1164 (9) (10) (11) (12) (13) (14) (15)
(16) (17) (16) (19) (20) (21) (22) (23) (24) (25) (26)
Anal. Chem. 1987, 59, 11641170 Barber, W. E.; Carr, P. W. J . Chromatogr. 1983, 260,6996. Barber, W. E.; Carr, P. W. J . Chromatogr. 1984, 301,2538. Barber, W. E.; Carr, P. W. J . Chromatogr. 1984, 316, 211225. Cassidy, R . M.; Fraser, M. Chromatographia 1984, 18, 369373. Fritz, J. S.;Gjede, D. T.; Becker, R. M. Anal. Chem. 1980, 52, 15 191 522. Hershcovitz, H.; Yarnitzkv, Ch.; Schmuckler, G. J . Chromatow. 1982, 244, 217224 Haddad, P R., Heckenberg. A L J Chromatogr 1984, 300, 357394   .   .. Haddad, P. R.; Heckenberg. A. L. J . Chromafogr. 1982, 252, 177184. Skelly, N. E. Anal. Chem. 1982, 54,712715. Okada, T.; Quwamoto, T. Anal. Chem. 1984, 56,20732076. Solms, D. J.; Smuts, T. W.; Pretorius, V . J . Chromatogr. Sci. 1971, 9. 600603. Scott, R. P. W.; Scott, C. G.; Kucera, P. Anal. Chem. 1972, 44, 100104. Slais, K.; Krejci, M. J . Chromafogr. 1974, 91, 161166. McCormick, R. M.; Karger, B. L. J . Chromatogr. 1980, 199,259273. Groh, R.; Halasz, I . J . Chromatogr. 1980, 199,2334. Mertens, J. J . Liq. Chromatogr. 1982, 5 , 14671480. Melander, W. R.; Erard, J. F.; Horvath, Cs. J . Chromafogr. 1983, 282, 229248. Dreux, M.; Laffose, M.; Pequignot, M. P. Chromatographla 1982, 15, 653656.
(27) Berek, D.; Bleha, T.; Pevna, 2 . J . Chromatogr. Sci. 1976, 14, 560563. (28) Berek, D.; Chalanyova, M.; Macko, T. J . Chromatogr. 1984, 286, 185 192. (29) Katime, I.; Campos, A.; Rivera, J. M. T. Eur. Polym. J . 1979, 75, 291293. (30) Campos, A.; Borque, L.; Fiqueruelo, J E. J . Chromatogr. 1977, 140, 2 19227. (31) Ng, T. L.; Ng S. J . Chromatogr. 1985, 329, 1324. (32), Sun, S. F.; Wong, F. Chromatographia 1985, 20,495499. (33) Geng. X.; Regnler, F. E. J . Chromatogr. 1985, 332, 147168. (34) Riedo, F.; Kovats, E. Sz. J . Chromatogr. 1982, 239, 126. (35) Le Ha, N.; Ungvaral, J.; Kovats, E. Sz. Anal. Chem. 1982, 54, 24102421. (36) Bidllngmeyer, B. A.; Deming, S. N.; Price, W. P., Jr.; Sachok, B.; Petrusek, M. J . Chromatogr. 1973, 186,419434. (37) Stranahan, J. J.; Deming, S.N. Anal. Chem. 1982, 54, 15401546. (36) McCormick, R. M.; Karger, B. L. Anal. Chem. 1980, 52,22492257. (39) Knox, J. H.; Kaliszan, R. J . Chromatogr. 1985, 349,211234. (40) Levin, S.;Grushka, E. Anal. Chem. 1985, 57, 18301835. (41) Pronaeus, S.Acta Chem. Scand. 1951, 5 ,659871. (42) Levin. S.;Grushka, E J . Chromatogr. 1987, 384,249258.
RECEIVED
for review August 1, 1986. Accepted December 1,
1986.
