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Elements and Chemical Bonds Contribution Estimation of Activity Coefficients in Nonideal Liquid Mixtures

Institute of Process System Engineering, College of Chemical Engineering, Qingdao University of Science and Technology, Qingdao 266042, China
College of Chemistry and Chemical Engineering, Qilu Normal University, Jinan 250200, China
Authors to whom correspondence should be addressed.
Processes 2022, 10(10), 2141;
Submission received: 21 September 2022 / Revised: 11 October 2022 / Accepted: 12 October 2022 / Published: 20 October 2022
(This article belongs to the Special Issue Research on Process System Engineering)


Based on the contribution of elements and chemical bonds, the UNICAC (Universal Quasi-Chemical elements and chemical bonds Activity Coefficient) method was proposed to estimate the activity coefficients of nonelectrolyte liquid mixtures. The UNICAC method defined 10 elements and 33 chemical bonds as contribution groups. The calculation of activity coefficients was divided into the combination term and the residual term. The combination term represents molecular size differences, and the residual term describes the interaction between molecules. The interaction energy parameters of 43 groups were regressed simultaneously with the experimental data of the vapor–liquid equilibrium of 1085 binary systems. According to the molecular structural information of compounds, the UNICAC method can accurately predict the activity coefficients of nonelectrolyte liquid mixtures. The vapor–liquid equilibrium of 16 groups of the binary system, which were not included in the parameters regress, was predicted using UNICAC. The average relative error of vapor composition was 1.53%. Compared with UNIFAC (2003), UNIFAC (Lyngby), UNIFAC (Dortmund), and ASOG (2011), the UNICAC model employs fewer parameters, provides a broader scope of application, and receives more precise predicted results of the vapor–liquid equilibrium. The UNICAC method would play an important reference role in the design of the chemical separation process.

