Heterogeneous Langmuir adsorption isotherms

Jaroniec, M., Gilpin, R. K., Kaneko, K. and Choma, J., Evaluation of energetic heterogeneity and microporosity of activated carbon fibers on the basis of gas adsorption isotherms, Langmuir, 1991, 7(1 1), 2719 2722. 

Mamleev, V. Sh. and Bekturov, E.A. (1996). Improved method for analysis of energetic heterogeneity of surfaces from adsorption isotherms. Langmuir, 12, 441-9. 

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 Langmuir adsorption isotherm provides a simple mechanistic picture of the adsorption process and gives rise to a relatively simple mathematical expression. It can also be used to obtain a crude estimate of specific surface areas. More important, from the viewpoint of the chemical engineer, it serves as a point of departure for formulating rate expressions for heterogeneous catalytic reactions. 

The Langmuir Equation for the Case Where Two or More Species May Adsorb. Adsorption isotherms for cases where more than one species may adsorb are of considerable significance when one is dealing with heterogeneous catalytic reactions. Reactants, products, and inert species may all adsorb on the catalyst surface. Consequently, it is useful to develop generalized Langmuir adsorption isotherms for multicomponent adsorption. If 0t represents the fraction of the sites occupied by species i, the fraction of the sites that is vacant is just 1 — 0 where the summation is taken over all species that can be adsorbed. The pseudo rate constants for adsorption and desorption may be expected to differ for each species, so they will be denoted by kt and k h respectively. 

CI2 evolution reaction, 38 56 electrochemical desorption, 38 53-54 electrode kinetics, 38 55-56 factors that determine, 38 55 ketone reduction, 38 56-57 Langmuir adsorption isotherm, 38 52 recombination desorption, 38 53 surface reaction-order factor, 38 52 Temkin and Frumkin isotherm, 38 53 real-area factor, 38 57-58 regular heterogeneous catalysis, 38 10-16 anodic oxidation of ammonia, 38 13 binding energy quantification, 38 15-16 Haber-Bosch atrunonia synthesis, 38 12-13... 

Adsorption Isotherm Measurements and Site-Selective Thermodynamics. For heterogeneous surfaces like CSPs, the adsorption isotherms are usually composite isotherms and often a Bi-Langmuir model (Equation 1.15) describes reasonably well the adsorption behavior [54]. 

Temkin (1941) approached the development of an adsorption isotherm by considering a heterogeneous surface (Section 6.8.3) where no molecular interactions exist. He divided the surface into different patches, and since there are no interactions between molecules, in each patch the Langmuir isotherm can be applied [see Eq. (6.196)] (Fig. 6.98). Thus, for the ith patch,... 

With this in mind, some important adsorption isotherms were introduced, and we found that each of them describes important characteristics of the adsorption process (Table 6.10). Thus, the Langmuir isotherm considers the basic step in the adsorption process the Frumkin isotherm was one of the first isotherms involving lateral interactions the Temkin is a surface heterogeneity isotherm and the Flory-Huggins-type isotherms include the substitution step of replacing adsorbed water molecules by the adsorbed entities (Fig. 6.98). 

In heterogeneous systems, the rate expressions have to be developed on the basis of (a) a relation between the rate and concentrations of the adsorbed species involved in the rate-determining step and (b) a relation between the latter and the directly observable concentrations or partial pressures in the gas phase. In consequence, to obtain adequate kinetic rate expressions it is necessary to have a knowledge of the reaction mechanism, and an accurate means of relating gas phase and surface concentrations through appropriate adsorption isotherms. The nature and types of adsorption isotherm appropriate to chemisorption processes have been discussed in detail elsewhere [16,17] and will not be discussed further except to note that, in spite of its severe theoretical limitations, the Langmuir isotherm is almost invariably used for kinetic interpretations of surface hydrogenation reactions. The appropriate equations are... 

A number of classic rate expressions are commonly used to characterize heterogeneous reactions. These include expressions for the Langmuir adsorption isotherm, competitive... 

Solid surfaces are usually heterogeneous therefore, since adsorption at the more active sites is favoured, heats of both monolayer physical adsorption and chemisorption might, in this respect, be expected to become significantly less exothermic as the surface coverage increases, as, for example, shown at low pressures in Figures 5.12a and 5.12b. This, in turn would cause the initial slope of an adsorption isotherm to be steeper than that predicted according to the Langmuir equation or the BET equation. 

In the limit of a - 0, the ideal Langmuir adsorption isotherm is obtained. See - Frumkin isotherm, and for the role of surface heterogeneity - Temkin isotherm. Refs. [i] Horanyi G (2002) Specific adsorption. State of art Present knowledge and understanding. In Bard A], Stratmann M, Gileadi M, Urbakh M (eds) Thermodynamics and electrified interfaces. Encyclopedia of electrochemistry, vol. I. Wiley-VCH Verlag, Weinheim, pp 349-382 [ii] Calvo EJ (1986) Fundamentals. The basics of electrode reactions. In Bamford CH, Compton RG (eds) Comprehensive chemical kinetics, vol. 26. Elsevier, Amsterdam, pp 1-78... 

Lipatov798 recognized that the sorption of Methylene Blue on starch is a heterogenous neutralization reaction of neutralization in which starch acts as an acid. Adsorption isotherms of Methylene Blue on starch indicate that it is an equilibrium process and that swelling as well as other internal structural factors are responsible for the sigmoidal shape of the isotherm.799-800 It is also known that the Langmuir isotherm is affected by equilibrium between the monomer and dimer forms of Methylene Blue, both of which are capable of adsorption on starch (Table XLV).801... 

In some cases, adsorption of analyte can be followed by a chemical reaction. The Langmuir-Hinshelwood (LH) and power-law models have been used successfully in describing the kinetics of a broad range of gas-solid reaction systems [105,106]. The LH model, developed to describe interactions between dissimilar adsorbates in the context of heterogeneous catalysis [107], assumes that gas adsorption follows a Langmuir isotherm and that the adsorbates are sufficiently mobile so that they equilibrate with one another on the surface on a time scale that is rapid compared to desorpticm. The power-law model assumes a Fre-undlich adsorption isotherm. Bodi models assume that the surface reaction is first-order with respect to the reactant gas, and that surface coverage asymptotically approaches a mmiolayer widi increasing gas concentration. 

Avir D, jaroniec M., An isotherm equation for adsorption on fractal surfaces of heterogeneous porous materials , Langmuir, 1989 5 (6)1431-1433... 

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 ... 

Derylo-Marczewska and Jaroniec [28] have reviewed the adsorption of organic solutes from dilute solutions and have provided a useful compilation of published experimental data for both single- and multisolute adsorption isotherms on carbonaceous adsorbents. They also presented a survey of theoretical approaches used to describe the solute adsorption equilibria, including the Polanyi adsorption model, the solvophobic interaction model, the Langmuir adsorption theory, the vacancy solution model, as well as considerations based on the energetic heterogeneity of the adsorbent. In particular, these authors emphasize the... 

The heterogeneity of the surface of the adsorbents used in chromatography has two important consequences. First, adsorption isotherm data do not fit well to the simple isotherm models designed for homogeneous srufaces, i.e., the Langmuir... 

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