Ind. Eng. Chem. Res. 1994,33, 965-974
965
Effect of Local Composition Enhancements on the Esterification of Phthalic Anhydride with Methanol in Supercritical Carbon Dioxide J. Bryan Ellington,+Kayleen M. Park,+and Joan F. Brennecke. Department of Chemical Engineering, University of Notre Dame, Notre Dame, Indiana 46556
We report the pressure effect on the bimolecular rate constants for the esterification of phthalic anhydride with methanol in supercritical carbon dioxide a t 40 and 50 "C. We observe as much as a 25-fold decrease in the bimolecular rate constant based on bulk concentrations when increasing the pressure from 97.5 to 166.5 bar. Calculations of the anticipated pressure effect on the rate constant according to transition-state theory suggest a less than 2-fold decrease for that increase in pressure. Therefore, we attribute the apparently large values of the rate constants a t lower pressures to an actual increase in the local concentration of the methanol around the phthalic anhydride a t those conditions. T o further support this interpretation we present solvatochromic measurements of the local composition of methanol around a solute in supercritical fluid solutions a t the temperatures and pressures used in the kinetic experiments. We present plausible evidence that the observed kinetic measurements can be explained by the local concentration of the methanol around the phthalic anhydride being significantly higher than the bulk concentration when operating in the supercritical fluid mixture.
Introduction Since supercritical fluids (SCFs) possess properties that can be varied between those of liquids and gases, they have received significant attention from scientists and engineers, especially in applications as solvents for extractions and separations. Several reviews and monographs are available that discuss advances in supercritical fluid technology (Williams, 1981; Paulaitis et al., 1982; McHugh and Krukonis, 1986;Subramaniam and McHugh, 1986; Eckert et al., 1986; Paulaitis, 1987; Johnston and Penninger, 1989;Brennecke and Eckert, 1989; Bright and McNally, 1992; Kiran and Brennecke, 1993). More recently, the potential of SCFs as reaction media has attracted additional attention (Subramaniam and McHugh, 1986; Sigman et al., 1987; Johnston and Haynes, 1987; Paulaitis and Alexander, 1987; Kim and Johnston, 1988; Hrnjez et al., 1989; Ikushima et al., 1991; Wu et al., 1991; O'Shea et al., 1991; Yokota and Fujimota, 1991; Shaw et al., 1991;Randolph and Carlier, 1992;Roberts et al., 1992, 1993a, 1993b; Brennecke, 1993). One of the reasons for this interest is the possibility of dramatically changing reaction rates with relatively small changes in temperature or pressure due to the thermodynamic pressure effect on the rate constant, which can be very large. The purpose of this work is to explore another potential mechanism for reaction rate enhancements in SCFs, i.e., the increased local concentration of solvent and cosolvent around a solute molecule, which can occur when operating over a wide range of intermediate densities. There is a growing body of experimental, theoretical, and simulation evidence that the local environment around the solute in a supercritical fluid is significantly different than that in the bulk. The purpose of this work is to take advantage of the local structure of the fluid to enhance reaction rates. There are two potential influences of local environments. First, there is evidence that the local density of a solvent around a solute molecule is greater than the bulk density when operating in regions of relatively low density, including the regions near and significantly below the critical point. Absorption and fluorescence spectroscopy
* Author to whom correspondence should be addressed.
+ Present address: Abbott Laboratories,1401 N. SheridanRd.,
North Chicago, IL 60064.
measurements (Kim and Johnston, 1987a; Brennecke et al., 1990a;Eckert and Knutson, 1993)show distinctly that the local density of the solvent molecules around a dilute solute can be as much as twice the bulk density. The most recent studies indicate that the local density augmentation exhibits a maximum somewhere between 0 . 4 and ~ ~ 0.8pc, where pc is the critical density (Knutson et al., 1992; Sun et al., 1992; Carlier and Randolph, 1993). Local density augmentation has been seen in molecular dynamics simulations (Petsche and Debenedetti, 1989; Knutson et al., 19921, as well as integral equation calculations (Wu et al., 1990). The experimental and simulation studies indicate that the local density enhancement is a solvation effect since it does not coincide with the maximum in the isothermal compressibility. This has been corroborated by a recent theoretical study (Chialvo and Cummings, 1993). Second, when using SCF mixtures the mole fraction of a cosolvent in the local solvation sphere can be significantly greater than that in the bulk. Using the same solvatochromic technique employed to measure local densities, Kim and Johnston (1987b) found that the local composition of a cosolvent (such as acetone or an alcohol) around a solute increased steadily with decreasingpressure, reaching values as high as 7 times the bulk value. These results were corroborated by Yonker and Smith (1988), using a slightly different solvatochromic probe. Although not quite as dramatic, increased local compositions do occur in liquid solutions and are the basis of a number of excess Gibbs free energy models. Therefore, like the local density enhancements, it appears that increased local compositions in supercritical fluids are due to short-range solvation effects, not long-range critical phenomenon. A special case of increased local cosolvent concentrations is increased local concentrations of one solute molecule about another, and there is some evidence of enhanced solute interactions in SCFs (Randolph et al., 1988; Wu et al., 1990;Chialvoand Debenedetti, 1992;Combes et al., 1992). In summary, these studies indicate that in the vicinity of a solute the density of the solvent can be greater than that in the bulk. Moreover,the local composition of a cosolvent about the solute can increase dramatically. It is the purpose of this paper to show that these augmented local densities and increased local cosolvent compositions represent enhanced reactant concentrations that result in 0 1994 American Chemical Society
966
Ind. Eng. Chem. Res., Vol. 33, No. 4, 1994
