Vapor-Liquid Equilibria of Nonideal Solutions Utilization of Theoretical e. e&
A E. l.
areaA&
P. e -
du Pont de Nemours & Company, Inc., Wilmington, Del.
Vapor-liquid equilibrium data may b e readily evaluated and extended when they are calculated as activity coefficients. The equations proposed by van Law, Margulas, and Scatchard and Hamer, which express the activity coefficients of both components of a binary mixture as functions of the liquid composition and empirical constants, are capable of fitting most of the available vaporliquid equilibrium dab. Experimental data may be examined for accuracy by noting the relation of the activity coefficients for the equimolal mixture t o the terminal values, and may b e smoothed by fitting the data with one or another of these equations. With these integrated forms of the Gibbr-Duhem equrtion, complete vapor-liquld equilibria may b e calculated from measurements of azeotropic composition, total pressure or boiling point curves, or liquid-liquid solubility. The terminal values of the activity Coefficients and the constants OF the van Laar equations for a homologous series of alcohols in binary mixtures with water or benzene are shown t o progress regularly with increasing molecular weight. The effect of pressure or temperature on the curves of equilibrium liquid and vapor composition may be calculated. Approximate procedures for estimating ternary equilibrium data from those on the three binary systems are given. Liquid-liquid distribution coefficients may be predicted from the activity coefficients.
DESPITE
the wide utilization of dis%illation and similar contacting apparatus and the need of reliable data on vapor-liquid equilibria to design such apparatus, relatively little attention has been paid to the important problem of evaluating and extending such data for nonideal solutions. Although considerable work has been done on the effect of deviation from the gas lbws under conditions of pressure distillation, this work is largely limited to those mixtures which form ideal solutions in the liquid, such as members of a homologous series. I n chemical industry a large share of distillation problems involves materials of dissimilar chemical nature, such as aqueous mixtures, and here large deviations from Raoult’s law are encountered. The practice has usually been to determine data experimentally for any system under the exact conditions of later use, but this is time consuming and sometimes too difficult to be practicable. Therefore, the purpose of this investigation was to study the theoretical background of nonideal mixtures and to derive convenient methods of evaluating and extending data on vapor-liquid equilibrium. 1
to Extend Data
When experimental data on vapor-liquid equilibrium are plotted in the customary manner of y, mole fraction of the more volatile component in the vapor, against 2, the mole fraction of the more volatile component in the liquid, curves of the types shown in Figure 1 are obtained. On such a plot there appears to be little rhyme or reason. If these same data are calculated in terms of deviation factors from Raoult’s law by the equations,
and the deviation factors are plotted on semilog paper again& the composition of one of the components in the liquid, characteristic ourves are obtained like those in Figure 2. These are orderly and consistent. Such plots obviously permit a ready detection of poor experimental data. They are more valuable than a mere graphical representation, however, as will be discuased below.
Theoretical Equations GIBBS-DWEM EQUATION. The term introduced as deviation factor from Raoult’s law is, under conditions to be deiined immediately, equivalent to the thermodynamic property, activity coefficient. As such, there is a definite relation between the values for the several components. This relation, according to Lewis and Randall ( I $ ) , follows from the Gibbs relation between the partial molal free energies, PI, F,, etc.: 21
Since
dF1
=
(3) dX1 , f s (3) dxl P.T + .... = 0 P T
+
= RT In fi constant, at constant temperature RT b lnfi, and Equation 2 becomes:
In this equation f, the fugacity, can be regarded as the “ideal” partial pressure, and is identical with partial pressure p for conditions under which the gas laws hold. Lewis and Randall (IS) define activity, a, as the relative fugacity, or the ratio between the fugacity of the substance in solution and its fugacity as a pure liquid. Thus, a1 = fl/f; and a2 = f2/fi1 where al and a2 are the activities of components 1 and 2, and f: and fi are the fugacities of t h e pure liquids. Activity coefficients y are simply the activities divided by their respective mole fractions, or y1 = a t / z ~ , 71 = a*/%*, etc. Thus:
Present address, University of Delaware. Newark. Del.
YI
581
=I
f1Kx1, etc.
(4)
582
Vol. 34, No. 5
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
In this symmetrical form, constants A and B have the property of being equal t o the terminal values of log y of the two curves. Thus, a t 21 = 0, log y1 = A , and a t $1 = 1 when 2 2 = 0, log yz = B. It is of interest, too, t o observe that at z1 = 0, log yz = 0 or y2 = 1, and a t z1= 1, log yl = 0 or y1 = 1, which satisfy the limiting condition that Raoult's law holds for a component whose concentration approaches 100 mole per cent. By performing the differential operations indicated by Equation 7 , the van Laar equations are readily proved to be a true solution of Equation 7 . The curves represented by Equations 8 and 9 have a peculiar property which is readily shown a t the mid-points on the curves. When x1 = 0.5,
YI
0.2
0
Figure Boiler
0.4
0.8
0.6
1.0
I.
E q u i l i b r i u m M o l e Fraction of Lowin the Vapor, yl, VI. M o l e Fraction i n
the L i q u i d , x1 Curve System A Chloroform-ethyl alcohol B n-Butyl alcohol-water C Isopropyl alcohol-isopropyl ether D Acetone-chloroform E n-Propyl alcohol-water
Where the vapors are perfect gases, fl that: 71 =
Conditions
3 5 O c. 1 atm. 1 atm. 35.17"C. 1 atm.
