A first-principles approach for calculating ion separation in solution through 2D membranes is proposed. Ionic energy profiles across the membrane are obtained first, where solvation effects are explicitly simulated by machine-learning molecular dynamics, electrostatic corrections are applied to remove finite-size capacitive effects, and a mean-field treatment of the electrochemical double layer charging is used. Entropic contributions are assessed analytically and through a thermodynamic integration scheme. Ionic separations are then inferred through a microkinetic model of the filtration process, accounting for steady-state charge separation effects across the membrane. The approach is applied to Li+, Na+, K+ sieving through a crown-ether functionalized graphene membrane, with a case study of the mechanisms for a highly selective and efficient extraction of lithium from aqueous solutions. This record contains the MD trajectories used to generate the energy and free energy profiles of Fig. 4.