The Column Radius

The radius of an analytical column is determined, among other factors, by the extra column dispersion of the chromatographic system. For preparative columns, however, the radius is determined by the sample load that is required to be placed on the column to obtain the necessary throughput. 

Now the maximum charge that can be placed on the column (Vj) has been shown to be, (Chapter 5 page 54), 

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), 

In practice the value of (w) will vary between about 2 and 5 ( i.e sample concentrations will lie between 2%w/v and 5%w/v) before mass overload becomes a significant factor In band dispersion. A numerical value for (g ) of 5 will be taken In subsequent calculations. The correct value of ( ), for the particular solute concerned, can be experimentally determined on an analytical column carrying the same phase system If so required. 

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 

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Equation (3) allows the calculation of the distance traveled axially by a solute band before the radial standard deviation of the sample is numerically equal to the column radius. Consider a sample injected precisely at the center of a 4 mm diameter LC column. Now, radial equilibrium will be achieved when (o), the radial standard deviation of the band, is numerically equal to the radius, i.e., o = 0.2 cm. 

The development of the function describing (tm) for a capillary column is similar to that for the packed column but (r), the column radius, replaces (dp), the particle diameter. 

It is seen that the value of (H) is completely dependent on the diffusivity of the solute in the mobile phase, the column radius and the linear velocity of the mobile phase. The simple uncoated open tube can clearly be used to determine the diffusivity of any solute in any given solvent (the mobile phase). This technique for measuring diffusivities will be discussed in a later chapter. 

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"). 

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. 

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. 

Rearranging equation (36), chapter 12, and substituting for (n) from the Purnell equation, an expression for the column radius can be obtained. 

It follows from equation (2) that the sample load will increase as the square of the column radius and thus the column radius is the major factor that controls productivity. Unfortunately, increasing the column radius will also increase the volume flow rate and thus the consumption of solvent. However, both the sample load and the mobile phase flow rate increases as the square of the radius, and so the solvent consumption per unit mass of product will remain the same. 

Where Q, is the minimum detectable amount, R the detector noise level and s the detector sensitivity [135,146,151,152]. For a concentration sensitive detector the minimum detectable concentration is the product of Q, and the volumetric gas flow rate through the detector. The minimum detectable amount or concentration is proportional to the retention time, and therefore, directly proportional to the column radius for large values of n. it follows, then, that very small quantities can be detected on narrow-bore columns. 

The average volumetric flow rate (ml/min) is calculated from the gas hold-up time, the column length (cm) and the column radius (cm). 

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. 

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)). 

It is seen that equation (15) is very similar to that for the optimum flow-rate for an analytical column except that (oe) Is replaced by the expression (IIOM/w) as the extra column dispersion no longer controls the column radius. 

If you have developed a chromatographic procedure to separate 2 mg of a mixture on a column with a diameter of 1.0 cm, what size column should you use to separate 20 mg of the mixture The most straightforward way to scale up is to maintain the same column length and to increase the cross-sectional area to maintain a constant ratio of sample mass to column volume. Because cross-sectional area is nr2, where r is the column radius, the desired diameter is given by... 

Narrow-bore columns of between 1.0 and 2.5 mm ID are available for use in specially designed liquid chromatographs having an extremely low extracolumn dispersion. For a concentration-sensitive detector such as the absorbance detector, the signal is proportional to the instantaneous concentration of the analytes in the flow cell. Peaks elute from narrow-bore columns in much smaller volumes compared to those from standard-bore columns. Consequently, because of the higher analyte concentrations in the flow cell, the use of narrow-bore columns enhances detector sensitivity. The minimum detectable mass is directly proportional to the square of the column radius (107) therefore, in theory, a 2.1-mm-ID column will provide a mass sensitivity about five times greater than that of a 4.6-mm-ID column of the same length. 

Side downcomer width (as measured along the column radius line)... 

Here column volume is defined as the volume of mobile phase in the column and may be calculated as Vm = O.lw L, where r and L are the column radius and length, respectively, and 0.7 is the approximate fraction of the empty tube occupied by mobile phase for a column packed with porous particles. 

Thus (m ), the mass sensitivity of the chromatographic system depends on the detector sensitivity, column dimensions, column efficiency and the capacity factor of the eluted solute. However, irrespective of the column properties, the mass sensitivity is still directly related to the detector sensitivity. It will also be seen that the column radius will depend on the extracolumn dispersion, much of which arises from the detector connecting tubes and sensor. It follows that the design of the detector and its sensitivity has a major influence on the mass sensitivity of the overall chromatographic system. 

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