Raoult's law

Vapor pressure of a binary solution that obeys Raoult's law. The black line shows the total vapor pressure as a function of the mole fraction of component B, and the two green lines are the partial pressures of the two components.

Raoult's law is a phenomenological law that assumes ideal behavior based on the simple microscopic assumption that intermolecular forces between unlike molecules are equal to those between similar molecules: the conditions of an ideal solution. This is analogous to the ideal gas law, which is a limiting law valid when the interactive forces between molecules approach zero, for example as the concentration approaches zero. Raoult's law is instead valid if the physical properties of the components are identical. The more similar the components are, the more their behavior approaches that described by Raoult's law. For example, if the two components differ only in isotopic content, then Raoult's law is essentially exact.

Comparing measured vapor pressures to predicted values from Raoult's law provides information about the true relative strength of intermolecular forces. If the vapor pressure is less than predicted (a negative deviation), fewer molecules of each component than expected have left the solution in the presence of the other component, indicating that the forces between unlike molecules are stronger. The converse is true for positive deviations.

For a solution of two liquids A and B, Raoult's law predicts that if no other gases are present, then the total vapor pressure ${\displaystyle p}$ above the solution is equal to the weighted sum of the "pure" vapor pressures ${\displaystyle p_{\text{A}}}$ and ${\displaystyle p_{\text{B}}}$ of the two components. Thus the total pressure above the solution of A and B would be

${\displaystyle p=p_{\text{A}}^{\star }x_{\text{A}}+p_{\text{B}}^{\star }x_{\text{B}}.}$

Since the sum of the mole fractions is equal to one,

${\displaystyle p=p_{\text{A}}^{\star }(1-x_{\text{B}})+p_{\text{B}}^{\star }x_{\text{B}}=p_{\text{A}}^{\star }+(p_{\text{B}}^{\star }-p_{\text{A}}^{\star })x_{\text{B}}.}$

This is a linear function of the mole fraction ${\displaystyle x_{\text{B}}}$, as shown in the graph.

Raoult's law was originally discovered as an idealised experimental law. Using Raoult's law as the definition of an ideal solution, it is possible to deduce that the chemical potential of each component of the liquid is given by

${\displaystyle \mu _{i}=\mu _{i}^{\star }+RT\ln x_{i},}$

where ${\displaystyle \mu _{i}^{\star }}$ is the chemical potential of component i in the pure state. This equation for the chemical potential may then be used to derive other thermodynamic properties of an ideal solution (see Ideal solution).

However, a more fundamental thermodynamic definition of an ideal solution is one in which the chemical potential of each component is given by the above formula. Assuming also that the vapor mixture acts as an ideal gas, it is then possible to re-derive Raoult's law as follows.

If the system is at equilibrium, then the chemical potential of the component i must be the same in the liquid solution and in the vapor above it. That is,

${\displaystyle \mu _{i,{\text{liq}}}=\mu _{i,{\text{vap}}}.}$

Assuming the liquid is an ideal solution and using the formula for the chemical potential of a gas gives

${\displaystyle \mu _{i,{\text{liq}}}^{\star }+RT\ln x_{i}=\mu _{i,{\text{vap}}}^{\ominus }+RT\ln {\frac {f_{i}}{p^{\ominus }}},}$

where ${\displaystyle f_{i}}$ is the fugacity of the vapor of ${\displaystyle i}$, and ${\displaystyle ^{\ominus }}$ indicates the reference state.

The corresponding equation for pure ${\displaystyle i}$ in equilibrium with its (pure) vapor is

${\displaystyle \mu _{i,{\text{liq}}}^{\star }=\mu _{i,{\text{vap}}}^{\ominus }+RT\ln {\frac {f_{i}^{\star }}{p^{\ominus }}},}$

where ${\displaystyle ^{\star }}$ indicates the pure component.

Subtracting the equations gives

${\displaystyle RT\ln x_{i}=RT\ln {\frac {f_{i}}{f_{i}^{\star }}},}$

which re-arranges to

${\displaystyle f_{i}=x_{i}f_{i}^{\star }.}$

The fugacities can be replaced by simple pressures if the vapor of the solution behaves ideally, i.e.

${\displaystyle p_{i}=x_{i}p_{i}^{\star },}$

which is Raoult's law.

Many pairs of liquids are present in which there is no uniformity of attractive forces, i.e., the adhesive and cohesive forces of attraction are not uniform between the two liquids, so that they deviate from the Raoult's law applied only to ideal solutions.

Negative deviation

Positive and negative deviations from Raoult's law. Maxima and minima in the curves (if present) correspond to azeotropes or constant boiling mixtures.

If the vapor pressure of a mixture is lower than expected from Raoult's law, there is said to be a negative deviation.

Negative deviations from Raoult's law arise when the forces between the particles in the mixture are stronger than the mean of the forces between the particles in the pure liquids. This is evidence that the adhesive forces between different components are stronger than the average cohesive forces between like components. In consequence each component is retained in the liquid phase by attractive forces that are stronger than in the pure liquid so that its partial vapor pressure is lower.

For example, the system of chloroform (CHCl₃) and acetone (CH₃COCH₃) has a negative deviation[6] from Raoult's law, indicating an attractive interaction between the two components that has been described as a hydrogen bond.[7]

The system hydrochloric acid – water has a large enough negative deviation to form a minimum in the vapor pressure curve known as a (negative) azeotrope, corresponding to a mixture that evaporates without change of composition.[8] When these two components are mixed, the reaction is exothermic as ion-dipole intermolecular forces of attraction are formed between the resulting ions (H₃O⁺ and Cl^–) and the polar water molecules so that ΔH_mix is negative.

The expression of the law for this case includes the van 't Hoff factor which is also known as correction factor for solutions

• Antoine equation
• Atomic theory
• Azeotrope
• DECHEMA model
• Dühring's rule
• Henry's law
• Köhler theory
• Solubility

Chapter 24, D. A. McQuarrie, J. D. Simon Physical Chemistry: A Molecular Approach. University Science Books. (1997)

E. B. Smith Basic Chemical Thermodynamics. Clarendon Press. Oxford (1993)