Anal. Chem. 1998, 70, 2104-2109
Helium Head Pressure Carbon Dioxide in Supercritical Fluid Extraction and Chromatography: Thermodynamic Analysis of the Effects of Helium Michal Roth*
Institute of Analytical Chemistry, Academy of Sciences of the Czech Republic, CZ-61142 Brno, Czech Republic
The Peng-Robinson equation of state (EOS) is employed to quantify the effects of entrained helium when helium head pressure carbon dioxide (HHPCO2) is used in supercritical fluid extraction (SFE) and supercritical fluid chromatography (SFC). Naphthalene serves as a model analyte, and the unlike interaction energy parameters of the EOS are obtained from the respective second crossvirial coefficients. The Peng-Robinson EOS provides a reasonable description of the vapor-liquid equilibrium in HHPCO2 storage tanks at room temperature. Variations in the content of helium in HHPCO2 within the relevant range (between 0 and 5 mol %) are predicted to produce significant changes in the density and solubility parameter of HHPCO2, in the solubility of naphthalene in HHPCO2, and in the chromatographic retention of naphthalene if HHPCO2 serves as the mobile-phase fluid. These results provide a computational indication that the use of HHPCO2 in SFE and SFC should be avoided. Carbon dioxide is the most common solvent in supercritical fluid extraction (SFE) and supercritical fluid chromatography (SFC). To achieve a smooth operation, the working cylinder of a syringe pump should be filled completely with liquid CO2. When using a storage tank with pure CO2, it is necessary to cool the pump cylinder to secure that the CO2 inside it will really be in the liquid state. Cooling-induced contraction of the piston seal may sometimes result in annoying leaks. Alternatively, storage tanks may be pressurized with helium; the liquid phase from the helium head pressure CO2 (HHPCO2) tanks can be transferred to the pump cylinder without cooling it. The practice of using HHPCO2 in SFE and SFC initially involved a tacit assumption that helium is essentially insoluble in liquid CO2 and that, therefore, the relevant thermodynamic properties of CO2 are not altered appreciably by the presence of helium. Earlier papers1-3 reported contradictory results with regard to the effects of entrained helium on reproducibility of * Fax: +420 5 41212113. E-mail:
[email protected]. (1) Porter, N. L.; Richter, B. E.; Bornhop, D. J.; Later, D. W.; Beyerlein, F. H. J. High Resolut. Chromatogr. Chromatogr. Commun. 1987, 10, 477-478. (2) Schwartz, H. E.; Barthel, P. J.; Moring, S. E. J. High Resolut. Chromatogr. Chromatogr. Commun. 1987, 10, 668-669. (3) Rosselli, A. C.; Boyer, D. S.; Houck, R. K. J. Chromatogr. 1989, 465, 1115.
2104 Analytical Chemistry, Vol. 70, No. 10, May 15, 1998
retention times and peak areas in SFC. More recent work, however, presents mounting indications that the above assumption is questionable at best and that the entrained helium acts as a “negative modifier” in both SFE and SFC. Go¨rner et al.4 observed a considerable increase in SFC retention times when using HHPCO2 as compared with pure CO2 at the same temperature, pressure, and flow rate. They ascribed their observation to a decrease in the density of HHPCO2 relative to that of pure CO2 and used the Lee-Kesler equation of state5 (EOS) to model the density decrease. Raynie and Delaney6 reported a reduced efficiency of SFE with HHPCO2 relative to that of SFE with pure CO2 at the same temperature and pressure. King et al.7 found that the presence of helium in the extraction fluid resulted in