V O L U M E 28, NO. 12, D E C E M B E R 1 9 5 6 sults. The composition of the solvent may vary widely between 60 and 1 0 0 ~ acetonitrile, o so long as the potassium tetraphenylborate remains in solution. I n previous methods which have been reported using nonaqueous solvents, the composition of the solvent had to be critically controlled (8) or be completely anhydrous (3). Results for titrations in dimethylformamide-water mixtures at -0.20 volt are shown in Table 11. These are not so precise as resiilts obtained in acetonitrile-water mixtures. A large number of titrations were performed with 0.01M silver nitrate in dimethylformamide-water mixture, but the results were from 2 to 5y0 high. Methyl Carbitol in the solvent system yielded results for titrations with 0.1M silve nitrate from 1 to 3.57, high. These titrations were performed at -0.1 volt us. the pool. Typical titration graphs are shown in Figure 2 for titrations in the acetonitrile-water and dimethylformamide-water system. This method should also be applicable for the precise determination of equivalent weights of organic bases, in case they form tetraphenylborate precipitates soluble in acetonitrile.
1901 ACKNOWLEDGMENT
The authors are indebted to the Atomic Energy Commission for funds under Contract No. AT( 11-1)-163. LITERATURE CITED
Dutoit, P., Friderich, L., B u l l .
SOC.
chim. France (3) 19, 321
(1898).
Findeis, A. F., De Vries, T., ANAL.CHEM.28, 209 (1956). Flaxhka, H., Chemist-Analyst 44, 60 (1945). Flaschka, H., Holasek, A.. Amin, A. A I . , Arzneirnittel-Forsch. 4, 38 (1954).
Hahn, F. L., Z . anal. Chem. 145, 97 (1955). Kemula. W.. Kornacki. J . . Roczniki Chem. 28. 635 (1954). Pflaum,'R.T., Popov,'h. S.,Goodspeed, S . ' C . , ANAL.CHEM. .
I
27, 253 (1955). Riidorf, W., Zanier, H. Z., 2. anal. Chem. 137, 1 (1952). Wittig, G., Angew. Chem. 62A,231 (1950). Wittig, G., Kucher, G., Ruckert, A . . Raff, P., Ann. 563,110, 126 (1949).
Wittig, G., Raff, P., Ibid., 573, 195 (1951). R E C E I V Efor D review November 17, 19%.
Accepted h u g u s t 15, 1956
Theory of Countercurrent Distribution in Solvent Systems near a Critical Point C. A. HOLLINGSWORTHand 1.1. TABER Department o f Chemistry, University of Pittsburgh, Pittsburgh, Pa.
B. F. DAUBERT Central Laboratories, General
foods Corp., Hoboken, N. 1.
By the use of the theory of regular solutions, equations are obtained expressing the behavior of partition ratios near the critical temperatures of complete miscibility of two-component solvent systems and near the plait point of three-component, symmetric solvent systems. These results are used with a criterion of separation to predict the optimum conditions for separation by countercurrent distribution in these systems. The theory is then applied to experimentally determined partition ratios of some triglycerides to predict the amount of separation that can be obtained. The theory leads to the following conclusions: In systems to which the theory applies, the relative behavior of a pair of solutes can be characterized by a constant, y, which is independent of the temperature or the third solvent component. Nearly optimum conditions for separation exist over a fairly w-ide range of temperature or solvent composition and far enough from the critical point so that extreme dependence on temperature or composition is not encountered.
W
I T H countercurrent distribution technique developed by Craig (2)using any given number of transfers, n, and any given ratio, T , of the two phases, the amount of separation obtained for two solutes depends upon the partition ratios (distribution coefficients) of the two solutes. I t often happens that t a o solutes are difficult to separate because both the partition ratios (or their reciprocals) are very large, say, 50 or more. When possible in such cases automatic equipment is used, which makes hundreds, perhaps even thousands, of transfers practical (9). When such equipment is not available, the value of T may be adjusted, but large adjustments are usually not convenient. A withdrawal procedure may be used, but the possibilities
here are limited. It may be possible to obtain the desired degree of separation by recycling, but when the equipment is not automatic this can be laborious and time-consuming. Sometimes the solvent system can be modified to improve the values of the partition ratios; there are various ways of doing this, and they depend upon the types of solutes and solvents that are involved. Only one type of system is investigated here-one in which the solutes and solvents are nonelectrolytes and the solvent system possesses a critical temperature or a plait point within operating range. Both partition ratios (or both reciprocals) are lowered as the system moves toward a critical point. As the critical point is approached, both partition ratios approach unity, and separation is again poor. There is, therefore, an optimum condition for separation. In a two-component solvent system there is an optimum temperature, and in a three-component solvent system with a plait point there is an optimum composition ( 6 ) , where optimum refers to that temperature, or composition, a t which the degree of separation is a maximum for a given n and r. If the optimum conditions are not so close to the critical point that there is extreme dependence upon temperature or composition, it may be practical to carry out the distribution under these conditions. It was the purpose of the work reported here to apply the theory of regular solutions to obtain theoretical expressions for these optimum conditions for separation and to apply them to experimentally determined partition ratios of some triglycerides in certain solvent systems. PARTITION RATIOS NEAR A CRITICAL POINT
Two-Component Solvent Systems near Critical Temperature Let x and y be the mole fractions of the two solvents, X and Y , and let w be the mole fraction of a solute, W . For a regular solution the Gibbs free energy of mixing is given by
