2372
J. Phys. Chem. 1996, 100, 2372-2375
Thermodynamics of Modifier Effects in Supercritical Fluid Chromatography Michal Roth Institute of Analytical Chemistry, Academy of Sciences of the Czech Republic, CZ-61142 Brno, Czech Republic ReceiVed: August 21, 1995; In Final Form: October 27, 1995X
Thermodynamic properties underlying the shift in solute retention that results from a change in composition of a binary mobile-phase fluid in supercritical fluid chromatography (SFC) are identified and discussed. It is shown that the composition change in the mobile-phase fluid affects the retention properties of the stationary phase even when interfacial adsorption is neglected, and bulk partitioning is supposed to be the only mechanism of solute retention. In the resultant relationship, the effects of modifier (cosolvent) on density are explicitly separated from the effects on intermolecular interactions. The relative contributions of the separate terms to the observed shift in retention are partly illustrated with the literature experimental data available at the moment. It appears that, in general, several corrections should be applied when using SFC to probe cosolvent effect in a solute-supercritical fluid mixture but that the corrections become less important as the cosolvent effect increases.
Introduction Retention of a solute in supercritical fluid chromatography (SFC) can be varied within wide limits by adding a suitable second component (variously termed as modifier, entrainer, or cosolvent) to a neat mobile-phase fluid.1-4 During the development of SFC, considerable effort has therefore been spent by seeking efficient instrumental designs to provide a stable source of a binary mobile phase of well-defined composition5-9 as well as by investigation of the sorption of binary mobile-phase fluids into the stationary phase.10-13 In contrast, it has been noted14 that relatively little attention has been paid to the theoretical background of the effect of modifier on solute retention10,15 although several qualitative explanations have been suggested.4,12 The thermodynamics of the modifier effect is mainly important for physicochemical measurements by SFC,16,17 in particular, for chromatographic studies of the cosolvent effect in solute-supercritical fluid mixtures.18 This contribution presents a thermodynamic analysis of the effect of a composition change in a binary mobile-phase fluid on retention of a solute. This paper therefore complements previous efforts describing the effects of temperature and pressure on retention in SFC with neat mobile-phase fluids.19-26 The treatment will focus on the chromatographic systems in which the stationary phase may be treated as a bulk phase, i.e., on the systems of open-tubular-column SFC wherein the principal component of the stationary phase is a high polymer. Therefore, the analysis below applies also to the effect of cosolvent on the distribution of a nonvolatile solute between a supercritical fluid and an amorphous polymer swollen with the fluid. Theory Consider a two-phase, four-component system composed of a solute (1), a principal component of the “stationary” phase (2), a principal component of the “mobile” phase (3), and a modifier (4), all components being nonelectrolytes. Suppose that component 2 of the system is nonvolatile so that it is confined to one of the two phases (the “stationary” phase); at this stage, no assumption regarding the molar mass of component 2 will yet be made. Suppose further that the amount of X
Abstract published in AdVance ACS Abstracts, January 1, 1996.
