91Ó°ÊÓ

Hess's law can be used to calculate reaction enthalpies for hypothetical processes that can't be carried out in the laboratory. Set up a Hess's law cycle that will let you calculate \(\Delta H^{\circ}\) for the conversion of methane to ethylene: \(2 \mathrm{CH}_{4}(g) \longrightarrow \mathrm{C}_{2} \mathrm{H}_{4}(g)+2 \mathrm{H}_{2}(g)\) You can use the following information: \(2 \mathrm{C}_{2} \mathrm{H}_{6}(g)+7 \mathrm{O}_{2}(g) \longrightarrow 4 \mathrm{CO}_{2}(g)+6 \mathrm{H}_{2} \mathrm{O}(l)\) \(\Delta H^{\circ}=-3120.8 \mathrm{~kJ}\) \(\mathrm{CH}_{4}(g)+2 \mathrm{O}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(l)\) \(\Delta H^{\circ}=-890.3 \mathrm{~kJ}\) \(\mathrm{C}_{2} \mathrm{H}_{4}(g)+\mathrm{H}_{2}(g) \longrightarrow \mathrm{C}_{2} \mathrm{H}_{6}(g) \quad \Delta H^{\circ}=-136.3 \mathrm{~kJ}\) \(\mathrm{H}_{2} \mathrm{O}(l) \quad \Delta H_{\mathrm{f}}^{\circ}=-285.8 \mathrm{~kJ} / \mathrm{mol}\)

Step by step solution

01

Identify the Target Reaction

The target reaction is \( 2 \text{CH}_4(g) \longrightarrow \text{C}_2\text{H}_4(g) + 2\text{H}_2(g) \). Our goal is to determine \( \Delta H^{\circ} \) for this reaction.

02

Write the Provided Reactions

We have the following reactions with their enthalpies:1. \( 2 \text{C}_2\text{H}_6(g) + 7 \text{O}_2(g) \longrightarrow 4 \text{CO}_2(g) + 6\text{H}_2\text{O}(l), \Delta H^{\circ} = -3120.8 \, \text{kJ} \).2. \( \text{CH}_4(g) + 2 \text{O}_2(g) \longrightarrow \text{CO}_2(g) + 2 \text{H}_2\text{O}(l), \Delta H^{\circ} = -890.3 \, \text{kJ} \).3. \( \text{C}_2\text{H}_4(g) + \text{H}_2(g) \longrightarrow \text{C}_2\text{H}_6(g), \Delta H^{\circ} = -136.3 \, \text{kJ} \).

03

Reverse Reaction 3

Reverse reaction 3 to align with our target reaction:\( \text{C}_2\text{H}_6(g) \longrightarrow \text{C}_2\text{H}_4(g) + \text{H}_2(g), \Delta H^{\circ} = +136.3 \, \text{kJ} \).

04

Use Reaction 2 for 2 Moles

Multiply reaction 2 by 2 to account for two moles of \( \text{CH}_4 \):\( 2 \text{CH}_4(g) + 4 \text{O}_2(g) \longrightarrow 2 \text{CO}_2(g) + 4 \text{H}_2\text{O}(l), \Delta H^{\circ} = 2 \times (-890.3) = -1780.6 \, \text{kJ} \).

05

Formulate a Hess's Law Equation

Combine reactions:1. \( 2 \text{C}_2\text{H}_6(g) + 7 \text{O}_2(g) \longrightarrow 4 \text{CO}_2(g) + 6 \text{H}_2\text{O}(l), \Delta H^{\circ} = -3120.8 \, \text{kJ} \).2. \( 2 \text{CH}_4(g) + 4 \text{O}_2(g) \longrightarrow 2 \text{CO}_2(g) + 4 \text{H}_2\text{O}(l), \Delta H^{\circ} = -1780.6 \, \text{kJ} \).3. \( 2 \text{C}_2\text{H}_6(g) \longrightarrow 2 \text{C}_2\text{H}_4(g) + 2 \text{H}_2(g), \Delta H^{\circ} = 2 \times 136.3 \, \text{kJ} = 272.6 \, \text{kJ}\).

06

Apply Hess's Law

The target reaction \( 2 \text{CH}_4(g) \longrightarrow \text{C}_2\text{H}_4(g) + 2 \text{H}_2(g) \) can be calculated using the reactions:\[ \Delta H^{\circ}_{\text{target}} = (-3120.8 + 1780.6 + 272.6) \, \text{kJ} = -1067.6 \, \text{kJ} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reaction Enthalpy

Understanding reaction enthalpy is key to mastering thermodynamics. It refers to the heat absorbed or released during a chemical reaction at constant pressure. For each reaction, we can express this energy change as \( \Delta H^{\circ} \), also known as the standard enthalpy change.

• Negative \( \Delta H^{\circ} \) means the reaction is exothermic (releases heat).
• Positive \( \Delta H^{\circ} \) indicates an endothermic process (absorbs heat).

When using Hess's Law, we can calculate \( \Delta H^{\circ} \) for reactions that are difficult to measure directly. By using known enthalpies from related reactions, we can piece together the overall enthalpy change.

Enthalpy Calculation

Calculating enthalpy changes accurately involves several steps, often using Hess's Law to piece together known reactions.First, identify your target reaction and break down provided reactions. For instance, in converting methane to ethylene:

• Use the enthalpies of combustion for methane and ethylene.
• Utilize the enthalpy for the formation of ethylene from ethane.

To calculate the desired \( \Delta H^{\circ} \):1. Reverse or multiply reactions to align them with the target.2. Sum the enthalpy changes:\[\Delta H^{\circ}_{\text{target}} = \text{sum of individual } \Delta H^{\circ}\]This approach allows combining smaller, known reactions to derive the desired enthalpy change.

Chemical Thermodynamics

Chemical thermodynamics explores the energetic changes and state transformations during chemical reactions. Central to this is Hess's Law, which states that the total enthalpy change for a reaction is the same, regardless of the pathway taken. This makes it possible to calculate enthalpy for complex reactions by using simpler, related processes.Key concepts include:

• State Functions: Properties like enthalpy that depend only on the current state of the system, not the path taken.
• Energy Conservation: The first law of thermodynamics ensures energy is conserved in chemical processes.

Using these principles, we can predict reaction behavior, calculate \( \Delta H^{\circ} \), and understand the feasibility and spontaneity of reactions.