Real Solutions and Azeotropes

Pour ethanol and water together and the mixture boils at 78.2 °C, not at any temperature you would predict from the pure components alone. This single fact ends a bottle of vodka's journey toward pure ethanol at 95.6% by mass, no matter how many times you redistill it. Real liquid mixtures deviate from Raoult's law, and when the deviation is strong enough a distillation column runs into a wall called an azeotrope. Understanding azeotropes matters wherever separation by boiling is the goal: fuel-grade ethanol production, HCl purification, refrigerant blending, and countless industrial distillations. The phenomenon is not a curiosity — it is the thermodynamic limit that distillation-column designers must engineer around. This article builds the molecular picture of positive and negative deviations from Raoult's law. It develops the mathematics linking activity coefficients to vapor pressure and boiling behavior. It shows how minimum- and maximum-boiling azeotropes arise, with fully worked numerical examples for ethanol–water and acetone–chloroform.

Key takeaways

Real mixtures deviate from Raoult's law because A–B interactions differ from A–A and B–B interactions.
Positive deviations (weaker A–B attraction) raise total vapor pressure and can produce a minimum-boiling azeotrope.
Negative deviations (stronger A–B attraction, e.g. H-bonding) lower vapor pressure and can produce a maximum-boiling azeotrope.
At an azeotrope the liquid and vapor compositions are identical, so distillation cannot cross it.
Activity coefficients $\gamma_i$ quantify the deviation; $\gamma_i = 1$ recovers ideal (Raoult's law) behavior.
Key idea: An azeotrope is the composition where competing non-ideal interactions make the mixture's total vapor pressure curve turn over, forcing liquid and vapor compositions to coincide.

The core insight

When the intermolecular forces between unlike molecules differ from those between like molecules, the vapor pressure of a mixture rises above (positive deviation) or falls below (negative deviation) the ideal Raoult's-law prediction; if the deviation is large enough, the total-pressure (or boiling-temperature) curve develops an extremum at some composition, and at that exact composition the vapor generated has the same composition as the liquid it boils from — an azeotrope — which acts as an impassable barrier to ordinary fractional distillation.

The physical picture

In an ideal solution, A and B molecules feel forces of identical strength whether they sit beside another A, another B, or each other. Raoult's law, $p_i = x_i p_i^*$ , follows directly: each component contributes to the vapor pressure in proportion to its mole fraction, unperturbed by its neighbors. Real mixtures rarely cooperate.

Positive deviation occurs when A–B interactions are weaker than the average of A–A and B–B interactions. Molecules escape the liquid more easily than Raoult's law predicts, so the observed vapor pressure exceeds the ideal value at every composition. Physically this happens when mixing disrupts favorable structure — for example, ethanol's hydrogen-bonded network is diluted and partially broken by nonpolar or weakly interacting water-ethanol contacts in certain concentration ranges, and acetone dilutes carbon disulfide's dispersion-dominated packing. If the positive deviation is large enough, the total vapor pressure curve rises above both pure-component vapor pressures somewhere in the middle of the composition range — producing a pressure maximum, which corresponds to a boiling-point minimum.

Negative deviation occurs when A–B interactions are stronger than the A–A/B–B average — typically because a new, favorable hydrogen bond forms between unlike molecules, as when chloroform's acidic C–H hydrogen bonds to acetone's carbonyl oxygen. Molecules are held in the liquid more strongly than ideality predicts, vapor pressure falls below the Raoult's-law line, and if the effect is strong enough the total pressure develops a minimum — a boiling-point maximum. At the extremum of either curve, the tangent to the $p$ – $x$ or $T$ – $x$ curve is horizontal in composition space, and by the Gibbs–Duhem relation this forces the vapor and liquid compositions to be equal there. That composition is the azeotrope.

Figure

Vapor pressure vs. liquid composition: ideal, positive-deviation, and negative-deviation cases

Figure

Boiling-temperature diagram for a minimum-boiling azeotrope: liquidus and vapor curves touch at x-az

The core relations

p_A = \gamma_A x_A p_A^*, \qquad p_B = \gamma_B x_B p_B^*

where $\gamma_i$ is the activity coefficient of component $i$ (dimensionless, $\to 1$ as $x_i \to 1$ in the Raoult's-law convention), $x_i$ the liquid mole fraction, $p_i^*$ the pure-component vapor pressure.

p_{\text{total}} = \gamma_A x_A p_A^* + \gamma_B x_B p_B^*

where $p_{\text{total}}$ is the total vapor pressure over the liquid mixture at fixed temperature.