=
+
Citation (20)
(23) (76) (44)
(6)
pl and f; = PI, so
PI/PIXI = Pyl/Plx1, etc.
log
+ +
=
=
11x2, log 7 2
AX:
(11)
(5)
It is thus apparent that the deviation factors defined by Equation 1 are equivalent t o values of activity coefficient under conditions that the vapors approximate perfect gases. Scatchard and Raymond (go) applied a slight correction for gas law deviation even a t atmospheric pressure. The Gibbs-Duhem equation, expressed by Equation 3, can be more conveniently used when in terms of activity coefficients. Substituting for fi,fi, etc., in Equation 3 their values from Equation 4,and noting that values off;, fi, etc., z2 are constant at constant temperature and that d ( z l . ) = d(1) = 0,
For binary mixtures, axl
If A = B , the term A B / ( A B ) 2 = 1/4. As A and B differ, the ratio decreases slightly; e. g., when A = 2B, A B / ( A + B ) ? = 219. According to Equation 10, the halfway value on one curve is approximately one fourth the end value on the other curve. Thus the curve having the higher end value will be the lower a t $1 = 0.5. This particular property can be readily seen on plots of experimental data similar to those of Figure 2 and permits a t once a qualitative check on the data. For the special case of A = B , Equations 8 and 9 become:
-ax2, so that
Equations 6 and 7 are of immediate value in studying experimental data on vapor-liquid equilibrium by relating the slopes of such curves as those of Figure 2. Inasmuch as the magnitudes of slopes are diflicult to determine precisely, however, studies have been made to obtain convenient mathematical solutions of this differential equation. The results of some of these studies are proving to be helpful and will be discussed below. VAN LAAR EQVATIONS. Perhaps the most useful of the solutions are the equations derived by van Laar (IO) from a thermodynamic study; they are given below in a slightly rearranged form :
On semilog paper these are ordinary parabolas passing through their respective origins, and the values a t z = 0.5 are one fourth the end values, as stated above. Many systems have values of A and B close together and, for a few, A and B are equal. MARGULESEQUATIONS.Margules (14) integrated the Gibbs-Duhem equation in terms of a pair of exponential series with unlimited numbers of terms and then derived the constants for one of the equations from those of the other by applying Equation 7 . He was hopeful that only two terms would be needed t o apply the equations because more would be unwieldy. We have slightly revised the constants of these equations in the same manner as for those of van Laar in order t o utilize the end values as constants. So expressed. the Margules two-term equations are:
- A ) 2%+ 2(A - B ) X; (2A - B ) X: + 2(B - A ) 1,9
log
7 1
= (2B
(12)
log
y2
=
(13)
As a check one readily finds that a t z1 = 0, log y1 = A and log y2 = 0; a t z1 = 1, log yl = 0 and log 7 2 = B. It is of interest that for these equations, a t zl= 0.5, log 71 = B/4 and log yz = A/4, regardless of the values of A and B. The Margules equations become identical with those of van Laar when A = B and are then also expressed by Equation 11. When A and B are different, however, the two sets trf equations represent different curves, with the difference growing as the ratio A / B departs more widely from unity. To distinguish between the two sets of equations, therefore, data are needed which will yield considerably different values of A and B. For other data either set of equations is satisfactory, but the van Laar equations are generally somewhat more convenient. QCATCWRD EQUATIONS. Scatchard and Hamer (19) extended the methods of van Laar to obtain equations which
May, 1942
INDUSTRIAL AND ENGINEERING CHEMISTRY
we have again revised so that the constants represent end values of the curves as follows : log y1 = A
(y-
AV logy2 = B ( 2 g
1) z;
- 1)z;
- 2A (A*2B V - 1) z: - 2 B ( ABVZ 7 - 1)z:
(14)
583
Scatchard (17) derived an equation in which all the constants represent physical properties for systems in which the change of entropy on mixing is the same as that for a n ideal mixture, designated a "regular solution" by Hildebrand :
(15)
where VI, V S = molar volumes zl,z2 = volume fractions of the components Noting that z1 = V1xl/(V1xl cases are found:
+ V2xJjthe following special
1. When the molar volumes are equal, i. e., VI = VL, then
et?., and Equations 14 and 15 reduce to the Margules equations given by Equations 12 and 13. 2. When the ratio A / B is equal to the ratio Vl/V*--i. e., when A V ~ / B V I= 1-Equations 14 and 15 reduce to those of van Laar, Equations 8 and 9.
z1 = zl,
When the ratio AVz/BVI lies between A / B and 1, the curves will lie between those of Margulea and van Law.
and AE is the change in internal energy for vaporization of the pure liquids. Scatchard, Wood, and Mochel ( d l ) , from a study of the three binary systems formed by benzene, cyclohexane, and carbon tetrachloride, have concluded that Equation 18 did not give values of b agreeing with those found by measurement of vapor-liquid equilibria. If b, V1, and V Zare regarded as empirical constants, Equations 16 and 17 may he arranged in a form equivalent to the van Laar equations.
Figure Q. ficients vs.
A
'
'
Coef-
Mole ct'v'? ractions
A.
Isopropyl ether in isopropyl alcohol. Curve calculated by van Ldar equation with A = 0.49 and &? = 0.60. Points are measured values of Miller and Bliss (76) at atmospheric pressure.
