Resolution equation

The fundamental resolution equation incorporates the terms involving the thermodynamics and kinetics of the chromatographic system ... 

Changes in heat capacity and measurement of T for blends have been used to determine components of copolymers and blends (126—129), although dynamic mechanical analysis has been found to give better resolution. Equations relating T of miscible blends and ratios of components have been developed from dsc techniques, eg, the Fox equation (eq. 1), where f the blend, or is the weight fraction of component 1 or 2,... 

The variables that control the extent of a chromatographic separation are conveniently divided into kinetic and thermodynamic factors. The thermodynamic variables control relative retention and are embodied in the selectivity factor in the resolution equation. For any optimization strategy the selectivity factor should be maximized (see section 1.6). Since this depends on an understandino of the appropriate retention mechanism further discussion. .Jll be deferred to the appropriate sections of Chapters 2 and 4. 

As with any separation technique, the desired goal is to maximize peak resolution at the fastest speed. Higher resolution in 2DLC is easier to achieve than when using onedimensional chromatography because selectivity differences between the two different columns can give a resolution enhancement. This is easily seen through the simplified resolution equation, discussed in Chapter 2,... 

The primary goal of any separation process is to achieve optimum resolution of the components. Resolution can be improved by varying the three terms a, N, or k in the resolution equation ... 

The resolution equation (19.11) may be expressed in a more useful form by introducing... 

The effectiveness of the separation (R ) in HPLC analysis is dependent on both thermodynamic factors (retention and selectivity) and kinetics factors (peak width and column efficiency)d° The relationship of resolution to other parameters can be expressed somewhat quantitatively in the resolution equation ... 

Scrutiny of the resolution equation indicates that is controlled by three relatively independent terms retention, selectivity, and efficiency (Figure 10). To maximize R, k should be relatively large. However, a value of k over 10 will approach a point of diminishing returns as the retention term of k /(l + k ) approaches unity. No separation is possible if k = 0, since R must equal zero if k is zero in the resolution equation. 

FIGURE 10 The resolution equation, which is governed by three factors retention, selectivity, and efficiency. 

Show that if the number of theoretical plates N is the same for two neighbouring compounds 1 and 2, then the classic expression yielding the resolution, equation (1) below, can be transformed. 

Temperature is another parameter that can be used for optimal analysis on RP columns. The overall resolution of a particular solute matrix can be improved and analysis time can be reduced because temperature affects every term in the resolution equation ... 

From these volumes, we can calculate three factors, kf, a, and N (Fig. 4.2), which will then be used to describe a resolution equation (Fig. 4.3).This equation predicts the effect of variations in these factors in controlling resolution within the FIPLC column. They are presented here to discuss the variables controlling each of them, their limits, and how you can use them to achieve your separations in a rational manner. 

All of these are combined in the resolution equation (Rs), which predicts how each factor will affect the separation. The derivation of the equation is not important to our work, but can be found in the Synder and Kirkland reference in Appendix G. In practice, the values used for the factors are empirically derived from chromatograms. For most uses, fairly crude measurements are sufficient, but care should be taken with peak widths in calculating efficiencies. 

Now, let us look at the variable controlling the various factors in the equation. We will return to the resolution equation when we get into column diagnostics and healing (Chapter 6) and, again, in scouting and methods development (Chapter 12). 

Resolution is a term used to describe the degree of separation between neighboring solute bands or peaks. It is affected by the selectivity (a), efficiency (N) and capacity (k ) of the column. The resolution equation [Eq. (1.3)] describes the relationship between those factors and indicates how they can be manipulated in order to improve the resolution between two peaks. 

By substituting Eq. (4.18) and the plate count equation [Eq. (4.13)] into the expression for resolution, the resolution equation may be rearranged to yield an expression that better describes the effect of various parameters on resolution ... 

Equation 1 is a fundamental resolution equation Q,2) which contains the measured terms a, k and n. Theoretical Plates (n) are a measure of a column s overall efficiency, the partition ration (k) is a measure of the amount of time a solute... 

The third factor in the resolution equation is the most vital one for the optimization of the separation. Since this factor involves the selectivity (a) we may talk about Selectivity optimization . We have seen in section 1.5 that Rs is very sensitive even to small changes in a if the components are difficult to separate (i.e. a close to 1). 

Once we have realized optimum capacity factors and optimized the selectivity, we can use the resolution equation to calculate the number of plates that is required to achieve baseline resolution (11,= 1.5). The required number of plates will to a large extent determine the kind of column and instrumentation needed to perform the separation. This will be briefly discussed in chapter 7. 

It appears from eqn.(4.68) that if the flow rate and the span of the gradient are kept constant, the gradient steepness parameter (6) is inversely proportional to the duration time (tG) of the gradient, and, hence, that the median capacity factor ( cg) is directly proportional to tG. Therefore, under these conditions, in gradient elution tG may take the place of the capacity factor kg in the resolution equation and eqn.(4.67) may be rewritten as... 

What are the best ways to optimize separations One way to answer this question is to consider a popular form of the resolution equation that is originally attributed to Purnell.1... 

B(C6Fs)3 works as an effective catalyst for the allylation of propargylic esters.217 Allyl and propargyl trimethylsilyl ethers bearing a 7r-electron-donating group at the a-position are smoothly allylated at the a- or 7-position under catalysis by McjSiOTf.218 Chiral titanium catalyst 29 enables highly enantioselective allylation of racemic benzyl trimethylsilyl ethers by a dynamic kinetic resolution (Equation (57)).219... 

By looking at the resolution equation (3-9), one can understand the interrelationship of selectivity and efficiency and the accompanying effect upon resolution. If our goal is to obtain a resolution value of 1 between two peaks (see Chapter 1, Figure 1-4) and we have a k value for the first peak of 2, we can illustrate the role of selectivity and efficiency by inserting those values into equation 3-9, which then simplifies to... 

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