Bohon, R. L., Claussen, W. F., J . Amer. Chem. SOC.73, 1571 (1951). Brian, P. L. T., IND. ENG.CHEM.,FUNDAM. 4, 100 (1965). Brusset, H., Gaynhs, J., C. R. Acad. Sci., 236, 1563 (1953). Carlson, H. C., Colburn, A. P., Ind. Eng. Chem. 34, 581 (1942). Franks, F., Gent, M., Johnson, H. H., J . Chem. SOC.2716 (1963). Horne, R. A,, “The Structure of Water and Aqueous Solutions,” in LLSurvey of Progress in Chemistry,” A. F. Scott, Ed., Vol. 4, Academic Press, New York, N. Y., 1968. Horyna, J., Collect. Czech. Chem. Commun. 24, 3253 (1959). Trmann. 37. 789 (196.5). . , F.. _ _Chem.-Tno.-Tech. , _ ... -.. = Kliment, V., Fried, V., Pick, J.,’Coilect. Czech. Chem. Commun. 29,2008 (1964). Kortum, G., Vogel, W., Andrussow, K., “Dissociation Constants of Organic Acids in Aqueous Solution,” Butterworths, London, 1961. Markuzin, N. P., J . Appl. Chem. USSR 34, 1121 (1961). Nelson, H. D., de Ligny, C. L., Red. Trav. Chim. Pays-Bas 87, 528 (1968). Pierotti, G. J., Deal, C. H., Derr, E. L., Ind. Eng. Chem. 51, 95 (1959). ~~~
\ - - - - I
Prigogine, I., Defay, R., “ChemicalThermodynamics,” translated by D. H. Everett, Chapter 23, Wiley, New York, N. Y., 1954. Renon, €Prausnitz, I., J. M., A.I.Ch.E. J . 14, 135 (1968). Renon, H., Prausnitz, J. M., Ind. Eng. Chem., Process Des. Develop. 8, 413 (1969). Rock, H., Sieg, L., 2. Phys. Chem. (Frankjurt am Main) 3, 355 (1955). Schreinemakers, F. A. H., Z. Phys. Chem. 35, 458 (1900). Schulek, E., Pungor, E., Trompler, J., Mikrochim. Acta 52 (1958). Tsonopoulos, C., Ph.D. Dissertation, University of California, Berkelev. 1970. .~ .~.~. Weimer, R.’F:, Prausnitz, J. M., J . Chem. Phys. 42, 3643 (1965). Weller, R., Schuberth, H., Leibnitz, E., J . Prukt. Chem. (4) 21, 234 (1963). Wilson, G. M., J . Amer. Chem. SOC.86, 127 (1964).
RECEIVED for review December 18, 1970 ACCEPTED June 7, 1971
Determination of the Thermodynamic Contribution to the Diffusion Coefficient Matrix of a Ternary Liquid System John P. Lenczyk and Harry T. Cullinan, Jr.” Department of Chemical Engineering, State University of New Yorlc at Buffalo,Bu$alo, N . Y . 14214
An experimental technique for the direct determination of the chemical potential composition derivatives of a ternary liquid system is described. The results of equilibrium sedimentation experiments conducted on an ultracentrifuge are reported for the system acetone-benzene-carbon tetrachloride. Equilibrium composition distributions, obtained from 1 3 separate initial ternary compositions, are used to determine the ternary interaction parameters of the four-suffix Scatchard equations. Consistency is confirmed by back-calculation of the individual distributions. The experimental values of the chemical potential composition derivatives are compared to the predictions of the Wilson equation which requires no ternary parameters. The resuits indicate that the values of the chemical potential composition derivatives obtained from the Wilson equation are in close agreement with those obtained from the Scatchard equation with the experimentally determined interaction parameters.
