Multi-path effect

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applying the Random Walk relationship, Le., o = then 

Dividing the total variance by the column length (1), the multi-path contribution (Hm) to the overall height of the theoretical plate (H) is obtained. 

Equation (5), however, would apply only to a perfectly packed column so Van Deemter introduced a constant (2X) to account for the inhomogeneity of real packing (for ideal packing (X) would take the value of 0.5). Consequently, his expression for the multi-path contribution to the total variance per unit length for the column (Hm) is 

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In a packed column the individual solute molecules will describe a tortuous path through the interstices between the particles and some will randomly travel shorter routes than the average and some longer. The multi-path effect is diagramatically depicted in Figure 5. 

Van Deemter considered peak dispersion results from four spreading processes that take place in a column, namely, the Multi-Path Effect, Longitudinal Diffusion, Resistance to Mass Transfer in the Mobile Phase and Resistance to Mass Transfer in the Stationary Phase. Each one of these dispersion processes will now be considered separately... 

It is seen that the composite curve is hyperbolic, consisting of the sum of a constant term value (A) that describes the multi-path effect, a... 

The A term represents eddy diffusion or multi-path effect and is proportional to (ip). 

The A term represents eddy diffusion or multi-path effect experienced by the various solute molecules as they transverse the packed bed. A term is proportional to dp and is smaller in well-packed columns. 

The van Deemter rate theory identified three major factors that cause band or zone broadening during the chromatographic process the eddy diffusion or the multi-path effect (A-term), longitudinal diffusion or molecular diffusion of the analyte molecules (B-term), and resistance to mass transfer in the stationary phase (C-term). The broadening of a zone was expressed in terms of the plate height, H, and was described as a function of the average linear velocity of the mobile phase, u. 

The dispersion of a solute band in a packed column was originally treated comprehensively by Van Deemter et al. [4] who postulated that there were four first-order effect, spreading processes that were responsible for peak dispersion. These the authors designated as multi-path dispersion, longitudinal diffusion, resistance to mass transfer in the mobile phase and resistance to mass transfer in the stationary phase. Van Deemter derived an expression for the variance contribution of each dispersion process to the overall variance per unit length of the column. Consequently, as the individual dispersion processes can be assumed to be random and non-interacting, the total variance per unit length of the column was obtained from a sum of the individual variance contributions. 

Although valence band spectra probe those electrons that are involved in chemical bond formation, they are rarely used in studying catalysts. One reason is that all elements have valence electrons, which makes valence band spectra of multi-component systems difficult to sort out. A second reason is that the mean free path of photoelectrons from the valence band is at its maximum, implying that the chemical effects of for example chemisorption, which are limited to the outer surface layer, can hardly be distinguished from the dominating substrate signal. In this respect UPS, discussed later in this chapter, is much more surface sensitive and therefore better suited for adsorption studies. 

PHI TRIFT IV ToF-SIMS (Physical Electronics, USA) employs three electrostatic analyzers in the ion path to filter the background and metastable secondary ions. Using liquid metal cluster ion guns (such as Aut ion beam for sputtering of sample surface) increased sensitivity compared to a Ga+ primary ion beam are obtained (www.phi.com). The application of dual primary ion guns is useful for an effective dual beam depth profiling on multi-layered samples. 

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