Problem 45 - Chemistry: Principles and Reactions

Question

Earlier civilizations smelted iron from ore by heating it with charcoal from a wood fire: $$ 2 \mathrm{Fe}_{2} \mathrm{O}_{3}(s)+3 \mathrm{C}(s) \longrightarrow 4 \mathrm{Fe}(s)+3 \mathrm{CO}_{2}(g) $$ (a) Obtain an expression for \(\Delta G^{\circ}\) as a function of temperature. Prepare a table of \(\Delta G^{\circ}\) values at \(100-\mathrm{K}\) intervals between \(100 \mathrm{~K}\) and \(500 \mathrm{~K}\) (b) Calculate the lowest temperature at which the smelting could be carried out.

Step-by-Step Solution

Answer

How would you find the lowest temperature at which smelting can be carried out? Answer: The expression for the Gibbs free energy change (\(\Delta G^{\circ}\)) at standard conditions for the smelting reaction as a function of temperature is given by \(\Delta G^{\circ}(T) = \Delta H^{\circ} - T\Delta S^{\circ}\). To find the lowest temperature at which smelting can be carried out, set \(\Delta G^{\circ} = 0\) and solve for the temperature \(T\) as follows: \(T = \frac{\Delta H^{\circ}}{\Delta S^{\circ}}\).

Step 1: Write the given reaction and find its \(\Delta G^{\circ}\) expression

The smelting reaction is given by: $$ 2 \mathrm{Fe}_{2} \mathrm{O}_{3}(s)+3 \mathrm{C}(s) \longrightarrow 4 \mathrm{Fe}(s)+3 \mathrm{CO}_{2}(g) $$ For the given reaction, the expression for the Gibbs free energy change at standard conditions (\(\Delta G^{\circ}\)) can be written as: $$ \Delta G^{\circ} = \Delta H^{\circ} - T\Delta S^{\circ} $$ where \(\Delta H^{\circ}\) is the enthalpy change at standard conditions and \(\Delta S^{\circ}\) is the entropy change at standard conditions.

Step 2: Calculate \(\Delta G^{\circ}\) values at 100 K intervals between 100 K and 500 K

As per the exercise, we assume that \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) are constant and we can use them to calculate \(\Delta G^{\circ}\) at different temperatures. However, the values of \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) are not provided. So, we cannot provide the actual values of \(\Delta G^{\circ}\) at the mentioned temperature intervals. But, we can leave the expression as: $$ \Delta G^{\circ}(T) = \Delta H^{\circ} - T\Delta S^{\circ} $$ where \(T\) is the temperature at which \(\Delta G^{\circ}\) will be calculated.

Step 3: Calculate the lowest temperature of smelting

Since we lack the values of \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\), we cannot calculate the numerical value of the lowest smelting temperature. However, we can provide the general approach to find it. At equilibrium, \(\Delta G = 0\), and the equilibrium constant \(K\) is given by: $$ K = e^{-\frac{\Delta G^{\circ}}{RT}} $$ where \(R\) is the universal gas constant and \(T\) is the temperature. As \(\Delta G\) approaches zero, the reaction approaches equilibrium. To find the lowest temperature for smelting, we can set \(\Delta G^{\circ} = 0\) and solve for the temperature \(T\): $$ 0 = \Delta H^{\circ} - T\Delta S^{\circ} $$ $$ T = \frac{\Delta H^{\circ}}{\Delta S^{\circ}} $$ The lowest smelting temperature can be calculated by finding the quotient of \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\).

Key Concepts

Enthalpy Change

Enthalpy change (\( \Delta H \) ) represents the heat absorbed or released during a chemical reaction at constant pressure. It is a critical component when calculating Gibbs Free Energy since it reflects the total energy change in the system. In the context of the smelting reaction:- Smelting involves breaking bonds of iron oxide (\( \mathrm{Fe}_{2} \mathrm{O}_{3} \) ) and forming new bonds to produce elemental iron (\( \mathrm{Fe} \) ) and carbon dioxide (\( \mathrm{CO}_{2} \) ).- \( \Delta H^{\circ} \) here represents the amount of thermal energy required to break these bonds during the process.When there's a positive enthalpy change, the reaction is endothermic, meaning it requires heat. If negative, it's exothermic and releases heat. In real-world practices like iron smelting, understanding \( \Delta H \) is crucial for determining energy efficiency and the necessary conditions to drive the reaction forward.

Entropy Change

The entropy change (\( \Delta S \) ) measures the disorder or randomness in a chemical system. In the smelting reaction:- The conversion of solid reactants to a gaseous product increases disorder, typically leading to a positive \( \Delta S \) .- Specifically, moving from \( 2 \mathrm{Fe}_{2} \mathrm{O}_{3}(s) \) and \( 3\mathrm{C}(s) \) to \( 4\mathrm{Fe}(s) \) and \( 3\mathrm{CO}_{2}(g) \) results in higher entropy due to the formation of gas molecules.Entropy is a driving force for many reactions. A positive \( \Delta S \) signifies increased spontaneity, favoring product formation at high temperatures. The interplay between \( \Delta H \) and \( \Delta S \) directly impacts \( \Delta G \) , dictating whether a reaction proceeds under given conditions or requires adjustments.

Smelting Reaction

The smelting reaction is a fundamental process used historically and industrially to extract metal from its ores. Here, the extraction of iron from iron ore:- Involves heating \( \mathrm{Fe}_{2} \mathrm{O}_{3} \) with carbon, often derived from charcoal, to produce elemental iron and carbon dioxide.- This process, simplified in the equation \[ 2 \mathrm{Fe}_{2} \mathrm{O}_{3}(s) + 3 \mathrm{C}(s) \rightarrow 4\mathrm{Fe}(s) + 3\mathrm{CO}_{2}(g) \] , requires overcoming the energy barriers through elevated temperatures.Smelting is energy-intensive, reflected in the significant roles of \( \Delta H \) and \( \Delta S \) in calculating Gibbs Free Energy (\( \Delta G \) ). A lower \( \Delta G \) suggests a more efficient reaction, indicating the least temperature needed for processes. Historical methods relied on optimizing conditions where \( \Delta G \) reaches zero, allowing iron to be economically and sustainably extracted.