Getting Started
Chemical kinetics is the study of reaction rates, exploring how quickly reactants are converted into products. On a macroscopic level, we can observe this by measuring how the concentration of a substance changes over time. This chapter focuses on a powerful graphical method for analyzing this change, allowing us to connect the observable decrease in reactant concentration to the underlying mathematical model, known as the rate law, that governs the reaction's speed.
What You Should Be Able to Do
After completing this section, you should be able to:
Analyze graphical data to determine if a reaction is zeroth, first, or second order with respect to a reactant.
Calculate the rate constant, k, for a reaction from the slope of the appropriate linear plot.
Define half-life and recognize that a constant half-life is a unique characteristic of first-order reactions.
Use the first-order half-life equation to solve for the rate constant or the half-life.
Identify radioactive decay as a common and important example of a first-order kinetic process.
Key Concepts & Analysis
We will explore how reactant concentrations change over time through the lens of Dynamics & Change. This approach helps us understand how a system evolves from an initial state by following a specific kinetic pathway.
Baseline Condition: The Start of the Reaction
Every reaction begins at time zero (t=0) with a specific initial concentration of reactants, which we denote as [A]₀. At this moment, the reaction has an initial rate that is determined by the rate law (e.g., Rate = k[A]₀ⁿ, where n is the reaction order). This initial state is our reference point for all subsequent changes.
The Process: Reactant Depletion Over Time
As the reaction A → Products proceeds, the concentration of reactant A, denoted [A]t, decreases. The crucial insight is that the functional form of this decrease depends on the reaction order. A simple plot of concentration versus time, [A] vs. t, will be a downward-sloping curve for first- and second-order reactions, but looking at this curve alone doesn't easily reveal the order. To determine the order, we must test the data to find a linear relationship. This is achieved by using the integrated rate laws, which are derived from the differential rate laws using calculus. You are not required to derive them, but you must know how to use their graphical forms.
The Resulting Change: Identifying Order Through Linearization
By manipulating the concentration data, we can create plots that will be linear for a specific reaction order. The plot that yields a straight line identifies the order of the reaction with respect to that reactant.
If a plot of [A] vs. time is linear, the reaction is zeroth-order.
The concentration decreases at a constant rate.
The slope of the line is equal to the negative of the rate constant (slope = -k).
If a plot of the natural log of [A], ln[A], vs. time is linear, the reaction is first-order.
The concentration decreases exponentially.
The slope of the line is equal to the negative of the rate constant (slope = -k).
If a plot of the inverse of [A], 1/[A], vs. time is linear, the reaction is second-order.
The concentration decreases, but more slowly over time than a first-order process.
The slope of the line is equal to the rate constant (slope = k). Note that the slope is positive in this case.
The First-Order Half-Life
A special and important characteristic of first-order reactions is the half-life (t₁/₂), defined as the time it takes for the reactant concentration to decrease to half of its initial value. For a first-order process, the half-life is constant and does not depend on the initial concentration. This means it takes the same amount of time for [A] to go from 1.0 M to 0.5 M as it does to go from 0.5 M to 0.25 M. This constant half-life is a definitive signature of a first-order reaction.
The half-life is related to the rate constant k by a simple equation:
t₁/₂ = 0.693 / k
This relationship allows for a quick calculation of k if the half-life is known, or vice-versa. Radioactive decay is the quintessential example of a first-order process. The half-life of Carbon-14, for instance, is about 5730 years, a constant value that is the basis for carbon dating.
Key Models & Representations
The relationship between reaction order and the corresponding graphical analysis is fundamental. The following table summarizes the key characteristics you must know to analyze kinetic data.
| Feature | Zeroth-Order | First-Order | Second-Order |
|---|---|---|---|
| Rate Law | Rate = k | Rate = k[A] | Rate = k[A]² |
| Linear Plot | [A] vs. time | ln[A] vs. time | 1/[A] vs. time |
| Slope of Line | -k | -k | +k |
| Half-Life (t₁/₂) | [A]₀ / 2k | 0.693 / k | 1 / (k[A]₀) |
| Half-Life Behavior | Decreases as [A] decreases | Constant | Increases as [A] decreases |
Key Terms, Quantities, & Concepts
Reaction Order: The exponent to which a reactant's concentration is raised in the rate law. It describes how the rate is affected by the reactant's concentration and must be determined experimentally.
