D I N E R I C DISTRIBUTION* BY DAYID BIRNEY HAND
I. The Distribution of a Consolute Liquid between Two Immiscible Liquids a.
Introduction: At present there is no adequate expression for the dis-
tribution of a solute between two phases. The distribution law of Kernst was advanced in 1891,~and may be stated in its simplest form: “If the molecular weight of the solute is the same in both solvents the ratio in which the solute distributes itself between the two solvents is constant for a given temperature.” That is, x1/x2 = const. in which x1 and x2 represent the concentrations of the solute in the two phases. The Sernst distribution law is limited not only to cases in which the molecular weight of the solute is the same in each phase, but also the two solvents must remain immiscible even after the addition of the common solute, I n so far as Henry’s Law is valid for the solubility of a gas in two immiscible solvents, the simple distribution law should hold for the division of this gas between the two liquids. I n such a case the distribution coefficient, K, would be the ratio of the solubilities of the gas in the two solvents. Nernst has expanded his distribution law to include cases of polymerization or dissociation with the statement: “If the dissolved substance exists as molecules in various states of aggregation, the constant distribution ratio holds for each individual type of molecule. If the molecular weight of the solute is twice as great in one solvent as it is in the other the ratio, x12/x2 is a constant .” This is on the assumptions that the square of the concentration of unpolymerized molecules is proportional to the concentration of double molecules and also that the concentration of unpolymerized molecules in the second phase is small enough to be neglected. One of the first systems investigated by Nernst was the distribution of acetic acid between water and benzene. The expression, cI2/c2was found to be much more constant than C ~ / C , . Xernst’s data for the system are given in Table I together with some of our own values for slightly more concentrated solutions. c1 is equal to the grams of acetic acid in 5.072 grams of water and CP equals the grams of acetic acid in 31.5 grams of benzene. I t can be seen that neither ratio is constant even for the low concentrations studied. The explanation for the failure of the acetic acid to distribute itself between * This work is part of the programme now being carried out at Cornell University under a grant to Professor Bancroft from the Heckscher Foundation for the Advancement of Research established by August Heckscher at Cornell University. W. Nernst: Z. physik. Chem., 8, I I O ( 1 8 9 1 ) .
1962
DAVID B I R S E Y H A S D
benzene and water according to a constant ratio is that polymerization is not complete enough in the benzene phase so that the concentration of the single acetic acid molecules can be neglected So for the distribution of acetic acid between benzene and water the Sernst distribution rquation has not been applied in a useful way over any range of concentrations. Tanm I Ilistribution of Acetic Acid between IVater and Benzene according to the Kernst Distribution Law C I = grams acetic acid in 5 . 0 7 2 grams water c? = grams acetic acid in 3 I , j grams benzene c,? )C?
CI
r2
0.24j
0.043
5.7
I
0,314
0.071
1.39
0.375
0.094 0 . I49
4.4 4.0
3.4
I
0.500
C1:
C?
.09
0.047 0.447
2.08
I .07
2.44 1.9
7.25
4.8;
1.5
0.242 I
5 . 2
.40
Experiments by Nernst
1.49
.67
I,25
2.6 4.0 10.8
Points recalculated from Table V
It has been suggested by a number of other authors that a reason for the deviation from the Sernst distribution law lies in the fact that addition of a consolute liquid always increases the solubility of the two immiscible liquids.' An empirical equation for distribution has been proposed by W. D. Bancroft and shown to apply excellently over wide ranges of concentrations for all the eases investigated. For the distribution of alcohol between benzene and water the relation,
has been found to hold over the whole possible range of concentrations. I n the equation the quantities of the liquids are expressed in grams, the subscripts I and z denoting the upper and lower phases respectively. Discussion of this equation and further evidence of its general validity are reserved for a later place in the paper. Hendrixson* has shown how the distribution law may be applied to cases in which polymerization and dissociation occur at the same time. The equation proposed by Hendrixson has been used by Dawson3 in describing the distribution of acetic acid between chloroform and water. The following discussion is taken almost directly from Dawson's article. S. F. Taylor: ,J. Phys. Chem., 1, 461 (1897); Timmermans: Z. physik. Chem., 5 8 , 129 (1907); Hem: Ahrens Sammlung, 15, I (1910). *IT.S.Hendrixson: Z. anorg. Chem., 13, ;3 (189;). 3H. M Dawson: J. Chem. Soc., 81, 521 ( 1 9 ~ 2 ) .
1963
DINERIC DISTRIBCTIOS
The ratio of distribution of acetic acid between chloroform and water increases very considerably with increase of the dilution. I t has been suggested that this is due to the gradual splitting up of the double acetic acid molecules into simple molecules. With the object of testing this assumption the equation of the dissociation isotherm has been applied to the acetic acid dissolved in the chloroform, the experimental data in the previous paper being used for the calculation.’ -1s will appear from what follows, the concentrations of acetic acid in chloroform which have been used correspond to a region of almost complete molecular transformation.
TABLE I1 Distribution of Acetic Acid between Chloroform and Water (Data from Dan-son) Concentration of aqueous layer C1
1.535 0.9084 0.6089 0.3178 0,2696 0,2515
0,1946 0.1591 0.1269 0.09594 0.06445
Concentration of Chloroform layer C2 0.227
q =
c1/c2
6.74
0.08904 0.04557 0.01576
20.17
0 .OI222
22.07
0.01109 o ,007586 0.005608 0.00406 I 0.002848 0.00174
22.67 25.68 28.36 31.23 33.68 37.04
10.21
‘3.37
“Let c1 and c2 represent corresponding concentrations of acetic acid in water and in chloroform. If a is the degree of dissociation of the acid in the aqueous solution at this concentration, then c1 (I-a) is the concentration of the undissociated molecules. If r is the ratio of distribution of the simple molecules between water and chloroform, ___ cl(l-a) will be the concentration of the simple acetic acid molecules in the chloroform, and c2 - -that r of the associated molecules. Applying the law of mass action, the result obtained is:
where K is a constant. Table 11, this paper.
I964
DAVID B I R N E Y HAKD
“By taking the second and the seventh experiments (see Table I1 for Dawson’s distribution data) as a basis, and assuming that the experimental data for these give the same value for K, the value of r is calculated to be 42.9. “For each of the eleven experiments on the distribution of acetic acid the value of the above expression for the constant Xi has been calculated by inserting the experimental numbers for c1 and c2, the value for r = 42.9, and that for a obtained from Ostwald’s dilution formula which, in the case of a2c acetic acid at zs0, is __ = o.oooo18. I--3.
“The following table [Table 1111 contains these values of K as well as the concentrations of the simple and double acetic molecules in the chloroform, these concentrations being expressed in gram-equivalents per liter. TABLE 111 Distribution of Acetic Acid between Chloroform and Water (Table from Dawson)
Concentration of undissociated acetic acid in water
Concentration of Simple molecules in CHCI,
Concentration of double molecules in CHCl, 0.1920
0.6052
0.03565 0.02109 0.01411
0.31j6
0.00735 5
0.2675
0.006234 0 . 0 05816 0,004495 0.003669 0.002922
1.530
0.9048
0.2496 0.1929 0.1574
0.1254 0.09469 o ,06336
0.002207
0.001477
0.06795 0.03146 0.008405 o.ooj986 0 , 0 0 5 2 74 0.003091 0.001939
0.001139 0.000641 0.000263
K
0.0066 0.006; 0.0063 0.0064 0.006; 0.0064 0.0065 0.0069 0.007j
0.0076 0.0083
“The constancy of the values of K in the last column leads to the conclusion that the assumption of a gradual dissociation of the double acetic acid molecules in the chloroform into simple molecules with increasing dilution is correct. The maximum and minimum values of K for the first eight experiments are 0.0069 and 0.0063 respectively. The limiting concentrations of acrkic acid in the chloroform which give this constant value of K are 0.23 and 0.006 gram-equivalents per liter, the ratio of these concentrations being approximately 40 to I . For the most dilute solutions which have been investigated, the deviations of K from the mean value are more considerable and increase with the dilution. At these very small concentrations, however, it must be remembered that the denominator in the first expression given above for K, representing as it does the concentration of the double acetic acid molecules, is the small difference between two gradually decreasing and approximating quantities, and a small error in the estimation of the amount of acetic acid in the water and chloroform layers would explain this gradually
DINERIC DISTRIBUTION
1965
increasing value of K. To show that this is the case, we may, for the experimental data a t the smallest concentration investigated, assume the true value of K to be the mean of the values found a t the higher concentrations, namely, 0.006j1 and calculate inversely the concentration of the acetic acid in the chloroform layer. We have then in the expression:
c1 = 0.0644;, a = 0.017,r = 42.9, from which the value of c2, the concentrawhereas. tion of the acetic acid in t,he chloroform, is calculated to be 0.00181, the number found by experiment is 0.00174. The difference is less than 4 per cent., and since in this experiment less than 4 cc. of K / ~ sodium o hydroxide were required for the titration of 50 cc. of the chloroform solution, it is obvious that an error of about 0.15cc. in the titration would account completely for the discrepancy between the numbers. “It is of interest to note the considerable difference in the relative proportions of the simple and double acetic acid molecules within the limits of concentration investigated. At the highest concentration, the acetic acid present in the form of double molecules is more than five times as large as that present in the form of simple molecules, whilst at the lowest concentration the proportion is less than one-fifth. The value 42.9 calculated for the ratio of distribution of the simple acetic acid molecules between water and chloroform a t zoo,when compared with the highest ratio of distribution determined experimentally, 37.04, would indicate that at concentrations only slightly less than the smallest actually investigated, the acetic acid in the chloroform would consist practically completely of simple molecules. I t was not considered advisable to carry out’ experiments at higher concentrations, for the addition of acetic acid to a mixture of chloroform and water increases the mutual solubility of these liquids, and this increase is only terminated by complete miscibility.” If they were not based on a false assumption these calculations would constitute a beautiful proof that deviation from a simple distribution ratio is due to association of the acetic acid and that the polymerization of the acetic acid increases with increasing concentration. The article was reviewed by W.D. Bancroft in 1903:’ “This is a beautiful illustration of the utter unreliability of the conclusions based on a hypothetical constant. In this particular case we know that acetic acid makes chloroform and water more miscible and that therefore the preliminary assumptions of the author are unjustifiable. I n spite of starting from inaccurate premises, he comes out with a result which is an apparent confirmation of an avowedly incorrect theory. If figures can be, and are, juggled in this way so as to lead to false results in a case where they can be ‘ W .D. Bancroft: J. Phys. Chem., 7,46 (1903).