w h e n a homogeneous single-phase mixture is subjected to a field of force, a distribution of concentrations sets in by reason of the differences among the sedimenting forces acting on the various species. The concentration gradients, as they are established, cause purely diffusive flows, and the process of redistribution continues until a balance is attained between the sedimenting forces and the diffusion forces. This balance corresponds to a state of true thermodynamic equilibrium. The diffusive fluxes are generally coupled so that the rate at which a given constituent approaches its equilibrium distribution depends on the instantaneous distribution of all species present. At equilibrium the distribution of the individual components is still coupled through the solution thermodynamics because the chemical potential of a given 600
Ind. Eng. Chem. Fundom., Vof. 10, No. 4, 1971
species depends on all of the component concentrations. The theory of sedimentation is well developed (Fujita, 1962). Recently it has been suggested (Cullinan, 1968) and demonstrated for a binary system (Cullinan and Lenceyk, 1969) that the composition derivatives of chemical potential of nonideal liquid systems can be determined from equilibrium sedimentation experiments. These quantities are of importance in the study of the multicomponent diffusion process, because the practical diffusion coefficient matrix is the product of a fundamental diffusion coefficient matrix and the matrix of chemical potential composition derivatives. I n this paper the previous equilibrium sedimentation work is extended to the study of a ternary liquid system in order to determine the thermodynamic contribution to the practical
diffusion coefficient matrix over the entire range of composition.
AXIS
OF
ROTATION
The Theory
The general principles of irreversible thermodynamics can be applied to a liquid system subjected to a centrifugal field. The equilibrium distribution is governed by (Cullinan, 1968) Figure 1 .
Sedimentation cell
and the time to reach a given degree of equilibrium, with an initially uniform system, is obtained from (Cullinan and Lenczyk, 1969)
For a multicomponent system, D in eq 2 should be taken as the smallest eigenvalue of the practical diffusion coefficient matrix. At a time corresponding to a = 0.99, the system will essentially be a t equilibrium. Equation 1 can be integrated for the case of radial sedimentation in a short ultracentrifuge cell to obtain the equilibrium distribution (Cullinan, 1968).
Gl(Am) = ((MI
-
P@))w2hR
(3)
I n eq 3, the matrix [ f i ] is evaluated a t the logarithmic average composition and the density, p , is an integral average value.
=-LIQUID
SAMPLE
Figure 2.
Sample removal
Procedure
A Beckman Model L ultracentrifuge with a titanium swinging bucket type head, Model SW 40, was used in this investigation. The SW 40 head had three buckets, each of which accept nitrocellulose liners that resemble 5 m l test tubes. An epoxy hemisphere, designed to fit the bottom of the liners, was used to eliminate the curvature a t the bottom of the liner. The cell could then be represented by a cylinder as is graphically shown in Figure 1. For the Model SW 40 head the distance from the center of rotation is large enough, compared to the distance for sedimentation, to justify the use of a cylinder. The constancy of the temperature of the system was experimentally verified. Of course, thermodynamic properties depend on temperature to some degree but more importantly a temperature gradient, if established in the centrifuge, could, upon stopping, generate a convection current t h a t would destroy the composition gradient. The results of several temperature measurement experiments indicate t h a t no temperature gradients exist in the cell and that the temperature of the cell is reproducible. The speed was carefully checked and found constant to better than 0.2% over the entire duration of each run. The technique was established on the basis of experimental results of a thermodynamically known binary of hexane and carbon tetrachloride. The experimental results of the binary are presented by Cullinan and Lenczyk (1969) along with a description of the technique. Only a brief discussion of the procedure will be given here. After the cells containing the liquid samples were centrifuged for a predetermined length of time, the buckets were carefully removed and secured upright in a holder. The bucket covers were removed to allow liquid sampling. A hypodermic syringe (diameter 0.030 in.) connected to a screw traverse was used for sampling. The syringe needle was lowered beneath the surface a given distance and all of the liquid would be carefully removed, as shown in Figure 2.
