Gaussian band broadening

Assuming a Gaussian profile, the extent of band broadening is measured by the variance or standard deviation of a chromatographic peak. The height of a theoretical plate is defined as the variance per unit length of the column... 

The efficiency, or plate count of a column N is often calculated as 5.54 (tr/a)2, where tr is the retention time of a standard and a is the peak width in time units at half-height.1 2 5 This approach assumes that peaks are Gaussian a number of other methods of plate calculation are in common use. Values measured for column efficiency depend on the standard used for measurement, the method of calculation, and the sources of extra-column band broadening in the test instrument. Therefore, efficiency measurements are used principally to compare the performance of a column over time or to compare the performance of different columns mounted on the same HPLC system. 

As readily observed in most chromatograms, peaks tend to be Gaussian in shape and broaden with time, where W, becomes larger with longer This is caused by band-broadening effects inside the column, and is fundamental to all chromatographic processes.The term, plate number (N), is a quantitative measure of the efficiency of the column, and is related to the ratio of the retention time and the standard deviation of... 

In terms of organization, the text has two main parts. The first six chapters constitute generic background material applicable to a wide range of separation methods. This part includes the theoretical foundations of separations, which are rooted in transport, flow, and equilibrium phenomena. It incorporates concepts that are broadly relevant to separations diffusion, capillary and packed bed flow, viscous phenomena, Gaussian zone formation, random walk processes, criteria of band broadening and resolution, steady-state zones, the statistics of overlapping peaks, two-dimensional separations, and so on. 

Fig. 26. Total and atom projected (P)DOS curves of the valence band region fo the M015O56H22 based vacancy cluster with an 0(3) surface vacancy, (a) Total DOS (thick solid) with decomposition into Mo (thin solid) and O (dashed) contributions, (b) PDOSs of the differently coordinated oxygen centers 0(1) (solid), 0(2) (dotted), and 0(3) (dashed). A gaussian level broadening of 0.4 eV is applied and the energetic position of the highest occupied cluster orbital Ehomo at -5.3 eV is marked by a thin vertical line.

Since the retention of the template on the imprinted polymers strongly depends on the sample load (see later Fig. 5.25), the theoretical models describing the various dispersion processes are not applicable. Nevertheless, on an imprinted CSP for L-PA, the least retained enantiomer, D-PA, elutes as a fairly Gaussian peak (Fig. 5.3), which should be governed by the same band broadening mechanisms that are considered in the models. A measure of the dispersion of a Gaussian peak is the deviation from its mean value, which is reflected in the reduced plate height as ... 

As previously indicated, this discussion is organized for chromatograms from very narrow polymer standards for which we can consider that the effect of molecular weight distribution is negligible and for which the unique separation process is size exclusion. With these limitations, the contribution to band broadening is conveniently separated into extra column effects, eddy dispersion, static dispersion, and mass transfer. In the most classical chromatographic interpretation, extra-column effects are not discussed and the three other contributions are considered as Gaussian, so there is simply the addition of their variances. The number of theoretical plates is defined as N = VJaY and the influence of v, the linear velocity of the eluent, is summarized by the so-called Van Deemter equation ... 

The retention time tRjin and the second moment for the Gaussian profile (Eq. 6.61) have been replaced by variables indexed with g . These parameters tg and og must be optimized by curve fitting. Equation 6.143 is only suitable for symmetric peaks. Analytical solutions of, for example, the transport dispersive model (which describes asymmetric band broadening only for a very low number of stages) are not suited to describing the asymmetry often encountered in practical chromatograms. Thus, many different, mostly empirical functions have been developed for peak modeling. A recent extensive review by Marco and Bombi (2001) lists over 90 of them. 

The study of the lumped kinetic models shows that, as long as the equilibration kinetics is not very slow and the column efficiency exceeds 25 theoretical plates (a condition that is satisfied in all the cases of practical importance), the band profile is a Gaussian distribution. We can thus identify all independent sources of band broadening, calculate their individual contributions to the variance of the Gaussian distribution, and relate the column HTTP to the sum of these variances. The method is simple and efficient. It has been used successfully for over 40 years [29]. We may want a more rigorous approach. 

Column efficiency refers to the amount of band broadening that occurs when a compound passes through the column. The typical Gaussian shape of the chromatographic band results from the random motions of the molecules making up the band as it moves down the column. The result of these random individual motions is a symmetric spread of velocities around the mean value. This spreading is referred to as molecular diffusion. 

The plate height lumps together the contributions of fluid dynamic nonidealities (e.g., axial dispersion), mass transfer resistances, and finite adsorption and desorption rates, which all contribute to undesired band broadening. It can be defined as the rate of the local gradient of the width of a Gaussian peak (Equation 2.31) ... 

Let us consider the impact of this on chromatography by imagining ourselves in three different situations travelling down an HPLC column along with a sample of solute that has just been injected onto the column. As the band-broadening effects described previously start to become obvious, a Gaussian distribution of solute concentrations starts to develop. If there is no effect of concentration on K (a linear sorption isotherm), then all that happens is that the band continues to broaden symmetrically as it travels down the column. A symmetrical Gaussian peak will elute from the column. 

As a solute is eluted, the distribution of its molecules along the longitudinal axes of the column changes, generating a Gaussian-like profile a band broadening effect whose maximum is known as time of retention as depicted in Fig. 1. This parameter, corrected for the time of retention of a solute, that is, not retained by the column (dead time, is related to the solute capacity factor k) by the relation ... 

This effect can be corrected by injecting a narrow MWD sample and measuring the variance of the peaks in each detector. Because the peak shape is nearly Gaussian, it should, ideally, be the same for all detectors. If it is not, the additional variance can be calculated for one of the detectors. In subsequent data analysis, the narrower peak can be digitally broadened using Gaussian band spreading to correct... 

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