757
Binary Liquid Diffusion Prediction in Diluted Systems
TABLE IV:Mobility Comparisons Ion
Mg2+ Ca2+ Sr2+ Ba2+ DiBu2+
RwM
R22'
0.64 0.59
0.98 0.98 0.97
0.60 0.57 1.16
nors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research (T.J.M. and T.L.B.), to the Computer Center, University of Maryland for providing computation facilities, to the Office of Saline Water Contract No. 14-01-001-1281. D.F.E. is supported by a NIH Career Development Award No. 5K4-AM-12972.
0.98
Supplementary Material Available: a listing of the somewhat better than that for univalent ions; however, the molar concentrations and corresponding equivalent conmagnitude is still too small. ductances (2 pages). Ordering information is available on Two additional comparisons of (bq)values for the alkaany current masthead page. line earth cations can be made; the first is the mobility in methanol relative to the value in water a t 25 "C, ( X o q ) ~ / References and Notes (X0q)w = RwM and the second is the temperature dependence of the mobility in methanol, (XOV)lO/(XO8)25= Rz51°. T. J. Murphy, Ph.D. Thesls, Rockefeller Unlversity, 1968. The comparisons are summarized in Table IV along with T. J. Murphy and E. G. D. Cohen, Bull. Am. Phys. SOC.,13, 647 (1968). similar data for DiBu2+ [(n-Bu)3N(CHz)sN(n-B~)3~+]. T. J. Murphy and E. G. D. Cohen, J. Chem. Phys., 53, 2173 (1970). R. M. Fuoss and L. Onsager, Proc. Nail. Acad. Sci. U.S.A.. 41, 274 RwM decreases slightly with increasing ionic size and cen( 1955). R. M. Fuoss and L. Onsager, J. Phys. Chem., 61, 668 (1957);66, 1722 ter around the value for Li+ (RwM= 0.635). The behavior (1962);67, 621 (1963);68, l(1964). of this latter ion is dominated in aqueous solution by elecR. M. Fuoss. L. Onsager, and J. F. Skinner, J. Phys. Chem., 69, 2581 trostriction. The large bolaform ion DiBu2+ has RwM > 1, (1965). J. P. Butler, H. I. Schiff, and A. R. Gordon, J. Chem. Phys., 19, 752 as is the case for tetraalkylammonium ions. The behavior (1951). of these latter monovalent ions is dominated by hydrophoT. L. Broadwater and D. F. Evans, J. Phys. Chem., 73,3985 (1969). bic effects in water. Similar behavior is found for the correT. L. Broadwater and D. F. Evans, J. Solution Chem., 3, 757 (1974). T. L. Broadwater, Ph.D. Thesis, Case-Western Reserve University, sponding ratios for the divalent ion in DzO. No difference 1968. is observed for R251° between the alkaline earth cations and R. L. Kay, private communication. T. J. Murphy, J. Chem. Phys., 56, 3487 (1972). DiBu2+ in MeOH. In summary, the agreement between the R. L. Kay and D. F. Evans, J. Phys. Chem., 70,2325 (1966). predictions of the Zwanzig equation and the experimentalR. A. Robinson and R. H. Stokes, "Electrolytic Solutions", 2nd ed, Butterworths, London, 1959,p 463. ly observed values is better for multivalent ions than for R. Zwanzig, J. Chem. Phys., 38, 1603 (1963). univalent ions. R. Zwanzig, J. Chem. Phys., 52, 3625 (1970). R. Fernandez-Prini and G. Atkinson, J. Phys. Chem., 75, 239 (1971). F. Perrin, J. Phys. Radium, 7, 1 (1936).
Acknowledgment. Acknowledgment is made to the do-
Binary Liquid Diffusion Prediction in Infinitely Diluted Systems Using the Ultimate Volume Approach Eli Grushka,* E. J. Kikta, Jr., and H. T. Cullinan, Jr. Department of Chemistry and Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14214 (Received October IO, 1975)
The diffusion of binary liquid mixtures is examined in terms of ultimate volumes using the fluidity concept. A diffusion coefficient expression is developed in terms of the solvent density, pj, a momentum related parameter, B , and a temperature independent constant aij which is characteristic of the solute-solvent pair. The diffusion of 12 phenones and benzene in heptane was analyzed using the developed expression. The quantity DijIRT is a linear function of the solvent density within a certain temperature range. The slope, which is a function of aij, of the line relating DiJRT to p j seems to be a function of the molecular weight of the phenones. This functionality allows perdiction of the diffusion coefficient with accuracy superior to the Wilke-Chang relationship.