The syringe was then disconnected from the needle and the sample was placed in a covered 5-ml vial for storage. A clean syringe was then attached to the needle and the needle again was lowered. The procedure was followed until the bottom of the cell was reached. The radial position of the bottom of the cell was given by the manufacturer and the radial position of the surfaces before and after sampling could be determined by back-calculating from the bottom. The radial position assigned to the sample was the average value of the radii of the surfaces. The true concentration profile is then determined from the stepwise sampling. Extreme care must be exercised during the sampling procedure to prevent systematic error. The previous work on a binary system (Cullinan and Lenczyk, 1969) demonstrates that this procedure is workable and that any systematic error can be minimized. It can be shown that for the binary system hexane-carbon tetrachloride, the experimental composition profiles, obtained by this technique, are within 296 of the theoretical curve calculated from eq 3 using known solution properties. There is also no significant trend of the data with respect to the predicted distribution, which indicates the absence of systematic error. The analysis of the ternary system acetone-benzene-carbon tetrachloride, which is the subject of the present investigation, was by gas chromatography. A Disc integrator was used to measure the area under the curve of the response from the thermal coflductivity bridge. The Hewlett-Packard 5754 gas chromatograph was calibrated before analysis of each run. The accuracy of analyzing the three comDonents is considered to be better than 1%. Experimental Results
The system chosen for study in this work is the nonideal liquid system acetone- benzene-carbon tetrachloride. The diflusion coefficient matrix and solution density have been previously determined (Cullinan and Toor, 1965). Ind. Eng. Chern. Fundam., Vol. 10, No. 4, 1971
601
l,5r X,"O2
X,"O.4
Xby04
I.oo c
2833ino 0
7
I .o
.5
0
0
1.5
A . 9 9 1
Figure 3.
Approach to equilibrium
P
164 -62
9
I
-58
B
L -54
-50
55 m 9
>
-46
5a
-I
-42
46
t
I 5
2
different rotational speeds do not alleviate this difficulty, because the resulting composition distributions would be simple multiples of the original ones. To circumvent this difficulty the four-suffix Scatchard (1937) equations are used to represent the elements of [*I in terms of solution properties, nine binary interaction parameters, Air, and three ternary parameters, Ci. The nine binary interaction parameters are determined from data for the activity coefficients of the three binary pairs (Bachman and Simons, 1952; Caldwell and Babb, 1964; and Canjar and Lonergan, 1956). According to eq 3, the two independent composition distributions vary linearly with the square of the radial position. A typical experimental result is shown in Figure 4. The data are linear, as expected, with each correlation coefficient in excess of 0.99. For each ternary experiment, the slopes of these lines were used to calculate the gradients of composition in eq 1. Duplicate experiments were performed a t 13 points in the ternary composition field. The four-suffix Scatchard equations were used to represent the elements of [ p ] in terms of the three ternary interaction parameters. Using a search program the best values, in the sense of least squares, of the three ternary interaction parameters were determined. These parameters were then used with the Scatchard equations to calculate the elements of the matrix [ p ] at each composition point. As a check of the consistency of the results, these ternary parameters were used at each composition point to calculate independently the slopes of the m us. r2 plots. These calculated slopes generally fell within the 95% confidence intervals for the slopes of the actual data. This also serves to confirm the validity of the use of the Scatchard equations to represent the system acetonebenzene-carbon tetrachloride. Finally the experimental values of the elements of the matrix [ p ] were compared to the predictions of the Wilson equation (1964) which requires no ternary parameters. The results show excellent agreement for the diagonal elements of [ p ] (an average deviation of 3%) and good agreement for the off-diagonal (or cross) terms (an average deviation of less than 10%). Conclusions
I 6
Figure 4.
