Equilibrium isotherms linear

Fig. 11. (a) Equilibrium isotherm and (b) dimensionless equilibrium diagram showiag favorable, linear, and unfavorable isotherms. 

Two variations of the technique exists isocratic elution, when the mobile phase composition is kept constant, and gradient elution, when the mobile phase composition is varied during the separation. Isocratic elution is often the method of choice for analysis and in process apphcations when the retention characteristics of the solutes to be separated are similar and not dramaticallv sensitive to vei y small changes in operating conditions. Isocratic elution is also generally practical for systems where the equilibrium isotherm is linear or nearly hnear. In all cases, isocratic elution results in a dilution of the separated produces. 

Fig. 9-7. Region for complete separation under Equilibrium Theory. Linear adsorption isotherms.

It should be stressed that in the case of linear isotherm, the peak broadening effect results from eddy diffusion and from resistance of the mass transfer only, and it does not depend on Henry s constant. In practice, such concentration profiles are observed for these analyte concentrations, which are low enough for the equilibrium isotherm to be regarded as linear. 

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] ... 

Due to the simplicity of this illustrative problem and of the linear equilibrium isotherm Y = aX+b we can reduce the two coupled two point boundary value differential equations (6.127) to one differential equation as follows. 

At sufficiently low concentrations on a homogeneous surface the equilibrium isotherm for physical adsorption will always approach linearity (Henry s law). The limiting slope of the isotherm [limp o(dq/dp)T] is referred to as the Henry constant K . It is evident that the Henry con-... 

FIGURE 6 (a) Equilibrium isotherms and (b) dimensionless equilibrium diagram showing distinction between favorable, unfavorable, and linear systems. (Reprinted with permission from Ruthven, D. M. (1984). Principles of Adsorption and Adsorption Processes, copyright John Wiley Sons, New York.)... 

If the equilibrium is linear, exact analytical solutions for the column response can be obtained even when the rate expression is quite complex. In most of the published solutions, axial dispersion is also neglected, but this simplification is not essential and a number of solutions including both axial dispersion and more than one diffusional resistance to mass transfer have been obtained. Analytical solutions can also be obtained for an irreversible isotherm with negligible axial dispersion, but the case of an irreversible isotherm with significant axial dispersion has not yet been solved analytically. 

The operation is most easily understood by reference to the equivalent true countercurrent system (Fig. 15). If we consider a feed containing two species A and B, with A the more strongly adsorbed, and a desorbent C, then in order to obtain separation the net flow directions in each section must be as indicated. With the equilibrium isotherms and the feed composition and flow rate specified, this requirement in effect fixes all flow rates throughout the system as well as the adsorbent recirculation rate or switch time. From simple theoretical considerations it can be easily shown that the affinity of the adsorbent for the desorbent should be intermediate between that for the strongly and weakly adsorbed feed compounds (i.e., a c > 1 -0, bc < 1 -0). The heights of the individualized bed sections are then determined by the requirement that each section contain sufficient theoretical plates to achieve the required purity of raffinate and extract products. For a linear system the analysis is straightforward since simple expressions for the concentration profile are available in terms of the kinetic and equilibrium... 

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) ... 

Figure 8. Langmuir plot for 2,4,5-T In accord with the Langmuir model for adsorption equilibrium, a linear trace is obtained for the isotherm for 2,4,5-T when (X) 1 is vlotted vs. (Ceq) 1. Intercept of plot is (Xm) 1, and the slope is (bXm) K...

Nonlinear optimization techniques have been applied to determine isotherm parameters. It is well known (Ncibi, 2008) that the use of linear expressions, obtained by transformation of nonlinear one, distorts the experimental error by creating an inherent error estimation problem. In fact, the linear analysis method assumes that (i) the scatter of points follows a Gaussian distribution and (ii) the error distribution is the same at every value of the equilibrium liquid-phase concentration. Such behavior is not exhibited by equilibrium isotherm models since they have nonlinear shape for this reason the error distribution gets altered after transforming the data... 

