Raoult's law and vapor pressure lowering: ideal solutions and Henry's law for gases

Intuition Beginner

Every liquid has a vapor pressure -- the pressure exerted by its vapor when the liquid and vapor are in equilibrium. Water at 25 degrees C has a vapor pressure of 23.8 torr. When you dissolve a non-volatile solute in water, the vapor pressure drops. This is vapor pressure lowering, and it is the simplest colligative property.

Why does it drop? At the surface of the liquid, some molecules escape into the vapor. Solute molecules take up space at the surface but do not contribute vapor molecules. Fewer solvent molecules are at the surface, so fewer escape. The result: lower vapor pressure.

Francois-Marie Raoult measured this effect systematically in the 1880s. He found that for many solutions, the vapor pressure of the solvent is proportional to its mole fraction. This proportionality is Raoult's law: the vapor pressure of the solvent above a solution equals the mole fraction of the solvent times the vapor pressure of the pure solvent.

For a gas dissolving in a liquid, a different rule applies. William Henry showed in 1803 that the amount of gas dissolved is proportional to the partial pressure of the gas above the liquid. This is Henry's law, and it governs everything from carbonated beverages to ocean chemistry.

Visual Beginner

   Vapor pressure vs mole fraction for a binary ideal solution
   (Raoult's law)
   P (torr)
   |
   | P_A*  o - - - - - - - - - - - - - - - - - - - o
   |       |  .                                .  |
   |       |    .        total P            .    |
   |       |      .      (linear)       .       |
   |       |        .                .          |
   |       |          .           .             |
   |       |     P_B   .      .    P_A          |
   |       |       o - - - - - - - o            |
   |       |     .      .      .     .          |
   |       |   .    .       .         .         |
   |       | .  .             .          .      |
   | P_B*  o - - - - - - - - - - - - - - - - - o
   |
   |_____________________________________________
       x_A = 0                              x_A = 1

   P_A (partial pressure of A) is a straight line
   from P_B* at x_A=0 to P_A* at x_A=1.
   P_B is a straight line from P_B* at x_A=0 to 0.
   The total pressure P = P_A + P_B is also linear
   for an ideal solution.

   Deviations from Raoult's law

   P
   |
   |      Positive deviation          Negative deviation
   |      (A-B weaker than            (A-B stronger than
   |       A-A and B-B)                A-A and B-B)
   |
   |        _                               _
   |      /   \       total P              /   \
   |     /     \     curves ABOVE         /     \
   |    /ideal  \    ideal line          / ideal  \
   |   /   line  \                     /  line    \
   |  /           \                   /            \
   |
   |_________________________________________________
         0       x_A            1          0       x_A            1

Worked example Beginner

Problem: At 25 degrees C, the vapor pressure of pure water is 23.8 torr and the vapor pressure of pure ethanol is 59.2 torr. A solution is prepared with mole fraction of water x_water = 0.70 and mole fraction of ethanol x_ethanol = 0.30. Assuming ideal behavior (Raoult's law), calculate (a) the partial vapor pressures, (b) the total vapor pressure, and (c) the mole fraction of each component in the vapor phase.

Solution:

(a) By Raoult's law, P_i = x_i * P_i*:

P_water = 0.70 * 23.8 torr = 16.7 torr

P_ethanol = 0.30 * 59.2 torr = 17.8 torr

(b) Total vapor pressure (Dalton's law):

P_total = P_water + P_ethanol = 16.7 + 17.8 = 34.5 torr

(c) Vapor-phase mole fractions (Dalton's law):

y_water = P_water / P_total = 16.7 / 34.5 = 0.484

y_ethanol = P_ethanol / P_total = 17.8 / 34.5 = 0.516

The vapor is enriched in ethanol (y_ethanol = 0.516) compared to the liquid (x_ethanol = 0.30). Ethanol is more volatile than water, so it preferentially enters the vapor. This is the basis of fractional distillation.