enhanced reaction rates for the reaction of phthalic anhydride with methanol in SC carbon dioxide. There have been a number of studies of the pressure effect on reaction rates (e.g., Johnston and Haynes, 1987; Paulaitis and Alexander, 1987; Brennecke et al., 1990b; Yoshimura and Kimura, 1991). As mentioned above, the primary emphasis of many of these studies has been the thermodynamic pressure effect on the rate constant. For instance, in the unimolecular decomposition of a-chlorobenzyl methyl ether,Johnston and Haynes (1987)found that the rate constant increased by an order of magnitude with a 20 bar increase in pressure. Transition-state theory (Evans and Polanyi, 1935) predicts that the thermodynamic pressure effect on the rate constant is the difference in partial molar volumes of the transition state and the reactants. Since partial molar volumes can be large, usually negative values in supercritical fluids, the thermodynamic pressure effect on the rate constant can be dramatic. Kinetic studies that explore the influence of increased local density of the solvent around a solute have centered on its effect on diffusion-controlledreactions with, thus far, rather diverse results. Bright and co-workers (Zagrobelnyet al., 1992;Zagrobelny and Bright, 1992,1993) found that pyrene excimer formation proceeded at the expected diffusion-controlled limit in SC C02 and C2H4 but was significantly slowed in SC CHF3 and mixtures of SC C02 with methanol or acetonitrile. Randolph and Carlier (1992) found that the Heisenberg spin exchange reaction between nitroxide free radicals occurred faster than diffusion control and attributed the increase to increased collision times due to increased local solvent density around the encounter pair. In contrast, Roberts et al. (1993a; 1993b) found that the triplet-triplet annihilation reaction of benzophenone and the benzyl radical recombination reaction both occur at the expected diffusion-controlled limit at all conditions and in all SCFs and fluid mixtures, which included SC CO2, C2H6, CHF3, and a CH3CN/C02 mixture. Troe and co-workers have found increased local solvent density to influence photoisomerization reactions (Schroeder et al., 1990; Gehrke et al., 1990). To our knowledgethe only other study looking explicitly at the influence of local cosolvent concentration on a kinetically controlled reaction is that of Roberts et al. (1992). For the hydrogen abstraction reaction of benzophenone triplet with 2-propanol or cyclohexadiene in supercritical C02 and C2H6 they observed bimolecular rate constants based on bulk concentrations that decreased dramatically with an increase in pressure. Since transitionstate theory predicted an increase in the rate constant with an increase in pressure, they attributed the results to local concentrations of the 2-propanol or cyclohexadiene around the benzophenone triplet being significantlyhigher than the bulk in the low-pressure region. The results presented below are meant to be a complement to that study, demonstrating the effect of increased local cosolvent concentrations for a ground-state, thermally activated reaction. Previously, we reported the preliminary observations of dramatic increases in the bimolecular rate constant for the esterification of phthalic anhydride with methanol in supercritical carbon dioxide at 40 OC (Ellington and Brennecke, 1993). Here we present complete results of the esterification reaction at both 40 and 50 OC and pressures ranging from 97.5 to 166.5 bar. This reaction was chosen as a representative reaction between a dilute solute (phthalic anhydride) and a typical cosolvent (methanol) in supercritical carbon dioxide. The reaction is kinetically controlled, and at low temperatures without
L--l---
UV-VIS spectrophotometer
compressor
Figure 1. Schematic of the experimental apparatus.
a catalyst, as will be studied here, it is a simple one-step addition (Hussain and Kamath, 1967). Also, we will present new solvatochromic measurements of local compositions of methanol in carbon dioxide, following the method of Kim and Johnston (1987b),at conditions similar to those used in the kinetics measurements and will use those local composition data to analyze the kinetic results. Experimental Section Materials and Apparatus. Carbon dioxide (Linde, Bone Dry Grade), methanol (J. T. Baker, -loo%), and phenol blue (Aldrich, -97%) were used as received. Phthalic anhydride (Aldrich, 99+%) was ground with a mortar and pestle before use. Carbon dioxide was compressed into 750-mL stainlesssteel high-pressure vessels with an air-driven gas compressor (Haskel AG-152). The vessels were sealed with cleaned silicone O-rings (Parker 2-151-S-604-70),and the pressure in each (sample and reference) was measured with a digital pressure gauge (Heise 901A). The temperature was monitored and controlled in the sample vessel with an RTD probe (Omega EI-1509101-1EA-PR-11-2100-118-6-E)and a PID controller (OmegaCS-6072A-P2), respectively. In the reference vessel, a thermocouple (Omega KTSS-18-G-12) and controller (Omega CS-6001K) were used. Both vessels were heated with strip heaters (Watlow G7126), and the solutions inside were mixed with magnetic stirrers (Dylastir 58935-250and Fisher Scientific ll-498-7SH). Methanol was pumped into the sample vessel with a 20-mL manual pressure generator (High Pressure Equipment 50-6-15). A schematic of the apparatus is shown in Figure 1. The high-pressure optical cells in which the reaction was carried out had a volume of -2.5 mL and a path length of 1.74 cm. Three faces of each stainless-steel cell held 114 in. thick, 112in. diameter quartz windows (Starna Suprasil) which were packed permanently into the cell with alternating brass and lead washers. Details of the optical cell design can be found elsewhere (Ellington, 1993). Pressure was measured with a digi';al pressure gauge (Heise 901A) and a needle gauge (Ashcroft Duralife) in the sample and reference cells, respectively. In each cell, temperature was measured and controlled with an RTD probe (Omega EI-1509101-1EA-PR-11-2-100-1/8-6-E)and a PID controller (Omega CS-6071A-P2 or CS-6001-P2). The cells were heated with cartridge heaters (Watlow C2A5). Kinetic data were collected and analyzed using a doublebeam UV-VIS spectrophotometer (Varian Cary-1). The