Measurement of Diffusion Coefficients in Supercritical Carbon Dioxide and Correlation with the Equation of Wilke and Chang Patrick R. Sassiat, Pierre Mourier, Marcel H. Caude, and Robert H. Rosset* Laboratoire de Chimie analytique, Ecole SupQrieure de Physique et de Chimie de Paris, 10, rue Vauquelin, 75231 Paris Cedex 05, France
Dlffuslon coefflclents of acetone, benzene, naphthalene, 1,3,5trImethylbenzene, phenanthrene, pyrene, and chrysene have been measured by a chromatographlc broadening technlque in an open capillary tube (950 X 0.103 cm) fllled wlth pure supercritical carbon dloxlde or, In the case of benzene, wlth C0,methanol mlxtures ranglng from 0 to 100% In methanol. I n pure supercrltlcal CO,, dlffuslon coefflcients decrease when denslty Increases; they Increase linearly vs. the reciprocal of the vlscoslty; a llnear relatlonshlp exists between the logarlthms of the dlffuslon coefflclents and the molar volumes wlth a slope of 0.6. Finally, In the range 0.60.9 g ~ c m  the ~ , Wllke and Chang equatlon for the calculatlon of dlffuslon coefflclents Is valld for supercritical C02. For methanolCO, mixtures there Is no dlscontlnuity of the dlffuslon coefficient of benzene when the methanol content varies from 0 to 100%. I n the usual supercrftical chromatographlc condltlons wlth a methanol content less than l o % , dlffuslon Coefficients are at least 4 times higher than In pure methanol.
Carbon dioxide is, nowadays, most commonly used in SFC because of its interesting properties (e.g., chemical stability, low critical temperature (31 "C), low critical pressure (73.8 bar), considerable inertness, low boiling point, no toxicity, no inflammability, low cost, and ready availability). Pure COP or mixture with polar modifiers such as alcohols (mainly methanol), methyl tertbutyl ether, chlorinated solvents, acetonitrile, etc., have good solvent properties. In chromatography, reduced parameters are valuable for testing column efficiency and the molecular diffusion coefficient of the solute in the solvent should be accurately known to calculate reduced velocities of the eluent. In liquid chromatography the diffusion coefficients are often approximated, in practice, by using the Wilke and Chang equation (6). Only few values of diffusion coefficients are found in the literature (7lo), so we decided to measure such coefficients for some solutes in supercritical COzand to study the influence of temperature, pressure, and addition of methanol. A correlation between these results and the Wilke and Chang equation for unassociated liquids (as C 0 2is an apolar solvent) was investigated.
Supercritical fluids and dense gas have found interesting applications in both chromatography (socalled supercritical fluid chromatography (SFC)) and extraction (socalled supercritical fluid extraction (SFE)). Recent interest in SFC is due to the possibilities of using capillary columns ( 1 ) and small particle columns ( 2 , 3 )in this mode of chromatography. Furthermore, the use of supercritical fluids may results in new detection possibilities such as Fourier transform infrared spectrometry ( 4 ) and slightly modified flame ionization detectors ( 5 ) .
THEORY The use of a chromatography method for the measurement of diffusion coefficients has been established in the early 1960s and has been used by several authors. The principle is to use an empty capillary tube of suitable length and diameter. As shown by Taylor (1115), the dispersion of a sharp pulse of solute introduced into a laminar flow of the mobile phase results in the combined action of convection along the axis of the tube and molecular diffusion in the radial direction.
