  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 data
for low and moderate salt concentration. However, Debye and H\"uckel (DH) later
developed a theory that prevailed, and this theory has a square root
dependence. The derivation of the DH equation is not easy or intuitive, and has
various uncertain elements, and the fit to data is not very good for salts with
non-unity valencies. We develop a model for the activity coefficient based on
the Coulombic forces between an ion and the nearest ions of opposite charge,
including the distribution of separations between these ions. For a symmetric
salt, we only have to analyse the interactions between one anion and one
cation, and we can derive an expression for the dilute limit that depends on
the Bjerrum length and the cube root of salt concentration, the same as an
expression put forward by Bjerrum. Our theory also analytically describes
higher salt concentrations and a non-zero ion size, not only for 1:1 but also
for 2:2 and 3:3 salts. For a 1:1 salt, the cube root scaling law describes data
of the activity coefficient very well, both at low and intermediate salt
concentrations, while the extended Bjerrum equation describes data even better
and up to higher concentrations, and also includes the effect of ion size. For
asymmetric salts (2:1 and 3:1 salts), a numerical procedure can be used which
considers all possible orbits of two or three monovalent ions around a central
multivalent ion of opposite charge, which also fits data accurately.