Ind. Eng. Chem. Res., Vol. 33, No. 4, 1994 967 sample compartment was modified to hold the optical cells, and the instrument was enclosed in an external housing to eliminate stray light. Local composition data were collected and analyzed with a single-beam UV-VIS spectrophotometer (SLM-AmincoSpectronic 3000 Array), also modified to hold optical cells. The entire apparatus was connected with 1/16-in. taper seal tubing and associated valves, fittings, and adapters (High Pressure Equipment). Procedure. In the kinetic experiments, the appropriate amounts of phthalic anhydride and methanol were loaded into the sample and reference vessels, respectively. Both vessels were pressurized with carbon dioxide and heated to the desired temperature and apressure somewhat higher than was needed for the reaction in the optical cell ( 145 bar at 40 "C and 190 bar a t 50 "C). At these conditions, the concentrations of methanol ranged from 0.415 to 3.14 mol % ' (0.073-0.58 M) while the concentration of phthalic anhydride was approximately 2.9 X 10"mole fraction (5.1 X 10"' M). The solutions were given 15 h to equilibrate with stirring before methanol was added to the sample vessel with the manual pressure generator to start the reaction. The methanol was allowed to mix for 30 min in the sample vessel, and then both vessels were opened to their respective preheated optical cells. After a 2-h equilibration in the optical cells,the absorbance of phthalic anhydride was recorded versus time to allow calculation of experimental rate constants. Since the experiments were run in a vast excess of methanol, the reaction was pseudo-first-order in phthalic anhydride; thus, the integrated form of the rate equation does not require knowledge of the initial time and concentration. Local composition measurements followed the method of Kim and Johnston (1987b). They required two experiments: determining wavelength shifts of phenol blue in pure carbon dioxide and in a binary solvent of carbon dioxide and 1mol % methanol (wavelength shifts in pure methanol were taken from the results of Kim and Johnston (1987b) at 35 "C after it was determined that in liquid methanol the shift was virtually insensitive to temperature and pressure). Phenol blue and methanol (for the C o d 1 mol 7% methanol mixture) were added to the sample vessel. The vessel was pressurized to approximately 180 bar and heated to 40 "C, where it was allowed to equilibrate with stirring for 12 h. At these conditions the methanol and phenol blue mole fractions were 0.010 (roughly in the middle of the concentration range used in the kinetics experiments) and 2.0 X 10-6, respectively. The vessel contents were then transferred to the optical cell where they equilibrated for 2 h. A t least three absorption spectra were recorded at each selected pressure throughout the desired pressure range, and maximum wavelengths of absorption were calculated by the instrument, which smoothed the curves and determined the zero of the first derivative. Pressure was changed by venting the homogeneous mixture from the cell, and 15 min were given for temperature equilibration a t each pressure. After filling the optical cells at 40 "C, the remaining vessel contents were heated to 50 "C and mixed for 6 h. Then the data acquisition procedure was repeated at this temperature. All experiments were performed in the one-phase region, based on the phase equilibrium data available in the literature (Gurdial et al., 1993). Analysis. The present research has confirmed that the esterification of phthalic anhydride with methanol in supercritical carbon dioxide is first order in both reactants. The experiments were performed in a vast excess of methanol (MeOH), and thus, the reaction was pseudofirst-order in phthalic anhydride (PA). Hence, absorbance versus time data was converted to concentration versus N
N
80
100
120
140
160
180
200
220
PRESSURE (bar) Figure 2. Comparison of the densities of a 2 mol 7% methanol/COz mixture, as calculated by eq 1 (see text), to the experimental data of Tilly et al. (1993) at 45 "C (0) and 55 "C (0).
time data with Beer's Law, and plots of the natural logarithm of this concentration versus time yielded pseudofirst-order, or observed, rate constants. Independent measurements confirmed that the extinction coefficient of phthalic anhydride changed by less than 3% with pressure over the pressure range investigated. However, any change in the extinction coefficient is not of concern since in the analysis of a pseudo-first-order reaction to obtain the rate constant the extinction coefficient cancels out. Since the rate = ~ B M [ P A I [ M ~ O H = Ik,b,[PA], subsequent plots of observed rate constant versus bulk methanol concentration at the same pressure resulted in bimolecular rate constants based on bulk concentrations. Pure C02 densities were needed to calculate the mole fraction of phthalic anhydride and methanol in the sample vessel. They were taken from the equation of state used to construct the IUPAC tables (Angus et al., 1976). In addition, the density of the COz/methanol mixtures were needed to calculate the molar concentration of methanol in the optical cell for each experiment at a given temperature and pressure. The mole fraction of methanol in the mixture was known accurately, since the solution was made up in the sample vessel, but the density of the mixture in the optical cell (which was at a lower pressure than the sample vessel) was needed to calculate the molar concentration of the reactant in the optical cell. The density of these COp/methanol mixtures can be significantly different than the density of pure C02 (Tilly et al., 1993) and were determined as Prnixture
-- ~IUPACpureC02(~BWRrnixtwe/~BWFtpureC02)(l)
which is the density of pure carbon dioxide from the accurate IUPAC equation of Angus et ai. (1976)multiplied by the ratio of the density of the mixture to the density of pure C02, both calculated by the Lee-Kesler modification of the Benedict-Webb-Rubin (BWR) equation of state (Benedict et al., 1951;Reid et al., 1987;Lee and Kesler, 1975)applied to mixtures (Plocker et al., 1978;Zhang and Brennecke, 1994). The BWR equation is known to be reasonably accurate for supercritical fluids but does require a binary interaction parameter. We used the value of 1.069 recommended for COz/methanol mixtures in the literature (Plocker et al., 1978). Densities of a 2 mol % methanol in COZ mixture at both 45 and 55 "C, as calculated by the method given in eq 1, are shown in Figure 2, along with the experimental density data of Tilly et al. (1993)at those same temperatures. Although not perfect, we believe the
968 Ind. Eng. Chem. Res., Vol. 33, No. 4, 1994 0.012 0.010
Bulk Composition = 2.1 mole% T=50C
0.008
.-C Figure 3. Schematic of the uncatalyzed esterification of phthalic anhydride with methanol.
