Mistake Master
Equilibrium concentrations
Knowing K, you can predict exactly where a reaction lands. Set up an ICE table, plug the equilibrium row into K, and solve for x — then check whether the shortcut you took was actually allowed.
§1
From K to concentrations.
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To find equilibrium concentrations from a known K, build an ICE table, write the equilibrium row in terms of x, and substitute it into the K expression. Solving that equation gives x, and hence every equilibrium concentration.
When K is small, the algebra can be simplified by the small-x approximation: where x is added to (or subtracted from) a much larger number, drop it. This turns a quadratic into a simple equation.
The approximation must be checked. It is valid only when x is small compared to the initial concentration — commonly, when x is under about 5% of it. If not, solve the full equation.
§2
Solving and checking.
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Solve for x, then verify the approximation.
- Set up the ICE table. Express the equilibrium row in terms of x.
- Substitute into K. Put the equilibrium expressions into the K equation.
- Solve for x. Use the small-x approximation if K is small to simplify.
- Check the approximation. Confirm x is under about 5% of the initial concentration; if not, solve the full equation.
§3
The pieces you'll meet.
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Approximate, then verify.
§4
Worked example: use the approximation.
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Setup. A weak dissociation has initial [HA] = 0.10 M and a small K. The ICE gives [H⁺] = x, [A⁻] = x, [HA] = 0.10 − x.
Approximate. Because K is small, x is tiny, so 0.10 − x ≈ 0.10. Then K ≈ x² / 0.10, giving x easily.
Check. Compute x / 0.10 × 100%. If it is under ~5%, the approximation was valid.
If it fails. If x is, say, 12% of 0.10, the approximation is not valid, and you must solve the full quadratic K = x² / (0.10 − x) instead.
§5
Mistakes that cost real points.
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"The small-x approximation is always valid."
It is valid only when x is small compared to the initial concentration (commonly under ~5%). For larger K or dilute solutions, x is not negligible and the approximation gives a wrong answer. It must be checked, not assumed.
Fix. Always check that x is under ~5% of the initial concentration; if not, solve the full equation.
"You can skip the check and keep the approximate answer."
Skipping the validity check risks reporting a wrong concentration. The whole point of the approximation is that it is a shortcut you are allowed to take only when it holds — verifying is part of the method.
Fix. Make the 5% check a required final step, not an optional one.
"If the approximation fails, there is no way to solve it."
If the approximation fails, you solve the full equation (often a quadratic) exactly — the answer is still obtainable, just with more algebra. Failure of the shortcut does not mean the problem is unsolvable.
Fix. When the approximation fails, solve the full (quadratic) equation for x.
§6
Skill Check.
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Ten scenarios. Pick the chips that match your answer, then check. A scenario marks complete the first time every part is right. Progress saves on this device.