TIE LINES IN
rnary Liquid Systems INARY liquid mixtures can often be separat economically by extraction with a third liquid; consequently, liquid-liquid extraction has found an ever-increasing use in industry. Recently there has been a consider4ble increase in the use of mixed-solvent extraction, particularly in petroleum refining. I n this operation the third liquid, or solvent, consists of two liquid components which are completely miscible with each other. Usually only one of them, however, is completely miscible with the mixture being only partially miscible with the stock mixture. Mixed-solvent extraction leads to the handling of four-component liquid systems and t o rather complicated stoichiometric calculations. As a result i t has not found general application in spite of its frequent advantages, for the experimental data available for four-component liquid systems are extremely limited, and the methods of representing them have been complex and inadequate. Brancker, Hunter, and Nash presented some data for systems involving lubricating oils ($), and rather complete data for the system chloroform-acetic acid-acetone-water (3), but outside
B
JULIAN C. SMITH' Wilmington, Del.
A method of representing equilibria in quaternary liquid systems by means of plane graphs is described. Graphs permit accurate interpolation and the rapid solution of problems in mixed solvent extraction.
of these, few quaternary systems have been studied. However, Hunter (6) described a method of calculating equilibrium data for quaternary systems from the data for the two ternary systems involved. If shown t o be general, this method will be extremely valuable, for a great many ternary liquid systems have been studied, Sources of data for ternary systems were compiled by Smith, Elgin, and others (4,7,8). The caloulations described by Hunter, however, are rather tedious, and require that for each point to be determined a complete set of calculations be made. When only one or two points are desired, this is not important; but if considerable work is t o be done on a given system, it is much more convenient t o have a means of representing the data such that interpolation can be made between a few known points, Figures 1, 2, and 3 are presented t o accomplish this, They are familiar types of plane graphs, and do not involve three-dimensional figures or orthogonal projections, such as are commonly used t o represent quaternary systems. They are based on the one system for which experimental data are availablechloroform-acetic acid-acetone-water, but are quite general in form and should be adaptable to most quaternary liquid systems. As mentioned above, their use is warranted
1 Prerent addreas, E. I. du Pout de Nemoum & Company, Ino., Wilmington. Del.
useful as a means of representing them. CONSTRUCTION OF CHARTS
It n shown (1,6) that the equilibrium distr of a liquid, D, between the layers of two partially miscible liquids, E and F , may usually be represented by an equation of the form: Figure 1. Equilibrium Distribution in Chloroform-AcetoneAcetic Acid-Water System at 25' C. 68
log &le. = B log
4/f/+ log k
(1)
January, 1944
where d , = wt. fraction of e, = wt. fraction of d/ = wt. fraction of wt. fraction of 8, f = constants
69
INDUSTRIAL A N D ENGINEERING CHEMISTRY
D in E-rich layer E in E-rich layer D in F-rich layer F in F-rich layer
By plotting log dele, against log d//jj, therefore, the equilibrium distribution for ternary systems may be represented by a single straight line; in Figure 1 straight lines AP1 and BPZ, respectively, indicate the distribution in the systems chloroform-acetone-water and chloroform-acetic acid-water, both of which have been shown to obey equations similar to Equation l. The quantity w equals the weight fraction of water, and c the weight fraction of chloroform. To make the chart cover both ternary systems and the quaternary system, and yet introduce only one more variable, m was made equal to the sum of the weight fractions of acetone, a,and acetic acid, h. For the system chloroform-acetone-water, therefore, m equals a; for the system chloroform-acetic acid-water m equals h. Equation 1 applies to nearly all ternary systems in which one component is completely miscible with the other two, which is the type of system most commonly used in extraction operations. A few systems are known which do not follow this equation but give a curve instead of a straight line when plotted as described above. Because of the curvature, extensive data for these systems are necessary; in the usual case a few accurately determined points completely define the equilibrium distribution. The system chloroform-acetone-water was studied by Hand (6), Brancker, Hunter, and Nash ($), and recently by Bancroft and Hubard ( I ) . All the data for mixtures containing high concentrations of acetone are in good agreement, but for low concentrations the agreement is not good. Only Brancker, Hunter, and Nash investigated mixtures containing very small amounts of acetone, and their data for this range are almost certainly inaccurate. However, as the method of calculation described by Hunter (6) is based on these data, and as they are the only data available for this system which cover a wide range of concentrations, they were used in the present example. They give the distribution indicated by line AP1 in Figure 1. Both Wright (10) and Brancker, Hunter, -1.6 and Nash (3) studied the system chloroformacetic acid-water. I n this case the agreement between the investigators is excellent; all the data may be accurately represented by straight line BPZ in Figure 1. The probable plait points for the ternary systems are shown in Figure 1 a t P I and P , Data for the two systems are given in Table I. Points in Figure 1 lying between the lines defining the ternary systems represent equilibria in the four-component system. The ternary equilibrium lines thus form boundaries for the quaternary system. Brancker, Hunter, and Nash (3) showed that for the system chloroform-acetone-acetic acid-water, when the quaternary system is represented by the usual tetrahedron, the quaternary tie lines lie in planes passing through tie lines in the ternary systems and through the opposite apex of the tetrahe-
Figure 2.
