*
4 w
= phaselag = relative yield =
literature Cited
frequency
SUBSCRIPTS
i j
k F B E
D
=
refers to ith reaction: i
=
1, 2, 3, . . .
B , C, D refers to k fraction of the semicontinuous cycle k = F , B , E, D = filling = batch = emptying = down = refers to jth component: j = A , =
Ark, R., Can. J. Chem. Eng. 40, 87 (1962). Codell, R. B., Enael, A. J.. A.I.Ch.E. J. 17. 220 (1971). Douglas, J., Ripp&,'D., Chem. Eng. Sci. 21,'305 (1966): Fang, M., Engel, A. J., paper presented a t 61st National AIChE Meeting. New Orleans. La.. 1967. Gillespie, %., Carberry, J., Chim. Eng. Sci. 21, 472 (1966). Lund, M., Seagrave, R., A.I.Ch.E. J. 17, 30 (1971). Van de Vusse, J., Chem. Eng. Sci. 19, 994 (1964). RECEIVED for review September 21, 1970 ACCEPTED June 4,1971 Presented a t the Division of Industrial and Engineering Chemistry, 159th National Meeting of the American Chemical Society, Houston, Tex., Feb 1970.
Calculation of Isothermal Vapor-Liquid Equilibrium Data for Binary Mixtures from Heats of Mixing Richard W. Hanks,' Avinash C. Gupta, and James J. Christensen Department oj Chemical Engineering and Center for Thermochemical Studies, Brigham Young University, Provo, Utah 84601
The Gibbs-Helmholtz relation is used together with two popular semitheoretical excess free energy relations (Wilson's equation and the NRTL equation) to calculate isothermal vapor-liquid equilibrium (x-y) data for six highly nonideal binary systems. This method is shown to produce reliable x-y curves directly from measured heat of mixing (AhM)data and pure component vapor pressures without the necessity of measuring any x-y data. This method, coupled with titration calorimetry techniques for measuring AhM easily and quickly, has considerable practical potential for phase equilibrium work. Although Wilson's equation and the NRTL equation were used, the calculation method is valid for any excess free energy function. The accuracy of the calculated x-y data appears to be limited only by the accuracy of the Ahw and vapor pressure data and the semitheoretical excess free energy model chosen.
Vapor-liquid equilibrium data are of considerable industrial and academic importance. Although these data can readily be calculated for ideal solutions, most solutions of interest are not ideal in behavior. A great deal of study has been perfumed on nonideal solutions, but a t present no way exists which will permit the accurate theoretical prediction of deviations from ideality. Therefore, one must turn to experimental measurement to determine nonidealities for a given system. I n this paper, we discuss a method of estimating isothermal vapor-liquid equilibrium which does not involve the experimental measurement of the composition of either the vapor or liquid in a vapor-liquid equilibrium mixture, a procedure which is often quite laborious. The method consists of using measured heats of mixing to evaluate the constants in semitheoretical thermodynamic equations which in turn are used to calculate equilibrium data from pure component vapor pressures. The conventional approach to vapor-liquid equilibrium involves a n attempt to predict heats of mixing from measured vapor-liquid equilibrium data and then to compare these values with measured heats of mixing determined by calorimetric techniques. 111general, the agreement between calculated and experimental heats of mixing has been poor. One To whom all correspondence should be addressed. 504 Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
reason for this poor agreement is inherent in the calculation methods employed. All of the theoretical equations used are based upon the concept of the excess free energy, G E , which is directly related to the activity coefficients, yi. The latter are defined by the expression
