Applicability of the Langmuir Isotherm

it is apparent that plots of against p, f against-V, and against should yield straight lines from which the values of V and b can be calculated. The last linear plot viz against jr is generally avoided becanse it lays too much emphasis on the low pressnre region of the isotherm and adsorption at low pressure is rarely representative of the entire surface. 

The adsorption isotherms of gases at tanperatures not far removed from then-condensation points show two regions for most of the adsorbents at low pressures the isotherms are concave, and at higher pressures convex, to the pressure axis. The higher pressure convex portion has been attributed to capillary condensation by some workers and to the formation of multimolecular layers by others. 

The formation of multimolecular layers was explained by assuming that the uppermost layer of the adsorbent induces dipoles in the first layer of adsorbate molecules, which in turn induces dipoles in the second layer, and so on until several layers are built up. Brunauer, Emmett, and TeUer were of the view that the polarization of the second layer of adsorbed gas molecules by the first layer gas molecules will be much too small to constimte a major portion of the binding energy between the two adsorbed layers, at least in those cases in which the gas molecules do not possess considerable permanent dipole moments. These workers proposed that the forces acting in mnltimolecnlar adsorption are the same as those acting in the condensation of vapors. Only the first layer of adsorbed molecules, which is in direct contact with the snrface of the adsorbent, is bound by adsorption forces originating from the interaction between the adsorbate and the adsorbent. Thus, the molecules in the second and subseqnent layer have the same properties as in the liqnid state. 

On this basis they derived an equation for multimolecular adsorption by a method that is a generalization of the Langmuir treatment of unimolecular adsorption. This equation is known as the Brunauer, Emmett, Teller equation or, more commonly, the BET equation. This equation has played a significant role in the studies of 

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Consider the application of the Langmuir isotherm to electrochemistry. For a simple charge transfer process, such as the formation of an adsorbed chlorine atom, we write... 

The reaction rate for adsorbed molecules is proportional to 6, and for molecules reacting by direct collision the rate is proportional to c. As an example of the application of the Langmuir isotherm to heterogeneous kinetics,... 

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.)... 

The fact that a Langmuir approximation works at all appears contradictory to the continuous decrease of with 0, demonstrating the existence of sites with a wide spectrum of values. However, the formal application of the Langmuir isotherm... 

Application of the Langmuir isotherm for Charge-Transfer Processes... 

In addition to the assumptions implicit in the use of the Langmuir isotherm the following assumption is applicable to all Hougen-Watson models the reaction involves at least one species chemisorbed on the catalyst surface. If reaction takes place between two adsorbed species, they must be adsorbed on neighboring sites in order for reaction to occur. The probability of reaction between adsorbed A and adsorbed B is assumed to be proportional to the product of the fractions of the sites occupied by each species (0A9B). Similar considerations apply to termolecular reactions occurring on the surface. 

Again, these structural results do not depend on specific values of the equilibrium constant K and the parameters of the Langmuir isotherm a , hi, as was shown in the appendix of Ref. [11]. Further, it was shown that the same patterns of behavior will arise if the chemical reaction is taking place in the solid phase instead of the fluid phase. The latter is of particular interest in applications where the adsorbent acts simultaneously as a catalyst. Practical examples will be discussed in the next section the interested reader is also referred to Chapter 6 of this book. In this context it is worth noting that the structural properties in Fig. 5.9 depend crucially on the stoichiometry of the system, which will be also discussed in the next section. 

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. 

Langmufr s work on gas adsorption and insoluble monolayers prepared the way for more progress to be made in the interpretation of adsorption from solution data. In the light of the Langmuir theory, it seemed logical to suppose that the plateau of a solute isotherm represented monolayer completion and that the monolayer capacity could be derived by application of the Langmuir equation. 

Regarding the application of the Langmuir equation to physical gas adsorption on homogeneous surfaces, good examples are difficult to find. Those systems that do obey the Langmuir equation appear to be microporous. Of course, the Initial parts of multilayer isotherms fulfil the Langmuir equation, and these will be dealt with in the appropriate section. The various ways of plotting and the Information obtainable from such plots have been discussed in sec. 1.4. 

Obstacles to practical application of the Langmuir and Freundlich isotherm theories include the following (1) These isotherms do not effectively address adsorption versus degradation and competitive adsorption (2) the conclusions are not all inclusive i.e., adsorption constants and coefficients do not hold true in all cases within similar oil types let alone across different oil types (3) the process has so many variables that the additive variance is commonly too great to prove any subtle difference between clays other than a vastly different level of activity (this problem is especially true when using log vs. log plots with incremental changes on the order of 0.1%) and (4) the adsorption constants and coefficients have limited use for the refiner. 

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...

Below in Section 2.4.4. and Appendix 2B, the physical background and derivation of the most frequently used adsorption isotherms will be discussed in more detail. Examples of the application of the Langmuir and Frumkin isotherms to experimental data are summarised in Appendix 5D. 

The approach of IAS of Myers and Prausnitz presented in Sections 5.3 and 5.4 is widely used to calculate the multicomponent adsorption isotherm for systems not deviated too far from ideality. For binary systems, the treatment of LeVan and Vermeulen presented below provides a useful solution for the adsorbed phase compositions when the pure component isotherms follow either Langmuir equation or Freundlich equation. These expressions are in the form of series, which converges rapidly. These arise as a result of the analytical expression of the spreading pressure in terms of the gaseous partial pressures and the application of the Gibbs isotherm equation. 

The Freundlich isotherm equation is a limiting form of the Langmuir isotherm and is applicable only in the middle ranges of vapor pressure. The equation is of greater significance for chemisorption, although some physical adsorption data have also been found to fit this equation. 

Before proceeding further a few words should be spent on the choice of the Langmuir isotherm. Obviously, the choice of this isotherm as local isotherm restricts the applications only to submonolayer localized adsorption with weak lateral interactions however, this seems the case where DR behaviour is observed. ... 

The second assumption makes possible the application of equation (14 10). It may be expected to break down either if the surface is appreciably heterogeneous or if there is appreciable attractive or repulsive interaction between the adsorbed molecules themselves. A third assui ption is also implicit in what follows, namely, that the gas does not dissociate on adsorption. However, this is not essential, and a form of the Langmuir isotherm can be obtained quite readily for the case where dissociation takes place. 

However, having acknowledged this paradox, there exists a set of rules and guidelines that allows assessment of the validity of the application of Langmuirian kinetics, i.e., the assumption of an ideal surface, to a particular set of kinetic data [26,31,32]. Criteria to evaluate whether rate parameters, such as the adsorption equilibrium constants appearing in the denominator of the Langmuir isotherm (and subsequently in the denominator of the rate... 

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. 

Despite these facts. Fig. 1 and the measurements published in the literature [5-7] prove that an extrapolation to infinite pressure [see Eqs (17) and (18)] is thermo-dynamieally inconsistent, that is, the application of the Langmuir and other classical isotherm equations at high pressures is incorrect and, therefore, these equations need some modification, which will be discussed later in this chapter. 

This value of is applicable to the Langmuir isotherm, and should therefore be compared with Eq. (15.19) with /= 0. The peak in the high-overpotential region is thus somewhat lower than in the reversible region, as shown in Figure 15.8. 

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... 

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