-1.2
Values of rnz/w* or L
- 0.4 L O G >(
-0.8
0
0.4
Figure 3. Values of hl/at or R
dron. The quaternary tie lines terminate in the quaternary equilibrium surface. Thus for any tie line in one of the ternary systems there is a series of quaternary tie lines which lie in the plane passing through the ternary tie line; in Figure 1 such a series is represented by a curve which passes through a point in the ternary equilibrium line, and which is apparently asymptotic to the other ternary boundary. Other planes may be passed through the tie lines in the other ternary system, giving rise to other series of quaternary tie lines, which are represented in Figure 1 by curves originating in the line defining the other ternary
zo
INDUSTRIAL AND ENGINEERING CHEMISTRY
system. The intersection of two planes in the tetrahedral representation completely defines a quaternary tie line; on Figure 1 the same tie line is defined by the intersection of two curves. Unfortunately the experimental data available even for the system chloroform-acetone-acetic acid-water are not sufficient t o test thoroughly this method of representation, since only two series of quaternary tie lines were determined. Quaternary equilibria were therefore calculated by the method described by Hunter, corresponding to six series in the system chloroformacetone-water and five in the system chloroform-acetic acidwater, and covering ranges on both sides of the experimental data. The experimental data are summarized in Table 11, and the results of the calculations are given in Table 111. I n these tables the number of the tie line refers t o the series or plane in which it lies. For example, tie line 2-5 is the intersection of the planes through tie line 2-0 in the system chloroform-acetone-water, and tie line 0-5 in the system chloroform-acetic acid-water. Thus tie lines 2-0 t o 2-6 lie in one plane, and define the line marked 2 on Figure 1; lines 0-5 t o 3-5 are likewise one series, represented by the line marked 5 on Figure 1. The values of X and Y obtained from Figure 1are not, however, sufficient t o defme the equilibrium completely, even for the ternary systems. Two more ratios are required, the most convenient of which are 7ptq/ws, or L, and hl/al, or R. Lines representing constant values of L are given in Figure 2; values of R may be obtained from Figure 3. These charts were constructed by first plotting the calculated values of ratios L and R as functions of X , for each of the series of tie lines originating in the system chloroform-acetone-water. From the resulting graphs the values of X corresponding t o even values of L and R were determined by interpolation. The corresponding values of Y were then read on Figure 1from the values of X and the lines representing the series of quaternary tie lines.
TABLIE I. EQUILIBRIA AT 25' C. Tie Line
--Upper
FOR
Two TERNARY SYSTEMS
Composition, Weight % layer-Lower ai
NO.
