TIE LINE CORRELATION DONALD F. OTHMER AND PHILIP E. TOBIAS Ternary solubility data from severalsources, as used in extraction calculations and design, have been plotted using a new and improved tie line plot. A straight line results from a plot of log (1 a,)/a,against b,)/b, where a, is the fraction of log (1 solvent in the solvent phase and b, is the fraction of diluent in the diluent or other phase. This plot and the corresponding equation derived from theoretical reasoning are shown to apply even to those systems in which there is considerable miscibility of the solvent and diluent phases; and by
-
-
H E use of tie lines in a ternary solubility system, where a solvent phase is in equilibrium with a diluent phase a third liquid or solute is distributed between these two phases, has considerable application in determining the quantitative distribution of the solute between two conjugate phases. While this method of representation has the advantage of easily yielding the weight ratios of the two conjugate phases separating from a mixture of known composition, it possesses the rather serious disadvantage of requiring considerable data for the entire range of concentrations. Unless an observed tie line coincides exactly with the composition of a mixture for which distribution data are desired, interpolation between adjacent tie lines must be resorted to. Since in the usual ternary system, the plait point is displaced from the apex of the isotherm, the slope of the tie lines (positive or negative) will increase with increasing concentrations of the consolute. Only in very unusual systems are the tie lines horizontal; in others, interpolation requires an approximation of the change in slope of adjacent tie lines. Except for systems for which a considerable number of tie lines have been determined, the interpolation of tie lines on ternary coordinates is rather unsatisfactory.
Previous Attempts a t Correlation Brancker, Hunter, and Nash (I)showed that by using a variable ordinate scale based upon the tie-line relation of a “standard” system, a plot of the relation between the weight percentages of the nonconsolute present in conjugate phases yields straight lines for thirty-one of thirty-three systems studied. Bachman (1) studied the results of Brancker, Hunter, and Nash, and found a method of plotting by which tie lines in many ternary liquid systems exhibit a straight-line relation if the proper functions of the concentrations a t the extremes of the tie lines are chosen as the dependent and independent variables. He found that if A and B are the nonconsolute components and C is the consolute component of a ternary liquid system, a plot of weight per cent A in the A-rich layer against weight per cent B in the B-rich layer will produce a line which has as its equation: al kbz = malbi (1)
means of this convenient and accurate tie line plot, only two experimentally determined points are required to establish the entire tie line or distribution curve. Only one set of coordinates is necessary for all systems. A simple nomograph or graphical construction on the log plot itself may be used to plot composition data without calculation and then to determine directly the value of the major component of each conjugate phase. From the ternary solubility diagram the values of the other constituents then follow immediately.
I n this and later equations all bl, and c1 are the respective weight fractions of A , B, and C in the A-rich layer; a,bs, and c2 are the corresponding weight fractions in the B-rich layer, and m and k are constants. Dividing Equation 1 by b2, al/b, = mal
-k
(1A)
Accordingly, a plot of al against aJb, will yield a straight line.
Modification of Bachman’s Equation Since this form of the equation has no apparent physical significance, some other type of correlation would be of interest. Furthermore, all of the ternary systems examined by Brancker, Hunter, and Nash and by Bachman have practically immiscible nonconsolute components in the absence of the consolute component. Thus, in the ternary diagrams to which this was applied, the tie line with zero solute or the extremities of the base line of the triangular plot would be satisfied approximately by al = 1 and bz = 1; consequently, -k = 1 - m
(2)
If this is substituted in Equation 1 and transposed:
+
al = ma&%
b2
(1
- m)
(3)
Subtracting albz from each side of Equation 3, there follows a1 - arbs = malbz - albz b t ( l - m),and factoring gives a l ( 1 - b2) = b2(l - m) ( 1 - al); or
+
1 - Ul al = l - n & = - - k
-
+ +
1 bz bz
Since a b c also be written.
=
(b,
(4)
1 in each liquid phase, Equation 4 may
+ U
Cd
( G T 3= l - m = - - k
(5)
b2
where cl, = weight per cent of consolute component in Arich and B-rich phases, respectively.
+
693
INDUSTRIAL AND ENGINEERING CHEMISTRY
694
Cross multiplication of Equation 5 gives:
Thus, in weight fractions, the ratio of the sum of the diluent and solute to the solvent in the solvent phase is proportional to the ratio of the sum of the solvent and solute to the diluent in the diluent phase. Taking the logarithm of both sides of Equation 6, log- (bl
+ c1) = log- + + a constant (7) + c,)/a, or log (1 - al)/al against log '2)
a1
b2
Plotting log (bl (az cZ)/bz or log (1 - b2)/b2 will yield a straight line of unit slope for those systems which have highly immiscible nonconsolute components in the absence of the consolute (i. e., all systems correlated by Brancker, Hunter, and Nash and by Bachman, as well as others in this same class).