Check your understanding Beginner

Formal definition Intermediate+

Raoult's law. For component i in an ideal solution at temperature T and total pressure P:

$$P_i=x_i{P}_i^{\ast }$$

where $P_i$ is the partial vapor pressure of component i above the solution, $x_i$ is its mole fraction in the liquid phase, and $P_i^{\ast }$ is the vapor pressure of pure component i at the same T and P.

The total vapor pressure above an ideal binary solution (components A and B) is:

$$P_{\text{total}}=x_A{P}_A^{\ast }+x_B{P}_B^{\ast }=x_A{P}_A^{\ast }+(1-x_A)P_B^{\ast }$$

which is a linear function of $x_A$. For a non-ideal solution, Raoult's law generalizes through the activity coefficient:

$$P_i={\gamma }_i{x}_i{P}_i^{\ast }$$

where ${\gamma }_i=1$ for an ideal solution.

Vapor pressure lowering. For a solution of a non-volatile solute (component 2) in a solvent (component 1), the vapor pressure lowering is:

$$\Delta P=P_1^{\ast }-P_1=P_1^{\ast }-x_1{P}_1^{\ast }=(1-x_1)P_1^{\ast }=x_2{P}_1^{\ast }$$

For dilute solutions, $x_2\approx n_2/n_1$, so the vapor pressure lowering is proportional to the mole fraction (and hence molality) of the solute.

Dalton's law applied to vapor composition. The mole fraction of component i in the vapor phase is:

$$y_i=\frac{P_i}{P_{\text{total}}}$$

For an ideal binary solution, $y_A=x_A{P}_A^{\ast }/(x_A{P}_A^{\ast }+(1-x_A)P_B^{\ast })$. The vapor is always enriched in the more volatile component relative to the liquid.

Henry's law. For a gas B dissolved in a solvent at low concentration:

$$C_B=k_H{P}_B$$

where $C_B$ is the concentration of dissolved gas, $P_B$ is the partial pressure of gas B above the solution, and $k_H$ is the Henry's law constant. In mole-fraction form:

$$P_B=K_H{x}_B$$

where $K_H$ is the Henry's law constant in pressure units. Unlike Raoult's law, where the proportionality constant is $P_i^{\ast }$ (the vapor pressure of the pure component), $K_H$ is an empirical constant that depends on both the gas and the solvent. For gases with favorable interactions with the solvent (e.g., NH3 in water), $K_H$ is small and solubility is high. For gases with unfavorable interactions (e.g., N2 in water), $K_H$ is large and solubility is low.

Key result Intermediate+

Connection between Raoult's law and Henry's law. In an ideal solution, Raoult's law applies to every component across the entire composition range. In real solutions, Raoult's law applies to the solvent (as $x_{\text{solvent}}\to 1$) and Henry's law applies to the solute (as $x_{\text{solute}}\to 0$). The two laws are complementary limiting cases: Raoult's law uses $P_i^{\ast }$ as the proportionality constant (the pure-component reference state), while Henry's law uses $K_H$ (a hypothetical pure-component reference state corrected for the different intermolecular environment at infinite dilution).

Boiling point elevation from vapor pressure lowering. A non-volatile solute lowers the solvent's vapor pressure. Since boiling occurs when the vapor pressure equals the external pressure, a higher temperature is needed to reach boiling. The quantitative relationship is:

$$\Delta T_b=K_{b}m$$

where $K_b$ is the ebullioscopic constant and $m$ is the molality of the solute. This is a direct thermodynamic consequence of Raoult's law combined with the Clausius-Clapeyron equation.

Deviations from Raoult's law. Real solutions deviate from Raoult's law because unlike-pair intermolecular forces differ from like-pair forces:

• Positive deviations (${\gamma }_i>1$): A-B interactions are weaker than the average of A-A and B-B interactions. The total vapor pressure is greater than the ideal prediction. Example: ethanol-water mixtures. The molecules escape more readily because the unlike neighbors do not hold them as tightly.

• Negative deviations (${\gamma }_i<1$): A-B interactions are stronger than the average of A-A and B-B interactions. The total vapor pressure is less than ideal. Example: acetone-chloroform mixtures, where hydrogen bonding between unlike pairs stabilizes the liquid phase.

