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

Determine the correct thermodynamic quantities for free expansion of an ideal gas under adiabatic conditions.

In an adiabatic process, no heat is exchanged with the surroundings, so $$q = 0$$.

Free expansion occurs when a gas expands into a vacuum (zero external pressure). The work done by the gas is given by $$w = \int P_{\text{ext}} dV$$, and since $$P_{\text{ext}} = 0$$, it follows that $$w = 0$$.

Applying the first law of thermodynamics, $$\Delta U = q + w = 0 + 0 = 0$$.

For an ideal gas, internal energy depends only on temperature: $$\Delta U = nC_v\Delta T$$. Setting $$\Delta U = 0$$ yields $$nC_v\Delta T = 0 \implies \Delta T = 0$$ (since $$C_v \neq 0$$ and $$n \neq 0$$).

Therefore, $$q = 0$$, $$\Delta T = 0$$, and $$w = 0$$.

The correct answer is Option D: $$q = 0, \Delta T = 0, w = 0$$.