\mu_i = \mu_i^{*} + RT\ln(\gamma_i x_i)

where $\mu_i$ is the chemical potential of $i$ in solution and $\mu_i^*$ that of the pure liquid at the same $T$ , $p$ .

G^{E} = RT\left(x_A \ln \gamma_A + x_B \ln \gamma_B\right)

where $G^{E}$ is the molar excess Gibbs energy, the thermodynamic measure of non-ideality (zero for an ideal solution).

RT\ln\gamma_A = w x_B^2, \qquad RT\ln\gamma_B = w x_A^2

where $w$ is the (temperature-dependent) interaction parameter of the simple regular-solution model; $w>0$ gives positive deviations, $w<0$ gives negative deviations.

\left(\frac{\partial p_{\text{total}}}{\partial x_A}\right)_{T} = 0 \quad \text{at the azeotrope}

where this stationarity condition, combined with Gibbs–Duhem, forces $y_A = x_A$ at that composition.

Worked examples

Derivation AFrom the regular-solution model to an azeotropic composition

Goal. Show how a single interaction parameter $w$ predicts a minimum-boiling azeotrope, and locate it.

Step 1 — Write the total pressure. Using the regular-solution (two-suffix Margules) expressions,

p_{\text{total}}(x_A) = x_A p_A^{*}\exp\!\left(\frac{w x_B^2}{RT}\right) + x_B p_B^{*}\exp\!\left(\frac{w x_A^2}{RT}\right)

with $x_B = 1-x_A$ .

Step 2 — Differentiate. Setting $dp_{\text{total}}/dx_A = 0$ and simplifying (using $dx_B = -dx_A$ ) gives the condition

p_A^{*}\exp\!\left(\frac{w x_B^2}{RT}\right)\left(1 - \frac{2w x_A x_B}{RT}\right) = p_B^{*}\exp\!\left(\frac{w x_A^2}{RT}\right)\left(1 - \frac{2w x_A x_B}{RT}\right)

Because the factor in parentheses is identical on both sides (it does not vanish at the physically relevant root), the extremum condition collapses to

p_A^{*}\exp\!\left(\frac{w x_B^2}{RT}\right) = p_B^{*}\exp\!\left(\frac{w x_A^2}{RT}\right)

This says: at the azeotrope the two partial activity-corrected vapor pressures per unit mole fraction become equal — the microscopic reason liquid and vapor compositions must coincide.

Step 3 — Relate to a measurable ratio. Taking logarithms,

\ln\!\left(\frac{p_A^{*}}{p_B^{*}}\right) = \frac{w}{RT}\left(x_A^2 - x_B^2\right) = \frac{w}{RT}(x_A - x_B)

since $x_A^2 - x_B^2 = (x_A-x_B)(x_A+x_B)$ and $x_A + x_B = 1$ .

Step 4 — Solve for the azeotropic composition. Rearranging with $x_B = 1-x_A$ ,

x_A - x_B = \frac{RT}{w}\ln\!\left(\frac{p_A^{*}}{p_B^{*}}\right) \quad\Longrightarrow\quad x_A = \frac{1}{2}\left[1 + \frac{RT}{w}\ln\!\left(\frac{p_A^{*}}{p_B^{*}}\right)\right]

Boxed result.

\boxed{x_A^{\text{az}} = \frac{1}{2}\left[1 + \frac{RT}{w}\ln\!\left(\frac{p_A^{*}}{p_B^{*}}\right)\right]}

An azeotrope exists only if this expression lands strictly between 0 and 1; if $|w|$ is too small relative to $RT\ln(p_A^*/p_B^*)$ , the curve never turns over and no azeotrope forms — the two components simply have overlapping but non-crossing vapor-pressure curves.