1966
DAVID B I R S E Y H A S D
detected, what reason have me to suppose that false results have not been so reached in many other similar cases?” There is actually little justification for the modified form of Sernst distribution equation proposed by Hendrixson. I t is not only theoretically inadequate and cumbersome, but at best can only be applied over a very narrow range of concentrations. The most Concentrated solution of acetic acid in chloroform to which the equation can be applied is slightly less than one percent. Since the liquids do not become completely miscible until the acetic acid is present in a concentration of about 40 percent, it can be seen that the modified form of distribution equation has not been successfully applied to the distribution of acetic acid between chloroform and water. So far as we know, this case of acetic acid is the most concentratcd solution of a consolute liquid in two immiscible liquids which is claimed to be described by the simple distribution law. One other objection to Dawson’s method of dealing with distribution is that it implies that concentrated solutions are too complicated to be handled and that all that can be done is to suggest that the same theory holds for all concentrations. We do not accept Dawson’s conclusion that the variations from the simple distribution law are to be accounted for solely on the ground that the polymerization of the acetic acid increases with increasing concentrations. The experiments of Beckmannl on freezing-points are often cited as independent evidence for the association of acetic acid in benzene. Beckmann suggests that the molecular weight of acetic acid varies from I I O to I j 3 in concentrations varying from 0.46jTc to 2 2 . 8 5 . His experiments certainly show that the acetic acid is abnormal in benzene. The freezing-point equation used by him was derived from the Clausius-C‘lapeyron and the Raoult equations. Although the abnormality can be accounted for by assuming the association of acetic acid, any other reason for deviation from the Raoult equation would do just as well. Furthermore Reckmann’s experiments can offer no clue to the effect that entrance of water into the benzene phase has on the partial pressure of the acetic acid. JYe have no independent indication as to whether the acetic acid becomes more or less abnormal in increasing concentrations. It appears experimentally that, instead of becoming more complicated with increasing concentrations, the distribution of a consolute liquid becomes more simple. Whereas the modified Nernst equation can only be applied to the distribution of a consolute liquid in exceedingly low concentrations, we have found an equation which describes the distribution in concentrated solutions up to the point at which the liquids become entirely miscible. 11-e hope in this paper to show how the increased solubility of the immiscible liquids affects the distribution of the consolute liquid between conjugate phases, When this miscibility is taken into account the general equation for distribution between phases becomes: E. Beckmann: Z. physik. Chem., 2 , 7 2 9 (1888).
DISERIC DISTRIBrTIOS
in which, A . = grams of consolute liquid, B and C = grams of immiscible components, I< = constant independent of the amounts of components, and the subscripts denote the upper and lower phases. I t is possible to make a simple assumption for distribution of a consolute component which will account for the general validity of this equation for distribution between phases, The consolute component is divided between the other two components according t o a simple ratio, so that in each liquid phase the ratio of the weights of the Consolute component held by unit quantities of the other components is a constant. The assumption can be stated mathematically by letting, A b = grams of consolute liquid . I held by B, A, = grams of X held by C, B and C = granis of the immiscible liquids.
Then for each phase, (Equation I)
If the immiscible liquids remained immiscible on addition of the consolute liquid, Equation I would be the general equation for distribution between phases, provided our assumption for distribution is true. K is possibly a measure of the relative physical affinities of A for B and C. If the value of K is 2, twice as much A is held by a gram of C as by a gram of B. I t is impossible to test Equation I by direct experiment because it is impossible to distinguish analytically between the amounts of the consolute liquid held by the two components in the same phase. All we can determine directly is the total amount of consolute liquid in each phase. However it is possible to derive from Equation I 3 form which can be tested experimentally in cases in which the immiscible components: are partially solubl:. Let, .Ale = amount A held by R amount h held by C = total .A in phase,. An A,, = total X in phase?.
+ +
Then if Equation I is correct,
Whence, and
+
1968
DAYID BIRNEY H A S D
Equation I1 is a general equation for distribution of a consolute liquid which was discovered empirically by us. I t supplies evidence that there is an actual distribution of the consolute liquid between the other two components even when they are both present in the same phase. Equations I and I1 approach identity with each other and wit,h the Xernst distribution equation in the special, limiting case in which the two immiscible liquids remain practically immiscible on addition of the consolute liquid. Then,
This is the way in which concentrations in the Xernst equation should be expressed and is the method which Kernst himself used. Since he did not explicitly define the concentration terms, other investigators have used volume concentrations, weight percents, mol fractions, etc. . I t is obvious that if
$/$ is a constant, then BI + CI / Bz + Ca will not be a constant. ___
~
We have been able to find only four cases of dineric distribution in which each phase has been analyzed accurately for all three components. For all of these cases Equation I1 describes the distribution of the consolute liquid between the conjugate phases in the region of high concentrations. For very dilute solutions neither Equation I1 nor the Kernst, equation will account for the distribution of the consolute liquid. b. Distribution of Acetic Acid between W a t e r and Benzene. We determined the tie-lines for the system acetic acid, water, and benzene by the method used by Wadde1l.l Our data for the isotherm a t z5OC. are given in Table IV. The distribution of acetic acid between the conjugate phases is shown in Table 1-and the tie-lines are plotted on a triangular graph in Fig. I . I n 1893Waddell studied the system benzene, water, and acetic acid. His method of determining the isotherm was to measure 5 cc. of acetic acid from a burette into a series of flasks submerged in a water bath at 25°C. A certain amount of water (or benzene) was next added, and the quantities of water or benzene needed to produce uniform clouding measured in. I n order to determine the distribution of tlic :icctic acid Waddell removed measured samples in a pipette from conjugnte phnsca :ind titrated for percent acetic acid. He plotted his data for the isotherm on n triangular graph and then drew in his tie-lines by finding the points on the isot,hcrm which corresponded to the acetic acid values found by nnnlysis. I n an effort to determine the manner in which the acetic acid influenced the solubility of the water and benzene, Lincoln repeated Waddell’s work on the isotherm.2 Lincoln’s chief criticism of Waddell’s work was in his selection of the point of uniform cloudiness as his end-point. Lincoln claimed correctly that over part of the curve the precipitate appears as globules, while over the J. Waddell: J. Phvs. Chem., 2, 233 (1898). * A . T. Lincoln: J. Phys. Chem., 8, 248 (1904).
D I S E R I C DISTRIBUTIOS
I969
rest it first produces either an opalescence or a cloudiness. Accordingly he took as his end-point the first distinct appearance of a second phase. Otherwise the methods of the two men were exactly alike, in that they measured their liquids volumetrically and added water and benzene to j cc. of acetic acid until a precipitate was produced. Lincoln rightly pointed out that Waddell, by taking a point of uniform cloudiness as his end-point, must have exceeded the exact point of saturation
/
\
FIG.I Tie-lines for Benzene, Water and Acetic Acid
when adding benzene or water. Our analyses confirm his statement that Waddell's points do not contain enough acetic acid. However every one of Lincoln's points when plotted in the same units lies on a curve in a region of even less acetic acid than Waddell's isotherm. Therefore our knowledge of the system benzene, water, and acetic acid is still a little dubious. I n determining the isotherm at ns"C., all but one of Waddell's points and all but three of Lincoln's required the measurement from a burette of less than I cc. of the third liquid. Due to evaporation from the tip during the titration and the error in reading so small a quantity, as well as finding a difficult cnd-point, accurate work must have been impossible by their method. Furthermore Lincoln criticised Waddell's results on the isotherm but did not make any measurements of distribution so we have felt obliged to repeat a complete study of the system.
