Problem 54 - Chemistry: An Atoms-Focused Approach

Question

Use the appropriate \(\Delta G_{f}^{\circ}\) data in Appendix 4 to calculate \(\Delta G_{\mathrm{rxn}}^{\circ}\) for the complete combustion of methanol: $$2 \mathrm{CH}_{3} \mathrm{OH}(g)+3 \mathrm{O}_{2}(g) \rightarrow 2 \mathrm{CO}_{2}(g)+4 \mathrm{H}_{2} \mathrm{O}(g)$$

Step-by-Step Solution

Answer

Question: Calculate the standard Gibbs free energy change for the complete combustion of methanol using the provided standard Gibbs free energies of formation. Answer: The standard Gibbs free energy change for the complete combustion of methanol is ΔG_rxn^° = -1380.4 kJ/mol.

Step 1: Write down the formula for calculating \(\Delta G_{\mathrm{rxn}}^{\circ}\)

To calculate the standard Gibbs free energy change for the reaction, we will use the formula: $$\Delta G_{\mathrm{rxn}}^{\circ} = \sum n_{prod} \Delta G_{f_{prod}}^{\circ} - \sum n_{react} \Delta G_{f_{react}}^{\circ}$$ where \(n_{prod}\) and \(n_{react}\) are the stoichiometric coefficients of products and reactants, respectively, and \(\Delta G_{f_{prod}}^{\circ}\) and \(\Delta G_{f_{react}}^{\circ}\) are their standard Gibbs free energies of formation.

Step 2: Find the standard Gibbs free energies of formation for each species

Consult Appendix 4 to obtain the standard Gibbs free energies of formation, \(\Delta G_{f}^{\circ}\), for each species involved in the reaction. The values are: $$\Delta G_{f}^{\circ}\, [\mathrm{CH}_{3} \mathrm{OH}(g)] = -161.4\, \mathrm{kJ/mol}$$ $$\Delta G_{f}^{\circ}\, [\mathrm{O}_{2}(g)] = 0\, \mathrm{kJ/mol}$$ $$\Delta G_{f}^{\circ}\, [\mathrm{CO}_{2}(g)] = -394.4\, \mathrm{kJ/mol}$$ $$\Delta G_{f}^{\circ}\, [\mathrm{H}_{2} \mathrm{O}(g)] = -228.6\, \mathrm{kJ/mol}$$

Step 3: Calculate the weighted sums of Gibbs free energies of formation for products and reactants

Compute the weighted sums of \(\Delta G_{f}^{\circ}\) for products and reactants by multiplying their stoichiometric coefficients with their respective Gibbs free energies of formation: $$\sum n_{prod} \Delta G_{f_{prod}}^{\circ} = 2 \times (-394.4) + 4 \times (-228.6) = -788.8 - 914.4 = -1703.2\, \mathrm{kJ/mol}$$ $$\sum n_{react} \Delta G_{f_{react}}^{\circ} = 2 \times (-161.4) + 3 \times 0 = -322.8\, \mathrm{kJ/mol}$$

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

Finally, apply the formula to calculate the standard Gibbs free energy change for the reaction: $$\Delta G_{\mathrm{rxn}}^{\circ} = \sum n_{prod} \Delta G_{f_{prod}}^{\circ} - \sum n_{react} \Delta G_{f_{react}}^{\circ} = -1703.2 - (-322.8) = -1380.4\, \mathrm{kJ/mol}$$ The standard Gibbs free energy change for the complete combustion of methanol is \(\Delta G_{\mathrm{rxn}}^{\circ} = -1380.4\, \mathrm{kJ/mol}\).

Key Concepts

Standard Gibbs Free Energy of Formation

The standard Gibbs free energy of formation, represented as \( \Delta G_{f}^{\circ} \), is a key concept in thermodynamics. It tells us about the stability of substances and is the energy change when one mole of a compound forms from its elements in their standard states.

This concept is vital because it helps us predict the direction and feasibility of chemical reactions. For example, a negative \( \Delta G_{f}^{\circ} \) means the formation of the compound from its elements is spontaneous under standard conditions.

Some standard states often considered are:

• Gases at 1 atmosphere pressure.
• Pure substances in their most stable form at 1 atmosphere and a specified temperature, usually 25°C.

By understanding \( \Delta G_{f}^{\circ} \), we can better grasp how chemical reactions transform matter.

Chemical Reactions

Chemical reactions are processes where substances, known as reactants, transform into different substances, called products. They involve breaking old bonds and forming new ones, rearranging atoms to give new structures.

In a balanced chemical equation, like the combustion of methanol, \( 2 \text{CH}_3\text{OH}(g) + 3 \text{O}_2(g) \rightarrow 2 \text{CO}_2(g) + 4 \text{H}_2\text{O}(g) \), all atoms must be accounted for on both sides of the equation.

Some key points about chemical reactions include:

• Conservation of Mass: Mass is neither created nor destroyed.
• Energy Changes: Reactions can release or absorb energy, often in the form of heat or light.
• Reaction Rates: Factors like temperature, pressure, and concentration affect how fast reactions occur.

Understanding chemical reactions helps us make sense of changes in matter.

Combustion of Methanol

The combustion of methanol is a chemical reaction where methanol reacts with oxygen to produce carbon dioxide and water. Here's a closer look at this type of reaction:

Methanol, \( \text{CH}_3\text{OH} \), is a simple alcohol used as a fuel. During its combustion, it combines with oxygen \( \text{O}_2 \). This reaction is exothermic, meaning it releases energy, specifically in the form of heat.

The balanced chemical equation for this combustion is: \[ 2 \text{CH}_3\text{OH}(g) + 3 \text{O}_2(g) \rightarrow 2 \text{CO}_2(g) + 4 \text{H}_2\text{O}(g) \]
Some important aspects include:

• Exothermic Nature: Releases a significant amount of energy, making it useful as a fuel.
• Products: Produces carbon dioxide and water, common products in combustion reactions.
• Gibbs Free Energy: The calculation of \( \Delta G_{\mathrm{rxn}}^{\circ} \) helps us assess the feasibility and energy changes in this reaction.

This reaction is crucial in understanding energy conversion processes.