Introduction Concerning the present state of knowledge of the process of molecular diffusion in liquid systems, a comparison of
* Address correspondence to this author a t the Department of Chemistry, State University of New York a t Buffalo.
the large volume of available data (at least for pure liquids and binary mixtures) with the level of development of useful theories of diffusive transport would appear to indicate an enormous gap. Although diffusion coefficients in liquid mixtures are generally much stronger functions of temperature than of relative composition, most available data are The Journal of Physical Chemistry, Vol. 80, No. 7, 1976
758
E. Grushka, E. J.
reported for only a single isotherm. Much effort has been directed at the characterization of composition variations of the diffusion coefficient because such behavior is of direct importance in understanding of the nature of liquid solutions. From a fundamental viewpoint the process of diffusion involves a constrained entropy increase. Thus, variable temperature observations are crucial for a complete understanding. From a practical viewpoint, many mass transfer operations are carried out over a range of temperatures far removed from the single temperature (usually 25 “C) at which data are reported. Most theories of the composition variation of liquid phase diffusion coefficients require a knowledge of the infinitely dilute values. However, the best available correlations for dilute diffusion coefficients are either restricted to very limited classes of solutes andlor solvents or can produce large errors in certain cases. More important, such correlations of dilute diffusion coefficients have generally not been extensively tested a t temperatures much different from 25 “C. The objective of the work reported here is to test directly the validity of some recently advanced theories of the temperature variation of transport properties in liquids.
Theory The theoretical basis of the analysis lies in Hildebrand’sl modification of Batchinski’s2 treatment of fluidity 4: 1 v - v, -=4 =B 9
(7)
For a pure liquid, V Ois the ultimate volume; i.e., the volume a t which fluidity ceases, and it is characteristic of the liquid. The quantity B is related to the capacity of the liquid molecules to absorb externally imposed momentum. V O appears to be a corresponding-state ratio of the critical volume V ,
VoIV, = 0.31
(2)
Equation 1 does not hold near the freezing point, as was shown by Dullien and c o - w o r k e r ~the , ~ ~reason ~ being that Vo is not directly related to the volume at that point. However, at reduced temperatures, T,, above 0.46 eq l is an excellent representation of the f l ~ i d i t y . ~ , ~ For the purpose of this paper, a relationship between diffusion coefficient and V Ois needed. Using the arguments of Kosanovich and C ~ l l i n a n which ,~ are based on Dullien’s a p p r ~ a c hthe , ~ following equation can be written: (3) is the diffusion coefficient of component i in solvent j; R is the gas constant, T is the absolute temperature, V is the molar volume of the solvent, Pij is a thermodynamic factor, and aij is a constant which is characteristic of the particular solute (i)-solvent (j) pair. When solute i is infinitely diluted in solvent j (then Pjj = l),making use of eq 1 results in the following expression: Djj
(4) The superscript zero indicates quantities a t infinite dilution, Mj is the molecular weight of the solvent, pjo is the ultimate density of the solvent, and pj is the density a t other temperatures. The Journal of Physical Chemistry, Vol. 80, No. 7, 1976
I2
I .6S
I
.6 I
Figure 1.
I
I
e
.6b
Kikta, and H. T. Cullinan
I
67
I
Self-diffusion coefficients of heptane as a function of the
density.
In eq 4 all quantities on the right-hand side, with the exception of ajj, are characteristic of the solvent. As previously mentioned the quantity aij is characteristic of the particular solute (i)-solvent (i) pair. The analogous expression for self-diffusion is
D -=RT
BV02/3 aM (PO - PI Here “a” is a universal constant. Recent analysis by Kosanovich and Cullinan7 indicates that aij in eq 4 is a temperature independent constant for a given i-j pair. Thus according to eq 4, DijIRT is a linear function of the density of the solvent, In principle eq 4 can form a basis for a prediction equation. Of great interest in this respect is the behavior of the constant aij as a function of the i-j pair. In addition the intercept a t Dij = 0 should converge to a single point, namely, po of the solvent.
