Reactions Matching Enthalpy of Reaction with Enthalpy of Formation

To determine the reaction(s) for which the standard reaction enthalpy ($\Delta_r H^\circ$) is equal to the standard enthalpy of formation ($\Delta_f H^\circ$), we refer to the definition of standard enthalpy of formation:

The standard enthalpy of formation ($\Delta_f H^\circ$) of a substance is defined as the enthalpy change that accompanies the formation of $1\text{ mole}$ of the substance from its constituent elements in their most stable standard states at $298\text{ K}$ and $1\text{ bar}$ pressure.

Let's evaluate each option based on this definition:

1. Option (A): $2\text{H}_2(\text{g}) + \text{O}_2(\text{g}) \rightarrow 2\text{H}_2\text{O}(l)$

   • In this reaction, $2\text{ moles}$ of liquid water ($\text{H}_2\text{O}$) are formed.
   • The standard enthalpy of formation is defined per mole of the product formed. Therefore: $\Delta_r H^\circ = 2 \cdot \Delta_f H^\circ[\text{H}_2\text{O}(l)] \neq \Delta_f H^\circ[\text{H}_2\text{O}(l)]$
   • Hence, Option (A) is incorrect.

2. Option (B): $2\text{C}(\text{g}) + 3\text{H}_2(\text{g}) \rightarrow \text{C}_2\text{H}_6(\text{g})$

   • Carbon is present in the gaseous state, $\text{C}(\text{g})$.
   • The reference (standard) state of elemental carbon at $298\text{ K}$ and $1\text{ bar}$ is graphite, i.e., $\text{C}(\text{s, graphite})$, not gaseous carbon.
   • Hence, Option (B) is incorrect.

3. Option (C): $\frac{3}{2}\text{O}_2(\text{g}) \rightarrow \text{O}_3(\text{g})$

   • Exactly $1\text{ mole}$ of ozone gas ($\text{O}_3$) is formed.
   • Its constituent element, oxygen, is in its standard reference state, which is dioxygen gas, $\text{O}_2(\text{g})$.
   • Therefore: $\Delta_r H^\circ = \Delta_f H^\circ[\text{O}_3(\text{g})]$
   • Hence, Option (C) is correct.

4. Option (D): $\frac{1}{8}\text{S}_8(\text{s}) + \text{O}_2(\text{g}) \rightarrow \text{SO}_2(\text{g})$

   • Exactly $1\text{ mole}$ of sulfur dioxide gas ($\text{SO}_2$) is formed.
   • Both sulfur in its standard state, rhombic sulfur $\text{S}_8(\text{s})$, and oxygen gas, $\text{O}_2(\text{g})$, are in their standard reference states.
   • Therefore: $\Delta_r H^\circ = \Delta_f H^\circ[\text{SO}_2(\text{g})]$
   • Hence, Option (D) is correct.

Correct Options: (C) and (D)