1. **State the problem:** We need to find the volume occupied by one mole of hydrogen gas at temperature $290\,K$ and pressure $2.0\,atm$. 2. **Formula used:** We use the Ideal Gas Law: $$PV = nRT$$ where: - $P$ is pressure, - $V$ is volume, - $n$ is number of moles, - $R$ is the ideal gas constant, - $T$ is temperature. 3. **Known values:** - $n = 1$ mole, - $T = 290\,K$, - $P = 2.0\,atm$, - $R = 0.0821\,L\cdot atm/(mol\cdot K)$ (common value for these units), - Initial volume at standard conditions (1 mole, 273 K, 1 atm) is $22.4\,L$. 4. **Calculate volume $V$:** Rearrange the ideal gas law to solve for $V$: $$V = \frac{nRT}{P}$$ 5. **Substitute values:** $$V = \frac{1 \times 0.0821 \times 290}{2.0}$$ 6. **Calculate numerator:** $$1 \times 0.0821 \times 290 = 23.809$$ 7. **Calculate volume:** $$V = \frac{23.809}{2.0}$$ 8. **Simplify fraction:** $$V = \cancel{\frac{23.809}{2.0}} = 11.9045\,L$$ **Final answer:** The volume occupied by the hydrogen gas sample is approximately $11.9\,L$.