J . Phys. Chem. 1990, 94, 495-500 of this cross section. We believe that the cross section employed is the most accurate presently available, taking into account exchange, binding energy, and relativistic effects. Any further improvement will require a detailed theoretical examination of the interaction processes (Le., of the generalized oscillator strength distribution) and a thorough comparison with experimental studies. The DOSDs used in our calculations are experimentally based and have been exhaustively tested, but a number of extrapolations were necessary in their compilation. It is to be expected that future studies will yield DOSDs with a greater experimental basis. The precision of our results is limited only by statistics and depends upon the number of times the Monte Carlo simulation is realized. We do not believe that any significant improvement is possible except at an unacceptable cost in terms of computer resources. This paper has been restricted to the distribution of energy deposition events along electron tracks. The next step toward an understanding of radiation chemistry processes is to determine
495
the relationship between the energy deposited and the reactive species produced. This information is the primary input into kinetic calculations and is especially important for stochastic calculations.I2 The distribution in the number of primary species is not necessarily proportional to the energy content of the entity. While some cross sections for the production of primary species are known for gaseous water, very little is known for the liquid phase. In the next article in this series, the distribution of ion pairs will be derived from the distribution of energy loss events as presented in this paper and from other experimentally observed cross sections.
Acknowledgment. The research described herein was supported by the Office of Basic Energy Sciences of the Department of Energy. This is Document No. NDRL-3201 from the Notre Dame Radiation Laboratory. Registry No. Water, 7732-18-5.
General Solution of the Ideal Model of Chromatography for a Single-Compound Band Sadroddin Golshan-Shirazi and Georges Guiochon* Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996- 1600, and Division of Analytical Chemistry, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 -61 20 (Received: March 27, 1989)
A general analytical solution of the ideal model of chromatography for a single-compound band is derived. This solution
is valid for any concave or convex isotherm but not for those that have an inflection point. This solution is applied to the case of a two-site adsorption model, with an equilibrium isotherm that is the sum of two Langmuir-typeexpressions, corresponding to each of the two sites. The isotherm for the higher energy site has a larger b coefficient (expressing the stronger solutestationary phase affinity) and a lower saturation capacity than the isotherm for the lower energy site. As predicted by Giddings long ago, the two-site model accounts very well for the tailing that appears on the experimental band profiles, in the cases examined. Thus, tailing cannot be explained only by a slow kinetics of the retention process. The asymptotic solution of the ideal model and its properties are also discussed.
Introduction The ideal model of chromatography derives from a very simple assumption, that the column efficiency is infinite.( This is certainly a “theoretical” model, in the sense that there is no such column in the real world. However, the efficiency of the modern high-performance chromatographic columns, available for about every mode of retention, except maybe bioaffinity chromatography, is very high. Most columns used for analytical or semipreparative applications typically have efficiencies in excess of 5000 theoretical plates, many have more than 10000 plates. Albeit, there is a wide difference between 10000 and infinity in the mathematical sense of the term, the results of the ideal model closely approximate experimental HPLC band profiles. As we have shown previously,2” the solution of the ideal model gives the limit toward which the profile tends when the column efficiency is increased indefinitely. This solution is a good caricature of actual band profiles and retains their essential traits. In two previous papers, we have derived the analytical solution of the ideal model of chromatography for the elution band profile at high concentration of a single-compound band and a singlecompound Langmuir isotherm2 and the analytical solution of the ideal model for the elution band profiles of a rectangular pulse of a two-component mixture with binary, competitive Langmuir isotherm^.^ We have shown that these solutions agree well with experimental result^.^^^ If the smoothing and broadening effects of a finite column efficiency are taken into account, the agreement becomes q~antitative.~,’ *Author to whom correspondence should be sent, at the University of
Tennessee.
