Step 1: Understanding the Concept:
The overall rate of a chemical reaction can be unambiguously expressed in terms of the rate of change of concentration of any participating reactant or product. To equate these individual species rates to one another, they must be divided by their respective stoichiometric coefficients from the balanced chemical equation. Furthermore, reactants are assigned a negative sign (since their concentration decreases over time), while products are assigned a positive sign.
Step 2: Key Formula or Approach:
For any general balanced reaction $aA + bB \longrightarrow cC + dD$, the relationship between the rates is defined as:
\[ \text{Rate} = -\frac{1}{a}\frac{\text{d}[A]}{\text{dt}} = -\frac{1}{b}\frac{\text{d}[B]}{\text{dt}} = +\frac{1}{c}\frac{\text{d}[C]}{\text{dt}} = +\frac{1}{d}\frac{\text{d}[D]}{\text{dt}} \]
Step 3: Detailed Explanation:
Given the balanced chemical equation for the synthesis of ammonia:
\[ \text{N}_{2(\text{g})} + 3\text{H}_{2(\text{g})} \longrightarrow 2\text{NH}_{3(\text{g})} \]
Let's apply the general formula to this specific reaction.
The overall rate of reaction can be expressed as:
\[ \text{Rate} = -\frac{\text{d}[\text{N}_2]}{\text{dt}} = -\frac{1}{3}\frac{\text{d}[\text{H}_2]}{\text{dt}} = +\frac{1}{2}\frac{\text{d}[\text{NH}_3]}{\text{dt}} \]
We need to evaluate the given options to find the mathematical equation that correctly represents a part of this fundamental relationship. Let's specifically look at the relationship between Hydrogen ($\text{H}_2$) and Ammonia ($\text{NH}_3$):
\[ -\frac{1}{3}\frac{\text{d}[\text{H}_2]}{\text{dt}} = \frac{1}{2}\frac{\text{d}[\text{NH}_3]}{\text{dt}} \]
To clear the fractions and find a match among the options, we can multiply the entire equation by the lowest common multiple of the denominators (which is 6):
\[ 6 \times \left( -\frac{1}{3}\frac{\text{d}[\text{H}_2]}{\text{dt}} \right) = 6 \times \left( \frac{1}{2}\frac{\text{d}[\text{NH}_3]}{\text{dt}} \right) \]
\[ -2\frac{\text{d}[\text{H}_2]}{\text{dt}} = 3\frac{\text{d}[\text{NH}_3]}{\text{dt}} \]
Rearranging this equation slightly gives:
\[ 3\frac{\text{d}[\text{NH}_3]}{\text{dt}} = -2\frac{\text{d}[\text{H}_2]}{\text{dt}} \]
This derived equation perfectly matches option (D). Let's briefly check why others fail: (A) relates a species to itself incorrectly, (B) mixes up the coefficients and signs, (C) lacks the necessary negative sign indicating disappearance.
Step 4: Final Answer:
The correct relationship is $3\frac{\text{d}[\text{NH}_3]}{\text{dt}} = -2\frac{\text{d}[\text{H}_2]}{\text{dt}}$.