J. Phys. Chem. 1986, 90, 2738-2146
2738
ratio is approximately 0.03 for the 0.5 wt % Pt/TiOz sample. Even for the hypothesis of a partial coverage of the pt, atoms by anions (note that we have ruled out this hypothesis on the basis of the radiotracer adsorption data), it follows that this would have contributed to the total charge of the Ti02 grains insignificantly, which is not in accord with the results: the electrophoretic curve of the powder containing 0.5 wt % Pt is clearly distinct from that of TiOz. Furthermore, the shifts of the other curves are not proportional to the Pt content (Figure 5), as would be the case for the adsorption of negative species on Pt, atoms.
Conclusion From (i) a comparison of the charge densities in different parts of the double layer for naked TiOz and a 10 wt % Pt/Ti02 sample, (ii) the electrophoretic mobilities of both these samples in NaCl ~
(33) In this paper the periodic group notation (in parentheses) is in accord with rexent actions by IUPAC and ACS nomenclature committees. A and B notation is eliminated because of wide confusion. Groups IA and IIA become groups 1 and 2. The d-transition elements comprise group 3 through 12, and the pblock elements comprise groups 13 through 18. (Note that the former Roman number designation is preserved in the last digit of the new numbering: e.&, I11 3 and 13.)
-
and in NaNO, solutions, and (iii) the distinct shift of the electrophoretic curve of a 0.5 wt % Pt/TiOz specimen with respect to that of naked Ti02 as compared with the very weak Pf/TiOH ratio, it can be inferred that the differences observed by potentiometry (Figure 4) and electrophoresis (Figure 5) for various Pt/Ti02 samples really stem from the effect of the deposited Pt particles on the Ti02 acid-base properties. As the Pt content increases, the acidity of the titania hydroxyl groups increases as shown by the PZC and PZZP variations (Table I). Consequently, for catalytic or photocatalytic reactions in which the aciditybasicity of the surface is involved, one should keep in mind that the properties of naked Ti02are altered by the platinum deposit. The increased acidity is consistent with a smaller number of free electrons in TiOZwith deposited Pt than in bare Ti026s7because of the alignment of the Fermi level of both materials. In other words, it is not unexpected that changes in the bulk properties of the semiconductor should be reflected in its surface properties. Acknowledgment. We thank Mrs. M.-N. Mozzanega for the preparation of the samples, Mrs. C. Carquille for her help with the potentiometry measurements, and Mrs. A. Chambosse and Mr. H. Urbain for the analyses of the Pt contents. Registry No. Ti02, 13463-67-7; Pt, 7440-06-4.
STATISTICAL MECHANICS AND THERMODYNAMICS Solute Partial Molal Volumes in Supercritical Fluids C. A. Eckert,* D.H.Ziger,+ Department of Chemical Engineering, University of Illinois, Urbana, Illinois 61 801
K. P. Johnston, and S . Kim Department of Chemical Engineering, The University of Texas, Austin, Texas 7871 2 (Received: July 22, 1985; In Final Form: October 18, 1985)
A novel technique is described for the measurement of the partial molal volume at infinite dilution of solutes in supercritical fluids. Results are reported for five systems from 2 OC above the solvent critical temperature up to 15 O C above, at pressures from just above the critical pressure to 350 bars. The solute partial molal volumes are small and positive at high pressures, but very large and negative in the highly compressible near-critical region. The results are interpreted in terms of solvent structure and intermolecular forces.
Introduction There is widespread interest in the development of theories to describe the phase behavior of supercritical fluid solutions, especially for supercritical fluid extraction, supercritical fluid A key benchromatography, and for enhanced oil eficial feature of supercritical fluid chemistry is that the chemical potential of a solute can be varied markedly by using only modest changes in pressure. This derivative, (6p2/6P)T,n,or the partial molal volume of the solute, 02, is a very fundamental solution for example, it can be expressed as a simple integral by using the direct correlation function (see eq 4 below). Prior to this study, only pressure-volumetemperaturwmposition data were used for developing and testing theories for supercritical fluid mixtures, although partial molal volume data, because of their differential nature, are particularly useful for this task. 'Present address: Western Electric, Princeton, NJ 08540.
Only very limited data for partial molal volumes in highly compressible systems exist, due in large part to the difficulty of taking such derivative data. The best known data are perhaps the classic and copious PVTy results of Sage and Lacey,' which ( 1 ) Paulaitis, M. E.; Krukonis, V. J.; Kurnik, R. T.; Reid, R. C. Reu. Chem. Eng. 1982,1, 179. (2) Johnston, K. P. In "Encyclopedia of Chemical Technology", 3rd ed.;
Wiley: New York, 1984. (3) Eckert, C. A.; Ziger, D. H.; Johnston, K. P.; Ellison, T. K. Fluid Phase Equilib. 1983, 14, 167. (4) Paulaitis, M. E.; Johnston, K. P.; Eckert, C. A. J . Phys. Chem. 1981, 85, 1770. (5) Chang, R. F.; Morrison, G . ;Levelt Sengers, J. M. H. J . Phys. Chem. 1984,88, 3389-3391. (6) Gubbins, K. E.; OConnell, J. P. J . Chem. Phys. 1974, 60, 3449. (7) B. H. Sage and R. H. Lacey co-authored over 35 papers between 1934-1942 in I d . Eng. Chem. on thermodynamic properties of hydrocarbons and common mixtures with hydrocarbons.
