Analytical Solution for the Ideal Model of Chromatography in the Case

J . Phys. Chem. 1989, 93, 4143-4157

4143

Analytical Solution for the Ideal Model of Chromatography in the Case of a Pulse of a Binary Mixture with Competitive Langmuir Isotherm 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: July 28, 1988; In Final Form: December 20, 1988)

An exact solution of the ideal model of chromatography is derived in the case of a binary mixture, when the equilibrium isotherms of the two compounds between the stationary and the mobile phases are competitive Langmuir isotherms. The concentration profiles of the two compounds are derived. Each compound profile is composed of discontinuities and continuous parts. There are two discontinuities (or shocks). The first one separates the pure mobile phase from a solution containing only the lesser retained component of the mixture. The second shock separates this solution from a solution of the two components. The composition of these solutions varies continuously between the shocks and behind the second shock. Thus, the concentration profile of the first component has two discontinuities; one that jumps from the base line to the band maximum constitutes the first shock. The second one, on its rear, jumps between two finite values of the concentration. The concentration profile of the second component has a single discontinuity, on its front. These last two discontinuities constitute the second shock. The profiles dbtained as analytical solutions of the ideal model are compared to those obtained as numerical solutions of the semiideal model. It is shown that as long as the mass-transfer kinetics between the two phases of the chromatographic system is fast enough and the column efficiency remains larger than a few thousand theoretical plates, the difference between the profiles is small, the shocks are replaced by shock layers whose thickness is proportional to the column height equivalent to a theoretical plate, and the shock layers move at almost the same speed as the ideal shocks.

Introduction The fundamental problems of nonlinear chromatography were formulated nearly 50 years Considerable attention has been devoted to these problems by many individual scientists and research groups a t various periods. Important progress has been made in the understanding of the mechanisms of band migration, band broadening, and band separation during the chromatographic process. Nevertheless, few authors have been able to report successful prediction of band profiles, even in the simplest case of a single-component p r ~ b l e m . ~ Accordingly, -~ little has been said about the multicomponent problems which are the most relevant ones, since they include the mechanism through which the bands separate progressively during their migration throughout a chromatographic column. This phenomenon is at the very heart of the chromatographic process. The popularity of preparative chromatography as a separation and purification method in the life sciences and in the pharmaceutical and biochemical industries requires significant progress in the understanding of nonlinear chromatography. The main roadblock in the solution of the rigorous, general model of chromatography is due to the requirement of this model for a proper description of the kinetics of mass transfer between phases. The great difficulties encountered in the derivation of proper theoretical expressions for the kinetics of mass transfer between the mobile and the stationary phases have led to the consideration of simplified models, where these mass transfers are accounted for either by a lumped rate c ~ n s t a n t or ~ ~by~ aJ ~calculation procedure. l In the first case, it is assumed that the concentration in the stationary phase tends toward the value a t equilibrium following a first-order kinetics. An exact solution has been derived in the case of a single component following a Langmuir-type kinetics, if the axial diffusion term can be neglected.I0 The solution is mathematically so complex that it does not seem possible to extend it to the case of a binary mixture, which is the simplest case of real practical interest. Furthermore, the assumption that the kinetics of phase equilibrium is of first order has never been demonstrated and is rather improbable when the kinetics of mass transfer between phases is slow. When it is fast, the simpler semiideal model gives the same results.14 'Author to whom correspondence should be addressed at the University of Tennessee.

0022-3654/89/2093-4143$01 S O / O

The alternative approach of importance consists in assuming that the kinetics of mass transfers between phases is so fast that there is constantly equilibrium between the two phases of the chromatographic system. This model has been called the ideal model.'l3 It is equivalent to assuming that the column efficiency is infinite. In practice, the efficiency of the real columns used in high-performance liquid chromatography (whatever the retention mechanism involved) is so high that the difference between the high concentration band profiles predicted by the ideal model ~ when and those obtained experimentally is very ~ m a l 1 . l Similarly, a high rate constant is used in the kinetic model, the band profiles calculated are very close to those derived from the ideal m0de1.I~ The properties of the ideal model have been extensively discussed. Numerical solutions have been derived in the case of single compoundsI1J2and of binary mixtures.16 These solutions may correct for the effect of the infinite efficiency assumption by adjusting the values of the time and space integration increments so that the effect of the numerical errors made while carrying out the numerical integration, following a finite difference algorithm, is exactly the same as that of the apparent axial dispersion created by the column.13 In the case of single-component bands, excellent agreement has been observed between experimental and predicted pr0fi1es.l~

(1) Wilson, J. N. J . A m . Chem. SOC.1940, 62, 1583. (2) De Vault, D.J . A m . Chem. SOC.1943, 65, 532. (3) Offord, A. C.; Weiss, J. Nature 1945, 155, 725. (4) Glueckauf, E. Discuss. Faraday Soc. 1949, 7, 12. (5) Guiochon, G.; Jacob, L. Chromatogr. Reu. 1971, 14, 77. (6) Ark, R.; Amundson, N. R. Mathematical Methods in Chemical Engineering; Prentice-Hall: Englewood Cliffs, NJ, 1973; Vol. 2. (7) Rhee, H. K.; Ark, R.; Amundson, N. R. Chem. Eng. Sci. 1974, 29, 2049. ( E ) Rouchon, P.; Schonauer, M.; Valentin, P.; Vidal-Madjar, C.; Guiochon, G. J . Phys. Chem. 1985, 89, 2076. (9) Thomas, H. C. J . A m . Chem. Sac. 1944.66, 1664. (10) Wade, J. L.; Bergold, A,; Carr, P. W. Anal. Chem. 1987, 59, 1286. (1 1) Rouchon, P.; Schonauer, M.; Valentin, P.; Guiochon, G. Sep. Sci. Technol. 1987, 22, 1793. (12) Guiochon, G.; Golshan-Shirazi, S.; Jaulmes, A. Anal. Chem. 1988, 60, 1856. (13) Lin, B. C.; Guiochon, G. Sep. Sci. Technol. 1988, 24, 32. (14) Lin, B. C.; Golshan-Shirazi, S.;Guiochon, G. J . Phys. Chem., in press. ( 1 5) Golshan-Shirazi, S.; Guiochon, G. Anal. Chem. 1989, 61, 462. (16) Guiochon, G.; Ghodbane, S. J . Phys. Chem. 1988, 92, 3682.

0 1989 American Chemical Society

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The Journal of Physical Chemistrj3, Vol. 93, No. 10, 1989

Recently, a solution was derived for the ideal model, in the case of one compound undergoing a Langmuir-type equilibrium isotherm.I8 This solution shows very little difference between the prediction of the ideal model and the numerical solution derived in the case of a 5000-plate column ( H E T P = 50 gum). This result is explained by the shock theory and by the consideration of shock layers." A shock layer is a physical discontinuity, with a small but finite thickness, as opposed to a mathematical discontinuity, which is thinner than a molecular diameter. However, shock layers migrate a t the same speed as shocks would under the same experimental condition^.'^ As a result, it has been possible to derive a procedure for the determination of isotherms, in the case where they are close enough to a Langmuir type, with a very small amount of experimental work.l5 It has also been possible to calculate the variation of column apparent efficiency with the column loading.I8 These results are in excellent agreement with experimental results and with independent o b ~ e r v a t i o n . ' ~ The first aim of this paper is to present the exact, analytical solution of the ideal model of chromatography in the case of a binary mixture, when the equilibrium isotherm of the two components is described by the competitive Langmuir formalism. The profiles of the two separate compounds are determined using the classical concepts of velocity associated to a concentration's2 and of shock p r o p a g a t i ~ n . ~They . ~ could also be derived, following the approach of Hellferich and Klein,2o through the h transform. Indeed, these authors have published distance-time diagrams which may be used to describe the migration of a two-component band and its progressive separation. Hellferich and Klein, however, have not derived the equations giving the elution profiles. They were more interested by displacement chromatography. Finally, the h-transform approach is valid only under the conditions of the ideal model of chromatography. No successful attempt has been made to extend the results to the case of a real column, with a finite efficiency.2' On the contrary, the shock theory can be extended by the consideration of shock layers and provides an attractive procedure for the correction of the results predicted by the ideal model." The second aim of this paper is to compare the exact solution of the ideal model to the profiles obtained by direct numerical integration of the semiideal model, which assumes a fast kinetics of mass transfers between phases resulting in a finite column efficiency. The results are in excellent agreement. The difference is extremely small in cases when the column efficiency is large.

