Problem 87 - Chemistry: An Atoms-Focused Approach

Question

Use standard heats of formation from Appendix 4 to calculate the standard heat of reaction for the following methane-generating reaction of methanogenic bacteria: $$4 \mathrm{H}_{2}(g)+\mathrm{CO}_{2}(g) \rightarrow \mathrm{CH}_{4}(g)+2 \mathrm{H}_{2} \mathrm{O}(\ell)$$

Step-by-Step Solution

Answer

Answer: The standard heat of reaction for the given methane-generating reaction is -252.9 kJ/mol.

Step 1: Write down the balanced chemical equation

The balanced chemical equation for the methane-generating reaction is given as: $$4 \mathrm{H}_{2}(g)+\mathrm{CO}_{2}(g) \rightarrow \mathrm{CH}_{4}(g)+2 \mathrm{H}_{2} \mathrm{O}(\ell)$$

Step 2: Identify the standard heats of formation for each species

Use Appendix 4 to find the standard heats of formation \((\Delta H^\circ_\mathrm{f})\) for each species in the reaction. The values are as follows (in kJ/mol): $$\Delta H^\circ_\mathrm{f} [\mathrm{H_{2}(g)}]=0$$ $$\Delta H^\circ_\mathrm{f} [\mathrm{CO_{2}(g)}]=-393.5$$ $$\Delta H^\circ_\mathrm{f} [\mathrm{CH_{4}(g)}]=-74.8$$ $$\Delta H^\circ_\mathrm{f} [\mathrm{H_{2}O}(\ell)]=-285.8$$

Step 3: Calculate the standard heat of reaction

The standard heat of the reaction \((\Delta H^\circ_\mathrm{rxn})\) can be calculated using the equation: $$\Delta H^\circ_\mathrm{rxn} = \sum \Delta H^\circ_\mathrm{f} (\mathrm{products}) - \sum \Delta H^\circ_\mathrm{f} (\mathrm{reactants})$$ Plugging in the values for the species involved in the reaction: $$\Delta H^\circ_\mathrm{rxn} = [\Delta H^\circ_\mathrm{f} (\mathrm{CH_{4}(g)})+2 \cdot \Delta H^\circ_\mathrm{f}(\mathrm{H_{2}O}(\ell))] - [4 \cdot \Delta H^\circ_\mathrm{f} (\mathrm{H_{2}(g)}) + \Delta H^\circ_\mathrm{f}(\mathrm{CO_{2}(g)})]$$ Substituting the values found in Step 2: $$\Delta H^\circ_\mathrm{rxn} = [(-74.8)+2 \cdot (-285.8)] - [4 \cdot (0)+(-393.5)]$$

Step 4: Calculate the final value for the standard heat of reaction

Perform the necessary calculations to find the value of \(\Delta H^\circ_\mathrm{rxn}\): $$\Delta H^\circ_\mathrm{rxn} = (-74.8 - 571.6) - (-393.5)$$ $$\Delta H^\circ_\mathrm{rxn} = (-646.4) - (-393.5)$$ $$\Delta H^\circ_\mathrm{rxn} = -252.9 \, \mathrm{kJ/mol}$$ The standard heat of reaction for the given methane-generating reaction is -252.9 kJ/mol.

Key Concepts

Standard Heats of Formation

The standard heat of formation, often denoted as \( \Delta H^\circ_\mathrm{f} \), is a crucial concept in understanding how energy is involved in chemical reactions. It is the amount of energy change that occurs when one mole of a compound is formed from its elements in their standard states. These standard states are typically at 1 atmosphere of pressure and a specified temperature, often 298.15 K (25°C).

For example:

• The standard heat of formation for hydrogen gas, \( \mathrm{H_2(g)} \), is \( 0 \) kJ/mol because it is an elemental form.
• For carbon dioxide, \( \mathrm{CO_2(g)} \), it is \( -393.5 \) kJ/mol, indicating a release of energy when CO2 is formed from carbon and oxygen.

These values are crucial as they allow us to calculate the energy changes in reactions by comparing products and reactants. A key principle is that lower energy states are more stable, hence products with lower standard heats of formation are generally more stable.

Enthalpy Change

Enthalpy change, represented as \( \Delta H \), is the heat change associated with a chemical reaction, at constant pressure. It includes not only internal energy but also the energy required to make room for the substance (pressure-volume work). In chemistry, it's a key concept, because it helps predict whether reactions are endothermic (absorb heat) or exothermic (release heat).

The formula for calculating the standard heat of reaction, \( \Delta H^\circ_\mathrm{rxn} \), is:\[ \Delta H^\circ_\mathrm{rxn} = \sum \Delta H^\circ_\mathrm{f} \text{(products)} - \sum \Delta H^\circ_\mathrm{f} \text{(reactants)} \]This means you sum up all the standard heats of formation for the products and subtract the sum of all the standard heats of formation for the reactants.

In the example of methanogenic bacteria generating methane, the reaction is exothermic with \( \Delta H^\circ_\mathrm{rxn} = -252.9 \) kJ/mol, showing a net release of energy.

Methanogenic Bacteria

Methanogenic bacteria are fascinating microorganisms that produce methane as a metabolic byproduct in anoxic conditions. They are part of the Archaea domain and are a key player in the carbon cycle, notably in wetlands, rice paddies, and the guts of ruminants.

They perform a reaction where carbon dioxide and hydrogen gas convert into methane and water, as shown in the reaction:\( 4 \mathrm{H_2(g)} + \mathrm{CO_2(g)} \rightarrow \mathrm{CH_4(g)} + 2 \mathrm{H_2O(l)} \).

This reaction is crucial for reducing carbon emissions and explains how methane, a significant greenhouse gas, is produced biologically. The energy change during this reaction is significant for understanding both microbial energy balances and broader environmental impacts. Understanding the enthalpy changes enables researchers to better grasp the energy production mechanisms of these bacteria and their ecological contributions.

Chemical Reactions

Chemical reactions are processes that transform one set of chemical substances into another. They involve breaking bonds in reactants and forming new bonds in products. The overarching goal is to reach a lower energy, more stable state.

Key Points about Chemical Reactions:

• Energy Changes: Reactions can either release energy to their surroundings (exothermic) or absorb energy (endothermic).
• Balance: Chemical equations must be balanced to obey the law of conservation of mass.
• Types: There are different types of reactions like synthesis, decomposition, single displacement, and double displacement.

In the example of the methane generation reaction, it is an exothermic synthesis reaction where new bonds form to create methane and water from simpler molecules. By knowing the energy involved in these reactions, chemists can predict reaction behavior, optimize conditions, and apply them in practical settings including industrial applications or biological systems.