1. Write the balanced chemical equation:

4 NH₃(g) + 5 O₂(g) → 4 NO(g) + 6 H₂O(g)

2. Determine the mole ratio between ammonia (NH₃) and oxygen (O₂):

The balanced equation shows that 4 moles of NH₃ react with 5 moles of O₂.

3. Calculate the moles of oxygen needed:

* Set up a proportion: (5 mol O₂) / (4 mol NH₃) = (x mol O₂) / (10.0 mol NH₃)

* Solve for x: x = (5 mol O₂ * 10.0 mol NH₃) / 4 mol NH₃ = 12.5 mol O₂

Therefore, 12.5 moles of oxygen gas are needed to react with 10.0 moles of ammonia.