a substantial reduction in soybean oil solubility relative to that in pure CO2. Leichter et al.8 used HHPCO2 as a mobile-phase fluid in SFC of polycyclic aromatic hydrocarbons and observed a systematic decrease in retention as the liquid phase was depleted from the storage tank. They also found an analogous effect in SFC separation of steroids with helium-pressurized, methanolmodified CO2. On the basis of their results, Leichter et al. concluded that the use of HHPCO2 should be avoided. Kordikowski et al.9 deduced from acoustic measurements that the composition of the liquid phase in helium-pressurized CO2 tanks did not necessarily correspond to thermodynamic equilibrium, and they used FT-IR spectroscopy to confirm that the presence of helium can have a profound effect on the density of HHPCO2 compared to pure CO2. Zhang and King10 employed a highpressure density meter to measure the densities of CO2 + He mixtures and evaluated the helium-induced reductions in solubilities of soybean oil and cholesterol in HHPCO2, concluding, again, that the use of HHPCO2 should be avoided whenever possible. It follows from the above account that there is a wealth of diverse experimental information regarding the use of HHPCO2 in SFE and SFC but that relatively little effort has been spent on thermodynamic modeling of the helium-containing systems to (4) Go ¨rner, T.; Dellacherie, J.; Perrut, M. J. Chromatogr. 1990, 514, 309-316. (5) Lee, B. I.; Kesler, M. G. AIChE J. 1975, 21, 510-527. (6) Raynie, D. E.; Delaney, T. E. J. Chromatogr. Sci. 1994, 32, 298-300. (7) King, J. W.; Johnson, J. H.; Eller, F. J. Anal. Chem. 1995, 67, 2288-2291. (8) Leichter, E.; Strode, J. T. B.; Taylor, L. T.; Schweighardt, F. K. Anal. Chem. 1996, 68, 894-898. (9) Kordikowski, A.; Robertson, D. G.; Poliakoff, M. Anal. Chem. 1996, 68, 4436-4440. (10) Zhang, Z.; King, J. W. J. Chromatogr. Sci. 1997, 35, 483-488. S0003-2700(97)01168-2 CCC: $15.00
© 1998 American Chemical Society Published on Web 04/14/1998
quantify the effects of helium on extraction efficiency and chromatographic retention. The purpose of this contribution is to develop a thermodynamic treatment of the effects of entrained helium in SFE and SFC that would extend the scope of the previous attempts in this field. The present treatment should be able to describe the vapor-liquid equilibrium in the HHPCO2 storage tanks, to provide the solubility parameter of a CO2 + He mixture of a selected composition, and to quantify directly the effects of entrained helium on solubility and retention of a model analyte in SFE and SFC, respectively. THEORY Equation of State. The mixtures of interest in SFE have often been modeled with the Peng-Robinson EOS,11
P)
RT a v - b v(v + b) + b(v - b)
∑∑y y a
(2)
∑y b
(3)
i j ij
i
j
and
b)
i i
i
where yi is the mole fraction of component i in the mixture, aij is the interaction energy parameter between components i and j, bi is the size parameter of component i, and the summations extend over all components of the mixture. For a pure component i, the parameter aii is given by
aii ) 0.45724
R2Tci2 {1 + κi[1 - (T/Tci)1/2]}2 Pci
(4)
where Tci is the critical temperature of component i, Pci is the critical pressure of component i, and κi may be expressed by a quadratic polynomial in the acentric factor, ωi, of component i,
κi ) 0.37464 + 1.54226ωi - 0.26992ωi2
component CO2 4He naphthalene
Tci (K)
Pci (bar)
ωi
304.136a 10.47c 748.4d
73.773a 6.76c 40.5d
0.225b 0 0.302d
a Reference 13. b Reference 14. c Effective critical constants from ref 15. d Reference 16.