I970
DAVID B I R S E Y H A S D
TABLE IT Isotherm at 2 5°C. for the System Benzene, Water, and Acetic Acid Run
Acetic .Acid
Water
Benzene
9.30
10.47
9.985 10.86 5.87 6.04 12,16 1 1 .64 5.89 4.13 5 .38 30.16 29.98 29.83
20..;4
11.20
20.57
I S .89
I
3 i ,46
2
36.26 30.88 33.2 31.29 36.39 36.74
3 4
5 6
8 9
2 6 . IO
IO
30.88
11
20.23
I3 I4 '5 16
14.21
17
20.32-20.45
I8 I9
20.80
20 21 22
23
Plait Point
I
7.09 14.43 16.56 4.10
6.44
22.23 =7.53 17.66 0.48 0.21 0.1s
I .50-1.61 0.98-1.06 28.38 0.89-0 90
.83
17.59
20.83 10.09-10.~1 9.85 10.45-10.85 10.19-10.33 5 2 ,3
2 .09-2 . I4
9.i4
11 .04
1.11
I
.31-1 .36 ,717 '
8.23 19.jo 31.53 40.5
404
7 . 2
TABLE Y Distribution of Acetic Acid between Benzene and Kater Data in grams per roo grams of solution Run
Upper Phase Acetic Benzene Acid a1
I1
0.15
7
I
8
3.2j
I
13.3
2
15.0
3
19.9 22.8 31 . o 35.3
I2
I4
r3
9 IO
Plait Point
11i
99.849 98.56 96.62 86.4 84.5 79.4 76.3j 67.1 62.2
R-ater
.Acetic Acid
e,
a?
0.001
0 .o'$ 0.11
0.4 0.j 0.7
0.85 1.9 2.j
37.8 44 ' 7
59.2
50.;
3.o 1.6
j2.3
40.;
7 . 2
Lower Phase Benzene Water hr
4.56 '7.7
0.04
29 . o 56.9
0.40
59.2
4.0
63.9 64.8 6j 8 64 5 63 4
6.5
95.4 82.1 jo.6 39.8 36.8 29.6
7.7
27.5
59.3
0.20
3.3
K
c2
31 8 I5 2
3 9 4 9 7
12
8 9
21.1
16.1 '1.4
'3.4 30 . o
13 . 2
8 9 9 9
10.;
8 0
18.1
0 7 2 0
DINERIC DISTRIBUTION
1971
By our method we were able to eliminate nearly all of the difficulties that beset Waddell and Lincoln. We started with the purest liquids available and fractionated and purified them as described by Lincoln. The benzene was thiophene-free at the beginning. The acetic acid was Kahlbaum's, for analysis and free from homologues, and was analyzed with barium hydroxide and found to be roo percent pure. T o escape volumetric errors all our measurements were made gravimetrically into light-weight, erlenmeyer flasks. These were fitted with ground-glass stoppers to prevent loss by evaporation. The experiments were made in a water bath whose temperature was maintained constant with the aid of a Beckmann thermometer standardized against a Bureau of Standards thermometer. Loss by evaporation due to opening the flasks never exceeded 0.03 grams during prolonged experiments. The greatest difficulty encountered by the previous workers was the finding of an exact end-point. We were able to eliminate this source of error entirely by recording first the quantity of the third liquid which was unmistakably insufficient and then definitely in excess. Another advantage of this method was that it showed the limit of accuracy of our determinations. There was no doubt about the points near the ends of the isotherm where very small quantities of one of the liquids were added. The exact limits between which the curve was to be drawn were obtained in this way. The procedure in detail was as follows: Quantities of acetic acid and of benzene (or water) were weighed into glass-stoppered flasks. The quantities chosen were such as to bring the total weight of liquids between 40 and 60 grams. The liquid to be added last was the one whose smallest quantity would yield the most voluminous precipitate. When a n amount of the third liquid had been added so that the solution became homogeneous a t z j"C., but clouded on cooling to PzOC., the weight of the third liquid was recorded. More of the liquid was then added until the precipitat,e remained clearly visible for hours at zs0C. This last weighing was also recorded but as a n excess of the third liquid. I n this manner P I points were located on the isotherm, 9 of the points being of the double variety and showing the exact limits of accuracy. The range covered by the double points lay between a solution on the benzene-rich end containing only one percent of water and a solution on the water-rich end containing 2 . 2 per cent of benzene. Beyond this point on the benzenerich end a drop of water more or less would amount to almost I O percent error in a total weight of 50 grams of liquid. Furthermore the end-point became more difficult to detect as the end of the isotherm was approached, because the volume of the second phase became smaller. The experiments were made over as wide a range as could be accomplished with uniform accuracy. The tie-lines were determined by the following method: Such quantities of benzene, water, and acetic acid as would separate into two layers were placed in small, cylindrical, glass-stoppered weighing flasks and brought to equilibrium for an hour in the water bath at P j"C. About z grams of each phase was carefully transferred with a P cc. pipette into a glass-stoppered, light-weight erlenmeyer. Duplicate quantities were weighed and titrated
19i2
DAT'ID BIRXEY HAND
with 0.208 normal barium hydroxide. The values for percent acetic acid were located on the isotherm and the tie-lines drawn in. The amount of the phases withdrawn and the strength of the barium hydroxide was so chosen that all the experiments could be done under identical conditions. The range of the experiments was as wide as possible when carried out with a uniform method. For example Run 9, in which the lower phase contained 63.4 percent of acetic acid, involved the use of IOO cc. of alkali to titrate the 1.963 grams of liquid withdrawn. However in Run 7, the lowest authentic tie-line, the upper phase contained 1.4 percent, acetic acid and only 2 . 0 5 cc. of barium hydroxide were required to neutralize the 1.880grams of liquid withdrawn. Four additional runs were made and not recorded in Table T.r because they duplicated tie-lines already determined. Run 14 was the most irregular tie-line and is probably in error. One interesting and unexpected source of error was encountered during the experiments. I t was found that the composition of saturated solutions in the neighborhood of the plait point was very sensitive to external pressure. For example when a portion of the clear, saturated phase was drawn up int'o a pipette it immediately clouded and separated into two phages. If two conjugate phases were stoppered in a test tube the pressure of stoppering caused them to become consolute at once. When the stopper was removed they separated immediately into two phases. This discovery was made by us after most of the work had been completed and no way was devised for regulating the external pressure. I t seems just as important that these experiments should be made under a constant pressure as under a constant temperature. Fig. I illustrates t,he general met'hod of describing ternary systems of two immiscible liquids and one consolute liquid by means of a triangular diagram. Any ternary system can be pictured in this fashion if the sum of the weights of the three components is kept constant at I O O grams. When the apex of fhe triangle represents I O O percent of the consolute liquid all the solutions in the region above the isotherm are unsaturated. Any point beneath the curve shows the composition of a mixture that will separate into two layers. The composition of each of these two saturated phases can be represented by points opposite each ot,her on the isotherm for saturated solutions. A straight line, passing through the point of total composition, connects these points on the isotherm which denote the compositions of conjugate layers. Such a line is called a tie-line and shows graphically the distribution of all the components between the two phases. With increasing amounts of the consolute liquid the two conjugate layers become more alike in composition. The uppermost tie-line, then, is a point on the isotherm a t which the upper and lower phases are identical in composition. This point is in a sense a critical point because, for a given system, its composition depends only on temperature and pressure. In referring to this point we will use the term, plait point, which was suggested by Lash Miller. Examination of the graph for the system benzene, water, and acetic acid shows that the isotherm is not symmetrical and that the plait point is not
D I S E R I C DISTRIBCTIOS
I973
on the maximum of the isotherm. The tie-lines are therefore not horizontal, with the exception of the limiting tie-line connecting pure benzene with pure water. It was suggested by Professor Bancroft that the tie-lines could be horizontal if the composition of one of the components was expressed in some unit other than the gram. For example if the unit for water was changed so as to bring the plait point to the maximum on the isotherm at least the top and bottom tie-lines would be horizontal. The top one because a continuation of it would be tangent to the maximum of the curve and the bottom tie-line because it coincides with the base of the triangle. I t was reasonable to hope that all the tie-lines might be horizontal if the top and bottom ones were made so by a proper selection of units. If the tie-lines of a ternary system of two immiscible liquids and one consolute liquid could be made horizontal by changing the units in plotting on a triangular graph, the complete diagramming of such a system would be very greatly simplified. Thus a knowledge of the isotherm and the plait point would give the composition of every possible liquid pair that existed in equilibrium. Or else a knowledge of the isotherm and the distribution of just one component between any two conjugate phases would give the distribution between every possible pair of conjugate phases. Accordingly the data for the tie-lines of the system benzene, water, and acetic acid were recalculated by multiplying all the water values by 9.0 and then bringing the total weight back to roo grams again for plotting. This was equivalent to expressing the water in units of 1 / 9 grams instead of in grams. As a result all the points on the diagram were shifted in the direction of higher percentage water. The ratio of benzene to acetic acid was unchanged for all the points. Before recalculation the upper phases always had percentages of acetic acid higher than the lower phases. After recalculation the tie-lines were nearly horizontal, i.e. the percentage of acetic acid was equal in conjugate phases. By horizontal tie-lines after a change of units we mean that, when the water values are multiplied by an empirical constant, the relation holds that, al
a1
a?
