Analytical Solutions of the Ideal Model for Gradient Liquid

Anal. Chem. 2006, 78, 7828-7840

Analytical Solutions of the Ideal Model for Gradient Liquid Chromatography Weiqiang Hao,*,† Xiangmin Zhang,† and Keyong Hou‡

Department of Chemistry, Fudan University, Shanghai 200433, China, and Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, China

The analytical solutions of the ideal model for gradient elution that ignores the influence of the solute concentration on the retention factor (k) were studied by using the method of characteristics for solving partial differential equations. It is found for any gradient profiles and solvent strength models used that the concentration of the solute will be discontinuous where the mobile-phase composition is. On a given characteristic curve, the product of the concentration and the retention factor is kept constant at the point where the concentration is continuous. At the point where the concentration is discontinuous, the product on the left side of this point is equal to that on the right side. We also discussed the basic equations to predict the retention time in gradient elution and introduced the injection time into them. For linear solvent strength stepwise and linear gradient elution, general expressions were proposed for the prediction and they can be used as the basis to derive others for specific gradient modes such as single linear, stepwise, and ladderlike gradients. For these modes, simple expressions to account for the band compression and the concentration change during the elution were also given. Gradient elution enhances considerably the separation and peak detection capabilities of liquid chromatography.1-4 It is based on the programmed change in mobile-phase composition, flow rate, and column temperature, with the most important mode the change in mobile-phase composition. Compared to isocratic elution, the theoretical studies for gradient elution are more complicated, because the retention behavior of the solute in this operation mode will also vary with the change of the mobile-phase composition. In the last decades, various approaches have been proposed to predict the retention time of the solute in gradient elution. However, even for simple gradient profiles, it seems that explicit expressions can only be obtained by using the so-called linear solvent strength (LSS) model.5-6 This model assumes that * To whom correspondence should be addressed. E-mail: [email protected]. † Fudan University. ‡ Institute of Chemical Physics, Chinese Academy of Sciences. (1) Freiling, E. C. J. Am. Chem. Soc. 1955, 77, 2067-2071. (2) Dorsey, J. G.; Cooper, W. T.; Siles, B. A.; Foley, J. P.; Barth, H. G. Anal. Chem. 1996, 68, 515-568. (3) Antos, D.; Seidel-Morgenstern, A. Chem. Eng. Sci. 2001, 56, 6667-6682. (4) Schellinger, A. P.;Stoll, D. R.;Carr, P. W. J. Chromatogr., A 2005, 1064, 143-156. (5) Jandera, P.; Churacek, J. J. Chromatogr. 1974, 91, 207-221. (6) Snyder, L. R.; Dolan, J. W.; Grant, J. R. J. Chromatogr. 1979, 165, 3-30.

7828 Analytical Chemistry, Vol. 78, No. 22, November 15, 2006

the logarithm of the retention factor (k) varies in an approximately linear manner with the mobile-phase composition (i.e., the volume fraction of strong solvent, φ) according to

lnk ) lnk0 - Sφ

(1)

where k0 is the retention factor of the solute in 100% of the starting (weak) solvent and S is the solvent strength parameter. The LSS model has been shown to be valid in most reversed-phase systems.7-10 However, Non-LSS (i.e., strongly curving plots of lnk vs φ) has also been found in some chromatographic techniques, especially in ion exchange chromatography.11-13 In the literature there, have already been various mathematical models to account for the mass balance during the chromatographic process.14-17 These models include the ideal model, the equilibrium dispersive (ED) model, the lumped kinetic (LK) model, the lumped pore model, and the general rate model. Among them, the ideal model is the simplest and usually preferred at the beginning of the studies. This model ignores the contributions of all mass-transfer processes to band broadening. Accordingly, the band profile predicted by it arises only from the characteristics of the equilibrium thermodynamics. The ideal model has the advantage of predicting the best possible results obtained: thermodynamics cannot be improved upon. Moreover, the analytical solution of it can be easily obtained, which in turn provides significant physical insight into the process. Note that in the literature there are many studies on gradient elution which are essentially based on the ideal model. In these studies, the eluent chromatographic zone is usually reduced to a point and the LSS model is adopted to simplify the mathematical analysis.6,7,18 (7) Schoenmakers, P. J.; Billiet, H. A. H.; Tijssen, R.; De Galan, L. J. Chromatogr. 1978, 149, 519-537. (8) Fitzpatrick, F.; Edam, R.; Schoenmakers, P. J. Chromatogr., A 2003, 988, 53-67. (9) Dolan, J. W.; Gant, J. R.; Snyder, L. R. J. Chromatogr. 1979, 165, 31-58. (10) Zhu, P. L.; Snyder, L. R.; Dolan, J. W.; Djordjevic, N. M.; Hill, D. W.; Sander, L. C.; Waeghe, T. J. J. Chromatogr., A 1996, 756, 21-39. (11) Quarry, M. A.; Grob, R. L.; Snyder, L. R. Anal. Chem. 1986, 58, 907-917. (12) Gallant, S. R.; Vunnum, S.; Cramer, S. M. AIChE J. 1996, 42, 2511-2520. (13) Natarajan, V.; Ghose, S.; Cramer, S. M. Biotechnol. Bioeng. 2002, 78, 365375. (14) Guiochon, G. J. Chromatogr., A 2002, 965, 129-161. (15) Felinger, A.; Guiochon, G. J. Chromatogr., A 1996, 724, 27-37. (16) Antos, D.; Kaczmarski, K.; Wojciech, P.; Seidel-Morgenstern, A. J. Chromatogr., A 2003, 1006, 61-76. (17) Kaczmarski, K.; Antos, D.; Sajonz, H.; Guiochon, G. J. Chromatogr., A 2001, 925, 1-17. (18) Nikitas, P.; Pappa-Louisi, A. Anal. Chem. 2005, 77, 5670-5677. 10.1021/ac061318y CCC: $33.50

