Question

Which of the following equations is correct regarding rate of disappearance of reactant and appearance of product for $\text{N}_{2(\text{g})} + 3\text{H}_{2(\text{g})} \longrightarrow 2\text{NH}_{3(\text{g})}$}

• A. $3\frac{\text{d}[\text{N}_2]}{\text{dt}} = \frac{1}{2} \frac{\text{d}[\text{N}_2]}{\text{dt}}$
• B. $\frac{1}{2} \frac{\text{d}[\text{N}_2]}{\text{dt}} = \frac{1}{3} \frac{\text{d}[\text{H}_2]}{\text{dt}}$
• C. $2\frac{\text{d}[\text{NH}_3]}{\text{dt}} = 3\frac{\text{d}[\text{H}_2]}{\text{dt}}$
• D. $3\frac{\text{d}[\text{NH}_3]}{\text{dt}} = -2\frac{\text{d}[\text{H}_2]}{\text{dt}}$

Hint:

Rate equation shortcut: Coefficient of species A $\times$ Rate of B = Coefficient of species B $\times$ Rate of A (ensure sign is correct for reactants).

Answer 1

The Correct Option is D

Step 1: Concept
The rate of reaction is expressed by dividing the rate of change of concentration of any species by its stoichiometric coefficient. Reactants have a negative sign (disappearance).

Step 2: Meaning
$\text{Rate} = -\frac{\text{d}[\text{N}_2]}{\text{dt}} = -\frac{1}{3}\frac{\text{d}[\text{H}_2]}{\text{dt}} = +\frac{1}{2}\frac{\text{d}[\text{NH}_3]}{\text{dt}}$.

Step 3: Analysis
To find the relationship between $\text{H}_2$ and $\text{NH}_3$: $-\frac{1}{3}\frac{\text{d}[\text{H}_2]}{\text{dt}} = \frac{1}{2}\frac{\text{d}[\text{NH}_3]}{\text{dt}}$. Cross-multiplying gives: $-2\frac{\text{d}[\text{H}_2]}{\text{dt}} = 3\frac{\text{d}[\text{NH}_3]}{\text{dt}}$.

Step 4: Conclusion
Option (D) matches this mathematical derivation. Final Answer: (D)