To solve this problem, we need to calculate the work done by the gas during expansion and use the first law of thermodynamics.$\\$The first law of thermodynamics is given by:$\\ \Delta U = q - W \\$where $\Delta U$ is the change in internal energy, $q$ is the heat exchanged, and $W$ is the work done by the system.$\\$Since the container is well-insulated, it is an adiabatic process, meaning $q = 0. \\$Therefore, the change in internal energy is:$\\ \Delta U = -W \\$The work done by the gas during expansion against a constant external pressure is given by:$\\ W = P_{ext} \cdot \Delta V \\$where $P_{ext}$ is the external pressure and $\Delta V$ is the change in volume.$\\$Given:$\\ P_{ext} = 2.5$ atm$\\ V_{initial} = 2.50$ L$\\ V_{final} = 4.50$ L$\\$Calculate the change in volume:$\\ \Delta V = V_{final} - V_{initial} = 4.50 - 2.50 = 2.00$ L$\\$Convert the pressure from atm to Pa (1 atm = 101325 Pa):$\\ P_{ext} = 2.5 \times 101325 = 253312.5$ Pa$\\$Convert the volume from L to $m^3$ (1 L = $10^{-3}$ m^3):} \\ \Delta V = 2.00 \times 10^{-3} $m^3 \\$Calculate the work done:$\\ W = 253312.5 \times 2.00 \times 10^{-3} \\ W = 506.625$ J$\\$Since $\Delta U = -W,$ the change in internal energy is:$\\ \Delta U = -506.625$ J$\\$Rounding to the nearest whole number, we get:$\\ \Delta U \approx -507$ J$\\$However, considering significant figures and rounding conventions, the closest option is:$\\ \Delta U = -505$ J$\\$Therefore, the correct answer is Option 2: $-505$ J.