Example 1 — Foundational: recognizing deviation from Raoult's law. At 50 °C the pure vapor pressures are $p_A^* = 350$ Torr (acetone) and $p_B^* = 200$ Torr (chloroform). An equimolar liquid mixture ( $x_A = x_B = 0.500$ ) is measured to have partial pressures $p_A = 110$ Torr and $p_B = 60$ Torr. Ideal (Raoult's law) predictions:

p_A^{\text{ideal}} = x_A p_A^{*} = 0.500 \times 350\ \text{Torr} = 175\ \text{Torr}

p_B^{\text{ideal}} = x_B p_B^{*} = 0.500 \times 200\ \text{Torr} = 100\ \text{Torr}

Activity coefficients from the measured values:

\gamma_A = \frac{p_A}{x_A p_A^{*}} = \frac{110}{175} = 0.629

\gamma_B = \frac{p_B}{x_B p_B^{*}} = \frac{60}{100} = 0.600

Both $\gamma_i < 1$ : the measured partial pressures fall well below ideal, confirming negative deviation. This matches the real acetone–chloroform system, where the C–H···O=C hydrogen bond stabilizes the mixed liquid relative to the ideal prediction. Sanity check: total measured pressure is $170$ Torr versus an ideal total of $275$ Torr — a 38% suppression, consistent with the known maximum-boiling azeotrope of this pair near $x_{\text{CHCl}_3} \approx 0.65$ , $T_b \approx 64.5\ ^\circ\text{C}$ (compare with acetone's own boiling point of 56.1 °C and chloroform's 61.2 °C, both lower than the azeotrope — as required for a maximum-boiling azeotrope).

Example 2 — Applied: ethanol–water azeotrope and a Margules-based estimate. The well-known ethanol–water azeotrope boils at $T_{az} = 78.2\ ^\circ\text{C}$ ( $351.35$ K) with composition $x_{\text{EtOH}} \approx 0.894$ (mole fraction; about 95.6% by mass). Take $R = 8.314\ \text{J K}^{-1}\text{mol}^{-1}$ . At $351.35$ K the pure-component vapor pressures (from the Antoine equation, standard tabulated values) are approximately

p_{\text{EtOH}}^{*} \approx 102.8\ \text{kPa}, \qquad p_{\text{H}_2\text{O}}^{*} \approx 43.6\ \text{kPa}

(Water's normal boiling point is 100 °C, so at 78.2 °C its vapor pressure is well below 1 atm; ethanol's normal boiling point is 78.4 °C, so its vapor pressure at 78.2 °C is close to, but just under, 101.3 kPa.) Using the boxed relation from the derivation to back out an effective interaction parameter $w$ at the azeotropic point, with $x_A = 0.894$ , $x_B = 0.106$ :

\ln\!\left(\frac{p_A^{*}}{p_B^{*}}\right) = \ln\!\left(\frac{102.8}{43.6}\right) = \ln(2.358) = 0.858

w = \frac{RT\ln(p_A^*/p_B^*)}{x_A - x_B} = \frac{(8.314)(351.35)(0.858)}{0.894 - 0.106}

w = \frac{2506\ \text{J mol}^{-1}}{0.788} = 3180\ \text{J mol}^{-1}

A positive $w$ of about $3.2\ \text{kJ mol}^{-1}$ correctly signals positive deviation — consistent with the physical picture that ethanol's hydrogen-bonded chains are disrupted upon dilution with water, weakening effective A–B cohesion relative to the A–A/B–B average. The activity coefficient of ethanol at infinite dilution predicted by this crude one-parameter model,

RT\ln\gamma_{\text{EtOH}}^{\infty} = w(1)^2 = 3180\ \text{J mol}^{-1} \;\Rightarrow\; \gamma_{\text{EtOH}}^{\infty} = e^{3180/2921} = e^{1.089} \approx 2.97

is in the right ballpark (experimental values from more complete models, e.g. van Laar or NRTL fits, give $\gamma^{\infty}_{\text{EtOH}} \approx 4$ – $5$ ), showing that the single-parameter regular-solution model captures the qualitative azeotrope but underestimates the true asymmetric deviation — real ethanol–water requires a two-parameter model (Margules, van Laar, or Wilson) for quantitative accuracy.

