The zeroeth moment is the area under the effluent curve, which is effectively the mass of the solute pulse fed. The first
moment is equal to the mean residence time. M
is the second moment of concentration with respect to time about the mean residence time. It describes the breadth of the
peak and can be calculated from the chromatogram and used to estimate the variance of Gaussian or near-Gaussian peaks. The
first and the second moments of a peak are independent of the amount of the component as they are normalized to the peak area.
It should be noted that Equations (11) thru (13) provide definitions for moments in the temporal time frame, whereas peak
analysis can be performed in either the temporal or spatial timeframe. These moments can be related through the migration
While the peak traverses the column, its variance increases due to longitudinal diffusion and other dispersion effects.6 The variance of the peak is an additive quantity. That means the total variance can be expressed as the sum of variances coming
from all the contributors to dispersion.7,8
To characterize skewness we suggest using the third moment about the mean residence time according to Equation (14).
Skewness also can be characterized by differences between the right- and left-hand half widths, but there is a major advantage
to using moments throughout. This is because M
, and M
are additive.9 The observed values of these three moments are just the sums of those for the column itself and those for all of the auxiliary
apparatus — tubing, headers, and even detectors. This is particularly important during scale-up as auxiliary apparatus can
have major effects on effluent curves.
Figure 4. Comparison of Techniques to Characterize Column Effluent Curves Experimental Conditions: Column: 1.6 × 8.8 cm with
ToyoPearl SP 650M. Mobile Phase: 150mM Citrate, Solute Pulse: 500 mL of 10 mg/mL globular protein.
Moments can be calculated by numerical integration, but care must be taken with the tails of the curves, especially for the
second (and even more, the third). Here small errors are magnified at large time through multiplication by the square or cube
of t. Extrapolations to infinite time are usually made from simple curve fits of the large-time data, and it often is sufficient
to assume exponential decay with respect to t. One property of M
is that it is independent of system size and so is the same for the column proper and the whole experimental system (See
Ref. 10, Ex. 23.6-3, for limitations on the use of moments. These are seldom important for our present purposes).
Table 1. ASTM Bases of N Determination