For a reaction \(2 \mathrm{NH}_{3} \rightarrow \mathrm{N}_{2}+3 \mathrm{H}_{2}\), it is observed that \(-\frac{\mathrm{d}\left[\mathrm{NH}_{3}\right]}{\mathrm{d} t}=K_{1}\left[\mathrm{NH}_{3}\right] ;+\frac{\mathrm{d}\left[\mathrm{N}_{2}\right]}{\mathrm{d} t}=K_{2}\left[\mathrm{NH}_{3}\right]\) and \(+\frac{\mathrm{d}\left[\mathrm{H}_{2}\right]}{\mathrm{d} t}=K_{3}\left[\mathrm{NH}_{3}\right]\) The correct relation between \(K_{1}, K_{2}\) and \(K_{3}\) is (a) \(K_{1}=K_{2}=K_{3}\) (b) \(2 K_{1}=3 K_{2}=6 K_{3}\) (c) \(3 K_{1}=6 K_{2}=2 K_{3}\) (d) \(6 K_{1}=3 K_{2}=2 K_{3}\)

In chemical kinetics, reaction rates refer to the speed at which reactants are converted into products. It's essential for students to understand that reaction rates can be affected by various factors, including concentration, temperature, and the presence of a catalyst.

In the given problem, the reaction rate is measured in terms of the time rate change of the concentration of ammonia (\textbf{NH}\(_3\)). Specifically, we look at the decrease in \textbf{NH}\(_3\) concentration, which is represented by the negative differential \textbf{-d[\textbf{NH}\(_3\)]/dt}. Understanding this concept is crucial, as it's a direct measure of how quickly the reaction is occurring. In kinetics problems, getting a grasp on the significance of these negative and positive signs in rate expressions is the first step. These indicate whether a species is being consumed (negative sign) or produced (positive prices) over time.

Stoichiometry is the branch of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. It allows chemists to predict the amounts of substances consumed and produced during reactions.

In the provided exercise, stoichiometry plays a critical role in linking the disappearance of ammonia to the formation of nitrogen and hydrogen. It's used to set up a proportional relationship between the respective rates of reactants and products based on their coefficients in the balanced equation. This means for every 2 moles of \textbf{NH}\(_3\) used, 1 mole of \textbf{N}\(_2\) and 3 moles of \textbf{H}\(_2\) are produced. Translating these stoichiometric ratios into rate expressions is vital in solving the problem.

Rate law is an expression that relates the reaction rate to the concentration of the reactants. It is generally represented as rate = k[\textbf{A}]\(^m\)[\textbf{B}]\(^n\)..., where 'k' is the rate constant, and 'm' and 'n' are the orders of the reaction with respect to reactants \textbf{A} and \textbf{B}, respectively. The rate law is determined experimentally and provides insight into the mechanism of the reaction.

In this case, the rate law is assumed to be directly proportional to the concentration of \textbf{NH}\(_3\). Thus, the rate constant 'k' for each substance (\textbf{K}\(_1\), \textbf{K}\(_2\), and \textbf{K}\(_3\)) will allow us to establish a relationship that uniquely defines the reaction rate. Understanding how to use the rate law and the significance of the rate constant is central to solving kinetic problems.