Choose the correct option for free expansion of an ideal gas under adiabatic condition:

🧠 Free expansion into vacuum — nothing pushes back, no work is done "Free expansion" means the gas expands into a vacuum: $P_{\text{ext}} = 0$. "Adiabatic" means no heat exchange with surroundings.

🗺️ Each variable $q = 0$: adiabatic, by definition. ✓

$w = -P_{\text{ext}} \cdot \Delta V = -0 \cdot \Delta V = 0$: no opposing pressure. ✓

First Law: $\Delta U = q + w = 0$.

For ideal gas, $\Delta U = C_v \Delta T$. Since $\Delta U = 0 \Rightarrow \Delta T = 0$. ✓

Option (d) matches: $q = 0$, $\Delta T = 0$, $w = 0$.

⚠️ Trap: Option (b) tempts students who confuse free expansion with reversible adiabatic expansion. In reversible adiabatic expansion, $w \neq 0$ and $\Delta T < 0$. Free expansion has no opposing force, so $w = 0$.

$\boxed{\text{Answer: (d)}}$