At what temperatures is the following reaction, the reduction of magnetite by graphite to elemental iron, spontaneous? Fe₃O₄(s) + 2 C(s, graphite) → 2 CO₂(g) + 3 Fe(s)

Verified step by step guidance

1. Identify the reaction and write it down in a balanced chemical equation format: \( \text{Fe}_3\text{O}_4(s) + 2\text{C}(s, \text{graphite}) \rightarrow 2\text{CO}_2(g) + 3\text{Fe}(s) \).
2. Determine the standard Gibbs free energy change (\( \Delta G^\circ \)) for the reaction using standard Gibbs free energies of formation for each compound involved in the reaction.
3. Use the equation \( \Delta G = \Delta G^\circ + RT \ln Q \) to relate the Gibbs free energy change at any temperature (T) and reaction quotient (Q). For spontaneity, \( \Delta G \) should be less than zero.
4. Since the reaction involves only solids and gases at standard state, the reaction quotient (Q) simplifies to 1 under standard conditions. Thus, the spontaneity depends solely on \( \Delta G^\circ \).
5. Calculate the temperature at which \( \Delta G^\circ \) changes sign using the relationship \( \Delta G^\circ = -RT \ln K \), where K is the equilibrium constant. The reaction is spontaneous at temperatures where \( \Delta G^\circ < 0 \).

Key Concepts

Gibbs Free Energy

Gibbs Free Energy (G) is a thermodynamic potential that helps predict the spontaneity of a reaction at constant temperature and pressure. A reaction is spontaneous when the change in Gibbs Free Energy (ΔG) is negative. The relationship between ΔG, enthalpy (ΔH), and entropy (ΔS) is given by the equation ΔG = ΔH - TΔS, where T is the temperature in Kelvin.

Entropy and Enthalpy

Entropy (S) is a measure of the disorder or randomness in a system, while enthalpy (H) is a measure of the total energy of a system, including internal energy and the energy required to make room for it. In the context of a reaction, an increase in entropy (positive ΔS) often favors spontaneity, especially at higher temperatures, while a decrease in enthalpy (negative ΔH) also promotes spontaneity.

Temperature's Role in Spontaneity

Temperature plays a crucial role in determining the spontaneity of a reaction through its influence on the Gibbs Free Energy equation. As temperature increases, the TΔS term becomes more significant, which can make a reaction spontaneous if ΔS is positive. Therefore, understanding how temperature affects the balance between enthalpy and entropy is essential for predicting the spontaneity of the reduction of magnetite by graphite.