Getting Started
In the intricate dance of chemical reactions, the overall balanced equation often hides the complex sequence of steps that molecules actually take. This sequence, called the reaction mechanism, usually involves a bottleneck—a single slow step that dictates the overall speed of the reaction. This chapter explores the scenario where this bottleneck is not the first step, creating a chemical puzzle: how to determine the reaction's rate law when the slow step involves a fleeting, unmeasurable species called a reaction intermediate.
What You Should Be able to Do
By the end of this section, you should be able to:
Explain why a valid rate law cannot include the concentration of a reaction intermediate.
Describe the conditions under which the pre-equilibrium approximation is used.
Derive the overall rate law for a multi-step reaction mechanism where a fast, reversible step precedes the slow, rate-determining step.
Connect the derived rate law to the elementary steps of the proposed mechanism.
Key Concepts & Analysis
Dominant Lens: Process & Causation
Deriving a rate law from a mechanism with a fast initial step is a logical process. We start with a proposed mechanism, identify a problem with the initial rate law, and apply a series of steps using the pre-equilibrium approximation to arrive at a valid, testable expression.
Let's analyze the process for the reaction:
2NO(g) + O₂(g) → 2NO₂(g)
A proposed mechanism for this reaction is:
Step 1: NO(g) + NO(g) ⇌ N₂O₂(g) (fast, reversible)
Step 2: N₂O₂(g) + O₂(g) → 2NO₂(g) (slow)
Inputs & Preconditions
Inputs: The primary inputs are the elementary steps of the proposed reaction mechanism and the identification of the slow, rate-determining step (RDS). In our example, Step 2 is the slow step.
Preconditions: The critical precondition for this method is that a fast, reversible step occurs before the slow step. This allows a dynamic equilibrium to be established, where the intermediate (N₂O₂) is formed and reverts back to reactants much more quickly than it is consumed in the slow step.
Key Steps in the Derivation Process
The core problem is that the rate-determining step involves N₂O₂, a reaction intermediate. A valid rate law can only include reactants or catalysts—species with measurable initial concentrations. The process below removes the intermediate from the expression.
Write the Initial Rate Law from the Slow Step: The overall reaction rate is governed by its slowest step. The rate for elementary Step 2 is based on its reactants.
Rate = k₂[N₂O₂][O₂]
Problem: This expression contains the intermediate [N₂O₂], which we cannot directly measure or control.
Apply the Pre-Equilibrium Approximation: Because Step 1 is fast and reversible, it quickly reaches a state of equilibrium. At equilibrium, the rate of the forward reaction equals the rate of the reverse reaction.
Let k₁ be the rate constant for the forward reaction of Step 1, and k₋₁ be the rate constant for the reverse reaction.
Rate_forward1 = Rate_reverse1
k₁[NO][NO] = k₋₁[N₂O₂]
k₁[NO]² = k₋₁[N₂O₂]
Solve for the Intermediate Concentration: Rearrange the equilibrium expression from Step 2 to isolate the concentration of the intermediate, [N₂O₂].
- [N₂O₂] = (k₁/k₋₁)[NO]²
Substitute and Simplify: Substitute the expression for [N₂O₂] from Step 3 back into the initial rate law from Step 1. This replaces the unmeasurable intermediate with measurable reactants.
Rate = k₂ * ( (k₁/k₋₁)[NO]² ) * [O₂]
Combine the individual rate constants (k₁, k₋₁, k₂) into a single observed rate constant, k_obs.
Rate = k_obs[NO]²[O₂]
Outputs & Effects
Output: The final derived rate law is Rate = k_obs[NO]²[O₂]. This expression is second-order with respect to NO and first-order with respect to O₂.
Effect: This derived rate law can now be compared to a rate law determined from experimental data. If the derived law matches the experimental law, it provides strong evidence that the proposed mechanism is plausible. If they do not match, the mechanism must be incorrect.
Controls & Limiting Factors
Control: The overall reaction rate is controlled by the slow step (Step 2), which acts as a kinetic bottleneck.
Limiting Factors: The rate is limited by the concentrations of the reactants that appear in the final rate law. Increasing the concentration of NO or O₂ will increase the reaction rate, as predicted by the derived expression.
Key Models & Representations
This flowchart models the logical process for applying the pre-equilibrium approximation.
| Step in Process | Action | Mathematical Representation |
|---|---|---|
| 1. Identify Bottleneck | Write the rate law for the slow, rate-determining step. | Rate = k_slow[Intermediate][Reactant] |
| 2. Apply Equilibrium | For the fast, reversible first step, set the forward reaction rate equal to the reverse reaction rate. | k_forward[Reactants] = k_reverse[Intermediate] |
| 3. Isolate Intermediate | Algebraically solve the equilibrium equation for the concentration of the intermediate. | [Intermediate] = (k_forward/k_reverse)[Reactants] |
| 4. Substitute & Finalize | Substitute the expression for the intermediate into the slow step's rate law and combine constants. | Rate = k_obs[Reactants] |
Key Terms, Quantities, & Concepts
Reaction Mechanism: The step-by-step sequence of elementary reactions by which an overall chemical change occurs.