1. Introduction

Products, process, and separation engineering design need complex fluids vapor–liquid equilibrium (VLE) data [1]. It can be determined with an experimental method. However, the experimental methods may be restricted by the experimental conditions, and they will consume a lot of human and material resources [2]. The vapor–liquid equilibrium data can also be predicted using phase equilibrium mathematical models based on the group contribution method and thermodynamic property prediction software based on molecular descriptors. Nowadays, the use of molecular descriptors to predict row phase equilibria is a new trend in the era of big data, and they have greatly improved in computational efficiency and convenience. Commercial software such as the COSMO-SAC model is available. As thermodynamic property prediction software also relies primarily on reliable mathematical models, developing and improving the underlying models are essential for thermodynamic property prediction. In terms of the group contribution method, the fugacity coefficient of vapor phases was usually calculated with an equation of state, and the activity coefficient of liquid phases was calculated with activity coefficient models [3].
The earliest activity coefficient models such as the Wohl, Margules and van Laar equations use ‘average’ or ‘overall’ compositions, and their theoretical basis is ‘random mixing’. However, due to intermolecular forces, the mixing of molecules is never entirely random. The models could be improved, considering the non-randomness, leading to better descriptions of phase behavior. Since the Wilson equation was proposed in 1964, local composition (LC) activity coefficient models have drastically changed the applicability range of liquid phase models. There exist several models that employ the LC concept, such as Wilson, NRTL, and UNIQUAC [4]. Although those models can give good predictions of liquid-phase activity coefficients, they need experimental VLE data to obtain model parameters [5]. However, the required experimental data are not always available.
One of the most successful methods presently used for calculating activity coefficients of the liquid phase is the group contribution (GC) method, in which the liquid phase is considered a mixture of structural groups [6]. The first GC method for the calculation of activity coefficients was ASOG (Analytical Solution of Groups), based on the Wilson equation, but an FH (Flory–Huggins model) combinatorial term was added. The ASOG model was proposed by Deal and Derr in 1969, and then it was modified by Tochigi (1990) [7] and Gmehling (2011) [8]. Meanwhile, the group interaction parameters were supplemented, and the number of groups was up to 47. Because of the absence of necessary groups or interaction parameters, the VLEs of some systems cannot be predicted, which restricts the application of the ASOG model. To improve the situation, Robles et al. (2016) [9] added 14 group-interaction parameters for the ASOG model, such as CH2/Imide and CH2/PF6.
Another well-known GC activity coefficient model is UNIFAC (Universal quasi-chemical Functional group Activity Coefficient). The UNIFAC model was proposed by Fredenslund et al. in 1975 [10]. The group-interaction parameters table of the UNIFAC model was revised and supplemented by Fredenslund et al. (1977) [11], Jørgensen et al. (1979) [12], Gmehling et al. (1982) [13], Macedo et al. (1983) [14], Tiegs et al. (1987) [15], Hansen et al. (1991) [16], and Wittig et al. (2003) [17].
In addition to revising and complementing the parameters, the combination and residual terms were improved using the UNIFAC (Lyngby) and UNIFAC (Dortmund) modified models. The modified UNIFAC (Lyngby) model divides 21 groups [18]. Although the modified model has a high estimation accuracy, the prediction range is relatively narrow. The UNIFAC (Dortmund) modified model is widely used in chemical design and production. With the expansion of the application range, the number of main groups and the required group-interaction parameters increase. The UNIFAC (Dortmund) parameter matrix was revised five times by Gmehling et al. (1993, 1998, 2002, 2006, and 2016) [19,20,21,22,23].
The UNIFAC method has become very popular because of its availability on commercial process simulators, its reliably predicted phase equilibrium results, and its simple usage. The UNIFAC method has several well-known limitations, for example, the problem of missing groups or group-interaction parameters [24]. According to the newly released group-interaction parameter matrix in 2016, the number of main groups has increased from 64 to 103. It can be seen that a total of 5356 parameters are needed, 1828 are obtained, and 65.87% of the parameters are still missing [23]. Because of missing major groups or interaction parameters, the vapor–liquid equilibrium data of some systems cannot be predicted. This limits the further application of UNIFAC (Dortmund). For example, Klauck et al. (2019) [25] determined the group parameters of ammonia and the interaction parameters between groups. Qin et al. (2019) [26] obtained six binary interaction parameters between C2H2F, CF4, CF3, and CH3 using the regression of experimental data. The VLE data of a methanol (1) + water (2) mixture from 298.15 K to 373.15 K was also regressed using the UNIFAC model by Nayak et al. (2022) [27].
In order to improve the problem of many groups and the lack of parameters in the traditional group contribution method, Xia et al. proposed the group division method based on elements and chemical bonds. Using the experimental data in the database manual of thermodynamics and the physical properties of compounds edited by Carl L. Yaws, a method based on the contribution of 10 elements and 33 chemical bonds was proposed to estimate the critical properties by Li (2016), which has high prediction accuracy [28]. The method based on the elements and chemical bonds contribution is successfully applied to estimate the properties of pure substances.
If fewer basic contribution groups are established in the GC activity coefficient model, the possibility of the group-interaction parameters missing will be less. Based on the above ideas, the UNICAC model (Universal Quasi-Chemical elements and chemical bonds Activity Coefficient) for estimating activity coefficients based on elements and chemical bonds will be proposed. The new model uses vapor–liquid phase equilibrium data (1085 data sets and 14,323 data points) for binary systems containing 14 types of compounds and regresses the interaction energy parameters between 10 elements and 33 chemical bonds using the sum of log errors squared of the experimental and calculated values of the activity coefficients as the objective function and a fitted Newtonian method to find their minimum values. Compared to the previous functional group contribution method, the UNICAC model has a relatively more complete set of necessary groups to split the compound, and a smaller scale of group interaction parameters. It also improves the prediction accuracy of aromatic hydrocarbon compounds by considering the benzene ring. The UNICAC model was compared with UNIFAC [10,11,12,13,14,15,16,17], UNIFAC (Dortmund) [19,20,21,22,23], UNIFAC (Lyngby) [18], and ASOG (2011) [7,8,9] to verify their accuracy and applicability range.

2. Method and Property Modeling

2.1. Data

The fundamental idea of the UNICAC model is to utilize the existing phase equilibrium data for predicting the phase equilibrium of systems for which no experimental data are available. The quality of the method strongly depends on the experimental data. The sources of vapor–liquid equilibrium data in this work were the literature [29]. This included alkanes, alkenes, alkynes, cycloalkanes, alcohols, ketones, esters, acids, halogenated hydrocarbons, nitrogenous compounds, etc. The data checked for thermodynamic consistency were used in this work.

2.2. Groups

The UNICAC model used elements and chemical bonds as the contribution group. The elements were divided into carbon (C), hydrogen (H), nitrogen (N), oxygen (O), silicon (Si), sulfur (S), fluorine (F), chlorine (Cl), bromine (Br), and iodine (I). The chemical bonds were divided into two categories, cyclic and non-cyclic. We divided chemical bonds based on each chemical bond at both ends that connect different atoms to different chemical bonds, such as C-C, C-H, C-O, C-S, and O-H. The structural group H2O describes water. According to this principle, the elements and chemical bond groups are seen in Table 1.

