Raoult's law

Established by François-Marie Raoult in 1882, Raoult's law states:

the vapor pressure of an ideal solution is dependent on the vapor pressure of each chemical component and the mole fraction of the component present in the solution.

File:Graph-for-raoults-law.jpg

Once the components in the solution have reached equilibrium, the total vapor pressure p of the solution is:

<math>p = p^{\star}_{\rm A} x_{\rm A} + p^{\star}_{\rm B} x_{\rm B} + \cdots</math>

and the individual vapor pressure for each component is

<math>p_i = p^{\star}_i x_i</math>

where

p_i is the partial pressure of the component i in mixture

p*_i is the vapor pressure of the pure component i

x_i is the mole fraction of the component i in solution (in mixture)

Consequently, as the number of components in a solution increases, the individual vapor pressures decrease, since the mole fraction of each component decreases with each additional component. 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.

This law is strictly valid only under the assumption that the chemical interactions between the two liquids is equal to the bonding within the liquids: the conditions of an ideal solution. Therefore, comparing actual measured vapor pressures to predicted values from Raoult's law allows information about the relative strength of bonding between liquids to be obtained. If the measured value of vapor pressure is less than the predicted value, fewer molecules have left the solution than expected. This is put down to the strength of bonding between the liquids being greater than the bonding within the individual liquids, so fewer molecules have enough energy to leave the solution. Conversely, if the vapor pressure is greater than the predicted value more molecules have left the solution than expected, due to the bonding between the liquids being less strong than the bonding within each.

The vapor pressure and composition in equilibrium with a solution can yield valuable information regarding the thermodynamic properties of the liquids involved. Raoult’s law relates the vapor pressure of components to the composition of the solution. The law assumes ideal behavior. It gives a simple picture of the situation just as the ideal gas law does. The ideal gas law is very useful as a limiting law. As the interactive forces between molecules and the volume of the molecules approaches zero, so the behavior of gases approach the behavior of the ideal gas.

Raoult’s law is similar in that it 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 will be equal to the vapor pressure of the pure substance <math>P_{0}</math> times the mole fraction in the solution. This is Raoult’s law.

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

<math>p = p^{\star}_{\rm A} x_{\rm A} + p^{\star}_{\rm B} x_{\rm B}</math>

Derivation

We define an ideal solution as a solution for which the chemical potential of component <math>i</math> is:

<math>\mu _i = \mu_i^{\star} + RT\ln x_i\,</math>,

where µ*_i is the chemical potential of pure i.

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,

<math>\mu _{i,{\rm liq}} = \mu _{i,{\rm vap}}\,</math>

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

<math>\mu _{i,{\rm liq}}^{\star} + RT\ln x_i = \mu_{i,{\rm vap}}^\ominus + RT\ln \fracTemplate:F i

Template:P^\ominus</math> where ƒ_i is the fugacity of the vapor of <math>i</math> and <math> ^\ominus </math> indicates reference state.

The corresponding equation for pure <math>i</math> in equilibrium with its (pure) vapor is:

<math>\mu _{i,{\rm liq}}^{\star} = \mu _{i,{\rm vap}}^\ominus + RT\ln \frac{{f_i^{\star}}}

Template:P^\ominus</math> where * indicates the pure component.

Subtracting both equations gives us

<math>RT\ln x_i = RT\ln \fracTemplate:F i{{f_i^{\star} }}</math>

which re-arranges to

<math>f_i = x_i f_i^{\star}</math>

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

<math>p_i \approx x_i p_i^{\star}</math>

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. We can see that from considering the Gibbs free energy change of mixing:

<math>\Delta_{\rm mix} G = nRT(x_1\ln x_1 + x_2\ln x_2)\,</math>

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.

<math>x_i = \frac{y_i p_{\rm total}}{p_{i,{\rm eqm}}}\,</math>

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 (<math>\phi</math>). The second, the activity coefficient (<math>\gamma</math>), is a correction for interactions in the liquid phase between the different molecules.

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

<math>p_{i} \phi_i = p_i^{\star} \gamma_i x_i\,</math>

Real solutions

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Many pairs of liquids are present in which there is no uniformity of attractive forces i.e. the adhesive & 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

When adhesive forces between molecules of A & B are greater than the cohesive forces between A & A or B & B, then the vapor pressure of the solution is less than the expected vapor pressure from Raoult's law. This is called as negative deviation from Raoult's law. These cohesive forces are lessened not only by dilution but also attraction between two molecules through formation of hydrogen bonds. This will further reduce the tendency of A and B to escape.

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For example, chloroform and acetone show such an attraction by formation of a hydrogen bond.

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 the 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, e.g. benzene & ethyl alcohol, carbon disulfide & acetone, chloroform & ethanol.

See also

• Atomic Theory
• Dalton's law
• DECHEMA model
• Dühring's rule
• Henry's law
• Köhler theory
• Solubility

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)

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