Computer simulation of the propagation of a large-concentration band

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Anal. Chem. 1988, 60,1856-1866

Computer Simulation of the' Propagation of a Large-Concentration Band in Liquid Chromatography Georges Guiochon,* Sadroddin Golshan-Shirazi, and Alain Jaulmes'

Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996-1600, and Division of Analytical Chemistry, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831

The Injectlon of large amounts of sample In a llquld chromatography column is oflen carrled out for preparatlve reasons, sometimes for trace analysis. I n all cases It results In the formatlon of a high-concentration band, the proflle of which is broader than that of a band of a much smaller amount of the same compound and Is most often asymmetrical. Thls phenomenon Is oflen descrlbed as a loss of efflclency taklng place when the column Is overloaded. I n fact, It arises from the nonllnearlty of the equlllbrlum Isotherm of the studled solute between the moMle and the statlonary phase. Uslng a theoretlcal model based on the mass balance equations of the solute and moblle phase In a column sllce, we describe the change In band profile durlng elution and the Influence of varlous parameters on the elutlon profile: sample slze, injection duration, shape of the equllibrlum Isotherm, slope and curvature of the Isotherm at the origin, and normal column efflclency. The most Important single factor that determlnes the elutlon profile Is the shape of the Isotherm around the origln, Le., the slgn and magnitude of the curvature at zero concentratlon, the rate of varlatlon of thls curvature wlth lncreaslng concentration, and the possible existence of an inflection point. As a consequence, extreme accuracy of the Isotherm Is required to relate quantltatlvely the equlllbrlum Isotherm and the elution proflles.

Although important work has been carried out in this area, there is still a considerable lack of theoretical understanding of the phenomena that control the migration of a large concentration zone along a chromatographic column (1-6). This is made more acute by the recent accumulation of experimental results, often incomplete, from which the derivation of general conclusions is attempted. The extreme complexity of the different phenomena involved in the chromatographic process and of their interaction (7) makes it very unlikely that an experimental study, however careful and detailed, can completely unravel the optimization procedures for the selection of the column design and operation parameters. This situation hampers the efforts of those who want to use liquid chromatography as a means to purify substances. It is well established that the shape of the equilibrium isotherm determines the direction of the band asymmetry, either fronting for an S-shaped isotherm or tailing for a Langmuir type isotherm (7). A direct quantitative relationship between the band profile and the isotherm has not been derived yet. In spite of efforts made by several groups (l-i'), there is still no general procedure for the accurate prediction of the band profile corresponding to a large sample size, taking into account the equilibrium isotherm, the kinetics of band broadening, the kinetics of adsorption-desorption, and the profile of the injection band. *Author to whom correspondence should be addressed at the University of Tennessee. Present address: Laboratoire de Chimie Analytique Physique, Ecole Polytechnique, Rte de Saclay, Palaiseau, France. 0003-2700/88/0360-1856$01.50/0

In a previous work we have derived a numerical algorithm for the calculation of a band profile in gas chromatography (8). Comparison between the results of the computations made and the experimental profiles obtained by using large injections of a compound whose equilibrium isotherm was known with accuracy were satisfactory (8). The only significant difference between profiles from experiments and from computer simulations was ascribed to the inability of the program either to account for the proper column efficiency or to adjust for it. This problem has been solved, and we present here a study of the influence of various parameters on the elution band profile of a large concentration sample. The dependence of the band profile on the sample size, the column efficiency, and the injection profile, as well as on the characteristics of the equilibrium isotherm, has been investigated in detail through computer simulation. The results obtained are reported here. They are in excellent qualitative agreement with experimental data obtained independently. This lends credibility to the approach and justifies an experimental investigation, now under way. THEORETICAL SECTION The complexity of the chromatographicprocess has a 2-fold origin. First, a number of different processes are involved in the separation. Some of them, such as molecular diffusion and solution equilibria, are reasonably well understood. Others are less well-known. Few of them, if any, can be modelized in detail. Secondly, an apparent paradox exists. On one hand retention times are related to equilibrium data (the thermodynamic equilibrium constant for the retention mechanism used), while on the other hand, it is recognized that the bandwidth of the concentration profile is related to the kinetics of mass transfers in the column. Since the pioneering work by Giddings (9),the interaction between the two independent phenomena has not been studied in great detail, in spite of the underlying kinetic nature of equilibrium. It has always been assumed, more or less implicitly, that in the general case, the thermodynamics does not affect the band profile, while the retention times are independent of the mass-transfer kinetics. In analytical chromatography the problem is solved by assuming that the two phenomena are independent and by treating separately the effect of a linear equilibrium isotherm and of fast kinetics. A linear isotherm results in a retention volume that is proportional to its slope, i.e., one that depends only on the column temperature and the phase ratio (for a given solute and phase system). The mass-transfer kinetics is accounted for by an apparent diffusion coefficient, leading to the concepts of column efficiency and height equivalent to a theoretical plate. The two sets of phenomena (thermodynamics and kinetics) are treated separately. Experimental results have systematically supported the theoretical conclusions derived by this analysis, as long as the concentration of the solute is small. The elution profile of a chromatographic band is Gaussian. The mean of the profile (Le., its first 0 1988 American Chemical Society

