Toth equation

No equation of state is known that leads to an adsorption isothenu wliich in general fits experimental data over the entire range of n from zero to full monolayer coverage. Isotlieniis tliat find practical use are often 3-parameter empirical extensions of the Langmuir isothenu. An example is the Toth equation ... 

Legend experimental data Henry s law -------Toth equation ------Langmuirequation n0... 

Equations (1) and (2) are applicable when the alumina is energetically homogeneous (isosteric heat of adsorption, q, is independent of water loading, n). The Toth equation can be used when the alumina is energetically heterogeneous (q decreases with increasing n) and the isotherm shape is Type I [7] ... 

The three-parameter Toth equation (1962) describes the pressure dependence of the amount of gas A that adsorbs as a monolayer on high-surface-area dispersed solid catalysts ... 

Answer Begin with the empirical relation between a and Pa given above, and take the f-power of the Toth equation. Rearrangement yields... 

The final form of the Toth equation that can be compared with experimental data via linear least-squares analysis is... 

Figure 3.2-10 Plot of the fractional loading versus bP for the Toth equation...

Being the three-parameter model, the Toth equation can describe well many adsorption data. We apply this isotherm equation to fit the isotherm data of propane on activated carbon. For example taking the isotherm data at 303 K in Table 3.2-1, the extracted optimal parameters are (using the ISO FITl routine) ... 

Several hundred sets of data for hydrocarbons on Nuxit-al charcoal obtained by Szepesy and Hies (Valenzuela and Myers, 1989) can be described well by this equation. Because of its simplicity in form and its correct behaviour at low and high pressures, the Toth equation is recommended as the first choice of isotherm equation for fitting data of many adsorbates such as hydrocarbons, carbon oxides, hydrogen sulfide, alcohols on activated carbon as well as zeolites. Sips equation presented in the last section is also recommended but when the behaviour in the Henry law region is needed, the Toth equation is the better choice. 

Like the other equations described so far, the temperature dependence of equilibrium parameters in the Toth equation is required for the purpose of extrapolation or interpolation of equilibrium at other temperatures as well as the purpose of calculating isosteric heat. 

The temperature-dependent form of the Toth equation (3.2-19) is used to simultaneously fit the isotherm data at 283, 303 and 333 K of propane onto... 

Table 3.2-6 Optimal parameters for the temperature dependent Toth equation...

This measure of heat of adsorption is much higher than that derived from the fitting of the data with the Sips equation earlier, where we have found a value of 28750 J/mol for Q. This large difference should cause no alarm as the parameter Q is only the measure of adsorption heat. For example in the case of the Sips equation, Q is the isosteric heat of adsorption at a fractional loading of 0.5, while the parameter Q in the case of the Toth equation is the isosteric heat of adsorption at zero fractional loading as we shall show in the next section. 

Like the Sips equation, the isosteric heat of adsorption is a function of pressure (or loading), and it takes a value of infinity at zero loading and minus infinity at very high loading, which limits the applicability of the Toth equation in its use in the calculation of isosteric heat at two ends of the loading. The meaning of the parameter Q in the Toth equation is now clear in eq. (3.2-19f). It is equal to the isosteric heat when the fractional loading is zero... 

Since the Sips and Toth empirical equations fit the equilibrium data reasonably well, it may be possible to use the isosteric heat as a function of loading as a criterion in choosing the correct isotherm equation. Using eq. (3.2-19g) and the optimal parameters in Table 3.2-6 for the Toth equation, we obtain the following equation for the isosteric heat at 283 K with values tabulated in Table 3.2-7 ... 

Table 3.2-7 Isosteric heat as a function of loading using the Toth equation...

The isosteric heats calculated by the Toth equation are lower than those calculated by the Sips equation (Table 3.2-5). The above table shows the percentage differences between the values calculated by the Sips and Toth equations. The difference is seen to be significant enough for the isosteric heat to be used as the criterion to better select the isotherm equation. 

