A \(0.500-g\) sample of steel is analyzed for manganese. The sample is dissolved in acid and the manganese is oxidized to permanganate ion. A measured excess of \(\mathrm{Fe}^{2+}\) is added to reduce \(\mathrm{MnO}_{4}^{-}\) to \(\mathrm{Mn}^{2+}\). The excess \(\mathrm{Fe}^{2+}\) is determined by titration with \(\mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{27}\). If \(75.00 \mathrm{~mL}\) of \(0.125 \mathrm{M} \mathrm{FeSO}_{4}\) is added and the excess requires \(13.50 \mathrm{~mL}\) of \(0.100 \mathrm{M} \mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7}\) to oxidize \(\mathrm{Fe}^{2+}\), calculate the percent by mass of \(\mathrm{Mn}\) in the sample.

Titration is an essential technique used in chemistry to determine the concentration of an unknown solution by allowing it to react with a standard solution of known concentration, known as the titrant. This process involves adding the titrant to the unknown solution from a burette, a process known as titration.

During the process, a color change or an indicator is often used to determine the end point, which signifies the completion of the reaction. In the context of the provided exercise, titration is used to determine the excess \text{\(\text{Fe}^{2+}\)} after it has reduced the \text{\(\text{MnO}_{4}^{-}\)} to \text{\(\text{Mn}^{2+}\)}. The excess \text{\(\text{Fe}^{2+}\)} is titrated with \text{\(\text{K}_{2} \text{Cr}_{2} \text{O}_{7}\)}, a redox titration where the end point can be identified by the change in color of the solution, without the need of an external indicator, as the reactants themselves act as self-indicators.

Molar mass is a fundamental concept in chemistry that refers to the mass of one mole of a substance, usually expressed in grams per mole (g/mol). To calculate the molar mass, we sum the atomic masses of all atoms in the molecule as provided in the periodic table of elements.

For instance, in our exercise, the molar mass of \text{\(\text{Mn}\)} (manganese) is given as 54.94 g/mol. Knowing the molar mass allows us to convert between mass and number of moles, a crucial step in stoichiometric calculations. In the problem, the mass of manganese present in the steel sample was calculated by first determining the moles of manganese and then using its molar mass to find the corresponding mass.

Redox reactions are chemical processes where the oxidation states of atoms are changed due to the transfer of electrons. Oxidation refers to the loss of electrons, while reduction refers to the gain of electrons. These reactions are key in many analytical techniques, including titration.

In the provided exercise, the titration involves a redox reaction between \text{\(\text{Fe}^{2+}\)} and \text{\(\text{MnO}_{4}^{-}\)}. The \text{\(\text{Fe}^{2+}\)} ions are oxidized to \text{\(\text{Fe}^{3+}\)} ions, while the \text{\(\text{MnO}_{4}^{-}\)} ions are reduced to \text{\(\text{Mn}^{2+}\)}. Understanding the stoichiometry of redox reactions is crucial as it determines the amount of titrant needed to reach the end point, which in turn gives us information about the unknown concentration or the quantity of a substance within a sample.