ANALYTICAL CHEMISTRY, VOL. 60, NO. 18, SEPTEMBER 15, 1988

moment) is related only to the equilibrium constant; its standard deviation (i.e., second moment), to the apparent diffusion coefficient (9-1 1). When the concentration increases, the band profile becomes wider and asymmetrical (12,13). Many chromatographers blame that effect on "the loss of column efficiency due to overloading". While it is a well-known experimental fact that bandwidth increases .steadily with increasing sample size, it does not seem that the loss of apparent column efficiency can result from an increasingly sluggish mass-transfer mechanism. Diffusion coefficients and the kinetics of adsorption-desorption do not vary much in the concentration range in which chromatography is carried out, i.e., for concentrations below 3-570. Rather, band spreading arises from the fact that, as soon as the equilibrium isotherm is not linear, the bandwidth results from a combination of kinetic and thermodynamic effects. A curved isotherm contributes to the profile of the elution band (12, 13). The exact origin of the change in band profile is important because the actual separation between two closely eluted compounds changes with increasing sample size in a way that is very different from what it appears to do when the column is overloaded with a pure compound (14). In analytical applications, resolution is important because we are looking for information, and it may be difficult or impossible to assess properly the relative importance of interfering peaks; i.e., quantitative information becomes less accurate. Compounds may even go unrecognized because they coelute with other ones that are already known. In preparative applications, resolution between bands is less important. Depending on the specifics of the problem, what counts is either the amount of compound that can be recovered at a certain degree of required purity or the recovery yield of this compound. A t any rate, however, some minor.losses may be accepted, and a very significant production may be achieved under such experimental conditions that the resolution between the two bands is very small, even when there is no valley between the bands (15). Since we are collecting the eluate in several distinct fractions, we can determine their purity by some independent analysis, and we do not need resolution to tell us what amount of each unresolved component we have isolated (16). The same theoretical approach that explains the origin and the mechanism of the change in band profile when the column is overloaded with a pure compound may tell us how the different components of the mixture are separated when the equilibrium isotherms are not linear (14). In the following report we first describe the equation system used to model the migration of a pure compound band of any injected shape under nonlinear conditions. Then we discuss the simplificationsmade to obtain a system that can be solved numerically, while retaining a realistic model of elution chromatography. I. Mass Balance Equations. The derivation of the system of the mass balance equations of chromatography has been discussed in detail (1,3,8,17). For the purpose of this work we merely need to discuss the assumptions made in this derivation, especially in the case of the applications to highperformance liquid chromatography (HPLC), and the simplifications required to formulate a problem tractable by the normal methods of numerical analysis. The mass balance of the solute in a cross section of the column, at abscissa z , is given by (19)

where C, is the solute concentration in the mobile phase, C, is the solute concentration in the stationary phase, z and t are the distance and time, respectively, u is the mobile-phase

1857

velocity, and D is the axial diffusion coefficient (see next section). In this equation, the following assumptions are made: The compositions of the mobile and stationary phases are constant in a column cross section. The mobile phase is not compressible; its velocity is constant and proportional to the pressure gradient (Darcy law). The partial molar volumes of the mobile phase and the solute are constant. More precisely, the partial molar volume of the mobile phase is equal to the molar volume of the pure solvent. That of the solute is the same in both phases and remains equal to its partial molar volume at infinite dilution. This is approximate, but the variation does not exceed a few percent (18). The diffusion coefficient of the solute is independent of the concentration (19). Equation 1is very general and should be valid at any point of the column. This is the fundamental equation that describes the migration of a solute band along a chromatographic column. It should be completed by two other equations, the mass balance of the mobile phase and a kinetic equation describing the mass transfers between phases, and by a proper set of boundary conditions. From a strict theoretical point of view, another mass balance equation should be written for the mobile phase. By a proper choice of the reference state, we may decide, however, that the mobile phase is not sorbed (20),on the condition that we remain consistent in writing the kinetic equation or the isotherm (see next section). Then the mass balance of the mobile phase vanishes. This assumption is valid only when the mobile phase is a pure solvent. If not, it is still valid for the weak solvent, but mass balance equations are required for the strong solvent and for all additives (21). In the case of binary mobile phases, the mass balance equation for the strong solvent may be omitted if the solute is much more strongly sorbed than the strong solvent. This may be dangerous, though, as it is tantamount to neglecting system peaks, which is rarely legitimate. The kinetic equation accounts for the rate of mass transfer between the mobile and the stationary phases. It is discussed in the next section. The boundary conditions describe the injection process of the sample. Generally, at t = 0 the column is empty (i.e., C = 0 everywhere); between t = 0 and t = t,, the concentration of solute at column inlet, Co, is given by the injection band profile (Co = C,(t)); for t larger than t,, Co = 0 all the time. It has been shown that for a system such as the one studied here, the solution, i.e., C at any time and for any value of the column abscissa, z , becomes 0 after a finite period of time (8). 11. Mass Transfer Kinetics. The two functions C, and C, in eq 1 are not independent. We need a relationship between them in order to proceed. Usually, the rate of variation of C, is related to the concentrations of the solute in both phases. The relationship accounts for the kinetics of mass transfer of the compound considered between the two phases. This kinetic equation has been discussed first by Lapidus and Amundson (22). Various forms of kinetic equations have been derived by Huber (23) and by Horvath and Lin (24),but those are of no use in the present case, as they are linear. The most simple kinetic equation can be written as d - c, = K(C,* - C,) dt where K is a rate constant and C,* is the value of C, at equilibrium between the two phases if the mobile phase concentration is C,. Other equations or combinations of equations may be written. They account for the various processes of mass

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transfer involved, axial diffusion, radial diffusion in the mobile-phase stream around the particles, diffusion in the stagnant mobile phase inside the particles, kinetics of adsorption-desorption, diffusion across boundary layers, etc. These kinetic equations incorporate the equilibrium constraints to which the mass-transfer-rate expressions must reduce after a sufficiently long time. For example, eq 2 reduces to the equilibrium isotherm, when dC,/dt becomes naught (equilibrium). Other kinetic equations could account for sorbate-sorbate interactions or for the formation and dissociation of a complex. The solution of the system of partial differential equations obtained by combining eq 1 and 2 is a t present beyond the realm of our possibilities, unless we assume a first-order mass-transfer kinetics and a linear isotherm (25). In this case the solution is a Gaussian profile, with a standard deviation resulting from the characteristics of the kinetic equation. Thus, Horvath and Lin and Huber have used this approach to derive plate height equations (23,24). There is no analytical solution in the case of a nonlinear equilibrium isotherm. The derivation of a numerical algorithm for the solutionof the system of eq 1 and 2 is still eluding our efforts. A simplification is necessary, which would nevertheless permit the solution of a problem sufficiently close to the real problem. Assuming instantaneous mass transfers and constant equilibrium between phases has been the most successful approach to this problem in the past. 111. Ideal and Semiideal Models of Chromatography. The ideal model, suggested long ago ( 1 , 2, 7), assumes that the kinetics of mass transfer between phases is infinitely fast. Then eq 2 vanishes, and C, in eq 1 is replaced by the concentration of the solute in the stationary phase derived from the equilibrium isotherm as follows:

(3) The derivative represents the retention associated with the corresponding value of the concentration in the mobile phase k'= dC,/dC, (4) In most cases, k ' decreases regularly with increasing concentration. For S-shaped isotherms, however, it goes through a maximum. More details regarding the isotherm functions used in this work are given later. For the present it suffices to know that f(C,) is a continuous function that can be differentiated at least once. In practice it will increase continuously toward a limit (saturation of the adsorbent surface in liquid-solid chromatography). Then the fundamental equation for a single component, eluted with a pure solvent, becomes dC d(uC) d2C -(1+ kq -= D dt dz dz2 where C is the solute concentration in the mobile phase, C,. Haarhof and Van der Linde (26) have shown that, if the mass-transfer kinetics is of the first order and is fast enough, the system of eq 1 and 2 can be replaced by eq 5 and the isotherm equation. Then, however, the diffusion coefficient in eq 5 is no more the axial diffusion coefficient, but the apparent diffusion coefficient (see below), which accounts for the fast but finite mass-transfer kinetics. Unfortunately, however, it is not possible to find an analytical function that solves the system of eq 3 and 5. Haarhof and Van der Linde (26) and Houghton (27) showed that, if the isotherm is replaced by the first two terms of its Taylor expansion (Le., for a parabolic isotherm), eq 5 becomes a Burgers equation (8)and an approximate solution, valid only for small solute concentrations can be derived (18). It should be underlined that the Burgers equation is not mass conservative. But, in the general case, this is not possible (8).

+

It is not even possible to fiid a numerical algorithm that would permit the derivation of a program permitting the calculation of numerical solutions of eq 5 (8). A further simplification is required. The model of ideal nonlinear chromatography assumes that the column efficiency is infinite. Thus, the apparent diffusion coefficient in the axial direction, D,,is equal to zero, and the partial differential equation (5) becomes

Equation 6 has to be resolved numerically. The properties of the solutions of eq 6 have been studied in detail (3). The profiles include concentration discontinuities or shocks; Le., at some time during the elution, the concentration may vary abruptly from a certain value to another one, without really assuming any of the intermediate values. This is due to the somewhat unrealistic assumption of an infinite rate of mass transfer, i.e., an infinite column efficiency. Although in many cases large column efficiencies can be achieved and extremely abrupt band profiles are often observed, axial diffusion and a finite kinetics of mass transfer still smooth the profiles. A concentration discontinuity means an infinite concentration gradient, which in turn means an infinite mass flux (see Ficks law). Thus, only a steep profile is physically possible, not a concentration shock. This steep profile, which moves at the same speed as the shock, is called a shock layer. Previous results have shown, however, that it is not possible to calculate numerically exact solutions of the ideal model, because computers cannot account properly for discontinuities (6,8). Previous attempts at using numerical schemes based on the method of characteristics were only partly successful, because the continuous parts of the profiles and the discontinuities have to be propagated separately, and it is difficult to locate discontinuities on the band profile (17). Finite difference methods permit the calculation of the profile as a whole, without requiring different procedures for the diffuse parts and for the discontinuities. The numerical schemes based on the Godunov method result in band broadening due to numerical diffusion (see next section). This diffusion is due to the finite nature of the space and time increments on which the numerical integration is based. Only a zero-width increment, resulting in an infinite computation time, could successfully simulate the ideal model. We have found that it is possible to use this feature to improve the ideal model, still assuming a very fast mass transfer kinetics but considering a column of finite efficiency. The semiideal model (28) reintroduces the concept of column efficiency in the calculation of the band profile. We assume that the mass-transfer kinetics is finite and does not change with increasing solute concentration in the mobile phase. A t infinite solute dilution, the analytical solution of the system of mass balance equations for a Dirac pulse injection is a Gaussian profile whose variance, C? (in length unit), is u2 = 2D,t0

(7)

where tois the liquid-phase hold-up time, i.e., the retention time of a nonretained compound. The variance of the Gaussian band is related to the height equivalent to a theoretical plate, H,and to the column length, L , by the following classical equation: u2 = H L

(8)

Combination of eq 7 and 8 gives the following relationship between D,and the column classical parameters, column plate height, mobile-phase velocity, and column capacity factor for a zero-concentration signal:


= -H=L -

D a

24)

Hu 2

If we assume that the molecular diffusion coefficient of the solute remains constant during the elution of a large concentration zone, the plate height contributions, which depend essentially on this coefficient, remain also constant, and so does the apparent diffusion coefficient. Although molecular diffusion coefficients in liquids are concentration dependent, the variation is rather small in the concentration range of a large chromatographic band, which rarely exceeds 5% at its maximum (19). The calculation of the numerical solution of eq 1for a very small Dirac pulse injection gives a Gaussian profile instead of the Dirac pulse predicted by the ideal model (8). This “numerical diffusion” may be used to simulate the band broadening due to the resistance to mass transfer and the axial diffusion. The procedure is described below. It consists essentially in adjusting the length of the space and time increments of the integrations. The ratio of these increments should satisfy the Courant-Friedrich-Lewy (CFL) condition (8). It has been demonstrated recently that by taking the Courant number equal to 2, the time increment equal to 2H/u,, and the space increment equal to H, the errors made during the numerical calculations are exactly equal to the right-hand side of eq 5 (29). In other words, while trying to calculate the numerical solutions of eq 6, because we do not know how to calculate the correct solutions of eq 5, we make a systematic error that results in our obtaining instead the solutions of eq 5, which is the equation that we really wanted to solve. IV. Numerical Solution. There is no analytical integration possible for the combination of the partial differential equation (6) with the first derivative of the isotherm given by eq 4 and the proper set of boundary conditions (representing the injection profile), if the isotherm is not linear. Little is known about its mathematical properties (17). In a previous publication we have discussed a procedure for the calculation of numerical solutions of a system of partial differential equations similar to eq 6, in the case of gas chromatography (because of the density profile of the carrier gas along the column, due to gas compressibility, a mass balance equation must be written for the mobile phase). Using this procedure, based on the Godunov algorithm, we have written a computer program that we have used to generate series of numerical simulations of the band profiles obtained by varying the experimental parameters that control them, such as the sample size, the curvature of the isotherm at zero concentration, the plate height, and the width of the injection band. Computer. Computations have been performed on a VAX 8650 (Digital Equipment Corp.) in the University Computer Center. Since the time and length increments chosen for the integration are both proportional to the column height equivalent to a theoretical plate (HETP) (see previous section and explanations below), the time needed for the central processing unit to calculate one profile increases as the square of the plate number of the column. For a 5000-plate column, the computer must run the main loop of the program about 25 million times, and it takes about 5 min to perform the calculation of the elution profile. Numerical Scheme. To perform the integration of the system of partial differential equations, one takes the injection band profile and propagates it in the time and the column length spaces. We use a finite difference method, based on the Godunov algorithm. Values are selected for the time and space increments, dt and dx, respectively, defining a grid in the time/column length space. The concentration of the solute in the mobile phase is calculated a t each point of the grid. A concentration profile along the column is the set of values of

