The balanced chemical equation for the reaction involving BaSO4, H2O, Cu, Cl, NaOH, and Mg$NO3$2 is as follows:

BaSO₄ + 2H₂O + CuCl₂ + 2NaOH + Mg$N{O}_3$₂ → Ba$OH$₂ + Cu$OH$₂ + 2NaCl + MgSO₄