Multiple paths

Schematics illustrating the contributions to band broadening due to (a) multiple paths, (b) longitudinal diffusion, and (c) mass transfer.

To determine how the height of a theoretical plate can be decreased, it is necessary to understand the experimental factors contributing to the broadening of a solute s chromatographic band. Several theoretical treatments of band broadening have been proposed. We will consider one approach in which the height of a theoretical plate is determined by four contributions multiple paths, longitudinal diffusion, mass transfer in the stationary phase, and mass transfer in the mobile phase. 

To minimize the multiple path and mass transfer contributions to plate height (equations 12.23 and 12.26), the packing material should be of as small a diameter as is practical and loaded with a thin film of stationary phase (equation 12.25). Compared with capillary columns, which are discussed in the next section, packed columns can handle larger amounts of sample. Samples of 0.1-10 )J,L are routinely analyzed with a packed column. Column efficiencies are typically several hundred to 2000 plates/m, providing columns with 3000-10,000 theoretical plates. Assuming Wiax/Wiin is approximately 50, a packed column with 10,000 theoretical plates has a peak capacity (equation 12.18) of... 

A distinction can be drawn between multiple paths to the same product(s) and simultaneous reactions that partition to different ones. Examples of both will be given. 

Our results show that the calculated potential energies for the TS obtained from the combined procedure are around 4 kcal/mol lower than the corresponding ones calculated with the CD method. In contrast, the calculated free energies of activation for all four paths differ by not more than 1.6 kcal/mol. These results show that the inclusion of the fluctuation of the MM environment dramatically improves the results of the calculations of enzymatic catalysis, even if the calculated PES is not highly accurate. In addition, the calculation of free energies for multiple paths using the QM/MM-FE method can serve as an alternative to more expensive sampling methods such as QM/MM-MFEP and QM/MM-MD. 

It is also possible for solute species to diffuse laterally (in a radial direction) across the column, and thus to move from one flow path into another. This effect reduces the amount of dispersion produced by the multiple path effect as it tends to equalise the speed of the solute species in the column. The longer the time a solute species spends in the column, the more lateral diffusion will occur, so that flow dispersion is reduced by using low flow rates of mobile phase. 

Colman, D., Lubetzki, C. and Reingold, S. Multiple paths towards repair in multiple sclerosis. Trends Neurosci. 26 59-61, 2003. 

A is the multiple path term which accounts for different portions of the mobile phase, and consequently the solute, travelling different total distances because of the various routes taken around the particles of stationary phase. The effect is minimized by reducing particle size but increases with length of column... 

The particle size of a solid support is critical in striking a compromise between column efficiency and speed of separation. Both the multiple path term A and the mass transfer term (CSl of equation (4.46) (p. 89)) are reduced by reducing particle size thus leading to increased efficiency. However, as particle size is reduced, the pressure drop across the column must be increased if a reasonable flow rate is to be maintained. The optimum particle sizes for 1/8 in columns are 80/100 or 100/120 mesh and for 1/4 in columns 40/60 or 60/80 mesh. 

Diffusion and mass transfer effects cause the dimensions of the separated spots to increase in all directions as elution proceeds, in much the same way as concentration profiles become Gaussian in column separations (p. 86). Multiple path, molecular diffusion and mass transfer effects all contribute to spreading along the direction of flow but only the first two cause lateral spreading. Consequently, the initially circular spots become progressively elliptical in the direction of flow. Efficiency and resolution are thus impaired. Elution must be halted before the solvent front reaches the opposite edge of the plate as the distance it has moved must be measured in order to calculate the retardation factors (Rf values) of separated components (p. 86). 

Some specifications fully determine the outcome of an operation. Our specification of squareRoot allows many implementations and even more than one result 2 2 are 4, but so are -2 -2. If we want to exclude the latter, we would add y>0. Similarly, schedule course constrains the new session to be assigned a qualified, available instructor but does not specify which one be assigned. And takeShortestPath does not say which path should be selected in the event there were multiple paths with the same length it says only that there is no path with a shorter length. 