+ bl + K.cl - a2 + bz + K.c2 -
in which a, b, and c, represent the masses of acetic acid, benzene, and water in the phases designated by the subscripts. The above equation can be written in a different form: (Equation 11.) Equation I1 is the general equation for distribution which we have found empirically t o describe the distribution of a consolute liquid between two immiscible liquids, and which takes into account the fact that addition of a consolute liquid to two immiscible liquids always increases their mutual solubility. It is obvious that points satisfying this general distribution equation
I974
DAVID BIRNEY HAND
will lie on horizontal tie-lines when plotted with the proper units for one of the immiscible liquids. Therefore the proof of the applicability of the distribution equation can rest either with a calculation of the constant, or with plotting horizontal tie-lines. Table VI gives the tie-lines for the system benzene water, and acetic acid when recalculated with the new units. Table Vgives thevalues of K calculated from Equation 11. Fig. z shows how closely the tie-lines are made horizontal
FIG.2 Horizontal Tie-lines for Benzene, Water and Acetic Acid
by a change of unit. The equation holds within one percent over the whole range. I n concentrated solutions this deviation may be within the experimental error. However in very dilute solutions, e.g. Run 7, a one percent deviation from Equation I1 amounts to 80 percent error in the value of the acetic acid percentage. This discrepancy cannot be attributed to experimental error and it must be stated that for those concentrations in which the mutual solubility of the benzene and the water has not been appreciably increased the simple form of distribution law fails to apply exactly, although the tie-lines are approximately horizontal. The data of Nernst, Waddell, Herz and Fischer,' and our own data all agree that for dilute solutions the distribution ratio for acetic acid between water and benzene increases with 'Her2 and Fischer: Ber., 38, 1 1 3 8 (1905).
I975
DINERIC DISTRIBUTIOS
dilution to a value of about 30 and is never constant. If Equation I1 were perfectly accurate for very dilute solutions, the ratio should approach a value of g with increasing dilution. If polymerization of the acetic acid in the benzene is the cause for failure of the simpIe distribution law in low concentrations, then it might be presumed that as the water enters the benzene phase in large amounts the acetic acid is changed back into normal molecules. Whatever the explanation, it may be said in summary that the general distribution equation, (Equation 11),applies accurately to the distribution of acetic acid between water and benzene when the acetic acid is present in high concentrations and applies approximately for low concentrations in which the water and benzene are only slightly miscible.
TABLE VI Horizontal Tie-lines for the System: Benzene, Water, and Acetic Acid. Data in weight percent with arbitrary unit for water. Acetic acid and benzene in grams, water in 1/9 grams. Run
Acetic Acid al
I1
0.1
7
1.4
8 I 2
3 I2
14
I3 9 IO
Plait Point
3.2
12.9 14.4 18.8 21.4 26.9 29.4 30 5 32.7 33.2
Upper Phase Benzene
Water
Acetic Acid
Lower Phase Benzene
bi
9.ci
a1
bi
90.8 98.3 95.8 83.6 81.3 75.2 71.4
0.1
0.528
0.02
0.3 I
.o
3.5 4.3 6 .o 7.2
58.2
14.9
51.8 47.8 37.0 25.7
18.8 21.7
30.3 41.1
2.34 4.36 13.5 I5 .o 18.9 20.3 28.7 29.9 30.8 32 .o
0.02 0.Oj
0.8 I
.o
1.9 2.4
7.9 9.8 11.4 16.2
Water 9.Q
99.452 97.64 95.57 85.7 84.0 i9.2 77.3 63.4 60.3 57.8 j1 . 8
c. General I.alidity of the Equation f o r Tie-lines in Ternary S y s t e m s . There are several other ternary systems described in the literature which can be tested to see whether Equation I1 applies to them as well as to the system benzene, water, and acetic acid. The first case on record is the distribution of alcohol between benzene and water. Table T’II gives the data of a. F. Taylor for this system together with the value of K calculated from the equation. Figs. 3 and 4 show how nearly the tie-lines are made horizontal by a change in the unit for water. There is no doubt that the deviation of the tie-lines from the horizontal position is well within the experimental error of Taylor’s analyses.
DAVID BIRSEY H A S D
FIG.3 Tie-lines for Benzene, Water and Alcohol
FIG.4 Horizontal Tie-lines for Benzene, Water and Alcohol
I977
DINERIC DISTRIBUTION
If Equation I1 applied perfectly even for dilute solutions the distribution ratio for alcohol between water and benzene should approach a value of 9 with increasing dilution. But according to the experiments of Morgan and Benson: the distribution ratio is 0.81 when 0.040 grams of alcohol are dissolved in one gram of water. Their experiments do not disagree with those of S.F. Taylor, because they report that the distribution ratio increases with increasing concentration of alcohol. TABLE VI1 Distribution of Alcohol between Water and Benzene Tie-lines for the system in weight percent Benzene hi
60.8 65.9 75'5 82.5 87.6 91.2
Upper Phase Water CI
7.1
Alcohol ai
Benzene h?
32.1 28.5
2.0
15.5
1.2
11.2
45.1 36.5 20.9 IO . 9 5.9
0.7
8.1
2.8
5.6 3'4
21.1
Lower Phase Alcohol Water a:! C?
12.6 16.7 26.6 37.2 46.2 55 .6
42.3 46.8 52.5
51.9 47.9 41.6
K
10.8 8.8 8.4 8.3 8.4 9.0
Horizontal Tie-lines for the System Benzene, Water, and Alcohol. Data in weight percent with arbitrary unit for water Benzene in grams, alcohol in 1li2 grams, water in 1/9 grams Benzene hi
32.2 38.0 jo.9 62.7 72.5 80.3
Upper Phase Water Alcohol 9x1 a1
33.8 29.1 20.6 13.7 8.9
5'5
34'0 32.9 28.5 23.6 18.6 14'2
Benzene h2
18.5 13.0 5.73 2.4 1.1
0.j
Lower Phase Water Alcohol 9x2 a?
46.7 53.6 65.5 74.5 80.5 8j.3
34.8 33.4 28.8 23.1 18.; 14.2
Another system on which direct analyses have been made is the distribution of acetic acid between chloroform and water.? Table VI11 gives the data for this system together with the calculated values of the constant. Figs. 5 and 6 show how nearly the tie-lines are made horizontal with a change in the unit for water. We know that the values for water in the chloroform phase are slightly inaccurate because they do not increase steadily as they should with increasing acetic acid. However the system supports the theory very nicely. The constant in Equation I1 shows signs of increasing in the region of small concentrations of acetic acid. This is confirmed by the d a t a of Rothmund and Wilsmore3 who show that the distribution ratio for acetic a
Morgan and Benson: Z. anorg Chem., 5 5 , 356 (1907). Wright, Thompson and Leon: Proc. Roy. Soc., 49, 74 (1891). Rothmund and Wilsmore: Z. physik. Chem., 40, 611 (1902).
DAVID BIRh’EY HAND
-CHLOROFORM
FIG.j
Tie-lines for Chloroform, Water and -4cetic Acid
FIG.6
Horizontal Tie-lines for Chloroform, Water and Acetic Acid
I979
DINERIC DISTRIBVTIOX
acid between water and chloroform reaches a value of 26. for a solution containing 0.024 grams of acetic acid in I gram of water. As would be expected they report that this ratio decreases with increasing concentration.
TABLE VI11 Distribution of Acetic Acid between Chloroform and Water Tie-lines for the system in weight percent Upper Phase Acetic Acid Chloroform Water a1 C, hi 0 0.84 99.16
0.92 0.79 1.21
1.85 2.97 7.30 12.82
15.11
92.62 81 5 2 73.69 70.42
63.32 48.j8 37.82 34.71
6.46 I 7 .69 25.IO
27.73 33,7' 44.12 49.36 50.18
Chloroform cz
99.01 98.24 94.98 91.85 91.23 87.82 80.00 72.86 70.13
Lower Phase Acetic Acid Water bz a2
0.99
0
0.72
1.04 3'83 6.77 7.95 11.05
1.19 1.38 0.82 1.13 2.28 3.62 4.12
17.72
23.52 25.75
K
6.91 5.77 4.95 4,77 4.42 4.52 4.58 4.59
Horizontal Tie-lines for the system Data in weight percent with arbitrary unit for water Chloroform and acetic acid in grams, water in 1/4.6 grams Chlorofarm Water 4.6b1 c1
0.18 0.3 0 . 2
0.3 0.5 0
.g
2
.6
5 .4
6.7
99.82 98.2 95.3 92 .8 91.7 88.9 81.4 73.8 7 1 .O
Acetic .4cirI a1
0
1.5 4.5 6.9 7.8 10.2
16 . o 20.8 22.3
Acetone c2
Water 4.6 bz
95.6 95.8 91.1 87.5 88.6 84.4 73.9 64.5 61.1
4.4 3.2 5.24 6 .o 3.7
5 .O 9.7 14.7 16.5
Acetone a2 0 I
.o
3.67 6.5
7.7 10.6 16.4 20.8 22.4
There are no other cases in the literature for which the tie-lines have been determined accurately enough to be tested with our distribution equation. W. D. Bonner' has reported the study of fifty ternary systems. However his data are not accurate enough for our use. In only a few of the fifty systems are more than two tie-lines given. For several of these systems, as one phase varies in composition the other phase is supposed to remain unchanged. For example Ronner reports that for the system propyl alcohol, water, and chloroform, pure water exists in equilibrium with four different conjugate phases. This is an obviously impossible situation. Bonner's tie-lines were determined 1 W.D. Bonner: J. Phys. Chem., 14, 738 (1910)
1980
DAVID B I R S E Y H A S D
by an indirect method of calculation from measurement of the densities and ratios of the volumes of the conjugate phases. For these reasons we are basing the proof of the validity of Equation I1 on data which have been determined by methods of direct analysis. However we felt it necessary to repeat the work of Bonner on a t least one of the systems which according to his data failed to support the distribution equation. T.4BLE
Ix
Distribution of Acetone between Chloroform and Water Tie-lines for the system in weight percent Chloroform c1
1.23 1.29 1.71 3.20 5.1
9.8
Upper Phase Water Acetone hi a1
82.97 73.11 62.29 54.1 45.6 34 5
Lower Phase Chloroform Water Acetone c2 112 a?
