To determine the net ionic equation, we need to identify the ions that actually participate in the chemical reaction and exclude the spectator ions, which do not change during the reaction.
### Step-by-Step Solution:
1. Write the Given Total Ionic Equation:
[tex]\[
Ba^{2+} + 2 NO_3^{-} + 2 Na^{+} + CO_3^{2-} \longrightarrow BaCO_3 + 2 Na^{+} + 2 NO_3^{-}
\][/tex]
2. Identify Spectator Ions:
- Spectator ions are those ions that appear on both sides of the equation unchanged.
- In this case, the spectator ions are [tex]\( NO_3^{-} \)[/tex] (nitrate) and [tex]\( Na^{+} \)[/tex] (sodium).
3. Remove the Spectator Ions:
- Remove [tex]\( 2 NO_3^{-} \)[/tex] and [tex]\( 2 Na^{+} \)[/tex] from both sides of the equation.
- This leaves us with the participating ions and the formation of the precipitate:
[tex]\[
Ba^{2+} + CO_3^{2-} \longrightarrow BaCO_3
\][/tex]
4. Write the Net Ionic Equation:
- The net ionic equation is the simplified form showing only the ions and the compounds that participate in the reaction:
[tex]\[
Ba^{2+} + CO_3^{2-} \longrightarrow BaCO_3
\][/tex]
From the options provided, the correct net ionic equation is:
[tex]\[
Ba^{2+} + CO_3^{2-} \longrightarrow BaCO_3
\][/tex]
Thus, the correct answer is the net ionic equation:
[tex]\[ \boxed{Ba^{2+} + CO_3^{2-} \longrightarrow BaCO_3} \][/tex]
