Consider the following reactions,
I. `N_(2) + O_(2) rarr 2NO, " "Delta H = x_1" "II. 2CO + O_(2) rarr2CO_(2), " "Delta H = x_(2)`
III. `2H_2O + O_(2) rarr 2H_(2)O_(2),Delta H = x_(3)" "IV. PCl_(3) + CL_(3) rarr PCl_(3) + Cl_(2) rarr PCl_(5) , " "Delta H = x_(4)`
In which case(s), `Delta H_(f) = (DeltaH)/(2)` ?

Options

A. I,II,III
B. IV
C. II,III
D. I

Explanation

To determine in which case(s) the enthalpy of formation (ΔH_f) is equal to ΔH/2, we need to analyze each of the given reactions based on the definition of enthalpy of formation. The enthalpy of formation is defined as the change in enthalpy when one mole of a compound is formed from its constituent elements in their standard states.

### Step-by-Step Solution:

1. **Reaction I: N₂ + O₂ → 2NO, ΔH = x₁**
- This reaction produces 2 moles of NO from nitrogen and oxygen in their standard states (N₂ and O₂).
- To find ΔH_f for NO, we can divide the reaction by 2:
\[ \frac{1}{2}N₂ + \frac{1}{2}O₂ → NO \]
- The ΔH for this reaction would be:
\[ ΔH_f = \frac{x₁}{2} \]
- Therefore, this reaction satisfies the condition ΔH_f = ΔH/2.

2. **Reaction II: 2CO + O₂ → 2CO₂, ΔH = x₂**
- This reaction produces CO₂ from CO and O₂. However, the formation of CO₂ should occur from its elements in their standard states (C in graphite form and O₂).
- Since CO is not in the standard state for the formation of CO₂, this reaction does not yield ΔH_f for CO₂. Thus, it does not satisfy the condition.

3. **Reaction III: 2H₂O + O₂ → 2H₂O₂, ΔH = x₃**
- This reaction produces hydrogen peroxide (H₂O₂) from water (H₂O) and O₂. However, the formation of H₂O₂ should occur from its elements (H₂ and O₂).
- Since H₂O is not in its standard state for the formation of H₂O₂, this reaction does not yield ΔH_f for H₂O₂. Thus, it does not satisfy the condition.

4. **Reaction IV: PCl₃ + Cl₂ → PCl₅, ΔH = x₄**
- This reaction produces PCl₅ from PCl₃ and Cl₂. However, PCl₅ should be formed from phosphorus (P) and chlorine (Cl₂) in their standard states.
- Since PCl₃ is not in the standard state for the formation of PCl₅, this reaction does not yield ΔH_f for PCl₅. Thus, it does not satisfy the condition.

### Conclusion:
Only **Reaction I** satisfies the condition where ΔH_f = ΔH/2.

### Final Answer:
The only case where ΔH_f = ΔH/2 is **Reaction I**.

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