Langmuir approach

The BET approach is essentially an extension of the Langmuir approach. Van der Waals forces are regarded as the dominant forces, and the adsorption of all layers is regarded as physical, not chemical. One sets the rates of adsorption and desorption equal to one another, as in the Langmuir case in addition, one requires that the rates of adsorption and desorption be identical for each and every molecular layer. That is, the rate of condensation on the bare surface is equal to the rate of evaporation of molecules in the first layer. The rate of evaporation from the second layer is equal to the rate of condensation on top of the first layer, etc. One then sums over the layers to determine the total amount of adsorbed material. The derivation also assumes that the heat of adsorption of each layer other than the first is equal to the heat of condensation of the bulk adsorbate material (i.e., van der Waals forces of the adsorbent are transmitted only to the first layer). If it is assumed that a very large or effectively infinite number of layers can be adsorbed, the following result is arrived at after a number of relatively elementary mathematical operations... 

Fig. 9.1. Sorption of selenate (SeO ) to a loamy soil, showing mass sorbed per gram of dry soil, as a function of concentration in solution. Symbols show results of batch experiments by Alemi et al. (1991 their Fig. 1) and lines are fits to the data using the reaction KA, reaction Freundlich, and Langmuir approaches.

We consider as a first example sorption of selenate (SeO ), as predicted by the reaction K(, reaction Freundlich, and Langmuir approaches (Sections 9.1-9.3). Alemi et al (1991) observed the partitioning of selenate in batch experiments between 10 g of a loamy soil and 20 ml of a pH 7.5 solution containing small amounts of Na2Se04 their results are shown in Figure 9.1. 

Brunauer et al. [57] extended the Langmuir approach to polymolecular adsorption, proposing the famous BET equation... 

In a 1922 paper written with W. O. Kermack, Robinson used the Lewis-Langmuir approach to give physical meaning to partial valency (so that the double bond [=] was equivalent to four-electron sharing to three-... 

Andre, C. and Guillaume, Y.C., Reanalysis of chiral discrimination of phen-oxypropionic acid herbicides on a teicoplanin phase using a bi-Langmuir approach, Chromatographia, 58, 201,2003. 

Indeed, a simple Langmuir approach, which considers a constant value of the adsorption energy, was not appropriate. This is consistent with the presence of distinct types of acid sites (Lewis and/or Bronsted) characterized by different acid strength and associated with V, W and Ti surface sites [54]. 

When Should One Use the Frumkin-Temkin Isotherms in Kinetics Rather than the Simple Langmuir Approach ... 

The assumption of monolayer adsorption in the Langmuir isotherm model is unrealistic in most cases, and a modification to multilayer adsorption should be considered. In 1938, Brunauer, Emmett, and Teller modified the Langmuir approach of balancing the rates of adsorption and desorption for the various molecular layers [Brunauer et al., 1938], This approach is known as the BET method. The BET isotherm assumes that the adsorption of the first layer has a characteristic heat of adsorption A Ha and the adsorption and desorption on subsequent layers are controlled by the heat of condensation of the vapor, A Hc. The derivation of the BET equation is beyond the scope of this book however, a common form of the BET equation is given as... 

The derivation again made use of the assumptions of the Langmuir approach, the following equilibria being assumed to hold ... 

The Langmuir approach described here is very suitable for analysis of results where surfactants are used, which may adsorb at the liquid-liquid interface in competition with the diffusant. 

Our proposed method of site energy distribution analysis can, of course, be applied using any function 0(P,T,Q) for the local isotherm (or even graphical representations not amenable to analytical replication). Nor need the function be explicit in 0. As an example of this last situation, it has been suggested (7) that lateral interaction be allowed for within the framework of the Langmuir approach by supposing Q to increase linearly with 0 ... 

An alternative approach which can be used to describe the adsorption phenomena of a multicomponent system is to assume that there is total competition between target polypeptides, proteins, and other components in the mixture for adsorption to all the accessible sites of the ligate. In this case, the Langmuir approach can be expanded to encompass binary (or more generalized examples) of adsorption isotherms8, A48-151,167175,227-236,365,369,372- for two or more... 

Equilibrium between solution and adsorbed or sorbed phases is a condition commonly used to evaluate adsorption or sorption processes in soils or soil-clay minerals. As previously stated, equilibrium is defined as the point at which the rate of the forward reaction equals the rate of the reverse reaction. Two major techniques commonly used to model soil adsorption or sorption equilibrium processes are (1) the Freundlich approach and (2) the Langmuir approach. Both involve adsorption or sorption isotherms. A sorption isotherm describes the relationship between the dissolved concentration of a given chemical species (adsorbate) in units of micrograms per liter (pg L 1), milligrams per liter (mg L-1), microequivalents per liter (pequiv L-1), or millimoles per liter (mmol L-1), and the sorbed quantity of the same species by the solid phase (adsorbent) in units of adsorbate per unit mass of adsorbent (solid) (e.g., pg kg-1, mg kg-1, peq kg-1, or mmol kg 1) at equilibrium under constant pressure and temperature. Sorption isotherms have been classified into four types, depending on their general shape (Fig. 4.13) ... 

