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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-
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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
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INDUSTRIAL A N D ENGINEERING CHEMISTRY
691
the other conjugate liquid (the diluent) will be some different function of its concentration in such solutions. As a first approximation, the following assumptions are made, the validity of which will be tested later: The partial pressure of the solute in a ternary mixture will be between the values of the two functions for the binaries; furthermore, the effect upon the partial pressure of the solute by A or B will be proportional to the amount of A or B present in the mixture. Thus, for a given weight fraction of C in a ternary mixture, the A pi Pii PB PA partial pressure of a mixture containing equal quantities of A and B will be midway beFIGURE l. LINES OF CONSTANTFIGURE2. LINES OF CONSTANT PRESSURE (ISOBARS) FOR tween the partial pressure of the same weight PARTIALPRESSURE (ISOBARS) FOR PARTIAL EACHOF THREECOMPONENTS OF EACHOF THREECOMPONENTS OF fraction of C in pure A and in pure B. In SYSTEMS OF THREE LIQUIDSWHICH SYSTEMS OF THREE LIQUIDS IN other words, the partial pressure of C in ARE COMPLETELY SOLUBLEWITH WHICHTwo OF THEM( A AND B) the ternary mixture may be considered as ONE ANOTHER ARE ONLY PARTIALLY SOLUBLE IN a weighted mean between the partial presEACHOTHER sures of the respective two binary solutions, each containing this same weight fraction of C. Expreased algebraically, the total pressure of an infinitely miscible mixture will be reflected only slightly in the behavior of the partial pressure. a The partial pressure of the solute, component C, in its Po = a-f’ (C) (1) +b a + b f” (C) binary system with the solvent, A , will be some function of the concentration of C in the binary solution: and the partial where pc a partial pressme of g from a ternary solution of Cornpressure of the solute component in its binary system with position u, b, c
+
2 5.
TEMPERATURE
VAPOR PRESSURE
-
Oc.
WATER- MM.
FIQURE 3. PARTIAL PRESSURE OF ACETIC ACIDOUT OF ITSAQUEOUSSOLUTIONS OF INDICATED STRENGTHS us. VAPORPRESSURE OF PUREWATERAT SAME TEMPERATURE, BY METHODPREVIOUSLY DESCRIBED (8) The diagonal lines represent values for eaoh Bolution of the indicated strength, and are drawn parallel to the line for pure acetic acid and through points corresponding t o values obtained from a vapor cornposition ourve.
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equal, respectively, t o partial pressures of all components of the conjugate phase. Since A is insoluble in B and is infinitely miscible in C, Equation 1 cannot be applied t o the partial pressures of A or‘ B. Instead, the partial pressures of A and B would appear t o have a relation as given by isobars P ’ A and P’B in Figure 2. If the partial pressure data for the two binary systems which include C are known at the temperature for which the ternary solubility curve has been determined, the position of the tie lines may be approximated. If addition of small amounts of the solute has little effect upon rendering the solvent and diluent more miscible-that is, if the ternary solubility line is close to the two sides of the triangle-Equation 1 may be used t o determine the position of the tie lines with greater accuracy. Thus, if A and B remained entirely insoluble in each other with additions of C, the tie lines would be given exactly by the relation: Pc = .f’(C) = f”(C)
FIGURE4. CHARTFOR GRAPHICAL DETERMINATIOX OF PARTIAL PRESSURE OF ACETICACID IN TERKARY MIXTURES VITH WATER AXD BENZENE AT 25” C. T h e acid-water binary curve is plotted from t h e intersections in Figure 3 of t h e lines of constant composition with t h e 25’ C . ordinate. The acid-benzene binary was plotted directly from published values. Intersections of ordinates with t h e acid-water curve are projected t o the right-hand border; these points are connected t o intersections of t h e same ordinates with the acid-benzene curve t o give lines of constant per cent acetic acid in ternary. T o find from t h e chart the partial pressure of acetic acid from any ternary solution, select t h e diagonal line corresponding t o the peroentage of acetic acid in t h e ternary, and t h e ordinate Corresponding t o the weight percentage of benzene on t h e top scale. T h e intersection of this diagonal and this ordinate indicates t h e desired partial pressure on t h e left-hand scale.
