Empirical isotherm equations

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

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],... 

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

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... 

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. 

There is little doubt that, at least with type II isotherms, we can tell the approximate point at which multilayer adsorption sets in. The concept of a two-dimensional phase seems relatively sterile as applied to multilayer adsorption, except insofar as such isotherm equations may be used as empirically convenient, since the thickness of the adsorbed film is not easily allowed to become variable. 

The Langmuir equation has a strong theoretical basis, whereas the Freundlich equation is an almost purely empirical formulation because the coefficient N has embedded in it a number of thermodynamic parameters that cannot easily be measured independently.120 These two nonlinear isotherm equations have most of the same problems discussed earlier in relation to the distribution-coefficient equation. All parameters except adsorbent concentration C must be held constant when measuring Freundlich isotherms, and significant changes in environmental parameters, which would be expected at different times and locations in the deep-well environment, are very likely to result in large changes in the empirical constants. 

Adsorption from liquids is less well understood than adsorption from gases. In principle the equations derived for gases ought to be applicable to liquid systems, except when capillary condensation is occurring. In practice, some offer an empirical fit of the equilibrium data. One of the most popular adsorption isotherm equations used for liquids was proposed by Freundlich 21-1 in 1926. Arising from a study of the adsorption of organic compounds from aqueous solutions on to charcoal, it was shown that the data could be correlated by an equation of the form ... 

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 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... 

For separation by adsorption, adsorption capacity is often the most important parameter because it determines how much adsorbent is required to accomplish a certain task. For the adsorption of a variety of antibiotics, steroids, and hormons, the adsorption isotherm relating the amount of solute bound to solid and that in solvent can be described by the empirical Freundlich equation. 

Sorption is most commonly quantified using distribution coefficients (Kd), which simplistically model the sorption process as a partitioning of the chemical between homogeneous solid and solution phases. Sorption is also commonly quantified using sorption isotherms, which allow variation in sorption intensity with triazine concentration in solution. Sorption isotherms are generally modeled using the empirical Freundlich equation, S = K CUn, in which S is the sorbed concentration after equilibration, C is the solution concentration after equilibration, and Kt and 1 In are empirical constants. Kd and K are used to compare sorption of different chemicals on one soil or sorbent, or of one chemical on several sorbents. Kd and K are also commonly used in solute leaching models to predict triazine interactions with soils under various environmental conditions. 

Geochemical models of sorption and desorption must be developed from this work and incorporated into transport models that predict radionuclide migration. A frequently used, simple sorption (or desorption) model is the empirical distribution coefficient, Kj. This quantity is simply the equilibrium concentration of sorbed radionuclide divided by the equilibrium concentration of radionuclide in solution. Values of Kd can be used to calculate a retardation factor, R, which is used in solute transport equations to predict radionuclide migration in groundwater. The calculations assume instantaneous sorption, a linear sorption isotherm, and single-valued adsorption-desorption isotherms. These assumptions have been shown to be erroneous for solute sorption in several groundwater-soil systems (1-2). A more accurate description of radionuclide sorption is an isothermal equation such as the Freundlich equation ... 

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. 

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. 

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). 

Another criterion for goodness of fit is to assess the residual distribution, that is, how well does the model predict values throughout the complete pattern of the data set Some models may fit some portions of the data well but not other portions, and thus the residuals (differences between the calculated and real values) will not be uniformly distributed over the data set. Figure 12.8 shows a set of data fit to an empirical model (Equation 3.1) and the Langmuir adsorption isotherm inspection of the fit dose-response curves does not indicate a great difference in the goodness of fit. However, an examination of the... 

Adsorption reactions by soil minerals and soils have been described historically using empirical adsorption isotherm equations. As their name implies, it is understood that they are used for experiments at constant temperature, which unless indicated, otherwise is standard temperature, T = 298 K. An adsorption isotherm is a plot of the concentration adsorbed to a solid surface versus the concentration in aqueous solution for different total concentrations of a chemical species. Typically, adsorption isotherm equations are very good at describing experimental data, despite their lack of theoretical basis. Popularity of these equations stems in part from their simplicity and from the ease of estimation of their adjustable parameters. 

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