Q.The thermal decomposition of calcium carbonate, $\text{CaCO}_3(s) \rightarrow \text{CaO}(s) + \text{CO}_2(g)$, has $\Delta H^\circ = +178\ \text{kJ mol}^{-1}$ and $\Delta S^\circ = +160\ \text{J K}^{-1}\text{mol}^{-1}$. Calculate the minimum temperature above which this decomposition becomes spontaneous, assuming $\Delta H^\circ$ and $\Delta S^\circ$ do not vary with temperature.

Gibbs free energy, $G$, is a state function defined as $G = H - TS$, constructed specifically to combine a system's enthalpy and entropy changes into a single quantity, calculable from properties of the system alone, whose sign directly predicts spontaneity at constant temperature and pressure — without requiring the surroundings' entropy change to be tracked separately, as the Second Law otherwise demands. For a process at constant $T$ and $P$:

This relation is derived directly from the Second Law, using $\Delta S_{surroundings} = -\Delta H_{system}/T$ (since the heat released by the system at constant $T,P$ is exactly the heat absorbed by the surroundings) substituted into $\Delta S_{total}=\Delta S_{system}+\Delta S_{surroundings}\geq 0$, then multiplied through by $-T$.

The resulting criterion is the most practically useful spontaneity rule in chemical thermodynamics: $\Delta G < 0$ signals a spontaneous (forward) process; $\Delta G > 0$ signals a non-spontaneous forward process (the reverse process is instead spontaneous); and $\Delta G = 0$ signals equilibrium. Because $\Delta G$ combines an enthalpy term and a temperature-scaled entropy term, its sign can be temperature-dependent whenever $\Delta H$ and $\Delta S$ share the same sign: a reaction with $\Delta H>0,\ \Delta S>0$ (such as the thermal decomposition of calcium carbonate) is non-spontaneous at low temperature but turns spontaneous above the threshold temperature $T=\Delta H/\Delta S$, where $\Delta G$ crosses zero. …