In the Classroom
An Analogue Column Model for Nonlinear Isotherms: The Test Tube Model Setsuro Sugata and Yoshihiro Abe Kyoritsu College of Pharmacy, Shibakoen 1-5-30, Minato-ku, Tokyo 105, Japan Asymmetric chromatographic peaks, “tailing” and “fronting”, have been explained well by the convex and concave isotherms, respectively (1, 2). However, it is not always easy for students to come to an instantaneous understanding of the theoretical basis of these peaks. There have been reports on the simulation study of the chromatographic process based on the discrete flow model (one of plate theories) for the linear isotherms (3, 4) and nonlinear (e.g. Langmuir) isotherms (5), as well. Of course these simulations can easily represent different types of chromatograms with changing parameters, but they alone can not give students a thorough understanding of the chromatographic basis, mostly because they involve intermediary equations and calculations. We devised an analogue column model to demonstrate chromatography. The model needs no calculations and is operated manually. It uses simple glass vessels (flat-bottomed test tubes) and involves simple procedures. Therefore, we have named this model the “test tube model” (TT model).
Table 1. Relationship between Combinations of Vessels and Isotherms Isotherm
Combination of Vessels
Linear Nonlinear (Langmuir)
Thick–Thick
Thin–Thin
Thick–Thin
Trumpet–Trumpet
Thick–Trumpet
Thin–Trumpet
TT Model for Linear Isotherms The discrete flow model, which is the theoretical basis of the TT model, has been well illustrated using ideal countercurrent chromatography (CCC) (1, 3). In CCC, column operations consist of two kinds of processes: the equilibration of solute molecules between the stationary and mobile phases in each tube, and the subsequent transfer of the series of mobile phases to the next tubes. Alternate repetitions of these two processes transfer the solutes at their intrinsic rates. Thus, the solutes are separated. Each tube is considered to be one theoretical plate according to the plate theory of chromatography (3). The equilibrium state of one kind of solute at each plate is illustrated in Figure 1. To simplify the description, we assumed that the volume of the stationary phase (VS) is equal to that of the mobile phase (VM). Under this assumption, the discussion about the concentration (CS and C M) is replaceable by discussion about the amount of solute (nS and n M). The partition coefficient, K, is defined as follows: K = CS / CM
(a)
(b)
Figure 1. Illustration of the equilibrium state of one kind of solute at each plate in the test tube model (TT model) of chromatography exemplified by a linear isotherm. S, stationary phase; M, mobile phase. (a) Actual state. (b) Image in the TT model. In the actual TT model, the solute in (b) is represented by water, and solvents and the capillary connection are omitted.
(a)
(1)
(b)
In linear isotherms this ratio is independent of the concentrations. At equilibrium, the solute is partitioned between two phases according to the value of the capacity factor, k9, for the phases expressed as follows: k9 = nS / nM = C SVS / CMVM = K(VS / VM)
(2)
Under the assumption that VS = VM, the value of k9 is equal to that of K.
Concept of TT Model In the TT model, the solute molecules are imagined separated from the solvents and gathered together tightly in shapes having intrinsic cross-sectional areas (AS and AM) as shown in Figure 1(b). The solute molecules can move through the capillary connection; at equilibrium the heights, h, of the shapes are equal.
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(c) Figure 2. Three kinds of flat-bottomed vessels used in the TT model: (a) thick test tube; (b) thin test tube; (c) trumpet-shaped vessel. A represents the averaged cross-sectional area of the column- or trumpet-shaped water emerging when water is poured into these vessels (as shown in Fig. 3).
Journal of Chemical Education • Vol. 74 No. 4 April 1997
In the Classroom In the actual TT model the gathered solute molecules are represented by “water”. Therefore, the volume of water denotes the amount of solute. A pair of shapes of water is formed by a pair of flat-bottomed vessels (Figs. 2 and 3); the pair represents an isotherm. At equilibrium, water is partitioned between the vessels according to the value of k9. This partition is accomplished by making the water levels in the two vessels the same (Figs. 1(b) and 3–5). Then the value of k9 is represented by the ratio of A of the test tubes as shown in eq 3 (see Figs. 2 and 3).
(a) Linear
k9 = nS / nM = (vol. of water)S / (vol. of water)M = ASh / AMh = AS / AM
(3)
This representation of the equilibrium state is similar to that in the hydrodynamic model of chemical reactions (6). In the TT model, this representation is applied not only to linear but also to nonlinear isotherms. (b) Nonlinear
Combinations of Vessels It is easy to imagine the shapes of isotherms from the combinations of vessels; this is one of the strong points of the TT model. Six combinations derived from three kinds of vessels (Fig. 2) are classified according to the corresponding isotherms (Table 1). The TT models of five kinds of isotherms (Fig. 3) derived from three of the combinations were studied in detail. Five kinds of isotherms correspond to five kinds of solutes.
