670
Anal. Chem. 1991, 63, 670-676
Liquid Mass-Transport Theories Applied to Molecular Diffusion in Binary and Ternary Supercritical Fluid Mixtures Susan V. Olesik* and Jerry L. Woodruff Department of Chemistry, The Ohio State University, 120 West 18th Avenue, Columbus, Ohio 43210
Dmuslon coetllclents for benzene and m c 1 d in supercrltkal flukls contaMng polar modlliers were determined by using the chromatographic band-broadening technique. The addition of low proportions of modifiers had a profound effect on the diffusion coefficlent of the solutes. The data were compared to diffusion coefficlents predicted from several different theories of mass transport. Significant devlations were found between the experimental and theoretical diffusion coefficients. For the binary supercritical solvents, the StokesEinstein expression was most applicable. The densegas Endtog hard-sphere theory showed considerable promlse for prediction of binary diffudon coeffkients in neat supercritical fluids.
I. INTRODUCTION Supercritical fluid are used increasingly in many areas of science. Examples of applications include the extraction of pollutants from soil (I), the extraction of flavoring material (I), the chromatographic separation of high molecular weight polymers (Z),and the separation of thermally labile compounds such as explosives (2). Due to its low critical temperature, low toxicity, and ease of removal from the analyte, carbon dioxide is the most commonly used fluid in both supercritical fluid extraction (SFE) and supercritical fluid chromatography (SFC). One disadvantage in the use of supercritical carbon dioxide is its low polarity. To increase the range of polar molecules suitable for SFE and SFC, small amounts of polar modifiers such as formic acid, methanol, and acetonitrile are often added. To best use SFC and SFE technology, an understanding of the diffusion process in pure and modified supercritical fluids is desirable. To date, only a few studies on the masstransport characteristics of supercritical fluids have been undertaken with the preponderance of the studies being either analyses of self-diffusion (3,4)or analyses of binary diffusion at infinite dilution (5-12).Almost no information is available on the effects of modifiers on the diffusion of solutes in supercritical fluids. The report of Sassiat et al. (12)briefly considers this problem, but experiments were conducted at subcritical conditions with methanol modifier. In addition, no consensus has been reached on how best to describe the diffusion process as a function of the properties of the supercritical fluid. In this paper, we report the results of measurements of binary and ternary diffusion coefficients for several supercritical mixtures. We have determined diffusion coefficients for benzene and m-cresol in two supercritical fluids with different modifiers and varying temperatures and pressures. We have also compared these data and data cited in the literature to values predicted by various liquid mass-transport theories.
11. EXPERIMENTAL SECTION The experimental setup is shown in Figure 1. The fluid or modifier/fluid mixture was delivered from the pump through 13 m of 250-pm-i.d. fused silica tube (Polymicro Technologies, 0003-2700/91/0363-0670$02.50/0
Phoenix, AZ)to the fluorescence detector (FD-300, Spectrovision, Inc., Chelmsford, MA). The pump was a syringe pump (Model 601 pump, Perkin-Elmer, Norwalk, CT) that was maintained at constant pressure by using a pressure controller manufactured from a Radio Shack Color Computer (Tandy Corp., Ft. Worth, TX) running under BASIC and a pressure transducer (PX9315KS V, Omega Engineering, Inc., Stamford, CT). The injector (ACI4W, Valco Instruments Co. Inc., Houston TX) had 60-nL and 200-nL injection loops and a Valco air actuator (A60) controlled by a Valco digital valve interface (DVI). The oven was an HP5790A gas chromatograph (Hewlett-Packard Company, Avondale, PA). The detector excitation wavelength was 254 nm. The emission wavelengths were 291 and 298 nm for benzene and m-cresol, respectively. On-column detection was achieved by removing approximately 2 cm of the polyimide coating from the capillary fused-silica tube and placing this clear section in the light path in the fluorimeter. An appropriate length of 18-pm4.d. fused silica tubing was used as a postdetector restrictor. The detector output was collected by using an IBM-AT compatible computer (System 1700C, Everex Systems Inc., Fremont, CA) equipped with an IBM data acquisition board with a 12-bit A/D converter and amplifier circuit built in-house. Propane, ethane, and