00032700/87/03591164$01.50/0
1987 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 59, NO. 8, APRIL 15, 1987
1165
Thus, in a straight open tube with laminar flow, a mathematical treatment gives the simplified equation for the plate height H
20,
H=+u
ro2u 240,
1II
c 0. CONTAINER
INJECTOR
I
where D, is the solute diffusion coefficient, u is the mobile phase linear velocity, and ro is the inner radius of the tube. H is defined by
!EXCHANGER
/I
I I I
H = a2(x)/L I;
here a2(x) is the peak variance (in length unit) and L is the length of the capillary tube. A similar relation is obtained from the Golay equation (16) when there is no coating inside the capillary tube and, consequently, no retention of the injected solute. The first term of this relation represents the spreading due to axial diffusion and the second one takes into account band broadening due to the parabolic flow profile and to radial diffusion. The contribution of axial diffusion (2D,/u) can be neglected (within 1%) for linear velocities greater than 70D,/ro. Therefore, except at extremely low flow velocities, which are hardly encountered in practice, eq 1 reduces to
H = r,2u/24Dm In a straight tube, the concentration profile becomes Gaussian when H < 0.02 (17). With our experimental conditions, this inequality is always valid. Sources of Errors. The main sources of errors will be only briefly discussed since the subject has been treated in full detail by other authors (1820). Dead Volume and Initial Dispersion. One of the difficulties encountered with diffusion coefficient measurements is the influence of injection, cell, and connecting tube volumes. In order to avoid these difficulties, Giddings and Seager (15)have used two columns of different lengths. A subtraction procedure is then used. In this way, the variance of the peak obtained with the first column (a short one) is subtracted from that obtained with the second and the resulting variance corresponds to a peak that would have been produced in the same column without dead volumes and initial peak dispersion. When this device,is not used, the injection mode must be carefully studied. If the sample loop is left in line with the mobile phase, the peak will exhibit serious tailing (21). On the contrary, if the sample is injected in the column by allowing the mobile phase to flow through the loop for only a very short time, the valve being then immediately rotated to allow the mobile phase to flow directly to the column, the dispersion effect of the sample is eliminated. The tailing is avoided and it results in a Diraclike injection. Adsorption. Adsorption can give large errors particularly for polar compounds in the low density range where the solvent power of the C 0 2 is reduced. Generally, elution profile with severe tailing is taken as an indicator for interaction between the species and the column wall. Therefore, calculation of diffusion coefficient is always done with symmetrical peaks (asymmetry factor A , determined by the ratio of halfpeak widths measured at 0.1 peak height (22)was always 11.2). Pressure Drop. The pressure drop disturbs the measurements only when the variation of mobile phase density vs. pressure is important, i.e., close to the critical point. The measured pressure drop was less than 1 bar and its effect can be neglected. Column Coiling. Equation 1 must be only used with straight tubes, in order to avoid the secondary flow effect characteristic of coiled tubes (2327) and inducing a depen
Figure 1. Apparatus scheme for the diffusion coefficient measurement.
dence between the measured diffusion coefficient and the linear velocity (28). This effect is due to centrifugal forces increasing the radial mass transport and, consequently, decreasing the solute zone dispersion. When coiled columns are chosen, this effect can be eliminated by using small linear flow velocity and large R / r o ratio where R is the coiling radius. In order to avoid these experimental difficulties, we have chosen to use a straight capillary tube. Injected Quantity. According to operating conditions and solute nature, measured D , values can vary with the injected quantity. For example, Timmermans (29) recommends measuring the limiting diffusion coefficients of solutes such as acetone, benzene, and ethanol in chloroform by extrapolation to zero concentration. On the contrary Ouano (30) and Lauer (9) do not observe any variations. This point will be discussed later. EXPERIMENTAL SECTION Apparatus. Figure 1shows the apparatus scheme. A syringe pump (Varian Model 8500) and a spectrophotometer Varichrom (Varian) were modified as follows for use with supercritical carbon dioxide: cooling of the syringe pump at 0 "C with a copper jacket system in order to realize an efficient filling with liquefied C02, thereby making it possible to obtain constant low flow rates, e.g., 20 mL.h' at 0 "C; modification of the optical flow cell (dead volume, 8 pL; path length, 10 mm) of the variablewavelength detector to withstand pressures of 350 bar; behind the spectrophotometer, a Tescom (Model 26172124082) back pressure regulator allows pressure control throughout the entire system. This regulator needs special heating (40 "C) to balance the endothermic decompression of the supercritical COz. Additional thermostating with a Tamson (Model TX 9) waterbath regulator ensures that temperature remains constant all along the chromatographic tube and within injector and connections at f0.2 "C. The injector is a Rheodyne Model 7413 equipped with a 1WL loop. We have verified that the D, values did not depend on the solvent nature used for injection (supercritical COz, methanol, isooctane) and thus we chose to inject the solutes dissolved in isooctane. When methanol is added to COP,a piston pump (Gilson Model 302) in conjunction with a dynamic mixing chamber is used. Diffusion Tube. We have used a 950 cm ( 1 0 . 5 % )straight stainless steel tube. For the mean diameter calculation, the tube volume was measured by two different procedures: (1)filling of the tube with a 0.1 M acid solution and subsequent titration; (2) weighing after filling the tube with C C 4 . The internal diameter is 0.103 f 0.001 cm. Chemicals and Reagents. Carbon dioxide (Air Liquide) was of high purity (99.995%). Methanol was of HPLC quality (Prolabo) and isooctane was of Chromasol quality (SDS Co.). Solutes (acetone, phenol, benzene, naphthalene, 1,3,5trimethylbenzene, phenanthrene, pyrene, and chrysene) were of analytical grade.