B. n-Propyl alcohol in water. Curve calculated by van Laar equation with A = 1.13 and B = 0.49. Points are the observed ddta of Gadwa (6) at atmospheric pressure.
I
C. Acetone in chloroform. Curve calculated by van Laar equation with A = -0.44 and B = -0.34. Points are the observed data of Zawidski ( 2 4 ) a t 35.17"C.
II
D. Chloroform in ethyl alcohol. Curve calculated by Mar des equation with A = 0.21 an35 = 0.71. Points are the observed data of Scatchard and Raymond (no) st 35" c.
E. Water in n-butyl alcohol. Curve calculated by van Ladr equation with A = 0.61 and B = 1.34. Points were measured by Stockhardt and H u l l (93) at atmospheric pressure. Two liquid phases are present for compositions between the t w o (dashed lines.
0.4
0.6 XI
INDUSTRIAL AND ENGINEERING CHEMISTRY
5&1
As stated previously, the van Laar and hlargules equations differ most when the ratio A / B departs considerably from unity. To compare the sets of equations, Figure 3 has been caloulated with the value of A = 3B for each set. One of the Margules curves has a maximum and the other a minimum with this large ratio of A / B . By their mathematical nature the van Laar equations cannot show such shapes. Figure 3 also shows curves for the Scatchard-Hamer equations for a value of AV2/BVl different from those listed under special cases 1 and 2.
8.0
6.0
5.0 4.0
2(
3.0
2.0 1.5
1.0
0.9;
' '
0.2
1 0.4
'
I 0.6
I
I
8.8
'
'
1.0
XI
Figure 3. Comparison of A c t i v i t y Coefficients Predicted b y v d n b a r , Margules, and Scatchard-Harner Equations Curves pass through the same terminal values and w i t h a ratio of the logarithms OF the terminal activity e., A = 38. The Scatchardcoefficients = 3-i. Hamer equation is for AVZ/BV, = 9. van Laar Margules- - - - Scatchard-Hamer -
__
COMPARISON OF EQUATIONS WITH DATA. Two sets of reliable experimental data, for which A values are considerably different from B values, are those of Gadwa (6) on npropyl alcohol-water and of Scatchard and Raymond (go) on chloroform-ethyl alcohol (Figure 2, B and D). The n-propyl alcohol-water data are represented closely by the van Laar equations, with A = 1.13 and B = 0.49, whence A / B = 2.31. In this case the Margules equations would not have been good. On the other hand, the chloroform-ethyl alcohol data agree with the Margules equations when A = 0.21 and B = 0.71 ( A / B = 0.296), while the van Laar equations do not apply. The clue as to which equations to use apparently follows from the values of molar volume. These values for wpropyl alcohol and water are 75 and 18, respectively (at room temperat.ure), which are in the ratio of 4.16/1 On the other hand, the values of molar volume for chloroform and ethyl alcohol are 81 and 58.5, respectively, which is a ratio of 1.38/1. The former case approaches Scatchard and Hamer's special case 2, and the latter approaches case 1 since the molar volumes are more nearly equal. In general, the van Laar equations have been found t o fit many systems well. Since many important binaries include water, the molar volumes are bound to be different and the van Laar equations are then indicated. The example of ohloroform-ethyl alcohol is the only one we have encountered
Vol. 34, No. 5
where the Margules equations were definitely called for, but they no doubt will have other application on systems with molar volumes which are nearly equal but where the values of A and B differ considerably. Further work should show whether the complete form of the Scatchard-Hamer equations should sometimes be employed. POSITIVEus. KEGATWE DEVIATIONS.Although the large proportion of nonideal systems shows positive deviations (meaning the values of log y are positive-i. e., the values of y are above unity), some have negative deviations (y is fractional) such as acetone-chloroform (Figure 2C). Negative deviations occur in electrolytes and other liquids where association or compound formation of some type reduces the volatility. Such systems encountered include, in addition t o electrolytes: chloroform-acetone, chloroform-benzene, chloroform-ethyl ether, acetylene-acetone, acetylene-ethanol, formic acid-water, methylamine-n-ater, hydrogen peroxidewater, and acetic acid-pyridine. LIMITATIONS.Although these equations represent satisfactorily a large share of existing reliable data, some precautions are necessary in their use. These have to be considered chiefly where the vapors depart appreciably from the ideal. Thus the experimental curves of values of log y us. x show unusual shapes for such systems as acetaldehyde-water, acetic acid-water, and acetic acid-benzene. Here the vapors are associated and one cannot use Equation 5 as an approximation for Equation 4. Similarly, for cases of high pressure the values of y must be defined by Equation 4, although at moderate pressures Equation .5 has proved satisfactory. One aqueous system has been found where no set of equations appears t o apply exactly-namely, butyl alcoholwater (Figure 2E). Here there is a wide difference between the end values of the two curves and also an immiscible region; hence the system is not common. There is need of considerable research on partially miscible mixtures. One must conclude at this time that although the van Laar equations fit many systems very closely, there are a few where the agreement is not satisfactory. However, they can be employed as B first approximation and thereby provide a powerful tool for the extension of data, as described below. Examination of Vapor-Liquid Equilibrium Data
In examining vapor-liquid equilibrium data to determine their reliability, several procedures have been proposed. Plots of y vs. z similar to Figure 1, or of the relative volatility, a,us. x can be made. These methods show only the spread of the experimental points from a smooth curve and do not give any idea of the thermodynamic consistency. The differential form of the Gibbs-Duhem equation permits a few qualitative conclusions to be drawn from an inspection of a plot of log y us. zl. If the slope of the activity coefficient curve for one component on this type of plot is zero, corresponding to a maximum, the slope of the curve for the other component must be zero a t the same composition; Figure 2 0 is an example. In some cases the activity coefficients for compositions approaching 2 = 1.0 are below unity without any indication of a maximum in the activity coefficient curve of the other component; this indicates some error in the data. The Gibbs-Duhem equation indicates that plots of log yl and log y~ us. z1should have slopes of opposite sign a t a given composition. (Xumerically, the slopes are in the ratio zl/xz.) Thus, if one curve starts a t log y p = 0 and constantly increases, the other curve must steadily decrease as we pass from x1 = 0 to z1= 1.0. If one component has activity coefficients always greater than unity and is without a maximum point, the activity coefficient curve for the other coniponent must always be greater than unity. If the activity
May, 1942
INDUSTRIAL AND ENGINEERING CHEMISTRY
585
Courtesy, Texas
High-pressure
Air Makes
Gull Sulphur Company
Possible the Efficient Production of Sulfur
Every day 1,500,000 cubic Feet of air at 500-600 pounds per square inch pressure are used to liFt the molten sulfur to the surface.
