Freundlich constants, values

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 calculation of Kd shown above assumes a linear relationship between the concentration sorbed and the concentration of the chemical in the aqueous phase (i.e., a linear isotherm). If sorption was known to be non-linear, the Freundlich constant, N, would need to be known or an estimate of it would need to be made. Generally values of N range from 0.7 to 1.2. 

Freundlich constants and ranges of Kd values for radionuclide sorption on the Rattlesnake Ridge sandstone are given in Table III. The constants K and N were calculated using linear regression. Linear sorption isotherms (N = 1.0) are observed only for strontium, selenium, and radium. 

Tin and americium were so extensively sorbed under all conditions that isotherm data could not be obtained. These elements are not significantly mobile in the Mabton Interbed aquifer. Values of Freundlich constants for technetium, radium, uranium, neptunium, and plutonium are given in Table IV. The Freundlich equation did not fit the selenium sorption data very well probably because of slow sorption kinetics or precipitation. Precipitation was also observed for technetium at 23°C for concentrations above 10 7M. This is about the same solubility observed for technetium in the sandstone isotherm measurements. Linear isotherms were observed only in the case of radium sorption. In general, sorption on the Mabton Interbed was greater than on the Rattlesnake Ridge sandstone. This is probably due to the greater clay content of the Mabton standard. 

Using the techniques of analytic geometry, let us derive the Freundlich constants in a little more detail than used in the derivation of the constants in the discussion of reverse osmosis treated previously. As mentioned, the straight-line form of the equation requires only two experimental data points however, experiments are normally conducted to produce not just two pair of values but more. Thus, the experimental resnlts must be reduced to just the two pairs of values required for the determination of the parameters therefore, assuming there are m pairs of values, these m pairs must be reduced to just two pairs. Once the reduction to two pairs has been done, the isotherm equation may be then be written to just the two pairs of derived values as follows ... 

A logarithmic form of this expression gives a linear relation from which values for the Freundlich constant, K, and the exponent, n, can be derived ... 

In a number of solutions of globular proteins this equation holds good with a constant value of s. The experiments show, however, that with flexible long-chain molecules the equation can only be maintained if s is assumed to be dependent on c. Now, since s represents the effective volume per gram of solute, Schulz assumes that Freundlich and Posnjak s equation holds good ... 

Initial benzene sorption on the soils was determined after 16 h of incubation. Table I shows the Freundlich constants which were determined for the adsorbents. The 1/n values are at unity or very close to it, indicating that sorption should be linear within the range of increasing benzene concentrations used for this study. This assumption was shown to be correct when plots of the Freundlich isotherms were made (Figures 1 and 2). 

A criterion often used to characterize the sorptivity of a poUutant on a sorbent is the Freundlich constant (A p), which can be normalized to the carbon mass (Me) of the target molecule (Me (PFOA and PFOS) 96 g carbon mol ). Based on the sorption affinity of different fi-actions of natural organic matter on activated carbon, Kc values of below 20 (mg C/g) (mg C/l) indicate a poor adsorbabihty, whereas compounds with Kc > 50 (mg C/g) (mg CA) can be considered as strongly adsorbable (derived from [19]). 

The Freundlich constant and 1/n are constants dependent on the relative adsorption capacity and intensity of adsorption, respectively. The values of and 1/n are determined from the intercept and slope of a linear plot of log versus log Cg. 

For an arbitrary value of the Freundlich constant, n, numerical calculation of the basic equations (Suzuki and Kawazoe, 1974c) gave... 

Concentration decrease in this case is calculated and given in Fig. S.8.a. For an arbitrary value of the Freundlich constant, n, numerical calculation of the set of basic equations is necessary. Typical examples are given in Fig. 5.8.b-5.8.d for n=1.5, 2, 5 (Suzuki and Kawazoe, 1974b). 

The function P(p) is constant, i.e., it is similar to functions shown in the lower part of Fig. 6. In this case the appropriate isotherm equation is the Freundlich equation however, it must be kept in mind that the F equation does not have a concrete and real physical meaning (infinitely great heterogeneity ) and, therefore, in most cases the constant value of function (/ ) relates to the fact that the number of measured data are not sufficient to determine the exact physical character of adsorption. So in this case it is recommended to extend the measurements towards lower or greater equilibrium pressures than the measured ones. 

Results of adsorption experiments for butylate, alachlor, and metolachlor in Keeton soil at 10, 19, and 30°C were plotted using the Freundlich equation. A summary of the coefficients obtained from the Freundlich equation for these experiments is presented in TABLE IV. Excellent correlation using the Freundlich equation over the concentration ranges studied (four orders of magnitude) is indicated by the r values of 0.99. The n exponent from the Freundlich equation indicates the extent of linearity of the adsorption isotherm in the concentration range studied. If n = 1 then adsorption is constant at all concentrations studied (the adsorption isotherm is linear) and K is equivalent to the distribution coefficient between the soil and water (Kd), which is the ratio of the soil concentration (mole/kg) to the solution concentration (mole/L). A value of n > 1 indicates that as the solution concentration increases the sorption sites become saturated, resulting in a disproportionate amount of chemical being dissolved. Since n is nearly equal to 1 in these studies, the adsorption isotherms are nearly linear and the values for Kd (shown in TABLE IV) correspond closely to K. These Kd values were used to calculate heats of adsorption (AH). 

The data of Loukidou et al. (2004) for the equilibrium biosorption of chromium (VI) by Aeromonas caviae particles were well described by the Langmuir and Freundlich isotherms. Sorption rates estimated from pseudo second-order kinetics were in satisfactory agreement with experimental data. The results of XAFS study on the sorption of Cd by B. subtilis were generally in accord with existing surface complexation models (Boyanov et al. 2003). Intrinsic metal sorption constants were obtained by correcting the apparent sorption constants by the Boltzmann factor. A 1 2 metal-ligand stoichiometry provides the best fit to the experimental data with log K values of 6.0 0.2 for Sr(II) and 6.2 0.2 for Ba(II). 

Using the Freundlich equation X/M) = find the values of constants k and, ... 

Because the equations are for straight lines, only two pairs of values of the respective parameters are required to solve the constants. For the Freundlich isotherm, the required pairs of values are the parameters ln(XIM) and ln[C for the Langmuir isotherm, the required pairs are the parameters C]I(XIM) and [C]. 

Sorption Modeling. Pesticide sorption is characterized by describing sorption isotherms using the Freundlich equation, S = Ay, where S is the pesticide sorbed concentration, Cis the pesticide solution concentration after equilibration, and and N are constants. Although other equations have been used, the Freundlich has satisfactorily described experimental sorption results for a wide range of pesticides in a variety of soils. The value of N is usually <1 and between 0.75 and 0.95, which indicates that pesticides are proportionally more sorbed at low solution concentration than at high solution concentration. 

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