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J . Phys. Chem. 1991, 95, 8-9
Relationship between Solute Retention in Supercritical Fluid Chromatography and Fluctuation Integrals In Dilute Supercritical Mixtures Michal Roth Institute of Analytical Chemistry, Czechoslovak Academy of Sciences, 61 1 42 Bmo, Czechoslovakia (Received: October 19, 1990)
Links are revealed between certain molecular quantities in dilute, binary supercritical mixtures and the solute retention in supercritical fluid chromatography (SFC). Use is made of the fact that the isothermal compressibility of the mobile-phase fluid and the infinite-dilution partial molar volume of the solute may be expressed in terms of correlation and fluctuation integrals. As a result, the mobile-phase portion of the pressure (or density) derivative of solute retention in SFC at a constant temperature is shown to be related closely to the direct correlation function integral and to the mean cluster size in a dilute mixture of the solute with the mobile-phase fluid. The possibility to determine these quantities by using open tubular capillary SFC is suggested.
Introduction Supercritical fluids have acquired considerable practical importance as solvents for various industrial extraction processes and as mobile phases for supercritical fluid chromatography (SFC). Most applications involve dilute solutions of nonvolatile solids in near- or supercritical fluids. The thermodynamics of such mixtures has been treated by a number of diverse approaches including, e.g., cubic equations of state with various mixing rules,’” mean field lattice-gas models,’-l’ decorated lattice-gas model^,^*-^^ and models based on the Kirkwood-Buff statistical mechanical theory of solution^.'^ A thermodynamic property of crucial importance here is the partial molar volume of the solute because this quantity controls the pressure quotient of the solute fugacity. O’ConnellI6 showed that the partial molar volume, Of),, of the solute 1 at infinite dilution in the solvent 3 may be expressed by O h
= RT6mdI - N A P m l C y l ( r ) dr]
(1)
where R is the molar gas constant, T is the thermodynamic temperature, NA is the Avogadro number, and cyl is the infinite-dilution limit of the direct correlation function. The direct correlation function” is a short-ranged quantity. Its range is comparable to that of the intermolecular pair potential. The symbols PmT and p m denote the isothermal compressibility and molar density of the pure solvent 3, respectively. Below, the solvent will be identified with the chromatographic mobile-phase fluid. The integral in eq 1 has a physical dimension of volume. Equation 1 is analogous to the compressibility equation for pure fluids.I8 ( I ) Kurnik, R. T.; Holla, S. T.; Reid, R. C. J . Chem. Eng. Data 1981,26, 47. (2) Won, K. W. Fluid Phase Equilib. 1983, 10, 191. (3) Deiters, U. K . Fluid Phase Equilib. 1985, 20, 275. (4) Schmitt, W . J.; Reid, R. C. J . Chem. Eng. Data 1986, 31, 204. (5) Benmekki, E. H.; Kwak, T. Y . ; Manswri. G. A. In Supercritical Fluids, Chemical and Engineering Principles and Applications; ACS Symposium Series, Vol. 329; Squires, T. G.,Paulaitis, M. E., Eds.; American Chemical Society: Washington, DC, 1987; pp 101-1 14. (6) Park, S. J.; Kwak, T. Y . ;Mansoori, G. A . Inf. J . Thermophys. 1987, 8, 449. (7) Vezzetti, D. J. J . Chem. Phys. 1982, 77, 1512. (8) Koningsveld, R.; Diepen, G.A. M. Fluidfhase Equilib. 1983, 10, 159. (9) Kleintjens, L. A. Fluid Phase Equilib. 1983, 10, 183. (IO) Vezzetti, D. J. J . Chem. Phys. 1984, 80, 868. ( I I ) Kleintjens, L. A.; van der Haegen, R.; van Opstal, L.; Koningsveld, R. J . Supercrit. Fluids 1988, I , 23. ( I 2) Wheeler, J . C. Ber. Bunsen-Ges. Phys. Chem. 1972, 76, 308. ( 1 3) Gilbert, S. W.; Eckert, C. A. Fluid Phase Equilib. 1986, 30, 41. (14) Nielson, G.C.; Levelt Sengers, J. M. H. J . Phys. Chem. 1987, 91, 4n7x .- _ .
( 1 5 ) Kirkwood, J . G.; Buff, F. P. J . Chem. Phys. 1951, 19, 774. (16) O’Connell, J. P. Mol. Phys. 1971, 20, 27. ( 1 7) Ornstein, L. S.;Zernike, F. Proc. Sect. Sci. K.Ned. Akad. Wet. 1914, 17, 793.