0022-3654/86/2090-2738$01 .50/0 0 1986 American Chemical Society
The Journal of Physical Chemistry, Vol. 90, No. 12, 1986 2739
Solute Partial Molal Volumes
S=L (pure 2)
UCEP
F u)
crilical
f"-.
LCEP
!
\ \.
.
C
-I-
t
&
I
2
f
9 7
\L=G \
\
e
critical
i .
b, Density Meter
Solvenf Inlet
c . Solute Cell d . Circulating Pump e , Volume Adjustment Valve
Temperalure
Figure 1. Pressure vs. temperature range of the partial molal volume data.
have been differentiated to yield B,. However, in the near-critical region such data represent somewhat questionable extrapolations of numerical derivatives and are probably only qualitatively correct. Hensel et a1.* reported limited data by a differential injection method. Ehrlich and co-workersgJOmade some measurements near the critical region at low but finite concentrations. Van Wasen and Schneider" calculated the solute partial molal volume at infinite dilution as a function of the isothermal pressure dependence of the measured solute retention time. However Paulaitis et al.4 showed that this calculational method assumes implicitly that (d82"/ay2)p.p,,r+.rc is negligible which is neither observedI0 nor expected based on theoretical arguments.12 In this work we describe a novel method for determining partial molal volumes a t infinite dilution capitalizing on the extreme precision of the new vibrating-tube type densitometer. Although data must be taken a t finite concentrations, the systems are sufficiently dilute that not only is behavior quite linear (see below, for example, Figure 6) but also the data can be taken far from the saturation boundary, so that never is there any danger of more than one phase being present. In fact, the region of application in PT space is shown in Figure 1.
Experimental Section Apparatus. Basically the partial molal volume of the solute at infinite dilution is determined by a series of very dilute, very precise density determinations of the supercritical solution, achieved by a vibrating-tube densitometer and a multiple-dilution high-pressure loop. The result may be found by a composition plot of the specific volume, u, or density from
where in a sufficiently dilute region, the plot to determine the last term is linear. This equation is the infinite dilution limit of the general expression where u u1 and y , au/ayl (au/ayl)". One advantage of this method is that for these data, the last term is dominant, and depends only on the precision of the density measurement, not on the absolute accuracy, thus ameliorating the problems of calibration at extreme conditions. A simplified schematic of the apparatus is shown in Figure 2, where multiple volumes (only two are shown) are in parallel in a loop with the densitometer b, a solute cell c, the pump d, and a variable volume e. The complete device is shown in Figure 3. The thermostat uses an insulated water bath controlled to better than f0.002 O C by a Bayley Instrument 123 precision controller; temperatures in both the bath and the densitometer flow loop were monitored with a Hewlett-Packard 2801A quartz crystal ther-
-
-
(8) Hensel, B. H.; Edmister, W. C.; Chao, K. C. AIChE J. 1967, 13, 785. (9) Ehrlich, P.; Fariss, R. J . Phys. Chem. 1969, 73, 1164. (10) Ehrlich, P.; Wu, P. C. AIChE J 1973, 19, 541. (11) Van Wasen, U.; Schneider, G. M. J . Phys. Chem. 1980, 84, 229. (12) Wheeler, J. C. Berg. Bunsenges. Phys. Chem. 1972, 76, 308.
Figure 2. Simplified schematic of partial molal volume experiment. 6
r--------
i I I I
-i; r--I --- 1 I
High
Pressure Solvent
'0I " E3
II
" E2
3
Constant -TemDera+ure
r--JrTL To Dead Welghl Tester
Figure 3. Detailed schematic of partial molal volume apparatus. Key: 1, storage vessel; 2, Heise pressure gauge; 3, gas-oil piston; 4a,b, differential pressure transducers; 5, volume adjustment valve; 6, densitometer; 7, solute cell; 8, circulation pump; 9, dilution cells.
mometer, with a precision of f104 OC, but calibration accuracy no better than f0.005 OC. Pressure determinations were made indirectly through a pressure transducer (Validyne Engineering DP-303) separating the supercritical fluid from the pressure measuring oil. The transducer had a sensitivity of f0.003 bar. Calibration and some pressure measurements were made with a high-precision dead-weight tester (Harwell Engineering, DT-1000) with a precision and accuracy of f0.01%, but such precision is tedious to achieve and unnecessary for this work, so the majority of pressure measurements were made with 16411. Heise gauges, giving a maximum uncertainty of f 0 . 3 bar to 135 bars and f0.7 bar above 135 bars. The densitometer is the Mettler-Paar DMA 512, whose design characteristics have been discussed by Kratky et al.I3 The sensitivity under optimal conditions is f g/cm3, but absolute accuracy is dependent on calibration, which is both pressure and temperature dependent. Because of limitations in calibration, the absolute accuracy of the density measurements here probably does not exceed f0.5% of the measurement in the range 12-45 OC and 48-350 bars. Because of vibrational interference, the densitometer was mounted just outside the thermostat bath. The vibrator was water-jacketed with circulation from the bath. The temperature of the outlet water from the jacket varied from the bath temperature by only a few millidegrees when measurements were made near room temperature. However, at the extremes of temperature (1 3) Kratky, 0.; Leopold, H.; Stabinger, H. In "Method in Enzymology"; Him, C. H. W., Timasheff, S. W., Eds.; Academic Press: London, 1973; Vol. XXVII, Part D.
