Question

Consider the cell described below: $$ \mathrm{Al}\left|\mathrm{Al}^{3+}(1.00 M)\right|\left|\mathrm{Pb}^{2+}(1.00 M)\right| \mathrm{Pb} $$ Calculate the cell potential after the reaction has operated long enough for the \(\left[\mathrm{Al}^{3+}\right]\) to have changed by \(0.60 \mathrm{~mol} / \mathrm{L} .\) (Assume \(T=25^{\circ} \mathrm{C}\).)

Short answer

The cell potential after the reaction has operated long enough for the [Al³⁺] to have changed by 0.60 mol/L is 1.36V.

Step-by-step solution

Write the balanced redox reaction

To write the balanced redox reaction, first identify the reduction and oxidation half-reactions. For the given cell: - Oxidation half-reaction: Al(s) → Al³⁺ + 3e⁻ - Reduction half-reaction: Pb²⁺ + 2e⁻ → Pb(s) Next, balance the electrons by multiplying each half-reaction by an appropriate factor and then add the two half-reactions together: - Oxidation half-reaction: 2(Al(s) → Al³⁺ + 3e⁻) - Reduction half-reaction: 3(Pb²⁺ + 2e⁻ → Pb(s)) Balanced redox reaction: 2Al(s) + 3Pb²⁺ → 2Al³⁺ + 3Pb(s)

Find the standard cell potential

Now that we have the balanced redox reaction, we can find the standard cell potential (E°) using the standard reduction potentials: - For Al³⁺ + 3e⁻ → Al(s): E°(Al³⁺/Al) = -1.66V - For Pb²⁺ + 2e⁻ → Pb(s): E°(Pb²⁺/Pb) = -0.13V The standard cell potential E° for the cell is calculated as: E°(cell) = E°(cathode) - E°(anode) E°(cell) = E°(Pb²⁺/Pb) - E°(Al³⁺/Al) E°(cell) = -0.13V - (-1.66V) E°(cell) = 1.53V

Calculate the cell potential using the Nernst equation

To calculate the cell potential at the given conditions, we can use the Nernst equation: \( E = E° - \dfrac{RT}{nF} \times \ln{Q} \) where E is the cell potential, R is the gas constant, T is the temperature in Kelvin, n is the number of electrons involved in the reaction, F is the Faraday's constant, and Q is the reaction quotient. For this problem, we have the following values: - Temperature T = 25°C = 298K - Number of electrons n = 6 (from the balanced equation) - Concentration changed Δ[Al³⁺] = 0.60 mol/L - [Al³⁺]final = 1.00 M - 0.60 M = 0.40 M (1.00 M initial concentration decreased by 0.60 M) - [Pb²⁺]final = 1.00 M - (0.60 M)(3/2) = 0.10 M (for every Pb²⁺, we need 2/3 the Al³⁺ ion) Now, let's calculate Q: \( Q = \frac{[\mathrm{Al}^{3+}]^{2}}{[\mathrm{Pb}^{2+}]^{3}} \) \( Q = \frac{(0.40)^{2}}{(0.10)^{3}} \) Now, plug in the values into the Nernst equation: \( E = 1.53\text{V} - \frac{(8.314 \text{J mol}^{-1} \text{K}^{-1})(298 \text{K})}{(6)(96485 \text{C mol}^{-1})} \times \ln{(0.40)^{2} \div (0.10)^{3}} \) Finally, calculate the cell potential E: E = 1.36V So, the cell potential after the reaction has operated long enough for the [Al³⁺] to have changed by 0.60 mol/L is 1.36V.

Key concepts

Understanding Electrochemistry

Electrochemistry is a branch of chemistry that deals with the interrelation of electrical currents or potentials and chemical reactions. It is fundamentally rooted in the principles that govern the transfer of electrons between substances—a process often referred to as oxidation-reduction, or redox, reactions.

In an electrochemical cell, such as the one described in the exercise, two half-reactions occur at separate electrodes, typically classified as the anode (where oxidation occurs) and the cathode (where reduction takes place). The movement of electrons from the anode to the cathode through an external circuit generates an electric current, while the corresponding charge balance is maintained by the movement of ions in the electrolyte.

Ions' concentration changes can affect the cell potential due to the relationship between ion activity and electrical potential energy. Understanding the principles of electrochemistry is crucial for innovating in fields such as battery technology, corrosion prevention, and electroplating.

Applying the Nernst Equation

The Nernst equation is vital in electrochemistry, allowing us to predict the behavior of cells under non-standard conditions. It's an equation that describes the relationship between the electromotive force of an electrochemical cell and the concentrations of the chemical species involved in the reaction.

The Nernst equation is given by:
\[ E = E° - \frac{RT}{nF} \times \ln{Q} \]
where \(E\) is the cell potential under non-standard conditions, \(E°\) is the standard cell potential, \(R\) is the ideal gas constant, \(T\) is the temperature in Kelvin, \(n\) is the number of moles of electrons exchanged, \(F\) is the Faraday constant, and \(Q\) is the reaction quotient, which reflects the concentrations of reactants and products at a given moment.

The reaction quotient, \(Q\), is poetry to scientists, as it narrates the story of the reaction's progress, letting us intuit where the reaction stands and predict which way it will proceed. Utilizing the Nernst equation, chemists and students alike can quantitatively dissect the dynamic landscape of cell potential in response to the nuanced shifts in concentration brought forth during a reaction.

The Dance of Redox Reactions

Redox reactions are fundamental processes where electrons are transferred between substances. They encompass two simultaneous events: oxidation, where a substance loses electrons, and reduction, where another gains those electrons. Learning this electrical pas de deux is critical for mastery in electrochemistry, as these reactions are at the heart of generating electricity in batteries, or the bane of it in corrosion.

In the context of the provided exercise, the oxidation involves aluminum (Al) losing electrons to transform into its ion, \(\text{Al}^{3+}\), while the reduction sees lead ions (\(\text{Pb}^{2+}\)) gaining electrons to become lead metal (Pb). The balanced redox equation offers the script detailing the stoichiometry between reactants and products, guiding us to glean the number of electrons involved—a key performer in calculating cell potential.

Remember too, that redox reactions don't just power our electronic devices—they're integral to the very fabric of life, playing leading roles in respiring cells and photosynthesizing plants. By becoming adept with redox equations, you're not just learning to predict the voltage of batteries; you're tracing the energetic pulse of the cosmos.