Problem 76 - Chemistry: Principles and Reactions

Question

In a nickel-cadmium battery (Nicad), cadmium is oxidized to \(\mathrm{Cd}(\mathrm{OH})_{2}\) at the anode, while \(\mathrm{Ni}_{2} \mathrm{O}_{3}\) is reduced to \(\mathrm{Ni}(\mathrm{OH})_{2}\) at the cathode. A portable CD player uses \(0.175\) amp of current. How many grams of \(\mathrm{Cd}\) and \(\mathrm{Ni}_{2} \mathrm{O}_{3}\) are consumed when the CD player is used for an hour and a half?

Step-by-Step Solution

Answer

The mass of Cd consumed in the battery is 0.5501 grams, and the mass of Ni2O3 consumed is 0.3502 grams when the CD player is used for 1.5 hours.

Step 1: Calculate the total charge passed through the battery

The current provided by the battery is 0.175 A. Using the formula Q = It, where Q is the charge, I is the current, and t is the time, calculate the charge: Current (I) = 0.175 A Time (t) = 1.5 hours = 1.5 × 3600 = 5400 s Q = It = 0.175 A × 5400 s = 945 C

Step 2: Calculate the moles of electrons passed

Using Faraday's constant (96485 C/mol), we can calculate the moles of electrons passed through the battery: Moles of electrons = Total charge / Faraday's constant = 945 C / 96485 C/mol = 0.00979 mol

Step 3: Calculate moles of Cd and Ni2O3 consumed

For the anode, the Cd half-reaction is: \(\mathrm{Cd} \rightarrow \mathrm{Cd}(\mathrm{OH})_{2} + 2\mathrm{e}^-\) 2 moles of electrons are required to oxidize 1 mole of Cd. Moles of Cd consumed = (0.00979 mol electrons) / 2 = 0.004895 mol For the cathode, the Ni2O3 half-reaction is: \(\mathrm{Ni}_{2}\mathrm{O}_{3} + 3\mathrm{H}_{2}\mathrm{O} + 6\mathrm{e}^-\rightarrow 2\mathrm{Ni}(\mathrm{OH})_{2} + 6\mathrm{OH}^-\) 6 moles of electrons are required to reduce 1 mole of Ni2O3. Moles of Ni2O3 consumed = (0.00979 mol electrons) / 6 = 0.001632 mol

Step 4: Convert moles to grams

Find the mass of Cd and Ni2O3 consumed by multiplying the moles consumed by their respective molar masses: Mass of Cd consumed = (0.004895 mol Cd) × (112.41 g/mol Cd) = 0.5501 g Mass of Ni2O3 consumed = (0.001632 mol Ni2O3) × (214.711 g/mol Ni2O3) = 0.3502 g

Step 5: Final Answer

When the CD player is used for an hour and a half, 0.5501 grams of Cd and 0.3502 grams of Ni2O3 are consumed in the battery.

Key Concepts

Electrochemical Cells

Electrochemical cells are fundamental components of batteries, like the nickel-cadmium (Nicad) battery mentioned in our exercise. They work on the principle of converting chemical energy into electrical energy through redox reactions. An electrochemical cell has two electrodes: an anode where oxidation occurs, and a cathode where reduction takes place. In the Nicad battery, cadmium at the anode donates electrons during oxidation, while at the cathode, nickel oxide accepts electrons during reduction. The flow of electrons from the anode to the cathode through an external circuit creates the electric current that powers devices such as the portable CD player.

In understanding battery chemistry, it's essential to recognize that the overall performance and capacity of the battery are affected by the materials used for the electrodes and the efficiency of the electrochemical reactions occurring within. The half-reactions at the anode and cathode are balanced, ensuring that the battery operates safely and effectively. Energy storage technology relies heavily on optimizing these components to improve energy density, longevity, and rechargeability.

Redox Reactions

Redox reactions are the heart of electrochemical cells and involve the transfer of electrons between chemical species. 'Redox' is a portmanteau of reduction (gain of electrons) and oxidation (loss of electrons). In the context of the Nicad battery, cadmium gets oxidized, losing electrons, and consequently, nickel oxide gets reduced, gaining those electrons. It's this transfer that facilitates the flow of electrical current.

To fully grasp these reactions, consider how atoms in redox processes move between different oxidation states. In the Nicad battery, Cd atoms are oxidized from a neutral state to a plus two oxidation state as \(\mathrm{Cd}(\mathrm{OH})_{2}\), while \(\mathrm{Ni}_{2}\mathrm{O}_{3}\) is reduced to \(\mathrm{Ni}(\mathrm{OH})_{2}\) at the cathode. Getting comfortable with the concept of redox reactions is not only vital in understanding batteries but also in many other fields, such as corrosion, metabolism, and even the functioning of cells in biology.

Faraday's Law of Electrolysis

Faraday's law of electrolysis provides a quantitative relationship between the amount of electric charge carried by electrons that pass through an electrochemical cell and the amount of substance that undergoes a redox reaction. It includes two laws; the first states that the mass of a substance altered at an electrode during electrolysis is proportional to the quantity of electricity that passes through the cell. The second law says that the mass of substance produced or consumed at an electrode is proportional to its equivalent weight.

For our exercise, Faraday's law helps us determine the moles of electrons that have passed through the Nicad battery by dividing the total charge by Faraday's constant (96485 C/mol). With this information, we can then calculate how much of the cadmium and nickel compound was consumed during operation. Faraday's laws are pivotal in fields like electroplating, electrosynthesis, and in devising battery management systems.

Mole Concept

The mole concept is a vital pillar of chemistry that provides a bridge between the atomic world and the macroscopic world we experience. One mole represents Avogadro's number, which is approximately \(6.022 \times 10^{23}\) entities, be they atoms, ions, molecules, or electrons. In the battery calculations from our exercise, the mole concept allows us to convert the electric charge passed into actual quantities of substances involved in the redox reactions – the moles of electrons, cadmium, and \(\mathrm{Ni}_{2}\mathrm{O}_{3}\).

Understanding the mole concept is crucial for balancing chemical equations, quantifying reaction yields, and calculating the masses of reactants and products involved in a reaction. For example, in the step where we converted the moles of cadmium and nickel oxide to grams, we utilized their molar masses, thereby linking the microscopic scale interactions with a measurable, macroscopic quantity—the weight of these substances consumed by the CD player.