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II-08: Kinetics, integrated rate laws and Arrhenius

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Module II-08 · Recommended before: Before kinetics problem sets

Kinetics, integrated rate laws and Arrhenius

Rate constants have units that depend on reaction order. First-order half-life depends on k, not initial concentration.

Learn the method

t₁/₂ = 0.693/k (first order); ln k = −Eₐ/(RT)+ln A

  1. Identify reaction order from the stated rate law or experimental evidence.
  2. Choose the appropriate integrated equation.
  3. Keep time units consistent with k.
  4. Use natural logarithms for first-order decay.
  5. For Arrhenius comparisons, use inverse absolute temperatures.

Common mistake: Do not use a first-order half-life equation for an unspecified zero- or second-order process.

Worked example

Problem: Calculate a first-order half-life when k = 0.0500 s⁻¹.

Step-by-step solution: t₁/₂ = ln 2 / k = 0.693147 / 0.0500 = 13.9 s.

Check the units and whether the answer is reasonable before continuing.

More applications of this mathematics

Try each extension before expanding its solution.

Rate-law order

If doubling [A] increases rate fourfold with all else unchanged, the reaction is second order in A (2² = 4).

Arrhenius temperature comparison

For Ea = 50.0 kJ/mol, T₁ = 300 K and T₂ = 310 K, ln(k₂/k₁) = −Ea/R(1/T₂−1/T₁) ≈ 0.646; k₂/k₁ ≈ 1.91.

Zero- and second-order plots

For a zero-order reaction, [A] vs t is linear with slope −k. For a second-order reaction in one reactant, 1/[A] vs t is linear with slope +k.

Independent practice — five questions

Suggested mastery threshold: four out of five. Incorrect answers display a worked explanation after submission.

Choose one response for each question, then select “Check my answers.” No timer. Hints are optional. You can retry freely. Questions are formative, not proctored.

Question 1. Compute half-life of a first-order reaction with k = 0.04 s⁻¹.
Hint for question 1

Half-life and rate constant use reciprocal time units.

Question 2. Compute half-life of a first-order reaction with k = 0.08 s⁻¹.
Hint for question 2

Half-life and rate constant use reciprocal time units.

Question 3. Compute half-life of a first-order reaction with k = 0.3 s⁻¹.
Hint for question 3

Half-life and rate constant use reciprocal time units.

Question 4. Compute half-life of a first-order reaction with k = 0.15 s⁻¹.
Hint for question 4

Half-life and rate constant use reciprocal time units.

Question 5. Compute half-life of a first-order reaction with k = 0.06 s⁻¹.
Hint for question 5

Half-life and rate constant use reciprocal time units.

Supporting reference

OpenStax Chemistry 2e / NIST — reference for this subject. This lesson’s wording, examples and questions are original.

W3C WCAG 2.2 accessibility guidance.

Applied practice: Integrated rate laws and half-life

These additional, ungraded problems cover skills beyond the scored five-question check above. Try each before expanding its solution. Your existing score and progress records are unchanged.

  1. Challenge 1. A first-order reaction decreases from 0.200 M to 0.0500 M with k = 0.0350 min⁻¹. Find the elapsed time.

    Show worked solution for challenge 1

    Use t = ln([A]₀/[A]ₜ) ÷ k = ln(4.00) ÷ 0.0350 min⁻¹ = 39.6 min.

  2. Challenge 2. A first-order process has a half-life of 12.0 min. What fraction remains after 36.0 min?

    Show worked solution for challenge 2

    Three half-lives have elapsed. Remaining fraction = (1/2)³ = 1/8 = 12.5%.

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