Since most solubility data are reported on the basis of weight per cent, a, b, and c will always be in weight per cent. The use of mole per cent throughout might, however, have some advantages. If partial pressure data are available for the binary systems A-C and B-C, the partial pressures of component C may be approximated for the entire system of ternary mixtures. It might thus be expected that Equation 1 and the methods following would be useful for all concentrations of completely miscible systems. If components A and B are immiscible-for example, the solvent and diluent in extraction processes-the partial pressure of C no longer follows the isobar given by Equation 1 in the region of two coexisting phases and plotted as the curved line L-M in Figure 2. I n such systems it is given by the tie line connecting the points of intersection of the isotherm with the theoretical isobar (straight solid line L-M in Figure 2) since the compositions of the two phases are constant along this line, although the relative amounts of the two phases vary. There is little doubt that the immiscibility of A and B may affect the shape of the isobar in the one-phase region also. It is believed, however, that this effect is small, and accordingly the isobar in the onephase region may be assumed t o be given by Equation 1. From this, the converse of the above statement that the tie line determines partial-pressure isobars follows. The points of intersection of the isobar with the ternary solubility diagram will indicate conjugate phase compositions, since partial pressures of all components of one phase must be
Approximation of Vapor Pressure Data
To apply Equation 1, partial pressurecomposition data would be required for the two binary systems solutediluent and solute-solvent. In addition, in order to check the validity of the results, the isothermal solubility curve and tie lines must be known for some representative standard systems. Several approximations may be made, however, which allow the use of the more voluminous vapor-liquid equilibrium composition data. A plot of the log of the vapor pressure of a solution against the log of the vapor pressure of the pure solvent has been shown in previous papers (6, 8) to give lines which are almost straight, when the two vapor pressures are always taken a t the same temperatures. In this method of plotting, the X axis of a sheet of logarithmic paper is calibrated according to the vapor pressure-temperature relation of a standard liquid (often water), and ordinates corresponding to the indicated temperatures are erected. On these temperature ordinates are plothed the given values of vapor pressures. As one example, the partial pressures of water from a solution of ammonia in water were plotted on logarithmic coordinates against the vapor pressure of pure water; straight lines were obtained which are parallel to one another as well as t o the 100 per cent water line of unit slope (6). Thus, with the assumption that these conditions prevail for other systems, the partial pressure of a solute in a solution may be found for a given composition of the liquid a t a certain temperature. Using this method of plotting, a line of unit slope is drawn through the point a t which the partial pressure, the composition, the temperature, and consequently the vapor pressure of the pure liquid are all known. The partial pressure of the solvent from this solution may then be read from the graph a t the vapor pressure of the pure liquid corresponding to the desired temperature.
INDUSTRIAL AND ENGINEERING CHEMISTRY
June, 1942
Example of Tie-Line Prediction To illustrate the method of tie-line prediction more clearly, one system, acetic acid-benzene-water, will be considered in detail. Acetic acid-water mixtures have some abnormalities with regard to vapor pressure relations which detract from the accuracy of the assumptions; nevertheless, the ternary system may be used as an illustration of the method under rather unfavorable conditions. This system serves to show another application of the above-indicated method of vapor pressure plotting to the solution of a problem relating to equilibria when only a minimum of data is available. It is first required to obtain the isothermal partial-pressure values from vapor composition and boiling point data. The log of the vapor pressure of acetic acid (the solute of the ternary) is plotted against the log of the vapor pressure of water a t the same temperature to give an approximately straight line (Figure 3) in the manner previously described (8). Vapor-liquid composition of the acetic acid-water system has been reported (7,10) and may be expressed as mole per cent water or weight per cent water in the respective phases a t the corresponding boiling points. (If the vaporcomposition data are few in number, it may be necessary to plot a composition-temperature curve as well as the vaporliquid composition curve, in order to determine the partial pressure and temperature for intermediate values of the liquid composition.) Since the mole per cent of acetic acid (difference of 100 minus per cent of water) present in the vapor is proportional to its partial pressure, multiplication by the atmospheric pressure gives the partial pressure of acetic acid a t the boiling point for a given liquid composition. Through these known points on Figure 3 lines are drawn parallel to the vapor pressure line of acetic acid. These parallel lines represent partial pressure lines on the assumption of no heat of solution. The ordinate corresponding to 25” C. (the