E
0.006
v
k 0.004
fit is satisfactory and did not believe a temperaturedependent binary interaction parameter was warranted for the small (10 "C) temperature range investigated in the kinetics studies. Note that the correction for the effect of the methanol on the density of the mixture was not made in calculating concentrations in our preliminary report, which gave the rate constants a t 40 "C (Ellington and Brennecke, 1993); thus, the rate constants reported here at 40 "C are slight improvements over the previously reported values. Kim and Johnston (1987b) used the solvatochromic probe phenol blue to estimate local composition enhancements by investigating the changes in the wavelength of maximum absorption of the probe. Here we extend those studies to the temperatures and pressures of interest for the kinetics experiments. Since the excited state of phenol blue is more polar than its ground state (Kim and Johnston, 1987b), the excited state is stabilized to a greater extent in a stronger solvent. As the excited state becomes more stabilized, the energy difference between the two states becomes smaller. Hence, A, shifts to longer wavelengths. In a mixed solvent, the wavelength shift may be due to one or both of the solvent components depending on the extent of their interactions with the probe at the conditions under study. If A, is known in both pure solvents comprising the mixed solvent, the shift in the mixture reveals information about the environment around the probe. If one assumes that in a mixed solvent the two solvents contribute linearly to the solvatochromic shift in proportion to their local compositions (Yonker and Smith, 19881, one obtains
+
AETmi, = x12AETl x3,AET, where AET is the residual energy of transition of phenol blue in a selected solvent and x12 and x32 are the local mole fractions of solvent and cosolvent about the probe. This analysis (eq 2) is equivalent to assuming negligible size difference between the C02 and the methanol in the method developed by Kim and Johnston (1987b). Using Kim and Johnston's raw data to calculate local compositions for COdMeOH with and without these size or packing effects verified that the assumption of negligible size differences is a reasonable one for this system. Thus, local compositions of methanol about phenol blue can be estimated if wavelength shifts are known in the mixture and in both pure solvents at the conditions of the mixture. Results
Kinetics Results. The reaction of phthalic anhydride with methanol is shown in Figure 3, and the kinetics of this reaction were measured at 40 "C from pressures of 97.5 to 145.8 bar and at 50 "C from pressures of 97.5 to 166.5 bar. A t 40 "C the methanol concentrations used ranged from 0.0 to 0.031 mole fraction, and at 50 " C the methanol concentrations used ranged from 0.0 to 0.021 mole fraction. At these concentrations of methanol the critical point of the mixture ranged from 31 "C and 73.98 bar to 36 "C and 79.7 bar (Gurdial et al., 1993).
Pressure (bar) Figure 4. Observed rate constants at 50 OC and 2.1 mol % methanol versus pressure. 0.0020-
0.0015
T=50C P = 111.3 bar
-
h
.-8
$0.0010w
k 0.0005 -
0.0000 0.0
0.1
0.2
0.3
Methanol Concentration (mol/L) Figure 5. Observed rate constants at 50 ' C and 111.3 bar versus methanol concentration.
Figure 4 shows a representative plot of the observed rate constants versus pressure for the esterification reaction at 50 "C and a bulk methanol composition of 2.1 mol % The rate constants decreased dramatically with an increase in pressure, from a value of 1.17 X lo-*min-l at 94.9 bar to 7.26 X 10" min-' at 166.3 bar. Figure 5 is an example of the observed rate constants plotted versus methanol concentration, at 50 "C and a pressure of 111.3 bar in this case. The methanol concentrations represent bulk methanol mole fractions of 0.0,0.0042,0.0083,0.016, and 0.021. The plot is relatively linear, and the slope of this line is the bimolecular rate constant based on bulk methanol concentrations at 50 "C and 111.3 bar. Thus, at these conditions, the experimental bimolecular rate constant was determined to be 6.47 X L mol-' min-l. Similar plots were constructed for all the pressures and both temperatures investigated to yield bulk-based bimolecular rate constants as a function of pressure at 40 and 50 "C, as shown in Figures 6 and 7, respectively. The average estimated uncertainty in the bimolecular rate constants is *12% at 50 "C and f28% at 40 "C. The uncertainty is larger at 40 "C because the rates are much slower at those conditions. There is a sharp decrease in the bulkbased bimolecular rate constants as the pressure is increased. At 40 "C, the rate constants exhibit a 4-fold decrease from 1.58 X to 3.65 X lo"' L mol-' min-l with an increase in pressure from 97.5 to 145.8 bar. Along the 50 "C isotherm, the decrease was 25-fold, from 3.48 X 10-2 L mol-' min-l at 97.5 bar to 1.38 X 103L mol-' min-1 at 166.5bar, and represents one of the largest decreases with increasing pressure ever reported for a reaction in a supercritical fluid. As will be discussed later, these rate constant decreases cannot be attributed to the thermo-
.