SFC-grade carbon dioxide were obtained from Matheson Gas Products. SpectrAR-grade benzene and AR-grade acetonitrile from Mallinckrodt and m-cresol from Kodak were also used. The water was laboratory distilled. Propylene carbonate was obtained from Aldrich. All chemicals were used as received. Benzene and m-cresol were chosen as analytes because they fluoresce, are easy to work with because they are both liquids at ambient temperature and pressure, and exhibit some difference in polarity. Acetonitrile (CH,CN) and propylene carbonate (C4H603)were used as polar modifiers in supercritical carbon dioxide, and distilled water was added to supercritical propane. Due to the limited solubility of propylene carbonate in liquid C02 (less than 5% v/v) and of water in liquid propane (less than 0.01% v/v), only one concentration level of each system was evaluated. Only m-cresol was studied in the C4H6o3/co2system because benzene had poor peak shapes. Fluid mixtures were prepared in the pump by pipeting the desired volume of modifier into the pump and then filling with COPor propane to the appropriate volume at 95 atm. Data were collected over the pressure range 95-218 atm. The upper pressure was determined by the capabilities of the high-pressure pump. The pump and the injection valve were held at room temperature throughout the study. Secondary flow in a coiled column can cause incorrect diffusion coefficients to be determined by using this method. To eliminate possible errors due to secondary flow, the aspect ratio (ratio of coil diameter to tube i.d.) should be as high as poeaible. The aspect ratio in this experiment was 680 for typical linear velocities of 2.5-6 cm/s. These conditions are well below those where secondary flow contributes to the flow pattern. The analyte of interest was placed into a 2-mL extraction cell and solvated with the pressurized mobile phase at room temperature. The solvated sample then flowed through a shut-off valve (Scientific Systems Inc., State College, PA) and into the injection loop. An 18-pm4.d. restrictor was connected to the waste line of the injector to avoid sample losses due to pressure drops across the injection valve. Data were collected at 10 points/s. Postrun analyses included baseline correction,smoothing via a Savitzky-Golay (13) routine (if necessary),calculation of the zeroth, fint, and second statistical moments of the eluted peak, and calculation of the diffusion coefficient. The data collection and analysis program was written 0 1991 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 63, NO. 7, APRIL 1, 1991 0.35 I
,pumppn , tube
8
Extraction cell
Injector
in our lab by using ASYST 2.1 (Macmillan Software Company, New York, NY). The second moment calculation was particularly sensitive to noise; therefore, the calculated values were checked by overlaying the experimental peak with a Gaussian peak of known variance and visually confirming the calculated peak variance. All lines in the Figures are smooth curves drawn through the data points unless designated differently in the figure caption. Fluid densities were measured by using a DMA 512 density meter (Anton Paar USA, Inc., Warminster, PA) and a TU16D constant-temperature bath (Techne Inc., Princeton, NJ). The density meter was calibrated at each temperature and pressure with carbon dioxide and ethane. Densities for COz at the various temperatures and pressures were interpolated from standard reference tables (14). Densities for ethane were calculated from published compressibility factors (15). The viscosities of the pure carbon dioxide and propane were determined by interpolating the data from Stephan and Lucas (16). Viscosities of the modified supercritical fluids were calculated by using an estimation method developed by Lucas (17). This method uses the critical constants of the components and their macroscopic mole fractions as the state variables, with appropriate mixing rules to derive values of mixture viscosities. This estimation technique provides viscosities that are typically accurate to within 110% error (17). 111. METHOD The chromatographic band-broadening procedure developed by Giddings (18,19) based on the work of Taylor (20) and Aris (21) was used to determine the experimental values of the diffusion coefficient. In this procedure, analyte is introduced into a capillary tube and eluted as a Gaussian peak with the desired mobile phase. The variance of this peak is related to the diffusion coefficient by the following equation: uL2
2DI2L d,2uL = -+ u
96D12