Figure 3. Representations of isotherms in the TT model. Under the assumption that VS = VM, the value of K is equal to that of k9 and the nS – nM graph is equal to that of CS – CM (see eqs 1–5). Each value of k9 is equal to the ratio of the values of A in Figure 2.
Procedures Procedures in the TT model are shown in Figure 4, in which N, j, and n represent the number of theoretical plates, the plate number, and the number of transfers (3), respectively, for one kind of solute. Each plate consists of a pair of thick and thin test tubes, representing the stationary and mobile phases, respectively. Some volume of colored water is poured into the first plate (j = 0 at n = 0). Equilibration is achieved using a graduated pipet (see appendix). Alternate repetitions of the equilibrations and the transfer (thin test tubes are supplied from the left side) result in the distribution of water. This distribution is called a “position peak” (3); it approaches a Gaussian distribution in a linear isotherm. At last, water is distributed in all plates (n = 4). After one more repetition, the mobile phase in the last plate (n = 4 and j = 4) leaves the column (n = 5). In this way, mobile phases leave the column successively (n = 5, 6, …); the series of various amount of water that can be transferred to thinner flat-bottomed test tubes (A = 2.18 cm 2) for the “higher sensitivity” is a “chromatogram” or an “exit peak” (3). If necessary, the volume of water at the “detector” can be measured. If a demonstration of the separation of two kinds of solutes is desired, then an additional series of paired vessels (accompanied by differently colored water) should be prepared. TT Model for Nonlinear Isotherms
Figure 4. Procedures in the TT model exemplified by a linear isotherm ( k9 = 2.63). The equilibrations (short arrows) and the transfer (long arrows) are repeated alternately.
In nonlinear isotherms the values of K and k9 depend on the concentrations, CS and C M (Fig. 3[b]). These isotherms are interpreted from the point of view of the linear isotherm as follows. The slope of a straight line imagined between the origin and any point on the curve corresponds to K in eq 1. As the concentrations increase, the value of K varies. According to this value, the solute is partitioned between two phases (eq 2). In the TT model of these isotherms, one of the paired test tubes is replaced by a trumpet-shaped vessel (Figs. 3[b] and 5). This vessel is designed to represent a Langmuir isotherm by pairing with a thin test tube.
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In the Classroom The Langmuir equations for convex and concave isotherms are eqs 4 and 5, respectively. nS = abnM / (1 + bnM)
(4)
nM = abnS / (1 + bnS)
(5)
In these equations, the values a = 77.4 and b = 0.0232 were obtained by representing n S and nM as the volume of water in milliliters. The averaged cross-sectional area of the trumpet-shaped water (A in Fig. 2[c]) corresponds to A in eq 3. As the volume of water increases, the value of A increases. Therefore, the value of k9 also varies with the total volume of water (see Fig. 3[b]).
Vessels The vessels were designed arbitrarily by referring to the results of computer simulations based on the equations in the literature (3, 5). The vessels produced by a glass blower differed slightly from the vessels designed and ordered; the characteristics of the actual vessels produced and used (Fig. 2) were obtained from five samples by measuring the volume of water at various heights. Twenty-five thick and 50 thin test tubes were prepared, and their shapes were good circular cylinders. Thirteen trumpetshaped vessels designed as the body of revolution of a quadratic equation were produced from the same carbon template. These trumpet-shaped vessels can be replaced by circular cone-shaped measuring glasses (and so on), though the isotherms in these cases don’t represent the Langmuir isotherms correctly.
Figure 5. Position peak on a convex isotherm at stage n = 4, in the TT model. This causes “tailing” of the chromatogram.
k 9 = 0.38 (212 mL)
k 9 = 1 (117 mL)
Results and Discussion TT models on each “solute” were done for both N = 5 and N = 10. Some of the results obtained for N = 10 are shown in Figures 6 and 7. The results were expressed in terms of recovery, which is equal to the ratio of the volume of water in the mobile phase at the “detector” to the total volume of water. In this expression, the total volume of water does not affect the results in linear isotherms, but does affect the results in nonlinear isotherms. All experimental results agreed well with the values simulated (3, 5). The chromatograms in linear isotherms exhibited “tailing” to various extents (Fig. 6). The degree of tailing decreases and the peak becomes more Gaussian as N and the value of k9 become larger (3). In the TT model, because N is a very small number, this effect is not negligible. The value of k9 relates to the so-called retention time (tR) as shown in Figure 6. Therefore, the tailing and fronting peaks in nonlinear isotherms are well-demonstrated in comparison with peaks having the same tR in a linear isotherm, i.e., when k9 = 1 (Fig. 7). In nonlinear isotherms, tR can be adjusted to some extent by changing the total volume of water. The operation of TT models at N = 10 is relatively laborious, whereas at N = 5 this is not so. TT models at N = 5 are sufficient to demonstrate chromatography, although their chromatograms showed a stronger tendency toward tailing.