Ind. Eng. Chem. Res., Vol. 33, No. 4, 1994 969 n
.-C
0.0020
Table 1. hand 50 OC
-E0
40 "C
T=MC
E 0.0015
P (bar)
2 Q y nnnin ".""
I
I
I"
for Phenol Blue in SCF Carbon Dioxide at 40
I
0
O
90
100
110
120
130
140
O
I
150
Pressure (bar) Figure 6. Experimental bimolecular rate constants for the esterification of phthalic anhydride with methanol in supercritical C02 based on bulk methanol concentrations at 40 "C versus pressure.
0.03
(nm) 530.8 530.5 530.3 529.8 528.9 528.4 527.3 525.4
(nm) 533.4 533.4 533.4 533.6 532.6 531.6 530.8
O O 0 0 0 0 0 01 110
130
150
170
Pressure (bar) Figure 7. Experimental bimolecular rate constants for the esterification of phthalic anhydride with methanol in supercritical COz based on bulk methanol concentrations at 50 "C versus pressure.
dynamic pressure effect on the rate constant from transition-state theory. Thus, we will show that the large decreases in the experimental rate constants are only apparent. The rate constants may be decreasing somewhat according to theory, but a significant portion of the large reaction rates a t low pressures is most likely due to local composition enhancements of methanol about phthalic anhydride. Hence, local compositions of methanol were estimated about a solvatochromic probe (phenol blue) at conditions mimicking the reaction. Local Composition Results. As was mentioned previously, local compositions can be estimated by eq 2 using measured wavelengths of maximum absorption in the pure solvent and the mixture. In this research, we attempted to estimate local compositions of methanol about phthalic anhydride, but we found that the wavelength shift was not sufficiently large to given reliable values. Thus, the probe phenol blue was chosen as a representative polar solute. The data of Kim and Johnston (198713) were only along the 35 "C isotherm: here we present measurements of local compositions at 40 and 50 "C. Table 1 shows ,A, of phenol blue as a function of pressure in pure supercritical carbon dioxide at both 40 and 50 "C. At the lower temperature, A, increased from 525.4 nm at 97.3 bar to 530.8 nm at 168.9 bar. At the higher temperature, ,A increased from 519.5 nm at 95.7 bar to 529.2 nm at 189.2 bar. Table 2 shows A, of phenol blue as a function of pressure in supercritical carbon dioxide/methanol (1 mol %) at both 40 and 50 "C. At 40 "C, ,A, increased from 530.8 to 533.4 nm with a pressure increase from 96.3 to 155.4 bar, while at 50 OC, ,A, increased from 523.1 to
(nm) 529.2 528.7 528.6 528.3 527.5 524.8 522.9 521.4 519.5
50 O C A,
1 .
0.00 90
,A
189.2 176.2 163.7 150.8 137.1 123.7 112.4 103.9 95.7
40 "C
P (bar)
0,021
0.01
P (bar)
Table 2..,k for Phenol Blue in SCF Carbon Dioxide/ Methanol (1 mol 5%) at 40 and 50 OC
155.4 144.5 134.6 125.1 113.2 105.1 96.3
T=50C
A,
168.9 157.6 146.1 134.1 124.3 112.0 105.0 97.3
50 "C
P (bar) 174.2 160.8 145.9 133.8 128.4 126.0 121.6 119.4 117.2 112.8 111.2 107.4 104.8 100.7 96.7
A,
(nm) 531.6 531.7 531.2 529.8 529.7 529.6 529.0 528.5 529.2 527.9 527.3 525.3 524.7 524.0 523.1
Table 3. Local Compositions of Methanol about Phenol Blue in SCF Carbon Dioxide/Methanol (1 mol 5% ) at 40 and 50 "C
P (bar) 155.4 144.5 134.6 125.1 113.2 105.1 96.3
40 "C local comp (mol %) 4.4 4.8 5.4 6.4 6.5 6.5 8.1
P (bar) 174.2 160.8 145.9 133.8 128.4 126.0 121.6 119.4 117.2 112.8 111.2 107.4 104.8 100.7 96.7
50 "C local comp (mol % ) 4.0 4.6 4.8 4.6 5.6 6.0 6.3 6.2 7.5 6.9 6.5 4.7 4.5 4.4 4.5
531.6 nm with a pressure increase from 96.7 to 174.2 bar. These values of A, along with the values in pure methanol from Kim and Johnston (198713) were used to calculate the local compositions of methanol about phenol blue, and the results are shown in Table 3. Clearly, at both temperatures and all pressures the local methanol compositions are much larger than the bulk composition of 1 mol % This is significant because it demonstrates that local composition enhancements exist at temperatures far removed from the critical point. Basically, the local compositions decreased with an increase in pressure throughout the entire pressure range at 40 "C,from 8.1 mol % at 96.3 bar to 4.4 mol % at 155.4 bar. Based on the variation in repetitive scans of the absorption spectrum of phenol blue in pure COn and the COdmethanol mixtures, we estimate that the uncertainty in the measured values of the local compositions is f l mol %. These local compositions are very similar in magnitude to those
.
970 Ind. Eng. Chem. Res., Vol. 33, No. 4, 1994
reported by Kim and Johnston (1987b)at 35 "C. Although we were mostly interested in the data at 40 and 50 "C, we did conduct some experiments at 35 "C and pressures between 76 and 90 bar and they reproduced Kim and Johnston's (1987b) data quite well. For instance, at 81.5 bar we measured a local composition of 7.4 f 1 mol % , while Kim and Johnston reported a value of 6.8 mol % for 80 bar. Also, as demonstrated by Kim and Johnston (1987b),we would expect the local compositionsto decrease toward the bulk value of 1mol % if we were to extend the experiments to higher pressures. At 50 "C our measurements of the local compositions exhibited a maximum, increasing from 4.5 mol % at 96.7 bar to 7.5 mol % at 117.2 bar and then decreasing to 4.0 mol 5% at 174.2 bar, with an experimental uncertainty comparable to that at 40 "C. Besides showing that local composition increases are prevalent even at temperatures significantly removed from the critical point, this is, to our knowledge, the first evidence of local compositions reaching a maximum value. We believe this is a potentially important result that warrants further investigation. However,most important for this research is the fact that the local composition enhancements are large enough to significantly affect the measurement of reaction rate constants.