2140
The Journal of Physical Chemistry, Vol. 90, No. 12, 1986
./1/4"
kl/16''
High
Pressure Filting
t
Taper Seal Filtinq
(a) Figure 4.
Eckert et al.
(a) Solute cell. (b) Solute transfer vcsscl.
in this work, the difference was sometimes as large as 10-20 millidegrees. The base of the vibrator and the lead tubes were not water-jacketed, but a special air jacket was constructed. The tubing to and from the densitometer was also water-jacketed. Nonetheless, the variations in temperature, Coupled with the very large coefficient of thermal expansion near the critical point, are a major contribution to uncertainties in the results. The solute cell, shown in Figure 4a, was basically a small stainless steel elbow, loaded with solute gravimetrically from the Pyrex solutetransfer vessel, Figure 4b. Typical amounts loaded were SIP200 mg with a precision of f0.5 mg. A specially designed noncontaminating magnetic circulation pump has been described previo~sly.~' The volume adjustment valve, Figure 5 , was a high-pressure screw injector made from an Autoclave micrometeringvalve body, in order to pennit volume variations up to 3 an3maximum to hold pressure constant. A precision dial was attached to permit volume change measurements to i 5 x IO4 cm3. Procedure. The apparatus was first cleaned from a previous run by flushing with 15-30 volume changes of supercritical solvent at 200 bars, with the procedure verified by the absence of solute in an outlet cold trap. Then pressure was reduced to atmospheric and the solute cell loaded. The solute cell was plugged and several volumes of solvent gas were vented to eliminate any t r a m of air. The system was then pressurized slowly (to protect the differential pressure diaphragm) with a charging system similar to that described previou~ly.'~During compression all valves except TI-TS and D1 (see Figure 3) were kept fully open. The top cell valves and DI were left closed to avoid solvent flowing through the solute cell and subsequently dissolving any solute. At the proper pressure the system was then left to equilibrate for - 5 h after which an initial pressure measurement was taken, along with a density measurement. Valve E l was closed at this point (14) Zigcr. D H.:Eckm. C. A. RN. Sd.Inrmrm. 1982. 53. 1296. (15) Johnston. K. P.: E c k n . C. A. AfChE J. 1981.27, 773. (16) Kim. S.; Johnston. K. P. N C h E Annual Meeting. San Franeirm. 1984.
Outlet Figure 5. Volume adjustment valve.
totally isolating the apparatus which was left overnight to ensure complete thermal equilibrium within the system. All cells but 9a were isolated by closing their respective T and B valves. Valves E2 and M were closed simultaneously trapping -20 cm3 of pure solvent between E2 and the inverting input (-) to the pressure transducer (dotted region). Since this volume was completely thermcstated and experimentally leak-free (as pmven by blank runs), its pressure was used as a reference throughout the run. Pumping was initiated at 30 cyclesfmin in the loop (100 cm3/min in a volume of 26 cm') for 20 min, then stopped to e l i n a t e pressure fluctuations for the initial density measurement. Then D2 was closed, D1 opened, and the pump run for 40 min to dissolve all solute. Volume changes were made to hold P amstant, and density measurements were made again. The volume changes gave a supplementary indication of excess volumes, and density measurements before and after the volume changes gave a rough measure of compressibility. Subsequently the other cells were valved into line one at a time and appropriate measurements made after each dilution. Chemicals. Carbon dioxide (Linde, 99.99%), ethylene (Air Products, 99.8%). and naphthalene (Baker Analyzed Reagent) were used as received. Camphor (Aldrich) was sublimed twice. saving the middle fraction. Carbon and hydrogen analyses (Perkin-Elmer 240 elemental analyzer) showed a probable purity of 99.9%. Attempts to sublime the CBr' (Aldrich) were not successful, but elemental analysis of the original material (bromine analysis hy burning in 02,absorption in alkaline peroxide, and the resulting material titrated with Hg(NO,), in 2-propanol solution with diphenyl carhazone indicator) showed a probable purity of 99.9%. Resulb
A blank run was conducted at pressure involving opening and shutting off all sections of the apparatus; the measured density was always constant to better than i1 ppm. A typical result for a data run is shown in Figure 6 at a temperature a b u t 5 OC above the critical, where the numbers on the points indicate the order
The Journal of Physical Chemistry, Vol. 90, No. 12, 1986 2141
Solute Partial Molal Volumes 6631
I
-
-
0
0.
Naphthalene in Carbon Dioxide
-1000-
u
-
\ pl
Ethylene Naphthalene
-
I
6581 -
al
2 -::
-
T=25*C
0
\
-2000-
ecu
I)
I>
Q
- 3000 , Solute Mole Fraction 6. Experimental density vs. composition dilution results for Figure naphthalene in COz.
