theoretical question band broadening

[Papatom]

Hi,

there is somting I do not understand:

During an analysis the different components of the sample are all an equal amount of time in the liquid phase. And they differ in the retention time because they are different times in the stationary phase. The band broadening due to longitudinal diffusion should therefor be the same for each component if the diffusion coefficient is the same. The peaks however get broader at the end of the chromatogram....

What is the explanation for this??

kind regards,

Papatom

[vmu]

At the column exit (the column length is L), the width of the analyte band, expressed as the standard deviation in length units, according to the Fick law is given by
Sigma(x) = (2*D*tR)^0.5 = (2*D*tM*(1+k))^0.5,
where D is the analyte longitudinal dispersion coefficient,
tR is the analyte retention time,
tM is the hold-up time (dead time),
k is the analyte retention factor.

The width of the analyte peak expressed as the standard deviation in time units is
Sigma(t) = Sigma(x)/uR = Sigma(x)*tR/L = Sigma(x)*tM*(1+k)/L
where uR is the band linear velocity, uR = L/tR.

Thus, Sigma(t) = [(2*D*tM*(1+k))^0.5]*tM*(1+k)/L.
If D is independent of the analyte (a wrong assumption), the higher k, the higher Sigma(t).

The assumption that D results only from the mobile phase diffusion and does not depend on the analyte (a wrong assumption) leads to D = DM/(1+k), where DM is the coefficient of diffusion of the analyte along the column when the analyte is in the mobile phase (the analyte spends 1/(1+k) fraction of tR in the mobile phase).

This results in Sigma(t) = [(2*DM*tM)^0.5]*tM*(1+k)/L.
Again, the higher k, the higher Sigma(t).

Actually, D depends on many parameters that are taken into account, e.g., in the Van Deemter equation for the plate height H (note that H = [Sigma(x)^2]/L = 2*D/uR). However, this does not generally change the trend, and more strongly retained analytes usually show wider peaks.

[Papatom]

Thank you for your response.

Can you actually replace the tR for tM*(1+k) in the formula below as you state?

Sigma(x) = (2*D*tR)^0.5 = (2*D*tM*(1+k))^0.5

I have my doubt because tM*1 is the time the molecules undergo diffusion in the mobile phase and the formula is correct.
However in the time tM*k the molecules are in the stationary phase and should not be in this formula. They should have a different diffusion behavior.

kind regards,

Tom

[vmu]

Can you actually replace the tR for tM*(1+k) in the formula below as you state?

Yes, sure. The analyte spends time tR in the column. This is the time for which we are attempting to find the resulting peak width. The coefficient D describes the overall axial dispersion of the analyte band migrating through the column. The equality tR = tM*(1+k) simply results from the definition of k.

The magnitude of D is governed by several processes in the column, such as the pure molecular diffusion along the column, the hydrodynamic dispersion ("eddy diffusion"), and the finite-rate equilibration between the eluent and the stationary phase. These contributions are taken into account by the A-, B-, and C-terms of the van Deemter equation (and other similar equations) for the plate height. Both the dispersion coefficient D and the plate height H are the two quantities related to each other and used to describe the band broadening. However, the use of H is a common practice in the field of chromatography, while the use of D is usual in chemical engineering.

Please note that your ideas are related only to the B-term broadening (pure molecular diffusion along the column), importance of which decreases with the increase in the eluent velocity. If we consider only this contribution to the band dispersion and make the assumption you suggest (no diffusion for the analyte in its retained state), we come to D = DM/(1+k), as I have already written. Then, Sigma(x) = (2*DM*tM)^0.5, and this is exactly what you expect.

Actually, the B-term contribution to the band broadening is significantly more complicated, and analytes can diffuse when they are in the stationary phase, though this diffusion is slower than the one in solution.

For advanced reading, I recommend the papers by Prof. Gert Desmet on the topic of band broadening in general and on the B-term in particular (e.g., you can search for the words "B-term" and "effective medium theory" in Journal of Chromatography A).

Below are a few links:
https://www.sciencedirect.com/science/a ... 7308002677
https://www.sciencedirect.com/science/a ... 7310014822
https://www.sciencedirect.com/science/a ... 7022005268