Raoult's_law - Wikiparency

# Raoult's law **Raoult's law** states that the partial [[Vapor_pressure|vapour pressure]] of each component of an ideal liquid [[Solution_(chemistry)|solution]] equals the vapour pressure of that component as a pure liquid multiplied by its [[Mole_fraction|mole fraction]] in the liquid. For a two-component mixture the total vapour pressure is a straight line between the two pure-liquid values, `p = x_A·p*_A + (1 − x_A)·p*_B`, where `x_A` is the mole fraction of component A in the liquid and `p*_A`, `p*_B` are the vapour pressures of the pure liquids at the same temperature. The law was stated by the French chemist François-Marie Raoult in 1887 on the basis of his measurements of the vapour pressures of solutions.[^raoult1887] It is the defining relation of an ideal solution and the starting point for the [[Colligative_properties|colligative properties]], for [[Fractional_distillation|fractional distillation]], and for the whole treatment of non-ideal [[Mixture|mixtures]].[^os-af-ch11] On the Chemistry flagship the article serves the section *Substance and mixture* (Part II — Modern principles › Matter): a pure [[Substance_(chemistry)|substance]] boils at a point, a mixture boils over a range, and Raoult's law is the equation that turns the second statement into numbers. In the microsim below the reader slides the liquid mole fraction `x_A` of an ideal benzene–toluene solution and reads two temperatures off the T–x "lens" drawn at 1 atm: the bubble temperature, at which `x_A·p*_A(T) + (1 − x_A)·p*_B(T) = 760 mmHg` and the first bubble of vapour forms, and the dew temperature, at which the last drop of liquid disappears. At the two ends of the slider the lens closes to the boiling points of pure benzene and pure toluene; everything between is the band over which the mixture boils. ## Principles Raoult's law is an equation about a liquid in [[Chemical_equilibrium|equilibrium]] with its own vapour. Above any liquid, molecules escape into the gas until the rate of escape equals the rate of return; the [[Pressure|pressure]] the vapour then exerts is the [[Vapor_pressure|vapour pressure]] of the liquid, and it depends only on the [[Temperature|temperature]]. For water it is 760.0 mmHg at 100 °C, which is why water boils at 100 °C under a standard atmosphere.[^averill-t104] In a mixture of two volatile liquids each component contributes a [[Partial_pressure|partial pressure]], and by [[Dalton's_law|Dalton's law]] the total pressure is the sum of the partial pressures.[^averill-dalton] Raoult's observation was that, for solutions of similar molecules, the partial pressure of each component is proportional to its share of the liquid: `p_A = x_A·p*_A` and `p_B = x_B·p*_B`, with `x_A + x_B = 1`.[^raoult1887] Adding the two gives the straight line `p = p*_B + (p*_A − p*_B)·x_A`, which is what the microsim draws as its vapour-pressure line.[^os-af-ch11] The vapour above the solution is not the same mixture as the liquid. Its composition follows from Dalton's law again: the mole fraction of A in the vapour is `y_A = p_A/p = x_A·p*_A/p`. Because the more volatile component has the larger `p*`, the vapour is always richer in it than the liquid was, and that asymmetry is the whole basis of [[Distillation|distillation]]. A worked case (ILLUSTRATIVE: the pure-liquid pressures are round values chosen to make the arithmetic visible, not measured data) shows how the two temperatures in the sim arise. Suppose that at some temperature the pure liquids have `p*_A = 1000 mmHg` and `p*_B = 400 mmHg`. A mixture with `x_A = 0.60` then has `p = 400 + 600 × 0.60 = 760 mmHg`, so that temperature is its bubble temperature at 1 atm. The first vapour to form has `y_A = 0.60 × 1000/760 = 0.79`: the liquid is 60 % A but its vapour is 79 % A. For benzene and toluene, whose normal [[Boiling_point|boiling points]] are about 80 °C and 111 °C,[^nist-webbook] the sim evaluates `p*_A(T)` and `p*_B(T)` from a stored vapour-pressure table and solves `x_A·p*_A(T) + (1 − x_A)·p*_B(T) = 760` for T. The dew temperature solves the companion equation for a vapour of composition `y_A`, `y_A/p*_A(T) + (1 − y_A)/p*_B(T) = 1/760`, which comes from writing `x_A = y_A·p/p*_A` and `x_B = y_B·p/p*_B` and demanding that the liquid mole fractions sum to one.