0022-3654/96/20100-2372$12.00/0
component 1 in the system is so small that the solute is at the state of infinite dilution in both phases, and suppose also that the temperature T and pressure P in the system are sufficiently removed from the vapor-liquid critical points of components 3 and 4 so that the distribution of the two components between the two phases is not disturbed by the presence of a trace amount of the solute. It will also be assumed that interfacial adsorption is absent from the system. The equilibrium distribution of the solute between the stationary (s) and the mobile (m) phase may then be characterized by the ratio of mole numbers of the solute in the two phases,
k1 ) (x1s/x1m)(ns/nm)
(1)
where the x1’s are the mole fractions of the solute in the two phases and the n’s are the total amounts of substance (mole numbers) in the two phases. Under conditions of ideal, linear chromatography,27 k1 defined by eq 1 is identical to the retention factor of the solute in a hypothetical, isothermal, isobaric column containing the same amount of component 2 as the static system mentioned above,
k1 ) (tR - t0)/t0
(2)
where tR is the retention time of the solute and t0 the retention time of an unretained marker. At equilibrium, fugacities of the solute are equal in the two phases, and substitution from the equilibrium condition into eq 1 yields ∞ ∞ - ln φ1s + ln ns - ln nm ln k1 ) ln φ1m
(3)
where the φ∞1 ’s are the infinite-dilution fugacity coefficients of the solute in the two phases. At a constant temperature and pressure, the derivative of ln k1 with respect to the mole fraction of modifier in the mobile phase, x4m, is therefore given by
( ) ∂ ln k1 ∂x4m
(
) ( ) ( ) ( )
∞ ∂ ln φ1m ) ∂x4m T,P,n2s,σ
∞ ∂ ln φ1s ∂x4m T,P
∂ ln ns ∂x4m
+
T,P,n2s,σ
-
T,P,n2s,σ
∂ ln nm ∂x4m
(4) T,P
where the subscript n2s indicates that the amount of component © 1996 American Chemical Society
Modifier Effects in Supercritical Fluid Chromatography
J. Phys. Chem., Vol. 100, No. 6, 1996 2373
2 in the stationary phase is constant (see above), and the subscript σ denotes that the partial derivatives are those along the coexistence curve between the two phases. Collecting the terms related to the stationary phase and rearranging, one obtains
( ) ( ) ( ) [( ) ( ) ]( ) ∂ ln k1 ∂x4m
∞ φ1m
∂ ln ) ∂x4m ,σ
∂ ln nm ∂x4m T,P
T,P,n2s
∂ ln ns ∂x4s
∞ ∂ ln φ1s ∂x4s T,P,n2s,σ
( )
1 ∂ ln nm Vm ∂ ln Vm
T,P,n2s,σ
∂x4s ∂x4m
(5) T,P,n2s,σ
dVm +
( ) ( ) T,P,Vm
∂ ln νm ∂x4m
[
( ) ∂x3s ∂x4s
T,P,n2s,σ
∂x4s ∂x4m
T,P,n2s,σ
( ) ( ) ∂µ4m ∂x4m
/
T,P
∂µ4s ∂x4s
(12) T,P,n2s,σ
)T,P,n2s,σ
( ) ( )
x4m ∂µ4s x3m ∂x4s
/
T,P,n2s,σ
∂µ3s ∂x3s
(13) T,P,n2s,σ
where x3m and x4m are the mole fractions of components 3 and 4 in the binary mobile-phase fluid (x3m + x4m ) 1). Equation 13 follows from the equilibrium conditions combined with the Gibbs-Duhem equation for the mobile phase. At infinite dilution of the solute (1), the stationary phase is composed of components 2, 3, and 4, and the derivative (∂ ln ns/∂x4s)T,P,n2s,σ in eq 5 is therefore given by
( ) ∂ ln ns ∂x4s
T,P
( ) ]( )
Vs ∂x3s ) ζ4m + ζ4s + ζ3s Vm ∂x4s T,P
)
where µ4m and µ4s are the chemical potentials of modifier in the mobile and the stationary phase, respectively. The derivatives on the right-hand side (rhs) of eq 12 can be calculated from a suitable EOS or a solution model. In a similar manner, modeling of the derivative (∂x3s/∂x4s)T,P,n2s,σ in eq 7 may be based on the relationship
dx4m (6)
where νm is the molar volume of the binary mobile phase. Obviously, (∂ ln nm/∂ ln Vm)T,P,x4m ) 1, and (∂ ln nm/∂ ln νm)T,P,Vm ) -1. Further, the total volume of the system, Vs + Vm, is constant, with Vs being the geometric volume of the stationary phase. Equation 6 then finally becomes