Limiting cases & common traps

Situation	Behavior	Example
$\gamma_A = \gamma_B = 1$ everywhere	Ideal solution, Raoult's law exact, no azeotrope possible	benzene–toluene
Weak positive deviation	$p_{\text{total}}$ above ideal line but no extremum	many hydrocarbon pairs
Strong positive deviation	Vapor-pressure maximum, boiling-point minimum azeotrope	ethanol–water (78.2 °C), 2-propanol–water
Strong negative deviation	Vapor-pressure minimum, boiling-point maximum azeotrope	acetone–chloroform (64.5 °C), HCl–water (110 °C)
Extreme positive deviation	Partial or complete liquid–liquid immiscibility (foreshadows liquid–liquid phase diagrams)	water–butanol

The chemist behind it

The mathematical treatment of azeotropes rests on the activity-coefficient framework built from Gilbert N. Lewis's concept of fugacity and activity, but the phenomenon itself was first systematically studied by the German chemist Ernst Wilhelm Erdmann and the French pioneer François-Marie Raoult, whose 1880s vapor-pressure law provides the ideal baseline against which every azeotrope is defined as a deviation. The term "azeotrope" itself (Greek for "boils unchanged") was coined in 1911 by the British chemists John Wade and Richard William Merriman, who catalogued dozens of constant-boiling mixtures and recognized that they violate the expectation, then novel, that boiling point should vary monotonically with composition.

Origins · 1 minFrançois-Marie Raoult and how this was discovered

François-Marie Raoult

1830–1901 · French chemist whose 1887 vapor-pressure law defines the ideal baseline from which every azeotrope is measured as a deviation

Raoult's systematic cryoscopic and vapor-pressure measurements on solutions established the proportionality between vapor pressure and mole fraction that bears his name. Ironically, his law's greatest legacy in industrial chemistry may be defining precisely the mixtures — like ethanol–water — that disobey it most dramatically.

Ask the tutor: How does adding a third component like benzene actually break the ethanol–water azeotrope, and what does that ternary phase diagram look like?

Why does a positive deviation from Raoult's law lower the boiling point?

A positive deviation means molecules escape the liquid more easily than ideality predicts, so the total vapor pressure at a given temperature is higher than ideal. Equivalently, at fixed external pressure (say 1 atm), the mixture reaches that vapor pressure at a lower temperature than either pure component would need. When this effect is strong enough to create a vapor-pressure maximum at some composition, the boiling-temperature curve correspondingly develops a minimum there — the minimum-boiling azeotrope.

Is every non-ideal mixture guaranteed to form an azeotrope?

No. Deviation from Raoult's law is necessary but not sufficient. The total vapor pressure (or boiling temperature) curve must actually develop an extremum somewhere between $x=0$ and $x=1$ . Many real mixtures show clear positive or negative deviations — visible as curvature in $p$ versus $x$ plots — without ever crossing the pure-component vapor pressure lines, so no azeotrope forms. Whether an azeotrope appears depends on how large the deviation is relative to the difference between the pure-component vapor pressures.

How does pressure affect an azeotrope's composition and boiling point?

Both shift with pressure, usually substantially, because the two pure-component vapor-pressure curves respond differently to pressure changes (different Clausius–Clapeyron slopes and enthalpies of vaporization). This pressure sensitivity is exploited industrially in pressure-swing distillation: distill at one pressure to approach the azeotrope, then switch to a second pressure where the azeotropic composition has moved, allowing further enrichment in an alternating sequence of columns.

Can a ternary or multicomponent mixture also form an azeotrope?

Yes, and this is common in solvent recovery: ethanol, water, and benzene famously form a ternary azeotrope near 65 °C, exploited historically to break the ethanol–water binary azeotrope by adding benzene as an entrainer. Ternary azeotropes are found the same way conceptually — by locating a stationary point of the multicomponent chemical-potential surface — though the geometry becomes a surface in composition space rather than a single point on a line.

How is an azeotrope different from a eutectic in a solid–liquid phase diagram?

Both are compositions where the system behaves as if it were a single "pure" substance in some respect, but they occur in different phase equilibria. An azeotrope is a liquid–vapor phenomenon: liquid and vapor share the same composition at a boiling-point extremum. A eutectic is a solid–liquid phenomenon: it is the lowest-melting liquid composition, where two solid phases crystallize together. The mathematics (a stationary point tied to equal chemical potentials of both phases) is analogous, but the phases involved and the physical context are distinct.