where uL2= variance of the Gaussian peak, cm2;L = length of the column, cm; DI2= binary diffusion coefficient of solute, cm2/s; u = average linear velocity, cm/s; and d, = internal diameter of the column, cm. Because diffusion coefficients in supercritical fluids are typically on the order of lo4 cm2/s, the first term is negligible except a t very low flow rates. Also the linear velocity, u , is equal to L/t,, where t, is the elution time for the center of mass of the peak. Therefore, for the determination of diffusion coefficients in supercritical fluids, eq 1 can be written as J272
All of the terms on the right side in eq 2 can be experimentally determined to yield a value for DIP Values oft, and aL2were determined by calculating the first and second statistical moments of the eluted peaks (22), respectively. Each experimental diffusion coefficient is the average of typically 15-20 independent determinations with a relative standard deviation of k6%. Ternary diffusion is a more complex process than binary diffusion. The diffusional flux of the solute depends on the
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Figure 1. Schematic diagram of equipment setup for diffusion coefficient measurements.
671
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-
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Figure 2. Diffusion coefficients of benzene in supercritical carbon dioxide. (m) This work, (A)ref 12, (0)ref 8, (0)ref 5, (+) ref 6. concentration gradients of both types of solvent molecules. To describe completely ternary diffusion, two “main” diffusion Coefficients and two “cross terms” are required (17). However, if the concentration of the solute is low, often the binary mixture is treated as though it is a pseudopure liquid (23,24). Then effective binary diffusion coefficients are determined by standard methods. The condition necessary for this simplification to apply is that the mixture be ”near ideal” such that a t any given concentration the solute effectively sees a single-component solvent with properties consisting of weighted averages of the two components. This approximation has been found appropriate to a broad range of ternary mixtures in liquids. Kim and Johnston recently showed (25) that, for modifier compositions comparable to those used in this study, the derivative of the local composition of the solute with pressure was constant over the entire modifier concentration range. This is a reasonable first approximation for many supercritical fluid mixtures, and we have used this approximation in the following analysis.
IV. RESULTS AND DISCUSSION A. Modifier Effects. T o check the performance of our experimental setup and to provide experimental binary diffusion coefficients for comparison with mass-transfer models, the binary diffusion coefficients for benzene in supercritical C02 at 40 “C were determined and compared with other published data (5,6,8,12). Figure 2 shows a compilation of these data as a function of carbon dioxide density. Our results are comparable to the other published results. The variation of the binary diffusion coefficient is a smooth function of density over the subcritical and supercritical regions. The slope of the curve is negative over the entire density range with the largest change in slope found in the subcritical region. At densities above the critical density, p,, the decrease in the diffusion coefficient with increasing density is more gradual. Diffusion coefficients for benzene for different concentrations of acetonitrile modifier are shown in Figures 3A. In general, benzene diffused more slowly in the CH3CN-modified solutions than in pure supercritical carbon dioxide of the same density (Figure 3A). Also, the diffusion coefficients for benzene decreased as the modifier content increased, although this effect diminishes with increasing solvent density. Benzene’s diffusion coefficients for all modifier concentrations and both temperatures tended to converge at densities 20.75 g/cm3. Also, as in pure carbon dioxide, benzene’s diffusion in the modified fluid increased with increasing temperature at constant density. Figure 3B shows the diffusion coefficients for m-cresol for the same temperature and modifier conditions as found in Figure 3A. Some similarities exist between the transport properties of cresol and benzene in supercritical solvents, but
872
ANALYTICAL CHEMISTRY, VOL. 63, NO. 7, APRIL 1, 1991 0.16
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DENSITY (g/cm3)
DENSITY (g/cm3)
Flgure 4. Diffusion coefficients of m-cresol in (0) COP, (V)1 %, C,H603/C02 at 40 "C and (0)COP, (V)1 % C,H8O3/CO2 at 50 "C.
