Isotherm Volmer

If the adsorption sites are not localised in space (as is the case for sorption to a fluid lipid membrane), then the Langmuir equation could be transformed to the Volmer isotherm [7],... 

Beside the theoretically derived Gibbs adsorption isotherm, a large number of models have been developed that empirically describe a relationship between the interfacial coverage, the surface tension, and the surfactant concentration in the bulk phase. These adsorption isotherms are known under the names of the authors that first described them—i.e., the Fangmuir, Frumkin, or Volmer isotherms. A complete mathematical description of these isotherms is beyond the scope of this unit and the reader is encouraged to consult the appropriate literature instead (e.g., Dukhin et al., 1995). 

Plgure 1.15. Comparison between the Langmuir and Volmer isotherm (a) and equation of state (b). In figure (a) the abscissa axis is scaled by a factor of e to let the two isotherms intersect at 0 = 0.5. 

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. 

The Henry constant dejjends on the units in which the concentration is expressed. Linear initial x(x) relations are also obtained from the Langmuir cind Volmer Isotherms (see below). Apart from its dimension. K... (or K. or K ) is a measure of the affinity of the molecule for the surface. The higher it is, the steeper -(dy/dx) jj or (dx/dx) Q. 

The major advantage of the Langmuir isotherm (Eq. 3.13) is that it can be inverted and solved for q in closed-form. This cannot be done with the Volmer isotherm (see next section), or with many other important isotherms, such as the Fowler or the virial isotherms (see below. Sections 3.1.6 and 3.2.3.1). Such isotherms are often called implicit isotherms. The other conditions of validity of the Langmuir model are the ideal behavior of the gas phase, the absence of adsorbate-adsorbate interactions, and the localized character of adsorption. 

If we make no assumptions regarding the magnitude of /3 compared to A, the direct integration of Eq. 3.12 gives the Volmer isotherm [18]... 

Now, let us apply the same general scheme, bnt this time to the derivation of the van der Waals isotherm, which corresponds to nonlocalized adsorption of interacting molecules. (Expressions corresponding to the Volmer isotherm can be obtained by setting p = 0 in the respective expressions for the van der Waals isotherm.) Now the adsorbed molecules are considered a two-dimensional gas. The corresponding expression for the canonical ensemble partition function is... 

In the special case of Volmer isotherm we have P = 0, and then = 1/(1 — 9). Finally, substituting Equation 5.24 into Equation 5.9 we derive the van der Waals adsorption isotherm in Table 5.2, with K defined by Equation 5.3. 

Table 1 lists the six most popular surfactant adsorption isotherms, i.e., those of Henry, Freundlich, Langmuir, Volmer (10), Frumkin (11), and van der Waals (9). For cj— 0 all other isotherms (except that of Freundlich) reduce to the Henry isotherm. The physical difference between the Langmuir and Volmer isotherms is that the former corre-... 

This equation is the fundamental equation relating gas pressure, spreading pressure and amount adsorbed. It is very useful in that if the equation of state relating the spreading pressure and the number of mole on the adsorbed phase is provided, the isotherm expressed as the number of mole adsorbed in terms of pressure can be obtained. We shall illustrate this with the following two examples linear isotherm and Volmer isotherm. 

Thus, for Langmuir isotherm the transport diffusivity increases as the loading inside the zeolite crystal increases. Stronger dependence with concentration can be observed if the isotherm takes the form of Volmer equation (Table 10.2-1). This stronger concentration dependence of the transport diffusivity in the case of Volmer isotherm equation is attributed to the mobility term in the Volmer equation. 

If we make no approximation concerning the relative magnitude of p/ and integrate Eq. (3.37) directly, setting 9 = p/s/, we obtain the Volmer isotherm equation ... 

Ion distribution around a spherical micelle can also be described with models that consider that polarizable and not very hydrophilic species (such as Br ) interact both coulombically and by a specific, noncoulombic, interaction [26]. This latter interaction allows the ion to intercalate at the micellar surface and to neutrahze an equivalent number of head groups. Ion distribution around a micelle is then calculated by solving the Poisson-Boltzmann equation (PBE) in the spherical symmetry with allowance for specific interactions via a Langmuir or Volmer isotherm [31]. The original kinetic treatment for a micelle of radius a, aggregation number iV in a cell of radius R yields [31] ... 

PBE equation was also solved numerically assuming water and ion-permeable hollow spheres treating specific ion adsorption using a Volmer isotherm [49]. The calculations suggest that the distribution of ions in the internal aqueous of (DODA)X vesicles is measurable and that the value of the electrical potential, ij/, at the vesicle center is not negligible at moderately low salt concentration. As could be expected the value of j/ at the vesicle center decreases with vesicle size and vesicle concentration (Fig. 4). Using realistic parameters, for a 265 nm (DODA)X vesicle, the value of the potential at the vesicle center can reach 100 mV [49]. The calculations yielded results that are consistent with measured values for external ion dissociation and zeta potentials of vesicles of synthetic amphiphiles. 

It should be noted the result mentioned earlier holds only for the van der Waals (or Volmer) isotherm. Instead, if the Frumkin (or Langmuir) isotherm is used, the value of a obtained from the surface tension fits is about 33% greater than that obtained from molecular size [44], A possible explanation of this difference could be the fact that the Frumkin (and Langmuir) isotherm is statistically derived for localized adsorption and is more appropriate to describe adsorption at solid interfaces. In contrast, the van der Waals (and Volmer) isotherm is derived for nonlocalized adsorption, and they provide a more adequate theoretical desaiption of the surfactant adsorption at liqnid-flnid interfaces. This conclnsion refers also to the calculation of the surface (Gibbs) elasticity by means of the two types of isotherms [44]. 

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