( )
∂x4s ∂x4m
T,P,x4m
∂ ln nm ∂ ln νm
∂ ln nm ∂x4m
( )
+
T,P
where x4s is the mole fraction of modifier in the stationary phase. The quotient (∂ ln φ∞1m/∂x4m)T,P is not related to any fundamental thermodynamic property;28 therefore, it cannot be handled without resorting to a particular solution model or an equation of state (EOS). The possibility to use eq 5 for estimating this quotient from the chromatographic data will be discussed in the next section. The derivative (∂ ln nm/∂x4m)T,P may be expressed as follows: At constant T and P, ln nm is a function of the geometric volume of the mobile phase in the system, Vm, and of the modifier mole fraction, x4m (because x2m ) 0 and, at infinite dilution of the solute, x1m f 0). The total differential of ln nm may therefore be written as
d ln nm )
Because of equilibrium conditions, the quotient (∂x4s/ ∂x4m)T,P,n2s,σ in eqs 5 and 7 may be obtained from
) T,P,n2s,σ
1 1 - x3s - x4s
[ ( ) ] 1+
∂x3s ∂x4s
(14)
T,P,n2s,σ
∞ /∂x4s)T,P,n2s,σ in In order to derive an expression for (∂ ln φ1s eq 5, it is expedient to start from the identity
(
)
∂ ln φ1s ∂x4s
) T,P,n1s,n2s,σ
( )
1 ∂µ1s RT ∂x4s
T,P,n1s,n2s,σ
( )
1 ∂x1s x1s ∂x4s
T,P,n2s,σ
(15) T,P,n1s,n2s,σ
(7) In a four-component system with dn1s ) 0 and dn2s ) 0, one obtains for the second term on the rhs of eq 15
where the quotient
ζ4m ) -
( )
1 ∂νm νm ∂x4m
(8) T,P
is sometimes termed “mixing expansivity” and the quantities ζ3s and ζ4s are respectively given by
ζ3s ) -
( )
T,P,n2s,x4s
( )
T,P,n2s,x3s
1 ∂Vs Vs ∂x3s
(9)
and
ζ4s ) -
1 ∂Vs Vs ∂x4s
(10)
where x3s is the mole fraction of component 3 in the stationary phase. In a binary mobile-phase fluid (3 + 4), ζ4m is related to isothermal compressibility, βmT, by
ζ4m ) -βmT
( ) ∂P ∂x4m
(11) T,νm
( )
1 ∂x1s x1s ∂x4s
)T,P,n1s,n2s,σ
[ ( )
∂x3s 1 1+ 1 - x3s - x4s ∂x4s
T,P,n1s,n2s,σ
] (16)
In the limit of infinite dilution of the solute, the above expression becomes identical to the rhs of eq 14, except for the minus sign. The composition derivative of the infinite-dilution chemical potential of the solute along the coexistence curve between the two phases may be written as
( ) ∞ ∂µ1s ∂x4s
( )
∞ ∂µ1s ) ∂x4s T,P,n2s,σ
( ) ( )
∞ ∂µ1s + ∂x3s T,P,n2s,x3s
T,P,n2s,x4s
∂x3s ∂x4s
T,P,n2s,σ
(17) Substitution from eqs 7 and 14-17 into eq 5 yields the following relationship for the change in ln k1 with the mole fraction of
2374 J. Phys. Chem., Vol. 100, No. 6, 1996
Roth
modifier in the mobile phase,
( ) ∂ ln k1 ∂x4m
)
( ) ∞ ∂ ln φ1m ∂x4m
Vs ∂x3s ζ + ζ3s Vm 4s ∂x4s
[( )
∞ 1 ∂µ1s RT ∂x4s
- ζ4m -
[ ( ) ]( )
T,P,n2s,σ
+
T,P,n2s,x3s
T,P
T,P,n2s,σ
∂x4s ∂x4m
T,P,n2s,x4s
∂x3s ∂x4s
-
( ) ( ) ]( ) ∞ ∂µ1s ∂x3s
T,P,n2s,σ
T,P,n2s,σ
∂x4s ∂x4m
T,P,n2s,σ
(18) In open-tubular-column SFC, component 2 typically is a cross-linked elastomer with a very high molar mass. In such a case, eq 18 may be formally simplified if the composition of the stationary phase is expressed in mole fractions. In a threecomponent (2 + 3 + 4) system,
( ) ∂x3s ∂x4s
) n2s
(x2s + x4s) dn3s - x3s dn4s
(19)
(x2s + x3s) dn4s - x4s dn3s
where x2s is the mole fraction of component 2 and n3s and n4s are the mole numbers of components 3 and 4, respectively. Since x2s f 0 in the limit of M2 f ∞, it follows from eq 19 that (∂x3s/∂x4s)n2s f -1. Consequently, the subscript σ becomes obsolete, and eq 18 may be rewritten as
( ) ( ∂ ln k1 ∂x4m
)
∞ ∂ ln φ1m ) ∂x4m T,P,n2s
( ) ( ) ( )
- ζ4m T,P
∞ 1 ∂µ1s RT ∂x4s
with ζ4s now defined by
ζ4s ) -
Vs ∂x4s ζ4s Vm ∂x4m
T,P,n2s
∂x4s ∂x4m
T,P,n2s,σ
(20)
T,P,n2s,σ
( )
1 ∂Vs Vs ∂x4s
(21)
T,P,n2s
