Dispersion Langmuir adsorption equation

For all these reasons, the mathematical aspects of the theory become much more complex. The mathematics of nonlinear chromatography are so complex that even for a single solute, there is no analytical, closed-form solution available, except with two simplified models, the ideal model and the Thomas model [120]. The ideal model is based upon the assumption of an infinite column efficiency. Its solutions are discussed in detail in Chapters 7 to 9. The Thomas model is based upon the assumptions that there is a slow Langmuir adsorption-desorption kinetics and that there are no other nvass transfer resistances, nor any axial dispersion. The system of equations of this model has been solved by Goldstein [121], and this general solution has been simplified for pulse injection by Wade et al. [122]. In aU other cases, the problem must be solved numerically. The Thomas model is discussed with other kinetic models in Chapter 14 and 16. 

The more dispersive solvent from an aqueous solvent mixture is adsorbed onto the surface of a reverse phase according to Langmuir equation and an example of the adsorption isotherms of the lower series of aliphatic alcohols onto the surface of a reverse phase (9) is shown in figure 9. It is seen that the alcohol with the longest chain, and thus the most dispersive in character, is avidly adsorbed onto the highly dispersive stationary phase, much like the polar ethyl acetate is adsorbed onto the highly polar surface of silica gel. It is also seen that... 

Knox and Piper (13) assumed that the majority of the adsorption isotherms were, indeed, Langmuir in form and then postulated that all the peaks that were mass overloaded would be approximately triangular in shape. As a consequence, Knox and Piper proposed that mass overload could be treated in a similar manner to volume overload. Whether all solute/stationary phase isotherms are Langmuir in type is a moot point and the assumption should be taken with some caution. Knox and Piper then suggested that the best compromise was to utilize about half the maximum sample volume as defined by equation (15), which would then reduce the distance between the peaks by half. They then recommended that the concentration of the solute should be increased until dispersion due to mass overload just caused the two peaks to touch. 

Before our theory was fully developed, extensive work by J. Koral in cooperation with R. Ullman (15) confirmed in detail and with considerable accuracy all previously known features. They ascertained, in addition, the particulars of the adsorption isotherms for a number of polymers and dispersed adsorbates and established the remarkable degree to which most isotherms could be approximated by 2-parameter equations, like Langmuir s isotherm for monolayers of small molecules. They found the dependence of the adsorption on MW to be weak and determined the area per adsorbed molecule. 

The solution of the simplest kinetic model for nonlinear chromatography the Thomas model [9] can be calculated analytically. The Thomas model entirely ignores the axial dispersion, i.e., 0 =0 in the mass balance equation (Equation 10.8). For the finite rate of adsorption/desorption, the following second-order Langmuir kinetics is assumed... 

In case of no axial dispersion, the right-hand side is zero. The adsorption equilibrium is represented by Langmuir equation ... 

Figure 8A. Adsorption isotherms of Pseudocyanine (No. 1, Circles) and of Astraphloxin (No. 2, Squares) in AgBr (Dispersion D) containing 0.2% gelatin at 23°C., pBr 3, pH 6.5. The data are expressed as the concentration of free dye (c) in equilibrium with dye adsorbed per mole of AgBr (a). Open data points and solid lines Results calculated from surface spectra. Solid data points and dashed lines Results obtained from phase-separation procedure B. Adsorption isotherms of Figure 8A expressed in terms of the Langmuir equation. See text...

All the balances have accumulation, convection, axial dispersion, and reaction terms. The equations include liquid holdup, Bi, and superficial liquid velocity, w. Langmuir-type rate equation, for the main reaction, Equation 15.4, included also an activity correction term a. Kst and in Equations 15.5-15.7 indicate the adsorption parameters for stearic acid and heptadecene, respectively. Equation 15.4 corresponds to a monomolecular transformation of stearic acid via the adsorption of the reactant to the main product. Adsorption terms for stearic acid and heptadecene were used, since both of these compounds contain functional groups enabling adsorption on the active sites of the catalyst Reaction rates were assumed not to be limited by heptadecane adsorp-UoiL Thus, the adsorption term of heptadecane was n ected. In line with the experimental observations indicating catalyst deactivation. Equation 15.4 (Table 15.2) was modified to incorporate the decrease in catalyst activity. In particular, the activity was assumed... 

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