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C for the points of the grid for which time is constant, while the elution profile is the set of values of C at the column end. From the injection profile (a Dirac pulse, a rectangular pulse, or any other shape, in which case it has to be digitalized), the profile along the column at time dt is derived by using the finite difference equation equivalent to eq 6. Then, the profile at time 2dt is derived from the profile at time dt, and so on. The elution profile is derived from the time variation of the concentration at the end of the column. The elution profiles of any column shorter than the one studied, but otherwise identical, are easy to calculate. Intermediate concentration profiles along the column may likewise be stored. Values of the Time and Space Increments. The numerical integration requires the selection of arbitrary values for the time and length increments, which must satisfy the CFL condition (8, 1 7 ) necessary for the convergence of the numerical calculation toward a stable solution of the system. The smaller these increments, the longer the computation time, but also the closer the result to the exact solution of eq 6. It is not the solution of this equation (ideal model) that we are looking for, however, as we have already explained. We are looking for the solution of the system of eq 3 and 5, and it is what the computer gives us. Program. The program used contains four modules, corresponding to the injection of the band, its propagation, its exit from the column, and to the equilibrium isotherm. The modular structure of the program permits its rapid adaptation to the solution of similar problems, such as the separation of two bands, displacement, or frontal analysis, which can be studied by using the same approach (14, 30). Injection Module. The program can be used to simulate the propagation of any type of injected band. Injection is represented by the profile of the input band introduced in the column. We have used essentially in this work Dirac pulses and rectangular plugs of various areas and widths. Tailing injections could also be used for a more realistic simulation of actual preparative chromatography. This would allow a study of the effect of valve dead volumes, the Poiseuille profile in empty connecting tubings, or sample/solvent mixing in the pump. Propagation Module. The propagation module contains the two loops corresponding to the integration in the time and the space domains. These two loops are written after the principles discussed previously. Since these loops are used millions of times, the program has been written to minimize calculation times. For example, the isotherm is introduced in the loop, instead of using a subprogram. Elution Module. The elution module operates after the principle that the fastest part of the profile gets to the column end and is eluted first. It permits the rear part of the profile to continue diffusing until it is also eluted. A modification at this level permits the determination of the concentration profiles inside the column at any instant, as well as the elution of the band at any intermediate length desired. Isotherm Module. An equilibrium isotherm must be assumed to carry out the calculations. The shape of the elution band profile depends greatly on the sign and magnitude of the isotherm curvature at the origin. It also depends, for very large sample sizes, on the rest of the isotherm. Thus, for example, inflection points of the isotherm are associated with sharp segments of the peak profile (they result from the discontinuities predicted by the ideal model, which are dampened by diffusion). We have used several isotherm functions belonging to the classes of Langmuir isotherms and S-shaped isotherms. Three of these isotherms are represented in Figures 1 (one Langmuir, two S-shaped isotherms with identical slopes at the origin but

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Flgurs 1. Examples of the isotherms used in this work (see numerical values of the parameters in Table I): 1, Langmuir isotherm (Table I, curve 1); 2,S-shaped isotherm (Table I, curve 3); 3, S-shaped isotherm (Table I, curve 2).

Flgure 2. Plot of the derivative of some of the isotherms used in this work (see Figure 1 and eq 4) versus the solute concentration in the mobile phase: 1, Langmuir isotherm (Table I, curve 1); 2,S-shaped isotherm (Table I, curve 3); 3, S-shaped isotherm (Table I, curve 2).

Table I. Parameters of the Isotherms Used

of the concentration of the injected band by its volume, the study of these results permits a discussion of the practical problem of the injection of large sample amounts: what is better, to inject a large dilute band or a narrow, concentrated one, and is there a compromise between concentration and volume of the injected sample? Knox and Pyper (31) have already shown that (for a given mass injected) the final peak shape is not affected by the injection volume when it is up to about half the peak elution volume, i.e., two standard deviations of the zero sample size band. We have also related the width of the eluted band to the parameters studied, in terms of the apparent plate height or plate number as conventionally determined in chromatography, in order to permit an easier comparison between our results and those of semiempirical (31) or experimental (32) studies previously published. We have not studied the influence of the mobile-phase flow velocity, the column capacity factor, the column length, nor the average particle size. The only effect of changing the particle size is in modifying the column plate number, whose effect has been studied. Changing the other factors (column length, flow velocity, capacity factor) modifies the plate number, an effect already studied, and the retention time. The change in retention time is merely a change in the abscissa scale, which is not very significant in itself as long as the production rate of a preparative chromatograph is not investigated. Our aim in this work is essentially to relate quantitatively the band profile to the parameters that determine it. The slope of the isotherm at the origin, i.e., the column capacity factor, determines the position of the band but has little effect on the elution profile. I. Influence of Sample Size on Band Profile. The data in Figures 3-5 show the variation in band profiles caused by a large increase of the sample size. Figure 3 corresponds to one of the four Langmuir isotherms studied; Figures 4 and 5 correspond to two of the S-shaped isotherms. Figure 6 shows

nature of figure curve isotherm

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Langmuir Q = 25C/1 + BC Langmuir Langmuir Langmuir S-shaped Q = 25C(1 + AC)/[I 2BC + ABC2] S-shaped A = 2B + 0.2

+

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0.020 0.100 0.250 2.500 0.050 0.080 0.250

Isotherm in Figure 1 and first derivative in Figure 2. different curvatures) and 2 (the derivatives of the isotherms shown in Figure 1). The equations of all the isotherms used and the numerical values of the parameters are given in Table

I. Practical Applications. The main drawback of computer simulation is that only the solution of a well-defined problem, where each parameter has a given numerical value, can be obtained at one time. Thus, numerous calculatioins are required for the study of even simple problems. In the present case, the column simulated has a length of 25 cm, the flow rate is 5 mL/min (for a conventional in. 0.d. column), the retention time of a nonretained compound is 40 s (u = 0.625 cm/s), the retention time of the studied compound is 290 s at zero concentration, and its column capacity factor is 6.25. The column plate height has been varied from 25 to 200 pm (Le. the plate number is between 10000 and 1250).

