Predicted values for N?

[LD]

I've been reading a few articles that discuss how measured column efficiencies (N) may fall short of their theoretically predicted values.

How do you calculate a predicted value for N?

[Uwe Neue]

To predict a plate count, you need to know a lot of things: the coefficients of the van-Deemter equation, including their dependence on the retention factor, and the diffusion coefficients of the analytes in the mobile phase at the temperature at which you are operating, which requires among other things a knowledge of the viscosity of the mobile phase. These things are not easy to estimate or measure, thus you will need to take such statements with a grain of salt, possibly with a lot of grains of salt...

You need: the van-Deemter equation:

H = A +B/u + C*u

For a well-packed column, A is about 2 times the particle size. B is proportional to the diffusion coefficient of the analyte in the mobile phase and the stationary phase and depends in a non-trivial way on the retention factor. C depends on the square of the volume-averaged particle size, and a function that depends on the retention factor and varies between different packings. The diffusion coefficient in the mobile phase can be estimated from either the Scheibel equation or the Wilke-Chang equation. If the size of the analyte is not negligible compared to the pore size, other complications come in.

I have listed a few conservative coefficients in my book on "HPLC Columns".

[LD]

Thank you very much!

We were discussing this at work, and I had a feeling such a prediction would be a non-trivial undertaking.

I'll stick with actual measurements...

[Victor]

For a well packed column operated at its optimum flow, the reduced plate height H/dp ~ 2. This is an experimental observation but it can be shown to have a theoretical basis by differentiating the Van Deemter equation (dH/du). Equating the differential to 0, the value of u(optimum) can be found and by substitution, the minimum value of H. Using estimates of the Van Deemter coefficients , the estimate above can be seen to be reasonable. No doubt this derivation is also found in Mr Neue's book.

Thus for a 5um column, H = 0.001cm

Since N= L/H, a 25cm column should have a maximum of 25,000 plates.

A 10cm column of 3um particles should have 16,667 plates.

These figures are what you might expect as maxima when using non-polar solutes under ideal conditions in RPHPLC.

Whether they are "theoretically predicted values" is another point, because the VD equation is an approximation in any case.

[Uwe Neue]

The issue in real life is that people are rarely working close to the minimum of the van Deemter curve.

[Veronika R. Meyer]

Victor's statement that the optimum plate height is at h = 2 (2 particle diameters) is too optimistic in many cases. Even a well-packed RP column may show h = 3 in its van Deemter optimum. This value would lead to 10000 theoretical plates with a 15 cm column packed with 5 um material. For ion exchangers and other special phases the h opt value is even worse, i.e. higher.
Veronika

[Uwe Neue]

These rules of thumb are a rough guidance only. Look at the certificate of analysis that comes with the column for the platecount and/or the specifications by the manufacturer. Depending on the packing, the surface chemistry, the particle size distribution and a few other things, it is sometimes easy to achieve a plate height of 2 dp, but I usually count on 2.5 dp as an expectation standard.

[Victor]

Veronika,

I agree with Mr Neue's statements. The original question asked for "theoretically predicted" values. I do not believe this is possible to answer, due to approximations in the Van Deemter equation and its parameters-such as the diffusion coefficient, which is usually approximated by the Wilke-Chang equation.

I did state that "these figures are what you might expect as maxima using non-polar solutes under ideal conditions in RPHPLC". I made no reference to ion exchangers or other special phases, but RPHPLC is by far the most widely used technique. If you are really getting h=3 for well packed RP columns, I would suggest a change of column supplier or examination of the dead volume of your system. Of course the latter is easily corrected for as you have shown yourself.