c1
1-0 2-0 El0 3-0 4-0 5-0
Chloroform-Acetone-Water (3) 3.0 96.0 1.0 90.0 8.3 90.5 75.0 1.2 85.0 66.4 13.5 1.5 17.4 81.0 60.0 1.6 22.1 76.1 55.0 1.8 31.9 66.0 45.0 2.1 44.5 51.0 35.0 4.5
6-0
W1
El
layerai
10s
9,0 23.7 32.0 38.0 42.5 50.5 57.0
1.0 1.3 1.6 2.0 2.5 4.5
8.0
Chloroform-Acetio Acid-Water (IO) -Upper layer-Lower
layerha WI 6.5 92.6 98.2 1.1 0.7 0-1 0.9 81.5 0.8 17.7 95.0 3.8 1.2 0-2 1.2 25.1 73.7 91.8 6.8 1.4 0-3 34.1 63.3 87.8 10.8 1.4 E2 b 2.6 48.6 7.3 44.1 80.0 17.7 2.3 0-4 50.2 34.7 70.1 25.8 0-5 15.1 4.1 0 The ternary tie line in experimentally determined series 1 Table 11. b The ternary tie line in the experimentally determined Berids 2. Table 11. c1
hi
M
C1
TABLE 11. EXPERIMENTALLY DETERMINED QUATERNARY EQUILIBRIA AT 25" C. (3) CI
W1
1.8 3.7 9.8 19.1
71.7 57.1 41.2 28.3
Composition, Weight % ' -Lower ai hi CZ Series 1 15.3 11.2 63.5 29.9 58.8 9.3 10.8 38.2 51.8 39.0 13.6 42.5
3.1 3.5 3.9 4.4 6.9
59.5 87.4 54.8 61.4 44.5
4.6 7.5 10.9 15.4 24.4
-Upper
layer-
Series 2 32.8 76.8 31.6 67.3 30.4 57.9 28.8 47.3 24.2 30.8
layerwa
as
ha
1.6 3.3 6.2 10.6
31.6 29.2 25.8 22.8
3.3 8.7 16.2 24.1
1.8 2.8 4.0 6.3 13.7
11.8 20.3 28.9 36.8 44.8
9.6 9.6 9.2 9.6 11.2
Vol. 36, No. 1
The values of X and Y for given values of L and R defme the curves on Figures 2 and 3. As a check, the operation was repeated using the series of tie lines originating in the system chloroformacetic acid-water. I n this case the ratios were plotted as functions of Y,and the corresponding values of X were read from Figure 1 as before. Approximately 180 points were used in constructing each of the figures to ensure a reasonable degree of accuracy. From X , Y , L , and R and the total weights of the four liquids, the entire quaternary equilibrium may be calculated by substitution in the following equations:
-
+ LY + Y )
w2 Y/(L ma = wzL ea = %/Y
-
-
V = T ( M - WX)/(m2 WZX) ( M - m2(T - V ) ) / V ml w1 = m l / X c1 = 1 m1 w1 hi = R m J U R) al = ml hl az = ( A - a l V ) / ( T - V) hz = m2 a2
- + -
Equation 2 is derived by the elimination of ing equations:
--
m2 = c2Y = (1 wz ms w2L
ms from the follow-
- m)/Y
(13) (14)
The derivation of all the other equations except 5, 6,and 11 is obvious from the definition of the quantities involved. These three are derived from the material balance equations:
mlV WiV alV
+ mz(T - V) = M + W Z ( T- V ) W + az(T - V) = A
(15) (16) (17)
Equation 5 is obtained by the substitution of wlX for m1 in Equation 15, followed by the elimination of W I from Equations 16 and 16. Equations 6 and €1are obtained by the rearrangement of Equations 16 and 17. The agreement between the experimental data and the values obtained from Figures 2 and 3 is only fair. This is t o be expected, however, as the method of representation illustrated by these charts is very sensitive. The ratio L involves percentages of water usually below 5 per cent, and a slight experimental error changes the ratio m/wa greatly. I n addition, inaccuracies in the data for the ternary systems cause errors in the calculated quaternary values. I n spite of this the charts are sufficiently accurate for nearly all engineering calculations. An idea of their accuracy F a y be gained from the following example, which is the same as that used by Hunter t o illustrate his method of calculation. EXAMPLE
A mixture consisting of 27.7 per cent by weight of acetone and 72.2 per cent of chloroform is to be batch-extracted at 25' C. in R single stage with a mixed solvent. The composition of the mixed solvent is 58.5 per cent by weight of water and 41.5 per cent of acetic acid. The ratio of the weight of mixed solvent t o the weight of treated mixture is t o be 0.93. Assuming that equilibrium is reached in the single-stage operation, what will be the composition of the two resulting layers? The conditions in this typical problem in mixed solvent extraction determine the tie lines in the two ternary systems involved. The equilibria in these systems must first be determined by calculating the over-all composition on an acetone-free and an acetic-acid-free basis, respectively, and obtaining the equilibrium distribution from the solubility diagrams for the ternary systems. From these and from Figures 1, 2, and 3 the values of X, Y , L, and R may be determined; substitution of these values in Equa-