Discussion The kinetics results of the esterification of phthalic anhydride with methanol in supercritical COz at 40 and 50 "C indicate that the bimolecular rate constants based on bulk concentrations decrease dramatically as the pressure is increased. Spectroscopic measurements of the local composition of methanol around phenol blue in SC COz confirm previous reports (Kim and Johnston, 1987b) that the local compositions of methanol around a solute in the SCF are significantly greater than the bulk concentrations and are largest at the lower pressures. This suggests that the apparently large values of the rate constants for the esterification reaction at low pressures are actually due to the increase in the local concentrations of methanol around the phthalic anhydride. However, we would like to explore the possibility that the apparent increases in the bimolecular rate constants are due solely to the thermodynamic pressure effect on the rate constant itself before proceedingwith the local composition analysis. The thermodynamic pressure effect on the rate constant, as predicted by transition-state theory (Evans and Polanyi, 1935)can be very pronounced (for example, Johnston and Haynes, 1987; Brennecke et al., 1990b). Transition-state theory assumes that the reactants are in equilibrium with a transition state and once the transition-state complex is formed it proceeds directly to products. With this analysis the pressure effect on the reaction rate constant for a bimolecular reaction can be given as follows: A
+B
- [TS]*
products
where k,, = bimolecular rate constant (M-' min-') Ag* = CTS
- i;,
- f,
pi = partial molar volume of component i
k, = mixture isothermal compressibility
o'oo20 C .-c
0.0015
-0 L
E
0.0010
2E
Q Y
0.0005
1
lo I 1
90
100
0 Experimental 0 Corrected
A Theoretical T=40C
0
0 0 O
110
120
O
130
0
140
0
150
Pressure (bar) Figure 8. Experimental ( O ) , corrected ( O ) , and theoretical (A) bimolecularrate constantsfor the esterificationof phthalicanhydride in supercritical COz at 40 O C versus pressure. Experimental points (0) are based on bulk methanol concentrations, corrected points (0) are the experimentalpoints based on local methanol concentrations, and theoretical points (A) are those predicted from the PengRobinson equation (see text).
The partial molar volumes and isothermal compressibility can be predicted from an appropriate equation of state. Therefore, it is possible to estimate the pressure effect on the rate constant from a thermodynamic model. We chose the Peng-Robinson (1976)equation for these calculations, and this is described in detail in the Appendix. The PengRobinson equation generally gives reasonable estimates of properties of supercritical fluids, but one would expect it to be somewhat less accurate than the Benedict-WebbRubin equation used earlier to calculate COdmethanol mixture densities. Unfortunately, the BWR parameters for phthalic anhydride and the transition state are not available and cannot be readily estimated so the PengRobinson equation was chosen as an alterative. It should be emphasized that the calculations of the pressure effect on the rate constant are estimates because the binary interaction parameters are not known and the critical properties of the phthalic anhydride and the transition state have to be estimated by the methods described in the Appendix. Nonetheless, the calculated estimates of the pressure effect on the rate constant, using the pure component parameters shown in the Appendix and the default value of the binary interaction parameter of zero for all binaries, are negative at all pressures and both temperatures. The values are largest in magnitude at lower pressures and tend toward zero as the pressure is increased. Thus, transition-state theory and the Peng-Robinson equation predict that the rate constants should be largest at the lower pressures and level out at higher pressures, as observed. However, the magnitude of the change predicted from the calculations is significantly less than the measured change in the bimolecular rate constants. A t each temperature the bimolecular rate constant at a high pressure is taken as a reference, and the calculated 8 In klaP values are integrated back to predict the rate constants at lower pressures. A t both temperatures, this predicts that the rate constants at low pressures should be approximately 1.2 times those at high pressures. This small decrease with increasing pressure is shown as the triangles in Figures 8 and 9. This can be contrasted with the actual measured bimolecular rate constants based on bulk compositions, shown as the circles in the same figures. Clearly, the thermodynamic pressure effect on the rate constant cannot account for the observed 25-fold change in the rate constants at 50 "C and 4-fold change at 40 "C. We believe that it is unlikely that there would be any cage effects since they are not observed in liquid solutions for
Ind. Eng. Chem. Res., Vol. 33, No. 4, 1994 971
-
Experimental Corrected A Theoretical 0 0
0.03
.-c
t E 0
i 0.02/
3
0.01
.
