J . Phys. Chem. 1985, 89, 5526-5530
5526
Temperature Dependence of Retention in Supercritical Fluid Chromatography C. R. Yonker,* B. W. Wright, R. C. Petersen, and R. D. Smith Chemical Methods and Kinetics Section, Pacific Northwest Laboratory,+Richland, Washington 99352 (Received: April 2, 1985)
The complex dependence of retention in supercritical fluid chromatography as a function of temperature is examined. A thermodynamic relationship is derived which gives the slope of solute retention as a function of temperature at constant pressure and is used to model retention data. This model is dependent on the partial molar enthalpy of transfer of the solute between the stationary and mobile phases at infinite dilution and on the volume expansivity of the solvent. The role of aim and solvent molar volume in retention is discussed based upon the comparison of this model with experimental data.
(a,")
Introduction The effect of pressure and density on retention in supercritical fluid chromatography (SFC) has been well Solute retention ( k ) as a function of pressure or density at constant temperature has been described by van Wasen and Schneider' and shown to depend on the partial molar volumes of the solute at infinite dilution in the mobile and stationary phases and on the isothermal compressibility of the fluid. The modeling of solute retention for such conditions is complicated due to the lack of accurate values for the former. The compressibility of the fluid could be predicted by a relatively simple two-parameter, cubic equation of state as proposed by Peng and R ~ b i n s o nbut , ~ the estimation of the partial molar volumes of the solute in the mobile and stationary phases under chromatographic conditions is difficult at b e ~ t . ~ , ~ The dependence of SFC retention on temperature at constant pressure has been studied e~perimentally,'-'~but the functional dependence of k with temperature has yet to be described. The simple thermodynamic approach outlined in this paper allows the prediction of the trend in In k vs. temperature at constant pressure from easily determined experimental quantities and a two-parameter, cubic equation of state. From this model, solute retention behavior can be examined through theory and experiment in order to gain some insight into the complicated dependence of retention on the thermodynamic and physical properties of the solute and the fluid, providing a basis for consideration of more subtle effects unique to SFC. Theory Beginning with the assumption of infinitely dilute solutions of the solute in the mobile and stationary phases, the concentration of the solute in these phases respectively is Ci = Xi/Vm,where Xi is the mole fraction of solute i and Vm is the molar volume of the pure mobile or stationary phase.' Solute retention is calculated from the retention factor, k , where
pustat
=
pyb = pLfWm+ R T In T t a t = p r b v m + R T In Fb(4)
where pmobis the chemical potential of solute i in the mobile phase and pf"'." is the chosen standard state at infinite dilution of solute i in the mobile phase. Therefore
Substituting eq 5 into eq 3 Ink=-
-A&" RT
+ In ( e b P t a tP$'Pob)) /(
(6)
The assumption can be made that P I a t , V2t, and P O b are independent of temperature at constant pressure (this may not be due to possible density dependent "swelling" strictly true for Pat of the stationary phase for some situations in supercritical fluid chromatography). Thus, differentiation of both sides of eq 6 with respect to temperature at constant pressure gives
The first term on the right-hand side of eq 7 can be evaluated through
-Api" - - -M-i,"ASi" T
T
(8)
aim
where Hi"and are the partial molar enthalpy and entropy of transferring the solute molecule from the stationary phase to the mobile phase at infinite dilution. Taking the derivative of equation 8 with respect to temperature at constant pressure, one obtains
cb
Here Gtatand are the concentration of solute i in the stationary and mobile phases and Phtand P O b are the volumes of the stationary and mobile phase. Substituting for concentration into eq 1
(1) van Wasen, U.; Schneider, G. M. Chromatographia 1975, 8, 274. (2) Gere, D. R.; Board, R.; McManigill, D. Anal. Chem. 1982, 54, 736. (3) van Wasen, U.; Swaid, I.; Schneider, G. M. Agnew Chem., Int. Ed. Engl. 1980, 19, 575. (4) Peng, D. Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976, IS, 59. (5) van Wasen, U.; Schneider, G. M. J . Phys. Chem. 1980, 84, 229. (6) Paulaitis, M. E.; Johnston, K. P.; Eckert, C. A. J . Phys. Chem. 1981, 85, 1770. (7) Schmitz, F. P.; Hilgers, H.; Leyendecker, D.; Lorenschat, B.; Setzer, U.; Klesper, E. J . High. Resolut. Chromatogr. Chromatogr. Commun. 1984, and taking the natural logarithum of both sides one obtains 7, 590. (8) Novotny, M.; Bertsch, W.; Zlatkis, A. J . Chromatgr. 1971, 61, 17. In k = In (qtat/Xf"ob) In (cbVStat/(PztPob)) (3) (9) Schmitz, F. P.; Leyendecker, D.; Klesper, E. Ber. Bunsenges. Phys. Chem. 1984,88, 912. When the standard definition for equilibrium of = is (10) Schmitz, F. P.; Leyendecker, D.; Klesper, E. J . Chromatogr. 1984, used, the following is obtained:" 315, 19. (1 1) Klotz, I. M.; Rosenberg, R. M. "Chemical Thermodynamics"; BenOperated by Battelle Memorial Institute. jamin: New York, 1972; 3rd ed.