coefficient of one component is less than unity and without an inflection point, the activity coefficient of the other component must always be less than unity. An integrated form of the Gibbs-Duhem equation, such as the van Laar equation, permits quantitative conclusions as t o the reliability of vapor-liquid equilibrium data more rapidly than the differential form of the Gibbs-Duhem equation. Comparison of the value of log y1 and log YZ measured at z1 = 0.5 and that usually extrapolated to XI = 0 and = 1.0 affords a rapid test of data. An earfier section of this paper showed that a t z = 0.5, log y2 is approximately one fourth of A , and log y1 is approximately one fourth of B. Measurement of the relative values of log y at the mid-point and end point on a semilog plot will give a rapid test of the data. Likewise the curve having the higher end value will be lower at the mid-point. Since Figures 2A, B, and C show that the van Laar or Margules equations are capable of fitting reliable determinations of activity coefficients for miscible systems, vaporliquid equilibrium data may be smoothed with some confidence by fitting the activity coefficient curves with van Laar or Margules equations. An inspection of data on a number of systems indicated that the majority of the determinations could be fitted better by the van Laar equation, as the type of curve shown in Figure 2 0 is relatively rare. Only a few determinations of vapor-liquid equilibria are so precise that they could not be improved by fitting the activity coefficients with an analytical equation, followed by calculation of equilibrium liquid and vapor compositions and boiling points from the smoothed activity coefficients. This method of smoothing activity coefficients rather than liquid and vapor compositions presupposes that the boiling points have been measured accurately and that the vapor pressures of the pure components are known. For convenience in calculating the constants of the van Laar and Margules equations, they have been solved simultaneously for A and B to give:
B = log yn (1
+ *>-xxz log
YZ
For the Margules equations written in terms of the terminal activity Coefficients:
xlg
B =
(21
Y2 I - w)log r:
2 log Y1
xz
(22)
Extension of Incomplete Data
In industrial work it is often necessary to estimate distillation apparatus on systems where complete equilibrium data for the conditions chosen are not available. Some of the types of data discussed below can sometimes be readily located, however, and, in lieu of complete information, will be found helpful. Estimation of Equilibrium Curve from Azeotropic Data
When complete measurements of the equilibrium liquid and vapor compositions are lacking, one may estimate the activity coefficients from the azeotropic compositions and boiling point. A wealth of azeotropic data has been tabulated by Lecat (11). Since the azeotropic liquid and vapor compositions are identical, the activity coefficients simplify to 7 1 = P/P1 (54 YZ = P/Pa (5B) One may solve for the constants in the van Laar or Margulea equations from these two values of the activity coefficient
INDUSTRIAL AND ENGINEERING CHEMISTRY
586
PRESSURE O N ETHYL.kCETATE-&HYL ALCOHOL AZEOTROPE
TABLEI. EFFECTOF Temp.,
18.7 40.5 56.3 71.8 83.1 91.4
C
P~/PI 0.598 0.725 0.827 0.921 0.994 1.049
Mole Fraction of Ethyl Acetate Calculated Measured (1.5: 0.787
0,734
0.602 0.539 0.498
0.601 0.539 0.490 0.461
0.677
0.480
Total Pressure, Mm. H g 77.4 220 423 760 1121 1476
0,660
by Equations 19 to 22. The whole activity coefficient curve and subsequently the x-y values may be calculated by substituting these constants in the van Laar or Margules equations. Allen (1) employed the original form of the Margules equation for this purpose. This procedure is most accurate when the azeotrope lies in the middle half of the composition range. When the azeotrope lies betv-een x = 0.75 and z = 1.0, only the value of B will be accurate. Inspection of many systems has shown that more cases are fitted by the van Laar equations than by the hfargules expansion.
