We describe an experimental technique to measure the chemical potential, $mu$, in atomically thin layered materials with high sensitivity and in the static limit. We apply the technique to a high quality graphene monolayer to map out the evolution of $mu$ with carrier density throughout the N=0 and N=1 Landau levels at high magnetic field. By integrating $mu$ over filling factor, $ u$, we obtain the ground state energy per particle, which can be directly compared with numerical calculations. In the N=0 Landau level, our data show exceptional agreement with numerical calculations over the whole Landau level without adjustable parameters, as long as the screening of the Coulomb interaction by the filled Landau levels is accounted for. In the N=1 Landau level, comparison between experimental and numerical data reveals the importance of valley anisotropic interactions and the presence of valley-textured electron solids near odd filling.
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