© 2006 American Chemical Society Published on Web 10/14/2006

Up to now, however, general solutions of the ideal model for gradient elution have not yet been discussed as far as we know. In this paper, we assume that the influence of the concentration of the solute on its retention behavior is negligible. Therefore, the discussion presented here is valid in the scope of analytical liquid chromatography. First, we discuss the general solutions of the ideal model for any gradient profiles and solvent strength models used. The basic equations to calculate the retention time and the concentration change during the elution are proposed. Then, because stepwise and linear gradients are the most widely used gradient modes in practice, we further give explicit expressions to predict the retention time in any forms of linear solvent strength stepwise and linear gradient elution (LSS-SLGE). Based on them, the expressions for specific gradient modes such as single linear, stepwise, and ladderlike gradients are given, which are also compared to those reported in the literature. For these modes, the changes of the bandwidth and the concentration during the elution are discussed. ASSUMPTIONS FOR THE IDEAL MODEL In this paper we will make the following assumptions: (1) The contributions of the mass-transfer processes to band broadening, such as axial dispersion in the mobile phase and molecular diffusions and sorption kinetics around and in the particles, are considered negligible. (2) The solid concentration of the solute depends linearly on the liquid concentration for a given value of φ; i.e., the influence of the concentration on the partition of the solute between mobile and stationary phases is negligible. (3) The mobile phase consists of two solvents. In order to simplify the mathematical analysis, the excess adsorption of these solvent components on the stationary phase is assumed to be negligible as suggested by Snyder,6 which has also been adopted in many other studies of gradient chromatography. However, it should be noted that this assumption is an approximation of practical situations. Recent study has shown that such adsorption is insignificant in the reversed-phase system but may deform the gradient shape in the normal-phase system.19 (4) The mobile-phase velocity is constant. The compressibility of the mobile phase is negligible. (5) The bed is packed uniformly. The concentration gradient in the radial direction of the bed is negligible. (6) The injection profile at the column inlet is rectangular. (7) The column is equilibrated with the mobile phase of starting composition, and there is no solute absorbed on the stationary phase before the injection. (8) The column temperature is kept constant. Moreover, in order to facilitate the analysis, the following dimensionless variables will be used:

y)

C , C0

Q)

q , C0

Z x) , L

τ)

tu L

(2)

All symbols used in eq 2 are explained in the Glossary below. IDEAL MODEL FOR THE MOBILE PHASE Prior to studying the retention behavior of the solute during the elution, it turns out to be necessary to first learn the (19) Piatkowski, W.; Kramarz, R.; Poplewska, I.; Antos, D. J. Chromatogr., A 2006, 1127, 187-199.

Figure 1. Analysis of the ideal model for the mobile phase by using the method of characteristics.

distribution of the strong solvent of the mobile phase across the column, because it essentially determines the migration velocity of the solute at any time and at any position inside the column. According to the above assumptions, the ideal model accounting for the change of the volume fraction of the strong solvent in the mobile phase can be written as14

∂φ ∂φ + )0 ∂τ ∂x φ(τ,x) ) φ(0,x) ) φ0 for τ e τD; φ(τ,0) ) Φ(τ - τD)

for

τ > τD;

(3) 0 < x < 1 (4) x)0

(5)

where eq 3 is the mass balance equation for the changes of the mobile-phase composition and eqs 4 and 5 account for initial and boundary conditions, respectively. In the above equations, φ denotes the volume fraction of the strong solvent at any spacetime point (τ, x), Φ is the gradient profile programmed into the pump, φ0 is the starting composition of the mobile phase, and τD is the dwelling time of the system that accounts for the time needed for a certain change in the mixer to reach the beginning of the column.7,18 Equation 3 is also equivalent to the following set of ordinary differential equations (ODEs):20,21

{

dx )1 dτ dφ )0 dτ

(6)

Generally eqs 4-6 can be solved by using the method of characteristics as depicted in Figure 1.19,20 For example, the Cartesian equation of the characteristic curve through the point P (τP, 0) is τ ) x + τP. According to the boundary condition (eq 5) it can be concluded that the value of φ along this characteristic curve is

φ(τ,x) ) φ(τP,0) ) Φ(τP - τD) ) Φ(τ - x - τD)

(7)

Similarly, the Cartesian equation of the characteristic curve through the point R (0,xR) is τ ) x - xR. According to the initial (20) Lin, B. C. Introduction of Chromatographic Models; Scientific Press: Beijing 2004; Chapter 3. (21) Smith, G. Numerical Solution of Partial Differential Equations: Finite Difference Methods, 2nd ed.; Oxford University Press: Oxford, 1978; Chapter 4.

Analytical Chemistry, Vol. 78, No. 22, November 15, 2006

7829

condition (eq 4), the value of φ along it is

φ(τ,x) ) φ(0,xR) ) φ0

(8)

In conclusion, the solution of the ideal model for the mobile phase can be written as

φ(τ,x) )

{

φ0 τ e x + τD (9a,b) Φ(τ - x - τD) ) Φ(τP - τD) τ > x + τD

where τP ) τ - x, and thus, Φ can also be taken as the function of one variable τP. Moreover, as shown below, it is also necessary to know the value of the partial differential ∂φ/∂τ to solve the ideal model for the solute. In the region where τ > x + τD, which is labeled as “II” in Figure 1, this value is given by

∂[φ(τ,x)] ∂[Φ(τ - x - τD)] d[Φ(τP - τD)] ∂τP ) ) ) ∂τ ∂τ dτP ∂τ d[Φ(τP - τD)] (10) dτP In the region where τ < x + τD, which is labeled as “I” in Figure 1, the value of ∂φ/∂τ is equal to zero according to eq 9a. Therefore, we have

{

τ < x + τD 0 ∂[φ(τ,x)] ) d[Φ(τP - τD)] τ > x + τ ∂τ D dτP

(11a,b)

Also it should be noted that the values of ∂φ/∂τ along the line τ ) x + τD may exist or not, which depends on the expression of Φ. For example, Φ can be expressed as the following expression for single linear gradient:

Φ(τ) ) φ0 + B1τ

(12)

where B1 denotes the slope of the gradient. At any point lying on τ ) x + τD, it can be easily seen from eq 11 that the value of ∂φ/∂τ is equal to zero on the left side of this point but is equal to B1 on the right side. Obviously, the value of ∂φ/∂τ does not exist at it. In general, it is usually found at the place where a new gradient begins that the value of ∂φ/∂τ does not exist. IDEAL MODEL FOR THE SOLUTE According to the assumptions made in this paper the ideal model for the solute can be written as12,14

∂y 1 - T ∂Q ∂y + + )0 ∂τ T ∂τ ∂x

(13)

Q ) K[φ(τ,x)]y y(0,x) ) 0 for τ ) 0; 0 e x e 1 y(τ,0) ) y′f for τ > 0; x ) 0

(14) (15) (16)

(with y′f )1 for τ ∈ [0,Tinj] and with y′f ) 0 for τ > Tinj) 7830


Figure 2. Analysis of the ideal model for the solute by using the method of characteristics.