Vol. 34, No. 5
perature the composition of the azeotropic mixture was found for which the ratio of the vapor pressures equded the ratio of the activity coefficients. Comparison of the predicted values with those measured by Merriman is shown in Table I. Calculation of x - y Data from Total Pressure or Boiling Point Curves
From an isothermal measurement of total pressure us. liquid composition or an isopiestic boiling point-composition curve, the van Laar or Margules equations may be employed to estimate equilibrium liquid and vapor compositions. Levy (12)reviewed the numerous procedures proposed in the past and developed one for use mith the Margules equation. The following method may be employed with either type of equation. Use of the method to calculate x-y values from boiling point curves is based upon the assumption that the activity coefficients do not change with temperature Whether working with total pressure or boiling point data, the procedure is the same. By definition,
As x1 approaches 1.0, y z approaches 1.0 so that we may calculate an approximate activity coefficient for component 1 by assuming y2 = 1.0. By plotting these apparent activity coefficients on semilog paper similar to Figure 2, we may extrapolate to find the terminal values of the activity coefficients, whose logarithms are the constants in the van Laar or Margules equations. The procedure is illustrated in Table I1 with the isothermal total pressure data of Hovorka, Schaefer, and Dreisbach (9) for the system dioxane-water a t 80" 0 t'C.
20 AND
40
MOLE PER
60 80 CENT ETHYL ACETATE
c.
100
Figure 4. Effect of Temperature on the Composition of the Ethyl AcetateEthyl A l c o h o l Azeotrope At a given temperature the azeotropic composition i s where Pz/Pi = Y I / Y ~ .
For example, Smith and Rojciechowski's measurement (29) of the dioxane-water azeotrope of 0.53 mole fraction water, boiling at 87.8" C. at atmospheric pressure, may be used to calculate that y1 = 1.57 and y~ = 1.55. The constants in the van Laar equation, A and B , may be calculated to be 0.68 and 0.88, respectively, which are close to the values calculated from total pressure data in a later section. Effect of Pressure on Azeotropic Composition
The change in composition of the azeotropic mixture with pressure may be estimated from the activity coefficients at one pressure and the vapor pressures of the pure components. By dividing Equation 5A by 5B,the ratio of the activity coefficients for the azeotropic mixture is the inverse of the vapor pressures of the components. To illustrate the procedure, Merriman's values (15) for the composition and boiling point a t atmospheric pressure of the ethyl acetateethyl alcohol azeotrope have been used to calculate the constants of .the van Laar equation, A = 0.39 and B = 0.374. Values of the activity coefficients were calculated at several compositions, and their ratio was plotted against composition on Figure 4. The ratio of the vapor pressures was plotted against temperature. Lacking any data on the effect of temperature on A and B, the ratio of the activity coefficients w5s assumed independent of temperature. At a given tem-
TABLE11. CALCULATION OF v h i LUR CONSTANTS FROM TOTAL PRESSURE DATA 0.1 0.2 0.3 0.4
476 526.5 556 571
35.5 71.0 106.5 142.1
345 307 268 229
0.6
575.6 669.5 550 601.5
213 248 284 319
153.5 115 76.6 38.3
0.7 0.8
0.9
..
3.69 3.08 2.70 2.40
.. 2.36 2.79 3.47 4.78
Extrapolation of the approximate values of 71 and y2 to = 0 and x1 =: 1.0, gives y1 = 4.36 and y~ = 7.4. The logarithms of these terminal values of the activity coefficients give the constants in the van Laar equation, A = 0.64 and B = 0.87. These constants have been used in the van Laar equation to recalculate the total pressure for comparison with the measured values (Table 111). If an acceptable fit is not obtained on the Grst try, approximate values of 72 may be substituted in Equation 23 for 71, leading to new values of A and B by extrapolation.
x1
TABLE zb
Yl 7 2
PI
P2 Poalcd. P0bS"d. IOO(Pobsvd.
111.
CALCULATED AND TOTAL PRESSURES
COMPARISON O F 0.1
3.51 1.012
124.6 358 482.5 476
- Ponlad.)/Pobsvd. -1.4
31EASURED
0.2 2.85 1.05 202 322 524 526.6
0.4 1.94 1.24 276 285 561 571
0.6 0.8 1.40 1.10 1.74 3.06 298 312 234 266 564 546 575.6 550
+0.5
+1.8
f2.0
+O.7
0 9
1 025 4.32 328 173 501 501.3 +O
1
May, 1942
587
INDUSTRIAL AND ENGINEERING CHEMISTRY
Prediction of Vapor-Liquid Equilibria for Partially Miscible Solutions
Binary liquid systems frequently form two liquid phases over a range of compositions. For systems said t o have an upper critical solution temperature, the immiscible zone decreases as the temperature rises until the liquids become completely miscible a t the critical solution temperature. Since the partial pressure or activity of a component is the same from each liquid phase at constant temperature, it is possible to calculate the vapor-liquid equilibria from measurements of liquid-liquid solubility. For systems which are mutually insoluble, the vapor pressures of the pure components are additive. For systems so slightly soluble that Raoult's and Henry's laws may be assumed, Carey (4) describes a procedure for calculating the equilibrium liquid and vapor compositions. For a system near the critical solution temperature, these assumptions are no longer valid. Scatchard and Hamer (19) employed Equations 12 to 15, calculating the parameters from the liquid-liquid solubility of aniline-water, phenol-water, anilinehexane, and platinum-gold. They found that Equations 14 and 15 in terms of volume fractions gave better agreement with the limited amount of experimental data than the Margules equations in terms of mole fractions. Equations for calculating the parameters of either the Margules or Scatchard-Hamer equations may be developed by denoting the mole fraction, volume fraction, and activity coefficients of component 1 in one of the liquid phases by xl,z,, and yl, respectively, and by 21,Zl, and in the Other liquid phase. Since the activities of component 1 in both phases are equal,
Figure
Plot of van Laar Constants cording to Equation 18
Ac-
For binary mixtures of benzene with methyl, ethyl nand isopropyl, sec- and tert-butyl alcohols. f h e squares are the van Laar constants OF the alcohols, and the circles are those for benzene! the number OF radiating lines denotes the number of carbon atoms in the alcohol.