2.3. Models

The UNICAC model contains a combinatorial part, essentially due to differences in the sizes and shapes of the molecules in the mixture, and a residual part, essentially due to energy interactions. The combinatorial part of the UNIQUAC activity coefficients and the residual part of the ASOG activity coefficients are used separately. The sizes and interaction surface areas of elements and chemical bonds are introduced from independently obtained pure-component molecular structure data. The resulting UNICAC model contains two adjustable parameters per pair of elements and chemical bonds.
In a multi-component mixture, the UNICAC equation for the activity coefficient of the (molecular) component i is
I n γ i = I n γ i C + I n γ i R
ln γ i C = ln ϕ i x i + z 2 q i ln θ i ϕ i + l i ϕ i x i j = 1 n c x j l j
ln r i R = k ν k ( i ) [ ln Γ k ln Γ k ( i ) ]
ln Γ k = ln l X l a l k + 1 l X l a k l m X m a l m
X l = i x i ν l i i x i k ν k i
where Γ k is the group residual activity coefficient, and Γ k ( i ) is the residual activity coefficient of group k in a reference solution containing only molecules of type i. In Equation (3), the term Γ k ( i ) is necessary to attain the normalization that the activity coefficient r i becomes unity as x i 1 . The activity coefficient for group k in molecule i depends on the molecule i in which k is situated.
l i = ( z 2 ) ( r i q i ) ( r i 1 ) ,   z = 10
θ i = q i x i j q j x j
ϕ i = r i x i j r j x j
In these equations, x i is the molar fraction of component i, and the summations in Equations (2) and (3) are overall components, including component i. θ i is the area fraction, and ϕ i is the segment fraction, which is similar to the volume fraction. The pure component parameters r i and q i are, respectively, measures of the molecular van der Waals volumes and molecular surface areas.
In the UNICAC method, the combinatorial part of the UNIQUAC activity coefficients in Equation (2) are used directly. Only pure component properties enter into this equation. Parameters r i and q i are calculated as the sum of the group volume and area parameters, R k , and Q k , which are given in Table 1:
r i = k ν k ( i ) R k
q i = k ν k ( i ) Q k
In Equations (4) and (5), ν k ( i ) , which is always an integer, is the number of groups of type k in molecule i. Group parameters R k and Q k are obtained from the van der Waals group volume and surface areas V k and A k , given by Bondi [30]:
R k = V k / 15.17
Q k = A k / ( 2.5 × 10 9 )
The normalization factors 15.17 and 2.5 × 109 are those given by Abrams and Prausnitz (1975) [10].

2.4. Group Interaction Parameters

It was necessary to first calculate the activity coefficients from the database to obtain the interaction parameters between the elements and chemical bonds. Only low-pressure phase equilibrium data were used. Vapor-phase nonidealities were not taken into account. One thousand eighty-five binary data sets (14,323 data points) representing vapor–liquid equilibrium were used as the base data in this work. The parameters were fitted simultaneously using the base data, which were checked for thermodynamic consistency.
The objective function, which is minimized by the parameter estimation program in the software MATLAB, is a sum of the squared deviations between the experimental and calculated activity coefficients. The deviations are normalized relative to a preset standard deviation (10−6), and the resulting expression for the objective function is then
F = k = 1 S j = 1 N i = 1 C ( I n γ i , j , k exp I n γ i , j , k c a l ) 2
In Equation (13), γ are the activity coefficients, k is summed over the number of systems, i is summed over the number of the constituent, and j is summed over the total number of data points.
Usually, the interaction parameters of elements and chemical bonds were 100 K used in calculating the initial parameters. The optimization algorithm applied in the parameter estimation program is the quasi-Newton method.
The parameters for 43 group combinations are in Table A1 (the UNICAC parameter table). In most cases, the parameters were estimated as indicated above, without difficulty. However, less than adequate data are available, and for numerous group interactions, no or very little data could be found. At the same time, it was our goal to calculate the group interaction parameters for all possible binary combinations of groups, as shown in Table A1. At present, it is impossible to reach this goal entirely because of a serious lack of reliable experimental data. The UNICAC method needed 946 parameters but only had 535 parameters, missing 43.4% of the group-interaction parameters. Figure 1 presents the interaction parameters matrix of UNICAC.

