Problem 53 - Chemistry: An Atoms-Focused Approach

Question

A mixture of \(\mathrm{CO}(g)\) and \(\mathrm{H}_{2}(g)\) is produced by passing steam over hot charcoal: $$\mathrm{H}_{2} \mathrm{O}(g)+\mathrm{C}(s) \rightarrow \mathrm{H}_{2}(g)+\mathrm{CO}(g)$$ Calculate the \(\Delta G_{\text {rxn }}^{\circ}\) value for the reaction from the appropriate \(\Delta G_{f}^{\circ}\) data in Appendix 4.

Step-by-Step Solution

Answer

Question: Calculate the Gibbs free energy change (\(\Delta G_{\text{rxn}}^{\circ}\)) of the following reaction using the provided Gibbs free energy of formation values: $$\mathrm{H}_{2} \mathrm{O}(g)+\mathrm{C}(s) \rightarrow \mathrm{H}_{2}(g)+\mathrm{CO}(g)$$ Answer: The Gibbs free energy change for the given reaction is \(\Delta G_{\text{rxn}}^{\circ} = 91.40 \ \text{kJ/mol}\).

Step 1: Write down the given reaction

The given reaction is: $$\mathrm{H}_{2} \mathrm{O}(g)+\mathrm{C}(s) \rightarrow \mathrm{H}_{2}(g)+\mathrm{CO}(g)$$

Step 2: Identify the reactants and products

In the reaction, the reactants are \(\mathrm{H}_{2}\mathrm{O}(g)\) and \(\mathrm{C}(s)\), and the products are \(\mathrm{H}_{2}(g)\) and \(\mathrm{CO}(g)\).

Step 3: Find the \(\Delta G_{f}^{\circ}\) values of each compound in Appendix 4

From Appendix 4, we obtain the following \(\Delta G_{f}^{\circ}\) values: - \(\Delta G_{f}^{\circ}(\mathrm{H}_{2}\mathrm{O \ (g)}) = -228.57 \ \text{kJ/mol}\) - \(\Delta G_{f}^{\circ}(\mathrm{C \ (s)}) = 0 \ \text{kJ/mol}\) (since elemental carbon is in its standard state) - \(\Delta G_{f}^{\circ}(\mathrm{H}_{2}\mathrm{(g)}) = 0 \ \text{kJ/mol}\) (since elemental hydrogen is in its standard state) - \(\Delta G_{f}^{\circ}(\mathrm{CO \ (g)}) = -137.17 \ \text{kJ/mol}\)

Step 4: Calculate the \(\Delta G_{\text{rxn}}^{\circ}\)

Using the equation \(\Delta G_{\text{rxn}}^{\circ} = \sum{ \Delta G_{f}^{\circ} \text{(products)}} - \sum{ \Delta G_{f}^{\circ} \text{(reactants)}}\), we can now calculate the Gibbs free energy change for the reaction: $$\Delta G_{\text{rxn}}^{\circ} = [0 + (-137.17)] - [-228.57 + 0] = -137.17 + 228.57 = 91.40 \ \text{kJ/mol}$$ So, the \(\Delta G_{\text{rxn}}^{\circ}\) value for the reaction is \(91.40 \ \text{kJ/mol}\).

Key Concepts

Chemical Thermodynamics

Chemical thermodynamics is the branch of physical chemistry that deals with the study of energy transformations in chemical reactions. It focuses on understanding how energy changes govern the direction and magnitude of chemical processes.
Thermodynamics explores how heat and work are related and often involves calculating properties like enthalpy, entropy, and Gibbs free energy.

• **Enthalpy (H)**: Represents the heat content of a system at constant pressure.
• **Entropy (S)**: Measures the disorder or randomness in a system.
• **Gibbs Free Energy (G)**: Combines enthalpy and entropy to determine a reaction's spontaneity.

This field provides the foundational principles to predict whether a chemical reaction will occur spontaneously or require energy input. By studying thermodynamic properties, chemists can design processes that are energy-efficient and environmentally friendly.
Understanding these concepts is essential for a range of fields like chemical engineering, pharmaceutical development, and environmental science.

Reaction Spontaneity

Reaction spontaneity is a critical concept that helps us understand whether a reaction will proceed on its own. It is primarily determined by the Gibbs free energy change, denoted as \(\Delta G\).

• If \(\Delta G < 0\), the reaction is spontaneous, meaning it will proceed without additional energy input.
• If \(\Delta G > 0\), the reaction is non-spontaneous and requires energy to proceed.
• If \(\Delta G = 0\), the reaction is at equilibrium, meaning no net change occurs.

In the example of steam passing over hot charcoal, determining \(\Delta G_{\text{rxn}}^{\circ}\) helps us know if the production of hydrogen and carbon monoxide will occur spontaneously.
Although the reaction's calculated \(\Delta G_{\text{rxn}}^{\circ}\) is positive, suggesting it is non-spontaneous under standard conditions, changing variables like temperature can potentially influence its spontaneity.

Standard Free Energy Change

The standard free energy change \(\Delta G_{\text{rxn}}^{\circ}\) is an essential concept for predicting reaction behavior under standard conditions (1 bar pressure, typically 298 K). It helps to quantify the driving force behind a chemical reaction.
The equation used to calculate \(\Delta G_{\text{rxn}}^{\circ}\) is:
\[ \Delta G_{\text{rxn}}^{\circ} = \sum{ \Delta G_{f}^{\circ} \text{(products)}} - \sum{ \Delta G_{f}^{\circ} \text{(reactants)}} \]
This equation sums up the free energies of formation (\(\Delta G_{f}^{\circ}\)) of the products and subtracts those of the reactants.
For the reaction given, this calculation revealed a positive \(\Delta G_{\text{rxn}}^{\circ}\) value of 91.40 kJ/mol, indicating non-spontaneity under standard conditions.

• The values used in this calculation are typically found in thermodynamic tables or appendices.
• A positive \(\Delta G_{\text{rxn}}^{\circ}\) suggests that conditions may need modifying (such as temperature or pressure) for the reaction to become favorable.

Understanding \(\Delta G_{\text{rxn}}^{\circ}\) is crucial for chemists in developing efficient processes and reacting strategies.