+
0022-3654185 12089-5526%01.50/0 0 1985 American Chemical Societv
The Journal of Physical Chemistry, Vol. 89, No. 25, 1985 5527
Retention in Supercritical Fluid Chromatography
TABLE I: Enthalpy of Interaction'
Substituting eq 9 in eq 7 one obtains
solute
AH, kcal/mol
n-heptadecane n-octadecane n-nonadecane 1 -pentadecanol 1-hexadecanol 1-octadecanol
-10.6 -11.2 -1 1.7 -9.9 -10.4 -11.2
'Obtained under SFC conditions for OV-17 stationary phase and carbon dioxide mobile phase at a constant density of 0.38 g cm-3
and upon rearangement
The second and third terms in the bracket on the right-hand side of eq 1 1 are equal and opposite because (aMi"/aT), = ACRm and (ahSim/aT)p = AcRm/T. Thus eq 1 1 becomes on substitution
Equation 12 is valid only over a limited temperature range and far from the critical point of the solution mixture. Therefore, for the case of typical gas or liquid chromatographic conditions, eq 12 becomes the limiting case where Mim is constant and independent of temperature. A further assumption which is valid for the two cases discussed above is that the second term on the right-hand side of eq 12 is small or negligible under normal operating conditions for liquid and gas chromatography. Therefore, eq 12 on rearrangement reduces to the familiar case described by Van't Hoffl*
Assuming Mi"is a function of t e m p e r a t ~ r e , one ' ~ obtains
AHi" = Ai?li"(To)+ I T f A c f i m d T To
(14)
where Acpi" is the difference in heat capacity of the solute in the mobile phase as compared to the solute in the stationary phase. Equation 14 is valid over the entire temperature range of interest for chromatographic separations, i.e., liquid, supercritical fluid, and gas chromatography. Substituting eq 14 into eq 12 one obtains
Equation 15 describes the relationship between temperature and solute retention at constant pressure over any range of temperatures.
A simplifying assumption which can be made in eqA5 is that at infinite dilution of the solute in the mobile phase ACRmin the integral can be approximated by the heat capacity of the mobile phase. The complicated dependence of heat capacity on tem(12) Horvath, C.; Melander, W.; Molnar, I. J . Chromatogr. 1976, 125, 129. (1 3) Moore, W. J. "Physical Chemistry";Prentice-Hall: Englewood Cliffs, NJ, 1972; 4th ed.
