Retention linear isotherm

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... 

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... 

Equation 3.9 is a linear isotherm. The concentration of the solute in the stationary phase is proportional to that in the mobile phase. This isotherm is widely used in analytical chromatography, where it gives results that are most often satisfactory. When the concentration becomes high, however, deviations from linear behavior take place, competitive interactions between the different components of the feed appear, and a more complex model becomes necessary to accormt for these experimental results. As an example. Figure 3.2 [20] shows a comparison between a linear isotherm and two nonlinear models (see below). The difference is small at concentrations below 0.05 mM, but significant deviations take place at 0.2 mM. They are sufficient to cause an important decrease in the band retention and a marked asymmetry of its profile [20]. 

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 second consequence of the assmnption of a linear isotherm is to make simple the mathematics of describing the migration of these independent, individual bands and of calculating their retention times and profiles. As we show later in this chapter, an analytical solution or, at least, a closed-form solution in the Laplace domain can be obtained with any model of linear chromatography. This is certainly not the case in nonlinear chromatography. 

Figure 13.9 shows the chromatogram calculated for the injection of a small vacancy in a chromatographic system when the mobile phase contains two additives at low concentrations (linear isotherms). The two peaks are imcoupled and each one elutes at a time corresponding to its retention factor at infinite dilution. The chromatogram is the same as the one obtained with a small injection of the same mixture on the same column, with a pure mobile phase, except that here the peaks are negative. 

Since, in this section, we consider only a linear isotherm, this velocity is independent of the concentration and equal to (uj + Ms)/(I + 9 where k( = Faj is the retention factor of component i. 

Ug linear isotherm equivalent retention volume in GSC (milliliters per grami ... 

The dependence of chromatographic separation on sample size is similar for all elution or bed development processes. Figure 4-1, serves as a general example, whether we are concerned with gas-liquid, partition, paper, ion exchange, or adsorption chromatography. For small amounts of sample, as in Fig. 4-1 (a), sample retention volumes (or Rf values) are constant for variation in sample size (linear isotherm separation). As the sample size is increased, however, a point is at last reached [Fig. 

Thus, for linear isotherms the retention times do not depend on concentration (Lapidus and Amundson, 1952 Van Deemter, Zuiderweg, and Klinkenberg, 1956). 

Band broadening effects such as dispersion and mass transfer resistance are represented by the number of tanks (or stages) N. This can be explained by evaluating the moments of the analytical solution of Equation 6.98. For linear isotherms and the injection of an ideal Dirac pulse of one component, this equation yields a gamma density function for the concentration profile. With the retention time Ir lin.i of Equation 6.49 one obtains the elution profile as the time dependence of the concentration in the last tank (k= N) ... 

It should be noted here that this chapter concentrates primarily on the retention mechanisms and the factors that affect retention levels. An extensive analysis of the effects of dynamic retention on polymer effluent profiles is not presented here since this is covered in Chapter 7 along with other polymer transport effects. Issues such as the effects of linear and non-linear isotherms and equilibrium and non-equilibrium adsorption on polymer core effluents are also discussed in more detail in Chapter 7, in which the appropriate polymer transport equations are developed. 

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. 

The simplest mode of IGC is the infinite dilution mode , effected when the adsorbing species is present at very low concentration in a non-adsorbing carrier gas. Under such conditions, the adsorption may be assumed to be sub-monolayer, and if one assumes in addition that the surface is energetically homogeneous with respect to the adsorption (often an acceptable assumption for dispersion-force-only adsorbates), the isotherm will be linear (Henry s Law), i.e. the amount adsorbed will be linearly dependent on the partial saturation of the gas. The proportionality factor is the adsorption equilibrium constant, which is the ratio of the volume of gas adsorbed per unit area of solid to its relative saturation in the carrier. The quantity measured experimentally is the relative retention volume, Vn, for a gas sample injected into the column. It is the volume of carrier gas required to completely elute the sample, relative to the amount required to elute a non-adsorbing probe, i.e. 

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