where eq 13 is the mass balance equation for the changes of the solute concentration in the mobile phase, eq 14 is the adsorption isotherm, and eqs 15 and 16 account for initial and boundary conditions, respectively. Note that in practice the sample is usually introduced onto the head of the column by using the six-port valve of which the sample loop lies between the pump and the column at position of injection. Therefore, the width of the injection profile, Tinj, should be part of the dwelling time τD, which gives Tinj < τD. By substituting eq 14 into 13 and using the definition of the retention factor (k ) (1 - T)/TK), we have

∂k[φ(τ,x)] ∂y ∂y {1 + k[φ(τ,x)]}‚ + ) -y ∂τ ∂x ∂τ

(17)

Equation 17 is equivalent to the following set of ODEs by taking both x and y as the function of τ21

{

1 dx ) dτ 1 + k[φ(τ,x)] ∂k[φ(τ,x)] y dy )‚ dτ ∂τ 1 + k[φ(τ,x)]

(18a,b)

The analysis of the ideal model for the solute by using the method of characteristics is depicted in Figure 2. In this figure, the characteristic curves 1 and 3 originate from the beginning of the injection (0,0) and the end (Tinj,0), respectively. The region between these two curves corresponds to the trajectory of the band of the solute. Out of this region, it can be concluded that the solute concentration y must be zero along the characteristic curves, such as the curves 4 and 5, because it is only y ) 0 that can satisfy eq 18b for which the initial values of y are zero (from initial and boundary conditions, eqs 15 and 16). Therefore, below we will only study the characteristic curves lying between the curves 1 and 3. 1. Retention of the Solute. In the literature, there are already two equations to predict the retention time in gradient elution, which have been discussed in detail by Nikitas and Pappa-Louisi recently.18 According to the opinions of the authors, these equations (corresponding to eqs 23 and 24 in ref 18) can be expressed as the following by using the dimensionless variables defined in this paper:

τD

∫ k(φ ) τ +∫ k(φ ) +

0

D

0

dτ )1 k(φ) dτC τR - τ1 )1 0 1 + k(φ) τR - τ D - 1

0

(19a)

τ1

∫

0

+ k(φ) dτ k(φ)

(21a)

dτ′ ∂τ′ ∂τ′ dx dx ) + )1dτ ∂τ ∂x dτ dτ

(21b)

where eq 21a takes x as the independent variable and eq 21b takes τ as that. It is also noted that τ′ is different from τP appearing in eq 9, in the definition of which τ and x may be independent of each other. However, τ′ can be considered as a special case of τP. Thus, it is reasonable to replace all τP appearing in the expressions by τ′ but τ′ cannot be replaced by τP. By substituting eqs 9 and 21a into eq 18a, we have

τ′ e τD

(22a)

and

for

τ′ > τD

(22b)

With initial condition τ′ ) τinj for x ) 0, eq 22a gives

τ ) τinj + [1 + k(φ0)]x for τ′ e τD By integrating eq 22b we obtain

dθ ) k[Φ(θ - τD)]

∫ dξ x

for

0

τ′ > τD (23b) where symbols θ and ξ are both dummy variables of integration. If further letting θ′) θ - τD, we obtain

τD - τinj

∫

τ-τD-x

+

0

dθ′ )x k[Φ(θ′)]

(24)

If the band of the solute is reduced to a point, i.e., letting τinj ) 0, and noting that τ ) τR for x ) 1, eq 24 becomes

τD

(23a)

+

k(φ0)

∫

τR-τD-1

0

dθ′ )1 k[Φ(θ′)]

(25)

which is identical to eq 19a. Equation 19b can be taken as the direct integration of eq 18a:

∫

τ′0

τinj

1 dθ + 1 + k(φ0)

∫

dθ ) 1 + k{ φ[θ,x(θ)]}

τ

τ′0

∫

x

0

dξ

(26)

where τ′0 is the time coordinate of the intersecting point (τ′0,x′0) of the lines τ ) τinj + [1 + k(φ0)]x and τ ) x + τD. The coordinates of the intersecting point are given by

τ′0 )

τD k(φ0)

x′0 )

-

τD k(φ0)

τinj k(φ0) -

+ τD

τinj k(φ0)

(27a)

(27b)

Note that in eq 26 x should be considered as the function of τ that is one solution of eq 18a. If letting θ′ ) θ - τ′0, we obtain

τD k(φ0)

-

τinj

+

k(φ0)

∫

τ-τ′0

0

dθ′ )x 1 + k{φ[θ′ + τ′0,x(θ′ + τ′0)]} (28)

If further letting τinj ) 0 and τ ) τR forx ) 1, eq 28 is reduced to:

τD

dτ′ ) dx k[Φ(τ′ - τD)]

τ′)τ-x

τD

(20)

dτ′ ∂τ′ dτ ∂τ′ dτ ) + ) -1 dx ∂τ dx ∂x dx

for

∫

1 dθ + k(φ0)

k(φ0)

Undoubtedly eqs 19a,b are both actually right. However, the current forms of them are still somewhat confusing. First, τ and τC are both dummy variables of integration for which the symbols chosen should not affect the values of the integrals. Thus it seems to be inappropriate to ascribe the differences between eqs 19a,b to these variables. Also, it is difficult to substitute eq 20 directly into eq 19b, which leads to an expression that is seldom used for the integral. Therefore, it turns out to be necessary to give further elaborations on these two equations. Below we will rewrite them according to eq 18a. Moreover, the injection time τinj (τinj ∈ [0,Tinj]) is introduced into the equations, and this will help to carry out further analysis of the ideal model. We first introduce a new variable τ′ ) τ - x, in which we assume that x and τ are related to each other and their relationship is accounted for by one solution of eq 18a. Therefore, τ′ can be considered as the function of one variable, either τ or x, and we have

dτ′ ) dx k(φ0)

τD

τinj

(19b)

where τ1 ) τD/(k(φ0)) + τD in eq 19b. It is also pointed out that the variables appearing in the integrals, τ and τC, are different. The relationship between them is accounted for by

τC )

∫

k(φ0)

+

∫

τR-τ′0

0

dθ′ ) 1 (29) 1 + k{φ[θ′ + τ′0,x(θ′ + τ′0)]}

which is identical to eq 19b. By comparing eq 25 with eq 29, we find that the differences between these equations lie in the different expressions to calculate the value of k(φ) in the integral, which in turn explains the differences between eqs 19a and 19b. Note that eq 29 is actually an implicit expression in which the relationship between x and τ is also determined by eqs 18a and 27. In general, the trajectory of the solute in the space-time map (such as Figure 2) can be obtained by solving eq 18a or 22 with appropriate initial conditions. Because there are already numerous analytical and numerical methods to solve these first-order ODEs, these equations can also be directly used in practice to predict Analytical Chemistry, Vol. 78, No. 22, November 15, 2006

