Henry-Langmuir model

Equation (2.113) does not take into account the influence of pressure on the permeability of the material and is only valid for dilute solutions. The Henry-Langmuir model takes into account the influence of pressure and works very well for amorphous thermoplastics. It is written as... 

Correspondence of experimental data to the Langmuir model does not mean that the stated assumptions are valid for the particular system being studied, because departure from the assumptions can have a canceling effect. An advantage of this model is that it can approach Henry s law at low concentrations. 

It is particularly interesting and instructive to note that application of Henry + Langmuir dual-mode sorption and diffusion models is not confined to glassy polymer-gas systems. Sorption and transport of high affinity ionic species, exemplified by anionic dyes, in charged polymers, exemplified by polyamides at low pH, has been treated in the same way. These systems are of considerable importance both from the bio-mimetic and from the textile processing point of view, but have received limited atten-... 

The Langmuir model also provides a convenient basis for estimating when Henry s law or saturation effects can be expected. If an individual attachment site has an area a 2 x 10 15 cm2, the order of atomic cross-sectional area (cf. Table 2.1), then Ns 5 x 1014 atoms/cm2 = 2 x 10 5cm3STP/cm2. Surface concentrations approaching this order of magnitude can be expected to exhibit saturation behavior. Conversely, much lower concentrations indicate 9 1 and lead to the expectation of Henry s law behavior. Possible adsorption effects important in noble gas geochemistry always involve much lower concentration than this illustrative value, which is one reason why Henry s law violation is not expected. 

This justifies the use of Henry constants for prehminary screening of selective adsorbents, but in practice a significant loading dependence of the separation factor is generally observed as a result of deviations from the binary (or multicomponent) Langmuir model. 

Propylene dissolves in two modes, a physical dissolution mode in the matrix and a binding mode resulting from a reversible chemical reaction with metal complexes. Henry s law commonly expresses the physical dissolution mode for small molecules, while the Langmuir model adequately describes the reversible binding mode, as shown in Eq. (9-2). Mathematically, a Langmuir adsorption isotherm for a small molecule in a porous media is identical to the expression of the olefin concentration bound to the metal complex. 

The total solubility is the sum of the contributions of both the dissolution mode based on Henry s law and the binding mode using the Langmuir model the dual mode sorption [11], At equilibrium, the total solubility of olefin is shown by Eq. (9-4), where C is the total concentration of the olefin gas absorbed in the sample, p is the applied olefin pressure, ku is the solubility coefficient of the olefin gas for Henry s law mode, K is the olefin binding equilibrium constant of Eq. (9-2), as defined in Eq. (9-3), and Q is the saturated amount of the olefin gas bound to the silver complex. 

In any case, Langmuir model did not hold in describing species—A thermodynamic features and Eq. 1.26 employed for species B did not apply. So, the entropy of CO adsorption at A sites was obtained by considering that, in the very early stage of adsorption, the coverage 0 increased linearly with the pco [18]. By employing the Eq. 1.29, based on the Henry-like isotherm ... 

This method consists to find the thermodynamic model (Henry, Langmuir, Fowler, Freundlich...) which gives the best fit with adsorption isotherms measured at various temperatures [7], The parameter of the model Kmodei is temperature dependent. It can be related to the dimensionless equilibrium constant of adsorption K(T) by the... 

In glassy polymers tire interactions of tire penetrant molecules witli tire polymer matrix differ from one sorjDtion site to anotlier. A limiting description of tire interaction distribution is known under tire name of tire dual-soriDtion model [, 60]. In tliis model, tire concentration of tire penetrant molecules consists of two parts. One obeys Henry s law and tire otlier a Langmuir isotlienn ... 

In the dual-mode sorption and transport model the pressure-dependence of a (= C/p), P and 0 in gas-glassy polymer systems arises from the pressure-dependent distribution of the sorbed gas molecules between Langmuir sites and Henry s law dissolution. Although k, Dg and are assumed to be constant, the average or effective solubility and diffusion coefficients of the entire ensemble of gas molecules change with pressure as the ratio of Henry s to Langmuir s population, C /C, changes continuously with pressure [eq. (14)]. 

Figure 2.28 An illustration of the two components that contribute to gas sorption in a glassy polymer according to the dual sorption model. Henry s law sorption occurs in the equilibrium free volume portion of the polymer. Langmuir sorption occurs in the excess free volume between polymer chains that exists in glassy polymers...

This equation has become known as the "dual sorption model", because obviously two separate sorption mechanisms are operative for gases in glassy polymers. One mode (first term on the right in Eq. (18.36)) follows the Henry s law the other mode (second term) follows a Langmuir form. This additional mode is attributed to sorption into micro-voids that apparently pre-exist in the glassy state of the polymer (and only there ) it disappears above Tg (see Fig. 18.9b). 

Early investigations of the dual-sorption model started from the assumption that only Henry s part of the sorbed gas contributed to the gas transport, whereas the Langmuir part would not contribute to it, due to immobilisation. Then the transport flux would be... 

Barrer (1984) suggested a further refinement of the dual-mode mobility model, including diffusive movements from the Henry s law mode to the Langmuir mode and the reverse then four kinds of diffusion steps are basically possible. Barrer derived the flux expression based on the gradients of concentration for each kind of diffusion step. This leads to rather complicated equations, of which Sada (1987, 1988) proved that they describe the experimental results still better than the original dual-mode model. This, however, is not surprising, since two extra adaptable parameters are introduced. 

From the scientific point of view, however, all approaches in the sense of the Kd concept (Henry, Freundlich or Langmuir isotherm) are unsatisfactory, since the complex processes on surfaces cannot be described by empirical fitting parameters. Boundary conditions like pH value, redox potential, ionic strength, competition reactions for binding sites are not considered. Thus results from laboratory and field experiments are not transferable to real systems. They are only advisable to provide a suitable prognosis model, if no changes concerning boundary conditions are to be expected and if no parameters for deterministic or mechanistic approach can be determined. 

The basic assumptions of the dual mode sorption theory as it applies to the transport model of Vieth and Sladek, have been stated by Vieth et al. in their excellent review of the subject The sorption isotherm was described by the combination of a Henry s law dissolved component, Cp, and a Langmuir hole filling term, Ch, i.e.. 

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.)... 

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