Discussion The measurement techniques as described by Grushka and Kikta8 provide an excellent basis for the thorough evaluation of eq 4. Using a flowing dispersion technique, which is capable of rapid determination of diffusion coefficients over a wide range of temperatures, these workers have reportedg diffusion coefficients, at infinite dilution, of 12 phenones in n-heptane at each of five different temperatures. Dullien3 reported the self-diffusion coefficients of heptane as a function of the temperature. This allows the calculation of po for that solvent. Figure 1 shows the self-diffusion coefficients of n-heptane, for the data reported by Dullien? plotted in the form of eq 5, Only data points where T,> 0.50 are included in the figure. It is seen that, as predicted by eq 5, DIRT is quite linear with the density. The correlatione coefficient was 0.993 and an F test indicated that well within 99% confidence level the plot is indeed linear. The data fit the following equation
DIRT = -1.287 x 10-14~+ 1.001x 10-14
(6)
The ultimate density of po = 0.778 is obtained by extrapolating to zero DIRT. This corresponds to Vo of 128.8 ml/ mol which agrees very well with the value of 129.1 ml/mol
759
Binary Liquid Diffusion Prediction in Diluted Systems TABLE I: Dij/RT X 10l6Values at Five Temperatures (K)n
Compound
313.15
323.15
333.15
343.15
353.15
Acetophenone Propiophenone n-Butyrophenone Isobutyrophenone Valerophenone Isovalerophenone Hexaphenone Heptaphenone Octaphenone Nonaphenone Decaphenone Myristophenone Benzene
8.69 8.30 8.00 7.77 7.46 7.15 7.15 6.96 6.80 6.57 6.50 6.31 17.11
9.67 8.83 7.97 7.79 7.60 7.26 7.38 7.23 7.04 6.74 6.59 6.30 18.81
10.73 10.08 9.32 8.64 8.71 8.17 8.31 8.13 7.88 7.62 7.44 7.16 19.91
11.79 10.98 9.47 9.30 9.23 8.74 8.77 8.53 8.28 8.03 7.44 7.23 22.14
13.79 12.47 11.22 10.67 10.81 9.92 10.26 9.78 9.34 8.90 8.76 8.42 23.86
The temperature is accurate to within f0.02 K. TABLE 11: Data from the Linear Regression.of DijlRTVs. ( p - po) Forced through the Origin
Compound Acetophenone Propiophenone n-Butyrophenone Isobutyrophenone Valerophenone
Isovalerophenone Hexaphenone Heptaphenone Nonaphenone Decaphenone Myristophenone Benzene
1015s -8.71 -8.11 -7.63 -6.96 -6.96 -6.51 -6.64 -6.18 -5.94 -5.74 -5.53 -16.04
quoted by Hildebrand and Lamoreaux.lo Using the value of 17.7 cP-l for B , given in the above reference, and 128.8 ml for Vo, the parameter “a” for heptane is found to be 3.5 X 10l6 which corresponds well with the theoretical value of 3.7 X 1016.7The present results reaffirm the temperature independence of “ a ” . Table I shows experimental Dij/RT for 1 2 phenones and for benzene (in n-heptane) each a t five different temperatures as obtained by Grushka and Kikta.g The data were then fitted to an expression given by eq 4, and since theoretically the plots should pass through zero the lines were forced through the origin. Table I1 shows the slopes, S, of the lines for the various phenones and benzene, together with the correlation coefficients in each case. An F test indicates 99% confidence level of a linear correlation between Dij/RT and ( p - PO). The magnitudes of the F values and the correlation coefficients show that even for dilute solutions made of solute i and solvent j, the constant aij is temperature independent. In principle aij can be extracted from the slope of the line, but from an operational point of view it is easier to manipulate the experimental slopes, S, since the constant B may not be known. Although aij values are given in Table 11the following argument, for the sake of generality, will be in terms of the slopes. Several observations can immediately be made from Table 11. (1) As the molecular weight of the phenones increases, the slope decreases (aij increases). (2) The slopes of the isophenones are smaller than those of the straight chain ones with the same carbon number. (3) As the phenones become larger the relative change in the magnitude of the slope decreases. The systematic increase in the slope, S, is in qualitative
Corr coef
F test
10’6aij
0.992 0.993 0.986 0.995 0.993 0.996 0.994 0.998 0.996 0.996 0.996 0.999
251 274 177 438 278 531 351 917 1140 450 558 1491
5.17 5.56 5.90 6.47 6.47 6.92 6.78 7.29 7.58 7.85 8.15 2.81