perature and pressure near the critical point of a fluid has been shown e~perimenta1ly.l~Therefore, an empirical approximation to the integral in eq 15 must be formulated to model the heat capacity as a function of temperature near the critical point of the fluid mobile phase. The second term on the right-hand side of eq 15 is known as the volume expansivity. The volume expansivity of the solvent can be calculated from a two-parameter, cubic equation of state (EOS) such as the Redlich-Kwong EOS.15 The Redlich-Kwong equation of state is not quite as accurate around the critical point of fluids4 as some of the later equations of state, such as the Soave modification to the Redlich-Kwong EOS16or the Peng-Robinson EOS: but simplifies the evaluation of the partial derivative in the volume expansivity. From the method of implicit differentiation with the Redlich-Kwong equation of state
+ y b R b + 0 . 5 y b a T 1 . 5- 0 . 5 ~ b T ' . ~ (16) 3 ( T b ) * P- 2 K b R T - Pb2 - RTb +
(*)2R
where R is the gas constant, P is pressure, and T i s temperature in K. The constants a and b of the Redlich-Kwong equation are
0.4278R2Tc2.5 a=
PC
(174
R TC b = 0.0867pc where P, and Tc are the critical pressure and temperature of the fluid, respectively. The determination of the derivative in eq 16 was accomplished by the use of the Peng-Robinson equation of state to solve for the compressibility (2)of the fluid. The compressibility of the fluid is defined as
The molar volume of the fluid mobile phase can be calculated from its compressibility. This value is used in eq 16 and then eq 15 can be used to determine the trend in retention as a function of temperature at constant pressure. The enthalpy of transfer of the solute between phases can be determined from experiment by a Van't Hoff plot of In k vs. 1 / T at constant density.17 Therefore, the right-hand side of eq 12 or 15 can be evaluated for a particular fluid mobile phasesolute combination and the behavior of the slope of In k against temperature at constant pressure can be predicted.
Experimental Section The experimental apparatus and techniques have been described in detail e l s e ~ h e r e . ' ~ *The ' ~ enthalpies of transfer for the n-alkanes (14) (15) (16) (17) 1370.
'Gas Encyclopedia"; Elsevier: New York, 1972. Redlich, 0.; Kwong, J. N. S.Chem. Rev. 1949, 44, 233. Soave, G. Chem. Eng. Sci. 1972, 27, 1197. Lauer, H.H.; McManigill, D.; Board, R. D. A d . Chem. 1983.55,
5528 The Journal of Physical Chemistry, Vol. 89, No. 25, 1985
Yonker et al.
l.61
1
L
1.2c I
\
X
X
C,,OH
A
o C,,OH
-2
I
x
X X
50
0
3 2.7 2.8 2.9 3.0 3.1
2.6
l i T x IO3 K
100 1 5 0 2 0 0 Temperature, OC
250
300
Figure 3. Experimental data (ref 9, Figure 2) and theoretical model (solid line) for chrysene with n-pentane, 35.5 atm, = -4.0 kcal/mol.
Figure 1. Van? Hoff plot of 1-alcohols,pentadecanol (C,,OH), hexadecanol (C,,OH), and octadecanol (CIROH), with the OV-17 column.
T10/ 20
m
K a s e2
10-
0
1-
1
"
'
:
J 37 __---
Y
C
0-
-li -1 -
-21
2.6
"
2.7
"
'
1
2.8 2.9 l i T x IO3 K
I
'
3.0
'
J 3.1
Figure 2. Van't Hoff plot of n-alkanes, heptadecane (C17),octadecane
(C18),and nonadecane (Clg), with the OV-17 column. (n-heptadecane, n-octadecane, and n-nonadecane, Alltech Associates) and 1-alkanols (1-pentadecanol, 1-hexadecanol, and 1 octadecanol, Wilmad Glass Co.) were obtained on a column containing a cross-linked OV-17 stationary phase over a range of temperatures at constant density (0.38 g/cm3). The effect of temperature on retention was studied by using n-hexadecane on a 20-m, 50-pm i.d. fused silica capillary column coated with an OV- 17 phase using FID detection. The OV-17 was cross-linked in situ to decrease its solubility in the supercritical fluid. The stationary phase film thickness was calculated to be -0.25 pm. A Varian 8500 syringe pump was operated under computer control providing precise and accurate control of the fluid density. The retention times of the solutes for the Van't Hoff plots and solute retention as a function of temperature were determined by a reporting integrator with an accuracy of 0.1 s.
Results and Discussion The Van't Hoff plots (In k vs. 1/T) for the 1-alkanols and n-alkanes are given in Figures 1 and 2, respectively. The slope of the plots in Figures 1 and 2 allows one to determine the partial molar enthalpy of transfer of the solute between the mobile and stationary phase at infinite dilution. These enthalpies are listed (18) Wright, B. W.; Smith, R. D. Chromatographia 1984, 18, 542. (19) Smith, R. D.; Kalinoski, H.T.; Udseth, H. R.; Wright, B. W. Anal. Chem. 1984, 56, 2416.
