For reducing \(1 \mathrm{~mol}\) of \(\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}\) to \(\mathrm{Cr}^{3+}\), the charge required is: (a) \(3 \times 96500\) coulomb (b) \(6 \times 96500\) coulomb (c) \(0.3 \mathrm{~F}\) (d) \(0.6 \mathrm{~F}\)

Understanding oxidation numbers is crucial when studying redox reactions in chemistry. Oxidation numbers, also known as oxidation states, are used to keep track of how many electrons are lost or gained by an atom in a chemical reaction. In essence, these numbers illustrate the degree of oxidation of an atom, which is represented by a positive or negative number.

For example, in the dichromate ion \rm\(Cr_2O_7^{2-}\), chromium is assigned an oxidation number of +6. Why is this? Oxygen generally has an oxidation number of -2, and since the ion has a -2 charge overall, calculating the sum of oxidation states will give us the chromium's state. In a reduction step, chromium changes to an oxidation state of +3 in the \rm\(Cr^{3+}\) ion. This signifies a decrease in oxidation number, indicating that chromium atoms have gained electrons during the reaction, which is a clear sign of reduction in a redox process.

To determine the change in electrons when reducing \rm\(Cr_2O_7^{2-}\) to \rm\(Cr^{3+}\), one must consider the change in oxidation numbers for chromium: from +6 in \rm\(Cr_2O_7^{2-}\) to +3 in \rm\(Cr^{3+}\), a difference of 3. Since the molecule contains two chromium atoms, we multiply this difference by 2 for a total change of 6 electrons.

Electrical Charge and Mole of Electrons

Faraday's constant is a fundamental quantity in electrochemistry representing the charge of one mole of electrons. It is named after the scientist Michael Faraday and is approximately \(96500\) coulombs. In the context of redox reactions, Faraday's constant allows us to relate the amount of substance reacted or produced at an electrode to the amount of electric charge passed through the electrochemical cell.

When solving problems related to the electrical charge required to drive a reaction, such as the reduction of \(Cr_2O_7^{2-}\) to \(Cr^{3+}\), Faraday's constant becomes immensely useful. Since reducing one mole of \(Cr_2O_7^{2-}\) requires 6 moles of electrons, we can use Faraday's constant to calculate the charge: \(6 \times 96500\) coulombs. Understanding Faraday's constant paves the way for accurately calculating the electrical energy involved in electrochemical processes.

Balancing Chemical Reactions

Stoichiometry is a section of chemistry that involves calculating the quantities of reactants and products in chemical reactions. It is based on the conservation of mass and the concept that the amount of product is directly proportional to the reactant consumed in a reaction, given a balanced chemical equation.

Using stoichiometry in our example, we know that the reduction reaction of \(Cr_2O_7^{2-}\) to \(Cr^{3+}\) involves a change of 6 electrons in total for the two chromium atoms. Stoichiometry enables us to understand the molar relationship between different substances in a reaction and, in the case of redox reactions, the flow of electrons. Mastery of stoichiometry is crucial because it guides chemists in predicting the outcome of reactions and determining the proportions of reactants needed to achieve a desired amount of product.

The Intersection of Electricity and Chemistry

Electrochemistry is the branch of chemistry that deals with the interrelation of electrical currents and chemical reactions. This field involves understanding how chemicals react to create electricity (as in a battery) and how electric currents can induce chemical changes (as in electroplating).

In electrochemistry, reactions are split into two half-reactions: oxidation, where a substance loses electrons, and reduction, where a substance gains electrons. The exercise provided deals with a reduction half-reaction, where \(Cr_2O_7^{2-}\) ions gain electrons to form \(Cr^{3+}\) ions. Through this reduction, 6 moles of electrons are transferred, and using the principles of electrochemistry, we can calculate the total charge needed for this transfer using Faraday's constant. These concepts form the foundational understanding of how batteries function, how metals are electroplated, and numerous other applications in science and industry.