critical properties of component i by
bi ) 0.07780RTci/Pci
(7)
(1)
where P is the pressure, R the molar gas constant, T the absolute temperature, and v the molar volume of the mixture. The interaction energy parameter a and the size parameter b are related to the composition of the mixture through conventional mixing rules:
a)
Table 1. Critical Properties and Acentric Factors in the Model System
(5)
All the thermodynamic properties of interest in the present work may be obtained by substituting eqs 1-7 into standard thermodynamic relationships.12 Model System and Parameter Sources. Among the parameters needed in the Peng-Robinson EOS, the empirical binary parameters kij are the least accessible, and they have to be known between any two components of the CO2(1)-He(2)-analyte(3) system. As thermodynamic data on the interaction between helium and nonvolatile analytes of interest in SFE and SFC are extremely rare, the choice of a model analyte has essentially been limited to naphthalene. The values and sources of the pure-component properties in the model system are compiled in Table 1. As a quantum gas, helium does not conform to the principle of corresponding states. Therefore, the respective entries in Table 1 are not the true critical properties of 4He; instead, they are the effective critical constants quoted by Prausnitz et al.15 The empirical binary parameter kij may be obtained, e.g., from fitting the Peng-Robinson EOS to the experimental values of the second cross-virial coefficient, Bij, for interaction between components i and j; one can easily show that the relationship between kij and Bij may be written as
kij ) 1 -
(
)
bi + bj RT - Bij 1/2 2 (aiiajj)
Applying eq 8 to the CO2(1)-He(2) second cross-virial coefficients compiled by Dymond and Smith17 and smoothing the results by a quadratic polynomial in the absolute temperature, one obtains that, within 303-370 K, k12 is given by
k12 ) (-1.98871 × 10-4)T2 + 0.1394T - 22.722 The unlike interaction energy parameter aij is related to the respective pure-component parameters by
aij ) (1 - kij)(aiiajj)1/2
(6)
where, in general, the empirical binary parameter kij has to be determined from suitable experimental data. The size parameter bi does not depend on temperature, and it is connected to the (11) Peng, D.-Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976, 15, 59-64.
(8)
(9)
(12) Prausnitz, J. M.; Lichtenthaler, R. N.; Gomes de Azevedo, E. Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed.; Prentice Hall: Englewood Cliffs, NJ, 1986; Section 3.4. (13) Duschek, W.; Kleinrahm, R.; Wagner, W. J. Chem. Thermodyn. 1990, 22, 841-864. (14) Tester, J. W.; Modell, M. Thermodynamics and Its Applications, 3rd ed.; Prentice Hall: Upper Saddle River, NJ, 1997; Appendix G, pp 934-936. (15) Reference 12, Tables 5-12, p 167. (16) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill: New York, 1986; Appendix A, pp 656-732. (17) Dymond, J. H.; Smith, E. B. The Virial Coefficients of Pure Gases and Mixtures. A Critical Compilation; Clarendon Press: Oxford, UK, 1980.
Analytical Chemistry, Vol. 70, No. 10, May 15, 1998
2105
In an analogous manner, the He(2)-naphthalene(3) second crossvirial coefficients17 yield the following expression for k23 within 305-347 K,
k23 ) (3.309 × 10-3)T + 0.0450
(10)
The use of the CO2(1)-naphthalene(3) second cross-virial coefficients to get k13 results in parameters that yield a rather poor representation of the solubility of naphthalene in supercritical CO2. Within this work, therefore, a temperature-independent value of k13 ) 0.11 has been employed, as quoted by Tester and Modell18 in connection with describing the phase behavior of the CO2naphthalene system. Calculations. Thermodynamic modeling of the vapor-liquid equilibrium in the HHPCO2 storage tanks starts from the condition that, at a given temperature and pressure, either component of the CO2 + He mixture should have equal fugacities in both phases. The fugacities in either phase are expressed from the PengRobinson EOS, and multiple Newton-Raphson technique is used to solve the system of nonlinear equations for the compositions and densities of both phases. The Peng-Robinson EOS has also been used to obtain the isothermal, isobaric composition derivative of a component’s chemical potential; this property is needed for considerations of thermodynamic stability19 of the CO2 + He mixtures. The solubility parameter of a CO2 + He mixture has been calculated according to its thermodynamic definition, i.e., as a square root of the cohesive energy density obtained from the Peng-Robinson EOS. Solubilities (equilibrium mole fractions) of naphthalene(3) in the CO2(1) + He(2) mixtures have been calculated from
y3 )
P3sat φ3sat exp P φ3
{∫
}
v30 dP P3satRT P
(11)
where P3sat is the vapor pressure of solid naphthalene, P the total pressure, φ3sat the fugacity coefficient of the pure naphthalene vapor, v30 the molar volume of pure solid naphthalene, and φ3 the fugacity coefficient of naphthalene in the CO2 + He mixture. The latter quantity has been computed from the Peng-Robinson EOS, P3sat has been calculated from the Antoine equation constants published by Fowler et al.,20 and v30 has been obtained from the density of solid naphthalene.21 Because of the very low vapor pressure of naphthalene at the temperatures considered (see below), the approximation φ3sat ≈ 1 has been retained throughout the solubility calculations. A semiquantitative assessment of the variation in the naphthalene retention in SFC with the content of helium in HHPCO2 may be obtained from a simplified form of a previous thermodynamic treatment of the modifier effects in SFC,22 (18) Reference 14, p 670. (19) Reference 14, pp 219-223. (20) Fowler, L.; Trump, W. N.; Vogler, C. E. J. Chem. Eng. Data 1968, 13, 209210. (21) Lide, D. R., Ed. CRC Handbook of Chemistry and Physics, 73rd ed.; CRC Press: Boca Raton, FL, 1992; p 3-327. (22) Roth, M. J. Phys. Chem. 1996, 100, 2372-2375.
