Column optimum radius

Equation (13) is the first important equation for open tubular column design. It is seen that the optimum radius, with which the column will operate at the optimum velocity for the given inlet pressure, increases rapidly as an inverse function of the separation ratio (cc-1) and inversely as the square root of the inlet pressure. Again it must be remembered that, when calculating (ropt)5 the dimensions of the applied pressure (P) must be appropriate for the dimensions in which the viscosity (r)) is measured. 

Although the optimum column radius increases linearly with the separation ratio of the critical pair, this simple relationship is moderated by the ratio of the square of the optimum radius to the optimum velocity, both of which are functions of (a). 

It Is seen that, in a similar manner to the packed column, the optimum mobile phase velocity is directly proportional to the diffusiv ty of the solute in the mobile phase, However, in the capillary column the radius (r) replaces the particle diameter (dp) of the packed column and consequently, (u0pt) is inversely proportional to the column radius. 

This allows the best possible performance of a capillary column to be determined for any solute, under optimum conditions (i.e., at best flow velocity). Again, this depends on column inner radius (rc) and... 

It is seen from equation (20) that the minimum detectable mass, or mass sensitivity of a chromatographic system, where the column has been designed to have the optimum radius for the detector employed, is directly proportional to the extra column dispersion and the detector concentration sensitivity. It follows that detector dispersion is as important as detector sensitivity in its influence on the overall chromatographic mass sensitivity where the chromatographic system has been optimized with respect to the radius of the column. The effect of extra column dispersion and in particular, detector dispersion on the overall mass sensitivity of the chromatogaphic system is not generally appreciated or completely understood. As the total extra column dispersion is the integral of a variety of sources, the distribution and nature of the various sources of dispersion will now be considered in some detail. 

Thus, for significant values of (k") (unity or greater) the optimum mobile phase velocity is controlled primarily by the ratio of the solute diffusivity to the column radius and, secondly, by the thermodynamic properties of the distribution system. However, the minimum value of (H) (and, thus, the maximum column efficiency) is determined primarily by the column radius, secondly by the thermodynamic properties of the distribution system and is independent of solute diffusivity. It follows that for all types of columns, increasing the temperature increases the diffusivity of the solute in both phases and, thus, increases the optimum flow rate and reduces the analysis time. Temperature, however, will only affect (Hmin) insomuch as it affects the magnitude of (k"). 

The optimum mobile phase velocity will also be determined in the above calculations together with the minimum radius to achieve minimum solvent consumption and maximum mass sensitivity. The column specifications and operating conditions are summarized in Table 4. 

Unfortunately, some of the data that are required to calculate the specifications and operating conditions of the optimum column involve instrument specifications which are often not available from the instrument manufacturer. In particular, the total dispersion of the detector and its internal connecting tubes is rarely given. In a similar manner, a value for the dispersion that takes place in a sample valve is rarely provided by the manufactures. The valve, as discussed in a previous chapter, can make a significant contribution to the extra-column dispersion of the chromatographic system, which, as has also been shown, will determine the magnitude of the column radius. Sadly, it is often left to the analyst to experimentally determine these data. 

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))-... 

The efficiency obtained from an open tubular column can be increased by reducing the column radius, which, in turn will allow the column length to be decreased and, thus, a shorter analysis time can be realized. However, the smaller diameter column will require more pressure to achieve the optimum velocity and thus the reduction of column diameter can only be continued until the maximum available inlet pressure is needed to achieve the optimum mobile phase velocity. 

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. 

Figure 7. Graph of Optimum Column Radius against Separation Ratio of the Critical Pair...

As the optimum column radius is inversely proportional to (a-1), and (uopt) is inversely proportional to (ropt)> the simple linear relationship between optimum velocity and the separation ratio is to be expected. The high velocities employed for... 

Equation (13) shows that the minimum value of (H) is solely dependant on the column radius (r) and the thermodynamic properties of the solute/phase system. As opposed to the optimum velocity, the minimum value of (H) is not dependent on the solute diffusivtty. 

It is interesting to note from equation (14) that when a column is run at its optimum velocity, the maximum efficiency attainable from a capillary column is directly proportional to the inlet pressure and the square of the radius and inversely proportional to the solvent viscosity and the diffusivity of the solute in the mobile phase. This means that the maximum efficiency attainable from a capillary column increases with the column radius. Consequently, very high efficiencies will be obtained from relatively large diameter columns. 

Equation (19) shows that the minimum radius will increase as the square root of the extra column dispersion and as the square root of (a-1) but, increase inversely with the square root of the particle diameter. (However,it will be shown later that, that if the column is packed with particles of optimum diameter for the particular separation then the column radius will become linearly related to the function (a-1)). 

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