15.8
70.0
1.3
2j.6 36.0
55.7
2.2
42.9
42.7 49.3
35.8 28.4
55.7
20.4
4.4 7.7 10.3 18.6
K
28.7 42.1
3.46
52.7
2.13
56.j 61.3 61.0
2.10
2.21
2.04 2.04
Horizontal Tie-lines Data in weight percent with arbitrary units. Chloroform in I , 2 . I grams, water and acetone in grams Chloroform 2.IC,
2.6 2.9
3.6 6.5 10.1
18.6
Upper Phase Kater hi
81.8 71 9 61.1 52.2
43.2 31.1
Acetone aI
15.6 2j.2
3j.3 41.3 46.7 jo.3
Lower Phase Chloroform Water rZcetone 2.IC? b2 a2 0.7 16.2 83.1 1.4 26.1 j 2 . j
61.2 53.9
3.0
45'4
7.9
3j.0
15.2
5'5
35.8 40.6 46.7 49.8
d. The Distrzbutzon of Acetone between Tl-ater and Chlorojorm. The system chloroform, water, and acetone was described by Bonner and was selected by us for our experiments because two of the three components could be determined by direct chemical analysis. Table I X gives our data for the distribution of acetone betwpen chloroform and water together with the values of K calculated from Equation 11. Figs. 7 and 8 show how nearly the tie-lines can be made horizontal by changing the unit for chloroform. The constant in Equation I1 appears from our data to increase for dilute solutions. Since K becomes identical with the distribution coefficient in very dilute solutions, the data of Herz and Levy' indicate that K approaches a value of 3.j in a solution containing 0.00186 grams of acetone in one gram of water. 'Hem and Levy: Z. Elektrochem 11, 6 1 6 (190j).
D I S E R I C DISTRIBUTIOS
FIG.7 Tie-lines for Chloroform, Water and Acetone
FIG.8
Horizontal Tie-lines for Chloroform, Water and Acetone
1981
1982
DAVID B I R S E I ' H h S D
The data for the distribution of acetone between chloroform and water were obtained by the following procedure. Kahlbaum's acetone (for analysis) was used and fractionated from calcium chloride and soda lime. The water was redistilled from barium hydroxide. h l l the liquids mere kept away from contact' with the carbon dioxide in the air. The chloroform was fractionated from calcium chloride and soda lime just before it was to be used because of the unstable nature of purified chloroform. The liquids were measured roughly into 6 cylindrical flasks fitted with ground-glass stoppers and having a capacity of j o cc. These were placed in the thermostat bath and allowed t o come t o equilibrium. Quantities of the three liquids had been chosen so that in all the flasks separation into two phases would occur and a graded series of conjugate phases would be obtained ranging from R low percentage acetone to a percentage near the plait point. The flasks w r e shaken frequently for two hours so that equilibrium would be established. Four samples from each phase were drawn into fragile glass bulbs for analysis. Small holes, about, a millimeter in diameter, had been previously drilled in the stopper of each weighing flask and kept filled with slivers of wood during the experiment. The small glass bulbs into which the samples were drawn were blown from 4 mm. tubing so that they weighed less than 0.2 grams and were 14 mm. in diameter with capillary tips about IOO mm. long. The bulbs were weighed empty and then filled by placing the tip through the hole in the flask cover and just below the surface of the phase to be withdrawn. The bulb was then alternately warmed and cooled, by which process a sample was withdrawn weighing about 100 mg. Since the withdrawing of samples tookseveral minutes for each flask it was thought advisable to avoid removing the covers from the flasks and for this reason the holes in the stoppers were bored. As a further precaution the temperature of the room mas brought t o z j"C. and the flasks dried with a cloth on removing from the bath so that the contents would not cool off while samples were being withdraJm. The samples were sealed in by melting apart the tip about z j mm. from the bulb. Both the tip and the bulb were then weighed. Analysis for chloroform was made by fusion in a Parr bomb' and then determining the chloride by the usual volumetric method. I t was found by previous trials that for samples containing as little as I O O mg. of chloroform the transformation of the chloroform was complete in nearly every case. Analyses for acetone were carried out, according t o the original Nessengrr method.* The bulbs were broken directly under the dilute sodium hydroxide, an excess standard iodine solution added, and the solutions backtitrated with standard thiosulphate. Duplicate runs were made in the majority of cases. The agreement was found to be within I and 2 percent of the total weight of liquid for both the chloroform and the acetone determinations. The chief sources of error were in withdraxing the samples and in the fusion of the chloroform. Instead of averaging the duplicate runs the points selected were the ones which lap on the smoothest curve when plotted on a triangular Lemp and Broderson: ,J. Am. Chern. Soc., 39, 2ohy (1917). J. Am. Chem. Soc., 42, 39 (1920).
* L. F. Goodwin:
I983
D I S E R I C DISTRIBUTIOS
diagram. However the agreement was satisfactory enough so that it made little difference which determinations were plotted. As a result of our analyses of the system acetone, chloroform, and water we believe we have shown that the new distribution equation is applicable to all the cases which have so far been investigated of the distribution of a consolute liquid. S o n e of the cases known observe the simple Kernst distribution law in low concentrations yet Equation I1 describes the distribution when the consolute liquid is present in high enough concentrations to cause an appreciable miscibility of the immiscible liquids.
TABLE X Distribution of Acetic Acid between Benzene and Water log (a2 c2) - o 79 log (a, b,) = const Run
H4c H 0
HAC CsHs
log%
loi:
CI
a h1
a? c j I1
7 8 I 2
3 I2
11 I3 9 IO
1 78 21
5
41
1
I41 161 216 236 408 447 480 551
3
bt
679
0
'5
9 176-10
I 333
I
12
0 Ij2
0
I
614
2
155
2
207
2
331 3J3 611
3 38 15 1 17 8 25
I88
I
250
I
400
I
I
476 662 754 805 945
2
6;o
2
681
2
713
88
2
529
I
1
29 9 A i 9 :6 8 63 8
2
0
I 1
I
I
e . A Second Distribution Eqiintion. The distribution of acetic acid between benzene and water is described by the equation,
log (a2IC?) -
0.j
9
log(al, bl)
=
const.
in which a represents grams of acetic acid b, = grams of benzene in the upper phase ce = grams of water in the lower phase. Table X gives the values for the ratios of acetic acid to benzene in the upper phase and acetic acid to water in the lower phase. Fig. 9 shows the data in Table X plotted on a logarithmic graph. Our experiments indicate that the equation applies perfectly to the distribution of acetic acid between benzene and water up to a concentration of 6; percent acetic acid in the lower phase. Above this concentration there is a discrepancy for certain of the runs. Run 14may lie off the line because of experimental error. This run also showed the greatest deviation from Equation 11. The points in the neighborhood of the plait point were most difficult to determine because the densities and optical properties of the two phases were nearly the same,
1984
D.4VID BIRXEY HAXD
thus causing difficulty in withdrawing samples. Another source of error in the region of high acetic adid concentrations was the sensitivity of the system to external pressure changes. I t was first suggested by IT. D. Bancroft' that the Mass Law should govern the equilibrium of a component between two phases. In the case of the distribution of a consolute liquid between two immiscible liquids, increasing concentration in one phase is accompanied by an increase in the other. In such a case applying the mass law to the equilibrium and expressing the concentrations in the two phases by Sland X?,
In this equation the exponents might not necessarily be integers.
FIG.9
Distribution of Acetic Acid between Benzene and Water
The equation could be put into a more usable form by taking the /3 root of each side. Then, letting = m, X,",Xz = K?
Letting b and c represent the grams of the immiscible liquids, and a) and a? the grams of the CGnSOlUte liquid in the upper and lower phases respectively, and expressing concentration of the consolute liquid in grams per gram of the immiscible liquid which characterizes the phase, we obtain the general form,
(;)72 =
const.
(Equation 111.)
