Question

The overall reaction and standard cell potential at \(25^{\circ} \mathrm{C}\) for the rechargeable nickel-cadmium alkaline battery is \(\mathrm{Cd}(s)+\mathrm{NiO}_{2}(s)+2 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow\) \(\mathrm{Ni}(\mathrm{OH})_{2}(s)+\mathrm{Cd}(\mathrm{OH})_{2}(s) \quad \mathscr{E}^{\circ}=1.10 \mathrm{~V}\) For every mole of Cd consumed in the cell, what is the maximum useful work that can be obtained at standard conditions?

Short answer

For every mole of Cd consumed in the cell, the maximum useful work obtained at standard conditions is \(212269\,\mathrm{J/mol}\).

Step-by-step solution

Identify the Gibbs free energy change formula

The formula to calculate the Gibbs free energy change under standard conditions is given by: \(\Delta G^{\circ} = -nFE^{\circ}\) Where \(\Delta G^{\circ}\) is the Gibbs free energy change under standard conditions, n is the number of moles of electrons transferred, F is the Faraday constant, which is approximately \(96485\,\text{C/mol}\), and \(E^{\circ}\) is the standard cell potential.

Determine the number of moles of electrons transferred

To find the number of moles of electrons transferred, we must first determine the balanced equation for the given redox reaction. The balanced redox reaction for the given nickel-cadmium alkaline battery is \(Cd(s) + 2NiO_2(s) + 2H_2O(l) \longrightarrow Cd(OH)_2(s) + 2Ni(OH)_2(s)\) From the balanced equation, we can see that each mole of Cd reacts with 2 moles of NiO₂, and 2 moles of electrons are transferred. So, the number of moles of electrons transferred (n) is 2.

Calculate the Gibbs free energy change

Now, we can plug in our values into the Gibbs free energy change formula: \(\Delta G^{\circ} = -nFE^{\circ} = -2 \times 96485\,\mathrm{C/mol} \times 1.10\,\mathrm{V}\) \(\Delta G^{\circ} = -212269\,\mathrm{J/mol}\)

Relate Gibbs free energy change to maximum work

The maximum useful work that can be obtained is equal to the Gibbs free energy change under standard conditions. Therefore, the maximum useful work that can be obtained for every mole of Cd consumed in the cell at standard conditions is: Maximum useful work = \(\Delta G^{\circ} = -212269\,\mathrm{J/mol}\) For every mole of Cd consumed in the cell, the maximum useful work obtained at standard conditions is \(212269\,\mathrm{J/mol}\).

Key concepts

Gibbs Free Energy

Gibbs free energy, denoted as \( \Delta G \), is a central concept in thermodynamics and electrochemistry, reflecting the amount of work a system can perform at constant temperature and pressure. It tells us if a process can occur spontaneously. The equation \( \Delta G^\circ = -nFE^\circ \) bridges the gap between electrochemical processes and thermodynamics by indicating that the energy produced in a battery or any electrochemical cell can do useful work. A negative value for \( \Delta G \) signifies a spontaneous reaction, which is essential for batteries, as they rely on spontaneous redox reactions to generate electricity. This value, when calculated for standard conditions, provides the maximum non-expansion work that the battery can perform during its operation.

Faraday Constant

The Faraday constant \( F \) plays a critical role in electrochemistry and is fundamental to calculating the work generated in redox reactions. Representing approximately \( 96485\ C/mol \), it relates charge to substance amounts in moles. In the context of an electrochemical cell, \( F \) corresponds to the charge on one mole of electrons. This constant is vital when converting between the electrical energy in coulombs and chemical energy in moles. By multiplying it with the standard cell potential \( E^\circ \) and the number of moles of electrons \( n \) transferred in the reaction, we can calculate the Gibbs free energy change, allowing us to understand the efficiency and possible work a chemical reaction within a battery can produce.

Redox Reactions

Redox reactions, or oxidation-reduction reactions, are the foundation of electrochemical cells, including batteries. These reactions involve the transfer of electrons between species, where oxidation refers to the loss of electrons, and reduction involves the gain of electrons. In the example of the nickel-cadmium battery, the cadmium is oxidized, and the nickel oxide is reduced during discharge. This electron transfer drives the electrical current flow in an external circuit. It's the careful balancing of these redox reactions within the battery that ensures the efficient production of electrical energy. Understanding the stoichiometry of redox reactions is key to determining the number of electrons transferred, which is a necessary step for calculating the cell's potential and its related Gibbs free energy.

Nickel-Cadmium Batteries

Nickel-cadmium batteries are a type of rechargeable battery that uses the redox reactions between nickel oxide and cadmium at the electrodes to generate electrical energy. These batteries are known for their long life cycle, ability to deliver a stable voltage, and perform well at low temperatures. They were widely used in portable electronics, power tools, and aviation, among other applications. However, environmental concerns due to the toxicity of cadmium and the evolution of battery technology have led to a decline in their usage in favor of more sustainable alternatives. Understanding the electrochemistry of nickel-cadmium batteries, including their standard cell potential, provides insight into their operation and efficiency.

Electrochemistry

Electrochemistry is the branch of chemistry that deals with the relationships between electrical potentials, or voltages, and chemical reactions. It encompasses the study of redox reactions, where electron transfer occurs, and includes the design and analysis of batteries, fuel cells, and electrolysis processes. The principles of electrochemistry are applied to calculate the standard cell potentials, like the \( 1.10\ V \) for nickel-cadmium batteries, to predict the thermodynamic feasibility and electrical output of reactions. By exploring the links between chemical energy and electrical energy, electrochemistry allows us to convert one form of energy into another efficiently, which is the cornerstone of operating devices that depend on battery power.