Theory The Ideal Model of Chromatography. If we assume that the effects of the axial dispersion on the elution profiles of the two components of a mixture are negligible and that the kinetics of mass transfer in the chromatographic column proceeds a t an infinitely fast rate, the system of partial differential equations that describes the band profile simplifies considerably.'-8 The concentrations of the two compounds in the mobile phase a t the column outlet are obtained by solving the following set of equations, with the proper initial and boundary conditions:

( 1 7 ) Golshan-Shirazi, S . ; Guiochon, G . Anal. Chem. 1988, 60, 2634. (18) Golshan-Shirazi, S . ; Guiochon, G. Anal. Chem. 1988, 60, 2364. (19) Lin, 8 . C.; Golshan-Shirazi, S.; Ma, Z . ; Guiochon, G. Anal. Chem. 1988, 60, 2647. (20) Helfferich, F.; Klein, G. Multicomponent Chromatography. A Theory of Interference; Dekker: New York, NY, 1970. (21) Helfferich, F. G . J . Chromarogr. 1986, 373, 45.

Golshan-Shirazi and Guiochon where Cm,,and Cm,2are the concentrations of the first and second solutes (components of the mixture) in the mobile phase, respectively, CS,'and Cs,2are the concentrations of the first and second solutes in the stationary phase, at equilibrium with the mobile phase, respectively, F is the phase ratio, u is the linear velocity of the mobile phase, and z and t are the abscissa along the column and the time, respectively. In the case more specifically discussed here, of a binary mixture whose components have Langmuir competitive isotherm, we have

and

where a,, a2, b,, and b2 are numerical coefficients. The first two coefficients, al and a2, are the slopes of the isotherms, a t infinite dilution of both compounds. The ratios a,/bl and a 2 / b 2are the saturation concentrations for each compound, separately. The Langmuir isotherm violates the Gibbs-Duhem equation, unless these two ratios are equal, which is rarely verified in practice. But it is often easy to obtain numerical values of these coefficients so that eq 5 and 6 fit correctly a set of experimental data. Accordingly, the Langmuir equations should be considered as an empirical model, not a physicochemical model. The first two equations of the system (eq I and 2) are the simplified mass-balance equations of the two components of the mixture studied (with no axial dispersion and infinitely fast mass-transfer kinetics). The last two equations of the system are the equilibrium isotherms of these compounds; both isotherms are functions of the concentrations of the two compounds in the mobile phase. This dependence of the concentration of each component in the stationary phase on the concentrations of all the components of the mobile phase accounts for their competition in the retention process (whatever the retention mechanism involved). The other assumptions made in writing the system of eq 1-4 above are that (i) the flow velocity is constant, i.e., the viscosity of the mobile phase does not change markedly with increasing concentration of the compounds studied, (ii) there is no volume change associated to the passage from the mobile phase to the stationary phase, and (iii) the mobile phase is a pure solvent, which is not sorbed by the stationary phase. A proper convention for the definition of the adsorption quantities can be chosen,22provided one of two conditions be met. Either the mobile phase must be a pure solvent or, when a mixture is used as mobile phase, the column capacity factors a t infinite dilution, ko', of all the mobile-phase components (strong solvent, organic modifier, additives, etc.) in the pure weak solvent must be a t least 5 times smaller than the capacity factors of the solutes under investigation." These assumptions are satisfied with most systems used in modern liquid chromatography (HPLC), except in the case of mixtures of biopolymers. The system of equations of ideal, nonlinear chromatography for a single-compound band (eq 1 and 5 , Cm,2= 0) was first discussed by Wilson' and then by De Vault.2 Wilson discovered that even a continuous injection profile could give rise to an elution band profile exhibiting a discontinuity and that the migration rate of the band cannot be derived simply from the mobile-phase velocity and the column capacity factor. It depends also on the solute concentration. De Vault made a rigorous study of the properties of this system and could derive a full qualitative description of the propagation of a large concentration band. Equations 1 and 2 are quasi-linear, hyperbolic, first-order partial differential equations. Their properties have been extensively discussed by mathematicians. Of special concern is their ability to propagate discontinuities. A numerical algorithm for the SOlution of eq 1 (single component) was discussed by Courant et (22) Kovats, E. The Science of Chromatography; Brunner. F., Ed.: Elsevier: Amsterdam, The Netherlands, 1985; p 205.

The Journal of Physical Chemistry, Vol. 93, No. IO, 1989 4145

Ideal Model of Chromatography in a Binary Mixture al.23 By assuming the existence of generalized Riemann invariants, Lax24 was able to extend the results of the simple wave theory and to develop a general theory of discontinuities and of their properties and behavior. These results are central to the solution of the band profiles for a two-component mixture that we present below. Accordingly, we first summarize the results of the shock theory that are relevant to the problem of ideal chromatography. Properties of Concentration Discontinuities in Ideal Chromatography. Concentration discontinuities, or shocks, are the parts of a profile where the concentration jumps instantaneously from one finite value to another one, for an infinitesimal increase of the time. The system of nonlinear hyperbolic partial differential equations that describes the migration of large concentration bands in chromatography can propagate such d i ~ c o n t i n u i t i e s . ~This *~~ result is valid for mixtures as well as for pure compounds. The total number of shocks is equal to the number of components of the mixture ~ t u d i e d .The ~ origin and properties of these discontinuities have been recognized first by De V a u k 2 They have been discussed by Guiochon and Jacob5 and by Rhee et aL7, using the method of characteristics, which explains the appearance, growth, decay and collapse of the discontinuities.18 De Vault2 has shown that a propagation velocity along the column can be associated to every concentration. For a pure solute, the propagation velocity in ideal, nonlinear chromatography, u,, is given by u, =

1

+ F dC,/dC,

(7)

where C, is given by the isotherm and F is the phase ratio of the chromatographic system. In linear chromatography (isotherm: Fa), with C, = aC,), eq 3 reduces to the classical U, = u / ( l F a = k:. If the isotherm is not linear, the different concentrations move at different velocities, since dC,/dC,,, depends on C ., For a convex isotherm, such as a Langmuir isotherm, the velocity associated to a certain solute concentration in the mobile phase, u,, increases with increasing concentration of the solute in the mobile phase (see eq 5 ) . However, it is not physically possible for the higher concentrations, which move faster than the lower ones, to pass them. Otherwise, the profile would adopt an "S" shape, and, as pointed out by De Vault,2 it is not possible to conceive of three different values of the concentration of a compound at any point in space. The physically meaningful solution is the appearance of a discontinuity.2-8*1' We can express this in another way.18 The trajectory of a certain concentration is a straight line of slope u,. Thus, the concentration profile of a pure compound band can be seen as propagating through the column along straight lines, of slope u,, parallel to the (z,t) plane. These lines are called characteristics. They are all parallel in the case of a linear isotherm. We can draw from the injection profile as many straight lines as needed, starting from points of this profile, with slopes u,. As long as they do not intersect, the solution of eq 1 and 2 may be obtained from these lines (with the additional condition of peak area conservation). If two straight lines intersect, this procedure does not give the solution, since a different value of the concentration is associated to each line. Thus, the concentration C should have two different values a t this intersection point. The solution to this problem is obtained by allowing discontinuities to be considered as part of the solution. At a discontinuity, eq 7 is no more valid. It must be replaced by another conservation equation, expressing propagation of the discontinuity at a speed such that there is no accumulation of matter or of energy at the discontinuity. This mass balance then

tells us that the rate of propagation of the discontinuity is given by U us = 1 + FAq/AC

where Aq and AC are the differences in the concentrations of the solute in the stationary and the mobile phase on both sides of the discontinuity, respectively; Le., Aq is the difference between the concentrations of the solute in the stationary phase immediately before and after the passage of the discontinuity. Equation 8 had already been derived and discussed by Guiochon and Jacob,5 Rhee et al.,7 and Rouchon et al.",26 In the case of a binary mixture, there will be two such discontinuities, one separating the pure mobile phase from a solution of the single first component and the other separating a solution of the first component in the mobile phase from a solution of the two components. The concentration of the first component in these two solutions is different, unless the two bands are totally resolved, in which case the concentration of the first component is zero both sides of the second shock. Accordingly, there are three successive parts in the incompletely resolved band of a binary mixture: (i) a zone containing the lesser retained pure component, (ii) a zone containing a mixture of the two components, with their concentrations functions of time and position in the column, and (iii) a zone containing the single more retained component. In the case of a binary mixture, there are two couples of equations such as eq 7 and 8, one for each of the two components 1 and 2. They give the propagation velocities associated to a given concentration of compound 1 or 2, on a continuous part of the profile:

+

(23) Courant, R.; Isaacson, W.; Rees, M. Commun. Pure Appl. Math. ..

1952, 5, 243.

(24) Lax, P. D. Comments Pure Appl. Math. 1957, 10, 5 3 1 . (25) Valentin. P.; Guiochon, G. SeD. Sci. Technol. 1975, 10, 245 (26) Rouchon, P.; Schonauer, M.. Valentin, P.; Guiochon, G. In The Science of Chromatography; Bruner, F., Ed.; Elsevier: The Netherlands, 1984; p 131.

(8)

U

(9)

and

and the velocities of the concentration shocks of compound 1 or 2:

and

As we see later, it is important to consider the concentrations of compounds 1 and 2 such that the velocities u,,] and uS2are equal. By differentiation of eq 3 and 4 we obtain

This equation was derived first by Offord and Weiss3 and later by Glueckauf4 using different approaches. It is extremely important, as it shows that the concentrations of the two compounds (27) Glueckauf, E. Proc. R. SOC.London, A 1946, 186, 35

4146

Golshan-Shirazi and Guiochon

Thi>Journal of Phjsiral Chemistrv, Vol. 93, No. 10, 1989

in the mobile phase are not independeot of each other. On the contrarg, if the isotherms are known, the coefficients of eq 15 can be calculated and we have a relationship between these two concentrations, which can be used for the determination of the tuo band profiles. I f the two isotherms are both convex or concave, the two derivatives aC,,2/C3Cm,,and aCs,,/C3Cm,2 have the same sign and eq I5 eq has two real roots, one positive and one negative. A different propagation velocity corresponds to each root. The positive root corresponds to a continuous part of the profile and the negative one to a shock. U'e show now how analytical equations for the band profiles can be determined by a proper use of the previous equations, in the case when the equilibrium isotherms of the components of a binary mixture are given by the classical competitive Langmuir model. We will derive successively the retention time of the more retained solute (Le., the position of the second concentration shock), the equations for the profiles of the two components in the mixed band. behind the concentration shock of the second component (ix., the second zone of the band), the equation for the profile of thc second component in the rear part of its band, when i t is pure ( i s . , the third zone of the band), and the equation for the profile of the first component, between the two concentration shocks. Boundary and Initial Conuitions. We consider the column to be empty at the beginning of the experiment. The mobile phase pumped into the column does not contain any solute, except during a brief period, of width tp, during which a mixture of constant composition is introduced. The concentrations of the two components of the binary mixture are C," and C2", respectivel). This can be summarized as follows: t

< 0 , C , ( t , z ) = C*(t,z) = 0

0 < t 5 tp, C,(t,O) = C , " , C,(t.O) = tp < t,

C,(t.O) = 0 , C,(t,O) = 0

first component by the second one. The previous discussion shows that the key to the solution of the problem of the determination of the band profiles is in determining first the position of the second shock. Mocement of the Front of the Band during the Erosion ofthe Pulse Top. The injection profile is a narrow rectangle. As shown above, its front is a stable shock that splits into two discontinuities, whose height and velocity must now be calculated. The rear of the injection profile is not a stable shock and i t collapses as a continuous profile. The top points of the continuous profiles of the two components, at the concentrations Clo and C,", respectively, move faster than the shocks. Thus the top of the rectangular pulse is progressively eroded. It shrinks until it disappears. The propagation velocity of the shocks remains constant as long as their height is constant. We calculate first the height and velocity of the front shock and then the velocity of the back of the pulse top. I t is easy to differentiate eq 5 2nd 6 with respect to the concentrations of the two solutes in the mobile phase and obtain the partial differentials of the isotherm required to calculate the coefficients of eq IS. Let dC,,, r=dCnl.2

and

Equation 15 becomes

cublC2r2c 2 0

(CY

- 1

+ cublC1- b2C2)r

-

b2C, = 0

(19)

which can be rewritten as

(16)

This set of conditions defines a rectangular injection, with a concentration discontinuity a t both ends. As for the injection of a pure compound band, one of the two shocks is stable, while the other one will collapse into a continuous profile. In the case of a Langmuir isotherm, the stable shock is the front one. Since there are two compounds, however, the stable shock will split and generate two different shocks. The first one separates the pure mobile phase from a solution of the single lesser retained component. The second shock separates this latter solution from a solution of the two components. The rear front collapses and moves as a continuous profile. In the second zone of the band, where the two components are present, the velocity depends on both concentrations (see eq 9 or IO). The point a t concentration C20 on this continuous profile moves faster than the second shock (compare the velocities given by eq I O and 12, using the differentials of eq 5 and 6), so the top of the rectangular pulse narrows. When it disappears, the maximum concentration of the continuous part of the profile keeps moving faster than the shock, so the shock erodes slowly (see following sections). The first shock moves between pure mobile phase and a solution of the first compound, as in the case of a pure compound band. Since its velocity depends on the concentration jump (see eq 1 I ) , its position cannot be derived by using the solution for the single compound band profile, because the concentration profile behind this shock does not decay as rapidly. Because of the concentration interaction introduced by the competitive isotherm, the positive jump in the concentration of the second compound, at the second shock. is accompanied by a negative jump in the concentration of the first compound; the sudden arrival of the molecules of the second compound in the stationary phase results in the expulsion of a large number of molecules of the first one. As long as the t h o bands are not completely separated, the presence of the second shock moving in the rear of the first band results in an increase of the concentration of the first component in the first zone of the band. between the two shocks, and in a slower decay of the first shock. which, accordingly, moves faster than in the case of the pure band. This is the basis of the displacement effect of the

CI = rC,

( a - I)r -

ab,r

+ b,

This is a nonlinear differential equation of the Clairaut form. Its solutions are derived by considering the family of characteristic straight lines obtained by making r constant in the differential eq 19. There are two roots for the corresponding second-degree algebraic equation. We call here the positive root rl and the negative root r2. Thus, eq 19 has two sets of characteristics. Both are tangent to the parabola of eq 21 [CY

- 1

+ ablCl

-

bzCJ2 + 4cublb2CIC; = 0

(21)

in the (C,,C2)plane. Through any point of the (C,,C,) plane external to the parcbola pass two characteristic lines, one for each set. Since we are studying what happens to the rear part of the profile, in the second zone, the boundary conditions for the integration of eq 19 are C, = C," and C, = C," at t = 0. The roots rl (positive) and r2 (negative) are solutions of the equation

ab1C2"r2 -

(CY

- 1

+ a b , C I o- b2Cz0)r

--

b2CIo= 0 (221

The injection is made as a rectangular pulse of width tp. It can be considered as the sum of two simultaneous rectangular pulse injections of the same width and heights CIoand e,', respectively. The common front of these two pulses is a stable shock, but, since the compounds are different, it splits into two shocks that move a t different speeds. As long as the top of the pulse has not been eroded away, both shocks have constant height and move a t constant speed. W e need first to determine their height and velocity. Consider in the (C2.C,)plane the point F representing the top of the injection pulse (see Figure 1). When the band passes a t a certain point of the column, the concentration increases a t first. The point representative of the composition of the mobile phase moves from 0 to F. The concentration shock for the first compound will be represented by a point on the C , axis, since it moves in a region of the column where C2 = 0. On the other hand, the solution to eq 19 must be on the characteristic (straight line of

