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Raoult's law

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Law of thermodynamics for vapour pressure of a mixture

Raoult's law (/ˈrɑːuːlz/ law) is a relation ofphysical chemistry, with implications inthermodynamics. Proposed by French chemistFrançois-Marie Raoult in 1887,^[1]^[2] it states that thepartial pressure of each component of anideal mixture ofliquids is equal to thevapor pressure of the pure component (liquid or solid) multiplied by itsmole fraction in the mixture. In consequence, the relative lowering of vapor pressure of a dilute solution of nonvolatilesolute is equal to the mole fraction of solute in the solution.

Mathematically, Raoult's law for a single component in an ideal solution is stated as

p_{i}=p_{i}^{\star }x_{i}

where $p_{i}$ is thepartial pressure of the component $i {\displaystyle i}$ in the gaseous mixture above the solution, $p_{i}^{\star }$ is theequilibrium vapor pressure of the pure component $i {\displaystyle i}$ , and $x_{i}$ is the mole fraction of the component $i {\displaystyle i}$ in the liquid or solid solution.^[3]

Where two volatile liquids A and B are mixed with each other to form a solution, the vapor phase consists of both components of the solution. Once the components in the solution have reachedequilibrium, the total vapor pressure of the solution can be determined by combining Raoult's law withDalton's law of partial pressures to give

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

In other words, the vapor pressure of the solution is the mole-weighted mean of the individual vapour pressures:

p={\dfrac {p_{\text{A}}^{\star }n_{\text{A}}+p_{\text{B}}^{\star }n_{\text{B}}+\cdots }{n_{\text{A}}+n_{\text{B}}+\cdots }}

If a non-volatile solute B (it has zero vapor pressure, so does notevaporate) is dissolved into a solvent A to form an ideal solution, the vapor pressure of the solution will be lower than that of the solvent. In an ideal solution of a nonvolatile solute, the decrease in vapor pressure is directly proportional to the mole fraction of solute:

p=p_{\text{A}}^{\star }x_{\text{A}},

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

If the solute associates or dissociates in the solution (such as an electrolyte/salt), the expression of the law includes thevan 't Hoff factor as a correction factor. That is, the mole fraction must be calculated using the actual number of particles in solution.^[4]

Principles

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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 relation that assumes ideal behavior based on the simple microscopic assumption that intermolecular forces between unlike molecules are equal to those between similar molecules, and that their molar volumes are the same: the conditions of an ideal solution. This is analogous to theideal 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 inisotopic 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 ofintermolecular 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 $p {\displaystyle p}$ above the solution is equal to the weighted sum of the "pure" vapor pressures $p_{\text{A}}$ and $p_{\text{B}}$ of the two components. Thus the total pressure above the solution of A and B would be

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,

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 $x_{\text{B}}$ , as shown in the graph.

Thermodynamic considerations

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Raoult's law was first observed empirically and ledFrançois-Marie Raoult^[1]^[2] to postulate that the vapor pressure above an ideal mixture of liquids is equal to the sum of the vapor pressures of each component multiplied by its mole fraction.^[5]^: 325 Taking compliance with Raoult's Law as a defining characteristic of ideality in a solution, it is possible todeduce that thechemical potential of each component of the liquid is given by

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

where $\mu _{i}^{\star }$ is the chemical potential in the pure state and $x_{i}$ is the mole fraction of component $i {\displaystyle i}$ in the ideal solution. From this equation, other thermodynamic properties of an ideal solution may be determined. If the assumption that the vapor follows the ideal gas law is added, Raoult's law may be derived as follows.

If the system is ideal, then, atequilibrium, the chemical potential of each component $i {\displaystyle i}$ must be the same in the liquid and gas states. That is,

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

Substituting the formula for chemical potential gives

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

as the gas-phase mole fraction depends on itsfugacity, $f_{i}$ , as a fraction of the pressure in the reference state, $p^{\ominus }$ .

The corresponding equation when the system consists purely of component $i {\displaystyle i}$ in equilibrium with its vapor is

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

Subtracting these equations and re-arranging leads to the result^[5]^: 326

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

For the ideal gas, pressure and fugacity are equal, so introducing simplepressures to this result yields Raoult's law:

p_{i}=x_{i}p_{i}^{\star }.

