# liquid chromatography column - Scott R.P.W.

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_2 _ 11 _ 2Dm r2u

°tu -Htu - u + 2^;.................................................(1)

where (u), (r) and (Dm) have the meanings previously ascribed to them,.

If u » Dm/r then equation (I) reduces to.

°t2u = Htu = ^ ................................................................ (2)

Now, from the Plate Theory, the peak Variance in volume units (02(v)) is,

(Tube Volume)2

°t(v) fT

2

È)

nVl2

(3)

Where (n) is the number of theoretical plates in the tube. Now, - = H.

n

Thus, replacing (^) by (H), and substituting for (H) from equation (2),

2 _ n2r4l2r2 °t(v) ¦ 24 Dm

n2r6l2 24 Dm

(4)

Bearing in mind that the flow rate Q = ÿã2è

2 _ Îÿã41 °t(v) " 24Dm ............................................. (5)

Fauation (5) clearly indicates the procedure that must be followed to reduce the dispersion that arises from connecting tubes. However, for maximum efficiency, the column should be operated at its optimum mobile phase velocity and consequently, the flow rate, (0), is already defined, and cannot be used to control tube dispersion In a similar manner the diffusivity of the

solute, (Dm), is determined by the nature of the sample and the mobile phase that is chosen and so this is also not a variable available for dispersion control. The major factor effecting dispersion, is, in fact, the tube radius. It is seen that the dispersion increases as the fourth power of the tube radius and thus, a reduction in the tube radius by a factor of 2 will reduce the dispersion by a factor of sixteen.

Unfortunately, there is a limit to the process of reducing (r) as, from Poiseuilie's equation the pressure drop across the tube is given by,

î 8rllu ÄÐ =

Thus, as Q - ëã2è,

r2

ÀÐ = ............................................. (6)

ËÃ

It is seen from equation (6), that the pressure drop across the connecting tube increases inversely as the fourth power of the tube radius. Thus, as it is inadvisable to dissipate a significant amount of the available pump pressure across a connecting tubing, there will be a lower limit to which (r) can be reduced in order to minimize dispersion.

It is also interesting to note that changing the length of the connecting tube has the same effect on both dispersion and pressure drop. Thus, reducing (1) will linearly reduce dispersion and at the same time proportionally reduce the pressure drop across the connecting tube. It follows that the length of the connecting tube is, by far, the best method of controlling dispersion and by making (l) as small as possible both the dispersion and the pressure drop can be minimized.

In practice the diameter of the connecting tube should not be made less than

0.012 cm, (0.005 in.l. D ), not merely because of the pressure drop that will occur across it, but for a more mundane, but very important reason, If tubes of less diameter are employed, they will easily become blocked. Employing equation (5) the volume variance and the volume standard deviation contribution from connecting tubes of different lengths were calculated and the results are shown in figure 1A and IB. The tube radius is assumed to be 0.012cm, the flow rate 1 ml per minute, and the diffusivity of the solute in the mobile phase 2.5 X 10~5 cmz/sec.

it is seen in figure 1A that, as would be expected, the variance increases linearly with the tube length. These values for variance must be added to

that of the column and other extra column dispersion variances to arrive at the final variance of the peak It might be possible to reduce the tube diameter to about 0.008cm (0.003 in) I.D., and so decrease the tube variance by about a factor of eight, but this would increase the chance of tube blockage very significantly The curve in figure IB is more informative

Figure 1A Graph of Tube Variance against Tube Length

4e-5

3e-5

Î

Î

e

î

Ã 2e-5 e > î

Ï

? 1e-5

0e+0

0 10 20 30

Tube Length (cm)

from a practical point of view as, although standard deviations are not additive as already discussed, they do give an idea of the actual band width that a tube alone can cause. It is seen that a tube 10 cm long and 0.012 cm

I.D. can result in a peak with a standard deviation of À ö.1. This would be equivalent to a peak with a base width of 16 |tl and, as it will be seen later, many short columns packed with particles, 3 öò (or less) in diameter will produce peaks of commensurate size. This means, that if high efficiency columns are to be used, with small plate heights, then connecting tubes should either be eliminated altogether, or reduced to the absolute minimum in length. In practice it is sometimes extremely difficult to achieve short lengths of a connecting tube, particularly for column detector connections. This is because manufacturers often design detectors such that

the sensor cells require significant lengths of tubing to connect them to the exterior union.

Figure IB

Graph of Tube Standard Deviation (ml) against Tube Length (cm)

n

î

L.

?

î

>

Ô

î

Î)

î

î

Tube Length (cm)

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