The Langmuir Adsorption Isotherms

The Langmuir adsorption isotherm equation may be written two ways  

The Langmuir isotherm is usually linearized by inversion and so used to test whether it is obeyed by experimental data. The inversion is often done incorrectly, producing an induced correlation in C, thus 

When C is very small, the isotherm reduces to the Freundlich isotherm with x/m = aC = KC, n = 1, and K = K. When C is very large, x/m = a/b = A ax. which is equivalent to the Freundlich isotherm with n = 0. 

Competition between adsorbing species for the same site on a sorbent has been modeled with the so-called competitive Langmuir model. In this model an individual isotherm equation with its own constants is written for each species sorbed by a given sorbent (see Table 10.7). 

Some examples of simple and complex Freundlich and Langmuir isotherm plots are shown in Fig. 10.12 from Domenico and Schwartz (1990). 

The Langmuir Adsorption Isotherm A type of adsorption isotherm commonly observed in adsorption from solutions of surfactants is the Langmuir-type isotherm (Langmuir, 1918), expressed by 

C = the concentration of the surfactant in the liquid phase at adsorption equilibrium, in mol/L, 

This type of adsorption is valid in theory only under the following conditions 

Both solute and solvent have equal molar surface areas. 

Both surface and bulk phases exhibit ideal behavior (e.g., no solute-solute or solute-solvent interactions in either phase). 

A simple model to describe adsorption was presented by Langmuir [370], Langmuir assumed that on the surface there are a certain number of binding sites per unit area S (fig. 9.5). S is in units of mol/m2 (or m-2). Of these binding sites 5) are occupied with adsorbate and S0 = S — Si are vacant. The adsorption rate in moles per second and per unit area 

Typical Langmuir adsorption isotherms are plotted in figure 9.6 for different values of the Langmuir constant. If adsorption from solution is considered, the pressure P has to be replaced by the concentration c and the Langmuir constant is given in units of L mol-1 instead of Pa 1. 

Alternatively, the Langmuir adsorption isotherm equation can be expressed by the number of adsorbed moles per gram or surface area 

rmon is the number of adsorbed moles per gram or per unit area of substrate, when all binding sites are occupied and a monolayer of molecules is bound. Tmon is related to the surface area occupied by one adsorbed molecule aA by rmon = / (N0aA) or Tmon = l/ N0aA). 

What is the significance of the constants kad and kde is the inverse of the adsorption time  

The following several sections deal with various theories or models for adsorption. It turns out that not only is the adsorption isotherm the most convenient form in which to obtain and plot experimental data, but it is also the form in which theoretical treatments are most easily developed. One of the first demands of a theory for adsorption then, is that it give an experimentally correct adsorption isotherm. Later, it is shown that this test is insufficient and that a more sensitive test of the various models requires a consideration of how the energy and entropy of adsorption vary with the amount adsorbed. Nowadays, a further expectation is that the model not violate the molecular picture revealed by surface diffraction, microscopy, and spectroscopy data, see Chapter VIII and Section XVIII-2 Steele [8] discusses this picture with particular reference to physical adsorption. 

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

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... 

This is simply a collection of constants in an exponential function format. The constants cannot be related to the interactions at a molecular level. In contrast, the refit of the data to the Langmuir adsorption isotherm... 

FIGURE 3.5 Major components of classical receptor theory. Stimulus is the product of intrinsic efficacy (s), receptor number [R], and fractional occupancy as given by the Langmuir adsorption isotherm. A stimulus-response transduction function f translates this stimulus into tissue response. The curves defining receptor occupancy and response are translocated from each other by the stimulus-response function and intrinsic efficacy. 

The Langmuir adsorption isotherm for radioligand binding [A ] to a receptor to form a radioligand-receptor complex [A R] can be rewritten in terms of one where it is not assumed that receptor binding produces a negligible effect on the free concentration of ligand ([A free]) ... 

As noted in Chapter 1, the most simple and theoretically sound model for drug-receptor interaction is the Langmuir adsorption isotherm. Other models, based on receptor behavior (see Chapter 3), are available. One feature of all of these models (with the exception of some instances of the... 

The fraction of sites occupied by A, designated a, is known as the Langmuir adsorption isotherm. It is given by... 

This precipitation process can be carried out rather cleverly on the surface of a reverse phase. If the protein solution is brought into contact with a reversed phase, and the protein has dispersive groups that allow dispersive interactions with the bonded phase, a layer of protein will be adsorbed onto the surface. This is similar to the adsorption of a long chain alcohol on the surface of a reverse phase according to the Langmuir Adsorption Isotherm which has been discussed in an earlier chapter. Now the surface will be covered by a relatively small amount of protein. If, however, the salt concentration is now increased, then the protein already on the surface acts as deposition or seeding sites for the rest of the protein. Removal of the reverse phase will separate the protein from the bulk matrix and the original protein can be recovered from the reverse phase by a separate procedure. 

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... 

Derive the Langmuir adsorption isotherm for the molecular adsorption of CO on a metal with equivalent adsorption sites. Do the same for the dissociative adsorption of H2, and, finally, for the case when CO and H2 adsorb together on the same surface. 

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... 

The basic assumption of the Langmuir adsorption isotherm is that the adsorbed molecules do not interact. This condition is not always fulfilled for adsorption, particularly on electrodes. The Frumkin adsorption isotherm includes interaction between molecules in the adsorption film,... 

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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