CALCULATING THE EFFICIENCY OF CHROMATOGRAPHIC SYSTEMS. THEORY

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:

 

N=5,545 Vr Vmr /σ2.

(1)

 

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_mr² /σ²), 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.
To avoid any allegations, I suggest considering in detail the proofs which have made such an important general conclusion possible.
It is common knowledge that in the classical chromatographic theory the following expression is conventionally used to describe the process of matter redistribution in the column:

(2)

This relationship describes the concentration at the m-th 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

(3)

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:

(4)

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.

(5)

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 co-factor. 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.

(6)

 

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 .

Our expression becomes even simpler:

or

 

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.
Curiously, a similar formula was suggested in 1957 by Golay (Golay M.J.E. / Anal.Chem. 1957. V.29. P.928). His idea found no support because he used extremely complicated logical formulations comparing chromatography with the functioning of the telegraph.
The advantage of the new formula can be clearly seen in the figure. It is worth noting that the straight line cuts off a segment of the ordinate axis which is equal to the squared volume of the injected sample. (The reasons why the linearization is made in this way are described in the work "Sample Volume Effect on Chromatographic Process".) In the old processing technique the physical meaning of the injected sample concept is lost because the squared sample volume cannot be a negative value. This fact demonstrates that the wrong relationship is used to describe the chromatographic process. For more detail on practical calculation of chromatographic parameters please see the work "Calculations in Chromatography".