Ideal mixing

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An ideal solution would follow Raoult's law, but most solutions deviate from ideality. Interactions between gas molecules are typically quite small, especially if the vapor pressures are low. However, the interactions in a liquid are very strong. For a solution to be ideal, the interactions between unlike molecules must be of the same magnitude as those between like molecules.^[6] This approximation is only true when the different species are almost chemically identical. One can see that from considering theGibbs free energy change of mixing:

\Delta _{\text{mix}}G=nRT(x_{1}\ln x_{1}+x_{2}\ln x_{2}).

This is always negative, so mixing is spontaneous. However, the expression is, apart from a factor $-T$ , equal to the entropy of mixing. This leaves no room at all for an enthalpy effect and implies that $\Delta _{\text{mix}}H$ must be equal to zero, and this can only be true if the interactions between the molecules are indifferent.

It can be shown using theGibbs–Duhem equation that if Raoult's law holds over the entire concentration range $x\in [0,\ 1]$ in a binary solution then, for the second component, the same must also hold.

If deviations from the ideal are not too large, Raoult's law is still valid in a narrow concentration range when approaching $x\to 1$ for the majority phase (thesolvent). The solute also shows a linear limiting law, but with a different coefficient. This relationship is known asHenry's law.

The presence of these limited linear regimes has been experimentally verified in a great number of cases, though large deviations occur in a variety of cases. Consequently, both its pedagogical value and utility have been questioned at the introductory college level.^[7] In a perfectly ideal system, where ideal liquid and ideal vapor are assumed, a very useful equation emerges if Raoult's law is combined withDalton's Law:

x_{i}={\frac {y_{i}p_{\text{total}}}{p_{i}^{\star }}},

where $x_{i}$ is themole fraction of component $i {\displaystyle i}$ in thesolution, and $y_{i}$ is itsmole fraction in thegas phase. This equation shows that, for an ideal solution where each pure component has a different vapor pressure, the gas phase is enriched in the component with the higher vapor pressure when pure, and the solution is enriched in the component with the lower pure vapor pressure. This phenomenon is the basis fordistillation.

Non-ideal mixing

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In elementary applications, Raoult's law is generally valid when the liquid phase is either nearly pure or a mixture of similar substances.^[8] Raoult's law may be adapted to non-ideal solutions by incorporating two factors that account for the interactions between molecules of different substances. The first factor is a correction for gas non-ideality, or deviations from theideal-gas law. It is called thefugacity coefficient ( $\phi _{p,i}$ ). The second, theactivity coefficient $\gamma _{i}$ , is a correction for interactions in the liquid phase between the different molecules.^[5]^: 326

This modified or extended Raoult's law is then written as^[9]

y_{i}\phi _{p,i}p=x_{i}\gamma _{i}p_{i}^{\star }.

Real solutions

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In many pairs of liquids, there is no uniformity of attractive forces, i.e., theadhesive (between dissimilar molecules) andcohesive forces (between similar molecules) are not uniform between the two liquids. Therefore, they deviate from Raoult's law, which applies only to ideal solutions.

Notably, when the concentration of A is small, its vapor pressure instead followsHenry's law, and likewise for substance B when its concentration is small.

Negative deviation

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When the adhesion is stronger than the cohesion, fewer liquid particles turn into vapor thereby lowering the vapor pressure and leading to negative deviation in the graph.

For example, the system ofchloroform (CHCl₃) andacetone (CH₃COCH₃) has a negative deviation^[10] from Raoult's law, indicating an attractive interaction between the two components that have been described as ahydrogen bond.^[11] The system HCl–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.^[12] 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.

Positive deviation

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When the adhesion is weaker than cohesion, which is quite common, the liquid particles escape the solution more easily that increases the vapor pressure and leads to a positive deviation.

If the deviation is large, then the vapor pressure curve shows a maximum at a particular composition and forms a positive azeotrope (low-boiling mixture). Some mixtures in which this happens are (1)ethanol andwater, (2)benzene andmethanol, (3)carbon disulfide andacetone, (4)chloroform and ethanol, and (5)glycine and water. When these pairs of components are mixed, the process is endothermic as weaker intermolecular interactions are formed so that Δ_mixH is positive.

