Work of adsorption

As discussed in Chapter III, the progression in adsoiptivities along a homologous series can be understood in terms of a constant increment of work of adsorption with each additional CH2 group. This is seen in self-assembling monolayers discussed in Section XI-IB. The film pressure r may be calculated from the adsorption isotherm by means of Eq. XI-7 as modified for adsorption from dilute solution ... 

Parameter k of Equation (4.10) is an expression of the breadth of the Gaussian distribution of the cumulative micropore volume IF over the normalized work of adsorption sfifi, and is therefore determined by the pore structure. Thus B also (cf. Equation (4.13)) is characteristic of the pore structure of the adsorbent, and has accordingly been termed the structural constant of the adsorbent. ... 

Jarvis, N.L. and Zisman, W.A. "Surface Activity of Fluorinated Organic Compounds at Organic-Liquid/Ari Interfaces Part III. Equation of State of Adsorbed Monolayers and Work of Adsorption of a Fluorocarbon Group," Naval Research Labs Report 5401, Surface Chemistry Branch, Chemistry Division, November 17, 1959. 

Equation (6.153) allows us to evaluate Z, but only if the values of AGf and AGj are known. What are these energy terms They can be considered to be built by three contributions (1) a chemical term related to the work of adsorption, represented by AGc, (2) an electrical or field term, and (3) a lateral interaction term. 

According to experimental data, and assuming that PSD is Gaussian, Dubinin and Radushkevich obtained an equation, which relates the degree of micropore filling (9) with the differential molar work of adsorption ... 

Thus the work of adsorption increases by a constant amount for each CH2 added to the hydrocarbon chain of the molecules. This must mean that each CH2 group is situated in the same relation to the surface as every other such group in the chain, and this can only be the case if the chains lie parallel to the surface. Hence Langmuir concluded that Traube s rule—for this is the name given to the effect of increasing length of chain... 

Considering other measurements by Traube, assuming the thickness of the surface layer to be 6 A., the work of adsorption of a large number of series was found to be approximately expressible by... 

Approximate work of adsorption at an air-water interface, for various organic groupings... 

In most actual cases, other forces besides simple electrostatic attraction enter into the work of adsorption of the ions, so that this relation is only approximate. 

Let us now consider the basic properties of the characteristic Equation 5, which is a Weibull distribution equation 13). Irrespective of the distribution order, n, the curves expressed by it have 2 points in common a point corresponding to maximal adsorption a = Uo 6 = 1) and a characteristic point at a = Oo/e, where e is the natural logarithm base ( = 0.368). If the differential molar work of adsorption for the char-... 

Traube rule, more often referred to as the Traube rule. In agreement with eq. (II. 12), the Ducleaux-Traube rule corresponds to a linear relationship between the surfactant chain length and the p0 - pj,s) value. The latter may be viewed as the work of adsorption performed under standard conditions. Indeed, let us assume that the work of adsorption, p0 - p(0s), is a linear function of the number of carbon atoms in the surfactant hydrocarbon chain, i.e ... 

Therefore, we have established once again that the linear dependence of the adsorption on concentration corresponds to the initial linear region in the surface tension - concentration dependence (see Fig. (II-7)). Since b is constant within the homologous series, it is the value of constant A that determines the steepness of the adsorption increase with increasing concentration. For this reason constant A is referred to as the adsorption activity. By comparing eqs. (II. 19) and (11.20) one establishes that A is related to the work of adsorption, p0 - p(0s), as... 

When water soluble surfactants adsorb at the interface between a liquid hydrocarbon and water, the trends in adsorption are very similar to those established for the air - solution interface (see Chapter II). The Traube rule remains valid, and the dependence of the surface tension on concentration can be described by Szyszkowski s equation (11.18). Moreover, at identical surfactant concentrations, the absolute values by which the surface tension is lowered at water - air and water - hydrocarbon interfaces are not that different. The surface tension isotherms for these interfaces are parallel to each other (Fig. III-6). That is due to the fact that the work of adsorption per CH2 group, given by eq. (II. 14), is determined mostly by the change in the standard part of the chemical potential of the solution bulk, q0. Similar to the air-water interface, the energy of surfactant adsorption from an aqueous solution at an... 

In the simple GCSG model already described, the charge densities are assumed uniformly smeared over each plane rather than in the form of discrete ions. Levine and co-workers [8] modified the GCSG model by introducing the discreteness of charge effect to account for various phenomena, mostly relating to adsorption at the mercury/water and silver iodide/water interfaces. Their theory shows that the electrostatic work of adsorption at the IHP is not zet /p but... 

Thus total electrostatic work of adsorption per 1 cm at full coverage W =icpHC -Go)+C,q,x (83)... 

