The reaction given is:

$2 \, \text{NH}_3(g) \rightarrow \text{N}_2(g) + 3 \, \text{H}_2(g)$

We are asked to relate the rate of change of $[ \text{H}_2 ]$ to the rate of change of $[ \text{NH}_3 ]$.

From stoichiometry, the relationship between the rates of change of the reactants and products is determined by their coefficients in the balanced chemical equation. Here, for every 2 moles of $\text{NH}_3$ consumed, 3 moles of $\text{H}_2$ are produced.

Thus, we can set up the following relation:

[ \frac{\Delta [\text{H}_2]}{3} = -\frac{\Delta [\text{NH}_3]}{2} ]

or equivalently:

[ \Delta [\text{H}_2] = -\frac{3}{2} \Delta [\text{NH}_3] ]

This means that the rate of change of $[ \text{H}_2 ]$ is $-\frac{3}{2}$ times the rate of change of $[ \text{NH}_3 ]$.

Thus, the correct answer is B:

[ \text{rate of change of } [\text{H}_2] = -\frac{3}{2} \left( \frac{\Delta [\text{NH}_3]}{\Delta t} \right) ]

Would you like further clarification or have any questions about this?

Here are a few related questions for deeper understanding:

1. How do the stoichiometric coefficients in a chemical equation influence the relationship between reactants and products' rates of change?
2. What is the significance of the negative sign in the rate equation?
3. How would this relationship change if the reaction were reversed?
4. How can the rate of a chemical reaction be experimentally determined?
5. What does it mean for a rate of change to be expressed in terms of concentration over time?

Tip: Always ensure you balance the chemical equation correctly before applying stoichiometry to rate relationships.