# SAMPLE VOLUME EFFECT ON THE CHROMATOGRAPHIC PROCESS. THEORY

Alexander M.Nemirovsky (novedu@yahoo.com)

Researchers do not like to go into detail on the sample volume effect on the chromatographic process, because with the present state of the question, calculations are only of theoretical interest. That is easy to understand: why waste effort for calculations if the dispersion of the chromatographic zone in the column cannot be satisfactorily described? Recently, however, some improvement has been achieved as it has become possible to describe the chromatographic process much better. In this connection, it is now important to consider the effect of the sample volume on the width of the chromatographic peak. This can be exemplified by a problem that could not have been resolved before. Suppose we have a histogram with peaks which are not completely separated. We have to make a conclusion whether the length of the chromatographic column should be increased, or the sample injector and other factors affecting the sample volume should be re-considered.
The most common and logical, from the point of view of the general chromatographic theory, is the following formula:

 σ 2 = σo2 + σin2 , where (1)

σ is the width of the peak affected by the sample amount sin;
σo is the width of the peak for a sample volume equal to the volume of one theoretical plate.

Application of this formula is restricted by the requirement that the concentration profile should originally have the Gaussian shape. This is possible, e.g., in gas-liquid chromatography, but generalizing this principle for all possible cases would be a mistake.
Undoubtedly, the more common situation when concentration is the same throughout the sample is of greater interest. Let us assume that a real sample consists of several parts, and that the volume of each part is equal to the volume of one theoretical plate. Then the chromatographic peak will result from additive contributions of consecutive peaks: (2)

where
C(max) is the height of the chromatographic peak;
Vmax is the interstitial volume of the column;
x is current eluent volume expressed in units of the width of the considered peak;
V is the volume of the sample expressed in units of the width of the considered peak;
N is the number of theoretical plates.

However, this formula implies an approximation which should be kept in mind. The point is that expression (2) assumes that the added peaks are equal, which can only be possible when the number of the plates in the column is large enough. The further conclusions cannot be applied to systems with low efficiency. The criterion is the shape of the chromatographic peak, as for low efficiency systems the peak will have a pronounced asymmetric shape.
It is quite difficult to use expression (2) in practice because the integral of the Gauss function does not make things better. Simplifying the expression through commonly used rules cannot bring us to simple calculation procedures either. In this connection one can take the easier heuristic way giving up accuracy for a simpler form of the final result. Mathematical investigation showed that the following formula would do:

 σ = σo (1 + 0,257 Vin2 / σo2 ), (3)

where Vin is the sample volume.

The error in the range Vin / σo <=2 does not exceed 1%. A smaller error (<0.1%) but within a narrower range of peak widths (Vin / σo <0,5) is achieved by the formula σ = σo (1 + 0,235 Vin2 / σo2 ).
Comparing (1) and (3) one can see that in the range of small σino values these expressions are very much similar, since (4)

It is easy to find the constant of proportionality between Vin and σin:

 0,257 (Vin / σo)2 K σin2 /(2σo2) , K 0,7 or σin 0,7 Vin. (5)

Thus we have come to a very important conclusion: at small sample volumes, the chromatographic peaks formed by different concentration profiles of the sample are practically undistinguishable. Calculations show that a discrepancy of, e.g., 5% between the types of dependencies appears at Vino 1,2. In this connection one does not need to bother considering the method of calculation of the sample volume effect on the chromatographic process up to Vino 1,2. However, at greater volumes this or that method has to be chosen.
Proceeding to practical estimations, let us consider the most common case of calculating the number of theoretical plates and the volume of the sample from the data on retention volume and peak widths. For the Gaussian concentration profile the calculation is not complex because the expression (6)

is easily linearized in the coordinates σ2 vs.VrVmr.

The figure shows an example of the calculation of a sample volume through linearization. It should be noted that the straight line sections a segment on the ordinate axis equal to squared volume of the ejected sample. (Details on chromatographic parameters calculation can be found in the work "Calculating the Efficiency of Chromatographic Systems".) For samples with homogenous concentration distribution, linearization is hardly possible. A solution can be found in performing calculations only in the range of small values, Vino <1,2. For such calculations, linearization in the manner of (6) can be harmlessly used. It is not difficult to recalculate the obtained value σin into Vin since σin 0,7Vin.

chromatographic chromatography theory theoretical gas-liquid gas ion