0.00 90
T=50C
110
130
150
170
Pressure (bar) Figure 9. Experimental (O), corrected ( O ) , and theoretical (A) bimolecularrate constants for the esterificationof phthalic anhydride in supercritical COz at 50 OC versus pressure. Experimental points (0) are based on bulk methanol concentrations, correctedpoints ( 0 ) are the experimentalpoints based on local methanol concentrations, and theoretical points (A) are those predicted from the PengRobinson equation (see text).
this reaction (for more discussion of SC solvent effects on reactions see Brennecke, 1993; Roberts et al., 1992). There is a significant uncertainty in the calculated values of the thermodynamic pressure effect on the rate constant mainly because the values of the binary interaction parameters between the species are not known and have been set to the default value of zero, as mentioned above. In order to quantify the uncertainty in the calculations, we varied the binary interaction parameters over reasonable values. Increasing all binary interaction parameters from 0.0 to 0.2 resulted in a small change from a 1.2-fold increase with decreasing pressure to a 1.14-fold increase with decreasing pressure at 50 "C. Selectively increasing the strength of the interaction between one or two pairs of molecules by increasing those specific kij's in some cases had a larger effect. For instance, increasing the strength of the interaction between the methanol and the transition state or between the methanol and the phthalic anhydride to a kij = 0.2 did not change the calculations from the approximate 1.2-fold increase with decreasing pressure at 50 "C. However, strengthening the transition state/carbon dioxide interaction to ki, = 0.2 changed the predictions to a 6-fold increase with decreasing pressure at 50 "C. It should be pointed out that this is still not enough to explain the experimentally observed 25-fold change in the bimolecular rate constant at 50 "C (based on bulk concentrations). Conversely, increasing the phthalic anhydride/ carbon dioxide interaction parameter to 0.2 changes the calculations to suggest that the reaction rate constant should actually be increasing with increasing pressure by a factor of 3, opposite to the trend that is observed experimentally. Actually, this trend, increasing rate constant with increasing pressure, is more frequently observed and expected in liquids for bimolecular reactions. Increasing the methanol/carbon dioxide interaction parameter to 0.2 also predicts an increase in rate constant with increasing pressure, but it is much less dramatic than for the phthalic anhydride/carbon dioxide case. In summary, using the default value of zero for the binary interaction parameters of all binary pairs suggests that the rate constants should decrease with increasing pressure, experiencing a 1.2-fold change at both 40 and 50 "C,as shown as the open triangles in Figures 8 and 9. Varying specific binary interaction parameters over reasonable values can change these predictions at 50 "C anywhere from a 6-fold increase with decreasing pressure to a %fold increase with increasing pressure. Nonetheless, no com-
bination of parameters examined predicts the 25-fold change that is observed experimentally at 50 "C when the rate constants are based on bulk concentrations. Thus, we conclude that it is likely that local composition increases are playing a significant role in increasing the reaction rate of phthalic anhydride with methanol in SC CO2. If the increased local concentration of methanol around the phthalic anhydride is contributing to the observed increase in the reaction rate, it would be useful to determine whether the local concentration increase is sufficient to account for the discrepancy between the observed rate constants and the trends predicted from the transitionstate theory. In other words, we are interested in comparing the experimental bimolecular rate constants to the calculated values from transition-state theory if we base the rate constants on local concentrations instead of bulk concentrations. To examine this quantitatively, we recognize that there are two factors contributing to the increase in the concentration of the methanol around the phthalic anhydride. First, the local mole fraction of the methanol is greater than the bulk mole fraction, by a factor as great as 7.5 times (see local composition results above). Second, based on results for solutes dissolved in pure solvents, one would expect the overall local density of the solvent mixture around the solute to be greater than that in the bulk. Therefore, to determine the local concentration of methanol around the phthalic anhydride we took both of these factors into account. As a conservative estimate, we assumed that the ratio of local density to bulk density for the COz/methanol mixture was the same as that for pure C02 at the same temperature and pressure. We used the values of Knutson et al. (1992) which are somewhat lower than those reported by other investigators. In this way, the largest ratio of local density to bulk density used was 1.21. Kim and Johnston (1987b) reported measurements of local compositionsat bulk concentrations ranging from 1 to 5.25 mol % . Their data indicate that the ratio of local to bulk mole fraction is relatively insensitive to bulk concentration. As a result, we used values of a smoothed curve through the local composition data in Table 3, which were taken for a 1 mol % bulk solution, as the ratio of local to bulk composition at that pressure and temperature. Therefore, the kinetics data were reanalyzed using values of the local concentration of methanol, determined as
[bulk composition 3 local composition
(4)
where the ratios of local density to bulk density were taken from Knutson et al. (1992) for pure C02 at the same temperatures and pressures and the ratios of local composition to bulk composition are the values in Table 3, also at the same temperatures and pressures. The results of this procedure, bimolecular rate constants based on local concentrations, are shown as the squares in Figures 8 and 9. Clearly, the local concentration increases have dramatic effects on the measured bimolecular rate constants at both temperatures. In fact, when the bimolecular rate constants are based on the local concentration of methanol around the phthalic anhydride, the pressure effect on the rate constant matches that predicted from the Peng-Robinson equation remarkably well. Thus, we conclude that the pressure effect on the rate of esterification of phthalic anhydride with methanol in SC COSis due to both a slight thermodynamic pressure effect on the rate constant and a dramatic increase in the local concentration of the methanol around the phthalic anhydride.
972 Ind. Eng. Chem. Res., Vol. 33, No. 4, 1994
We do not expect the local concentration based rate constants to match the predicted pressure effect on the rate constant exactly for avariety of reasons. As mentioned above, some of the pure component parameters for the Peng-Robinson equation are estimated and reasonable variation of the binary interaction parameters results in significant uncertainty in the predicted values of the thermodynamic pressure effect on the rate constant from transition-state theory. One could argue that the rate constants based on local concentrations should be compared to new calculations of the thermodynamic pressure effect on the rate constant using the local methanol concentration. However, changing the methanol concentration in the calculations has very little influence on the end values of the pressure effect on the rate constant. For instance, using a 2 mol ?6 methanol solution rather than a 1 mol % solution, with all the binary interaction parameters set equal to zero, results in a 1.19-fold change in the rate constant, barely distinguishable from the 1.2fold change reported above for the 1 mol 7% solution. Secondly,the local composition measurements were taken around the solvatochromicprobe phenol blue. Actual local compositionsaround phthalic anhydride may be somewhat different than those around phenol blue. Thirdly, the estimates of the local density increase used were those for pure COZ around the fluorescent probe pyrene rather than a COz/methanol mixture around phthalic anhydride. Nonetheless, we believe that we have shown that the pressure effect on the esterification of phthalic anhydride in SC COS cannot be explained solely by the thermodynamic pressure effect on the rate constant but can reasonably be attributed to the increase in the local concentration of methanol around the phthalic anhydride.
