To solve this problem, we need to analyze the process that occurs when the stopcock is opened.$\\$Given:$\\$• Cylinder $A$ contains an ideal gas at standard temperature and pressure (STP).$\\$• Cylinder $B$ is completely evacuated.$\\$• The entire system is thermally insulated.$\\$When the stopcock is opened, the gas from cylinder $A$ expands into cylinder $B. \\$Since the system is thermally insulated, there is no heat exchange with the surroundings.$\\$Let's analyze the process:$\\$• The expansion of gas occurs without any heat exchange, which is the definition of an adiabatic process.$\\$• In an adiabatic process, the system is insulated, so $Q = 0$ (no heat transfer).$\\$Therefore, the process is adiabatic.$\\$Correct Option: 1 (Adiabatic)