agreement with the measured values is rather poor. These methods may be expected to give only rough estimates of the deviations from Raoult's law. Interpretation of x-y Data, Lacking Temperature Measurements
rl
It is frequently necessary to estimate an z-y curve for reduced pressure when only z-y data are available a t atmospheric pressure. A procedure similar to that for calculating activity coefficients from total pressure-composition data may be used to calculate the constants in the van Laar or Taking the logarithms of both sides of these equations and Margules equations. Employing these constants, assuming substituting Equations 14 and 15, no variation with temDerature or bv making some estimate*of the change, log = log 3 A (I' (25) the x-y curve may be calculated a t 7 1 XI reduced pressure. l o g 2 = log-x2 x2 = B [(2$ - 1) (2," - 2:) - 2 - 1) ( I : - 5,9)] (26) The trial-and-error procedure of cal72 culating the activity coefficient is started by assuming Raoult's law for Dividing Equation 25 by Equation 26, and solving for A / B , the component present in excess, calculating the boiling point to satisfy log - .i; - 2; i : ) (log ( r : - z: - 22: 2 4 the known x,y, and pressure by Raoult's AJB = (27). law, This boiling point permits cal+ 5;) (log (%I - i: - 22; f 2ig culation of an amroximate activity coefficient for the c-omponent present Substitution of A / B in Equation 25 or 26 will solve for A or small quantities. These approximate activity coefficients may B. When VI = Vz,this will give the solution to the Margules be plotted on semilog paper and extrapolated to obtain A Equations 12 and 13. and B. Activity coefficients and values of z and y are calBy a similar procedure, a solution for the van Laar conculated by the van Laar or Margules equations. Further tants may be obtained as: adjustment of A and B may be required to obtain a good representation of the xy curve. This second trial may be + ig) (log w x 1 -2 carried out using approximate activity coefficients of the A 52 log x2/& component present in excess in calculating the boiling point. (28) = 3 a, 2x1 5 log % / X I 2 2 + h - 5 2 2 2 log x2/& Homologous Series A = log % / X I It is occasionally desired to extrapolate or interpolate to find the deviations from Raoult's law for a member of a homologous series of compounds with another type of compound. For instance, one might want to predict the activity From the mutual solubility of n-butyl alcohol-water a t coefficients for n-propyl alcohol-water mixtures, knowing 100" C. (XI = 0.9765 and Z1 = 0.677), the constants in the those for methyl and ethyl alcohol with water. van Laar equations may be calculated to be A = 0.334 and The similarity of Scatchard's equation for regular solutions B = 1.60. Since A V*/BVl = 1.07for the system, the paramwith the van Laar equation suggested that Equation 18 eters of the Scatchard-Hamer equations are the same. might serve a.a a basis for correlating the constants of the van Comparison of these values with Figure 2E shows that Laar equation. Thus, one would plot A and B against
:
iE
[(% ')
2)(.:
("
(E
z
*
5.
+
-)
")
(2 (g
+ +
2)
5)
'11
) ("
+
)
INDUSTRIAL AND ENGINEERING CHEMISTRY
588
Vol. 34, No. 5
ferent. Hildebrand (7) noted that isomeric nitrobenzenes have the same activity coefficients in benzene solutions. Further improvements in the method are needed before the correlation of the isomers on Figures 5 and 6 will be satisfactory. Effect of Temperature on Vapor-Liquid Equilibria
Figure 6. Plot of Activity Coefficients According to Equation
18
A t infinite dilution for mixtures of methyl, ethyl, n- and isopropyl, n-, iso-, sec-, and tert-butyl alcohols in water. The squares give values for water,, and the circles represent those For alcohol; the number of radiating lines denotes the number of carbon atoms in the alcohol.
