Langmuir competitive

The competitive adsorption isotherms were determined experimentally for the separation of chiral epoxide enantiomers at 25 °C by the adsorption-desorption method [37]. A mass balance allows the knowledge of the concentration of each component retained in the particle, q, in equilibrium with the feed concentration, < In fact includes both the adsorbed phase concentration and the concentration in the fluid inside pores. This overall retained concentration is used to be consistent with the models presented for the SMB simulations based on homogeneous particles. The bed porosity was taken as = 0.4 since the total porosity was measured as Ej = 0.67 and the particle porosity of microcrystalline cellulose triacetate is p = 0.45 [38]. This procedure provides one point of the adsorption isotherm for each component (Cp q. The determination of the complete isotherm will require a set of experiments using different feed concentrations. To support the measured isotherms, a dynamic method of frontal chromatography is implemented based on the analysis of the response curves to a step change in feed concentration (adsorption) followed by the desorption of the column with pure eluent. It is well known that often the selectivity factor decreases with the increase of the concentration of chiral species and therefore the linear -i- Langmuir competitive isotherm was used ... 

Also for the bi-Langmuir isotherm, that describes a surface which is covered with two different kinds of sites, we can account for the competitive behavior of a mixed sample by using the bi-Langmuir competitive isotherm [109, 112] ... 

Figure 4.2 illustrates the best competitive adsorption isotherm model for benzyl alcohol and 2-phenylethanol [16]. The whole set of competitive adsorption data obtained using Frontal Analysis was fitted to obtain the Langmuir parameters column saturation capacity qs =146 g/1), equilibrium constant for benzyl alcohol bsA = 0.0143) and the equilibrium constant for 2-phenylethanol (bpE = 0.0254 1/g). The quality of the fit obtained with this simple model is in part explained by the small variation of the activity coefficients of the two solutes in the mobile phase when the solute concentrations increased from 0 to 50 g/1. The Langmuir competitive adsorption isotherm simplifies also in the case where activity coefficients are of constant value in both phases over the whole concentration range [17]. 

Figure 4.3 Experimental competitive isotherm data (symbols) and best bi-Langmuir competitive isotherm (solid lines) for bradykinin and kallidin. Reproduced with permission from D. Zhou, K. Kacztmrski, G. Guiockon, ]. Chromatogr. A, 1015 (2003) 73 (Fig. 1).

This equation is a correct Langmuir competitive isotherm only if qg i = qg 2- The second-order isotherm is probably the most useful. The common value of qs is then given by... 

Figure 4.8 Comparison of the two-term and three-term LeVan-Vermeulen isotherms and the Langmuir competitive isotherm. Solid line two-term LeVan-Venneulen isotherm dotted line three-term LeVan-Vermeulen isotherm dashed Une Langmuir competitive isotherm. F = 0.25, qg,l = 1 mmol/ml, 2 = 2 mmol/ml kg =3 = 2/ 01 ...

Figure 4.10 Experimental competitive adsorption isotherm data of ds- and fraws-andro-sterone. Same phase system as in Figure 4.9. Comparison of the Langmuir competitive model (bottom) and the two-term expansion of the LeVan-Vermeulen isotherm (top). In both cases, the best-fit parameters are used to calculate the Unes. Experimental data A ds-androsterone o trafjs-androsterone. Theory czs-androsterone (dotted lines) trans-androsterone (solid Hnes). a and d 2 1 mixture b and e 1 1 mixture and c and f 1 2 mixture. Reproduced with permission from S. Golshan-Shirazi, J.-X. Huang and G. Guiochon, Anal. Chem., 63 (1991) 1147 (Figs. 1 and 2), ( )1991 American Chemical Society.

IAS theory and the LeVan-Vermeulen model were able to explain the reversal of the elution order observed at high concentrations the Langmuir competitive model was imable to do so. This problem is discussed in more detail in Chapter 11 (Section 11.2.5). 

When the system follows Langmuir competitive equilibrium behavior, the coherence condition defines a grid of coherent composition paths to which the system is restricted once the coherence condition is satisfied. Knowing the feed history, i.e., the boxmdary condition, one can use this grid, find the composition routes for the column and predict the column effluent history. 

The Hybrid Method of Mass Balance (HMMB) This method is a modification of the MMB method in which, instead of measuring a series of values of C which is cumbersome, these concentrations are estimated through Eqs. 4.85 and 4.86 by using the isotherm parameters, , and determined by the MCV method. This hybrid approach employs MCV only to estimate the mezzanine concentrations, so it is more acciurate than MCV when the actual isotherm deviates from the Langmuir competitive isotherm. 

Figures 4.26A and 4.26B compare the results of the experimental determination of isotherms using the traditional mass balance method (MMB) and those obtained with MMC. The adsorption isotherm predicted by MMC deviates significantly from the isotherm data obtained by MMB. This may be due to the limited applicability of the Langmuir competitive model for the modeling of the adsorption behavior even of such simple systems as p-cresol and phenol in reversed-phase chromatography. Figures 4.26C and 4.26D compare the results obtained by MMB and HMMB for the same system. Over most of the concentration range, the agreement between the experimental data and the results of these two methods is...