The Journal of Physical Chemistry, Vol. 93, No. 10, 1989 4147

Ideal Model of Chromatography in a Binary Mixture

C1axis, it must be represented by the segment OA. The second shock is represented by the segment AF. We note that since the corresponding root, r, is negative, the concentration C I Ais larger than Clo. There is a concentration increase of the first compound at its front, under the influence of the second compound. The positive concentration jump of the second compound at the second shock is accompanied by a negative jump of the first compound concentration, as we have explained earlier. The concentration at point A can be calculated by using eq 20, with C2A= 0 and r = r2:

A n

h

5"

E

Since the product of the roots of eq 22 is -b2Clo/cyblCz0,the value of rz is given by

v c

V

bZC1O rz = --ab I C20r I

N

where r l is the positive root of eq 22. Hence CIA =

r

a-1

a b l ( C z 0 r I / C l 0- 1)

= Clo(1

+A) ablrl

(25)

Equations 1 1 and 12 gives the velocities of the two shocks: 0

U

-

U

=

1 + FACs,2/ACm,z 1

+ F a z / ( l + blClo + b2Cz0)

0

(26) and

In-

U

A

B

us30A

=

1

+ Fal/(l + blCIA)

Equations 26 and 27 permit the calculation of the position of the shocks when time passes by, as long as the shock height remains constant, Le., until the width of the pulse top becomes zero. Movement of the Rear of the Band during the Erosion of the Pulse Top. On the band rear, the path of return to the pure solvent is from F to B and 0 (see Figure 1). This time, however, the shocks are not stable, so the pathways are through continuous profiles; Le., the point representative of the composition of the mobile phase in a given point of the column does not jump from F to B to 0 (as it does from 0 to A to F on the band front), but moves progressively, as a function of time on the segments F B and BO. The concentration of component 2 in B can be obtained from eq 20, with C1 = 0 and r = r l :

t'

r)

n

2E

..-

v

0

N

Equation 9 gives the velocity associated to a concentration Cl.on the rear of the elution band of the first component. Combination with eq 1 3 and 5 gives

-

-~ -G , -I al(1 + M C Z- C I / ~ I ) ) 0

0 0.0

1.0

2.0

3.0

40

I

cz (mol/) Figure 1. Schematics of the representation of the composition of the mobile phase in a slice of the chromatographic column in the (C2,Cl) plane. F, point representing the height of the injection profile (rectangular pulse); r , . r2, characteristics of eq 20 (the slopes of these straight lines are given by eq 22); A, point representing the height of the front shock originating from the injection profile; B, point representing the

+ b,Cl + bzcz)2

(1

dCm,l

B

(29)

and using eq 28, we obtain U

u, =

1+

Fa,(l (1

-

+ b2C2') -

(30)

_

+ b]Cl + b2C2)2

height of the plateau on the rear part of the second component profile. Isotherm coefficients are given in Table 11. Concentration ratio of the injection pulse: (A) 9/1; (B) 1/9.

which is the velocity associated to the concentrations C1and Cz, on a continuous profile part. Retention Time of the Front of the More Retained Solute. At time t , the second shock has moved the distance z, and we have the relationship

slope r l or r2) passing through F. As we have seen above, there are two such lines: one has a positive slope and intersects the Cz axis in B and the other has a negative slope and intersects the C, axis in A. Since the first shockis represented by a point on the

In same time. the rear Doint. a t the tOD of the continuous -. the .. -

The Journal of Physical Chemistry, Vol. 93, No. IO, 1989

4148

profile, has moved a distance z', such that

-

t = rP

(I

-

+ b2C2B)

F a l (1

+ b,C10+ b2C20)2

Comparison between eq 31 and 32 (or eq 27 and 30) shows that the rear end of the pulse top moves faster than the concentration shock. Thus, the pulse top shrinks. It will eventually disappear, a t the point where z = z'. The coordinates of that point, I, are (1

z , = utp

Fa2(l

+ b l C I o+ b2C2'),

+ b,C1° + b2C2'

- (1

+ b2C2B)/~) (33)

and

Fa2

1

U

+ b,C1° + b2C2'

)

(34)

The velocity of the last point of the profile ( C , = C2 = 0, on the band rear) is lower than that of the shock (cf. eq 27 and 30). Thus, the band broadens and dilutes, which is in agreement with the second principle of thermodynamics. W e need now to find the path of the shock. The slope of the shock trajectory, dzldt, is given by eq 26. This equation can be rearranged as

-

d(t - t, - Z / U ) d(z/u)

Fa2 1

(35)

+ blCl + b2C2

The velocity associated to a concentration C2on a continuous part of the profile is given by eq 30. The time a t which this concentration passes a t point z is

-

+ b2C2B) (1 + b , C, + b&)2 Fa,( 1

-

m2

(t

+ bQ)

- 1, - z / u )

=

z/u

(+ 1

= tp+

2 U

),

Fa2 b,C1 + b2C2

(37)

+

~ [ ( F a ~ z / u )-'i/b ~l C l o t p+ b2C20tp)l/2 +

c2 = b, + 1ab1rl

+

Combination between eq 35 and 37, rearrangement, and integration, following a classical procedure and deriving the integration constant from the coordinates of point I, leads to t

where N2is the number of moles of component 2 injected in the column, t is the column total porosity, and S is the cross-section area of the column. Equations 40 and 42 are very similar to the corresponding equations obtained for the retention time of a band of a single component and for the loading factor, to which they reduce when C,' = 0. Equation 40 gives the retention time of the second shock of the band. In front of it, a zone of pure first compound is eluted. Behind it, there are two zones, one containing a mixture of the two components and the other one, the pure second component. W e calculate first the continuous parts of the elution profiles of the two components behind the second shock. Continuous Elution Projlles of the Two Components in the Mixed Zone. The retention time associated to a concentration C2 of the second component, after the second shock, is given by eq 36, where z is replaced by the column length, L, and z / u by the holdup time, to. The two concentrations, C , and C,, in this equation are related by eq 20, where r is the root rl (positive) of eq 22. Replacing C, in eq 36 by its value taken from eq 20 and solving for the concentration of the second component gives the elution profile, Le., the variation of the concentration of the second compound in the mixed zone, as a function of time, a t column exit:

This equation is very similar to the one giving the profile of a single-component band, except for the replacement of u2 by ya2 and of b2 by b, ab,rl.'s The concentration of the first component is obtained by combining eq 20 and 43:

which can be rewritten as

(1


tp((r- l)/r)I2 (38)

c, =

1

b,

+ b,/ar,

Y=

- ablrl

-

I

+b2~2B

hr,

+ b2

+ b2

tB

(39)

Equation 38 can be used to determine the elution time of the front of the second component, which is also its retention time, merely by replacing z by L. Equation 38 can then be rearranged to where r,,, is the retention time of the second shock, to is the holdup time of the column ( z / u ) , tR.2' is the retention time of the second component of the mixture, when analyzed as a zero sample size plug (Le., under conditions of linear chromatography), and L f is the loading factor, defined as

L f = tp

C,'b,

+ Cz'b, + (Y - I ) / Y ?R.2'

-

(Y_- 'R,I0 CY

t -

-

(44)

t p - to

where tRlo is the retention time of the first component under linear conditions. This equation is also very similar to the one obtained for the rear profile of a single-component band, except for the replacement of a, by a , y / a and of bl by ( b , + b 2 / a r l ) . Equations 43 and 44 give the elution profiles of the two components of the binary mixture behind the second shock. They are valid until the concentration of the first component becomes zero. The corresponding time is

with a

[

= tp

+ t o + -YcY( t R , ] '

