > water (1) + ethanoic acid (2) > methanol (1) + ethanoic acid (2). The absolute maximum value of GE/RT is approximate at an equimolar fraction in three binary systems. Figures 3–5 show that the comparison of the predicted vapor-phase and experimental liquid-phase compositions with those of the literature [10, 11, 32, 33]. Comparing with the values of vapor-phase and liquid-phase component from the literatures, the values of those from the paper are in good agreement with the literature, as shown in Figures 3–5. The results have demonstrated that the methods for Tpx and HmE are appropriate for representing the experimental data of the three binary systems. The optimum model interaction parameter of liquid activity coefficient and the absolute average deviations for the different models, and from Tpx and HmE are listed in Table 4. Herein, we obtained the results by the four different types of correlations for the prediction of activity coefficients in these systems, which reveal that the deviations of NRTL, Wilson, Margules, and van Laar equations are reasonably small in Table 4. For comparison, the mean deviations obtained by Gmehling and Onken [32, 33, 45, 46] are also shown in the Table 4. It can seem that two sets of deviation values are comparable. Since the superiority of one method over the others is not always obvious, practice must rely on experience and analogy. The comprehensive comparisons of four of the methods (NRTL, Wilson, van Laar, and Margules) were made in Table 4. From the data analysis, the temperature deviations between the experimental and calculated values of four different types of model are very similar in the three binary systems, and the vapor-phase mole fraction deviations between calculated values from Tpx and HmE, and from the model are very similar. Therefore, the activity coefficient models are appropriate for representing the experimental data of the three binary systems. In Table 4, the absolute average deviations dT of the difference between boiling point temperature from experiment and bubbling point temperature from calculation by NRLT model parameters for the three binary systems are 0.28°C, 0.51°C, and 0.33°C, respectively. And the absolute average deviations dy of the difference between vapor-phase mole fraction from Tpx and HmE calculation and from NRTL model calculation are 0.0072, 0.0054, and 0.0096, respectively.Table 3
VLE data for the methanol(1) + water (2), methanol (1) + ethanoic acid (2), and water (1) + ethanoic acid (2). Three binary systems at 101.325 kPa: liquid-phase mole fraction x1, experimental boiling point temperature Texp, calculated bubble point temperature Tcal, vapor-phase mole fraction y1, activity coefficients γ1 and γ2 using NRTL equation correlation, fugacity coefficients φ^1V and φ^2V, and molar excess enthalpies of mixing of the liquid phase HmE.
x
1
T
exp/K
T
cal/K
y
1
γ
1
φ
^
1
V
γ
2
φ
^
2
V
H
m
E/J·mol−1
Methanol(1) + water (2)
0.0000
373.15
373.15
0.0000
1.0000
0.9882
0.00
0.0484
366.00
365.54
0.2704
2.1436
0.9852
1.0034
0.9875
225.74
0.0539
365.05
364.88
0.3009
2.1147
0.9849
1.0042
0.9875
248.74
0.0607
364.40
364.12
0.3213
2.0799
0.9846
1.0053
0.9874
276.88
0.0696
363.50
363.19
0.3492
2.0358
0.9843
1.0069
0.9874
312.53
0.0806
362.25
362.13
0.3866
1.9835
0.9839
1.0092
0.9873
354.36
0.0911
361.35
361.21
0.4126
1.9360
0.9836
1.0117
0.9872
392.94
0.1056
360.05
360.05
0.4492
1.8739
