CALCULATIONS IN FRONTAL CHROMATOGRAPHY. THEORY

I cannot state that frontal chromatography and related calculations are frequently used in the world of chromatography. The application field of frontal chromatography is limited to preparative studies. But unpopular does not mean useless.
To calculate the efficiency of chromatographic columns from frontal chromatography data the following formula is usually used:

The notation is obvious from the figure.

This expression has a major disadvantage: it has no connection with the rule of matter redistribution in chromatographic systems at normal conditions, i.e. at small sample volume. Indeed, it is rather difficult to find a strict logical connection between the following formulas:

But there must be a connection, since frontal chromatography is a special case of normal conditions chromatography. In this sort of a discussion, the following formula would look more logical:

.

My search through scientific literature gave no result. And if the connection mentioned above does not exist, then formula (1) becomes much less valuable.
To improve the situation, I took certain steps that allowed me to derive a formula which did not contradict the general chromatographic considerations.
Since in frontal chromatography it is assumed that sample volume is unrestrictedly large, the migration of the chromatographic zone is considered as an overall result of the motion of a multitude of chromatographic peaks formed by the volume of one sample. The frontal curve can then be described in the following way:

where
C _max , σ₀ are the height and the width of the chromatographic peak obtained for a sample volume equal to the volume of one theoretical plate;
Vm is the interstitial volume of the column;
x is the axial coordinate whose unit is equal to the width of the peak at Cmax/2 above its base.
Undoubtedly, C_maxσ₀N/V_m=0,94C_in, because the product of the height of the peak by its width is proportional to the initial sample concentration value. Consequently,

Using this expression in practice does not seem to be possible because the integral of the Gauss function is not resolved in the indefinite form. A way out could be provided by an approximate solution of the problem. On this way, the following rearrangement is helpful:

The reasoning that follows is going to be quite surprising.

At a closer look, the second term is very much similar to the half-width of the peak, provided the sample volume is 2x. What can this observation give us? Quite a lot, as we are going to see. Since the effect of the volume of the peak width was studied in detail in the previous work, the peak height can be found by dividing the amount of matter in the sample by the peak width.

where
2x is the sample volume;
C ^'_max , σ are the height and the width of the peak formed by the sample volume 2x.
(Of course, it is arguable that the amount of matter is equal to the product of the peak height by peak width under conditions of a large sample volume, but I suggest that this should be temporarily neglected. If final results based on this assumption are satisfactory, then putting these problems into one's head is not worth it.)
The previous work "Sample Volume Effect on the Chromatographic Process" showed that

σ = 0,257(2x)² +1.

Frankly, however, the factor of 0,257 is only used in this formula in order to expand the working interval of sample volumes. Actually, for moderate sample volumes the factor is equal to 0,235. Combining (6) and (7) we get

Turning back to expression (5) we get a function describing the frontal curve:

Having made such an important conclusion we must consider the legitimacy of the assumptions that had been made before. In other words, to what extent does the obtained function correctly describe the frontal curve. This can be easily checked. The graph obtained by solving function (3) numerically has to be compared to the graph of the new function. The comparison is shown in the figure. In the range of x between -1 to 1 the approximation error does not exceed 0,5 10^-2 C_in. Consequently, the new function satisfactorily describes 95% of the frontal curve height.

However, we have to solve the main problem: how can a formula for the calculation of chromatographic system efficiency be obtained from frontal chromatography data? I am going to proceed form my own achievements:

N = 5,545 V_rV_mr / σ₀ ² ,

which are discussed in detail in the previous work "Calculating the Efficiency of Chromatographic Systems".

What should we rearrange in this formula in order to obtain an expression for efficiency calculation in frontal chromatography? First of all, such a segment of the frontal curve should be found which would be equal to σ₀/(5,545)^1/2, a step which would eliminate the factor 5,545. Using our new formula it is easy to find C/Cin having x= ±1/(5,545)^1/2. The obtained value amounts to 0,159, therefore

One can obtain quite a family of such expressions. For example, in practice it is more convenient to use a formula with (V_mr - V_0,25):

Further information about the work is available in the paper by Nemirovsky A.M. Calculations in Frontal Chromatography / Rasschety vo frontalnoy khromatografii (Zavodskaya laboratoriya 1996. N3. P. 13-15.)