Explain why the coefficients of an elementary step equal the reaction orders of its rate law but those of an overall reaction do not.

An elementary step is a fundamental process in a chemical reaction that occurs in a single stage and cannot be broken down into simpler processes. Each elementary step represents a specific collision or interaction between reactant molecules. They are characterized by their stoichiometric coefficients, which indicate the number of molecules or atoms participating in the step. For example, in the elementary step involving molecules A and B forming product C, the reaction can be written as:

\[ A + B \rightarrow C \]

This step means one molecule of A reacts with one molecule of B to produce one molecule of C. The coefficients (in this case, both 1) directly tell how many molecules are involved in the reaction at that specific moment. This specificity makes understanding elementary steps crucial when analyzing reaction mechanisms and formulating rate laws.

A reaction mechanism is a detailed, step-by-step description of how a chemical reaction occurs. It includes all the elementary steps and intermediates formed along the way from reactants to products. Each step in the mechanism provides insight into the molecular interactions and transformations taking place.

A reaction mechanism for a complex reaction may look something like this:

\[ A + B \rightarrow C \]
\[ C + D \rightarrow E \]
The overall reaction would then be the sum of these elementary steps. Understanding the reaction mechanism is essential because it helps identify key intermediates and the sequence of events, which ultimately determines the reaction rate and pathway. It also aids in troubleshooting unexpected results during experiments and designing better catalysts.

The rate-determining step (RDS) in a reaction mechanism is the slowest elementary step that limits the overall reaction rate. This step is crucial because the entire reaction cannot proceed faster than this slowest step. Think of it as the bottleneck of a reaction mechanism.

For instance, consider a two-step reaction mechanism:

\[ \text{Step 1: } A + B \rightarrow C \text{ (slow)} \]
\[ \text{Step 2: } C + D \rightarrow E \text{ (fast)} \]
Since Step 1 is slower, it determines the rate at which the overall reaction can occur. The rate law for the overall reaction is generally derived from this slow step. Even though the fast step might complete quickly once the intermediate (C) is formed, the overall reaction cannot exceed the rate at which C is produced in the slow step. Identifying the RDS helps chemists understand and control the reaction rate more effectively.

Stoichiometric coefficients are the numbers placed before reactants and products in a balanced chemical equation to indicate their relative quantities. In elementary steps, these coefficients also represent the molecularity, which is the number of molecules coming together to react in that step.

For example, in the reaction:

\[ 2A + B \rightarrow 3C \]
The stoichiometric coefficients are 2 for A, 1 for B (often omitted but implied), and 3 for C. These coefficients are essential for stoichiometric calculations, allowing chemists to predict how much of each reactant is needed or how much product will be formed. However, in the context of rate laws, stoichiometric coefficients of the overall reaction do not necessarily translate directly into reaction orders. The reaction orders are instead determined by the rate-determining step in the reaction mechanism.