Phase equilibria for the binary mixtures containing hydrogen fluoride and non-polar compound

Ž .Fluid Phase Equilibria 154 1999 139–151

Phase equilibria for the binary mixtures containing hydrogen fluorideand non-polar compound

Yun Whan Kang )

DiÕision of EnÕironmental and CFC Technology, KIST, P.O. Box 131, Cheongryang, Seoul, Korea

Received 2 December 1997; accepted 24 September 1998

Abstract

Isothermal vapor–liquid equilibria for propaneqhydrogen fluoride have been measured. The experimentaldata are correlated with the association model proposed by Lencka and Anderko for the mixtures containinghydrogen fluoride and the relevant parameters are presented. The recalculated parameters of the associationmodel for pure hydrogen fluoride are presented. The problems occurred in the applications of the associationmodel for the mixtures containing hydrogen fluoride are discussed. The correlation was found to be in goodagreement with the experimental data. However, the calculated equilibrium pressures at very diluted composi-tions of hydrogen fluoride below about 0.01 were shown rather higher than the experimental values. q 1999Elsevier Science B.V. All rights reserved.

Keywords: Vapor–liquid equilibria; Equation of state; Chemical association; Hydrogen fluoride; Hydrogen bonding

1. Introduction

The molecules of hydrogen fluoride are known to be linked together by strong hydrogen bondingthat considerably affects its thermodynamic properties. Many researchers have made their efforts to

w xexplain the vapor-phase nonideal behavior for hydrogen fluoride 1–6 . In general, the nonideality ofhydrogen fluoride in the liquid phase is much stronger than that in the vapor phase. Several modelsapplicable to both phases have been developed to describe the vapor–liquid equilibria for the mixtures

w xcontaining hydrogen fluoride 7–12 . However, it was not sufficient to verify the validity of theirmodels because the very limited experimental data have been reported in the literatures. Even thougha number of systems concerning the mixtures containing hydrogen fluoride have been reportedw x13–16 , none of the systems were able to ignore the interaction between the organic compound andhydrogen fluoride due to the polarities of their compounds. Therefore, there are some difficulties to

) Ž . Ž .Tel.: 82 2-958-5877; fax: 82 2-958-5809; e-mail: [email protected]

0378-3812r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0378-3812 98 00442-7

( )Y.W. KangrFluid Phase Equilibria 154 1999 139–151140

verify the developed models that can be applied to the mixture containing hydrogen fluoride and oneor more non-associating components.

In this study, the isothermal vapor–liquid equilibria for the binary system propaneqhydrogenfluoride were measured at 20.08C and 30.08C and correlated with the association model of Lencka and

w x w xAnderko 9 . In order to find out the physical meanings for the parameters of the association model 9w xcombined with the Peng–Robinson 17 equation of state, the recalculated parameters for pure

hydrogen fluoride are presented. The problems occurred in the applications of the association modelfor the mixtures containing hydrogen fluoride are also discussed.

2. Experimental

2.1. Chemicals

Ž . Ž .Propane Matheson Gas Products and anhydrous hydrogen fluoride Ulsan Chemical were ofguaranteed reagent grade and were used without any further purification. The purity of propane wasmore than 99.0%.

2.2. Apparatus and procedure

w xA static equilibrium apparatus, which is described by Kang and Lee 18 , was used to determine theisothermal vapor–liquid equilibria. The equilibrium cell, which was manufactured with 316 stainlesssteel pipe of 65 mm diameter, was placed in an isothermal water bath controlled with an accuracy of

Ž ."0.18C using a Haake circulator Model F3-K . The liquid phase was mixed with a magnetic stirrer.The volume of the equilibrium cell was 545 cm3. The equilibrium temperature was measured with a

Ž .T-type thermocouple converter Yokogawa Electric, Model STED-210-TT)B having an accuracy ofŽ"0.18C. The equilibrium pressure was determined by a gauge pressure transmitter Yokogawa

.Electric, Model UNE43-SAS3)B and a barometer having an accuracy of "0.5 kPa and "0.05 kPa,respectively.

