Consider a system formed by two phases, liquid and gas. The composition of the liquid phase is given by the mole fractions $x_i$ and that of the gas phase by $x_{iv}$. The partial pressures exerted by the components in the gas phase are $P_i$.
Once equilibrium is reached, the chemical potential of each component in the liquid and vapor phases equalize. \begin{equation} \mu_{i,v}=\mu_{i,l} \end{equation} We write the chemical potential of a component $i$ in the vapor phase, assuming that it behaves ideally.
\begin{equation} \mu_{i,v}=\mu_{i,v}^{0}+RTln\frac{P_i}{P^0} \end{equation}
The chemical potential of a component $i$ in the liquid, assuming ideal solution, it will be given by:
\begin{equation} \mu_{i,l}=\mu_{i(l)}^{\ast}+RTlnx_i \end{equation}
Equating both chemical potentials
\begin{equation} \mu_{i,v}^{0}+RTln\frac{P_i}{P^0}=\mu_{i,l}^{\ast}+RTlnx_i \label{130}\end{equation}
We now consider the equilibrium of the pure liquid with its vapor
\begin{equation} \mu_{i,v}^{\ast}(T,P_{i}^{\ast})=\mu_{i,l}^{\ast} (T,P_{i}^{\ast}) \end{equation}
Since the chemical potential of pure $i$ in the vapor phase is given by:
\begin{equation} \mu_{i,v}^{\ast}(T,P_{i}^{\ast})=\mu_{i,v}^{0}(T)+RTln\frac{P_{i}^{\ast}}{P^0} \end{equation}
Substituting
\begin{equation} \mu_{i,v}^{0}(T)+RTln\frac{P_{i}^{\ast}}{P^0}=\mu_{i ,l}^{\ast}(T,P_{i}^{\ast})\label{131} \end{equation}
Since chemical potential changes little with pressure for liquids, $\mu_{i,l} ^{\ast}(T,P_{i}^{\ast})=\mu_{i,l}^{\ast}(T,P)$, substituting (\ref{131}) into equation (\ref{130})
\begin{equation} \cancel{\mu_{i,v}^{0}}+RTln\frac{P_i}{P^0}=\cancel{\mu_{i,v}^{ 0}}+RTln\frac{P_{i}^{\ast}}{P^0}+RTlnx_i \end{equation}
Simplifying the normal potentials in the gas phase and the RT we get:
\begin{equation} ln\frac{P_i}{P^0}-ln\frac{P_{i}^{\ast}}{P_0}=lnx_i \end{equation}
Applying properties of natural logarithms
\begin{equation} ln\frac{P_i} {P_{i}^{\ast}}=lnx_i \end{equation}
Taking the inverse of the Neperian and solving for the pressure, gives us:
\begin{equation} P_i=x_iP_{i}^{\ast} \end{equation}
This last equation is known as Raoult's law
$P_i$: Vapor pressure of component i.
$P_{i}^{\ast}$: Vapor pressure of pure component i.
$x_i$: Mole fraction of component i in the solution.