where P is the total system pressure, yi, G ~ j ,z o L , and x i are, respectively, the vapor phase mole fraction, the vapor phase fugacity coefficient, the pure liquid fugacity, and the liquid phase mole fraction of component i in the equilibrium mixture. Since this expression involves the ratio yi/x2, errors in y t a t a given x L are reflected directly in the G E data derived therefrom, especially a t low values of z, where this ratio is essentially the first derivative of the equilibrium curve. The G E data are used to determine adjustable constants in semitheoretical thermodynamic equations. Finally, the heats of mixing, Ah", are calculated from the G E data by differentiation of the semitheoretical thermodynamic equations using the well known Gibbs-Helmholtz relation
Thus, the error magnification inherent in two differentiations
of the basic experimental equilibrium data can result in poor agreenieiit between calculated aiid measured heats of mixing. \Ye have found, however, that if experimentally deteriiiined heat of mixing data are iiit,egrated cia eq 2 using a given semitheoretical equation for g E and combined with pure component vapor pressures to compute vapor-liquid equilibrium data, the above inaccuracies are largely avoided because of the error smoothiiig characteristics inherent in integrations. Good agreement can be obtained between calculat,ed and measured z-y data by this technique. This method of determiiiation of vapor-liquid equilibrium curves by integrating heat' of mixing measurements has great practical potential. Recent developments in heat of mixing calorimet'ry have made it much simpler to measure heats of riiixiiig than to determine the 2-y data experimentally. Heat's of mixing can be rapidly and accurately measured as a fuiictioii of cornposition by the technique of titrat'ion calorimetry. Titration calorimeters have been developed by Mrazek and Van Kess (1961), Becker, et al. (1963), and Christensen, et al. (1968) , which allow binary isothermal heat of mixing data to be determined over t,he entire raiige of composition in two experimental runs. X commercial isothermal calorimet'er based on t'he tit'ratioii technique has recently beeii marketed by Tronac, Iiic., Orem, Utah. The availability of procedures and equipment for measuring heats of mixing rapidly aiid accurately nieaiis that n-ith the method described in this paper good estimates of z-y data can be made 1%-ith much less difficulty than is involved in t,heir direct experimental measurement.
aiid proposed the following expression for the Gibbs energy of a binary mixture
The activity coefficients derived from this equation are
(6) Wilson's equation contains two adjustable parameters, X U and hZ1, which in his original derivation were relat'ed t o the pure component molar volumes, vi, aiid t o certain characteristic molecular energy differences, gij. Orye aiid Prausnit'z (1965) give these expressions in the form (7)
As an approximation, it, can be assumed that the differences in t,he characteristic energies are temperature independent, a t least over a modest temperature interval. With this assurnption the heat of mixing for binary solut'ioii can be calculated from eq 2 aiid 4 as
Thermodynamic Models
The most useful thermodynamic concept for expressing the nonideality of a liquid mixt,ure is the excess Gibbs energy, G E , which was originally introduced by Scatchard (1937). This quaiitit'y is usually expressed as a fuiictioii of liquid composition a t coilstant temperature aiid pressure and from it the activity coefficieiits may be calculated by using the relation
