Retention fundamental equation

The slope of a plot of the partition coefficient vs. the reciprocal of the temperature (in Kelvin) is PUR. This is the fundamental equation of gas and liquid chromatography. In our laboratory, we coat a capillary column with a polyurethane of interest and measure the retention time of chemicals passing through it. The retention time is colinear with the partition coefficient. 

Eqn.(1.6) is the fundamental equation for retention in chromatography. Throughout this book, extensive use will be made of the capacity factor as a convenient means to describe retention. A major advantage of the use of k for this purpose is the fact that it is a dimensionless quantity. It follows from eqn.(1.6) that... 

This expression for resolution may be written in an alternative form in terms of the relative retention Fj/Kj (Section 24-1) and the partition ratio k. Thus it can be shown that (24-34) is equivalent to the following fundamental equation involving three nearly independent factors ... 

The subscripts n and a in the above equation represent a molecule in a nonsorbed and adsorbed phase, respectively. In other words, retention in adsorption chromatography involves a competition between sample and solvent molecules for sites on the adsorbent surface. A variety of interaction energies are involved, and the various energy terms have been described in the literature [7,8], One fundamental equation that has been derived from the displacement model is... 

The fundamental equation for any chromatographic process, relating the retention volume Fr to other quantities, is... 

Equation (4.16) has been called the fundamental equation for chromatography. Each and every analyte of interest that is introduced into a chromatographic column will have its own capacity factor, k. The column itself will have a volume Fq. The retention volume for a given analyte is then viewed in terms of the number of column volumes passed through the column before the analyte is said to elute. A chromatogram then consists of a plot of detector response versus tg, where each analyte has a unique retention time if sufficient chromatographic resolution is provided. Hence, with reference to a chromatogram, the capacity factor becomes... 

This equation is one of the fundamental equations of analytical chromatography and offers the possibility to determine linear adsorption coefficients from analytical retention times. 

The resolution can also be predicted by substituting in the expressions for retention times and the standard deviations. Assuming that the N values are the same for the two conponents (a reasonable assunption since resolution is usually calculated for similar conpounds) the resulting fundamental equation of chromatography fGiddings. 1965 Wankat. 19901 is... 

The fundamental equations describing the particle retention behavior in a deep-bed filtration are the continuity equation, the rate equation, and the expression for pressure drop. The removal rate of suspended solids as a function of solid concentration is written as... 

Equation (5) is regarded as a fundamental equation of column chromatography as it relates the retention volume of a solute to its distribution ratio. Planar separations (PC and TLC). Separations are normally halted before the mobile phase has travelled completely across the surface, and solutes are characterized by the distance they have migrated relative to the leading edge of the mobile phase (solvent front). A solute retardation factor, Rf, is defined as... 

In fundamental SEC studies retention is often described in terms of a distribution coefficient. The theoretical distribution coefficient Kg is defined as the ratio of solute concentration inside and outside of the packing pores under size exclusion conditions. The experimental distribution coefficient as defined in Equation 1, is a measurable quantity that can be used to check the theory. 

During their passage through the column, sample molecules spend part of the time in the mobile phase and part in the stationary phase. All molecules spend the same amount of time in the mobile phase. This time is called the column dead tine or holdup time (t.) and is equivalent to the tine required for an unretained solute to reach the detector frsolute retention time (t,) is the time between the instant of saiq>le introduction and when the detector senses the maximum of the retained peak. This value is greater than the column holdup time by the amount of time the solute spends in the stationary phase and is called the adjusted retention time (t, ). These values lead to the fundamental relationship, equation (1.1), describing retention in gas and liquid chromatography. 

First, we will explore the three fundamental factors in HPLC retention, selectivity, and efficiency. These three factors ultimately control the separation (resolution) of the analyte(s). We will then discuss the van Deemter equation and demonstrate how the particle diameter of the packing material and flow rate affect column efficiencies. 