0
0
0
4
-
150
100
50
250
200
PRESSURE (Bar 1 Figure 8. 02-vs. pressure for naphthalene in ethylene at 25 "C. 0
082200 - 1 002 I75 - Gm"'
-1000 I
al 0
Oe2 I 50 1
u
-
E
\ U
Ethylene Naphthalene
-2000
e-w
Om2 I 25 -
I>
\
m
P
-3000
P = 53,8Bar
0,2050* 0
I
I
I
0,004
00002 0.0004 Solute Mole Fraction
F w e 7. Experimental uncertainties in' density and solute mole fraction for camphor in ethylene.
I
I
- 4000
I
0,008
0,012
0,016
p (molekc I Figure 9. 02- vs. density for naphthalene in ethylene at 25 OC. 0
I
I
I
I
I
TABLE I: Selected Experimental Compressibility Results for Ethylene at 25 O C
8, bar-' P. bar 61.1 63.2 66.2 83.6
V7
0 0.000516 0 0.000695 0 0.000393 0.000556 0 0.001073 0.002176
exvtl 0.050 0.050 0.053 0.052 0.055 0.054 0.055
lita 0.049
--
0.053
-
-5,000
OI 0
: -10,000 U
0.054
?>N
-15,000
0.0095 0.0102 0.0101
Differentiated and interpolated from Douslin and Harri~0n.l~ in which data were taken-an initial point, five dilutions, and a final backmix as a check. The linearity demonstrates complete dissolution of the solvent, good mixing, and typical reproducibility. However, closer to the critical point, the higher compressibility caused greater uncertainties. Even with vastly increased mixing and circulation times and greater attention to temperature control, there was more scatter in the results. A typical set of data just 2 "C above criticality is shown in Figure 7. The uncertainty in y z is due primarily to uncertainties in the amount of solute dissolved, due to weighing errors and losses in transfer. The uncertainty in density is really not attributable to the limitations of calibration, because only density differences are important, but rather is due to temperature variations in the vibrator loop.
-20.000
01 5
0.009
0.013
p (moleiccl Figure 10. d2- vs. density for naphthalene in ethylene at 12 O
C
The u2 data which were obtained by the densitometer method were more accurate than those obtained by the excess volume method. The agreement was typically within 20%, although it was poorer as the temperature approach T,. Compressibility results generally agreed rather better than one might expect with literature data (see Table I). Systems measured include naphthalene (12 and 25 " C ) ,CBr, (12 and 25 "C),and camphor (12 "C)in ethylene and naphthalene (35 and 45 " C ) and CBr4 (35 " C ) in carbon dioxide. The data
2142
The Journal of Physical Chemistry, Vol. 90, No. 12, 1986
Eckert et al.
0
I
-5,000
--
Ethylene Tetrobromomet hone
E
-10,000
8
I>N
-15,000
-20,000
- os04
- 8 0 ~ 1
-10000
* 2 ,,
1-
I
-0.08
2
I I
-
u
k
1
70.12
I
Carbon Nophtholene Dioxide
T ~35~23OC
00016
0.006
0.009 0.013 p (mole I c c )
Figure 11. 17," vs. density for tetrabromomethane in ethylene at 12 "C.
I,/
-
0.005
-
f
0)
-2
I
I
p (molelcc)
Figure 13. 0,- for naphthalene in carbon dioxide cross plotted against the solvent compressibility ( T = 35.23 " C ; 0,0,"; - -,p).
I
0
cc
.:: ?
-IC t@oo~l,
I
Ethylene
,
Camphor
>N
I;: 72
80
88
96
104
112
Pibar)
Figure 12. 0-, and solubility vs. pressure for naphthalene in carbon dioxide at 35.23 OC.
are given in Table 11. Some typical results are shown in Figures 8-1 1. In Figure 8 note that B2- for the naphthalene-ethylene system at 25 OC is small and positive in the dense, far-supercritical region above -200 bars, but becomes large and negative in the near-critical region. Figure 9 is a density plot of the same data, showing experimental uncertainties, always greatest where the solvent is most compressible. Figure 10 shows data for the same system much closer to the critical point, with even more negative values of fi2- and greater experimental uncertainities. To a large measure, these very great uncertainities are due in part to the low solubility at the lower densities; for CBr4 somewhat higher concentrations were used without danger of approaching the phase boundary. The plot for CBr4 in ethylene a t 12 OC is shown in Figure 11. Of course, the minimum in V2- corresponds in the greatest rate of increase in solubility, as predicted by thermodynamics, and a comparison of 9," data with solubility results is shown in Figure 12. It is also useful, as will be discussed below, to recognize that a plot of 02- and solvent compressibility @ on the same axes shows excellent correspondence; two examples are shown in Figures 13 and 14. Discussion Extrema in the Solubility and the Partial Molal Volume. The location of extrema in both the first and second derivatives of y 2 vs. P is a strong function of B2. The slope of the solubility vs. pressure isotherms can be related to o2 by4*"
0.005
0.009
08013
p (mole/cc)
Figure 14. 0,- for camphor in ethylene cross plotted against solvent compressibility ( T = 12 OC;0,n2-; --,6).