The curve for Equation I11 when plotted on logarithmic coordinates takes the form of a straight line. The equation was first tested by S.F. Taylor with the system benzene, water, and alcohol. His data are given in Table XI. Fig. I O shows the points Phys. Rev., 3,
2 1 , 114,
193 (189j).
I985
DINERIC DISTRIBUTION
TABLE XI Distribution of Alcohol between Benzene and Water. (S. F. Taylor) 1 . 2 log(aa/c2) = log (al/bl) const.
+
log 2 c2
Alc/HnO dc2
7.45 6.22 4.40 3 .IO 2 ,30 I .66
,872
,794 ,644 ,491 ,362 ,222
Alc/C6C6
a1/h 10.6 8.65 5.60
log 3
log K
I ,025
.02 I
bi
,937 ,748 ,575 ,407 ,248
3,76 2.55
1.77
,011
,024 .0x4 ,027
,019
TABLE XI1 Distribution of Acetone between Water and Methyl Alcohol across a Rubber Membrane. (D. S.Morton) log (al/bl) - 1.09 log (aa/ca) = const. % Acetone in water IO 20
30 40
50 60 to
Acetone in alcohol
Ac/Alc a h
Ac/W found adc2
17 32 46 59 69.5 77.5 85
0.205
0.111 0.250 0.429 0.666 I .oo
0.239 0.412 0.667
I .SO
I
2.34
2.341
0.471
0.850 1.44 .28 3.44 5.66 2
Ac/W calc. adc2 0.111
I ,017
,482
log K
0,357 0,330 0.331 0.35' 0.358 0,345 0.354
TABLE XI11 Distribution of Acetic Acid between Water and Chloroform (Wright, Thompson, and Leon) 1.1 log (al/bl) - log (az/cz) = const. HAC CHCI,
log &2 CI
a2ic2
HAc/H20
log K
0.844 1.336 1 ' 533 1.596 I ,726 1.958 2.114 2,161
,902 ,865 ,820 ,815 ,800 808 ,816
adbl
I 06 4 03 7 38 8 71 12 6
0.205
6.98
0.605 0.868 0.940
21.7
22.2
I
32 3 36 7
a1
log bi
I ,I O 0
,346 1.509 1.565
34.1 39.4 53.2 90.7 130. 145,
,815
1986
DAVID B I R S E Y HAND
TABLE XIV Distribution of Acetone between Chloroform and R a t e r log (adbl) - 1.07 log (adc?) = c,onst. Acetone,lH20
ah: I ,90
3.50 5.78 7.89 10.80 1 6 .IO
log 81 bi
'279 '541
Acetone?CHCI, al,b2 4. IO
,762
i .56 12.28
,897
15.80
1.033 1.207
21.60 29,90
log .'i
log K
,613 ,879
- ,397
c2
1.090
1.199 1.331 1,476
- ,377 - ,404 - ,387 - ,395 - ,372
on a logarithmic diagram The data for three other systems are available: D S Morton' studied the distribution of acetone across a rubber membrane between methyl alcohol and water He reported that the equation W R P followed accurately u p to j 6 mol per0 8 cent of acetone. His data are shoan in Table XI1 without further expla06 nation (Fig 1 1 ) ThedataofWright, 0 4 Thompson and Leon for the dlstribution of acetic acid between O.? chloroform and water are given in 1 L O G1 1 ' 1 'I Table XI11 and shown graphically ob 0 4 06 OB 10 in Fig. 12. Our own data for the FIG.I O distribution of acetone between Distribution of Alcohol between Water chloroform and water also follows and Benzene Equation I11 over the entire range. These data are calculated in TableXIT'. We conclude that the empirical equation proposed by W.D. Bancroft t o describe the distribution of a consolute liquid between two immiscible liquids is applicable within the limits of the experimental error over the whole range of concentrations for all systems so far investigated. f . The Cse of Equations 11 and 111 in calculating the Compositions o.f Conjugate Phases. I t has been stated that if the isotherm for saturated solutions and just one tie-line are known the compositions of all possible conjugate pairs can be calculated from the equation: ALCOYO'/RENZCff€
This calculation can best be made graphically since we have no general, simple equation for the isotherm. D. S. Morton: J. Phys. Chem., 33, 384 (1929)
DINERIC DISTRIBUTIOX
0.5
I O
FIG.I I Distribution of Acetone between Alcohol and Water across a Rubber Membrane
FIG.1 2 Distribution of hcetic Acid between Water and Chloroform
I987
1988
DAVID BIRNEY HAND
If a line is drawn on a triangular diagram so that two points on the line satisfy Equation 11, all the points on that line will fit into Equation I1 to give the same constant, K. I n the accompanying graph (Fig. 14), all the points a on the upper line represent values of ___ = 0.2 joo. All the points on b 10.c a = 0.1I I . The lines are made horithe lower line fit the equation, b 10.c zontal if the amounts of component, c, are expressed in a unit one tenth the eize shown in the graph. The lines are thus hypothetical tie-lines for a system or which K in Equation I1 equals I O . I t is obvious that if the isotherm can be plotted and the composition of one phase is known as well as the constant, K, a line may be drawn fitting the equation for the tie-lines and the composition of the unknown conjugate phase will be the point a t which the tie-line intersects the isotherm. All possible pairs of conjugate phases are represented by the intersections with the isotherm of a series of lines having the same value for K in Equation 11. The fact that Equation I1 holds FIG.13 for all the points on a tie-line and Distribution of Acetone between Kater and Chloroform not only for points representing conjugate phases makes it possible to . . calculate the constant K for a tie-line very simply. Instead of substituting in the equation it is sometimes more convenient to extend the tie-line until it intersects the sides of the triangle. Since Equation I1 holds all along the tie-line, a t the edges of the graph, al/bl = a2,/K.c2. From this K can be readily calculated. By a similar line of reasoning it can be shown how Equation I11 can be used to calculate the composition of a conjugate phase when one phase and the isotherm are known. For example if a / b in the phase rich in b is known, the value of a/c in the conjugate phase can be calculated. Graphically this value falls anywhere on a straight line extending from 100%; b to a point on the ac side of the triangle. The exact composition of the unknown phase can only be known as the point a t which the isotherm crosses the line for a given a/c value. Neither Equation I1 or I11 is sufficient to calculate the composition of a conjugate layer in equilibrium with a known phase. However the two equat'ions can be used together when the isotherm is not known and one wishes to calculate the composition of a phase in equilibrium with a given liquid layer. A point on the triangular diagram can be found which cor-
+
~
+
~
DINERIC DISTRIBUTIOK
1989
responds to the composition of the known phase. According to Equation 11, the composition of the unknown phase must lie on the line connecting equal Kc). According to Equation I11 the value of a/c for the values of a / ( b second phase can be calculated and the point representing the unknown phase must fall somewhere on the line for the calculated values of a/c. The intersection of the two lines gives the composition of the unknown phase.
+
FIG.14 Graph of equations for straight lines on triangular diagram
The Effect of a Consolute Liquid on the Solubility of Two Immiscible Liquids a, The M a s s Law Equation and Physical Equilibria. We have seen how the distribution of a consolute liquid between two immiscible liquids is altered 11.
by the increase in mutual solubility of the immiscible liquids. If we knew also the way in which addition of a consolute liquid increases the solubility of two immiscible liquids we would have a very complete knowledge of ternary liquid systems. For example from the equation for the isotherm and either equation of the distribution of the consolute liquid, (Equations I1 or 111) we would be able to calculate the compositions of all existing conjugate phases. In 1894, W. D. Bancroft' pointed out that the Mass Law could be applied to certain physical equilibria and he advanced an equation for the isotherm for saturated solutions of a consolute liquid and two immiscible liquids. IieepW. D. Bancroft: Am. Acad., 30, 324 (1894);Phys. Rev., 3, P I (1895).
I990
DAVID BIRSEY HAXD
ing the amount of the consolute liquid constant, and letting C1 represent the amount of one non-miscible liquid and C z the amount of the other, an application of the mass law a t equilibrium would mean that,
Cla.C20 = const. In this equation the exponents are not necessarily integers. The equation can be written in a slightly simpler form by substituting n for alb. . . Then,
C?C2 or
n log C,
=
const.
+ log C 2
=
log const.
(Equation IT).
This last form is the equation of a straight line. Plotting the points obtained by analysis on a logarithmic graph offers a convenient way of testing the theory. The isotherms of a number of systems were found to yield straight lines on logarithmic graphs within the accuracy of the experiments. However in certain systems two straight lines were needed to describe all the saturated solutions. I t was suggested that these two lines corresponded with the two series of solutions, one being the upper phase and the other being the lower. The two lines in these cases were thought to intersect at the plait point. The assumption that the law of Mass Action governs the composition of saturated solutions of one consolute and two immiscible liquids can be stated in a different way. For example if we have a solution of acetic acid, water, and benzene which is saturated with benzene, adding either water or benzene will precipitate a phase rich in benzene. Holding the acetic acid quantity constant, and varying the amount of water, we may assume that the percentage decrease in amount of benzene is proportional to the percentage increase in amount of water. Then,
-db_ b
- n - dc C
in which b and c represent the amounts of the benzene and water. From this equation we obtain by integration, log b
+ n log c = log const.