-

to)

(45)

The point representative of the composition of the mobile phase in a (C2,C,)plane reaches B (see Figure 1). The concentration of the second component is equal to C2B(see eq 28), which is also the value obtained by combining eq 43 and 45. After the elution of the first component is finished, eq 43 is no more valid, because there is no first compound in the solution. The concentration profile of pure component 2 is given by the classical equation, derived from the velocity associated to a certain concentration of the second compound alone

The corresponding profile is obtained by solving for C2:lS

to

Inserting y from eq 39 and using eq 28 give

Lf = (1 with18

+ $)Lf,2

I t turns out that the velocity associated to the concentration C2 on the profile given by eq 47, which is the elution profile of the second component when it is eluted pure (in the third zone of the


Ideal Model of Chromatography in a Binary Mixture chromatogram), is lower than the velocity associated to the same concentration C2B(eq 26) on the elution profile of the second component, when in the presence of the first one (in the second zone of the chromatogram), in spite of the fact that the concentration of the first component has just become zero when the concentration of the second one becomes equal to CZB. Thus, during the migration of the second zone, a plateau of the second component at the concentration C2Bappears on the rear of the band and grows slowly. This phenomenon is a result of the coupling of the elution profiles through the competitive isothen"' The time a t which the concentration plateau disappears is obtained by writing that the concentration of the second component in eq 47 is equal to C2B:

correspond to the boundary conditions, CIo and C20,anymore. The point representing it moves from F. The coordinates of that point (Le., the maximum concentrations C, and C2 at the shock) are given by eq 51 and 52, respectively. These are the values that correspond to the second shock and that we should use for the calculation of the properties of the first shock. The relevant equations are now eq 19 and 25. In the latter, however, Clo and C20 should be replaced by Cl,Mand C2,M, respectively, as given by eq 51 and 52. The concentration Cl,A'of the first component on the front side of the second shock, at the end of the column, becomes

(54) Combining with eq 51 gives

The width of the plateau is given by

(55)

When the plateau is eluted, the concentration profile of the rest of the band (zone 3) is given by eq 47. The elution ends when the concentration of the second component becomes equal to 0. This time is derived from eq 47:

t, = tp

+ tR,2O

(50)

The continuous parts of the profile in the second zone of the band are given by eq 43 and 44, which can be used to calculate the heights of the bands a t the shock. Concentrations of the Two Components in the Second Shock. We want now to calculate the concentrations of the first and second components a t the shock, on the high retention time side of it (in the second zone of the band). For reasons of continuity, they are obtained by combining eq 43 and 44, which give the variations of the concentrations of the two components on the continuous parts of the profile, with the retention time of the shock, itself given by eq 40. We obtain

and

Equation 54 relates the concentrations of the first component on both sides of the second shock. Continuous Elution Profile of the First Component between the Two Shocks. A solution of the first component in the mobile phase is eluted behind the first shock. We now derive the relationship between the elution time of a concentration C, and that concentration. First, all concentrations in the continuous part of the profile of the first compound which is eluted before the second shock move at a velocity given by eq 9, where Cl,sis a function of C,,? only, since C2,,, is zero. Hence the time at which the concentration C, is located a t the abscissa z of the column is given by the relationship

U

(1

]

+ blCl')2

(56)

where td and zd characterize the trajectory of second shock and C,' is the concentration of the first component of the mixture in the first zone of the chromatogram. Equation 56 is valid for all concentrations in the continuous part of the profile of the first component in the first zone of the band, between the two shocks. The position and the time of the second shock are related by eq 38, which can be rewritten as

The ratio of these two concentrations is

-C2,M - -

LI' J 2

CI.M r,(I - a

+ aLf1/2)

(53)

Concentration of the First Component at the Front of the Second Shock. We have shown above that immediately after the injection of a rectangular pulse into the column, the stable front shock splits into two shocks, one front shock between the pure mobile phase and a solution of the sole first component and a second shock between this last solution and a solution containing the two components. The points representing these two concentration shocks in a (C2,C,)plane are A and F, respectively (see Figure 1). They are both on one of the characteristics of the differential eq 20. This characteristic line goes through point F (C2O,CIo) and has a slope r2, solution of eq 22. This equation is used to calculate the root r2 and the value of the concentration CIA. It is important to note that the concentration of pure component 1 on the front side of the second shock depends on the concentrations of both components on the rear side of the second shock. Equation 15, hence eq 20, requires that the points representing the two shocks in the plane (C2,C,)be on the same characteristic line of eq 20. When the plateau at the top of the injection pulse is eroded, the height of the second shock starts to decrease. It does not

As we have shown above, the concentration of the first compound on the back side of the second shock is given by eq 44. In the case of the position zd in the column, instead of the column end, eq 44 is rewritten as

In the same way as we have derived eq 54, it can be shown that . the concentrations Cl' and C, are related by (59) Substituting eq 59 into eq 58 gives

From eq 57 and 60, we can derive

The Journal of Physical Chemistry, Vol. 93, No. 10, 1989

4150

+ b2Czo + (7 - I ) / y ) ( l + bjC1’)’ (61) Fa2((cy - l ) / c y + b,C,’)2 ub2tPCzo( I + b l r l / b 2 )1( + bICI’)’ (61a) Fa2((cy - I ) / m + BIG,')'

ut,(bIClo Id

=

and t d is given by eq 60. zd and td are the coordinates of the second shock in the column and give its trajectory. Inserting eq 60 and 61a in eq 56 yields = tp

+ 2 + Fa, U

u(l

+ blC1’)2

+ -

-I

At the end of the column, z / u is equal to b , r , / b 2 ) we , have t=t,+t,+

+

- ])/(I

to.

N

Since y - cy = -(cy

This is the equation of the concentration profile of the first component of the mixture, in the first zone of the chromatogram, when it is eluted pure. Retention Time of the First Component. As we have shown previously, the front of the rectangular injection profile splits into two shocks. The amplitude of the first shock remains constant until the top of the rectangular injection disappears and the continuous profile of pure component 1 intersects the front shock. The rear continuous part of the profile of the first component is given by eq 62. In the case when the top of the rectangular injection profile has not been totally eroded, the equation is obtained by rewriting eq 62, with C,’ = C I A : t =

t,

+ 5 + Fa, U

Z

u(l

+ b,CIA)2 cy-

rbCo cy2[(.

- l)/a

1

+

Uz (

I +

1

1

+ b,CIA

The coordinates of the intersection point, J, are ut,(l

+ blC,A)2

b2C20(cy - I )

i =

-1

Fa, b ,C I A

d [ ( a- I ) / a

of eq 62, the integration of the differential equation becomes difficult and we have been unable to derive its exact solution. The h-transform approach is also unable to supply an algebraic equation for the retention time of the first shock.20 There is an alternative, an easy numerical solution. The retention time of the first shock is evaluated by integrating the two parts of the concentration profile of the first component, in the second zone of the chromatogram (finite integral with known boundaries) and in the first zone (finite integral with one floating boundary), and by determining the floating boundary assuming mass conservation of the first component. This is faster than determining a numerical solution of the differential equation. Case of Complete Resolution between the Two Bands. When the relative retention between the two components of the mixture is large enough, or when the column length is sufficient, the two components become completely separated. Let assume that we have a long enough column, and call K the point a t which the base line between the two profiles is reached for the first time. The concentration of the first component a t this point is obviously equal to 0, while that of the second component is C2B. The trajectory of the second shock is given by eq 38. The continuous profile of the second component is given by eq 36, as long as both components are present simultaneously in the column. Their intersection is the position of the second shock, as we have derived under Movement of the Rear of the Band during the Erosion of the Pulse Top. Equations 36 and 38 remain valid until point K is reached, since we are still in the region where the two components are simultaneously present in the column. Introducing the proper values of the concentrations (C, = 0, C2 = CZB)in eq 36 gives

Inserting this value of the time in eq 38 and solving for z k gives the abscissa of K, Le., the column length a t which a complete resolution is achieved: u(./(a

