Langmuir sites

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)]. 

For very high values of bp, such as at high pressures, when Langmuir sites are saturated and thus do not participate in transport. Equation 23 reduces to the simple Henry form.13... 

For energetically heterogeneous adsorbents one may choose to represent the isotherm as the sum of the contributions from two independent sets of Langmuir sites ... 

When two gas species are present, competition can restrict both the solubility within the polymer matrix and the amount adsorbed in the Langmuir free volume. Competition in the former case is best modelled by adjustments to based on Equations (11.20) or (11.22). To accoimt for changes to the occupancy of the Langmuir sites for a binary mixture of gases A and B, the mobile concentration of gas A becomes [22] ... 

At elevated pressures, the dependence of the solute concentration on the ideal gas pressure can be represented as a sum of linear terms, bjp (for sites j where bjp 1), and Langmuir-like terms, bjP/(l -i- bjp) (for sites] where the term bjp is at least comparable to unity). If a system consists only of linear sites and a set of identical Langmuir sites, the well-known dual-mode-sorptiorr model [2] is obtained [54],... 

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... 

Various functional forms for / have been proposed either as a result of empirical observation or in terms of specific models. A particularly important example of the latter is that known as the Langmuir adsorption equation [2]. By analogy with the derivation for gas adsorption (see Section XVII-3), the Langmuir model assumes the surface to consist of adsorption sites, each having an area a. All adsorbed species interact only with a site and not with each other, and adsorption is thus limited to a monolayer. Related lattice models reduce to the Langmuir model under these assumptions [3,4]. In the case of adsorption from solution, however, it seems more plausible to consider an alternative phrasing of the model. Adsorption is still limited to a monolayer, but this layer is now regarded as an ideal two-dimensional solution of equal-size solute and solvent molecules of area a. Thus lateral interactions, absent in the site picture, cancel out in the ideal solution however, in the first version is a properly of the solid lattice, while in the second it is a properly of the adsorbed species. Both models attribute differences in adsorption behavior entirely to differences in adsorbate-solid interactions. Both present adsorption as a competition between solute and solvent. 

The derivation that follows is essentially that given by Langmuir [9] in 1918, in which one writes separately the rates of evaporation and of condensation. The surface is assumed to consist of a certain number of sites S of which S are occupied and Sq = S - S arc free. The rate of evaporation is taken to be proportional to 5, or equal tokiSi, and the rate of condensation proportional to the bare surface So and to the gas pressure, or equal to k PSo. At equilibrium. 

This difference looks large enough to be diagnostic of the state of the adsorbed film. However, to be consistent with the kinetic derivation of the Langmuir equation, it was necessary to suppose that the site acted as a potential box and, furthermore, that a weak adsorption bond of ifi corresponding to 1 /tq was present. With these provisions we obtain... 

A variety of experimental data has been found to fit the Langmuir equation reasonably well. Data are generally plotted according to the linear form, Eq. XVn-9, to obtain the constants b and n from the best fitting straight line. The specific surface area, E, can then be obtained from Eq. XVII-10. A widely used practice is to take to be the molecular area of the adsorbate, estimated from liquid or solid adsorbate densities. On the other hand, the Langmuir model is cast around the concept of adsorption sites, whose spacing one would suppose to be characteristic of the adsorbent. See Section XVII-5B for an additional discussion of the problem. 

As a simple model of a heterogeneous surface, assume that 20% of it consists of sites of Q= 2.5 kcal/mol 45% of sites Q = 3.5 kcal/mol and the remainder, of sites of Q= 4.5 kcal/mol. Calculate Q(P, T) for nitrogen at 77 K and at 90 K, assuming the adsorption to follow the Langmuir equation with bo given by Eq. XVII-15. Calculate qsi for several 6 values and compare the result with the assumed integral distribution hinction. 

