AP Chemistry: Reaction Rates and Rate Laws (Differential and Integrated)

AP Chemistry – Reaction Rates and Rate Laws (Differential and Integrated)

AP Chemistry: Reaction Rates and Rate Laws (Differential and Integrated) – Exam-Ready Study Guide

What This Is

Reaction rates measure how fast reactants disappear or products form, while rate laws mathematically describe how reactant concentrations affect speed. This topic is high-yield on the AP exam—expect 1–2 FRQs and 3–5 MCQs on kinetics, including experimental design, rate law determination, and integrated rate law graphs. Real-world example: The decomposition of hydrogen peroxide (H?O?) is slow at room temperature but speeds up dramatically with a catalyst (like manganese dioxide in elephant toothpaste demos). Without understanding rate laws, we couldn’t design efficient industrial reactions (e.g., Haber process for ammonia) or explain why some reactions (like rusting) take years while others (like explosions) happen in milliseconds.

Key Terms & Concepts

• Reaction rate: Change in concentration of a reactant or product per unit time (units: M/s or mol/L·s).

• Example: If [A] decreases from 0.50 M to 0.30 M in 20 s, rate = (0.50 – 0.30) M / 20 s = 0.01 M/s.

• Differential rate law: Shows how rate depends on reactant concentrations.

• General form: rate = k[A]?[B]?, where:

  • k = rate constant (units vary with reaction order).
  • [A], [B] = reactant concentrations.
  • n, m = reaction orders (determined experimentally, not from coefficients!).

• Reaction order: Exponent in the rate law (e.g., zero-order, first-order, second-order).

• Zero-order: Rate = k (rate doesn’t depend on [A]).
• First-order: Rate = k[A] (rate doubles if [A] doubles).

• Second-order: Rate = k[A]² (rate quadruples if [A] doubles).

• Rate constant (k): A proportionality constant that depends on temperature and activation energy (not concentration!).

• Units: Vary by order (e.g., M?¹s?¹ for second-order, s?¹ for first-order).

• Integrated rate laws: Relate concentration to time for a given order.

• First-order: ln[A]? = –kt + ln[A]? (linear plot: ln[A] vs. t).
• Second-order: 1/[A]? = kt + 1/[A]? (linear plot: 1/[A] vs. t).

• Zero-order: [A]? = –kt + [A]? (linear plot: [A] vs. t).

• Half-life (t?/?): Time for reactant concentration to drop to half its initial value.

• First-order: t?/? = ln(2)/k (constant, independent of [A]?).
• Second-order: t?/? = 1/(k[A]?) (depends on initial concentration).

• Zero-order: t?/? = [A]?/(2k) (depends on initial concentration).

• Collision theory: Reactions occur when molecules collide with sufficient energy (activation energy, E?) and proper orientation.

• Example: NO and O? react only if they collide with enough energy to break/form bonds.

• Arrhenius equation: Relates rate constant to temperature and activation energy.

• k = Ae^(–E?/RT), where:

  • A = frequency factor (collision frequency).
  • E? = activation energy (J/mol).
  • R = 8.314 J/mol·K.
  • T = temperature (K).

• Catalyst: Speeds up a reaction by lowering E? (not consumed in the reaction).

• Example: Enzymes (e.g., catalase) break down H?O? 10? times faster than uncatalyzed.

• Reaction mechanism: Step-by-step sequence of elementary reactions that sum to the overall reaction.

• Rate-determining step (RDS): The slowest step in the mechanism (determines the rate law).

Step-by-Step: Solving Rate Law Problems

1. Determine the Rate Law from Experimental Data

Problem: Given initial rates for different reactant concentrations, find the rate law and k. Steps:
1. Identify pairs of trials where only one reactant’s concentration changes. - Example: Compare Trial 1 and 2 if [A] changes but [B] stays constant.
2. Compare rates to find the order for that reactant. - If [A] doubles and rate doubles-first-order in A. - If [A] doubles and rate quadruples-second-order in A. - If [A] changes but rate stays the same-zero-order in A.
3. Repeat for all reactants to find all orders.
4. Write the rate law (e.g., rate = k[A]²[B]¹).
5. Solve for k by plugging in data from any trial (include units!).

2. Use Integrated Rate Laws to Find Concentration or Time

Problem: Given a first-order reaction with k = 0.02 s?¹, find [A] after 30 s if [A]? = 0.50 M. Steps:
1. Pick the correct integrated rate law based on order (here, first-order).
2. Plug in known values: - ln[A]? = –(0.02 s?¹)(30 s) + ln(0.50 M).
3. Solve for [A]?: - ln[A]? = –0.6 + (–0.693) = –1.293. - [A]? = e^(–1.293)-0.27 M.

3. Determine Reaction Order from Graphs

Problem: Given concentration vs. time data, identify the reaction order. Steps:
1. Plot [A] vs. t-if linear-zero-order.
2. Plot ln[A] vs. t-if linear-first-order.
3. Plot 1/[A] vs. t-if linear-second-order.
4. Slope of the line = –k (zero/first-order) or +k (second-order).