where the term in brackets is positive, u2s is the molal volume of the solid, and c denotes saturation. At low pressures o2 >> u2s and the solubility decreases with pressure. At a pressure below the critical of the solvent, P,,, O2 approaches u2s and the solubility goes through a minimum." At P c l , D2- approaches negative infinity.12 At a pressure significantly above PcI,D2 exceeds u2s due to repulsive forces and the solubility goes through a maximum. The location of the extrema in y 2 depends upon the strength of the intermolecular forces which effect B2 relative to uZs. The partial molal volume can be separated into two terms using the triple-product relationship 02- = upn(dP/an,),,,-
(3)
where /3 is the isothermal compressibility. Since the absolute value of the pressure derivative of p (see Figures 13 and 14) is much larger than that of n(aP/an2)- (see Figure 15), the minimum in D2- occurs at approximately the same pressure where p is a maximum.l6>l8 This pressure is located on the critical isochore. Molecular Thermodynamic Models for the Partial Molar Volume of the Solute. The asymptotic property ij2- can be normalized by the isothermal compressibility of the pure solvent by using eq 3 to give a new property n(aP/an2)T,v;which is nearly constant in the highly compressible near-supercritical region. At infinite dilution, the isothermal compressibility becomes that of the pure solvent. Using experimental values for all of the other properties in eq 3, we can obtain an experimental value of n(17) Kurnik, R. T.; Reid, R. C. AIChE J . 1981,27, 861. (18) Gitterman, R. B.; Procaccia, I. J . Chem. Phys. 1983, 78,2648
The Journal of Physical Chemistry, Vol. 90, No. 12, 1986 2743
Solute Partial Molal Volumes TABLE II: Partial Molal Volume Data
temp, OC 35.23
45.00
82- f
ADZ-, P9
P,bar cm3/mol Naphthalene in Carbon Dioxide
mol/cm'
74.6 76.0 78.5 79.7 82.5 84.8 87.8 105.6 137.9 206.8 206.8 309.5 310.2 373.0 87.3 93.0 101.4 126.5 188.2 261.9
0.00636 0.00683 0.00764 0.008 8 3 0.01230 0.01367 0.01478 0.01666 0.01820 0.01977 0.02002 0.02150 0.02152 0.02222 0.00694 0.00867 0.01 176 0.01553 0.01835 0.01991
-21 10 -2300 -5100 -7800 -4140 -1390 -596 -123 20 47 56 93 96 103 -1163 -2420 -1310 -170 40 79
950 910 1500 1200 470 220 21 17 18 12 9 12 9 8 89 460 230 7 9 5
temp, OC
25.00
45.00
53.4 53.5 53.7 54.1 54.5 54.8 56.5 59.2 62.7 248.8 61.1 63.2 66.2 68.0 69.7 76.3 83.6 103.4 136.1 250.2 251.9
-4800 -1 2600 -15200 -11500 -7600 -2860 -1671 -996 -594 40 -1740 -2660 -3220 -3030 -2320 -1086 -544 -184 -5 1 54 45
1500 7000 3100 2700 3900 800 76 46 21 18 300 150 540 340 360 54 13 14 14 11 12
0.00640 0.00684 0.00698 0.00787 0.00876 0.00918 0.01016 0.01074 0.01 125 0.01563 0.00508 0.00563 0.00664 0.00730 0.008 17 0.00956 0.01054 0.01 194 0.01308 0.01507 0.01402
(aP/an2)"19.20 (see Figure 15). Consider a pure gas in the near-critical region where the reduced density pr < 1.3. At constant temperature and volume, the addition of a solute molecule causes a decrease in the pressure because of the attractive dispersion forces; thus n(8P/an2)- is negative. Next suppose the pressure is restored to the original pressure. Since the compressibility is large in this region, this requires a pronounced decrease in the volume; thus D2- is an unusually large negative number. WheelerI2 predicted this phenomenon using a decorated lattice-gas model. These experimental results agree with the predictions. At higher reduced densities, e.g., greater than 1.6, the solvent is sufficiently dense such that the addition of solute at constant volume increases the pressure due to the repulsive forces. At these conditions, ij2- is positive and relatively small as 0 is small. In both density regimes, the factor n(aP/anz)describes the effects of the intermolecular forces on the properties of the mixture at constant volume and is therefore not as sensitive to /3 as is 02- (see Figures 13-15). The property Bi can be expressed as a simple integral involving the direct correlation function c(r) Bi = kT@(1 - pxy,lci,(r) dr)
Ab2", P9
35.23
77.7 79.6 80.7 81.1 83.3 94.4 137.5 250.5
12.00
53.0 53.7 54.5 55.9 59.6 64.9 75.1 83.5 249.5 59.7 61.2 62.7 66.9 70.4 78.1 87.0 103.9 130.5 247.7
-3020 -5100 -7030 -5500 -1810 -273 -150 96
860 1100 740 1200 210 55 82 30
0.00723 0.00873 0.01022 0.01072 0.01296 0.01558 0.018 18 0.02069
Tetrabromomethane in Ethylene
25.00
Naphthalene in Ethylene 12.00
02-
P,bar cm3/mol mol/cm3 Tetrabromomethane in Carbon Dioxide
-11220 -16550 -7710 -2180 -828 -475 -199 -127 59 -920 -1480 -3441 -1770 -2340 -809 -460 -159 -29 72
900 440 520 270 34 82 63 46 21 160 120 91 370 340 30 120 56 52 31
