Adsorption, coefficient linear isotherms

The values of q are plotted as a function of the equilibrium concentration. For constituents at low or moderate concentrations, the relationship between q and C can be generated. If n = 1, the (q-C) relationship will be linear (Eq. 9), and the slope of the line (i.e.,ITd) defines the adsorption distribution of the pollutant. Kd is generally identified as the distribution or partition coefficient, and is used to describe pollutant partitioning between liquid and solids only if the reactions that cause the partitioning are fast and reversible, and if the isotherm is linear. For cases where the partitioning of the pollutants can be adequately described by the distribution coefficient (i. e.,fast and reversible adsorption, with linear isotherm), the retardation factor (R) of the subsurface environment can be used as follows ... 

Axial Dispersion Effects In adsorption bed calculations, axial dispersion effects are typically accounted for by the axial diffusionhke term in the bed conservation equations [Eqs. (16-51) and (16-52)]. For nearly linear isotherms (0.5 < R < 1.5), the combined effects of axial dispersion and mass-transfer resistances on the adsorption behavior of packed beds can be expressed approximately in terms of an apparent rate coefficient for use with a fluid-phase driving force (column 1, Table 16-12) ... 

The distribution coefficient assumes that adsorption is linear (i.e., the amount of adsorption is directly proportional to the concentration of the compound in solution) and is actually a special case of the Langmuir and Freundlich isotherms, which are nonlinear.31 32... 

The macroscopic proton coefficient may be determined by graphical analysis of observed system variables according to two different procedures fractional adsorption edge linearization (6) and isotherm analysis (7 ). The procedures for calculating the macroscopic proton coefficients according to these two methods are discussed in detail below, as are their relative advantages and disadvantages for use in semi-empirical descriptions of adsorption. 

TABLE 4.6. Freundlich Adsorption Constants (1In and K) and Distribution Coefficients (Kd) for Adsorption of Alachlor (Generally Linear Isotherms), Imazethapyr (Nonlinear Freundlich Isotherms), and Rimsulfuron (Langmuir Isotherms) on Humic Acids (HAs) Isolated from Two Sewage Sludges (SSI and SS2), a Soil Amended with 10tha 1yr 1 of SSI for 2 Years (SOI + SSI), and a Soil Amended with 40(ha 1yr 1 of SS2 for 2 Years (S02 + SS2), with the Corresponding Unamended Soils (SOI and S02, Respectively) (from Senesi et al., 2001)... 

Approximation of the linear form is not necessary for the Langmuir isotherm, and the first plot of the adsorption data will determine whether or not the model is applicable, and also will allow calculation of the adsorption coefficients. Usually a single model will not be satisfactory for a wide range of adsorbate concentrations but will only serve in narrow range of concentration. At low concentrations, C/C 1, the BET model reduces to a Langmuir model. 

Distributed Reactivity Model. Isotherm relationships observed for natural systems may well be expected to reflect composite sorption behavior resulting from a series of different local isotherms, including linear and nonlinear adsorption reactions. For example, an observed near-linear isotherm might result from a series of linear and near-linear local sorption isotherms on m different components of soft soil organic matter and p different mineral matter surfaces. The resulting series of sorption reactions, because they are nearly linear, can be approximated in terms of a bulk linear partition coefficient, KDr that is... 

We compare in Figure 6.20 two profiles that were calculated as numerical solutions of the equilibrium-dispersive model, using a linear isotherm. The first profile (solid line) is calculated with a single-site isotherm q = 26.4C) and an infinitely fast A/D kinetics (but a finite axial dispersion coefficient). The second profile (dashed line) uses a two-site isotherm model q — 24C - - 2.4C), which is identical to the single-site isotherm, and assumes infinitely fast A/D kinetics on the ordinary sites but slow A/D kinetics on the active sites. In both cases, the inverse Laplace transform of the general rate model given by Lenhoff [38] (Eqs. 6.65a to h) is used for the simulation. In the case of a surface with two t5q>es of adsorption sites, Eq. 6.65a is modified to take into accoimt the kinetics of adsorption-desorption on these two site types. 

Adsorption. The solution used to evaluate the pesticide transport equation, Equation 4a, assumes a linear adsorption isotherm that is constant with depth. However, linearity may not be the case for some pesticides and the adsorption coefficient will almost never be constant with depth. The rationale for using a linear model is initially based on the Freundlich isotherm... 

Sorption Distribution Coefficient. Measurements of DBCP adsorption on soils from the Kunia site at solution concentrations ranging from about 0.25 yg/ml to 290 y g/ml indicated that a linear isotherm described sorption reasonably well when sorption data were fitted with the Freundlich equation, S = KfCeN, the values of N on soils from three depths were 0.92, 0.76 and 0.95 (1 ). Subsequently,... 

Fig. 10. UOP Sorbex operation with linear isotherms. Slope of = Klm, slope of (D = K2. Conditions for separation Kl > K2i Lzj S > Kly k2 < l2/s < k19 k2 Ll — L2 = F where K = adsorption coefficient, L = net liquid flow rate, S = net solids flow rate, and...

There are certain conditions that must be fulfilled if Eqs. (2.2), (2.3) and (2.4) are to be used to calculate partition coefficients. The basic assumption is that the individual retention mechanisms are independent and additive. This will be true for conditions where the infinite dilution and zero surface coverage approximations apply or, alternatively, at a constant concentration with respect to the ratio of sample size to amount of liquid phase. The infinite dilution and zero surface coverage approximations will apply to small samples where the linearity of the various adsorption and partition isotherms is unperturbed and solute-solute interactions are negligible. The constancy of the solute retention volume with variation of the sample size for low sample amounts and the propagation of symmetrical peaks is a reasonable indication that the above conditions have been met. For asymmetric peaks, however, the constant concentration method must be employed if reliable gas-liquid partition coefficients are to be obtained [191]. It is difficult to state absolutely the conditions for which contributions to retention from the structured liquid phase layer can be neglected. This will occur for some minimum phase loading that depends on the support surface area, the liquid phase... 

The linear adsorption isotherm curve can be described by the linear adsorption coefficient K. Peak shape and peak position are independent of the concentration. The influence of kinetic effects can be treated independently from the thermodynamic equilibrium. 

Ka is the mass transfer coefficient and / is the function describing a particular isotherm providing a relation between solute concentration in stationary phase Cs) and equilibrium composition of solute in mobile phase. Equation 8.16, 8.20, and 8.21 can be solved simultaneously with the following boundary conditions to obtain breakthrough curves C versus t) for adsorption. This is illustrated in the following example for linear isotherm. 

In chapter 4, we discuss the adsorption of linear and branched alkanes in the zeolite Silicalite. We have used the simulation techniques described in the previous chapters for this. Silicalite has a three dimensional channel structure which consists of straight and zigzag channels that cross at the intersections (see figures 1.1 en 4.1). To compute the adsorption behavior, we have fitted a force field which is able to reproduce the Henry coefficient (adsorption isotherm at low pressure) and the heat of adsorption. From CBMC simulations it turns out that linear alkanes can occupy all channels of Silicalite. For u-Cg en u-C/, the length of the molecule is almost identical to the length of the zigzag channel. In literature, this process is called commensurate freezing and causes an inflection in the adsorption isotherm of these molecules. This effect has also been observed experimentally. 

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