agreement with the contention of Kosanovich and Cullinan7 that aij is a function of the size and shape of the i-j pair. The greater the difference in size, the larger is the deviation from the universal “a” constant for self-diffusion. The fact that the relative change in the slope decreases as the size of the solute molecule increases can be rationalized in terms of segmental diffusion as discussed by Van Geet and Adams0n.l’ For a long solute the diffusion is limited by the movement of its segments rather than the whole molecule. Since the size of the segment will be independent of the solute (but not the solvent), the slopes of the DIRT vs. ( p - po) lines should approach a limiting value. The value of eq 4 lies in the fact that it could be used as a predictive e q u a t i ~ n .It~ ?is,~ therefore, useful to relate the slopes, Sij, in Table I1 to a parameter such as the molecular weight of the solutes, a t least for the straight chain phenones. The behavior of the slopes suggests some sort of exponential dependence on the molecular weight, and regression analysis yields the following empirical relationship: In (4) = -2.82 X 10-3(M) - 32.10
(7) The correlation coefficient was 0.943 and an F test value of 64 indicated 99% confidence level of a straight line relation. This empirical equation allows calculation of the diffusion coefficients of the straight chain phenones in heptane. It reproduces the measured slopes within an average error of 3%, with the largest error being 9%. The error in predicting the diffusion coefficients is roughly similar. Hence using eq 7, the diffusion coefficients of undeca- and dodecaphenone, missing in Table 11, could be easily predicted. Moreover, a fairly good estimation can be made by using diffusion data of only two solutes to obtain the empirical constants in an The Journal ofPhysical Chemistry, Vol. 80, No. 7, 1976
760
E. Grushka, E. J.
Kikta, and H. T. Cullinan
TABLE 111: Data from the Linear Regression of DijlRTVs. ( p - PO)” Compound
lOI4S
Intercept X 10l6
Corr coef
F test
1016aij
Acetophenone Propiophenone n -Butyrophenone Isobutyrophenone Valerophenone Isovalerophenone Hexaphenone Heptaphenone Octaphenone Nonaphenone
-1.31 -1.19 -1.06 -0.826 -0.942 -0.794 -0.860 -0.784 -0.714 -0.673 -0.607 -0.582 -1.90
-5.97 -5.09 -4.37 -1.77 -3.33 -1.94 -2.67 -1.95 -1.30 -1.06 -0.449 -0.392 -4.05
0.994 0.990 0.960 0.956 0.965 0.972 0.967 0.976 0.980 0.982 0.935 0.937 0.995
265 144 46 32 40 50 43 60 72 82 21 27 321
3.44 3.79 4.25 5.45 4.78 5.67 5.24 5.75 6.31 6.69 7.42 7.74 2.37
Decaphenone
Myristophenone Benzene
(p
- PO) at Dij = 0 -0.0456 -0.0428 -0.0412 -0.0214 -0.0354 -0.0244 -0.0310 -0.0249 -0.0182 -0.0158 -0.00740 -0.00674 -0.0213
p
at Dij = 0 0.732 0.735 0.737 0.757 0.743 0.754 0.747 0.753 0.760 0.762 0.771 0.771 0.757
Lines are not forced through the origin. expression such as eq 7. In this case, the relative errors in the predicted diffusion coefficients are about 10-20%. Equation 7 can be used to yield an estimate of the slope of the DIRT vs. p plot for self-diffusion of heptane. From eq 6, the slope is -1.29 X Equation 7 predicts the value of -8.59 X Le., about 33% relative error. Although there is no a priori reason to expect the empirically fitted eq 7 to extrapolate accurately to the self-diffusion limit, part of the error limit lie in the fact that the data in Table I1 were obtained by forcing the lines to pass through the origin. I t is consequently instructive to obtain the slopes of DijIRT vs. p - po without fixing the intercept. The data are shown in Table 111. Also shown are the correlation coefficients, the F-test values, the intercepts, the value of ( p - po) a t D = 0 and the corresponding p. It is seen that the correlation coefficients here are not as good as in Table 11. However, the F test values still indicate a high degree of linear correlation (better than 99% in all but three phenones). The trends and explanations of the behavior of the slopes are as described previously. The behavior of the constant aij is to be noted. When the sizes of the i-j pair are roughly comparable, the value of aij is close to that obtained for the self-diffusion of heptane. However, as the size differences