To identify the net ionic equation for the reaction, we need to focus only on the species that undergo a chemical change. In other words, we will eliminate the spectator ions, which are ions that appear unchanged on both sides of the equation.
Let's start with the total ionic equation:
[tex]\[ Ba^{2+} + 2 NO_3^{-} + 2 Na^{+} + CO_3^{2-} \rightarrow BaCO_3 + 2 Na^{+} + 2 NO_3^{-} \][/tex]
To find the spectator ions, look for the ions that appear in the same form on both sides of the equation. Here, [tex]\(2 NO_3^{-}\)[/tex] and [tex]\(2 Na^{+}\)[/tex] appear on both sides without changing. This means these ions are spectators and can be eliminated from the total ionic equation.
After removing the spectator ions, we get the net ionic equation:
[tex]\[ Ba^{2+} + CO_3^{2-} \rightarrow BaCO_3 \][/tex]
So, the correct net ionic equation for the reaction is:
[tex]\[ Ba^{2+} + CO_3^{2-} \rightarrow BaCO_3 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{Ba^{2+} + CO_3^{2-} \rightarrow BaCO_3} \][/tex]