Azeotropes. When deviations from Raoult's law are large enough, the vapor-pressure-composition curve develops an extremum. At this point, the vapor and liquid compositions are identical: $y_i=x_i$. This is an azeotrope. A minimum-boiling azeotrope (from positive deviations) boils at a temperature lower than either pure component. A maximum-boiling azeotrope (from negative deviations) boils at a temperature higher than either pure component. Azeotropes cannot be separated by simple distillation because the vapor and liquid have the same composition.

Derivation of Raoult's law from chemical potential Master

Raoult's law follows from the equality of chemical potentials at liquid-vapor equilibrium. For component i in an ideal solution, the chemical potential in the liquid phase is:

$${\mu }_i^{\text{liquid}}={\mu }_i^{\ast ,\text{liquid}}+RT\ln x_i$$

where ${\mu }_i^{\ast ,\text{liquid}}$ is the chemical potential of pure liquid i at the same T and P. At equilibrium with an ideal-gas vapor phase:

$${\mu }_i^{\ast ,\text{liquid}}+RT\ln x_i={\mu }_i^{\circ ,\text{vapor}}+RT\ln (P_i/P^{\circ })$$

For the pure component ($x_i=1$):

$${\mu }_i^{\ast ,\text{liquid}}={\mu }_i^{\circ ,\text{vapor}}+RT\ln (P_i^{\ast }/P^{\circ })$$

Subtracting the pure-component equation from the mixture equation:

$$RT\ln x_i=RT\ln (P_i/P^{\ast })$$

Therefore $P_i=x_i{P}_i^{\ast }$, which is Raoult's law. The critical assumption is that the intermolecular interactions between unlike pairs (A-B) are identical to those between like pairs (A-A and B-B), which gives $\Delta H_{\text{mix}}=0$ and $\Delta V_{\text{mix}}=0$. This defines an ideal solution.

Ideal-solution mixing thermodynamics. The thermodynamic quantities for forming an ideal solution from pure components are:

$$\Delta H_{\text{mix}}=0\text{(no enthalpy change)}$$ $$\Delta S_{\text{mix}}=-R(n_A\ln x_A+n_B\ln x_B)>0$$ $$\Delta G_{\text{mix}}=RT(n_A\ln x_A+n_B\ln x_B)<0$$

The driving force for mixing is purely entropic. Real solutions have $\Delta H_{\text{mix}}\ne 0$, leading to deviations from Raoult's law.

Derivation of vapor pressure lowering. For a single non-volatile solute (component 2) dissolved in a solvent (component 1), Raoult's law gives $P_1=x_1{P}_1^{\ast }$. The vapor pressure lowering is:

$$\Delta P=P_1^{\ast }-P_1=P_1^{\ast }(1-x_1)=x_2{P}_1^{\ast }$$

Since $x_2=n_2/(n_1+n_2)\approx n_2/n_1$ for dilute solutions, the vapor pressure lowering is proportional to the number of solute particles. This is the first colligative property and the thermodynamic basis for boiling point elevation and freezing point depression.

Connection to boiling point elevation. The Clausius-Clapeyron equation relates vapor pressure to temperature. Raoult's law lowers the vapor pressure at every temperature. To restore boiling ($P_{\text{vapor}}=P_{\text{external}}$), the temperature must increase. Combining Raoult's law with the Clausius-Clapeyron equation and expanding to first order in $x_2$:

$$\Delta T_b=\frac{R{T}_b^{\ast 2}}{\Delta H_{\text{vap}}}{x}_2=K_{b}m$$

where $K_b=R{T}_b^{\ast 2}{M}_{\text{solvent}}/(1000\Delta H_{\text{vap}})$ is the ebullioscopic constant.