increase, so does aij, and eventually that constant seems to approach an asymptote. Perhaps, again, the segmental approach to diffusion can be applied here. In the case of long solute molecules, the smaller heptane molecules do not “see” the movement of the solute; rather they only see segments of the phenones. The behavior of ( p - po) a t Dij = 0, as shown in Table 111, is noteworthy; rather than being zero the magnitude of p a t zero diffusion coefficient tends to an asymptote which is close in value to po. In other words, as the size of the solute molecule increases relative to heptane, the intercept approaches zero. At this point, we can offer the following speculation explaining this phenomenon. According to eq 4, pj? = MjjVO, the ultimate density, is not only the theoretical intercept of a plot of DijIRT vs. p, but it also contributes to the slope of any such plot. Statistically, this results in a high covariance between the slope and intercept of a linear regression line through the data. This “compensation” effect may be the cause of the slight trends which are noted. The effect is analogous to the compensation effect which exists in the treatment of chemical kinetics data.12 It should be pointed out, perhaps, that one is dealing here with rather small numbers, and errors generated in the linear regression analysis are within the deviation of p from the value po at Dij = 0 (6% at most). However, the The Journal of Physical Chemistry, Voi. 80, No. 7, 1976
persistance of the dependence on the size of the phenones tends to indicate a real trend This trend will have to be checked with additional systems, and work in that direction is now underway. As with the data in Table 11, the log of the slopes of the regression lines of the straight chain phenones can be related to the molecular weights of the phenones. The empirical equation obtained here is given by In (-S) = -5.29 X 10-3(M) - 31.42
(8)
The correlation coefficient was 0.955 and an F-test value of 84 indicated better than 99% confidence level in a linear relation between In ( 4and ) the molecular weight. Equation 8 allows the prediction of the diffusion coefficients with relative percent error of at most 20% (in the case of myristophenone), and typically within about 10%. In addition, eq 8 predicts the slope of DIRT vs. p for heptane within 3%. If the data of acetophenone and myristophenone are fitted to an expression of the form of eq 8, then the diffusion coefficient of the rest of the phenones can be predicted typically within 20-30% (the largest error being 37%). The Wilke-Chang and similar estimating equations predicts the diffusion of phenones in heptane with much larger errors (as high as 55%). It is felt that the data treatment where the lines were not forced through zero is more realistic due to the uncertainty in the relationship between the slope and ( p - P O ) . In addition the points a t 40-80 OC are far removed from the temperature at which Dij is zero. Hence the regression analysis through the origin fits essentially a line between two points, one a t Dij = zero and the second at Dij = 2 X 10-5. This is the reason for the excellent correlation coefficients in Table 11. In conclusion it can be said that the approach suggested here, which is based on the ultimate volume concept, can provide a physically sound basis for a predictive equation. Due to the nature of aij a t infinite dilution, it is possible to relate diffusion coefficients to size and shape differences of an i-j pair. Thus, it is conceivable that from the data presented here, the diffusion of, e.g., myristophenone in octane can be predicted to within about 20% (provided p of the solvent as a function of temperature is known). This is so because the aij for this i-j pair is probably not much different from the constant when j is heptane, since as far as the solute is concerned, the solvent is still rather small. This speculation remains to be validated experimentally. Current efforts are directed at obtaining a better under-
761
Solute-Membrane Interactions under Osmotic Pumping standing of the constant aij for various solute (i)-solvent G) systems with the hope that the size and shape effect can be better quantitized.
