Generalized sorption isotherm

Sorption is measured by recording sorption isotherms, which themselves are a way to express the amount of surfactant sorbed as function of the concentration of the compound in the solution. The Freundlich isotherm (Equation II) is a general sorption isotherm which describes sorption behavior and often is used in studies of surfactant sorption. KF is the Freundlich sorption coefficient which expresses the affinity of a surfactant for a given solid... 

The water uptake data are presented in Figure 51.1, which shows that polymorph A adsorbs greater quantities of water vapor than polymorph B under all conditions of temperature and RH investigated. In general, sorption isotherms of form A also exhibit more pronoimced hysteresis than form B. A notable exception is the 45°C isotherm, where the two crystal forms show similar and quite pronounced hysteresis. In fact, the profiles obtained at 45°C are suggestive of condensation taking place at that temperature. 

Figure 4. Generalized sorption isotherm for plasticizing penetrants in glassy polymers. (Schematic.)...

Figure 7.41 General sorption Isotherm of food In a wide range of water contents w/d = grams of water per gram of dry matter,...

Generally the above expressions adequately describe sorption isotherms of the type reported in Fig. 1, giving good agreement with experimental results in non-... 

Qads.(max) = 5.7 molecules by unit cell). Generally speaking, Qacis.(max) is closely related to the molecular size, as it is observed for the other molecular species. Secondly, as shown on Figure 5, sorption isotherm sub-step observation could be another signature of zeolite inner surface influence. Such isotherm sub-step reflects a phase transition existence between a fluid phase and a solid phase stabilized by the inner surface sorption sites. 

FIG. 15 A comprehensive stability map illustrating the general relationship between the occurrence of various reactions important in foods, as a function of water activity, superimposed on a sorption isotherm. M, mold Y, yeast B, bacteria. The isotherm is plotted as moisture content (left y axis) versus water activity, whereas all other curves are plotted as relative reaction rate (right y axis) versus water activity. Additional information corresponding to regions I, II, and III is given in Table IV. 

Nevertheless, surfactant sorption isotherms on natural surfaces (sediments and biota) are generally non-linear, even at very low concentrations. Their behaviour may be explained by a Freundlich isotherm, which is adequate for anionic [3,8,14,20,30], cationic [7] and non-ionic surfactants [2,4,15,17] sorbed onto solids with heterogeneous surfaces. Recently, the virial-electrostatic isotherm has been proposed to explain anionic surfactant sorption this is of special interest since it can be interpreted on a mechanistic basis [20]. The virial equation is similar to a linear isotherm with an exponential factor, i.e. with a correction for the deviation caused by the heterogeneity of the surface or the energy of sorption. 

Steroid hormones form a group of pollutants that includes natural hormones such as estradiol, testosterone, and their metabolites as well as several synthetic analogues. Steroid hormones used as growth promoters have already been found in water and sediments (Lai et al. 2000 Thorpe et al. 2003), and their adsorption properties on earth materials have been considered. Lee et al. (2003) report batch experiments where simultaneous sorption of three hormones (17- 3-estradiol, 17-a-ethyl estradiol, and testosterone) on four midwestem U.S. soils and a freshwater sediment were performed. Apparent sorption equilibria were reached within a few hours. Sorption isotherms generally were linear for the chemicals studied on one of these soils (Drummer soil), ranged from 23.4 to 83.2 L kg and log ranged... 

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... 

The discussion directly following Eq (6) provides a simple, physically reasonable explanation for the preceding observations of marked concentration dependence of Deff(C) at relatively low concentrations. Clearly, at some point, the assumption of concentration independence of Dp and in Eq (6) will fail however, for our work with "conditioned" polymers at CO2 pressures below 300 psi, such effects appear to be negligible. Due to the concave shape of the sorption isotherm, even at a CO2 pressure of 10 atm, there will still be less than one CO2 molecule per twenty PET repeat units at 35°C. Stern (26) has described a generalized form of the dual mode transport model that permits handling situations in which non-constancy of Dp and Dh manifest themselves. It is reasonable to assume that the next generation of gas separation membrane polymers will be even more resistant to plasticization than polysulfone, and cellulose acetate, so the assumption of constancy of these transport parameters will be even more firmly justified. 