3. Results and Discussion

With the parameters in Table A1, it is now possible to predict the activity coefficients and vapor–liquid equilibriums (VLEs) using the UNICAC model and Equation (14), for a large variety of binary and multicomponent systems.
p y i ϕ i V = x i γ i p i S φ i S exp p i S p V i L R T d p y i = 1 ; x i = 1
In Equation (14), for the binary VLE data, T is the temperature, P is the pressure, xi is the liquid mole fraction of component i, and yi is the vapor mole fraction of component i. ϕ i V is the fugacity coefficient of component i, p i S is the saturated vapor pressure of the component i, ϕ i S is the fugacity coefficient of the component i at p i S , V i L is the liquid molar volume at T, and the Poynting factor is 1.
Vapor–liquid phase equilibrium data are usually the vapor–liquid phase equilibrium data of the system measured via experimentation under isobaric or isothermal conditions, namely pressure, temperature, liquid phase composition, and vapor phase composition (T, P, x, y), which can be divided into isobaric and isothermal vapor–liquid phase equilibrium data according to different measurement conditions.
For the isothermal vapor–liquid phase equilibrium data of the system measured experimentally under isothermal conditions, since the measurement temperature is unchanged, the calculation type of the vapor–liquid phase equilibrium is as follows: Given the temperature T and the liquid phase composition X, find the bubble point pressure P and vapor phase composition y, and the calculation expressions are shown in Equations (15) and (16).
p c a l = i x i γ i p i S y i
y i c a l = x i γ i p i S y i p c a l
For the isobaric vapor–liquid phase equilibrium data of the system experimentally measured under isobaric conditions, the calculation type of the vapor–liquid phase equilibrium is as follows: given the pressure P and the liquid phase composition x, find the bubble point temperature T and the vapor phase composition y. Since, γ i and p i S are functions of temperature, the results cannot be obtained directly, and so the method of trial and error is used to solve this. The calculation block diagram of bubble point temperature T and vapor phase composition y is shown in Figure 2.
The temperature under the experimental conditions was taken as the initial temperature, and the convergence accuracy was set to 10−6. Equations (17) and (18) are used for iterative calculation, and T is constantly adjusted to make the iterative calculation meet the convergence criteria.
T k + 1 = T k f ( T k ) f ( T k )
f ( T ) = y i 1
The ultimate test of the UNICAC method lies in its ability to predict activity coefficients for systems that were not included in the database, that is, the data set used to determine the parameters in Table A1. Therefore, we distinguish between the calculated results for systems contained in the database and predicted results for systems not contained in the database. The discussion of predictions is divided into three parts:
  • Binary systems included in the database;
  • Binary systems not included in the database;
  • Ternary systems.

3.1. Binary Systems Included in the Database

The predicted and experimental vapor mole fraction, and the temperature or the pressure for binary VLEs included in the database used in the proposed method were compared.
The results of the UNIFAC method, UNIFAC (Dortmund), UNIFAC (Lyngby), ASOG method, and the new method are shown in Table 2 and Table 3, respectively. The systems can be divided into different categories to illustrate the use of the new estimation method. ARDP stands for the relative deviation for pressure to experimental values, ARDT stands for the relative deviation for temperature to experimental values, and ARDy stands for the relative deviation for vapor-phase composition to experimental values.
The result shows a total of 1085 binary systems participating in the UNICAC model’s bivariate interaction parameter regression. The vapor–liquid equilibria of some systems cannot be predicted by ASOG (2011), UNIFAC (2003), UNIFAC (Dortmund), and UNIFAC (Lyngby), due to the lack of group interaction parameters or compounds that cannot be resolved by the corresponding groups.
As can be seen from Table 3, UNIFAC (Dortmund), UNIFAC (Lyngby), and ASOG (2011) could not predict systems containing silicon compounds, such as dichlorosilane + trichlorosilane, etc. Only UNIFAC can predict a part of silica-containing organics, because it contains SiH2 and SiO groups. Still, UNIFAC cannot resolve all silica-containing organics such as trichlorosilane and silicon tetrachloride.
In addition, UNIFAC (Lyngby) could not predict some systems containing fluorine and chloromethane, such as difluoromethane + propanol, difluoromethane + methanol, difluoromethane + ethanol, and other systems. Moreover, the model cannot predict the system containing sulfur, bromine, and iodine.
As seen in Table 2 and Table 3, because of the small parameter matrices of modified UNIFAC (Lyngby), this model allows for calculations only for approximately 76% of those data that UNICAC can predict.
Furthermore, because many new groups have been introduced into the parameter matrix, UNICAC can predict the activity coefficients for systems that contain various mono- and dialkylated amides, carbonates, anhydrides, sulfones, epoxides, silicon, refrigerants, etc. Table 3 shows the improvement in the range of applicability for UNICAC. In all cases, the scope of UNICAC’s applicability is larger than that of modified UNIFAC (Lyngby).
The UNICAC model has an ARDy of 5.18%, which is better than the ASOG model, and slightly worse than UNIFAC and its modified model. Meanwhile, the ARDP/ARDT for the UNICAC model is 2.69%, which is a better forecast. It is close to the predictive power of UNIFAC (Dortmund), and UNIFAC (Lyngby). When predicting the vapor–liquid phase equilibriums of binary systems, the new model can predict both polar and non-polar systems, and is particularly suitable for predicting aromatic systems, silicon containing systems, sulfur systems, bromine containing systems, and iodine containing systems. However, the UNIFAC and UNIFAC (Dortmund) models have a much better prediction capability for the watery, aldehyde containing system, ketone containing system, ester containing system, and chloride containing system.

