DAT General Chemistry: Elementary Steps and the Rate-Determining Step

DAT High-Yield Concept

Elementary Steps: Individual molecular events that constitute a reaction mechanism. Unlike overall reactions, the rate law order of an elementary step directly equals its stoichiometric coefficients.
Molecularity: The exact number of reactant molecules colliding in an elementary step (unimolecular = 1, bimolecular = 2, termolecular = 3). Termolecular steps are extremely rare due to low collision probability.
Rate-Determining Step (RDS): The slowest elementary step in a multistep mechanism. It acts as the bottleneck and determines the overall reaction rate law.
Intermediates vs. Transition States: Intermediates are real chemical species generated in one step and consumed in another; transition states are high-energy maxima that cannot be isolated. Intermediates must never appear in an overall rate law.

Elementary Steps and Molecularity

Most chemical transformations do not occur in a single collision. Instead, they proceed through a sequence of discrete single-step molecular events called elementary steps. The detailed collection of elementary steps that describes how reactants convert into products is the reaction mechanism.

The molecularity of an elementary step is defined by the number of reactant particles that must simultaneously collide or react:

Unimolecular: A single reactant species rearranges or decomposes ( $\ce{A -> products}$ ).
Bimolecular: Two reactant species collide ( $\ce{A + B -> products}$ or $\ce{2A -> products}$ ).
Termolecular: Three reactant species collide simultaneously ( $\ce{2A + B -> products}$ ). These steps are exceptionally rare because three-body simultaneous collisions are statistically highly improbable.

A simple reaction coordinate diagram (Source: Chem540grp1f08 · CC BY-SA 3.0 · Wikimedia Commons)

Writing the Rate Law of an Elementary Step

For an overall reaction, rate law exponents must be determined experimentally (a concept covered in depth in differential rate laws). However, for an elementary step, the rate law can be written directly from its stoichiometry because it describes an actual single physical event.

Elementary Step	Molecularity	Elementary Rate Law
$\ce{A -> products}$	Unimolecular	$\text{Rate} = k[\ce{A}]$
$\ce{A + B -> products}$	Bimolecular	$\text{Rate} = k[\ce{A}][\ce{B}]$
$\ce{2A -> products}$	Bimolecular	$\text{Rate} = k[\ce{A}]^2$

Worked Example: Elementary Rate Law Determination

Question: Given the elementary step $\ce{2NO2 -> NO3 + NO}$ , write its rate law.

Reasoning: Because this step is explicitly identified as an elementary step, its molecularity determines its reaction order. The step is bimolecular in $\ce{NO2}$ , meaning the stoichiometric coefficient 2 becomes the exponent in the rate law.

Answer: $\text{Rate} = k[\ce{NO2}]^2$

Exam Trap: Do not write the rate law of an overall reaction using its stoichiometric coefficients! This direct relationship is valid only for individual elementary steps.

The Rate-Determining Step (RDS)

In a multistep mechanism, individual elementary steps proceed at varying rates. The slowest elementary step is called the rate-determining step (RDS). The RDS acts as a bottleneck, capping the overall rate of product formation.

Unimolecular Elimination Reaction Coordinate (Source: Kkukla · CC BY-SA 4.0 · Wikimedia Commons)

On a reaction coordinate energy diagram, the RDS corresponds to the step with the highest activation energy barrier ( $E_a$ ) relative to its starting materials.

Key Distinction: Intermediates vs. Transition States

Reaction Intermediate: A distinct molecule or ion formed in an early elementary step and consumed in a subsequent step. It sits in a local energy minimum (valley) on a reaction coordinate diagram and has a finite lifetime.
Transition State (Activated Complex): A temporary arrangement of breaking and forming bonds at a potential energy maximum (peak). It cannot be isolated.

Deriving Overall Rate Laws from the Slow Step

The rate law of the overall reaction is dictated by the rate law of the rate-determining step.

Case 1: The First Step is Rate-Determining

If the first step of a mechanism is the slowest, the overall rate law is simply the rate law of Step 1.