0.9832
1.0155
0.9871
443.59
0.1197
359.05
359.04
0.4762
1.8172
0.9829
1.0197
0.9870
489.29
0.1354
358.60
358.02
0.4885
1.7584
0.9826
1.0248
0.9869
538.32
0.1436
358.00
357.53
0.5047
1.7293
0.9824
1.0277
0.9869
562.20
0.1586
356.95
356.70
0.5323
1.6788
0.9822
1.0334
0.9868
602.83
0.1815
355.55
355.56
0.5677
1.6083
0.9819
1.0428
0.9867
659.52
0.2092
354.35
354.35
0.5969
1.5325
0.9815
1.0555
0.9865
719.43
0.2202
354.10
353.91
0.6031
1.5049
0.9814
1.0609
0.9865
741.25
0.2380
353.25
353.24
0.6237
1.4633
0.9812
1.0701
0.9864
772.62
0.2648
352.05
352.32
0.6519
1.4066
0.9809
1.0847
0.9862
812.77
0.2918
351.35
351.48
0.6681
1.3561
0.9807
1.1005
0.9861
846.82
0.3166
350.30
350.76
0.6923
1.3148
0.9805
1.1157
0.9860
870.15
0.3442
349.05
350.02
0.7197
1.2740
0.9803
1.1336
0.9859
889.06
0.4015
347.95
348.61
0.7431
1.2038
0.9800
1.1730
0.9856
909.80
0.4282
347.90
348.00
0.7443
1.1767
0.9798
1.1924
0.9854
911.28
0.4644
347.20
347.21
0.7613
1.1449
0.9796
1.2194
0.9853
903.04
0.5240
346.20
345.98
0.7860
1.1026
0.9793
1.2656
0.9850
869.55
0.6091
344.80
344.33
0.8231
1.0595
0.9789
1.3338
0.9845
783.83
0.6899
343.31
342.85
0.8633
1.0325
0.9785
1.3995
0.9840
666.92
0.8213
341.20
340.56
0.9221
1.0086
0.9779
1.5038
0.9832
419.31
1.0000
338.10
337.66
1.0000
1.0000
0.9771
0.00
Methanol(1) + ethanol acid (2)
0.0000
390.15
391.04
0.0000
1.0000
0.9703
0.00
0.0359
387.32
386.23
0.1574
0.9388
0.9765
0.9999
0.9685
77.28
0.0410
386.21
385.56
0.1664
0.9393
0.9768
0.9999
0.9682
86.70
0.0648
383.89
382.68
0.2719
0.9419
0.9779
0.9997
0.9666
134.05
0.0741
382.73
381.90
0.3133
0.9428
0.9782
0.9996
0.9660
149.55
0.0859
381.20
380.60
0.3333
0.9441
0.9787
0.9995
0.9652
171.14
0.1040
380.03
379.06
0.3830
0.9460
0.9791
0.9992
0.9639
205.37
0.1280
378.46
377.35
0.4602
0.9485
0.9797
0.9988
0.9622
245.43
0.1629
375.62
374.70
0.5213
0.9521
0.9802
0.9981
0.9595
299.80
0.1919
373.21
372.43
0.5873
0.9551
0.9805
0.9973
0.9572
341.32
0.2510
369.31
368.44
0.6495
0.9609
0.9808
0.9954
0.9528
411.44
0.3391
364.35
363.56
0.7538
0.9692
0.9809
0.9916
0.9461
490.73
0.4401
358.54
357.71
0.8232
0.9776
0.9805
0.9857
0.9388
536.56
0.5829
351.77
350.75
0.8999
0.9874
0.9797
0.9748
0.9292
526.43
0.7050
346.47
346.39
0.9389
0.9936
0.9790
0.9633
0.9217
449.37
0.7509
344.90
344.06
0.9533
0.9955
0.9787
0.9583
0.9189
401.45
0.8109
342.81
342.35
0.9693
0.9974
0.9783
0.9515
0.9155
330.61
0.8711
341.60
340.75
0.9766
0.9988
0.9779
0.9442
0.9122
241.41
0.9161
339.65
340.00
0.9859
0.9995
0.9776
0.9384
0.9097
163.78
1.0000
338.15
337.66
1.0000
1.0000
0.9771
0.00
Water(1) + ethanoic acid (2)
0.0000
390.15
391.04
0.0000
1.0000
0.9703
0.00
0.0665
387.22
387.67
0.1140
1.4161
0.9899
1.0025
0.9693
245.31
0.0749
387.00
387.31
0.1288
1.4046
0.9898
1.0031
0.9692
273.17
0.0855
386.48
386.88
0.1603
1.3903
0.9897
1.0040
0.9691
307.50
0.0995
385.86
386.34
0.1938
1.3722
0.9896
1.0054
0.9689
350.32
0.1507
384.16
384.60
0.2802
1.3117
0.9892
1.0121
0.9683
489.63
0.1722
383.44
383.96
0.3155
1.2888
0.9891
1.0156
0.9681
540.35
0.1926
382.99
383.39
0.3375
1.2684
0.9890
1.0193
0.9679
585.57
0.2168
382.39
382.77
0.3676
1.2458
0.9890
1.0242
0.9677
632.34
0.2502
381.53
381.98
0.4101
1.2171
0.9889
1.0316
0.9674
689.92
0.2936
380.94
381.06
0.4407
1.1838
0.9887
1.0425
0.9670
750.61
0.3303
380.20
380.36
0.4815
1.1589
0.9887
1.0528
0.9667
788.98
0.3756
379.40
379.57
0.5253
1.1318
0.9886
1.0666
0.9663
821.39
0.4465
378.27
378.48
0.5890
1.0965
0.9885
1.0908
0.9658
