Zone spreading

Zone spreading is characterized quantitatively by a height equivalent to a theoretical plate, H [16] 

The complex relationship for has been discussed in detail [17]. In the limit when X tends to zero, it holds simply [16] 

We now obtain a generalized equation for R by substituting Eqs. 9.4 and 9.7 into Eq. 9.5, which gives 

Our remedy is based on the observation that if component molecules diffuse or otherwise transfer rapidly between fast and slow velocity states, zone broadening will be reduced. The reduction occurs because rapid diffusion will quickly shuttle molecules between high and low velocity states, thus preventing them from getting very far behind or ahead of neighboring molecules occupying different states. 

A rapid diffusional exchange is encouraged by reducing the dimensions 

We additionally note that for effective F( + ) operation—for an ability to separate multicomponent mixtures—the exchange must occur so rapidly that each component is virtually in a state of equilibrium between the velocity states. Thus near-equilibrium distributions can be assumed in most cases. 

The transfer of molecules back and forth between velocity states is a two-way random process thus step distance / may be positive or negative. The random sequence of positive and negative steps constitutes a random walk (see Section 5.3). The total number n of such steps, forward and backward, is the total time (retention time) of the process tr over exchange time t q 

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Tijssen, R., The mechanism and importance of zone-spreading, In Katz, E., Eksteen, R., Schoenmakers, P. and Miller, N., Eds., Handbook of HPLC, Marcel Dekker, New York, 1998. 

The apparent dispersion coefficient in Equation 10.8 describes the zone spreading observed in linear chromatography. This phenomenon is mainly governed by axial dispersion in the mobile phase and by nonequilibrium effects (i.e., the consequence of a finite rate of mass transfer kinetics). The band spreading observed in preparative chromatography is far more extensive than it is in linear chromatography. It is predominantly caused by the consequences of the nonlinear thermodynamics, i.e., the concentration dependence of the velocity associated to each concentration. When the mass transfer kinetics is fast, the influence of the apparent axial dispersion is small or moderate and results in a mere correction to the band profile predicted by thermodynamics alone. 

We will look at the three variables that may cause zone spreading, that is, ordinary diffusion, eddy diffusion, and local nonequilibrium. Our approach to this discussion will be from the random walk theory, since the progress of solute molecules through a column may be viewed as a random process. 

Equation 2. 59 states that zone spreading is proportional to step length but not to the number of steps. For instance, movement is random it takes 16 steps to give a displacement 4 times the average length of each step. 

We know from statistical treatment that standard deviations are not additive. However, variances, the square of the standard deviation, are additive. In terms of the chromatographic process three diffusive process variables contribute to zone spreading. Thus, we can sum these variables in terms of variances to give... 

The reader should keep in mind when developing a theory of zone spreading that we must have a point of reference to show how the spreading develops. This point of reference is the zone center. 

This equation states that eddy diffusional effects on zone spreading increase with the square root of zone displacement and particle size. 

In a follow-up study, the same authors examined the applicability of the same device for relevant protein samples and investigated the main contributions to band broadening [82]. As a consequence of the small depth of the beds, zone spreading caused by Joule heating was shown to be negligible (see Sect. 3.1.1). Cross fields of up to 100 V/cm were applied for the separation of human serum albumin, ribonuclease A and bradykinin. The feasibility of fraction collection was demonstrated with four collected fractions of a whole rat plasma sample. Off-line analysis of these four isolated fractions by CE indicated the separation of serum albumins and globulins. 

Plate tectonic activity, which is responsible on Earth for subduction zones, spreading centres and obducted ophiolites, as well as associated ore deposits of Cu, Cr and Ni described in 8.6, appears to have been less significant on other terrestrial planets. As a result, local enrichments of these and other transition elements (apart from Fe and Ti) are probably absent on the Moon, Mercury, Venus, Mars and the asteroids. Since Fe and Ti minerals are predominant on terrestrial planets, electronic spectra of Fe2+ and Fe3+ in silicates and oxides influenced by Ti4+ and Ti3+ are expected to dominate remote-sensed spectra of their surfaces. 

The degree of zone spreading for random processes depends on the length and frequency of the random steps and on the length of time over which they occur. A simplified model which describes the nature of this dependence is the random walk model [4,7,8]. 

It is a fundamental statistical law that when several independent processes contribute to zone spreading, the variances are additive (4). Thus the total variance [Pg.94]

Several kinds of random events other than molecular diffusion contribute to zone spreading in separation systems. With each increment in zone spreading there is a corresponding loss of resolution it is thus important to understand these processes in order to minimize them. The random processes described below are responsible for effective diffusion in many engineering and chemical systems as well as in separations. 

The random nature of the second term on the right assures us, through the central limit theorem, that it contributes an effective diffusion term to zone spreading [11]. Thus, this term must have the equivalent form... 

An extended discussion of zone spreading and plate height in nonuniform separation systems is beyond the scope of the present chapter. However, later we account for flow nonuniformities due to gas compressibility in gas chromatography. More generally, the treatment developed by the author [8,16] for chromatographic columns can be expanded to describe most other zonal systems. 

The equations of this section show that resolution and peak capacity are inversely proportional to a and w (usually reflected in H and N). These equations illustrate how the capacity for separation is diminished, using any reasonable measure, by increases in zone width. This conclusion reemphasizes our deep concern with zone spreading phenomena and the fundamental transport processes that underlie them. 

Clearly, departures from equilibrium—along with the resultant zone spreading—will decrease as means are found to speed up equilibrium between velocity states. One measure of equilibration time is the time defined in Section 9.4 as teq, equivalent to the transfer or exchange time between fast- and slow-velocity states. Time teq must always be minimized this conclusion is seen to follow from either random-walk theory or nonequilibrium theory. These two theories simply represent alternate conceptual approaches to the same band-broadening phenomenon. Thus the plate height from Eqs. 9.12 and 9.17 may be considered to represent simultaneously both nonequilibrium processes and random-walk effects. 

The above phenomena have been incorporated into a theoretical approach which explicitly associates zone spreading with flow velocity and the rates of various equilibration processes. This is the generalized nonequilibrium theory developed by the author [5]. While the theoretical details are too lengthy to develop here, some semi-quantitative reasoning can be used to understand the nature of nonequilibrium-induced zone spreading and the parameters that control it. 

In accord with the earlier discussion, zone broadening occurs because solute velocity at the front and back of the zone is such as to pull these parts further and further from the center. An instructive way to look at this is to note that when Acm is positive, as it is ahead of center, solute in amounts greater than the equilibrium value are being transported through each unit area of column in each second. That is, with respect to equilibrium, extra solute in proportion to Acm is transported forward thus solute at the front of the zone is out-pacing all other solute. At the rear, with Acm negative, solute transport falls behind. Thus the zone spreads out. 

The effective diffusion coefficient for nonequilibrium zone spreading, obtained by equating the last term of Eq. 10.36 to Eq. 10.37, is... 

Despite such complications of detail, the random-walk model describes the essence of chromatographic zone spreading. It properly accounts for the way in which all major experimental parameters influence the broadening process. 

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