Computer analysis of asymmetric peaks in gas chomatography
A digital computer was used to measure accurate gas chromatographic peak symmetry, position and dispersion by central statistical moments. Benzene samples were chromatographed on squalane for columns of 3, 6 and 12 foot lengths and 1/8", 1/ 4" and 1/2" diameters as a function of sample size. Peak symmetry was monitored by measuring skew, γ₁ , and "excess," γ₂ two quantities derived from the higher central moments.
Skew was found to increase in a positive manner for tailing peaks, pass through a maximum and approach a limiting step form for extremely overloaded columns. Skew could be used to indicate saturation of the liquid phase when it passes through a maximum. Negative skew for fronting peaks also approached a zero limiting form.
Excess, γ₂ was found to decrease rapidly for all columns. A few microliters of sample were sufficient to cause significant negative values of excess. Excess provides a semi-quantitative measure of column capacity.
Three general types of peak shapes were observed with increasing sample sizes: 1) gaussian behavior at very low sample sizes; 2) distorted peaks suitable characterized by central moments at normal analytical size samples; and 3) highly distorted peaks at larger sample sizes where central moments no longer reflect the step shapes observed. Moments can be used to set limits on sample sizes which will produce these highly distorted peaks.
Two moment related measures of skewness were also calculated. Pearson’s skew, (Mean - Mode)/(variance)½, along with Pearson’s β, γ measure of skewness, were found to qualitatively reflect peak shape behavior only in the region of analytical sample sizes. Pearson's skew is subject to difficult interpretation due to equivalent modal values for large samples and the β, γ measure was insensitive to fronting peaks.
Third and fourth central moments were observed to have regular behavior as a function of sample size.
A moment definition of resolution was derived
R = 0.5 (M(1)₂ - M(1)₁) /( √M(2)₁ + √M(2)₂ )
where M(1) = mean and M(2) = variance. This definition was used to compare solvent efficiency for the separation of benzene and cyclohexane on three liquid phases, squalane, dinonalphalate and TRIS. In terms of equivalent throughput, defined as moment resolution per unit time per gram of sample, TRIS was found to be 175 times more selective than squalane.
A preparative chromatograph was built with four thermal conductivity detectors at 50' intervals in a 200' x 3/8" column. Column efficiency was measured by comparison of moment parameters at the end of each 50' section. The column was operated both at normal, high pressure drop (ambient outlet pressure) and low differential pressure (constricted outlet). The condition of high pressure drop caused acceleration of samples through the last two column sections and resulted in much poorer column efficiency. The low differential pressure column, inlet 350 psi, outlet approximately 150 psi produced a more linear velocity and greater column efficiency. In fact 100 feet of the low differential pressure column generated the same resolution as 200 feet of the high pressure column.
For most chromatographic peaks manual methods of peak evaluation are subject to significant operator errors due to the subjective nature of assigning base width and peak retention time. The use of moments greatly increases the accuracy of two important measurements; (1) retention time as measure from the first moment and (2) resolution for preparative scale samples as measured from the first and second moments. The method of moments provides an accurate means of measuring retention time, dispersion, resolution and preparative scale equivalent throughput.