Problem 90 - General Chemistry: Principles and Modern Applications

Question

A compound contains only Fe and O. A \(0.2729 \mathrm{g}\) sample of the compound was dissolved in \(50 \mathrm{mL}\) of concentrated acid solution, reducing all the iron to \(\mathrm{Fe}^{2+}\) ions. The resulting solution was diluted to \(100 \mathrm{mL}\) and then titrated with a \(0.01621 \mathrm{M} \mathrm{KMnO}_{4}\) solution. The unbalanced chemical equation for reaction between \(\mathrm{Fe}^{2+}\) and \(\mathrm{MnO}_{4}^{-}\) is given below. \(\begin{aligned} \mathrm{MnO}_{4}^{-}(\mathrm{aq})+& \mathrm{Fe}^{2+}(\mathrm{aq}) \longrightarrow \mathrm{Mn}^{2+}(\mathrm{aq})+\mathrm{Fe}^{3+}(\mathrm{aq}) \quad(\text { not balanced }) \end{aligned}\) The titration required \(42.17 \mathrm{mL}\) of the \(\mathrm{KMnO}_{4}\) solution to reach the pink endpoint. What is the empirical formula of the compound?

Step-by-Step Solution

Answer

The empirical formula for the compound is \(Fe_2O\).

Step 1: Balance the Reaction

First, balance the half-reactions in acidic solution. For \(\mathrm{MnO}_{4}^{-}\to \mathrm{Mn}^{2+}\), the balanced half-reaction is \(5\mathrm{e}^{-}+8\mathrm{H}^{+}+\mathrm{MnO}_{4}^{-}\) gives \(\mathrm{Mn}^{2+} + 4\mathrm{H}_{2}\mathrm{O}\). And for \(\mathrm{Fe}^{2+}\) to \(\mathrm{Fe}^{3+}\), the balanced half-reaction is \(\mathrm{Fe}^{2+}\to \mathrm{Fe}^{3+}+\mathrm{e}^{-}\). For the overall balanced reaction, multiply the second half-reaction by 5 and add them to get 5\(\mathrm{Fe}^{2+}+ \mathrm{MnO}_{4}^{-}+ 8\mathrm{H}^{+}\) gives \(5\mathrm{Fe}^{3+}+ \mathrm{Mn}^{2+}+ 4\mathrm{H}_{2}\mathrm{O}\).

Step 2: Find the Moles of \(\mathrm{KMnO}_{4}\)

The moles of \(\mathrm{MnO}_{4}^{-}\) can be calculated using the formula \(\mathrm{Moles}\ =\ \mathrm{Molarity} \times \mathrm{Volume}\). Here, molarity of \(\mathrm{KMnO}_{4}\) is 0.01621 M and volume used is 42.17 mL or 0.04217 L. Therefore, moles of \(\mathrm{MnO}_{4}^{-}\) = \(0.01621 \mathrm{M} \times 0.04217 \mathrm{L}\) = \(684.0 \times 10^{-6} \mathrm{mol}\).

Step 3: Find the Moles of \(\mathrm{Fe}^{2+}\)

From the balanced chemical equation, five moles of \(\mathrm{Fe}^{2+}\) react with one mole of \(\mathrm{MnO}_{4}^{-}\). Thus, the moles of \(\mathrm{Fe}^{2+}\) is five times the moles of \(\mathrm{MnO}_{4}^{-}\), which is \(5 \times 684.0 \times 10^{-6} \mathrm{mol}\) = \(3420 \times 10^{-6} \mathrm{mol}\) or \(3.42 \times 10^{-3} \mathrm{mol}\).

Step 4: Calculate the Empirical Formula

Now, find the mass of Fe in the sample by using the moles of Fe and its molar mass (\(55.85 \mathrm{g/mol}\)). The mass of Fe = \(3.42 \times 10^{-3} \mathrm{mol} \times 55.85 \mathrm{g/mol}\) = \(0.191 \mathrm{g}\). Subtracting this mass from the total mass of the compound, one gets the mass of oxygen, \(0.2729 \mathrm{g} - 0.191 \mathrm{g}\) = \(0.0819 \mathrm{g}\). Dividing this mass by the molar mass of oxygen (\(16.00 \mathrm{g/mol}\)) gives its number of moles, which is \(0.00512 \mathrm{mol}\). So, the mole ratio of Fe to O is about \(3.42 / 0.00512 = 668\). As the ratio needs to be integer, this is approximately \(2:1\), resulting in an empirical formula of \(Fe_2O\).

Key Concepts

Redox Reactions

Redox reactions, short for reduction-oxidation reactions, are fundamental to understanding how chemical changes occur. In these reactions, one species gains electrons (reduction) while another loses electrons (oxidation). In our exercise, iron (Fe) is oxidized from \( \mathrm{Fe}^{2+} \) to \( \mathrm{Fe}^{3+} \). Conversely, the permanganate ion \( \mathrm{MnO}_{4}^- \) is reduced to \( \mathrm{Mn}^{2+} \).
The importance of balancing redox reactions is highlighted by ensuring that the number of electrons lost in oxidation equals the number gained in reduction. This maintains charge balance and accurately represents the chemical changes occurring.
In the context of our specific problem, balancing these half-reactions properly is critical before proceeding with subsequent calculations such as titration.

Titration

Titration is a laboratory technique used to determine the concentration of a solute in a solution. It involves slowly adding a titrant of known concentration to a solution with an unknown concentration until a reaction is complete. In our exercise, the permanganate solution acts as the titrant.
The endpoint of the titration is visually identified by a persistent color change, usually pink in this case, signifying all the \( \mathrm{Fe}^{2+} \) has reacted. Here, 42.17 mL of \( 0.01621 \\mathrm{M \ KMnO}_{4} \) was needed, providing data to calculate the moles of substances and helping derive the empirical formula.
Titration not only needs careful volumetric measurement but also an understanding of stoichiometry for accurate results.

Stoichiometry

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It allows chemists to predict the amounts of substances consumed and produced. In the provided exercise, stoichiometry is used to relate the moles of \( \mathrm{Fe}^{2+} \) to those of \( \mathrm{MnO}_4^- \).
Using the balanced equation, five moles of iron react with one mole of permanganate ion. This stoichiometric relationship ensures that we can determine the amount of iron present in the original sample based on the titration data.
This concept lets us correctly calculate the mass of iron and then derive the sample’s empirical formula by comparing the actual masses of iron and oxygen in the compound.

Iron Compounds

Iron compounds often involve variations of iron's oxidation states, commonly \( \mathrm{Fe}^{2+} \) and \( \mathrm{Fe}^{3+} \). Iron oxides are the result of reactions between iron and oxygen, forming compounds with varying compositions like \( \mathrm{FeO} \), \( \mathrm{Fe}_2\mathrm{O}_3 \), or mixed states found in \( \mathrm{Fe}_3\mathrm{O}_4 \). These compounds are widely pertinent in fields ranging from chemistry to geology.
The problem at hand requires determining the empirical formula for an unknown iron-oxygen compound. Based on the stoichiometry from titration, and the balance of masses and moles for Fe and O, we identified \( \mathrm{Fe}_2\mathrm{O} \) as the most likely empirical formula.
This understanding of iron compounds is essential for diverse applications, from material science to environmental studies, allowing for accurate development of product formulations.