Adsorption isotherms Volmer isotherm

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). 

Assuming the Butler-Volmer equation and Langmuir adsorption isotherm hold for this reaction, the charge-transfer rate in the presence of CO adsorption is... 

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 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. 

Each surfactant adsorption isotherm (that of Langmuir, Volmer, Frumkin, etc.), and the related expressions for the surface tension and surface chemical potential, can be derived from an expression for the surface free energy, F, which corresponds to a given physical model. This derivation helps us obtain (or identify) the self-consistent system of equations, referring to a given model, which is to be applied to interpret a set of experimental data. Combination of equations corresponding to different models (say, Langmuir adsorption isotherm with Frumkin surface tension isotherm) is incorrect and must be avoided. 

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. 

The parameters F, and K are the same as in Tables 5.2 through 5.4. Setting Q = 0 (assuming equilibrium surface-subsurface), from each expression in Table 5.5 we deduce the respective equilibrium adsorption isotherm in Table 5.2. In addition, for P = 0 the expressions for Q related to the Frumkin and van der Waals model reduce, respectively, to the expressions for Q in the Langmuir and Volmer models. For Fj F both the Frumkin and Langmuir expressions in Table 5.5 reduce to the Henry expression. 

The Langmuir adsorption isotherm was developed by Irving Langmuir in 1916 from kinetic considerations to describe the dependence of the surface fractional coverage of an adsorbed gas on the pressure of the same gas above the adsorbent surface at a constant temperature. The Langmuir isotherm expression was re-derived thermodynamically by Volmer and statistically mechanically by Fowler. In his original treatment, Langmuir made several assumptions for his model ... 

These adsorption isotherms are known as Henry (1801), von Szyszkowski (1908), Langmuir (1916), Frumkin (1925), Volmer (1925) or HOckel-Cassel-isotherm (Huckel 1932, Cassel 1944), respectively. The constants in these isotherms refer to kinetic models of adsorption/desorption, interactions between adsorbed molecules and/or to the minimum area of adsorbed species. 

Langmuir s basic treatment for an adsorption isotherm was set out in his paper "The condensation and evaporation of gas molecules" (1917). Langmuir (1917, 1918) and Volmer (1925) derived a similar formula for the interpretation of the kinetics of adsorption/desorption. In these papers the process of adsorption was proposed as a dynamic situation for the first time. The basic relationship describing the adsorption as a kinetic mechanism results from the balance of the adsorption flux and desorption flux... 

Another adsorption isotherm was derived by Volmer (1925). It assumes a self area of the adsorbed molecules... 

For an ideal monolayer and nj= 1, this expression is reduced to the adsorption isotherm (2.157) derived using Pethica s equation. It can be concluded therefore, that the two approaches lead to similar results. At the same time, Butler s equation (2.7) always leads to a logarithmic form of the equation of state for mixed monolayers, which often disagrees with the experimental results. For these systems, Volmer s or van der Waals equations of state are more appropriate [58, 98]. Therefore the method based on Pethica s equation is advantageous, enabling one to apply semi-empirical model equations of state for mixed monolayers. 

As aggregation of the insoluble component occurs only when its surface concentration is sufficiently high, the description of the two components based on Volmer s equation seems to be more appropriate than that based on the Szyszkowski-Langmuir equation. If a first-order phase transition does not occur in the monolayer, i.e. no aggregates are formed, then the simultaneous solution of Volmer s equation (2.159) for the components 1 and 2, and Pethica s equation (2.152) yields the adsorption isotherm for the soluble component 2 (see [156])... 

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-... 

We have addressed the various adsorption isotherm equations derived from the Gibbs fundamental equation. Those equations (Volmer, Fowler-Guggenheim and Hill de Boer) are for monolayer coverage situation. The Gibbs equation, however, can be used to derive equations which are applicable in multilayer adsorption as well. Here we show such application to derive the Harkins-Jura equation for multilayer adsorption. Analogous to monolayer films on liquids, Harkins and Jura (1943) proposed the following equation of state ... 

In the last chapter, we discussed the description of pure component adsorption equilibrium from the fundamental point of view, for example Langmuir isotherm equation derived from the kinetic approach, and Volmer equation from the Gibbs thermodynamic equation. Practical solids, due to their complex pore and surface structure, rarely conform to the fundamental description, that is very often than not fundamental adsorption isotherm equations such as the classical Langmuir equation do not describe the data well because the basic assumptions made in the Langmuir theory are not readily satisfied. To this end, many semi-empirical approaches have been proposed and the resulting adsorption equations are used with success in describing equilibrium data. This chapter will particularly deal with these approaches. We first present a number of commonly used empirical equations, and will discuss some of these equations in more detail in Chapter 6. 

The local adsorption isotherm equations of the form Langmuir, Volmer, Fowler-Guggenheim and Hill-de Boer have been popularly used in the literature and are shown in the following Table 6.3-1. The first column shows the local adsorption equation in the case of patchwise topography, and the second column shows the corresponding equations in the case of random topography. Other form of the local isotherm can also be used, such as the Nitta equation presented in Chapter 2 allowing for the multisite adsorption. 

It might be noticed that the Volmer and Heyrovsky reactions are electrochemical while the Tafel reaction is chemical, without an exchange of electrons. Assuming a Langmuir adsorption isotherm for H, the rates, Vi, are written as... 

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. 

Expressions for the rates of these mechanistic reactions can be derived by assuming a reasonable adsorption isotherm, such as that proposed by Langmuir. For a PEMFC with Pt as the catalyst, it is generally agreed that the HOR process involves the Tafel and Volmer reactions, with the Tafel reaction being the rate-determining step. In this case, the rate of the reaction can be expressed in the form of the well-known Buder-Volmer (B-V) equation [8] ... 

For the derivation of Volmer s and Fowler Guggenheim isotherms from the gas phase equations of state and Gibbs adsorption isotherms, we will refer the reader to Ruthven (1984). 

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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