For simplicity, up to now, first-order kinetics have been assumed, but obviously other rate laws may apply. Further complications can be generated by the presence of multiple paths for M on a variety of sites exhibiting different kinetics [5,11] or sequential enzymatic processes [100], Some complexes, labelled as lipophilics , have been shown to cross the membrane without the need for specific pre-adsorption sites [5,11,18,19,50] see also Chapters 5, 6 and 10 in this volume. Fortin and Campbell [76] have recently reported the accidental uptake of Ag+ induced by thiosulfate ligand. 

A multiple-path mechanism has been elaborated for dissociation of the mono- and binuclear tris(hydroxamato)-iron(III) complexes with dihydroxamate ligands in aqueous solution. " Iron removal by edta from mono-, bi-, and trinuclear complexes with model desferrioxamine-related siderophores containing one, two, or three tris-hydroxamate units generally follows first-order kinetics though biphasic kinetics were reported for iron removal from one of the binuclear complexes. The kinetic results were interpreted in terms of discrete intrastrand ferrioxamine-type structures for the di-iron and tri-iron complexes of (288). " Reactivities for dissociation, by dissociative activation mechanisms, of a selection of bidentate and hexadentate hydroxamates have been compared with those of oxinates and salicylates. ... 

Note 1 Usually, and in all systems that exhibit rubber elasticity, the number of distinct paths is very high, but some constitutional units exist, in most cases, which are connected by a single path only. Sometimes, a structure without any multiple path has also been called a network. 

There are multiple paths to achieving the state when a product and a process are in control. A pictorial representation of this concept is shown in Figure 1.2. Simpler molecules may achieve a state of control (predictable state) early in the development process, while more complex molecules may retain a high state of variability until late in the process. The goal for development must be a development path that is documented and performed by qualified scientists, equipment, facilities, instruments, etc. Development paths that can be followed are varied, but the final outcome, when a project is transferred to manufacturing, is a product and a process that are in a well-characterized state of control. 

In packed columns, all three terms contribute to band broadening. For open tubular columns, the multiple path term, A, is 0, so bandwidth decreases and resolution increases. In capillary electrophoresis (Chapter 26), both A and C go to 0, thereby reducing plate height to submicron values and providing extraordinary separation powers. 

One reason why small particles give better resolution is that they provide more uniform flow through the column, thereby reducing the multiple path term, A, in the van Deemter equation (23-33). A second reason is that the distance through which solute must diffuse in the mobile and stationary phases is on the order of the particle size. The smaller the particles, the less distance solute must diffuse. This effect decreases the C term in the van Deemter equation for finite equilibration time. The optimum flow rate for small particles is faster than for large particles because solutes diffuse through smaller distances. 

Capillary electrophoresis provides unprecedented resolution. When we conduct chromatography in a packed column, peaks are broadened by three mechanisms in the van Deemter equation (23-33) multiple flow paths, longitudinal diffusion, and finite rate of mass transfer. An open tubular column eliminates multiple paths and thereby reduces plate height and improves resolution. Capillary electrophoresis reduces plate height further by knocking out the mass transfer term that comes from the finite time needed for solute to equilibrate... 

Multiple path term Mass transfer term... 

While there remain open questions on a few experimental aspects of early work on the stereomutations of 1 -phenyl-2-d-cyclopropanes and 1,2-d2-cyclopropanes, the preponderance of data and theory now provides a consistent understanding of the thermal stereomutations of cyclopropanes multiple paths and three types of diradical transition structures are involved. Evolving theory relevant to cyclopropane stereomutations and to vinylcyclopropane to cyclopentene isomerizations, and to other 1,3-carbon shifts283 288, may well provide more detailed insights, rationales and predictions. 

The reflectances are correct only after the patch with the maximum reflectance has been passed. In order to obtain the correct estimate of the reflectance for all the rectangles, Land and McCann assume that the paths are very long. Also, the estimated reflectances from multiple paths are averaged. For the example in the preceding text (with two paths, one forward and one backward) one would obtain... 

Let us assume for a moment that the illumination is uniform, then the threshold operation is not needed. If we take the current point as a reference and consider multiple paths from the surrounding points that pass through the current point, we obtain... 

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