Chemistry / Unit 10 Chem 1771 worked example, 60 to 90 seconds
The Small-x Approximation
The idea
Solving an equilibrium expression by the small-x approximation means assuming x is negligible next to a large initial concentration wherever it is added or subtracted, solving the simplified expression directly, then checking that x is under about five percent of that initial concentration before accepting the answer. This shortcut avoids the quadratic formula whenever the underlying K is small enough for the approximation to hold.
In practice
For a weak acid at 0.200 M with Ka = 1.8 × 10⁻⁵, approximating (0.200 − x) as 0.200 gives x² = (1.8 × 10⁻⁵)(0.200), so x ≈ 1.9 × 10⁻³ M, which is under one percent of 0.200 M and so is confirmed valid without further work.
Where people slip
Skipping the five-percent check risks accepting an answer where x was never actually small enough, in which case the full quadratic must be solved instead, since the approximation quietly fails for larger Ka values.
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