After the cell was evacuated to 1 Pa, a known mass of one component was introduced into the celland then a known mass of the other component was added. The mass was determined with a digitalbalance having an accuracy of "0.01 g. To change the compositions in the cell, the additional massof the second component was added. The compositions of the vapor phase were not measured becausehydrogen fluoride is toxic and very reactive with the packing materials of the gas chromatographycolumn. The compositions of the liquid phase were calculated by subtracting the mass of eachcomponent existing in the vapor phase from the total mass of each component introduced into the cell.The densities of both phases and the vapor-phase compositions were estimated by using the

w xassociation model of Lencka and Anderko 9 with the binary interaction parameters, which wereobtained by data reduction of the experimental data. The accuracy of the liquid-phase compositionswas affected by the factors such as the masses of each components introduced into the cell, theestimated densities of both phases, and the estimated vapor-phase compositions. The effect of 10%error in the estimated densities of the liquid and vapor phase gives an error of approximately 0.0001and 0.0005 in the respective mole fraction. The error in the estimated vapor-phase compositions alsogives a similar result. Therefore, it can be said that the estimated densities and vapor-phasecompositions hardly affects the calculated liquid-phase compositions. The standard deviation of theliquid-phase mole fractions was "0.001.

( )Y.W. KangrFluid Phase Equilibria 154 1999 139–151 141

3. Phase equilibria for the mixtures containing hydrogen fluoride

The equation of vapor–liquid equilibrium for any component i with the equation of state is givenby:

xVfV sx Lf L 1Ž .i i i i

where x is the mole fraction, f is the fugacity coefficient, and the superscript V and L indicate ai i

vapor phase and a liquid phase, respectively. The fugacity coefficient f in both phases wasi

calculated by the association model, which is combined the chemical theory with the Peng–Robinsonw x w xequation of state 17 , proposed by Lencka and Anderko 9 . The complexity of the phase behavior for

the mixtures containing hydrogen fluoride increases because of its unusually strong association.w xAnderko 19,20 have been shown that the compressibility factor Z can be determined from the

following equation:

ZsZ ph qZch y1 2Ž .where the chemical contribution Zch to the compressibility factor is defined as the ratio of the numberof moles of the true species in an associated mixture n to the number of moles of the apparentT

species in the absence of association n .0nTchZ s 3Ž .n0

From the material balances with respect to the definitions of n and n , the following two0 T

equations can be obtained in terms of the true mole fraction of the j-mer Z .Aj`n0

s jz 4Ž .Ý AjnT js1`

1s z 5Ž .Ý Ajjs1

The physical contribution Z ph to the compressibility factor can be expressed by a commonequation of state for the true monomeric species. For the consecutive association reactions:

A qA sA js1,2, . . . ,` 6Ž .j 1 jq1

Ž .the equilibrium constants of the reaction 6 can be expressed as the following relationship:

K s f j K 7Ž . Ž .j, jq1

Ž . Ž .where f j is the distribution function with f 1 s1 and K is the dimerization equilibrium constant.The equilibrium constants of the i-merization reactions K can be related to the dimerizationi

equilibrium constant K as follows:iy1

iy1K s f j K 8Ž . Ž .Łi ž /js1

where K s1. The i-merization equilibrium constant can be expressed in terms of the true mole1ch w xfractions of i-mer Z and the Z term 21,20 .Ai

1yichf z zZ RTA A Ai i iK s s 9Ž .i i iy1 i iž /f P z Õ zA A Ai i i


w x Ž . chAs shown by Anderko 20 with known distribution function f j , the Z for the pure associatingŽ . Ž . Ž . Ž .component A that can be obtained from the solutions of Eqs. 4 , 5 , 8 and 9 is only function of

qsRTKrÕ. The Zch term for mixtures containing one associating component A and r non-associat-Ž .ing components B ks1, . . . ,r can be obtained from the following relationship:k

rRTKxAchZ sx F q x 10Ž .ÝA Bkž /Õ ks1

Ž .where x is the analytical apparent mole fraction, R is the ideal gas constant, T is the absolutetemperature, and Õ is the molar volume. F denotes the algebraic function that depends only on the

Ž .distribution function f j . The fugacity coefficient from the definition of the classical thermodynam-ics was separated into the physical and chemical contribution terms as follows:

ln f Z s ln f phZ ph q ln f chZch 11Ž . Ž .Ž . Ž .i i i

The coefficient f ph is obtained from the equation of state representing the physical contribution Z phi

Ž ch ch.to the compressibility factor. The coefficient f Z for hydrogen fluoride is only function of qi

where qsRTKx rÕ, whereas its coefficient for the other non-associating components is 1.0.A

4. Association model for pure hydrogen fluoride

w xThe distribution function proposed by Lencka and Anderko 9 to describe the phase behavior ofhydrogen fluoride was based on the Poisson-like distribution function as follows:

k jy1

f j s 12Ž . Ž .jy1 !Ž .

where the parameter k roughly indicates the location of the maximum. For the physical contributionph w xZ , the Peng–Robinson equation of state 17 was used:

Õ a T ÕŽ .phZ s y 13Ž .

