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The activity coefficient of ions in solution was proposed by Bjerrum (1916)to depend on the cube root of concentration, because of a good fit with datafor low and moderate salt concentration. However, Debye and H\"uckel (DH) laterdeveloped a theory that prevailed, and this theory has a square rootdependence. The derivation of the DH equation is not easy or intuitive, and hasvarious uncertain elements, and the fit to data is not very good for salts withnon-unity valencies. We develop a model for the activity coefficient based onthe Coulombic forces between an ion and the nearest ions of opposite charge,including the distribution of separations between these ions. For a symmetricsalt, we only have to analyse the interactions between one anion and onecation, and we can derive an expression for the dilute limit that depends onthe Bjerrum length and the cube root of salt concentration, the same as anexpression put forward by Bjerrum. Our theory also analytically describeshigher salt concentrations and a non-zero ion size, not only for 1:1 but alsofor 2:2 and 3:3 salts. For a 1:1 salt, the cube root scaling law describes dataof the activity coefficient very well, both at low and intermediate saltconcentrations, while the extended Bjerrum equation describes data even betterand up to higher concentrations, and also includes the effect of ion size. Forasymmetric salts (2:1 and 3:1 salts), a numerical procedure can be used whichconsiders all possible orbits of two or three monovalent ions around a centralmultivalent ion of opposite charge, which also fits data accurately.

Derivation of an extended Bjerrum equation for the activity coefficient of ions based on an analysis of Coulombic forces