Henry's law: thermodynamic derivation. Henry's law applies in the dilute-solute limit. For component 2 at low $x_2$, the chemical potential in the liquid is:

$${\mu }_2^{\text{liquid}}={\mu }_2^{\circ }+RT\ln ({\gamma }_2{x}_2)$$

At infinite dilution, ${\gamma }_2$ approaches a constant ${\gamma }_2^{\infty }$ (the infinite-dilution activity coefficient, using the Raoult's-law reference state). Setting ${\mu }_2^{\text{liquid}}={\mu }_2^{\text{vapor}}$ and rearranging:

$$P_2={\gamma }_2^{\infty }{x}_2{P}_2^{\ast }=K_H{x}_2$$

where $K_H={\gamma }_2^{\infty }{P}_2^{\ast }$. The Henry's law constant $K_H$ differs from $P_2^{\ast }$ because the solute at infinite dilution experiences a different molecular environment than in the pure state: ${\gamma }_2^{\infty }\ne 1$ in general. For gases like N2 in water, ${\gamma }_2^{\infty }$ is enormous (the gas does not want to dissolve), so $K_H\gg P_2^{\ast }$. For highly soluble gases like NH3 in water, ${\gamma }_2^{\infty }$ is small, and $K_H\ll P_2^{\ast }$.

Non-ideal solutions: activity coefficients. For a real binary solution, each component's partial pressure is:

$$P_1={\gamma }_1{x}_1{P}_1^{\ast }{P}_2={\gamma }_2{x}_2{P}_2^{\ast }$$

The activity coefficients satisfy the Gibbs-Duhem equation at constant T and P:

$$x_{1}d(\ln {\gamma }_1)+x_{2}d(\ln {\gamma }_2)=0$$

This constraint means the activity coefficients are not independent: if ${\gamma }_1$ is measured across the composition range, ${\gamma }_2$ can be calculated (and vice versa). The Gibbs-Duhem equation provides a rigorous consistency check on vapor-liquid equilibrium data.

Regular solution model. The simplest model for non-ideal solutions assumes ideal entropy of mixing but non-zero enthalpy of mixing:

$$\Delta G_{\text{mix}}=RT(x_A\ln x_A+x_B\ln x_B)+\Omega x_A{x}_B$$

where $\Omega$ is an interaction parameter. When $\Omega >0$, unlike interactions are unfavorable and the system shows positive deviations from Raoult's law. When $\Omega >2RT$, the solution phase-separates into two liquid phases at low temperature (upper critical solution temperature behavior). The activity coefficients in the regular solution model are:

$$RT\ln {\gamma }_1=\Omega x_2^{2}RT\ln {\gamma }_2=\Omega x_1^2$$

Both activity coefficients approach 1 as the component becomes the majority species (Raoult's law limit), as required.

Azeotropes in detail. An azeotrope occurs when the vapor and liquid compositions are equal ($y_i=x_i$ for all i). From the definition $y_i={\gamma }_i{x}_i{P}_i^{\ast }/P_{\text{total}}$, the azeotropic condition requires:

$${\gamma }_1{P}_1^{\ast }={\gamma }_2{P}_2^{\ast }$$

at the azeotropic composition. Since ${\gamma }_1$ and ${\gamma }_2$ are functions of composition, this equation determines the azeotropic composition. A minimum-boiling azeotrope arises from positive deviations (the total pressure curve has a maximum, producing a boiling-point minimum). A maximum-boiling azeotrope arises from negative deviations. The ethanol-water system at 1 atm forms a minimum-boiling azeotrope at 95.6 wt% ethanol and 78.2 degrees C, which is why simple distillation cannot produce pure ethanol from aqueous solution.

Breaking an azeotrope requires changing the conditions so that the azeotropic composition shifts or disappears. Pressure-swing distillation exploits the pressure dependence of the azeotropic composition: operate at two different pressures, and the different azeotropic compositions allow separation. Azeotropic distillation adds a third component (entrainer) that forms a new azeotrope with one component. Extractive distillation adds a high-boiling solvent that selectively interacts with one component, altering the relative volatility.

Temperature dependence of Henry's law constant. The van't Hoff equation gives:

$$\frac{d(\ln K_H)}{dT}=\frac{\Delta H_{\text{sol}}}{R{T}^2}$$

where $\Delta H_{\text{sol}}$ is the enthalpy of solution. For most gases dissolving in water, $\Delta H_{\text{sol}}<0$ (dissolution is exothermic), so $K_H$ decreases with increasing temperature and solubility decreases. This is why warm water holds less dissolved oxygen than cold water -- a fact with major environmental consequences for aquatic ecosystems.

