Gaussian profile, chromatography

Figure 6.2 Comparison of the chromatogram given by the equilibrium-dispersive model of chromatography with a Gaussian profile. Dimensionless plot of = Ctn/Ap versus frf = f/tj . Solid line Gaussian profile with N theoretical plates. Dotted line equilibrium-dispersive model with an "open-open" boundary condition and (2D )-...

Using the results of Lapidus and Amundson [3], Van Deemter et al. [4] demonstrated that a simplification of considerable importance can be made to the solution derived by these authors if we assume that the mass transfer kinetics is not very slow, which is almost always the case in analytical or preparative applications of chromatography. Then, Eqs. 6.46 and 6.47 can be reduced to a Gaussian profile ... 

Figure 6.6 Comparison of the chromatogram given by the film mass transfer-pore diffusion model of chromatography with a Gaussian Profile. Dimensionless plot of versus f. Solid line Gaussian profile. Dotted line Carta s solution [34]. (a) Nap = N = 25 theoretical plates, (b) Nap = N = 100. Reprinted by permission of Kluwer Academic Publishing, from S. Golshan-Shirazi and G. Guiochon, NATO ASI Series C, vol 383, 61 (Fig. 4), with kind permission of Springer Science and Business Media.

The apparent plate munber can be calculated from the experimental profiles [27]. However, this number depends on the fractional height at which the bandwidth is measured. The value of Nth is calculated from the profiles predicted, under the same experimental conditions, by the ideal model. Finally, Nion is derived from the band profiles recorded in linear chromatography, e.g., with a very small sample size, using the relationships valid for Gaussian profiles. From Eqs. 7.24 and 7.26, we can derive the band width at half height, Wi/2, and the retention time of the band profile, ty, obtained with an infinitely efficient column. In the case of a Langmuir isotherm, we obtain [31]... 

The solute velocities are measured easily by analytical chromatography. When a small pulse is injected, the peak exits with a Gaussian profile but the peak maximum moves through the column at a velocity given by Eq. (14.1-5). The velocity is determined easily as... 

The analysis of linear chromatography with small poises usually is based on analysis of a Gaussian profile. The striution can be written... 

The theory of chromatography shows that there should be no noticeable difference between a recorded peak shape and a Gaussian profile so long as the number of theoretical plates of the column exceeds 100 [16]. The difference is small even for 25 plates, and careful experiments would be needed to demonstrate that difference. However, it is common to experience in practice peak profiles that are not truly Gaussian. 

As discussed previously (Chap. 10.4), the band profile in linear chromatography is Gaussian. The relative standard deviation of this Gaussian profile is used as a way of characterizing column efficiency. By definition... 

The solutions of the equilibrium-di.spersive model exhibit the same features as those of the ideal model at high conceniraiions. where thermodynamic effects are dominant and dispersion due to finite column efficiency merely smoothes the edges. When concentrations decrease, the solutions tend toward the Gaussian profiles of linear chromatography. 

For example, a Gaussian peak model is used to derive many fluently used fundamental equations in chromatography (17), but the Gaussian function rarely provides an accurate model for real chromatographic peaks. Convectional effects introduced by the flow cell can cause asymmetry in chromatographic peaks even if a Gaussian profile has been established prior to detection (18). One model diat is well accepted to represent a real chromato phic peak is an exponentially modified Gaussian (EMG) function. The FIA profile is characterized primarily by a dispersion coefficient (19) which offers information about die manifold, but direct information about the peak parameters such as second moment or variance can not be readily obtained. Recently, the EMG was used to describe FIA profiles to obtain the second moment and characterize the peak (20-22). 

The profile of the concentration of a solute in both the mobile and stationary phases is Gaussian in form and this will be shown to be true when dealing later with basic chromatography column theory. Thus, the flow of mobile phase will slightly displace the concentration profile of the solute in the mobile phase relative to that in the stationary phase the displacement depicted in figure 1 is grossly exaggerated to demonstrate this effect. It is seen that, as a result of this displacement, the concentration of solute in the mobile phase at the front of the peak exceeds the equilibrium concentration with respect to that in the stationary phase. It follows that there is a net transfer of solute from the mobile phase in the front part of the peak to the... 

The area of a peak is the integration of the peak height (concentration) with respect to time (volume flow of mobile phase) and thus is proportional to the total mass of solute eluted. Measurement of peak area accommodates peak asymmetry and even peak tailing without compromising the simple relationship between peak area and mass. Consequently, peak area measurements give more accurate results under conditions where the chromatography is not perfect and the peak profiles not truly Gaussian or Poisson. 

Since the concentration profile of a solute in the effluent from a chromatography column can be approximated by the Gaussian error curve, the peak width W (m ) can be obtained by extending tangents at inflection points of the elution curve to the base line and is given by... 

In analytical chromatography the column is typically run with very dilute sample mixtures. Therefore the chromatographic parameters generally remain within the range of the linear isotherm and are independent of the mass of sample loaded. The concentration profiles are symmetric and Gaussian (Fig. 2.18a). The sample mass m is given by Eq. 2.58,... 

In most cases, chromatography is performed with a simple initial condition, C(f = 0,z) = q t = 0,z) = 0. TTie column is empty of solute and the stationary and mobile phases are under equilibrium. There are some cases, however, in which pulses of solute are injected on top of a concentration plateau (see Chapter 3, Section 3.5.4). The behavior of positive concentration pulses injected xmder such conditions is similar to that of the same pulses injected in a column empty of solute and they exhibit similar profiles. Even imder nonlinear conditions (high plateau concentration), a pulse that is sufficiently small can exhibit a quasi-linear behavior and give a Gaussian elution profile. Its retention time is linearly related to the slope of the isotherm at the plateau concentration. Measuring this slope is the purpose of the pulse method of measurement of isotherm data. Large pulses may also be injected and they will give overloaded elution profiles similar to those obtained with a column empty of solute. 

---