841.59
0.4946
377.61
377.82
0.6274
1.0768
0.9885
1.1087
0.9654
836.35
0.5615
376.80
376.99
0.6777
1.0543
0.9884
1.1355
0.9649
802.60
0.6036
376.24
376.51
0.7137
1.0427
0.9884
1.1533
0.9645
767.63
0.6517
375.59
376.00
0.7546
1.0315
0.9884
1.1746
0.9642
714.78
0.7067
375.11
375.46
0.7849
1.0213
0.9884
1.1998
0.9637
639.01
0.7045
375.11
375.48
0.7848
1.0217
0.9884
1.1988
0.9638
642.76
0.7507
374.71
375.05
0.8173
1.0148
0.9883
1.2207
0.9634
568.24
0.8008
374.31
374.62
0.8501
1.0091
0.9883
1.2451
0.9630
475.20
0.8723
373.80
374.05
0.9011
1.0035
0.9883
1.2809
0.9624
323.52
0.9116
373.54
373.76
0.9315
1.0016
0.9883
1.3009
0.9620
230.13
0.9546
373.24
373.45
0.9630
1.0004
0.9883
1.3231
0.9616
121.95
1.0000
373.15
373.15
1.0000
1.0000
0.9882
0.00Table 4
Correlation parameters for activity coefficients, average deviation for studied systems and vapor-phase composition mean absolute deviation from the literatures.
Equation
Parameters or deviations
Methanol(1) + water (2)
Methanol(1) + ethanoic acid (2)
Water(1) + ethanoic acid (2)
(
g
12−g11)/J·mol−1
−436.50
−85.50
−352.42
(g21−g22)/J·mol−1
1159.55
0.98
715.43
NRTLa
α
12
0.241
0.30
0.23
dT/K
0.28
0.51
0.33
dy
0.0072
0.0054
0.0096
Wilsona
Λ
12/J·mol−1
389.37
50.46
644.71
Λ
21/J·mol−1
368.45
95.58
−87.59
dT/K
0.43
0.47
0.64
dy
0.0036
0.0059
0.0053
Margulesb
A
12
0.95
0.12
0.29
A
21
0.65
−0.46
0.57
dT/K
0.37
0.32
0.56
dy
0.0049 (0.0057)c
0.0075 (0.0234)c
0.0086 (0.0123)c
van Laarb
A
12
0.86
0.24
0.32
A
21
0.56
0.03
0.55
dT/K
0.27
0.58
0.47
dy
0.0069
0.0078
0.0098
aWilson’s interaction parameters (J·mol−1), NRTL’s interaction parameters (J·mol−1). bMargules and van Laar interaction parameters (dimensionless). dT=∑|Texp-Tcal|/N; N: number of data points; Tcal: calculated bubble point from model, K; Texp: experimental boiling point temperature, K. dy=∑|ycal-ymod|/N; N: number of data points; ycal: calculated vapor-phase mole fraction fromTpx; ymol: calculated vapor-phase mole fraction from model. cThe values in parentheses from the literatures [38–41].Figure 3
T
-
x
1-y1 diagram for methanol (1) + water (2) at 101.325 kPa: □, vapor-phase mole fraction y1 from Tpx and HmE; ■, vapor-phase mole fraction y1 from the literature [32, 33]; °, liquid-phase experimental temperature; ●, liquid-phase mole fraction x1 from the literature [32, 33]; —, NRTL correlation temperature.Figure 4
T-x1-y1 diagram for methanol (1) + ethanoic acid (2) at 101.325 kPa: □, vapor-phase mole fraction y1 from Tpx and HmE; ■, vapor-phase mole fraction y1 from literature [10]; –-, dot line, vapor-phase mole fraction y1 with Block diagram for the calculation y by T, p, x, and HmE neglected from our previous work [15]; °, liquid-phase experimental temperature; ●, liquid-phase experimental temperature from literature [10]; -·-·-, dash dot, liquid-phase experimental temperature from our previous work [15]; —, NRTL correlation temperature.Figure 5
T-x1-y1 diagram for water (1) + ethanoic acid (2) at 101.325 kPa: □, vapor-phase mole fraction y1 from Tpx and HmE; ■, vapor-phase mole fraction y1 from literature [11]; ●, liquid-phase experimental temperature from the literature; °, liquid-phase experimental temperature [11]; —, NRTL correlation temperature.Figure 6
Excess Gibbs energy functions (GE/RT) versus liquid-phase mole fraction of component 1 (x1) diagram: °, methanol (1) + water (2); ●, water (1) + ethanoic acid (2); □, methanol (1) + ethanoic acid (2).


