Other Isotherm Equations

From nearly a century ago, from the earlier works of Langmuir and Freundlich, many different models and isotherm eqnations have been proposed and compared (Parsons 1964 Delahay 1965 Kinnibnrgh 1986 Toth 1995 Aranovich and Donohue 1996 Rocha et al. 1997 Donohue and Aranovich 1998 Marczewski 2002), especially in food science (Van den Berg and Bruin 1981 Iglesias and Chirife 1982). 

FIGURE 4.28 Plots of the Langmuir-Freundlich isotherm for different values of the exponent n (a) linear activity scale and (b) logarithmic activity scale. 

Localized adsorption, identical sites, no lateral interactions Localized adsorption, identical sites, lateral interactions with random distribution Localized adsorption, identical sites, no lateral interactions, multilayer, liquid-like from second layer up Localized adsorption, identical sites, no lateral interactions, multilayer not liquid-like from second layer up Mobile adsorption, no lateral interactions, except for excluded volume 

Mobile adsorption, with lateral interactions and excluded volume 

Langmuir isotherm with quasi-Gaussian surface heterogeneity (energy distribution) 

---

Other isotherm equations begin as an alternative approach to the developed equation of state for a two-dimensional ideal gas. As mentioned earlier, the ideal equation of state is found to be... 

In the past, much attention was given to the study of dye and iodine adsorption by active carbons (Bmnauer, 1945 Orr and Dalla Valle, 1959). Many studies have been made with dye molecules of well-known size, shape and chemical properties, but the results have not been easy to interpret (Giles et al., 1970 McKay, 1982, 1984). In a systematic study of iodine adsorption (from aqueous solution) on a carbon black and four activated carbons (Femandez-Colinas etal., 1989b), it was found that the iodine isotherms could be analysed by the as-method. In this way it was possible to assess values of the available volume in pores of effective width of 0.5-1.5 nm. The adsorption of iodine was also featured in a recent study by Ziolkowska and Garbacz (1997), who applied the Langmuir, Freundlich and other isotherm equations. 

Further discussion of [1.7.71 and other isotherm equations with their derivations can be found. A variety of other empirical Isotherms with their "derivations" on the basis of 11.7.11 can be found in the books by Rudzinski and Everett and by Jaroniec and Madey, mentioned in sec. 1.9c. 

A problem often encountered in nonlinear optimization is the necessity to provide suitable initial guesses. Therefore, it should be tested if different initial guesses lead to drastically different sets of isotherm parameters. This is connected to the sensitivity problem, which is pronounced in the case of multi-component systems where several parameters need to be fitted at once. Substantial initial guesses are often difficult to find and the sensitivity is often low, which demands much experimental data or leads to the selection of other isotherm equations. 

Like other isotherm equations developed in this section, the micropore size is assumed to take a certain form and then the isotherm equation derived from such assumed pore size distribution is fitted with equilibrium data to obtain the relevant structural parameters. This method must be used with some precaution because if the true micropore size distribution does not conform to the assumed form the derived micropore size distribution can be erroneous. 

Second, the function i/ (0) is required to calculate the function P) numerically if the isotherm equation eannot be expressed in terms of 0 = /(P). In Fig. 6 (bottom, left), the functions y4 (0) calculated analytically or numerically by Eq. (121) are represented. In Fig. 6 (bottom, right) the functions Af.(P ) calculated similar to the function i/ (P) are shown. In particular, to every value of P = P/Pm calculated by Eqs. (115) and (120) are attributed the values of Af.( >), so we obtain the functions Af.(P ). These two types of function in the bottom of Fig. 6 characterize thermodynamically the adsorption process and thus seem to complete the imiform interpretation of the mL and other isotherm equations. The thermodynamic consistency is best reflected by the functions A i ) A XPr.m) because both functions have finite values at 0 = 1 or at P = 1. 

It is also easy to see that integration of Eq. (63) with infinite upper limit, i.e., /jjjj = 00, leads to the original L equation (56), which is thermodynamically inconsistent as is shown above. This statement can be generalized and stated in oflier words the classic isotherm equations (see Table 1) and others suppose the fulfilment of initial conditions, but they apply in Eq. (54) or (55) to an infinite upper limit of integration (Pjjj = oo), which contradicts the initial conditions mentioned above. The thermodynamic inconsistence of the original L and other isotherm equations in Table 1 can also be proved with Eq. (43), namely, for the L equation ... 

The preceding derivation, being based on a definite mechanical picture, is easy to follow intuitively kinetic derivations of an equilibrium relationship suffer from a common disadvantage, namely, that they usually assume more than is necessary. It is quite possible to obtain the Langmuir equation (as well as other adsorption isotherm equations) from examination of the statistical thermodynamics of the two states involved. 