Sorption coefficients generally are determined from an isotherm, a diagram that depicts the distribution of the test chemical between a solid sorbent and the solution in equilibrium with it over a range of concentrations at constant temperature. These isotherms can be linear or nonlinear, depending on the properties of the test chemical and solid and on the aqueous phase concentration of the chemical. In many cases, sorption isotherms are linear at low concentration but tend to become nonlinear (sorption tends to decrease) as the concentration of chemical in the aqueous phase increases, especially for polar or ionizable... 

Sorption is most commonly quantified using distribution coefficients (Kd), which simplistically model the sorption process as a partitioning of the chemical between homogeneous solid and solution phases. Sorption is also commonly quantified using sorption isotherms, which allow variation in sorption intensity with triazine concentration in solution. Sorption isotherms are generally modeled using the empirical Freundlich equation, S = K CUn, in which S is the sorbed concentration after equilibration, C is the solution concentration after equilibration, and Kt and 1 In are empirical constants. Kd and K are used to compare sorption of different chemicals on one soil or sorbent, or of one chemical on several sorbents. Kd and K are also commonly used in solute leaching models to predict triazine interactions with soils under various environmental conditions. 

Tin and americium were so extensively sorbed under all conditions that isotherm data could not be obtained. These elements are not significantly mobile in the Mabton Interbed aquifer. Values of Freundlich constants for technetium, radium, uranium, neptunium, and plutonium are given in Table IV. The Freundlich equation did not fit the selenium sorption data very well probably because of slow sorption kinetics or precipitation. Precipitation was also observed for technetium at 23°C for concentrations above 10 7M. This is about the same solubility observed for technetium in the sandstone isotherm measurements. Linear isotherms were observed only in the case of radium sorption. In general, sorption on the Mabton Interbed was greater than on the Rattlesnake Ridge sandstone. This is probably due to the greater clay content of the Mabton standard. 

The permeation of a gas through a porous polymer is generally described by equations based on the kinetic theory of gases. The sorption isotherm described by Eq. 1 is concave to the pressure axis and is commonly observed for a penetrant gas in a glassy polymer. It is composed of Henry s law and Langmuir-terms [20] ... 

Equilibrium between solution and adsorbed or sorbed phases is a condition commonly used to evaluate adsorption or sorption processes in soils or soil-clay minerals. As previously stated, equilibrium is defined as the point at which the rate of the forward reaction equals the rate of the reverse reaction. Two major techniques commonly used to model soil adsorption or sorption equilibrium processes are (1) the Freundlich approach and (2) the Langmuir approach. Both involve adsorption or sorption isotherms. A sorption isotherm describes the relationship between the dissolved concentration of a given chemical species (adsorbate) in units of micrograms per liter (pg L 1), milligrams per liter (mg L-1), microequivalents per liter (pequiv L-1), or millimoles per liter (mmol L-1), and the sorbed quantity of the same species by the solid phase (adsorbent) in units of adsorbate per unit mass of adsorbent (solid) (e.g., pg kg-1, mg kg-1, peq kg-1, or mmol kg 1) at equilibrium under constant pressure and temperature. Sorption isotherms have been classified into four types, depending on their general shape (Fig. 4.13) ... 

The hydration product occupies more space than the cement from which it is formed, and the capillary pores were regarded as the remnants of the initially water-filled space. Their volume thus decreases, and that of the gel pores increases, as hydration proceeds. Evidence from water vapour sorption isotherms indicated that the hydration product was composed of solid units having a size of about 14 nm, with gel pores some 2 nm across (P34). The width of the capillary pores could not be determined from the available data, but they were considered to be generally much wider than the gel pores, though tending to become narrower as the water-filled space was used up, and thus in some regions indistinguishable from gel pores. 

It is possible, and perhaps generally believed, that the high reproducibility of an isotherm justihes the extraction of thermodynamic values from data that show hysteresis. However, hysteresis would still be a source of systematic error in the values. There is a poorly documented impression that small samples or thin films display less hysteresis. Hysteresis was not found for the heat capacity isotherm (Yang and Rupley, 1979), which may he taken as support for the view that meaningful free-energy information also can be derived from sorption isotherms. 

The above picture should be general. As noted, globular proteins are closely similar in their sorption isotherms, and heat capacity measurements in the water-poor region, for several proteins, are consistent with the data of Fig. 38. 

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