Since in chemisorption systems it is reasonable to suppose that the strong adsorbent-adsorbate interaction is associated with specific adsorption sites, a situation that may arise is that the adsorbate molecule occupies or blocks the occupancy of a second adjacent site. This means that each molecule effectively requires two adjacent sites. An analysis [106] suggests that in terms of the kinetic derivation of the Langmuir equation, the rate of adsorption should now be... 

The Langmuir-Hinshelwood picture is essentially that of Fig. XVIII-14. If the process is unimolecular, the species meanders around on the surface until it receives the activation energy to go over to product(s), which then desorb. If the process is bimolecular, two species diffuse around until a reactive encounter occurs. The reaction will be diffusion controlled if it occurs on every encounter (see Ref. 211) the theory of surface diffusional encounters has been treated (see Ref. 212) the subject may also be approached by means of Monte Carlo/molecular dynamics techniques [213]. In the case of activated bimolecular reactions, however, there will in general be many encounters before the reactive one, and the rate law for the surface reaction is generally written by analogy to the mass action law for solutions. That is, for a bimolecular process, the rate is taken to be proportional to the product of the two surface concentrations. It is interesting, however, that essentially the same rate law is obtained if the adsorption is strictly localized and species react only if they happen to adsorb on adjacent sites (note Ref. 214). (The apparent rate law, that is, the rate law in terms of gas pressures, depends on the form of the adsorption isotherm, as discussed in the next section.)... 

Another limitation of tire Langmuir model is that it does not account for multilayer adsorption. The Braunauer, Ennnett and Teller (BET) model is a refinement of Langmuir adsorption in which multiple layers of adsorbates are allowed [29, 31]. In the BET model, the particles in each layer act as the adsorption sites for the subsequent layers. There are many refinements to this approach, in which parameters such as sticking coefficient, activation energy, etc, are considered to be different for each layer. 

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 present study we try to obtain the isotherm equation in the form of a sum of the three terms Langmuir s, Henry s and multilayer adsorption, because it is the most convenient and is easily physically interpreted but, using more a realistic assumption. Namely, we take the partition functions as in the case of the isotherm of d Arcy and Watt [20], but assume that the value of V for the multilayer adsorption appearing in the (5) is equal to the sum of the number of adsorbed water molecules on the Langmuir s and Henry s sites ... 

Substituting (6) to (5) and assuming there are M < N — 1 types of primary sites for which << 1 (sites without saturation, Henry s sites) and neglecting the heterogeneity of the Langmuir s and Hemy .s sites we can... 

The BET treatment is based on a kinetic model of the adsorption process put forward more than sixty years ago by Langmuir, in which the surface of the solid was regarded as an array of adsorption sites. A state of dynamic equilibrium was postulated in which the rate at which molecules arriving from the gas phrase and condensing on to bare sites is equal to the rate at which molecules evaporate from occupied sites. 

Short-Chain Organics. Adsorption of an organic dispersant can reduce polarizabiHty attraction between particles, ie, provide semisteric stabilization, if A < A.p < A or A < A.p < A (T = dispersant) and the adsorption layer is thick. Adsorption in aqueous systems generally does not foUow the simple Langmuir profile because the organic tails on adsorbed molecules at adjacent sites attract each other strongly. 

In synthetic fibers the number of ionic groups or dye sites is relatively small, and may have been introduced dehberately to make the base polymer dyeable. The restrictions on dye absorption are therefore very great the dye molecule must find an available specific site from among the limited number of sites in the fiber. This situation follows a Langmuir isotherm, where the reciprocal of dye in fiber 1 /DF is direcdy proportional to the reciprocal of dye in the dyebath 1 /HT. A plot of 1/against 1/H therefore gives a straight line. 

Langmuir isotherms are typically found with ionic synthetic fibers and ionic dyes, eg, dyeing polyacrylonitrile with modified basic dyes, and on hydrophilic fibers in situations when the number of sites becomes very low. This may arise when the internal pH is such that only a small number of sites ionise. 

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