We have seen how to calculate q for the isochoric and isobaric processes. We indicated in Chapter 1 that q = 0 for an adiabatic process (by definition). For an isothermal process, the calculation of q requires the application of other thermodynamic equations. For example, q can be obtained from equation (2.3) if AC and w can be calculated. The result is... 

In the mechanism illustrated by scheme B, significant inhibition is only realized after equilibrium is achieved. Hence the value of vs (in Equations 6.1 and 6.2) would not be expected to vary with inhibitor concentration, and should in fact be similar to the initial velocity value in the absence of inhibitor (i.e., v, = v0, where v0 is the steady state velocity in the absence of inhibitor). This invariance of v, with inhibitor concentration is a distinguishing feature of the mechanism summarized in scheme B (Morrison, 1982). The value of vs, on the other hand, should vary with inhibitor concentration according to a standard isotherm equation (Figure 6.5). Thus the IC50 (which is equivalent to Kfv) of a slow binding inhibitor that conforms to the mechanism of scheme B can be determined from a plot of vjv0 as a function of [/]. 

On the other hand, the data for some organic compounds are often fitted into the Frumkin isotherm equation in a modified form... 

The adsorption data is often fitted to an adsorption isotherm equation. Two of the most widely used are the Langmuir and the Freundlich equations. These are useful for summarizing adsorption data and for comparison purposes. They may enable limited predictions of adsorption behaviour under conditions other than those of the actual experiment to be made, but they provide no information about the mechanism of adsorption nor the speciation of the surface complexes. More information is available from the various surface complexation models that have been developed in recent years. These models represent adsorption in terms of interaction of the adsorbate with the surface OH groups of the adsorbent oxide (see Chap. 10) and can describe the location of the adsorbed species in the electrical double layer. 

The assumption of linear chromatography fails in most preparative applications. At high concentrations, the molecules of the various components of the feed and the mobile phase compete for the adsorption on an adsorbent surface with finite capacity. The problem of relating the stationary phase concentration of a component to the mobile phase concentration of the entire component in mobile phase is complex. In most cases, however, it suffices to take in consideration only a few other species to calculate the concentration of one of the components in the stationary phase at equilibrium. In order to model nonlinear chromatography, one needs physically realistic model isotherm equations for the adsorption from dilute solutions. 

Probably the reason why the (1 — 0) concept has received such widespread credence is that Langmuir was able to derive his famous adsorption isotherm on the basis of this concept. Since the Langmuir isotherm equation has been experimentally verified in many cases, it was felt that the (1 — 0) concept must be essentially correct. This again is fallacious reasoning, since in the derivation two other assumptions are necessarily made which are not in accord with recent experiments. These are ( ) the rate of evaporation is proportional to 0 and (2) one can treat the experimental data as if the surface were homogeneous. Because of this situation, it is desirable that someone derive the Langmuir isotherm equation on more realistic assumptions. (See ref. 10a.)... 

The theoretical lines in Figure 2 are calculated assuming constant values of D0 with the derivative d In p/d In c calculated from the best fitting theoretical equilibrium isotherm (Equation 8). The theoretical lines give an adequate representation of the experimental data suggesting that the concentration dependence of the diffusivity is caused by the nonlinearity of the relationship between sorbate activity and concentration as defined by the equilibrium isotherm. The diffusivity data for other hydrocarbons showed similar trends, and in no case was there evidence of a concentration-dependent mobility. Similar observations have been reported by Barrer and Davies for diffusion in H-chabazite (7). 

The virial isotherm equation, which can represent experimental isotherm contours well, gives Henry s law at low pressures and provides a basis for obtaining the fundamental constants of sorption equilibria. A further step is to employ statistical and quantum mechanical procedures to calculate equilibrium constants and standard energies and entropies for comparison with those measured. In this direction moderate success has already been achieved in other systems, such as the gas hydrates 25, 26) and several gas-zeolite systems 14, 17, 18, 27). In the present work AS6 for krypton has been interpreted in terms of statistical thermodynamic models. 

Equation 3 was verified experimentally (3) over wide ranges of temperature and equilibrium pressure for the adsorption of various vapors on active carbons with different parameters for the microporous structure. For adsorption on zeolites, this equation fitted the experimental results well only in the range of high values of 0 (4, 5, 6, 7). Among other equations proposed for the characteristic curve (4, 5, 8, 9, 10) we chose to use the Cohen (4) and Kisarov s (10) equation, which starts from the following adsorption isotherm equation ... 

