Resolution fundamental equation

A more fundamental equation for resolution than equation (4.45) is given by the expression... 

Another method is to optimize those chromatographic parameters which affect resolution Although this paper is directed at a pragmatic approach to accomplishing this task, a brief detour into the mathematics is needed before we proceed The fundamental equation describing chromatographic resolution, Rg is... 

This expression for resolution may be written in an alternative form in terms of the relative retention Fj/Kj (Section 24-1) and the partition ratio k. Thus it can be shown that (24-34) is equivalent to the following fundamental equation involving three nearly independent factors ... 

The solution of a protein crystal structure can still be a lengthy process, even when crystals are available, because of the phase problem. In contrast, small molecule (< 100 atoms) structures can be solved routinely by direct methods. In the early fifties it was shown that certain mathematical relationships exist between the phases and the amplitudes of the structure factors if it is assumed that the electron density is positive and atoms are resolved [255]. These mathematical methods have been developed [256,257] so that it is possible to solve a small molecule structure directly from the intensity data [258]. For example, the crystal structure of gramicidin S [259] (a cyclic polypeptide of 10 amino acids, 92 atoms) has been solved using the computer programme MULTAN. Traditional direct methods are not applicable to protein structures, partly because the diffraction data seldom extend to atomic resolution. Recently, a new method derived from information theory and based on the maximum entropy (minimum information) principle has been developed. In the immediate future the application will require an approximate starting phase set. However, the method has the potential for an ab initio structure determination from the measured intensities and a very small sub-set of starting phases, once the formidable problems in providing numerical methods for the solution of the fundamental equations have been solved. 

When examining the theoretical basis of rapid separations, it is important to balance the considerations of temporal resolution with well-resolved separations. The fundamental equations defining electrophoresis show that it is possible to increase both the speed and efficiency of a separation by increasing the voltage. - The migration time (fmig) is given by... 

Equation (4.16) has been called the fundamental equation for chromatography. Each and every analyte of interest that is introduced into a chromatographic column will have its own capacity factor, k. The column itself will have a volume Fq. The retention volume for a given analyte is then viewed in terms of the number of column volumes passed through the column before the analyte is said to elute. A chromatogram then consists of a plot of detector response versus tg, where each analyte has a unique retention time if sufficient chromatographic resolution is provided. Hence, with reference to a chromatogram, the capacity factor becomes... 

The resolution can also be predicted by substituting in the expressions for retention times and the standard deviations. Assuming that the N values are the same for the two conponents (a reasonable assunption since resolution is usually calculated for similar conpounds) the resulting fundamental equation of chromatography fGiddings. 1965 Wankat. 19901 is... 

Therefore, it is possible to increase throughput, and thus the speed of analysis without affecting the chromatographic performance. The advent of UPLC has demanded the development of a new instrumental system for LC, which can take advantage of the separation performance (by reducing dead volumes) and consistent with the pressures (8000-15,000 psi, compared with 2500 to 5000 psi in HPLC). Efficiency is proportional to column length and inversely proportional to the particle size [41], Smaller particles provide increased efficiency as well as the ability to work at increased linear velocity without a loss of efficiency, providing both resolution and speed. Efficiency is the primary separation parameter behind UPLC since it relies on the same selectivity and retentivity as HPLC. In the fundamental resolution (Rs) equation [38], resolution is proportional to the square root of N. 

This is a fundamental equation in chromatography. The resolution of two peaks can be improved by increasing a, N, or k. 

Let us now review the fundamental resolution expression [equation (2.13)], which is repeated here for convenience, and examine each term on the right-hand side ... 

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

It is important to recognize that a number of assumptions were made in deriving Eq. (15.17) to arrive at the simplified Eq. (15.19). A more fundamental form of resolution expression is given below (see Eq. (15.22) in Section 15.10.1). To get a more accurate equation, the actual values of the peak widths and their respective capacity factors should be used however, for most practical purposes the above equation or its original form, Eq. (15.16), is satisfactory. 

First, we will explore the three fundamental factors in HPLC retention, selectivity, and efficiency. These three factors ultimately control the separation (resolution) of the analyte(s). We will then discuss the van Deemter equation and demonstrate how the particle diameter of the packing material and flow rate affect column efficiencies. 

In the operation of preparative columns, it is necessary to obtain the maximum mass throughput per unit time and, at the same time, achieve the required resolution. Consequently, the column will be operated at the optimum velocity as in the case of analytical columns. Furthermore, the D Arcy equation will still hold and the equation for the optimum particle diameter can be established in exactly the same way as the optimum particle diameter of the analytical column. The equation is fundamentally the same as that given for the optimum particle diameter for a packed analytical column, i.e. (18) In chapter 12, except that (a) and (k ) have different meanings. 

A more fundamental expression for resolution is given by the following equation ... 

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 equations of this section show that resolution and peak capacity are inversely proportional to a and w (usually reflected in H and N). These equations illustrate how the capacity for separation is diminished, using any reasonable measure, by increases in zone width. This conclusion reemphasizes our deep concern with zone spreading phenomena and the fundamental transport processes that underlie them. 

Now that the fundamental parameters of LC have been defined and calculated, the focus of the experiment can be directed to the primary objective resolution. Resolution is the measure of how well two compounds are separated by the LC column. The quantitative measurement of resolution between the two bands of color takes into account the separation of the band centers as well as the width of each colored band according to the equation... 

The overall quality of the separation of two solutes is measured by their resolution Rs), a combination of the thermodynamic factors causing separative transport and the kinetic factors causing dispersive transport and is an index of the effectiveness of the separa-tion. Defined by Rs = (r, - tr,a)l (w,b + where a and b refer to the two solutes, is the retention time of solute X, and is the peak width at the base of solute X in units of time, it is frequently estimated by use of the fundamental resolution equation. 

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