Ideal Monolayer Langmuir Isotherm

The Langmuir model is based on some fundamental assumptions  

These sites are all equivalent and independent, meaning that the adsorbate-surface interaction energy is the same for all sites, and it is not affected by the state (empty or occupied) of neighboring sites. 

FIGURE 4.21 Schematic picture of the adsorption of a gas onto a solid. White circles indicate (some) empty sites. Molecule 1 will bounce against an already adsorbed molecule (4) so that it cannot adsorb. Molecule 2 could adsorb. Molecule 3 has just desorbed. Molecule 5 is still adsorbed. 

Adsorption of an incoming molecule can proceed only on empty sites. 

From the kinetic theory, it can be found that the number of gas molecules hitting the surface per unit of time and area is (Hill 1986 Adamson and Gast 1997 Hiemenz and Rajagopalan 1997 McQuarrie and Simon 1997 Levine 2008) 

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Figure 5.19 shows an idealized form of the adsorption isotherm for physisorption on a nonporous or macroporous solid. At low pressures the surface is only partially occupied by the gas, until at higher pressures (point B on the curve) the monolayer is filled and the isotherm reaches a plateau. This part of the isotherm, from zero pressures to the point B, is equivalent to the Langmuir isotherm. At higher pressures a second layer starts to form, followed by unrestricted multilayer formation, which is in fact equivalent to condensation, i.e. formation of a liquid layer. In the jargon of physisorption (approved by lUPAC) this is a Type II adsorption isotherm. If a system contains predominantly micropores, i.e. a zeolite or an ultrahigh surface area carbon (>1000 m g ), multilayer formation is limited by the size of the pores. 

For an ideal gas Equation 5.22a may be considered as elementary probability of cavity i occupation by molecule J. This is one of the most useful equations in the method of hydrate prediction, and it may also be recognized as the Langmuir isotherm. If the equation were written for one guest component J, it would contain the Langmuir constant Cjj as the only unknown for a given pressure and fraction of the cavities filled (or fraction of monolayer coverage). 

The Langmuir isotherm describes equilibrium adsorption when there are no lateral interactions between the adsorbed molecules, the limiting surface coverage is dictated simply by the size of the adsorbate, all adsorption sites on the surface are equivalent and adsorption is fully reversible. As discussed in Section 5.5.1 below, spontaneously adsorbed monolayers that fulfill these criteria will exhibit ideal voltammetric responses at slow scan rates, thus including a linear dependence of the peak current, zp, on the scan rate, v, an FWHM of 90.6In mV, where n is the number of electrons transferred, and a formal potential that is independent of the surface coverage. 

If, in an ideal case, the probability of desorption of an adsorbed molecule from the surface is independent of the surface coverage (i.e. there are no lateral interactions between the adsorbed molecules), then the value of E is constant for a particular adsorption system. Equation (4.10) is then applicable over the complete range of monolayer coverage. By rearrangement and simplification of Equation (4.10), we arrive at the familiar Langmuir isotherm equation,... 

Because of the existing dangling bonds on the surface of the solid, ideally every atom can act as an adsorption site for the gas particles and saturation is obtained with a monolayer formation when all accessible sites are occupied. This process is very well described through the Langmuir isotherm and, in the case of dissociative chemisorption, the equilibrium value of the surface coverage is proportional to the square root of pressure. Equation (6.5) is therefore modified to give Eq. (6.7)... 

Note that the treatment above is based on the totally reversible electrochemical adsorphon and desorption of H+ (or an ideal Langmuir isotherm model with a monolayer adsorption) on a metal surface. 

As we have explained in the previous sections, the Langmuir model has been established on firm theoretical groimd for gas-solid adsorption, a case where there is no competition between the adsorbate and the mobile gas phase. On the contrary, in liquid-solid adsorption, there is competition for adsorption between the molecules of any component and those of the solvent. Although we can choose a convention canceling the apparent effect of this competition on the isotherm [30,36], the conditions of validity of Eq. 3.47 are not met. These conditions are (i) the solution is ideal (ii) the solute gives monolayer coverage (iii) the adsorption layer is ideal (iv) there are no solute-solute interactions in the monolayer (v) there are no solvent-solute interactions. These conditions cannot be valid in liquid-solid adsorption, especially at high concentrations. 

The second and fourth assumptions are seldom valid, as solvent molecules do adsorb to the monolayer of solvent attached to the aaivated adsorbent — forming layers. But, in the same way that the ideal gas law is useful, so also is the Langmuir Equation (isotherm). It is often the first choice for a model of the adsorption process because it has been shown to be valid for, and produces information about, the number of activated sites on (not within) a monolayer of an adsorbent. 

Extending the classical Langmuir adsorption isotherm (7.1) from monolayer to ideal multilayer adsorption and considering the limiting case of infinite many layers, Brunauer, Emmett, and Teller derived in 1938 the AI [7.1-7.5, 7.42]... 

On the assumption that the forces that produce condensation are chiefly responsible for the binding energy of multilayer adsorption, ey proceeded to derive an isotherm equation for multilayer adsorption by a method that was a generalization of Langmuir s treatment of the unimolecular layer. The generalization of the ideal localized monolayer treatment is effected by assuming that each first layer adsorbed molecule serves as a site for the adsorption of a molecule into the second layer and so on. Hence, the concept of localization prevails at all layers and forces of mutual interaction are neglected. 

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