Langmuir s isotherm

In the course of the study, we found that a family of monosaccharides exhibited a definite adsoi ption selectivity (Langmuir s isotherm adsoi ption constant), reflecting their molecular stmctures. 

But Langmuir s isotherm for the solute entails the generalized form of Raoult s law (Eq. 13) as a necessary thermodynamic consequence. This can best be seen from the Gibbs-Duhem equation,... 

Before our theory was fully developed, extensive work by J. Koral in cooperation with R. Ullman (15) confirmed in detail and with considerable accuracy all previously known features. They ascertained, in addition, the particulars of the adsorption isotherms for a number of polymers and dispersed adsorbates and established the remarkable degree to which most isotherms could be approximated by 2-parameter equations, like Langmuir s isotherm for monolayers of small molecules. They found the dependence of the adsorption on MW to be weak and determined the area per adsorbed molecule. 

In 1938, Brunauer, Emmett and Teller(12) and Emmett and de Witt(13) developed what is now known as the BET theory. As in the case in Langmuir s isotherm, the theory is based on the concept of an adsorbed molecule which is not free to move over the surface, and which exerts no lateral forces on adjacent molecules of adsorbate. The BET theory does, however, allow different numbers of adsorbed layers to build up on different parts of the surface, although it assumes that the net amount of surface which is empty or which is associated with a monolayer, bilayer and so on is constant for any particular equilibrium condition. Monolayers are created by adsorption on to empty surface and by desorption from bilayers. Monolayers are lost both through desorption and through the adsorption of additional layers. The rate of adsorption is proportional to the frequency with which molecules strike the surface and the area of that surface. From the kinetic theory of gases, the frequency is proportional to the pressure of the molecules and hence ... 

An adsorption isotherm describes the way in which the concentration of the species adsorbed on the electrode, r(, depends on the activity, a(, of the species in solution. Langmuir s isotherm is the simplest equation for adsorption phenomena, but it must be kept in mind that it is generally valid only for non-ionic species. 

We start with Langmuir s isotherm. In this case the adsorption speed is k, having the characteristic size fciR and k — k-[s.k. For the second parameter we set fc CR = 2, where kz is a dimensionless positive constant. 

The driving force of the polymerization reaction is the monomer concentration at the surface of the polymer. In this case Langmuir s isotherm (equation 5.4-7) could be used to describe the monomer concentration quantitatively or, more simply, Henry s sorption rule (Eqn 5.4-8) holds true... 

Since the constancy of enthalpy with coverage is analogous to the constancy of enthalpy with pressure in an ideal gas, the adsorbed state in a system following Langmuir s isotherm is sometimes called an ideal adsorbed state. 

For simple adsorption of two adsorptives A and B competing for the same sites, Langmuir s isotherm takes the form... 

By then, Polanyi had renewed laboratory experiments on adsorption with several co-workers, [23] substituting for his original adsorption theory a picture that covered both unimolecular and multimolecular adsorption. [24] Polanyi began to argue that Langmuir s formula represented too simple an idealization, which was not obeyed in all cases, and that his own revised theory could be used to derive Langmuir s isotherm as a special case. [20, 25]... 

Intracrystalline sorption is normally of Type 1 in Brunauer s classification ( V7) and isotherm contours therefore resemble those according to Langmuir s isotherm equation. This can describe actual isotherms well enough (18) to be of value in predicting, through Equations 5 or 6, some features of zeolite chemistry. 

The maximum value of the relative pressure, x, is unity and this value will be closely approached for the aqueous phase where the guest is water. Then for Langmuir s isotherm... 

When sorption in each cavity can be represented by Langmuir s isotherm with the same or a different guest in each cavity. 

AdsorptionIdesorption kinetics is generally a simple rate equation based on Langmuir s isotherm derivation [see Equation (14b) in Table 14.3] ... 

