The Linear Isotherm

Shapes of different equilibrium adsorption isotherms 8.1.1 The Linear Isotherm 

The simplest isotherm is the linear isotherm. At very low concentrations the molecules adsorbed are widely spaced over the adsorbent surface so that one molecule has no influence on another. For these conditions an isotherm similar to Henry s law 

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The simplest relationship between sohd-phase and fluid-phase concentrations is the linear isotherm... 

FIGURE 2.1 The linear isotherm of adsorption and the corresponding Gaussian distribution of the analyte s concentration in the chromatographic band. 

Use of densitometric detection provides an insight into the concentration profiles of chromatographic bands, thus furnishing an indispensable prerequisite, needed for proper assessment of the retention mechanisms in the preparative adsorption TLC. Figure 2.4 shows three types of the band eoncentration profiles. The Gaussian peak (a) in this figure represents the linear isotherm of adsorption of a given species, peak... 

The asymptotic behavior of transitions under the influence of mass-transfer resistances in long, deep beds is important. The three basic asymptotic forms are shown in Fig. 16-2. With an unfavorable isotherm, the breadth of the transition becomes proportional to the depth of bed it has passed through. For the linear isotherm, the breadth becomes proportional to the square root of the depth. For the favorable isotherm, the transition approaches a constant breadth called a constant pattern. 

Once the transport step is defined, it is necessary to take into account which is the adsorption isotherm preceding the internalisation (the latter is always assumed to be first order). By extending results from Section 2, analytical solutions can be written for steady-state for both linear and Langmuir isotherms, as well as for the transient case with the linear isotherm. Langmuirian isotherms require numerical approaches for the transient case. 

For the linear isotherm model, the parameter (Kd) that relates both sorbate and solute is called the partition coefficient. A number of studies have developed empirical relationships for partition coefficients in natural solid phases and several of these studies are summarized in Table 1.Various theoretical-based methods of partition coefficient estimations also exist (Table 1, Eqs. a- f). 

The speed of the adsorption wave can be readily derived by introducing the linear isotherm assumption and the chain mle derivative of q with respect to t. The wave speed results because the assumptions turn Eq. (9.10) into a kinematic wave equation and the wave speed W is instantly recognized as ... 

Here K is the slope of the linear isotherm. Eor non-trace systems or for rather high loadings the true slope K may be replaced by the slope of the operating line or chord of the isotherm, which quantity is simply Aq/Ac. The equivalent expression in molar loadings and gas phase mole fractions is ... 

Alkaline earths Distribution coefficients for Sr(II) and Ba(II) as a function of concentration of Ca(II) are summarized in Figures 13a and 13b. The loadings in the case of Sr(It) (Figure 2) are in the linear isotherm range or only slightly out of it. Slopes of log D vs log calcium concentration are close to the ideal value, -1. Invariance of between 0.01 M and 0.1 M acetate buffer indicates that there is no interference from acetate complexing. At the same Ca(II) concentration, is a factor of... 

Figure 9.19 Aldrich humic acid (AHA)-water distribution ratio (Adoc) of TBT (A) and TPT ( ) as a function of pH. Each point was determined from a sorption isotherm. Error bars are the standard deviation of the slope of the linear isotherm and of pH. The lines were calculated using the model described by Arnold et al. (1998). The insert shows the speciation of TBT and TPT as a function of pH. Adapted from Arnold et al. (1997 and 1998).

The linear isotherm is obtained when the ratio of the concentration of substance adsorbed per unit mass and the concentration of the substance in solution remains constant. This means the partition coefficient or distribution constant, K (see Section 1.4) is a constant over all working concentration ranges. Thus, the front and rear boundaries of the band or zone will be symmetrical. 

This equation reduces to the linear isotherm at low concentrations, to the Freundlich isotherm at high concentrations, and to the Langmuir isotherm when 0=1. However, the equation cannot be linearized for easy estimation of isotherm parameters. 

Figure 3 summarizes the typical equilibrium data for fluoride sorption on 0.150-0.300 mm surface-tailored zeolite F9. Only the low-concentration range was considered where the Henry s law is applicable. Co-incidentally, fluoride ions exist in natural systems, such as groundwater, at low concentrations. It is observed that the data fit well to the linear isotherm suggesting sorption onto sites with high capacity for fluoride. From Fig. 3, the linear isotherm constant, K ( = 2.337 L/g), was obtained and coupled into the diffusion model, equation (16). 

It was clear that an equilibrium desorption time of more than 100 h is required for 1,3-DCB. Accordingly, three triplicate samples each of the PPI soil at three different initial soil concentrations were kept well-mixed with 40 ml of distilled water for 180 h. The desorption isotherm thus obtained is plotted in Fig. 6. It is clear that this desorption isotherm indicates that a portion of the compound remains bound to the soil. The linear isotherm representing this would consist of (i) a reversible portion obeying the conventional linear isotherm, Wrev= KSWCW, where Wrev is the soil concentration (pg/g), Ksw is the partition coefficient (1/g), and Cw is the aqueous phase concentration (pg/1), and (ii) a desorption... 

Figure 24. Neutralization isotherms (adsorption isotherm of OH") for acidic surface groups on a carbon black The lower plot represents the linearized isotherms according to a modified Langmuir adsorption model.

In analytical chromatography the column is typically run with very dilute sample mixtures. Therefore the chromatographic parameters generally remain within the range of the linear isotherm and are independent of the mass of sample loaded. The concentration profiles are symmetric and Gaussian (Fig. 2.18a). The sample mass m is given by Eq. 2.58,... 

In ideal chromatography, we assume that the column efficiency is infinite, or in other words, that the axial dispersion is negligibly small and the rate of the mass transfer kinetics is infinite. In ideal chromatography, the surface inside the particles is constantly at equilibrium with the solution that percolates through the particle bed. Under such conditions, the band profiles are controlled only by the thermodynamics of phase equilibria. In linear, ideal chromatography, all the elution band profiles are identical to the injection profiles, with a time or volume delay that depends only on the retention factor, or slope of the linear isotherm, and on the mobile phase velocity. This situation is unrealistic, and is usually of little importance or practical interest (except in SMB, see Chapter 17). By contrast, nonlinear, ideal chromatography is an important model, because the profiles of high-concentration bands is essentially controlled by equilibrium thermodynamics and this model permits the detailed study of the influence of thermodynamics on these profiles, independently of the influence of the kinetics of mass transfer... 

The linear isotherm, which is empirical, takes the form... 

Linear The constant K in the linear isotherm expression was determined by fitting that equation to the data. This was accomplished by minimizing the sum of the errors squared, X[q(data)-q(calculated)]T The resulting equation is... 

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