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
January, 1944
tions 2 t o 12 permits the direct calculation of the quaternary equilibrium. The calculations are as follows: Basis:
100 lb. stock mixture, 93 lb. mixed aolvent T = 193 lb. A = 0.278 X 100 = 27.8 Ib. C = 0.722 X 100 = 72.2 lb. W = 0.585 X 93 = 54.4 lb. H = 0.415 X 93 = 38.6 lb. M = 27.8 38.6 = 66.4 Ib. EQUILIBRIUM IN SYSTEM CnLoRoEQUILIBRIUM IN SYSTEMCHLOROFORM-ACETIC ACID-WATER FORM-ACETONE-WATER Over-a11 composition (acetoneOver-all composition (acetio-acidfree basis) free basis)
+
%:;$
3 2 : $ $ chloroform water 23.4% acetic acid Upper layer 6 3 . 0 7 water (wt) 3 . 0 2 chloroform (ct) 34.0% acetic acid (ht) log hi7lL.i = -0.264-. Lower layer 2.0% water (WZ) 8 6 . 6 7 chloroform (cz) 1 1 . 4 9 acetic acid (hz) log hz9cz = -0.879
water chloroform 18.0% acetone Upper layer 90.0% water (wt) 1 07' chloroform (ct) 9 acetone (a) log al/wt -1 .OOO Lower laver 1.4%- water (wz) 75.2% Chloroform ( c 3 23.4% acetone ( a d log ~ Z / C I = -0.504
.'OCA'
-
The logarithms of the ratios calculated above determine points M and N in Figure 1. The intersection of the two curves through these points determines the values of log X and log Y for the quaternary system; from these and from Figures 2 and 3 the values of L and R are obtained. The results are as follows: l o g X = -0.170; X = 0.676 log Y = -0.340; Y = 0.457
L
= 11.0
R = 5.0
Substitution in Equations 2 to 12 gives the quaternary equilibrium values shown below. The calculated and experimental values of Hunter are included for comparison: -Composition, Phase Upper layer Present calcn. Calcd., Hunter Exptl Hunter Lower 1lyer Present calcn. Calcd., Hunter Exptl., Hunter
W
0
Wt. %a
h
57.5 57.8 57.4
3.6 3.8 3.5
6.5 6.6 7.5
32.4 31.8 31.6
2.8 3.3 2.8
66.7 66.5 67.3
21.3 20.2 20.3
9.2 10.0 9.6
An interesting mathematical analysis of problems in mixed solvent extraction was recently made by Wiegand ( 9 ) , which likewise reduces the time and labor involved in their solution. The present method is perhaps more general than the one he proposes, for it is not necessarily limited to quaternary liquid systems. It may prove useful in the solution of other problems involving the distribution of four components between two phases. On large graph paper the three figures might be superimposed SO as to show the entire quaternary equilibrium on a single chart, but for convenience and illustration they were drawn as separate figures. This method of representing four-component liquid systems, as shown by the above example, is at least as accurate as the method of calculation described by Hunter. It does not, of course, do away with the need for extensive data, either experimental or calculated, for the quaternary system under consideration. It does, however, provide a ready and usable means of representing such data, and permits easy, rapid, and accurate interpolation between relatively few known points. NOMENCLATURE
a = weight fraction of acetone A = total weight of acetone c = weight fraction of chloroform h = weight fraction of acetic acid L = ratio rnz/we rrt = sum of weight fractions of acetone and acetic acid M = total weight of acetone plus acetic acid R = ratio hl/al. T = total weight of both layers V = total weight of water layer w = weight fraction of water W = total weight of water X = ratio rnl/wl Y = ratiomlq Subscript 1 = water layer Subscript 2 = chloroform layer LITERATURE CITED
(1) Bancroft, W. D., and Hubard, S. S., J . Am. Chem. SOC.,64, 347 (1942). (2)
Brancker, A. V., Hunter, T. G., and Nash, A. W., IND.ENO.
