Summation of variances

Peak dispersion can happen in any part of the chromatographic system, from the 

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Maximum Sample Volume that Can Be Placed on a Chromatographic Column 

Having established that a finite volume of sample causes peak dispersion and that it is highly desirable to limit that dispersion to a level that does not impair the performance of the column, the maximum sample volume that can be tolerated can be evaluated by employing the principle of the summation of variances. Let a volume (Vi) be injected onto a column. This sample volume (Vi) will be dispersed on the front of the column in the form of a rectangular distribution. The eluted peak will have an overall variance that consists of that produced by the column and other parts of the mobile phase conduit system plus that due to the dispersion from the finite sample volume. For convenience, the dispersion contributed by parts of the mobile phase system, other than the column (except for that from the finite sample volume), will be considered negligible. In most well-designed chromatographic systems, this will be true, particularly for well-packed GC and LC columns. However, for open tubular columns in GC, and possibly microbore columns in LC, where peak volumes can be extremely small, this may not necessarily be true, and other extra-column dispersion sources may need to be taken into account. It is now possible to apply the principle of the summation of variances to the effect of sample volume. 

the acceptable degree to which the performance of the column is denigrated is basically an arbitrary decision. As long ago as 1957 Klinkenberg [9] recommended 

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Equation (1) is the algebraic expression of the principle of the summation of variances. If the individual dispersion processes that take place in a column can be identified, and the variance that results... 

Before progressing to the Rate Theory Equation, an interesting and practical example of the use of the summation of variances is the determination of the maximum sample volume that can be placed on a column. This is important because excessive sample volume broadens the peak and reduces the resolution. It is therefore important to be able to choose a sample volume that is as large as possible to provide maximum sensitivity but, at the same time insufficient, to affect the overall resolution. 

Let a volume (Vi) be injected onto a column resulting in a rectangular distribution of sample at the front of the column. According to the principle of the Summation of Variances, the variance of the final peak will be the sum of the variances of the sample volume plus the normal variance of a peak for a small sample. 

To determine the band dispersion that results from a significant, but moderate, sample volume overload the summation of variances can be used. However, when the sample volume becomes excessive, the band dispersion that results becomes equivalent to the sample volume itself. In figure 10, two solutes are depicted that are eluted from a column under conditions of no overload. If the dispersion from the excessive sample volume just allows the peaks to touch at the base, the peak separation in milliliters of mobile phase passed through the column will be equivalent to the sample volume (Vi) plus half the base width of both peaks. It is assumed in figure 10 that the efficiency of each peak is the same and in most cases this will be true. If there is some significant difference, an average value of the efficiencies of the two peaks can be taken. 

SUMMATION OF VARIANCE FROM SEVERAL DATA SETS... 

Any sample placed on to an LC column will have a finite volume, and the variance of the injected sample will contribute directly to the final peak variance that results from the dispersion processes that take place in the column. It follows that the maximum volume of sample that can be placed on the column must be limited, or the column efficiency will be seriously reduced. Consider a volume Vi, injected onto a column, which will form a rectangular distribution on the front of the column. The variance of the peak eluted from the column will be the sum of the variances of the Injected sample plus the normal variance of the eluted peak. The principal of the Summation of Variances will be discussed more extensively in a later chapter, at this time it can be stated that,... 

Individual variances. This is how the Rate Theory provides an equation for the final variance of the peak leaving the column. As an. example the principle of the summation of variances will be applied to extra column dispersion... 

In this introduction, one further important basic concept needs to be discussed as it is used extensively in detector design and in particular, the design of detector connections and detector sensor geometry and that is the principle of the summation of variances. 

The above equation is the algebraic description of the principle of the summation of variances and is fundamentally important. If the individual dispersion processes that are taking place in a column can be identified, and the variance that results from each dispersion determined, then the variance of the final band can be calculated from the sum of all the individual variances. An example of the use of this principle is afforded by the calculation of the maximum extracolumn dispersion that can be tolerated for a particular column. This... 

Here, the variance of the IBW is the summation of variances from the injector, the detector, and the connection tubing ... 

Consider the separation depicted in figure 12.1 the retention parameters are labeled according to the plate theory as discussed in chapter 1. It is clear that the column could be heavily overloaded, to allow the peaks to disperse until they touched at the base, before resolution would be lost. Under these conditions the summation of variances cannot be used, as when the sample volume becomes excessive, the dispersion of the peak becomes, to the first approximation, the volume of the sample itself Presented in a different way, the sample volume acts as a part of the mobile phase and contributes to the elution process in the same manner. 

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