In this work, a magnetic resonance (MR) imaging method for accelerating the acquisition time of two dimensional concentration maps of different chemical species in mixtures by the use of compressed sensing (CS) is presented. Whilst 2D-concentration maps with a high spatial resolution are prohibitively time-consuming to acquire using full k-space sampling techniques, CS enables the reconstruction of quantitative concentration maps from sub-sampled k-space data. First, the method was tested by reconstructing simulated data. Then, the CS algorithm was used to reconstruct concentration maps of binary mixtures of 1,4-dioxane and cyclooctane in different samples with a field-of-view of 22mm and a spatial resolution of 344μm×344μm. Spiral based trajectories were used as sampling schemes. For the data acquisition, eight scans with slightly different trajectories were applied resulting in a total acquisition time of about 8min. In contrast, a conventional chemical shift imaging experiment at the same resolution would require about 17h. To get quantitative results, a careful weighting of the regularisation parameter (via the L-curve approach) or contrast-enhancing Bregman iterations are applied for the reconstruction of the concentration maps. Both approaches yield relative errors of the concentration map of less than 2mol-% without any calibration prior to the measurement. The accuracy of the reconstructed concentration maps deteriorates when the reconstruction model is biased by systematic errors such as large inhomogeneities in the static magnetic field. The presented method is a powerful tool for the fast acquisition of concentration maps that can provide valuable information for the investigation of many phenomena in chemical engineering applications.