A number of attempts have been made to estimate the HLB from the comparison of the work of surfactant-molecule transfer not between the adjacent bulk phases, but from bulk phases into the surface layer, that is, to use the adsorption work as basis for such estimates. The adsorption work from the aqueous phase is w(z) — -u, while the adsorption work from the oil phase is given by the relation w(z) — w. Here, z is the coordinate of the hydrophilic-lipophilic center (HLC) of a surfactant molecule, which corresponds usually to the minimum of w(z) (261). This minimum work of transfer of a surfactant molecule corresponds to the maximum at the plot of the concentration distribution in the surface layer. Usually the location of the minimum of (i.e., HLC) is displaced from the geometric interface towards the aqueous phase. It is seen from the above relations that the adsorption work, unlike the work of transfer from bulk phases, is not a definite and unambiguous characteristic. Usually the local value of the work w z) is substituted by the mean (integral) work -uP, related to the entire adsorption layer. Due to the inhomogeneity of the interface region, the adsorption work cannot be calculated additively from the work of adsorption for particular groups of the molecule, because these work values depend on the location of... 

Some adsorptive kinetic models were used in different works of adsorption of organic compounds in organophilic clays, including models of Lagergren, also called models of pseudo-first order and pseudo-second order, represented by Equations 13 and 14 (Akgay, 2004, 2005 Cavalcanti et al., 2008 Yilmaz Yapar, 2004). 

The fundamental quantity in the micropore filling is the differential molar work of adsorption, given by ... 

The characteristic energy is a measure of the adsorption strength between adsorbate and adsorbent. The function f is regarded as the distribution function of filling of micropores 0 over the differential molar work of adsorption, and n is the parameter associated with the distribution function. 

Thus, the Dubinin and Radushkevich equation states the distribution of the adsorption space W according to the differential molar work of adsorption. A t)q)ical plot of the adsorption potential versus the reduced pressure is shown in Figure 4.2-1 for T = 77, 273 and 473 K. For low reduced pressure, the adsorption potential is high, while it is low for high reduced pressure. The latter means that less molar work is required for adsorption via micropore filling when the gas is approaching the vapour pressure. 

The DA equation (4.2-5) is obtained by assuming the temperature invariance of the adsorption potential at constant loading and a choice of the Weibull s distribution to describe the filling of micropore over the differential molar work of adsorption. It can be shown to be a special case of an isotherm equation derived from the statistical mechanical principles when the loading is appreciable (Chen and Yang, 1994). They derived the following isotherm... 

With any form of the distribution and a particular choice of the local isotherm, eq. (4.4-2) can be in general integrated numerically to yield the overall adsorption isotherm equation. The local fractional loading 0 can take the form of either the DR or DA equation. As discussed earlier the DA equation with n = 3 describes well solids having narrow micropore size distribution, and hence this makes this equation a better candidate for a local isotherm equation rather than the DR equation. However, since the selection of a distribution function is arbitrary, this does not strictly enforce the local isotherm to reflect the intrinsic local isotherm for a specific characteristic energy. Moreover, the DR or DA itself stems from a Weibull s distribution function of filling of micropore over the differential molar work of adsorption. Thus, the choice of the local isotherm is empirical, and in this sense the procedure of eq. (4.4-2) is completely empirical. The overall result, however, provides a useful means to describe the equilibrium data in microporous solids. 

The last section shows the analysis of an inhomogeneous solid when the local isotherm taking the form of the DR equation and the distribution in terms of 1/Eq taking the Gaussian distribution (eq. 4.4-5). In this section, the local isotherm is taking the form of DA (n = 3) as discussed earlier it describes narrow pore solids better than the DR equation, and hence it should be a better choice for the local isotherm. But this DA, like the DR equation, does not truly represent the intrinsic local isotherm for the reason that the DA comes from the Weibull s distribution of micropore filling over the differential molar work of adsorption. 

This has led to the development of the theory of volume filling of micropores (TVFM theory). The theory provides a satisfactory description of the shapes of adsorption isotherms where the adsorption takes place largely in micropores. It is based on the assumption that the characteristic adsorption equation expressing the degree of filling of micropores (i.e., the volume filling of micropores) is a function of the differential molar work of adsorption. The adsorption process in this case involves dispersion forces as the main component in the adsorption-adsorbent interactions. 

The theory has been developed mainly by Dubinin and coworkers over a period of time to describe adsorption of gases and vapors by microporus solids in general and active carbons in particular. Dubinin defined the molar work of adsorption as the change in the Gibbs free energy rather than as adsorption potential (Polanyi). Accordingly, the molar work of adsorption A is given by... 

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