The Freundlich Isotherm

One of the most popular adsorption isotherm equations used for liquids was proposed by Freundlich in 1926. This isotherm has been used to describe a wide range of of pharmaceutical products hke antibiotics, steroids, and hormones [87]. This isotherm is given by the following equation 

of course, unrealistic to expect real surfaces to always behave as the ideal surface defined by the previous assumptions, and many adsorption studies have reported data that do not fit a Langmuir isotherm. Among the early efforts to correlate such data was the empirical isotherm proposed by Freundlich in 1926 [7], i.e.. 

Remember from equation 5.38 that K = where Kq is a constant. 

Substituting this into equation 5.61 and rearranging gives  

If one postulates that Qad 5 ttRT, which is satisfied once Qad exceeds about three times the thermal energy, which is reasonable (this value is less than 1.9 kcal mole at room temperature), equation 5.62 can be integrated to give  

It can be shown that when 0 = noQadm this corresponds to maximum adsorption, i.e., 0 = 1, so this equation can be normalized to (KoP) , which is the form of the empirical Freundlich isotherm given by equation 5.56. 

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If the differential distribution function is exponential in Q (Section XVII-14A), the resulting Q(P, T) is that known as the Freundlich isotherm... 

For different acceptor particle adsorption isotherms expressions (1.85) - (1.89) provide various dependencies of equilibrium values of <7s for a partial pressure P (ranging from power indexes up to exponential). Thus, in case when the logarithmic isotherm Nt InP is valid the expression (1.85 ) leads to dependence <75 P" often observed in experiments [20, 83, 155]. In case of the Freundlich isotherm we arrive to the same type of dependence of - P" observed in the limit case described by expression (1.87). 

Spirodela intermedia, L. minor, and P. stratiotes were able to remove Pb(II), Cd(II), Ni(II), Cu(II), and Zn(II), although the two former ions were removed more efficiently. Data fitted the Langmuir model only for Ni and Cd, but the Freundlich isotherm for all metals tested. The adsorption capacity values (K ) showed that Pb was the metal more efficiently removed from water solution (166.49 and 447.95 mg/g for S. intermedia and L. minor, respectively). The adsorption process for the three species studied followed first-order kinetics. The mechanism involved in biosorption resulted in an ion-exchange process between monovalent metals as counterions present in the macrophytes biomass and heavy metal ions and protons taken up from water.112... 

Data for adsorption isotherms can often be correlated by the Freundlich Isotherm Equation. For adsorption from... 

The Langmuir and Freundlich equations have often been employed to model the sorption of metal ions by bacteria. Mullen et al. (1989) used the Freundlich isotherm to describe the sorption of Cd and Cu by B. cereus, B. subtilis, E. coli and P. aeruginosa over the concentration range of 0.001-lmM. The respective values of the Freundlich constant (Kf) indicated that E. coli was most efficient at sorbing Cd2+ and Cu2+. 

The Freundlich isotherm (or Freundlich model) is an empirical description of species sorption similar to the K, approach, but differing in how the ratio of sorbed to dissolved mass is computed. In the model, dissolved mass, the denominator in the ratio, is raised to an exponent less than one. The ratio, represented by the Freundlich coefficient Kf, is taken to be constant, as is the exponent, denoted f, where 0< f <1. As before, the masses of dissolved and sorbed species are entered, respectively, in units such as moles per gram of dry sediment and moles per cm3 fluid. Since the denominator is raised to an arbitrary exponent rif, the units for Kf are not commonly reported, and care must be taken to note the units in which the ratio was determined. 

Geochemical models, as with the approach, are commonly formulated with a variant of the Freundlich isotherm based on a chemical reaction, like Reaction 9.1. In this approach, known as the reaction Freundlich model or the activity Freundlich model, the extent of sorption by the reaction can be expressed,... 

Fig. 5.4.4. Sorption isotherms of AE homologues on sediment (EPA-12), showing the fits obtained with the Freundlich isotherm (from Brownawell et al. [6]).

As pointed out by Sposito (1984) this equation initiated the surface chemistry of naturally occurring solids. Maarten van Bemmelen published this equation (now referred to as the Freundlich isotherm) more than 100 years ago and distilled from his results, that the adsorptive power of ordinary soils depends on the colloidal silicates, humus, silica, and iron oxides they contain. 