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Gubbins and O’Connelllg deduced that, for dense polyatomic fluids, the direct correlation function in eq 1 may be replaced by its isotropic part. Debenedetti20 recast the original expres~ions’~ for the solute and solvent partial molar volumes at infinite dilution of the solute in terms of concentration fluctuations. It follows from his argument that the normalized covariance between the solute and solvent concentration fluctuations equals the excess number of solvent molecules surrounding a solute molecule, with respect to a uniform distribution in an ideal gas at the prevailing density. Debenedetti denoted this normalized concentration covariance as a mean cluster size, {; in the present notation, the result may be written as Unlike the direct correlation function integral in eq 1, { is a long-ranged quantity because it is related to the infinite-dilution limit of the unlike pair correlation function.21 The kind of mixtures encountered in applications of supercritical solvents all belong to that class20.22of dilute, binary mixtures in which the infinite-dilution partial molar volume of the solute diverges to negative infinity as the solvent’s critical point is approached. Since, concomitantly, Pmrdiverges to positive infinity, it follows from eq 2 that rather high positive values of { may be expected in this kind of mixtures at the temperatures and pressures approaching the solvent’s critical point. By using experimental values23of ijTm and isothermal compressibilities calculated from an accurate equation of state, DebenedettiZ0 found that { may reach values of ca. 100. Thus, clustering seems to provide a plausible molecular interpretation of the large, negative partial molar volumes observed e ~ p e r i m e n t a l l y . ~ ~ * ~ ~ The purpose of the present contribution is to show the relationship between the molecular quantities of eqs 1 and 2 and the solute retention in SFC. This relationship suggests the possibility of using SFC as a method to determine these molecular quantities. Theory The pressure derivative of the logarithm of the solute capacity ratio at a constant temperature may be decomposed into mobile-phase and stationary-phase portions, (a In k l / d P ) , = K , + K, (3) (18) Rowlinson, J. S.; Swinton, F. L. Liquids and Liquid Mixtures, 3rd ed.; Butterworths: London, 1982; pp 238-239. (19) Gubbins, K. E.; O’Connell, J. P. J . Chem. Phys. 1974, 60, 3449. (20) Debenedetti, P. G. Chem. Eng. Sci. 1987, 42, 2203. (21) Debenedetti, P. G.; Kumar, S. K. AIChE J . 1988, 34, 645. (22) Debenedetti, P. G.;Mohamed, R. S. J . Chem. Phys. 1989.90,4528. (23) Eckert, C. A.; Ziger, D. H.; Johnston, K. P.; Kim, S. J . Phys. Chem. 1986, 90, 2738. (24) Eckert, C. A.; Ziger, D. H.; Johnston, K. P.; Ellison, T. K. Fluid Phase Equilib. 1983, 14, 167.
0 199 1 American Chemical Society
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J. Phys. Chem. 1991, 95, 9-10
In principle, therefore, eqs 3-8 provide a way to estimate the molecular characteristics of the dilute mixture of the solute 1 with thc mobile-phase fluid 3 from the pressure (or density) course of the solute retention in SFC. The respective procedure may be broken down to three basic steps, namely (a) calculation of (a In k l / a P ) , or (a In k , / d p , ) , from experimental retention data, In eqs 3-5, P is the pressure, x3sis the equilibrium mole fraction (b) application of appropriate expressions for K, or Q, to obtain of the mobile-phase fluid in the stationary phase, I& is the infiK , or Q,, respectively, and (c) use of eqs 7 or 8 to calculate the nite-dilution fugacity coefficient of the solute in the stationary direct correlation function integral or mean cluster size, respecphase, 07, is the infinite-dilution partial molar volume of the solute tively. As the column pressure drop has to be minimized in order in the stationary phase, PsTis the isothermal compressibility of to obtain meaningful values of (a In k , / a P ) , from the experimental the stationary phase, and V, and V, are the volumes of the staretention data, an open tubular capillary column is to be preferred tionary and the mobile phases in the column, respectively. The to a packed column. A major problem in applying the above ~ derivation and discussion of eqs 3-5 may be found e l s e ~ h e r e . ~ ~ , ~procedure lies in the term (a In &/dx3,) in eq 5 . This term is It should only be emphasized here that k , in eq 3 is a local value not accessible experimentally so that a molecular or an equaat some particular position within the column rather than the tion-of-state model is required for a theoretical evaluation. experimentally observed, average value over the pressure drop across the column. Conclusion In a similar way, the derivative with respect to the molar density Relationships have been shown between the correlation and of the mobile phase may be written as fluctuation integrals in dilute, binary, supercritical mixtures and the solute retention in SFC. Ensuing from these relationships is (a In kl/dPm)T = Qm + Qs = Km/(PmPmT) + Ks/(PmPmT) (6) a possibility of using SFC as a potential technique to determine The quantities r ? t and PmTin eq 4 combine in such a way that the correlation and fluctuation integrals. However, the interesting straightforward relationships result between K,,, (or Q,) and the information hidden in the pressure (or density) course of solute molecular quantities of eqs 1 and 2. After simple rearrangements, retention cannot be exploited until significant progress is made one obtains in the understanding of the effect on solute retention of the absorption of the mobile-phase fluid into the stationary phase. To this end, more experimental data on equilibrium absorption of supercritical fluids by chromatographic stationary phases are needed, and improved models for the effect of composition on the solute fugacity coefficient in the stationary phase will have to be (25) Roth, M. J . Phys. Chem. 1990, 94,4309. (26) Roth, M. J . Supercrit. Nuids, in press. developed.