V
AE
[(+)I
/
AE2 '/ ' - (K)
'1
J
placing VI outside the bracket
when plotting against A and Siz when against B. Figures 5 and 6 are plots of this type for the homologous series of aliphatic alcohols with benzene and water. The binary vapor-liquid equilibria for mixtures of benzene with the alcohols taken from measurements between 25" C. and the normal boiling point were fitted with van Laar equations. Since the activity coefficients of the higher alcohols in water cannot be fitted exactly with a van Laar equation, terminal values of log y, obtained by extrapolation, were used in Figure 6. Although Figure 6 shows that log y is roughly proportional to Vb for the aqueous solutions of the alcohols, log y is five to seven times greater than predicted by Equation 18. The values of A and B in Figure 5 for the benzene solutions are not proportional t o Vb, although the relation is linear. Extrapolation or interpolation on this type of plot should enable a rough prediction'of activity coefficients to be made. It should be noted that the mathematical form of Equation 18 will not predict negative deviations for Raoult's law. Inspection of Figures 5 and 6 shows that activity coefficients of the aqueous solutions increase with increasing molecular weight of the alcohol, while those in benzene solutions decrease with increasing molecular weight. The original equations of van Laar ( I O ) relating the constants to those of the van der Waals equation give no better correlation, and the van der Wads constants would usually have to be predicted from the critical data. Butler (2) measured the partial pressures of esters, nitriles, amines, alcohols, and acids in dilute solutions in water a t 25" C. Butler and Harrower (3) presented similar data for alcohols and alkyl halides in benzene, carbon tetrachloride, and cyclohexane. They noticed that, while the activity coefficients change rather rapidly with the molecular weight of the solute, the quantity RT In ( p / s ) increases or decreases regularly. With benzene, carbon tetrachloride, or cyclohexane solvents, RT In ( p / z ) decreases 600 calories/mole for each carbon atom added to the alkyl halide or alcohol. In aqueous solutions RT In ( p i % ) increases between 100 and 240 calories/mole for each carbon atom added to the radical. As Figures 5 and 6 shorn, isomeric alcohols do not have the same van Laar constants, although they are not widely dif-
When s-y data are available at atmospheric pressure and an estimate of the x-y curve a t reduced pressure is desired, the effect of temperature on the activity coefficients needs to be considered. There is a similar problem in the fitting of a van Laar or Margules equation to isopiestic data covering a range of 2 0 4 0 " C., because the Gibbs-Duhem equation and its integrated forms are valid only a t constant temperature. The change of the activity coefficient with temperature is related to the relative partial molal enthalpy, L , by the thermodynamic equation,
h ._
RT2
L1 is the partial molal enthalpy of component 1 in solution minus the enthalpy of the pure liquid a t the same temperature, and may be visualized as the heat absorbed on adding a mole of component 1 to an infinite quantity of solution. When heat is evolved on mixing two liquids, Lis negative and the activity coefficient rises with the temperature. The labor of calculating the effect of temperature on the 2-y diagram is considerably reduced when the activity coefficient curves can be fitted with van Laar or Margules equations. These equations have been written in the form so that the constants are terminal values of log y. Thus, the change of A and B with temperature may be related to L a t z1= 0 and z1 = 1.0, respectively. An inspection of the values of L a t infinite dilution calculated from heats of mixing liquids in the International Critical Tables reveals values as high as 2000 calories/mole for mixtures of organic liquids. For mixtures of alcohols with water, the partial molal enthalpy may reach 5000 caloriesimole. As a general rule, systems of organic liquids having positive deviations from Raoult's law have positive values of L; and for systems having negative deviations, L is negative. As a result, activity coefficients for systems of organic liquids, having either positive or negative deviations from Raoult's law, approach unity as the temperature rises and thus approach Raoult's law as a limit. For aqueous solutions the change is not so simple inasmuch as the value of L frequently reverses its sign in the ordinary temperature range. Activity coefficients for immiscible systems with an upper critical solution temperature must decrease with rise in temperature. To give an idea of the order of magnitude of the change in the activity coefficient with temperature, a change of 6 per cent in the activity coefficient results from a 30" C. temperature change with L = 500 calories/mole. If the lefthand side of the various equations had been written RT In y instead of log y, and the resulting parameters had been regarded as independent of temperature, the activity coefficients would approach unity as the temperature increased, agreeing with the general rule for organic liquids. Unfortunately, most of the available data on L have been measured only a t room temperature, and there are indications that L changes rapidly with temperature. Ternary Vapor-Liquid Equilibria
Methods of predicting ternary equilibrium dsts from known data on the three binary systems and of extending the available ternary data are of increasing importance t o the
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
May, 1942