Over a narrow range of x4m, a derivative with respect to x4m in eq 20 may be replaced by the ratio of the respective differences. Therefore, in a hypothetical, isothermal, isobaric column with a high polymer as a principal component of the stationary phase, the shift in solute retention, ∆ ln k1, that results from a (small) change in composition of the binary mobile-phase fluid, ∆x4m, may be approximated by ∞ - ζ4m∆x4m ∆ ln k1 ≈ ∆ ln φ1m
Vs ζ ∆x Vm 4s 4s
( )
∞ 1 ∂µ1s RT ∂x4s
experiments, components 2, 3, and 4 were, respectively, a poly(methyl-n-octylsiloxane) elastomer, carbon dioxide, and methanol. The cosolvent effects were evaluated for anthracene and 2-naphthol as solutes in a carbon dioxide-methanol mixture (x4m ) 0.035) relative to pure CO2 at 35 °C and pressures within 90-350 bar. The cosolvent effect was expressed as the ratio of the infinite-dilution fugacity coefficient of the solute in pure CO2 to that in the CO2-methanol mixture. Ekart et al. obtained the cosolvent effects from a truncated form of eq 22, retaining only the first term on the rhs, and found a good agreement of their results with those from solubility measurements. In light of these findings, it is natural to inquire about the remaining terms in eq 22. Within the three terms, ζ4m is the quantity which is relatively easiest to estimate; at present, the use of a lattice-fluid model to evaluate Vs/Vm, ζ4s, ∞ /∂x4s)T,P,n2s, and ∆x4s is precluded by the lack of experi(∂µ1s mental data to obtain pure-component parameters for poly(methyl-n-octylsiloxane) and the energy parameters for interaction of the polymer with the other components. In the particular experimental conditions, βmT in eq 11 is the isothermal compressibility of pure CO2, and the quotient (∂P/∂x4m)T,νm may be replaced by its value in the limit x4m f 0 at the temperature and pressure of the critical point of CO2. This limiting value has been termed the Krichevskii parameter and shown to be of central importance in the thermodynamics of dilute solutions near the solvent’s critical point.29 In the present notation, the Krichevskii parameter for methanol in CO2 is related to the limiting slopes of the critical line (subscript CRL) by29-31
∆x4s (22) T,P,n2s
where ∆x4s is the shift in modifier mole fraction in the stationary phase induced by the change in composition of the mobilephase fluid. The first term on the rhs of eq 22 accounts for the modifier effect on interactions of the solute with the mobilephase fluid, and the second term accounts for the modifier effect on the density of the mobile-phase fluid. The third and fourth terms reflect the effects of modifier on density and intermolecular interactions in the stationary phase, respectively. Application of the Final Equation Ekart et al.18 employed open-tubular-column SFC to measure cosolvent effects in dilute supercritical mixtures. In their
lim x4mf0
( ) ∂P
∂x4m
c3
)
[
T,νm
lim x4mf0
() ∂P ∂T
CRL
( )] ( ) dPsat 3 dT
c3
lim
x4mf0
∂T
∂x4m
(23) CRL
For the limiting slope (∂P/∂T)CRL at x4m f 0, a value of 0.607 bar K-1 has been obtained by fitting the initial portion of the critical line32 with a cubic polynomial. The limiting slope of the vapor pressure curve of CO2 at the critical point, c3 (dPsat 3 /dT) , has been estimated from the equation of Duschek et al.33 to be 1.712 bar K-1; this value compares favorably with an estimate for the limiting slope of the critical isochore, 1.722 bar K-1, from the Jacobsen-Stewart EOS for CO2 with the parameters by Ely et al.34 An estimate of 90 K for the limiting slope (∂T/∂x4m)CRL at x4m f 0 has resulted from a cubic polynomial fit of the experimental data.35 These values combine to a Krichevskii parameter