2106 Analytical Chemistry, Vol. 70, No. 10, May 15, 1998
( ) ( ∂ ln k′ ∂y2m
≈
T,P
)
∂ ln φ3m∞ ∂y2m
T,P
+
( )
1 ∂vm vm ∂y2m
(12)
T,P
where k′ is the capacity factor of naphthalene, y2m the mole fraction of helium in the mobile-phase fluid (HHPCO2), φ3m∞ the infinitedilution fugacity coefficient of naphthalene in the mobile-phase fluid, and vm the molar volume of the mobile-phase fluid. Equation 12 only takes into account the contributions to (∂ ln k′/∂y2m)T,P that arise in the mobile phase; the contributions arising in the stationary phase have been neglected. Both derivatives on the right-hand side of eq 12 may be expressed from the PengRobinson EOS. The computer codes needed to calculate the properties mentioned within this section have been written in Turbo Pascal and run on an IBM-compatible PC. RESULTS AND DISCUSSION Vapor-Liquid Equilibrium in Helium Head Pressure Carbon Dioxide Tanks. Figure 1 shows the calculated composition of the coexisting phases in the CO2 + He system at 19.98 °C, together with the pertinent experimental data.23 The apparently reasonable reproduction of the equilibrium composition data justifies some confidence in the parameters specified above. The overall performance of equations of state deteriorates with increasing density; in the liquid phase, therefore, the relative deviations between the experimental and calculated compositions exceed those in the vapor phase. Figure 1 also shows the calculated limits of thermodynamic stability of the system, i.e., the spinodal curves (2 and 3). The region between curves 1 and 2 and the region between curves 3 and 4 are metastable regions, while the region between curves 2 and 3 is unstable. For application of HHPCO2 in SFE and SFC, the metastable region between curves 3 and 4 is relevant. It suggests that the liquid phase in a HHPCO2 tank may become temporarily supersaturated with helium because of, e.g., a sudden decompression. Therefore, there is a computational indication that the deviations from equilibrium in the composition of the liquid phase in a HHPCO2 tank can occur also in the direction opposite to that observed experimentally by Kordikowski et al.9 Figure 2 presents a comparison of the experimental23 and calculated densities of both phases at 19.98 °C. Although the calculations result in underestimating the experimental liquid-phase densities by nearly 8%, the qualitative trend of increasing liquid-phase density with an increase in pressure is reproduced correctly. The trend is the outcome of two opposing effects. First, the equilibrium mole fraction of helium in the liquid phase increases with increasing pressure; this effect alone tends to decrease the liquid-phase density as the pressure increases. Second, as the temperature is rather close to the critical temperature of CO2 and CO2 is the major component of the liquid phase, the liquid phase of the system is highly compressible. The compressibility effect turns out to prevail over the composition effect in determining the pressure course of the liquid-phase density at the given temperature. Effects of Helium in SFE. To illustrate the accuracy of predictions based on the Peng-Robinson EOS at the operating conditions of SFE, Table 2 presents a comparison of the calculated (23) Burfield, D. W.; Richardson, H. P.; Guereca, R. A. AIChE J. 1970, 16, 97100.