## 4.3. Correlation and Prediction of VLE of Ternary System

The binary interaction parameters of the NRTL, Wilson, Margules, and van Laar model given in Table4 were used to correlate and predict the VLE data of the ternary system. VLE data for methanol (1) + water (2) + ethanoic acid (3) at 101.325 kPa included liquid-phase mole fraction x1, x2, and x3, experimental boiling point temperature Texp, calculated bubble point temperature Tcal, calculated vapor-phase mole fraction y1, y2, and y3, activity coefficients γ1, γ2, and γ3, the average deviation in the bubble temperatures of the ternary system using NRTL equation correlation listed in Table 5. The absolute average and maximum deviation between the boiling point from experimental data and the bubble point from NRTL model calculation are 0.77°C and 1.94°C, respectively. The average and maximum deviations using Wilson, Margules, and van Laar equation individually are 0.79°C, 1.90°C, 0.85°C, 1.89°C, and 0.96°C, 1.91°C. Diagram of VLE for the ternary system methanol (1) + water (2) + ethanoic acid (3) at 101.325 kPa is shown in Figure 7.Table 5
VLE data for the methanol(1) + water (2) + ethanoic acid (3) Ternary system at 101.325 kPa: liquid-phase mole fraction x1, x2, and x3, experimental boiling point temperature Texp, calculated bubble point temperature Tcal, vapor-phase mole fraction y1, y2, and y3, activity coefficients γ1, γ2, and γ3 using NRTL equation correlation.
x
1
x
2
x
3
T
exp/K
T
cal/K
y
1
y
2
y
3
γ
1
γ
2
γ
3
0.7919
0.2080
0.0001
341.08
340.94
0.9108
0.0870
0.0022
1.0152
1.5155
0.9163
0.7502
0.1967
0.0531
342.79
342.12
0.9024
0.0882
0.0094
1.0156
1.5068
0.9294
0.6923
0.1816
0.1261
344.60
343.90
0.8886
0.0872
0.0242
1.0147
1.4953
0.9444
0.6623
0.1841
0.1536
345.45
344.77
0.8783
0.0910
0.0307
1.0152
1.4838
0.9493
0.5917
0.1645
0.2438
348.36
347.27
0.8561
0.0895
0.0544
1.0114
1.4719
0.9632
0.4619
0.4554
0.0827
349.12
348.30
0.7566
0.2242
0.0191
1.1037
1.2749
0.9612
0.4122
0.3196
0.2682
350.38
352.17
0.7339
0.1929
0.0732
1.0461
1.3352
0.9808
0.5333
0.1483
0.3184
350.67
349.55
0.8338
0.0881
0.0781
1.0072
1.4623
0.9722
0.4263
0.4203
0.1534
351.53
350.19
0.7359
0.2254
0.0387
1.0874
1.2853
0.9763
0.4946
0.1375
0.3679
351.93
351.19
0.8167
0.0869
0.0964
1.0040
1.4558
0.9772
0.3166
0.4774
0.2060
352.84
354.29
0.6481
0.2893
0.0627
1.1178
1.2310
1.0116
0.3959
0.3903
0.2138
353.19
351.89
0.7162
0.2258
0.0580
1.0733
1.2939
0.9847
0.2537
0.5812
0.1651
354.90
355.93
0.5880
0.3565
0.0555
1.1972
1.1675
1.0536
0.2148
0.6454
0.1398
356.16
357.08
0.5479
0.4013
0.0508
1.2669
1.1308
1.0932
0.1750
0.4020
0.4230
363.36
363.66
0.4673
0.3513
0.1814
1.0664
1.2364
1.0246
0.0585
0.9055
0.0360
364.18
364.83
0.2965
0.6797
0.0238
1.9547
1.0150
1.5148
0.1563
0.3589
0.4848
365.42
365.67
0.4367
