Net Ionic Equation of Four Solutions

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The problem statement

The solutions of ##NaC_2H_3O_3 + BaCl_2 +NH_4NO_3 + K_2CrO_4## are mixed together.

1. Write the Net Ionic Equation.

2. What is the driving force of the reaction?

The attempt

I'm not sure how to solve for the net ionic equation without a balanced equation. My professor showed us a way that involves only finding the ions and discarding the unneeded ones, but I haven't yet grasped the concept.


As of now, I have the following net ionic equation: $$Na{_{}}^{+}+C_2H_3O_3{_{}}^{-}+CrO_4{_{}}^{2-}\rightarrow Na_2CrO_{4(s)}+H_2O_{(l)}+CO_{2(g)}$$

Assuming this NIE is correct (I'm sure it isn't), then the driving forces would be all three products.

Any insight into the proper approach to this and similar problems would be greatly appreciated.
 
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Hint: think about solubility rules (sodium chromate is well soluble, so it is not a product here).

Most of the ions are just spectators.