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The "test tube model" (TT model), an analogue column model manipulated by hand, was devised for the explanation of asymmetric peaks[8]. The theoretical basis of the TT model is the Craig plate model, which has been well illustrated using ideal countercurrent distribution [1, 3, 6]. In the TT model, solute molecules are imagined separated from the solvents and gathered together tightly, and are represented by "

Therefore, a computer program was developed to draw graphically the TT model accompanied by some modifications for better presentation; the modified model was named the "double-glazed vessel model" (DGV model). Then, simulations of chromatography using the DGV model were performed.

In linear isotherms, this ratio is independent (in nonlinear isotherms, dependent) of solute concentrations. The capacity factor,

Figure 1. Concept of the "double-glazed vessel model" (DGV model) exemplified by a linear isotherm. A and B represent the same equilibrium state of one kind of solute at a given plate in a column. In the DGV model, solvents are omitted and the *solute* (full lines) is in a vessel (broken lines). The *solute* levels (*h*) in the two compartments are the same (equilibrium state). The volume of *solute* in each compartment denotes the amount of solute (*n*), not phase volume (*V*); also, each compartment does not mean the whole of each phase and is drawn with adequate height. Other symbols: *C*, concentration of solute; *A*, averaged cross-sectional area of *solute*; *r*_{b}, radius at the bottom. For details, see the text.

where *V _{S}* and

Figure 2. Isotherms ordinarily represented (left) and the corresponding isotherms represented by the DGV model (right). Under the assumption that *V _{S}* =

This is a fundamental equation for the TT and DGV models, and suggests that many models can be used for one isotherm; for example, in the TT model [8], a linear isotherm (

In this isotherm, because the value of

Equation 5 is identical to the inverse function of Eq.4:

For nonlinear isotherms (Figure 2B and Figure 2C), although it is difficult to estimate the value of

The amounts of

For the drawing of outlines of

To draw outlines, relationships between the volume of

where

For linear isotherms, the relationships on the outer outlines are represented as

For the convex isotherm, the relationship on the outer outline is obtained from Eqs. 4 and 7 by replacing

where

where

Each of the inner and outer outlines of

The migration velocity (

where c is a proportional constant.

Figure 3. Computer simulations of chromatography using the DGV model for linear isotherms. *n*_{t} (total amount of *solute*) = 360 ml. A: Position peaks (equilibrated states) represented by the DGV model at stage *n* = 7. Vessels are omitted. *r*_{b,out} = 1.63cm. *j* : the plate number. Arrows (neglected in plates with less than 1% of *n*_{t}) represent band velocities, *v* (Eq.11). B: Position peaks (ordinary graphs) corresponding to A. C: Chromatograms with *N* (number of theoretical plates) = 30. *n*: the number of transfers.

The characteristic retention behaviors for the linear isotherms are as follows, and are directly understood by referring to Figures 2, 3 (if necessary, see Eqs. 3 and 11). As the value of *k'* is independent of the amount of *solute*, each position peak has its constant value of *v* in all parts of the band. Therefore, the early asymmetric position peak approaches a Gaussian (symmetric) distribution with increasing value of *n* [1, 6]; the constancy of *v* is a strong reason for the process toward a Gaussian distribution. As the tendency toward tailing is weakened, the chromatogram (Figure 3C) also approaches a Gaussian distribution with increasing value of *N* [6, 8]. The differences between the exact Gaussian profiles [1] and the chromatograms (Figure 3C) are very small. From the comparison of two isotherms (*k'* = 0.5 and 1), as the value of *A _{M}* increases relatively to

Figure 4. Computer simulations of chromatography using the DGV model for nonlinear isotherms. Langmuir coefficients are in Figure 2. *n*_{t} = 360 ml. A: Position peaks (equilibrated states) represented by the DGV model at stage *n* = 7. *r*_{b,in} = 1.15cm. Vessels are omitted. Arrows (neglected in plates with less than 1% of *n*_{t}) represent relative band velocities, *v*_{rel} (Eq.12). B: Position peaks (ordinary graphs) corresponding to A. C: Chromatograms with *N* = 30.

Asymmetric peaks caused by nonlinear isotherms have been explained well by the difference of *v* between the positions of the band [2, 3, 5, 8]. In this paper, these peaks are explained by the DGV model likewise. The following can be directly understood by referring to Figures 2, 4 (if necessary, refer to Figure 3 also). In these isotherms, the value of *k'* is dependent on the amount of *solute*; in the DGV model, *A _{S}* is independent of the amount of

where

The sample size (

The position peak represented by the DGV model (in Figure 3A and Figure 4A) is a kind of bar graph with a special function, and explains why one early asymmetric position peak (the peak of

Computer simulations of chromatography using the DGV model with changing parameters were performed easily. Position peaks for the linear isotherms (Figure 3) at an appropriate stage are useful to understand the process toward a Gaussian distribution and the fundamental retention behaviors directly. Those of nonlinear isotherms (Figure 4) are also useful to understand asymmetric peaks directly, because the following explanation can be directly understood. The part of the

[ 2] Horvath, Cs. and Melander, W.R.,

[ 3] Gaucher, G.M.,

[ 4] McDonald, P.D. and Bidlingmeyer, B.A.,

[ 5] Scott, R.P.W.,

[ 6] Fritz, J.S. and Scott, D.M.,

[ 7] Sundheim, B.R.,

[ 8] Sugata, S. and Abe, Y.,

[ 9] Sugata, S.,

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