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Imagine you are tasked with testing the purity of an antacid tablet made of calcium carbonate, but the powder dissolves far too slowly to titrate directly with an acid.
If a direct reaction is too slow, insoluble, or lacks a sharp indicator color change, analytical chemists use a two-step technique called a back-titration.

The Logic of Back-Titrations

In a back-titration, you flood the analyte with a known, measured excess of standard reagent so the analyte reacts completely.
You then titrate the leftover (unreacted) excess reagent with a second standard solution to find out exactly how much excess remains.
The core relationship relies on conservation of matter: the moles of reagent initially added equal the moles consumed by the sample plus the moles left over in excess.
ninitial added=nreacted with analyte+nunreacted excessn_{\text{initial added}} = n_{\text{reacted with analyte}} + n_{\text{unreacted excess}}
How do we apply this logic to calculate the purity percentage of a solid sample? Let's walk through a complete laboratory analysis.

Worked Example: Purity of Calcium Carbonate

A 1.250 g1.250\text{ g} impure sample of chalk (containing CaCO3\text{CaCO}_3, molar mass 100.09 g/mol100.09\text{ g/mol}) is treated with 50.00 mL50.00\text{ mL} of 0.5000 M HCl0.5000\text{ M }\text{HCl}. The unreacted acid requires 18.40 mL18.40\text{ mL} of 0.2500 M NaOH0.2500\text{ M }\text{NaOH} for complete neutralization.
First, calculate the total moles of HCl\text{HCl} initially added to the chalk using n=C×Vn = C \times V:
n(HCl)initial=0.5000 mol/L×0.05000 L=0.02500 moln(\text{HCl})_{\text{initial}} = 0.5000\text{ mol/L} \times 0.05000\text{ L} = 0.02500\text{ mol}