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The position or retention time of a chromatographic peak is governed largely by the fundamental thermodynamics of solute partitioning between mobile and stationary phases. This retention time is called the capacity factor, k',and it is measured experimentally as
where tr is the retention time of the peak (usually measured at the peak maximum, but most rigorously measured as the first statistical moment or centroid), and t0 is the dead time of the column (the time required for an unretained solute to traverse the length of the column).
In chromatographic peaks, band broadening arises from a wide array of real-world phenomena that are generally categorized as “non-idealities”. These band broadening processes include:
• Axial diffusion
• Dispersion (varying flow paths through a packed column)
• Resistances to mass transfer in both mobile and stationary phases
• Kinetic resistances to adsorption and desorption
• Extracolumn instrument response effects from detector, tubing, and electronics
All of these non-idealities are present to widely varying degrees with the different forms of chromatography. The width and distortion parameters in a chromatographic model may account for only a portion of these effects, or they may all be dealt with in a combined way. The physical meaning attributed to the broadening parameter is highly dependent upon the chromatographic model from which it was derived, and the assumptions which underlie that model.
Unlike spectroscopy where bands generally broaden symmetrically, chromatographic band broadening processes are often asymmetric in nature because of the directional constraints of column dynamics.
Instrumentally, a detector cannot sense a component prior to its arrival, but if the detector has a slow response, it may record that component as having arrived further along in time. Many chromatographic effects can also be viewed as having directional constraints. In affinity chromatography, for instance, components tend to desorb very slowly, thereby “drizzling” off the column with a long extended tail. Column overload, which is a non-linear effect, causes peaks to elute earlier than they otherwise would because the column adsorption capacity has been effectively decreased by the overload. Peaks come out with a sharp right-shifted skew which is commonly called “tailing”, and in extreme cases, they can even take on a right triangular profile.
A more unusual case occurs in gas-liquid chromatography, where column overload, rather than causing peaks to elute earlier, causes them to elute later because the solute has effectively increased the column adsorption capacity. This phenomenon is called “fronting”. In strictly thermodynamic terms, tailing results from a Langmuir adsorption isotherm, while fronting results from an anti-Langmuir isotherm.
The essential point is that chromatography exhibits a very wide variety of asymmetric processes. This explains both the diversity and complexity of the various mathematical models used to describe the process.
The PeakLab chromatographic models each offer an a parameter which relates to the retention time of the peak. More advanced users can relate this directly to the thermodynamic capacity factor (k’) by applying an X=X/t -1 transformation to the time axis of the data prior to the data fit, where t is the dead time value.
Band broadening is modeled by the a2 parameter in PeakLab's chromatographic models.
Peak asymmetry is modeled by the a3 parameter and possibly by additional parameters. Certain models also address the kurtosis or fatness of the tails of the peaks.
These parameters relate to different physical broadening and peak skewing processes depending upon the peak model which is chosen. These are covered in the separate function descriptions.
Chromatographers often characterize band width by the common half-maxima and extrapolated baseline widths, as well as the second statistical moment of the peaks. Also used is the concept of column efficiency as a function of the number of “theoretical plates”.
where µ1 =moment1 (centroid), µ2 =moment2
For a Gaussian, this translates to:
While PeakLab does report the historical Gaussian form of this computation in its numeric analysis, its use in column efficiency determinations is not recommended, except for nearly symmetrical peaks.
Reduced Plate Height
When comparing columns, a more useful figure of merit is the “reduced plate height”:
where L=column length and dp =sorbent particle size.
This calculation will allow you to directly compare the efficiency of columns which have different lengths, and different sorbent particle sizes—factors which can dramatically affect the number of theoretical plates in the column. A value of 2 is the theoretical minimum of the reduced plate height (the maximum possible efficiency).
Resolution is defined between two adjacent peaks. This is why PeakLab’s chromatographic analysis reports no resolution for the first peak. Resolutions are computed by:
where tr1 and tr2 are the retention times of the second and first peaks, and W1 and W2 are their full widths at peak base.
Peaks that are right skewed are commonly referred to as “tailed” and those left-skewed as “fronted”. Traditionally, chromatographers measure the peak asymmetry at 10% of the peak maximum while statisticians do so at 50% of maximum. An asymmetry is a ratio of the width to the right of the mode to that left of the mode (the mode is the position of the peak apex). As such tailed peaks have asymmetries > 1, and fronted peaks have asymmetries < 1.
The chromatographic capabilities within PeakLab owe much to the kind contributions of Dr. James L. Wade. Dr. Wade built upon the earlier work of H. C. Thomas, and derived the NLC function you find in PeakLab's chromatographic function set. Jim has also provided us some exceptional chromatographic data for the research we made in developing PeakLab. With the enhancements in this current version, PeakLab extensively builds upon Dr. Wade’s NLC function and mathematical work. We wish to thank Dr. Wade for assisting us in determining the analytical needs of chromatographers, and for those portions of these chromatography and function notes he furnished to us.
J. C. Giddings, Dynamics of Chromatography, Part I, Marcel Decker, New York, 1965
J. R. Conder and C. L. Young, Physicochemical Measurement by Gas Chromatography, John Wiley and Sons, 1989
P. R. Brown and R. A. Hartwick (eds.), High Performance Liquid Chromatography, John Wiley and Sons, 1989
J. C Sternberg, in Advances in Chromatography, Vol. 2, Eds. J. C. Giddings and R. A. Keller, Marcel Decker, New York, 1966