7831

the retention time for any gradient profiles and solvent strength models used. Accordingly, the equivalent integral forms of them are eqs 28 and 24, respectively. Finally, it should be noted that the difference between eq 19, which is presented in the literature, and eq 24, which is obtained in this paper by using the method of characteristics, lies in that the latter also involves the injection time, τinj. With this mathematical treatment, it is not necessary to take the band zone of the solute as a point any more as done by many other previous studies. Therefore, eq 24 can be further used instead of eq 19 to investigate the band compression during gradient elution, which turns out to be an important feature of this operation mode but seems to become a dispute in some recent publications.22,23 Moreover, as shown in the following section, eq 24 is also necessary to derive the expressions to calculate the solute concentration in the ideal model, especially when there are discontinuities in the gradient profiles, such as stepwise gradients. This derivation can also not be accomplished by using eq 19. 2. Concentration Change of the Solute. The concentration along the characteristic curve can be calculated by using eq 18b. In Figure 2, before the characteristic curve reaches the line τ ) x + τD, it can be easily concluded from eqs 11a and 18b that the concentration along this curve is always equal to the feed concentration. After the characteristic curve crosses τ ) x + τD, we first obtain the following according to eqs 9b and 11b:

migration of a mass point in the injection zone of the solute inside the column; i.e., the curve is the trajectory of the mass point moving in the space-time map. Therefore, the physical meaning of eq 32 is that it accounts for the propagation of the solute concentration upon this point during its migration. In order to understand better the meaning of this equation, we will choose two points on a characteristic curve, such as A and B labeled in Figure 2, between which the value of ∂φ/∂τ at any point is assumed to be always existent. The values of y at A and B are denoted by yA and yB, and the mobile-phase compositions are denoted by φA and φB, respectively. According to eq 32 we have

d[Φ(τP - τD)] ∂k[φ(τ,x)] dk ∂φ(τ,x) dk ) ) ‚ ) ∂τ dφ ∂τ dτP d[Φ(τP - τD)]

yA k[Φ(τ′B - τD)] k(φB) ) ) yB k[Φ(τ′ - τ )] k(φA)

dk[Φ(τP - τD)] (30) dτP where k is taken as the function of φ (also see eq 1), φ is the function of τ and x, Φ is the function of τP, and the relationship between φ and Φ is accounted for by eq 9b. By substituting eqs 9b, 18a, 21b, and 30 into eq 18b and letting τP ) τ′ we have

{

}

dy dy 1 dy dτ′ ) ‚ ) 1) dτ dτ′ dτ 1 + k[Φ(τ′ - τD)] dτ′ -

dk[Φ(τ′ - τD)] y (31) dτ′ 1 + k[Φ(τ′ - τD)]

Equation 31 gives the following in which the concentration y may be taken as the function of τ′:

1 1 d (ln y) ) d ln ) d ln (32) k[Φ(τ′ - τD)] k[Φ(τ - x - τD)] In the above equation, the relationship between τ and x is simultaneously determined by eq 18a; i.e., eq 32 is valid for the coordinates of the points on the same characteristic curve. Note that in the ideal model the characteristic curve accounts for the (22) Neue, U. D. J. Chromatogr., A 2005, 1079, 153-161. (23) Neue, U. D.; Marchand, D. H.; Snyder, L. R. J. Chromatogr., A 2006, 1111, 32-39. (24) Department of mathematics of Tongji University. Advanced Mathematics, 3rd ed.; High Education Press: Shanghai 1993; Chapter 9.

7832


Figure 3. Calculation of the concentration at the point where the value of ∂φ/∂τ does not exist.

A

(33)

D

That is to say, for any two points on the same characteristic curve between which the value of ∂φ/∂τ also always exists, the product of the concentration and the retention factor at one point is equal to that of the other. Below we will discuss the value of y at the point where the value of ∂φ/∂τ does not exist, such as the point on the line τ ) x + τD in Figure 2 if a single linear gradient (eq 12) is assumed. In this case, eq 18b cannot be directly used to calculate the concentration. Instead, this value should be obtained according to the principle of mass conservation. An illustration for such analysis is shown in Figure 3. In this figure, we choose the characteristic curve 1 as the object. It is assumed that this curve originates from the point G(τinj,B,0) and intersects the line τ ) x + τE at point E, which is the first point on the curve where the value of ∂φ/∂τ does not exist. In order to obtain the value of the concentration at this point, we choose a point B on the curve that is close to the point E and below the lineτ ) x + τE. Then, a line parallel to the τ axis is plotted and it intersects τ ) x + τE at point A. We assume that it is the characteristic curve 2 that passes through point A and originates from point F(τinj,A,0). According to the principle of mass conservation, we obtain

∫

τB

τA

y(τ, x) dτ )

∫

τinj,B

τinj,A

y′f dτ

(34)

It should be noted that y′f appearing in eq 34 can be any function of time, although it is assumed to be a rectangular profile in this paper. According to eq 24 we may define

F(τinj,τ,x) )

τD - τinj k(φ0)

+

∫

τ-τD-x

0

dθ′ -x)0 k[Φ(θ′)]

(35)

in which either one of the variables τinj, τ and x can be taken as the function of the other two. By considering τ as the function of τinj and x, we have

( )

(∂F/∂τinj)τ,x k[Φ(τ - τD - x)] ∂τ )) ∂τinj x (∂F/∂τ)τinj,x k(φ0)

( )

(37)

k(φD+) k(φD+) yD+ ) y + k(φC) k[Φ(τC - x - τD)] D

k(φ+) k(φ+) y+ ) y k(φ) k[Φ(τ - x - τD)] +

(38)

By substituting eqs 37 and 38 into eq 34 we have

∫

τinj,B

τinj,A

y+

k(φ+) k(φ0)

dτinj )

∫

τinj,B

τinj,A

y′f dτ

(39)

According to the integral mean value theorem, an alternative to eq 39 is

yξ+

k(φξ+) k(φ0)

(τinj,B - τinj,A) ) y′f(ξ′)(τinj,B - τinj,A)

k(φξ+)

(41)