This rate equation must satisfy die boundary conditions imposed by the equilibrium isotherm and it must be thermodynamically consistent so that the mass transfer rate falls to zero at equilibrium It may be a linear driving force expression of the form... 

To illustrate this integration analytically, Eq. (22-48) becomes Eq. (22-49) for the case of a stripping column removing a colligend which is subject to the linear-equilibrium isotherm E = KC. 

Table 3.24 shows the computed data for k, for both solid/liquid ratios and the mean values if we consider the hypothesis of a linear equilibrium isotherm. 

Dual-temperature based theory for linear equilibrium isotherms has been developed [26]. From the results of this analysis the optimal value of flow ratio in the simplest case of nonselective conditions for concentrating the least strongly sorbed component can be found with the following expression ... 

Equilibrium Data. Finally, equilibrium isotherms were established from the measurements made in this study. These are plotted in Figure 4. Clearly, the results indicate the isotherms to be essentially linear except at 25.2 °C. Subsequent work at higher ethane concentrations has con-... 

The strong dependence of the isotherm parameters on the composition of the mobile phase conJfirms that adsorption of small molecules on chemically bonded Cis silica is more complex than is usually beUeved. Most prior work has been based on data acquired under linear conditions, that is, at infinite dilution. Chro-matographers have long ignored the way in which consideration of the whole equilibrium isotherm and its modeling may inform on the detail of the retention mechanisms involved [131]. 

All cases of practical importance in liquid chromatography deal with the separation of multicomponent feed mixtures. As shown in Chapter 2, the combination of the mass balance equations for the components of the feed, their isotherm equations, and a chromatography model that accounts for the kinetics of mass transfer between the two phases of the system permits the calculation of the individual band profiles of these compounds. To address this problem, we need first to understand, measure, and model the equilibrium isotherms of multicomponent mixtures. These equilibria are more complex than single-component ones, due to the competition between the different components for interaction with the stationary phase, a phenomenon that is imderstood but not yet predictable. We observe that the adsorption isotherms of the different compounds that are simultaneously present in a solution are almost always neither linear nor independent. In a finite-concentration solution, the amount of a component adsorbed at equilib-... 

The importance of linear chromatography comes from the fact that almost all analytical applications of chromatography are carried out xmder such experimental conditions that the sample size is small, the mobile phase concentrations low, and thus, the equilibrixim isotherm linear. The development in the late 1960s and early 1970s of highly sensitive, on-line detectors, with detection limits in the low ppb range or lower, permits the use of very small samples in most analyses. In such cases the concentrations of the sample components are very low, the equilibrium isotherms are practically linear, the band profiles are symmetrical (phenomena other than nonlinear equilibrium behavior may take place see Section 6.6), and the bands of the different sample components are independent of each other. Qualitative and quantitative analyses are based on this linear model. We must note, however, that the assumption of a linear isotherm is nearly always approximate. It may often be a reasonable approximation, but the cases in which the isotherm is truly linear remain exceptional. Most often, when the sample size is small, the effects of a nonlinear isotherm (e.g., the dependence of the retention time on the sample size, the peak asymmetry) are only smaller than what the precision of the experiments permits us to detect, or simply smaller than what we are ready to tolerate in order to benefit from entertaining a simple model. 

The hypothesis of linear behavior of the equilibrium isotherm in analytical chromatography has three important consequences. First, the different components contained in a sample of a mixture behave independently of each other. They do not compete for interaction with the stationary phase because the sample size is small and the solutions are dilute. Therefore, the elution profiles and the retention times of the various components of a mixture are independent of the presence of other solutes and of their relative concentrations. Each band profile is the same as if the corresponding solute were alone, pure. As a consequence and in contrast with nonlinear chromatography, there is only one problem to solve in linear chromatography, the determination of the peak profile of a single component. 

If the isotherm is supposed to be linear, the equilibrium isotherm does not intervene in the band profile and the global effect is derived from the flow properties. The characteristic method applies. It shows that, in linear gas-chromatography, although the isotherm is linear, the sorption effect causes the velocity associated with a given concentration to decrease with increasing concentration. [22]. A slice of mobile phase having a given mole fraction, X, moves with the velocity... 

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