RESULTS AND DISCUSSION We have studied the influence on the elution band profile of the following parameters: sample size, injection bandwidth, column efficiency, sign and intensity of the curvature of the isotherm. Since the sample amount is equal to the product


1861

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TIME. Figure 3. Band profiles generated for samples of increasing size for third Langmuir isotherm (see Figures 1 and 2 and Table I): column efficiency, 2500 plates (zero sample volume and mass). Sample size is given in millimoles (and in percentage of column saturation capacity): 1, 0.083 (0.10%); 2, 0.415 (0.50%); 3, 0.83 (1%); 4, 1.66 (2%); 5, 2.49 (3%); 6, 4.15 (5%).

Figure 4. Band profiles generated for samples of increasing size for third S-shaped isotherm (see Figures 1 and 2 and Table I): column efficiency, 2500 plates (zero sample volume and mass). Sample size is given in millimoles (and in percentageof column saturation capacity): 1, 0.083 (0.10%); 2, 0.415 (0.50%); 3, 0.83 (1%); 4, 1.66 (2%); 5, 2.49 (3%); 6 , 4.15 (5%).

the variation in band profiles obtained by increasing the isotherm curvature at constant sample size. In the case of the Langmuir isotherm the retention time decreases steadily with increasing sample size. As observed in many experimental studies (23,24),when the retention time is large and the curvature of the Langmuir isotherm at the origin is moderate, or when the degree of overloading is moderate, the peak looks like a rectangular triangle lying with one side on the base line and the retention time of the band maximum decreases linearly with increasing sample size (Figure 3). When the curvature of the isotherm is large (see Figure 6), or the column is strongly overloaded, the profile tail becomes convex toward the time axis. The retention time of the almost vertical front decreases more slowly with increasing maximum band concentration because it becomes close to the solvent hold-up time. All the peaks corresponding to a given Langmuir isotherm, except the smallest ones, have the same tailing end. The intensity of all the effects increases with increasing curvature of the isotherm (see Figure 6). For S-shaped isotherms, the same general trend is observed, with the intensity of the effects increasing with increasing curvature of the isotherm at the origin (cf. Figures 4 and 5). What characterizes an S-shaped isotherm (see Figures 1and 2) is that the amount of compound sorbed by the stationary phase at equilibrium increases faster than the concentration in the solvent at first, until an inflection point is reached. Beyond the inflection point, the amount of solute sorbed increases more slowly than the concentration in the mobile phase and tends toward saturation. The retention time of the band maximum varies accordingly. When the sample size is increased, starting from a very low value, the retention time increases first, goes through a maximum, the value of which corresponds to the slope of the inflection tangent of the isotherm, and decreases with further increase in sample size.

The Langmuir and S-shaped isotherms have the same slope at the origin. The difference between each Langmuir isotherm used and the corresponding S-shaped one (see Figure 1) is relatively inconspicuous. Nevertheless, the band profiles are extremely different (compare Figures 3 and 4). We have already reported the considerable influence of the curvature of the isotherm at the origin on the band profile when the column is overloaded (33). The existence of an inflection point causes a major change in the profiles of large-concentration bands. This may have important consequences in preparative chromatography, because the separation of bands such as those in Figure 5 from the band(s) of closely eluted compound(s) is easier than the separation of triangular zones (see Figures 3 or 6). The corresponding band spreads in a smaller eluent volume than when the isotherm is Langmuir; the resolution from its neighbors may be better and its concentration in the collected eluate larger. Thus, in some cases the amount of interference with neighboring bands may not change much and the production may be greatly improved by a large increase in sample size (from 2.49-16.2 mmol in Figure 5). Then, the sharp front of the large-concentration bands (n = 4.15-16.2 mmol; Le., A = 1.6-6.4% of column loading capacity) moves back with increasing sample size, covering the slowly trailing front of the low-concentration bands (A = 0.16-1’70 of column loading capacity). Finally, the comparison between Figure 1on one hand, and Figures 3-6 on the other hand, points to the serious problem that all attempts at using isotherms to predict the elution band profile will meet: the extreme sensitivity of the profile of large-concentration bands to minor variations in the isotherm at low concentration. A high degree of accuracy in the determination of these isotherms will be required. 11. Influence of Curvature of the Isotherm at the Origin. The sign and intensity of the isotherm curvature at

1862

ANALYTICAL CHEMISTRY, VOL. 60,NO. 18,SEPTEMBER 15, 1988

1

2

4

IO

240

260

'0

TIME.

Band profiles generated for samples of increasing size for second S-shaped isotherm (see Figures 1 and 2 and Table I): column efficiency, 2500 plates (zero sample volume and mass). Sample size is given in millimoles (and in percentage of column saturation capacity): Figure 5.

1, 0.415(0.160%);2,0.83(0.32%);3,2.49 (0.96%);4,4.15(1.6%); 5, 8.3 (3.2%); 6, 16.3 (6.4%).

the origin determine the band profile, as illustrated in Figures 3-6. If the curvature is positive (i.e., with an S-shaped isotherm), the band becomes unsymmetrical with a smooth front and a sharp tail (see Figures 1,4, and 5). If the curvature is negative (e.g., with a Langmuir isotherm), the opposite takes place (Figures 1, 3, and 6). Figure 6 shows the progressive change in the shape of the band of a sample of constant size when the isotherm curvature changes from a linear isotherm to the Langmuir isotherms 1, 2, 3, and 4 (Table I). Because elution chromatography is a dilution process, the maximum concentration reached in preparative chromatography is moderate. Even when high-concentration, narrow bands are injected, they collapse very rapidly, broaden considerably, and dilute to a large extent over a very short fraction of the column length (34). Thus, the isotherm region sampled during most of the band migration corresponds to an arc rather close to the origin. This explains why the curvature of the isotherm at the origin has such a considerable influence on the band profile. It is only when the sample size is very large, when the isotherm has an inflection point (i.e., its curvature changes very rapidly), or when the curvature is very strong (see the fifth profile in Figure 6) that the rest of the isotherm plays a significant role in the band profile. From a quantitative standpoint, however, if an accurate prediction of the band profile is needed, the isotherm must be accurately known. Replacing it by the parabola having the same slope and curvature at the origin gives satisfactory results only for very small sample sizes (18, 33). 111. Influence of Width of the Injection Profile. We have simulated the elution of rectangular injection profiles of increasing widths. A t constant sample size the peak broadens progressively, while the retention varies, usually increasing with increasing bandwidth. To account for this variation, it is preferable to use the time frame of the mass

Band profiles generated for samples of equal size (4.15 mmol) for five different isotherms (see Table I): column efficiency, 2500 plates (zero sample volume and mass); 1, linear Isotherm; 2,first Langmuir isotherm, sample amount 0.4% of column saturation limit; 3, second Langmuir isotherm, sample amount 2% of column saturation limit: 4,third Langmuir isotherm, sample amount 5 % of column saturation limit; 5, fourth Langmuir isotherm, sample amount 50% of column saturation limit. Flgure 6.

center of the zone, i.e., to take as the time origin on the chromatogram the moment when the mass center of the injection profile enters the column, not the time when the injection begins. This is the convention used to present Figures 7-10.