1902
ANALYTICAL CHEMISTRY
+
+
+
+
+
G = azy bxw cyw RT(x1nz ylny wlnzc) (1) \\-here a, b, and c are constants related to the interaction energy of the components. Meijering has shown (8) that when Equation l holds, and w is small, the distribution of IP between tn-o phases is given by
K=g (15) If there are two solutes, A and B , each satisfying Equation 10 for the same solvent system, then the following relationship exists beheen the partition ratios K A and KB: = k7.tY
(16)
-,= k a / k a
(17)
K B
where n-here xl and WI refer to the X-rich phase and zz and us2refer to the Y-rich phase. When W I and 202 are small, x 1 and 2 2 are the roots of the Van Laar equation (3) Since Equation 3 is symmetrical, its solutions can be expressed by 1 zl = $1 E) (4) 1 22 = - (1 - E) 2
+
where 0 5 E < 1, and E = 0 at the critical temperature, T, = a / 2 R . Equation 3 becomes by use of 4
and where k A and k B are the constants given by Equation 12 for the two solutes. As the amount of separation of A and B that can be obtained by a given number of transfers in countercurrent distribution is a function of K A and K B , it follows from Equation 16 that the amount of separation that can be obtained a t the optimum temperature is a function of y alone. For regular solutions the minimum amount of information required to predict the separation as a function of the temperature, in the neighborhood of the critical temperature, is the value of the critical temperature and the values of the two partition ratios at one temperature not too far from the critical temperature. In terms of the solubility parameters 15) of the solvents and the two solutes-Le., 6 ~ 6 y, , 6 ~and , 6 ~ respectively-the , constant y is given by
01'
RT In 5
or = aE
(6)
22
The logarithm in Equation 5 can be expanded as the power series +
E =
In I - €
2 e (
+ 1 + 1 + ...) -e5
-E3
3
5
(7)
n-hich converges rapidly for all values of e that are of interest here. The use of Equation 5 with Equation 7 , neglecting powers of E greater than the cube, leads to e =
4
(gT -
I/ 2
1)
or
Thus from a knowledge of the solubility parameters one can, in principle, predict the amount of separation that can be obtained at the optimum temperature. However, in practice such a prediction is not very reliable unless the system satisfies closely the theory of regular solutions, and unless the solubility parameters are known accurately. Three-Component, Symmetric Solvent Systems near Plait Point. The Gibbs free energy of mixing of a four-component regular solution is given by
G = azy
+ bxz + cyt +RT(z1nx rxui + s y w + tzw + + ylny + t l n z + wlnw)
(20)
where z,y, z , and w are the mole fractions of the componente X , Y , Z, and JV, and a, b, c, r , s, and tare constants. The chemical potential of solute w is given by
From Equations 2: 6, and 9 one obtains
where (11) i; =
~ B ( -c b)/RTca"
(12)
n-hich, with Equation 20, gives PTP = rx
and
+ sy + t z - ( m y +~ xbxzw++sywcyz++t zw) + RTlnw
(221
When solute W is distributed between two phases, the condition for equilibrium gives for small values of w For small values of O/T,, say, 0 < 10 when T, is near room temperature, TJT can be taken as unity. Partition ratios are usually defined by
RTlnz
us1
;
+
+
y(zl - x2) s(yl a ( z 1 ~ 1- zzyz) i- bixlzl
I/?)
i t(tl - 22)
- xztz
)
=
+ ~ ( y l-~ l
YZZZ)
(23)
n-here zl, yl, etc., are the mole fractions in the first phase and 2 2 , etc., are the mole fractions in the second phase. The condition for equilibrium m-ith respect to component 2 gives, for the case w ~ O , 211,
where K is the partition ratio and c1 and c1 are the concentrations (grams per milliliter) of solute in the upper and lower layers, respectively. When ZUI and w z are small, and the compositions of the two phases are approximately equal, as they will be near the critical point, then
a(w1
- ny2)
+ b(z1zl -
~
RTln(zI/zd
2
+ c(yltl - = + b ( r ~- + c(yl - YZ) )
From Equations 23 and 24 one obtains
~ZZZ)
ZZ)
(24)
1903
V O L U M E 28, NO. 12, D E C E M B E R 1 9 5 6 In terms of the symbols used by Sichols (9) T
Further treatment is greatly simplified by the assumption that h = c, and, therefore, the system is symmetric, so that 21
=
22
= z;
51
= y*; x* = y1
a(r,
=
- Q)
(27)
L-se of Equations 25 and 26 gives rn =
-c~ix~
where
tp/d/n
T
11
p(dGr)
12
2
2
one has after a little algebra
The plait point of a symmetric system is given by (8)
, - , = 1 - - 2RT a
RT = yc =
(37)
=
100
[f + 4(di.)]
(38)
where +(t) is the area between the peak of the normal curve and the abscissa, t. Equation 35 has been given for the case in which r = l-i.e., for the case in which the volumes of the upper and lower phases are equal. If T # 1, then K A and K B must be replaced by T K A and T K Bin Equation 35. Only the case in which r = 1 is considered in detail here. From K B = KAY and with Equation 35 it is possible to calculate the values of K A for which 7 is a maximum for any given value of y. These values of K A are designated Ka* and are shown as a function of y in Figure 1 along with the corresponding maximumvalues, .*and p*(dg7 ) . The values of K A * increase as y decreases, and the limiting value of KA* as y goes to unity is approximately 11.
+ + =1 + y2 + =“ 1 YI
(36)
I n Equation 36 to is approximately the abscissa of the intersection of the two normal distribution curves when A and B are present in equal amounts; in Equation 37 eS is the distance between the peaks in units of the arithmetic mean of the two standard deviations. It can be shown, also, that T has the following significance: When the distribution of mixture A and B is divided into two portions such that the percentage of A in one portion is equal to the percentage of B in the second portion, then t/n T ( = t o ) is the abscissa corresponding to the point of division, and the percentage of the major component in either portion is p ( & ~ ) , where
ST.
From Equations 26 and 28 and the approximations
114s
= es/2d