3.2. Binary Systems Not Included in the Database

In this section, the UNICAC model was assessed using binary mixtures not included in the database. One of the main advantages of the UNICAC method is its ability to predict activity coefficients for systems of this type from experimental information on normal systems, that is, those with only a few different functional groups. The results are shown in Table 4, and Table 5 illustrates this ability. Table 4 shows the calculated mean deviations in mole fraction, temperature, and pressure for various binary systems. The parameters (see Table 1 and Table A1) allow for accurate predictions of systems containing alcohols such as methanol, ethanol, or 2-methyl-1-propanol. As shown in Table 5, the modified UNIFAC (Dortmund), modified UNIFAC (Lyngby), and ASOG cannot describe the system containing silicide; the usage of UNICAC leads to much better results. Table 6 shows the compounds that cannot be described in the binary systems not included in the database.
Because of the small parameter matrices of modified UNIFAC (Lyngby), Figure 3, Figure 4, Figure 5 and Figure 6 are mainly limited to relatively simple compounds (alkanes, alkenes, alcohols, arenes, ketones, etc.) The complete VLE data sets (x, y, T, and P are given) can be described with all selected group contribution methods and which have passed consistency tests. Besides the predicted results using the five group contribution methods: UNICAC, UNIFAC modified UNIFAC (Dortmund), modified UNIFAC (Lyngby), and ASOG, a comparison of the predicted results from the five methods with the experimental values was made. Figure 3 for the ethanol + 2-methyl-1-propanol system indicates that the UNICAC method predicts vapor compositions well. The calculated and experimental vapor compositions are in good agreement for similar systems. Figure 4 shows good predictions for the systems toluene + 3-methyl-1-butanol; these favorable results suggest that the UNICAC method applies to systems containing toluene, and perhaps, other arenas. As Figure 5 shows, the results of the five methods for hexane+ 1-pentanol are the same. Figure 6 shows typical examples of toluene and 4-methyl-2-pentanone. The UNICAC leads to much more reliable results than UNIFAC and ASOG.

3.3. Ternary Systems

The UNICAC method is directly applicable to multicomponent systems. Table 7 gives the vapor mole fraction, and the temperature or the pressure for binary VLEs calculated from the UNICAC method for ternary systems. No ternary systems were included in the database. There is an excellent agreement between the calculated and observed vapor–liquid equilibriums for these systems.

4. Conclusions

A generalized model for the prediction of the activity coefficients of nonelectrolytes is proposed. Phase equilibrium data were used to generate many elements/chemical bonds-interaction parameters. These are useful for predicting activity coefficients in binary and multicomponent systems where little or no experimental information exists. The method gives good predictions for a large variety of systems, and should therefore provide a valuable tool for solving practical phase equilibrium problems as encountered in the chemical process design.

Author Contributions

H.L. and X.L.: Formal analysis, Validation, Visualization, Writing—original draft; X.S.: Resources, Supervision, Validation, Project administration; Y.W. and W.Z.: Formal analysis, Investigation; L.X. and S.X.: Conceptualization, Methodology, Software, Writing—review and editing. All authors have read and agreed to the published version of the manuscript.


This work was supported by the National Natural Science Foundation of China (No. 22178190, 22008129).

Data Availability Statement

The data presented in this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare no conflict of interest.


nThe number of data sets for VLEs
NThe number of systems can be predicted using the functional groups of the selected method
qPure component area parameter
QGroup area parameter
rPure component volume parameter
RGas constant (without subscript)
RkGroup volume parameter (with subscript)
aklUNICAC binary interaction parameter
xLiquid-phase mole fraction
XLiquid-phase group fraction
yVapor-phase mole fraction
zLattice coordination number, a constant here set, equal to ten
γ i Activity coefficient of component i
Γ k The group residual activity coefficient
Γ k ( i ) The residual activity coefficient of group k in a reference solution containing only molecules of type i
x i The mole fraction of component i
θ i The area fraction
ϕ i The segment fraction
i, j, kComponents i, j, and k
ΔPThe mean absolute deviation for pressure to experimental values
ΔTThe mean absolute deviation for temperature to experimental values
ΔyThe mean absolute deviation for vapor-phase composition to experimental values
ARDPThe mean relative deviation for pressure to experimental values
ARDTThe mean relative deviation for temperature to experimental values
ARDyThe mean relative deviation for vapor-phase composition to experimental values

