General resolution equation

[m32311]

Hello,

I was looking through a couple of books and papers and have found a discrepancy among general resolution equations quoted, but I do not understand where the difference comes from.

Sometimes it is stated as:

eq. 1. Rs = (N/16)^1/2 * (a-1/a) * (k'/1+k')

whereas sometimes I have found it as:

eq. 2. Rs = (N/16)^1/2 * (a-1) * (k'/1+k') --> note the missing "a" in the denominator

(a = selectivity factor, k' = capacity factor).

Just wondering if anyone knew the reason (or in which circumstance) either equation could be used. Thanks for your help.

[Molly]

Resolution is defined as Rs=2(tr2-tr1)/(w1+w2). If you use tr=to(1+k) and W=4tr/N^0.5 then the equation become
Rs=(N^0.5/4)*(k2-k1)/[1+(k1+k2)/2]

The k' in denominator of the general equation is the average of k1 and k2.

For the k' in the numerator, if you use k2=k1*alpha, you get equation without alpha in the denominator, and k' in the numerator is k1 in the general equation.

If you use k1=k2/alpha, you get the other equation and the k' in the numerator of the general equation is k2.

Since we typically only care resolution between close related compound, alpha is close to 1, so both equation will give you practically the same number.

Hope this helps. Molly

[THNDRacket]

I too have recently been confused by this. I have come up with quite a number of resolution equations.

Rs1 = sqrt(N/4)*(α-1)/(α+1)*(kbar/(1+kbar))
= sqrt(N)/2*(k2-k1)/(2+k2+k1)
= sqrt(N)/2*(tr2-tr1)/(tr2+tr1)
= sqrt(N/4)*(α-1)*kbar/((α+1)*(kbar+1))
= sqrt(N)/2*((α-1)/(α+1+2/k1))

exact equation sqrt(N/4)*ln(1+(kf2-kf1)/(1+kf1))
since ln(1+x) = x for small x, if kf term small compared to 1,
sqrt(N/4)*(k2-k1)/(1+kf1)

RsKnox sqrt(N/4)*(α-1)*(k1/(1+k1))
RsPurnell sqrt(N/4)*(α-1)/α*(k2/(1+k2))

Rs Gaussian 1.18*(tr2-tr1)/(W1+W2)
most of these from
http://ull.chemistry.uakron.edu/analyti ... atography/

Phenomenex (1/4)*(α-1)*sqrt(N)*kbar/(1+kbar)


USP (tr2-tr1)/([W2]/2 + [W1]/2) W extrapolated from tangent

Mac-Mod (N^0.5)/4*[(a-1)/a]*[(kbar/(kbar-1)]
Mac-Mod LabNotes Aug 17, 2006

Some of these are certainly not compatible since sqrt(N/4)<>sqrt(N)/4

Here are some results that I have obtained for two peaks on the same run:


exact 4.13
Knox 4.25
Purnell 4.01
Guassian 2.24
Phenomenex 2.13
Mac-Mod 2.74
Rs1 2.06

[Uwe Neue]

All the ones that gave you the smaller number have a factor of 2 error in it. Resolution is meant to be the distance between the peaks, divided by the sum of half the peak width of the first peak and half the peak width of the second peak. The USP definition is essentially the original thought process. Everything else are derivatives of this (unless of course they contain the 2x error, as some of these equations do).

[tom jupille]

THNDRacket, to elaborate on Uwes post, it looks like the big (2x) difference is simply a case of misplaced parentheses:

(√N)/4 ≠ √(N/4)

[THNDRacket]

Evidently. I'm just surprised that reputable companies servicing the chromatography field could publish incorrect equations and not have them corrected by now.

[ravenwork]

Thank providence that the scales have fallen from your eyes!

Good science requires skepticism, and verification. Don't take things on faith, and if you can't check it out for yourself then you probably don't have enough knowledge. Get more!

And, I ask you, what is more fun than finding out you were right and a multi-billion dollar company is WRONG?



Evan L. Cooper, Ph.D.