Raoult's law states that the vapor pressure of each component in an ideal solution is dependent on the vapor pressure of the individual component and the mole fraction of the component present in the solution.

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

$\backslash \; P\_\{solution\}=\; (P\_\{1\})\_\{pure\}\; X\_1\; +\; (P\_\{2\})\_\{pure\}\; X\_2\; \backslash cdots$

and the individual vapor pressure for each component is

$\backslash \; P\_\{i\}=(P\_\{i\})\_\{pure\}\; X\_i$

where

• (P_i)_pure is the vapor pressure of the pure component
• X_i is the mole fraction of the component in solution

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 bond 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.

Deduction of Raoult’s Law (or Raoult’s Equation)

We define an ideal solution, as the solution for which the chemical potential of the component $i$ is

$\backslash mu\; \_i\; =\; \backslash mu\; \_i\; ^\backslash circ\; +\; RT\backslash ln\; x\_i$

Where the reference state is the pure substance at work $P$ and $T$.

Experimentally, an ideal solution has a zero enthalpy of solution and verifies volume additivity.

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,

$\backslash mu\; \_\{i,L\}\; =\; \backslash mu\; \_\{i,V\}\backslash ,$

If the liquid is an ideal solution, and using the formula for a gas’ chemical potential

$\backslash mu\; \_\{i,L,P\}\; ^\backslash circ\; +\; RT\backslash ln\; x\_i\; =\; \backslash mu\; \_\{i,1\; \backslash mathrm\{bar\}\}\; ^\backslash circ\; +\; RT\backslash ln\; \backslash frac$ (1)

(where $f$ is the fugacity of the vapor of $i$)

If we study the component i in its pure state, we would have

$\backslash mu\; \_\{i,L\}^*\; =\; \backslash mu\; \_\{i,V\}^*\backslash ,$

Where * indicates that we study a pure component.

$\backslash mu\; \_\{i,L,P\_i^*\}\; ^\backslash circ\; +\; RT\backslash ln\; x\_i\; =\; \backslash mu\; \_\{i,1\{\backslash mathrm\{\; bar\}\}\}\; +\; RT\backslash ln\; \backslash frac$

But now, $x\_i=1$, so

$\backslash mu\; \_\{i,L,P\_i^*\}\; ^\backslash circ\; =\; \backslash mu\; \_\{i,1\{\backslash mathrm\{\; bar\}\}\}\; +\; RT\backslash ln\; \backslash frac$ (2)

Substracting (1)-(2) gives us

$(\backslash mu\; \_\{i,L,P\_i\}^\backslash circ\; -\; \{\backslash mu\; \_\{i,L,P\_i^*\}\; ^\backslash circ\; \})\; +\; RT\backslash ln\; x\_i\; =\; RT\backslash ln\; \backslash frac$

which can be written as

$\backslash frac\; +\; \backslash ln\; x\_i\; =\; \backslash ln\; \backslash frac$

or

$e^\{\backslash frac\; \}x\_i\; =\; \backslash frac$

This equation is valid for the ideal solution.

Now, let’s suppose the vapor of the solution behaves as an ideal gas. In this case, fugacity and pressure are identical, and we get

$e^\{\backslash frac\; \}\; x\_i\; =\; \backslash frac$

For most substances we have $\backslash Delta\; \backslash mu\; \backslash approx\; 0$, and then

$x\_i\; \backslash approx\; \backslash frac$

Finally,

$P\_i\; \backslash approx\; x\_i\; P\_i^*$

This last equality is what is known as Raoult’s Law. However, in physical chemistry and other sciences, a scientific law is a generalization of an empiric observation, and must be therefore verifiable in every experiment.

However, this equation is not verified by most solutions, and then cannot be considered a law. We can clearly see that not only the solution must be ideal; we also have to suppose that the vapor has an ideal gas behavior and that for the substance it is true that $\backslash Delta\; \backslash mu\; \backslash approx\; 0$. Raoult’s Law should be called Raoult’s Equation, as its range of validity is strictly limited.

Moreover, an ideal solution cannot be defined as the one that follows Raoult's Equation. As shown above, an ideal solution follows approximately the equation.

See also

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• Henry's law
• Dalton's law

Solutions | Thermodynamics

Loi de Raoult | Wet van Raoult | ラウールの法則 | Prawo Raoulta | Lei de Raoult | Raoultov zákon