B
A
Y n DENSITY (g/cm3)
0.1 3 0.30
Flgure 3. Diffusion coefficients of benzene (A) and m-cresol (B)in (0)GO,, (V),1% CH,CN/CO,, (0)5 % CH,CN/CO, at 40 "C and (0) CO,, (V)1 % CH,CN/CO,, (m) 5 % CH,CN at 50 "C.
many differences are apparent. In both cases, the diffusion coefficients decrease with increasing solvent density, and in the high density range (0.75-0.90 g/cm3), the diffusion coefficients converge to a constant value. However, the extent of decrease of Dl2 with density increase is much greater for benzene than for m-cresol. (Note: The y-axis scale on Figure 3B is expanded compared to that of Figure 3A). The variation of the diffusion coefficient with modifier concentration is also much greater for benzene than for m-cresol. Another interesting dissimilarity is the effect of CH3CN concentration on the measured diffusion coefficient. For m-cresol, the diffusion coefficients in 1 and 5% CH3CN/C02are lower than those in pure COzand statistically the same for the 50 "C data, while in the 40 "C data, the diffusion coefficients of m-cresol for 0, 1, and 5% COP are all statistically the same. Figure 4 shows diffusion coefficients measured for m-cresol in 1% C4H603-modifiedcarbon dioxide. The general shapes of the curves from C4H603are very similar to those obtained using CHRCNmodifier. The graph also shows that at 40 "C the addition of C4H603increases the diffusion rate of m-cresol. Again, the diffusion coefficients decrease rapidly a t densities greater than approximately 0.75 g/cm3. A similar sudden decrease in the diffusion coefficient was observed in numerous dense liquid systems which was attributed to significant back-scattering interactions between the solute and its nearest-neighbor molecules (26). Back-scattering is a correlated interaction where a solute surrounded closely by a solvation sphere is likely to have its velocity reversed by collision with its nearest neighbors. This correlated interaction then causes the observed decrease in diffusion coefficient (26). Parts A and B of Figure 5 shows the diffusion coefficients of benzene and m-cresol in HzO-modified propane solutions as a function of density. The addition of water increases the rate of diffusion of m-cresol In supercritical propane a t both
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(B)in
(0) propane, ( 0 )H,O/propane at 110 "C and (0)propane,
(e)
H,O/propane at 130 "C.
40 and 50 "C. However, for benzene, the addition of HzO
causes increased diffusion only a t 50 "C. B. Transport Models. Many theoretical and empirical relations have been developed to correlate the viscosity and diffusivity in liquids. In the following sections of this paper, the most successful of these relations are applied to supercritical fluids. Specifically the applicability of the StokesEinstein (27), Wilke-Chang (28), power-law dependence of D,, and Enskog hard-sphere (26) theories is considered. Stokes-Einstein is most applicable in liquids where the molecular diameter of the solute is much greater than that of the solvent. Wilke-Chang is the most commonly used theory to describe diffusion in liquids. The power-law function is routinely used when both the Stokes-Einstein and Wilke-
ANALYTICAL CHEMISTRY, VOL. 63, NO. 7, APRIL 1, 1991
Chang do not correctly describe the diffusive behavior of a solute in liquids. Enskog hard-sphere theory has been recently applied to liquids. A reasonable correlation was found between its theory and experiment for the diffusion of small molecules. We therefore thought it was most appropriate to consider its possible application to diffusion in supercritical fluids. All four theories were applied to the binary diffusion of trace quantities of benzene and m-cresol in carbon dioxide and propane. However, only Stokes-Einstein, Wilke-Chang, and power-law dependence were considered for the ternary diffusion of the solutes in CH3CN/COP, H20/C3H8, and C4H603/C02* C. Stokes-Einstein Theory. The Stokes-Einstein (SE) law is a mass-transport theory based on linearized NavierStokes fluid mechanics (29). This theory was originally developed to describe the slow diffusion of large spherical solute molecules that were much larger than the solvent molecules. Under these conditions, solvent is treated as a continuous fluid. This expression corresponds to the “stick” boundary condition of fluid flow. The fluid sticks to the surface of the diffusing particle. This theory also assumes that the solute diffusional motion is solely random translational motion. Rotational interactions between the solute and neighboring molecules are not included in this theory. This relation also makes no provisions for association between solute and solvent molecules. Under these boundary conditions, the StokesEinstein law (eq 3) predicts that the diffusion coefficient is controlled primarily by the macroscopic viscosity and the hydrodynamic radius of the diffusing species
