Question

Determine \(E^{\circ}\) for the reaction \(\mathrm{Cr}^{2+}(a q)+2 \mathrm{e}^{-} \rightarrow \mathrm{Cr}(s)\) from the one-electron reduction potential for \(\mathrm{Cr}^{3+}(a q)\) and the three-electron reduction potential for \(\mathrm{Cr}^{3+}(a q)\) given in Table 11.1 (see Appendix B).

Short answer

The standard reduction potential for the reaction \(\mathrm{Cr}^{2+}(aq)+2 \mathrm{e}^- \rightarrow \mathrm{Cr}(s)\) can be calculated using the given one-electron and three-electron reduction potentials for Cr³⁺(aq). After balancing the reactions and applying the cell potential equation, we find that \(E^{\circ}_{Cr^{2+}(aq)} = 0.08 \:V\).

Step-by-step solution

Write the reduction reactions

We are given two reduction potentials: 1. The one-electron reduction potential for Cr³⁺(aq): \[ \mathrm{Cr}^{3+}(aq) + \mathrm{e}^- \rightarrow \mathrm{Cr}^{2+}(aq) \] 2. The three-electron reduction potential for Cr³⁺(aq): \[ \mathrm{Cr}^{3+}(aq) + 3 \mathrm{e}^- \rightarrow \mathrm{Cr}(s) \]

Balance reactions

To solve for the reduction potential of Cr²⁺(aq), we need to subtract the first reaction from the second reaction. To do this, we need to balance the reactions in terms of electrons. We multiply the first reaction by 2 to balance the number of electrons: \[ 2\mathrm{Cr}^{3+}(aq) + 2\mathrm{e}^- \rightarrow 2\mathrm{Cr}^{2+}(aq) \] The second reaction remains the same: \[ \mathrm{Cr}^{3+}(aq) + 3\mathrm{e}^- \rightarrow\mathrm{Cr}(s) \] Now, we can subtract the balanced first reaction from the second reaction.

Calculate the standard reduction potential for Cr²⁺(aq)

Using the equation for the cell potential: \[E_{cell} = E_{cathode} - E_{anode}\] Let's call the one-electron reduction potential E¹ and the three-electron reduction potential E³. The balanced equations look like this: \[2\mathrm{Cr}^{3+}(aq) + 2\mathrm{e}^- \rightarrow 2\mathrm{Cr}^{2+}(aq) \qquad E = 2E¹\] \[\mathrm{Cr}^{3+}(aq) + 3\mathrm{e}^- \rightarrow\mathrm{Cr}(s) \qquad E = E³\] Now we can calculate E° for the desired reaction, which corresponds to E², by subtracting E³ - 2E¹: \[E^{\circ}_{Cr^{2+}(aq)} = E^{\circ}_{Cr^{3+}(aq)} - 2E^{\circ}_{Cr^{3+}(aq) - Cr^{2+}(aq)}\] Refer to Table 11.1 (Appendix B) and substitute the given values: \[E^{\circ}_{Cr^{2+}(aq)} = (- 0.74 \:V) - 2(- 0.41\:V)\] \[E^{\circ}_{Cr^{2+}(aq)} = - 0.74 \:V + 0.82\:V\] Finally, we have the standard reduction potential for Cr²⁺(aq). \[E^{\circ}_{Cr^{2+}(aq)} = 0.08 \:V\]

Key concepts

Reduction potential

Reduction potential is like a measure of how willing a chemical species is to gain electrons, turning itself into a reduced form. Think of it as the eagerness of an ion or molecule to become a different particle by accepting electrons. This concept is pivotal in electrochemistry, where oxidation and reduction reactions occur.

Every reduction potential is measured against a standard, called the Standard Hydrogen Electrode (SHE), which is assigned an arbitrary potential of 0 volts. This helps create a reference point, making it easier to compare different half-reactions. The values are typically given in volts (V) and can predict how likely a reaction is to proceed. A positive reduction potential means a strong tendency to gain electrons, while a negative value indicates a lower likelihood.

In our exercise, the challenge is determining the reduction potential for the conversion of chromium ions. Knowing specific potentials for other chromium reactions allows us to calculate the necessary one using redox principles.

Balancing redox reactions

Balancing redox reactions is a fundamental skill in electrochemistry, helping to maintain the principle of conservation of charge and mass. It involves ensuring that both sides of a chemical equation have an equal number of atoms for each element and a balanced total charge. This is crucial for accurately predicting the flow of electrons and how reactions will occur.

In a redox process, there are two half-reactions: oxidation, where electrons are lost, and reduction, where electrons are gained. To balance these half-reactions, particularly in reactions occurring in acidic or basic solutions, one can use methods like the ion-electron method.

In the provided example, we have reactions for chromium transitioning from its +3 oxidation state to +2 and then to solid chromium. For this particular exercise, correctly balancing the electron flow between these reactions by multiplying them appropriately was essential to finding the final reduction potential.

Standard electrode potential

Standard electrode potential is a specific type of reduction potential measured under standard conditions, typically 25°C, 1 atm pressure, and 1 M concentration for all reactants and products. It is a critical tool in determining cell potentials and the feasibility of electrochemical reactions.

These potentials are usually compiled in tables, allowing for easy comparison between various half-reactions. They serve as the building blocks for calculating the potential of electrochemical cells, where a positive cell potential indicates a spontaneity in the reaction.

In the context of our example, we estimated the standard reduction potential for chromium reactions by using the given potentials for similar reactions and manipulating these to achieve a desired balanced equation. By using calculations based on the standard electrode potential values, one can easily deduce the converting voltage necessary for different reactions. This highlights the importance and utility of standard potentials in electrochemistry.