(Equation I V )
The first careful verification of this equation was reported by Lincoln' for the system benzene, water, and alcohol. Lincoln carried out his experiments by adding the consolute liquid to a flask containing weighed amounts of the two immiscible liquids until the solution just became homogeneous at 2 5 ' . He claimed that the exponent, 1.9 in equation I V described the isotherm over the whole range and that the equation was accurate to one third of I per cent for all saturated solutions of benzene, water, and alcohol. Lincoln's determination of the isotherm agrees beautifully with thosc of other authors. Fig. I j shows the points obtained by Lincoln, Taylor, and .4.T. Lincoln: J. Phys. Chem.,
4, 160 ( 1 2 0 0 ) .
1991
DIXERIC DISTRIBUTION
Barbaudy’ plotted on the same diagram. We have begun working on the system again a t Cornell with the hope of obtaining data by direct analysis for the distribution of the alcohol between the benzene and water layers. The average value of our constant in Equation IT obtained in our fircq t seven experiments confirmed Lincoln’s within one tenth of one percent.
BENZENE
WAT6R
FIG.I j Data of Barbaudy, Taylor and Lincoln for the Isotherm for Benzene, Water and Alcohol 0 = Barbaudy 0 = Lincoln 0 = Taylor
The fact that the most accurate work done on the isotherm agrees perfectly with the theory and also that Lincoln’s work is so well confirmed by other authors is strong evidence that Equation IV is theoretically accurate over the whole range. However during the 36 years since the equation was first advanced nobody has been able to give it a theoretical basis. The equation has not been recognized because it is against the instincts of physical chemists for a n equation to apply to solutions over all concentrations. Severtheless there are certain weaknesses in the evidence for the theoretical validity of Equation IT. In the first place it is purely an empirical equation. There might be other equations which would fit the data just as well. I n the accompanying graphs (Figs. 16 and I;) are plotted two entirely different equations which describe very nearly the same curve. If an experimental isotherm fell between these two curves either equation might be selected to fit the facts. 1
J. Barhaudy: Bull., 39 A, 371 (1926).
1992
DAVID BIRNEY H A S D
The second weakness in the evidence is due to the nature of the isotherm. Over most of the isotherm one of the liquids is present in exceedingly small amount while the quantity of the other liquid is large. A small error in endpoint might amount to a large percentage error for the liquid present in least quantity. Over most of the isotherm, especially on the ends, Lincoln’s chief difficulty lay in finding the end-point of his titration. He added alcohol last,
FIG.16 Curve for Equations, XY = ez and X+Y+Z =
IO
which was a mistake. Reference to the graph shows that adding alcohol changes the total composition of the solutions along a line almost tangent to the isotherm on the two ends. Therefore the addition of alcohol would have only a gradual effect in making the solution unsaturated and the end-point would be difficult to ascertain. A more logical method would have been to add one of the immiscible liquids last and the composition would be shifted almost a t right angles to the isotherm. I n this way the smallest quantity of liquid would result in the separation of the largest possible precipitate. If we assume that Lincoln’s error in determining the end-points amounted to 0.1 grams of the liquid present in smallest quantity, on the benzene-rich end of the isotherm the error would be 2 5 percent of the value for water in a total of 30 grams of solution. In other words if a man is allowed a leeway of 0.1gram in the amount of the liquid present in smallest amount he can prove almost anything for the solutions at the ends of the isotherm where one of the liquids is present in amount less than one percent.
I993
D I X E R I C DISTRIBUTION
The only reliable way to determine the isotherm for a ternary system would be to eliminate the uncertainty of the error in finding the end-point. This could be done by either of two methods: Quantities of the third liquid which are unmistakably insufficient and also amounts definitely in excess could be reported. Or else the liquid present in least amount should be determined by direct analysis. I n this latter way the accuracy of the experiment would be limited by the known accuracy of the analysis.
FIG.1 7 Curve for Equations, XY
=
2Z2 and X + Y + Z
= 100
In all of the cases so far studied one of the immiscible liquids is water. It would be of great benefit in the study of these ternary systems if a method could be developed for the direct determination of small percentages of water. K e have tried two methods and both were unsuccessful. To mixtures of benzene, water, and alcohol of known composition we added weighed amdunts of anhydrous copper sulphate. The flasks were allowed to stand for several hours. We tried to drive off the benzene and alcohol from the partially hydrated copper sulphate at 60' t o 80" in an electric oven but all the water came off too. T o other known mixtures weighed amounts of calcium oxide were added and the flasks were heated to 100' protected from the COS of the air by Bunsen valves. After seven hours of heating, much of the liquids in addition to the water still remained with the calcium oxide. However we hope to find some method for determining water in the presence of various organic liquids. We thought for awhile that data for the distribution of the
I994
D A V I D B I R S E Y HAND
alcohol between the benzene and water layers could be obtained by direct analysis of one component (e.g. the water) and then locating the total composition of each phase on Lincoln’s graph for the isotherni. We abandoned the effort because Lincoln’s isotherni was not entirely unquestionable and we did not feel certain we could get data by this method which were any more accurate than those already supplied by Taylor. In the light of what we now know it would be unprofitable to repeat Lincoln’s work on benzene, water, and alcohol unless one could get accurate distribution data by the same experiments. Po we turned our efforts to the study of another system, that of benzene, water, and acetic acid. This system was first investigated by Waddelll who reported that the isotherm did not conform to Equation IY. His logarithmic graph for the isotherm resembled an elongated S. Waddell also studied the distribution of the acetic acid and claimed that for this case Equation I11 did not describe the facts. Lincoln? repeated Waddell’s work on the isotherm as we have described earlier in this paper. He claimed that Waddell was mistaken in his selection of an endpoint, implying that all of Waddell’s solutions had too little acetic acid in them. Lincoln reported that the isotherm for benzene, water, and acetic acid could be described on a logarithmic graph by two straight lines. However he was not entirely convincing because his isotherm lay in a region of even less acetic acid than did Kaddell’s. Lincoln suggested that his two straight lines intersected at the plait point but he made no experiments to show that this was the case. Actually the plait point, as calculated from the tie-lines, would fall nearly in the center of one of Lincoln’s straight lines. Due to the fact that Lincoln‘s only interest was in championing the equation for the isotherm, he made no analyses for the distribution of the acetic acid between the two phases. Waddell’s data for distribution had been obtained by analyzing for acetic acid and then determining the total composition of each phase by locating points on the isotherm which corresponded to the found values of acetic acid. Therefore Lincoln’s claim that Waddell’s isotherni was incorrect would also throw out his data for the distribution of acetic acid. K e set out to confirm Lincoln’s isotherm for the system and then t o repeat Kaddell’s experiments on the distribution of the acetic acid. Our method is described earlier in the paper. Our results for the isotherm are given in Table IV. +As stated before, no attempt was made to determine the exact end-point in 9 of the experiments but double points are recorded showing both an insufficient amount and an excess of one of the liquids. Holding the acetic acid constant at ten grams the recalculated values for benzene and water are given in Table I T 7 . The logarithmic values in the smooth curve is obtained which passe3 table are plotted in Fig. 18. easily through the narrow limits set for it by the 9 “double” poirlts. Although this curve is similar in some mays t o the isotherm reported by Lincoln, it practically coincides with Waddell‘s isotherm. The isotherm for the system benzene, water, and acetic acid when plotted with logarithmic COA\
1 2
J. Waddell: J. Phys. Chem., 2 , 2 3 3 (1898). A . T.Lincoln: J. Phys. Chem., 8,248 (1904).
I995
D I S E R I C DISTRIBUTIOS
ordinates might be considered as falling approximately on two straight lines over a narrow range of concentrations in the center but the curve deviates unmistakable from the theory for all concentrations.
TABLE XV Logarithmic Diagram for the Isotherm for the System Benzene, Water, and Acetic Acid Run
CsHs HAC
c a
h a
If ater HAC
c/a
4249 4771 2;88 2601 5888
,3945 ,2900
Log c a
5
3 88
48 1 95 4 67 4 99 1 3'
6
3
20
1
77
505 I
8
I
2050
I
60 73
6 06
IO
5 71
2380
I
2
66 00
2
3
3 4
I 90 I
82
2
6
72
15
j6 5 45
0
i3I
'93' 7364
0
16
7
0
784 477
8876
9
I
72
0 jlj
17
T.og b!a
Water/H4c
0
90
0
89
I8
8 44
19
4 67
'3 94 13 86 o 428 0 432
,6693 , 6 9 8I '"73 ,2480 ,7825 . 7 j66
,8274 - ,1361 - .IOjj - ,321.; - ,2882
- 04j8 - 0482
I , 1440
9263
- ,3686 - ,3645
I .I420
. 00 IO
6693
I 002 I 02j
20
I
IO
09 1 33 1 38 o 686 o 661 1
21
22
23
0 0
397 391
95 81 8 37
10 IO
18 65 I8 00 30 90 3 0 50
,0107
0414 036; '239 I399 - 1640
-
1800
- 4010 - 4080
,0395 1,0338
I
,9227
I ,2710
1.2545 I I
,4900 ,4840
1Ye have shown that Lincoln's isotherm for benzene, water and acetic acid is inaccurate. 11-e can postulate that his error arose from his determination of end-point in adding the third liquid. If we concede Lincoln an error of only 0 . 1 gram for the liquid present in least quantity when the total weight of liquid is 30 g. vie can completely account for the discrepancy between his isotherm and the one reported by Xaddell and ourselves. In other words by unconsciously making an error of 0.1gram in the amount of one of his liquids Lincoln was able to make the system fit the theory. If his trouble lay in the
I996
DAVID BIRNEY HAND
selection of an end-point, the same criticisms should apply to his data for the system benzene, water, and alcohol. The only data that are accurate enough to distinguish between a curve of the form of an elongated integral sign and a straight line show that the isotherm for a ternary liquid system only approximately follows Equation ITr. Another exception to the theory has been reported by Walton and Jenkins.' They have determined the isotherm for acetone, toluene, and water by titration into open test tubes. They concluded that Equation I V could not be applied to this system.