Take the example of propane on activated carbon, we plot the isosteric heat (eq. 3.2-19f) versus the fractional loading for the Toth equation using the parameters obtained in Table 3.2-6 (Figure 3.2-11). Just like the case of the Sips isotherm, the isosteric heat decreases with loading, and it shows a weak temperature dependence. 

Although the Toth equation, like Sips and Freundlich, is an empirical equation, it has an advantage over the other two in that it has the following Henry constant at zero loading ... 

Using the energy distribution concept of eq. (3.2-17a) with Langmuir equation describing the local isotherm. Sircar and Myers (1984) provided the following energy distribution which gives rise to the Toth equation ... 

Eq. (3.2-21) is the energy distribution of the Toth equation. Having an energy distribution only means that the system is not homogeneous, and it does not point to the source of heterogeneity as well as the physical meaning of the parameters in the energy distribution. 

Like the Sips and Toth equation, the Unilan equation is a three-parameter isotherm equation, and they are commonly used to correlate adsorption equilibrium data of many solids, such as activated carbon and zeolite. 

The reference temperature Tq used in the above fitting is 283 K. The degree of fitting between the Unilan equation and the experimental data is comparable to that for the cases of Sips and Toth equations. The discrimination between these three empirical equations now rests on the behaviour of the isosteric heat. This is what we will do next. 

We see a distinction between the isosteric heat calculated from the Toth equation and those calculated from the Sips and Unilan equations, despite of the fact that the three isotherm equations describe the equilibrium data fairly well. This distinction in the isosteric heat curve could be utilised in the matching between the... 

Recently, Keller and his co-workers (1996) proposed a new isotherm equation, which is very similar in form to the original Toth equation. The differences between their equation and that of Toth are that ... 

Fitting the Keller et al s equation with the isotherm data of propane on activated carbon at three temperatures 283, 303 and 333 K, we found the fit is reasonably good, comparable to the good fit observed with Sips and Toth equations. The optimally fitted parameters are ... 

Various practical isotherm equations have been presented and they are useful in describing adsorption data of many adsorption systems. Among the many equations presented, the Toth equation is the attractive equation because of its correct behaviour at low and high loading. If the Henry behaviour is not critical then the Sips equation is also popular. For sub-critical vapours, multilayer isotherm equations are also presented in this chapter. Despite the many equations proposed in the literature, the BET equation still remains the most popular equation for the determination of surface area. When condensation occurs in the reduced pressure range of aroimd 0.4 to 0.995, the theory of condensation put forward by Kelvin is useful in the determination of the pore size as well as the pore size distribution. 

In this theory of Ideal Adsorption Solution, adsorption isotherm equation for pure components can take any form which fits the data best (Richter et al., 1989). Two isotherms commonly used are Toth and Unilan equations (Chapter 3), although Toth equation is the preferable equation from the computational point of view because it usually gives faster convergence than the Unilan equation does. When DR equation is used to describe pure component data, the application of the LAST in this case has some special features (Richter et al., 1989) and it was used by Lavanchy et al. (1996) in the description of equilibria of chlorobenzene and tetrachloride on activated carbon. 

Example 5,3-2 Ethane/ ethylene /activated carbon Using the pure component data of ethane and ethylene at 293 K tabulated in Valenzuela and Myers (1989), we fit the data with the Toth equation and the following table lists the optimized parameters. 

When N=2 in eq. (6.2-1) we have a dual Langmuir isotherm occasionally used in the literature. This semi-empirical equation contains 4 parameters, and if the purpose is to fit experimental data the three-parameter models learnt in Chapter 3 such as the Toth equation is a better choice because it contains one less parameter and the four-parameter dual Langmuir equation very often does not provide any better fit than the three-parameter Toth equation. 

Appendix 9.5 lists a number of kinetic rate expressions which at equilibrium will give rise to Sips and Toth equations. 

We finish this section by considering another isotherm equation the Toth equation, which is a popular equation in describing adsorption isotherm of numerous practical systems. It has the form ... 

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