ANALYTICAL CHEMISTRY, VOL. 59, NO. 8, APRIL 15, 1987
1166
4 1
1
"L
 A,&
~
10 '0
to

, IO
10 7
9
Inwted q~ainlity (",ole)
Figure 2. Variation of the measured diffusion coefficient with injected quantity for naphthalene (A) benzene (B): column, 950 X 0.103 cm i.d.; flow rate, 20 cm3.h' (liquefied COS at 0 "C); pressure, 155 bar; temperature, 40 OC;detection, UV at 210 nm; injected volume, 1 pL.
,
100
140
120
160
200
300
240
+
I
density,u solute
acetone phenol
benzene naphthalene 1,3,5trimethylbenzene phenanthrene pyrene chrysene a
molar vol," cm3.mol'
mol wt, g
gem+
58 94
0.79
78 128
0.88
88.6
1.14
120 178
0.86
111.8 138.9 150.8
202
228
1.07
1.18 1.27 1.27
73.4 87.8
0.1
0.1
0.6
Table I. Characteristics of the Studied Test Solutes
P(bar1
09
P(g.cm3)
Figure 3. Variation of the diffusion coefficient with the CO, density for various solutes: temperature, 40 "C;solutes, (A) benzene, (B) naphthalene, (C) 1,3,5trimethylbenzene, (D) phenanthrene. Other conditions are as given in Figure 2 and Table 11.
159.0
179.0
Densities and molar volumes are given at room temperature (25
"C).
RESULT AND DISCUSSION Injected Quantity. We have measured the D, values for benzene and naphthalene at several injected quantities. Figure 2 shows that for benzene D , values are constant up to 3 x mol injected. On the contrary there is an important decrease of naphthalene diffusion coefficient for injected quantities greater than mol. An explanation of this behavior may be a difference in the solvatation of the two aromatic hydrocarbons in carbon dioxide. The extrapolated m 2 4 ) is in agreement with the value value of D (11.7 X m2s'). of Lauer et al. (9) (11.5 X Pure Carbon Dioxide. Seven solutes were studied and their properties of interest for D, value discussion are given in Table I. Phenol was also used but the D, values have been discarded due to asymmetric peak profile ( A , > 1.2). SFC with COz is interesting within the following boundaries: 74 < pressure P < 400 bar; 30 < temperature T < 100 "C; 0.1 < density p < 1 g . ~ m  ~Between . the nearly liquid state (30 "C, 400 bar) and the nearly gaseous state (100 "C, 74 bar) D, varies greatly and the influence of the parameters ( P , T , p ) has to be studied. The density p has been calculated from P and T with a suitable state equation (31). The overall results of our measurements are given in Table 11. Influence of the Density at Constant Temperature. Figure 3 represents D , vs. COS density at a temperature of 40 "C. As expected a large dependence of D, on density is observed. The diffusion coefficients are greater than m2.s1 and decrease with increasing solute molecular weight and size. ~ 40 "C) gives Raising the density from 0.65 to 0.9 g . ~ m (at a solute diffusion coefficient decrease of about 40%. But, in all cases, these values are about 520 times greater than with classical LC solvents. Influence of the Viscosity at Constant Temperature. COz viscosity varies according to pressure and is calculated by using
L__,, 13
I1
IS
c 11
19
I/1 ( 1 0 . 4 ~1 ~
)
Flgure 4. Variation of the diffusion coefficient with the reciprocal of the CO, viscosity for benzene (A) and naphthalene (B); temperature, 40 "C. Other conditions are given in Figure 2 and Table 11.
