Column efficiency: plates, band broadening and resolution

Two compounds have a selectivity of only α = 1.05 — say, isomers that differ in the position of one methyl group. Separating them to baseline at k = 3 takes about 28 000 theoretical plates. An old-fashioned 2 m packed GC column with a few thousand plates is helpless; a 30 m capillary with more than a hundred thousand does it easily. Liquid chromatography shrank its packing from 10 µm to 1.7 µm chasing the same quantity: narrower bands.

As a band travels, each molecule meets its own string of random events: a wider channel here, a longer stay in the stationary phase there. The sum of random steps is Gaussian, and its variance grows in proportion to the distance travelled: $\sigma^2 = H L$. The constant, the plate height $H$, is smaller for narrower bands and a more efficient column. Column length divided by plate height is the number of theoretical plates $N = L/H$ — the counterpart of the last lesson’s tubes.

N can be read straight from a chromatogram: for a Gaussian peak $N = (t_\mathrm{R}/\sigma)^2$. In practice one uses the base width $w_\mathrm{b}$ (between the tangents’ intersections with the baseline, 4σ) or the half-height width $w_{1/2}$ (2.355σ).

In 1956 the Dutch chemical engineer van Deemter split the plate height into three terms:

• A (eddy diffusion): in a packed bed, molecules take channels of different lengths between particles; the term does not depend on flow and scales with particle diameter. Open capillaries have no A term.
• B/u (longitudinal diffusion): the band diffuses back and forth along the column; the longer it stays, the wider it gets, so it matters most at low velocity u. Diffusion in gases is about 10 000 times faster than in liquids, so this term matters most in GC.
• C·u (mass transfer): getting in and out of the stationary phase and through the pores takes time; if the flow is too fast, equilibrium lags and the band is smeared. Smaller particles and thinner films shrink C.

The sum falls and then rises, and its minimum marks the optimum velocity.

Whether two peaks separate is measured by the resolution Rs: the difference in retention time divided by the average base width. At Rs = 1.0 the peaks overlap by about 2 %; at Rs = 1.5 they are essentially baseline-separated. Substituting $t_\mathrm{R} = t_\mathrm{M}(1+k)$ and $\sigma = t_\mathrm{R}/\sqrt{N}$ gives the Purnell equation, which splits resolution into three independent factors: efficiency $\sqrt{N}/4$, selectivity $(\alpha-1)/\alpha$ and retention $k/(1+k)$.

The levers are very unequal. Doubling N buys only 41 %; beyond k ≈ 5 the factor $k/(1+k)$ hardly grows and the run just gets longer; but raising α from 1.05 to 1.10 nearly doubles Rs. That is why a chromatographer’s first move is usually to change the stationary or mobile phase to change the selectivity.