- Identify the reaction as endothermic (\(\Delta H > 0\)) with an increase in entropy (\(\Delta S > 0\)).
- Apply the Gibbs free energy equation \(\Delta G = \Delta H - T \Delta S\), noting that the process becomes spontaneous when \(\Delta G < 0\).
- Recognize that as temperature \(T\) increases, the \(-T \Delta S\) term becomes more negative, facilitating the reaction.
- Conclude that temperature is the most important factor in driving this non-spontaneous endothermic process, thus the correct option is \(\boxed{\text{b}}\).
Explanation
- Analyzing the Reaction Thermodynamics
The decomposition reaction of magnesium oxide is represented by the equation:
\[MgO(s) + 601.7 \text{ kJ} \rightarrow Mg(s) + \frac{1}{2} O_2(g)\]
From the chemical equation, we observe that the reaction is endothermic, as energy is listed as a reactant (\(\Delta H > 0\)). Additionally, there is a clear increase in entropy (\(\Delta S > 0\)) as the system transitions from a solid state to produce a gaseous product.
- The Role of Temperature in Spontaneity
The spontaneity of a reaction at constant temperature and pressure is determined by the Gibbs free energy change, defined by the equation:
\[\Delta G = \Delta H - T \Delta S\]
For a reaction to be spontaneous, the Gibbs free energy change must be negative (\(\Delta G < 0\)). Since this reaction is endothermic (\(\Delta H > 0\)), the term \(-T \Delta S\) must be sufficiently negative to overcome the positive \(\Delta H\) value. Given that \(\Delta S > 0\), increasing the temperature \(T\) makes the \(-T \Delta S\) term more negative, effectively lowering the overall \(\Delta G\). Therefore, temperature is the primary factor that dictates whether the energy requirements for the reaction can be met and the process rendered spontaneous.
- Conclusion on the Determining Factor
Surface area, pressure, and concentration can affect the rate of a reaction or the position of equilibrium, but they do not fundamentally alter the thermodynamic feasibility (the spontaneity) of a reaction in the same way temperature does for this specific endothermic process. The requirement to supply enough energy to overcome the enthalpy barrier (\(\Delta H\)) necessitates a sufficiently high temperature to change the sign of \(\Delta G\). Thus, temperature is the critical variable.
Examples
In industrial metallurgy, temperatures often exceeding thousands of degrees are required to extract magnesium from its oxide. Engineers rely on the thermodynamic principle that at high enough temperatures, the entropy gain from producing oxygen gas offsets the energy required to break the magnesium-oxygen bond, enabling the production of pure metal. This illustrates how understanding thermodynamic factors allows us to manipulate conditions to force non-spontaneous processes to occur.