(64) bIClA]*

The trajectory of the shock O A follows the equation t = -


+ blCIA12

and

dz

u

1

) :()::‘

+ b2C2B)2 = t , + -

I+-

(71)

The trajectory of the shock of the second component when pure is given by

The coordinates of the intersection point, L, where the plateau disappears, are given by

+ b,C,’

The continuous part of the band profile of pure component 1, in the first zone of the chromatogram, is given by eq 62. Eliminating C,’ between eq 62 and 68 and integrating the differential equation obtained should give the retention time of the front shock of the first component (the integration constant is obtained by evaluating the integral at point J ) . However, because of the second term

b2G0(1 + b1r,/b,)f,

(70) Fa2 Until point K is reached, the plateau on the rear part of the second component profile, a t concentration C2B,remains stable. At point K, the maximum concentration of the second component band is equal to C2B,and this plateau constitutes the flat top of the band. Then, beyond point K, the plateau begins to erode. It disappears a t the intersection between the continuous profile of pure component 2 and the front shock of the pure component 2. At point K, the concentration shock of the second component becomes a pure compound shock. While the plateau is disappearing, the rear continuous profile of component 2 is given by the equation (1

where C I Ais given by eq 25. Beyond the point I ( t l , z J ,see eq 33 and 34, the height of the shock A F decreases, while the shock O A remains constant until point J([,,z,). After this point, the amplitude of the front shock decreases. Since it is a concentration shock of the pure component I , its trajectory is described by

-

zk =

(73) and (74) When the column length exceeds zl, the bands of the two com-



TABLE 11: Numerical Values of the Coefficients of Eq 5 and 6 for the Two Compounds Studied Fieures 1-7 Fieures 8 and 10 Fieure 9

TABLE I: Equations Used for the Determination of the Elution Profiles of the Two Components of a Binary Mixture

isotherms

~~

a,

alCm,I = 1

"*I

(5)

+ b,C,,, + b2C,,,,

a2

b,

and

b2

24 28.8 (a = 1.2) 2.5 3.0

24 26.4 ( a = 1.1) 2.5 2.15

pounds are completely separated beyond the point of abscissa z , in the column and elute as resolved bands. The band profile of the second component can then be described as that of a pure compound. The continuous part of the profile of the second component is described by eq 47. Its retention time is given by1*

loading factors

+ %)Lf,z

Lf = ( I

24 43.2 ( a = 1.8) 2.5 4.5

tR,,

where rl is the positive root of eq 22 and L , , is given by

= t p + to

+ (tR.20 - t,)(l

- Lf,21/2)2

(75)

where Lf,2is given by eq 42.

Results and Discussion First Zone of the Chromatogram elution profile of the first component t = tp + to + (tR,I' - to) x

with

[

1

(1

a-1

+ blCI')2- L f , 2 ( y

1

( ( a- I ) / a

+ blCl')2

Y = (ablrl + b t ) / ( b f l + b,) concentration of the first component on the front side of the second shock (55)

Second Zone of the Chromatogram position of the second shock tf.2 = tp + 20 + Y(tR.2' - to)(1 - Lf1/2)2 (40) concentrations of the two components at the rear of the second shock rl 1 - a + aLf1I2 (51) = b2 + ablrl 1 - ~ f 1 / z and Lr' 12 1 (52) c2*M = b2 ablrI 1 - La12 elution profiles of the two components in the second zone

+

c,

=

1

b2 + ablrl

and

end of the second zone and beginning of the rear plateau Y tg = t p + t o + i(ZR,1' - t o )

(45)

Third Zone of the Chromatogram length of the rear plateau

concentration of the rear plateau of the second component C2B =

a-I

b2 + ablrl elution profile of the second component behind the rear plateau (47) end of the second-component profile t, = t p + tR.2'

We have applied the equations derived above and summarized in Table I to the determination of the elution profiles from columns of increasing lengths of the bands of the two components of a binary mixture. The column efficiency is assumed to be constant. The values of the coefficients of the isotherm used in this case (eq 5 and 6 ) are summarized in Table 11. Change in the Band Profile during Its Migration along the Column. Figures 2-8 show solutions of the ideal model derived as described in the previous section. These solutions are presented in order of increasing time spent in the column. They are not profiles inside the column, however, but elution profiles, corresponding to columns of increasing lengths. For the sake of clarity of the figures, only the individual elution profiles of the two compounds are shown. The real chromatogram, Le., the sum of the two profiles, is not represented. However, each profile is completed by the solution of the semiideal model corresponding to a 5000 theoretical plate column, in order to show the effect of the mass-transfer kinetics and axial diffusion (16). Figure 2 shows the elution profile of a 9/1 mixture for a very short column (less than 1 cm, shorter than z,, eq 33). The time spent in the column, smaller than t,, is too short to permit complete erosion of the top of the rectangular injection band. The profiles of the two components exhibit the beginning of a separation, however. The two front shocks are slightly separated. The profile of the first component exhibits two plateaus. The first one corresponds to the concentration CIA(see eq 25, with rl being the positive root of eq 22) and is limited to the times of the first and second shocks. The second plateau has a concentration C,'. The profile of the second component on Figure 2A has also two plateaus. The first one has the concentration C20. The second one (not visible on Figure 2A, but seen on Figure 2B) corresponds to the concentration C2B(eq 28, with rl the positive root of eq 22). Figure 2C shows the profiles obtained with a 1/9 mixture. Now, the second component strongly displaces the first one and a very thin, tall spike appears a t the front of the profile of the first component. This is illustrated on Figure 2D, which shows an enlargement of this profile. Figure 3 shows the elution profiles of the binary band from a slightly longer column (about 1 cm, larger than z,, but less than zl, eq 66). The top of the rectangular injection pulse has completely disappeared. Each profile has still one plateau, however. The profile of the first component has a plateau at concentration CIAwhich has not been completely eroded yet, although the second shock is slowing down its migration rate, the maximum concentration C2,,, of the concentratiori shock of the second component decreasing since the injection plateau has disappeared. The profile of the second component has a plateau at the concentration C2B, which remains stable until the two bands are completely separated. Figure 4 shows an intermediate solution, corresponding to a column of moderate length (2.6 cm for Figure 4A, 4 cm for Figure 4B). The plateaus of the first-component profile have all disappeared. The concentration shocks are moving at a decreasing velocity. The second shock moves more slowly than the first one, and the distance between them increases. The profiles slowly

The Journal of Physical Chemistry, Vol, 93, No. 10, 1989

4152

Golshan-Shirazi and Guiochon 3

1

u Q o

x

II

1

2

3

4

5

6

7

8

9

Time ( s e d

Time ( s e d

I I

3

5

7

9

11

13

Time ( s e c )

2

3

Time ( s e c )

Figure 2. Profiles calculated for the elution of a large concentration band on a very short column. The top of the rectangular injection pulse has not yet disappeared. Flow rate: 5 mL/min. Column diameter: 4.6 mm. Efficiency, 5000 plates. Injection time, tp. 1 s. Isotherm coefficients are given in Table 11. ( A ) Concentration ratio of the injection pulse: 9/1. Column length: 0.44 cm. CIo = 4.5 M; Czo= 0.5 M. 1, elution profile of the first component, ideal model; 2, elution profile of the first component, semiideal model; 3, elution profile of the second component, ideal model; 4, elution profile of the second component, semiideal model. (B) Enlargement of the profile of the second component shown on (A), to show the profile tail and the plateau at the rear. Numbers on curves, see (A). (C) Concentration ratio of the injection pulse: 1/9. Column length: 0.78 cm. C , O = 0.5 M; C,' = 4.5 M. Numbers on curves, see (A). (D) Enlargement of the profile of the first component shown on (C), to show the profile front and its very sharp spike. Note that the maximum on the semiideal profile (see text) is at 3.9 mol/L only. Numbers on curves, see (A).

broaden. The second-component profile still has a plateau at C2B. The formation of this plateau and its stability during the entire process of separation between the bands of the two components explains what has been called the tag-along effect.I6 The coelution of the two components accelerates markedly the migration of the second component, when its relative concentration in the studied

mixture is small, and results, through the stability of the plateau at C2B,in the spread of the second component over a long fraction of the column. It is important to note that, for a given mixture and phase system, the height of the intermediate plateau on the second-component profile increases and the length of this plateau decreases with increasing proportion of the second component in


3

7

5

9


13

11

0

Time ( s e d

0

0

B 0

'?