When B f E, ξ f E, and ξ′ f G. We obtain

yE+ )

k(φ0)

y′f(G)

k(φE+)

yE- )

k(φ0)

y′f(G)

k(φE-)

(42)

(40)

where ξ and ξ′ denote the points on the segments AE and FG, respectively, yξ+ and φξ+ denote the concentration and the mobilephase composition on the right side of the point ξ, respectively, and y′f(ξ′) denotes the feed concentration at the point ξ′. Equation 40 gives

(43)

From eqs 42 and 43 it can be seen that the continuity of y at the point where the value of ∂φ/∂τ does not exist is determined by that of φ. If φ is continuous (i.e., φE- ) φE+), y is also continuous and the product of y and k at this point is equal to that at any other point where the value of ∂φ/∂τ exists. If φ is discontinuous, which is usually found in stepwise gradients, y is also discontinuous at the same point. However, the product of y and k on the left side of the point is equal to that on the right side:

yE-k(φE-) ) yE+k(φE+)

where yC and φC denote the concentration and the mobile-phase composition at point C, and yD+ and φD+ denote those on the right side of point D, respectively. Note that point C can be at any position on the segment AB. Thus, we can replace τC, φC, and yC by τ, φ, and y, respectively. Accordingly, we use y+ and φ+ to denote the concentration and the mobile-phase composition on the right side of the corresponding point lying in the segment AE, respectively:

y)

k(φ0)

On the left side of point E, the concentration of the solute (yE-) can be directly obtained by using eq 33:

Now we choose a point C on AB through which the characteristic curve 3 passes. This curve intersects τ ) x + τE at point D. According to eq 33 we have

yC )

y′f(ξ′)

)

(36)

where (∂F/∂τ)τinj,x is obtained by using the Leibnitz formula for the variable τ appears in the upper limit of the integral.24 Therefore, we have the following expression for any point (τ, x) on the segment AB where dx ) 0:

k[Φ(τ - x - τD)] ∂τ dτ ) dτ ) dτinj ∂τinj x inj k(φ0)

yξ+

(44)

The above results can be easily extended to other points on curve 1, where the values of ∂φ/∂τ do not exist. In conclusion, for any gradient profiles and solvent strength models used in the ideal model, the concentration will be discontinuous where the mobile-phase composition is. On a given characteristic curve, the product of the concentration and the retention factor is kept constant at the point where the concentration is continuous. At the point where the concentration is not, the product on the left side of this point is equal to that on the right side. PREDICTION OF THE RETENTION TIME IN LSS-SLGE In practice, stepwise and linear gradients are the most widely used operation modes in gradient elution.25-27 In the literature, there are some expressions to predict the retention time of the solute in specific stepwise or linear gradient elution. However, these expressions are still not enough for practical use, because stepwise and linear gradients can be combined in any forms and this will lead to various gradient profiles. Moreover, although the prediction can also be implemented by solving eq 18a or 22 with numerical methods as discussed earlier, the analytical solutions have the advantage of showing some features of gradient elution more clearly. Therefore, it is still necessary to study the general expressions for the prediction of the retention time in LSS-SLGE. First we will introduce the numbering used in this paper of the segments in the gradient profile, which is illustrated in Figure (25) Cela, R.; Lores, M. Comput. Chem. 1996, 20, 175-191. (26) Kaliszan, R.; Baczek, T.; Cimochowska, A.; Juszczyk, P.; Wisniewska, K.; Grzonka, Z. Proteomics 2005, 5, 409-415. (27) Fitzpatrick, F.; Staal, B.; Schoenmakers, P. J. Chromatogr., A 2005, 1065, 219-229.


7833

φ(τ,x) ) φK-1 + BK(τ - x - τP,K-1 - τD)

(46)

By substituting eqs 1 and 46 into eq 18a, and noting that τ′K-1 ) x′K-1 + τP,K-1 + τD, we obtain the solution

τ ) x + τP,K-1 + τD + 1 ln[1 + BKSk(φK-1)(x - x′K-1)] (47a) BKS

Note that eq 47a is only valid when BK * 0. When BK ) 0, we find in this case that the value of τ is equal to the limit of eq 47a, where BK f 0: Figure 4. Gradient profile consisting of both stepwise and linear gradients.

τ ) x + τP,K-1 + τD + lim

1

ln[1 + BKSk(φK-1)(x - x′K-1)] )

BKf0BKS

x + τP,K-1 + τD + k(φK-1)(x - x′K-1) (47b)

Figure 5. Analysis of the ideal model for LSS-SLGE.

4. In the gradient profile, every turning point as well as the starting and the end points is numbered from the beginning value of zero. The index of the segment is defined to be equal to that of its end point and is denoted in the subscript of the slope B. In order to facilitate the analysis, the index of the point is also denoted in the subscript of the variables such as τP and φ. According to this numbering the change in the mobile-phase composition in the segment K can be expressed as

Φ(τ))φK-1 +BK(τ - τP,K-1)

for

τP,K-1 e τ e τP,K (45)

In the above equation, if BK is zero, the segment corresponds to isocratic elution. If BK is not, the segment corresponds to linear gradient elution. For the segment that is perpendicular to the time axis, such as segment 4, there is no explicit expression of Φ for it. However, the change in the mobile-phase composition in this segment can be reflected by the differences between the end value of φ of the previous segment and the starting value of φ of the next segment. Figure 5 illustrates the analysis of the ideal model for LSSSLGE. We first choose the segment K as the object. In this segment, the corresponding initial condition for eq 18a is τ ) τ′K-1 for x ) x′K-1, where τ′K-1 and x′K-1 are the coordinates of the intersecting point of the characteristic curve and the line τ ) x + τP,K-1 + τD. From eqs 9 and 45 we obtain 7834


When BK ) ∞, i.e., τP,K ) τP,K-1 and thus the lines τ ) x + τP,K-1 + τD and τ ) x + τP,K + τD overlap, it can be concluded that x′K ) x′K-1 and τ′K ) τ′K-1. This is because, if it did not, the solute will move along the line τ ) x + τP,K-1 + τD and this leads to k ) 0 in eq 18a. However, this cannot happen in the LSS model because the value of k should always be beyond zero. Therefore, the position of the solute inside the column will not change when BK ) ∞. Below we will calculate the value of x′K-1 appearing in eq 47. For the intersecting point (τ′K,x′K) there are

