Relative Volatility

The shortcut α_AB = P_A^sat/P_B^sat assumes ideal-solution behavior — that is, Raoult's law holds and the activity coefficients γ_A and γ_B are both equal to 1. This is a good approximation for chemically similar components like benzene and toluene, hexane and heptane, or isomers of the same molecule. It is a poor approximation when the components have different polarity, different hydrogen-bonding capability, or when one is an associating species (water, alcohols, amines). For non-ideal systems, the rigorous formula is α_AB = (γ_A·P_A^sat)/(γ_B·P_B^sat), and the activity coefficients must be computed from a model like NRTL, UNIQUAC, or Wilson, or taken from measured VLE data. Ethanol-water is the classic example where the vapor pressure shortcut fails badly — it does not predict the azeotrope at 96 wt%.

Because temperature varies from the top to the bottom of the column, and because α depends on temperature, α also varies — this is why column design uses an average α. The simplest approach is to compute α at the top and bottom of the column and take the geometric mean: α_avg = √(α_top · α_bottom). This is the value that goes into the Fenske equation. For most industrial systems, α changes modestly with temperature (say 10–20% across the column), and a geometric mean is adequate. For very tall columns or for systems with strong temperature-dependent α, the Fenske equation is applied stage-by-stage in a rigorous simulation. Composition also matters for non-ideal systems, so α at the top may reflect a very different activity coefficient than α at the bottom.

When α = 1, the equilibrium compositions of the vapor and liquid phases are identical — the vapor has the same composition as the liquid at every point. Distillation works by exploiting a difference in composition between the phases: vapor is enriched in the light component, and when that vapor condenses, the liquid it produces is also enriched. If there is no difference (as at α = 1), the process cannot enrich anything, no matter how many stages you use. At an azeotrope, α = 1 at that specific composition, so the column "pins" at that composition and no further separation is possible. This is why ethanol cannot be concentrated past about 96 wt% by ordinary distillation, and why separating azeotropic systems requires tricks like adding an entrainer (extractive distillation), operating at reduced pressure (pressure-swing distillation), or pervaporation.

The reflux ratio R is set by an economic balance between capital cost (more stages → taller column) and operating cost (more reflux → more reboiler duty). The Underwood equations give the minimum reflux ratio R_min for a given separation and feed condition; the design reflux ratio is typically 1.2–1.5 times R_min. As α decreases, R_min increases sharply — a column separating components with α = 5 might operate at R = 1.5, while one with α = 1.2 might need R = 8 or higher. The Gilliland correlation relates R, R_min, N, and N_min so that once you have α, N_min from Fenske, and R_min from Underwood, you can estimate the actual stage count for any chosen R.

For a new separation, α can be: (1) Measured directly — take samples of vapor and liquid from a laboratory still at equilibrium and analyze by GC; (2) Predicted from vapor pressures — use the Antoine equation at the column temperature, valid only for ideal solutions; (3) Predicted from an activity model — NRTL, UNIQUAC, Wilson, or UNIFAC, which require binary interaction parameters but handle non-ideal systems; (4) Taken from published VLE data — the Dortmund Data Bank, DECHEMA Chemistry Data Series, and Perry's Handbook all have extensive tables; (5) From a process simulator — Aspen Plus, ChemCAD, and Pro/II all have extensive VLE databases and flash calculations. For a real project, the choice of method depends on available data and the required accuracy.

The Fenske equation — N_min = ln[(x_D/(1−x_D))·((1−x_B)/x_B)] / ln(α_avg) — gives the theoretical minimum number of stages required to achieve a given separation at total reflux. Total reflux means no distillate is withdrawn, all condensed vapor is returned to the column, and the column operates at its theoretical best. Real columns operate at finite reflux and need more stages than N_min. The Fenske equation is useful because: (1) it gives a hard lower bound on column height; (2) together with the Underwood equations (for R_min) and Gilliland correlation (for actual N), it forms the standard shortcut design method; (3) it directly shows how sensitive stage count is to α — a small change in α can change N_min by a factor of two or more, which is why accurate α data is critical.