Langmuir adsorption isotherm application

The equation can be fitted to a wide range of isotherms with quite small residual errors over the entire range of concentrations of interest. It is a liquid liquid analogue of the Langmuir adsorption isotherm applicable to... 

Brunauer, Emmett, and Teller (1938) successfully extended the Langmuir adsorption isotherm applicable to monolayer adsorption to multilayer adsorption. While the assumptions made contain some serious deficiencies, the reproducibility of the BET method has made it the standard procedure for determining total surface area. In addition to the assumptions involved in the Langmuir adsorption, the BET method assumes that heats of adsorption for the multilayer adsorption are the same for the second and subsequent layers and are equal to the heat of liquefaction of the adsorbate. Adsorption and desorption rate constants are assumed to be the same for all layers except for the first layer, which is in contact with the... 

Reaction rates for the start-of-cycle reforming system are described by pseudo-monomolecular rates of change of the 13 kinetic lumps. That is, the rates of change of the lumps are represented by first-order mass action kinetics with the same adsorption isotherm applicable to each reaction step. Following the same format as Eq. (4), steady-state material balances for the hydrocarbon lumps are derived for a plug-flow, fixed bed catalytic reformer. A nondissociation, Langmuir-Hinshelwood adsorption model is employed. Steady-state material balances written over a differential fractional catalyst volume dv are the following ... 

What is the Langmuir adsorption isotherm, and when is it applicable Describe its limitations. 

For example, the Langmuir adsorption isotherm specifically describes adsorption of a single gas-phase component on an otherwise bare surface. When more than one species is present or when chemical reactions occur, the functional form of the Langmuir adsorption isotherm is no longer applicable. Thus, although such simple functional expressions are very useful, they are not generally extensible to describe arbitrarily complex surface reaction mechanisms. 

A final area of difficulty is in the application of data analysis to specific models of adsorption isotherms. This difficulty results from the fact that different models for adsorption isotherms generate plots of surface versus dissolved concentration that have characteristic shapes. If a plot of observational data results in a curve with a shape similar to that generated by a model, this result is often taken as proof that the particular model applies. Unfortunately, this assumption has been made for situations where many of the basic requirements of the model are violated in the system under study. The Langmuir adsorption isotherm model has suffered considerable abuse by geochemists in this regard. It should be remembered that "shapes" of adsorption isotherms are far from proof that a specific model applies. 

A frequently used adsorption model that allows for adsorption in multilayers has been introduced by Brunauer, Emmett and Teller [10] and is known as the BET equation. With the exception of the assumption that the adsorption process terminates at monolayer coverage, these authors have retained all the other assumptions made in deriving the Langmuir adsorption isotherm. Hence all objections to the application of the Langmuir equation apply here, too. 

The isotherm that is most easily understood theoretically and widely applicable to experimental data is, known as the Langmuir isotherm. This is also the simplest model of a nonlinear isotherm. It accounts well for the adsorption of single components on homogenous surfaces at low to moderate concentrations, or for the adsorption to isolated type-I or type-II sites [103, 110, 111]. The Langmuir adsorption isotherm equation is written ... 

The signal intensity increased with the stagnation pressure of the molecular beam and finally leveled off. This behavior is well represented by the Langmuir adsorption isotherm which is applicable to chemical adsorption. A monolayer on the substrate is photodissociated to generate Ga atoms. 

The application of mass action law to heterogeneous elementaiy reactions is somewhat refined by treating the adsorption of the species S on unoccupied site a(0) as bimolecular combination of S and elementary reaction is thus expressed as proportional to 0 0(0), where <7 is the concentration of S in gas and 0(0) the probability of the site a being unoccupied. The approximation of Langmuir adsorption isotherm is precisely in line with this refinement of the application of the mass action law, which will be called the extended mass action law in what follows. The application of the extended mass action law to the hydrogen electrode reaction leads now to the value of a, which decreases from 2 to 0 with the increase of rj passing through the observed value ea 0.5 (6). The value of tj at a = 0.5 is however far too low and the interval of ij, in which a stays near 0.5, far too short as compared with observations. 