+
Vol. 34, No. 6
systems plotted yield approximately straight lines; a number of those shown indicate a slope substantially different from unity. Plotting these systems by the other equations which are shown to be reducible to Equation 9 without exponent TZ (or with n always equal to unity) would not give straight lines. Thus, when data for these systems are plotted according to Bachman's method, straight lines are not obtained. It is apparent, therefore, that the exponential parameter of Equations 9 and 10 is a necessary addition. A further generalization is that for immiscible systems the exponent will be equal to unity (i. e., lines are at 4 5 O ) , while for miscible systems the exponent will vary from unity by an undetermined amount which will depend on the degree of immiscibility.
1.8
General Equation for Tie Line Data Equation 7 and the corresponding method of plotting reduces the straight-line plots of Bachman, with different slopes, to straight-line plots with the same slope (unity) through the use of a log plot. It is desired to change the form of Equation 7, add to it, or develop some new relation to include other systems than the ones with substantially immiscible, nonconsolute liquids. Nernst (7) states that, if the two liquids remain immiscible on addition of the consolute liquid and the molecular weight of the consolute is the same in either of the other two components, the distribution which results is given by:
I.AMYL ALCOHOL 2.97%ACETIC ACID 3. 98.1%ACETIC ACID 4. B E N Z E N E 5. FURFURAL -6. TOLUENE
-08.1
.2 .3 A .5.6 .8 I
I-b 2 b
3
FIGURE1. PLOTon LOGARITHMIC COORDINATES OF
Equation 6 resembles Equation 8 if the sum of the solute and the minor nonconsolute component is regarded as obeying the Nernst law of distribution of a consolute in an immiscible pair of liquids. Equation 6 reduces t o Equation 8 if bl and a2 are equal to zero. Hand ( 3 ) revised Equation 8 to a form which he shows t o be useful:
By applying the suggestion of Equation SA, a more general equation was formulated with an additional parameter, exponent n, of one of the ratios of Equation 7; thus, log
'1
~
where S
+ 1'
a1 =
=
log
[TI + "
= n log a2 + cz ~
b2
+s
(9)
a constant
Also, since 1 - al
=
bl
+ c1 and 1 - bz
I--a 1 l o g 2 = nlog-
- bz b2
=
+s
a?
+ cf, (10)
Data from a previous paper (8) were plotted in Figure 1 on logarithmic coordinates, with the ratio of one minus the fraction of solvent in the solvent-rich phase to the fraction of solvent in the solvent-rich phase plotted against the ratio of one minus the fraction of water in the conjugate phase to the fraction of water in the conjugate phase. The points yield straight lines with substantially unit slopes; a deviation of the slope from unity will be seen in some systems. I n Figure 2 several other systems were taken a t random from the literature and plotted in the same fashion. All the
(I--a)/-a
AGAINST
(l-b)/b
FROM
PREVIOUS DATA (8)
a is fraction of solvent in solvent phase and b is fraction of diluent in oonjugate phase. Sy.atems i , 4, 6 , and 6 are for acetaldehyde, water, and indicated solvents; systems 2 and 3 are for toluene and nheptane with acetic acid of t h e given strengths (with water) as solvent.