0.00589 0.00694 0.00859 0.00986 0.01082 0.01 149 0.01222 0.01268 0.01578 0.00475 0.00509 0.00549 0.00690 0.00813 0.00985 0.01086 0.01 190 0.01293 0.01 505
790 2900 4500 420 180 68 41 77 4
0.0061 1 0.00693 0.00735 0.008 8 6 0.00983 0.0 1023 0.01079 0.01302 0.01575
Camphor in Ethylene 12.00
53.1 53.7 53.8 54.6 55.8 57.0 59.5 93.6 248.4
-49 10 -17800 -16200 -6970 -2370 -1596 -756 -57 102
where 0 is the isothermal compressibility and y j is the mole fraction of component j . This expression is analogous to the compressibility equation for pure fluids. For dense polyatomic fluids, Gubbins and O'Connel16 demonstrate that q ( r ) can be replaced by its isotropic part cip(r). Thus eq 4 involves the pair potential only implicitly via c.o(r) and is applicable without considering the nonadditivity of intermolecular forces, nor the orientation correlation between molecules. As a result, bi data can be used to improve the understanding of the relationship used between the strength of intermolecular forces in the supercritical phase and the phase behavior. It is useful to focus on low solubilities that approach infinite dilution where solutesolute forces are insignificant. For a binary mixture at infinite dilution, eq 4 may be simplified to give Bzm = kT@(1 - p I ~ c I z m (dr) r)
Substitution of eq 3 into eq 5 yields
The Percus-Yevick approximation can be used for c12"(r) to give (4)
(19) Douslin, D. R.; Harrison, R. H. J . Chem. Thermodyn. 1976,8, 301. (20) Tsekhanskaya, Y.V.; Iomtev, M.B.;Mushkina, E. V. Russ. J. Phys. Chem. 1962, 36, 1177. (21) Ziger, D.H.Phd. D. Dissertation, University of Illinois, 1983.
where u(r) and g(r) are the pair potential and radial distribution function, respectively. For an ideal gas n(aP/dnz)" equals plkT
2744
The Journal of Physical Chemistry, Vol. 90, No. 12, 1986
TABLE III: Unlike-Pair Attractive Interaction Constant Optimized from n(dP/dn$' Data
1i
OR :I 1
-
Eckert et al.
b,,
0 005
I
-0005
-
-001 0
system EOS naphthalene-C02 HS-VDW HS-VDW PR PR
1
2
I
i
I
I
I
naphthalene-
HS-VDW HS-VDW PR PR
ethylene
T, cm3/ cm3/ mol mol
0
I
I
4
8
I 12
.,
I
I
16
20
mol2
(rjZm)
13.8 13.7 26.7 26.7
60.2 60.2 119.0 119.0
1.22 1.21 1.44 1.42
0.12 0.16 0.17 0.10
12 25 12 25
18.5 18.5 36.2 36.2
60.2 60.2 119.0 119.0
1.29 1.34 1.57 1.60
0.12 0.15 0.12 0.24
24
Density (mol/l)
15
Figure 15. Correlation of n(dP/an2)- for naphthalene in carbon dioxide at 35 OC using VDWl mixing rules: experimental values using eq 3; - -,Peng-Robinson equation of state.
or P. For the virial equation of state truncated at the second coefficient B g12-(r) = e-buiz(r)
(8)
and
n(aP/an2)" = p,kT(I
+ 2p,B12)
+ n(aP/an2),-
I-
8 8' (II
/
h N
10
2a
--I a-
Y
(9)
Since B I 2is negative at the temperatures of interest, there is a maximum in the low-density region (see Figure 15). The supercritical fluid state may be described by using hightemperature expansion perturbation theory such that n(aP/an2)- = n(aP/an2),"
( ~ m ' ) ~ / AAD
35 45 35 45
t
-7
a I 2x io-', bar
OC
20 -2
bz,
C
5
/ /
(10)
where the subscripts r and a denote repulsion and attraction, respectively. The repulsive part also includes the ideal gas contribution. The attractive term is obtained from the second term in eq 7 where the attractive part of the potential is used for uI2(r). Assuming gI2"(r) is insensitive to density, the integral may be expressed as a constant a! that depends upon temperature only, and
n(aP/an,),- = -p& (11) The fundamental nature of 02- or n(aP/an2)- is apparent as no mixing rules were required to obtain this expression. This van der Waals approximation is reasonable for nonpolar systems in the dense supercritical region where the range of dispersion forces exceeds the intermolecular separation significantly. In the near-supercritical region where the density is lower, the attractive forces cause fluctuations in the structure of the fluid by compressing the molecules into energetically favorable locations. This was demonstrated for square-well clusters in a mixture with hard spheres of identical size.u Here, g120D(r) is expected to be a strong function of density and eq 11 is not valid. Experimental light scattering data or computer simulation data would be extremely useful to observe the structure of a highly compressible solvent about a strongly polarizable solute at infinite dilution. The range of applicability of eq 11 can be estimated from experimental data for the nonpolar systems. The property n(aP/an2)," can be calculated accurately for a mixture of hard spheres.23 The size parameters, b, = 2/sru?,were obtained from (22) Henderson, D.J . Chem. Phys. 1974, 61, 926. (23) Mansoori, G. A,; Leland, T. W. J . Chem. SOC.,Faraday Trans I 1972, 323.
