1. Write the balanced chemical equation:

3 NO₂(g) + H₂O(l) → 2 HNO₃(aq) + NO(g)

2. Determine the mole ratio between nitrogen dioxide (NO₂) and nitrogen monoxide (NO):

From the balanced equation, we see that 3 moles of NO₂ react to produce 1 mole of NO.

3. Convert molecules of NO to moles:

* Avogadro's number: 6.022 x 10²³ molecules/mole

* Moles of NO = (5.00 x 10²² molecules) / (6.022 x 10²³ molecules/mole) = 0.0830 moles NO

4. Calculate moles of NO₂ needed:

* Moles of NO₂ = (0.0830 moles NO) * (3 moles NO₂ / 1 mole NO) = 0.249 moles NO₂

5. Convert moles of NO₂ to grams:

* Molar mass of NO₂ = 14.01 g/mol (N) + 2 * 16.00 g/mol (O) = 46.01 g/mol

* Grams of NO₂ = (0.249 moles NO₂) * (46.01 g/mol) = 11.4 g NO₂

Therefore, 11.4 grams of nitrogen dioxide must react with water to produce 5.00 x 10²² molecules of nitrogen monoxide.