Nitrogen monoxide, NO, reacts with hydrogen, H_2, according to the following equation: \(2 \mathrm{NO}+2 \mathrm{H}_{2} \longrightarrow \mathrm{N}_{2}+2 \mathrm{H}_{2} \mathrm{O}\) What would the rate law be if the mechanism for this reaction were: \(2 \mathrm{NO}+\mathrm{H}_{2} \longrightarrow \mathrm{N}_{2}+\mathrm{H}_{2} \mathrm{O}_{2}(\text { slow })\) \(\mathrm{H}_{2} \mathrm{O}_{2}+\mathrm{H}_{2} \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}\) (fast)

When studying chemical kinetics, one concept that frequently arises is the 'rate-determining step' (RDS). It's pivotal to understand that the RDS is the slowest step in a reaction mechanism; akin to the narrowest point in a funnel. No matter how fast the preceding or succeeding steps may be, the overall reaction can only be as fast as this bottleneck. In the exercise provided, the RDS is the first step involving nitrogen monoxide and hydrogen.

Because the RDS controls the reaction pace, it directly shapes the rate law, which mathematicall represents the reaction rate. Thus, to determine the rate law for a chemical reaction, focus on the stoichiometry of this critical step. For the given reaction, the rate law is derived from the stoichiometry of the RDS: Rate = k[NO]^2[H_2]. Here, k denotes the rate constant, and the square brackets represent the concentrations of reactants, raised to powers matching their stoichiometric coefficients in the RDS.

Understanding the RDS not only informs us about the rate law but also about the reaction's kinetics and can shed light on potential ways to increase the reaction rate, such as by altering concentrations or conditions favoring the rate-limiting step.

The 'reaction mechanism' is the step-by-step sequence through which reactants transform into products. This concept is like a detailed instruction manual for a chemical reaction, illustrating each intermediate state and transition. The provided exercise reveals a two-step mechanism for the reaction between nitrogen monoxide and hydrogen:

• 2 NO + H2 → N2 + H2O2 (Slow step)
• H2O2 + H2 → 2 H2O (Fast Step)

Here, the first step is the rate-determining step, indicating its crucial role in the kinetics of the reaction. The stepwise nature of reaction mechanisms often showcases the creation and consumption of intermediate species, such as H2O2 in this example. These intermediates are paramount in connecting the dots between the initial and final states of a reaction. They may be fleeting and difficult to detect, but they hold the narrative of the transformation taking place and can significantly affect the kinetics and pathways of the reaction.

Delving into 'stoichiometry', we encounter the quantitative relationships between reactants and products in a chemical reaction. Returning to the equation provided in the exercise, stoichiometry dictates that two moles of nitrogen monoxide react with two moles of hydrogen to produce one mole of nitrogen and two moles of water.

These ratios are not just mere numbers; they're the backbone of chemical equations, providing insights on the amounts of substances involved. By expressing the rate law as Rate = k[NO]^2[H_2], it's clear how stoichiometry of the rate-determining step is used to formulate the rate law. Each reactant's exponent corresponds to its stoichiometric coefficient, reflecting how changes in the concentration of each reactant will influence the reaction rate.

It's also important to recognize that while stoichiometry is based on the balanced equation, the rate law includes only those reactants involved in the rate-determining step, and not necessarily all reactants in the overall balanced equation. Hence, understanding stoichiometry not only assists with predicting product formation but also serves as a guide in the crafting of rate laws from reaction mechanisms.