13C NMR Study of the C60 Cluster in the Solid State: Molecular Motion and Carbon Chemical Shift Anisotropy C. S. Yannoni,* R. D. Johnson, G. Meijer, D. S. Bethune, and J. R. Salem IBM Research Division, AImaden Research Center, San Jose, California 951 20-6099 (Received: November 12, 1990)
I3C N MR spectra of solid buckminsterfullerene, the soccerball-like cluster of 60 carbon atoms, have been obtained at temperatures down to 77 K. The ambient spectrum shows rapid isotropic rotational motion. The motion is sufficiently slow at 77 K that a measurement of the chemical shift tensor of the carbon nucleus can be made. The tensor components (220, 186,40 ppm) have values that are typical for an aromatic carbon. Spectra at intermediate temperatures suggest the possibility of either growth of a low-temperature phase in which C60 rotation is inhibited or a distribution of rotational correlation times.
The recent success in generating macroscopic quantities of the C60 cluster'-3 has stimulated intense interest and a ~ t i v i t y .A ~ variety of spectroscopic studies have been made to explore the Infrared1?*and vibrational properties of the C60 moiety.'~~*~-' ( I ) Kratschmer, W.; Fostiroplous, K.; Huffman, D. R. Chem. Phys. Lett. 1990. 170. 167-170. (2) Kratschmer, W.; Lamb, L. D.; Fostiropoulos, K.; Huffman, D. R. Norure 1990, 347, 354-358. (3) Meijer, G.; Bethune, D. S. J . Chem. Phys. 1990, 93, 7800-7802. (4) For an overview of some of the current work, see: Baum, R. M. Chem. Eng. News 1990, Ocr 29, 22-25. (5) Bethune, D. S.; Meijer, G.; Tang, W. C.; Rosen, H. J . Chem. Phys. Lett. 1990, 174, 219-222. (6) (a) Johnson, R. D.; Meijer, G.; Bethune, D. S. J . Am. Chem. Soc. 1990, 112. 8983-8984. (b) Taylor, R.; Hare, J. P.; Abdul-Sada, A. K . ; Kroto, H. W. Chem. Commun. 1990, 20, 1423-1425. (7) Ajie, H.; Alvarez, M. M.; Anz, S. J.; Beck, R. D.; Diederich, F.; Fostiropoulos, K.; Huffman, D. R.; Kratschmer, W.; Rubin. Y.; Schriver, K. E.; Sensharma, D.; Whetten. R. L. J . Phys. Chem. 1990, 94, 863043633,
Raman spectras of C60 strongly support the soccerball geometry for this molecule, and solution N M R spectra6 have shown that the molecule has icosahedral point group symmetry. The pseudospherical symmetry of this molecule suggests potential for facile isotropic motion in the solid state,5 and solid-state N M R spectroscopy is uniquely suited as a probe of this kind of molecular dynamics. The sample was prepared by toluene extraction of soot produced by arc heating graphite in 100 Torr of He.* The dried powder consists mainly of C60 and C70 in a ratio of 1O:l and also shows a strong, sharp (1.543 width) EPR signal at g 2. I3Cspectra of approximately 1 g of sample have been obtained at 1.4 T (15 MHz). Figure 1 shows spectra at several temper-
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(8) Haufler, R. E.; Conceicao, J.; Chibante, L. P. F.; Chai, Y.; Byrne, N . E.; Flanagan, S.; Haley, M. M.; OBrien, S. C.; Pan, C.; Xiao, Z.; Billups, W. E.; Ciufolini, M . A,; Hauge, R. H.; Margrave, J. L.; Wilson L. J.; Curl, R. F.; Smalley, R . E. J . Phys. Chem. 1990, 94, 8634-8636.
0022-365419 112095-0009$02.50/0 0 1991 American Chemical Society
