Solution
Step 1: Find Temperature of the Room (Air)
Given RMS speed of air molecules $v_{rms} = 500 \text{ m/s}$ and Molar mass $M_{air} = 29 \text{ g/mol} = 0.029 \text{ kg/mol}$.
$$ v_{rms} = \sqrt{\frac{3RT}{M}} \implies T = \frac{M v_{rms}^2}{3R} $$
$$ T = \frac{0.029 \times (500)^2}{3R} = \frac{0.029 \times 250000}{3R} $$
Step 2: Internal Energy of Hydrogen
Since the balloon is in thermal equilibrium, $T_{hydrogen} = T_{air}$.
Hydrogen is a diatomic gas ($f=5$). Mass $m = 1.0 \text{ g}$. Moles $n = \frac{1}{2} = 0.5 \text{ mol}$.
Internal Energy $U = \frac{f}{2} n R T$.
Substitute $RT$ from Step 1:
$$ U = \frac{5}{2} (0.5) \left( \frac{0.029 \times 250000}{3} \right) $$
$$ U = \frac{5}{4} \times \frac{7250}{3} = \frac{36250}{12} \approx 3020.8 \text{ J} $$
Correct Option: (b) 3021 J