of -100 bar. Ekart et al.18 found that ∆ ln k1 ≈ -0.34 for anthracene at 35 °C and 90 bar, and ∆ ln k1 ≈ -1.7 for 2-naphthol at 35 °C and 100 bar. Employing the above estimate of the Krichevskii parameter and the isothermal compressibility of CO2 calculated from the EOS of Ely et al.,34 one finds that ζ4m equals 1.12 at 35 °C and 90 bar, and 0.53 at 35 °C and 100 bar. Substitution of the Krichevskii parameter for (∂P/∂x4m)T,νm in eq 11 at these conditions, rather far from the critical point of CO2, is partly justified by the fact that this quotient has no strong near-critical anomalies.29,31 Therefore, the term -ζ4m∆x4m in eq 22 contributes by ∼12% to ∆ ln k1 of anthracene at 35 °C and 90 bar, and by ∼1% to ∆ ln k1 of 2-naphthol at 35 °C and 100 bar. As one can expect, the relative contribution of ζ4m∆x4m to |∆ ln k1| decreases with increasing strength and/or specificity of the solute-cosolvent interaction and with decreasing compressibility of the mobile-phase fluid. The above findings indicate that, in general, the correction terms in eq 22 are not
Modifier Effects in Supercritical Fluid Chromatography negligible, at least in the systems with a weak interaction between solute and cosolvent. The good agreement between chromatographic and static results on the cosolvent effect suggests that, in the systems studied by Ekart et al.,18 ∆ ln φ∞1m makes the most important contribution to ∆ ln k1 while there may be a partial cancellation among the other three terms on the rhs of eq 22. In addition to the ∆ ln k1 values for other systems, a detailed assessment of the significance of the correction terms would require experimental mass-fraction data on sorption from binary, near-critical fluids into siloxane polymers, and the use of lattice models to estimate the ∞ /∂x4s)T,P,n2s as well as the complicated quotients ζ4s and (∂µ1s quantities Vs/Vm and ∆x4s. Conclusion Thermodynamic analysis has been presented of the shift in solute retention that results from a change in composition of a binary mobile-phase fluid in supercritical fluid chromatography. The purpose of this investigation is to strengthen the theoretical basis for the use of chromatography to measure cosolvent effects in dilute supercritical mixtures. Bulk partitioning of the solute between the stationary and the mobile phase has been assumed to be the only mechanism of solute retention. The resultant relationship explicitly separates the effects of cosolvent (modifier) on density and on intermolecular interactions in both phases of the chromatographic system. Comparison of the relationship with the limited, pertinent experimental data18,32,35 shows that, in the particular model systems and experimental conditions, the effect of modifier on intermolecular interactions in the mobile phase makes the major contribution to the observed shift in solute retention and that the relative importance of this effect increases with increasing strength and/or specificity of the solute-modifier interaction. It also appears that, in the model systems, there is a partial cancellation among the effects of modifier on density of the mobile phase, on density of the stationary phase, and on intermolecular interactions in the stationary phase. Further experimental and theoretical work will be needed to validate these conclusions for a broader class of systems and experimental conditions. Acknowledgment. The author gratefully acknowledges the financial support of this work by the Grant Agency of the Academy of Sciences of the Czech Republic under Project No. A4031503. References and Notes (1) Wright, B. W.; Kalinoski, H. T.; Smith, R. D. Anal. Chem. 1985, 57, 2823.
