## As Easy as ABC: How to Use the Van Deemter Equation to Optimize Your Chromatography

I have spoken in previous blogs about various chromatographic methods and discussed their advantages and disadvantages. Regardless of which method you choose, be it flash, prep HPLC , or prep SFC , the aim is to achieve an efficient separation with high resolution. There are many factors to consider, from column selection to the nature of the sample, the solid and mobile phase used, and the flow rate. In the past, trial and error led to the development of more efficient techniques; however, one physicist and engineer wanted a definitive solution to ensure the highest efficiency and resolution possible. That physicist was Jan Josef van Deemter, who considered a separation’s physical, kinetic, and thermodynamic properties and formed an equation to predict the optimal separation conditions. In this blog, I would like to shed light on his work and apply his equation to various methods so that you, too, can optimize your chromatography and make your process as easy as ABC!

As my friends and I found out, when we competed to see who could chill their beer the quickest, there is usually an optimal way to achieve most tasks. The same can be said for chromatography, and the Deemter equation that I have spoken about in previous blogs is a hyperbolic function that attempts to optimize each process . Today I would like to take a deep dive into the equation and explain how it relates to different methods such as HPLC and SFC.

To ensure maximum efficiency, the Deemter equation relates the variance per unit length of a separation column to the linear mobile phase velocity. The equation involves analyzing the diffusion coefficient of the solute in the mobile phase, the mass transfer kinetics between the mobile and stationary phases, and the thickness of the stationary phase.

The equation is written as:

### HETP = A + (B / u) + Cu

Where:
– HETP is the Height Equivalent to a Theoretical Plate.
– u is the linear velocity of the mobile phase.
– A, B, and C are the Van Deemter coefficients that correspond to:
o A – The multitude of paths the solute can take (“eddy diffusion”).
o B – The longitudinal diffusion of the solute.
o C – The mass transfer of the solute between the mobile and stationary phases.

Although the equation seems complex, each variable understood on its own is relatively simple, and once the data has been plotted on a curve (known as a Deemter curve), it is straightforward to read the data and know the optimal separation conditions. So, let’s break it down:

To understand the first part of the equation (HETP), you need to understand what Theoretical Plates (TP) are in column chromatography. The efficiency and resolution of the chromatography process are directly proportional to the square root of theoretical plates. Meaning that every time the number of plates is increased by four, the resolution will double. So, TP represents the distance needed for every absorption-desorption step; what about Height (H)? Well, the number of plates (N) over the column length (L) depends on the plate height (H).

N = L/H

Using a staircase analogy, if you had to get up a height of 20 feet and only have 2 stairs, each step must be 10 feet high. A smaller plate height implies a large number of plates (stairs) in the column and greater efficiency. As the equation shows, the column length directly affects the number of plates. You can therefore achieve greater efficiency by opting for lengthier columns. So, HETP looks at height equivalency, also referred to as the plate’s thickness.

Many factors influence the number of theoretical plates , such as: ### Efficiency and resolution are directly proportional to the square root of theoretical plates. Meaning that every time the number of plates is increased by four, the resolution will double.

Next up in the equation is u, which is the linear velocity of the mobile phase – essentially, the flow rate. This is the speed at which the mobile phase (liquid, gas, supercritical fluid) moves through the column and is measured in cm per minute (cm/min) or mm per second (mm/s). The optimal linear velocity ensures the highest resolution. A slow linear velocity can result in overlapping peaks, whereas too fast a velocity can cause the peaks to elute too quickly, reducing resolution. The linear velocity is calculated with the following formula:

v = F/A

F is the flow rate (typically in mL/min or mL/s), and A is the column’s cross-sectional area (typically in cm2 or mm2). The area for a circular cross-section is calculated using the following formula:

A = π * (d/2)2

Where d is the internal diameter of the column (Remember to convert units appropriately when calculating!)

### The optimum flow rate ensures the highest resolution. A slow linear velocity can result in overlapping peaks, whereas too fast a velocity can cause the peaks to elute too quickly, reducing resolution.

Finally, the van Deemter coefficients A, B, and C relate to the factors contributing to band broadening in chromatography. An ideal chromatogram has sharp peaks that clearly indicate the specific compounds present and their respective concentrations in the sample. Broad peaks are a sign of low resolution and an inefficient separation.

The first of the Deemter coefficients (A) is what is known as Eddy Diffusion, also referred to as multiple-path diffusion. Eddy diffusion relates to molecules’ variable path lengths as they navigate a packed column. In an ideal situation, each molecule would travel along the same path; however, the stationary phase creates a labyrinth of different paths a molecule can take. This leads to different molecules reaching the end of the column at different times and broadens the peak of analyte eluting from the column, lowering the resolution. The influence of Eddy diffusion is minimized by using smaller particles for the mobile phase reducing the range of paths the molecules can take.

### An ideal chromatogram has sharp peaks that clearly indicate the specific compounds present and their respective concentrations in the sample. Broad peaks are a sign of low resolution and an inefficient separation.

The longitudinal diffusion (B) deals with the tendency of molecules to spread from regions of high concentration to regions of lower concentration. It is crucial to understand as it also affects band broadening, especially at lower velocities when the mobile phase moves more slowly.

The last coefficient is the mass transfer (C) which relates to the time it takes for a solute to equilibrate between the mobile and stationary phases. It takes a finite amount of time for the sample injected into the column to partition between the phases and reach equilibrium. If the mobile phase moves too quickly, some molecules may be swept along before being efficiently separated, resulting in band broadening. So, the influence of B on band broadening is more prevalent at lower velocities, and the influence of C is more significant at higher velocities.

### To optimize chromatographic separations, you must minimize the HETP (H) by finding an appropriate speed (u) that minimizes the value of H given the specific values of A, B, and C for the system giving you the optimal velocity.

Although each factor is relatively simple on its own, there are still several factors to consider leading to confusion. Luckily for us then, the Van Deemter equation is used to plot a curve that offers a graphical representation of the data. Linear velocity (u) is plotted on the x-axis, and Height (H) is plotted on the y-axis. The resulting curve usually has a U-shape, with the lowest point on the curve representing the optimal velocity that will offer the most efficient separation with the highest resolution. Black line – HETP
Blue line – Cu
Green Line – A
Red Line – B/u

The flow rate that gives the lowest HETP has the highest efficiency. Too slow or too fast will result in less efficiency. The curve’s shape depends on the system’s properties, such as the specific mobile and stationary phases and the nature of the analyte. Regarding the stationary phase, particle size has a considerable impact as smaller particles will result in higher resolution; however, the pressure required for optimal velocity increases by the inverse of the particle diameter squared. So, switching to twice as small particles while keeping the column length the same means the pressure required is increased by a factor of four.

### Supercritical carbon dioxide has a low viscosity and high diffusion coefficient enabling analysis with a higher linear velocity than HPLC or flash chromatography.

Chromatography often involves a trade-off between column length, particle size, flow rate, run time, and pressure. When it comes to the mobile phase, some of the trade-offs can be mitigated as specific methods allow the flow rate to be increased without drawbacks. Supercritical Fluid Chromatography (SFC) uses supercritical carbon dioxide with both low viscosity and a high diffusion coefficient enabling analysis with a higher linear velocity than HPLC or flash chromatography. Supercritical fluids also penetrate packing material pores more efficiently, resulting in lower mass transfer diffusion values even for high linear velocities allowing for shorter run times without loss of resolution. As you can see, the chosen method can significantly impact the speed and resolution of chromatographic separations. I hope this explanation of the factors that influence the efficiency of your separations is helpful and enables you to speed up your workflow and increase the precision of your methods.

Till next time, 