0.3420
0.2213
1.0459
1.2516
1.0186
0.0478
0.9228
0.0294
366.09
365.86
0.2627
0.7164
0.0209
2.0514
1.0104
1.5759
0.1413
0.3246
0.5341
366.99
367.35
0.4109
0.3321
0.2570
1.0318
1.2630
1.0145
0.0404
0.9348
0.0248
367.10
366.66
0.2359
0.7454
0.0187
2.1259
1.0077
1.6231
0.0350
0.9435
0.0215
367.41
367.30
0.2142
0.7689
0.0169
2.1843
1.0059
1.6602
0.1290
0.2963
0.5747
368.56
368.80
0.3888
0.3218
0.2895
1.0214
1.2720
1.0116
0.0241
0.9592
0.0167
368.82
368.78
0.1626
0.8230
0.0144
2.2986
1.0032
1.7357
0.0203
0.9657
0.0140
369.63
369.35
0.1425
0.8449
0.0126
2.3510
1.0023
1.7687
0.0175
0.9704
0.0121
370.14
369.79
0.1266
0.8622
0.0112
2.3906
1.0017
1.7938
0.0154
0.9740
0.0106
370.54
370.13
0.1141
0.8759
0.0100
2.4220
1.0013
1.8136
0.0884
0.2313
0.6803
371.87
373.49
0.3017
0.3004
0.3979
1.0004
1.2864
1.0068
0.0779
0.2038
0.7183
373.44
375.08
0.2771
0.2814
0.4415
0.9933
1.2933
1.0050
0.0690
0.1805
0.7505
375.20
376.50
0.2548
0.2629
0.4822
0.9878
1.2987
1.0037
0.0625
0.1635
0.7740
375.65
377.59
0.2376
0.2480
0.5144
0.9841
1.3022
1.0030
0.0413
0.1802
0.7785
379.37
379.33
0.1656
0.2871
0.5473
0.9859
1.2885
1.0044
0.0359
0.1567
0.8074
379.87
380.59
0.1486
0.2617
0.5897
0.9812
1.2942
1.0032
0.0314
0.1369
0.8317
381.03
381.69
0.1338
0.2380
0.6282
0.9775
1.2986
1.0023
0.0280
0.1222
0.8498
381.64
382.55
0.1220
0.2192
0.6588
0.9749
1.3016
1.0018
Deviations
d
T
=
0.77Figure 7
Diagram of VLE for the ternary system methanol(1) + water (2) + ethanoic acid (3) at 101.325 kPa: °, liquid-phase mole fraction; ●, vapor-phase mole fraction.


## 5. Conclusions

VLE data for the ternary system methanol + water + ethanoic acid and three constituent binary systems at 101.325 kPa: methanol + water, methanol + ethanoic acid, and water + ethanoic acid were determined at different liquid-phase compositions using a novel pump ebulliometer. The equilibrium composition of the vapor phase was calculated fromT, p, x, and HmE based on the Q function of excess Gibbs free energy by the indirect method. The experimental data were correlated using the NRTL, Wilson, Margules, and van Laar equations. It was shown that the deviations of NRTL, Wilson, Margules, and van Laar equations are reasonably small. The VLE data of ternary system were predicted by NRTL, Wilson, Margules, and van Laar equation. The calculated bubble points accorded well with the experimental data. The results show that the calculated bubble point is fitted by the models, which satisfy the need for the design and operation of separation process in chemical industry. Moreover, the method will provide theoretical guidance for the research of VLE data of strongly associating system of vapor and liquid phase in nonideal behavior.



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*Source: 641251-2012-10-10.xml*