0022-3654/90/2094-0495$02.50/0
In some cases, however, we have observed a less than satisfactory agreement between experimental results and the band profiles derived through the use of our numerical program that calculates the numerical solution of the ideal model, while accounting for the finite column efficiency.’** The difference observed between these profiles is in their tail and exceeds what could be ascribed to experimental errors. The experimental profile returns to base line more slowly than predi~ted.~.’Since the program uses isotherm data determined on the same column by frontal analysis or ECP6 and since the agreement is quantitative in a number of cases, the most probable explanation is that we used an erroneous isotherm model to account for the isotherm data. This is all the more reasonable as, for experimental reasons, neither in frontal analysis nor with the ECP method are equilibrium data taken at very low concentrations. Thus, the explanation that some “active sites” are playing an important role in retention at low concentration comes to mind. In order to test it, and to solve one of the last problems of the ideal model of chromatography, we need the solution to the band profile problem in the general case, when the equilibrium isotherm ( I ) Wilson, J. N. J . A m . Chem. SOC.1940, 62, 1583. (2) Golshan-Shirazi, S.; Guiochon, G . Anal. Chem. 1988,60, 2364. (3) Golshan-Shirazi, S.; Guiochon, G. Anal. Chem. 1989, 61, 462. (4) Golshan-Shirazi, S.; Guiochon, G. J . Phys. Chem. 1989, 93, 4143. (5) Golshan-Shirazi, S.; Guiochon, G . Submitted for publication in J . Chromatogr. ( 6 ) Golshan-Shirazi, S.; Ghodbane, S.; Guiochon, G. Anal. Chem. 1988, 60, 2630. (7) Golshan-Shirazi, S . ; Guiochon, G. Anal. Chem. 1988, 60, 2634. (8) Guiochon, G.;Golshan-Shirazi, S.; Jaulmes, A. Anal. Chem. 1988.60, 1852.
0 1990 American Chemical Society
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The Journal of Physical Chemistry, Vol. 94, No. 1, 1990
is not represented by the classical Langmuir equation. It is the aim of this paper to present such a derivation and the properties of the solution obtained from it, including those of the asymptotic solution, i.e., of the limit band profile when the column length becomes infinitely long.
Theory In the ideal model of chromatography, we assume infinite column efficiency. The coefficient of axial dispersion is naught, and the two phases of the chromatographic system are constantly in equilibrium, all along the column.2 Thus, the concentration of the compound studied in the stationary phase is related to the one in the mobile phase by the equilibrium isotherm. Under such a set of assumptions, the mass balance of the studied compound is written
ac
aq at
- + Fat
+ u-ac aZ = 0
where t and z are the time and space coordinates, respectively; u is the velocity of the mobile phase, u = L/to, where L is the column length and to the column dead time; F is the phase ratio, F = u,/u, = (1 - e)/€, where u, and u, are the volumes available to the stationary and mobile phases, respectively, and t is the porosity of the column packing; and C and q are the concentration of the sample compound in the mobile and the stationary phases, respectively. q is a function of C alone, the equilibrium isotherm, which can be differentiated. q is a function o f t and z only through the time and space dependence of the mobile-phase concentration of the sample compound. The properties of the band profiles propagated by eq 1 have been discussed. It has been demonstrated that eq 1 can propagate concentration discontinuities or shock^.^^'^ In fact, if the sample concentration is large enough, a shock will always take place on one side or the other of the band profile and, under some particular conditions, on both sides (Le., the isotherm has an inflection point). On the other side of the band, the profile is diffuse, Le., continuous. The velocity associated to a concentration C on the diffuse boundary of a chromatographic band profile is given by the following equation:I0-" u, =
dq 1+FdC
+ -L = t p + t o 4
numerical solution can still be obtained easily as shown previou~ly.~ Finally, eq 3 does not permit the calculation of the maximum concentration of the band nor its retention time. They must be found otherwise. The end of the diffuse profile in the case of a convex upward isotherm and the beginning of the diffuse part of the profile in the case of a concave upward isotherm are given by the limit of eq 3 for C = 0, which is t = f p + to(1 + Fq'(0)) = tp + tR,0 (4) where tR,0 is the analytical retention time of an extremely small pulse of the studied compound and q'(0) is the value of dq/dC for C = 0. In the ideal model, depending on the sign of the isotherm curvature, the entire sample is eluted before 1 = tR.0 (convex isotherm) or after that time (concave isotherms). Since the column efficiency is assumed to be infinite, the elution profile of a sample pulse of very small amount (Le., with a linear isotherm) would be the same as the injection profile. Since the sample mass is constant, the area of the band profile must remain constant during the band migration along the column. Therefore, if we assume for the sake of convenience that the sample is introduced as a rectangular pulse of finite width, the retention time of the shock is the boundary of the mass conservation integral
( l y ' R o Cdtl = Cotp= N / F ,
(5)