Example: Consider the two-step mechanism for $\ce{NO2 + CO -> NO + CO2}$ :

Step 1 (slow): $\ce{NO2 + NO2 -> NO3 + NO}$

Step 2 (fast): $\ce{NO3 + CO -> NO2 + CO2}$

Because Step 1 is the RDS, the overall rate law is: $\text{Rate} = k_1[\ce{NO2}]^2$

Notice that $\ce{CO}$ is zero-order because it reacts in a fast step after the RDS.

Case 2: Slow Step Preceded by a Fast Equilibrium Step

When the RDS is preceded by a fast reversible step, the tentative rate law of the RDS will contain a reaction intermediate. Because reaction intermediates cannot appear in an overall rate law, they must be substituted using the fast equilibrium step.

Worked Example: Rate Law with Fast Pre-Equilibrium

Proposed Mechanism:

Step 1 (fast equilibrium): $\ce{2NO <=> N2O2}$ (forward rate constant $k_1$ , reverse rate constant $k_{-1}$ )

Step 2 (slow): $\ce{N2O2 + O2 -> 2NO2}$ (rate constant $k_2$ )

Overall Reaction: $\ce{2NO + O2 -> 2NO2}$

Step-by-Step Derivation:

Write the rate law for the slow step (Step 2): $\text{Rate} = k_2[\ce{N2O2}][\ce{O2}]$
Identify intermediate: $\ce{N2O2}$ is formed in Step 1 and consumed in Step 2. It must be eliminated.
Set forward rate equal to reverse rate for Step 1 at equilibrium: $k_1[\ce{NO}]^2 = k_{-1}[\ce{N2O2}] \implies [\ce{N2O2}] = \frac{k_1}{k_{-1}}[\ce{NO}]^2$
Substitute $[\ce{N2O2}]$ into the RDS rate law: $\text{Rate} = k_2 \left(\frac{k_1}{k_{-1}}[\ce{NO}]^2\right) [\ce{O2}] = k_{\text{obs}}[\ce{NO}]^2[\ce{O2}]$

Answer: $\text{Rate} = k_{\text{obs}}[\ce{NO}]^2[\ce{O2}]$ where $k_{\text{obs}} = \frac{k_1 k_2}{k_{-1}}$ .

Exam Trap: Never leave an intermediate in the final rate law expression on the DAT! Always substitute it in terms of starting reactants.

Elementary Steps vs. Overall Reactions

Feature	Elementary Step	Overall Reaction
Rate Law Source	Stoichiometric coefficients	Experimental data or RDS mechanism
Molecularity	Defined (unimolecular, bimolecular)	Not applicable
Energy Profile	Single transition state (1 peak)	Multiple peaks & valleys (if multistep)
Intermediates	Can form or consume intermediates	Contains no intermediates in net equation

DAT Memory Trick — The Bottleneck Rule:

The Rate-Determining Step (RDS) is like a 1-lane toll bridge on a 4-lane highway. No matter how fast cars drive before or after, the bridge caps the total vehicle throughput! Also remember: Intermediates are In-and-Out — formed in one step, consumed in another, and NEVER permitted in the overall rate law.

Critical Thinking Questions

Why are termolecular elementary steps extremely rare in gas-phase chemical kinetics compared to unimolecular and bimolecular steps?
If a multi-step reaction mechanism features a fast reversible first step followed by a slow second step, how does doubling the concentration of a reactant present only in the first step affect the overall reaction rate?
How does adding a heterogeneous or homogeneous catalyst alter the activation energy of the rate-determining step without changing the net energy difference ( $\Delta H$ ) between reactants and products?

References

Sources returned by Google Search grounding during lesson verification.

Reaction mechanism and rate law (article) | Khan Academy — khanacademy.org
3.2.3: Rate Determining Step - Chemistry LibreTexts — chem.libretexts.org

Elementary Steps and the Rate-Determining Step

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