Because Al is trivalent and Mg is divalent, the excess charge associated with the dissolved Al must be compensated by the formation of oppositely charged Mg vacancies. This is an energetically costly process and, therefore, very little alumina will dissolve in magnesia. In fact, it is generally the case that the aliovalent impurities have low solubilities in ceramics. There are, however, interesting and noteworthy exceptions including CaO•ZrO₂ and Bi₂O₃•Nb₂O₅. In the latter case, Bi₂O₃ dissolves 22.5 m/o Nb₂O₅. The goal of this experiment is determine how the excess charge associated with the dissolved pentavalent Nb is compensated in Bi₂O₃. This problem was originally solved experimentally by Takahashi et al.; the original account of their work can be found at: T. Takahashi, H. Iwahara, and T. Esaka, "High Oxide Ion Conduction in Sintered Oxide of the System Bi2O3-M2O5" J. Electrochem. Soc. 124 [10] (1977) 1563-1569.