in Table I for the different solutes at a fluid C 0 2 density of 0.38 To) g/cm3. The data in Table I shows, as expected, that ARim( becomes more negative with increasing chain length. Lauer et al. report average enthalpy values of -6 kcal/mol for solutes with supercritical C 0 2 and N 2 0 at a density of 0.80 g/cm3 with a PRP- 1 column (styrene-divinylvenzene copol~mer).'~For reverse phase liquid chromatography, Grushka et al. have reported enthalpy values ranging from -2.0 to -6.0 kcal/mol,20 while Knox and Vasvari have reported a wider range of -0.9 to -5.9 kcal/mol on permaphase ODS with methanol-water (40:60) as the mobile phase.2' In gas chromatography, Meyer et a1.22and Tewari et al.23s24have reported enthalpies of the n-alkanes, hexane, heptane, and octane, of -6.8 to -9.1 kcal/mol on n-tetracosane between 76 and 88 O C . The enthalpy values reported here are quite reasonable when compared to these literature values. The work of Schmitz and co-workers9J0 with various solvents and four aromatic solutes show distinct maxima in retention as a function of temperature at constant pressure. This same type of retention behavior has been seen by others8,25,26 for pentane on different stationary phases. Data from Figure 2 of Schmitz et aL9 has been replotted in Figure 3. Equation 12 was used to predict the trend in retention of chrysene with n-pentane as the fluid at a constant pressure of 35.5 atm. In Figure 3 the enthalpy of transfer, ARlm(To)was assumed constant over the experimental temperature interval; therefore we assume ACpm = 0. One can see that eq 12 qualitatively predicts the trends in retention for this solute. The solid (20) Grushka, E.; Colin, H.; Guiochon, G. J . Chromatogr. 1982, 248, 325. (21) Knox, J. H.; Vasvari, G. J . Chromatogr. 1973, 83, 181. (22) Meyer, E. F.; Stec, K. S.; Hotz, R. D. J . Phys. Chem. 1973, 77,2140. (23) Tewari, Y. B.; Martire, D. E.; Sheridan, J. P. J . Phys. Chem. 1970, 74, 2345. (24) Tewari, Y. B.; Sheridan, J. P.; Martire, D. E. J . Phys. Chem. 1970, 74, 3263. (25) Sie, S. T.; Rijnders, G. W. A. Sep. Sci. 1967, 2, 729. (26) Sie, S . T.; Rijnders, G. W. A. Sep. Sci. 1967, 2, 1 5 5 .
Retention in Supercritical Fluid Chromatography
; : : o.8 0.4
The Journal of Physical Chemistry, Vol. 89, No. 25, 1985 5529 7."
3.2 2.4 -
I Y
-C
-0.41
-0.8
L 0 50 100 1 5 0 200 2 5 0 3
T e m p e r a t u r e , OC Figure 5. Experimental data (ref 9, Figure 6) and theoretical model (solid line) for chrysene in n-pentane at 69.1 atm, mi"= -3.0 kcal/mol.
-n-pentane
......... n-butane 1.2
-- iso-butane
h\
0
- a
-.-
n-propane
200 300 400 500 Temperature, OC Figure 6. Simulated retention dependence on temperature for n-pentane (35.5 atm), n-butane (38.5 atm), isobutane (36.5 atm), and n-propane (42.4 atm); A@- = -4.0 kcal/mol. 100
line in Figure 3 is In k as a function of temperature at constant pressure calculated from the slope determined from eq 12. For these conditions the model predicts a decrease in retention in the subcritical or liquid region coupled with a sharp rise in retention near the critical point of pentane (196 "C), which increases to a maxima followed by a decline in retention. The absolute position on the ordinate is unknown, unless eq 12 is integrated with respect to temperature and the integration constant is determined. Therefore, this constant is selected to obtain the best fit for a specific enthalpy value. One of the factors affecting the retention maxima seen in SFC near the critical point of the solvent is the volume expansivity term of eq 12. The effect of temperature on the volume expansivity of the solvent is shown in Figure 4. The differential of solvent molar volume with respect to temperature is plotted for n-pentane a t 35.5 (case 1 ) and 69.1 atm (case 2). One can see that near the critical point for n-pentane (33.3 atm, 196.4 "C) the differential term becomes dominant. At pressures further removed from the solvent critical pressure the differential becomes negligible, as seen in Figure 4. Therefore, density plays a direct role in the onset of the increase in retention (through its affect on solvent molar volume). The decline in In k after the retention maxima is similarly controlled through the volume expansivity of the solvent. Figure 5 contains the data for chrysene from Figure 6 of Klesper et plotted against the theoretical prediction from eq 12. The data in Figures 3 and 5 are with n-pentane as the fluid, the (27) One should note that the captions for Figures 6 and 7 in Klesper et
al. (ref 9) are reversed.
