Equilibrium isotherms Freundlich isotherm

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. 

Free gel was added to definite concentrations of the MB solutions (2-20 mg/1) at room temperature and was noted for its adsorption. It is clear from Figure 13.3 that the dye adsorption increases sharply with an increase in the initial dye concentration. When Cq was reached at 5 ppm and 10 ppm, the was reached at 10.04 and 20.81 respectively, which were much higher than reported Qe values of other adsorbents (Table 13.3). Equilibrium adsorption isotherm is an important criterion to determine the mechanism of dye adsorption on hydrogel. The Langmuir and Freundlich models are widely used to examine the adsorption isotherms. Freundlich isotherm models are based on the assumption that the surface of the adsorbent is not homogeneous. The experimental data in Figure 13.3 was also analyzed with the Freundlich isotherm model, which describes a heterogeneous system with multilayer adsorption. The linear form of Freundlich isotherm 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. 

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). 

Adsorption and desorption. The user can choose to handle this using either temperature-corrected first order reaction kinetics, in which case the concentrations are always moving towards equilibrium but never quite reach it, or he can use a Freundlich isotherm, in which instantaneous equilibrium is assumed. With the Freundlich method, he can elect either to use a single-valued isotherm or a non-single-valued one. This was included in the model because there is experimental evidence which suggests that pesticides do not always follow the same curve on desorption as they do on adsorption. 

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). 

Loukidou et al. (2005) fitted the data for the equilibrium sorption of Cd from aqueous solutions by Aeromonas caviae to the Langmuir and Freundlich isotherms. They also conducted, a detailed analysis of sorption rates to validate several kinetic models. A suitable kinetic equation was derived, assuming that biosorption is chemically controlled. The so-called pseudo second-order rate expression could satisfactorily describe the experimental data. The adsorption data of Zn on soil bacterium Pseudomonas putida were fit with the van Bemmelen-Freundlich model (Toner et al. 2005). 

Fig. 4.18 shows the result of Cd2+ adsorption on illite in presence of Ca2+ (Comans, 1987). The data are fitted by Freundlich isotherms after an equilibration time of 54 days. It was shown in the experiments leading to these isotherms that adsorption approaches equilibrium faster than desorption. Comans has also used 109Cd to assess the isotope exchange he showed that at equilibrium (7-8 weeks equilibration time) the isotopic exchangeabilities are approximately 100 % i.e., all adsorbed Cd2+ is apparently in kinetic equilibrium with the solution. The available data do not allow a definite conclusion on the specific sorption mechanism. 

Adsorption-desorption equilibrium for Cd(II) on illite after 54 days of equilibration. The solution contains HCO3, 2 x 10 3 M Ca2+ and has a pH = 7.8. Freundlich isotherms based on separate adsorption ( ) and desorption (O) data are given from Comans (1987). 

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 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] ... 

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... 

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

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). 

It can be easily seen that if adsorption equilibrium follows the Freundlich isotherm, then r will be proportional to 0". According to (143), (145), and (160),... 

If the equilibrium isotherm can be expressed by the Freundlich equation and fresh adsorbent is used in each stage (Y0 = 0), the total amount of adsorbent used for a two-stage crosscurrent adsorption unit (Figure 10.9) is... 

Manganese oxides As(lll) and As(V) removal by Mn02(s) is similar, up to 5 mmol As mol-1 Mn at micromolar As equilibrium solution concentrations. Freundlich isotherm obeyed. As(lll) oxidized to As(V). Rapid oxidation (minutes) and adsorption of As(lll). Monitored Mn release and effect of pH, Ca, phosphate, and sulfate Driehaus, Seith and Jekel (1995)... 

Freundlich isotherm An equation for an adsorption isotherm in the form of Cads = Kf(Cso n)n (if n = 1, the distribution is linear, see Chapter 2 for details), which describes the distribution of a chemical species between an adsorbent and an associated solution under equilibrium conditions. Some adsorption isotherms are better described with the Langmuir isotherm. 

Rate-limited sorption can also be modeled assuming a kinetic rate expression coupled with a nonlinear equilibrium expression. If we assume a Freundlich isotherm and a first-order rate expression, we can use the following equation to model sorption kinetics [21] ... 

The extent of adsorption can have a profound effect on the rate of the surface reaction. Equilibrium isotherms of many kinds have been reported for adsorption from solution and have been classified by Giles et al. [24-27], The shapes of these adsorption curves often furnish qualitative information on the nature of the solute-surface interactions. Several of the types of isotherm observed in dilute solution are represented reasonably well by three simple and popular isotherm equations, those of Henry, Langmuir, and Freundlich. Their shapes are illustrated in Fig. 1. Each of these isotherms relates the surface concentrations cads (mol m"2) to the bulk equilibrium concentration c of the solute species in question. When few surface sites are occupied, Henry s law adsorption... 

The surface area of Black Pearls 2 carbon was sufficiently large (850 m2g 1) for the rates and extents of adsorption to be easily determined from the initial decreases in either optical rotation (see Fig. 2) or optical absorbance. The equilibrium amounts of Co(en)3+ and of I adsorbed per gram of carbon fitted Freundlich isotherms. The rates of adsorption exhibited first-order behaviour and led to half-lives of adsorption of 2 min for Co(en)3+ and of 3min for I" [223], These were much faster than the rate of racemisation or the rate of the slow accompanying carbon-catalysed redox reaction. 

Many studies indicated that in the presence of DOM, the metal sorption capacity decreased markedly for most soils, and the effect on the calcareous soil was greater than on the acidic sandy loam. Figure 10.4 shows the metal sorption equilibrium isotherms onto soils with or without the addition of 400 mg C/l of DOM. The equilibrium isotherms could be better depicted according to the linear Freundlich equation with the high value for the correlation coefficient of determination (r2) ... 

In the cases of Langmuir and Freundlich isotherms, it has been assumed that the total disappearance rate of substrate per imit surface area, Vj, follows a pseudo-first-order kinetics with respect to the substrate concentration which is expressed by its fractional coverage. The same assumption is made for R-P isotherm however, as the R-P isotherm relates an adsorbed amount (and not a fractional coverage) with the equilibrium concentration... 

It is also possible that, because the surface of the catalyst is inhomogeneous, the equilibrium between H3VO4 vapor and surface vanadium may not follow the Langmuir isotherm. In this case, the vapor pressure can be modeled by a Freundlich isotherm. 

The adsorption capacity of activated carbon may be determined by the use of an adsorption isotherm. The adsorption isotherm is an equation relating the amount of solute adsorbed onto the solid and the equilibrium concentration of the solute in solution at a given temperature. The following are isotherms that have been developed Freundlich Langmuir and Brunauer, Emmet, and Teller (BET). The most commonly used isotherm for the application of activated carbon in water and wastewater treatment are the Ereundlich and Langmuir isotherms. The Freundlich isotherm is an empirical equation the Langmuir isotherm has a rational basis as will be shown below. The respective isotherms are ... 

In this paper, we present the unusual adsorption properties of ETS-10 towards heavy metal ions Pb and Cu. Adsorption equilibrium and kinetic data are reported. Fitting of the experimental equilibrium results to both Langmuir and Freundlich isotherms and the kinetic data to both pseudo-first- and pseudo-second-order kinetic models is described. 

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