An extension of the Langmuir approach to multilayers adsorption was made by Brunauer, Emmett, and Teller, BET [17]. They assumed that the Langmuir equation applies to each layer. The heat of adsorption of the first layer was assumed to have a special value, but for the subsequent layers, the heat of adsorption was assumed to be equal to the heat of condensation of the gas. The volume adsorbed is then a summation of the adsorbed volumes of each layer. Upon evaluation of the summation, the BET equation results ... 

The transient response experiments were analyzed by a dynamic isothermal PFR model, and estimates of the relevant kinetic parameters were obtained by global nonlinear regression over all runs. It was found that a simple Langmuir approach could not represent the data accurately, and surface heterogeneity had to be invoked. The best fit was obtained using a Temkin-type adsorption isotherm with coverage-dependent desorption energy ... 

Within the Langmuir approach, the general expression for the fractional coverage by component A in the case of multicomponent adsorption is given by... 

Measured isotherms of the type shown in Fig. 6-2 are frequently described using the Langmuir approach. In fact, several investigators indicated that one-di-... 

The consequence of the above is that based on goodness of fit of the sorption data, both the Freundlich and the Langmuir approaches described the data equally well for all reaction times (see Table 6-1). It is also conceivable that such a finding is expected since both approaches result from the general isotherm equation (Kinniburgh, 1986)... 

Interpretation of the decomposition mechanism along the lines of the Knudsen-Langmuir approach is based on the difference between the real vaporization rate of a reactant from a free surface (after Langmuir) and that of the same reactant in practically equilibrium conditions from effusion cells (after Knudsen). The ratio of these rates is called the vaporization coefficient, Oy. Its value turns out in many cases to be smaller (quite frequently, by several orders of magnitude) than unity. As pointed out by Somorjai and Lester [3] All the information on the evaporation mechanism is hidden into a correction factor which is applied to adjust the deviation of the evaporation rate far from the maximum evaporation rate in equilibrium. ... 

In view of Langmuir approach, we observe excess of propylene oxide (PO) in the surface for both random copolymer and blend (Figure 2). This corresponds to Equation (17) of II). As one expects, depletion of PEO in the surface is more pronounced for the blend. With = 31.5 mN/m and = 43.5 mN/m at 25 °C,... 

The last section dealt with the basic Langmuir theory, one of the earliest theories in the literature to describe adsorption equilibria. One should note that the Langmuir approach is kinetic by nature. Adsorption equilibria can be described quite readily by the thermodynamic approach. What to follow in this section is the approach due to Gibbs. More details can be found in Yang (1987) and Rudzinski and Everett (1992). 

Because of its simplicity and wide utility, the Langmuir isotherm has found wide applicability in a number of useful situations. Like many such classic approaches, it has its fundamental weaknesses, but its utility generally outweighs its shortcomings. The Langmuir isotherm model is based on the assumptions that adsorption is restricted to monolayer coverage, that adsorption is localized (i.e., that specific adsorption sites exist and interactions are between the site and a specific molecule), and that the heat of adsorption is independent of the amount of material adsorbed. The Langmuir approach is based on a molecular kinetic model of the adsorption-desorption process in which the rate of adsorption (rate constant /ca) is assumed to be proportional to the partial pressure of the adsorbate (p) and the number of unoccupied adsorption sites (N - n), where N is the total number of adsorption sites on the surface and n is the number of occupied sites, and the rate of desorption (rate constant d) is proportional to n. 

The Langmuir approach was extended to multilayer adsorption by Brunauer, Emmett, and Teller in the form of the BET equation with two constants. The linearized form of the BET equation is important in the measurement of total surface areas of porous solid catalysts [5] ... 

A consideration of Figure 4 demonstrates the inadequacy of the Langmuir approach to most biomolecular systems. For one, the saturation is approached much more slowly than the exponential behavior predicted by equation 2. Secondly, by setting Cs=0, equation 1 would predict a complete desorption during a rinse. Instead, desorption of only a small fraction of the adsorbed molecules results. Finally, equation 1 predicts a linear relationship between adsorption rate and adsorbed amount. In fact, most systems demonstrate a nonlinear relationship. An example is shown in Figure 5 for transferrin adsorption onto silica-titania (81). Despite these and other drawbacks, the Langmuir model continues to find use in a number of instances. 

Consideration of the rates of processes of adsorption and desorption of ions in the state of a dynamic equilibrium in the framework of the Langmuir approach results in the following equation for monolayer adsorption ... 

---