(3) where k = Boltzmann constant; T = temperature, K; 7 = viscosity of the solvent, P; and r = hydrodynamic radius of the solute, cm. The Stokes-Einstein expression holds over a broad range of solvents and solutes. It has even been used to predict self-diffusion coefficients in liquid benzene (30) and methanol (31). For this application, the “slip” boundary condition (fluid slips over the surface of the diffusing particle) was used, changing the constant in eq 3 from 6 to 4. There are also numerous examples of liquid systems where the SE expression does not apply. For example, the diffusion of solutes in liquid hydrocarbons is not well described by SE. In parts A and B of Figure 6, D I Pis plotted versus 1/77 for benzene and cresol, respectively, in the COPsystem to test the applicability of SE. The nonlinear variation of the diffusion coefficients with reciprocal viscosity and the lack of a zeropoint intercept indicate deviation from the Stokes-Einstein model. The dashed line in each curve shows the expected variation of the binary diffusion coefficient of benzene or m-cresol as a function of 1-l using Lennard-Jones diameters (17) of the solute molecules. The curvature in the plots corresponds to variation in the hydrodynamic radius of the diffusing species. Attractive forces at the highly compressible (low-density) region of the supercritical state are known to cause considerable clustering around solutes (32). With increasing density, the repulsive force begins to dominate and to decrease the size of the solvent cluster. Debenedetti recently predicted theoretically that the solvent clusters can contain as many as 100 solvent molecules/solute molecule in the region of highest compressibility, decreasing to values as low as 5-10 solvent molecules/solute molecule in the highdensity region (33). This is exactly the trend seen in the curves in Figure 3. In all of the curves, the data points that more closely approach the Stokes-Einstein line are those a t high viscosity (high density). As the solvent association with the solute increases with decreasing density (lowered viscosity), then the deviation from Stokes-Einstein becomes more acute.
673
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Flgure 6. Diffusion coefficients of benzene (A) and m c r e s o l (B) versus reciprocal viscosity for CH,CN/CO,. The dotted line is the curve predicted by using the Stokes-Einstein law. Symbols represent the same conditions as in Figure 3.
Figure 6 also shows the variation of DI2for CH3CN/C02 for the aforementioned solutes. In all cases, the curvature of the plots is greatest for benzene compared to m-cresol. (To make visualization of these trends more apparent, solid lines were drawn through the data points for 5% CH3CN/C02in Figure 6.) On a microscopic scale, this corresponds to greater variation in the hydrodynamic radius of benzene with variation in density (pressure). For benzene, the extent of curvature increases with CH3CN modifier concentration with the 5% CH3CN/C02plot having the greatest curvature. However, for m-cresol, the variation of curvature of the plot is opposite of that seen for benzene. At low densities, the 1% CH,CN/C02 plot is similar to the curve of the binary diffusion coefficient and the 5% CH3CN/C02solutions are well described by the Stokes-Einstein law. This behavior of the CH,CN-modified system can be understood by the following simple description of the microscopic structure. The attractive forces between benzene and acetonitrile in the cluster are predominantly the weak interactions of the r-electron cloud of benzene and the dipole of acetonitrile. For m-cresol, these interactions can occur as well as hydrogen bonding between the hydroxyl group and acetonitrile, a shorter range, stronger interaction. Therefore, a t low densities, both benzene and m-cresol have a well-developed solvation sphere predominantly composed of acetonitrile molecules. As the density (pressure) is increased, the repulsive forces disrupt the weaker solvation sphere of benzene while that of m-cresol is affected minimally. Parts A and B of Figure 7 show similar plots that test the applicability of the SE law to the diffusion of benzene and m-cresol in propane in H,O/propane systems. Figure 8 is a
874
ANALYTICAL CHEMISTRY, VOL. 03, NO. 7, APRIL 1, 1991
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Flgwe 7. Diffusion coefficients of benzene (A) and m-cresoi (B) versus reciprocal viscosity for H,O/C,H,. The dotted line is the cuve predicted by using the Stokes-Einstein law. Symbols represent the Same conditions as in Figure 5.