Appendix A

Table A1. The interaction parameters of elements and chemical bonds/K.
Table A1. The interaction parameters of elements and chemical bonds/K.
benzene ring−1449.9631199.247321.987518.333456.047100.000−504.803−20.526132.280
benzene ring100.000200.904−543.78941.236157.386−292.539965.465−185.123−850.751
benzene ring−504.800−286.526132.279100.000100.000132.231279.280194.465100.000
benzene ring100.000100.000100.000−130.231100.000490.002−836.341100.000214.480
kBenzene Ringc–C–Cc–C=Cc–C–Oc–C–Sc–C–Nc–C=N
benzene ring0.000625.329583.208642.880100.00071.630−19.934


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Figure 1. The interaction parameters matrix of elements and chemical bonds.
Figure 1. The interaction parameters matrix of elements and chemical bonds.
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Figure 2. The diagram of calculation of bubble point temperature T and vapor composition y.
Figure 2. The diagram of calculation of bubble point temperature T and vapor composition y.
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Figure 3. Comparison of the predicted results from five methods with the experimental values of ethanol(1) + 2-methyl-1-propanol(2) at 1 atm.
Figure 3. Comparison of the predicted results from five methods with the experimental values of ethanol(1) + 2-methyl-1-propanol(2) at 1 atm.
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Figure 4. Comparison of the predicted results from five methods with the experimental values of toluene (1) + 3-methyl-1-butanol (2) at 1 atm.
Figure 4. Comparison of the predicted results from five methods with the experimental values of toluene (1) + 3-methyl-1-butanol (2) at 1 atm.
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Figure 5. Comparison of the predicted results from five methods with the experimental values of hexane (1) + 1-pentanol (2) at 323.15 K.
Figure 5. Comparison of the predicted results from five methods with the experimental values of hexane (1) + 1-pentanol (2) at 323.15 K.
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Figure 6. Comparison of the predicted results from five methods with the experimental values of toluene (1) + 4-methyl-2-pentanone (2) at 323.15 K.
Figure 6. Comparison of the predicted results from five methods with the experimental values of toluene (1) + 4-methyl-2-pentanone (2) at 323.15 K.
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Table 1. The volume and surface of elements and chemical bonds.
Table 1. The volume and surface of elements and chemical bonds.
Chemical BondsNo. R k /(cm3·mol−1) Q k /(109 cm2·mol−1) Group Assignment Examples
H2O10.921.4water: 1(1)
C212.392.19butane: 4(2),10(3),3(12),10(18)
H34.361.09ethyl alcohol: 2(2),6(3),1(4),1(12),1(15),1(17), 5(18)
O48.511.70methyl alcohol: 1(2),4(3),1(4), 1(15),1(17), 3(18)
N59.391.82propylamine: 3(2),9(3),1(5),1(15),1(17)
S614.702.45n-butyl mercaptan: 4(2),10(3),1(6),3(12),10(18),1(23),1(29)
F78.011.63perfluorobenzene: 6(2),6(7),6(19),1(37)
Cl813.512.32chloroform: 1(2),1(3),3(8),1(18),3(20)
Br915.962.59bromoethane: 2(2),5(3),1(9),1(12),5(18),1(21)
I1019.572.97ethyl iodide: 2(2),5(3),1(10),1(12),5(18),1(22)
Si1123.353.34silicontetrachloride: 4(8),1(11),4(25)
C-C12−3.86−1.02isopentane: 5(2),12(3),4(12),12(18)
C=C13−5.46−1.24isoprene: 5(2),8(3),2(12),2(13),8(18)
C≡C14−4.84−1.34vinylacetylene: 4(2),4(3),1(12),1(13),1(14),4(18)
C-O15−4.54−1.05propanoicacid: 3(2),6(3),2(4),2(12),1(15),1(16),1(17),5(18)
C=O16−4.96−1.16allyl acetate: 5(2),8(3),2(4),2(12),1(13),2(15),1(16),8(18)
O-H17−2.92−0.80l-hexanol: 6(2),14(3),1(4),5(12),1(15),1(17),13(18)
C-H18−3.35−0.98benzene: 6(2),6(3),6(18),1(37)
