Langmuir type equation

As stated earlier, the retardation term owing to the product adsorption or the constant K in the Langmuir-type equation, v = k/(l + /<[P]) (Equation 13.4), where P is the condensable product), is related to its adsorption equilibrium driven by the concentration term in solutions. Under boiling conditions, the reaction products are desorbed not only to the solution bulk but also into the bubble formed at the catalyst surface. Because the molecular translational entropy is far larger in the gas phase than in the liquid phase, the... 

The Exchange Isotherm. For adsorption at constant pH the amount of selenite on the surface is related to the solution concentration by an equation of the Langmuir type (Equation 1). 

The partial occupaneies (O.j) can be calculated through a Langmuir-type equation ... 

In UTCHEM, the Langmuir-type equation is used to describe pH-dependent adsorption. To include pH effect, we can use the following empirical equation to modify the input parameter as in Eq. 7.154 (with permeability correction omitted here) ... 

As the mass transfer resistance at the external surface of the silicalite crystal particle has negligible effects on the overall mass transfer in the present LC system (13), local equilibrium at the external surface thus can be assumed to simplify the mathematics involved in the numerical solution. Since the adsorption isotherm data of alcohols from dilute aqueous solution in silicalite, as reported by Milestone and Bibby (2), can be described by a Langmuir type equation, the following Langmuir type equation is used as the boundary condition at the external surface of the silicalite crystal particle ... 

If we assume a Langmuir type equation of state (2.37) to describe the equilibrium adsorption behaviour of the system then the following exchange of matter fimction is obtained,... 

The experimental data obtained by the modified Hummel-Dreyer method were analyzed using Scatchard plots. Binding affinity values were calculated from a Langmuir-type equation. When a compound binds to only one binding site on a protein, the model ean be represented by the following equation ... 

It will be obvious that equations (35) and (37) are not necessarily restricted to the special model used. In fact, equations of this type will always be obtained when mixing is random and interaction is confined to nearest neighbours.. With small values of the concentration A o becomes proportional to cpr, L e., a small quantity of a higher order than the concentration itself, a result which evidently is not affected by corrections of the Langmuir type (equation 37). In section 6 b. 3, p. 70 this result will be derived from a general thermo-, dynamic consideration. It is of such great importance in its application to the properties of dilute solutions, that we shall also indicate its molecular origin. 

The local adsorption isotherm is assumed to take the form of the Langmuir-type equation... 

Once the above procedure is established, the adsorption isotherm can be determined, whereby the amount of adsorption in moles per unit area, F, is plotted as function of equilibrium concentration, C2. With many surfactants, where adsorption reversible, the results can usually be fitted with a Langmuir-type equation, i.e. r increases gradually with increasing C2 and, eventually, reaches a plateau value. 

One of the assumptions made by Horvath and Kawazoe [35] in their original derivation was that the adsorbate behaved as a two-dimensional ideal gas. This implies that the isotherm obeys Henry s law and is therefore linear in nature. The equation of state shown in Eq. (30) was thus substituted in Eq. (29), which caused the term T/d) dli/dT) = RT to cancel out with the other RT term in the expression, resulting in Eq. (31). However, the isotherms for the typical sorbates used in HK analysis, such as N2 at 77 K and Ar at 87 K, clearly show a type 1 adsorption behavior. An example of such an isotherm is shown in Fig. 10. As is obvious from the figure, the assumption of linearity is only valid for the steeply rising portion of the isotherm, whereas the concave portion of the steep rise may also provide useful information. For this reason, Cheng and Yang [40] proposed the use of a Langmuir-type equation of state in place of Eq. (30) because it is known to represent type 1 isotherms in the best manner ... 

Figure 8.3b shows that the initial polymerization rate (Rpo) is not linearly dependent on the monomer concentration (here, the initii polymerization rates were estimated by extrapolating the polymerization rate data to t = 0). The initial rate levels off for the initial monomer concentration larger than about 2.0mol/l, possibly because of a rapid buildup of polymer around the silica-supported catalyst particles. The initial polymerization rate profile can be fitted experimentally using the following Langmuir-type equation where it is assumed that the monomer sorption equilibrium is established between the solid phase and the liquid phase ... 

This equation can be applied even under low NH3 pressure. When various catalysts are examined below a total pressure of 1 atm, the activity (or NH3 pressure on the catalyst) is sometimes quite low. Under these conditions, the Langmuir type equation is quite often used. 

A Langmuir-type equation with NH as the main adsorbed species. 

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