Equilibrium isotherm sigmoidal

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. 

Adsorption isotherms are used to quantitatively describe adsorption at the solid/ liquid interface (Hinz, 2001). They represent the distribution of the solute species between the liquid solvent phase and solid sorbent phase at a constant temperature under equilibrium conditions. While adsorbed amounts as a function of equilibrium solute concentration quantify the process, the shape of the isotherm can provide qualitative information on the nature of solute-surface interactions. Giles et al. (1974) distinguished four types of isotherms high affinity (H), Langmuir (L), constant partition (C), and sigmoidal-shaped (S) they are represented schematically in Figure 3.3. 

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... 

High-resolution isotherms of the equilibrium catalyst are represented in Figure 2. The observed sigmoid curves demonstrate the microporous character of the catalyst, but the relative pressure corresponding to the inflexion point depends on both the adsorbate and the adsorption temperature (Ar at 77 K or 87 K). Thus, mesopores and micropores are qualitatively evidenced from each physisorption isotherm but quantitative differences exist. 

We now cite the types of experimental data in the literature, by which an analysis of surface adsorption effects is carried out. One common experiment involves measuring adsorption isotherms. By weighing or by volumetric techniques one determines as a function of equilibrium gas pressure the amount of gas held on a given surface at a specified temperature. Usually this quantity varies sigmoidally with rising pressure P, as sketched in Fig. 5.2.1 for a variety of temperatures 7). By standard methods that rely on the Brunauer, Emmett, Teller isotherm equa-tion one can determine the point on the isotherms at which monolayer coverage of the surface is complete it is usually is located fairly close to the knee of the isotherm. From the cross sectional area of the adsorbate molecules and from the amount needed for monolayer coverage one may then ascertain more or less quantitatively the surface area of the adsorbent. As-... 

The relationship between water activity and moisture content for most foods at a particular temperature is a sigmoidal-shaped curve called the sorption isotherm (Figure 3.10). The term equilibrium moisture content curve is also used. Sorption... 

The absorption isotherms of keratin from human hair shows a sigmoidal form as did those for collagen which are classified as type II with a hysteresis over the entire humidity range as shown in figure 3. A difference exists in the kinetics of absorption when observed by successive increments. Longer times are required to attain equilibrium for the keratin in the first increment than for those above P/Pq = 0.6 (figure 4). 

Consequently, the continuous variation of specific volume of the vapor-liquid mixture at fixed temperature and pressure is a result of the continuous change in the fraction of the mixture that is vapor. The conclusion, then, is that an isotherm such as that shown in Fig. 7.3-2 is an approximate representation of the real phase behavior (shown in Fig. 7.3-3) by a relatively simple analytic equation of state. In fact, it is impossible to represent the discontinuities in the derivative dP/dV)T that occur at and v with any analytic equation of state. By its sigmoidal behavior in the two-phase region, the van der Waals equation of state is somewhat qualitatively and crudely exhibiting the essential features of vapor-liquid phase equilibrium historically, it was the first equation of state to do so. 

Figure 7.19 Dependent multiple-site variable affinity cooperative binding equilibria, (a) Classical sigmoidal binding isotherms indicative of ligand-receptor interactions that involve strong positive cooper-ativity. Curves move to right as composite equilibrium constant ff increases, (b) Linear Hill plots derived from sigmoidal data illustrated in (a). Gradients and intercepts define values of n and K respectively.

The relationship between relative humidity and the water absorbed at equilibrium by a sample of purified wool fibers, at a fixed temperature, is sigmoid [290]. The relationship shows hysteresis (Figure 5.13). The lower curve represents the equilibrium water content (EWC) of samples of wool that were originally dry and have reached equilibrium at different relative humidities (RH). It is known as the absorption isotherm. The upper curve represents the EWC of originally wet samples that have reached equilibrium at different RHs it is the desorption isotherm. 

Figure 5 demonstrates the kinetic evaluation of these results. On the left-hand side, the initial velocity Vp (intersection points with the ordinate) is plotted versus the concentration of the Al component while the Ti concentration was kept constant. The result is the Al isothermal curve of the second equilibrium reaction generating the active species. It demonstrates in the initial part the demanded sigmoid curve course according to the location of the two successive equilibria. This induction period is enlarged in Fig. 6. It can be clearly seen how sensitively the initial polymerization rate responds to the ratio Al/Ti. With a ratio >1 (i.e., here [AlEtCl2] > 3x10 mol L ), the formation of the active species increases drastically.

Physically this means that the water in the fully swollen wool tends to be Squeezed out of the gel by the forces acting in the gel structure, and is, for this reason, in equilibrium with a vapour of relative vapour pressure 100% instead of 50%. Incidentally we observe that the sigmoid shape of the isotherm does not persist in the corrected curve. It is to be noted, however, that the application of stress-strain data to the swelling process is of an extremely approximate nature, and the results recorded here are not to be considered as final. 

Where the adiabatic equilibrium curve is sigmoidal, with a point of inflection, the operating line must be drawn from the initial or plateau point to the point of tangency. Under these conditions a composite front, partly sharp and partly diffuse, is obtained as shown in Figure 8.6 for an isothermal system with a BET isotherm. 

---