C-F19−4.48−0.97dichlorodifluoromethane: 1(2),2(7),2(8),2(19),2(20)
C-Cl20−3.56−0.991,1-dichloroethane: 1(2),2(7),2(8),2(19),2(20)
C-Br21−3.72−0.90bromoform: 1(2),1(3),3(9),1(18),3(21)
C-I22−2.17−1.01methyl iodide: 1(2),3(3),1(10),3(18),1(22)
C-S23−4.45−1.02propyl mercaptan: 3(2),8(3),1(6),2(12),7(18),1(23),1(29)
C-Si24−5.47−1.11tetra-ethylsilane: 8(2),20(3),1(11),20(18),4(24)
Si-Cl25−4.07−1.19dichlorophenylsilane: 6(2),6(3),2(8),1(11),5(18),1(24),2(25),1(26),1(27)
Si-H26−3.70−0.99trichlorosilane: 1(3),3(8),1(11),3(25),1(26)
Si-O27−2.81−1.28hexamethyldisiloxane: 6(2),18(3),1(4),2(11),18(18),6(24),2(27), 1(26),1(27)
C=S28−0.342−0.464carbon disulfide: 1(2),2(6),2(28)
S-H29−2.05−0.69ethyl mercaptan: 2(2),6(3),1(6),1(12),6(18),1(23),1(29)
S=O30−3.19−1.04dimethylsulfoxide: 2(2),6(3),1(4),1(6),6(18),2(23),1(30)
C-N31−3.05−0.87diethylamine: 4(2),11(3),1(5),2(12),10(18),2(31),1(34)
C=N32−5.29−1.35methyl isocyanate: 2(2),3(3),1(4),1(5),1(16),3(18),1(31),1(32)
C≡N33−5.22−1.28methacrylonitrile: 4(2),5(3),1(5),2(12), 1(13),5(18),1(33)
N-H34−3.32−0.77aniline: 6(2),7(3),1(5),5(18),1(31),2(34),1(37)
N-O35−3.16−0.82methylethylketoxim: 4(2),9(3),1(4),1(5),3(12),1(17),8(18),1(32), 1(35)
NO236−9.07−2.46nitromethane: 1(2),3(3),2(4),1(5),3(18),1(31),1(36)
benzene37−29.93−7.58toluene: 7(2),8(3),1(12),8(18),1(37)
c-C-C38−4.47−1.12cyclohexane: 6(2),12(3),12(18),6(38)
c-C=C39−5.53−1.31furan: 4(2),4(3),1(4),4(18),1(38),2(39),2(40)
c-C-O40−4.54−1.031,4-dioxane: 4(2),8(3),2(4),8(18),2(38),4(40)
c-C-S41−4.02−0.99thiophene: 4(2),4(3),1(6),4(18),1(38),2(39),2(41)
c-C-N42−2.87−0.95morpholine: 4(2),9(3),1(4),1(5),8(18),1(34),2(38),2(40), 2(42)
c-C=N43−5.89−0.79pyridine: 5(2),5(3),1(5),5(18),2(38),2(39),1(42),1(43)
Table 2. Comparison between predicted VLEs of experimental data for 1085 binary systems included in the database, and other group contribution methods.
Table 2. Comparison between predicted VLEs of experimental data for 1085 binary systems included in the database, and other group contribution methods.
UNIFAC (Dortmund)10484.943.75
UNIFAC (Lyngby)8305.714.22
Table 3. Comparison of the predicted vapor mole fraction of experimental data for different categories included in the database.
Table 3. Comparison of the predicted vapor mole fraction of experimental data for different categories included in the database.
watery system744.3510.051.756.715.206.141.949.298.4212.51
aromatic system2911.833.962.904.652.744.672.697.384.095.30
silicon containing system122.822.391.660.68n.a.n.a.n.a.n.a.n.a.n.a.
sulfur system181.041.935.483.265.082.22n.a.n.a.1.372.57
acid system783.228.9813.9416.3311.7714.4323.4711.3810.7928.68
ester containing system1262.614.993.314.822.854.203.853.822.925.36
aldehyde containing system212.945.650.464.681.424.330.1112.084.068.45
ketone containing system1442.384.791.633.251.444.034.953.301.913.53
fluorine containing system272.624.052.576.200.468.53n.a.n.a.4.686.28
chloride containing system2213.025.843.915.
bromine containing system92.311.6513.212.723.131.87n.a.n.a.22.302.92
iodine containing system82.092.7111.321.4311.231.30n.a.n.a.1.022.06
nitrogen containing system1194.424.707.446.474.965.043.045.4310.9815.27
alcohol containing system3992.074.192.864.623.064.742.644.652.774.42
n.a. (Not available): The compounds in the mixture cannot be described using the functional groups of the selected method.
Table 4. Deviations of predicted values from observed vapor–liquid equilibriums for binary systems not included in the database using the UNICAC method.
Table 4. Deviations of predicted values from observed vapor–liquid equilibriums for binary systems not included in the database using the UNICAC method.
