Here's how to determine the net ionic equation for the precipitation reaction between silver nitrate (AgNO₃) and sodium sulfide (Na₂S):

1. Write the balanced molecular equation:

2 AgNO₃(aq) + Na₂S(aq) → Ag₂S(s) + 2 NaNO₃(aq)

2. Write the complete ionic equation:

This involves breaking down all soluble ionic compounds into their respective ions:

2 Ag⁺(aq) + 2 NO₃⁻(aq) + 2 Na⁺(aq) + S²⁻(aq) → Ag₂S(s) + 2 Na⁺(aq) + 2 NO₃⁻(aq)

3. Identify and cancel out spectator ions:

Spectator ions are those that appear on both sides of the equation and do not participate in the reaction. In this case, Na⁺ and NO₃⁻ are spectator ions.

4. Write the net ionic equation:

This is the equation that only includes the ions directly involved in the precipitation reaction:

2 Ag⁺(aq) + S²⁻(aq) → Ag₂S(s)

This net ionic equation represents the formation of the black precipitate, silver sulfide (Ag₂S), from the reaction of silver ions (Ag⁺) and sulfide ions (S²⁻).