A sample of polyphosphoric acid was dissolved in water and titrated with dilute NaOH(aq). Two stoichiometric end points were observed at 30.4 and 45.6 cm³. What is the chain length of polyphosphoric acid (polymer of how many phosphoric acids)?

We need to determine the ratio of the two different types of OH group. The strongly acidic OH groups are titrated by the first 30.4cm3. The two terminal OH groups are titrated by the remaining 45.6 – 30.4 cm3=15.2 cm3. Because the concentrations of analyte and titrant are such that each OH group requires 7.6 cm3 of the titrant (because 15.2 cm3 is used to titrate two such groups), we conclude that there are (30.4 cm3)/(7.6 cm3)=4 strongly acidic OH groups per molecule. A molecule with two terminal OH groups and four further OH groups is a tetrapolyphosphate.