Answer :
To solve this problem and find the enthalpy change ([tex]\(\Delta H^{\circ}\)[/tex]) for the reaction [tex]\(2 \text{S}(s) + 3 \text{O}_2(g) \rightarrow 2 \text{SO}_3(g)\)[/tex], we will use Hess's Law. This principle states that the total enthalpy change for a chemical reaction is the same, no matter how many steps the reaction is carried out in. With the given reactions and their respective enthalpy changes, we can determine the desired [tex]\(\Delta H^{\circ}\)[/tex].
### Step-by-Step Solution:
1. Identify Given Reactions:
- Reaction 1:
[tex]\[
\text{S}(s) + \text{O}_2(g) \rightarrow \text{SO}_2(g) \quad \Delta H^{\circ} = -296.1 \, \text{kJ/mol}
\][/tex]
- Reaction 2 (reverse):
[tex]\[
2 \text{SO}_3(g) \rightarrow 2 \text{SO}_2(g) + \text{O}_2(g) \quad \Delta H = 198.2 \, \text{kJ}
\][/tex]
2. Calculate the Enthalpy Change for the Reverse Formation of [tex]\(\text{SO}_3(g)\)[/tex]:
Since Reaction 2 involves the decomposition of [tex]\(\text{SO}_3(g)\)[/tex], we will reverse it to match the formation from [tex]\(\text{SO}_2(g)\)[/tex]:
[tex]\[
2 \text{SO}_2(g) + \text{O}_2(g) \rightarrow 2 \text{SO}_3(g)
\][/tex]
When reversing a reaction, the sign of [tex]\(\Delta H\)[/tex] changes:
[tex]\[
\Delta H = -198.2 \, \text{kJ}
\][/tex]
3. Apply Hess's Law:
We need to construct the target reaction:
[tex]\[
2 \text{S}(s) + 3 \text{O}_2(g) \rightarrow 2 \text{SO}_3(g)
\][/tex]
This can be done by combining Reaction 1 twice and the reversed Reaction 2:
- Two moles of [tex]\(\text{SO}_2\)[/tex] are formed from two moles of [tex]\(\text{S}\)[/tex]:
[tex]\[
2 \left( \text{S}(s) + \text{O}_2(g) \rightarrow \text{SO}_2(g) \right) \quad \Delta H = 2(-296.1) \, \text{kJ} = -592.2 \, \text{kJ}
\][/tex]
- Then, combine with the reversed Reaction 2 (formation of [tex]\(\text{SO}_3\)[/tex]):
[tex]\[
2 \text{SO}_2(g) + \text{O}_2(g) \rightarrow 2 \text{SO}_3(g) \quad \Delta H = -198.2 \, \text{kJ}
\][/tex]
4. Calculate the Overall Enthalpy Change:
Add the enthalpy changes from the steps above to find the enthalpy change for the overall reaction:
[tex]\[
\Delta H^{\circ} = -592.2 \, \text{kJ} + (-198.2) \, \text{kJ} = -790.4 \, \text{kJ}
\][/tex]
Therefore, the enthalpy change for the reaction [tex]\(2 \text{S}(s) + 3 \text{O}_2(g) \rightarrow 2 \text{SO}_3(g)\)[/tex] is [tex]\(-790.4 \, \text{kJ}\)[/tex].
The correct answer is D) -790.4 kJ.
### Step-by-Step Solution:
1. Identify Given Reactions:
- Reaction 1:
[tex]\[
\text{S}(s) + \text{O}_2(g) \rightarrow \text{SO}_2(g) \quad \Delta H^{\circ} = -296.1 \, \text{kJ/mol}
\][/tex]
- Reaction 2 (reverse):
[tex]\[
2 \text{SO}_3(g) \rightarrow 2 \text{SO}_2(g) + \text{O}_2(g) \quad \Delta H = 198.2 \, \text{kJ}
\][/tex]
2. Calculate the Enthalpy Change for the Reverse Formation of [tex]\(\text{SO}_3(g)\)[/tex]:
Since Reaction 2 involves the decomposition of [tex]\(\text{SO}_3(g)\)[/tex], we will reverse it to match the formation from [tex]\(\text{SO}_2(g)\)[/tex]:
[tex]\[
2 \text{SO}_2(g) + \text{O}_2(g) \rightarrow 2 \text{SO}_3(g)
\][/tex]
When reversing a reaction, the sign of [tex]\(\Delta H\)[/tex] changes:
[tex]\[
\Delta H = -198.2 \, \text{kJ}
\][/tex]
3. Apply Hess's Law:
We need to construct the target reaction:
[tex]\[
2 \text{S}(s) + 3 \text{O}_2(g) \rightarrow 2 \text{SO}_3(g)
\][/tex]
This can be done by combining Reaction 1 twice and the reversed Reaction 2:
- Two moles of [tex]\(\text{SO}_2\)[/tex] are formed from two moles of [tex]\(\text{S}\)[/tex]:
[tex]\[
2 \left( \text{S}(s) + \text{O}_2(g) \rightarrow \text{SO}_2(g) \right) \quad \Delta H = 2(-296.1) \, \text{kJ} = -592.2 \, \text{kJ}
\][/tex]
- Then, combine with the reversed Reaction 2 (formation of [tex]\(\text{SO}_3\)[/tex]):
[tex]\[
2 \text{SO}_2(g) + \text{O}_2(g) \rightarrow 2 \text{SO}_3(g) \quad \Delta H = -198.2 \, \text{kJ}
\][/tex]
4. Calculate the Overall Enthalpy Change:
Add the enthalpy changes from the steps above to find the enthalpy change for the overall reaction:
[tex]\[
\Delta H^{\circ} = -592.2 \, \text{kJ} + (-198.2) \, \text{kJ} = -790.4 \, \text{kJ}
\][/tex]
Therefore, the enthalpy change for the reaction [tex]\(2 \text{S}(s) + 3 \text{O}_2(g) \rightarrow 2 \text{SO}_3(g)\)[/tex] is [tex]\(-790.4 \, \text{kJ}\)[/tex].
The correct answer is D) -790.4 kJ.