30
40
SO
BO
Tm~p~rdl~rI I C)
Figure 5. Variation of the diffusion coefficient with the temperature for various solutes: C02 density, 0.8 g . ~ m  ~solutes ; (A) benzene, (B) naphthalene, (C) 1,3,54rirnethylbenzene, (D) phenanthrene. Other conditions are given in Figure 2 and Table 11.
the equation given by Altunin (32). Figure 4 shows the variation of D, vs. the reciprocal of the viscosity for benzene and naphthalene. As with a liquid phase and within experimental errors a quasilinear relationship is obtained and will be discussed further. Influence of Temperature at Constant Density. As the COP density is kept a t constant values, the D , variations vs. temperature are small (Figure 5). D , values only increase by about 10% when temperature increase from 30 to 60 "C. Moreover a linear relationship is obtained. This observation
ANALYTICAL CHEMISTRY, VOL. 59, NO. 8, APRIL 15, 1987
1,
14
00
~ _
02
11
Figure 8. Variation of log D, vs. log (TMS1"/q)for various solutes (A, acetone; B, naphthalene: C, phenanthrene).
~
_ .
~
06
Ob
16
1167
OB
10
'2
P(L!Crn
')
Figure 6. Variation of the carbon dioxide viscosity TJ with density for 400 K; ( ) 500 K; (. .) various temperatures: ( ) 320 K; () 600 K.



0.1
1
0.6:
0.7 1 1.0
0.9
0.8
0.7
 0.6
1)
0.5
Figure 9. Variation of log ( D J T ) vs. log ( T J V ~ . ~ I Mfor, "all ~ )our data; see Table 11.
30.4
8
\
I_; I 1.8
1
ac*Ion.
1:9
1
benzene
2:O
Q
 hu
1
naphlhalen.
2:l
1
ph.nanthr.n.
I
!
1.3.5trimathyl
2.2 I
IOgv'
chryssn.
pyren.
benzen.
Figure 7. Variation of log D, vs. log V for various CO, densities at various temperatures: (1) 0.51 g.cm3, 60 O C ; (2) 0.65 g.cm3, 60 O C ; (3) 0.80g . ~ r n  ~60 , O C ; (4) 0.80 g . ~ m  ~ 40, O C ; (5) 0.88 g ~ c m  ~ 40, O C ; (6) 0.80 g.cm3, 30 O C ; (7) 0.88~ c r n  30 ~ , O C . Other condklons are given in Figure 2 and Table 11.
is consistent with the Wilke and Chang equation, whereas, for supercritical COP,at constant density, viscosity does not vary with temperature. The Altunin equation has the following form: 77
m n
VT
i=lj=O
In  = CCaijx1jyli being expressed in poise, where y1 = p / p , and x1 = T,/T (p, and T , are the C 0 2 critical density and the COz critical temperature). T J expressed ~ in poise is equal to
TJ
+
TT = ~1'/'(2722.46461  1663.46048~1 466.920566~1')
Table I11 gives the ai, values. Figure 6 represents the variation o f t with density at various temperatures. It shows that, at constant density and in the supercritical range (0.4 < p < 1 g ~ m  ~7)variations , with T are very small. Influence of the Molar Volume. D, values decrease when the solute molar volume V is increasing and a linear relationship is obtained between log D , and log V at constant T and p (Figure 7 ) . For COPdensities greater than 0.6 g ~ m  ~ , the slope is constant whatever the operating conditions and equal to 0.6 f 0.05, a value in good agreement with the Wilke and Chang equation for unassociated liquids. For COPden~ , slope decreases (e.g., 0.45 for sities lower than 0.6 g  ~ m the ) . have no satisfactory explaa COz density of 0.5 g ~ m  ~ We nation for this phenomenon. Correlation with the Wilke a n d Chang Equation. The Wilke and Chang equation for unassociated liquids is
D, = (7.4 x 1 0  ~ 5 ) ~ ~ , 1 ~ 2 ~ ~(1)v ~ ~ 0 ~ ~ where T is the temperature in Kelvin, M , is the molecular weight (g) of the solvent (here M , = 44 g), TJ is the solvent viscosity (Pas) which can be calculated, for COP,by using the equation given by Altunin (32) as described above, and Veb is the molar volume (~m~.moll) at the ebullition temperature and under atmospheric pressure. We can assume that COz, which is an apolar solvent ( 3 ) , is also an unassociated fluid. The variation of log D, vs. log ( T M , " 2 / ~for ) three solutes is represented in Figure 8; it shows that a linear relationship with a slope equal to 1 is obtained only for C 0 2 densities
1168
ANALYTICAL CHEMISTRY, VOL. 59, NO. 8, APRIL 15, 1987
Table 111. a,, Values for the Calculation of COz Density According to the Altunin Equation (32) j
i = l
i=2
i=3
i=4
0 0.248 566 120 0.373 300 660 0.363 854 523 0.063 907 755 1 0.004 894 942 1.227 534 488 0.774 229021 0.142 507049
Flgure 10. Variation of V , vs. V for various solutes (data for V taken from ref 34; V,, values calculated according Lebas (33).
