Empirical isotherms

Several of the often used empirical adsorption isotherms (Als) are presented for single- and multicomponent systems. Hints to possible physical interpretation of parameters are given. Examples of systems to which the isotherms successfully may be applied are mentioned. Many more empirical Als referring to anorganic, organic, and biological sorbent materials respectively can be found in the literature, cp. for example [7.3, 7.26, 7.32, 7.33]. The isotherms discussed here in brief are  

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We should not be too surprised that the Langmuir equation often yields only an empirical isotherm. There are several reasons why real systems are likely to deviate from the theoretical model ... 

In practice, the following Freundlich-type empirical isotherm can be used for many liquid-solid adsorption systems ... 

Over the valid range of the empirical isotherm equation, the heat of the sorption process is given by ... 

In this equation AH is the heat of the process, R is the gas constant, T is the absolute temperature, and the remaining symbols a, b, C are those appearing in the empirical isotherm. The equation arises from the empirical isotherm and the following thermodynamic relationships Ap = H — TAS, Ap = RT InC, and... 

The most common way to present adsorption data is an adsorption isotherm. The amount adsorbed is plotted as a function of the partial pressure of the adsorbate, p, at a constant temperature. The first empirical isotherm was proposed by Freundlich in 1906 [Freundlich, 1926], and since then, a number of empirical adsorption isotherms have been reported. The Freundlich isotherm is usually written in the form... 

Now, to conclude this section, it is necessary to affirm that any one of the equations described here to correlate the relation between the amount adsorbed, ct, with the equilibrium concentration in solution, Ci, corresponds to a particular model for adsorption from solutions. That is, these should be considered as empirical isotherm equations [2],... 

Equations of this type appear to fit Isotherms of type II (fig. 1.13) quite well, sometimes better than BET theory does. The exponent -1/3 stems from 11.5.51). In practice, values between -1/3 and -1/2 are usually found. From the viewpoint of dispersion forces this is difficult to account for. Retardation does not play a role and. even if it did, this would further reduce the exponent. Rather, the sum effect of all "hand-waving" approximations (including the assumption of surface homogeneity) leads to a semi-empirical Isotherm of the form j 1.5.54] in which the constants and exponent are, within certain limits, adjustable. Because of this, the equation is often written in the more general form... 

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. 

Boedeker [68] proposed the following empirical isotherm equation for the adsorption of polar compoxmds on polar adsorbents ... 

This empirical isotherm is a combination of two classical models, the Langmuir and the Freimdlich models discussed earlier... 

These equations contain seven parameters instead of four in the Langmuir isotherm. These parameters include the two column satiuation capacities in the case of an empirical isotherm. However, like the competitive Langmuir isotherm, this model is thermodynamically consistent only if these saturation capacities are equal. 

The FA method gives isotherm data. To be useful in preparative chromatography, these data must be fitted to an isotherm model. There are presently no numerical procedures available to smooth the data from multidimensional plots, similar to the 2-D splines or French curves and obtain purely empirical isotherms. Therefore, the major difficulty is the selection of adequate models. The Langmuir isotherm is too simplistic in most cases, and the LeVan-Vermeulen isotherm is complicated and difficult to use as a fitting fimction. Several methods have been described to extract the "best" set of Langmuir parameters which could accormt for a set of competitive adsorption data [108]. These methods have been compared. The most suitable method seems to depend on the aim of the determination and on the deviation of the system from true Langmuir behavior [108]. 

Finally, to illustrate one case of band profiles calculated with a convex downward isotherm, we have selected a simple, but completely empirical, isotherm model, q = a C + C P), with p < 1. The result is shown in Figures 7.10a and 7.10b, with p = 0.3 and 0.6, respectively. The band front is diffuse and a rear shock appears. The direction of the convexity of the band profile itself depends on the... 

Although for the sake of clarity the previous discussion was limited to the case of a binary mixture, these results are easily generalized to the study of an n-component mixture. Because of the coupling between the mobile phase components, the velocity eigenvalues are related to the slopes of the tangents to the n-dimensional isotherm surface, in the n composition path directions. These slopes can be calculated when the isotherm surface is known. Conversely, systematic measurement of the retention times of very small vacancy pulses for various compositions of the mobile phase may permit the determination of competitive equilibrium isotherms, but only if a proper isotherm model is available. Least-squares fitting of the set of slope data to the isotherm equations allows the calculation of the isotherm parameters. If an isotherm model, i.e., a set of competitive isotherm equations, is not available, the experimental data cannot be used to derive an empirical isotherm (see Chapter 4). 

Good agreement was achieved between the experimental results and the solutions calculated with the above empirical isotherm equations. The simultaneous concentration and separation of the two feed components was achieved by gradient elution. Figure 15.10 compares the experimental and calculated results obtained with a linear gradient of 10 to 40% ACN in 20 minutes, with a 2.4 mL feed... 

Though this interpretation is now commonly accepted in the interpretation of the classic empirical isotherms (Eq. (23), Chapter 6), it must be mentioned that other approaches have been proven to account for them [34]. 

In a system containing a solution and an adsorbent, solute and adsorbate molecules are considered to be in dynamic equilibrium. It appears that a solute molecule is adsorbed at an adsorption site from the solution but an adsorbed molecule is dissolved into the solution by the solvation power of the solvent. This idea is supported by the fact that fairly satisfactory empirical isotherms have been presented by H. Freundlich and others. 

Nearly all reactive transport models used in the regulatory environment, however, are based on empirical isotherms. The models take into account the effects of hydrologic processes due to the movement of groundwater such as advection and dispersion, but the effects of chemical reactions are typically described by an isotherm linking the concentrations of i in solid mass to that in groundwater,... 

While the mechanistic treatment of chemical reactions in the coupled multi-component, multi-species mass transport has obvious advantages over the empirical isotherm-based transport models, we can also easily compile a long list of shortcomings for coupled reactive transport models. We choose a few and list them here. 

Freundlich Empirical isotherm which can be made to fit most equilibrium data, 27... 

Ihe empirical isotherm method considers equilibrium distribution of only one component and requires the experimental determination of its sorption isotherms in specific conditions. However, ground water contains multiple ions, which compete for the location in the exchange capacity. For this reason, ion exchange equilibrium directly correlates with the ground water composition and adsorption capacity of its components. This effect of ions on one another may be identified only computational methods based on exchange coefficients. 

The temperature dependence of the Unilan equation is shown in eqs.(3.2-23) assuming the maximum and minimum energies are not dependent on temperature. Like the last two empirical isotherm equations (Sips and Toth), we assume that the saturation capacity follows the following temperature dependence ... 

The following table summarises the various empirical isotherm equations. 

Table 3.2-12 Summary of commonly used empirical isotherm equations...

The adsorption isotherm of eq. (6.9-21) is complicated than many empirical or semi-empirical isotherm equations dealt with in Chapter 3. Because of its limited testing against experimental data, eq. (6.9-21) does not receive much applications. 

Redlich-Peterson (1959) KrC, l + fl C It approaches the FrermcUich model at high concentrations and is in accord with the low concentration limit of the Langmuir equation. Furthermore, the R-P equation incorporates three parameters into an empirical isotherm and, therefore, can be applied in either homogeneous or heterogeneous systems due to its high versatility. 

Zeldowitsch (1934), assuming an exponentially decaying function of site density with respect to Q, obtained the classical empirical isotherm ... 

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