How do I calculate the number of theoretical plates in gas chromatography?

The number of theoretical plates #n#, is the number of discrete distillations that would have to be performed to obtain an equivalent separation.

Gas chromatography columns normally have #10^3# to #10^6# theoretical plates.

The number of theoretical plates is related to the retention time, #t_r#, and the width of the peak containing the compound.

If the peaks are reasonably symmetric, it can be assumed that they have a Gaussian shape. Then

#n = 5.54(t_r/w_(1/2))^2#

where #w_(1/2)# is the peak width at half-height.

You find the peak width at half height by drawing a line vertically from the peak maximum to the baseline, measuring half-way up the peak, drawing a horizontal line, and measuring the length of the horizontal line.

You measure the retention time (designated avove as #V_e# for elution volume) at the point where the vertical line drawn through the maximum intersects the baseline.

Both #t_r# and #w_(1/2)# must be measured in the same units.

EXAMPLE

A chromatogram from a certain column has a peak with a #w_(1/2)# of 12 mm and a #t_r# of 650 mm as measured on the chart. What is the number of theoretical plates?

Solution

#n = 5.54(t_r/w_(1/2))^2 = 5.54((650 cancel("mm"))/(12 cancel("mm")))^2 = "16 250 theoretical plates"#