Historical notes Master

Raoult's measurements (1887). Francois-Marie Raoult was professor of chemistry at the University of Grenoble. Beginning in the 1870s, he systematically measured the freezing points and vapor pressures of solutions. His 1887 paper in Comptes Rendus (vol. 104, pp. 1430-1433) reported the proportionality between vapor pressure lowering and mole fraction for a wide range of solute-solvent pairs. Raoult was careful to note that the law worked best for solutions that were dilute and for solvents whose molecules were chemically similar to those of the solute. The law was not presented as universal; Raoult himself documented numerous exceptions. The generalization to all ideal solutions and the recognition that ideality is a limiting behavior rather than a universal rule came from later workers, particularly van Laar and Margules, who developed the first activity coefficient models around 1910.

Raoult's work provided the experimental foundation for van't Hoff's theory of dilute solutions (1886), which showed that the osmotic pressure of a dilute solution follows an equation formally identical to the ideal gas law. Van't Hoff received the first Nobel Prize in Chemistry (1901) in part for this work. Raoult, whose measurements made van't Hoff's theory possible, was never awarded the Nobel Prize -- a recurring pattern in which the experimentalist who generates the data that enables a theoretical breakthrough receives less recognition than the theorist.

Henry's law (1803). William Henry, a Manchester physician and chemist, published his measurements of gas solubilities in water in the Philosophical Transactions of the Royal Society (vol. 93, pp. 40-62, 1803). Henry's law states that the mass of gas dissolved is proportional to the partial pressure of the gas above the solution. Henry noted the law was approximate and broke down at high pressures. The law was later placed on a thermodynamic foundation by the recognition that it describes the dilute-solution limit of the general phase-equilibrium condition.

Henry's law has gained modern urgency through its role in ocean acidification. The atmosphere's CO2 partial pressure has risen from 280 ppm (pre-industrial) to over 420 ppm (2024). Henry's law predicts a proportional increase in dissolved CO2, which forms carbonic acid in seawater, lowering the pH. The solubility of CO2 in seawater is further modulated by the carbonate buffer system, but the thermodynamic driving force is Henry's law.

Dalton's law of partial pressures (1801). John Dalton established that in a mixture of ideal gases, each component exerts a pressure independently of the others. Applied to vapor-liquid equilibrium, Dalton's law gives $y_i=P_i/P_{\text{total}}$, connecting the liquid-phase composition (through Raoult's law) to the vapor-phase composition. The combination of Raoult's law and Dalton's law provides the complete description of ideal vapor-liquid equilibrium.

G.N. Lewis and the activity concept (1907-1923). The recognition that Raoult's law and Henry's law are limiting cases of a general framework came from Gilbert Newton Lewis. In his 1907 paper and his 1923 textbook Thermodynamics and the Free Energy of Chemical Substances, Lewis introduced the activity $a_i={\gamma }_i{x}_i$ as the effective thermodynamic concentration. With this substitution, all solution equilibrium equations retain their ideal form: $P_i=a_i{P}_i^{\ast }$. The activity coefficient ${\gamma }_i$ encodes all non-ideality. Lewis's approach was pragmatic: rather than deriving ${\gamma }_i$ from molecular theory, he provided a framework for organizing experimental data. Molecular-theory-based prediction of activity coefficients came decades later through statistical mechanics and the local-composition models of Wilson (1964), Renon and Prausnitz (NRTL, 1968), and Abrams and Prausnitz (UNIQUAC, 1975).

Azeotropes and distillation. The word "azeotrope" was coined by Wade and Merriman in 1911 from Greek: a- (without) + zeo (boil) + tropos (change) -- "boiling without change." The existence of constant-boiling mixtures had been known since the early 19th century (Dalton noted the phenomenon), but the recognition that azeotropes prevent complete separation by distillation drove the development of azeotropic and extractive distillation methods in the early 20th century. The petroleum industry's need to separate close-boiling and azeotropic mixtures was the major practical motivation for the development of modern solution thermodynamics.