0

0

0

1-

2

h

E?

-0

d

8 0 ,-

0

E C

C

4

6

8

10

12

14

16

18

20

22

Time ( s e d

Figure 3. Profiles calculated for the elution of a large concentration band on a short column (see Figure 2). The top of the rectangular injection

pulse has disappeared on the profiles of the two components, but the height of the first shock is still equal to CIA. Same conditions as for Figure 2. (A) Concentration ratio of the injection pulse: 9 / 1 . Column length: 1.0 cm. Numbers on curves, see Figure 2A. (B) Concentration ratio of the injection pulse: 1 /9. Column length: 1.4 cm. The differences between the profiles predicted by the two models are so small that we could not distinguish between profiles 1 and 2 nor between profiles 3 and 4. the mixture (compare Figure 4A,B and see eq 49). Thus, t h e tag-along effect becomes very important for a trace amount of the second component. Purification of early eluted components will require the injection of moderate sample sizes; otherwise the

C

-

3

Figure 4. Profiles calculated for the elution of a large concentration band on a normal length column. Same conditions as for Figure 2. Note the plateau on the rear profile of the band of the second component, at C2B.

The front plateau at CIAhas disappeared on the first component profile. The two bands have considerably broadened and their height is much smaller than at injection. (A) Concentration ratio of the injection pulse: 9/1. Column length: 2.6 cm. Numbers on curves, see Figure 2A. (B) Concentration ratio of the injection pulse: 1/9. Column length' 4.0 cm. Numbers on curves, see Figure 2A. separation from later eluted impurities may become very difficult and costly. Figure 5 corresponds t o the shortest column permitting total separation of the two bands ( L equals zk,eq 70, 4 c m for Figure

The Journal of Physical Chemistry, Vol. 93, No. 10, 1989

4154


(D

0

A

2

I

m 0

q 0

1 h

i t"

-0

u

5

0 N

0

-

0

3

0

30

20

40

50

6C

73

150

200

2i0

Time (sec)

300

350

400

450

503

Time ( s e d

m 0 1

3

25

50

75

100

125

150

Time (sec)

Figure 5. Profiles calculated for the elution of a large concentration band on a long column. Same conditions as for Figure 2. The separation at the column end is just total. The plateau on the top of the second component profile at C2Bremains. (A) Concentration ratio of the injection pulse: 9 / l . Column length: 4.1 cm. Numbers on curves, see Figure 2A. (B) Concentration ratio of the injection pulse: 1/9. Column length: 6.31 cm. Wumbers on curves, see Figure 2A. 5 A , 6.3 cm for Figure 5B). There is an exact return to the base

line, the second concentration shock of the first component has just completely vanished, in the same time that its tail under the second component has disappeared. However, the plateau a t CzB on the profile of the second compound is still present, since this

75

100

'25

Time (set) Figure 6. Profiles corresponding to experimental

conditions such that the plateau at the top of the second component profile has just vanished ( L = z,, eq 73). Same conditions as for Figure 2. (A) Concentration ratio of the injection pulse: 9/1. Column length: 25 cm. Numbers on curves, see Figure 2A. ( B ) Concentration ratio of the injection pulse: 1/9. Column length: 6.7 cm. Numbers on curves, see Figure 2A. plateau begins when the concentration of the first component becomes zero, and it has a finite length. As long as the plateau at CzBcontinues to exist, the front of the second component moves at the same velocity. Figure 6 shows the chromatogram obtained a t the end of a column of length 2 , (see eq 73, 25 cm for Figure 6A, 6.37 cm for



Figure 6B). The plateau on the top of the second-component profile has just disappeared. The time it takes for this plateau to disappear increases rapidly with decreasing proportion of second component of the mixture (for a given phase system and pair of compounds). When the last plateau on the profile of the second component has vanished, the bands continue to drift apart (see Figure 7). The velocities of both shocks decrease monotonically, since their heights decrease (see eq 8). The profile of the second compound becomes rapidly identical with the profile that would be obtained for the same amount of the second compound if injected pure.I8 On the other hand, the profile of the first component is eluted earlier and is narrower than the profile of the same amount of material injected pure. This results from the displacement effect of the first component by the second one, which we have mentioned earlier. Comparison between the Solutions ofthe Ideal and the Semiideal Models. The exact integration of the system of eq I , 2, 5 and 6 was long considered to be impossible, so we derived an algorithm for the computer calculation of numerical solutions of this ~ y s t e m . " - ~Using ~ * ~ ~a finite difference method, we obtained diffused solutions. It is possible to show that the proper choice of the column length and time increments used for the numerical integration permits the simulation of the effect of the column finite efficiency on the pr0fi1es.I~ The errors introduced by replacing the partial differential eq 1 and 2 by finite difference equations are of the second order. They are equivalent to the contribution of an apparent diffusion term. In order to determine the influence of the column efficiency, we have also carried out numerical calculations using the numerical solution of the semiideal model derived previously.l'-'3s16 W e have plotted the resulting profiles on Figures 2-7. The agreement between the sets of profiles is impressive. The retention times of the shocks of the ideal model is nearly identical with that of the shock layers of the real profiles.I8 Further comparison between the results of the two approaches is instructive. Figure 8 shows the profiles obtained with the ideal model (infinite column efficiency) and with columns having IOOOO, 5000. and 2500 theoretical plates, respectively. Several important observations must be made. First, there is a very substantial agreement between the different profiles. They obviously are highly similar and have common features. This demonstrates that the ideal model, in spite of its extreme simplification, when it assumes the column to have an infinite efficiency, still keeps the essential features of nonlinear chromatography a t large concentrations. Second, the concentration shocks of the ideal model are relaxed by the apparent diffusion in the semiideal model, but only to a moderate extent. The shocks are replaced by extremely steep fronts. The steepness of these fronts increases with increasing height of the concentration shock and with decreasing value of the apparent diffusion coefficient. This confirms the validity and the interest of the concept of shock layer.18 The shock layer replaces the shock concept of the ideal model. A similar concept exists in aerodynamics. The shock theory would demand that the composition of the eluent jumps from one value to another one over a distance equal to one molecular diameter, clearly a physical impossibility in a continuous fluid. The front of the shock layer moves at the same speed as the true shock of the ideal model, but the shock layer has a finite thickness, a t the difference of the ideal shock. Thus, the foot of the profile, the inflection point of the front, and the band maximum do not coincide anymore, and the retention time of the band maximum depends on the apparent diffusion coefficient, Le., on the kinetics of the mass transfer between phases. Third, the rear shock layer of the first component profile is much thicker than the front shock layers of the two profilcs. because in the case of a rear shock layer. the concentration is decreasing, and the drive to apparent diffusion, i.e.. the differences between the chemical potentials of the compound at the liquid-solid i n terface, in the mobile phase inside the particles and i n thc bulk solution decreases and becomes rapidly small.

2 2

IL /

200

1

250

3(

400

350

450

Time (sec)

Figure 7. Profiles calculated for the elution of a large concentration band on a long column. The separation between the two bands is increasing. The profile of the second component is now identical with the one that would be obtained with the same amount of pure material. This is not true for the first one. Same conditions as for Figure 2C. Concentration ratio of the injection pulse: 1/9. Column length: 25 cm. Numbers on curves, see Figure 2A.

3 /

1

240

260

280

300

320

340

Time (sec)

Elution profiles of a two-component mixture with a competitive Langmuir isotherm. Column length: 25 cm. Column i.d.: 4.6 mm. Flow rate: 5 mL/min. Isotherm parameters, see Table 11. 1 , exact solution for the ideal model; 2, numerical solution for a 10000 theoretical plate column: 3, numerical solution for a 5000 theoretical plate column; 4, numerical solution for a 2500 theoretical plate column. Figure 8.

