Adsorption isotherms Fowler isotherm

One important direetion of study has been to use empirieal adsorption data, together with the preassumed model for loeal adsorption, and attempt to extraet information about the form of x(e) [13,14]. The ehoiee of the model for loeal adsorption, whieh is an important input here, has been eustomarily treated quite easually, assuming that it has rather limited influenee on the form and properties of the evaluated EADFs. Usually, one of so many existing equations developed for adsorption on uniform surfaees is used as the loeal adsorption isotherm. The most often used forms of 0 p, T,e) are the Langmuir [6] and the Fowler-Guggenheim [15] equations for loealized adsorption. Ross and Olivier [4] extensively used the equation for mobile adsorption, whieh results from the two-dimensional version of the van der Waals theory of fluids. The most radieal solution has been... 

The adsorption isotherm was modeled in order to deduce a numerical value for A. Following the model of Frumkin and Fowler reported in [21], is given by the following set of equations ... 

F. Fowler. A Statistical Derivation of Langmuir s Adsorption Isotherm. Proc. Camb. Phil. Soc., 31 260-280,1935. 

This equation is sometimes called the Frumkin-Fowler-Guggenheim (FFG) isotherm [374— 376], For j3 = nEP/RT < 4 lateral interactions cause a steeper increase of the adsorption isotherm in the intermediate pressure range. Characteristic of all Langmuir isotherms is a saturation at high partial pressures P/Po —> 1. 

Figure 9.7 Left Frumkin-Fowler-Guggenheim (FFG) adsorption isotherms (coverage 0 versus the pressure in units of Kj 1). The curves were calculated using Eq. (9.35) with ft 0,2,4,6. For ft 6 the physically correct adsorption curve is plotted as a continuous curve while the one calculated with Eq. (9.35) is plotted as a dotted curve. Right Adsorption isotherms for krypton adsorbing to the (0001) plane of graphite at two different temperatures. The dotted curves were fitted using Eq. (9.35) with ft = 4.5. Experimental results were taken from Ref. [377],...

Fig. 6. A Frumkin-Fowler s adsorption isotherm for uniform surfaces. B, C Adsorption isotherm for heterogeneous surfaces with condensation [after Ref. 89]...

Statistic mechanics were used by Kiefer and Wilson104 to calculate adsorption isotherms of ionic surfactants on charged solid-water interface. The effect of coulombic repulsions between the ionic heads of the surfactant species are considered, as well as the van der Waals attractions of their hydrocarbon tails. Using the method of Fowler and Guggenheim93 they obtained the equation for an adsorption isotherm ... 

Langmuir-Type and Fowler-Guggenheim-Type Adsorption Isotherm Equations... 

Figure 3. Adsorption isotherms calculated as described in text forC = 4 and 5 ----------------Fowler-Guggenheim isotherm...

Fowler-Frumkin isotherm - Frumkin adsorption isotherm... 

In geological surfaces, the solid-gas and solid-liquid interfaces are important, so the correct thermodynamic adsorption equation (Gibbs isotherm) cannot be used. Instead, other adsorption equations are applied, some of them containing thermodynamic approaches, and others being empirical or semiempirical. One of the most widespread isotherms is the Langmuir equation, which was derived for the adsorption of gas molecules on planar surfaces (Langmuir 1918). It has four basic assumptions for adsorption (Fowler 1935) ... 

Explicit expressions for i(f) and i s/Me" r Quire the knowledge of the adsorption isotherm (cf. Section 3.3). For example, 2D Meads phase formation according to the Frumkin-Fowler adsorption isotherm gives by combining eqs. (3.18) and (3.20) ... 

The classical theory of the Gibbs adsorption isotherm is based on the use of an equation of state for the adsorbed phase hence it assumes that this adsorbed phase is a mobile fluid layer covering the adsorbent surface. By contrast, in the statistical thermod)mamic theory of adsorption, developed mainly by Hill [15] and by Fowler and Guggenheim [12], the adsorbed molecules are supposed to be localized and are represented in terms of simplified physical models for which the appropriate partition function may be derived. The classical thermodynamic fimctions are then derived from these partition fimctions, using the usual relationships of statistical thermodynamics. 

Besides the heterogeneity of the adsorbent surface, the second major reason for the adsorption of a compound to deviate from Langmuir isotherm behavior is that the adsorbed molecules interact. In this category, we find the Fowler isotherm, the anti-Langmuirian isotherm, and several S-shaped isotherm models, including the quadratic isotherm, the extended BET isotherm models, and the Moreau model. 

Figure 3.22 Illustration of the Fowler isotherm. x = 0,l, and 2 fe = 1.61 mL/mg qs = 300 mg/mL. (a) Fowler adsorption isotherm, (b) First derivative of the Fowler isotherm.

Figure 3.23 Use of the Fowler isotherm to account for experimental data. The adsorption isotherms of 2-phenylethanol and 3-phenylpropanol between spherical ODS silica (Vydac, Hesperia, CA), 10 fim, and a methanol/water 50 50 solution. Column dimensions, 2.1 x 250 mm. Data from J. Zhu, AM. Katti and G. Guiochon, J. Ghromatogr., 552 (1991) 71.

Figure 4.14 Comparison between the experimental adsorption data of 2-phenylethanol (a, c) and 3-phenylpropanol (b, d) and the best competitive Fowler isotherms derived from the best single-component parameters. For mixtures, the mobile phase concentration is expressed via the relative composition of PE and PR All concentrations in mg/ml. Symbols data for single component, o data for 3 1 mixtures (3/1 PE and PP), x data for 1 1 mixtures (1/1 PE and PP), + and data for 1 3 mixtures (1/3 PE and PP),. The RSS are the residual sum of squares calculated for each set of experiments performed with a constant relation of the mobile phase concentrations of the phenylalcohols. ODS silica from Vydac and water-methanol mixture (50 50) at room temperature. I. Quinones, G. Guiochon, Langmuir, 12 (1996) 5433 (Figs 1 and 2) and J. Zhu, A. Katti and G. Guiochon,. Chromatogr. 552 (1991) (Figs 7 and 8)71.

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 ... 

In the special case w = 0, this problem was solved kinetically by Langmuir (26) and statistically by Fowler (27), the adsorption isotherm being just the well-known Langmuir isotherm. 

Using the van t Hoff equation (2.2-11), we can obtain the following heat of adsorption for the Fowler-Guggenheim adsorption isotherm ... 

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 ... 

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. 

FIGURE 14.5 Frumkin-Fowler-Guggenheim adsorption isotherms for different interactions between solvent and adsorbate molecnles. 

At first the BET equation was derived from the kinetic considerations analogous to those proposed by Langmuir while deriving the monomolecular adsorption isotherm. First, statistical thermodynamic derivation was carried out by Cassie [123]. Lately, a slightly modified derivation has been proposed by HiU [124-126], Fowler and Guggenheim [127]. 

FigOFe 7 Distribution of adsorption energies for cyclohexane adsorbed on wide-pore silica gel at 126.3 °C. The solid line denotes the function F (U) calculated by using the Fowler-Guggenheim local adsorption isotherm the dashed line denotes F (U) for the Langmuir isotherm... 

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