References and No.tes (1) (2) (3) (4) (5)
J. L. Hildebrand, Science, 174, 490 (1971). A. J. Batschinski, Z. Phys. Chem.. 84, 643 (1913). F. A. L. Dullien, AlChE. J., 18, 62 (1972). H. Erti and F. A. L. Duliien, AlChE. J., 19, 1215 (1973). H. T. Culllnan and G. M. Kosanovlch, 77th National AlChE Meeting,
Pittsburgh, Pa., 1974. (6) H. T. Cullinan and G. M. Kosanovich, AlChE J.,21, 195 (1975). (7) G. M. Kosanovich and H. T. Cullinan, lnd. fng. Chem., Fundam., 15, 41 (1976). ( 8 ) E. Grushka and E. J. Kikta, Jr., J. Phys. Chem., 78, 2297 (1974). (9) E. Grushka and E. J. Kikta, Jr., J. Am. Chem. SOC.,accepted for publication. (101 . . J. H. Hildebrand and R. H. Lamoreaux, Proc. Nati. Acad. Sci. U.S.A.,69, 3428 (1972). (11) A. L. Van Geet and A. W. Adamson, J. Phys. Chem., 68, 238 (1964). (12) M. Boudart "Kinetics of Chemical Phenomena", Prentice-Hall, New York, N.Y., 1968.
Adsorption, Desorption, and Permeation of Methoxychlor on Semipermeable Membranes under Osmotic Pumping R. Seevers Department of Chemistry, Southern Oregon State College, Ashiand, Oregon 97520
and M. De/nzer* Environmental Health Sclences Center, Department of Agricultural Chemistry, Oregon State University, Corvallis, Oregon 9733 1 (Received Ju/y 16, 1975) Publication costs assisted by the Envlronmental Health Sciences Center
The characteristics of cross-linked polyethyleneimine-toluene-2,4-diisocyanatemembranes were investigated with regard to adsorption, desorption, and permeation of the pesticide methoxychlor during osmotic pumping using water and ethanol as cosolvents. The rejection for ethanol was determined to be 0.35 and for methoxychlor to be 0.995. An investigation of the adsorption-desorption kinetics for methoxychlor in ethanol-water system was conducted and shows that the presence of ethanol plays a significant role in these processes. A mechanism consistent with the rate studies is proposed which indicates that when a molecule of methoxychlor adsorbs to the membrane two ethanol molecules are released, one from the membrane and one from the solvation sheath surrounding each methoxychlor molecule. The rate constants for methoxychlor adsorption and desorption were found to be 725 mol-l min-1 and 5.35 x 10-5 1. m o k 2 min-l, respectively. The equilibrium constant was calculated to be 1.36 X 10' M which was then used to obtain an expression for the adsorption isotherm relating the quantity of adsorbed methoxychlor to its concentration in solution and to the concentration of ethanol in solution. The membrane adsorption density a t saturation mol of methoxychlor per cm2. was found to be 3.56 X
Introduction Within the past few years, considerably increased emphasis has been directed toward the use of semipermeable membrane technology for concentrating or removing trace amounts of organic contaminants in aqueous systems. The separation of these organic compounds from water by reverse osmosis is potentially useful for purification of drinking water supplies and for removal of chemicals and other wastes in effluents from industry and other sources prior to their discharge. Hindin and co-workers1 first showed the potential of reverse osmosis for purifying water contaminated with such diverse organic compounds as pesticides, detergents, petroleum products, and humic acids. Recently, because of important desirable characteristics for environmental studies, reverse osmosis was investigated as a technique for concentrating and removing organic contaminants from drinking water supplies for toxicity studies2 and identification.3
Investigations to elucidate solute transport mechanisms have produced relationships between membrane solute transport and hydrogen bonding of alcohols and phenol^,^ molar cohesive energy or aqueous solubility of hydrocarb o n ~ pK, , ~ of carboxylic acids: and general solubility parameters for organic solutes.' Systematic efforts to elaborate on factors which influence solute transport have shown an inverse relationship between diffusivity of the solute and its partition coefficient in cellulose acetate membranes.8 In view of the fact that solute permeability is the product of its solubility and diffusivity within the membraneg these findings provide considerable encouragement for the use of membranes as a means of concentrating trace organic contaminants in water supplies for water quality analyses. One serious drawback, however, is the tendency for many organic compounds to strongly sorb to the membrane. This is a disadvantage when membrane techniques are used in environmental studies for concentrating trace The Journal of Physical Chemistry. Vol. 80, No. 7, 1976
