Langmuirs Equilibrium Adsorption Isotherm

Generally speaking, the single-gas flux through supported zeolite membranes, for a given temperature, depends on the sorption capacity of the gas on the zeolite pores and its equilibrium adsorption constant (Langmuir isotherm is often used to describe the relationship between the amount adsorbed and the gas-phase pressure), the gas diffusion coefficient, the thickness of the zeolite layer, the porosity of the support, and the pressure at the feed and permeate sides. 

The parameters F, and K are the same as in Tables 5.2 through 5.4. Setting Q = 0 (assuming equilibrium surface-subsurface), from each expression in Table 5.5 we deduce the respective equilibrium adsorption isotherm in Table 5.2. In addition, for P = 0 the expressions for Q related to the Frumkin and van der Waals model reduce, respectively, to the expressions for Q in the Langmuir and Volmer models. For Fj F both the Frumkin and Langmuir expressions in Table 5.5 reduce to the Henry expression. 

From (3.11) the Langmuir-Hinshelwood equilibrium adsorption isotherm is obtained ... 

Studies of dynamic adsorption layers require the use of equilibrium adsorption isotherms, as will be demonstrated in detail in Chapter 4. The use of Langmuir s isotherm is restricted to large molecular areas of adsorbing molecules because specific interaction is left out of his account. This isotherm is however the most frequently-used relationship because of its simple form and comparatively good agreement with many experimental data. 

For discrete nonuniformity the total surface coverage is determined by summation of equilibrium adsorption isotherms of Langmuir type... 

The batch adsorption is assumed to be visualized by the application of adsorption isotherm models. The different isotherms give an indication of how the contaminants from MDEA solution interact with the PAAM functional groups and distribute themselves over hydrogel surface. The equilibrium adsorption isotherms of total metal ions as well as organic acid anions were determined at 23°C. The equilibrium MDEA concentration (Ce) for varions uptake capacities of heavy metal ions and HSS anions (qe) were fitted to four standard isotherm models and it was observed that Langmuir isotherm best fitted among all. The maximum adsorption capacity (qmax) valnes obtained from the isotherm model using PAAM... 

Franses, E.I., Siddiqui, FA., Ahn, D.J., Chang, C.-H., and Wang, N.-H.L., Thermodynamically consistent equilibrium adsorption isotherms for mixtures of different-sized molecules, Langmuir, 11, 3177, 1995. 

Free gel was added to definite concentrations of the MB solutions (2-20 mg/1) at room temperature and was noted for its adsorption. It is clear from Figure 13.3 that the dye adsorption increases sharply with an increase in the initial dye concentration. When Cq was reached at 5 ppm and 10 ppm, the was reached at 10.04 and 20.81 respectively, which were much higher than reported Qe values of other adsorbents (Table 13.3). Equilibrium adsorption isotherm is an important criterion to determine the mechanism of dye adsorption on hydrogel. The Langmuir and Freundlich models are widely used to examine the adsorption isotherms. Freundlich isotherm models are based on the assumption that the surface of the adsorbent is not homogeneous. The experimental data in Figure 13.3 was also analyzed with the Freundlich isotherm model, which describes a heterogeneous system with multilayer adsorption. The linear form of Freundlich isotherm equation... 

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. Batch equilibrium adsorption isotherms showing different types of isotherms, a is the adsorption constant in the Langmuir model and is related to the binding energy, and 0 refers to the maximum amount of solute that can be sorbed by the solids. Af is a constant in the FreundUch model, and and Kc are the slopes of the curves.

Langmuir adsorption isotherm A theoretical equation, derived from the kinetic theory of gases, which relates the amount of gas adsorbed at a plane solid surface to the pressure of gas in equilibrium with the surface. In the derivation it is assumed that the adsorption is restricted to a monolayer at the surface, which is considered to be energetically uniform. It is also assumed that there is no interaction between the adsorbed species. The equation shows that at a gas pressure, p, the fraction, 0, of the surface covered by the adsorbate is given by ... 

The preceding derivation, being based on a definite mechanical picture, is easy to follow intuitively kinetic derivations of an equilibrium relationship suffer from a common disadvantage, namely, that they usually assume more than is necessary. It is quite possible to obtain the Langmuir equation (as well as other adsorption isotherm equations) from examination of the statistical thermodynamics of the two states involved. 

For nonlinear systems, however, the evaluation of the flow rates is not straightforward. Morbidelli and co-workers developed a complete design of the binary separation by SMB chromatography in the frame of Equilibrium Theory for various adsorption equilibrium isotherms the constant selectivity stoichiometric model [21, 22], the constant selectivity Langmuir adsorption isotherm [23], the variable selectivity modified Langmuir isotherm [24], and the bi-Langmuir isotherm [25]. The region for complete separation was defined in terms of the flow rate ratios in the four sections of the equivalent TMB unit ... 

Table 9-3. Operating conditions for complete separation under Equilibrium Theory. Langmuir adsorption isotherms (see Fig. 9-8).

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 ... 

Adsorption, like extraction, depends on equilibrium relationships. Isothermal adsorption is projected by Langmuir isotherms. The model is shown in Figure 7.14, which is based on the linear model of the following equation ... 

Assuming adsorption to behave according to the Langmuir adsorption isotherm, we get Eq. (1.22b). Both the rate constant of proton activation and the equilibrium constant of adsorption K q depend on cavity details. 

The Langmuir adsorption isotherm is easy to derive. Again we assume that the catalyst contains equivalent adsorption sites, and that the adsorbed molecules do not interact. If the adsorbed molecules are in equilibrium with the gas phase, we may write the reaction equation as... 

Figure 2.10. Langmuir adsorption isotherm for associative adsorption as a plot of coverage versus pressure for three values of the equilibrium constant, K.

In this assumption, the singlet oxygen adsorption on a surface may be considered a quasistationary process and described by the Langmuir Adsorption Isotherm. Then the equilibrium constant is... 

This is the important Hill-Langmuir equation. A. V. Hill was the first (in 1909) to apply the law of mass action to the relationship between ligand concentration and receptor occupancy at equilibrium and to the rate at which this equilibrium is approached. The physical chemist I. Langmuir showed a few years later that a similar equation (the Langmuir adsorption isotherm) applies to the adsorption of gases at a surface (e g., of a metal or of charcoal). 

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