FIG.18 Logarithmic Isotherm for Benzene, Water, and Acetic Acid
The conclusions to be drawn from this section of the paper are: N o satisfactory experiments have thus far been made which prove the theoretical validity of the mass law equation for the isotherm in ternary systems. The only experiments whose extent of possible error is definitely known indicate that Equation IT' does not describe the isotherm for the special system investigated. There is a need for data on the isotherms of ternary systems which should be determined by direct analysis of the liquid present in smallest amount. b. Interrelations avaong Equations I I , 111, and I I-. We have shown earlier in the paper that a knowledge of the isotherm and either equation for distribution makes it possible to calculate all possible conjugate phases. Knowing the constants in any two of the equations should make it possible to calculate t'le constants in the third equation providing all three held simultaneously and exactly. Accordingly we set out to calculate the exponent in Equation IIl from the constants in Equations I1 and IT'. We have taken the case of Walton and Jenkins: J . Am. Chem. Yoc., 45,
Zjjj (1923).
I997
DISERIC DISTRIBUTIOS
benzene, water, and alcohol and assumed that the isotherm lay on the theoretical straight line, log b 1.9 log c = log const. that Lincoln reported. We next selected three pairs of points on this line which would yield horizontal tie-lines if plotted on a triangular diagram. These points represented three theoretical pairs of conjugate phases. X h e n tested with Equation I11 we found that the points did not fit the equation at all. Our method for making these calculations was as follows: -4line was drawn on a rectangular graph with a slope of 1.9 (Fig. 19). I t has been previously shown by IT. D. Bancroft that the equation for the isotherm holds with the
+
-.
I LOG
-0.5
h 0.5
0
FIG.19 Points giving Horizontal Tie-lines and lying on Corrected Isotherm
same exponent no matter in what units the quantities are expressed. Therefore we would still have the theoretical curve for the isotherm no matter what absolute values were chosen for the coordinates. Arbitrary values were then assigned to the coordinates and pairs of points representing horizontal tielines were selected on the isotherm. The requirement for a horizontal tieb c = 100. Thereline is that a, = as on a triangular diagram when a fore if a is kept constant and the points are plotted on a rectangular graph the requirement for horizontal tie-lines would be bi c1 = bS cS. I n this way we obtained three pairs of points representing theoretical conjugate phases in accordance to Equations I1 and IV. The values for log al/bl are plotted against log a ? j b 2 in the accompanying graph (Fig. 20). I t is apparent that points corrected to give theoretical agreement with Equation I1 and Equation I P are not described by Equation 111. We consider these calculations as evidence if not proof that the three equations are inconsistent with each other. However the equations might agree and a slight error in choosing the theoretical constant for Equation I V might have resulted in the apparent disagreement with Equation 111. Furthermore we have no proof that the three equations would be inconsistent if Equation IV had two sets of constants corresponding to two straight lines for
+ +
+
+
I998
DAVID RIRSET H A S D
the logarithmic graph of the isotherm. All we claim to have proven by the above calculation is that it is impossible t o derive the constants of one equation from the known constants of the other two equations using the present available data. The calculations we have made have confronted us with the following questions: Are two of the equations proposed for concentrated ternary solutions theoretically correct and the third inaccurate'? Or are all three of the equations only approximately correct? If a choice must be made of two of the three equations it should be pointed out that we have most reason t o doubt the validity of Equation Is', as described earlier in the paper. The evidence in support of Equation 111 is much more convincing. Equation 111
Flti. 2 0 Distriliution for Points corrected t o fit Equations
IT and I V
concerns itself with the liquids present in largest amounts in the two phases. I n this respect the data in favor of Equation I11 are uniformly accurate over practically the whole range. TVe have no indication that Equation I11 may not be theoretically exact for all concentrations and for all systems. Also Equation I1 is supported by direct analyses of the component present in least amount. The equation has been shown to apply to four different systems in concentrated solutions. In every case there is an unmistakable deviation from the theory for very dilute solutions, but the equation may be theoretically accurate for high concentrations. There is only one possible way to answer the questions concerning the validity of the three equations and that is through the study of a system for which data could be obtained which would be equally reliable in testing all the equations. It is useless for this purpose to determine the isotherm by adding the three liquids together until saturation and then determine the distribution by analyzing for the consolute liquid in conjugate phases and locating the points on the isotherm. In order to get reliable data it will be necessary to study a system for which all three components can be determined by direct analysis. In finding the isotherm the immiscible liquids present in smallest amounts
I999
DISERIC DISTRIBUTIOS
must be analyzed for directly with a known experimental accuracy. I n obtaining data on distribution the consolute liquid and the immiscible liquid present in largest amount in each phase must be analyzed by direct experiment. To make this important study a method for the determination of water is needed because water is usually one of the immiscible liquids in twophase ternary liquid systems.
TABLE XVI Horizontal Tie-lines on Theoretical Isotherm bi
CI
1.9 2.19 3.16 . ;. O I
I
.o
I .;6 0.38 0.16
+
hi
CI
2.90 2.95 3.54 5.17
.66
I
I .20
2
0.86
4.27
I
.29 .36
2.95 3.56 :. . 7 3
Distribution of the Consolute Liquid calculated from the above Theoretical Points. a1
log ai
a2
52.6 45.6 31.6
1.72'
IO0
I
20.0
I
.6j9 ,301
2
77.5
I
.4
I
42
I ,SO
log ar
23 . 5
.o
,889 ,627 1.369
Summary I. The tie-lines for ternary systems of one consolute and two immiscible liquids become horizontal when the systems are plotted with properly selected units for one of the immiscible liquids. This is accurate for all the cases known within one percent. The discovery makes it possible to predict the compositions of all pairs of conjugate phases of a system if the isotherm and one tie-line are known. 2. The fact that tie-lines become horizontal has been shown to be equivnlent to the distribution equation:
(Equation 11.) This equation describes the distribution of 3 consolute liquid when in concentrated solution in conjugate phases. The equation accounts for the increasing solubility of the immiscible liquids. Equation I1 holds for concentrated solutions within the limit of experimental error, but for very dilute solutions there is an unmistakable deviation from the theory. 3 . In order to account for the validity of Equation I1 it can he assumed that there is a distribution of the consolute component between the other two components according to a simple ratio even when the immiscible components
DAVID BIRKET H A S D
2 000
are in the same phase Xathematically stated, the ratio of the weights of the consolute component held by unit quantities of the other componcnts is a constant for each phase (
Eqiiation 1.1
4. Equation I anti Eqwtion I1 becorn? idcritical with the Sernst distribution equation in the limiting casc in which thc. immiscible liquids are entirely in separate phases. .Iceording to this theory of distribution we would expect the Sernst equation t o apply to the distribution of consolute liquids in very dilute solutions. Yet none of the four cases which have been studied support the simple form of the Sernst distribution equation. Only by an assumption of association of the consolute liquid in one phase is it possible to make the distribution of acetic acid between chloroform and water apparently obey the Sernst distrihution law up to concentrations of one percent acetic acid in chloroform. j. Our work indicates that it is a mistake to approach the study of actual solutions through an extension of the approximate laws for dilute solutions. This method of attack has only brought us to the conclusion that concentrated solutions are highly complicated. As a matter of fact the concentrated solutions appear to be the simple case, 6 . -111 the cases so far investigated for dineric distribution in high concentration follow the empirical equation,
(:>,/’:
m
=
const.
(Equation 111.)
By using Equations I1 and Equation I11 it is possible to calculate the composition of a second phase when the composition of the first phase is kn0n.n. 7 . The isotherm for saturated solutions of benzene, water, and acetic acid does not follow the mass law equation, log
($) + n log (f)
=
log const.
(Equation IT)
\Ve have criticised the evidence for the rheoretical validity of Equations 11. IIJ, and IT without arriving at any definite conclusion. I n order to compare the accuracy of all three equations and to establish the truth of Equation IV in particular it will be necessary to study a system in which all three components can be determined by direct analysis. Acknowledgment The xork described in this paper \vas suggested by Professor Kilder D. Bancroft and has been carried on under his direction. I am deeply indebted to him for suggestions and criticism, It has been a great privilege to have been associated with Professor Bancroft in the study of physical chemistry. Cornell rnii,ersity. !
\
