Freundlich Equations

The Langmuir Equation, Freundlich Equation, or the BET Equation can be used to adequately represent experimental data. However, only the Langmuir or Freundlich equations are recognized as being reliable at very low solvent concentrations in the influent gas. 

Another typical example of the isotherms frequently employed is the Freundlich type equation (Freundlich, 1926). 

The Freundlich equation, unlike the Langmuir one, does not become linear at low... 

An adsorbed film obeys a modified Amagat equation of state, t(t = qkT (see Eq. ni-107). Show that this corresponds to a Freundlich adsorption isotherm (Eq. XI-12) and comment on the situation. 

The Freundlich equation is defective as a model because of the physically unrealistic/((2) consequences of this are that Henry s law is not approached at low P, nor is a limiting adsorption reached at high P. These difficulties can be patched by supposing that... 

Solubility is also affected by particle size, small crystals (<1 pm say) exhibiting a greater solubility than large ones. This relationship is quantified in the Gibbs-Thomson, Ostwald-Freundlich equation (see Mullin, 2001)... 

The isotherms for the two enantiomers of phenylalanine anilide were measured at 40, 50, 60 and 70 C, and the data fitted to each of the models given in Equations (1-3) [42]. The isotherms obtained by fitting the data to the Langmuir equation were of a quality inferior to the other two. Fittings of the data to the Freundlich and to the bi-Langmuir equations were both good. A comparison of the residuals revealed that the different isotherms of d-PA were best fitted to a bi-Langmuir model, while the... 

However, in the pH range 1-4, the effect of the OH ion predominates to such an extent that corrosion rates are similar in the presence of many other anions at concentrations less than 0.1 M. Since an adsorption process is involved in the mechanism, the corrosion rate in the pH range 1-4 may be represented by the Freundlich equation ... 

The dissolution of passive films is, in the main, controlled by a chemical activation step in contrast to film-free conditions at. Many protective anodic films are oxides and hydroxides whose dissolution depends upon the hydrogen ion concentration, and the rate follows a Freundlich adsorption equation ... 

The arbitrary division of behaviour has been made because of the extreme behaviour of some chemicals that initiate small areas of attack on a well-passivated metal surface. The form of attack may manifest itself as stress-corrosion cracking, crevice attack or pitting. At certain temperatures and pressures, minute quantities of certain chemicals can result in this form of attack. Chloride ions, in particular, are responsible for many of the failures observed, and it can be present as an impurity in a large number of raw materials. This has led to the development of metals and alloys that can withstand pitting and crevice corrosion, but on the whole these are comparatively expensive. It has become important, therefore, to be able to predict the conditions where more conventional materials may be used. The effect of an increase in concentration on pitting corrosion follows a similar relationship to the Freundlich equation where... 

Another useful model for isothermal adsorption is the Freundlich model, which is presented by the following equation. 

P the total pressure, aHj the mole fraction of hydrogen in the gas phase, and vHj the stoichiometric coefficient of hydrogen. It is assumed that the hydrogen concentration at the catalyst surface is in equilibrium with the hydrogen concentration in the liquid and is related to this through a Freundlich isotherm with the exponent a. The quantity Hj is related to co by stoichiometry, and Eg and Ag are related to - co because the reaction is accompanied by reduction of the gas-phase volume. The corresponding relationships are introduced into Eqs. (7)-(9), and these equations are solved by analog computation. 

In a study of adsorption of organic herbicides by montmorillonite, Bailey and colleagues138 found that none of the compounds conformed to the Langmuir adsorption equation. Of the 23 compounds tested, only a few did not conform well to the Freundlich equation. 

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. 

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

Sorption and desorption are usually modeled as one fully reversible process, although hystersis is sometimes observed. Four types of equations are commonly used to describe sorption/desorption processes Langmuir, Freundlich, overall and ion or cation exchange. The Langmuir isotherm model was developed for single layer adsorption and is based on the assumption that maximum adsorption corresponds to a saturated monolayer of solute molecules on the adsorbent surface, that the energy of adsorption is constant, and that there is no transmigration of adsorbate on the surface phase. 

The adsorption coefficients (K) were determined using the equation for the Freundlich adsorption isotherm ... 

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). 

TABLE IV. Adsorption Coefficients for Butylate, Alachlor, and Metolachlor in Keeton Soil at Various Temperatures Obtained Using the Freundlich Equation. ... 

In coal bed capacity estimation, the Langmuir equation provides a simple and efficient relation for single layer low-pressure conditions. In the case of high pressure and high temperature, Bi Langmuir, extended Langmuir, Sips, Langmuir-Freundlich, Toth, UNILAN, two-dimensional... 

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