Raoult's law

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Raoult's law is a law of thermodynamics established by French physicist François-Marie Raoult in 1882. ^[1] The partial vapor pressure of each component of an ideal mixture of liquids is equal to the vapor pressure of the pure component multiplied by its mole fraction in the mixture. The total vapor pressure of the ideal solution is then directly dependent on the vapor pressure of each chemical component and the mole fraction of the component present in the solution.^[2]

Once the components in the solution have reached equilibrium, the total vapor pressure p of the solution is: $\scriptstyle p=p_{{{\rm {A}}}}^{{\star }}x_{{{\rm {A}}}}+p_{{{\rm {B}}}}^{{\star }}x_{{{\rm {B}}}}+\cdots$ and the individual vapor pressure for each component is $\scriptstyle p_{i}=p_{i}^{{\star }}x_{i}$ where p_i is the partial vapor pressure of the component i in the gaseous mixture (above the solution), p*_i is the vapor pressure of the pure component i, and x_i is the mole fraction of the component i in the mixture (in the solution).

If a pure solute which has zero vapor pressure (it will not evaporate) is dissolved in a solvent, the vapor pressure of the final solution will be lower than that of the pure solvent.

Principle

Vapor pressure of a binary solution which 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 which assumes ideal behavior based on a simple picture just as the ideal gas law does. The ideal gas law is very useful as a limiting law. The validity of this law requests the microscopic assumption regarding intermolecular forces between unlike molecules to be equal to those between similar molecules: the conditions of an ideal solution.

As the interactive forces between molecules and the volume of the molecules approach zero, so the behavior of gases approach ideality. Raoult’s law similarly assumes that 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 the vapor pressure of each component (i) will be equal to the vapor pressure of the pure substance $p_{i}^{{\star }}$ times the mole fraction in the solution.

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.

Using the example of a solution of two liquids, A and B, if no other gases are present, then the total vapor pressure p above the solution is equal to the weighted sum of the "pure" vapor pressures of the two components, p_A and p_B. Thus the total pressure above solution of A and B would be

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

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

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

which is a linear function of the mole fraction x_B as shown in the graph.

Thermodynamic considerations

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 is given by

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

where $\mu _{i}^{{\star }}$ 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 theoretical thermodynamic definition of an ideal solution is one in which the chemical potential of each component is given by the above formula. 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,

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

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

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

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

The corresponding equation for pure $i$ in equilibrium with its (pure) vapor is:

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

where * indicates the pure component.

Subtracting both equations gives us

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

which re-arranges to

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

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

$p_{i}\approx x_{i}p_{i}^{{\star }}$

which is Raoult’s Law.

Ideal mixing

An ideal solution can be said to follow Raoult's Law but it must be kept in mind that in the strict sense ideal solutions do not exist. The fact that the vapor is taken to be ideal is the least of our worries. Interactions between gas molecules are typically quite small especially if the vapor pressures are low. The interactions in a liquid however are very strong. For a solution to be ideal we must assume that it does not matter whether a molecule A has another A as neighbor or a B molecule. This is only approximately true if the two species are almost identical chemically. One can see that from considering the Gibbs free energy change of mixing:

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

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 Δ_mixH must be equal to zero and this can only be if the interactions U between the molecules are indifferent.

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

If the deviations from ideality are not too strong, Raoult's law will still be valid in a narrow concentration range when approaching x = 1 for the majority phase (the solvent). The solute will also show a linear limiting law but with a different coefficient. This law is known as Henry's law.

The presence of these limited linear regimes has been experimentally verified in a great number of cases. In a perfectly ideal system, where ideal liquid and ideal vapor are assumed, a very useful equation emerges if Raoult's law is combined with Dalton's Law.

$x_{i}={\frac {y_{i}p_{{{\rm {total}}}}}{p_{{i,{{\rm {eqm}}}}}}}\,$ ,

where $x_{i}$ is the mole fraction of component $i$ in the solution and $y_{i}$ is its mole fraction in the gas phase. This equation shows that, for an ideal solution where each pure component has a different vapor pressure, the gas phase will be enriched in the component with the higher pure vapor pressure and the solution will be enriched in the component with the lower pure vapor pressure. This phenomenon is the basis for distillation.

Non-ideal mixing

Raoult's Law may be adapted to non-ideal solutions by incorporating two factors that will account for the interactions between molecules of different substances. The first factor is a correction for gas non-ideality, or deviations from the ideal-gas law. It is called the fugacity coefficient ( $\phi _{{p,i}}$ ). The second, the activity coefficient ( $\gamma _{i}$ ), is a correction for interactions in the liquid phase between the different molecules.

This modified or extended Raoult's law is then written:

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

Real solutions

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 show deviation from the Raoult's law which is 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. 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 which 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^[3] from Raoult's law, indicating an attractive interaction between the two components which has been described as a hydrogen bond.^[4] 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 which evaporates without change of composition.^[5]

Positive deviation

When the cohesive forces between like molecules are greater than the adhesive forces, the dissimilarities of polarity or internal pressure will lead both components to escape solution more easily. Therefore, the vapor pressure will be greater than expected from the Raoult's law, showing positive deviation. If the deviation is large, then the vapor pressure curve will show a maximum at a particular composition and form a positive azeotrope. Some mixtures in which this happens are (1) benzene and methanol, (2) carbon disulfide and acetone, and (3) chloroform and ethanol.

Electrolytes solutions

The expression of the law for this case includes the van't Hoff factor.

References

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)

↑ Barnett, Martin K. (2001), A Dictionary of the History of Science, Informa Health Care, p. 287 . Extract.
↑ A to Z of Thermodynamics by Pierre Perrot. ISBN 0-19-856556-9
↑ P. Atkins and J. de Paula, Physical Chemistry (8th ed., W.H. Freeman 2006) p.146
↑ K. Kwak, D.E. Rosenfeld, J.K. Chung and M.D. Fayer, J. Phys. Chem. B (2008), 112, 13906–13915
↑ Atkins and de Paula, p.184

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

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