For a linear isotherm, the band profile, which was Gaussian for an infinitely narrow sample plug, becomes progressively flatter and broader. The retention time of the band measured from the injection time of the mass center of the band remains constant (see Figure 7). The band profile is given by an equation derived independently by Reilley (35)and by Wicke (36),which expresses the fact that if a rectangular injection profile is the integral during the injection time of an infinite number of Dirac injections (i.e., of injection plugs of infinitely narrow widths), the response signal is the integral during the corresponding time of the Gaussian profiles corresponding to each of the Dirac pulses. The limits of the profile derived by Reilley and Wicke are a Gaussian profile for a narrow pulse injection (injection pulse width equivalent to 0) and a plateau ending with an error function (erf) at both ends, for a rectangular pulse having a large width compared to the base width of the Gaussian profile. The results of our calculations are in agreement with these predictions (Figure 7). Because axial diffusion continues to act on the part of the band that is not yet eluted, elution profiles tail slightly. This effect could be accounted for by assuming that the band variance increases linearly with time, after the classical Einstein equation for diffusion. In practice, the effect is quite negligible. For a nonlinear isotherm, the retention time varies with increasing injection bandwidth, even though the data are presented with the time origin placed when the mass center of the injection profile enters the column. This effect is illustrated Figures 8 and 9, corresponding to a Langmuir and

ANALYTICAL CHEMISTRY, VOL. 60, NO. 18,SEPTEMBER 15, 1988

240

260

280

300

1863

.O

320

T-Ti/2

T-Ti/2 Flgure 7. Change in band profile with the width of the injection band at constant sample size: column efficiency, 2500 plates; linear isotherm. Volume of the injection bandwidth is given in milliliters (and in units of the standard deviation of the profile for a Dirac pulse injection): 1, 0.02 (0.04~); 2, 0.5 (la);3, 1.0 ( 2 4 ; 4, 2.0 (40); 5, 4.0 (8a).

Figure 8. Change in band profile with the width of the injection band at constant sample size for third Langmuir isotherm (see Figures 1 and 2 and Table I): column efficiency, 2500 plates (zero sample volume and mass); sample size, 4.15 mmol (5% of column saturation limit). Volume of the injection bandwidth is given in milliliters and in units of the standard deviation of the profile for a Diiac pulse injection): 1, 0.02

an S-shaped isotherm, respectively. Since the sample size injected in these experiments is constant, its height decreases in proportion to its width and the arc of isotherm sampled by the band during its migration becomes narrower. Thus, for a Langmuir isotherm (Figure 8) and an S-shaped isotherm, with a large sample size (Figure 9) the retention time increases with increasing injection bandwidth, since smaller concentrations are more retained, so to speak, than larger ones. For an S-shaped isotherm at low or moderate sample sizes, or with a moderate curvature, the converse is true. The retention time decreases with increasing injection bandwidth (Figure 10). In the same time, the width of the elution profile increases with increasing bandwidth. These figures show how the sample size does not affect peak shape up to a certain value, as suggested by Knox and Pyper (31). Figure 11shows a plot of the apparent plate number calculated as is conventional in HPLC, versus the injection bandwidth, for three isotherms, one linear, one Langmuir, and one S-shaped. It is remarkable that while the apparent plate number decreases with increasing bandwidth, as expected, in the first two cases, the linear and the Langmuir isotherms, it actually increases first with the S-shaped isotherm, before eventually falling to very low values. This is easily explained by the fact that the retention time of the elution profile increases faster than the bandwidth at low concentrations. This demonstrates that the origin of the “loss of column efficiency” observed by many experimentalists when they “overload” chromatographic columns is to be found in the nonlinear behavior of equilibrium isotherms at large concentrations. This phenomenon has nothing to do with the kinetics of mass transfer, which is properly described by the relative bandwidth at very low concentrations (linear equilibrium isotherm) but not by the bandwidth at high concentrations. IV. Influence of the Apparent Diffusion Coefficient.

(0.04~); 2,0.5 (la);3, 1.0(24;4,2.0 (44;5,4.0(84;6,6.0(124. Table 11. Relationship between the Apparent Diffusion Coefficient and the Column HETP

0,“ 0.000 39 0.000 78 0.00156 0.003 12 0.006 24 0.012 5 0.025 a

H,b pm

H exp, pm

N

12.5 25 50 100 200 400 800

NA

20 000 10 000 5000 2500 1250 625 312

24.5 49 99 198 397 796

Apparent diffusion coefficient in cm2/s. Equation 9: L = 25

cm; t

= 40.

As explained above, the apparent diffusion coefficient is an empirical factor taking into account the kinetics of mass transfers between the two phases during elution. The definition of this empirical coefficient is related to the Einstein equation for molecular diffusion and is given by eq 9. It takes into account the whole set of kinetic phenomena responsible for band broadening and assumes that their contribution does not change with the concentration of the compound under study in the mobile phase. As this concentration rarely exceeds 0.1 M, this is a reasonable assumption. The numerical relationship between the values of the apparent diffusion coefficient used in this work and the HETP of the column is illustrated in Table 11. It is seen that the value most used in this work corresponds to columns of rather low efficiency (H is equal to 100 pm, 2500 theoretical plates). At present, such values of the plate height are typically obtained with 20- or 40-wm particles. Accordingly,these results are directly applicable to preparative separations currently carried out by HPLC. Much better efficiencies can be and have been obtained by using 3- to 10-pm particles. Calcula-

1864

ANALYTICAL CHEMISTRY, VOL. 60, NO. 18, SEPTEMBER 15, 1988 t

3 2

2 1

1 0 r

m 0

0'

2

0

u

'4 0

t

0

N

A

01

so

270

290

3iO

330

T-Ti/2

250

2j5

300

325

350

5

4 0

T-Ti/Z

Flgure 9. Change in band profile with the width of the injection band at constant sample size for third Sshaped isotherm (see Figures 1 and 2 and Table I): column efficiency, 2500 plates (zero sample volume and mass); sample size, 4.15 mmol (5 % of column saturation limit). Volume of the injection bandwidth is given in milliliters (and in units of the standard deviation of the proflle for a Dirac pulse Injection): 1, 0.02 ( 0 . 0 4 ~ )2, ; 0.5 (lo);3, 1.0 ( 2 4 ; 4, 2.0 (4a); 5, 4.0 (80); 6, 6.0 ( 1 2 4 .