[^averill-ch13] ## Thermodynamic considerations Raoult's law looks like an empirical rule about pressures, but it is a statement about [[Gibbs_free_energy|Gibbs free energy]]. A component in a liquid is in equilibrium with the same component in the vapour when its [[Chemical_potential|chemical potential]] is the same in both phases. For an ideal vapour the chemical potential rises with the logarithm of the partial pressure, `μ_A(gas) = μ_A° + RT·ln(p_A/p°)`, so the question of what `p_A` a solution produces is the question of what the chemical potential of A in the liquid is.[^averill-ch13] ### Ideal mixing An ideal solution is defined as one in which the chemical potential of every component is `μ_A(liquid) = μ*_A + RT·ln x_A`, where `μ*_A` is the value for the pure liquid.[^os-af-ch11] Setting the liquid and vapour chemical potentials equal, and using the same equality for the pure liquid at its own vapour pressure `p*_A`, the standard terms cancel and leave `RT·ln(p_A/p*_A) = RT·ln x_A`, which is Raoult's law. The definition therefore contains the law, and the physical content is in the assumption: the molecules of A and B attract one another with the same strength that A attracts A and B attracts B, so a molecule does not care what its neighbours are. The consequences for mixing follow in one line each. The Gibbs energy of mixing `n_A` mol of A with `n_B` mol of B is `ΔG_mix = RT·(n_A·ln x_A + n_B·ln x_B)`, which is negative for every composition because each mole fraction is less than one. Since the [[Enthalpy|enthalpy]] of mixing is zero for an ideal solution (nothing changes in the [[Intermolecular_force|intermolecular forces]]), the whole of `ΔG_mix` is entropic: `ΔS_mix = −R·(n_A·ln x_A + n_B·ln x_B)`, the [[Entropy_of_mixing|entropy of mixing]], and the volume does not change either.[^averill-ch13] Mixtures of chemically similar [[Molecule|molecules]], such as benzene with toluene or one alkane with the next, come close to this ideal and are the systems on which the microsim's straight line is an honest model. The sim states this on its header: the lens is ILLUSTRATIVE for any pair whose molecules are not near-identical. ### Non-ideal mixing When unlike molecules attract each other more or less strongly than like molecules, the chemical potential departs from the ideal form. The departure is written with an activity, `μ_A = μ*_A + RT·ln a_A`, and the activity is written as a corrected mole fraction, `a_A = γ_A·x_A`, where `γ_A` is the activity coefficient.[^averill-ch13] Raoult's law then generalises to `p_A = γ_A·x_A·p*_A`. Everything that is not ideal about the solution is packed into `γ_A`, which equals one in the ideal case and tends to one for any component as `x_A → 1`, because a nearly pure liquid behaves like the pure liquid. The simplest model of the departure is the [[Regular_solution|regular solution]] of Hildebrand, in which the entropy of mixing keeps its ideal form but the enthalpy of mixing is `ΔH_mix = W·x_A·x_B` per mole, with a single interaction parameter `W` that is positive when unlike contacts are unfavourable and negative when they are favourable.[^hildebrand1929] The activity coefficients that follow are `RT·ln γ_A = W·x_B²` and `RT·ln γ_B = W·x_A²`, so a single sign of `W` bends the whole vapour-pressure curve the same way at every composition. Two-parameter forms, the Margules equations, allow the curvature to differ at the two ends.