Figure 8.7 Two adsorption isotherms according to Freundlich s formula with (1) k = 3 and n = 0.8 and (2) k = 5 and n = 0.3. A Langmuir s isotherm is shown by line 3, with a = 0.2 and b = 30. The scales have arbitrary units. Experiments must be done in order to find the best relation.

Thus we have an expression relating the current density (/) to the cell potential (V) in terms of the equilibrium surface coverage (0o) and the concentration of oxygen at the surface of the electrode (r. o2o ) To complete the derivation, Langmuir s isotherm for chemisorption is introduced to relate the equilibrium surface coverage to the concentration of the adsorbed species ... 

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. 

This kinetic behavior indicates a rate-determining surface reaction of weakly adsorbed nitrous oxide on a surface which adsorbs ammonia according to Langmuir s isotherm. The adsorption energy of ammonia on the active surface was found to be 2.2 kcal./mol. 

An equation of more interest in catalytic kinetics is Langmuir s isotherm... 

Langmuir s isotherm is based on the assumption that pomts of valency exist on the surfoce of the adsorbent and that each of these sites is capable of adsorbing one molecule thus, the adsorbed layer will be one molecule thick. Furthermore, it is assumed that all the adsorption sites have equal affoiities for molecules of the adsorbate and that the presence of the adsorbed molecules at one site will not affoct the adsorption of molecules at an adjacent she. The Langmum piation is commonly written as follows (36) ... 

The dependence of N, with the partial gas pressure p is not always achieved and the measured isotherms are often different from Langmuir s isotherm. This difference can be explained by the interaction between adsorbed molecules or by the nonhomogeneous adsorption on the surface (Wolkenstein 1991). 

Consider in detail what happens at the reacting surface, since the condition (6.8) is, in fact, a boundary condition for the solution of diffusion equations. In the course of heterogeneous reaction, the molecules of reacting substance get adsorbed at the surface. If the surface is homogeneous, and the adsorbed substance forms a monolayer on this surface, then the process of adsorption is described by a Langmuir s isotherm, which relates the quantity of substance adsorbed at the surface to concentration C of this substance in the solution near the surface ... 

In this chapter, we will discuss how the chemical and physical properties of substances at interfaces differ from those in the bulk. For quantitative description, quantities like surface tension and surface energy have to be introduced. With the help of these quantities, phenomena known from everyday life like the lotus effect can be explained. However, perhaps you are more interested to learn how detergents clean Then have a look at Sect. 16.3 which deals with the adsorption on liquid surfaces. The next section covers the adsorption on solid surfaces and the variation of the extent of coverage with pressure or concentration of the substance to be adsorbed. Langmuir s isotherm, the simplest description of such an adsorptiOTi process, is deduced by kinetic interpretation of the adsorption equilibrium. Alternatively, it can be derived by introducing the chemical potential of free and occupied sites and cmisideiing the equilibrium condition. In the last part of the chapter, some important applications such as surface measurement and adsorption chromatography are discussed. 

We can also derive Langmuir s isotherm with the help of the chemical potential. In order to do this, we consider the following sequence of simple processes ... 

In general chemistry, Langmuir s isotherm fits the experimental data in a very large number of cases where the adsorbed layer is unimolecular. However, some other patterns of adsorption are known and a convenient general classification has been made, as follows, by Giles (1960). 

The Langmuir s isotherm is one of the most common models used to represent equilibrium behavior in the adsorption process. Its main concept proposes the uniformity in terms of surface adsorption sites with adsorption monolayers. Also considered independent interactions of neighboring sites and possibilities of dynamic equilibrium of adsorption and desorption (Fogler, 2002 Ruthven, 1984). 

When two substances are undergoing reaction at a surface there are two possibilities. The reaction may be an interaction between two molecules that are adsorbed side-by-side on a surface, or it may be an interaction between an adsorbed molecule of one kind and a gas molecule of the other. The kinetics are satisfactorily interpreted by an extension of Langmuir s isotherm [Eq. (22)], which for two gases A and B gives the following expressions for the fractions of surface covered by A and B respectively ... 

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