(3) where ( j i E is t'he part'ial molal excess Gibbs energy of species i and n i represents t'he iiuniber of moles of species i in the solution. Numerous aiialytical equations have beeii proposed for expressing the composition dependence of t'he excess Gibbs energies of binary mixtures by AIargules (1895) Van Laar (1913)!Scatchard-Hamer (1935) ,Kohl (1946) ,Redlich-Kister (1948), Wilson (1964), and Renon-Prausnitz (1968). Although these equatioiis have some theoretical basis they all involve empirical constants which must be determined by fit,ting the equations to experimental data. Of the several equations proposed for relating g E t o the composition of the mixture, those given by Kilson (1964) aiid Renoii and Prausnitz (1968) currently enjoy popularity as they rather closely approximate data for a large iiuniber of systems. For this reason these t'wo equatioiis were used in this study to deiiioiistrate the method of relating heat of mixing data to x-y data. ~
where the expressions (g12- gI1) and (gI2 - 9 2 2 ) are the two adjustable parameters. JYiIsoii's equation appears to provide a good representation of g E for a variety of completely miscible mixtures and is particularly useful for highly symmetrical systems such as solutions of polar or associating components in nonpolar solvents. Orye (1965) shows that for approximately 100 miscible binarj- mixtures of various chemical t,ypes Kilson's equation is quite adequate to represent' the nonideal behavior. In esseiit,iallg all the cases, this representation was as good as, and in many cases bet'ter than, the representation given by three suffix (two-constaiit) Xargules equations and by the Van Laar equation. Kilson's equation has t'wo disadvantages: first, eq 5 and 6 are not strictly applicable for the syst'ems where the logarithms of the activity coefficient,s, when plotted against,composition, exhibit maxima or minima; and second, it does not predict the occurrence of part,ial miscibility. When Wilson's equation is subst.ituted in t,he equations of thermodynamic st'ability for a binary system, no values of the parameters XI2 and XPI can be found which indicate the existence of two stable liquid phases. Wilson's equation, therefore, can be reliably used only for liquid systems which are complet,ely miscible or else for those limited regions of immiscible systems where only one liquid phase is present. NRTL Equation
Wilson's Equation
On the basis of molecular considerations, Wilson (1964) took g E to be a logarithmic fuiictioii of liquid composition
The essential contribution by IT'ilson (1964) was the introduction of the concept of local composition. This same concept was used by Renon and Prausnitz (1968) in their Ind. Eng. Chern. Fundam., Vol. 10, No. 3, 1971
505
Table 1. Parameters for Wilson's Equation (Eq 9 ) and Related Vapor-liquid Equilibrium Dala
Temp,
System
Carbon tetrachloride Acetone
Wilson parameters, 912 922, 912 gll, col/mole
'C
45
-
Errorsquare suma
963.62 -179.5
49.91
Molar volumes, v1, v2, cm*/mole
Vapor pressures,
PI, P2, mm 258.84 512,72
Source of exptl vopor-liquid equilibria data
Source of exptl heot of mixing data
99.6 76.2
Brown and Smith (1957a) Benzene 45 330.1 21.91 223.66 91.6 Brown and Acetone -118.4 512.72 76.2 Smith (1957a) a Error-square sum based on all AhM data reported by author. In some cases, these include a few extrapolated points.
Brown and Smith (1957b) Brown and Smith (1957b)
80
a
-
60
'ij
E
e
40-
g
L 40 31'
c
a
20
0
0.2
0.4
0.6
0.8
I.o
I.o
0.8
0.6
r
Calculated o Experimental
0.4
0.2 X2
C
I 0.2
I 0.4
I
I
0.6
0.8
xz
Figure 2. Comparison of calculated and experimental data for the benzene (1)-acetone (2) system at 45°C. Solid curves are calculated from Wilson's equation
Figure 1 . Comparison of calculated and experimental data for the carbon tetrachloride (1)-acetone (2) system at 45OC. Solid curves are calculated from Wilson's equation
derivation of the nonrandom two-liquid (NRTL) equation. The KRTL equation, unlike Wilson's, is applicable to partially miscible as well as completely miscible systems. Renon and Prausnitx's equation for the excess free energy is
where (912 - q l 1 ) , (g12 - 922)' and a12 are three adjustable parameters. If the NRTL equation parameters, a12, r12, and 721 are assumed to be temperature independent, then the heat of mixing for a binary solution calculated from eq 10 by using eq 2 is AhM
