Van Deemter Equation

Imagine the following scenario: we’re cleaning out the back of the fume hood and realize we have a bottle labeled 2 ethyl 1 butanol and 2 ethyl 1 hexanol (and who knows what else!). We don’t want to be wasteful, so we decide to run it through our gas chromatography setup after our routine samples to see if we can’t figure out exactly what’s in there. The peak we saw in our analysis ended up looking like this:

 

Ah! We’d really want to have better resolution than this. If only we had a magic wand to show us how to read a chromatogram that looked like this and tell us how to get better peak separation! Not all is lost though, since we have the next best thing: the Van Deemter Equation (J.J. van Deemter, F.J. Zuiderweg, A. Klinkenberg, Chemical Engineering Science, Volume 5, Issue 6, 1956, Pages 271-289. http://dx.doi.org/10.1016/0009-2509(56)80003-1). Most of the time the Van Deemter Equation is written in the following way:

 

HETP stands for the Height Equivalent of a Theoretical Place, which is a topic for another day. For now, it suffices to say that it’s related to the broadness of the peaks in our chromatogram. Substituting the letters for words, the equation becomes something like:

Peak Spreading = Eddy Diffusion + Axial Diffusion / speed + Mass Transfer * speed.

Eddy diffusion in chromatography is simply the fact that the analytes have to travel around the particles in our column. Since each analyte molecule will be following different paths, the time it takes for each to reach the end is a little bit different. According to the Van Deemter Equation, we can’t do much about this term by simply varying our velocity, since the “A” term in the equation is not coupled to the velocity in any way.

We do have some control over the other terms though. The axial diffusion term (B) arises from normal diffusion of the particles as they travel down the column. Here axial refers to the particles moving randomly back and forth in the same direction as the flow of the mobile phase. The displacement due to diffusion is mostly dependent on the diffusion constant of the molecules, and the amount of time it takes for them to move down the column. If it takes less time to go through the column, diffusion will be less significant and the resulting peaks will be narrower. This explains why the axial diffusion term in the Van Deemter equation is inversely proportional to velocity. We can utilize this by simply increasing the flow rate to increase the velocity of the particles, making them reach the end faster and in tighter groupings.

The last term is the so-called Mass Transfer term (or finite rate of mass transfer, depending on who you’re asking). As the analytes move through the column, some of them will get stuck interacting with the stationary phase. Not all of them will interact the same amount of time, so some of them will keep travelling downstream while others are stuck effective in place! If the velocity of our analytes is faster, they will travel farther while some are stuck interacting with stationary phase, and the broadening increases.

Of course, all of these terms are present in our chromatograph. Here’s an example of a Van Deemter Plot, showing the effects of the different terms of the equation visually.

 

The dashed line shows the optimal velocity to reduce our peak broadening. How does this help us separate our 2 ethyl 1 butanol and 2 ethyl 1 hexanol from one another? Well, in order to make optimal use of our new-found information, we should probably do a series of chromatograms at different analyte velocities to figure out the optimal choice, and we may get something at the end that looks a bit like the following chromatogram.

 

With any luck, we’ll be able to see two distinct peaks, one for 2 ethyl 1 butanol and one for 2 ethyl 1 hexanol. All we had to do was control the velocities of our analytes by adjusting flow rates to adjust our peak broadening.

So, at the end of the day we can properly characterize our back of the fume hood mixture. By using insight from the Van Deemter Equation, we may be able to achieve better separation in our chromatographs in the future as well. We just have to keep in mind the three relevant phenomena that contribute to the broadening of our peaks: the effects of Eddy Diffusion, (Finite Rate of) Mass Transfer, and the diffusion of analytes in the axial direction. That’s not to say that there aren’t other ways to increase our resolution, but there’s something to be said for being able to adjust a single knob on our instrument t