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α = i siαi, where si is the stoichiometric coefficient of ion i in the salt. The Born energy will simply be added to the Debye-Huckel excess chemical potential and our new formula will be ex (zie)2 (zie)2 1 1 μi =kBTlnγi=−8πεε(a+λ)+8πεa ε −ε (6.3) 0c0cw To our knowledge this formula has not been tested in literature; however, for our purposes it will simply serve as a reference for how the following somewhat more complex model performs. 6.2.2 Modified Debye-Huckel Activity Formula The mathematical model introduced here is very similar in derivation to that of the Debye-Huckel activity formula. In essence, these activity models are based on a statistical description of electrolytes or ionic liquids to allow the description of a macroscopic phenomenon (i.e. activity coefficient) based on microscopic/local ion- ion interactions. It is important to note that the important length scale below which discrete ion-ion interactions become important is the Bjerrum length in typical dilute electrolytes and the ion size in very densely charged systems (i.e. ionic liquids). The Bjerrum length is the length scale at which the thermal fluctuation energy of a charge is balanced by the bare Coulomb interaction energy between two charges. e2 e2 4πεεl ∼kBT⇒lB=4πεεkT (6.4) 0cB 0cB At length scales much larger than the Bjerrum length (0.7 nm in water at room temperature) use of the ’mean-field approximation’ to describe the diffuse part of the double layer (at least at low salt concentrations) is justified, since thermal fluctuations are strong enough to overcome ion-ion Coulomb forces and the Debye length is much larger than the Bjerrum length. To get a statistical description of the electrolyte, we simply need to calculate the probabilities of finding different local ionic configurations. If we consider a solution 143PDF Image | Shock Electrodialysis for Water Purification and Electrostatic Correlations
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