1. Given process: free expansion of an ideal gas under adiabatic condition

   We analyze each thermodynamic quantity:

   • Adiabatic condition means no heat exchange: $q=0$

   • Free expansion means expansion against zero external pressure, so work done is: $w=-P_{\text{ext}}\Delta V=0$ since $P_{\text{ext}}=0$.

2. Apply the first law of thermodynamics

   $\Delta U=q+w$

   Substituting the above values: $\Delta U=0+0=0$

3. For an ideal gas

   Internal energy depends only on temperature. Therefore, $\Delta U=nC_V\Delta T$

   Since $\Delta U=0$, we get $\Delta T=0$

4. Conclusion

   For free expansion of an ideal gas under adiabatic conditions: $q=0,\quad w=0,\quad \Delta T=0$

5. Check options

   • A: $q=0,\ \Delta T=0,\ w=0$ ✅
   • B: $q=0,\ \Delta T\neq 0,\ w=0$ ❌
   • C: $q\neq 0,\ \Delta T=0,\ w=0$ ❌
   • D: $q=0,\ \Delta T<0,\ w\neq 0$ ❌

Hence, the correct option is A.