By Anonymous on Tuesday, June 8, 2004 - 05:31 pm:

Over the years I have seen the following stated:

N proportional to L/dp.

This can be broken down into 2 equations.

1, N proportional to L
2, N proportional to 1/dp

The first of these equations, of course, is true and just comes from the basic defention of N. However the second of these equations I don't quite understand. It must be a simplification of some type, because the true relationship between N and dp is given by the van deemter; and this is a complex equation which cannot be reduced to a simple proportionality.

Can anyone offer any insight here. If it is a simplification, what are the conditions under which it applies?

One of the reasons I am asking is that if it's true (that N prop. to L/dp) it means that you can maintain the efficiency of a given method but increase the speed by reducing column length and particle diamter by the same factor.

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By tom jupille on Tuesday, June 8, 2004 - 11:12 pm:

It's only a rule of thumb. Accurate to maybe +/- 30% (as a guess). It applies about as generally as "men are taller than women".

Like many generalizations, it provides a reasonable description of reality. If you look at the evolution of HPLC, you'll see that today, we get about the same plate count from a 10-cm column packed with 3 micron material that we got 30 years ago from a 30-cm column packed with 10 micron material.

The bottom line is that it approximately applies all the time, and it exactly applies almost never!

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By Uwe Neue on Wednesday, June 9, 2004 - 05:17 pm:

It depends how you measure N (i.e. the operating conditions) and what the purpose of the comparison is.

At the minimum of the van Deemter curve, N is proportional to 1/dp. This means that the maximum plate count that you can get out of a column is simply proportional to L/dp.

This also means that I can scale this maximum performance in a very straight-forward way. A 15 cm 5 micron column, a 10 cm 3.3 micron column and a 5 cm 1.7 micron column all have the same maximum plate count, but the shorter column with the smaller particles generates this same plate count in a much shorter time.

In this case, the short column is 9 times faster: 3 times faster because it is 3x shorter, and 3 times faster because the velocity for the minimum of the van Deemter increases by a factor of 3, and 3x3 = 9.

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By Anonymous on Wednesday, June 9, 2004 - 06:24 pm:

Uwe

I would like to understand better. I have a pretty in-depth knowledge of chromatography theory, but I was not able to understand the statement: "at the minimum of the van Deemter curve, N is proportional to 1/dp"

At the van Deemter minimum, there are significant contributions from all three terms: A, B, and C. So how do we arrive at the above-mentioned simplification. Could you please explain mathematically how you get there (or else suggest a good reference).

I think this would be of interest to many people as these simplified relationships are infinitely easier to utilize than the actual van Deemter.

Thanks very much

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By Uwe Neue on Thursday, June 10, 2004 - 08:14 pm:

The best thing to do is to read some really ancient literature. There are three publications by Guiochon, Eon and Martin in J. Chrom. 99 (p357ff), 108 (p229ff) and 110 (p213ff) that explain everything one ever wanted to know about column design.

The next thing is to take the first derivative of the van Deemter, and then determine the HETP at the minimum. It is indeed proportional to the particle size, dp.

The math does not work well on a website, but if you insist, I will show it.

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By Uwe Neue on Friday, June 11, 2004 - 12:13 pm:

I'll try to show the math. It's not complicated.

you start off with the van Deemter equation:

H = A +B/u + C*u

Take the first derivative:

dH/du = - B/u^2 + C = 0

and you get the velocity at the minimum:

u(min) = sqrt*(B/C)

Put this back into the v.D.:

H(min) = A + 2*sqrt(B*C)

A = 1.5 dp (the particle size)
B = Dm (the diffusion coefficient)
C = dp^2/(6*Dm)

H(min) ~ 2.3 dp
The HETP at the minimum of the van Deemter depends only on the particle size. Thus the maximum plate count that one can get out of a column is nothing but column length devided by 2.3 x dp:

N(max) = L/(2.3*dp)

(QED)

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By Anonymous on Friday, June 11, 2004 - 04:29 pm:

Thanks. I think I've got it.