The Maximum Sample Volume

The expression for the maximum sample volume can be found in Chapter 12 equation (45) and is as follows, 

It is seen that the maximum sample volume ranges from a fraction of a microliter when (a=1.01) to several ml when (a) = 1.2. The small sample volume results from 

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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, 

Now the maximum Increase In band width that can be tolerated due to any extraneous dispersion process is obviously a matter of choice but Klinkenberg (5) suggested a 5% increase in standard deviation (ora 10% 

Increase In peak variance) was the maximum extra-column dispersion that could be tolerated without serious loss In resolution. This criteria is now generally accepted. 

Consider a volume (Vj), injected onto a column and forming a rectangular distribution at the front of the column. 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. Now the variance of the rectangular 

It is seen that the that the maximum sample volume that can be tolerated can be calculated from the retention volume of the solute concerned and the the efficiency of the column. A knowledge of the maximum sample volume that can be place on a column is important where the column efficiency available is only just adequate and the compounds of interest are minor components of the mixture to be analyzed and are only partly resolved. 

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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. 

Equation (22) allows the maximum sample volume that can be used without seriously denigrating the performance of the column to be calculated from the retention volume of the solute and the column efficiency. In any separation, there will be one pair of solutes that are eluted closest together (which, as will be seen in Part 3 of this book, is defined as the critical pair) and it is the retention volume of the first of these that is usually employed in equation (22) to calculate the maximum acceptable sample volume. 

Now, the maximum sample volume (Vi) that can be placed on the column that would restrict the increase to less than 5% has been shown to be. 

The effect of sample volume on peak width has been considered and treated theoretically in Chapter 6 however, it is of interest to determine the maximum sample volume that can be tolerated with modern columns packed with small particles. The maximum sample volume is defined by the following equation,... 

Equation (5) was used to calculate the maximum sample volume for a series of columns having different lengths and internal diameters and packed with particles 3 p... 

The use of 5 pm particles permits the use of much longer columns due to the increased permeability. This, in turn, permits the use of much larger sample volumes. In fact, for a column 20 cm long, eluting a solute at a (k ) of 5, the maximum sample volume that can be used without increasing the peak variance by more than 10% will... 

Extra-column dispersion can arise in the sample valve, unions, frits, connecting tubing, and the sensor cell of the detector. The maximum sample volume, i.e., that volume that contributes less than 10% to the column variance, is determined by the type of column, dimensions of the column and the chromatographic characteristics of the solute. In practice, the majority of the permitted extra-column dispersion should... 

There remains the need to obtain expressions for the optimum column radius (r(opt)), the optimum flow rate (Q(opt)), the maximum solvent consumption (S(sol)) and the maximum sample volume (v(sam))-... 

It is seen from equation (46) that, as would be expected, the maximum sample volume decreases as the separation ratio decreases, i.e., with difficulty of the separation. 

In a packed column the HETP depends on the particle diameter and is not related to the column radius. As a result, an expression for the optimum particle diameter is independently derived, and then the column radius determined from the extracolumn dispersion. This is not true for the open tubular column, as the HETP is determined by the column radius. It follows that a converse procedure must be employed. Firstly the optimum column radius is determined and then the maximum extra-column dispersion that the column can tolerate calculated. Thus, with open tubular columns, the chromatographic system, in particular the detector dispersion and the maximum sample volume, is dictated by the column design which, in turn, is governed by the nature of the separation. 

Employing the data from Table 1 the values already calculated for (ropt) and (Lmin) the maximum sample volume was calculated for a range of values for (a) and the results are shown as a graph in Figure 10. 

The expression defines the maximum sample volume is given in chapter 13, equation (21) and is reproduced as follows. 

Employing equation (25), the effect of the separation ratio of the critical pair on the maximum sample volume was calculated using equation (25) and the results are shown in Figure 17. 

It is seen that, in all cases, the maximum sample volume is extremely small ranging from about 100 pi for a very difficult, lengthy separation to less than a tenth of a... 

The capacity ratios, efficiencies and separation ratios are given in Table 1, together with the maximum sample volume as calculated from equation (1). Comparing the maximum sample volume used for benzene given in Table 1, with the actual sample volumes for benzene shown in Figure 3, it would appear that equation (1) can be used with confidence for calculating (Vl). In a similar manner, using the same... 

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. 

Now the total volume (Vc) of a column radius (r) and length (1) will be nr l. Furthermore, the volume occupied by the mobile phase will be approximately 0.6Vc (60% of the total column volume is occupied by mobile phase).Thus as a general rule the maximum sample volume that can be employed without degrading the resolution of the column is... 

The analyst can calculate the maximum sample volume (Vi) in a simple manner from the efficiency of a peak eluted close to the dead volume and the dimensions of the column. 

Knox and Piper (13) assumed that the majority of the adsorption isotherms were, indeed, Langmuir in form and then postulated that all the peaks that were mass overloaded would be approximately triangular in shape. As a consequence, Knox and Piper proposed that mass overload could be treated in a similar manner to volume overload. Whether all solute/stationary phase isotherms are Langmuir in type is a moot point and the assumption should be taken with some caution. Knox and Piper then suggested that the best compromise was to utilize about half the maximum sample volume as defined by equation (15), which would then reduce the distance between the peaks by half. They then recommended that the concentration of the solute should be increased until dispersion due to mass overload just caused the two peaks to touch. 

Unfortunately, the magnitude of the variance contribution from each source will be different and the ultimate minimum size of each is often dictated by the limitations in the physical construction of of the different parts of the apparatus and consequently not controllable. It follows that equipartition of the permitted extra column dispersion is not possible. It will be seen later that the the maximum sample volume provides the maximum chromatographic mass and concentration sensitivity. Consequently, all other sources of dispersion must be kept to the absolute minimum to allow as large a sample volume as possible to be placed on the column without exceeding the permitted limit. At the same time it must be stressed, that all the permitted extra column dispersion can not be allotted solely to the sample volume. 

It is seen from equation (34) that for a fully optimized column the maximum sample volume depends solely on the extra column dispersion (oe) This again emphasizes the importance of not only using equipment with low extra column dispersion but, also, knowing the value of (oe) for the particular chromatograph being used. 

The properties of open tubular columns shown in figures (I) to (6) indicate that the areas where such columns would have practical use is very restricted. At pressures in excess of 10 ps.i., and whatever the nature of the separation, whether simple or difficult, the optimum column diameters are so small that they would be exceedingly difficult to fabricate or coat with stationary phase. The maximum sample volumes and extra column dispersion that couid be tolerated would also be well below that physically possible at this time. At relatively low pressures, that Is at pressures less than 10 p.s.l. the diameter of the optimum column is large enough to fabricate and coat with stationary phase providing the separations required are difficult i.c. the separation ratio of the critical pair must be less than 1.03. However, even under these conditions the sample volume will be extremely small, the extra column dispersion restricted to an almost impossibly low limit and the analysis time would be very long Nevertheless, open tubular columns used for very difficult separations... 

It Is seen from equation (9) that the maximum sample volume depends on the square of the radius and inversely on the square root of the column inlet pressure. Now, although (r) and (P) are not mathematically interdependent, there is a practical dependance of (r) on (P). The column must, physically, be able to withstand the the pressure (P) and thus, the column walls must be sufficiently thick to accommodate the pressure for any given radius (r). The aspect of column strength, and weight will be discussed further in due course. Now, if the mass of the selected solute that is required per separation is (M) and is placed on the column in the maximum permissible sample volume (Vj),... 

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