Langmuir equilibrium

Then, the relationship between tfA B and the Langmuir equilibrium parameter (La) is... 

The Langmuir equilibrium approach was developed in 1918 by Langmuir to describe vapor adsorption on a homogeneous surface. It incorporates several assumptions when employed to model adsorption of chemical species in soil-solution suspensions. These assumptions are that... 

Table V. Comparison of Langmuir Equilibrium Constants for Organic Pesticides and Selected Adsorbates...

Hiester and Vermeulin [39] extended the model of Thomas by deriving a breakthrough curve for a slow diffusion rate-limiting step with a nonlinear Langmuir equilibrium isotherm. The model was further adapted by Arnold and Blanch [40] to describe affinity chromatographic experiments ... 

The slopes and linear coefficients of lines allow to determine the Ng and values corresponding to these runs these values are almost the same as those determined by means of Equation (19) (error percentage, +4%). Eigure 8 also reports the adsorption results obtained in the absence of irradiation, that is, in the dark. A least-squares best fitting procedure allows to determine the values (R > 0.99) of the Langmuir equilibrium constant and the maximum adsorption capacity in the absence of irradiation, that is, fCL = 350 and Ns = 4.57 x 10 mol/g of catalyst. 

The Langmuir equilibrium isotherm model can be extended to multicomponent systems [7,8]. However, when several components are simultaneously present in the solution, these compounds interfere. The amount of each of them that is adsorbed at equilibrium is smaller than if this compoimd were alone. Although... 

We will assume a Langmuir equilibrium isotherm. The general equation of the characteristic line that begins in a point of this boundary condition, i.e., from the point X = Q,t = Tjtfi, (0 < / < 1) is... 

We note that this last experimental result is in contradiction to the predictions made on the basis of competitive Langmuir equilibrium behavior. Crommen et al. [21] suggested that complexation could take place between the analyte and one of the mobile phase components, and that the free analyte and the complex would compete for interaction with the stationary phase. If this correct, the isotherms would be written... 

A constant in the Langmuir equilibrium equation Moles adsorbed at time, t Moles adsorbed at equilibrium Equivalent moles error due to pressure error Equivalent moles error due to volume error Equivalent moles error due to temperature error Moles adsorbed at steady state (constant)... 

It is evident from Eq. (8.3) that the equilibrium. factor is in general concentration dependent. However, in the special case of a Langmuir equilibrium isotherm... 

TABLE 8.6. Summary of Asymptotic Constant-Pattern Solutions for Breakthrough Curve for Langmuir Equilibrium... 

FIGURE 8,22. Combined effects of axial dispersion and mass transfer resistance for a favorable Langmuir equilibrium system ( ==0.33). a) Constant pattern breakthrough curves for various values of (fluid film resistances + axial dispersion), (h) The same curves plotted on a modified time scale with 7 = t/(1 + B). [Reprinted with permission from Chem. Eng. Sci 30. Garg and Ruthvcn (ref. 54). Copyright 1975, Pcrgamon Press, Ltd.)... 

With the development of improved numerical methods for solution of differential equations and faster computers it has recently become possible to extend the numerical simulation to more complex systems involving more than one adsorbable species. Such a solution for two adsorbable species in an inert carrier was presented by Harwell et al. The mathematical model, which is based on the assumptions of plug flow, constant fluid velocity, a linear solid film rate expression, and Langmuir equilibrium is identical with the model of Cooney (Table 9.6) except that the mass transfer rate and fluid phase mass balance equations are written for both adsorbable components, and the multicomponent extension of the Langmuir equation is used to represent the equilibrium. The solution was obtained by the method of orthogonal collocation. 

We can outline the procedure used to solve the diffusion problem under the Langmuir equilibrium adsorption isotherm. A diffusion field in the solution (5.29) together with Fick s law gives... 

Note that even if the adsorption reaction is in equilibrium, the diffusion field is time-dependent. Here q is the mass flux of the species a in the solid, and k is a diffusion coefficient of a species a in the solid. Differentiating (5.112) of the Langmuir equilibrium adsorption isotherm with respect to time gives... 

Fig 7 4), which corresponds to the Langmuir equilibrium with separation factor rl in the Freundlich isotherm system In these cases, a constant profile of mass transfer zone is established while adsorption proceeds in a column (Fig 7 5) This is in contrast to linear isotherm systems where mass transfer zone continues to spread with increase of traveling time The reason for the formation of a constant pattern is explained as follows The speed of movement of the point on the mass transfer zone whose concentration is C, F(C), can be related to the equilibrium through the basic equation... 

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