Unlike in the single-site case, we now include an (un)cooperativity term that follows from the fact that: 1) the cycle is closed (i.e., K 1 + K 3 = K 2 + K 4 ), and 2) there is an "interaction free energy", w, associated with the second (de)protonation event given the first. Given that we can define the standard free energy of deprotonation as: