To determine the correct thermodynamic conditions for a spontaneous reaction at all temperatures, we need to consider the Gibbs free energy change $(\Delta G). \\ \\$The Gibbs free energy change is given by the equation:$\\ \Delta G = \Delta H - T\Delta S \\ \\$For a reaction to be spontaneous at all temperatures, $\Delta G$ must be negative.$\\ \\$Let's analyze the conditions:$\\$• If $\Delta H < 0$ and $\Delta S > 0: \\ \quad$Both terms contribute to making $\Delta G$ negative, as $\Delta H$ is negative and $-T\Delta S$ is also negative.$\\ \quad$Thus, $\Delta G$ is negative at all temperatures. This condition is correct for spontaneity at all temperatures.$\\ \\$• If $\Delta H < 0$ and $\Delta S < 0: \\ \quad$At low temperatures, $\Delta G$ could be negative, but at high temperatures, $-T\Delta S$ becomes positive, potentially making $\Delta G$ positive.$\\ \quad$This condition does not guarantee spontaneity at all temperatures.$\\ \\$• If $\Delta H < 0$ and $\Delta S = 0: \\ \quad$Here, $\Delta G = \Delta H$, which is negative, so the reaction is spontaneous. However, this condition is not as general as the first one.$\\ \\$• If $\Delta H > 0$ and $\Delta S < 0: \\ \quad$Both terms contribute to making $\Delta G$ positive, so the reaction is non-spontaneous at all temperatures.$\\ \\$Therefore, the correct condition for a spontaneous reaction at all temperatures is:$\\ \Delta H < 0$ and $\Delta S > 0 \\ \\$This corresponds to Option 1.