Conclusions Experimental bimolecular rate constants for the esterification of phthalic anhydride with methanol in supercritical carbon dioxide based on bulk concentrations were shown to decrease dramatically with an increase in pressure at both 40 and 50 O C . Theoretical predictions of the rate constants predicted much smaller changes in the rate constants with pressure. Hence, the large rate constants at low pressures were attributed to increased local compositions of methanol about the phthalic anhydride at those conditions. These local compositionswere estimated experimentally using a solvatochromic probe, and the experimental rate constants were corrected to reflect the enhancements. The corrected, or local composition based, bimolecular rate constants showed much better agreement with theory than the raw experimental values. Therefore, we conclude that the increased local concentration of methanol about phthalic anhydride significantly enhances the rate of this reaction. In addition, the new local composition measurements yield two important results. First, the local composition increases are prevalent even along an isotherm significantly removed from the critical point of the solution. Thus, the local solvation, which results in increased local concentration of one reactant around the other, has the potential of enhancing the reaction rates of kinetically controlled bimolecular reactions over a wide range of temperatures and pressures in compressed gases and supercritical fluids. Second, at 50 "C the local composition of methanol around the phenol blue in COz passes through a maximum of about 7.5 times the bulk mole fraction, decreasing at both lower and higher pressures. Acknowledgment is made to the National Science Foundation (Grants NSF-CTS 90-09562 and NSF-CTS
Table 4. Pure Component Critical Parameters (Smith and Van Ness, 1987 and Estimates) component phthalic anhydride methanol carbon dioxide methyl hydrogen phthalate
Tc(K) 806.2 512.6 304.2 867.5
Pc(bm) 47.8 81.0 73.8 35.8
w
0.611 0.559 0.225 0.878
91-57087)and the donors of the Petroleum Research Fund, administered by the American Chemical Society,for partial support of this work. We thank Jianwei Zhang for use of the program for the BWR equation and Christopher B. Roberts for the use of the Peng-Robinson equation program to calculate the thermodynamic pressure effect on the rate constants.
Appendix Applying transition-state theory (Evans and Polanyi, 19351, the thermodynamic pressure effect on the rate constant can be explained as
A+B
- [TSI*
products
where
kbm= bimolecular rate constant (M-' min-') A?* = 3,s - V- A - V- B
ai = partial molar volume of component i I t , = mixture isothermal compressibility By definition, the partial molar volume can be written as
where
n = total number of moles
V = total volume of mixture umix = molar volume of mixture
In - order to evaluate the right-hand side of eq A l , - ( b-~ Y A - VB) - RTkT, the partial molar volumes and k~ can be calculated from an equation of state. For simplicity, the Peng-Robinson equation of state (Peng and Robinson, 1976)was used in calculating rough estimates of the partial molar volumes of phthalic anhydride, methanol, and the transition state in the quaternary mixture (carbon dioxide being the fourth component). The Peng-Robinson equation of state is given as P=
RT umix - 'mix
-
amix
+
~ m i x ( ~ m i x 'mix)
+ 'mix(vmix
where
bmix= &bi i
bi = (0.07780)RTc/Pc
- 'mi,)
Ind. Eng. Chem. Res., Vol. 33,No. 4,1994 973
+
ai = [(0.45724)R2T~/Pcl[1p(1- T:'2)12
0= 0.37464 + (1.54226)~ - (0.26992)~~ and
T, = TIT,
xi and kij represent the species mole fractions and binary interaction parameters, respectively, while w is the acentric factor. Clearly, the equation requires the critical temperature, T,,the critical pressure, P,,and the acentric factor, w, of the pure components. The properties that could not be found in the literature, such as those for phthalic anhydride and the transition state, were estimated using Joback's modification of Lyderson's method for T,and P,and Lee and Kesler's method for the acentric factor (Reid et al., 1986). The transition state was assumed to be the product, methyl hydrogen phthalate. The values of the properties used in the Peng-Robinson equation of state for estimation of the partial molar volumes are shown in Table 4. Due to a lack of necessary information, the binary interaction parameters for all pair combinations were set to zero. We now present a brief description of the method of calculation of the partial molar volumes using the PengRobinson equation of state. It is difficult to obtain
directly from the equation because it is explicit in pressure. From the triple product rule,
The derivatives in the numerator and denominator can be obtained directly from the equation of state. Thus, the partial molar volume of each component can be calculated from this rule. The isothermal compressibility is defined as
and, hence, can also be calculated from the equation of state. The right-hand side of eq A1 can now be calculated by substituting in the appropriate partial molar volumes and k ~ .This description follows that of Roberts et al., 1992;however, please note that there are several typographical errors in the appendix of that paper. Thus, it has been repeated here for clarification. Literature Cited Angus, S., Armstrong, B., de Reuck, K. M., Eds. International Thermodynamic Tables of the Fluid State: Carbon Dioxide; Pergamon Press: Oxford, 1976. Benedict, M.; Webb, G. B.; Rubin, L. C. An Empirical Equation for Thermodynamic Properties of Light Hydrocarbons and Their Mixtures. Chem. Eng. Bog. 1951,47,419.
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* Abstract published in Advance ACS Abstracts, February 15, 1994.
