Every separation in this chapter rests on one fact: put a liquid mixture in a warm vessel and the vapour standing over it is not the same mixture. Three relations describe that gap, and picking the wrong one is the first mistake a paper will offer you.

Raoult's law, $P = x\,P^{0}$ — read aloud P equals x times P-nought. $P$ is the partial pressure one component puts up over the liquid, in kPa; $x$ is that component's mole fraction in the liquid, a bare number between 0 and 1; and $P^{0}$ — P-nought — is the vapour pressure of that component pure, at the same temperature. It is the law for the component that makes up most of the liquid, and it assumes an ideal mixture: molecules that do not much care which neighbour they have.

Henry's law is its opposite number, for the component that is barely there at all: $C = H P$, C equals H times P, where $C$ is the concentration dissolved in the liquid in mol/m³, $P$ is the gas's partial pressure above it in pascals, and $H$ is the Henry solubility constant for that gas in that solvent at that temperature. Be warned about $H$: it is tabulated in at least six different forms, some of them inverted, and a number copied without its units is worth nothing. In absorption work the same law is usually written as a straight equilibrium line in mole fractions, $y^{*} = m x$, where the slope $m = H/P$ uses a Henry constant in pressure units and $P$ is the column pressure. Same physics, different clothes.

Relative volatility is the one that decides whether a column is possible: $\alpha = \dfrac{y\left(1 - x\right)}{x\left(1 - y\right)}$, alpha equals y times one-minus-x, over x times one-minus-y. Here $x$ and $y$ are the mole fractions of the light component in the liquid and in the vapour standing in equilibrium with it, and $\alpha$ — alpha — compares how much each component is enriched on the way up. Rearranged, it draws the equilibrium curve itself: $y = \dfrac{\alpha x}{1 + \left(\alpha - 1\right)x}$.

Carry one number out of this lesson. At $\alpha = 1$ there is no separation, at any reflux, on any number of trays, for any money. Near 1.1 the tower is a monster; benzene and toluene at 2.4 is a comfortable teaching column; above about 4 a couple of stages will do. Alpha is not a detail of the design — it is the question of whether there is a design at all.