Elementary Step: An individual reaction in a mechanism that describes a single molecular event, such as a collision. The rate law for an elementary step can be written directly from its stoichiometry.
Reaction Intermediate: A chemical species that is formed in one step of a mechanism and consumed in a subsequent step. Intermediates do not appear in the overall balanced chemical equation.
Rate-Determining Step (RDS): The slowest elementary step in a reaction mechanism. It limits the overall rate of the reaction, much like the narrowest section of a funnel limits the flow of liquid.
Pre-Equilibrium Approximation: A kinetic assumption that a fast, reversible step preceding the rate-determining step achieves equilibrium. This allows the concentration of a reaction intermediate to be expressed in terms of reactant concentrations.
Rate Law: An equation that links the rate of a reaction to the concentrations of reactants (and sometimes catalysts). It must be determined experimentally or derived from a proposed mechanism.
Rate Constant (k): A temperature-dependent proportionality constant in the rate law. In a multi-step mechanism, the observed rate constant (k_obs) is often a composite of the rate constants from several elementary steps.
Skill Snapshots
Causation
Cause: The first step of a mechanism is fast and reversible.
Effect: A dynamic equilibrium is established, allowing the forward and reverse rates to be set equal to each other.
Cause: A reaction intermediate appears in the rate law written from the slow step.
Effect: The pre-equilibrium approximation must be used to substitute the intermediate's concentration with an expression involving only reactants.
Cause: The second step of the mechanism is significantly slower than the first.
Effect: The second step becomes the rate-determining step, and its kinetics govern the overall reaction rate.
Comparison
A rate law derived from a slow first step is written directly from that step's reactants, whereas a rate law derived using the pre-equilibrium approximation is initially written from the slow second step and then modified.
The rate constant
kfor an elementary step is a fundamental constant for that specific molecular collision, whereas the observed rate constantk_obsin the final rate law is often a composite of multiple elementary rate constants (e.g., k_obs = k₂k₁/k₋₁).Reactants are present at the beginning of a reaction and their concentrations decrease over time, whereas intermediates are produced and then consumed during the reaction, keeping their concentrations low and relatively steady.
Change and Continuity Over Time (CCOT)
Baseline: A fast, reversible reaction establishes a dynamic equilibrium, creating a small but relatively constant concentration of the intermediate (N₂O₂).
Change 1: The intermediate is steadily siphoned off by the slow second step to form the final products (NO₂).
Change 2: As the intermediate is consumed, the equilibrium of the first step shifts to produce more intermediate, attempting to compensate for the loss.
Continuity: Throughout the reaction, the ratio of the intermediate's concentration to the reactants' concentrations (e.g., [N₂O₂]/[NO]²) remains approximately constant, as dictated by the equilibrium condition of the first step.
Common Misconceptions & Clarifications
Misconception: The rate law for a reaction can be determined from the coefficients of the overall balanced equation.
- Clarification: The rate law is determined by the rate-determining step of the reaction mechanism. The exponents in the rate law (reaction orders) only match the stoichiometric coefficients if the reaction is an elementary step, which is rarely the case for an overall reaction.
Misconception: It is acceptable to have a reaction intermediate in the final rate law expression.
- Clarification: A valid rate law must be experimentally testable. Since the concentrations of intermediates are typically transient and too low to measure easily, the rate law must be expressed only in terms of species whose concentrations are known and controllable, such as the initial reactants.
Misconception: At equilibrium, the reactions have stopped.
- Clarification: Chemical equilibrium is a dynamic state. In the pre-equilibrium step, the forward reaction (forming the intermediate) and the reverse reaction (breaking the intermediate down into reactants) are occurring at equal rates, resulting in no net change in concentration.
One-Paragraph Summary
When a reaction mechanism involves a slow step that is preceded by a fast, reversible step, the rate law cannot be written directly from the slow step because it will contain a reaction intermediate. To resolve this, the pre-equilibrium approximation is used. This method assumes the initial fast step reaches dynamic equilibrium, allowing the rate of its forward reaction to be set equal to the rate of its reverse reaction. By solving this equality for the intermediate's concentration, we can substitute it into the rate law for the slow step. The result is a valid, composite rate law expressed solely in terms of measurable reactants, which can then be compared with experimental data to test the plausibility of the proposed mechanism.