The following mechanism has been proposed for the reaction of \(\mathrm{NO}\) with \(\mathrm{Br}_{2}\) to form \(\mathrm{NO} \mathrm{Br}\) $$ \begin{aligned} &\mathrm{NO}(\mathrm{g})+\mathrm{Br}_{2}(\mathrm{~g}) \rightleftharpoons \mathrm{NOBr}_{2}(\mathrm{~g}) \\ &\mathrm{NOBr}_{2}(\mathrm{~g})+\mathrm{NO}(\mathrm{g}) \longrightarrow 2 \mathrm{NOBr}(\mathrm{g}) \end{aligned} $$ If the second step is the rate determining step, the order of the reaction with respect to \(\mathrm{NO}(\mathrm{g})\) is: (a) 1 (b) 0 (c) 3 (d) 2

In a multi-step chemical reaction, the rate determining step (RDS) is the slowest step that dictates how fast the overall reaction will proceed. It's akin to the slowest runner in a relay race; no matter how fast the other runners are, the team's overall speed is limited by the slowest member. In chemistry, the rate of product formation is governed by this slowest step, and therefore, it becomes the focus for studying the reaction's kinetics and determining the reaction order.

Understanding the RDS is crucial in chemical kinetics because it holds the key to controlling reaction rates, and hence, the efficiency of many industrial processes. When solving problems related to RDS, as we saw in the provided exercise, it's essential to recognize that the overall reaction rate is dependent on the concentrations of reactants in the RDS, rather than the entire reaction mechanism.

The study of chemical kinetics involves analyzing how different factors such as concentration, temperature, and catalysts affect the speed of chemical reactions. The core idea is to understand the reaction mechanism, which is the step-by-step sequence of elementary reactions that lead to product formation. Chemical kinetics allows us to derive rate laws, which articulate how the rate depends on the concentration of species involved in the reaction.

To make these concepts more digestible, let's consider a pizza baking analogy. The rate at which pizzas are baked (analogous to product formation rate) is influenced by the oven temperature (akin to reaction temperature), the quality of the yeast (similar to a catalyst), and the number of pizza chefs working (related to the concentration of reactants). Thus, kinetics is the study of how these 'factors' influence the 'baking rate' of reactions in the chemical 'oven'.

The term stoichiometry is derived from the Greek words 'stoicheion', meaning 'element,' and 'metron', meaning 'measure.' In essence, stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It's like a recipe that dictates the proportions of ingredients needed to bake a cake, except the 'ingredients' are chemical substances, and the 'cake' is the chemical product.

The use of stoichiometry is paramount when trying to determine the reaction order, as it's the coefficients in the balanced chemical equation that guide us. For example, when we explore the relationship with respect to \(\mathrm{NO}\) in the rate determining step of the exercise, we rely on stoichiometry to deduce there is one molecule of \(\mathrm{NO}\) reacting, resulting in the reaction order of 1. It's these stoichiometric coefficients that provide the framework for constructing the rate law and thus connect directly back to chemical kinetics and the rate determining step.