=
xlxz
where r12= (g12 - g2d/RT;n1= (glZ - gI1)/RT;G12 = exp. (--mz); and GU = exp(-a12T?]). The expressions for the activity coefficients are
7izGiz
+ xzGz1
721G21
{XI
2 RT
[
xz
+ x&
Xi~'izGzi (xi x~GzJ'
+
+
~27'12G12
+
( ~ 2
I}
(13)
~iGi2)~
Theoretical Calculations
Vapor-liquid equilibrium (x-y) data were calculated from measured heat of mixing data and pure component vapor 506 Ind. Eng. Chcm. Fundam., Vol. 10, No. 3, 1971
Table
II. Parameters for NRTL Equation (Eq 13) and Related Vapor-Liquid Equilibrium Data NRTL parameters g12 - 922, Temp, OC
System
Toluene Acetonitrile Toluene Acetone Toluene Nitroethane Cyclopentane Tetrachloroethylene
45
g12
- gll,
cal/mole
-446.3 1173.0 207.1 30.3 -169.2 570.6 81.91 86.31
Vapor pressures,
Errorsquare rum
a12
0.1
Molar volumes
PI, P2,
8.44
Vlr
Source of exptl heat of mixing data
v2,
mm
cma/mole
74.2 208.35 74.2 512.72 74.2 57.89 317.5 18.4
111.0 54.4 111.0 76.2 111.0 72.6 96.0 88.0
Source of exptl vapor-liquid equilibria data
Orye (1965) 45 0.1 19.03 Orye (1965) 45 0.1 2.7 Orye (1965) 25 -2.22 3.30 Polak, et al. (1970) Error-square sum based on all AhM data reported by author. I n some cases these included a few extrapolated points.
pressure data by the following procedure. Either eq 9 or 13 was fitted to the experimental heat of mixing data and the parameters in the equation were evaluated. A nonlinear leastsquares fitting program was used which minimized the sum of squares of relative deviations in the experimental and calculated Ah-M values for all data points. A computer program LETAGROP developed by Sillen (1962) was used for calculating the M‘ilson parameters while a subroutine FIND developed by Free (1970) which is essentially based on Powell’s nongradient technique of minimizing a function using the concept of conjugate direction was used to evaluate the N R T L parameters. These parameters were then used to compute g E and 7 1 , y 2 values from either eq 4, 5, and 6 or 10, 11, and 12. The isothermal vapor phase compositions were thus calculated (assuming ideal vapor phase behavior) from the relation y1 =
XlPlY1
ZlPlYl
+ (1 -
0
Orye (1965) Orye (1965) Orye (1965) Polak,
et al. (1970)
Experimental
I .o
(14) Xl)P2Y2
where P1 and Pz are the saturation vapor pressures of pure liquid components 1 and 2. Consequently, it should be noted t h a t the accuracy of the y1 values depends directly on t h a t of the vapor pressure data used. Isothermal heats of mixing data were used to determine the constants in the Wilson or Y R T L equations giving constants which apply strictly only to isothermal vapor-liquid equilibria. For this reason, the method was only applied to systems where both isothermal AhM and x-y data were available.
0.8
0.6 h U
0.4
1
Results
Tables I and I1 give, respectively, the values of the parameters obtained by fitting experimental Ah-+’ with eq 9 and 13 for six highly nonideal systems. The number of systems available was limited by the requirement that for a given system both isothermal vapor-liquid equilibrium data and heats of mixing data must have been determined. The six systems presented here were chosen because they are all very nonideal systems and thus represented a severe challenge t o the method. It was found that eq 9 gave a satisfactory fit of the AkJf data for only the carbon tetrachloride-acetone and benzene-acetone systems and eq 13 was used to fit the Ah.+’ data for the other four systems. How well eq 9 and 13 fit the experimental heat of mixing data is indicated by the errorsquare sum values reported in the tables. The error-square sum is given as 21” (AhMcalod - AhJfeXptJ2where N is the number of experimental data points. The number of data points, W , was the same for each system; thus the magnitudes of the error-square sum values are a direct indication of the fit of
I
1
I
I
0.2
0.4
0.6
0.8
I
I
1.0
XL