7
8 ?(IN2)
9
IO
II
Equilibrium distribution
The approach to sedimentation equilibrium was investigated by conducting independent runs of 71, 91, and 120 hr duration at 30,000 rpm, starting with solutions of 20 mole % acetone, 40 mole % benzene, 40 mole % carbon tetrachloride. Using eq 2 with the smallest eigenvalue of the diffusion coefficient matrix, the calculated time to reach 99% of equilibrium is 78 hr under these conditions. The results are shown in Figure 3 in the form of normalized composition us. the ratio of real run time to the calculated time to reach 99% of equilibrium. On the basis of these results, run times in subsequent experiments were generally set at t = 1 . 5 t 0 . 9 9 to ensure equilibration. The matrix, [I], in eq 3 is symmetric. Thus, for a given ternary starting composition, there are three independent elements of [ p ] to be determined from the two independent composition distributions attained at a given rotational speed. Separate experiments at the same initial composition but 602 Ind. Eng. Chem. Fundam., Vol. 10, No. 4, 1971
The use of equilibrium sedimentation to determine directly the chemical potential composition derivatives for a nonideal, multicomponent, nonelectrolyte, liquid system has been investigated. The equations governing the unsteady-state motion have been developed and verified for both a binary and a ternary system. The techniques for sample removal and analysis can be applied in any system regardless of the number of components. The binary system hexane-carbon tetrachloride is thermodynamically well known and has been successfully measured by this technique. A comparison of the values of the composition derivative of chemical potential determined by equilibrium sedimentation to the literature values is excellent (Cullinan and Lenczyk, 1969). Equilibrium sedimentation results in the direct determination of the composition derivative of chemical potential over a wide range of composition in a single experiment for a binary system. It is possible t o characterize the nonideal behavior of a binary completely with very few experiments because a range of composition is investigated. This is contrasted with the standard vaporliquid cell measurements where a single composition point is explored.
The ternary system acetone-benzene-carbon tetrachloride has been completely characterized thermodynamically by using equilibrium sedimentation. Unlike the binary case where an experiment directly results in values of the composition derivatives of chemical potential, the ternary requires a model of the thermodynamic nonideality. The Scatchard equations with the interaction parameters determined in this work are considered to be the best of the previously used expressions for the composition derivatives of chemical potential. The equations describing sedimentation for the ternary system can be applied to a system containing four or more components. The experimental technique used for the ternary system can easily be extended to include more components. However, the equations describing the behavior of the composition deriyative of chemical potential are much more complicated, but there is no difficulty, in principle, in extending the present results.
m i = molality of species i (m) = column vector of molalities molecular weight of species i *$i = f$ ( M )= column vector of molecular weights r = radial coordinate Ri = radial position of the boundaries of the cell i AR = radial length of the cell, RZ - R1 R = average radial distance (R1 R2)/2 t = time t7( = partial molar volume of species i (0) = column vector of partial molar volumes
=
1, 2
+
GREEKLETTERS CY = fractional approach to equilibrium pi = chemical potential of species i [ p ] = matrix of composition derivatives of chemical potential p i j = an element of [ p ] p = total density w = angular velocity literature Cited
Ac knowledgment
Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this research. J. P. L. was the recipient of an NSF Traineeship. Nomenclature
A
Ci D
= = =
binary interaction parameters ternary interaction parameters smallest eigenvalue of diffusion coefficient matrix
Bachman, K. C., Simons, E. L., Ind. Eng. Chem. 44, 202 (1952). Caldwell, C. S., Babb, A. L., J. Phys. Chem. 60, 51 (1956). Canjar, L. N., Lonergan, T. E., A.I.Ch.E. J. 2,280 (1956). Cullinan, H. T., IND. END.CHEM.,FUNDAM. 7,317 (1968). 8, Cullinan, H. T., Lenczyk, J. P., IND.ENG.CHEM.FUNDAM. 819 (1969). Cullinan, H. T., Toor, H. L., J.Phys. Chem. 69,3941 (1965). Fujita, H., “Mathematical Theory of Sedimentation Analysis,” Academic Press, New York, N. Y., 1962. Scatchard, G., Trans. Faraday SOC.33, 160 (1937). Wilson, G. M., J.Amer. Chem. SOC.86, 127 (1964). RECEIVED for review September 8, 1970 ACCEPTED June 14, 1971
Ind. Eng. Chem. Fundam., Vol. 10, No. 4, 1971
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