Mixed deviation

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It is possible to have mixed deviations, which are positive for one component and negative for the other, and which switch between positive and negative while moving from $x=0$ to $x=1$ . These are not merely theoretically possible, as actual examples of mixed deviation exist.^[13] The possible physical deviations are not entirely arbitrary however, as they are constrained by theDuhem–Margules equation: for example, if one component has positive deviation over the entire range then the other component cannot have negative deviation over the entire range.^[13]

References

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^^a ^bRaoult, F.-M. (1886)."Loi générale des tensions de vapeur des dissolvants" [General law of vapor pressures of solvents].Comptes rendus (in French).104:1430–1433.
^^a ^bRaoult, F.-M. (1889)."Recherches expérimentales sur les tensions de vapeur des dissolutions" [Experimental research on the vapor pressures of solutions].Journal de Physique Théorique et Appliquée (in French).8 (1):5–20.doi:10.1051/jphystap:0188900800500.
^A to Z of Thermodynamics by Pierre Perrot.ISBN 0-19-856556-9.
^"Raoult's Law".Libretexts. 2 October 2013.
^^a ^b ^cChao, Kwang-Chu; Corti, David S.; Mallinson, Richard G. (2008)."Thermodynamics of Fluid Phase and Chemical Equilibria". In Albright, Lyle F. (ed.).Albright's Chemical Engineering Handbook.CRC Press. pp. 255–392.ISBN 9780824753627.
^Rock, Peter A.Chemical Thermodynamics (MacMillan 1969), p. 261.ISBN 1891389327.
^Hawkes, Stephen J. (1995)."Raoult's Law Is a Deception".J. Chem. Educ.72 (3):204–205.Bibcode:1995JChEd..72..204H.doi:10.1021/ed072p204.S2CID 95146940.
^Felder, Richard M.; Rousseau, Ronald W.; Bullard, Lisa G. (2004-12-15).Elementary Principles of Chemical Processes. Wiley. p. 293.ISBN 978-0471687573.
^Smith, J. M.; Van Ness, H. C.; Abbott, M. M. (2005),Introduction to Chemical Engineering Thermodynamics (seventh ed.), New York: McGraw-Hill, p. 545,ISBN 0-07-310445-0
^P. Atkins and J. de Paula,Physical Chemistry (8th ed., W. H. Freeman 2006) p. 146.
^Kwak, Kyungwon; Rosenfeld, Daniel E.; Chung, Jean K.; Fayer, Michael D. (6 November 2008)."Solute-Solvent Complex Switching Dynamics of Chloroform between Acetone and Dimethylsulfoxide Two-Dimensional IR Chemical Exchange Spectroscopy".J. Phys. Chem. B.112 (44):13906–13915.CiteSeerX 10.1.1.560.3553.doi:10.1021/jp806035w.PMC 2646412.PMID 18855462.
^Atkins and de Paula, p. 184.
^^a ^bMcGlashan, M. L. (1963). "Deviations from Raoult's law".Journal of Chemical Education.40 (10): 516.Bibcode:1963JChEd..40..516M.doi:10.1021/ed040p516.

External links

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v t e Distillation
Principles	Raoult's law Dalton's law Reflux Fenske equation McCabe–Thiele method Theoretical plate Partial pressure Vapor–liquid equilibrium
Industrial processes	Batch distillation Continuous distillation Fractionating column Spinning cone
Laboratory methods	Alembic Kugelrohr Rotary evaporator Spinning band distillation Still
Techniques	Azeotropic Catalytic Destructive Dry Extractive Fractional Reactive Salt-effect Steam-based Vacuum-based

v t e Chemical solutions
Solution	Ideal solution Aqueous solution Solid solution Buffer solution Flory–Huggins Mixture Suspension Colloid Phase diagram Phase separation Eutectic point Alloy Saturation Supersaturation Serial dilution Dilution (equation) Apparent molar property Miscibility gap
Concentration and related quantities	Molar concentration Mass concentration Number concentration Volume concentration Normality Molality Mole fraction Mass fraction Isotopic abundance Mixing ratio Ternary plot Total dissolved solids
Solubility	Solubility equilibrium Solvation Solvation shell Enthalpy of solution Lattice energy Raoult's law Henry's law Solubility table (data) Solubility chart Miscibility
Solvent	(Category) Acid dissociation constant Protic solvent Polar aprotic solvent Inorganic nonaqueous solvent Solvation List of boiling and freezing information of solvents Partition coefficient Polarity Hydrophobe Hydrophile Lipophilic Amphiphile Lyonium ion Lyate ion