AP® Chemistry · Unit 7 · Topic 7.7
The Small-x Approximation
When K is very small, almost none of the reactant converts to product, so x is tiny compared to the initial concentration and can be dropped from the denominator, turning a quadratic into simple algebra.
An ICE table for a reaction like A <→ B with initial concentration c0 gives K = x2/(c0 - x). Solving that exactly means the quadratic formula, every time. But if K is small, say 1.0 x 10−6, the reaction barely moves forward. Almost all of the A stays as A, so x ends up being a tiny sliver of c0. That means c0 - x is essentially just c0. Swap in that approximation and K = x2/c0, which you solve by rearranging to x = √(K times c0) — no quadratic formula needed.
The justification is proportional: the smaller K is relative to c0, the smaller x has to be to satisfy the equilibrium expression, which makes the approximation more trustworthy. The common check is the 5 percent rule — if x comes out to less than 5 percent of c0, the approximation is usually treated as valid. But that rule is a heuristic, not a guarantee. It was built for simple single-equilibrium problems. In systems with multiple coupled equilibria, common ions, or two reactions sharing a species, an x under 5 percent can still introduce real error, because small errors can compound or interact with the other equilibrium. The only airtight check is solving the full quadratic and comparing.
This topic sits right after ICE tables because it's the shortcut that makes most weak acid, weak base, and small-K problems solvable by hand instead of by quadratic formula. It connects forward to percent ionization and common-ion problems, where knowing when the shortcut fails matters as much as knowing how to use it.
Key terms
- Small-x approximation
- Treating x as negligible compared to the initial concentration when K is small, simplifying the equilibrium expression.
- ICE table
- A table tracking initial, change, and equilibrium concentrations used to set up equilibrium calculations.
- 5 percent rule
- A heuristic check that the approximation is valid if x is less than 5 percent of the initial concentration.
- Quadratic solution
- The exact algebraic solution to the equilibrium expression without dropping any terms.
Watch it explained
- How it worksHow does a very small K justify dropping x from the initial concentration term?
- Worked exampleFind the equilibrium concentration for a reaction with K of 1.0e-6 and 0.100 M initial reactant.
- Common mistakeWhy isn't the 5 percent rule a guarantee that the approximation is safe?
- Going furtherWhat happens to the answer when you solve the same problem with the full quadratic instead?
Check yourself
End of topic.