Question

Given that \(\Delta G_{\mathrm{f}}^{\circ}\left(\mathrm{Fe}_{2} \mathrm{O}_{3}\right)=-741.0 \mathrm{~kJ} / \mathrm{mol}\) and that \(\Delta G_{\mathrm{f}}^{\circ}\left(\mathrm{Al}_{2} \mathrm{O}_{3}\right)=-1576.4 \mathrm{~kJ} / \mathrm{mol},\) calculate \(\Delta G^{\circ}\) for the following reactions at \(25^{\circ} \mathrm{C}:\) (a) \(2 \mathrm{Fe}_{2} \mathrm{O}_{3}(s) \longrightarrow 4 \mathrm{Fe}(s)+3 \mathrm{O}_{2}(g)\) (b) \(2 \mathrm{Al}_{2} \mathrm{O}_{3}(s) \longrightarrow 4 \mathrm{Al}(s)+3 \mathrm{O}_{2}(g)\)

Short answer

(a) 1482.0 kJ; (b) 3152.8 kJ

Step-by-step solution

Understanding the Reaction

First, identify what is required for both reactions. For reaction (a), we need the Gibbs free energy change for the decomposition of \(2 \mathrm{Fe}_{2}\mathrm{O}_{3}\). For reaction (b), it's the decomposition of \(2 \mathrm{Al}_{2}\mathrm{O}_{3}\).

Formula for Gibbs Free Energy Change

Use the formula for the change in Gibbs free energy: \[ \Delta G^{\circ} = \sum \Delta G_{\text{products}}^{\circ} - \sum \Delta G_{\text{reactants}}^{\circ} \] Since both reactions decompose into elements, the \(\Delta G_{\text{products}}^{\circ}\) (for elements in their standard state) is 0.

Apply the Formula to Reaction (a)

For reaction (a), calculate \(\Delta G^{\circ}\) as follows: \[ \Delta G^{\circ} = 0 - [2(-741.0)] = 1482.0 \text{ kJ} \]

Apply the Formula to Reaction (b)

For reaction (b), calculate \(\Delta G^{\circ}\) as follows:\[ \Delta G^{\circ} = 0 - [2(-1576.4)] = 3152.8 \text{ kJ} \]

Conclusion of the Calculations

The \(\Delta G^{\circ}\) for reaction (a) is \(1482.0\ \text{kJ}\), and for reaction (b), it is \(3152.8\ \text{kJ}\). This indicates the energy required for each reaction to proceed at standard conditions.

Key concepts

Thermodynamics

The field of thermodynamics is a branch of physics that explains how energy is transferred and transformed in natural processes. When studying thermodynamics, we often come across concepts like energy, heat, work, and entropy. In the context of chemical reactions, these processes are crucial in understanding how and why reactions occur.

One of the key aspects considered in thermodynamics is the Gibbs free energy, denoted as \( \Delta G \), which helps predict the spontaneity of a reaction. A negative \( \Delta G \) indicates a spontaneous reaction, while a positive \( \Delta G \) suggests non-spontaneity, meaning energy input is needed. The Gibbs free energy combines enthalpy, temperature, and entropy to assess the free energy change during a reaction:

• Enthalpy (\( H \)) - the heat content of a system at constant pressure.
• Entropy (\( S \)) - a measure of disorder or randomness.
• Kelvin temperature (\( T \)) - the absolute temperature in the formula \( \Delta G = \Delta H - T\Delta S \).

By understanding thermodynamics, scientists can predict whether a chemical reaction will occur under given conditions, and calculate the amount of energy needed or released during these reactions.

Chemical Reactions

Chemical reactions are transformative processes where reactants are converted into products. They involve breaking and forming chemical bonds, which results in an energy change. The exercise given demonstrates the decomposition of iron (III) oxide and aluminum oxide.

These reactions are analyzed to understand the energy changes associated with decomposing compounds into their elemental states. In these particular reactions, compounds such as \( \text{Fe}_2\text{O}_3 \) and \( \text{Al}_2\text{O}_3 \) are broken down into iron, aluminum, and oxygen, which are in their elemental forms. The Gibbs free energy calculation is based on the differences between the sum of energies of products and reactants.

Important points to note about chemical reactions include:

• Reactants - substances that start the reaction.
• Products - substances that are formed as a result of the reaction.
• Conservation of Mass - matter is neither created nor destroyed.
• Catalysts - substances that can speed up a reaction without being consumed.

Understanding these fundamental concepts can help in manipulating and predicting the outcome of chemical reactions, which is essential in fields like pharmaceuticals, environmental science, and materials engineering.

Standard State Conditions

The concept of standard state conditions is crucial in thermodynamics and chemistry. When considering reactions or calculating thermodynamic quantities like Gibbs free energy, it's important to understand what 'standard state' means.

Standard state conditions refer to a set of conditions where substances exist in their most stable form at 1 atmosphere of pressure and a specified temperature, commonly taken as 25°C (298 K). For solutions, a standard concentration of 1 molar is also assumed. These conditions provide a reference baseline from which calculations and comparisons can be made.

The importance of standard state conditions includes:

• Consistency in data comparison - ensuring reactions evaluated at different times and places can be compared fairly.
• Simplifying calculations - with known standard states, the calculations of parameters like \( \Delta G^{\circ} \) become more straightforward.
• Facilitating thermodynamic assessments - comparing how different conditions influence chemical behavior.

By utilizing standard state conditions, chemists and engineers can reliably calculate and predict the energy changes in reactions, thereby designing systems that optimize the energy efficiency and effectiveness of industrial and laboratory processes.