Question

A \(1.012-g\) sample of a salt containing \(\mathrm{Fe}^{2+}\) is titrated with \(0.1201 \mathrm{M} \mathrm{KMnO}_{4}\). The end point of the titration is reached at \(22.45 \mathrm{~mL}\). Find the mass percent of \(\mathrm{Fe}^{2+}\) in the sample. The unbalanced redox reaction that occurs in acidic solution during the titration is: $$ \mathrm{Fe}^{2+}(a q)+\mathrm{MnO}_{4}{ }^{-}(a q) \longrightarrow \mathrm{Fe}^{3+}(a q)+\mathrm{Mn}^{2+}(a q) $$

Short answer

The mass percent of \(\mathrm{Fe}^{2+}\) in the sample can be calculated using the titration data and the balanced chemical equation, resulting in the mass of \(\mathrm{Fe}^{2+}\) divided by the initial mass of the sample, multiplied by 100 to convert to percentage.

Step-by-step solution

Balance the redox reaction

Before calculating the mass percent of \(\mathrm{Fe}^{2+}\) in the sample, the redox reaction must be balanced. This involves assigning oxidation numbers, separating the reaction into half-reactions, balancing the atoms other than oxygen and hydrogen, balancing the oxygens by adding water, balancing the hydrogens by adding protons (H+), and balancing the charge by adding electrons. Finally, multiply the half-reactions by appropriate factors so that the number of electrons gained and lost is the same when the half-reactions are combined.

Determine the moles of \(\mathrm{KMnO}_4\) used

Use the molarity of the \(\mathrm{KMnO}_4\) solution and the volume used at the end point to find the moles of \(\mathrm{KMnO}_4\) that reacted. The formula for moles is: \(\text{moles} = \text{molarity} \times \text{volume in liters}\).

Determine moles of \(\mathrm{Fe}^{2+}\) in the sample

Use the stoichiometry of the balanced chemical equation to find the moles of \(\mathrm{Fe}^{2+}\) that reacted with the \(\mathrm{KMnO}_4\).

Calculate the mass of \(\mathrm{Fe}^{2+}\)

Multiply the moles of \(\mathrm{Fe}^{2+}\) by its molar mass to determine the mass of \(\mathrm{Fe}^{2+}\) in the sample.

Calculate the mass percent of \(\mathrm{Fe}^{2+}\)

Finally, use the mass of \(\mathrm{Fe}^{2+}\) and the initial mass of the salt sample to calculate the mass percent of \(\mathrm{Fe}^{2+}\). The formula for mass percent is: \(\text{mass percent} = (\text{mass of solute} / \text{mass of solution}) \times 100\%\).

Key concepts

Redox Reaction Balancing

Balancing redox reactions is a critical step in many chemical processes, including titrations. A redox reaction involves the transfer of electrons between species, resulting in changes to their oxidation states. To balance such reactions, we undertake a systematic approach that includes assigning oxidation numbers to determine which species are oxidized and which are reduced. We then separate the reaction into half-reactions, each representing either the loss or gain of electrons.

In balancing the atoms, we add water molecules to balance oxygen atoms and hydrogen ions (H+) to balance hydrogen atoms. Finally, electrons are added to one side of each half-reaction to balance the charge. These half-reactions are adjusted by multiplying them by appropriate factors so that the number of electrons transferred in both reactions is equal when combined. This ensures that the overall reaction is balanced in terms of both mass and charge, which is essential for accurately calculating reaction stoichiometry during titrations.

Molarity and Concentration

Understanding molarity and concentration is fundamental when dealing with titrations. Molarity (\textbf{M}) is a measurement of the concentration of a solution, defined as the number of moles of solute per liter of solution. It provides a quantifiable way of describing how concentrated a solution is.

To calculate molarity, we use the formula: \( \text{Molarity (M)} = \frac{\text{moles of solute}}{\text{liters of solution}} \). This formula allows us to determine the number of moles of solute in a given volume of solution, which is a key step in titration calculations. Knowing the molarity of the titrant (the solution added during titration) and the volume it takes to reach the \textendash endpoint allows us to calculate the number of moles of titrant that has reacted, which can then be related to the moles of the substance being titrated.

Stoichiometry

Stoichiometry is the study of the quantitative relationships between the amounts of reactants and products in a chemical reaction. It involves using the balanced chemical equation to determine the mole ratios of reactants and products. In the context of a titration, once the redox reaction has been balanced, we use stoichiometry to relate the moles of one reactant to the moles of another.

The stoichiometric coefficients from the balanced equation denote the ratio in which reactants and products react and are formed, respectively. These ratios allow us to convert between moles of different substances in the reaction. For instance, in a titration exercise, once we know the number of moles of titrant, we can use the stoichiometry of the balanced equation to determine the number of moles of the substance being analyzed.

Mass Percent Determination

Mass percent determination is a method used to express the concentration of an element or compound in a mixture. It represents the mass of the solute (the substance being measured) divided by the total mass of the solution and then multiplied by 100 to get a percentage.

In a titration, once we know the mass of the substance of interest (solute), we compare it to the total mass of the sample to calculate mass percent. The formula is: \( \text{mass percent} = \frac{\text{mass of solute}}{\text{total mass of sample}} \times 100\% \). This calculation is particularly useful for determining the purity of a sample or the concentration of a particular component in a mixture. In educational exercises, such as titrations, calculating the mass percent helps students apply theoretical knowledge to practical, real-world scenarios.