10 Chromatographic theory

Introduction

Chromatography is the most frequently used analytical technique in pharmaceutical analysis. An understanding of the parameters which govern chromatographic performance has given rise to improvements in chromatography systems, so the ability to achieve high-resolution separations is continually increasing. The system suitability tests which are described at the end of this chapter are now routinely included in chromatographic software packages so that the chromatographic performance of a system can be monitored routinely. The factors determining chromatographic efficiency will be discussed first in relation to high-pressure liquid chromatography (HPLC).

Void volume and capacity factor

Figure 10.1 shows an HPLC column packed with a solid stationary phase with a liquid mobile phase flowing through it.

Fig. 10.1 Schematic diagram of bands for three different compounds travelling through an HPLC column. The compound with the largest capacity factor emerges last.

If a compound does not partition appreciably into the stationary phase, it will travel through the column at the same rate as the solvent. The length of time it takes an unretarded molecule to flow through the column is determined by the void volume of the column (V_o). The porous space within a silica gel packing is usually about 0.7 × the volume of the packing; a typical packing volume in a 0.46 × 15 cm column is ca 2.5 cm³. Thus, in theory, it should take solvent or unretarded molecules, flowing at a rate of 1 ml/min, ca 1.8 min to pass through the void volume of such a column (the internal space is likely to be reduced where the silica gel has been surface coated with stationary phase). The length of time it takes a retarded compound to pass through the column depends on its capacity factor (K′), which is a measure of the degree to which it partitions (adsorbs) into the stationary phase from the mobile phase:

where V_o is the void volume of the column; V_r is the retention volume of the analyte; t_o is the time taken for an unretained molecule to pass through the void volume; and t_r is the time taken for the analyte to pass through the column. In the example shown in Figure 10.1, compound B has a larger capacity factor than compound A. For example, if a compound had a K′ of 4, the V_o of a column was 1 ml and the solvent was flowing through the column at 1 ml/min, the total time taken for the compound to pass through the column would be 5 min, i.e. for the 1 min required to pass through the void space in the column 4 min would be spent in the stationary phase. This is a simplification of the actual process but it provides a readily understandable model. As can be seen in Figure 10.1 the peaks produced by chromatographic separation actually have width as well as a retention time and the processes which give rise to this width will be considered later.

Self-test 10.1

Calculate the time taken for the following compounds to emerge from a chromatographic column under the specified conditions.

K′ of compound	V_o of column (ml)	Solvent flow rate (ml/min)
6	1	1
10	1	2

Answers: 7 min and 5.5 min

Calculation of column efficiency

The broader a chromatographic peak is relative to its retention time, the less efficient the column it is eluting from. Figure 10.2 shows a chromatographic peak emerging at time t_r after injection; the efficiency of the column is most readily assessed from the width of the peak at half its height W_1/2 and its retention time using Equation 1:

Fig. 10.2 Determination of retention time and peak width at half height.

Equation 1

where n is the number of theoretical plates.

Column efficiency is usually expressed in theoretical plates per metre:

where L is the column length in cm.

A stricter measure of column efficiency, especially if the retention time of the analyte is short, is given by Equation 2:

Equation 2

where N eff is the number of effective plates and reflects the number of times the analyte partitions between the mobile phase and the stationary phase during its passage through the column and t′_r = t_r−t_o.

Another term which is used as a measure is H, the ‘height of a theoretical plate’:

where H is the length of column required for one partition step to occur.

Self-test 10.2

A standard operating procedure states that a column must have an efficiency > 30 000 theoretical plates/m. Which of these 15 cm columns meets the specification?

Retention time of analyte (min)	W_1/2 (min)
1. 6.4	0.2
2. 5.6	0.2
3. 10.6	0.6

Answer: Column 1

Origins of band broadening in HPLC

Van Deemter equation in liquid chromatography

Chromatographic peaks have width and this means that molecules of a single compound, despite having the same capacity factor, take different lengths of time to travel through the column. The longer an analyte takes to travel through a column, the more the individual molecules making up the sample spread out and the broader the band becomes. The more rapidly a peak broadens the less efficient the column. Detailed mathematical modelling of the processes leading to band broadening is very complex.¹ The treatment below gives a basic introduction to the origins of band broadening. The causes of band broadening can be formalised in the Van Deemter equation (Equation 3) as applied to liquid chromatography:

Equation 3

H is the measure of the efficiency of the column (discussed above); the smaller the term the more efficient the column.

u is the linear velocity of the mobile phase, simply how many cm/s an unretained molecule travels through the column, and A is the ‘eddy’ diffusion term; broadening occurs because some molecules take longer erratic paths while some, for instance those travelling close to the walls of the column, take more direct paths, thus eluting first. As shown in Figure 10.3, for two molecules of the same compound, molecule X elutes before molecule Y. In liquid chromatography the eddy diffusion term also contains a contribution from streaming within the solvent volume itself, i.e. A (see the C_m term) is reduced if the diffusion coefficient of the molecule within the mobile phase is low because molecules take less erratic paths through not being able to diffuse out of the mainstream so easily.

Fig. 10.3 Eddy diffusion around particles of stationary phase.

B is the rate of diffusion of the molecule in the liquid phase, which contributes to peak broadening through diffusion either with or against the flow of mobile phase; the contribution of this term is very small in liquid chromatography. Its contribution to band broadening decreases as flow rate increases and it only becomes significant at very low flow rates.

C_s is the resistance to mass transfer of a molecule in the stationary phase and is dependent on its diffusion coefficient in the stationary phase and upon the thickness of the stationary phase coated onto silica gel:

where d² thickness is the square of the stationary-phase film thickness and D_s is the diffusion coefficient of the analyte in the stationary phase.

Obviously the thinner and more uniform the stationary-phase coating, the smaller the contribution to band broadening from this term. In the example shown in Figure 10.4, molecule Y is retarded more than molecule X. It could be argued that this effect evens out throughout the length of the column, but in practice the number of random partitionings during the time required for elution is not sufficient to eliminate it. As might be expected, C_s makes less contribution as u decreases.