[^margules1895] The sim's pending non-ideal preset uses the one-parameter form as a display fit (ILLUSTRATIVE): it reproduces the direction and rough size of a deviation, not the measured curve of any particular pair. ## Real solutions Real solutions are compared with Raoult's law by plotting the measured total pressure against `x_A` on the same axes as the ideal straight line, the vapour-pressure half of the [[Phase_diagram|phase diagram]] that the sim's lens turns into temperatures. Three outcomes are possible: the curve lies below the line, above it, or crosses it. A useful check on any such curve is the Gibbs–Duhem relation, which ties the two activity coefficients together, `x_A·d(ln γ_A) + x_B·d(ln γ_B) = 0`, so that if one component obeys Raoult's law over some composition range the other must obey [[Henry's_law|Henry's law]] there, `p_B = K_B·x_B`, with a constant `K_B` that is not the pure-liquid vapour pressure.[^averill-ch13] The nearly pure solvent follows Raoult; the dilute solute, whose [[Solubility|solubility]] in a gas–liquid system Henry's law also governs, follows Henry; the two laws are the two ends of the same curve. ### Negative deviation A negative deviation means the measured vapour pressure is lower than Raoult's law predicts, so `γ < 1` for both components. It arises when unlike molecules attract one another more strongly than like molecules, which holds the molecules in the liquid more tightly and lowers their escaping tendency. The classic example is acetone with chloroform, where the hydrogen of chloroform forms a [[Hydrogen_bond|hydrogen bond]] to the carbonyl oxygen of acetone that neither pure liquid can form on its own; another is hydrogen chloride or nitric acid in water, where the mixing is strongly exothermic.[^averill-ch13] In the regular-solution language, `W < 0` and mixing releases [[Heat|heat]]. A strong enough negative deviation produces a maximum-boiling azeotrope: a composition whose vapour has the same composition as the liquid, so that it distils unchanged and boils above either pure component. In the sim a negative `W` pulls the vapour-pressure line down into a sagging curve and pushes the lens upward on the temperature axis. ### Positive deviation A positive deviation means the measured pressure is higher than the ideal line, so `γ > 1`: unlike molecules attract one another less than like molecules, each component is effectively squeezed out of the other, and escaping into the vapour is easier than in either pure liquid. Ethanol with [[Water|water]] is the familiar case. Pure water is a dense hydrogen-bonded network; pure ethanol hydrogen-bonds through its hydroxyl group; mixed, the ethyl groups disrupt water's network and the mixture holds its molecules less tightly than the ideal model assumes, so the vapour pressure runs above the line and mixing absorbs heat.[^averill-ch13] Benzene with methanol, or an alkane with an alcohol, behaves the same way. A large enough positive deviation produces a minimum-boiling azeotrope, and the ethanol–water azeotrope is why simple distillation of a fermented liquor cannot pass a fixed ethanol content no matter how many times it is repeated; the composition at which the T–x lens pinches shut is the ceiling. In the sim a positive `W` lifts the vapour-pressure line into a hump and pulls the lens down toward lower temperatures. ### Mixed deviation Some pairs show a positive deviation over one range of composition and a negative deviation over the rest, so that the measured curve crosses the ideal line. Thermodynamically this requires the excess Gibbs energy, `G^E = RT·(x_A·ln γ_A + x_B·ln γ_B)`, to change sign somewhere between `x_A = 0` and `x_A = 1`. The one-parameter regular solution cannot do this, because `G^E = W·x_A·x_B` has the sign of `W` at every composition; a two-parameter Margules form can, and that is one reason the two-parameter equations were introduced.