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Figure 8. Diffusion coefficients of m-cred versus reciprocal viscosity for C,H,O,/CO,. The dotted line is the curve predicted by using the Stokes-Einstein law. Symbols represent the Same conditions as in Figure 4.
similar plot for m-cresol in C4H6O3/CO2For three systems, the absolute values of the diffusion coefficients vary with the addition of modifier, but the slopes of the curves remain approximately constant. This indicates that addition of the modifier changed the hydrodynamic radius, but once the solvation sphere is formed, increases in density do not affect the size of the solvation sphere. In summary, considering both solutes in all the supercritical solvents, significant trends are shown in Figures 5-7 that can be described qualitatively in terms of SE hydrodynamic theory. In every solvent at comparable densities, the diffusion coefficient for m-cresol is smaller than for benzene. This trend is expected with the modified fluids because the hydrodynamic
radius of m-cresol should be larger due to the association of the hydroxyl substituent with polar modifiers. The difference between the diffusion coefficients of benzene and m-cresol in pure carbon dioxide is larger than in the modified solvents. The most likely cause of this difference is quadrupolrudipolar interaction between carbon dioxide and m-cresol, but the extent of this effect is surprising especially at low densities. Finally, the effect of CH3CN concentration on the diffusion of benzene and m-cresol shows similar trends as found in liquids. For example, as mentioned earlier, the diffusion coefficient of benzene decreases monotonically with increasing CH&N concentration. Apparently, as the bulk CHBCN concentration increases, the local composition of CH3CN near the benzene also increases, which causes the hydrodynamic radius of the diffusing benzene molecule to increase. However, for m-cresol, the addition of CH3CN, H20, and C4H603 modifiers often caused an increase in the diffusion coefficient rather than a decrease. Fuoss (34) found similar variation in the diffusion coefficients of polar compounds in two-component liquids under conditions where the predominate component was nonpolar. He proposed that a braking effect was caused by the orientation of the solvent dipoles in front of the solute and relaxation of these dipoles behind the ion. This effect was predicted and found to be more apparent with decreasing dielectric constant of the primary component of the solvent. If this behavior can be extrapolated to polar solutes in supercritical solvents, then the diffusion coefficient of the polar solute should decrease as the local dielectric constant of the solvent decreases. For m-cresol, this is exactly the trend found. D. D a 7-p Relation. When the classical Stokes-Einstein relation does not apply to the diffusion of a solute in a liquid, eq 4 often applies (35)
D12= Aq-J'
(4)
where A = solute-dependent constant, q = viscosity, and p = solute-dependent constant. For liquids, the value of the exponent, p , typically varies from 0.6 to 0.7 (35-37) although exponents as low as 0.45-0.56 have been reported for the diffusion of noble gases in liquids. No theoretical basis for this relation has been found to date, but it does seem to apply to a broad range of solvent systems. The application of this relationship was considered for the supercritical solvent systems in this study. For a few of the data sets, this relation fit very well. For example, the diffusion of benzene in propane and H,O/propane at 130 "C was well fit with exponents of 0.34 and 0.30, respectively, and the diffusion of benzene in carbon dioxide was well described by the relation with an exponent of 0.65. However, in general, this relation did not hold for the other data sets. E. Wilke-Chang Relation. Another way to model diffusion data is a modified Stokes-Einstein relation developed by Wilke and Chang (28). This expression (eq 5) is purely empirical. It was developed by measuring the diffusion of 25 organic compounds in water and then determining an expression that would fit those data well
