The Fundamentals of the Rate Law
At its core, the rate law is an algebraic equation that quantitatively links the reaction rate to the concentrations of the reactants. It is determined experimentally; it cannot be predicted from the stoichiometry of the balanced chemical equation alone. The general form of a rate law for a reaction where substances A and B react to form products is:
Rate = k [A]^m [B]^n
Each component of this equation holds specific meaning:
- Rate: The speed of the reaction, typically expressed in units of concentration per time (e.g., M/s, mol/L·s).
- k: The rate constant, a proportionality constant that is specific to a particular reaction at a given temperature. Its units vary depending on the overall reaction order.
- [A], [B]: The molar concentrations of the reactants A and B, respectively.
- m, n: The reaction orders with respect to A and B, respectively. These are typically small positive integers (0, 1, 2) but can sometimes be fractions or negative numbers. They define how the rate is affected by the concentration of each reactant.
The overall reaction order is the sum of the individual orders: (m + n). A reaction with an overall order of 1 is called first-order, an overall order of 2 is second-order, and so on.
The power of the rate law lies in its predictive capability. Once the values of k, m, and n are determined through experiment, the rate of the reaction can be calculated for any set of reactant concentrations. This is indispensable for controlling industrial processes, understanding metabolic pathways in biochemistry, and predicting the behavior of pollutants in the environment.
The Critical Distinction: Stoichiometry vs. Reaction Order
A common point of confusion arises from assuming the exponents in the rate law (m and n) are the same as the coefficients in the balanced equation. This is generally not true. The stoichiometric coefficients indicate the proportional amounts of reactants consumed and products formed, while the reaction orders describe the kinetic dependence of the rate on concentration.
For example, consider the balanced equation for the decomposition of nitrogen dioxide:
2 NO₂(g) → 2 NO(g) + O₂(g)
The experimentally determined rate law for this reaction is:
Rate = k [NO₂]²
The reaction is second-order with respect to NO₂ and second-order overall. The exponent (2) matches the coefficient (2) in this specific case, but this is coincidental. Contrast this with the reaction of nitrogen monoxide with chlorine:
2 NO(g) + Cl₂(g) → 2 NOCl(g)
The experimentally determined rate law is:
Rate = k [NO]² [Cl₂]
Here, the order with respect to NO (2) matches its coefficient, but the order with respect to Cl₂ is 1, not the coefficient of 1. This illustrates that the reaction mechanism—the step-by-step pathway by which the reaction occurs—dictates the rate law, not the overall stoichiometry.
Determining Reaction Order: The Method of Initial Rates
The most reliable and straightforward method for determining the reaction order is the Method of Initial Rates. This technique isolates the effect of each reactant’s concentration on the initial rate of the reaction, minimizing complications from reverse reactions or product interference.
The procedure involves a series of controlled experiments:
- Conduct Multiple Trials: The reaction is run several times, with each trial starting with different known concentrations of the reactants. Crucially, only the concentration of one reactant is changed between a pair of trials, while the concentrations of all others are held constant.
- Measure the Initial Rate: For each trial, the instantaneous rate is measured at the very beginning of the reaction (t ≈ 0). This is often done by monitoring the change in concentration of a reactant or product over a very short initial time interval using techniques like spectrophotometry, pressure measurement, or titration.
- Analyze the Ratio of Rates: By comparing the initial rates from two trials where only one reactant’s concentration changes, the order with respect to that reactant can be determined mathematically.
A Detailed Example: Determining the Rate Law for a Reaction
Consider a hypothetical reaction: A + B → C
Experimental Data:
| Experiment | Initial [A] (M) | Initial [B] (M) | Initial Rate (M/s) |
|---|---|---|---|
| 1 | 0.100 | 0.100 | 4.0 x 10⁻⁵ |
| 2 | 0.200 | 0.100 | 8.0 x 10⁻⁵ |
| 3 | 0.100 | 0.200 | 16.0 x 10⁻⁵ |
Step 1: Find the order with respect to A (m).
Compare Experiments 1 and 2, where [B] is held constant and [A] doubles (from 0.100 M to 0.200 M).
- The rate changes from 4.0 x 10⁻⁵ M/s to 8.0 x 10⁻⁵ M/s. This is also a doubling of the rate.
- Set up the ratio of the rate laws for the two experiments:
- Rate₂ / Rate₁ = (k [A₂]^m [B₂]^n) / (k [A₁]^m [B₁]^n)
- Since k and [B] are constant, they cancel out:
- Rate₂ / Rate₁ = ([A₂] / [A₁])^m
- Plug in the values:
- (8.0 x 10⁻⁵) / (4.0 x 10⁻⁵) = (0.200 / 0.100)^m
- 2 = (2)^m
- Therefore, m = 1. The reaction is first-order in A.