Table 2. Comparison of Experimental10 and Calculated Densities of CO2 + He Mixtures T (°C)
P (bar)
mol % He
Fexp (g cm-3)
Fcalc (g cm-3)
100(Fcalc - Fexp)/Fexp (%)
50
344.7
0 4.9 0 4.9 0 4.9 0 4.9 0 4.8 0 4.8 0 4.8 0 4.8 0 4.6 0 4.6 0 4.6 0 4.6
0.8968 0.8457 0.9564 0.9159 0.9795 0.9420 1.0164 0.9827 0.8609 0.8135 0.9276 0.8882 0.9526 0.9153 0.9926 0.9593 0.8242 0.7752 0.8981 0.8598 0.9254 0.8898 0.9686 0.9372
0.9157 0.8452 0.9975 0.9378 1.0284 0.9722 1.0776 1.0263 0.8716 0.8016 0.9623 0.9031 0.9962 0.9405 1.0496 0.9987 0.8274 0.7605 0.9272 0.8702 0.9640 0.9103 1.0215 0.9725
2.1 -0.06 4.2 2.4 5.0 3.2 6.0 4.4 1.2 -1.5 3.7 1.7 4.6 2.7 5.7 4.1 0.3 -1.9 3.2 1.2 4.2 2.3 5.5 3.8
482.6 551.6 686.0 60
344.7 482.6
Figure 1. Composition of the coexisting phases in the CO2 + He system at 19.98 °C. Points represent experimental equilibrium composition data;23 calculated lines represent (1) equilibrium composition of the vapor phase, (2) vapor-phase stability limit, (3) liquidphase stability limit, and (4) equilibrium composition of the liquid phase.
551.6 686.0 70
344.7 482.6 551.6 686.0
Figure 2. Experimental23 (points) and calculated (lines) equilibrium densities of the coexisting phases in the CO2 + He system at 19.98 °C: (1) vapor phase, (2) liquid phase.
densities of HHPCO2 with the recent experimental results of Zhang and King.10 Considering the very high pressures involved, the decreasing performance of equations of state with increasing density, and the fact that no high-pressure data have been used to optimize the parameters of the EOS employed, the relative deviations of the calculated values from the experimental densities are reasonable. The solvating power of supercritical fluids has often been discussed in terms of the density or, more accurately, the solubility parameter. Figures 3 and 4 present the calculated densities and solubility parameters, respectively, for CO2 + He mixtures within the relevant composition range at 35 °C. Figures 5 and 6 show the corresponding plots at 100 °C. It appears that the strong isothermal correlation between the density and the solubility parameter, known in pure supercritical fluids, applies also to the CO2 + He mixtures. Further, Figure 4 indicates that the effect of helium on the solubility parameter of HHPCO2 can be very significant. At 35 °C and 85 bar, for example, the calculated solubility parameter of HHPCO2 containing 5 mol % of helium amounts to only 52% of the value for pure CO2 at the same conditions. This dramatic drop reflects the relative proximity of the particular conditions to the critical point of CO2, and, therefore, it is not directly relevant to everyday applications of SFE, as these have to be performed at much higher temperatures to secure
Figure 3. Density of CO2 + He mixtures as a function of pressure at 35 °C. Labels indicate the mole percent of helium.
sufficient volatility of the analytes. However, Figure 6 implies that, even within the working range of temperature and pressure of most applications of SFE, the presence of helium in HHPCO2 still has important consequences. To retain sufficient solvating power (solubility parameter) of the fluid, the elevated temperature requires elevated pressures, and it is just at the elevated pressures where the helium-induced reduction in the solubility parameter becomes significant. To provide a direct measure of the effect of helium in SFE, Figures 7 and 8 present the calculated solubilities of solid naphthalene in the CO2 + He mixtures containing up to 5 mol % of helium at 35 and 60 °C, respectively. Figure 7 reflects the profound influence of helium on the solvating power that was already indicated in Figure 4. Calculations based on the PengRobinson EOS suggest that, at 35 °C and 85 bar, a switch from pure CO2 to HHPCO2 containing 5 mol % of helium results in an 11-fold reduction in the solubility of naphthalene. As the temperature increases further above the critical temperature of CO2, the maximum impact of a particular content of helium decreases but still remains important (see Figure 8). Analytical Chemistry, Vol. 70, No. 10, May 15, 1998
2107
Figure 4. Solubility parameter of CO2 + He mixtures as a function of pressure at 35 °C. Labels indicate the mole percent of helium.
