Problem 47 - Chemistry: Principles and Reactions

Question

Two possible ways of producing iron from iron ore are (a) \(\mathrm{Fe}_{2} \mathrm{O}_{3}(s)+\frac{3}{2} \mathrm{C}(s) \longrightarrow 2 \mathrm{Fe}(s)+\frac{3}{2} \mathrm{CO}_{2}(g)\) (b) \(\mathrm{Fe}_{2} \mathrm{O}_{3}(s)+3 \mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{Fe}(s)+3 \mathrm{H}_{2} \mathrm{O}(g)\) Which of these reactions proceeds spontaneously at the lower temperature?

Step-by-Step Solution

Answer

To determine which reaction proceeds spontaneously at a lower temperature, follow these steps: 1. Calculate the enthalpy change (\(\Delta H\)) for both reactions using the enthalpy of formation values found in a standard enthalpy table. 2. Calculate the entropy change (\(\Delta S\)) for both reactions using the entropy values found in a standard entropy table. 3. Calculate the Gibbs free energy change (\(\Delta G\)) for both reactions using the formula \(\Delta G = \Delta H - T\Delta S\) and a certain temperature in Kelvin. 4. Compare the \(\Delta G\) values between reactions. The reaction with the lower \(\Delta G\) value at the desired temperature will proceed spontaneously at a lower temperature.

Step 1: Calculate the enthalpy change (\(\Delta H\)) for both reactions

To calculate the enthalpy change, we need to know the enthalpy of formation for each compound involved in the reactions. Using a standard enthalpy table, we can find these values and use them to calculate \(\Delta H\) for each reaction. (a) \(\Delta H_{a} = [2 \times \Delta H_f(\mathrm{Fe}_{(s)}) + \frac{3}{2} \times \Delta H_f(\mathrm{CO}_{2_{(g)}})] - [\Delta H_f(\mathrm{Fe}_2\mathrm{O}_{3_{(s)}}) + \frac{3}{2} \times \Delta H_f(\mathrm{C}_{(s)})]\) (b) \(\Delta H_{b} = [2 \times \Delta H_f(\mathrm{Fe}_{(s)}) + 3 \times \Delta H_f(\mathrm{H}_{2}\mathrm{O}_{(g)})] - [\Delta H_f(\mathrm{Fe}_2\mathrm{O}_{3_{(s)}}) + 3 \times \Delta H_f(\mathrm{H}_{2_{(g)})]\)

Step 2: Calculate the entropy change (\(\Delta S\)) for both reactions

Similarly, for calculating the entropy change, we need the entropy values for each compound involved in the reactions. Using a standard entropy table, we can find these values and use them to calculate \(\Delta S\) for each reaction. (a) \(\Delta S_{a} = [2 \times S_f(\mathrm{Fe}_{(s)}) + \frac{3}{2} \times S_f(\mathrm{CO}_{2_{(g)}})] - [S_f(\mathrm{Fe}_2\mathrm{O}_{3_{(s)}) + \frac{3}{2} \times S_f(\mathrm{C}_{(s)})]\) (b) \(\Delta S_{b} = [2 \times S_f(\mathrm{Fe}_{(s)}) + 3 \times S_f(\mathrm{H}_{2}\mathrm{O}_{(g)})] - [S_f(\mathrm{Fe}_2\mathrm{O}_{3_{(s)}) + 3 \times S_f(\mathrm{H}_{2_{(g)})]\)

Step 3: Calculate the Gibbs free energy change (\(\Delta G\)) for both reactions

Using the calculated enthalpy and entropy changes, we can calculate the Gibbs free energy change for both reactions at a certain temperature. Be sure to convert the temperature into Kelvin. (a) \(\Delta G_{a} = \Delta H_{a} - T\Delta S_{a}\) (b) \(\Delta G_{b} = \Delta H_{b} - T\Delta S_{b}\)

Step 4: Compare the Gibbs free energy change (\(\Delta G\)) between reactions

Finally, we can compare the Gibbs free energy change between reactions. The reaction with the lower \(\Delta G\) value at the desired temperature will proceed spontaneously at a lower temperature. If \(\Delta G_{a} < \Delta G_{b}\), then reaction (a) proceeds spontaneously at a lower temperature. If \(\Delta G_{a} > \Delta G_{b}\), then reaction (b) proceeds spontaneously at a lower temperature. After determining which reaction has a lower \(\Delta G\) value at the desired temperature, we can conclude which reaction proceeds spontaneously at a lower temperature.

