5.3 Concentration Changes Over Time

Understanding the Math Behind Rate Laws in Kinetics

What Are Rate Laws?

In chemical kinetics, we often encounter the phenomenon that increasing the concentration of a reactant accelerates the reaction rate. But exactly how much faster? This is where the rate law comes in.

A rate law describes the relationship between the rate of a chemical reaction and the concentrations of its reactants. It’s expressed as:

$$R = k[A]^n[B]^m$$

where:

• R is the reaction rate.
• k is the rate constant.
• [A] and [B] are reactant concentrations.
• n and m are reaction orders that indicate how changes in concentration affect the rate.

The reaction order tells us how the concentration of a reactant impacts the reaction rate. If the reaction order for A is 2, doubling [A] will quadruple the reaction rate. The overall reaction order is the sum of individual orders (e.g., if n = 2 and m = 1, the overall order is 3).

Key Concepts for the AP Exam

• Reaction orders (n): These can be integers or fractions but are most commonly 0, 1, or 2 for AP Chemistry.
• Rate constant (k): This is unique to each reaction and varies with temperature.
• Rate law determination: This is always done experimentally; you cannot deduce it solely from the chemical equation.

Calculating Reaction Orders Using Experiments

Rate laws can only be found by experimenting. Chemists measure the reaction rate with varying concentrations of reactants. Consider this example reaction:

$$2NO + 2H_2 \rightarrow N_2 + 2H_2O$$

Given experimental data for different concentrations, we can determine the order of each reactant by observing how rate changes with changes in concentration.

Example Calculation

• Compare Experiments 1 and 2: Doubling [NO] while keeping [H₂] constant quadruples the rate. Thus, the reaction is second-order with respect to NO.
• Compare Experiments 2 and 3: Doubling [H₂] doubles the rate. The reaction is first-order with respect to H₂.

The rate law becomes:

$$R = k[NO]^2[H_2]$$

Understanding the Rate Constant (k)

k is a proportionality constant that quantifies reaction speed. It depends on temperature and changes with the overall reaction order.

Units of k Based on Reaction Order:

• Zeroth Order: Units are M/s.
• First Order: Units are s⁻¹.
• Second Order: Units are M⁻¹s⁻¹.

Integrated Rate Laws: How Concentration Changes Over Time

Integrated rate laws describe how the concentration of a reactant changes over time. For the AP exam, focus on the following key integrated rate laws:

• Zeroth-Order Reaction: $[A] = [A]_0 – kt$
  • Graph: $[A]$ vs. time is linear with slope $-k$.
• First-Order Reaction: $\ln[A] = \ln[A]_0 – kt$
  • Graph: $\ln[A]$ vs. time is linear with slope $-k$.
• Second-Order Reaction: $\frac{1}{[A]} = \frac{1}{[A]_0} + kt$
  • Graph: $1/[A]$ vs. time is linear with slope $k$.

Example Problem

For the reaction $A \rightarrow B$:

Given data shows:

• Linear relationship for $1/[A]$ vs. time.

Solution:

• a) The rate law is R = k[A]² (second-order reaction).
• b) Calculate k using slope from graph data points.
• c) Use the second-order integrated rate law to find concentration after 30 seconds.

Half-Life of First-Order Reactions

The half-life of a first-order reaction (time it takes for half of the reactant to be consumed) is given by:

$$t_{1/2} = \frac{0.693}{k}$$

For first-order processes, half-life remains constant regardless of the initial concentration.