Let's analyze what happens during a free (or unrestrained) expansion of an ideal gas under adiabatic conditions:

• Adiabatic process : By definition, in an adiabatic process, no heat is transferred between the system and its surroundings. Thus, $ q = 0 $.


• Free expansion : In a free expansion, the gas expands into a vacuum or against zero external pressure. Since work ($ w $) is defined as the product of pressure ($ P $) and volume change ($ \Delta V $), and since the external pressure is zero, we conclude that $ w = 0 $.


• Temperature change ($ \Delta T $) : Since work is zero and no heat is transferred, the first law of thermodynamics tells us that the change in internal energy ($ \Delta U $) is zero as well. For an ideal gas, internal energy is a function of temperature only; thus, if the internal energy does not change, the temperature does not change. Therefore, $ \Delta T = 0 $.

Based on the above points, the free expansion of an ideal gas under adiabatic conditions is characterized by no heat transfer ($ q = 0 $), no work done ($ w = 0 $), and no change in temperature ($ \Delta T = 0 $). Therefore, the correct option is:

Option A : $ q = 0, \Delta T = 0, w = 0 $