Whenever describing purely diffusive behavior in multicomponent mixtures, the Maxwell-Stefan equations pose an important framework for various engineering applications, which include polymers, plasmas, ultrafiltration, electrolysis and even diffusion processes in the human lung. In this thesis, the link between molecular diffusion and the continuum-mechanical description in the Maxwell-Stefan equation is made plausible by following the recent exposition by Boudin, which illustrates the physical assumptions under which the Maxwell-Stefan equations pose a valid model for describing multi-component diffusion on a macroscopic scale. In addition, several current efforts of generalizing some of the simplifications made in the model are stated. Furthermore, an important result for the existence of weak solutions to the Maxwell-Stefan equations (as presented by Jüngel and Stelzer) is discussed in detail, by hinting links to the more general framework of the "Boundedness-By-Entropy'' method developed by Jüngel. Finally, a new conforming lowest-order Finite Element discretization in space and a semi-implicit Euler discretization in time is discussed, which employs an "entropy-variable'' formulation of the Maxwell-Stefan equations using techniques from Jüngel. For this purpose, Python code employing this method has been written to solve 1D and 2D problems even on complex polygonal geometries. The performance of the code is then investigated by a benchmark, which employs the Method of Manufactured Solutions, yielding a novel model problem with a given analytic solution.