CALCULATING THE EFFICIENCY OF CHROMATOGRAPHIC SYSTEMS. THEORYAlexander M.Nemirovsky (novedu@yahoo.com)
In the present work, a new formula for the calculation of the efficiency of chromatographic systems is suggested. In contrast to other formulas of this type, the suggested formula is not an empiric one but it follows from the main chromatographic equation describing redistribution of eluted matter. The new formula has the following form:
At first glance it may appear that there is no significant difference between
using the new formula or the old one (N=5.545 V_{mr}^{2} /σ^{2}),
yet this is not true. Using the old formula gives no reliable way of calculating
the parameters of a chromatographic process, e.g. the volume of an injected
sample, the number of theoretical plates, etc.
This relationship describes the concentration at the mth chromatographic plate after passing n portions of the solvent (eluent). k is the factor of matter extraction from the mobile phase to the immobile phase (sorbent). In further discussion it is more convenient to use the terminology commonly accepted in the chromatographic practice. In that case, m is the interstitial volume of the column Vm, as we are going to consider the contents of the eluted matter at the column exit, and n is the retention volume Vmr.
; Then
To proceed to describing the chromatographic peak in the form of equation (1) we have to rearrange the main redistribution relationship using the following expressions:
where σ is the width of the chromatographic peak. In other words, we must rearrange the expression to such a form where the magnitude of the peak maximum would appear instead of the initial concentration value.
Frankly, I have to admit that rearranging such a mathematical expression does not give much pleasure. A good way out could be provided by a transform that would produce an expression containing the factorial of a term of the sum as a cofactor. I have found such a transform! To understand its meaning let us consider an abstract example of the factorial of two numbers (a+b)!, where a>>b.
(a+b)! = a! (a+1)(a+2)+...+(a+b); (a+b)! = a! a^{b}(1+1/a)(1+2/a)...(1+b/a); ln((a+b!)=ln(a!)+b ln(a) + ln(1+1/a)+ln(1+2/a)+...+ln(1+b/a).
Since a>>b, that ln(1+b/a)=b/a. Then ln((a+b)!) = ln(a!) + b ln(a) + 1/a + 2/a+ ... +b/a; ln((a+b)!)=ln(a!) + b ln(a) + b(b+1)/2a; or (a+b)! = a! a^{b }exp (b(b+1)/2a).
Now we can see with satisfaction that, having factorized the factorial of the sum, we have simplified the expression (5) considerably.
Turning back to expression (1) and using the Stirling's formula (n!=(n/e)^{n}√(2πn))
for the factorial, it is easy to find the position of the peak maximum (Vmr)
as the extremum of the function, having k=Vr / Vmr .
where N is the number of theoretical plates in the chromatographic column. If we want to measure the FWHM (Full Width at Half Magnitude) of the peak,
then 8(ln(Cmax/C))=5.545. Thus, having made the whole way through the demonstration,
we obtain formula (1), as was required.
Further information on this work is available in the paper by Nemirovsky A.M., Sukhoruchko V.I. 'Calculation of the Efficiency of Chromatographic Systems' (Raschet effektivnosti khromatograficheskikh sistem, Zavodskaya laboratoriya, 1994, V.60, N6, P.14).
