Integrated rate laws and half-life - Chemistry AP Study Notes
Overview
# Integrated Rate Laws and Half-Life: Summary This lesson covers the mathematical relationships between reactant concentration and time for zero-order, first-order, and second-order reactions, including their graphical representations and half-life expressions. Students learn to determine reaction orders from concentration-time data, calculate rate constants, and apply half-life concepts to solve kinetics problems—skills essential for AP Chemistry exam free-response questions on chemical kinetics. Mastery of integrated rate laws enables prediction of reactant/product concentrations at any time and is fundamental for understanding radioactive decay, pharmacokinetics, and real-world reaction monitoring.
Core Concepts & Theory
Integrated rate laws mathematically relate the concentration of reactants to time, derived from differential rate laws through integration. These equations allow us to predict concentrations at any point during a reaction.
Zero-Order Reactions: Rate = k, independent of concentration.
- Integrated form: [A] = -kt + [A]₀
- Linear plot: [A] vs time (slope = -k)
- Units of k: mol dm⁻³ s⁻¹
First-Order Reactions: Rate = k[A]
- Integrated form: ln[A] = -kt + ln[A]₀ or [A] = [A]₀e⁻ᵏᵗ
- Linear plot: ln[A] vs time (slope = -k)
- Units of k: s⁻¹
Second-Order Reactions: Rate = k[A]²
- Integrated form: 1/[A] = kt + 1/[A]₀
- Linear plot: 1/[A] vs time (slope = k)
- Units of k: dm³ mol⁻¹ s⁻¹
Half-life (t₁/₂) is the time required for reactant concentration to decrease to half its initial value.
- Zero-order: t₁/₂ = [A]₀/2k (depends on initial concentration)
- First-order: t₁/₂ = 0.693/k or ln2/k (constant, concentration-independent)
- Second-order: t₁/₂ = 1/(k[A]₀) (inversely proportional to [A]₀)
Key Mnemonic: "Z-F-S" - Zero order plots [A], First order plots ln[A], Second order plots 1/[A]
The constant half-life characteristic uniquely identifies first-order kinetics, crucial for radioactive decay and many biological processes.
Detailed Explanation with Real-World Examples
Real-World Applications:
Pharmaceutical Industry: Drug metabolism typically follows first-order kinetics. If aspirin has t₁/₂ = 20 minutes in blood plasma, after 20 minutes 50% remains, after 40 minutes 25% remains, and after 60 minutes 12.5% remains. This constant half-life helps doctors calculate dosing schedules.
Radioactive Dating: Carbon-14 dating uses first-order decay (t₁/₂ = 5,730 years). Archaeologists measure remaining ¹⁴C to determine artifact age. If a bone contains 25% of original ¹⁴C, it's approximately 11,460 years old (two half-lives).
Zero-Order Analogy: Imagine a tap dripping at constant rate into a bucket—the water level decreases linearly with time regardless of how much water remains. This mimics enzyme-catalyzed reactions at substrate saturation, where enzymes work at maximum capacity.
First-Order Analogy: Consider a rumor spreading where 50% of people who haven't heard it learn about it each hour. The rate depends on how many don't know (concentration), creating exponential decay.
Second-Order Analogy: Think of dancers pairing up at a party—the rate depends on finding two available partners simultaneously, proportional to [dancers]².
Environmental Chemistry: Pesticide degradation in soil often follows first-order kinetics. Understanding half-lives helps predict environmental persistence. DDT's long half-life (15 years) explains its bioaccumulation problems.
Cambridge Context: Exam questions frequently test ability to identify reaction order from concentration-time data or half-life patterns—essential skills for practical investigation analysis.
Worked Examples & Step-by-Step Solutions
**Example 1**: A first-order reaction has [A]₀ = 0.80 mol dm⁻³. After 120 s, [A] = 0.20 mol dm⁻³. Calculate (a) k and (b) t₁/₂. **Solution**: (a) Use: ln[A] = -kt + ln[A]₀ - ln(0.20) = -k(120) + ln(0.80) - -1.609 = -120k + (-0.223) - -1.609 + 0.223 = -120k - **k = 0.0116 s⁻¹** (b) t₁/₂ = 0.693/k =...
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Key Concepts
- Integrated Rate Laws: Equations that show how the concentration of a reactant changes over time during a chemical reaction.
- Half-Life (t1/2): The specific amount of time it takes for half of a reactant to be used up in a chemical reaction.
- Reaction Order: A number (like zero, first, or second) that tells us how the rate of a reaction depends on the concentration of its reactants.
- Rate Constant (k): A specific number for each reaction at a certain temperature that tells us how fast the reaction generally proceeds.
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Exam Tips
- →Always identify the reaction order first; this dictates which integrated rate law and half-life formula you should use.
- →Pay close attention to units, especially for the rate constant 'k' as they change with the reaction order.
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