10.4 Pre-equilibrium and rate-limiting steps

Pre-equilibrium and Rate-limiting Steps in Multi-step Reaction Mechanisms

Pre-equilibrium and rate-limiting steps

Most reactions don't happen in a single step. Instead, they proceed through a series of elementary steps, and two concepts help you make sense of these mechanisms: pre-equilibrium and rate-limiting steps.

A pre-equilibrium step is a fast, reversible step that reaches equilibrium long before the overall reaction finishes. The forward and reverse reactions in this step interconvert reactants and products so rapidly that their concentrations settle into a fixed ratio, even as the rest of the mechanism is still playing out. A classic example is the rapid formation and dissociation of an enzyme-substrate complex before the slower catalytic step occurs.

A rate-limiting step (also called the rate-determining step) is the slowest step in the mechanism. Because everything else waits on this step, it controls the overall reaction rate. It typically has the highest activation energy of any step in the mechanism, which is why it's the bottleneck.

Characteristics of pre-equilibrium steps

Pre-equilibrium steps have a low activation energy barrier in both directions, so molecules shuttle back and forth quickly. The result is that the concentrations of species involved in that step stay roughly constant over the timescale of the overall reaction, even though the reaction as a whole hasn't finished.

Because this step is fast and balanced, it doesn't limit the overall rate. What it does do is set the concentration of any reactive intermediate produced in that step. That's the real payoff: you can write an equilibrium expression for the fast step and use it to express the intermediate's concentration in terms of stable reactant concentrations. This is the equilibrium approximation, and it's what makes deriving rate laws for complex mechanisms tractable.

Identification of rate-limiting steps

To find the rate-limiting step, look for the step with the highest activation energy barrier. On an energy diagram, it's the tallest hill the reaction must climb. If you're given rate constants for each step, the rate-limiting step is the one with the smallest forward rate constant.

Why does this matter so much?

• The overall rate law mirrors the rate law of this single step (after substituting out any intermediates).
• Changes to conditions that speed up or slow down this step have the biggest effect on the overall rate. Adding a catalyst that lowers the activation energy of the rate-limiting step accelerates the whole reaction; doing the same for an already-fast step barely matters.
• Raising temperature increases all rate constants, but the rate-limiting step benefits the most (Arrhenius behavior hits hardest where $E_a$ is largest).

Rate equations for multi-step reactions

Deriving the overall rate law from a mechanism that includes a pre-equilibrium step follows a clear procedure:

1. Write out every elementary step with its rate constant (forward and reverse for reversible steps).
2. Identify which step is the pre-equilibrium (fast, reversible) and which is the rate-limiting step (slow).
3. Write the rate law for the rate-limiting step directly from its molecularity. For example, if the slow step is $I + C \rightarrow P$ with rate constant $k_2$, then $rate = k_2[I][C]$, where $I$ is an intermediate.
4. Eliminate the intermediate. Because the pre-equilibrium step is in equilibrium, set the forward and reverse rates equal: $k_1[A][B] = k_{-1}[I]$. Solve for the intermediate: $[I] = \frac{k_1}{k_{-1}}[A][B]$. Equivalently, you can write the equilibrium constant $K_{eq} = \frac{k_1}{k_{-1}}$ and express $[I] = K_{eq}[A][B]$.
5. Substitute back into the rate-limiting step's rate law: $rate = k_2 \cdot K_{eq}[A][B][C] = k_{obs}[A][B][C]$ where $k_{obs} = k_2 K_{eq}$ is the experimentally observed rate constant.
6. Check your result. The final rate law should contain only species that appear in the overall balanced equation (reactants and possibly products), with no intermediates remaining.

This approach works whenever there's a clear separation of timescales: one or more fast steps that stay in equilibrium, followed by (or surrounding) a single slow step. When that separation isn't clean, you may need the steady-state approximation instead, which is covered elsewhere in this unit.