τ′K ) x′K + τP,K-1 + τD +

1 ln[1 + BKSk(φK-1)(x′K - x′K-1)] BKS

and

τ′K ) τP,K + τD + x′K

The above equations give

x′K - x′K-1 )

exp[BKS(τP,K - τP,K-1)] - 1 BKSk(φK-1)

(48a)

When BK * 0, by substituting eqs 1 and 45 (noting that Φ(τP,K) ) φK) into eq 48a, we have

x′K - x′K-1 )

exp[S(φK - φK-1)] - 1 BKSk(φK-1)

[

)

]

1 1 1 for BK * 0 (48b) BKS k(φK) k(φK-1)

When BK ) 0, we have

x′K - x′K-1 ) lim

exp[BKS(τP,K - τP,K-1)] - 1 BKSk(φK-1)

BKf0

τP,K - τP,K-1 k(φK-1)

)

for BK ) 0 (48c)

where it is defined τP,0 ) 0. Obviously, when BK ) ∞, there is

x′K - x′K-1 ) 0

BK ) ∞

for

(48d)

In order to facilitate the writing, we will define the function ∆(x′K,x′K-1)to account for the right-hand sides of eqs 48b-d:

x′K - xK-1 ) ∆(x′K,x′K-1)

(48e)

Note that eq 48e is also suited for other segments in the gradient profile. Therefore, by summing up all of the expressions for the segments together, we have K

x′K ) x′0 +

∑ ∆(x′,x′

i i-1)

i)1

By substituting eq 27b into the above equation we obtain

x′K )

τD

-

k(φ0)

τinj

K

+

k(φ0)

∑∆(x′, x′ i

i-1)

(49)

i)1

Now, with the above expressions, we can calculate the retention time in any forms of LSS-SLGE. First, the axial coordinate of every intersecting point in the map like Figure 5 are calculated by using eqs 48 and 49. Then, the index of the segment (N) in the gradient profile in which the solute is eluted from the outlet (x ) 1) is found out by using the following criterion:18,28

x′N-1 < 1

and

x′N g 1

(50)

Finally, the retention time (τR) is calculated by substituting the value of x′N-1 into eq 47 and letting x ) 1:

τR ) 1 + τP,N-1 + τD +

1 ln[1 + BNSk(φN-1)(1 - x′N-1)] BNS for

BN * 0 (51a) in practice. Such optimization can be implemented by means of various algorithms, for example, the genetics algorithm.28,29

or

τR ) 1 + τP,N-1 + τD + k(φN-1)(1 - x′N-1)

Figure 6. Specific gradient profiles: (a) single linear gradient; (b) stepwise gradient; (c) ladderlike gradient.

for

BN ) 0 (51b)

The above strategy provides a way to figure out the segment of the gradient profile in which the solute is eluted from the column. Also, this strategy can be easily programmed into the computer and then used to design the optimal gradient profiles (28) Nikitas, P.; Pappa-Louisi, A. J. Chromatogr., A 2005, 1068, 279-287.

SOLUTIONS OF THE IDEAL MODEL IN LSS-SLGE FOR SPECIFIC GRADIENT PROFILES In order to show the use of the above general expressions, we will derive those in LSS-SLGE for some specific gradient profiles as illustrated in Figure 6. Also, for convenience in discussion, we assume that the solute will be eluted from the column in the last segment of the gradient profile. (29) Nikitas, P.; Pappa-Louisi, A.; Agrafiotou, P. J. Chromatogr., A 2006, 1120, 299-307.


7835

1. Expressions To Predict the Retention Time. For single linear gradient there is N ) 1. Thus eq 51a becomes

τR ) 1 + τ D +

1 ln[1 + B1Sk(φ0)(1 - x′0)] ) 1 + τD + B 1S

[

{

]}

τD - τinj 1 ln 1 + B1Sk(φ0) 1 B1S k(φ0)

(52a)

If the band is reduced to a point, i.e., τinj ) 0, eq 52a becomes the one proposed by Schoenmakers.7 If further assuming k(φ0) is large enough, we have

τR ≈ 1 + τD +

1 ln[1 + B1Sk(φ0)] B 1S

τD

τinj

-

k(φ0)

τP,1

+

k(φ0)

∑

P,2L-1

- τP,2L-2

(53)

k(φ2L-2)

L)2

By substituting eq 53 into eq 51b we obtain

[

τR ) 1 + τP,N-1 + τD + k(φN-1) 1 τP,1

-

τD

+

k(φ0)

(N-1)/2τ P,2L-1

k(φ0)

∑

τinj

-

k(φ0)

]

- τP,2L-2

k(φ2L-2)

L)2

τR ) 1 + k(φ2) + τP,1 + τD -

k(φ2) (τ + τD - τinj) (55) k(φ0) P,1

τD k(φ0)

-

τinj k(φ0)

+

τP,1 k(φ0)

+

[

]

1 1 1 B2S k(φ2) k(φ0)

(56)

By substituting eq 56 into eq 51b we obtain (30) Synder, L. R.; Dolan, J. W. J. Chromatogr., A 1996, 721, 3-14. (31) Nikitas, P.; Pappa-Louisi, A.; Papachristos, K. J. Chromatogr., A 2004, 1033, 283-289.

7836


}

(58a)

If B1STinj , 1 + B1S[k(φ0) - τD], i.e., Tinj , k(φ0) - τD for B1 > 0, we obtain

Tinj ≈ 1 + B1S[k(φ0) - τD] ∆τR

(58b)

Because the slope B1 is usually positive in practice, the bandwidth at an outlet will be narrower than the injection width; i.e., band compression occurs during the gradient elution. Also, because the ideal model only considers the effects of the equilibrium thermodynamics of chromatography and the results obtained by it can be taken as the best, eq 58b is the expression to calculate the maximum of the compression factor, Gideal, in single linear gradient elution. Similarly, for single stepwise and ladderlike gradients we have

Gideal )

For the gradient profile as illustrated in Figure 6c, which is also called a ladderlike gradient in this paper, we have the following from eqs 48b,c and 49:

x′2 )

{

B1STinj 1 ln 1 + B1S 1 + B1S[k(φ0) - τD]

(54)

If letting τinj ) 0, eq 54 becomes the one proposed by Nikitas, which has also been pointed out to be suited for the cases where nonlinear strength solvent models are used.31 For single stepwise gradient, there is N ) 3. From eq 54, noting that τP,2 ) τP,1, we have

]