Õyb RT Õ Õqb qb ÕybŽ . Ž .where

0.45724R2T 2c

a T s a T 14Ž . Ž . Ž .Pc

0.07780RTcbs 15Ž .

Pc

with20.5a T s 1qm 1yT 16Ž . Ž .Ž .Ž .r

ms0.37464q1.54226vy0.26992v 2 17Ž .Ž .The parameters a T and b are related to the energy of intermolecular interactions and the excluded

volume, respectively. In order to complete the equation of state for pure hydrogen fluoride, theŽ .unknown four parameters, which are consisted of k and K T to the chemical contribution term and


Ž . w xa T and b to the physical contribution term, must be characterized. Lencka and Anderko 9 haveproposed an analytical function for the chemical contribution Zch to avoid the repetitive numericalcalculation. As shown in Fig. 1, the vapor–liquid coexistence curves calculated with their parametersfor pure hydrogen fluoride showed some different results not only in comparison to their reportedvalues but also in comparison between the numerical and analytical solution. Furthermore, thecorrelation for the vapor-phase compressibility factor was scarcely improved by using their proposed

Ž .form for the parameter a T . To simplify the application of the equation of state to the mixturesŽ .containing hydrogen fluoride, the original equations for the parameters a T and b were used.

Therefore, the physical contribution Z ph to the compressibility factor is characterized by the apparentcritical temperature T ), critical pressure P ), and acentric factor v ) of a hypothetical compound,c c

which has non-specific interactions identical with those in the associating component but is unable toform associates. The temperature dependence of the dimerization equilibrium constant K is given by:

yDh0 yDc0 TyT D s0 qDc0ln TrT 0Ž . Ž .p 0 pln Ks q 18Ž .

RT R

where the reference temperature T 0 is 273.15 K. The objective function OF used to obtain the1

unknown parameters is given by:2 2 2V L sZ Z Pj,calcd j ,calcd j ,calcd

OF s 1y q 1y q 1y 19Ž .Ý Ý Ý1 V L sž /ž / ž /Z Z Pj,exptl j ,exptl j ,exptlj j j

In data reduction, the vapor pressure and saturated vapor and liquid densities, which are compiledw x w xby Kao et al. 12 , and the superheated vapor densities of Fredenhagen 22 were used. To obtain the

optimum parameters for pure hydrogen fluoride, the procedure has been progressed in three steps.Ž . Ž . Ž ch ch.Firstly, for a given k value of the distribution function f j , the values of F q and f Z were

calculated by using the numerical methods for different values of q. In further calculations, their two

Ž . w x Ž .Fig. 1. Vapor–liquid coexistence curve for hydrogen fluoride: ` Franck and Spalthoff 4 ; —PP—PP— with analyticalch w x Ž . ch w xsolution Z given by Lencka and Anderko 9 ; PPPP with numerical solution Z given by Lencka and Anderko 9 ;

Ž . Ž . Ž . Ž .- - - - - with b s0.0 in Eq. 18 ; with b s11.0 in Eq. 20 .


Table 1Ž . Ž .Vapor–liquid equilibria for propane 1 qhydrogen fluoride 2 at 208C and 308C

L V Ž .Prbar x x calcd1 1

ts208C1.006 0.0000 0.00001.355 0.0007 0.09171.745 0.0014 0.16372.497 0.0029 0.27523.230 0.0044 0.35433.881 0.0059 0.41504.553 0.0075 0.46305.198 0.0092 0.50055.805 0.0108 0.53116.410 0.0126 0.55757.055 0.0148 0.58327.756 0.0174 0.60718.443 0.0204 0.62749.031 0.0234 0.64339.505 0.0275 0.65979.527 0.0343 0.67139.535 0.0481 0.67139.538 0.0703 0.67139.538 0.1037 0.67139.540 0.1494 0.67139.540 0.1905 0.67139.559 0.4485 0.67139.552 0.5185 0.67139.538 0.6036 0.67139.528 0.6858 0.67139.501 0.7641 0.67139.456 0.8325 0.67139.411 0.8803 0.67139.355 0.9122 0.67139.301 0.9346 0.68609.210 0.9515 0.72989.074 0.9689 0.78958.961 0.9784 0.83088.816 0.9859 0.87008.647 0.9913 0.90398.530 0.9963 0.94248.414 1.0000 1.0000

