1. Write the balanced chemical equation:

CO₂(g) + 4H₂(g) → CH₄(g) + 2H₂O(g)

2. Determine the mole ratio between hydrogen gas (H₂) and water vapor (H₂O):

From the balanced equation, we see that 4 moles of H₂ react to produce 2 moles of H₂O.

3. Calculate the moles of water vapor:

* Molar mass of water (H₂O): 18.015 g/mol

* Moles of water vapor: (30.6 g) / (18.015 g/mol) = 1.699 mol

4. Calculate the moles of hydrogen gas:

* Use the mole ratio: (1.699 mol H₂O) * (4 mol H₂ / 2 mol H₂O) = 3.398 mol H₂

Therefore, you would need 3.398 moles of hydrogen gas to react with excess carbon dioxide and produce 30.6 grams of water vapor.