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II-06: Approximations and checking assumptions

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Module II-06 · Recommended before: Before weak acid and equilibrium calculations

Approximations and checking assumptions

Small-change approximations simplify equilibrium algebra but must be checked. A common classroom criterion is that change x is no more than 5% of the original concentration.

Learn the method

percent change = (x/C) × 100

  1. Write the exact denominator such as C−x.
  2. Solve the approximate equation.
  3. Calculate 100x/C.
  4. If the percent is too large, solve the exact equation.
  5. Check physical constraints.

Common mistake: The 5% rule is a practical guideline, not a universal mathematical law.

Worked example

Problem: For an initial concentration of 0.20 M and an approximate change x = 0.0040 M, test the 5% check.

Step-by-step solution: 100(0.0040 / 0.20) = 2.0%, below 5%; the chosen approximation passes this conventional test.

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

More applications of this mathematics

Try each extension before expanding its solution.

When small-x fails

For x = 0.012 M and C = 0.100 M, (x/C)×100 = 12%; do not neglect x using the conventional 5% criterion.

Check physical constraints

For initial reactant 0.100 M, an ICE root x = 0.130 M is impossible because the remaining concentration would be negative.

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. An ICE-table approximation neglects x=0.012 M relative to C=0.6 M. What is 100x/C?
Hint for question 1

Compute change divided by initial concentration.

Question 2. An ICE-table approximation neglects x=0.016 M relative to C=0.8 M. What is 100x/C?
Hint for question 2

Compute change divided by initial concentration.

Question 3. An ICE-table approximation neglects x=0.02 M relative to C=1 M. What is 100x/C?
Hint for question 3

Compute change divided by initial concentration.

Question 4. An ICE-table approximation neglects x=0.03 M relative to C=0.3 M. What is 100x/C?
Hint for question 4

Compute change divided by initial concentration.

Question 5. An ICE-table approximation neglects x=0.045 M relative to C=0.9 M. What is 100x/C?
Hint for question 5

Compute change divided by initial concentration.

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: When is the small-x approximation valid?

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

  1. Challenge 1. An approximate equilibrium calculation predicts x = 0.0020 M from an initial concentration of 0.100 M. Test the 5% criterion.

    Show worked solution for challenge 1

    100x/C = 100(0.0020/0.100) = 2.0%, which is below the common 5% screening threshold. The approximation is acceptable by this criterion.

  2. Challenge 2. For an idealized weak-acid equilibrium with Kₐ = 1.0 × 10⁻⁵ and C = 0.200 M, estimate x using the small-x formula and test its validity.

    Show worked solution for challenge 2

    x ≈ √(KₐC) = √(2.0 × 10⁻⁶) = 1.41 × 10⁻³ M. The ratio x/C × 100 = 0.707%, below 5%. This supports the approximation.

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