Sigmoidal isotherms

The Langmuir equation (Eq. XI-4) applies to many systems where adsorption occurs from dilute solution, but some interesting cases of sigmoid isotherms have been reported [54-56]. In several of these studies [54,55] the isotherms... 

Sigmoid, the characteristic S-shaped curves defined by functions such as the Langmuir isotherm and logistic function (when plotted on a logarithmic abscissal scale). 

Reference to the decomposition of KMn04 has already been made in the discussion of chain branching reactions (Chap. 3, Sect. 3.2) in which the participation of a highly reactive intermediate was postulated. This work provided a theoretical explanation of the Prout—Tompkins rate equation [eqn. (9)]. Isothermal decomposition in vacuum of freshly prepared crystals at 473—498 K gives symmetrical sigmoid a time curves which are described by the expression... 

Rienaecker and Werner [907] suggest that Ba3(Mn04)2 is the initial decomposition product, by analogy with results for KMn04. Isothermal a—time curves for vacuum decomposition at 413—463 K were sigmoid... 

Isothermal a—time curves were sigmoid [1024] for the anhydrous Ca and Ba salts and also for Sr formate, providing that nucleation during dehydration was prevented by refluxing in 100% formic acid. From the observed obedience to the Avrami—Erofe ev equation [eqn. (6), n = 4], the values of E calculated were 199, 228 and 270 kJ mole"1 for the Ca, Sr and Ba salts, respectively. The value for calcium formate is in good agreement with that obtained [292] for the decomposition of this solid dispersed in a pressed KBr disc. Under the latter conditions, concentrations of both reactant (HCOJ) and product (CO3") were determined by infrared measurements and their variation followed first-order kinetics. 

As a result of fliese temperature - programmed runs, additiond experiments were conducted isothermally at temperatures between 723 - 873 K. Based on tbe second phase peak (peak 2), the conversion - time profiles at different temperatures exhibited a characteristic Sigmoid shape and may therefore be used to interpret the solid carburization kinetics. As detailed in Brown [5], the transient convrasion data may be d cribed by... 

Many models have been developed that deal with the sorption properties of wood in the presence of moisture these have been discussed in a number of works (e.g. Skaar, 1972 Siau, 1984). They can be approximately divided into sorption models, such as the Brunauer-Emmett-Teller (BET) model, or solution models (such as the Hailwood-Horrobin, H-H, model). The sigmoidal shapes of sorption or desorption isotherms can be deconvoluted into two components. These are often taken to represent a monomolecular water layer (associated with the primary sorption sites, OH groups), and a multilayer component where the cell wall bound water molecules are less intimately associated with the fixed cell wall OH groups. 

Of particular interest in the case of the displacement separation of the melanotropins, acetonitrile was mixed with both the carrier and the displacer. The purpose was to suppress the sigmoidal isotherm of MSH which was then found to underlie the Langmuirian isotherm of the displacer in the area of operational interest. The idea of isotherm suppression through use of acetonitrile/displacer mixtures surfaced again later in the Horvath laboratory (see below). 

Diffusion-type models are two-parameter models, involving kt or Ds and La, while BDST models are one-parameter models, involving only 0, as gmax is an experimentally derived parameter. The determination of La requires the whole experimental equilibrium curve, and in case of sigmoidal or other non-Langmiur or Freundlich-type isotherms, these models are unusable. From this point of view, BDST models are more easily applied in adsorption operations, at least as a first approximation. 

The sorption behaviour of a number of dairy products is known (Kinsella and Fox, 1986). Generally, whey powders exhibit sigmoidal sorption isotherms, although the characteristics of the isotherm are influenced by the composition and history of the sample. Examples of sorption isotherms for whey protein concentrate (WPC), dialysed WPC and its dialysate (principally lactose) are shown in Figure 7.13. At low aw values, sorption is due mainly to the proteins present. A sharp decrease is observed in the sorption isotherm of lactose at aw values between 0.35 and 0.50 (e.g. Figure 7.13). This sudden decrease in water sorption can be explained by the crystallization of amorphous lactose in the a-form, which contains one mole of water of crystallization per mole. Above aw values of about 0.6, water sorption is principally influenced by small molecular weight components (Figure 7.13). 

In milk powders, the caseins are the principal water sorbants at low and intermediate values of aw. The water sorption characteristics of the caseins are influenced by their micellar state, their tendency towards self-association, their degree of phosphorylation and their ability to swell. Sorption isotherms for casein micelles and sodium caseinate (Figure 7.14) are generally sigmoidal. However, isotherms of sodium caseinate show a marked increase at aw between 0.75 and 0.95. This has been attributed to the... 

Another case that is less frequently encountered involves the situation in which previously sorbed molecules lead to a modification of the sorbent which favors further sorption (Fig. 9.3e). Such effects have been seen in studies involving anionic or cationic surfactants as sorbates. In some of these cases, a sigmoidal isotherm shape (Fig. 9.3/) has been observed, indicating that the sorption-promoting effect starts only after a certain loading of the sorbent. 

A sigmoidal isotherm (type D) indicates cooperative effects. A molecule binds to the surface better if it can interact with a neighboring adsorbed molecule. As a consequence of this lateral interaction two-dimensional condensation occurs. In order to observe sigmoidal isotherms, flat and homogeneous adsorbents are required. 

Type II isotherms (e.g. nitrogen on silica gel at 77 K) are frequently encountered, and represent multilayer physical adsorption on non-porous solids. They are often referred to as sigmoid isotherms. For such solids, point B represents the formation of an adsorbed monolayer. Physical adsorption on microporous solids can also result in type II isotherms. In this case, point B represents both monolayer adsorption on the surface as a whole and condensation in the fine pores. The remainder of the curve represents multilayer adsorption as for non-porous solids. 

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