Summary: We extend the Van der Waals-Cahn-Hilliard theory of phase transitions to the case of a mixture of n non-interacting fluids. By describing the state of the mixture as given by a vector density function \(u=(u_ 1,...,u_ n)\), the problem consists in studying the asymptotic behaviour as \(\epsilon \to 0^+\) of minimizers of the energy functionals: \[ E_{\epsilon}(u)=\int_{\Omega}| \epsilon^ 2| Du|^ 2+W(u)| dx \] under the volume constraint \(\int_{\Omega}u(x)dx=m\), with \(m\in {\mathbb{R}}^ n\) fixed. The function W, which represents the Gibbs free energy, is non-negative and vanishes only in a finite number of points \(\alpha_ 1,...,\alpha_ k\in {\mathbb{R}}^ n\). The result is that the minimizers asymptotically approach a configuration which corresponds to a partition of the container \(\Omega\) into k subsets whose boundaries satisfy a minimality condition.