The Langmuir adsorption Isotherm may now be regarded as a classical law In physical chemistry. It has all the Ingredients of a classical equation it is based on a clear and simple model, can be derived easily from first principles, is very useful now, about 50 (now 60) years after it was first derived and will probably be useful for many years to come, and is rarely ever applicable to real systems, except as a first approximation. 

The Langmuir adsorption isotherm is of limited application since for real surfaces the energy is not the same for aU sites and interactions between adsorbed molecules carmot be ignored. 

The classical Langmuir adsorption isotherm and several of its generaUzations are presented. The physical background is enUghtened and hints for engineering applications are given. The generalizations refer to... 

FIG U RE 2.9 To the derivation of Langmuir adsorption isotherm (a,b). Application of the Langmuir isotherm to the analysis of the experimental F(c) (c). (Data from Shchukin, E.D. et al.. Colloid and Surface Chemistry, Elsevier, Amsterdam, the Netherlands, 2001.)... 

Various boundary conditions limit each of the theories, hence a range of equations have been developed to cover the various phenomena equation developed by Brunauer, Emmett and Teller commonly known as the BET equation. This equation is for multilayer adsorption, but is based upon the Langmuir equation where adsorption is restricted to a monolayer. Both of these equations are developed below, although the application of the Langmuir equation to gas adsorption is restricted to adsorption in micropores where adsorption is limited to a monolayer due to pore geometry. Langmuir adsorption isotherms are common in adsoiption of solute from solution. 

It would be difficult to over-estimate the extent to which the BET method has contributed to the development of those branches of physical chemistry such as heterogeneous catalysis, adsorption or particle size estimation, which involve finely divided or porous solids in all of these fields the BET surface area is a household phrase. But it is perhaps the very breadth of its scope which has led to a somewhat uncritical application of the method as a kind of infallible yardstick, and to a lack of appreciation of the nature of its basic assumptions or of the circumstances under which it may, or may not, be expected to yield a reliable result. This is particularly true of those solids which contain very fine pores and give rise to Langmuir-type isotherms, for the BET procedure may then give quite erroneous values for the surface area. If the pores are rather larger—tens to hundreds of Angstroms in width—the pore size distribution may be calculated from the adsorption isotherm of a vapour with the aid of the Kelvin equation, and within recent years a number of detailed procedures for carrying out the calculation have been put forward but all too often the limitations on the validity of the results, and the difficulty of interpretation in terms of the actual solid, tend to be insufficiently stressed or even entirely overlooked. And in the time-honoured method for the estimation of surface area from measurements of adsorption from solution, the complications introduced by... 

The major surfactant in the foam may usually be considered to be present at the bubble surfaces in the form of an adsorbed monolayer with a substantially constant F, often of the order of 3 X 10" (g mol)/ cm", for a molecular weight of several hundred. On the other hand, trace materials follow the linear-adsorption isotherm Tj = KiCj if their concentration is low enough. For a wider range of concentration a Langmuir or other type of isotherm may be applicable (Davies and Rideal, loc. cit.). 

Such a model should be as simple as possible, without however missing any of the underlying thermodynamic and physicochemical factors which cause electrochemical promotion. In particular it will be shown that even the use of Langmuir-type adsorption isotherms, appropriately modified due to the application of potential (or equivalently by the presense of promoters) suffice to describe all the experimentally observed rules G1 to G7 as well as practically all other observations regarding electrochemical promotion including the effect of potential on heats of adsorption as well as on kinetics and reaction orders. 

Another concept is that the electronic work function changes linearly with the amount adsorbed or that the dipole moment is independent of the concentration. The (1 — 0) concept states that the rate of adsorption for a constant arrival rate is proportional to the fraction of the surface which has not yet been covered. The last two concepts permitted Langmuir to derive his famous adsorption isotherm, which has been verified by experiment in many systems (see the discussion in section IV). Langmuir s experimental work for Cs on W convinced him that the (1 — 0) law was not applicable in this system. This work also led to the concept that the energies involved in surface migration were much smaller than the energies involved in evaporation. 