Figure 7. Decadic logarithm of the equilibrium mole fraction of naphthalene in CO2 + He mixtures as a function of pressure at 35 °C. Labels indicate the mole percent of helium in the CO2 + He mixture.
Figure 5. Density of CO2 + He mixtures as a function of pressure at 100 °C. Labels indicate the mole percent of helium. Figure 8. Decadic logarithm of the equilibrium mole fraction of naphthalene in CO2 + He mixtures as a function of pressure at 60 °C. Labels indicate the mole percent of helium in the CO2 + He mixture.
Figure 6. Solubility parameter of CO2 + He mixtures as a function of pressure at 100 °C. Labels indicate the mole percent of helium.
Effects of Helium in SFC. The general discussion of a fluid’s solvating power in terms of the solubility parameter (Figures 4 and 6) applies to SFC as well as to SFE. More specifically to SFC, Figures 9 and 10 illustrate the sensitivity of retention of naphthalene to the content of helium in the mobile-phase fluid, as measured by the isothermal, isobaric derivative given by eq 12. This derivative makes it possible to evaluate the shift in naphthalene retention that results from a change in composition of the binary (CO2 + He) mobile-phase fluid. For a small change, ∆y2m, in the mole fraction of helium in the binary mobile-phase fluid, one can assume that the derivative (∂ ln k′/∂y2m)T,P is relatively 2108 Analytical Chemistry, Vol. 70, No. 10, May 15, 1998
Figure 9. Sensitivity of the natural logarithm of the naphthalene capacity factor to the mole fraction of helium in HHPCO2 as a function of pressure at 35 °C. Labels indicate the mole percent of helium in the mobile-phase fluid.
constant (i.e., independent of composition), and the shift in the capacity factor may be estimated from a simple proportionality:
∆ ln k′ ≈ (∂ ln k′/∂y2m)T,P∆y2m
(13)
At 100 °C and 180 bar, for example, (∂ ln k′/∂y2m)T,P for pure CO2
Figure 10. Sensitivity of the natural logarithm of the naphthalene capacity factor to the mole fraction of helium in HHPCO2 as a function of pressure at 100 °C. Labels indicate the mole percent of helium in the mobile-phase fluid.
is about 9.8 (see Figure 10). Therefore, even at these conditions, rather far from the critical point of CO2, replacing pure CO2 with HHPCO2 containing 1 mol % of helium is predicted to result in an increase in the capacity factor of naphthalene by around 10%. At temperatures and pressures closer to the critical point of CO2, the change in the naphthalene capacity factor would be much more significant, as suggested by Figure 9. CONCLUSION The calculations based on the Peng-Robinson EOS indicate that variations in the content of helium in HHPCO2 within the
relevant range,23 i.e., between 0 and 5 mol %, have important effects in both SFE and SFC of naphthalene. As naphthalene typifies the chemical character of an important class of analytes in applications of SFE and SFC, it is likely that, in other aromatic analytes, the effects of entrained helium will be of the same order of magnitude as those calculated here for naphthalene. Further, there are both experimental9 and calculated (this work) indications that the composition of the liquid phase drawn from the HHPCO2 storage tanks need not necessarily be the equilibrium composition. Consequently, it appears that, in both SFE and SFC, the complications induced by the use of HHPCO2 can actually outweigh its benefits. Therefore, the results of this study provide computational support for the conclusion of Leichter et al.8 and Zhang and King10 that the use of HHPCO2 should be avoided, at least in the cases when reproducibility of solubilities in SFE and/or reproducibility of retention times in SFC is of primary concern. ACKNOWLEDGMENT This work was supported by the Grant Agency of the Academy of Sciences of the Czech Republic under Project No. A4031503 and by the Grant Agency of the Czech Republic under Project No. 203/98/0635.
Received for review October 22, 1997. Accepted February 19, 1998. AC971168E
Analytical Chemistry, Vol. 70, No. 10, May 15, 1998
2109