The P-x-y diagram. The graphical representation of vapor-liquid equilibrium as pressure-composition diagrams at constant temperature (P-x-y) and temperature-composition diagrams at constant pressure (T-x-y) was systematized by Ostwald and Roozeboom in the 1890s. These diagrams encode the complete equilibrium behavior of a binary system and remain the standard engineering tool for designing distillation processes.

Connections Master

• Thermodynamics (14.06): Raoult's law and Henry's law are direct consequences of the chemical-potential equality at phase equilibrium, which is itself a consequence of the second law. The ideal-solution mixing entropy $\Delta S_{\text{mix}}=-R\sum x_i\ln x_i$ is the same expression that appears in the statistical-mechanical definition of entropy.
• Colligative properties (14.09.03): Vapor pressure lowering via Raoult's law is the thermodynamic foundation for all four colligative properties — boiling-point elevation, freezing-point depression, and osmotic pressure follow by combining Raoult's law with the Clausius-Clapeyron equation or the pressure dependence of chemical potential.
• Electrochemistry (14.11): The activity-coefficient formalism developed for non-ideal solutions (Debye-Huckel theory) applies directly to electrolyte solutions, where ionic activities replace concentrations in the Nernst equation for cell potentials.

Bibliography Master

• Raoult, F.-M. "Loi generale des tensions de vapeur des dissolvants." Comptes Rendus Hebdomadaires des Seances de l'Academie des Sciences 104 (1887): 1430-1433. The original paper establishing the proportionality between vapor pressure lowering and mole fraction.

• Henry, W. "Experiments on the quantity of gases absorbed by water, at different temperatures, and under different pressures." Philosophical Transactions of the Royal Society of London 93 (1803): 40-62. The original statement of Henry's law for gas solubility.

• Atkins, P. W. and de Paula, J. Physical Chemistry, 12th ed. Oxford University Press, 2023. Chapter 5 provides a thorough treatment of Raoult's law, Henry's law, and activity coefficients at the intermediate level.

• Zumdahl, S. S. and DeCoste, D. J. Chemical Principles, 8th ed. Cengage, 2017. Chapters 11.1-11.4 cover vapor pressure lowering, Raoult's law, and boiling point elevation at the introductory level.

• Engel, T. and Reid, P. Thermodynamics, Statistical Thermodynamics, and Kinetics, 4th ed. Pearson, 2019. Chapter 8 develops the thermodynamic formalism for solutions and phase equilibria.

• Smith, J. M., Van Ness, H. C., and Abbott, M. M. Introduction to Chemical Engineering Thermodynamics, 8th ed. McGraw-Hill, 2018. Chapters 10-12 provide the engineering treatment of vapor-liquid equilibrium, including bubble-point and dew-point calculations.

• Prausnitz, J. M., Lichtenthaler, R. N., and de Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria, 3rd ed. Prentice Hall, 1999. The authoritative reference for activity coefficient models and phase equilibrium calculations.

• Gmehling, J. and Kolbe, B. Thermodynamik, 3rd ed. Wiley-VCH, 2017. A comprehensive German-language treatment of solution thermodynamics with extensive data tables for Henry's law constants and activity coefficients.

• Sander, R. "Compilation of Henry's law constants (version 4.0) for water as solvent." Atmospheric Chemistry and Physics 15 (2015): 4399-4981. The most comprehensive compilation of Henry's law constants, essential for environmental chemistry applications.

• Wilson, G. M. "Vapor-liquid equilibrium. XI. A new expression for the excess free energy of mixing." Journal of the American Chemical Society 86 (1964): 127-130. The original Wilson equation for activity coefficients.

• Renon, H. and Prausnitz, J. M. "Local compositions in thermodynamic excess functions for liquid mixtures." AIChE Journal 14 (1968): 135-144. The NRTL equation.

• Abrams, D. S. and Prausnitz, J. M. "Statistical thermodynamics of liquid mixtures: a new expression for the excess Gibbs energy of partly or completely miscible systems." AIChE Journal 21 (1975): 116-128. The UNIQUAC equation.