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Raoult's Law
Raoult's Law states that the partial pressure of each component of an ideal mixture of liquids is equal to the vapour pressure of the pure component (liquid or solid) multiplied by its mole fraction in the mixture.
It also implies that the relative lowering of vapour pressure of a dilute solution of non-volatile solute is equal to the mole fraction of solute in the solution.
Raoult's Law is only valid for liquids that are almost pure solutions, while for sufficiently dissolved solutions, Henry's Law is used.
- Raoult's law: \({\displaystyle \lim _{\chi\to 1}\left({\frac {p}{\chi}}\right)=p^{*}}\)
- Henry's law: \({\displaystyle \lim _{\chi\to 0}\left({\frac {p}{\chi}}\right)=H_{\rm {v}}^{p\chi}}\)
Formulas
Partial Pressure of Vapours in a Solution
\({\displaystyle p_{i}=p_{i}^{\star }\chi_{i}}\),
where \({\displaystyle p_{i}}\) is the partial pressure of the component \({\displaystyle i}\) in the gaseous mixture above the solution, \({\displaystyle p_{i}^{\star }}\) is the equilibrium vapour pressure of the pure component \({\displaystyle i}\), and \({\displaystyle \chi_{i}}\) is the mole fraction of the component \({\displaystyle i}\) in the liquid or solid solution.
Combined with Dalton's Law of Partial Pressures, this gives:
\({\displaystyle p=p_{\text{A}}^{\star }\chi_{\text{A}}+p_{\text{B}}^{\star }\chi_{\text{B}}+\cdots}\)
or
\({\displaystyle p={\dfrac {p_{\text{A}}^{\star }n_{\text{A}}+p_{\text{B}}^{\star }n_{\text{B}}+\cdots }{n_{\text{A}}+n_{\text{B}}+\cdots }}}\) (i.e. the vapour pressure of the solution is equal to the mole-weighted mean of individual vapour pressures of the pure components)
Ideal Solution
If a non-voltatile solute \(B\) dissolves in a solvent \(A\) to form an ideal solution, then the vapour pressure of the solution will be lower than that of the solvent. In an ideal solution with a non-volatile solute, the decrease in vapour pressure will be proportional to the mole fraction of the solute.
\({\displaystyle p=p_{\text{A}}^{\star }\chi_{\text{A}}}\) (just \(\displaystyle p\) because only the solvent is vaporized)
\({\displaystyle \Delta p=p_{\text{A}}^{\star }-p}\)
\({\displaystyle \Rightarrow \Delta p=p_{\text{A}}^{\star }(1-\chi_{\text{A}})=p_{\text{A}}^{\star }\chi_{\text{B}}}\)
In non-ideal solutions, if the solute associates with the solvent, the van 't Hoff factor is introduced as correction.