The Resolution Equation

If all the parameters described above are combined, namely plate number, capacity factor and selectivity, we obtain finally a general formula for the resolution R  

The aim of every gas-chromatographic analysis is to separate the individual substances from each other. A measure of the quality of the separation 

Equation 2.2, the resolution is equal to the product of the capacity term 

In order to improve the resolution R, each term may be optimized. 

According to Equation 2.3, the partition constant Kj) is equal to the product of k and B. Resolving in terms of k and substituting m/ s for B, we obtain  

The degree of separation or resolution (Rs) between two solutes is dependent on both thermodynamic factors (retention, k, and selectivity, a) and kinetic factors (peak width and column efficiency, n).1 3,6 712 Resolution is controlled by three somewhat independent factors (retention, selectivity, and efficiency) as expressed quantitatively in the resolution equation  

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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. 

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... 

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 ... 

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 ... 

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... 

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... 

TABLE 4-4. Parameters that can be Changed to Influence the Terms in the Resolution Equation... 

The numerator in the resolution equation is the distance between bands, which is related to the selectivity, and the denominator is the average band width, which is proportional to the efficiency of the column. For experiments A-C resolution can be calculated from the already measured values and compared for both column types. 

Resolution. Maximizing selectivity, as part of the early stages of methods development, will result in the fastest methods since as flow rate is increased or column length is decreased, resolution will decrease. The resolution equation describes the key parameters. In equation (17-24), k is substituted for the isocratic retention factor, k, to give... 

The resolution achievable by a quadrupole filter depends on the selection of the U/ Vo ratio and can thus be adjusted throughout the mass range. For this reason the resolution varies throughout a scan of the mass range, with AM in the resolution equation remaining constant. 

The effect of the EOF on migration time and selectivity depends on the mutual signs of the mobilities of analytes and EOF, respectively. Concerning the change in separation selectivity, we refer to the expression of the selectivity term in the resolution equation. The difference between the mobilities of the two separands, i and j, will not be influenced by the EOF. However, the mean mobility is larger for the case of comigration. This means that the selectivity term in the expression for the resolution is always reduced in this case. In practice, selectivity is lost for cation separation when the EOF is directed, as is usual in uncoated fused-siUca capillaries, toward the cathode. For this reason, cationic additives are applied in the BGE to reverse the EOF direction. 

These effects stem from thermodynamic principles. If in a separation with a given purity requirement the peak resolution is not high enough due to these effects there is no way to overcome this problem by increasing column efficiency, which means optimizing the fluid-dynamic term of the resolution equation (Eq. 2.33). If a tag-along effect occurs the use of a smaller particle diameter has no effect, as long as the... 

The driving force behind the creation of these many choices emerges from the resolution equation, which in the following form can be used to estimate the number of theoretical plates required for a given degree of separation of any two solutes (1 ) ... 

The different parameters that allow such separation are to be found in the resolution equation, which is the basic equation of liquid chromatography ... 

Future symposia will address this problem. In the interim it is patently clear that, individually, none of the terms (N, a and k ) in the resolution equation Is sufficient to describe resolution (1) ... 

The Resolution Equation (effect of efficiency, retention and selectivity)... 

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