Question

Which from following is a correct representation of reaction rate for reaction stated below?
\( \text{N}_{2(\text{g}) + 3\text{H}_{2(\text{g})} \rightleftharpoons 2\text{NH}_{3(\text{g})} \)}

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

Hint:

Rate = $\frac{1}{coefficient} \times \frac{d[Concentration]}{dt}$. Use $(-)$ for reactants.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The rate of a chemical reaction must be a positive value and must be the same regardless of which reactant or product is used to measure it.
Because reactants are consumed, their concentration changes (\( d[Reactant] \)) are negative. We add a negative sign to make the overall rate positive.
Because products are formed, their concentration changes (\( d[Product] \)) are positive.
To equate the rates across all species, we divide the rate of change of each species by its stoichiometric coefficient from the balanced equation.
Step 2: Key Formula or Approach:
For a general balanced chemical equation:
\[ a\text{A} + b\text{B} \rightarrow c\text{C} + d\text{D} \]
The unique average rate of the reaction is defined as:
\[ \text{Rate} = -\frac{1}{a}\frac{\Delta[\text{A}]}{\Delta t} = -\frac{1}{b}\frac{\Delta[\text{B}]}{\Delta t} = +\frac{1}{c}\frac{\Delta[\text{C}]}{\Delta t} = +\frac{1}{d}\frac{\Delta[\text{D}]}{\Delta t} \]
Using instantaneous rates (derivatives):
\[ \text{Rate} = -\frac{1}{a}\frac{\text{d}[\text{A}]}{\text{dt}} = -\frac{1}{b}\frac{\text{d}[\text{B}]}{\text{dt}} = +\frac{1}{c}\frac{\text{d}[\text{C}]}{\text{dt}} = +\frac{1}{d}\frac{\text{d}[\text{D}]}{\text{dt}} \]
Step 3: Detailed Explanation:
The specific reaction given is the Haber process:
\[ 1\text{N}_{2(\text{g})} + 3\text{H}_{2(\text{g})} \rightarrow 2\text{NH}_{3(\text{g})} \]
Here, Nitrogen (\( \text{N}_2 \)) and Hydrogen (\( \text{H}_2 \)) are reactants.
- Their stoichiometric coefficients are 1 and 3, respectively.
- Their rate expressions will be negative.
- Rate with respect to \( \text{N}_2 = -\frac{1}{1}\frac{\text{d}[\text{N}_2]}{\text{dt}} = -\frac{\text{d}[\text{N}_2]}{\text{dt}} \)
- Rate with respect to \( \text{H}_2 = -\frac{1}{3}\frac{\text{d}[\text{H}_2]}{\text{dt}} \)
Ammonia (\( \text{NH}_3 \)) is the product.
- Its stoichiometric coefficient is 2.
- Its rate expression will be positive.
- Rate with respect to \( \text{NH}_3 = +\frac{1}{2}\frac{\text{d}[\text{NH}_3]}{\text{dt}} \)
Equating all these expressions gives the correct representation of the overall reaction rate:
\[ \text{Rate} = -\frac{\text{d}[\text{N}_2]}{\text{dt}} = -\frac{1}{3}\frac{\text{d}[\text{H}_2]}{\text{dt}} = \frac{1}{2}\frac{\text{d}[\text{NH}_3]}{\text{dt}} \]
Comparing this with the given options, Option (C) matches exactly.
Step 4: Final Answer:
The correct representation is Option (C).