\begin{equation} \Delta G_{mez}=G-G^{\ast}=\sum_{i}n_i\underbrace{(\mu_i-\mu_{i}^{\ast})}_{RTlnx_i}=RT \sum_{i}lnx_i \end{equation}

$\Delta G_{mez}<0$ for a spontaneous and irreversible process.

\begin{equation} \Delta V_{mez}=\left(\frac{\partial\Delta G_{mez}}{\partial P}\right)_{n_j,P}=0 \end{equation}

At no depend $\Delta G_{mez}$ on the pressure $\Delta V_{mez}$ is null. When mixing two components that form an ideal solution, there is no increase or decrease in volume with respect to the pure components.

\begin{equation} \Delta S_{mez}=-\left(\frac{\partial\Delta G_{mez}}{\partial T}\right)_{n_j,T}=-R\sum_{i} n_ilnx_i \end{equation}

$\Delta S_{mez}$ is usually positive since when the components are mixed the system becomes disordered, increasing its entropy.

\begin{equation} \Delta G_{mez}=\Delta H_{mez}-T\Delta S_{mez} \end{equation}

From this equation we conclude that, $\Delta H_{mez}=0$

\begin{equation} \Delta H_{mix}=\Delta U_{mix}+P\Delta V_{mix} \end{equation}

Given that $\Delta H_{mix}$ and $\Delta V_{mix}$ are zero, we conclude that $\Delta U_{mez}=0$