The Journal of Physical Chemistry, Voi. 93, No. 10, 1989

4156


N

N

0

0

.1

n

m 0 a

0

3

a

0

m

x

1h

2-

h

i

i

ze

EX

- C

- 0

ti

U C

0

0

0

0 a *

C

a

0

0 h

C

C

4 a c

33

C

-

C

220

240

260

280

300

Time ( s e c )

Figure 9. Comparison between band profiles predicted by the ideal model and profiles calculated for a 5000 theoretical plate column. Same conditions as for Figure 8, except isotherm coefficients of the second component (see Table 11). Note the steepness of the shock layers and the smoothing out of the plateau on the rear of the second component band profile. Relative concentration of the two components: 119. I , elution profile of the first component, ideal model; 2, elution profile of the first component, semiideal model; 3, elution profile of the second component, ideal model; 4, elution profile of the second component, semiideal model. Lr.1 = 0.52%: Lf.2 = 4.68%; L f = 5.15%.

Finally, the plateau on the rear of the profile of the second component is entirely smoothed away by diffusion in the case of a 1/9 mixture, because it is very high (fast diffusion) and very short. I n the case of a 9 / 1 mixture it remains well stable and only the ends are dampened. Naturally, because of the finite column efficiency, the band profile of the second component does not end up abruptly as predicted by the ideal model, but a short tail appears, which ends as a Gaussian profile and corresponds to the classical band broadening phenomenon observed in linear, nonideal chromatography (in linear, ideal chromatography the injection profile is conserved all along the elution process). Figures 9 and I O compare the solution of the ideal model to the numerical solution of the semiideal model, only in the case of a 5000 theoretical plate column for the sake of clarity, under different simulated experimental conditions. These figures give the elution profiles of a 1/9 and a 9/1 mixture of the same two components. In the former case (Figure 9), the ideal and the semiideal ( N = 5000 theoretical plates) profiles are again very close. The thicknesses of the shock layers are very small, even for the rear shock layer of the first component. The only significant difference between the ideal and semiideal profiles is the disappearance of the plateau on the rear of the second-component profile. The plateau has vanished into what has already been observed as a strong change in the profile slope and curvature occurring when the elution of the profile of the first component ends ( 1 6). In the case of the 9 / 1 mixture (Figure IO), the differences between the profiles predicted by the ideal and the semiideal models are much more important. In spite of its efficiency (5000 theoretical plates), the column produces a front shock layer for the second-component band which is not steep at ali. Similarly, the rear concentration shock on the first component band profile is a mere inflection of the profile, which would hardly be noticed bq an unforewarned experimentalist. This is certainly

IO

200

220

240

260

280

300

320

-0

Time (sec)

Figure 10. Comparison between band profiles predicted by the ideal

model and profiles calculated for a 5000 theoretical plate column. Same conditions as for Figare 8, except relative concentration of the two components: 9/ 1. 1 , elution profile of the first component, ideal model; 2, elution profile of the first component, semiideal model; 3, elution profile of the second component, ideal model; 4, elution profile of the second component, semiideal model. Le, = 4.68%; Lf,2 = 0.52%; Lf = 4.42%. (N.B. Lf< L , , ,see eq 41). Note the softness of the shock layers on the rear of the first component band profile and the front of the second component band profile. due to the relatively small concentration jump corresponding to the shock observed on the ideal profile. On the other hand, the long plateau on the rear of the second-component profile is almost entirely preserved, except for the smoothing of its tail. Conclusion The availability of an exact solution for the ideal model and the closeness between the profiles derived from this model and those obtained by numerical integration, using the semiideal model, permit the detailed study of the influence of the various experimental conditions on the profiles of the two compounds, hence on the maximum theoretical yields and production rates of a preparative column for the two compounds. The agreement between experimental results and those predicted by the numerical, semiideal model gives validity and relevance to the profiles calculated as solution of the ideal model. Certainly, the replacement of the true concentration shocks of the ideal model by shock layers introduces a loss in both the yields and production rate of the two components of a binary mixture. It is very useful, however, to be able to compare the actual values of these characteristics for a practical application to those derived from the more theoretical ideal model. These latter results constitute the theoretical, upper limit which can never be exceeded in practice. If the difference is moderate, for example, it might be unreasonable to try and improve the performance of the process by using a more efficient column. If the difference is large, an estimate of the possible gain permits to assess rapidly the economics of an increase in the column efficiency. Comparison between the profiles obtained for a given mixture and those predicted by the ideal model permits a rapid estimate of the best parameters for a competitive Langmuir isotherm, accounting for the profiles obtained. The values of a, and a2 can be derived from the chromatogram obtained with a very small



injected amount (linear chromatography). The values of bl and

b2 could be derived from the determination of some characteristic feature of the double band profile. For example, if the proportion of the second component in the mixture is lower than SO%, the height of the plateau on the rear of its profile can be measured accurately, with an efficient column. Several determinations of CzB, for different values of CIoand C20,would permit the calculation of bl and b2. All these applications are under investigation. Results will be reported elsewhere.

Acknowledgment. This work has been supported in part by Grant CHE-8715211 of the National Science Foundation and by the cooperative agreement between the University of Tennessee and Oak Ridge National Laboratory.

CI'

coefficients (origin slope) in the Langmuir isotherm (eq 5) coefficients in the Langmuir isotherm (eq 5) concentrations of the first and second solutes (components of the mixture) in the mobile phase, respectively (eq 1 and 2) concentrations of the first and second solutes in the stationary phase at equilibrium with the mobile phase, respectively (eq 1 and 2) concentration of the first component in the first zone of the chromatogram (eq 56) concentrations of the two components in the sample plug introduced in the column (eq 16) concentration of the first component in point A (Figure 1) concentration of the second component in point B (Figure 1)

cl .A'

concentration of the first component on the front side of the second shock (eq 54 and 55) concentrations of the two components at the rear of the second shock (eq 51-53) phase ratio (eq 1) isotherms of the two components (eq 3 and 4)

tP tR.lOj IR,2'

S Z CY

Y Aq, AC At c

amounts of the two components in the injected pulse (eq 42) ratio dC,,,l/dCm,2 (eq 17) positive and negative roots of eq 22, respectively linear velocity of a concentration shock (eq 8) linear velocity of the mobile phase (eq 1) velocity associated to a concentration (eq 7) time (eq 1) elution time of a concentration C2*on the band tail (eq 45); this is also the time when the elution of the first component band ends time when the elution of the plateau of the second component zone ends, at concentration C2B(eq 48) trajectory of the second discontinuity of the chromatogram (eq 56, 60, and 61) retention time of the second concentration shock (eq 40) coordinates of the point I, where the top width of the injected sample pulse of the second component shrinks to 0 (eq 33 and 34) coordinates of the point J, where the top width of the injected sample pulse of the first component shrinks to 0 (eq 66 and 67) coordinates of the point K, where the zones of the two components are just resolved (eq 69 and 70) coordinates of the point L, where the concentration plateau on the tail of the second component at C2Bjust disappears (eq 73 and 74) holdup time of the colunn (L/u) width of the injected pulse (eq 16) retention time of the two components under linear conditions cross-section area of the column (eq 42) abscissa along the column (eq 1) relative retention of the two components at infinite dilution (eq 18) convenient combination of parameters (eq 39) amplitude of the concentration shocks in the stationary and the mobile phase, respectively (eq 8) time width of the plateau at concentration C2Bon the tail of the second component zone (eq 49) column total porosity (eq 42)

Subscripts column capacity factor at infinite solute dilution (k,' = Fa) loading factor for the mixed zone (eq 41) loading factor for the second component (eq 42)

1, 2

m, s

the first and the second eluted components of the sample, respectively the mobile and the stationary phase, respectively

Analytical Solution for the Ideal Model of Chromatography in the Case

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