The cause of this exponential variation between various systems may be explained tentatively by the formation of either associated or dissociated moleculeti in solution. Such a fact would, in the case of systems in which the Nernst equation is applicable, be reflected by an exponential parameter of the composition ratio of consolute to nonconsolute. As mentioned above, this has been shown by Hand. Reasoning in parallel manner, mainly because of the similarity of the derived equation to the Kernst equation, the association or dissociation of the molecules of any component might be considered the cause of the deviation of the exponent of Equation 9 from unity. The plotted data give fairly straight lines even when the systems are fairly miscible (i. e., those in which the two branches of the solubility curve are removed from the two sides and the curve itself is close to the bottom of the triangle). The Bachman plot will give lines of considerable curvature for these systems of appreciable miscibility. I n Figure 3 several systems investigated by Othmer, White, and Trueger (9) are plotted. Here again, for even fairly miscible systems the data fall on straight lines almost within their probable experimental accuracy. It is apparent that all the systems studied fit easily on one sheet of log-log paper using only two cycles. The Bachman plot of the ratio of the major nonconsolute com-
INDUSTRIAL AND ENGINEERING CHEMISTRY
June, 1942
I
1.0
.e
.8 .6 .5
.6
.S
A
.3
A .3
.2
.e
.I
.I
el0 :::
.OB
. WBUTANOL ..n-BUTANOL CHLOROFORM . CHLOROFORM
.05 .O 4
-03
.06 .O 5 .O 4
1. AMYL ALCOHOL 2.ME.I-BUT. KETONE 3.CHLOROBENZENE 4. FENC H 0 NE 5.BUTYL ETHER 6. OCTYL ACETATE 7. BUTYL ETHER
.03 .02
.o2
.o I
695
.07 .I
.2
.3 4.5 8.7.8 I
I-b
2
3 4 567610
.01 I I I 0 4 .07
I I I IIll/ -3 1.5.6 .8 l
I
.2
.I
I
a 2
I
I l l
3 4 567
b b FIGURE 2. PLOTON LOGARITHMIC COORDINATES OF ( l - a ) / a FIGURE3. PLOT ON LOGARITHMIC COORDIXATES OF AGAINST (1-b)/b FOR SYSTEMS OF VARIOUSIKVESTIGATORS (1 -a)/a AGAINST (1-b)/b FOR SYSTEMS OF OTHMER, WHITE, Curve System Temp., C. Citation AND TRUEGER (9) 1 2
3 4
5 6 7 8
Methanol, water, n-butanol Methanol, water, n-butanol Acetic acid, water, chloroform Acetone water chloroform Ethanol: water: benzene Acetic aoid, water, toluene Ethanol water, isoamyl alcohol Acetone: water, furfural
60
(6)
15 60
(6)
Curve No. 1 2
3
4
25
15
25
5
IW
6 7
System Ethanol, water n-amyl alcohol Acetone, water,' methyl isobutyl ketone Acetone water, monochlorobenzene Acetio Loid, water fenchone Acetone water d;butyl ether Acetio abid, wa'ter, octyl acetate Acetic acid, water, dibutyl ether
ponents of conjugate phases against the numerator of this fraction requires almost as many different ordinate scales as there are systems. Thus the new method has the additional advantage of requiring but one set of coordinates and allowing comparison of different systems directly on the same plot.
3
Graphical Conversion from and to Composition Units Another useful advantage of this method is the ease with which a nomograph may be constructed so that the values of the compositions of the nonconsolute components may be read from the graph directly (rather than -by subtraction or addition of unity before or after determining a reciprocal by slide rule or table). Thus, two or more points (representing an equal number of known tie lines) are determined for any new system and plotted by a simple nomogram to give a straight line; then other values representing other undetermined tie lines, in the same units as the original data, may be readilg read from the same nomogram. Figure 4 shows this graphical method of converting composition data (in per cent) to the values plotted, and these plotted values back into per cent composition. The tie line data for one system are plotted as log ( l - u ) / a against log (1-b)/b and give a straight line. First drawn, however, is the plot representing the reciprocal function, y = l/x, which on log paper is a straight line through s = 1, y = 1 and has a slope of -1. Since (l-u)/a is equal to (l/a-l), the value of (1-a)/a may be found graphically as follows, for an example where fraction of water in water layer is 0.47 = b when the fraction of solvent in solvent layer is 0.65 = a: Starting with the value of b = 0.47 on the vertical axis and proceeding horizontally to the y = 1/x or log y = -log x line, the
I-b b
FIGURE 4. GRAPHICAL METHODOF DETERMINIKG AND PLOTTINQ VALUESOF (1-a)/a AND (1-b)/b FROM VALUESOF a AND b
-
I n the example shown b 0.47 a n d is read on the vertical axis, projected t o point 1 t o give l / b : unity is subtracted t o give point 2, which has the desired value of ( I / b ) 1 or (1 - b ) / b . Similarly, from a given point a 0.65 read on the horizontal axis, the reciprocal is obtained a t point 5 , and unity is subtracted t o give point 4, which has the desired value of (I/a) 1 or (1 a ) / a . Point 3, resulting for given values of b and a , follows from points 2 and 4.
-
-
-
696
INDUSTRIAL AND ENGINEERING CHEMISTRY
value of the reciprocal is read immediately, t o give in this case 2.13 at point 1. Unity is subtracted, to obtain 1.13 at 1. In a similar point 2, which thus has the value of (l/b) manner, a value for ( l / a ) - 1 is obtained from the value of a = 0.65, taken on the horizontal axis, rojected to the y = 1/x line t o give 1.54 at point 5, from w h d unity is subtracted to give 0.54 at point 4. From point 4 and point 2 is obtained point 3, one value on the tie line curve. Other points are plotted similarly.
