Band broadening eddy diffusion

Considering a chromatographic process controlled by a partition equilibrium and neglecting extracolumn effects (i.e., band broadening caused by factors outside the column, e.g., tubings, detector etc.), several factors can contribute to the overall solute band broadening eddy diffusion, longitudinal diffusion, and resistance to mass transfer in mobile and stationary phase. 

Fig. 14.2. van Deemter plot showing contributions of eddy diffusion, molecular diffusion and mass transfer to the rate of band broadening. Picture courtesy of Prof. Harold McNair. 

CE is a technique with a very high power of resolution. This is attributed to low diffusion and high plate numbers obtained from the absence of band-broadening factors (e.g., eddy diffusion, equilibrium dynamics, etc.) other than diffusion, which is also minimized by short analysis time. 

The plate theory assumes that an instantaneous equilibrium is set up for the solute between the stationary and mobile phases, and it does not consider the effects of diffusional effects on column performance. The rate theory avoids the assumption of an instantaneous equilibrium and addresses the diffusional factors that contribute to band broadening in the column, namely, eddy diffusion, longitudinal diffusion, and resistance to mass transfer in the stationary phase and the mobile phase. The experimental conditions required to obtain the most efficient system can be determined by constructing a van Deemter plot. 

The velocity-independent term A characterises the contribution of eddy (radial) diffusion to band broadening and is a function of the size and the distribution of interparticle channels and of possible non-uniformiiies in the packed bed (coefficient A.) it is directly proportional to the mean diameter of the column packing particles, dp. The term B describes the effect of the molecular (longitudinal) diffusion in the axial direction and is directly proportional to the solute diffusion coefficient in the mobile phase, D, . The obstruction factor y takes into account the hindrance to the rate of diffusion by the particle skeleton. 

In the frontal analysis experiment described in Section 5.3.2, the transport model of chromatography was used to fit the experimental data [40]. Neglecting axial and eddy diffusion, band broadening was accounted for by one single mass transfer rate coefficient. The mass transfer rate coefficients estimated were small and strongly dependent on the temperature and solute concentration, particularly the rate coefficients corresponding to the imprinted L-enantiomer (Fig. 5.12). Above a concentration of ca. 0.1 g/L the mass transfer rate coefficients of the two enantiomers are similar. 

Figure 4.1 Band-broadening processes in porous irregular microparticles, (a) eddy diffusion analyte molecules take different routes to circumnavigate the particles. They also move more quickly through wide channels than through narrow channels, (b) diffusion in the mobile phase. The short bracket indicates initial band width, the long bracket indicates final band width, (c) mass transfer. On the left is shown mass transfer in stagnant mobile phase in pores, and that due to the adsorption/desorption process. The narrow band represents initial band width, the broad band final band width. On the right is shown mobile phase mass transfer caused by laminar flow.

Broadening of Chromatographic Bands as a Resuit of Eddy Diffusion and Resistance to Mass Transfer... 

The most characteristic feature of chromatographic bands is that the longer the development time and the greater the distance from the start, the greater become their surface areas. This phenomenon is not restricted to planar chromatography—it occurs in all chromatographic techniques. Band broadening arises as a result of eddy and molecular diffusion, the effects of mass transfer, and the mechanism of solute retention. 

The contribution of eddy diffusion and other factors to band broadening in liquid chromatography can be quantitatively described by the following equation, which relates the column plate height H to the linear velocity of the solute, ju ... 

The first term in Eq. (3), includes the contribution of eddy diffusion to band broadening as well as that of mass transfer of the solute through the mobile phase. This contribution of this mobile-phase mass transfer to this term, Hj, increases with the square of the stationary-phase particle diameter dp. It is also in-... 

The size of the particles used in the stationary phase has also been shown to exert a significant effect on resolution. Small particles result in fast equilibration of solutes, as well as smaller contributions of eddy diffusion to band broadening. There is a practical limitation in the use of small particles, however. The smaller... 

Eddy diffusion Diffusion of solutes that contributes to broadening of chromatographic bands, the result of differences in the pathways for solutes as they traverse a column. 

The sources of band broadening of kinetic origin include molecular diffusion, eddy diffusion, mass transfer resistances, and the finite rate of the kinetics of ad-sorption/desorption. In turn, the mass transfer resistances can be sorted out into several different contributions. First, the film mass transfer resistance takes place at the interface separating the stream of mobile phase percolating through the column bed and the mobile phase stagnant inside the pores of the particles. Second, the internal mass transfer resistance controls the rate of mass transfer between this interface and the adsorbent surface. It is composed of two contributions, the pore diffusion, which is molecular diffusion taking place in the tortuous, constricted network of pores, and surface diffusion, which takes place in the electric field at the liquid-solid interface [60]. All these mass transfer resistances, except the kinetics of adsorption-desorption, depend on the molecular diffusivity. Thus, it is important to study diffusion in bulk liquids and in porous media. 

In contrast to the equilibrium-dispersive model, which is based upon the assumptions that constant thermod3mamic equilibrium is achieved between stationary and mobile phases and that the influence of axial dispersion and of the various contributions to band broadening of kinetic origin can be accounted for by using an apparent dispersion coefficient of appropriate magnitude, the lumped kinetic model of chromatography is based upon the use of a kinetic equation, so the diffusion coefficient in Eq. 6.22 accounts merely for axial dispersion (i.e., axial and eddy diffusions). The mass balance equation is then written... 

Equation 16 shows that the peak variance or band broadening is comprised of individual contributions from different aspects of the separation process. The first term in equation 16 represents the contribution of the width of the feed band to the peak variance. The second term represents the contribution to band broadening from dispersion due to eddy diffusion. The third term represents the contribution of mass transfer effects external to the particles while the fourth term represents the contribution of diffusional resistances within the stationary phase. The significance of each term relative to the total variance depends upon the operating parameters, the column and packing dimensions and the size of the solute. 

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