Rate Law: An equation that mathematically links the rate of a reaction to the concentrations of the reactants. For example, Rate = k[A]ⁿ.
Rate Constant (k): The proportionality constant in the rate law. Its value is specific to a reaction at a given temperature and its units depend on the overall reaction order.
Integrated Rate Law: An equation that expresses the concentration of a reactant as a function of time. These are the equations that give rise to the linear plots used for analysis.
Half-life (t₁/₂): The time required for the concentration of a reactant to decrease to one-half of its initial value.
First-Order Reaction: A reaction whose rate is directly proportional to the concentration of a single reactant. These reactions are characterized by a constant half-life.
Second-Order Reaction: A reaction whose rate is proportional to the square of a single reactant's concentration or the product of the concentrations of two reactants.
Radioactive Decay: The spontaneous process by which an unstable atomic nucleus loses energy by radiation. This is a classic real-world example of a first-order kinetic process.
Skill Snapshots
Causation
Cause: A reaction is determined to be first-order. Effect: A plot of the natural logarithm of its concentration versus time will be a straight line with a slope of -k.
Cause: The half-life of a process is observed to be constant regardless of the starting amount. Effect: The process must follow first-order kinetics.
Cause: A reaction is second-order. Effect: The time required for the concentration to halve will double as the concentration is halved.
Comparison
A first-order reaction's half-life is constant, whereas a second-order reaction's half-life increases as the reactant is consumed.
The slope of the linear plot for a first-order reaction is negative (-k), while the slope for a second-order reaction is positive (+k).
In a zeroth-order reaction, the rate is independent of reactant concentration, while in a first-order reaction, doubling the concentration doubles the rate.
Change and Continuity Over Time
Baseline: At t=0, a first-order reaction begins with an initial concentration [A]₀.
Change 1: After one half-life has passed, the concentration will be exactly [A]₀ / 2.
Change 2: After a second half-life has passed (for a total time of 2 × t₁/₂), the concentration will be [A]₀ / 4.
Continuity: Throughout the entire reaction, the rate constant k and the half-life t₁/₂ remain constant, provided the temperature does not change.
Common Misconceptions & Clarifications
Misconception: You can determine the reaction order from the coefficients in the balanced chemical equation.
Clarification: Reaction orders are not related to stoichiometric coefficients. They must be determined experimentally by analyzing how reaction rate changes with concentration, for example, by using the graphical methods described in this chapter.
Misconception: The rate of a reaction is constant.
Clarification: Only for a zeroth-order reaction is the rate constant. For first- and second-order reactions, the rate is dependent on concentration and therefore decreases as reactants are consumed over time.
Misconception: Half-life is always a constant value for any reaction.
Clarification: A constant half-life is the unique and defining characteristic of a first-order reaction only. For all other orders, the half-life depends on the concentration of the reactants.
Misconception: A steeper slope on a concentration vs. time graph always means a larger rate constant, k.
Clarification: This is only true when comparing reactions of the same order. More importantly, the slope is directly related to k only on the appropriate linearized plot (e.g., ln[A] vs. t for first-order), not the direct [A] vs. t curve (unless it's zeroth-order). Remember that the slope is -k for zeroth/first order and +k for second order.
One-Paragraph Summary
To determine a reaction's order and rate constant from experimental data, we can monitor how reactant concentration changes over time. By plotting this data in three different ways—[A] vs. t, ln[A] vs. t, and 1/[A] vs. t—we can identify the reaction as zeroth, first, or second order, respectively, based on which plot yields a straight line. The slope of this linear graph is directly related to the rate constant, k, a fundamental descriptor of the reaction's intrinsic speed. First-order reactions, such as radioactive decay, are uniquely characterized by a constant half-life, which is independent of the initial concentration and provides a direct way to calculate the rate constant using the relationship t₁/₂ = 0.693/k. This graphical method provides a powerful bridge between macroscopic concentration data and the microscopic details of the rate law.