Problem 88 - Chemistry: An Atoms-Focused Approach

Question

Use standard enthalpies of formation from Appendix 4 to calculate the standard enthalpy of reaction for the following methane-generating reaction of methanogenic bacteria, given \(\Delta H_{f}^{\circ}\) of \(\mathrm{CH}_{3} \mathrm{NH}_{2}(g)=-22.97 \mathrm{kJ} / \mathrm{mol}:\) $$4 \mathrm{CH}_{3} \mathrm{NH}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(\ell) \rightarrow 3 \mathrm{CH}_{4}(g)+\mathrm{CO}_{2}(g)+4 \mathrm{NH}_{3}(g)$$

Step-by-Step Solution

Answer

Question: Calculate the standard enthalpy of reaction for the methane-generating reaction: \(4\,\mathrm{CH}_{3}\mathrm{NH}_{2}(g) + 2\,\mathrm{H}_{2}\mathrm{O}(\ell) \rightarrow 3\,\mathrm{CH}_{4}(g) + \mathrm{CO}_{2}(g) + 4\,\mathrm{NH}_{3}(g)\). Answer: The standard enthalpy of reaction for the given methane-generating reaction is \(-138.00\,\mathrm{kJ/mol}\).

Step 1: List the standard enthalpies of formation and the given reaction

First, let's list the standard enthalpies of formation for each reactant and product in the reaction. We are given the value for \(\mathrm{CH}_{3}\mathrm{NH}_{2}(g)\), and we can find the others from Appendix 4: - \(\Delta H_{f}^{\circ}(\mathrm{CH}_{3}\mathrm{NH}_{2}(g)) = -22.97\,\mathrm{kJ/mol}\) - \(\Delta H_{f}^{\circ}(\mathrm{H}_{2}\mathrm{O}(\ell)) = -285.83\,\mathrm{kJ/mol}\) - \(\Delta H_{f}^{\circ}(\mathrm{CH}_{4}(g)) = -74.81\,\mathrm{kJ/mol}\) - \(\Delta H_{f}^{\circ}(\mathrm{CO}_{2}(g)) = -393.51\,\mathrm{kJ/mol}\) - \(\Delta H_{f}^{\circ}(\mathrm{NH}_{3}(g)) = -45.90\,\mathrm{kJ/mol}\) Now, let's write down the given methane-generating reaction: $$4\,\mathrm{CH}_{3}\mathrm{NH}_{2}(g) + 2\,\mathrm{H}_{2}\mathrm{O}(\ell) \rightarrow 3\,\mathrm{CH}_{4}(g) + \mathrm{CO}_{2}(g) + 4\,\mathrm{NH}_{3}(g)$$

Step 2: Apply the formula for the standard enthalpy of reaction

The standard enthalpy of reaction can be calculated using the formula: $$\Delta H_{rxn}^{\circ} = \sum n\Delta H_{f}^{\circ}(\text{products}) - \sum n\Delta H_{f}^{\circ}(\text{reactants})$$ where \(n\) is the stoichiometric coefficient of each substance in the reaction. Using this formula, we can calculate the standard enthalpy of reaction for the given methane-generating reaction: $$\Delta H_{rxn}^{\circ} = [3(-74.81) + (-393.51) + 4(-45.90)] - [4(-22.97) + 2(-285.83)]$$

Step 3: Calculate the standard enthalpy of reaction

Now, let's calculate the standard enthalpy of reaction by performing the operations in the equation: $$\Delta H_{rxn}^{\circ} = [-224.43 - 393.51 - 183.60] - [-91.88 - 571.66]$$ $$\Delta H_{rxn}^{\circ} = (-224.43 - 393.51 - 183.60) - (-91.88 - 571.66)$$ $$\Delta H_{rxn}^{\circ} = -801.54 + 663.54$$ $$\Delta H_{rxn}^{\circ} = -138.00\,\mathrm{kJ/mol}$$ So, the standard enthalpy of reaction for the given methane-generating reaction is \(-138.00\,\mathrm{kJ/mol}\).

Key Concepts

Standard Enthalpies of Formation

Understanding standard enthalpies of formation is crucial when calculating the enthalpy change of reactions. A standard enthalpy of formation, denoted as \( \Delta H_{f}^{\circ} \), is the heat change that occurs when one mole of a compound is formed from its elements in their standard states. Standard states typically refer to the most stable form of an element at 1 atm pressure and a specified temperature, usually 25°C or 298 K.
For many reactions, you can find the standard enthalpies of formation for common compounds in tables, usually found in appendices of textbooks. These values help you determine how much energy is absorbed or released during chemical changes.
The enthalpy of reaction can be calculated using the formula:

• \( \Delta H_{rxn}^{\circ} = \sum n\Delta H_{f}^{\circ}(\text{products}) - \sum n\Delta H_{f}^{\circ}(\text{reactants}) \)

This equation effectively describes that the total enthalpy change for a reaction is the difference between the sum of the enthalpies of the products and the reactants. This method simplifies tracking energy changes throughout a chemical process.

Stoichiometry

Stoichiometry is the area of chemistry that involves calculating the quantities of reactants and products in a chemical reaction. Understanding stoichiometry is essential in determining the exact amount of reactants needed and predicting the amount of products formed in reactions.
In any balanced chemical equation, the stoichiometric coefficients represent the number of moles of each substance involved. These coefficients are key in applying the standard enthalpies of formation to find enthalpy changes. For example, if a balanced equation shows "4 CH₃NH₂(g) + 2 H₂O(l) → 3 CH₄(g) + CO₂(g) + 4 NH₃(g)", these coefficients are used in the enthalpy change calculation.
When plugging into the enthalpy-of-reaction formula,

• Each coefficient (like the 4 in \(4 \mathrm{CH}_3\mathrm{NH}_2(g)\)) tells you to multiply the \(\Delta H_{f}^{\circ}\) value by this number.
• This multiplication correctly scales the enthalpy changes to match the actual amount of substance reacting or being produced.

By using stoichiometry in combination with thermodynamic data, we can more accurately predict the energy dynamics of chemical reactions.

Chemical Thermodynamics

Chemical thermodynamics is the branch of chemistry that deals with the relationship between the heat changes and the energy transfer that occur during a chemical reaction. When dealing with reactions like methane generation, understanding thermodynamics helps explain how energy is conserved and transferred as heat, work, and internal energy.
At the heart of chemical thermodynamics lies the concept of "enthalpy," which measures the total heat content of a system. Reacting substances often absorb or release energy, leading to a change in this total heat content, measured as the enthalpy change, \(\Delta H\).
This change indicates whether a reaction is endothermic (absorbs heat) or exothermic (releases heat).

• A negative \(\Delta H\) means the reaction is exothermic, releasing heat into the surroundings.
• A positive \(\Delta H\) indicates an endothermic reaction, where heat is absorbed from the surroundings.

In the case of the methane-generating reaction given, a \(\Delta H_{rxn}^{\circ} = -138.00\,\mathrm{kJ/mol}\) shows exothermic behavior, meaning it produces heat, which can significantly impact how such reactions are utilized or controlled especially in industrial and environmental contexts.