/
0
0
5
10
15
20
25
Density (mol/ I ) Figure 16. Test of the van der Waals 1 mixing rule for naphthalene in COz at 35.23 O C . a previous (see Table 111). Although the molecules of interest are not spherical, an "effective hard sphere" size parameter can be correlated from PVT data.24 The property n(8P/an2),was calculated from eq 10 by using experimental data for n( a P / b 2 ) - and the above method for n(dP/dn2),". As expected from eq 11, the behavior is linear at high density where g12(r)is relatively insensitive to density (see Figure 16). At lower densities ( p , C 1.3) where fluctuations are prevalent, the van der Waals approximation is less satisfactory. The n(dP/dn2)- data in Figure 15 were correlated by using the popular semiempirical Peng-Robinson (PR) equation of state (EOS), and a hard-sphere van der Waals (HS-VDW) EOS. The VDWl mixing rule a = Cyfjaij ij
(12)
was used in both cases. The HS-VDW EOS consists of the accurate expression for a mixture of hard spheres discussed above23 along with the original van der Waals attractive term. Differentiation of this EOS including eq 12 yields n(dP/&12)a- = -2a,,p2
(13)
(24) Oellrich, L. R.; Knapp, H.; Prausnitz, J. M. Fluid Phase Equilib.
1978, 2, 163-171.
The Journal of Physical Chemistry, Vol. 90, No. 12, 1986 2145
Solute Partial Molal Volumes
tested using the physically meaningful attraction constants obtained from the 02- data. In the supercritical phase, the fugacity of the solute may be described as a compressed gas
fZV= Yz42P
(14)
where the fugacity coefficient In 42 =
A'( Dz
I) d P
RT - P
or as on expanded liquid
where y2is the activity coefficient, and the superscript O means reference. In either approach, 0,is the key property of the mixture that influences the solubility. The solid-phase fugacity is
-12
Ir.
P(bari
8,;
6
14
1;;
"p
It can be seen that the two approaches are identical by equating eq 14 and 16 at the reference pressure, Po, to give
22
42(P0) Po = Y2(P0J2) f20L(P0)
Density (mol/ I ) Figure 17. Calculation of 02- for naphthalene in C 0 2 at 35.23 O C using an a12which is optimized for reduced densities above 1.3 (0, experimental data; -: HS-VDWEOS using literature values of the density and compres~ibility;~~ --, PR EOS using literature values of the density and compre~sibility;~~ - -,PR EOS using calculated values of the density and compressibility).
-
which is in agreement with the fundamental expression, eq 11, which does not depend upon mixing rules. Since both equations of state depend upon the van der Waals approximation, the unlike pair attraction constants were obtained by limiting the optimization to pr > 1.3. The average absolute deviation (AAD) in the correlation of n(aP/an2)- or equivalently of 02- (see eq 3) was within the experimental uncertainty in the dense supercritical region which is consistent with the expectations based on Figure 16 (see Table 111). The unlike-pair attraction constant is the key property of the mixture which influences n(aP/an,)- and ultimately yz. The n(aP/an2)" data at infinite dilution have been used to obtain a physically meaningful value of aI2without the need for mixing rules nor other parameters, such as az2. Neither of these simplifications can be accomplished by using solubility data. The success of the PR EOS in the near-critical region is fortuitous. A better physical description of this region would include a density-dependent g12(r),higher order terms in the high-temperature expansion perturbation t h e ~ r y ~ and ~ - ~a' rigorous expression for the mixture which avoids the VDWl mixing rules (eq 12). However, given the common utilization of the above equations of state for the prediction of supercritical phase behavior, it is important to note their success in the dense supercritical region. Although the PR EOS is considered to be one of the best cubic EOS, it does not predict the density nor the isothermal compressibility of the pure fluid accurately.21 The U2 calculated by eq 3 with the PR EOS is inaccurate as shown by the dashed line in Figure 17, even though n(aP/an2)" was correlated accurately in Figure 15. The results are improved when the literature value of p1 and fi are used instead of calculated values. The same is true for the HS-VDW EOS. Thermodynamic Models for Solid-Fluid Equilibria. Thermodynamic models for solubility isotherms may be developed and (25) Alder. B. J.: Hecht. C. E. J . Chem. Phvs. 1969. 50. 2032 (26j Alder; B. 5.1 Young, D. A,; Mark, M. A.J . Chem. Phys. 1972, 56. (27) Johnston, K. P.; Ziger, D. H.; Eckert, C. A. Ind. Eng. Chem. Fundam. 1982, 21, 191-197.
