Distillation and Fractional Distillation

Every chemistry student has heated a flask of wine-like mixture. They watch vapor curl into a condenser. That simple act — boiling a liquid mixture and collecting the vapor — is distillation. It is one of the oldest separation methods in chemistry. It also happens to be a beautiful, quantitative application of the two-component liquid-vapor phase diagram. Distillation matters because most liquid mixtures are not pure. Petroleum refining, ethanol purification, air separation, and countless laboratory purifications rely on one key fact. Vapor above a boiling mixture is richer in the more volatile component than the liquid below it. Repeating that enrichment step, over and over, is the essence of fractional distillation. This article builds on the temperature-composition diagram and the lever rule. From these, it derives how a single distillation step enriches vapor. It then formalizes repeated enrichment using the McCabe-Thiele idea of theoretical plates. You will see the boiling-point diagram again. Then you will follow a step-by-step vaporization-condensation cycle. Finally you will count plates needed for a real separation. This sets up the later discussion of azeotropes, where this simple picture breaks down.

Key takeaways

Distillation separates volatile liquids because vapor in equilibrium with a boiling mixture is enriched in the lower-boiling component.
A temperature-composition (T-x-y) diagram, read with tie-lines and the lever rule, predicts the vapor composition at any boiling temperature.
Simple distillation gives one vaporization-condensation step; fractional distillation stacks many such steps inside a column.
Each theoretical plate is one equilibrium vaporization-condensation event; the Fenske equation counts how many plates a given separation needs.
Real columns need more actual trays than theoretical plates because equilibrium is never perfectly reached — this gap is the column efficiency.
Key idea: repeated vaporization-condensation steps, each governed by the liquid-vapor equilibrium curve, progressively concentrate the more volatile component until the desired purity is reached.

The core insight

Because the vapor in equilibrium with a boiling liquid mixture is always richer in the more volatile component (Raoult's-law behavior), condensing that vapor and reboiling it repeatedly — as happens continuously inside a fractionating column — drives the top product toward the pure low-boiling component and the bottom product toward the pure high-boiling one, with the number of theoretical plates setting the achievable purity.

The physical picture

Consider an ideal binary mixture of a volatile component A (lower boiling point) and a less volatile component B, obeying Raoult's law. At any temperature between the two pure boiling points, the liquidus and vapor curves on the temperature-composition diagram enclose a two-phase region. A horizontal tie-line at a given boiling temperature connects the liquid composition $x_A$ (on the lower, liquidus curve) to the vapor composition $y_A$ (on the upper, dew-point curve). Because A is more volatile, $y_A > x_A$ everywhere between the pure endpoints — vapor is always enriched relative to its parent liquid. Simple distillation exploits exactly one such tie-line: the liquid boils, the vapor is drawn off, and it condenses to a distillate richer in A than the original liquid. But that distillate is not pure. Fractional distillation repeats the trick: condense the vapor, reboil the condensate, draw off a new vapor richer still. In an idealized column, this cascade happens automatically along the length of a fractionating column packed with trays or packing, where liquid trickling down meets vapor rising up in countercurrent contact. Each theoretical plate represents one complete equilibration — one vaporization step followed by condensation — equivalent to one horizontal-then-vertical step traced on the phase diagram.

Figure

Stepping across the T-x-y diagram: each horizontal tie-line plus vertical drop is one theoretical plate

The zig-zag construction on the diagram — horizontal tie-line to the vapor curve, then vertical drop to the liquid curve at the next plate down — is the graphical version of the plate-counting method (McCabe-Thiele in spirit), and it is the clearest way to see why more plates buy more purity.

The core relations

y_A = \frac{x_A P_A^{*}}{x_A P_A^{*} + (1-x_A) P_B^{*}}

where: $y_A$ is vapor mole fraction, $x_A$ liquid mole fraction, $P_A^{*}$ and $P_B^{*}$ the pure-component vapor pressures at that temperature (Raoult's law).

\alpha_{AB} = \frac{y_A / x_A}{y_B / x_B} = \frac{P_A^{*}}{P_B^{*}}

where: $\alpha_{AB}$ is the relative volatility of A with respect to B, constant for an ideal mixture at fixed temperature range.

N = \frac{\ln\!\left[\dfrac{x_{D}}{1-x_{D}}\cdot\dfrac{1-x_{W}}{x_{W}}\right]}{\ln \alpha_{AB}} - 1

where: $N$ is the number of theoretical plates (Fenske equation, total reflux), $x_D$ distillate composition, $x_W$ still-pot (bottoms) composition.

n_{\min}=\frac{\ln\!\left[\left(\dfrac{x_D}{1-x_D}\right)\left(\dfrac{1-x_W}{x_W}\right)\right]}{\ln\alpha}

where: $n_{\min}$ is the minimum number of equilibrium stages excluding the reboiler, an equivalent bookkeeping convention to the Fenske count above.

