Carminic acid, a naturally occurring red pigment extracted from the cochineal insect, contains only carbon, hydrogen, and oxygen. It was commonly used as a dye in the first half of the nineteenth century. It is 53.66% C and 4.09% H by mass. A titration required 18.02 mL of 0.0406 M NaOH to neutralize 0.3602 g of carminic acid. Assuming that there is only one acidic hydrogen per molecule, what is the molecular formula of carminic acid?

In a reaction of neutralization, an acid and a base must react to form salt and water. The ions OH⁻ of the base must react with the ions H⁺ of the acid. If there is only one hydrogen acid, it means that the number of moles of OH⁻ and H⁺ must be the same, or the number of moles of NaOH and the acid is the same.

If the concentration of NaOH is 0.0406 mol/L, to find the number of moles we need to multiply it by the volume (18.02 mL = 0.01802 L)

n = 0.0406x0.01802 = 7.32x10⁻⁴ mol

The molar mass of the carminic acid is the mass divided by the number of moles, so:

MM = 0.3602/7.32x10⁻⁴

MM = 492 g/mol

The composition is 53.66% of C, 4.09% of H and 42.25% of O, so it means that the mass corresponding of these atoms in one mol are:

C = 0.5366x492 = 264 g

H = 0.0409x492 = 20 g

O = 0.4225x492 = 208 g

Knowing the molar masses (C = 12 g/mol, H = 1 g/mol, O = 16 g/mol), the number of atoms on one mol of the acid is:

C = 264/12 = 22

H = 20/1 = 20

O = 208/16 = 13

So the molecular formula is C₂₂H₂₀O₁₃