Flgure 10. Change in band profile with the width of the injection band at Constant sample size for first S-shaped isotherm (see Table I): cdumn efficiency, 2500 plates (zero sample volume and mass); sample size, 4.15 mmol (1% of column saturation limit). Volume of the injection bandwidth is given in milliliters (and in units of the standard deviation of the profile for a Dirac pulse injection): 1, 0.02 ( 0 . 0 4 ~ ) ; 2, 0.5 (la);3, 1.0 ( 2 ~ )4, ; 2.0 (40); 5, 4.0 (8u);6, 6.0 (12a).

tions have not been made with values of the apparent diffusion coefficient corresponding to the plate height achieved with these particles (ca. 7-20 pm). The computation time required to obtain a series of elution profiles would be very long, and we felt it was more important to check first to what extent our results agree with experimental band profiles. Figure 12 shows four profiles obtained for the same set of parameters, except the column efficiency. The values of the HETP corresponding to these profiles are 25,50,100, and 200 pm, respectively. The four profiles differ only slightly, confirming previous independent experimental results that the band profiles eluted from strongly overloaded columns having different efficiencies are almost identical. Of course, for s m d sample sizes, Gaussian profiles are generated, and the HETP derived from the calculated profiles agrees very well with those derived from eq 9 and the value of the apparent diffusion coefficient used (see Table 11). Similar results were obtained with all the other isotherms investigated in this work. It does not seem useful to elaborate on what seems to be a minor effect. The fact that the column efficiency, i.e., the rate of radial mass transfer in the column, does not influence the width of large-concentration bands in chromatography very much is another illustration of the thermodynamic nature of the band broadening due to overloading. The band profile is directly related to the shape and equation of the equilibrium isotherm (31). This fact should not be construed, however, as a demonstration of the lack of importance of column efficiency in preparative applications of liquid chromatography. The aim now is not to produce thin bands, nor to study band profiles, but to separate mixtures. The nonlinear nature of the phenomenon makes it impossible to predict the elution profile of the individual components of a binary band from the elution

profiles of pure samples of the same two compounds, in the same amounts. In fact, there seems to be a strong interaction between the components of unresolved bands, and the column efficiency strongly influences the concentration profies of the two components in the region where the mixture is still unresolved (14). V. Sample Size and Apparent Column Efficiency. We have discussed above the influence of sample size on the band profile. Since the elution band broadens with increasing sample size, we may expect the apparent column efficiency to decrease. Experimentalists define the apparent column plate number as proportional to the square of the ratio of the retention time of the band maximum to the bandwidth at half-height (31,32,37,38).Unless the retention time increases very fast, the apparent efficiency will decrease with increasing bandwidth, and thus, with increasing sample size. For a linear isotherm the plate number is independent of the sample size. For a nonlinear isotherm, the band broadens as explained above. The extent of the phenomenon depends on the isotherm curvature. Thus, curves 1 and 2 in Figure 13 correspond to two Langmuir isotherms. The decrease in plate number is much less important in the former case (B = 0.02),where it takes an almost 10 times larger sample than in the latter case ( B = 0.25) to observe the same reduction in apparent plate number. Finally, the third curve gives the apparent efficiency of the same column for a compound having an S-shaped isotherm. Remarkably, in this last case, the efficiency actually increases slightly at first with increasing sample size before decreasing rapidly, because for small sample sizes, the retention time of the band maximum increases slightly faster than the bandwidth. At large concentrations, however, the curvature of this S-shaped isotherm is comparable to the curvature of the second Langmuir isotherm (see


1865

1

1

i.5

2.0

2.5

3.0

3.5

250

230

270

LOG Vi Figure 11. Plot of the logarithm of the apparent plate number (5.54 times the square of the ratio of the retention time of the maximum of the elution profile to its width at half-height) versus the logarithm of the injection band volume (In microliters): column efficiency, 2500 plates (zero sample volume and mass); sample slze: 4.15 mmol; 1, linear isotherm; 2, third Langmuir isotherm; 3, third S-shaped isotherm; 4, first S-shaped isotherm.

Table I), and the apparent efficiencies of the column for the two compounds become comparable. Curves 1 and 2 in Figure 13 are very similar to those found in many experimental reports (32), especially to those of De Jong et al. (37). Similarly, our numerical results confirm earlier reports that the apparent efficiency decreases monotonously with decreasing value of the column efficiency at zero sample size (16). This means that the most efficient column a t infinite dilution remains so under overloading conditions, although ita advantage becomes vanishingly small when the sample size becomes very large. As an example, the apparent column efficiency observed for different sample sizes, for various compounds with Langmuir isotherms having different curvatures (i.e., column saturation limit) has been plotted on Figure 14 versus the product of the sample size and the curvature of the isotherm. A universal plot is obtained, quite reminiscent of the one obtained by Eble et al. (38).This result confirms the possibility of scaling the Langmuir cases, as suggested by Knox and Pyper (31).

CONCLUSION The results described above are in excellent agreement with experimental results obtained by many investigators during the last few years (31, 32, 16, 21, 39). The influence of the experimental parameters on the band profile is correctly described on a qualitative basis. This agreement remains qualitative, however. A quantitative comparison between predicted and recorded profiles is required to test the value of our model and numerical solution. This requires an accurate determination of the equilibrium isotherms. One of the results of this work is to show how sensitive the profiles of large-concentration chromatographic bands are to slight changes of the isotherm. Consequently, it is to be expected that in a number of cases

290

3

TIME.