Step 2: Find the order with respect to B (n).
Compare Experiments 1 and 3, where [A] is held constant and [B] doubles (from 0.100 M to 0.200 M).
- The rate changes from 4.0 x 10⁻⁵ M/s to 16.0 x 10⁻⁵ M/s. This is a quadrupling of the rate (a factor of 4).
- Set up the ratio:
- Rate₃ / Rate₁ = ([B₃] / [B₁])^n
- Plug in the values:
- (16.0 x 10⁻⁵) / (4.0 x 10⁻⁵) = (0.200 / 0.100)^n
- 4 = (2)^n
- Therefore, n = 2. The reaction is second-order in B.
Step 3: Write the complete rate law.
Rate = k [A]¹ [B]²
The overall reaction order is 1 + 2 = 3 (third-order).
Step 4: Calculate the rate constant (k).
Use the data from any single experiment. Using Experiment 1:
- Rate = k [A] [B]²
- 4.0 x 10⁻⁵ M/s = k (0.100 M) (0.100 M)²
- 4.0 x 10⁻⁵ M/s = k (0.100 M) (0.0100 M²)
- 4.0 x 10⁻⁵ M/s = k (0.00100 M³)
- k = (4.0 x 10⁻⁵ M/s) / (0.00100 M³)
- k = 4.0 x 10⁻² M⁻²s⁻¹
The units for the rate constant (M⁻²s⁻¹) confirm the overall third-order reaction.
Integrated Rate Laws: Connecting Concentration and Time
While the differential rate law (Rate = k[A]^m) expresses how rate depends on concentration, the integrated rate law is a different form that shows how the concentration of a reactant changes over time. This is extremely useful for predicting how long a reaction will take to reach a certain point. Each reaction order has a unique integrated rate law that produces a characteristic straight-line plot.
First-Order Reactions
For a reaction that is first-order in a single reactant A: Rate = k [A]
The integrated rate law is: ln[A]ₜ = -kt + ln[A]₀
Where:
- [A]ₜ is the concentration of A at time t
- [A]₀ is the initial concentration of A at time zero
- k is the rate constant
This equation has the form of a straight line (y = mx + b). A plot of ln[A]ₜ versus time (t) will yield a straight line with a slope equal to -k and a y-intercept equal to ln[A]₀. The half-life (t₁/₂)—the time required for the concentration of A to decrease to half of its initial value—is constant for a first-order reaction and is given by: t₁/₂ = ln(2) / k ≈ 0.693 / k. Radioactive decay is a classic example of a first-order process.
Second-Order Reactions
For a reaction that is second-order in a single reactant A: Rate = k [A]²
The integrated rate law is: 1/[A]ₜ = kt + 1/[A]₀
A plot of 1/[A]ₜ versus time (t) will yield a straight line with a slope equal to k and a y-intercept equal to 1/[A]₀. The half-life for a second-order reaction is not constant; it depends on the initial concentration: t₁/₂ = 1 / (k [A]₀).
Zero-Order Reactions
For a zero-order reaction, the rate is independent of the concentration of the reactant: Rate = k
The integrated rate law is: [A]ₜ = -kt + [A]₀
A plot of [A]ₜ versus time (t) will yield a straight line with a slope equal to -k and a y-intercept equal to [A]₀. The half-life is given by t₁/₂ = [A]₀ / (2k). Zero-order kinetics are often observed in surface-catalyzed reactions, such as the decomposition of nitrous oxide on a hot platinum surface, where the reaction rate is limited by the available surface area, not the gas concentration.
The Link to Reaction Mechanisms
The ultimate goal of kinetic studies is often to propose a plausible reaction mechanism—a series of simple elementary steps that sum to the overall reaction. The rate law provides the most critical clue for this task.
For an elementary step (a single, simple reaction event), the rate law is directly determined by its molecularity (the number of molecules colliding). The exponent for a reactant in an elementary step is equal to its stoichiometric coefficient in that step.
- Unimolecular step (A → products): Rate = k [A]
- Bimolecular step (A + B → products): Rate = k [A][B]
The rate law for the overall reaction is determined by the slowest step in the mechanism, known as the rate-determining step (RDS). If a proposed mechanism is correct, the rate law derived from it must match the experimentally determined rate law. For instance, the reaction 2 NO₂(g) + F₂(g) → 2 NO₂F(g) has the experimental rate law: Rate = k [NO₂][F₂]. This suggests a mechanism where the RDS is a single bimolecular collision between one NO₂ molecule and one F₂ molecule, which is consistent with the observed kinetics.