When B2 f ∞, it can be found that eq 57 is the same as eq 55. 2. Change of the Bandwidth. The general expression to calculate the change of the bandwidth at an outlet can be easily derived from eq 51 by subtracting the value of τR obtained by letting τinj ) 0 from that obtained by letting τinj ) Tinj. Here, in order to understand better this change, we will only discuss the cases of single linear, single stepwise, and ladderlike gradients. For single linear gradient, we have the following from eq 52a:

Gideal )

k(φ0)

[

k(φ2) k(φ2) 1 1(57) (τP,1 + τD - τinj) B2S k(φ0) k(φ0)

∆τR )

+

(N-1)/2τ

τD -

(52b)

which is the expression proposed by Snyder.6,30 For multistepwise gradient elution we have the following from eqs 48c,d and 49:

x′N-1 ) x′N-2 )

τR ) 1 + k(φ2) + τP,2 +

Tinj k(φ0) ) ∆τR k(φ2)

(59)

Equations 58 and 59 confirm the band compression occurring in gradient elution as the result of the equilibrium thermodynamics of chromatography. In practice, there are also the mass-transfer processes, such as axial dispersion in the mobile phase and the mass-transfer resistances around and in the particles, that lead to band broadening.14 Therefore, two different effects influencing the bandwidth of the solute should be considered in gradient chromatography: one is the band compression resulting from the equilibrium thermodynamics of chromatography, and the other is the band broadening caused by the mass-transfer kinetics. This situation is somewhat different from that of isocratic chromatography, in which it is usually the mass-transfer kinetics only that is needed to be taken into account. In order to learn the contributions of the mass-transfer kinetics to band broadening in gradient chromatography, the rate models of chromatography such as the ED or the LK model should be used, and this work is underway in our group. Moreover, by comparing experimental results with those obtained from the above expressions accounting for the band compression effects, it can also be helpful from another point of view in practical gradient operations to assess the contributions of the mass-transfer kinetics to band broadening.

Figure 7. Analysis of the characteristics for the ideal model in single stepwise gradient. The insets show the chromatograms obtained at different positions of the column.

Figure 8. Analysis of the characteristics for the ideal model in single linear gradient. The insets show the chromatograms obtained at different positions of the column.

3. Concentration Change. According to the general conclusion on the concentration change in the ideal model, and the expressions presented above to predict the retention time, we can easily plot the chromatograms obtained in the ideal cases. As an example, Figure 7 illustrates the results obtained for single stepwise gradient elution. In this figure, the mobile-phase composition is φ ) φ0 on the left side of the line τ ) x + τP,1 + τD, and φ ) φ2 on its right side (also see Figure 6b and eq 9). Together with the assumption that the injection profile is rectangular and the feed concentration y′f ) 1, we can calculate the outlet concentration by using the following expression according to the general conclusion

y)

k(φ0) k(φ0) y′f ) k(φ2) k(φ2)

(60)

Equation 60 shows that the outlet concentration profile is also rectangular. At the position x ) xM inside the column, it can be found that the mobile-phase composition is discontinuous at the point B that lies on the line τ ) x + τP,1 + τD. Therefore, the solute concentration is also discontinuous at this point, and there is y ) 1 on its left side while y ) k(φ0)/k(φ2) on its right side, according to the general conclusion. Moreover, by using eqs 48c,d and 49, we can calculate the axial coordinate of the point F, which is the intercepting point of the line τ ) x + τP,1 + τD and the characteristic curve 1 that originates from the beginning of the injection (0,0):

xF )

τD k(φ0)

+

τ)

{

x + k(φ2)x + τp,1 + τD -

intercepting point of the line τ ) x + τP,1 + τD and the characteristic curve 2 that originates from the end of the injection (Tinj,0), is given by

xG )

τD k(φ0)

+

τP,1 k(φ0)

-

Tinj k(φ0)

(61)

k(φ0)

x e xF k(φ2) (τ + τD) x > xF k(φ0) p,1

(62)

Similarly, the axial coordinate of the point G, which is the

(63)

The Cartesian equation for the characteristic curve 2 is

{

τ)

x + k(φ0)x + Tinj

τP,1

By using eq 47b we obtain the Cartesian equation for the characteristic curve 1:

x + k(φ0)x

Figure 9. Analysis of the characteristics for the ideal model in the ladderlike gradient. The insets show the chromatograms obtained at different positions of the column.

x + k(φ2)x + τp,1 + τD -

x e xG k(φ2) (τp,1 + τD - Tinj) x > xG

k(φ0)

(64)

With eqs 62 and 64, one can easily calculate the retention time of the beginning and the end of the band profile obtained at any position inside the column. Figure 8 illustrates the results obtained for single linear gradient elution. There is φ ) φ0 on the left side of the line τ ) x + τD and φ ) φ0 + B1(τ - τD - x) on its right side (also see Figure 6a and eqs 9 and 12). When the characteristic curve, such Analytical Chemistry, Vol. 78, No. 22, November 15, 2006

7837

as the one originating from (τinj,0), reaches the outlet x ) 1, we can calculate the corresponding value of φ at the intercepting point, such as the point G, by using

φ ) φ0 + B1(τR - τD - 1)

(65)

The outlet concentration can be calculated by using the following expression, which is derived from the LSS model (eq 1) and 1k(φ0) ) yk(φ) according to the general conclusion

y)

k(φ0) k(φ)

k0 exp(-Sφ0)

)

k0 exp{-S[φ0 + B1(τR - τD - 1)]}

) exp[B1S(τR - τD - 1)]

(66)

where τR ∈ [τF,τH], τF and τH denoting the retention time of the beginning and the end of the injection profile when they reach the outlet, respectively. Equation 66 shows that the outlet concentration will increase with increasing retention time. In the chromatogram that is obtained at x ) xM, τA and τB denote the retention time of the beginning and the end of the injection profile when they reach this position, respectively, and τE denotes that of the intercepting point of the lines x ) xM and τ ) x + τD. Due to φ ) φ0 on the segment AE we have

y)1

τ ∈ [τA,τE]

for

When τ ∈ (τE,τB], there is

φ ) φ0 + B1(τ - τD - xM) and thus we obtain

y)

k(φ0) k(φ)

)

k0 exp(-Sφ0) k0 exp{-S[φ0 + B1(τ - τD - xM)]}

) exp[B1S(τ - τD - xM)]

for

τ ∈ (τE,τB]

In Figure 8, the characteristic curves 1 and 2 that originate from (0,0) and (Tinj,0) intercept the line τ ) x + τD at the points C and D, respectively. The axial coordinates of these points can be calculated by using eq 27b:

xC ) xD )

τD

(67)

k(φ0)