Figure 13.6 shows a schematic for IGC operation. Inverse, in this instance, refers to the observation that the powder is the unknown material, and the vapor that is injected into the column is known, which is inverse to the conditions that exist in traditional gas chromatography. After the initial injection of the known gas probe, the retention time and volume of the probe are measured as it passes through the packed powder bed. The gas probes range from a series of alkanes, which are nonpolar in nature, to polar probes such as chloroform and water. Using these different probes, the acid-base nature of the compound, specific surface energies of adsorption, and other thermodynamic properties are calculated. The governing equations for these calculations are based upon fundamental thermodynamic principles, and reveal a great deal of information about the surface of powder with a relatively simple experimental setup (Fig. 13.6). This technique has been applied to a number of different applications. IGC has been used to detect the following scenarios ...

Bartu [615] uses a different relationship to describe the retention vs. temperature relationship. His equation is also not fundamentally linear and requires a minimum of three parameters ... 

Equation (8) is a fundamental relationship for retention in LSC as a function of the solvent strength of the mobile phase. It states that log A values for different solutes will yield linear plots against values of for different mobile phases, and the slopes of these plots will be proportional to the molecular size A, of the solute. Numerous data are summarized or referenced in Ref. /. showing the validity of Eq. (8) when applied to LSC systems where the solute and solvent molecules are nonlocalizing (nonpolar or moderately polar compounds—see Section II,B below). Similar data showing the applicability of Eq. (8) to amino-phase polar-bonded-phase columns are given in Ref. 17. 

In addition to molecular weight, thermal FFF is used to measure transport coefficients. For example, the measurement of thermodiffusion coefficients is important for obtaining compositional information on polymer blends and copolymers (see the entry Thermal FFF of Polymers and Particles). Thermal FFF is also used in fundamental studies of thermodiffusion because it is a relatively fast and accurate method for obtaining the Soret coefficient, which is used to quantify the concentration of material in a temperature gradient. However, the accuracy of Soret and thermodiffusion coefficients obtained from thermal FFF experiments depends on properly accounting for several factors that involve temperature. In order to understand the effect of temperature on transport coefficients, as well as the effect on thermal FFF calibration equations, a brief outline of retention theory is given next. 

The overall quality of the separation of two solutes is measured by their resolution Rs), a combination of the thermodynamic factors causing separative transport and the kinetic factors causing dispersive transport and is an index of the effectiveness of the separa-tion. Defined by Rs = (r, - tr,a)l (w,b + where a and b refer to the two solutes, is the retention time of solute X, and is the peak width at the base of solute X in units of time, it is frequently estimated by use of the fundamental resolution equation. 

The accurate determination of the column void time, 0, is of fundamental importance in chromatography [1]. This is explained by the fact that a reliable estimation of this quantity is essential for the correct calculation of the retention factors (some refer to this as the capacity factor), k, which serves as the fundamental parameter for the comparison of retention data and for the interpretation of the physicochemical phenomena taking place within a chromatographic column. However, the determination of this parameter is very sensitive to the estimated value of the column void time, as can be seen from the equation... 

The principle of this pulse method and its general equations are easily extended to the case of several components in a mixture. The method was used by Lindholm et al. [24] to determine the quaternary isotherms of the enantiomers of methyl- and ethyl-mandelate on the chiral phase Chiral AGP. One of the serious roadblocks encountered in the use of the pulse tracer method is that the amplitudes of most of the system peaks decrease rapidly when the plateau concentration increases. Since the signal noise increases in the same time, it becomes rapidly impossible to make any accurate measurements of the retention time of these peaks. On the basis of fundamental work by Tondeur et al. [114], the origin of this variation of the relative intensity of the system peaks was explained by Forss n et al. [47], who then derived an effective rule to determine the composition of a perturbation pulse that generates system peaks that are detected easily. The concentrations of the components in the injected perturbation pulse should... 

Equation 10.115 has a considerable fundamental and practical importance. It combines parameters of fimdamentally different origins, the plate number at infinite dilution, N, which characterizes the intensity of axial dispersion taking place in the column and two parameters of thermod5mamic origin, the retention factor at infinite dilution, ICg, related to the initial slope of the isotherm, and the loading factor, proportional to the sample size and related to the saturation capacity of the isotherm. Accordingly, Eq. 10.115 indicates the extent to which the self-sharpening effect on the band profile due to the nonlinear thermodynamics is balanced by the dispersive effect of axial and eddy diffusion and of the mass transfer resistances. 

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