Langmuir equilibrium relation

This is called a Langmuir adsorption isotherm for a species A, and the function 9a( Pa) isi shown in Figure 7-23. The KjS are the adsorption-desorption equilibrium constants for species A and B. By historical convention we call these the adsorption isotherms. Before i we proceed let us note that this is a true thermodynamic equilibrium relation so that... 

Assume that typical time scales of both direct and reverse reactions in (3.3.2) are much shorter than any other time scale in the system. Then the reaction (3.3.2) yields the Langmuir s local equilibrium relation between Ci and 0 of the form... 

The Langmuir constant related to heat of adsorption, L/mg The equilibrium concentration of substance in solution, mg/L Concentration of species A and B in the solution phases, meq/L Aqueous phase concentration at the particle surface, in equilibrium with q, mole/L... 

During the adsorption step, the binding of solute by the affinity ligand is often very tight, essentially irreversible. For Langmuir, or constant separation factor adsorption, the equilibrium relation is... 

The number of adsorption sites per unit area for bound Na+ ions equals 1/A. Assuming a simple Langmuir-type expression for the entropic term and making use of Equation 9, the adsorption isotherm of the bound counterion is given by the equilibrium relation ... 

The equilibrium relations that must be expected for a mixed sorbate, are relatively simple for localized sorbate particles on identical independent sites in a zeolite. The pure components (indices i, j) obey Langmuir isotherms, Eq. (11), and the sorption equilibrium of the mixture will be described by... 

Batch adsorption is often used to adsorb solutes from liquid solutions when the quantities treated are small in amount, such as in the pharmaceutical industry or other industries. As in many other processes, an equilibrium relation such as the Freundlich or Langmuir isotherms and a material balance are needed. The initial feed concentration is Cf and the final equilibrium concentration is c. Also, the initial concentration of the solute adsorbed on the solid is qf and the final equilibrium value is q. The material balance on the adsorbate is... 

Adsorption equilibrium relation of benzene on two types of activated carbon are plotted by the Langmuir plot and the Dubinin plot in Fig. 3.9(a) and (b). Apparently, the Dubinin equation gives a better regression to the data of Hasz (1969). 

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... 

For an initially empty column, AiARlinmai = Au/I initial = (Note In determining dxAn/dxAw), the overbar has no relevance since the equilibrium relation is independent of column cross section. However, there may be variations along the cross section of xaw, therefore Xaw is relevant) In such a conteict, the Langmuir ion exchange isotherm (3.3.121n) for counterion A may be written as... 

Eq. (5.13.2) can be assumed to be at equilibrium and can be given by the following Langmuir adsorption equilibrium relation ... 

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 ... 

To understand the chemisorptive property of lr/ln/H-ZSM-5 in a comparison with that of ln/H-ZSM-5, the chemisorption data were analyzed in detail. The data shown in Fig. 3 are relatively well fitted to the Langmuir isotherm, as shown in Fig. 5. From these relations, the equilibrium constant (K) and the amount of NO2 adsorbed at saturation (Vo) were determined according to the following equations ... 

From this we see that the relation between the concentration of A and the amount of it that is bound should follow the Hill-Langmuir equation. Ke, the macroscopic dissociation equilibrium constant, is given by ... 

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... 

A variety of models have been derived to describe the kinetics of semiconductor photocatalysis, but the most commonly used model is the Langmuir-Hinshel-wood (LH) model [77-79]. The LH model relates the rate of surface-catalyzed reactions to the surface covered by the substrate. The simplest representation of the LH model [Eq. (7)] assumes no competition with reaction by-products and is normally applied to the initial stages of photocatalysis under air- or oxygen-saturated conditions. Assuming that the surface coverage is related to initial concentration of the substrate and to the adsorption equilibrium constant, K, tire initial... 

The Langmuir adsorption isotherm describes the equilibrium between a single-component gas A and adsorbed species A(s) at a surface [237]. The expression relates the fraction of the surface 6a covered by adsorbed species as a function of the partial pressure pa exposed to the surface. The usual form of the Langmuir adsorption isotherm is... 

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