Free expansion of a gas occurs when it expands into a vacuum, where the external pressure ($p_{ext}$) is zero. The work done during expansion is given by the formula $W = -p_{ext} \Delta V$. Since $p_{ext} = 0$, the work done $W = 0$ . Since the condition is given as adiabatic, it means there is no heat exchange between the system and its surroundings, so $q = 0$ . According to the First Law of Thermodynamics, the change in internal energy is given by $\Delta U = q + W$. Substituting the values, we get $\Delta U = 0 + 0 = 0$ . For an ideal gas, the change in internal energy is a function of temperature only ($\Delta U = nC_v\Delta T$). Since $\Delta U = 0$, it implies that the change in temperature $\Delta T$ is also $0$ .