Figure 3. Comparison of calculated and experimental data for the toluene (1)-acetonitrile (2) system at 45OC. Solid curves are calculated from the NRTL equation
the experimental data by eq 9 and 13. The sources for both heat of mixing data and vapor-liquid equilibrium data are given in Tables I and I1 for each system studied. Figures 1 through 6 show experimental and calculated values of AhM and vapor composition as functions of liquid composition for the six systems studied using the parameters given in Tables I and 11. I n general, good to very good agreement between calculated and experimental %-y data was obtained. The calculation method is also valid for systems which Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
507
:
¶ *
c Q
Table 111. Selected Parameters for NRTL Equation (Eq 13) for Toluene-Acetonitrile System at 45OC
1 OO
NRTL parameters Ex Calculated per imenta I
612
- 622
812
Error-square sum
- 611
(AhMdod
a12
- AhMmptl)*
1
0.4
0.2
0.6
0.8
1.0
6 7 . 6 497.7 -446.3 1173.0
373 8.44
0.2 0.1
XI
1.0)
I
I
I
-Calculated
C
0
1
Experimental
XP
Figure 4. Comparison of calculated and experimental data for the toluene (1)-acetone (2) system a t 45°C. Solid curves are calculated from the NRTL equation
0
form azeotropes, as is illustrated in Figure 5 for the toluenenitroethane system, where the calculated azeotropic vapor composition is 0.70 mole fraction toluene, which is in excellent agreement with the experimental value of 0.71. The data for the cyclopentane-tetrachloroethylene systems shown in Figure 6 are especially interesting. This system is quite volatile, values of y < 0.6 being diffiult to obtain. I n the region x < 0.1 which corresponds to the range y < 0.7, the slope of the x-y curve is very steep. Thus yl/xl, and consequently 71, is essentially equal to dyl/dxl. This means that very precise x-y data are required in order to obtain even moderately precise g E data. The present technique, which capitalizes on the inherent error smoothing property of integration, provides a simple and reliable way of determining x-y data. I n highly volatile systems these data, especially in the dilute ranges, are probably as good as or better than can be measured. The better the fit of the experimental A h M data by either eq 9 or 13, the more precisely can the experimental vaporliquid equilibrium data be predicted. I n Table I11 two sets of
v
Experimental
OO
0.2
0.4
0.6
0.8
1.0
XI
f
-
*-
s
-
Calculated
0
Experimental
-
0.4 Calculated Experimental
0.2
o*2OO *
0.2
0.4
XI
0.6
0.8
I
-
-
*
OO
0.2
0.4
0.6
0.8
1.0
XI
Figure 5. Comparison of calculated and experimental data for the toluene (1)-nitroethane (2) system at 45°C. Solid curves are calculated from the NRTL equation 508
Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
Figure 6. Comparison of calculated and experimental data for the cyclopentane (1 )-tetrachloroethylene (2) system at 25°C. Solid curves are calculated from the NRTL equation
I d; 40
o Experimental \, ---- Calculated; Error Square Sum = 3 7 3 -Calculated; Error Square Sum ‘8.44
I
0.2
0.8
I
0.4
I
0.6
I
0.8
I .o
Conclusions
-
0.6
-
0.4
-
rapidly and easily than by the conventional vapor-liquid equilibrium methods. For example, a complete heat of mixing curve can be obtained in a minimum of t n o experimental runs b y titration calorimetry, each run taking from approximately 1 hr using a continuous titration calorimeter to 12 hr using an increment titration calorimeter. A conventional bulbbreaking calorimeter, on the other hand, will provide d a t a sufficient for a heat of mixing curve in a period of several days to several weeks, which time is comparable to the time required to determine the vapor-liquid equilibrium data experimentally. An additional benefit of evaluating x-y data from heat of mixing data is that it is relatively easy to obtain heat of mixing data over a wide temperature range thus in turn giving x-y data of a function of temperature.