temperature of the solubility isotherm) is drawn. This intersects the several lines representing partial pressures of acetic acid for the several compositions expressed and thus indicates the isothermal partial pressures. From the values obtained from these intersections, the partial pressure-composition curve at the constant temperature of 25’ C. may be constructed-for example, the lower curve, “Acid-Water Binary”, in Figure 4. The corresponding partial pressures a t 25” C. for acetic acid in binary solutions having different amounts of benzene were taken directly from the literature (4) and are plotted as the upper curve, “Acid-Benzene Binary”, in Figure 4 on the same calibration of the horizontal axis. Using Equation 1, a simple method for determining the ternary compositions a t which the partial pressure is constant was developed. The sum of a (per cent benzene in ternary), 6 (per cent water in ternary), and c (per cent acetic acid in ternary) is equal to 100 per cent. If c is constant, (a b) is also constant. Singe f ’ ( C ) and f”(C)are functions of the values of c, these functions are constant for given values of c. The quotients f ’ ( C ) / ( a -t 6) and f”(C)/a-t 6) are constant; and Equation 1 becomes
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ternary mixtures. For a h e d value of c in each binary system, the partial pressure a t the given temperature of C in the respective binary system is given by the intersection of the corresponding curves of the binary systems in Figure 4 with a vertical line a t the given concentration. Thus, the partial pressure of acetic acid in B 20 per cent mixture with water a t 25’ C. is found from the acid-water binary curve to be 1.2 mm.; in a 20 per cent mixture with benzene it is found from the acid-benzene binary curve to be 8.9 mm. A line is drawn from this point of intersection at 8.9 on the acidbenzene binary curve to the point which is the projection of the corresponding point of intersection of the acid-water binary onto the 100 per cent line-in this case 1.2 on the right-hand ordinate. This last line will then indicate the linear variation of the partial pressure of acetic acid with the amount of benzene, or water, present in all ternary mixtures having a constant c equal to 20 per cent acetic acid. These constant c lines may be drawn for as many values as desired; each represents a horizontal line on the ternary diagram. The values of a (per cent of benzene) for the ternary may then be calibrated directly on the upper scale, since it is equal t o 100 minus the value of c for the binary system of acetic acid and benzene. From this top calibration the percentage of benzene corresponding to any given partial pressure of acetic acid from a ternary mixture (always containing the given percentage of acetic acid) may be read directly, I n the given case, the line for all ternary mixtures containing 20 per cent acetic acid has been indicated; and by following this line down to 6 mm., the ternary mixture which has such a vapor pressure has 50 weight per cent of benzene (and, of course, 30 per cent of water).
+
+
+
P o = K’u L’ [(a b) - U ] or P , = Ka L (since b is equal to a constant minus a ) where K, L constants for constant values of c
-+
This is the equation of a straight line. Thus, based on these assumptions, the partial pressure of c will vary linearly with a or b, from its value of f’(C) a t a = 1 - c, or b = 0, to its value of f”(C) a t a = 0, or b = 1 - c. Since a t zero per cent a this line coincides with f’(C), and a t zero per cent 3 the pressure line coincides with f”(C), the position of this line can be determined. This linear relation suggests a convenient method for determining graphically the partial pressures of C in various
FIGURE 5. TERNARY DIAGRAM AND SOLUBILITY CURVE OF BENZENE-WATER-ACETIC ACIDSYSTEM Observed tie lines are solid; lines of constant partial pressures of acetio acid (determined from Figure 4) are plotted in broken curved lines, and the predicted tie lines are drawn in broken lines between t h e two intersections of these isobars with the solubility curve.
Thus, any horizontal line on Figure 4 (i. e., a constant partial-pressure line) intersects the constant acetic acid composition lines (the diagonals) a t points for which the benzene content can be read off by projecting to the upper margin. All ternary compositions corresponding to the same partial pressure of acetic acid may be found in this manner.
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INDUSTRIAL A N D ENGINEERING CHEMISTRY
These compositions of constant partial pressure of acetic acid were plotted on the ternary solubility diagram as the curved dashed lines of Figure 5. The isobars (partial) are drawn in completely to act as construction lines in determining the points of intersection with the ternary solubility curve. These intersections are the ends of tie lines, predicted in the manner indicated from vapor pressure data which had been, in turn, predicted from vapor composition data. That part of the curve of partial pressure in the two-phase region is meaningless, since the tie line itself (as indicated above) is the isobar between the intersections. Outside, in the single phase region, the curves should indicate the lines of constant partial pressure fairly closely.
FIGURE 6. DISTRIBUTION DATAFOR SEVERAL SYSTEMS WITH WATER AS DILUEXT Solid lines represent values obtained from tie lines which have been determined by others: dotted lines represent values predicted from partial pressures of solute calculated by present method.