where 4 = dimensionless association factor for the solvent; M = molecular mass of the solvent, g/mol; q = viscosity of the solvent, cP; and = molar volume of the solute at its boiling point, cm3/mol. Association factors of 1 for unassociated solvents, 1.5 for ethanol, 1.9 for methanol, and 2.6 for water were assigned. To model diffusion in liquid mixtures, Perkins and Geankoplis (23) showed that the apparent molecular mass, 4M, should be determined by taking the mole fraction weighted average
ANALYTICAL CHEMISTRY, VOL. 63, NO. 7, APRIL 1, 1991
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Accordingly, for binary diffusion at infinite dilution of the solute, the Enskog theory scales the low-pressure diffusion coefficient by using the following expression:
where the numerator, D12, is the low-pressure binary diffusion coefficient at infinite dilution of the solute and g(alz)is the radial distribution function. The low-pressure diffusion coefficient was calculated by using the solution of the Boltzmann equation
[kTml+mzlz
Dlzo = - - 8n~r,,~2* mlmz
(7)
where n is the number density (molecules/volume) of the solvent, a12is the average effective hard-sphere diameter of the solute and the solvent, (al + a2)/2, and m land m2 are the masses of the solute and solvent molecules, respectively. The infiite dilution radial distribution function was approximated by using the standard expressions shown below (assuming n2 = 0) (39)
where
and Figure 9. Test of the Wilke-Chang relation for benzene (A) and mcresol (e) in CH,CN/CO, supercritical solvents. Symbols represent the same conditions as in Figure 3.
of the mass of the solvents. For the case of CH3CN/C02,the molecular weights are similar. Therefore, the molecular weight of carbon dioxide was used in the calculation. Figure 9 shows a plot of experimental diffusion coefficients versus viscosity compared to that predicted by the WilkeChang expression. The data for both binary and ternary diffusion are found to be predominantly below the line predicted by the Wilke-Chang expression. The Wilke-Chang expression comes closest to correctly describing the diffusion in the supercritical fluids at high density. Olander showed that the Wilke-Chang expression also overestimates the diffusion coefficient for water in hydrocarbon solvents (38). He proposed that the deviation was caused by the water diffusing as an associated tetramer species instead of a monomer. Since we know that solutes in supercritical fluids have large solvation spheres, similar justification for the deviation from the Wilke-Chang expression can be given for solutes in supercritical fluids. Therefore, if this empirical expression is to be used throughout the supercritical region, a variable must be added to the denominator of the Wilke-Chang expression that allows for the variation of the size of the diffusing species as a function of density. The diffusion coefficients calculated by using the Wilke-Chang equation showed no improvement over those from Stokes-Einstein. F. Enskog Hard-Sphere Fluid Theory. Another successful liquid-transport theory is the Enskog dense-gas hard-sphere theory. Dymond (26) has recently reviewed this theory and recent improvements made in it. The Enskog theory treats a liquid as though it is a system of hard spheres that interact with their neighbon at high pressure just as they would at low pressure, only more frequently. Collisions are assumed to be completely random in velocity and direction.
