Nonlinear isotherms

The peak symmetry, resolution, and detector response are directly dependent on the concentration of the sample. As the concentration of a sample increases, the retention time, separation, and peak symmetry generally decrease. These phenomena are due to isotherm nonlinearity. The detector response may also be nonlinear above or below certain concentrations. In some cases, small amounts of a dilute component are irreversibly adsorbed to the column, leading to reduced recovery. Above some concentration, the response of any detector will cease to be linear. The UV-VIS is one of the most linear detectors, generally exhibiting at least three decades of linearity, while RI, electrochemical, and fluorimetric detectors have a markedly narrower range of linearity. 

Comparing the values of the two terms in the sum in the numerator indicates that DS is mainly partitioning into near-surface water due to its hydrophobicity the sum in the denominator implies that the dissolved DS concentration is already large enough to cause isotherm nonlinearity. 

Huang, W., T. M. Young, M. A. Schlautman, H. Yu, and W. J. Weber Jr., A distributed reactivity model for sorption by soils and sediments. 9. General isotherm nonlinearity and applicability of the dual reactive domain model , Environ. Sci. Technol., 31, 1703-1710 (1997). 

Wilson, J. N. First theoretical paper on chromatography. Assumed complete equilibration and linear sorption isotherms. Qualitatively defined diffusion, rate of adsorption and isotherm nonlinearity. 

Several papers have shown that isotherm nonlinearity is indeed an important factor in MIP HPLC [31-35]. The consequences will be dealt with in the next sections. Mass transfer kinetics, on the other hand, has not yet been sufficiently studied. Its role has been proven in some cases [36] but it is not known how general these observations are and how important the effects are generally. 

Any types of adsorption isotherm functions do not have any region with constant derivative, and natural existence of this region would mean the break in the continuous variation of system properties, which could not exist. This quasi-linear region essentially means the region where the dispersion processes (diffusion, band broadening) significantly exceeded the effects of isotherm nonlinearity, and chromatographic peaks appear almost symmetrical (within the accuracy of our detection and data acquisition system). 

Most determination methods finally lead to discrete loading versus concentration data that have to be fitted to a continuous isotherm equation. For this purpose it is advised to use a least-squares method to obtain the parameters of the isotherm. Nonlinear optimization algorithms for such problems are implemented in standard spreadsheet programs. To select an isotherm equation and obtain a meaningful fit,... 

Composite Sorption Magnitude and Isotherm Nonlinearity. Accurate assessment of the extent to which the global isotherm for a system is nonlinear is important for accurate portrayal of sorption processes in that system. From a practical point of view, the extrapolation of linear approximations of weakly nonlinear or near-linear sorption isotherms to concentration ranges beyond which they are valid can result in significant errors in projections of contaminant fate and transport (1). From a conceptual point of view, observations of isotherm nonlinearity over specific concentration ranges may be employed in conjunction with models such as the DRM to probe and evaluate the extent to which multiple sorption mechanisms are operative in a particular system. 

The sorption of TCB by another Michigan soil designated Ann Arbor II, which was collected closer to the surface, is also shown in Figure 7. The carbon-normalized sorption here is equal to or lower than that predicted for partitioning into an organic phase, using the average value of 1752 cm3/g from Table I. The extent of isotherm nonlinearity in = 0.86) is lower for this near-surface soil than for the isotherms of the subsoil and shale isolate. 

Interaction between adjacent adsorbed sample molecules (2) as a contribution to isotherm nonlinearity is relatively unimportant, particularly in liquid-solid chromatography. These interactions are normally possible "fenly at relatively high surface coverages, where linearity is already limited by surface saturation [Eq. (4-1)] and by adsorbent heterogeneity. Furthermore, interactions between adsorbed sample molecules are largely canceled... 

Sample reaction or chemisorption on the adsorbent bed (3) is generally associated with the strong adsorption sites or small pores which are responsible for isotherm nonlinearity (1) and excessively large sample retention volumes (2). The use of adsorbent deactivation or homogeneous surface adsorbents hence serves as a simultaneous answer to all three problems (1-3). Adsorbent standardization (4) is still a significant problem in both GSC and LSC, but much progress has been made recently (see Section 6-3). Once a satisfactory GSC column has been prepared, moreover, its lifetime will generally exceed that of a GLC column many times over. 

The Langmuir isotherm is by definition free from interactions between adsorbate molecules (2) and surface heterogeneity (3). It is therefore an ideal model for illustrating the contribution of surface saturation (1) to isotherm nonlinearity. Beginning with the Langmuir isotherm for liquid-solid systems [Eq. (3-5a)], we can define a distribution coefficient K equal to S/iVa , so that ... 

In nonlinear adsorption systems where parallel diffusion mechanisms hold, the mass balance equation given in eq. (9.2-lb) is still valid. The difference is in the functional relationship between the concentrations of the two phases, that is the local adsorption isotherm. In general, this relationship can take any form that can describe well equilibrium data. Adsorption isotherm such as Langmuir, Unilan, Toth, Sips can be used. In this section we present the mathematical model for a general isotherm and then perform simulations with a Langmuir isotherm as it is adequate to show the effect of isotherm nonlinearity on the dynamics behaviour. The adsorption isotherm takes the following functional form ... 

It is important to note that the mass balance equation (9.2-45a) contains the concentration-independent surface diffusivity. In general this surface diffusivity is a function of concentration when the isotherm is nonlinear. We shall postpone the treatment of concentration-dependent D, and assume its constancy for the study of the effects of isotherm nonlinearity on the system dynamic behaviour. 

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