Relative Volatility (Binary)

Relative volatility is the whole question of whether a mixture can be distilled, compressed into one number. Take a vapour at 60% light key sitting over a liquid at 40%: $\alpha = (0.6/0.4)/(0.4/0.6) = 2.25$. For an ideal mixture it is simply the ratio of the two pure-component vapour pressures, so Antoine's equation feeds this page directly. Benzene over toluene comes out near 2.4, which is why that pair is every textbook's worked example.

The number to have a feel for is how quickly separations get expensive as $\alpha$ approaches 1. At $\alpha = 2.5$ a 30% liquid produces a 51.7% vapour in a single equilibrium stage, and a handful of trays does real work. At $\alpha = 1.2$ the same liquid produces about 34%, and you need dozens of trays and a large reflux ratio to achieve anything. Below roughly 1.1 the column becomes tall, energy-hungry and commercially unattractive, and engineers start looking at extractive distillation, adding a third component that shifts $\alpha$, or at membranes and crystallisation instead.

The hard stop is $\alpha = 1$, an azeotrope, where liquid and vapour have identical composition and no number of stages will separate them. Ethanol and water hit this at 95.6% ethanol, which is why every ordinary still tops out there and why fuel-grade ethanol needs molecular sieves or an entrainer to go further. Note too that $\alpha$ varies along a column as temperature and composition change; the single value used in shortcut methods like Fenske is a geometric mean of the top and bottom values, not a constant of nature.