SystemSystem NumberP or TnΔyΔT or ΔP
Component 1 + Component 2
ethanol + 2-methyl-1-propanol1101.33 kPa140.00520.72 K
toluene + 3-methyl-1-butanol2101.33 kPa150.01960.58 K
acetone + methanol3328.15 K110.01321.36 kPa
1-hexene + 2-butanol4333.15 K120.01603.59 kPa
hexane + 2-butanol5333.15 K110.01302.25 kPa
hexane + 1-pentanol6323.15 K130.00790.42 kPa
benzene + 2-methyl-1-propanol7101.33 kPa310.01490.19 K
toluene + 1-pentanol8383.15 K230.01261.92 kPa
toluene + 4-methyl-2-pentanone9323.15 K260.00650.32 kPa
acetone + benzene10318.15 K110.00910.72 kPa
methyl formate + dimethyl carbonate11101.33 kPa140.00360.32 K
methanol + epoxy chloropropane12101.33 kPa100.01030.87 K
trichlorosilane + silicon tetrachloride1398.70 kPa110.00860.50 K
Diethyl disulfide + 2,2,4-trimethylpentane14368.15 K80.03897.59 kPa
1,1-difluoroethane+ methanol15101.33 kPa40.381525.16 K
water + propanoic acid16343.2 K90.309023.98 kPa
Table 5. Deviations of predicted values from observed vapor–liquid equilibriums for binary systems not included in the database using different methods.
Table 5. Deviations of predicted values from observed vapor–liquid equilibriums for binary systems not included in the database using different methods.
Table 6. The compounds cannot be described in the binary systems not included in the database.
Table 6. The compounds cannot be described in the binary systems not included in the database.
11methyl formate××
11diethyl carbonate××
13silicon tetrachloride××××
14ethyl disulfide×××
16propanoic acid×
√: The compounds can be described using the functional groups of the selected method; ×: the compounds cannot be described.
Table 7. Deviations of predicted values from observed vapor–liquid equilibriums for ternary systems using elements and chemical bonds contribution model.
Table 7. Deviations of predicted values from observed vapor–liquid equilibriums for ternary systems using elements and chemical bonds contribution model.
SystemP or TnΔy1Δy2ΔT or Δp
Component 1 + Component 2 + Component 3
acetone + methanol + ethanol101.33 kPa830.02280.01231.09 K
methanol + ethanol + 1-propanol101.33 kPa450.02240.01220.75 K
acetone + methanol + 2-propanol328.15 K270.02060.01632.02 kPa
2,3-dimethyl-butane + methanol + acetone101.33 kPa270.02470.02362.36 K
methanol + 2-methylbutane+ isoprene101.33 kPa130.04890.05340.69 K
methanol + heptane + toluene101.33 kPa80.00650.00490.21 K
acetone + ethanol + hexane328.15 K210.02710.014112.20 kPa
hexane + ethanol + benzene328.15 K430.02060.02534.81 kPa
ethanol + benzene + heptane53.33 kPa500.02320.01761.08 K
ethanol + benzene + heptane101.33 kPa470.02640.01990.99 K
benzene + heptane + 1-propanol348.15 K770.02730.01531.83 kPa
acetone + 2-methyl-butane + isoprene101.33 kPa150.02510.02052.22 K
2-butanone + 3-pentanone + 4-methyl-2-pentanone101.33 kPa640.02140.00850.54.K
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Liu, H.; Li, X.; Wang, Y.; Sun, X.; Zhao, W.; Xia, L.; Xiang, S. Elements and Chemical Bonds Contribution Estimation of Activity Coefficients in Nonideal Liquid Mixtures. Processes 2022, 10, 2141.

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Liu H, Li X, Wang Y, Sun X, Zhao W, Xia L, Xiang S. Elements and Chemical Bonds Contribution Estimation of Activity Coefficients in Nonideal Liquid Mixtures. Processes. 2022; 10(10):2141.

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Liu, Haodong, Xinyu Li, Yuxin Wang, Xiaoyan Sun, Wenying Zhao, Li Xia, and Shuguang Xiang. 2022. "Elements and Chemical Bonds Contribution Estimation of Activity Coefficients in Nonideal Liquid Mixtures" Processes 10, no. 10: 2141.

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