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# Reaction mechanism and rate law

AP.Chem:
TRA‑5 (EU)
,
TRA‑5.A (LO)
,
TRA‑5.A.1 (EK)
,
TRA‑5.A.2 (EK)
,
TRA‑5.A.3 (EK)
,
TRA‑5.B (LO)
,
TRA‑5.B.1 (EK)

## Key points

• A reaction mechanism is the sequence of elementary steps by which a chemical reaction occurs.
• A reaction that occurs in two or more elementary steps is called a multistep or complex reaction.
• A reaction intermediate is a chemical species that is formed in one elementary step and consumed in a subsequent step.
• The slowest step in a reaction mechanism is known as the rate-determining step.
• The rate-determining step limits the overall rate and therefore determines the rate law for the overall reaction.

## Introduction: Multistep reactions

One of the most important applications of kinetics is to the study of reaction mechanisms, or the sequences of steps by which chemical reactions occur. For example, consider the reaction of nitrogen dioxide with carbon monoxide:
$\ce{NO2}(g) + \ce{CO}(g) \rightarrow \ce{NO}(g) + \ce{CO}_2(g)$
Based on the balanced equation, we might hypothesize that this reaction occurs by a single collision between a molecule of nitrogen dioxide and a molecule of carbon monoxide. In other words, we might hypothesize that this reaction is an elementary reaction.
If that were the case, then the rate law would be based on the reactant coefficients in the balanced chemical equation:
$\text{rate} = k[\ce{NO2}][\text{CO}]$
However, when this reaction is studied experimentally, the rate law is in fact observed to be
$\text{rate} = k[\ce{NO2}]^2$
Since the experimental rate law does not match the one derived by assuming an elementary reaction, we know immediately that the reaction must involve more than one step. Reactions that occur in two or more elementary steps are called multistep or complex reactions. As we'll see in the next section, we can use the experimental rate law to help deduce the individual steps that might be involved in a multistep reaction mechanism.

## Multistep reaction mechanisms

Once the experimental rate law for a reaction is known, chemists can begin to devise and investigate possible reaction mechanisms. At minimum, a possible reaction mechanism must meet the following two conditions:
1. The equations for the elementary steps in the mechanism must add up to the overall equation for the reaction.
2. The mechanism must be consistent with the experimental rate law.
Let's use these conditions to evaluate a proposed mechanism for the reaction between $\ce{NO2}$ and C, O. It is generally believed that this reaction occurs through two elementary steps:
\begin{aligned}\text{Step 1:}&\quad \ce{NO2}(g) + \ce{NO2}(g) \xrightarrow{} \ce{NO}(g) + \ce{NO3}(g) \\\\[-0.50em] \text{Step 2:}&\kern1.5em \ce{NO3}(g) + \ce{CO}(g) \xrightarrow{} \ce{NO2}(g) + \ce{CO2}(g)\end{aligned}
First, let's check that the equations for these two steps add up to the overall reaction equation:
\begin{aligned} \text{Step 1:}&\quad \ce{NO2}(g) + \blueD{\cancel{\ce{NO2}(g)}} \xrightarrow{} \ce{NO}(g) + \maroonD{\cancel{\ce{NO3}(g)}} \\\\[-0.50em] \text{Step 2:}&\kern1.5em \maroonD{\cancel{\ce{NO3}(g)}} + \ce{CO}(g) \xrightarrow{} \blueD{\cancel{\ce{NO2}(g)}} + \ce{CO2}(g) \\\\[-0.75em] &\kern1.4em \overline{\phantom{\ce{NO2}(g) + \ce{CO}(g) \xrightarrow{} \ce{NO}(g) + \ce{CO2}(g)}} \\\\[-2em] \text{Overall:}&\kern1.5em \ce{NO2}(g) + \ce{CO}(g) \xrightarrow{} \ce{NO}(g) + \ce{CO2}(g) \end{aligned}
Check! Note that one molecule, $\ce{NO3}$, appears in both steps of the mechanism but doesn't show up in the overall equation. In this case, $\ce{NO3}$ is a reaction intermediate, a species that is formed in one step and consumed in a subsequent step.
Next, let's determine if the two-step mechanism is consistent with the experimental rate law. To do so, we need to know which of the two steps is the rate-determining step, or the slowest step in the mechanism. Because a reaction can occur no faster than its slowest step, the rate-determining step effectively limits the overall rate of a reaction. This is analogous to how a traffic jam limits the overall rate at which cars can move along a highway, even if other parts of the highway are clear.
In our proposed mechanism, the rate-determining step is believed to be step 1:
\begin{aligned} \text{Step 1:}&\quad \ce{NO2}(g) + \ce{NO2}(g) \xrightarrow{slow} \ce{NO}(g) + \ce{NO3}(g) \\\\[-0.50em] \text{Step 2:}&\kern1.5em \ce{NO3}(g) + \ce{CO}(g) \xrightarrow{fast} \ce{NO2}(g) + \ce{CO2}(g) \end{aligned}
Since step 1 limits the overall rate of the reaction, the rate law for this step will be the same as the overall rate law. The predicted rate law for the overall reaction is therefore
$\text{rate} = k[\ce{NO2}]^2$
This rate law is in agreement with the experimentally-determined rate law we saw earlier, so the mechanism also meets the second condition (check!). Since the reaction mechanism meets both conditions, we can safely say that it is a valid mechanism for the reaction.

## Practice: Analyzing a reaction mechanism

The elementary steps of a proposed reaction mechanism are represented below.
\begin{aligned} \text{Step 1:}&\kern5.0em \ce{2NO}(g) \xrightleftharpoons{fast} \ce{N2O2}(g) \\\\[-0.50em] \text{Step 2:}&\quad \ce{N2O2}(g) + \ce{H2}(g) \xrightarrow{slow} \ce{N2O}(g) + \ce{H2O}(g) \\\\[-0.50em] \text{Step 3:}&\kern1.5em \ce{N2O}(g) + \ce{H2}(g) \xrightarrow{fast} \ce{N2}(g) + \ce{H2O}(g) \end{aligned}
Based on this information, try to answer the following questions:
1. What is the overall equation for the reaction?

2. What are the reaction intermediates?

3. What is the rate-determining step?