Problem 38 - Chemistry: An Atoms-Focused Approach

Question

Laptops The first generation of laptop computers was powered by nickcl-cadmium (NiCad) batterics, which generated \(1.20 \mathrm{V}\) based on the following cell reaction: \(\mathrm{Cd}(s)+2 \mathrm{NiO}(\mathrm{OH})(s)+2 \mathrm{H}_{2} \mathrm{O}(\ell) \rightarrow\) $$ \mathrm{Cd}(\mathrm{OH})_{2}(s)+2 \mathrm{Ni}(\mathrm{OH})_{2}(s) $$ What is the value of \(\Delta G_{\text {cell }} ?\)

Step-by-Step Solution

Answer

Based on the given chemical reaction representing the cell reaction in a laptop battery using NiCad batteries, the Gibbs free energy change (ΔG_cell) is approximately -231,564 J/mol. This was calculated using the balanced chemical equation, identifying the oxidation and reduction half-reactions, determining the number of electrons transferred and the cell potential, and applying the formula ΔG = -nFE.

Step 1: Balance the chemical reaction

Before we can calculate ΔG_cell, we need to balance the given chemical reaction by adding coefficients such that the number of atoms of each element is the same on both sides of the reaction. The balanced equation is: $$ \mathrm{Cd}(s)+2 \mathrm{NiO}(\mathrm{OH})(s)+2 \mathrm{H}_{2} \mathrm{O}(\ell) \rightarrow \mathrm{Cd}(\mathrm{OH})_{2}(s)+2 \mathrm{Ni}(\mathrm{OH})_{2}(s) $$

Step 2: Identify the half-reactions

Now that we have a balanced equation, we can identify the half-reactions, which represent the oxidation and reduction processes: Oxidation: \(\mathrm{Cd}(s) \rightarrow \mathrm{Cd}(\mathrm{OH})_{2}(s) + 2 e^-\) Reduction: \(2 \mathrm{NiO}(\mathrm{OH})(s) + 2 e^- + 2 \mathrm{H}_{2} \mathrm{O}(\ell) \rightarrow 2 \mathrm{Ni}(\mathrm{OH})_{2}(s)\)

Step 3: Calculate the number of electrons (n) transferred

In the half-reactions, we can see that 2 electrons are being transferred. So, the number of electrons transferred n is equal to 2.

Step 4: Determine the cell potential (E)

The cell potential, E, is given as 1.20 V in the problem statement.

Step 5: Calculate ΔG_cell using the formula

Now we are ready to calculate ΔG_cell, using the formula ΔG = -nFE: ΔG = - n * F * E, where F is the Faraday constant (approximately 96,485 C/mol) ΔG = - (2 mol) * (96,485 C/mol) * (1.20 V) ΔG ≈ -231,564 J/mol Therefore, the value of ΔG_cell is approximately -231,564 J/mol.

Key Concepts

Cell Potential

Cell potential, often denoted as \(E_{cell}\), reflects how effective an electrochemical cell is at generating an electric current. Essentially, it tells us the driving force of the reaction occurring in the cell. This potential difference arises due to the movement of electrons across the cell.
In a galvanic cell setup, cell potential can be measured in volts (V). The potential indicates the eagerness of electrons to move from one electrode to another:

• Anode: where oxidation (loss of electrons) takes place.
• Cathode: where reduction (gain of electrons) occurs.

Ionic movement within the cell complements electron flow, maintaining charge balance.
Given the problem's details, the cell potential is provided directly as 1.20 V, this reflects the energy per unit charge the electrons have as they move through the circuit.

Electrochemistry

Electrochemistry is the branch of chemistry that deals with the relationship between electrical energy and chemical changes. It's the fundamental science behind batteries and fuel cells. In our discussed example, a nickel-cadmium (NiCad) battery is a classic application of electrochemical principles.
In an electrochemical reaction, two half-reactions occur:
The oxidation half-reaction occurs at the anode and involves the loss of electrons, whereas the reduction half-reaction happens at the cathode, involving the gain of electrons.

• Oxidation involves cadmium (\( \mathrm{Cd}(s) \rightarrow \mathrm{Cd}(\mathrm{OH})_{2}(s) + 2 e^- \)).
• Reduction involves nickel oxyhydroxide (\( 2 \mathrm{NiO}(\mathrm{OH})(s) + 2 e^- + 2 \mathrm{H}_{2} \mathrm{O}(\ell) ightarrow 2 \mathrm{Ni}(\mathrm{OH})_{2}(s) \)).

These half-reactions collectively contribute to the overall cell potential by creating an environment where electrons flow freely, thus producing electricity.

Redox Reactions

Redox reactions are a vital component of electrochemical cells and involve two simultaneous but opposite processes: oxidation and reduction.
In simple terms, oxidation refers to the loss of electrons from a substance, while reduction involves the gain of electrons. They always occur together — when one substance is oxidized, another is reduced.
Within the NiCad battery:

• Cadmium (Cd) undergoes oxidation, losing electrons, which is described by:\( Cd(s) \rightarrow Cd(OH)_{2}(s) + 2 e^- \).
• Nickel oxyhydroxide \((NiO(OH))\) undergoes reduction, gaining electrons:\( 2 NiO(OH)(s) + 2 e^- + 2 H_{2}O(l) \rightarrow 2 Ni(OH)_{2}(s) \).

These chemical changes are paramount in generating energy, allowing devices such as laptops to operate efficiently.