Key Concepts

Enthalpy Change

Enthalpy change, denoted as \( \Delta H \), is a crucial concept in thermodynamics. It refers to the amount of heat absorbed or released during a chemical reaction at constant pressure. Here’s how it works:

The enthalpy change depends on the difference in enthalpy between the products and the reactants. If a reaction releases heat, it is called exothermic, and \( \Delta H \) will be negative. In contrast, if a reaction absorbs heat, it is endothermic, and \( \Delta H \) will be positive.

To determine \( \Delta H \), we use the standard enthalpy of formation for each compound involved. It is defined at a standard state (usually 1 atm and 25°C). The formula to calculate the enthalpy change for a reaction is:

• For reaction \( a \): \( \Delta H_{a} = [2 \times \Delta H_f(\mathrm{Fe}_{(s)}) + \frac{3}{2} \times \Delta H_f(\mathrm{CO}_{2_{(g)}})] - [\Delta H_f(\mathrm{Fe}_2\mathrm{O}_{3_{(s)}}) + \frac{3}{2} \times \Delta H_f(\mathrm{C}_{(s)})] \)
• For reaction \( b \): \( \Delta H_{b} = [2 \times \Delta H_f(\mathrm{Fe}_{(s)}) + 3 \times \Delta H_f(\mathrm{H}_{2}\mathrm{O}_{(g)})] - [\Delta H_f(\mathrm{Fe}_2\mathrm{O}_{3_{(s)}}) + 3 \times \Delta H_f(\mathrm{H}_{2_{(g)})] \)

The change in enthalpy gives us insight into the energy requirements or outputs of a reaction, informing us whether heat is needed or released.

Entropy Change

Entropy change, expressed as \( \Delta S \), is an indicator of the disorder or randomness in a system during a chemical reaction. It helps us understand how the arrangement of atoms or molecules changes.

Entropy is often related to the phase and organization of the products and reactants. Generally, gases have higher entropy than liquids, and liquids have higher entropy than solids. When a reaction involves a transition from gas to liquid, or liquid to solid, \( \Delta S \) may be negative because the system becomes more ordered.

To calculate \( \Delta S \), we use the standard entropy values for all compounds. The equations used in reactions are:

• For reaction \( a \): \( \Delta S_{a} = [2 \times S_f(\mathrm{Fe}_{(s)}) + \frac{3}{2} \times S_f(\mathrm{CO}_{2_{(g)}})] - [S_f(\mathrm{Fe}_2\mathrm{O}_{3_{(s)}) + \frac{3}{2} \times S_f(\mathrm{C}_{(s)})] \)
• For reaction \( b \): \( \Delta S_{b} = [2 \times S_f(\mathrm{Fe}_{(s)}) + 3 \times S_f(\mathrm{H}_{2}\mathrm{O}_{(g)})] - [S_f(\mathrm{Fe}_2\mathrm{O}_{3_{(s)}) + 3 \times S_f(\mathrm{H}_{2_{(g)})] \)

The entropy change gives us insights into the potential spontaneity of reactions, especially when combined with enthalpy change to calculate Gibbs free energy.

Gibbs Free Energy

Gibbs free energy, denoted as \( \Delta G \), is a vital concept that combines both enthalpy and entropy changes to predict the spontaneity of a reaction. It ties these concepts together with temperature, giving a comprehensive picture of a reaction's feasibility.

Gibbs free energy assesses whether a reaction can occur on its own without added energy. The formula used to compute \( \Delta G \) is:

• \( \Delta G = \Delta H - T\Delta S \)

Where:

• \( \Delta G \) is Gibbs free energy change
• \( \Delta H \) is enthalpy change
• \( T \) is temperature in Kelvin
• \( \Delta S \) is entropy change

A reaction is considered spontaneous at a given temperature if \( \Delta G \) is negative. This negativity indicates that the reaction will proceed without needing extra energy input.

For the reactions given:

• Reaction \( a \) has \( \Delta G_{a} = \Delta H_{a} - T\Delta S_{a} \)
• Reaction \( b \) has \( \Delta G_{b} = \Delta H_{b} - T\Delta S_{b} \)

By comparing \( \Delta G_{a} \) and \( \Delta G_{b} \) at the desired temperature, you can determine which reaction proceeds spontaneously at lower temperatures. This understanding is crucial in selecting the most energy-efficient process in industrial applications.