organic chemical industry. Where experimental data on a ternary system are available, they may be tested by calculating the activity coefficients of the three components and making three separate plots of log y of one component vs. x of that component. On each plot two reference curves can be drawn representing the given component with each of the other two in a binary mixture. On such plots many of the existing data on ternary systems have been found inconsistent and therefore unreliable. Lacking precise data, to suggest a general method of predicting intermediate values of y from those in binary mixtures only, a few possibilities were studied theoretically. At first glance i t might be expected that the logarithm of the activity coefficient for, say, component 1 could be interpolated linearly between the two binary curves, depending upon the relative amounts of components 2 and 3. While this procedure is workable aqd has been used in the absence of ternary data, nevertheless in one hypothetical case the resulting curves were not in agreement with the Gibbs-Duhem equation ; z1d In y I
+ z2d In yz + z3d In ys = 0
Scatchard (18) developed the equations for the activity coefficients of a regular ternary mixture, and Scatchard, Wood, and Mochel ( M )applied a slightly modified form to predict ternary data for benzene-carbon tetrachloridecyclohexane mixtures. These equations for the regular ternary solutions obey the Gibbs-Duhem equation. Their use is limited because there are few regular binary solutions or even solutions whose values of A and B are in the same ratio as the molal volumes. By substituting arbitrary values of the constants in one case, we found that the curve of activity coefficients for component 1 plotted against for equal amounts of components 2 and 3 crossed the lower binary curve. I n this case a linear interpolation would have been greatly in error. Calculation of Extraction Equilibria
When values of y1 are available for the solute in each of two solvent liquids, partition coefficients may be calculated by employing the principle that the activity of the solute is the same in both solvents. Denoting the activity coefficient of component 1 in solvents 2 and 3 as ylzand 713, respectively, we may express the equality of activities as: YlZzl2
=
?‘I3213
are hopeful that the paper may stimulate further research in the field of vapor-liquid equilibrium, particularly of homologous series of compounds, of ternary systems, and of heats of solution above room temperature. Above all, i t is hoped that the paper will show the importance of the measurement and publication of the complete data on compositions, prFssure, and boiling point required for calculation of a c t m t y coefficients for binary and ternary systems. Nomenclature
A
= arbitrary constant in van Laar, Margules, and Scatchard-
a
= activity referred to pure substance as the standard state = arbitrary constant in van Laar, Margules, and ScatchardHamer equations, and equal to log yz at zz = 0 = constant in Equations 16 and 17, defined by Equation 18
B
(32)
where z12and 2 1 3 are the mole fractions of component 1 in the liquid phases, composed mainly of components 2 and 3, respectively. The ratio of 212/213, recognized as a partition coefficient, is then equal to y13/yl2. A convenient method of evaluating equilibrium values of x12 and 213, as shown by Hildebrand (8),is to plot ylzzI2against xIzJand 7 1 8 5 1 3 against xla. From the same ordinate, values of zls and 2 1 3 as abscissas are read from the two curves. When the mutual solubilities are small, the activity coefficients may be taken from the binary activity coefficient curves. For systems of higher mutual solubility, knowledge of the activity coefficients in ternary systems would be necessary. Conclusion
This paper has been offered to point out to the engineer ways of evaluating and extending the available data relating to vapor-liquid equilibria. It is hoped that some of the methods reviewed will also be found useful by the experimenter in smoothing and presenting new data. The authors
H-er
equations, and equal t o log y1 at
21
= 0
b AE = change in internal energy on vaporization of pure com-
F
ponent, cal./mole
= free energy, cal./mole =
$0
L (31)
589
fugacity
= fugacity of pure component = partial molal enthalpy referred to pure liquid, cal./mole
P = total pressure, mm. Hg. PI, Pa = vapor pressures of pure components, mm. Hg = partial pressure = Py, mm. Hg = gas law constant, 1.9870cal./(mole)(0K.) T = absolute temperature, K. V = molal volume of pure liquid, usually evaluated at room
H
temperature, cc./mole
x
= mole fraction in liquid
y z
= =
mole fraction in va or in equilibrium with z volume fraction in {quid
a
= relative volatility, ylsr/y~zl
y
=
activity coefficient
Subscripts 1 = component of binary mixture with lower boiling point = components of progressively higher boiling points
Bib1iography Allen, C., IND.ENO.CHEM.,22, 608-9 (1930). Butler, J. A. V., Trans. Faraday SOC.,33, 229-36 (1937). Butler, J. A. V., and Harrower, P., Ibid.. 33, 171-5 (1937). Carey, J. S.,in Perry’s Chemical Engineers’ Handbook, 1st ed., p. 1157, New York, McGraw-Hill Book Co., 1934. Carey, J. S., and Lewis, W. K., IND.ENG.CHEM.,24, 882-3 (1932). Gadwa, T. A., in Perry’s Chemical Engineers’ Handbook, 2nd ed., p. 1367, New York, McGraw-Hill Book Co., 1941. Hildebrand, J. H., “Solubility of Non-Electrolytes”, 2nd ed.. p. 83, New York, Reinhold Publishing Corp., 1936. Ibid., p. 184. Hovorka, F., Schaefer, R. A., and Dreisbach, D., J . Am. C h m . SOC.,58, 2264-7 (1936); 59,2753 (1937). Laar, J. J. van, Z . physik. Chem., 72, 723-51 (1910); 83, 599608 (1913). Lecat, M.,“La tension de vapeur des m6langes de liquidea. L’Azeotropisme”, Brussels, Henri Lamertin, 1918; subsequent articles in Ann. SOC. sei. Bruxelles. Levy, R. M., IND.ENG.CHEM.,33, 928-31 (1941). Lewis, G. N., and Randall, M., “Thermodynamics and the Free Energy of Chemical Substances”, New York, McGraw-Hill Book Co., 1923. Margules, M., Sitzber. Akad. W i s s . Wien, Math. natutw. Klasse, 11, 104, 1243-78 (1895). Merriman, R. W.,J . Chem. SOC.,103, 1801-16 (1913). Miller, H.C., and Bliss, H., IND. ENQ.CHEM.,32,123-5 (1940). Scatchard, G., Chem. Reu., 8, 321-33 (1931). Scatchard, G.,Trans. Faraday SOC.,33, 160-6 (1937). Scatchard, G., and Hamer, W. J., J . A m . Chem. SOC., 57, 1805-9 (1935). Scatchard, G., and Raymond, C. L., Ibid., 60, 1278-87 (1938). Scatchard, G., Wood, S. E., and Mochel, J. M., Ibid., 62, 71216 (1940). Smith, E. R., and Wojciechowski, M., J . Research Natl. Bur. Standards, 18,461-5 (1937). Stockhardt, J. S., and Hull, C. M., IND. ENO.CHEM.,23. 143840 (1931). Zawidski, J. von, Z . physik. C h m . , 35, 129-203 (1900).
