Question

Nitric oxide reacts with \( \text{H}_2 \) according to the reaction: 
\[ 2\text{NO}_{\text{(g)}} + 2\text{H}_2{}_{\text{(g)}} \rightarrow \text{N}_2{}_{\text{(g)}} + 2\text{H}_2\text{O}_{\text{(g)}} \] Identify the correct relation for the disappearance of reactants and appearance of products.

• A. \(\frac{1}{2} \frac{\text{d}[\text{NO}]}{\text{dt}} = -\frac{\text{d}[\text{H}_2]}{\text{dt}}\)
• B. \(\frac{\text{d}[\text{N}_2]}{\text{dt}} = \frac{1}{2} \frac{\text{d}[\text{H}_2\text{O}]}{\text{dt}}\)
• C. \(-\frac{\text{d}[\text{N}_2]}{\text{dt}} = \frac{1}{2} \frac{\text{d}[\text{H}_2\text{O}]}{\text{dt}}\)
• D. \(\frac{\text{d}[\text{H}_2]}{\text{dt}} = -\frac{1}{2} \frac{\text{d}[\text{N}_2]}{\text{dt}}\)

Hint:

For rate relations, always divide by stoichiometric coefficients and use negative sign for reactants, positive sign for products.

Answer 1

The Correct Option is B

Concept:
For a balanced chemical reaction: \[ aA+bB \rightarrow cC+dD \] the rate relation is: \[ -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = \frac{1}{c}\frac{d[C]}{dt} = \frac{1}{d}\frac{d[D]}{dt} \]

Step 1: Write the balanced reaction.
\[ 2\text{NO} + 2\text{H}_2 \rightarrow \text{N}_2 + 2\text{H}_2\text{O} \]

Step 2: Write the rate expression.
\[ -\frac{1}{2}\frac{d[\text{NO}]}{dt} = -\frac{1}{2}\frac{d[\text{H}_2]}{dt} = \frac{d[\text{N}_2]}{dt} = \frac{1}{2}\frac{d[\text{H}_2\text{O}]}{dt} \]

Step 3: Match with the correct option.
The relation that matches directly is: \[ \frac{d[\text{N}_2]}{dt} = \frac{1}{2}\frac{d[\text{H}_2\text{O}]}{dt} \] Hence, the correct answer is:
\[ \boxed{(B)\ \frac{d[\text{N}_2]}{dt} = \frac{1}{2}\frac{d[\text{H}_2\text{O}]}{dt}} \]