Adsorption Langmuir

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 ... 

Various functional forms for / have been proposed either as a result of empirical observation or in terms of specific models. A particularly important example of the latter is that known as the Langmuir adsorption equation [2]. By analogy with the derivation for gas adsorption (see Section XVII-3), the Langmuir model assumes the surface to consist of adsorption sites, each having an area a. All adsorbed species interact only with a site and not with each other, and adsorption is thus limited to a monolayer. Related lattice models reduce to the Langmuir model under these assumptions [3,4]. In the case of adsorption from solution, however, it seems more plausible to consider an alternative phrasing of the model. Adsorption is still limited to a monolayer, but this layer is now regarded as an ideal two-dimensional solution of equal-size solute and solvent molecules of area a. Thus lateral interactions, absent in the site picture, cancel out in the ideal solution however, in the first version is a properly of the solid lattice, while in the second it is a properly of the adsorbed species. Both models attribute differences in adsorption behavior entirely to differences in adsorbate-solid interactions. Both present adsorption as a competition between solute and solvent. 

Derive Eq. XI-IS, assuming a Langmuir adsorption process described in Eq. XI-2, where ka and kd are the adsorption and desorption rate constants. Treat the solution... 

One may choose 6(Q,P,T) such that the integral equation can be inverted to give f Q) from the observed isotherm. Hobson [150] chose a local isotherm function that was essentially a stylized van der Waals form with a linear low-pressure region followed by a vertical step tod = 1. Sips [151] showed that Eq. XVII-127 could be converted to a standard transform if the Langmuir adsorption model was used. One writes... 

Make a plot of Eq. XVII-69 as 6 versus P, and, for comparison, one of a Langmuir adsorption isotherm of same limiting or Henry s law slope. Comment on the comparison. 

The straight line in Figure A 1.7.8 shows the relationships between S and 0 expected for various models, with the straight line indicating Langmuir adsorption. 

If Langmuir adsorption occurs, then a plot of 9 versus p for a particular isothenn will display the fonn of equation (Al.7.3). Measurements of isothenns are routinely employed in this manner in order to detennine adsorption kinetics. 

Another limitation of tire Langmuir model is that it does not account for multilayer adsorption. The Braunauer, Ennnett and Teller (BET) model is a refinement of Langmuir adsorption in which multiple layers of adsorbates are allowed [29, 31]. In the BET model, the particles in each layer act as the adsorption sites for the subsequent layers. There are many refinements to this approach, in which parameters such as sticking coefficient, activation energy, etc, are considered to be different for each layer. 

Hyperbolic, based on the Langmuir adsorption principle for instance,... 

A solvent can be adsorbed from a solvent mixture on the surface of silica gel according to the Langmuir adsorption isotherm as previously discussed. 

Figure 5. Langmuir Adsorption Isotherms for Benzene, Butyl Chloride and Chloroform...

This equation is the mono-layer adsorption isotherm of solvent (B) and is exactly the same as the previously derived Langmuir adsorption equation but somewhat differently expressed. 

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).

Correlation between experimental and calculated concentrations is obtained using a modified Langmuir adsorption isotherm with the following parameters ... 

This is the Langmuir adsorption isotherm in its original form. In pharmacological nomenclature, it is rewritten in the convention... 

Affinity can be depicted and quantified with the Langmuir adsorption isotherm. 

As with the Langmuir adsorption isotherm, which in shape closely resembles Michaelis-Menten type biochemical kinetics, the two notable features of such reactions are the location parameter of the curve along the concentration axis (the value of Km or the magnitude of the coupling efficiency factor) and the maximal rate of the reaction (Vmax). In generic terms, Michaelis-Menten reactions can be written in the form... 

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