ts308C1.461 0.0000 0.00002.096 0.0008 0.09792.400 0.0012 0.13702.950 0.0020 0.20163.777 0.0034 0.29505.115 0.0058 0.39906.527 0.0086 0.47487.908 0.0117 0.5303


Ž .Table 1 continuedL V Ž .Prbar x x calcd1 1

ts308C9.148 0.0150 0.5694

10.312 0.0183 0.597311.265 0.0219 0.618312.052 0.0250 0.631812.500 0.0284 0.643112.521 0.0328 0.649012.534 0.0418 0.649012.546 0.0565 0.649012.556 0.0847 0.649012.557 0.1239 0.649012.565 0.1707 0.649012.545 0.5313 0.649012.531 0.5893 0.649012.520 0.6825 0.649012.480 0.7640 0.649012.478 0.8333 0.649012.418 0.8872 0.649012.337 0.9206 0.672412.243 0.9385 0.710312.122 0.9530 0.748812.002 0.9635 0.782311.872 0.9722 0.815311.730 0.9789 0.844011.527 0.9859 0.878811.354 0.9905 0.905611.228 0.9935 0.925611.063 0.9963 0.947910.982 0.9987 0.977410.906 1.0000 1.0000

values for a given q were calculated from the interpolation of the neighboring data. Comparing thisw x Ž .method with that of Lencka and Anderko 9 which is calculated with the analytical function F q ,

this method could dramatically reduce the computing time. Secondly, since the original temperatureŽ .dependence function of the parameter a T did not describe over the whole temperature range, the

w xparameters were obtained by the BSOLVE algorithm of Marquardt 23 with the limited data below390 K. The optimum value of k to minimize the objective function OF is 4.55. The values of the1

other parameters obtained with its k value are T )s124.85 K, P )s83.568 bar, v )sy0.4488,c c

Dh0 sy31.800 kJrmol, D s0 sy123.6 Jrmol K, and Dc0 s7.577 Jrmol K. The vapor–liquidp

coexistence curve for pure hydrogen fluoride with these parameters was shown as the dotted line inFig. 1. This figure was shown in good agreement between the calculated and experimental values ofthe compressibility factors of both phases in the temperature range below 410 K, but it presented largedeviations over the temperature range from 420 K to critical. In these parameter regressions, theaverage absolute deviation between the experimental and correlated values was 0.8% for vaporpressures, 0.9% for saturated vapor densities, 1.1% for superheated vapor densities, and 1.3% for


saturated liquid densities. Thirdly, in order to correlate the data over the whole temperature range, theŽ .correction term in the original function of the parameter a T is added as follows:

20.50.5a T s 1qm 1yT yb 1y Tr390 , T)390 K 20Ž . Ž . Ž .Ž . ž /ž /r

The optimum value of the parameter b with the previously obtained parameters is 11.0. In Fig. 1,some different results of the vapor-phase compressibility factor in comparison to those of Lencka and

w xAnderko 9 were mainly caused by using the different temperature dependence function of theŽ .parameter a T .

5. Results and discussion

Ž .The experimental vapor–liquid equilibrium data for the binary system propane 1 qhydrogenŽ .fluoride 2 at 20.08C and 30.08C are shown in Table 1 and in Figs. 2 and 3. In comparison to the

w xother binary systems such as chlorodifluoromethaneqhydrogen fluoride 16 , difluoromethaneqw x w xhydrogen fluoride 15 , 1,1,1,2-tetrafluotoethaneqhydrogen fluoride 13 , and 1,1-difluoroethaneqw xhydrogen fluoride 14 , the bubble point pressure of the binary system propaneqhydrogen fluoride

was more steeply increased with decreasing the liquid-phase composition of propane in the rangeabove about 0.97. The binary systems that show a similar tendency to the system propaneqhydrogen

w xfluoride are chlorineqhydrogen fluoride and dichlorodifluoromethaneqhydrogen fluoride 15 . Thedipole moment of the organic components for these binary systems is relatively smaller than that for

w xthe other binary systems. As explained by Economou and Peters 11 , the hydrogen fluoride moleculesprefer to be in a polar environment rather than in a non-polar one and its self-association is verystrong. So it exclude non-polar molecules from their immediate neighborhood resulting in theformation of two liquid phases. Thus, the composition range forming the immiscible liquid-phaseincreases with decreasing the polarity of the organic compound for the binary system containingorganic compound and hydrogen fluoride. To correlate the vapor–liquid equilibrium data for the

