Question

Glycerol \(\left(\mathrm{C}_{3} \mathrm{H}_{8} \mathrm{O}_{3}\right)\) is commonly used as an additive to moisturizers. The amount of glycerol added can be determined by titration with permanganate in basic solution. $$ \begin{array}{r} 14 \mathrm{MnO}_{4}^{-}(a q)+\mathrm{C}_{3} \mathrm{H}_{8} \mathrm{O}_{3}(a q)+20 \mathrm{OH}^{-}(a q) \longrightarrow \\ 14 \mathrm{MnO}_{4}^{2-}(a q)+3 \mathrm{CO}_{3}^{2-}(a q)+14 \mathrm{H}_{2} \mathrm{O} \end{array} $$ How many milliliters of \(6.15 \mathrm{M} \mathrm{KMnO}_{4}\) are required to completely react with \(7.25 \mathrm{~mL}\) of an aqueous solution of glycerol \((d=1.144 \mathrm{~g} / \mathrm{mL})\) that is \(51.6 \%\) by mass?

Short answer

Answer: 105.8 mL

Step-by-step solution

Calculate the mass of glycerol in the solution

The mass of glycerol in the solution can be determined by multiplying the mass percentage of glycerol by the total mass of the solution. First, we need to find the total mass of the solution using the given volume and density: $$ \text{mass of solution} = 7.25\ \text{mL} \times 1.144\ \frac{\text{g}}{\text{mL}} = 8.296\ \text{g} $$ Now we can find the mass of glycerol in the solution: $$ \text{mass of glycerol} = 0.516 \times 8.296\ \text{g} = 4.28096\ \text{g} $$

Find the number of moles of glycerol

To find the number of moles of glycerol in the solution, we need to divide the mass of glycerol by its molar mass. The molar mass of glycerol is: $$\ce{C3H8O3}\: :\: 3(12.01\ \mathrm{g/mol}) + 8(1.01\ \mathrm{g/mol}) + 3(16.00\ \mathrm{g/mol}) = 92.09\ \mathrm{g/mol}$$ Now, we can find the moles of glycerol in the solution: $$ \text{moles of glycerol} = \frac{4.28096\ \text{g}}{92.09\ \tfrac{\text{g}}{\text{mol}}} = 0.04650\ \text{mol} $$

Find the number of moles of \(\ce{KMnO4}\) required for the complete reaction

Using the stoichiometry of the balanced chemical equation, we can determine the number of moles of \(\ce{KMnO4}\) required to completely react with the glycerol: $$ \text{moles of }\ce{KMnO4} = \frac{14\ \text{moles of }\ce{MnO4-}}{\text{1 mole of }\ce{C3H8O3}} \times 0.04650\ \text{mol} = 0.651\ \text{mol} $$

Calculate the volume of \(\ce{KMnO4}\) solution needed

Now that we know the number of moles of \(\ce{KMnO4}\) required for the reaction, we can calculate the volume using the given molarity: $$ \text{volume of }\ce{KMnO4\ \text{solution}} = \frac{\text{moles of }\ce{KMnO4}}{\text{M of }\ce{KMnO4\ \text{solution}}} = \frac{0.651\ \text{mol}}{6.15\ \text{M}} = 0.1058\ \text{L} $$ Converting the volume from liters to milliliters: $$ \text{volume of }\ce{KMnO4\ \text{solution}} = 0.1058\ \text{L} \times 1000\ \frac{\text{mL}}{\text{L}} = 105.8\ \text{mL} $$ So, 105.8 mL of \(6.15\mathrm{M}\) \(\ce{KMnO4}\) are required to completely react with \(7.25\mathrm{~mL}\) of an aqueous solution of glycerol that is \(51.6 \%\) by mass.