Gaussian zones

Equation 3.42 is a partial differential equation that will, upon solution, yield concentration as a function of time and distance for any sample pulse undergoing uniform translation and diffusion. In theory, we need only specify the initial conditions (i.e., the mathematical shape of the starting peak) along with any applicable boundary conditions and apply standard methods for solving partial differential equations to obtain our solutions. These solutions tend to be unwieldy if the initial peak shape is complicated. Fortunately, a majority of practical cases are described by a relatively simple special case, which we now describe. 

We begin, then, by writing concentration c as a Gaussian (or normal distribution) profile along coordinate y 

This equation—which is in fact the basic differential equation dc/dt = D(d2c/dy2) applied to our assumed solution—must be valid at all points of space that is, the two sides must be equal for all values of space coordinate y. Therefore terms containing various powers of y can be grouped to form individual equations. First, the coefficients of y2 are equated 

However, since g(t) must approach infinity at t = 0 to meet the postulated 8-function starting condition, the constant of integration must be zero. Therefore, g(t) is simply 

The second group of terms is free of dependence on y. The collection of these terms yields 

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FIGURE 2.6 Segmenting the concentration signal over a Gaussian zone with 1,3,10, and 20 samples taken in phase to the main peak. The area under these segmented curves is equal. This figure is taken from Murphy (1998). Reprinted with permission from the American Chemical Society. 

The standard deviation of the Gaussian zones expresses the extent of dispersion and corresponds to the width of the peak at 0.607 of the maximum height [24,25]. The total system variance (ofot) is affected by several parameters that lead to dispersion (Eq. 17.22). According to Lauer and McManigill [26] these include injection variance (of), longitudinal (axial) diffusion variance (of), radial thermal (temperature gradient) variance (of,), electroosmotic flow variance (of,), electrical field perturbation (electrodispersion) variance (of) and wall-adsorption variance (of ). Several authors [9,24,27-30] have described and investigated these individual variances further and have even identified additional sources of variance, like detection variance (erf,), and others... 

Figure 5.3. Gaussian zone showing standard deviation a (from the center line to the point of inflection), 2cr, and effective zone width tv.

Figure 5.5. A Gaussian zone emerges from a column. In the time r the zone is displaced one standard deviation or through the column end. Time r is thus the standard deviation in time units.

Figure 5.8 shows pairs of Gaussian zones at different resolution levels. The top row shows how peaks of equal height disengage from one another as Rs increases. Two distinct maxima are found only when / ,>0.5. Baseline resolution is found only for Rs > 1.5. 

Figure 5.8. Left to right sequence gives the profile of a pair of Gaussian zones of equal a at increasing levels of resolution, as shown. Top row shows this sequence for two zones of equal peak height middle and bottom rows show the sequence for 2 1 and 5 1 peak height ratios (profiles courtesy of Joe M. Davis).

We should add that while resolution and peak capacity are excellent criteria of merit for the separation of multicomponent mixtures into discrete zones, other criteria exist, some very general, for judging the efficacy of separation and purification in any separative operation (see Section 1.4). Various terms such as impurity ratio and purity index abound. Rony has developed a criterion termed the extent of separation [22]. Stewart, as well as de Clerk and Cloete, have shown that entropy can be formulated as a very general measure of separation power, as we might expect from the discussion of Section 1.6 [23,24]. An excellent discussion of separation indices, with an emphasis on non-Gaussian zones (below), is found in Dose and Guiochon [25]. 

A Gaussian zone is the model around which most discussions of zonal separation methods revolve. However, there are frequent departures from... 

The Gaussian exponential function, exp(-y2/2cr2), remains finite for all finite values of coordinate y. If interpreted literally, this means that absolute purification is not possible with Gaussian zones because, no matter what their separation, each is contaminated by the residual finite concentration of the other. To get perspective on this problem, assume that the concen-... 

It is clear that as one moves outward from the center of a Gaussian zone, concentration falls off. It is less obvious that this concentration falls off at an accelerating pace. Imagine moving out from y = 0 in steps of length a so that after it steps, y = nobserved after taking step n +1 compared to the concentration c encountered in the previous (or nth) step is given by... 

Figure 6.1. Steady-state Gaussian zone formed in methods such as isoelectric focusing and isopycnic sedimentation by the opposing interplay of a focusing force and diffusion. Different components focus at different locations to give separation.

For albumin, assume that k- 7.5 x 105 cal mol"1 cm"2 (refer to previous problem) and that molecular diffusion alone causes zone broadening (DT = D). Calculate the effective width 4a of the resulting Gaussian zone at 20° C. 

A small spot containing exactly 1 fig of leucine on a thin-layer plate has evolved into a two-dimensional Gaussian zone with <7X = 1.25 mm and spot length a and width 6. Compare these values to the width, 4o of the underlying Gaussians. 

It was shown in Section 6.1 that a component focused at y = 0 by the velocity expression W-U --ay would form a Gaussian zone centered at y- 0. Equation 6.18 shows that the molecular diffusion coefficient D, yielding... 

Factors leading to non-Gaussian zones in separation systems were described generally in Section 5.9. One source of zone asymmetry identified was the variation of local solute velocity W with solute concentration, described as overloading. The way in which overloading causes zone asymmetry in chromatography is explained below. 

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. 

A large number of random steps creates a Gaussian zone having a variance... 

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