The Langmuir adsorption equation

Consider the adsorption of water vapor on a solid surface. The adsorption may be written in the form of a reaction 

The fractional coverage 0 may be expressed as a ftmction of partial pressure from Equation (7.33) as 

The equilibrium constant (this case is analogous to Equation (7.33)) will be 

Replacing the activities of S and V by their surface excesses and defining a Henrian standard state for S yields 

It should be noted that Equations (7.36) and (7.40) may be arranged, respectively, to 

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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. 

In a study of adsorption of organic herbicides by montmorillonite, Bailey and colleagues138 found that none of the compounds conformed to the Langmuir adsorption equation. Of the 23 compounds tested, only a few did not conform well to the Freundlich equation. 

The adsorption isotherms of Astraphloxin and Pseudocyanine in Figure 8A were also expresed by the Langmuir adsorption equation. The resulting linear relation of Figure 8B demonstrates excellent agreement betwen the data obtained from phase separation and spectral techniques. 

The most notable criticism of the Langmuir adsorption equation concerns the simplifying assumption that the heat of adsorption is independent of surface coverage, which, as discussed in the next section, is not likely to be the case. Nevertheless, many experimental adsorption isotherms fit the Langmuir equation reasonably well. 

It may be noted that the denominator of the heterogeneous term resembles the format expected from the Langmuir adsorption equation (8) even though Scheme (XLVI) has not stipulated equilibrium between bulk and adsorbed concentrations. As eqn. (82) predicts, kobs was indeed found to increase linearly with the mass of carbon (Spheron 6) added. The rate constant also fell with rising binaphthyl concentration (cf. Fig. 14) but the equation fitted these results only qualitatively. However, the fall in rate with added inhibitor could be fitted better, a plot of 1 l(kohs — kohB unc) rising linearly with increasing concentration of inhibitor. The relative inhibitory properties in acetone increased with aromatic size in the order benzene, nil naphthalene,... 

Here S is the total emulsifier concentration, a the constant in the Langmuir adsorption equation. the total monomer droplet surface, and the specific surface per unit amount of emulsifier. The expression for A is... 

It follows from this that an empirical approach relies heavily upon exhaustive experimentation. An example of an empirical approach often applicable to studies of ion exchange equilibria is a modified form of the Langmuir adsorption equation ... 

Gentry et al. (1976) have recently shown, on the basis of isotopic analyses, that uranium introduction may have occurred far more recently than was previously supposed. However, they find also that, in some instances, the uranium was introduced before coalification was complete since the haloes have been compressed with the coal as it increased in rank. These results are consistent with laboratory and field work by Szalay (1964) who showed that the insoluble humic acids in peat are capable of concentrating uranium from very dilute solutions in natural waters. Sorption occurs as uranyl humate , the process following the normal kinetics of the Langmuir adsorption equation. (Where uraninite occurs in association with peat or other carbonaceous matter, the uranium may thus have been initially sorbed as a uranyl compound which was later reduced to uraninite.)... 

The preceding equation is the Langmuir adsorption isotherm. The constants a and b are characteristic of the system under consideration and are evaluated from experimental data. Their magnitude also depends on the temperature. At any one temperature the validity of the Langmuir adsorption equation can be verified most conveniently by first dividing both sides of this equation by p and then taking reciprocals. The result is... 

Both the Rideal and Langmuir-Hinshelwood rate laws are based upon the Langmuir adsorption equation, which is applicable for gas-solid as well as liquid-solid systems where diffusion of the sorbate to the solid surface is not rate limiting (generally true). The basic assumption of the Langmuir adsorption is that adsorption occurs at adsorption sites and all these sites are equivalent. For gas-solid systems, the rate of adsorption, fo, of the gas A is proportional to the gas pressure, pa, and the number of vacant sites, i.e.. 

Some molecules are dissociated upon adsorption, and this type of adsorption is called dissociative adsorption. The adsorption reaction can be shown as [(XY)gas + 2 Surface <-> Xads-Surface + Yads-Surface], and the Langmuir adsorption expression must be modified because two sites on the adsorbent are consumed per adsorbate molecule. The probability of desorption is also different. When these differences are considered, the Langmuir adsorption equation for the dissociative adsorption becomes... 

Regarding the quantity of boron adsorbed, adsorption maxima calculated with the Langmuir adsorption equation 16,17, 25-27) vary from approximately 10 up to 100 figrams of B/gram of soil. Soils derived from volcanic ash deposits adsorb unusually large amounts of boron 16, 18) because of their enriched contents of amorphous materials with affinity for anions. 

Thus putting the Langmuir adsorption equation in the form... 

According to the Langmuir adsorption equation, the amount (weight) of material deposited per unit area and per unit time is expressed as... 

Laszlo et al. studied the adsorption of phenol and 2.3,4 trichlorophenol from dilute aqueous solutions on a granular activated carbon prepared from PAN by a two-step physical activated process. The adsorption isotherms were Type 1 of the BET classification and followed the Langmuir adsorption equation. The adsorption capacity and the adsorption constant K values, obtained using the Langmuir equation (Table 7.6), were found to depend on the pH of the solution. The results were discussed in terms of the acid-base character of the carbon surface and the acidic character of the two phenols. The effect of pH is more significant in the... 

The rate of removal of surface-active contaminants from treated water using flotation depends upon many factors. In particular, it depends on the surfactant concentration in the water bulk, the magnitude of equilibrium constant of surfactant adsorption at the air-water interface, the adsorption rate of surfactants, air flow, and air bubble dimensions. The analytical equation that relates the volume and the surface concentration of the surfactant in adsorption process is the Langmuir adsorption equation (see Appendix 8.A) ... 

Equation (28.18) can also be derived from the Langmuir adsorption equation, because 6 is the fraction of sites filled. 

The adsorption of amphiphiles from solution on to a solid surface is described by the Langmuir adsorption equation. It is sometimes known as the Langmuir isotherm, since it refers to adsorption as a function of concentration (or pressure, when dealing with gases) at constant temperature. The Langmuir adsorption equation assumes that the surface is homogeneous, adsorption cannot occur beyond monolayer coverage, and all adsorption sites are equivalent. In addition, the equation applies to dilute solutions where there are no snrfactant-surfactant or surfactant-solvent interactions. 

EinaUy, from the value of the solute-surface affinity parameter in the Langmuir adsorption equation, it is possible to determine the Gibbs energy of adsorption ... 

In a special case, when the variation in b is attributed entirely to the variation in the heat of adsorption, the solution to the Langmuir adsorption equation can be reduced to the Freundlich adsorption equation [12] ... 

The measurements of rupture times for Newton black Aims were found to be in good agreement with Eq. (4) [17]. However, it has been remarked that above the Szykowski concentration s 2FI, where H is the surface pressure of the monolayer (FI = yo % To being the surface tension of water) [20]. Because n is usually well described by the Langmuir adsorption equation, n = bTF ln(c/a), it follows that if the rupture time is an exponential function of e, the bulk concentration dependence of t is formally identical to that of Eq. (4). We have simultaneously measured the him rupture time and the surface elasticity and found that t exp(e), as predicted by Eqs. (5) and (6) [21]. 

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