Computer simulations of polyatomic liquids (40) indicate that there is significant correlated molecular motion in such systems. Thus, diffusion coefficients calculated from the Enskog dense-gas hard-sphere theory, DlzE,must be multiplied by a correction factor, C12,that accounts for this correlation. Various methods have been developed to calculate a realistic correction factor. A polynomial expression developed by Easteal et al. (41) that describes the correction factor as a function of V / Vo,where Vois the close-packed hard-sphere molar volume, V is the molar volume, and a is the effective hard-sphere diameter of the solvent molecule was used in this study. The value of Voused in the calculations corresponded to high-density conditions of C02at 987 atm and 240 K, which should approximate the close-packed value. The corrected diffusion coefficient, D12(sb), assumes collisions between smooth hard spheres (shs) with no rotation. A more realistic value is based on a rough hard-sphere (rhs) model (42). Dlz(,b)= A1$112(shs)where Alz is a coupling factor that accounts for rotational transfer of momentum in a collision of unlike molecules. Thus, the experimentally determined diffusion coefficient, D12,which should correspond to D12(rb),is given by
DlZ = A1zC1zD12E
(11)
From these calculations and the experimentally determined diffusion coefficients, values of Alz were calculated for diffusion of benzene m-cresol in supercritical COP The effective hard-sphere diameters used in these calculations were 5.03 (43),5.4 (estimated from effective diameters of molecules of comparable size) (26, 44, 45), and 3.6 A (3) for benzene, mcresol, and carbon dioxide, respectively. Alz varied considerably with density (Figure 10). The trends in the translational-rotational coupling factor can be understood by comparing these values with those of liquids
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ANALYTICAL CHEMISTRY, VOL. 63, NO. 7, APRIL 1, 1991
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0.84
collection program, which we used in modified form. We I thank the Chemical Instrumentation Support Group for their 1 assistance in designing and building the pump control interface.
-
LITERATURE CITED
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0.80
0.90
DENSITY ( g / c m 3 )
,*
Figure 10. Dependence of A correction factor on density in CO, for benzene at (0)40 OC this work, (+) 40 OC ref 6, and (m) 50 O C and for cresol at (0)40 O C and (0)50 OC.
and considering the local structure in the supercritical fluid as a function of density. Low values of A12(0.3-0.4) are found in liquids that are highly associated such as methanol (46). As mentioned earlier, as the density is increased, the degree of ordering in the supercritical solvent decreases drastically. Therefore, the increase in A12 with density increase is as predicted. Other trends in the data found in Figure 10 are in accord with our present use and understanding of the translational-coupling term, Alz. For example, the degree of ordering around the more polar m-cresol is expected to be larger than that around benzene, which would cause a decrease in the A12term for m-cresol compared to benzene. This trend is shown in Figure 10. Also, an increase in temperature in supercritical fluids and liquids causes the size of the solvation sphere to decrease; therefore, the Alz term would be expected to be larger at higher temperature, which is exactly the trend seen in Figure 10.
V. CONCLUSION The diffusion coefficients of benzene and m-cresol in supercritical solvents were found to be highly dependent on the supercritical solvent. Also, the addition of the polar hydroxyl group had a profound impact on both the absolute value of the diffusion coefficient and the variation of the diffusion coefficient with density in a specific supercritical solvent. In general, the Stokes-Einstein, Wilke-Chang, and 9-P models were unable to predict well the binary and ternary diffusion coefficients for benzene and cresol across the entire range of solvent strengths studied. In the case of the Stokes-Einstein relation, the observed deviations were similar to those found in some liquid solvents. Interestingly, in the CH3CN/C02 system, the SE law best described the solvents with the higher concentration of modifier, and in all systems, the SE expression worked best at the higher densities. More solutes and modifiers will need to be analyzed before a predictive model can be formulated to allow ready calculation of binary and ternary diffusion coefficients. However, the preliminary results on the applicability of the dense-gas hard-sphere Enskog theory to binary diffusion in supercritical fluids looks quite promising as a possible predictor for binary diffusion coefficients.
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ACKNOWLEDGMENT We thank Luther Giddings for his assistance in some of the preliminary work in setting up the experimental apparatus and Curtis Miller and Tina Engel for their assistance in some of the data collection. We thank Lars Pekay for his data
RECEIVED for review August 30, 1990. Accepted January 7, 1991. Financial support of this research was through the Petroleum Research Fund, administered by the American Chemical Society.