Equation 8.2 states that the two components compete for interaction with the stationary phase, following the Langmuir competitive model. Since the stationary phase concentration of each component at equilibrium is a fimction of both concentrations in the mobile phase, the two partial differential Eqs. 8.1a and 8.1b are coupled. This coupling increases considerably the complexity of the mathematical problem, compared to the single-component case. 

Thus, for a given value of the relative composition of the feed, the intensity of the tag-along effect, like the intensity of the displacement effect, depends strongly on the ratio of the two column saturation capacities. Note, however, that the solution of the ideal model for a binary mixture that is discussed in this chapter assumes that the Langmuir competitive model is valid. But the Langmuir competitive model is truly valid only if 5,1 = qsg-... 

This section summarizes the analytical solutions of the ideal model with a Langmuir competitive isotherm in the different cases identified . 

Assume that the isotherm of each component, i, is given by the Langmuir competitive isotherm model (Eq. 4.5) ... 

This theory was originally developed for the cases in which the retention mechanism is accoimted for by a stoichiometric reaction such as ion-exchange. However, Helfferich [9] has shown that it can also be applied in the case of adsorption, if assuming a Langmuir competitive isotherm (see Eq. 9.45), by introducing a dummy species, p, that is used to convert the non-stoichiometric adsorption of an n-component system into the stoichiometric reaction of an n -I- 1-component system. The Langmuir competitive isotherm is given by... 

In this equation, the solutes are listed in order of their decreasing affinity for the stationary phase. The less retained solute is component n and the more retained is component 1. If all the components follow thermodynamically consistent Langmuir competitive isotherm behavior (in practice, an unusual case, unfortimately). 

Figure 11.5 Comparison between the results of the calculations of the individual band profiles of a 3 1 binary mixfure using the CXZFE method and two finite difference methods (Eqs, 10.78 and 10.80). Solid fine profile calculated with the OCFE method, with Sz = 0.050 cm, St = 0.15 s. Dotted line profile calculated with the forward-backward method (Eq. 10.78), Sz = 0.0050 cm and St = 0.33 s. Dash-dotted line profile calculated with the backward-forward method (Eq. 10.80), Sz = 0.0050 cm and St = 0.033 s. Calculation parameters column length 15 cm. Phase ratio 0.25. Mobile phase flow velocity 0.15 cm/s. Column capacity factors (Cg j = 4, fcpj = 5. Np = 3. 3 1 binary mixture. Langmuir competitive isotherms parameters = 20, fl2 = 16/ = 5 62 = 4 Input profile ...

Figures 13.4a and 13.4b compare chromatograms published by Heme et al. [23] and chromatograms calculated from the data formd in their paper, assuming Langmuir competitive isotherms (Eq. 13.3) and numerical values of the other parameters that permit the best approximation of the experimental results. In this experiment, a sample containing acetonitrile, N,N-dimethylformamide, ethyl formate, fso-butanol, ethyl acetate and 1-pentanol is eluted on a C18 chemically bonded silica phase (Nucleosil C18), using a water-methanol solution containing 0.00025 M salicylamide, the UV-absorbing additive. The agreement between experimental and calculated chromatograms is excellent. It demonstrates the validity of the theoretical approach. The system peaks associated with the first five components, those that are eluted before the additive system peak (peak 6), are...

In this case, the system of equations of a model of chromatography (Chapter 2) must be solved numerically. These models are very general. They apply to all modes of chromatography, independently of the model of competitive isotherms selected. Most theoretical studies used the equUibrium-dispersive model of chromatography because the mass transfer kinetics are fast xmder the experimental conditions employed in the study of system peaks. These theoretical studies also used the Langmuir competitive model because it is both general and convenient. 

For Langmuir competitive isotherms, the optimum flow rate ratios are [43]... 

At the optimum value of m2 and m3, the flow rate of desorbent that is necessary to operate the SMB decreases and the concentrations of the compounds 1 and 2 in the products increase with decreasing values of mj and increasing values of m. So, for Langmuir competitive isotherm behavior, we have... 

Equations 4.48 through 4.51 are different forms of the Langmuir competitive adsorption isotherm it can easily be extended to more than two adsorbates. They have been written for gas phase, but equally they can apply to adsorption from solution, replacing the pressure by activity or (approximately) concentration of each solute. 

Further, this model is based on adsorption modeling (Section 11.1) thus, the binding of ions at each plane is treated as Langmuir competitive adsorption, with consideration for electrostatic interactions in the adsorption equilibrium constants as in Equations 11.16 and 11.17. Thus, the charge at the 0 plane is given by the balance between adsorbed protons and adsorbed hydroxide ions... 

In equations (6) and (7) all the terms are known or can be experimentally determined, except that of the Langmuir competitive adsorption coefficients bi. Thus n equations can be used to determine the unknown bi (i = 1,2, -n). 

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