Zone spreading diffusion

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. 

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... 

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... 

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

We will return now to the three independent random processes that underlie migration and induce zone spreading molecular diffusion, sorption-desorption, and flow-diffusion processes in the mobile phase. 

Also, the theory is much more complex than just presented. Bly36 has classified the process into three mechanisms steric exclusion, restricted diffusion, and thermodynamic considerations, and the process has been thoroughly studied. The rate equation is also different for SEC. For polymers, the longer retained peaks have smaller peak dispersivities, H, than early peaks, in direct contrast to normal LC expectations. In part this is due to the fact that the smaller molecules that elute last have higher diffusion coefficients and therefore less mass transfer zone spreading. 

In FFF, however, the fractogram is recorded in dependence of time so that a correction via extrapolation to infinite time in order to eliminate diffusion effects is not possible. A different strategy may be used for the correction of zone spreading which suffers from a number of assumptions and restrictions. A number of authors, reviewed by Janca [459], have dealt with the methods of correction for zone spreading which was found to be particularly extensive at high flow rates or low retentions. The results are summarized below. 

The non-equilibrium effect is decreased by decreasing the solvent flow velocity V. So low solvent flow rates are used in high resolution work, but not so low that the diffusion term becomes large. The best compromise is V =k2DI[R —R). In practice the values of the variables are such that flow rates of a few ml/hr cm (sometimes written cm hr ) cross sectional area of the resin bed will give low zone spreading with excellent resolution. In many applications where speed is more important than resolution, higher flow rates are used. 

Peak broadening is a result of ordinary diffusion, eddy diffusion (due to flow along longer or shorter paths in packed columns), and local non-equilibrium. The eddy diffusion is absent in capillary columns. The zone spreading due to the ordinary diffusion od can be expressed by the formula ... 

The second term,. represents the zone spreading that each sample component exhibits due to diffusion along the column axis. Diffusion coefficients in aqueous solution are generally low, so the contribution of this term is relatively small unless the retention time is quite long due to a very slow flow rate or a high retention factor. 

In the ultimate limiting case, if all other mechanisms of zone spreading can be rendered insignificant, zone broadening in zone electrophoresis will be dominated by a seemingly unpreventable mechanism, longitudinal diffusion. As diffusion in liquids is rather slow, this may result in rather narrow zones. 

Since SEC is an essentially linear system, the theories developed in Sections 14.1-1 and 14.1-2 are applicable. A chromatographic separation thus looks like Fig. 14.1-2 with zone spreading added on. Since the mass transfer terms in Eq. (14.1-12) are inversely proportional to the diffusivity, large solutes thee penetrate the porea have high H values. Desalting is an easy separation and has a relatively low M since the large molecules do not penetrate the potes. 

In linear systems the variances (o ) from different sources add. This is equivalent to stating that the amount of zone spreading from different sources is additive. Mathematically, this ability to add variances is the reason we can use an effective diffusion coefficient to model a system where mass transfer resistances are inportant. 

Axial diffusion may cause substantial zone broadening for late-eluting sample ions or when a very slow eluent flow rate is employed. The magnitude of zone spreading by axial diffusion is a function of the time spent in the column and the diffusion coefRcient of the particular ion. 

Comparison of zone sizes gives information on the quantitative ratios of identified zones (see Chapter 10). Because zone spreading always takes place during development, it is valid only to compare zones with the same Rf values. The zone size also depends on the detection reaction used an insensitive detection will yield small, apparently well-defined zones, whereas a sensitive (e.g., fluorescence) detection often produces large, diffuse spots. 

The classic Van Deemter equation and its modifications have been used to describe zone spreading in GC and HPLC in terms of eddy diffusion, molecular diffusion, and mass transfer. The efficiency of a zone in HPTLC is given by the equation (34)... 

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