Ž . Ž . chbinary system propane 1 qhydrogen fluoride 2 , the Z term for this binary system, which is

Ž . Ž . Ž . Ž .Fig. 2. Equilibrium curve for propane 1 qhydrogen fluoride 2 at 20.08C: ` experimental; - - - model predictions withŽ .u s0.0, model correlation with u s0.119.12 12


Ž . Ž . Ž . Ž .Fig. 3. Equilibrium curve for propane 1 qhydrogen fluoride 2 at 30.08C: ` experimental; - - - calculated withŽ .u s0.0, calculated with u s0.132.12 12

Ž .consisted of an associating component and a non-polar component, can be well described by Eq. 10 .Furthermore, the classical quadratic mixing rule can be well applied to the Zch term, and it isrepresented as follows:

1r2as x j a a 1yu 21Ž . Ž .Ž .Ý Ý i i i j i j

i j

bs x b 22Ž .Ý i ii

w xwhere u is the binary interaction parameter. The physical properties from Reid et al. 24 fori jw xpropane and hydrogen fluoride, Sato et al. 25 for dichlorodifluoromethane, and Braker and Mossman

w x26 for chlorine used in this work are given in Table 2. The binary interaction parameter u wasi jw xevaluated by a non-linear regression method based on maximum-likelihood principle 27 , as

w ximplemented in the computer programs published by Prausnitz et al. 28 , with the following objectivefunction OF :2

2 2 2j j j j j jP yP T yT x yxž / ž / ž /exptl calcd exptl calcd 1,exptl 1,calcd

OF s q q 23Ž .Ý2 2 2 2s s s� 0P T xj 1

where s is the estimated standard deviation of each of the measured variables, i.e., pressure,temperature, and liquid-phase mole fraction. In data reduction, s s1.0 kPa, s s0.1 K, andP T

Table 2Physical properties of the pure components

Component MW T rK T rK P rbar v Dipmrdebyeb c c

Propane 44.09 231.1 369.8 42.50 0.153 0.00Chlorine 70.91 238.7 417.2 77.10 0.069 0.00Dichlorodifluoromethane 120.91 243.4 385.0 41.29 0.179 0.50Hydrogen fluoride 20.01 292.7 461.0 64.80 0.372 1.90


Table 3Binary interaction parameters and standard deviations of the measured variables for the binary systems

System tr8C u Standard deviations12

PrkPa tr8C 100 x1

Ž . Ž .Propane 1 qHF 2 20.0 0.119 0.6 0.13 0.1230.0 0.131 0.3 0.14 0.11

Ž . Ž .Chlorine 1 qHF 2 30.0 0.192 1.2 0.31 0.08Ž . Ž .Dichlorodifluoromethane 1 qHF 2 30.0 0.058 1.5 0.32 0.10

s s0.001 are selected. The binary interaction parameters obtained by data reduction and thex1

standard deviations of the measured variables for the binary systems are presented in Table 3. Thecalculated vapor-phase mole fractions at the given experimental liquid-phase mole fractions for the

Ž . Ž .system propane 1 qhydrogen fluoride 2 are also presented in Table 1. The comparisons of theexperimental and calculated values are also shown in Figs. 2 and 3. The equilibrium data arecalculated withoutrwith an adjustable binary parameter. As shown in these figures, the calculatedresults with an adjustable binary parameter are in good agreement with the experimental values.However, the predicted phase behavior without a binary parameter, that is, with the information ofpure component data alone, shows large deviations from the experimental values. Vapor–liquid

