Iron(II) can be oxidized by an acidic \(\mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7}\) solution according to the net ionic equation \(\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}+6 \mathrm{Fe}^{2+}+14 \mathrm{H}^{+} \longrightarrow\) \(2 \mathrm{Cr}^{3+}+6 \mathrm{Fe}^{3+}+7 \mathrm{H}_{2} \mathrm{O}\) If it takes \(26.0 \mathrm{~mL}\) of \(0.0250 \mathrm{M} \mathrm{K}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7}\) to titrate \(25.0 \mathrm{~mL}\) of a solution containing \(\mathrm{Fe}^{2+},\) what is the molar concentration of \(\mathrm{Fe}^{2+} ?\)

Redox titration is a type of titration based on a redox reaction between the titrant and the analyte. It's a powerful technique used to determine the concentration of an unknown solution by allowing it to react with a standard solution of known concentration, also known as the titrant. During the titration, the transfer of electrons occurs from the reducing agent to the oxidizing agent until the reaction reaches the endpoint, which is often indicated by a color change with the help of an indicator or by using an instrument.

In the exercise provided, iron(II) ions, \(\mathrm{Fe}^{2+}\), act as the reducing agent, and the dichromate ions, \(\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}\), from the \(\mathrm{K}_{2}\mathrm{Cr}_{2}\mathrm{O}_{7}\) solution act as the oxidizing agent. The acidic condition is necessary for the redox reaction to proceed. By carefully measuring the volume of the titrant needed to react completely with the analyte, we can calculate the concentration of the iron(II) ions in the solution.

The net ionic equation represents the chemical species that are directly involved in the chemical reaction, excluding the spectator ions which do not participate in the reaction. It allows us to focus on the essence of the chemical change. To write a net ionic equation, one must first balance the full chemical equation, then remove the spectator ions that appear on both sides of the reaction.

In the provided redox titration example, the net ionic equation shows the reaction between \(\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}\), \(\mathrm{Fe}^{2+}\), and \(\mathrm{H}^{+}\) ions, forming \(\mathrm{Cr}^{3+}\), \(\mathrm{Fe}^{3+}\), and water. Potassium ions \(\mathrm{K}^{+}\) from \(\mathrm{K}_{2}\mathrm{Cr}_{2}\mathrm{O}_{7}\) are spectator ions and are not included in the net ionic equation because they do not participate in the redox reaction. The balanced net ionic equation helps us identify the stoichiometry of the redox partners, which is crucial for stoichiometric calculations.

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It is derived from the balanced chemical equation and allows us to predict the amounts of substances consumed and produced. By understanding the molar ratios of the reactants and products (the coefficients in the balanced equation), we can calculate the amount of one substance that will react with a given amount of another substance.

In our exercise, the stoichiometric ratio of \(\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}\) to \(\mathrm{Fe}^{2+}\) is 1:6. This ratio means that one mole of \(\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}\) is required to react with six moles of \(\mathrm{Fe}^{2+}\). By measuring the moles of \(\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}\) used in the titration, we can use stoichiometry to calculate the moles of \(\mathrm{Fe}^{2+}\) present in the solution, as demonstrated in steps 2 and 3 of the provided solution.

Molarity, also known as molar concentration, is a measure of the concentration of a solute in a solution, defined as the number of moles of solute per liter of solution (\(M = \frac{mol}{L}\)). It provides a convenient way to communicate how much solute is present in a given volume of solution and is vital in preparing solutions for chemical reactions and titrations.

The final step of the solution process involves calculating the molarity of the \(\mathrm{Fe}^{2+}\) ions in the unknown solution. Once the moles of \(\mathrm{Fe}^{2+}\) have been determined through stoichiometric calculations, we divide this value by the volume of the iron(II) solution that was titrated (expressed in liters) to find the molarity. As in step 4, after finding the moles to be 0.0039, and knowing the volume is 25.0 mL or 0.0250 L, the molarity is calculated to be 0.156 M, revealing the concentration of \(\mathrm{Fe}^{2+}\) ions in the analyzed solution.