0.8 0-
-1.21 - 1.6
1.6-
-0.8-
-1.6-
I
I
1
Figure 7. Experimental data (ref 9, Figure 2) and theoretical model = -5.0 kcal/mol, (solid line) for chrysene with n-pentane, 35.5 atm, a = 0.04.
mi-
difference between the sets of data being the experimental pressure. For the solute chrysene, Figures 3 and 5 show that the model predicts the trend in retention for both cases. Figure 6 depicts the theoretical prediction for retention in four different solvents; n-propane, n-butane, isobutane, and n-pentane. The retention values merge at high temperatures which would be the case for gas chromatography. Figure 6 is qualitatively comparable to Figure 7 of Klesper et al.10927and rationalizes both the retention maximum seen with temperature and the relative heights of the retention maxima which are both solvent dependent. The was assumed independent of enthalpy of transfer (ART(To)) temperature and equal to -4.0 kcal/mol for all the solvents modeled in Figure 6. The thermodynamic relationship outlined above predicts all the major features for the temperature dependence of retention. However, further discussion is needed regarding the apparent correlation between the retention maximum and molecular weight evident in the data of Klesper and c o - w o r k e r ~ The . ~ ~ ~assumption ~ is invariant with temperature is poor near the critical that u,point of the solvent. Equation 15 incorporates a correction term to film which dominates near the critical point of the fluid. Therefore, two contributions compete near the solvent's critical point: (a) the volume expansivity of the solvent and (b) ACpm. Density effects both of these terms through its effect on the solvent molar volume and the number of intermolecular interactions occurring in the critical region. As stated earlier, with conditions of solute infinite dilution which are approximated by SFC, ACpm can be assumed to be equal to the heat capacity of the mobile phase. For carbon dioxide at its critical point, the heat capacity follows a sharply increasing Gaussian function.14 Therefore the term in brackets on the right-hand side of eq 15 could be approximated as [AI?,-(T0) a T c b ] where a is an is the empirical parameter which is solute dependent and heat capacity of the mobile phase. Since no simple mathematical at the pressures relevant to relationship presently exists for S F C we roughly approximated this term as (dcb/C3Z')p25. Figure 7 shows the model vs. experimental data from ref 9 for chrysene using the approximation discussed above for the functional dependence of ACpmnear the critical temperature. The model predicts the trend in retention quite well. The addition of the correction term to AI?: making it dependent on temperature also allows simulation of the effect of the change in the number of intermolecular interactions between the solute and the solvent molecules as the density decreases rapidly near the critical point of the solvent. The pressure (and density) drop across the chromatographic column resulted in uncertainty in the interpretation of the results of previous studies.I0 For capillary columns, however, the pressure drop is typically negligible.18-'9The capillary SFC study in this work was with supercritical carbon dioxide and the solute hexa-
+
cb
cb
5530
J . Phys. Chem. 1985,89, 5530-5536 a narrow temperature range (ACfim = 0) and the effect of the volume expansivity of the mobile phase is negligible. For this case, solute retention is dominated by the partial molar enthalpy of The volume expansivity solute transfer at infinite dilution and ACpmbecome dominant near the critical temperature of the solvent, contributing to the physical and chemical phenomenon controlling retention in the SFC regime. In deriving this relationship we have made the simplifying assumption that ARim( To) is the same in both the high-temperature (gas chromatography) and low-temperature (liquid chromatography) regions. While this assumption is almost certainly incorrect, and will be addressed in future studies, it provides a basis for treatment of the intermediate supercritical fluid regime. Further refinements in the model which are being undertaken include using the Peng-Robinson EOS instead of the RedlichKwong EOS for evaluation of the partial derivative in the volume expansivity, because of its greater accuracy near the critical point. Deiters and SchneideP have added an adjustable parameter into the Peng-Robinson EOS to account for large size disparities between solvent and solute molecules. Alternatively Kurnik et aLZ9have shown that a single adjustable binary interaction parameter can be added to the Peng-Robinson EOS to account for specific interactions between the solute and solvent, resulting in a relation that adequately models the solubility of a solid in a fluid. Both of these parameters would impact the volume expansivity in eq 15 and better describe the experimental data. A mathematical description of the heat capacity of the fluid near its critical point is being undertaken and will be discussed in a future work. These changes and further investigations into the variation of AHiover the relevant ranges of temperature and pressure are in progress and will serve to guide future experimental and theoretical developments.
