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II-03: Natural logarithms and exponential models

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Module II-03 · Recommended before: Before integrated rate laws

Natural logarithms and exponential models

Natural logarithms undo exponential functions with base e. These operations solve kinetic equations for time or concentrations.

Learn the method

Aₜ = A₀ e^(−kt)

  1. Identify an equation containing e^(−kt).
  2. Divide to isolate the exponential term.
  3. Take ln of both sides.
  4. Solve for time or rate constant.
  5. Verify that decay produces a smaller amount.

Common mistake: ln(2) is approximately 0.693, not 0.301 (which is log₁₀2).

Worked example

Problem: Find time when a first-order process decreases to half its original amount with k = 0.100 min⁻¹.

Step-by-step solution: t = ln(2)/k = 0.693147/0.100 = 6.93 min.

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

More applications of this mathematics

Try each extension before expanding its solution.

Time from first-order concentration

For [A]₀ = 0.100 M, [A]ₜ = 0.0250 M and k = 0.200 min⁻¹, t = ln(4)/0.200 = 6.93 min.

Log base matters

ln(2)=0.6931, whereas log₁₀(2)=0.3010. They are not interchangeable without the correct conversion factor.

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. A first-order quantity halves with k = 0.25 min⁻¹. Calculate half-life.
Hint for question 1

Use natural logarithm for exponential decay.

Question 2. A first-order quantity halves with k = 0.04 min⁻¹. Calculate half-life.
Hint for question 2

Use natural logarithm for exponential decay.

Question 3. A first-order quantity halves with k = 0.3 min⁻¹. Calculate half-life.
Hint for question 3

Use natural logarithm for exponential decay.

Question 4. A first-order quantity halves with k = 0.15 min⁻¹. Calculate half-life.
Hint for question 4

Use natural logarithm for exponential decay.

Question 5. A first-order quantity halves with k = 0.06 min⁻¹. Calculate half-life.
Hint for question 5

Use natural logarithm for exponential decay.

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: Natural logarithms and exponential change

These ungraded questions extend the scored five-question check. Work each problem before opening its solution; progress records are not changed.

  1. Challenge 1. A first-order reaction starts at [A]₀ = 0.200 M with k = 0.100 min⁻¹. Find [A] after 10.0 min.

    Show worked solution for challenge 1

    [A] = [A]₀ e⁻ᵏᵗ = 0.200e⁻¹·⁰⁰ = 0.0736 M.

  2. Challenge 2. Compute ln(5.00) and log₁₀(5.00). Why do they differ?

    Show worked solution for challenge 2

    ln(5.00) ≈ 1.609; log₁₀(5.00) ≈ 0.6990. Natural logs use base e; common logs use base 10. Match the equation’s required base.

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