Ž . Ž .equilibria were also calculated for the two binary systems chlorine 1 qhydrogen fluoride 2 andŽ . Ž . w xdichlorodifluoromethane 1 qhydrogen fluoride 2 15 . As shown in Table 2, the dipole moments

of the more volatile components for these two systems have relatively small values. The binaryinteraction parameters and the standard deviations of the measured variables for these binary systemsare presented in Table 3. The comparisons of the experimental and calculated equilibrium data arealso shown in Figs. 4 and 5. For these two binary systems, their results are similar to those of thesystem propaneqhydrogen fluoride. Therefore, it is concluded that the binary adjustable parametersare required to calculate the phase behavior of the systems containing hydrogen fluoride. In the range

Ž . Ž . Ž . w x Ž .Fig. 4. Equilibrium curve for chlorine 1 qhydrogen fluoride 2 at 30.08C: ` Kang 15 ; - - - calculated with u s0.0,12Ž . calculated with u s0.192.12


Ž . Ž . Ž . w x Ž .Fig. 5. Equilibrium curve for dichlorodifluoromethane 1 qhydrogen fluoride 2 at 30.08C: ` Kang 15 ; - - - calculatedŽ .with u s0.0, calculated with u s0.058.12 12

of the liquid-phase composition of hydrogen fluoride below about 0.01, the calculated pressures wereshown higher than the experimental values. In the above composition range, the compressibility factorZ is becoming more dominated by the Z term than the Zch term with decreasing the liquid-phaseph

composition of hydrogen fluoride. This implies that the calculated pressure obtained by using only theZ ph term, that is, the vapor pressure of hypothetical hydrogen fluoride in the absence of the chemicalassociations is higher than the reasonably acceptable value. As a results, it can be estimated that thereare some errors in the apparent parameters, T ), P ), and v ) which are used to characterize thec c

physical contribution to the compressibility factor for pure hydrogen fluoride. Such an estimation maybe supported by the fact that the apparent acentric factor v )sy0.4488 for hypothetical hydrogenfluoride is much lower than zero. The acentric factor represents the geometric shape of the molecule.It is almost equal to zero for monatomic gases and rises with the size and polarity of a molecule.

w xFrom the literature’s data 24 of the acentric factors, dipole moments, and molecular weights for theŽ .similar compounds such as hydrogen chloride, hydrogen bromide, and hydrogen iodide to hydrogen

fluoride, it may be estimated that the apparent acentric factor v ) for hypothetical hydrogen fluorideis in the range zero to 0.1. Using this estimated apparent acentric factor for hypothetical hydrogenfluoride, the apparent critical temperature and pressure will be changed. Considering the results so fardiscussed, in order to describe well the phase behavior not only for pure hydrogen fluoride but alsofor mixtures containing hydrogen fluoride, it may be concluded that the different type to thePoisson-like distribution function should be introduced.

6. List of symbols

a, b Parameters of the Peng–Robinson equation of stateA Associating component


Dipm Dipole momentŽ .f Distribution function defined by Eq. 12

Ž . Ž .F q Algebraic function defined by Eq. 10K Chemical equilibrium constant

Ž .m Defined by Eq. 17MW Molecular weightn Number of moles of the apparent species in an associated mixtureT

n Number of moles of the true species in the absence of the association0Ž . Ž .OF ,OF Objective functions defined by Eqs. 19 and 23 , respectively1 2

P PressureP s Vapor pressureq RTKx rÕA

R Ideal gas constantT TemperatureT Normal boiling pointb

Õ Molar volumeŽ .x Analytical apparent mole fraction

z True mole fractionZ Compressibility factor

Greek lettersa Temperature dependence parameter of the Peng–Robinson equation of state

Ž . Ž .defined by Eqs. 16 and 20Ž .b Parameter used in Eq. 20

0 0 0 Ž .Dc ,Dh ,D s Parameters used in Eq. 18p

f Fugacity coefficientŽ .k Parameter of the distribution function defined by Eq. 12

u Binary interaction parameterv Acentric factor

Superscriptsch Chemical contributionL Liquid phaseph Physical contributionV Vapor phase) Hypothetical property of the associating component in the absence of the

chemical associations

SubscriptsA Associating componentA i-mer of the associating component Ai

B Non-associating component kk

c Critical statei Component icalcd Calculated valueexptl Experimental value


Acknowledgements

The author would like to thank Yeong Jun Kim for his assistance in the experimental works andJongcheon Lee and Jeong-Ho Yun for his valuable suggestions.

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Documents

Phase equilibria for the binary mixtures containing hydrogen fluoride and non-polar compound