Chromatography Forum

I'm trying to find a formula that relates the measured retention time "tr" to the relative retention (or capacity factor) k* in a gradient elution. put simply I want to estimate k* from a chromatogram.
For isocratic this is a simple case ot k = (tr- to)/to but clearly k is changing as %B increases (in a typical reversed pahse gradient) so cannot be used for gradient elution.
Whenever I try to find an equation that covers this I come up with K* = tg F / D%B Vm S (D here being delta) or k* =1/[1.15b + (1/k0)] neither of which relate back to retention time or volume.
Can anyone help?

The short answer is that it can't be done from a single gradient run. You need to run two gradients with different steepness (%B/min), and even then the math is complex.

If you want to wallow in the math, it's all laid out in the Snyder & Dolan book, High-Performance Gradient Elution: The Practical Application of the Linear-Solvent-Strength Model. Here's a link to it on Amazon: http://tinyurl.com/77jrf54

If you have access to one of the chromatography modeling programs like DryLab or ChromSword, you can enter retention data and get k* values that way (you still need two runs).

Thanks Tom
I had a feeling I might get pointed in the direction of "that book" since that's where the equations I have are quoted as coming from. It was a the vain hope that someone might have the magic formula to hand to save me a trip to the book shop.

By definition the k* of most compounds in gradient elution will be similar, i.e. independent of retention time. From memory, k* is the retention factor a compound when it reaches the mid-point of the column.

tR = (t0/b) log (2.3 k0 b + 1) + t0 +tD
b = Vm *delta phi / (tG F)

Delta phi is change in organic %
tD is grad delay time
t0= column dead time
tR= analyte retention time
k0= theoretical retention factor in 0% organic
Vm= column dead volume = to*F
F=flow rate

S and k0 are regression coefficients from a plot of logarithm of the retention factor as a function of %
organic from several isocratic runs. Ko is the Y intercept, S is the slope. By assuming S = 4 (safe estimate for small molecules), and doing 1 gradient run, you can get an approximation for retention time under any isocratic or gradient conditions.

Yes, but she asked the other way around: estimating k* from retention time. By assuming a value for S, you are, in effect, indirectly assuming the value for k*.

Carls how is the point at which the analyte reaches the mid point of the column determined?
DJ thanks for this explanation almost a "Eureka" moment there.
Tom so I if I'm geting you right it's not possible or at least not accurate to say if an analyte is retained on a column based on a single gradient chromatogram and in order to determine if the analyte is retained we need to run >2 chromatograms (isocratic) in order to determine S?
PS the books on order!

Tom so I if I'm geting you right it's not possible or at least not accurate to say if an analyte is retained on a column based on a single gradient chromatogram and in order to determine if the analyte is retained we need to run >2 chromatograms (isocratic) in order to determine S?

I may have misunderstood your original question.

It *is* possible to tell if a compound is *retained* using a single gradient -- in fact a "screening" gradient is the typical starting point for method development. What is *not* possible is to calculate an exact value of k* using a single gradient (that's where you need the S value, which requires two runs -- which, by the way, can be either isocratic or gradient).

The concept of k* in a gradient is primarily a conceptual tool that ties together gradient and isocratic LC. The key concept being that steepness (%B/min) in a gradient plays the same role that solvent strength (%B) plays in an isocratic separation.

Tom I dont think you did misunderstand my initial question I guess I was just asking the wrong question.

So how do I determine if my analyte is being retained (interacting with the stationary phase efficeintly) on the column when running a single gradient screen? i.e. that k* is between 1 and 10 if we cannot measure k*?

Do I just estimate S as being ~4 (or 5 as I've also seen in the literature) as DJ suggests and use the equation to find K*?

It's probably easier just to evaluate the peak shape and retention times. If you have a well-defined sharp peak at a reasonable retention time then it's being efficiently retained. You can still apply the isocratic rule of thumb that you want your first peak of interest to be at least twice the dead time.

Thanks Peter however I was asked to come up with a spread sheet that will calculate k* from a gradient chromatogram though now I suspect I might as well have been asked to fetch a a "left handed screw driver" or a "tin of elbow grease".

Ah, management trying to get into science!

I think Tom summed it up nicely, you need to do two scouting runs to really calculate k* values, otherwise apply a best guess value for S and call it an approximate model.

Yes. What you do is to run an initial "scouting" gradient. Choose the conditions to give a reasonable k* value (somewhere around 5 is a good choice). You can estimate the gradient time from the equations you mentioned at the beginning:
tG = k* x S x Vm x ΔΦ / F
where ΔΦ is the gradient range expressed as a decimal fraction (e.g. 5% - 95% would make ΔΦ = 0.9). I usually use S = 5 as a convenient value (this is only an approximation in any case). Vm is the column internal volume (for reversed phase columns you can estimate this as 0.5 x L x dc^2 where L is length and dc is column internal diameter) and F is the flow rate.

As Peter said, if you get a reasonably shaped peak somewhere in the gradient, then your compound was retained.

You can take things a step further: by looking at the difference in retention time between the first peak of interest and the last peak, you can make an informed choice about whether or not you need a gradient. If you need a gradient, you can make another informed choice about the range (initial and final %B). If you can do it isocratically, you can make a reasonable guess as to a %B to start with for method development. This is all stuff that we cover in our Advanced HPLC Method Development course (shameless plug here! ).

We've actually put together a spreadsheet that does all the calculations. I just set up an auto-reply e-mail address for the download link, so if anyone is interested, send an e-mail to:
gradient-screening-request@lcresources.com
You can put anything you want in the subject or the body. You will get an automated reply with the download link. Obviously, please make sure that your spam filter is set to "whitelist" e-mail from lcresources.com so that it gets through. The "quid pro quo" here is that you will be added to our mail list to receive announcements about upcoming courses (you can unsubscribe from the list at any time).

I'm wondering-- What is the meaning of K* in gradient HPLC? Rather, why would one desire an estimate of K* in gradient HPLC chosen over, say, retention time, or logKw, which is directly related to hydrophobicity,log P.,, both of which are excellent indicators of relative retention time.

LogKw, for instance, is independent of column geometry, flow rate, type of isocratic or gradient program, and, rather surprisingly, has a pretty impressive level of consistency, reproducibility on like-columns from different manufactures.

K* from a gradient may be meaningful... on that column, system, and precise mobile phase program. Change column geometry, flow rate, gradient program, etc, and that value of K* determined from the initial separation is meaningless.

Tom I dont think you did misunderstand my initial question I guess I was just asking the wrong question.

So how do I determine if my analyte is being retained (interacting with the stationary phase efficeintly) on the column when running a single gradient screen? i.e. that k* is between 1 and 10 if we cannot measure k*?

Do I just estimate S as being ~4 (or 5 as I've also seen in the literature) as DJ suggests and use the equation to find K*?

How do you determine if your analyte is being retained? Shoot some uracil on your column. If you compound has the same retention time, it's not being retained. (or did I mess the question )