Usually specific surface areas are determined from adsorption experiments. To illustrate this let us assume that adsorption of a specific sample is adequately described by the Langmuir Eq. (9.22). From fitting experimental results we obtain Tmon in units of mol/g. Then we assume a reasonable value for the cross-section area of a gas molecule a a, and obtain the specific surface from J2 = rmon ANA- In most practical applications the BET adsorption isotherm is used instead of the Langmuir Eq. (9.22) because it fits better. From a fit with the BET isotherm we get Tmon or nmon. Some cross-sectional areas for suitable gases in A2 are N2 16.2 02 14.1 Ar 13.8 n-C4Hi0 18.1. 

The adsorption capacity of activated carbon may be determined by the use of an adsorption isotherm. The adsorption isotherm is an equation relating the amount of solute adsorbed onto the solid and the equilibrium concentration of the solute in solution at a given temperature. The following are isotherms that have been developed Freundlich Langmuir and Brunauer, Emmet, and Teller (BET). The most commonly used isotherm for the application of activated carbon in water and wastewater treatment are the Ereundlich and Langmuir isotherms. The Freundlich isotherm is an empirical equation the Langmuir isotherm has a rational basis as will be shown below. The respective isotherms are ... 

We saw earlier (cf. Section 19) that the potential dependence of adsorption of intermediates formed by charge transfer affects the kinetics of electrode reactions. We have worked out the kinetic parameters for a few mechanisms under so-called Langmuir and Temkin conditions (i.e., when the Langmuir and the Temkin isotherms are applicable, respectively). Here we shall derive the appropriate kinetic equations for the combined adsorption isotherm. 

In the discussion of the hydrogen and oxygen evolution reactions, we saw that the current-potential relationship is influenced, and sometimes determined completely, by the potential dependence of the coverage 9. Thus, one may expect that the kinetic parameters will depend on the adsorption isotherm, which relates the surface concentration to the bulk concentration, and more importantly in electrochemistry, to the potential. In the preceding derivations it was tacitly assumed that the Langmuir isotherm applies. In Section 19 we discuss the limitations of this assumption and show how the kinetic parameters change when different isotherms are applicable. 

Fig. 2 highlights the isotopic effect for H2 and D2 by showing the adsorption isotherms measured under both sub- and supercritical conditions (Tcrii(H2>=33.19 K, Tcrit(D2)=38.34 K). The lower part illustrates the evaluation of the experimental data according to the Langmuir equation. It becomes clear that the Langmuir fit works very well for nitrogen, shows an acceptable performance for the hydrogens, but is not applicable to helium. The variation of the results is shown in Table 1. 

This isotherm model has been used successfully to accoimt for the adsorption behavior of numerous compounds, particularly (but not only) pairs of enantiomers on different chiral stationary phases. For example, Zhou et ah [28] foimd that the competitive isotherms of two homologous peptides, kallidin and bradyki-nine are well described by the bi-Langmuir model (see Figure 4.3). However, most examples of applications of the bi-Langmuir isotherm are found with enantiomers. lire N-benzoyl derivatives of several amino acids were separated on bovine serum albumin immobilized on silica [26]. Figure 4.25c compares the competitive isotherms measured by frontal analysis with the racemic (1 1) mixture of N-benzoyl-D and L-alanine, and with the single-component isotherms of these compounds determined by ECP [29]. Charton et al. foimd that the competitive adsorption isotherms of the enantiomers of ketoprofen on cellulose tris-(4-methyl benzoate) are well accounted for by a bi-Langmuir isotherm [30]. Fornstedt et al. obtained the same results for several jS-blockers (amino-alcohols) on immobilized Cel-7A, a protein [31,32]. 

Figures 4.26A and 4.26B compare the results of the experimental determination of isotherms using the traditional mass balance method (MMB) and those obtained with MMC. The adsorption isotherm predicted by MMC deviates significantly from the isotherm data obtained by MMB. This may be due to the limited applicability of the Langmuir competitive model for the modeling of the adsorption behavior even of such simple systems as p-cresol and phenol in reversed-phase chromatography. Figures 4.26C and 4.26D compare the results obtained by MMB and HMMB for the same system. Over most of the concentration range, the agreement between the experimental data and the results of these two methods is...

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