Langmuir affinity constants

The effects of organic molecules and phosphate on the adsorption of acid phosphatase on various minerals, and kaolinite in particular, have been investigated by Huang et al. [97]. The Langmuir affinity constant for AcP adsorption by kaolinite follows the series tartrate (K — 97.8) > phosphate (K= 48.6) > oxalate (K — 35.6) > acetate (K= 13.4). At low concentration, acetate even promoted the adsorption of acid phosphatase. It was considered that competitive interactions between anionic adsorbates can occur directly through competition for surface sites and indirectly through effects of anion adsorption on the surface charge and protonation. 

C h is the maximum adsorption capacity while b is the ratio of rate coefficients of adsorption and desorption, or the Langmuir affinity constant, defined as ... 

Figure 11.1 Langmuir affinity constants (h) for a range of gases in various polymeric membranes relative to gas critical temperature. Reprinted with permission from Separation Purification Reviews, Effects of minor components in carbon dioxide capture using polymeric gas separation membranes, by Scholes, C. A., S. E. Kentish, and C. W. Stevens, 38 1-44, Copyright (2009) Taylor and Francis...

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. 

Figure 4.2 shows the Langmuir isotherms of (a) phenobarbital and (b) mephobar-bital adsorption on activated carbon. The maximum adsorption capacity (K,/K2) of activated carbon for mephobarbital is lower than that for phenobarbital. The affinity constant (1/K2) for mephobarbital is higher than that for phenobarbital. This is expected because higher solubility gives a lower adsorption affinity. Table 4.1 illustrates the Langmuir constants of various drugs on carbon black. 

A plot of C/-SC versus C will produce a straight line with slope 1/ST and y intercept of 1/KlSt (Fig. 4.21). Such a plot will allow one to determine the so-called adjustable parameters, ST and KL. Note that both the Freundlich equation and the Langmuir equation produce a surface adsorption affinity constant, but the latter (Langmuir) also produces the often-sought adsorption maximum (ST). 

In the Henry region the Langmuir and Volmer isotherms differ only with respect to their affinity constants this is the only way in which the two modes of adsorption can be distinguished here. In practice the distinction is not always easy because of surface heterogeneity the most energetical sites are filled first. However, the equations of state are identical (both reduce to nA = NkT) because such equations do not include adsorbent-adsorbate interactions. 

Figure 4.6. The term cq accounts for Henry s law, while Ch is the Langmuir term with b being the hole affinity constant (bar )...

FIGURE 9.4 Dependence of constants (a, b, and c present Henry constant, sorption affinity constant, and Langmuir sorption capacity respectively) of the model of dual-mode sorption of hydrocarbons by glassy polyphenylene oxides on boiling temperatures of hydrocarbons Z), is pDMePO, poly-2,6-dimethyl-l,4-phenylene oxide o is pDPhPO, poly-2,6-diphenyl-l,4-phenylene oxide is pDMePO/pDPhPO copolymer (97.5/2.5% mol) v is pDMePO/pDPhPO copolymer (75/25% mol). (From analysis of results presented in Lapkin, A.A., Roschupkina, O.P., and Ilinitch, O.M., J. Membr. Sci., 141, 223, 1998.)... 

If the sorption curves pictured in Fig. 9 are caused by heterogeneity, what distribution of site properties would give rise to such curves Sposito [64] reported that a Freundlich curve could be derived if the log of the affinity constants of a series of Langmuir equations had a distribution which was similar to a normal distribution. 

During the adsorption step, the binding of solute by the affinity ligand is often very tight, essentially irreversible. For Langmuir, or constant separation factor adsorption, the equilibrium relation is... 

Ch = solubility in the Langmuir type sites Ch = hole saturation constant b = hole affinity constant... 

The solubility in glassy polymers is usually described by the so-called dual-mode model, which implies that there is a need for a more detailed definition of the sorption, c, in the flux Equation 7.1. The following Equations 7.20 and 7.21 illustrate this and can be referred to Figure 7.6. The term Cg accounts for Henry s law, while Cu is the Langmuir term with b being the hole affinity constant (bar ) and is the saturation constant (cm [STP]/cm ) ... 

Here Q is the heat of adsorption and is equal to the activation energy for desorption, Ej. The parameter b is called the affinity constant or Langmuir constant. It is a measure of how strong an adsorbate molecule is attracted onto a surface. The preexponential factor b , of the affinity constant is ... 

The difference between the Volmer equation and the Langmuir equation is that while the affinity constant remains constant in the case of Langmuir mechanism, the apparent affinity constant in the case of Volmer mechanism decreases with loading. This means that the rate of increase in loading with pressure is much lower in the case of Volmer compared to that in the case of Langmuir. 

Here b is the adsorption affinity constant at infinite temperature, bo is that at some reference temperature Tq, Uq is the parameter n at the same reference temperature and a is a constant parameter. The temperature dependence of the affinity constant b is taken from the that of the Langmuir equation. Unlike Q in the Langmuir equation, where it is the isosteric heat, invariant with the surface loading, the parameter Q in the Sips equation is only the measure of the adsorption heat. We shall discuss its physical meaning in Section 3.2.2.2.I. The temperature-dependent form of the exponent n is empirical and such form in eq. (3.2-18c) is chosen because of its simplicity. The saturation capacity can be either taken as constant or it can take the following temperature dependence ... 

Figure 4.7. Single component isotherms of (top) proteins and (bottom) synthetic tri-block copolymers measured at room temperature for Octyl-sepharose stationary phase. The affinity constants calculated from an isotherm fit according to the Langmuir model seemed to indicate that the proteins bind more strongly than the copolymers and hence could not be displaced by them [reproduced with permission from reference 14].

C -.Langmuir sorption capacity cm STP/cm polym. kp Henry s law solubility coefficient cm STP/ (cm polym. cmHg). b affinity constant of Langmuir site cmHg... 

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