Freundlich relationship

The equilibrium partial pressure of water vapor over silica gel containing various concentrations of adsorbed water is shown in Figure 12-7. This figure is based upon the data of Taylor (1945) and Hubard (1954), extrapolated to cover the very low water range by application of the Freundlich relationship ... 

Freundlich isotherm The empirical relationship between the amount of a substance adsorbed and the concentration of the solute... 

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)... 

For many metals and alloys the determination of /p is complex, and its magnitude is governed by many factors such as surface finish, rate of formation, alloying constituents, and the presence of those anions, such as halides, that promote localised breakdown. In many instances the attack on passive films by halide ions shows a temperature and concentration dependence similar to the effect of hydrogen ions, i.e. the rate of film dissolution increases with concentration in accordance with a Freundlich adsorption relationship... 

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... 

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 the simplest case, a linear relationship may exist between solutes in soil solution and those on the solid pha.se, such that C, = hCr, where C, is the. solute concentration on the solid phase and h defines the buffer power. In perhaps more typical situations, the relationship might be defined by a more complicated, concentration-dependent relationship such as a Freundlich or Langmuir expression here the instantaneous buffer power may be defined as the tangent to the exchange isotherm i.e., h = dCJdCi. Some care is needed when defining this buffer power (7). 

Example 10.4 A gas mixture with a flowrate of 0.1 m3 s-1 contains 0.203 kg m-3 of benzene. The temperature is 10°C and the pressure 1 atm (1.013 bar). Benzene needs to be separated to give a gas stream with a benzene concentration of less than 5 mg m-3. It is proposed to achieve this by adsorption using activated carbon in a fixed bed. The activated carbon is to be regenerated using superheated steam. The experimental adsorption isotherms cannot be adequately represented by Freundlich isotherms and, instead, can be correlated at 10°C by the empirical relationship ... 

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... 

The linear equilibrium isotherm adsorption relationship (Eq. 11) requires a constant rate of adsorption, and is most often not physically valid because the ability of clay solid particles to absorb pollutants decreases as the adsorbed amount of pollutant increases, contrary to expectations from the liner model. If the rate of adsorption decreases rapidly as the concentration in the pore fluid increases, the simple Freundlich type model (Eqs. 8 and 9) must be extended to properly portray the adsorption relationship. Few models can faithfully portray the adsorption relationship for multicomponent COM-pollutant systems where some of the components are adsorbed and others are desorbed. It is therefore necessary to perform initial tests with the natural system to choose the adsorption model specific to the problem at hand. From leaching-column experimental data, using field materials (soil solids and COMs solutions), and model calibration, the following general function can be successfully applied [155] ... 

The dSp/dC term can be found from the equilibrium relationship of Freundlich isotherms, expressed as... 

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). 

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... 

Equation 1 is given in spherical coordinates, thus assuming a spherical shape for the carbon particle, an assumption which accords reasonably well with microscopic observations of the geometry of particles of the experimental carbon. In Equation 1, C represents the H30+ activity in solution t, time r, the radial distance from the particle center D, the diffusion coefficient and S, the H30+ concentration at the surface of the carbon. For the present experiments, the equilibrium relationship between S and C is described in terms of the Freundlich expression... 

Isotherms. When a fiber is immersed in a dyebath, dye moves from the external phase into Lhe fiber. Initially the rate is quick but with time this slows and eventually an equilibrium is reached between the concentration of dye in the fiber and the concentration of dye in the dyebath. For a given initial dyebath concentration of a dye under given dyebath conditions, e.g.. temperature, pH, and conductivity, there is an equilibrium concentration of dye in fiber, D, and dye in the dyebath external solution, D,. Three models describe this relationship simple partition isotherm. Freundlich isotherm, and Langmuir isotherm. 

For nonlinear isotherms, the Freundlich model most often is used to describe the relationship between the sorbed (Cs) and the solution phase concentrations (Cw) ... 

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. 

Several different mathematical relationships (referred to as isotherms) have been developed to describe the relationship of fractional surface coverage with respect to the adsorbing species. Work by Langmuir, Freundlich, Tempkin, and others have attempted to describe the above mentioned pressure differential to fractional surface coverage. The most widely used and accepted isotherm is the BET equation, named after its orignators, Brunauer, Emmett, and Teller, and is as follows 29... 

Ostwald (1900), using the thermodynamic relationship between vapor pressure of spherical drops and the curvature of surface, derived an equation giving the relation between solubility and particle size. Ostwald s equation was later modified by Freundlich (1909), and is as follows ... 

Equilibrium between solution and adsorbed or sorbed phases is a condition commonly used to evaluate adsorption or sorption processes in soils or soil-clay minerals. As previously stated, equilibrium is defined as the point at which the rate of the forward reaction equals the rate of the reverse reaction. Two major techniques commonly used to model soil adsorption or sorption equilibrium processes are (1) the Freundlich approach and (2) the Langmuir approach. Both involve adsorption or sorption isotherms. A sorption isotherm describes the relationship between the dissolved concentration of a given chemical species (adsorbate) in units of micrograms per liter (pg L 1), milligrams per liter (mg L-1), microequivalents per liter (pequiv L-1), or millimoles per liter (mmol L-1), and the sorbed quantity of the same species by the solid phase (adsorbent) in units of adsorbate per unit mass of adsorbent (solid) (e.g., pg kg-1, mg kg-1, peq kg-1, or mmol kg 1) at equilibrium under constant pressure and temperature. Sorption isotherms have been classified into four types, depending on their general shape (Fig. 4.13) ... 

The modeling of the obtained isotherms was carried out by applying the Freundlich equation. This is an empirical relationship based on the assumption of a logarithmic decrease in adsorption heat with adsorption surface coverage. The Freundlich equation is commonly used in the form ... 

Equations (5) and (6) have been confirmed qualitatively by several investigators. A summary of the literature will be found in Freundlich s book. The equations have recently been tested in a quantitative manner by W. C. M C. Lewis. He investigated the surfaces between two non-miscible liquids, such as mercury and an aqueous solution, to which the same relationships apply. In some cases he obtained an entire confirmation of equation (6). In other cases, particularly when electrolytes were present, he found deviations, which he was able to explain, however, as being due to electrocapillarity. 

Figure 9.2. Plot of adsorption data in a double logarithmic plot. In a Langmuir isotherm the initial slope is unity. A Freundlich isotherm shows in a double log plot a slope of n < I. Such a Freundlich isotherm is obtained if the adsorbent is heterogeneous (decreasing tendency for adsorption with increasing q). (Adapted from Morel, 1983.) Inset Observed relationship between the concentrations of a chemical in the sorbed state r or and the dissolved state [A] or C, .

It was stated, that macroporous carbons of plant origin show better properties in dynamic than in static conditions. It is cormected with relationship of quantity of macropores and diffusion rate. Greater differences in classsification are observed at lower equilibrium concentrations (0,1 and 0,01 mg/dm ) in static adsorption. Therefore direct factors of Freundlich s isotherms influencing adsorption capacities at different concentrations should be considered. It was stated that angle of slope, adsorption rate in static conditions, shape of breaktrough curves in dynamic conditions... 

For homologous series, adsorption from solution increases as the series is ascended and the molecules become more hydrophobic. There is, for example, a good correlation between the Freundlich adsorption constant, 1/n (related to the extent of adsorption) and the molecular weight of the local anaesthetics discussed above (Fig. 6.18). The data for pheno-barbital deviated from this linear relationship possibly as a result of the difficulty of adhesion... 

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