where t, is the duration of the rectangular pulse of the sample; N is the mass of the compound injected in the column, in mole; Co is the concentration of the compound injected in this pulse; F, is the volumetric flow rate of the mobile phase; to and tR,Oare the column dead time and the compound retention time at infinite dilution, respectively, and tR is the retention time of the shock of the band profile. Equation 5 relates the retention time of the shock to the equilibrium isotherm through the time dependence of the concentration given by eq 3. There are two ways to solve eq 5, either by solving eq 3 for C and eliminating C between eq 3 and 5 or by differentiating eq 3 and eliminating dt. In this case, we obtain
~
Equation 6 can be integrated by parts, and we obtain the following algebraic equation
With use of this general equation, it is possible to derive directly the band profile for any isotherm function. ( I ) Derivation of the General Band Profile Equations. By definition of the velocity and the elution time, a concentration Con a diffuse profile appears after a time given by the following equation t = tp
Golshan-Shirazi and Guiochon
(3)
where t , is the width of the rectangular sample pulse introduced in the column. This equation represents the diffuse, Le., continuous, part of a band profile. It has been known for a very long and it is the basis, among other uses, of the classical ECP method of isotherm determination.6 It is very general, and its use suffers only three minor limitations. First, one must know the isotherm to apply eq 3. Secondly, the band profile must equation, q =Ac), be continuous on the band side studied, which is simple if the isotherm has no inflection points. When the isotherm has an inflection point, the determination of the equation of the band profile becomes very complicated and is not discussed here. The (9) De Vault, D. J . Am. Chem. SOC.1943,65, 532. (10) Lin, B.; Golshan-Shirazi, S.; Ma, 2.; Guiochon, G. Anal. Chem. 1988, 60, 2641. ( I I ) Aris, R.; Amundson, N. L. Mathematical Methods in Chemical Engineering; Prentice-Hall: Englewood Cliffs, NJ, 1973: Vol. 2 .
which relates CM,, to the equation of the isotherm and can be solved in closed form in certain cases. This equation is equivalent to one previously derived by Glueckauf,12 using a different approach and a complex system of notations. The combination of eq 3, 4, and 5 or 7 gives the exact solution of the ideal model of chromatography, for any concave or convex equilibrium isotherm equation, provided it can be differentiated once. Depending on the isotherm equation, the retention time of the band is obtained either as an analytical expression (if eq 5 can be integrated or eq 7 solved analytically for CMax). or as the result of a numerical calculation. In both cases, the diffuse part of the profile is given by the analytical eq 3 and 4. As an example, we derive now the retention time equation for several classical isotherm equations. (11) Application to Some Classical Isotherm Equations. If we introduce in eq 3 the Langmuir isotherm equation ( q = aC/( 1 + bC)) and eliminate C between the resulting eq 3 and 5 , we obtain the following equation for the retention time in the case of a Langmuir isotherm where LI is the loading factor, the dimensionless parameter that (12) Glueckauf, E J Chem Soc 1947, 1302
Ideal Model of Chromatography for Single-Compound Band
The Journal of Physical Chemistry, Vol. 94, No. 1, 1990 491
characterizes the sample amount injected
where b is the second coefficient of the Langmuir isotherm and S the cross-section area available to the mobile phase in the
column. Equation 8 is identical with the one previously derived through a different approach.2J0 If, instead of a Langmuir isotherm, we use a similar equation for a concave isotherm (q = aC/( 1 - bC)), we obtain an equation similar to eq 8, which has also been derived previously: Similarly, in the case of a Freundlich isotherm ( q = a C , n # I ) , we obtain
In a previous work,3 we have reported excellent agreement between the experimental band profiles recorded for large samples of various compounds on different chromatographic systems and the profiles predicted by the ideal models2 The agreement becomes quantitative6*' when the theoretical profiles are derived by numerical integration of eq l,*with use of values of the integration increments selected in order to account exactly for the effect of the column e f f i c i e n ~ y . ' ~ In ? ~the ~ reversed-phase experiments, however, there exists a slight tailing of the high-concentration bands that is not accounted for by the model. This residual looks very much like the tailing profiles that chromatographers often ascribe to "active-site effects". In order to check the validity of this assertion, we have investigated the profile obtained with a two-term Langmuir isotherm. In liquid-solid adsorption equilibria, a two-site model corresponds to an adsorbent surface covered by a patchwork of two different types of chemical moieties. In reversed-phase chromatography, for example, one type of sites is made of the chemically bonded alkyl groups, while the other type is made of the unreacted silanols. The former sites account for the larger fraction of the adsorbent surface and have the larger saturation capacity, but they have a relatively low adsorption energy. The other sites are fewer, but their adsorption energy for polar, basic, or electrodonor molecules is large. Their contribution to the retention volume at low concentration is large, but their saturation capacity is low. If all sites adsorption processes are independent, such an adsorption model would be properly accounted for by the following isotherm15
where the numerical coefficients of the isotherm, a l , b , , a2,and b2 are such that b2 >> bl and a 2 / b 2