Derivation AFrom Raoult's law to the Fenske plate-counting equation

Goal. Show how successive equilibrium vaporizations enrich a volatile component, and derive a closed-form equation for the number of theoretical plates needed to go from a still-pot composition $x_W$ to a distillate composition $x_D$ .

Step 1 — Single equilibrium stage. For an ideal solution, Raoult's law gives the partial pressures $P_A = x_A P_A^{*}$ and $P_B = x_B P_B^{*}$ . Dalton's law converts these into vapor mole fractions,

y_A = \frac{x_A P_A^{*}}{x_A P_A^{*} + x_B P_B^{*}}.

Because $P_A^{*} > P_B^{*}$ (A is more volatile), algebra confirms $y_A > x_A$ whenever $0 < x_A < 1$ : the vapor is always enriched in A relative to the liquid it came from.

Step 2 — Define relative volatility. Divide the vapor-to-liquid ratio of A by that of B:

\alpha_{AB} = \frac{y_A/x_A}{y_B/x_B} = \frac{P_A^{*}}{P_B^{*}}.

This ratio depends only on the two pure vapor pressures (hence weakly on temperature) and is essentially constant over the boiling range of an ideal mixture. It measures how easy the separation is: $\alpha_{AB}=1$ means no separation is possible at all (an azeotrope-like limit), while large $\alpha_{AB}$ means easy separation.

Step 3 — One theoretical plate. Rearranging Step 2 gives the operating relation used to hop from one equilibrium stage to the next:

\frac{y_A}{1-y_A} = \alpha_{AB}\,\frac{x_A}{1-x_A}.

At total reflux (all condensed vapor is returned as liquid to the plate below, the idealization used for counting minimum plates), the vapor leaving plate $n$ becomes the liquid entering plate $n+1$ , i.e. $y_A^{(n)} = x_A^{(n+1)}$ . Substituting repeatedly,

\frac{x_A^{(n+1)}}{1-x_A^{(n+1)}} = \alpha_{AB}\,\frac{x_A^{(n)}}{1-x_A^{(n)}}.

Step 4 — Iterate over $N$ plates. Applying this relation $N$ times from the still pot (composition $x_W$ ) to the top plate (composition $x_D$ ) multiplies the volatility ratio by itself $N+1$ times (the $+1$ accounts for the reboiler acting as the first equilibrium stage):

\frac{x_D}{1-x_D} = \alpha_{AB}^{\,N+1}\,\frac{x_W}{1-x_W}.

Step 5 — Solve for $N$ . Take logarithms of both sides and isolate $N$ :

\boxed{N = \frac{\ln\!\left[\dfrac{x_D}{1-x_D}\cdot\dfrac{1-x_W}{x_W}\right]}{\ln \alpha_{AB}} - 1}

This is the Fenske equation. It says the number of plates needed grows only logarithmically with the desired purity ratio, but grows rapidly as $\alpha_{AB}\to 1$ — precisely the situation that arises near an azeotrope, foreshadowing why real solutions can defeat this idealized picture entirely.

Worked examples

Example 1 — Reading one tie-line (simple distillation). An ideal mixture of benzene (A) and toluene (B) is boiling at 90 °C. At this temperature the pure vapor pressures are $P_A^{*} = 136$ kPa (benzene) and $P_B^{*} = 54$ kPa (toluene). Suppose the liquid in the pot is $x_A = 0.40$ . Step 1: compute vapor composition from Raoult/Dalton:

y_A = \frac{(0.40)(136)}{(0.40)(136)+(0.60)(54)} = \frac{54.4}{54.4+32.4}=\frac{54.4}{86.8}=0.627.

Step 2: relative volatility at this temperature:

\alpha_{AB} = \frac{136}{54} = 2.52.

Sanity check: $y_A = 0.627 > x_A = 0.40$ , consistent with vapor enrichment in the more volatile benzene. A single simple distillation of this mixture, condensed and collected, gives distillate at 62.7 mol % benzene — a real enrichment, but far from pure. This is exactly why simple distillation, one tie-line, one lever-rule read, is only useful for coarse separations or for liquids with very different boiling points.

Example 2 — Plates needed for a sharp separation (fractional distillation). A chemist must separate the same benzene/toluene mixture into a distillate that is 98 mol % benzene ( $x_D = 0.98$ ) from a still pot residue that is only 2 mol % benzene ( $x_W = 0.02$ ), using the relative volatility found above, $\alpha_{AB}=2.52$ (treated as constant over the boiling range, a reasonable approximation for this near-ideal pair). Step 1: form the two composition ratios.

\frac{x_D}{1-x_D} = \frac{0.98}{0.02} = 49.0, \qquad \frac{1-x_W}{x_W} = \frac{0.98}{0.02} = 49.0.

Step 2: multiply them inside the logarithm.

\frac{x_D}{1-x_D}\cdot\frac{1-x_W}{x_W} = 49.0 \times 49.0 = 2401.