-
Once the tie line curve is obtained in this manner, the corresponding values of a and b can be found for any number of unknown tie lines by reverse steps. Thus, in Figure 4 let us assume it was desired to find the other extremity, corresponding to a tie line having a given value of b; this value of b would be plotted as shown on the vertical axis, the points 1 and 2 obtained as before, and the point 3 obtained by vertical projection from 2 to the known tie line curve. Point 3 is then projected to the left until the vertical axis is reached a t point 4’. Unity is added to the value of 4’to give point 5’, which is projected to the y = l/a: line t o give point 5. Projecting from point 5 down to the horizontal axis gives the value of a, which determines the other extremity of the tie line in question. (These directions are carried out in less time than it takes to read them.) Thus the disadvantages that the Bachman plot possessesi. e., the necessity of further calculations of experimental values in order to plot them or, in using the resulting plot, the necessity to transform the units of the coordinate axes back into the units of the original data-have been obviated by the simple mechanical plotting method described. Ob-
Vol. 34, No. 6
viously this requires no more than the drawing of the y = 1/z line as a construction line on the logarithmic paper used for plotting. Acknowledgment Thanks are due to Irvin Bachman, now attached to the Chemical Warfare Service of the United States Army, for the helpful suggestions in arrangement of the material in this and other papers of this series. Literature Cited (1) Bachman, I., IND. E N G . CIIEM.,ANAL,. ED., 12, 38 (1940). (2) Brancker, A. V., Hunter, T. G., and Nash, A. W., Ibid., 12, 35 (1940). (3) Hand, D. B., J . Phys. Chem., 34, 1961 (1930). (4) International Critical Tables, Vol. 111,p. 405 (1928). ( 5 ) Lloyd, Thompson, and Ferguson, Can. Y. Research, 15B. 98 (1938). (6) Muller, A. J., Pugsley, L. I., and Ferguson, J. B., J . Phys. Chem., 35, 1313 (1931). ( 7 ) Nernst, 2.physik. Chem., 8 , 110 (1891). (8) Othmer, D. F., and Tobias, P. E., IND. ENG.CHEM.,34, 690 (1942). (9) Othrner: D. F., White, R. E., and Trueger, E., Ibid., 33, 1240 (1941). (10) Varteressian, K. A., and Fenske, M. R., Ibid., 28, 928 (1936). (11) Woodman, R. M., J . Phya. Chem., 30, 1283 (1926). (12) Wright, C. R. A., Proc. Roy. SOC.(London), 49, 174 (1891). PRESENTED before the Division of Industrial rmd Engineering Chemistry at the 103rd Meeting of the AMERICAN CHEMICAL SOCIBTY. Memphis, Tenn.
PARTIAL PRESSURES OF TERNARY LIQUID SYSTEMS AND THE PREDICTION OF TIE LINES DONALD F. OTHMER AND PHILIP E. TOBIAS
A n equation is derived, relating the partial pressure of the solute in a ternary solubility system to the composition of the mixture. In the one-phase region of solubility the following equation may be assumed:
Through the use of this equation, of solubility data, and of partial pressure data for the two binary systems A-C and B-C, the tie lines of ternary systems may be predicted. Less accurately, partial pressure data alone yield an index which may be used to determine approximately the distribution characteristics of a solvent to be employed in a solvent-extraction system. Where partial pressure data are not available, they may be approximated by the use of vapor composition data and the log plot for vapor pressures previously described.
I”
MIXTURE of three compounds which are mutually completely miscible, each component will exert a partial pressure determined by its concentration, the ratio of the amounts of the other two components present, the ternperature, and the total pressure. From the phase rule, the number of degrees of freedom is equal t o two more than the difference between the number of components and the number of phases. Thus, for a ternary system with air as a fourth component and one liquid and one vapor phase present, there will be four degrees of freedom. If the pressure and temperatures are kept constant, the number reduces to two. If the partial pressure of one of the components is considered as an independent variable, fixing this partial pressure will yield a univariant system. Accordingly, constant partial pressure conditions for any component may be indicated on a ternary diagram by an isobar (of partial pressure) for each of the three components, as shown in Figure 1. These lines of constant partial pressure should not be confused with similar isobars for total pressure. The latter may yield a maximum or minimurn in either a binary or a ternary mixture, and thereby account for a constant-boiling mixture. On the other hand, the partial pressure of any one component will always increase with increasing fractions of that component, the ratio of the other two components remaining constant. Thus, the abnormal characteristics of
