Toth isotherm

Several isotherms combine aspects of both the Langmuir and Freudhch equations. One that has been shown to be effective in describing data mathematically for heterogeneous adsorbents is the Toth isotherm [Acta Chim. Acad. Sci. Hung., 69, 311 (1971)]... 

As many other models, even the Toth isotherm was originally derived for the study of gas-solid equilibria. However, it has been extended to the description of liquid-solid system. This isotherm assumes a continuous and wide adsorption energy distribution. The equation of the Toth isotherm is... 

FIGURE 10.13 Experimental adsorption data (symbols) and the best Toth isotherm (solid line) for porcine insulin, human insulin Lispro. (Reproduced from Liu, X. et al., Biotechnol. Prog., 18, 796, 2002. With permission from John Wiley Sons.)... 

One drawback of Langmuir-type adsorption isotherms is the conjunction between the initial slope of the isotherm and its curvature. This can be overcome by the Toth isotherm model (Toth, 1971) (Eq. 2.42) ... 

Actual surfaces are heterogeneous and the adsorption data on real adsorbents are usually not well accounted for by one of the two models discussed in the section above. The distribution of the adsorption energy (AED) on a homogeneous surface is a Dirac distribution. On an actual surface, the adsorption energy distribution has a finite width. It can be unimodal or multi-modal. The isotherm models discussed in this section correspond to such surfaces. Each of them can be related to a different AED. In all cases, however, there are no adsorbate-adsorbate interactions. In this group, the most useful models are the bi-Langmuir and multi-Langmuir isotherms, the Toth isotherm, and the Freundlich isotherm. 

The competitive equivalent to the single-component Toth isotherm model (see Chapter 3, Section 3.2.22) is written... 

Zhou et al. measured by FA the single-component isotherms of two related peptides, bradykinin and kallidin (MW 1060 and 1188, respectively) on a Zorbax SB Ci8 microbore column (Agilent Technologies), using a 20% aqueous solution of ACN (with 0.1% TFA) as the mobile phase [81]. The isotherm data were fitted to the Langmuir, the bi-Langmuir, the Langmuir-Frermdlich and the Toth isotherm... 

The Toth isotherm allows one to manipulate both the numerator and denominator on the right side of the preceding equation... 

Eqs.(9.2-47) are in the form of nonlinear partial differential equation and are solved by a combination of the orthogonal collocation method and the Runge-Kutta method. Appendix 9.2 shows the details of the collocation analysis, and a computer code ADSORBIA.M is provided to solve the cases of Langmuir and Toth isotherms. 

For Toth isotherm, the relevant functions for this model are ... 

Vasanth Kumar, K. Monteiro de Castro, M. Martinez-Escandell, M Molina-Sabio, M Rodriguez-Reinoso, F. (2011). A site energy distribution fimction from Toth isotherm for adsorption of gases on heterogeneous surfaces. Physical Chemistry Chemical Physics, 13,5753-5759. 

Biosorption capacity by weight at time t [mgg ] R = Universal gas constant (8.314 Jmol K ) r,a = Radke-Prausnitz isotherm model constants T =Temperature [K] t =Toth isotherm constants. 

From the derived Toth isotherms, the isosteric heat of adsorption as a fiinction of coverage can be calculated by the following equation [11] ... 

The relationship to the Freundlich isotherms is important for two reasons. First the question as to whether the % theory can predict isotherms such as the Freundlich (of which = 1 is a special case), Dubinin-Astakov, Dubinin-Radushkevich and Toth isotherms All but the Toth isotherm will be referred to as the Dubinin-Polanyi (DP) isotherm. Second, the reason for the observation that in most cases P appears to approach 0 as approaches 0. Even though there are cases where P approaches a finite value, thus disproving the universal application of Henry s law , this is not convincing without an explanation as to why it is observed in many cases. 

The comparison to isotherms, when there is a distribution, comes back to the Henry s law question. Why is it that sometimes one observes the Freundlich isotherm and thus at least the appearance that the pressure and adsorbate amount simultaneously approach zero. As demonstrated in Ch ter 4 a log-normal distribution in yields the Dubinin-Polanyi (DP) set of isotherms of which the Freundlich isotherm is a subset. The Toth isotherm is similar but mathematically not in this class. The question becomes, are the generated isotherms, and not just the energy distributions, similar. 

For these isotherms, especially the DP and Toth isotherms, not only must the distribution in be considered but also the distribution in the micropore sizes. The reason for this is that these two distributions are close enough to overlap somewhat, thus interacting to change the values of the parameters. 

Consider a factory air space of 10 m , which is regularly contaminated with 10 ppm by volume of hydrogen sulfide, a toxic and foulsmelling gas. It is proposed to use adsorption to purify the air. Activated carbon does not lend itself to this purpose because of its low capacity, but the zeolite mordenite has a suitably high loading factor represented by the following Toth isotherm ... 

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