The computed values of WQ lie near those calculated from crystallographic data for synthetic zeolites. The value Wo = 0.195 cm3/gram estimated here for zeolite 5A also compares favorably with the mean value Wo = 0.20 cm3/gram previously obtained 21) for the adsorption of various adsorbates on the same adsorbent, without reference to any isotherm equation. For the synthetic zeolite, the preparation method may lead to variations in adsorption properties, and this may explain the difference between values of W0 shown in the Table I. Finally, for Cecalite, where no theoretical value is known, the values obtained here for W0 with two different adsorbates are consistent with each other. Thus, the proposed method gives realistic values for W0. 

Because the Langmuir isotherm is not an adequate description of most systems, Equation (2.9) is not used much for area measurement. A number of other isotherm formulations utilize adsorption in surface area measurements, however (cf. Young Crowell, 1962, for example). The best known and most widely used is the BET (Brunauer, Emmett Teller, 1938) theory, a generalization of the Langmuir model to multilayer absorption. Assuming that for the second and succeeding molecular... 

One more isotherm equation that could be helpful for the determination of the micropore volume is the osmotic isotherm of adsorption. Within the framework of the osmotic theory of adsorption, the adsorption process in a microporous adsorbent is regarded as the osmotic equilibrium between two solutions (vacancy plus molecules) of different concentrations. One of these solutions is generated in the micropores, and the other in the gas phase, and the function of the solvent is carried out by the vacancies that is, by vacuum [26], Subsequently, if we suppose that adsorption in a micropore system could be described as an osmotic process, where vacuum, that is, the vacancies are the solvent, and the adsorbed molecules the solute, it is possible then, by applying the methods of thermodynamics to the above described model, to obtain the so-called osmotic isotherm adsorption equation [55] ... 

Equation 5.21a is known in literature [10] as the Sips or Bradleys isotherm equation. This isotherm equation describes fairly well the experimental data of adsorption in zeolites and other microporous materials [26], The linear form of the osmotic equation can be expressed as follows... 

The Davies and Jones derivation makes some fundamental assumptions concerning the surface concentrations of the lattice ions and the BCF theory is only applicable to very small supersaturations. Thus, both theories have limitations which affect the interpretation of the results of growth experiments. Nielsen [27] has attempted to examine in detail how the parabolic dependence can be explained in terms of the density of kinks on a growth spiral and the adsorption and integration of lattice ions. One of the factors, a = S — 1, comes from the density of kinks on the spiral [eqns. (4) and (68)] and the other factor is proportional to the net flux per kink of ions from the solution into the lattice. Nielsen found it necessary to assume that the adsorption of equivalent amounts of constituent ions occurred and that the surface adsorption layer is in equilibrium with the solution. Rather than eqn. (145), Nielsen expresses the concentration in the adsorption layer in the form of a simple adsorption isotherm equation... 

Although the simple Langmuir equation is more applicable to some forms of chemisorption, the underlying theory is of great historical importance and has provided a starting point for the development of the BET treatment and of other more refined physisorption isotherm equations. It is therefore appropriate to consider briefly die mechanism of gas adsorption originally proposed by Langmuir (1916, 1918). 

Equation (4.47) and obtain an isotherm equation in which the distribution function, (B) was expressed in an analytical form (Huber et al., 1978 Bansal et al., 1988). In principle, f(fl) provides an elegant basis for relating the micropore size distribution to the adsorption data. However, it must be kept in mind that the validity of the approach rests on the assumption that the DR equation is applicable to each pore group and that there are no other complicating factors such as differences in surface heterogeneity. 

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) ... 

To complete the Stem theory, a model is required to determine the specifically adsorbed charge, adsorption isotherm equation. A number of such Isotherms have been derived for uncharged... 

The methods depend on the theoretical treatment which is used. A majority of them are based on the Generalised Adsorption Isotherm (GAI) also called the Integral Adsorption Equation (LAE). The more recent approaches use the Monte Carlo simulations or the density functional theory to calculate the local adsorption isotherm. The analytical form of the pore size distribution function (PSD) is not a priori assumed. It is determined using the regularization method [1,2,3]. Older methods use the Dubinin-Radushkevich or the Dubinin-Astakhov models as kernel with a gaussian or a gamma-type function for the pore size distribution. In some cases, the generalised adsorption equation can be solved analytically and the parameters of the PSD appear directly in the isotherm equation [4,5,6]. Other methods which do not rely on the GAI concept are sometimes used the MP and the Horvath-Kawazoe (H-K) methods are the most well known [7,8]. 

---