Step 3: take logarithms.

\ln(2401) = 7.784, \qquad \ln(2.52) = 0.9243.

Step 4: apply the Fenske equation.

N = \frac{7.784}{0.9243} - 1 = 8.42 - 1 = 7.42.

Rounding up (a fractional plate count means the last plate does not fully complete its job), this separation needs about 8 theoretical plates at total reflux. Sanity check: with only 1 plate (simple distillation), Example 1 showed we barely reach 62.7% from 40% in a single pass; reaching 98% from 2% is a far bigger composition swing, so needing roughly 8 stacked equilibrations is physically reasonable. A real packed column, since trays never achieve 100% equilibrium efficiency, would need somewhat more than 8 actual trays to deliver this performance — commonly 12-15 for a column with 60-70% plate efficiency.

Limiting cases & common traps

Situation	What actually happens	Common mistake
$\alpha_{AB} \to 1$	Fenske equation gives $N \to \infty$ ; no amount of plates achieves separation efficiently	Assuming more plates always overcomes any volatility difference
Azeotropic mixture	Vapor and liquid compositions become equal at the azeotropic point; distillation stalls there	Forgetting that Raoult's law (and thus this whole derivation) assumes ideal solutions
Simple distillation of close-boiling liquids	Only modest enrichment per pass; distillate still impure	Expecting simple distillation to purify miscible liquids with similar boiling points
Total reflux (Fenske limit)	Gives the minimum possible plates; no product is actually withdrawn	Using the Fenske number directly as the plates needed for a working column with real reflux ratio and takeoff
Actual trays vs. theoretical plates	Actual trays = theoretical plates ÷ efficiency (efficiency $<1$ always)	Treating theoretical plate count as the number of physical trays to build

The chemist behind it

The theoretical-plate concept that underlies fractional distillation was formalized for engineering use by Merritt Fenske in 1932, building on chromatographic and distillation plate theory developed through the 1920s by chemical engineers refining Sorel's earlier graphical stage-by-stage method (1893). The plate-counting equation used above still bears Fenske's name in nearly every unit-operations and physical chemistry textbook.

Origins · 1 minMerritt Fenske and how this was discovered

Merritt Fenske

1904-1971 · American chemical engineer who derived the total-reflux plate equation still used to size distillation columns

Fenske worked at Pennsylvania State University, where his 1932 analysis of distillation at total reflux gave a simple algebraic way to estimate the minimum number of theoretical plates for a target separation, directly from relative volatility and desired purities — the equation used throughout this article.

Ask the tutor: How does an azeotrope form thermodynamically, and what tricks let chemists distill past one?

What is the difference between simple and fractional distillation?

Simple distillation is one vaporization-condensation step: boil the mixture, condense the vapor, collect it. It gives one point read off a single tie-line on the temperature-composition diagram, so it only modestly enriches the more volatile component. Fractional distillation repeats this step many times inside a column, with each theoretical plate acting as another equilibration. The cumulative effect, captured by the Fenske equation, can drive the distillate toward much higher purity than a single simple distillation ever could, provided the mixture is far from an azeotrope.

Does the Fenske equation give the actual number of trays to build a column?

No. Fenske's result gives the minimum number of theoretical plates, valid strictly at total reflux (no product withdrawn) with a constant relative volatility. Real columns operate with a finite reflux ratio and always fall short of full equilibrium on each real tray. Engineers divide the theoretical plate count by an empirical plate efficiency (often 50-80%) to estimate actual trays needed, then add margin for practical operation and non-ideal behavior.

Why does relative volatility matter more than boiling-point difference alone?

Boiling points are a rough guide to separability, but relative volatility $\alpha_{AB} = P_A^{*}/P_B^{*}$ is the quantity that directly controls the enrichment per theoretical plate. Two liquids can have quite different boiling points yet a modest $\alpha_{AB}$ if their vapor pressure curves converge somewhere in the operating range, making separation harder than boiling points alone would suggest. This is why relative volatility, not just the boiling-point gap, is the number engineers actually design around.

What happens to distillation near an azeotrope?

As composition approaches an azeotropic point, the liquid and vapor curves on the temperature-composition diagram touch, meaning $y_A \to x_A$ and $\alpha_{AB} \to 1$ . The Fenske equation then predicts an infinite plate requirement, reflecting the physical fact that no ordinary distillation, however many plates, can cross that composition. Real solutions and azeotropes, covered next, explain why this happens and how techniques like pressure-swing or extractive distillation get around it.

Can this analysis be used for more than two components?

The two-component Fenske equation extends conceptually to multicomponent mixtures by tracking a "key pair" of components (light key and heavy key) and their relative volatility, but the full multicomponent problem needs additional balances for every other component's distribution between top and bottom. The core logic, multiplicative enrichment per plate converted to plates via a logarithm, still underlies the more elaborate multicomponent design equations used in industrial distillation.