To solve this problem, we need to understand the concept of free expansion of an ideal gas under adiabatic conditions.$\\$In a free expansion:$\\$• The gas expands into a vacuum without doing any work on the surroundings.$\\$• There is no heat exchange with the surroundings, as it is adiabatic.$\\$Let's analyze the given options:$\\$Option 1: $q=0, \Delta T < 0$ and $w > 0 \\$• $q=0$ is correct because it is adiabatic (no heat exchange).$\\$• $\Delta T < 0$ is incorrect because in free expansion, the temperature does not change.$\\$• $w > 0$ is incorrect because no work is done in free expansion.$\\$Option 2: $q < 0, \Delta T=0$ and $w=0 \\$• $q < 0$ is incorrect because $q=0$ in adiabatic processes.$\\$• $\Delta T=0$ is correct as there is no change in temperature.$\\$• $w=0$ is correct because no work is done.$\\$Option 3: $q > 0, \Delta T > 0$ and $w > 0 \\$• $q > 0$ is incorrect because $q=0$ in adiabatic processes.$\\$• $\Delta T > 0$ is incorrect because there is no change in temperature.$\\$• $w > 0$ is incorrect because no work is done.$\\$Option 4: $q=0, \Delta T=0$ and $w=0 \\$• $q=0$ is correct because it is adiabatic (no heat exchange).$\\$• $\Delta T=0$ is correct as there is no change in temperature.$\\$• $w=0$ is correct because no work is done.$\\$Therefore, the correct option is Option 4: $q=0, \Delta T=0$ and $w=0.$