14.2 Reaction mechanisms and elementary steps

Chemical reactions often proceed through multiple molecular events rather than a single transformation. Reaction mechanisms map out these individual events, revealing how bonds break and form along the path from reactants to products. Understanding mechanisms is essential for predicting rate laws, identifying bottlenecks, and ultimately controlling reaction outcomes.

Elementary steps are the irreducible building blocks of any mechanism. Each one represents a single molecular event, such as a bond dissociation or a collision between two species. By analyzing these steps, you can identify the rate-determining step and derive the rate law for the overall reaction.

Reaction Mechanisms and Overall Reactions

Reaction Mechanisms: Detailed, Step-by-Step Descriptions

A reaction mechanism is a proposed sequence of elementary steps that describes, at the molecular level, how reactants become products. It includes every intermediate species and transition state along the way, specifying which bonds break, which form, and how atoms rearrange at each stage.

The overall (net) reaction is what you get when you sum all the elementary steps together:

• Intermediate species appear in individual steps but cancel out when you add the steps, so they don't show up in the net equation.
• The stoichiometry of the overall reaction follows directly from the stoichiometry of the individual steps. If the steps don't sum to give the known overall equation, the proposed mechanism is inconsistent.

Rate-Determining Step and Overall Reaction Rate

In a multi-step mechanism, one step is typically much slower than the others. This rate-determining step (RDS) acts as a bottleneck: no matter how fast the other steps are, the overall reaction can't proceed faster than this slowest step.

Why is it the slowest? The RDS has the highest activation energy barrier among all the elementary steps, so molecules need the most energy to get through it.

Because the RDS limits the overall rate, the rate law for the overall reaction can be derived from the rate law of the RDS alone. There's an important caveat, though: if the RDS rate law contains the concentration of an intermediate (a species produced in an earlier step), you can't leave it that way. Experimentally measurable rate laws are written only in terms of reactant concentrations. To eliminate the intermediate, you apply either a pre-equilibrium assumption or the steady-state approximation (discussed below).

Elementary Steps and Rate Laws

Characteristics of Elementary Steps

Elementary steps are the simplest possible molecular events in a mechanism. Each one occurs in a single collision or molecular rearrangement and cannot be broken down further. Key types include:

• Unimolecular dissociation: A single molecule breaks apart. $AB \rightarrow A + B$
• Bimolecular collision: Two species collide and react. $A + B \rightarrow AB$
• Unimolecular isomerization: A single molecule rearranges its internal structure without breaking into separate fragments.

Every elementary step passes through exactly one transition state on the potential energy surface.

Rate Laws for Elementary Steps

Here's a critical distinction: for elementary steps (and only for elementary steps), you can write the rate law directly from the stoichiometry. The reaction order with respect to each reactant equals its stoichiometric coefficient in that step.

This works because an elementary step represents an actual molecular event. If two molecules of $A$ must collide with one molecule of $B$ in a single step, the rate naturally depends on $[A]^2[B]$.

You cannot do this for an overall reaction. The rate law for a multi-step overall reaction must be determined experimentally or derived from the mechanism. Writing rate laws from stoichiometric coefficients is a privilege reserved for elementary steps.

Rate-Determining Step in Mechanisms

Identifying the Rate-Determining Step

To identify the RDS in a proposed mechanism:

1. Compare the activation energies (or equivalently, the rate constants) of each elementary step.
2. The step with the highest activation energy (smallest rate constant) is the RDS.
3. Verify that the rate law derived from this step is consistent with the experimentally observed rate law.

If the derived and experimental rate laws don't match, the proposed mechanism (or your choice of RDS) needs to be revised.

Steady-State Approximation for Intermediates

When the rate law for the RDS includes an intermediate, you need a way to express that intermediate's concentration in terms of measurable reactant concentrations. The steady-state approximation assumes that, after a brief induction period, the intermediate's concentration stays roughly constant because its rate of formation equals its rate of consumption.

Consider this two-step mechanism:

• Step 1: $A + B \rightleftharpoons C$ (fast, reversible)
• Step 2: $C + D \rightarrow E$ (slow, rate-determining)

The RDS rate law is $\text{Rate} = k_2[C][D]$, but $C$ is an intermediate. Applying the steady-state approximation to $C$:

$\frac{d[C]}{dt} = k_1[A][B] - k_{-1}[C] - k_2[C][D] \approx 0$

Solving for $[C]$:

$[C] = \frac{k_1[A][B]}{k_{-1} + k_2[D]}$

Substituting back gives a rate law entirely in terms of reactants. Notice that if $k_{-1} \gg k_2[D]$ (the back reaction of step 1 is much faster than step 2), this simplifies to $\text{Rate} = \frac{k_1 k_2}{k_{-1}}[A][B][D]$, which is the same result you'd get from a pre-equilibrium approximation on step 1.

Molecularity vs Reaction Order

Molecularity of Elementary Steps

Molecularity is the number of reactant molecules (or ions) that participate in a single elementary step. It's a theoretical, integer property of that step:

• Unimolecular (molecularity = 1): A single molecule dissociates or isomerizes. $A \rightarrow B + C$
• Bimolecular (molecularity = 2): Two species collide and react. $A + B \rightarrow C$
• Termolecular (molecularity = 3): Three species collide simultaneously. $A + B + C \rightarrow D$

Termolecular steps are rare in practice. The probability of three molecules colliding at the same instant with the correct orientation and sufficient energy is very low. When a reaction appears to require three bodies, it usually proceeds through two successive bimolecular steps instead.

Reaction Order and Rate Laws

For elementary steps, molecularity directly determines the rate law:

The distinction between molecularity and reaction order matters most for overall reactions. Molecularity is defined only for elementary steps and is always a positive integer. Reaction order, on the other hand, is an empirical quantity for overall reactions and can be fractional, zero, or even negative with respect to a given species. When someone asks you to compare the two, the key point is: they coincide for elementary steps but can diverge for complex, multi-step reactions.