4. Calculate Half-Life

Problem: For a first-order reaction with k = 0.05 s?¹, find t?/?. Steps:
1. Use the first-order half-life formula: t?/? = ln(2)/k.
2. Plug in k: t?/? = 0.693 / 0.05 s?¹-13.9 s.

Common Mistakes

• Mistake: Assuming reaction orders match stoichiometric coefficients.

• Correction: Orders are experimentally determined (e.g., 2NO + O?-2NO? is second-order in NO and first-order in O?, not third-order overall).

• Mistake: Forgetting units for k.

• Correction: Units depend on order (e.g., M?¹s?¹ for second-order, s?¹ for first-order). Always include them!

• Mistake: Mixing up integrated rate law plots.

• Correction:

  • ln[A] vs. t-first-order (slope = –k).
  • 1/[A] vs. t-second-order (slope = +k).
  • [A] vs. t-zero-order (slope = –k).

• Mistake: Confusing half-life dependence on [A]?.

• Correction:

  • First-order: t?/? is constant (independent of [A]?).
  • Second/zero-order: t?/? depends on [A]?.

• Mistake: Misapplying the Arrhenius equation.

• Correction: T must be in Kelvin, and E? must match R’s units (J/mol, not kJ/mol).

AP Exam Insights

1. FRQ Hot Topics:
2. Experimental design: Given data, determine the rate law, k, or reaction order (e.g., “Explain how you would experimentally find the order with respect to A”).
3. Graph interpretation: Identify reaction order from a plot (e.g., “Which graph would be linear for a first-order reaction?”).

4. Mechanisms: Given a mechanism, identify the RDS and write the rate law (e.g., “If the first step is slow, what is the rate law?”).

5. MCQ Traps:

6. Units for k: AP loves testing if you know k’s units (e.g., “What are the units for k in a second-order reaction?”).
7. Half-life dependence: “Which order has a half-life independent of initial concentration?” (Answer: first-order).

8. Catalysts vs. intermediates: Catalysts appear first as reactants, intermediates appear first as products (e.g., “Is Cl a catalyst or intermediate in the ozone depletion mechanism?”).

9. Tricky Distinctions:

10. Rate vs. rate constant: Rate changes with concentration; k is constant at a given temperature.

11. Differential vs. integrated rate laws: Differential = rate vs. concentration; integrated = concentration vs. time.

12. Lab-Based Questions:

13. AP often includes spectrophotometry data (absorbance vs. time) to test integrated rate laws. Remember: Absorbance-concentration for dilute solutions.

Quick Check Questions

1. Multiple Choice

For the reaction 2A + B-C, the rate law is rate = k[A]²[B]. If [A] is doubled and [B] is halved, the rate will: (A) Stay the same (B) Double (C) Quadruple (D) Decrease by a factor of 2

Answer: (B) Double Explanation: Rate = k(2[A])²(0.5[B]) = k·4[A]²·0.5[B] = 2k[A]²[B]-rate doubles.

2. Short FRQ

The decomposition of dinitrogen pentoxide (N?O?) is first-order with k = 0.075 min?¹ at 45°C. (a) Write the integrated rate law for this reaction. (b) Calculate the concentration of N?O? after 10.0 minutes if [N?O?]? = 0.100 M.

Answers: (a) ln[N?O?]? = –kt + ln[N?O?]? (b) ln[N?O?]? = –(0.075 min?¹)(10.0 min) + ln(0.100) = –0.75 – 2.303 = –3.053-[N?O?]? = e^(–3.053)-0.047 M.

3. Multiple Choice

Which of the following plots would be linear for a second-order reaction? (A) [A] vs. time (B) ln[A] vs. time (C) 1/[A] vs. time (D) [A]² vs. time

Answer: (C) 1/[A] vs. time Explanation: The integrated rate law for second-order is 1/[A]? = kt + 1/[A]?, which is linear.

Last-Minute Cram Sheet

1. Rate = ?[ ]/?t (units: M/s).
2. Rate law = k[A]?[B]? (orders not from coefficients!).
3. Zero-order: [A] vs. t linear (slope = –k).
4. First-order: ln[A] vs. t linear (slope = –k); t?/? = ln(2)/k (constant).
5. Second-order: 1/[A] vs. t linear (slope = +k); t?/? = 1/(k[A]?).
6. Units for k:
7. Zero-order: M/s.
8. First-order: s?¹.
9. Second-order: M?¹s?¹.
10. Arrhenius equation: k = Ae^(–E?/RT) (T in Kelvin!).
11. Catalysts lower E? but don’t change ?H or equilibrium.
12. RDS = slowest step in a mechanism (determines rate law).
13. Half-life for first-order is constant; for others, it depends on [A]?.