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value in the consolute-favored phase is generally less than 10 per cent, which is within the range of accuracy of extraction equipment design. While the method appears rather inT7olved for the determination of the lines from vapor pressure data, which is often a more difficult determination, it is immediately useful when the vapor pressure data are available. Of‘ more interest, possibly, is the demonstration by this method of the interelation of solubility and partial pressure phenomena and the interdependence of the two diffusional processes of rectification and extraction. Furthermore, the suggestion follows, from this demonstrated relation of solubility and vapor pressures, that vapor pressures might be predicted from solubilities rather than vice versa, as above. If vapor composition data alone are available, it may be concluded that the distribution of the solute between solvent and diluent can be indicated approximately, The composition of the solute in pure water yielding a partial pressure of the solute equal t o the partial pressure of a given composition of the solute in pure solvent will evidently be an index of the distribution properties of the considered solvent. This is apparent upon examining the curves of the respective binary systems of Figure 4. For a partial pressure of acetic acid equal to 6 mm. the weight percentages of acetic acid in the benzene and the water binary systems are, respectively, 11.7 and 65.5, the ratio being 5.6. If this ratio was unity, the acetic acid would be distributed equally between the two phases. Since the ratio is greater than unity, it is then indicated that the distribution favors the aqueous phase. In finding the proper solvent for a solvent extraction process, this index may be found for different solvents and its value used for the preliminary choice of a suitable solvent. Generally, a large difference in the distribution between the solvent and aqueous phases implies a comparatively large region of two phases in the ternary diagram. The smaller the index, the greater will be the usefulness of the solvent in the solvent extraction system considered.
Nomenclature A = solvent a = weight per cent of solvent
B = diluent
The predicted tie lines cross the observed tie lines ( I ) at higher compositions of acetic acid. The deviation of the predicted results from the observed values can be more easily seen on Figure 6. The distribution values follow directly from the tie lines and allow a more ready comparison.
Application to Other Systems Other systems were studied, and the tie lines and distribution data determined. The latter for four such systems are plotted in Figure 6. They all have water as the diluent and include benzene and acetaldehyde (Q), ethanol and benzene (4, IO, II), acetone and chloroform (2, 3, 6), and ethanol and cyclohexane (4, IO, fa). The equilibrium relations for the binary solutions of all these systems are far from ideal. Thus, minimum-boiling azeotropes are formed with water and ethanol, acetic acid and benzene, ethanol and cyclohexane, and ethanol and water. A maximum-boiling mixture is formed with acetone and chloroform. The ternary azeotrope is formed with benzene, ethanol, and water. I n spite of these abnormalities, the prediction of the tie lines based upon the partial pressure of the solute in the ternary mixture gives fairly close results. For the range studied in each system, the deviation from the observed
b = weight per cent of diluent C = solute c = weight per cent of solute f ’ ( C ) = function C = partial pressure of C in binary system A-C f”(C)= function C = partial pressure of C in binary system B-C K,K’, L, L’ = constants P , P’, P” = partial pressures of components indicated by subscripts
,
Literature Cited
(1) Hand, D. B., J. Phys. Chem., 34, 1961 (1930). (2) Hausbrand. E., “Principles and Practice of Industrial Diatillation”, London, Chapman & Hall, 1925. (3) International Critical Tables, Vol. 111, p. 426 (1928). (4) LandoltrBornstein, Physik.-Chem. Tabellen (1931). (5) Nernst, 2.physik. Chem., 8, 110 (1891). (6) Othmer, D. F., Chem. & Met. Eng., 47, 551 (1940). (7) Othmer, D.F., IND.ENQ.CHEM.,20, 743 (1928). (8) Ibid., 32,841 (1940). (9) Othmer, D. F., and Tobias, P. E . , Ibid., 34, 690 (1942). (10) Perry, J. H., Chemical Engineer’s Handbook, 2nd ed., 1941. (11) Varteressian, K. A., and Fenske, M. R., IND.ENG. CHEM.,28, 928 (1936). (12) Vold, R. D., and Washburn, E. R., J. Am. Chem. SOC.,54, 4217 (1932).
P~EBENTED before the Division of Industrial and Engineering Chemistry at. the 103rd Meeting of the AMERICAN CHEMICAL SOCIETY. Memphis, Tenn.
