The concentration of a hydrogen peroxide solution can be conveniently determined by titration against a standardized potassium permanganate solution in an acidic medium according to the following unbalanced equation: $$\mathrm{MnO}_{4}^{-}+\mathrm{H}_{2} \mathrm{O}_{2} \longrightarrow \mathrm{O}_{2}+\mathrm{Mn}^{2+}$$ (a) Balance the above equation. (b) If \(36.44 \mathrm{~mL}\) of a \(0.01652 M \mathrm{KMnO}_{4}\) solution are required to completely oxidize \(25.00 \mathrm{~mL}\) of a \(\mathrm{H}_{2} \mathrm{O}_{2}\) solution, calculate the molarity of the \(\mathrm{H}_{2} \mathrm{O}_{2}\) solution.

The balancing of chemical equations is a fundamental skill in chemistry. When a chemical reaction is described, the number of atoms for each element should be equal on both sides of the equation. This reflects the conservation of mass and the fact that atoms are neither created nor destroyed in chemical reactions.

For the reaction between potassium permanganate and hydrogen peroxide, it involves the transfer of electrons, meaning it's a redox reaction. The initial unbalanced equation given is \(\mathrm{MnO}_{4}^{-} + \mathrm{H}_{2} \mathrm{O}_{2} \longrightarrow \mathrm{O}_{2} + \mathrm{Mn}^{2+}\). To balance it, we need to ensure that the number of atoms of each element is the same on both sides. Additionally, the charge should be balanced, considering the ions involved.

When balancing, we typically start by balancing atoms that appear in the fewest substances and then balance the hydrogens and oxygens, which appear in water and as ions. In the solution, it was found that balancing required five hydrogen peroxide molecules and two permanganate ions to form five oxygen molecules and two manganese ions. Thereby, the balanced equation is \(2 \mathrm{MnO}_{4}^{-} + 5 \mathrm{H}_{2} \mathrm{O}_{2} \longrightarrow 5 \mathrm{O}_{2} + 2 \mathrm{Mn}^{2+}\).

Molarity, a term for molar concentration, is a measure of the concentration of a solute in a solution. It is defined as the number of moles of solute per liter of solution (mol/L). Molarity can be calculated using the formula:\[\text{Molarity} = \frac{\text{Moles of solute}}{\text{Volume of solution in liters}}\]

In our specific exercise, the molarity of a potassium permanganate (\(\mathrm{KMnO}_4\right)\) solution was already provided, and we needed to use it to find the concentration of hydrogen peroxide (\(\mathrm{H}_2\mathrm{O}_2\right)\) in another solution. By understanding the molarity of one solution and the balanced chemical equation, we can find the molarity of the other solution, given the information about volumes and the stoichiometry of the reaction. The stoichiometry is derived from the balanced equation, which gives us the ratio of how many moles of hydrogen peroxide react with potassium permanganate.

Redox titration is a type of titration based on a redox reaction between the analyte and the titrant. It involves the transfer of electrons from one substance to another. In our exercise, potassium permanganate (\(\mathrm{KMnO}_4\right)\) is the titrant, and hydrogen peroxide (\(\mathrm{H}_2\mathrm{O}_2\right)\) is the analyte.

In a redox titration, the end point is often indicated by a color change, which occurs when the titrant has completely reacted with the analyte. In the case of potassium permanganate, it acts as a self-indicator with its distinct purple color disappearing when the reaction is complete. The amount of titrant used at this point allows us to calculate the amount of the analyte present using the stoichiometry from the balanced chemical equation.

Potassium permanganate titration is widely used to determine the concentration of oxidizable substances, such as hydrogen peroxide, through redox titration. The clear advantage of using \(\mathrm{KMnO}_4\right)\) is its strong purple color, which eliminates the need for an external indicator. As the \(\mathrm{KMnO}_4\right)\) is added to the analyte solution, it will react and the purple color will fade.

When all the \(\mathrm{H}_2\mathrm{O}_2\right)\) has been oxidized, excess \(\mathrm{KMnO}_4\right)\) will not change color anymore, indicating the end-point of the titration. This method is particularly useful as it provides a clear and visible sign of completion, which helps to obtain accurate and reliable results. It is important to perform this titration in an acidic medium to ensure the proper reaction pathway and to obtain correct stoichiometric ratios as seen in the exercise.