.4.,
0 IOJ
0 007
" I
Flgure 11. Variation of log D, vs. log V , from data taken in ref 34: mobile phase, 5% ethyl acetate In nhexane; temperature, 25 OC.
greater than 0.6 g  ~ m  ~In. the range 0.60.9 g  ~ m  Figure ~, 9 shows that a linear relation is obtained when log (D,/T) is plotted vs. log (qv0.6/M,1/2), V being the molar volume at ambient temperature. The calculated slope is 0.97 f 0.03 and the ordinate intercept is (1.13 f 0.02). Thus, in this range of density, the solute diffusion coefficients can be calculated from the following equation: 0
Y
1 e
m A W
1%
D, = (7.4 x
1 0  1 5 ) ~ ~ , 1 ~ 2 / ~ ~ . 6
(2)
In this equation we have preferred to use the molar volume a t ambient temperature V as it can be calculated easily by using the molecular weight on density ratio. In the original equation of Wilke and Chang the solute molar volume V,, is the value at ebullition and under atmospheric pressure, which is often difficult to obtain. Lebas (33)proposed a method for the v&calculation consisting of obtaining the sum of atomic volumes for the different molecule elements. However this method can be applied to only a few of the solutes we use for the D, calculation. Fortunately, it is easy to show that v& is a linear function of V, molar volume at room temperature.
ANALYTICAL CHEMISTRY, VOL. 59, NO. 8, APRIL 15, 1987
1169
~109
(rn?s':
15
10
c
0
? .P
2 . l 
23
u
Y
w
v
w
zp
w
''Qv
5
Flgure 12. Variation of log D, vs. log V from data taken in ref 34; mobile phase, 5 % ethyl acetate in nhexane; temperature, 25 O C .
Figure 10 shows the variation of v e b with V for many different compounds. A linear relation is obtained and it is veb
= 5.31
+ 1.30V
1.3V
as molar volumes lie between 100 and 400 for most compounds ( V and v&are expressed in cm3.mol'). Figures 11 and 12 show the variation of log D, with log Veb and log V in the case of liquids (data from Katz and Scott (34)). It is established in Wilke and Chang's equation that log D, is a linear function of log v e b with a slope equal to 0,595. Figure 12 shows that log D, is also a linear function of log V with a slope equal to 0.619. From these results the following equation, valid for liquids, can be proposed:
D,
= (6.9 x 1 0  1 5 ) ~ ( + ~ ~ ) 1 / 2 / ~ ~ 0 ~ (3) 3
# being the association factor. This equation is much easier to use because V is the molar volume at ambient temperature and not a t the ebullition temperature as in the original Wilke and Chang equation. We notice that the constant with V (6.9 X is not very different from the constant with v e b (7.4 X w 5 ) . Thus eq 2 for the calculation of D, in COPbecomes, when using v e b instead of V
D, = (8.6
X 10'5)TM,'/2/vVeb0.6
(4)
This equation is similar to the Wilke and Chang equation but the constant value is 8.6 X instead of 7.4 X MethanolCarbon Dioxide Mixtures. A polar modifier such as methanol is often added to carbon dioxide in SFC in order to change the retention and, consequently, the selectivity (higher the methanol content, lower the retention). Therefore, it would be interesting to know how diffusion coefficients vary against methanol content in COD Various methanol contents were obtained by mixing COz and methanol which are pumped separately at known mass rates. The calculation of the critical parameters of the C02methanol mixtures has been realized by several authors and curves giving the variations of the critical pressure and temperature as a function of the mole fraction of methanol can be found (35). We have chosen to measure the benzene diffusion coefficients for various methanol contents in carbon dioxide at a pressure of 150 bar and a temperature of 40 "C. Under these operating conditions, the mixture is supercritical only for a methanol content lower than 4% (mole fraction