(3)
Brancker, A. V., Hunter, T. G., and Nash, A. W., J . Phys.
TABLE111. CALCULATEI? QUATERNARY EQUILIBRIA AT 25' C.
--
c
Tie Line NO.
1-1 1-2 1-3 1-4 1-5
2-1 2-2 2-3 2-4 2-5 3-1 3-2 3-3 3-4 4-1 4-2 4-3 4-4 5-1 5-2 5-3 6-1 6-2 6-3
CI
0.8 1.5 1.7 7.4 17.2 1.0 1.6 2.2 8.9 21.9 1.3 2.0 3.0 12.2 1.6 2.5 3.9 14.3 2.6 4.2 6.0 5.5 8.1 12.7
Upper wi 89.8 79.2 71.8 47.2 33.1 85.0 75.4 68.3 43.9 27.3 76.6 68.3 61.8 37.3 72.5 64.5 58.1 33.7 62.5 55.1 49.1 47.6 40.7 32.6
Composition, Weight % layer--Lower QI hi cz w2 2.9 6.5 89.3 1.0 2 . 8 16.5 86.5 1.1 2.5 24.0 84.4 1.2 2 . 4 43.0 73.2 2.6 3 . 3 46.4 62.2 5.9 8.0 6.0 74.3 1.2 7 . 4 15.6 72.4 1.4 6 . 9 22.6 1.9 70.4 6.7 40.5 60.3 4.5 9 . 6 41.2 48.0 9.6 5.4 59.9 1.6 16.7 1 5 . 3 14.4 57.4 2.8 14.8 20.4 3.2 56.1 15.4 35.1 7.5 46.3 21.1 4.8 54.6 2.7 1 9 . 5 13.5 52.8 3.4 1 8 . 9 19.1 51.3 3.9 20.1 31.9 41.2 9.2 30.7 4.2 44.4 4.6 29.2 11.5 42.5 5.8 28.4 16.5 41.0 6.8 3.3 43.6 34.5 8.2 8.7 42.5 32.2 10.1 43.1 11.6 29.8 12.0
layeraz h, 8.6 1.1 8.6 3.8 8.4 6.0 7 . 5 16.7 6.4 25.5 23.6 0.9 22.9 3.3 22.2 5.5 19.4 15.8 1 6 . 0 26.4 37.7 0.8 36.7 3.1 35.9 4.8 30.6 15.6 41.9 0.8 40.8 3.0 39.9 4.9 33.4 16.2 50.2 0.8 48.7 3.0 47.4 4.8 56.5 0.8 54.3 3.4 52.2 6.0
.----,-
CEEM.. , 33., 880 f1941). ~~~
The solution of the problem is thus obtained in a matter of minutes, whereas several hours of graphical computations are necessary when the method of Hunter is used. The only supplementary graphs needed for this solution are the solubility diagrams for the ternary systems, which may easily be constructed from the data in Table I.
71
Chem.. 44. 683 (1940). (4) Elgin, J.' C.,' Chemical Engineers' Handbook, 2nd ed., p. 1239,
New York, McGraw-Hill Book Co.,
1941.
( 5 ) Hand, D. B., J. Phys. Chem., 34, 1961 (1930). (6) Hunter, T. G., IND. ENG.CHEW,34, 963 (1942).
(7) Othmer, D. F., White, R. E., and Trueger, E., Ibid., 33, 1240 (1R41). ,- - - -,. ( 8 ) Smith, J. C., Ibid., 34, 234 (1942). ENG.CHEM.,ANAL.ED.,15, 380 (1943). (9) Wiegand, J. H., IND. (10) Wright. C. R. A., Proo. Roy. Soo. (London), 49, 183 (1891): 50, 375 (1892). I
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