McCabe and Thiele's construction is the only version of this problem you can see. Draw the equilibrium curve on a plot of vapour $y$ against liquid $x$, add the 45° diagonal, then add three straight lines. Step between the curve and the lines, and each triangle is one theoretical stage.

Before the lines, one distinction that decides every question in this lesson. Equilibrium pairs a vapour with the liquid it is sitting on, on the same tray. An operating line pairs a vapour with the liquid it is passing, between trays. Two different pairings, and mixing them up is the classic error here.

Rectifying line, above the feed: $y = \dfrac{R}{R+1}\,x + \dfrac{x_D}{R+1}$. $R$ is the reflux ratio and $x_D$ the distillate composition. Its slope $R/(R+1)$ is always below 1, so it lies shallower than the diagonal, and setting $x = x_D$ gives $y = x_D$ — it passes through $(x_D, x_D)$ on the diagonal, which is how it gets drawn. Its intercept $x_D/(R+1)$ measured off a drawing is the classic way to read a column's reflux ratio.

Stripping line, below the feed: $y = \dfrac{V_B+1}{V_B}\,x - \dfrac{x_B}{V_B}$, with $V_B$ the boilup ratio and $x_B$ the bottoms composition. Its slope is always above 1 — steeper than the diagonal — and it passes through $(x_B, x_B)$. Shallower above, steeper below: that pair of facts will catch most sign errors before they leave the page.

The q-line: $y = \dfrac{q}{q-1}\,x - \dfrac{z_F}{q-1}$, passing through $(z_F, z_F)$. $q$ is the feed's thermal condition — the moles of liquid added to the stripping section per mole of feed. Five cases: $q > 1$ subcooled liquid, $q = 1$ saturated liquid, $0 < q < 1$ part vapour, $q = 0$ saturated vapour, $q < 0$ superheated vapour. The slope behaves oddly on purpose: vertical at $q = 1$, flat at $q = 0$, and negative — leaning backwards — for a flashing feed.

Both operating lines must cross on the q-line, and where all three meet is the optimum feed stage. That is what q is for commercially: preheating the feed is not free, and it shifts work between the reboiler and the condenser. Two footnotes worth carrying. All three lines are straight only because of constant molar overflow — every mole condensing boils exactly one mole — which holds for chemically similar pairs and fails for methanol and water. And the reboiler counts as a stage: the last step on the diagram is not a tray, and forgetting it buys a column one tray too tall.