The energy cost is zero. In desalination, you take the ions that were in one body of water and move them to a different body of water, but the energy is the same no matter where the ions are. Desalination is about paying the entropy cost, not the energy cost. We imagine the desalination process part-way complete. On the left we have the ocean with a constant concentration c 0 c 0 of ions. On the right we have water we're trying to desalinate, which currently has concentration c c of ions. How much does the entropy change if we move an ion from the right to the left? You may know the formula for entropy, S = ln Ω S = ln Ω where Ω Ω is the number of possible states of the system. We need a way to estimate Ω Ω . Imagine each ion living in its own little house, with all the houses jammed up next to each other to take up all the space in the water. When the concentration of ions is low the houses are big and when the concentration is high the houses are small. The volume of