Figure 12. Profiles generated for the same sample slze and Isotherm (third S-shaped isotherm, Table I) for different column efficiencies: 1, 10000 plates; 2, 5000 plates; 3, 2500 plates; 4, 1250 plates. Sample size was 4.15 mmol, Le., 5% of the column saturation capacity.

x e

:: Z 0

9

x

b

R 3

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

LOG (n) Figure 13. Plot of the logarithm of the apparent column efficiency versus the logarithm of the sample size (millimoles): column efficiency, 2500 plates (zero sample volume and mass); curve 1, first Langmuir isotherm; curve 2, third Langmuir isotherm; curve 3, third S-shaped isotherm (see Figures 1 and 2 and Table I).

the simple Langmuir model will prove unsuitable to adequately represent the isotherms, leading to unacceptabIe model errors. Work is in progress in this area and will be

1866


LITERATURE CITED Glueckauf, E. Discuss. Faraday SOC. 1040, 7 ,12. Glueckauf, E. Trans. Faraday Soc. 1055, 57,1540. Jacob, L.;Guiochon, G. Chromatogr. Rev. 1971, 74, 77. Rhee, H. K.; Aris, R.; Amundson, N. Chem. Eng. Sci. 1074, 29,2049. Huber, J. F. K.;Gerritse, R. E. J. Chromatogr. 1071, 58, 138. Guiochon, G.;Colin, H. Chfomatogr. Forum, 1086, 1(3), 21. Schay, G. Grundlagen der Chromatographle; Akademle Verlag: Berlin, 1962, (8) Rouchon, P.; Valentin, P.; Schonauer, M.; Guiochon, G. Sep. Sci. Technol. 1987, 22, 1793. (9) Giddings. J. C. Dynamics of Chromatography; Dekker: New York. 1966. (10) Grushka, E.; Myers, M.; Giddings. J. C. Anal. Chem. 1070, 42, 21. (11) Grubner, 0.I n Advances in Chromatography;Giddings, J. C., Keller, R. A., Eds.; Dekker: New York, 1968; Vol. 6, p 173. (12) Phillips, C. S.G. Discuss. Faraday SOC.1040, 7,241. (13) Littlewood, A. B. I n Gas Chromatography, 2nd ed.; Academic: New York, 1970; pp 11-14 and 37-42. (14) Guiochon, G.; Ghodbane, S . J. Phys. Chem. 1088, 92,3682. (15) Roz, B.; Valentin, P.;Schonauer, M.; Guiochon, G. I n The Science of Chromatography; Bruner, F., Ed.; Elsevier: Amsterdam, 1985; p 131. (16) Newburger, J.; Guiochon, G. Sep. Scl. Technol. 1087, 22, 1033. (17) Rouchon. P.; Valentin. P.; Schonauer, M.;Gulochon, G. I n The Science of Chromatography; Bruner, F., Ed., Elsevier: Amsterdam, 1985; p 131. (18) Jaulmes, A.; VidaCMadjar, C.; Colin, H.; Guiochon, G. J. Phys. Chem. 1086, 90,207. (19)Bird, R.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960. (20) Kovats, E. I n The Science of Chromatography; Bruner, F., Ed.; Elsevier: Amsterdam, 1985; p 205. (21) Golshan-Shirazi, S.;Ghodbane, S.;Guiochon, G. submitted for publication in Anal. Chem . (22) Lapidus, L.; Amundson, N. R. J. Phys. Chem. 1052, 56, 984. (23) Huber, J. F. K. Ber. Bunsen-Ges. Phys. Chem. 1073, 77,179. (24) Horvath, C.; Lin, H. J. J. Chromatogr. 1078, 749,43. (25) Villermaux, J. Chem. f n g . Scl. 1972, 27, 1231. (26) Haarhof, P. C.: Van der Llnde, H. J. Anal. Chem. 1966, 38, 573. (27) Houghton. G. J. Phys. Chem. 1063, 67,84. (28) Guiochon,. G.; Ghodbane, S. Bull. SOC.Chlm. Fr., in press. (29) Lin, Bing Chang; Guiochon, G. Sep. Sci. Technoi., In press. (30) Katti, A.; Guiochon, G. J. Chromatogr., in press. (31) Knox, J. H.; Pyper, H. M. J. Chromatogr. 1088, 363, 1. (32) Eisenbeisss, F.; Ehlerding, S.; Wehrli, A,; Huber, J. F. K. Chromatographia 1985, 20,657. (33) Jaulmes, A.; Vidal-Madiar, C.; Ladurelli, A.; Guiochon, G. J. Phvs. Chem. 1084, 88, 5379: (34) Rouchon, P.;Schonauer, M.:Vaientin, P.;Vidal-Madjar, C.; Guiochon, G. J. Phys. Chem. 1085,89,2076. (35) Reilley, C. N.; Hildebrand, G. P.; Ashley, J. W., Jr. Anal. Chem. 1962, 34. 1198. (36) von Wicke, E. Ber. Bunsen-Ges. Phys. Chem. 1065, 69, 761. (37) de Jong, A. W. J.; Poppe, H.; Kraak, J. C. J. Chromatogr. 1981, 209, 432. (38) Eble, J. E.; Grob, R. L.; Antle, P. E.; Snyder, L. R. J. Chromatogr. 1987, 384, 25. (39) Colin, H. Sep. Sci. Technol. 1987, 22, 19. (1) (2) (3) (4) (5) (6) (7)

I

.5 -40 -3.5 -3.0 -2.5

-2.0

-15

,

-1.0

-0.5

0.0

LOG (nb)

Figure 14. Plot of the logarithm of the apparent column efficiency (N) versus the logarithm of the product of the sample size ( n , in millimoles) and the second coefficient (b) of the Langmuir isotherm. Column efficiency was 2500 plates (zero sample volume and mass).

reported soon (21). Before going further and investigating in detail the relationships between apparent column efficiency and the variation of the experimental conditions we find it necessary to study carefully the predictive value of our model. The results obtained with the Langmuir isotherm are especially simple. Since the Langmuir equation uses only two parameters, it seems possible to describe simply the dependence of the band profile on the sample size and the isotherm curvature by using dimensionless parameters. The plot shown Figure 14 is an example of what could be obtained. This would permit predictions based on the determination of a single band profile, without requiring the exact determination of an isotherm. The method would be impossible to extend to other isotherms, however.

RECEIVED for review September 30,1987. Accepted April 19, 1988. This work was supported in part by a grant from the National Science Foundation (Grant CHE-85-19789).

Computer simulation of the propagation of a large-concentration band

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