Isotherms S’ shaped

In Section XVII-16C there is mention of S-shaped isotherms being obtained. That is, as pressure increased, the amount adsorbed increased, then decreased, then increased again. If this is equilibrium behavior, explain whether a violation of the second law of thermodynamics is implied. A sketch of such an isotherm is shown for nitrogen adsorbed on a microporous carbon (see Ref. 226). 

In most cases, the formation of complexes in molten salts leads to an increase in the molar volume relative to the additive volume. This phenomenon is usually explained by an increase in bond covalency. Nevertheless, the nature of the initial components should be taken into account when analyzing deviations in property values, as was shown by Markov, Prisyagny and Volkov [314]. In particular, this rule applies absolutely when the system consists of pure ionic components. The presence of initial components with a significant share of covalent bonds leads to an S-shaped isotherm [314]. 

The S-shaped isotherm has an initial slope that increases with increasing equilibrium solute concentration and has two causes. Giles et al. (1974) attributed the S-shape to cooperative adsorption due to solute-solute interactions. These interactions stabilized the solute at the solid surface, and therefore the first adsorbed molecules enhance the adsorption of the next solute molecules. At high concentration, when the sites of the solid surface are saturated with solute the slope of adsorption isotherm start to decrease again. Sposito (1984) explained the S-shaped isotherm by a competing reaction within the solution. Solution ligands compete with surface... 

Besides the heterogeneity of the adsorbent surface, the second major reason for the adsorption of a compound to deviate from Langmuir isotherm behavior is that the adsorbed molecules interact. In this category, we find the Fowler isotherm, the anti-Langmuirian isotherm, and several S-shaped isotherm models, including the quadratic isotherm, the extended BET isotherm models, and the Moreau model. 

Figure 3.28 Adsorption data of butylbenzoate measured by FA on Cig-Chromolith, with methanohwater, 65/35 (v/v) at 296 K. A) Plot of the isotherm data and best extended liquid-solid BET isotherm.. B) Plot of the experimental isotherm chord. Note the S-shaped isotherm behavior (fc < bs/2).

Depending on the relative values of the numerical coefficients, the isotherm may be convex, concave or exhibit an inflection point. An isotherm similar to the one given in Eqs. 3.23 and 3.24, but simplified by omission of the term 2 30x02 in the denominator, has been used by Svoboda for the calculation of the individual elution band profiles of mixtures with an S-shaped isotherm [72]. 

Figure 10.25 Comparison of the experimental (solid line) and calculated (dashed and dotted lines) band profiles of the two enantiomers of Troger s base on Chiralpak AD. (Left) ( )-enantiomer sample sizes 1.31, 2.62, 3.93, and 5.23 mg. Dashed lines, profiles calculated with an S-shaped isotherm model. (Right) (-l-)-enantiomer sample sizes 3.62, 7.25, 10.87, and 14.5 mg. Dashed lines, profiles calculated with a Langmuir isotherm model. Dotted Unes, profiles calculated with the three-layer competitive isotherm model. Reproduced with permission from K. Mihlbachler, K. Kaczmarski, A. Seidel-Morgenstem, G. Guiochm,. Chromatogr. A, 955 (2002) 35 (Figures 9 and 10).

The isotherm of adsorption from solution may be determined by frontal chromatograms [6-8]. In [6] it was shown that if to take into consideration the broadening the band owing to diffusion, the adsorption isotherms determined by the column chromatography and in a batch process are coincided. This method can be used for the calculation of S-shape isotherm of adsorption. The isotherm of adsorption can be determined by frontal chromatography if the adsorbents have not fine pores [7]. 

The case of the S-shaped isotherm, representing cooperative adsorption, is a special situation with a similar derivation. Suppose that there is a tendency for the adsorbate, A, to adsorb in pairs on the surface. Then, in addition to reaction 10.5, the following adsorption reaction must be considered ... 

Equation 10.20 describes the S-shaped isotherm commonly observed in studies of organic molecule adsorption on clays from aqueous solution. 

S-shaped isotherm, on the other hand, will always be the result of attractive interactions between sorbate particles. 

Equation 2.73 is called the simple or infinity form of the BET equation and gives an S-shaped isotherm. The equation can be tested by writing it in the linear form as... 

A two-branch S-shaped isotherm in which the two branches usually have different sizes... 

U-shaped or S-shaped isotherms are more common. Although U-shaped isotherms correspond to Type (a) of the Ostwald-de Izaguirre, S-shaped isotherms may replace almost all of the remaining types. Linear isotherm (Type c) is generally obtained for adsorption by molecular sieves when only one component of the solution can enter the pores of the adsorbent while the entry of the second component is inhibited. 

Nagy and Schay recognize five different types of composite isotherms that are variations of U-shaped and S-shaped isotherms of Kipling. Three isotherm types of Nagy and Schay are U-shaped, and the other two are S-shaped. These five types are presented in Figure 3.10. The S-shape of the isotherms indicates that the surface shows a preference for both the components of the solution but over different... 

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