Error Square
0.2 -
0
373
0
8.44
It has been shown for six binary systems that heat of mixing data coupled with pure component vapor pressures can be used to predict vapor-liquid equilibrium data accurately. The parameters in the U’ilson and X R T L equatioiis were evaluated from heat of mixing data and in turn were used to calculate x-y data for the binary systems studied. In most cases excellent agreement was obtained between calculated and experimental equilibrium data and in all cases the agreement was good. This method together with the rapid and accurate measurement of heats of mixing data by titration calorimetry can be used for the prediction of vapor-liquid equilibrium data with much less difficulty in many cases than by direct measurements. Furthermore, in volatile systems the x-y data for small values of x can probably be estimated more reliably by this method than they can be measured.
a
OO
0.2
0.4
0.6
0.8
1.0
y z (Calculated)
Figure 7 . Illustration of the fit of Ahdv1 data on x-y data. Upper graph shows two different fits of AhM data for the toluene (1)-acetonitrile (2) system at 45°C using the NRTL equation. Bottom graph shows corresponding comparisons of yz(experimenta1) with yz(calcu1ated).
parameters are shown for the K R T L equation for the tolueneacetonitrile system. These parameters were selected to give various degrees of agreement between the calculated and experimental AhM data as indicated by the error-square sum. The Ah’M and y values calculated using these parameters are shown in Figure 7 and demonstrate that more accurate x-y values are calculated using parameters resulting in the smallest error-square sum in the fit of the AhM data. I n the application of the present method to the calculation of x-y data it is imperative that the experimental Ah.+f data be accurate and that the thermodynamic model chosen give as nearly as possible the proper representation of the actual phenomena. The method is independent of the source of the experimental heat of mixing data in that such data are equally valid whether determined b y titration, incremental, or conventional calorimetry. However, when the method is used in conjunction with heat of mixing data evaluated b y titration calorimetry, and especially continuous titration calorimetry, x-y data can be evaluated for a given binary system more
l i t e r a t u r e Cited
Becker, F., Schmahl, N. G., Pflug, H. D., 2. Phys. Chem. (Frankfurt am Main) 39, 306 (1963). Brown, I., Smith, F., Aust. J . Chem. 10, 417 (1957a). Brown, I., Smith, F., Aust. J . Chem. 10, 423 (1957b). Christensen, J. J., Johnston, H. D., Izatt, R. M., Rev. Sci. Znstrum. 38, 1356 (1968). Free, J. C., Brigham Young University, private communication, 1970.
Margules, &I., Sitzungsber. Akad. Wiss. Wien, Math. Naturwiss. KZ. Abt. 2A 104, 1243 (1895). Mrazek, R. V., Van Ness, H. C., A.Z.Ch.E. J . 7, 190 (1961). Orye, R. V., Ph.D. Dissertation, University of California, Berkeley, Berkeley, Calif., 1965. Orye, R. V., Prausnitz, J. M., Ind. Eng. Chem. 5 7 , 18 (1965). Polak, J., Murkami, S., Lam, V. T., Benson, G. C., J . Chem. Eng. Data 15, 323 (1970). Redlich, O., Kister, A. T., Znd. Eng. Chem. 40, 345 (1948). Renon, H., Prausnitz, J. M., A.I.Ch.E. J . 14, 135 (1968). Scatchard, G., Trans. Faraday SOC. 33, 160 (1937). Scatchard, G., Hamer, W. J., J . Amer. Chem. SOC.5 7 , 1805 ( 1 93.5). \ - - - - ,
Sillen, L. G., Acta Chem. Scand. 16, 159 (1962). Van Laar, J. J., 2. Phys. Chem. 83, 599 (1913). Wilson, G . At., J . Amer. Chem. SOC.8 6 , 127 (1964). Wohl, K., Trans. Am. Inst. Chem. Eng. 42, 215 (1946). RECEIVED for review Xovember 5, 1970 ACCEPTED May 18, 1971 Contribution Xo. 17 from the Center for Thermochemical Studies, Brigham Young University, Srovo, Utah 84601.
Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
509
