Experimentally it is observed that two liquids that are very similar to each other form an ideal solution and present a variation in the Gibbs free energy during the mixing process that is given by:
\begin{equation} \Delta G_{mez}=RT\ sum_{i}n_ilnx_i \label{100}\end{equation}
In the particular case of a binary solution, we are left with:
\begin{equation} \Delta G_{mez}=RT(n_Alnx_A+n_Blnx_B)\label{101} \end{equation}
Chemical potential of an ideal solution
\begin{equation} \Delta G_{mez}=G-G^{\ast}=\sum_{i}n_i\mu_i - \sum_{i}n_i\mu_{i}^{\ast} \end{equation}
Equating this last equation to (\ref{100})
\begin{equation} \sum_{i}n_i\mu_i - \sum_{i}n_i\mu_{i}^{\ast}=RT\sum_{i} n_ilnx_i \end{equation}
Grouping Terms
\begin{equation} \sum_{i}n_i\mu_i=\sum_{i}n_i(\mu_{i}^{\ast}+RTlnx_i) \end{equation}
So that satisfy the equation it is necessary that:
\begin{equation} \mu_i=\mu_{i}^{\ast}+RTlnx_i \end{equation}
In thermodynamics, the rigorous definition of ideal solution is one in which all co Component obeys the equation $\mu_i=\mu_{i}^{\ast}+RTlnx_i$ for all compositions