J. Phys. Chem., Vol. 100, No. 6, 1996 2375 (2) Yonker, C. R.; Smith, R. D. J. Chromatogr. 1986, 361, 25. (3) Fields, S. M.; Markides, K. E.; Lee, M. L. HRC & CC, J. High Resolut. Chromatogr. Chromatogr. Commun. 1988, 11, 25. (4) Janssen, J. G. M.; Schoenmakers, P. J.; Cramers, C. A. J. High Resolut. Chromatogr. 1989, 12, 645. (5) Hirata, Y.; Kawaguchi, Y.; Funada, Y.; Katoh, S. J. High Resolut. Chromatogr. 1993, 16, 601. (6) Pyo, D.; Ju, D. Anal. Lett. 1993, 26, 2009. (7) Schweighardt, F. K.; Mathias, P. M. J. Chromatogr. Sci. 1993, 31, 207. (8) Francis, E. S.; Lee, M. L.; Richter, B. E. J. Microcolumn Sep. 1994, 6, 449. (9) Pyo, D. J.; Ju, D. W. Anal. Sci. 1994, 10, 171. (10) Lochmu¨ller, C. H.; Mink, L. P. J. Chromatogr. 1989, 471, 357. (11) Yonker, C. R.; Smith, R. D. Anal. Chem. 1989, 61, 1348. (12) Strubinger, J. R.; Song, H.; Parcher, J. F. Anal. Chem. 1991, 63, 104. (13) Yonker, C. R.; Smith, R. D. J. Chromatogr. 1991, 550, 775. (14) Janssen, H.-G.; Cramers, C. A. J. Chromatogr. 1990, 505, 19. (15) Martire, D. E.; Boehm, R. E. J. Phys. Chem. 1987, 91, 2433. (16) Conder, J. R. In Theoretical AdVancement in Chromatography and Related Separation Techniques; NATO ASI Series C, Vol. 383; Dondi, F., Guiochon, G., Eds.; Kluwer: Dordrecht, 1992; pp 315-337. (17) Roth, M. J. Microcolumn Sep. 1991, 3, 173. (18) Ekart, M. P.; Bennett, K. L.; Eckert, C. A. In Supercritical Fluid Engineering SciencesFundamentals and Applications; ACS Symposium Series, Vol. 514; Kiran, E., Brennecke, J. F., Eds.; American Chemical Society: Washington, DC, 1993; Chapter 18. (19) van Wasen, U.; Schneider, G. M. Chromatographia 1975, 8, 274. (20) van Wasen, U.; Swaid, I.; Schneider, G. M. Angew. Chem., Int. Ed. Engl. 1980, 19, 575. (21) Brown, B. O.; Kishbaugh, A. J.; Paulaitis, M. E. Fluid Phase Equilib. 1987, 36, 247. (22) Kelley, F. D.; Chimowitz, E. H. AIChE J. 1990, 36, 1163. (23) Roth, M. J. Phys. Chem. 1990, 94, 4309. (24) Shim, J.-J.; Johnston, K. P. AIChE J. 1991, 37, 607. (25) Shim, J.-J.; Johnston, K. P. J. Phys. Chem. 1991, 95, 353. (26) Roth, M. J. Phys. Chem. 1992, 96, 8548. (27) Giddings, J. C. Dynamics of Chromatography, Part 1; Marcel Dekker: New York, 1965; Vol. 1, p 8. (28) Prausnitz, J. M.; Lichtenthaler, R. N.; Gomes de Azevedo, E. Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed.; PrenticeHall: Englewood Cliffs, NJ, 1986; Chapter 10. (29) Levelt Sengers, J. M. H. J. Supercrit. Fluids 1991, 4, 215. (30) Krichevskii, I. R. Russ. J. Phys. Chem. 1967, 41, 1332. (31) Levelt Sengers, J. M. H. In Supercritical FluidssFundamentals for Application; NATO ASI Series E, Vol. 273; Kiran, E., Levelt Sengers, J. M. H., Eds.; Kluwer: Dordrecht, 1994; pp 3-38. (32) Brunner, E. J. Chem. Thermodyn. 1985, 17, 671. (33) Duschek, W.; Kleinrahm, R.; Wagner, W. J. Chem. Thermodyn. 1990, 22, 841. (34) Ely, J. F.; Haynes, W. M.; Bain, B. C. J. Chem. Thermodyn. 1989, 21, 879. (35) Brunner, E.; Hu¨ltenschmidt, W.; Schlichtha¨rle, G. J. Chem. Thermodyn. 1987, 19, 273.
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