τD k(φ0)

-

Tinj

(68)

k(φ0)

With eqs 23a and 47a, the Cartesian equation for the characteristic curve 1 is given by

{

x + k(φ0)x

τ)

[

{

]}

τD 1 ln 1 + B1Sk(φ0) x x + τD + B1S k(φ0)

x e xC x > xC

(69)

Similarly, the Cartesian equation for the characteristic curve 2 is

{

x + k(φ0)x + Tinj

τ)

x + τD +

{

[

]}

τD - Tinj 1 ln 1 + B1Sk(φ0) x B1S k(φ0)

x e xD x > xD

(70)

Figure 9 illustrates the results obtained for the ladderlike gradient elution. According to Figure 6c and eq 9, the mobile-phase composition is given by

{

φ0 τ e x + τP,1 + τD φ(τ,x) ) φ0 + B2(τ - x - τP,1 - τD) x + τP,1 + τD < τ e x + τP,2 + τD φ2 τ > x + τP,2 + τD 7838


(71)

At the outlet, it can be easily found that y ) k(φ0)/k(φ2), which is similar to that of single stepwise gradient elution as illustrated in Figure 7. At the position x ) xM, the chromatogram becomes similar to that of single linear gradient elution, in which τA and τC denote the retention time of the beginning and the end of the injection profile, respectively, and τB denotes that of the intercepting point of the lines x ) xM and τ ) x + τP,1 + τD. When τ ∈ [τA,τB] there y ) 1 due to φ ) φ0. When τ ∈ (τB,τC], we have the following from eq 71

φ ) φ0 + B2(τ - xM - τP,1 - τD) and thus obtain

y)

k(φ0) k(φ)

)

k0 exp(-Sφ0)

) exp[B2S(τ - xM - τP,1 - τD)]

k0 exp{-S[φ0 + B2(τ - xM - τP,1 - τD)]}

{

for

τ ∈ (τB,τC]

The Cartesian equation for the characteristic curve 1 that originates from the point (0,0) can be obtained by using eqs 47-49:

x + k(φ0)x

τ)

[

{

x e xG

]}

τP,1 + τD

1 ln 1 + B2Sk(φ0) x B2S k(φ0) k(φ2) k(φ2) 1 x + k(φ2)x + τP,2 + τD 1(τ + τD) B2S k(φ0) P,1 k(φ0)

x + τP,1 + τD +

[

]

xG < x e xH

(72)

x > xH

where the axial coordinates of the intercepting points G and H can be obtained by using eqs 48 and 49:

xG ) xH )

τD k(φ0)

{

+

τD k(φ0)

τP,1 k(φ0)

+

τP,1

+

(73)

k(φ0)

[

]

1 1 1 B2S k(φ2) k(φ1)

(74)

Similarly, the Cartesian equation for the characteristic curve 2 that originates from (Tinj,0) is given by

x + k(φ0)x + Tinj

[

{

]}

τP,1 + τD - Tinj 1 ln 1 + B2Sk(φ0) x x + τP,1 + τD + B S k(φ0) τ) 2 k(φ2) k(φ2) 1 x + k(φ2)x + τP,2 + τD 1(τ + τD - Tinj) B2S k(φ0) P,1 k(φ0)

[

]

x e xI xI < x e x J

(75)

x > xJ

where the axial coordinates of the intercepting points I and J can be calculated by

xI ) xJ )

τD k(φ0)

+

τD k(φ0)

τP,1 k(φ0)

-

+

τP,1 k(φ0)

Tinj k(φ0)

+

-

Tinj

(76)

k(φ0)

[

]

1 1 1 B2S k(φ2) k(φ1)

(77)

Finally, according to the above expressions to calculate the retention time and the concentration, it can be proven that the amount of solute eluted from the outlet and passing through position x ) xM is equal to that inputted at the inlet for all these gradient modes. These results justify the general conclusion on the concentration change in the ideal model for gradient elution which is given at the beginning of the discussion. CONCLUSIONS In this paper, we discuss the analytical solutions of the ideal model of chromatography for gradient elution, which ignores the influence of the concentration on the retention factor. By introducing the injection time into the fundamental equations of gradient elution, it is possible to investigate the band compression and the concentration change of the solute during the elution process. It is found that the concentration is discontinuous where the mobile-phase composition is. On a given characteristic curve, the product of the concentration and the retention factor is kept constant at the point where the concentration is continuous. At the point where the concentration is Analytical Chemistry, Vol. 78, No. 22, November 15, 2006

7839

discontinuous, the product on the left side of it is equal to that on the right side. This conclusion can largely simplify the calculation of the concentration in the ideal model. We also present explicit expressions to predict the retention time in LSS-SLGE, which can be used as the basis to derive others for specific gradient profiles. For single linear, stepwise, and ladderlike gradients, we give the expressions to calculate the retention time and the changes of the bandwidth and the concentration during the elution. These results will also help to understand the contributions of masstransfer processes to band broadening in gradient elution.

x′

axial coordinate of intersecting point

y

dimensionless concentration of solute in the mobile phase

y′f

dimensionless concentration at inlet

Z

axial coordinate, cm

Greek Letters ∆(x′i,x′i-1)

function accounting for eqs 48b-d

T

total porosity

φ

mobile-phase composition, i.e., volume fraction of strong solvent

GLOSSARY B

slope of linear gradient

Φ

gradient profile programmed into the pump

C

concentration of solute in the mobile phase, mg/ mL

φ0

starting composition of the mobile phase

t

dimensionless time

C0

feed concentration, mg/mL

τ′ ) τ - x

Gideal

ideal compression factor

where τ is the function of x that is one solution of eq 18a

k

retention factor

τD

K

partition coefficient

dimensionless dwelling time of the chromatographic system

k0

retention factor of solute in 100% of the starting (weak) solvent

τinj

dimensionless injection time

Tinj

dimensionless width of rectangular injection profile

L

column length, cm

τP ) τ - x

where τ may be independent of x

N

index of the segment in the gradient profile in which solute appears at outlet

τR

dimensionless retention time of the solute when it reaches the outlet

q

concentration of solute in the stationary phase, mg/ mL

Subscripts

Q

dimensionless concentration of solute in the stationary phase

i

S

solvent strength parameter

t

time, min

u

interstitial velocity, cm/min

Received for review July 20, 2006. Accepted September 7, 2006.

x

dimensionless axial coordinate

AC061318Y

7840


index

Analytical Solutions of the Ideal Model for Gradient Liquid

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