[^margules1895] Mixed deviations are rarer than either pure case; like a [[Miscibility_gap|miscibility gap]] they are the signature of two competing effects, such as a specific attraction between unlike molecules that dominates when one component is dilute and a size or shape mismatch that dominates when the two are present in comparable amounts. The sim does not attempt them: its single parameter can be pushed positive or negative, and a crossing curve is left as a redlink to the pair's measured data. ## See also - [[Mixture]] - [[Substance_(chemistry)]] - [[Solution_(chemistry)]] - [[Alloy]] - [[Colligative_properties]] - [[Fractional_distillation]] - [[Vapor_pressure]] - [[Henry's_law]] - [[Mole_fraction]] - [[Dalton's_law]] ## References [^raoult1887]: Raoult, F.-M. (1887). "Loi générale des tensions de vapeur des dissolvants." *Comptes rendus hebdomadaires des séances de l'Académie des sciences* 104: 1430–1433. [^os-af-ch11]: Flowers, Paul; Neth, Edward; Robinson, William, et al. (2019). *Chemistry: Atoms First 2e*. OpenStax. Chapter 11, "Solutions and Colloids", §11.4 "Colligative Properties" (page to pin). Portal Book 051. https://openstax.org/details/books/chemistry-atoms-first-2e [^averill-ch13]: Averill, Bruce; Eldredge, Patricia (2011). *General Chemistry: Principles, Patterns, and Applications*. Chapter 13, "Solutions" (page to pin). Portal Book 050. https://open.umn.edu/opentextbooks/textbooks/general-chemistry-principles-patterns-and-applications [^averill-t104]: Averill, Bruce; Eldredge, Patricia (2011). *General Chemistry: Principles, Patterns, and Applications*. Chapter 10, "Gases", Table 10.4, p. 934 (vapour pressure of water, 760.0 mmHg at 100 °C). Portal Book 050. [^averill-dalton]: Averill, Bruce; Eldredge, Patricia (2011). *General Chemistry: Principles, Patterns, and Applications*. Chapter 10, "Gases", pp. 923–928 (Dalton's law of partial pressures, `P_A = X_A·P_t`). Portal Book 050. [^nist-webbook]: NIST Chemistry WebBook, SRD 69. Phase-change data for benzene and for toluene (normal boiling points). National Institute of Standards and Technology. https://webbook.nist.gov/chemistry/ [^hildebrand1929]: Hildebrand, Joel H. (1929). "Solubility. XII. Regular Solutions." *Journal of the American Chemical Society* 51 (1): 66–80. [^margules1895]: Margules, Max (1895). "Über die Zusammensetzung der gesättigten Dämpfe von Mischungen." *Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien, Mathematisch-Naturwissenschaftliche Klasse*, Abt. IIa, 104: 1243–1278. [^ball-ch11]: Ball, David W. (2011). *Introductory Chemistry*. Chapter 11, "Solutions", pp. 518–570 (page to pin). Portal Book 056. https://open.umn.edu/opentextbooks/textbooks/introductory-chemistry ## Further reading - Flowers, Neth, Robinson et al., *Chemistry: Atoms First 2e* (OpenStax, 2019), Chapter 11 — Portal Book 051. - Averill and Eldredge, *General Chemistry: Principles, Patterns, and Applications* (2011), Chapter 13 — Portal Book 050. - Ball, *Introductory Chemistry* (2011), Chapter 11 — Portal Book 056.[^ball-ch11] ## External links - [Chemistry: Atoms First 2e](https://openstax.org/details/books/chemistry-atoms-first-2e), OpenStax — the open text behind Portal Book 051 - [NIST Chemistry WebBook](https://webbook.nist.gov/chemistry/) — vapour-pressure and boiling-point data for the pure liquids in the sim - For the pair's other external links, see the Wikipedia article's *External links* section.  **Microsim — three.js (Wikitube framework), pending deploy:** *Raoult's law* will play here once `https://wikitube-3d-microsims.netlify.app/matter/Raoult's_law.html` is live.   ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Raoult%27s_law) : [Wikitube](https://en.wikitube.io/wiki/Raoult's_law) · pinned revision [1355427560](https://en.wikipedia.org/w/index.php?oldid=1355427560) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Chemistry]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Chemistry row K12 · sim pending (matter/Raoult's_law).*