When the Freundlich isotherm n values approximate one, that indicates a linear relationship between the amount sorbed and the equilibrium concentration in solution. Thus, the distribution of any organic pollutant in the aqueous-solid... 

A number of attempts have been made to modify the IAS model (Eqs. 22-29) to improve its accuracy and reduce computational efforts. Using the IAS model, DiGiano et al. [80] derived a Simplified Competitive Equilibrium Adsorption Model (SCAM). This model, which is based on the Freundlich isotherm, assumes the single-solute isotherms of all the components are equal and it utilizes average isotherm constants when this assumption is not valid. The IAS model equations have been reduced to a single expression ... 

The early experimental data of Freundlich Eapillarchemie) indicated that the adsorption of a solute from a solution followed the general laws governing the adsorption of vapours by solids, i.e. the Freundlich isotherm... 

The adsorption of a particular solute from a solution is however dependent upon the nature of the solvent, a point not envisaged by the Freundlich isotherm. The influence of the solvent on the amount of adsorption of a particular solute has been investigated in the case of iodine and charcoal by Davis (J.C.8. xci. 1666,1911) who found that the amount of iodine adsorbed by charcoal from solutions of equimolecular concentrations decreased with the nature of the solvent in the following order ... 

Gustafson Zeit. Physik, Ohem. cxi. 405, 1916) has attempted to extend the conception of Williams that solvent and solute are both adsorbed with the aid of. the Freundlich isotherm in the following way. 

There exists an equilibrium at the interface between a charged suspension and its solution consisting of a dynamic interchange of both cation and anion. If the suspension be positively charged the adsorption of the positive ion is greater than that of the negative at that particular concentration. On application of the Freundlich isotherm equation to each ion... 

Rampey AM, Umpleby RJ, Rushton GT, Iseman JC, Shah RN, Shimizu KD. Characterization of the imprint effect and the influence of imprinting conditions on affinity, capacity, and heterogeneity in molecularly imprinted polymers using the Freundlich isotherm-affinity distribution analysis. Anal Chem 2004 76 1123-1133. 

This adsorption follows the Freundlich isotherm. AFM images of the films obtained in 10 mM ethanolic solution of the studied acid exhibited dramatically different morphologies, which were ascribed to the formation of multilayered aggregates. 

Freundlich Adsorbents that follow the Freundlich isotherm equation are assumed to have a heterogeneous surface consisting of sites with different adsorption potentials, and each type of site is assumed to adsorb molecules, as in the Langmuir equation (Perry and Green, 1999) ... 

However, there is no explicit relationship for the Freundlich isotherm. In this case, for a desired value of X, several Xi values should be checked, and thus an extra calculation step is needed, i.e. an iteration procedure (Figure 4.26). 

Figure 4.27 Examples of theoretical breakthrough curves calculated from the analytical solutions for the Freundlich isotherm (Fr = 0.5).

In Figure 4.27, some examples of theoretical breakthrough curves calculated from the analytical solutions for the Freundlich isotherm (Fr = 0.5) are presented. As is clear, the curve corresponds to the case of equal and combined solid and liquid-film diffusion resistances ([ = 1) which is between the two extremes, i.e. solid diffusion control (l = 10,000) and liquid-film diffusion control ( = 0.0001). 

Figure 9.4 Graphic representation of the Freundlich isotherm Eq. 9-2 for the three cases ni > 1, , = 1, and n, < 1. Note that n, and log Klf are obtained from the slope ( , ) and intercept (log Kif indicated by the points at log C,w = 0) of the regression line.

A very common mathematical approach for fitting experimentally determined sorption data using a minimum of adjustable parameters employs an empirical relationship known as the Freundlich isotherm ... 

The exponent n usually is less than unity. Both gas and liquid adsorption data are fitted by the Freundlich isotherm. Many liquid data are fitted thus in a compilation of Landolt-Bornstein (II/3, Numerical Data and Functional Relationships in Science and Technology, Springer, New York, 1956, pp. 525-528), but their gas... 

The Freundlich isotherm, where the dye in fiber D, is directly proportional to (D,)- and a plot of log D against log D, gives a straight line, is generally found with cellulosic and other ionic hydrophobic fibers. 

Langmuir equation for a uniform surface, but by the Zel dovich and Rogin-skii equation or by the Bangham equation. Adsorption equilibrium is described not by the hyperbolic Langmuir isotherm, but by the Freundlich isotherm or the logarithmic isotherm (40). 

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