(aim).
-2,6h jJ
-3.4
-4.21
0
'
I
20
w
'
, I 1 I 40 60 80 Temperature, OC
I
I
100
1
I
120
Figure 8. Experimental data for hexadecane on OV-17 with C02, 76.5 = -10.6 kcal/mol, a = 0.90. atm, and theoretical model (solid line) ai"
decane; modeling was undertaken using a ARim(To)value of -10.6 kcal/mol (see Table I) obtained from the Van? Hoff plot of heptadecane on OV-17. These data are plotted in Figure 8, with a fit parameter value for hexadecane of a = 0.90. Once again the fit of the experimental data by the model is quite good.
Conclusion The thermodynamic relationship developed in this work has been shown to describe the features of solute retention as a function of temperature at constant pressure quite well for conditions which include gas, liquid, and supercritical fluid chromatography. The dependence of retention upon temperature can apparently be ascribed to a combination of two effects. These effects include the rapid change in the number of intermolecular interactions of the solutesolvent molecules as one progresses through the critical p i n t for the solvent (ACpm)and on the volume expansivity of the solvent. The right-hand side of eq 15 reduces to the limiting case for is a constant over liquid and gas chromatography, where Mim
Acknowledgment. The authors acknowledge the financial support of the U S . Department of Energy, Office of Basic Energy Science, under Contract DE-AC06-76RLO-1830. (28) Deiters, U.; Schneider, G . M . Ber. Bunsenges. Phys. Chem. 1976, 12, 1316. (29) Kurnik, R. T.; Holla, S. J.; Reid, R. C. J . Chem. Eng. Dara 1981, 26, 47.
Electrochemical Behavior of Dispersions of Spherical Ultramicroelectrodes. 1. Theoretical Considerations Martin Fleischmann,+Jamal Ghoroghchian,*and Stanley Pons* Department of Chemistry, University of Southampton, Southampton SO9 5NH, England, and Department of Chemistry, University of Utah, Salt Lake City, Utah 841 1 2 (Received: April 8, 1985: In Final Form: July 30, 1985)
The bipolar electrolysis of suspensions of spherical ultramicroelectrodes is discussed. It is shown that the reactions at the surface will be rate controlling over a wide range of conditions in view of the high rates of mass transfer to the electrodes. The effects of diffusion can be taken into account in a straightforward manner in view of the absence of discontinuities in the spherical coordinate system. Bipolar electrolyses on ultramicroelectrodes can also be used for kinetic measurements and synthesis in solutions containing no deliberately added support electrolyte. Estimates are made of the effects of the coupling of diffusion and migration; exact predictions of the asymmetric polarization of the particles under these conditions will require numerical analysis.
Introduction The construction and behavior of microdisk (see, e.g., ref 1-19), m i c r ~ r i n g , ' and ~ - ~microsphere (see,e.g., ref 19,21-23) electrodes have been discussed extensively. The development of spherical diffusion fields in the bulk of the solution surrounding these University of Southampton. 'University of Utah.
0022-3654/85/2089-5530$01.50/0
electrodes leads to high steady-state rates of mass transfer to the electrode surfaces so that measurements can be made on fast (1) Z . G. Soos and P. J. Lingane, J . Phys. Chem., 68, 3821 (1964). (2) J.-L. Ponchon. K. Cesuuelio. F. Gunon. M . Jouvet. and J.-F. Puiol. Anal.' Chem., 51, 1483 (1979'). (3) M . A. Dayton, A. G. Ewing, and R. M . Wightman, A n d . Chem., 52, 2392 (1980).
0 1985 American Chemical Society
