The Optimum Column Radius

The optimum column radius has been discussed in chapter (1) but the expression obtained was for any column and not specifically an optimized column. Consequently a slightly different derivation will be employed here. 

Starting with equation (9) from chapter (10) The maximum value the extra column dispersion, (oe), that can take place is given by, 

The limitation of (oe) to (0.32oc ), allows the variance of the peak eluted from the column to be increased by a maximum of 10% as a result of the extra column dispersion and the width of the peak by a maximum of 5%. 

Substituting for (Vn) from equation (I) and for (Hmjn) from equation (12) In equation (26), 

Graph of Optimum Column Radius against Separation Ratio 

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 maximum value for the extra-column dispersion that is acceptable, (oe), is given by (chapter 9) 

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Thus for a GC column, substituting for (Hmin) from equation (24) in equation (36), 

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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. 

Taking a value for (oe) of 2.5 pi (which would be typical for a well-designed column detector system) and using equation (8), values for (ropt) are shown plotted against separation ratio in Figure 7. It is seen that the optimum column radius increases linearly with the separation ratio of the critical pair (ranging from 0.1 mm... 

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 (19) shows that the optimum column length is inversely proportional to the third power of the function (ot-1) and thus, will increase very rapidly with the difficulty of the separation It is also seen that the length is inversely proportional to the square root of the available inlet pressure and, consequently, has a similar sensitivity to pressure as the optimum column radius. 

The optimum column radius for the separation of solute mixtures ranging in difficulty from (a= i.Ol) to ( - 1.12 ana for inlet pressures of 1,10, 100. arid 1000 p s i. respectively, were calculated employing equation (11) The results obtained are shown as curves relating the optimum column radius to separation ratio in figure I,... 

It is seen that the optimum column radius for an open tubular column varies widely with inlet pressure arid the difficulty of the separation. Considering a separation of some difficulty, for example ( a = 1.02), it is seen that at an inlet pressure of lOOOp.s.f, the optimum column diameter would be about 4 micron whereas, at an inlet pressure of only 1 psi, it would be about 43 micron. The former would be quite difficult to coat with stationary phase and would demand detectors and injection systems of almost impossibly low dispersion A column of 43 micron in diameter, on the other hand, would be piactical from the point of view of both ease of coating and an acceptable system extra column dispersion. However, the lengths of such columns arid the resulting analysis times remains to be determined and may preclude their- use. 

Employing equation (12) the optimum column radius can be calculated that would be required to separate 25 g of solute for a range of separation ratios for the critical pair of l.OI to 1.50. Curves were obtained relating optimum column radius to separation ratio for inlet pressures of 1,10,100,1000, and 10,000 p.s.i. respectively. The resulting curves are shown in figure 4... 

Having determined the optimum column radius, the optimum velocity being known then the optimum flow-rate volume is given by,... 

The optimum column length, 2/ The optimum column radius. 

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