(18)
Combination of eq 14, 16, and 18 gives the Poynting correction
The expanded liquid model has a practical advantage in that Po need not be sufficiently low such that the gas is ideal. The advantage of choosing PO in the dense supercritical region is evident from Figure 17. Most theories give reasonable values for ijZmin this region where pr > 1.3, despite the errors which occur at lower densities where the fluid is highly compressible. Suppose the reference pressure were 250 bar for COz ( p = 0.02 gmol/cm3). The integral of iiz d P is accurate in the dense supercritical region, and the calculated solubility isotherms in this region are not affected by errors at lower density. This would not be the case for the compressed gas model where the integral in eq 16 always traverses the highly compressible region. In essence, the ideal gas reference state is quite far away from supercritical conditions where the density and cohesive energy density approach that of a liquid. The fugacity coefficient is typically on the order of lo4 and the calculated yz is extremely sensitive to a12.z The compressed gas model has been used successfully by numerous investigators for the correlationzJ7 and in some cases, the p r e d i ~ t i o of n ~solubility ~ ~ ~ ~ data. ~ ~ ~ The a12which is optimized from the data must compensate for errors in the integration of i$ d P in the near-critical region. Therefore, the optimized a12is not in agreement with the value listed in Table 111. Conceivably, the errors in the near-critical region may be similar for a series of hydrocarbons such that the optimized values of aI2may be related to other physical properties such as the polarizability. The dense supercritical reference fugacity can be determined from a single solubility data point x20 at the reference pressure
Here, the vapor pressure of the solid, which is often unavailable, is eliminated with the combination of eq 16, 17, and 20. In effect, P2" is included implicitly in the experimental data point xzo. Here y2(P0,xz) has been absorbed intofioL(PO), and it is assumed that y2 does not change with composition for low solubilitie~.~~ (28) Wong, J. M.; Pearlman, R. S.; Johnston, K. P. J . Phys. Chem. 1985, 89, 267 1 .
(29) Mackay, M. E.; Paulaitis, M. E. Ind. Eng. Chem. Fundam. 1979.18, 149.
2746
The Journal of Physical Chemistry, Vol. 90, No. 12, 1986 80
I
Eckert et al. highly complex near critical region.
I
\ o
60 70:
-
I N
L
401
P
0 301 20
-
IO
\
Ok
I
I
60 70 80 AH vaP(IGl/mol) Figure 18. Correlation for the standard-state fugacity of several hydrocarbons in ethylene at 250 bar and 45 OC, data ref 15 and 27.
50
If the experimental data p i n t x20is unavailable, the reference fugacity can be obtained by using the following correlation. It is important to choose PO in a region where x2is relatively constant such t h a t h o Lis insensitive to Po (see eq 20). For ethylene, and carbon dioxide at 250 bar, the liquid is only slightly expanded and is relatively incompressible. We choose to correlate the reference fugacity hoL, which was obtained from experimental data using eq 20, vs. the enthalpy of vaporization of the solute at the melting point. Figure 18 shows that the correlation is linear for a series of hydrocarbons in ethylene a t 45 OC. Johnston et al.27 showed that the unlike-pair attraction parameter can also be correlated with the enthalpy of vaporization. The dense supercritical reference state is sufficiently close to the conditions of interest that the calculated y 2 is not nearly as sensitive to uI2as above for the ideal gas reference. Thus, the solubility isotherms can be predicted accurately by using the constants in Table I11 provided a satisfactory approach is used to obtain u22.28The differences in the results of the two reference states will diminish as better theories become available for the (30) Angus, S.; Armstrong, B.; de Reuck, K. M. 'IUPAC International Tables of the Fluid State-Carbon Dioxide"; Pergamon Press: New York, 1916.
Conclusions A new technique has been developed for measurement of f i 2 = as close as 2 OC to the critical temperature in a supercritical fluid. Results are good because of the precision of the vibrating-tube densitometer coupled with a differential mode of operation that precluded the necessity for extremely accurate calibration in a region where such standards are generally unavailable. Data are reported for eight isotherms in the range 40-350 bars for three solutes in two solvents. An examination of the Liz" data reveals that the seemingly complex effect of pressure in a supercritical fluid mixtures is a result of two basic thermodynamic properties. These include the isothermal compressibility of the solvent, p, and n(dP/dn2)T,vm which is a direct measure of the strength of the unlike-pair attractive forces. The latter factor is relatively constant in the critical region, in contrast with p, y2,and Bzm. A dense supercritical region can be identified where the solvent structure about the solute is insensitive to density, and ulz can be obtained from n(aP/an2)" without the need for mixing rules nor u22. As a result, a quantitative relationship may be described between the strength of the intermolecular forces in the supercritical phase and the solution behavior. This provides a basis for the development and testing of thermodynamic models for the solubility isotherms. Using these results, a new dense supercritical reference state can be defined which avoids the complexities of the highly compressible near critical region. Acknowledgment. This material is based in part on work supported by the National Science Foundation under Grant No. CPE-8306327. Any opinions, findings, and conclusions or recommendations expressed in this publication do not necessarily reflect the views of the National Science Foundation. This project has also been financed in part with federal funds as part of the program of the Advanced Environmental Control Technology Research Center, University of Illinois, at Urbana-Champaign, which is supported under cooperative agreement CR 8068 19 with the Environmental Protection Agency. The contents do not necessarily reflect the views and policies of the Environmental Protection Agency nor does the mention of trade names or commercial products constitute endorsement or recommendation for use. Additional funding was provided by the U S . Department of Energy, under grant DOE DEFG22-84 PC70801 and by the Illinois Coal Research Board, through the Center for Research on Sulfur in Coal. Registry No. C02, 124-38-9; naphthalene, 91-20-3; ethylene, 74-85-1; acetylene, 74-86-2; tetrabromomethane, 558-13-4; camphor, 76-22-2.