Retention Behavior and Symmetry (Linear Isotherms) In linear isotherms, the difference in the values of k9 is easily understandable from Figures 3(a) and 4. As the value increases, the volume of transferred water decreases: therefore, the band velocity decreases, and the value of tR increases (Fig. 6). The approach of the position peak (chromatogram also) to Gaussian distribution (“symmetry”) with increasing number of n (or N) that accompanies peak broadening and decreasing height is easily understood as follows. After each transfer, at both edges of the band, equilibrations
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k 9 = 2.63 (212 mL)
n
Figure 6. Chromatograms obtained from the TT model at N = 10 (linear isotherms). Open circles and solid circles are experimental values. Lines are drawn between the simulated values. Volumes in parentheses are the total volumes of water in the TT model.
concave (192 mL) (–• –)
linear (k 9 = 1) (–• –)
°
convex (192 mL) (– –)
n Figure 7. Chromatograms obtained from the TT model at N = 10 (nonlinear isotherms). Data for the linear isotherm (k9 = 1) are reproduced from Figure 6 for reference. For others, see Figure 6.
Journal of Chemical Education • Vol. 74 No. 4 April 1997
In the Classroom are done between the water-containing and -noncontaining tubes (Fig. 4). Therefore, at both edges the water levels decrease drastically. As the position is nearer to the center, the degree of decrease becomes smaller. In addition to this phenomenon, the equality of the band velocities in all watercontaining plates leads to a Gaussian distribution.
Tailing and Fronting (Nonlinear Isotherms) In nonlinear isotherms, TT models are also easily understandable. An example of a convex isotherm is shown in Figure 5. From this it is easily recognized that the values of k9 at the center of the band are smaller than those at either edge. Therefore, because the center moves faster than the edges, the chromatogram exhibits tailing (Fig. 7). Conversely, in a concave isotherm the center moves more slowly than the edges, and so the chromatogram exhibits fronting (Fig. 7). We think these explanations based on TT models are much more understandable than other explanations (1, 2).
(a)
(b)
Figure 8. (a) Siphon filled with water. (b) “Equilibration” using the siphon.
Conclusions The TT model is described for cases of linear and nonlinear (e.g., Langmuir) isotherms. The model accurately produces the results expected from the theory of CCC. As discussed above, the TT model is a convenient model to illustrate the concept of chromatography directly. This is because (i) this model needs no equations and no calculations and is operated manually; (ii) two kinds of processes (the “equilibration” and the “transfer of the series of mobile phases”) are very simple and reasonable; and (iii) the shapes of isotherms can be imagined easily from the combinations of vessels. Appendix Equilibration using a siphon is more convenient and more reasonable than equilibration using a graduated pipet. N pieces of siphon filled with water and closed off by clips are prepared (Fig. 8[a]). Of course the filled water is not counted as “solute”. The siphons are dropped into each
two-phase pair, which should be equilibrated, and the clips are removed. Then water moves spontaneously through the siphons and equilibration is achieved (Fig. 8[b]). After this the siphons are broken again. After one transfer of the series of mobile phases, the siphons are pulled out and dripped into newly paired “two phases”. In this method the effective cross-sectional areas of vessels should be diminished to that of the glass tube, A9 in Figure 8(a). If students are not confused, this method is recommended. Literature Cited 1. Gaucher, G. M. J. Chem. Educ. 1969, 46, 729–733. 2. Scott. R. P. W. Liquid Chromatography Column Theory; John Wiley & Sons: Chichester, 1992; p 41. 3. Fritz, J. S.; Scott, D. M. J. Chromatogr. 1983, 271, 193–212. 4. Hara, S.; Dobashi, Y.; Dobashi, A. Kagaku no Ryoiki 1982, 36, 752– 760; Muraki, A.; Sasamura, Y. Kaiho-Kagaku PC Kenkyukai 1989, 11, 27–32. 5. Sundheim. B. R. J. Chem. Educ. 1992, 69, 1003–1005. 6. Lago, R. M.; Wei, J.; Prater, C. D. J. Chem. Educ. 1963, 40, 395–400.
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