# Theories of column chromatography

This article is about the two theories of chromatography: Plate Theory and Rate Theory. By now, we have learnt and we fully understand the importance of column chromatography in analytical separation science. We are also aware of the diversity present in stationary phase packings for a chromatographic column but can we control the separation efficiency of this column? What factors affect the overall chromatographic separation? How can these factors be controlled in order to achieve a desirable outcome? Can these factors be mathematically interpreted? Keep reading to get answers to all these questions and many more.

## What is the Plate Theory of chromatography

This theory was proposed by Martin and Synge, in order to predict the efficiency of a column used in a liquid chromatographic process. According to the plate theory, the solute components present in the analyte mixture equilibrate between the stationary phase and the mobile phase, at various positions throughout the length of the column. The equilibration occurs at the point where the solute molecules are equally distributed within the two phases. In other words, the numerical value of partition/distribution coefficient k (refer to formula 1) is equals to 1 at an equilibrium point. However, Cs may not be equals to Cm at this equilibrium point.

$k=\frac{Cs}{Cm}$          Formula 1

where k=partition/distribution coefficient, Cs= molar concentration of a solute in stationary phase and Cm= molar concentration of solute in mobile phase.

Each equilibrium point can be represented by using a hypothetical concept called a ‘theoretical plate’’. Greater the number of theoretical plates, higher the column efficiency. The analyte components travel down the column by moving from one plate to the next. The distance between two adjacent theoretical plates is known as height equivalent to a theoretical plate (HETP). HETP can also be defined as the length of the column required for one equilibration of solute between the stationary phase and the mobile phase.

Keeping the length (L) of the column constant, HEPT is in an inverse relationship with the number of theoretical plates (N) as shown in the formula 2 given below. Smaller the HETP value, greater the number of theoretical plates, better the column efficiency thus a high-resolution chromatogram will be achieved. The chromatographic resolution is directly related to the value of N as shown in the numerical expression 1.

$HETP=\frac{L}{N}$               Formula 2

Resolution α $\sqrt{N}$        Expression 1

Throughout a single chromatographic experiment, the length of the column will obviously be kept constant but the length L can vary from one experiment to another by using a different column in both cases. Henceforth, we should be aware of the fact that a longer column will automatically lead to more chances for the solute molecules to equilibrate between the two phases thus a greater N value and consequently a higher chromatographic resolution, refer to the numerical expression 2.

Resolution $\alpha\sqrt{N}\alpha&space;\sqrt{L}$   Expression 2

The numerical value of N for a chromatographic column whatsoever can be determined using Formula 3.

$N=\frac{5.55&space;(tr)2}{(w1/2)2}$              Formula 3

where tr represents retention time while w1/2 stands for the width of the peak at half height.

The plate theory successfully accounts for the Gaussian (bell-shaped curve) distribution of the chromatographic peaks. The detector signal spikes up as the molecules of a particular solute component (say A) elute out of the column at a fast pace until it reaches a ‘peak’ value. The subsequent downfall of the curve then illustrates the decrease in the elution of A till all the molecules of A get eluted out and the curve returns back to the baseline.

However, the plate theory cannot explain why in some chromatographic separations we achieve a single, sharp peak (as desired) while in another, the peak gets broadened. The concept of peak broadening is thus explained by the second important theory of chromatography i.e., the rate theory.

## What is the Rate Theory of chromatography

Good resolution in chromatographic peaks stands for baseline stable, single, sharp peaks for individual analyte components which are adequately separated from their adjacent peaks. Peak broadening is the least favourable outcome that a chemist would like to come across while performing column chromatography. Different factors may lead to peak broadening while performing a chromatographic separation and these factors can be investigated by using the chromatographic rate theory.

The rate theory is based on Van Deemter Equation as given below.

HETP=A + B/µ +Cµ Equation 1

Equation 1 relates HETP with three factors A, B and C where A represents multiple path effect also known as eddy diffusion. B represents longitudinal diffusional. C stands for equilibration time or mass transfer rate while µ is representative of mobile flow rate.

Let us unwind all these concepts in detail, one-by-one.

### Multiple Path Effect/Eddy Diffusion

This factor influences chromatographic separations in which the column is not uniformly packed. As a consequence of non-uniform column packing, different flow paths are available to the molecules of an analytical component (again refer to analyte A). All the molecules of A may not take the same path while their elution, as you can see in the figure given below. Accordingly, the molecules injected into the column at the same time, do not reach the detector simultaneously.

The molecules of A following the simpler and shorter path 1 will reach the detector first followed by those travelling through 2 and the longest time will be required for the molecules travelling through the column via the most complicated path i.e., 3. Until all the molecules of A reach the detector, the peak will continue to get broadened which leads to a poor chromatographic resolution.

Smaller the size of stationary phase particles, more compactly packed the column thus a weaker effect of factor A will be witnessed. The effect of eddy diffusion can be completely eliminated by employing more sophisticated columns such as the commercially packed columns as used for HPLC and/or the open tubular columns available for GC.

For a uniformly packed column, A=0 thus equation 1 will be reduced to equation 2.

HETP= B/µ +Cµ Equation 2

### Longitudinal Diffusion

The factor B/µ represents longitudinal diffusion. The effect of longitudinal diffusion is more visible in open tubular columns where the stationary phase is coated along the walls of the column while it stays hollow from the middle. If a thin, disk-shaped band of solute is applied at the center of the column, the solute molecules from this band does not stay concentrated at the center. Rather these diffuse towards the peripheries, down a concentration gradient along with the flow of the mobile phase. This diffusional effect ultimately leads to peak broadening.

In accordance with the mathematical principles, faster the mobile phase flow, greater the diffusional peak broadening.

If µ (the denominator) increases, the overall fraction decreases, therefore the value of B must increase to raise the numerical value of the fraction to its original. Gases diffuse faster than liquids thus the longitudinal effect is more pronounced in gas chromatography than in liquid chromatography.

### Mass Transfer Rate

The factor Cµ represents the finite rate of transfer of solute molecules between the stationary phase and the mobile phase. If the mobile phase flows through the column at a very high rate, there will be lower chances for the solute components to equilibrate between the two phases which means a greater influence of factor C on peak broadening. Thus, most of the solute molecules will elute out of the column with the mobile phase with negligible separation. This will lead to overlapped, poorly separated, broad chromatographic peaks.

Mass transfer rate thus can be controlled by controlling the rate of flow of mobile phase. Internal diameter of the column and the column temperature also affects the distribution of the solute components between the two chromatographic phases i.e., affecting mass transfer rate.

### Van Deemter Plot

The cumulative effect of all the three factors can be represented in the form of a Van Deemter Plot as shown in the figure below.

The net influence of the triplet is shown by the blue curve where the point P marks the optimum flow rate F which should be maintained in order to achieve a desirable HETP. A mobile flow rate above F increases the effect of longitudinal diffusion while a flow rate lower than F suppresses mass transfer rate undesirably. On the other hand, mobile flow rate has no positive or negative influence on eddy diffusion.

To revise the concepts learnt in this article, watch a video tutorial on chromatographic theories.

References

1. C.Harris, D. (2010). Quantitative Chemical Analysis.

2. Cazes, J. (2009). Encyclopedia of Chromatography.