Question

(a) Write the reactions for the discharge and charge of a nickel-cadmium (nicad) rechargeable battery. (b) Given the following reduction potentials, calculate the standard emf of the cell: $$ \begin{array}{r}{\operatorname{Cd}(\mathrm{OH})_{2}(s)+2 \mathrm{e}^{-} \longrightarrow \mathrm{Cd}(s)+2 \mathrm{OH}^{-}(a q)} \\\ {E_{\mathrm{red}}^{\circ}=-0.76 \mathrm{V}}\end{array} $$ $$ \begin{array}{r}{\mathrm{NiO}(\mathrm{OH})(s)+\mathrm{H}_{2} \mathrm{O}(l)+\mathrm{e}^{-} \longrightarrow \mathrm{Ni}(\mathrm{OH})_{2}(s)+\mathrm{OH}^{-}(a q)} \\\ {E_{\mathrm{red}}^{\circ}=+0.49 \mathrm{V}}\end{array} $$ (c) A typical nicad voltaic cell generates an emf of \(+1.30 \mathrm{V}\) . Why is there a difference between this value and the one you calculated in part (b)? (d) Calculate the equilibrium constant for the overall nicad reaction based on this typical emf value.

Short answer

(a) Discharge reactions: Cadmium reduction: \( \mathrm{Cd(OH)_2}(s) + 2 \mathrm{e^-} \rightarrow \mathrm{Cd}(s) + 2 \mathrm{OH^- }(aq) \) Nickel oxidation: \( \mathrm{NiO(OH)}(s) + \mathrm{H_2 O}(l) + \mathrm{e^-} \rightarrow \mathrm{Ni(OH)_2}(s) + \mathrm{OH^-}(aq) \) (b) Standard emf: \(E_\text{cell}^\circ = 1.25\,\mathrm{V}\) (c) The difference in emf values is due to factors like temperature, concentration, and nonreaction losses not accounted for in standard emf calculations. (d) Equilibrium constant for the overall nicad reaction, based on the typical emf value: \(K \approx 2.6 \times 10^9\)

Step-by-step solution

Part (a): Discharge and Charge Reactions

We need to identify the reactions that occur during the discharge and charge process in a nickel-cadmium (ni-cad) battery. From the given information, we can identify the following half-cell reactions: 1. Cadmium reduction half-reaction: $$ \mathrm{Cd(OH)_2}(s) + 2 \mathrm{e^-} \rightarrow \mathrm{Cd}(s) + 2 \mathrm{OH^- }(aq) $$ 2. Nickel oxidation half-reaction: $$ \mathrm{NiO(OH)}(s) + \mathrm{H_2 O}(l) + \mathrm{e^-} \rightarrow \mathrm{Ni(OH)_2}(s) + \mathrm{OH^-}(aq) $$ Discharge Reaction: During discharge, the battery is providing power to a device, so the cadmium reduction reaction occurs at the negative electrode while the nickel oxidation reaction occurs at the positive electrode. Charge Reaction: During charging, the battery is receiving power to store energy, and the reactions reverse. The cadmium oxidation reaction occurs at the negative electrode and the nickel reduction reaction occurs at the positive electrode.

Part (b): Standard Emf Calculation

To find the standard electromotive force (emf) of the cell, we use the following formula: $$ E_\text{cell}^\circ = E_\text{red, cat}^\circ - E_\text{red, an}^\circ $$ where: \(E_\text{cell}^\circ\) is the standard emf of the cell, \(E_\text{red, cat}^\circ\) is the standard reduction potential of the cathode, \(E_\text{red, an}^\circ\) is the standard reduction potential of the anode. The nickel reduction potential is given to be \(0.49\,\mathrm{V}\), and the cadmium reduction potential is given to be \(-0.76\,\mathrm{V}\). Therefore, the standard emf of the cell can be calculated as: $$ E_\text{cell}^\circ = (0.49\,\mathrm{V}) - (-0.76\,\mathrm{V}) = 1.25\,\mathrm{V} $$

Part (c): Difference in Emf Values

The experimental emf obtained by a typical nicad voltaic cell is \(+1.30\,\mathrm{V}\), whereas the calculated standard emf is \(1.25\,\mathrm{V}\). The discrepancy between the two values arises due to factors not taken into account when calculating the standard emf, such as temperature, concentration, and nonreaction losses (e.g., internal resistance).

Part (d): Equilibrium Constant Calculation

To calculate the equilibrium constant, \(K\), we can use the Nernst equation: $$ E_\text{cell} = E_\text{cell}^\circ - \frac{RT}{nF} \ln K $$ Since the Nernst equation was derived assuming ideal conditions, it is reasonable to use the experimental emf value (\(1.30\,\mathrm{V}\)) rather than the standard emf calculated in part (b), as it provides a more accurate representation of the equilibrium constant of the reaction in a real nicad voltaic cell. Rearrange the Nernst equation to solve for \(K\): $$ K = e^{\frac{nF}{RT}(E_\text{cell} - E_\text{cell}^\circ)} $$ where: \(R = 8.314\,\mathrm{ J\,mol^{-1}\,K^{-1}}\) (gas constant), \(T = 298.15\,\mathrm{K}\) (standard temperature), \(n = 3\) (number of electrons transferred — 2 from the Cd half-reaction and 1 from the Ni half-reaction), \(F = 96485\,\mathrm{ C\, mol^{-1}}\) (Faraday's constant), \(E_\text{cell} = 1.30\,\mathrm{V}\) (given experimental emf), \(E_\text{cell}^\circ = 1.25\,\mathrm{V}\) (calculated standard emf). Substitute the values and calculate \(K\): $$ K = e^{\frac{3 \times 96485\,\mathrm{C\, mol^{-1}}}{8.314\,\mathrm{J\, mol^{-1}\,K^{-1}} \times 298.15\,\mathrm{K}} (1.30\,\mathrm{V} - 1.25\,\mathrm{V})} $$ $$ K \approx 2.6 \times 10^9 $$ The equilibrium constant for the overall nicad reaction, based on the typical emf value, is \(K \approx 2.6 \times 10^9\).

Key concepts

Electrochemical Cells

Electrochemical cells are the foundational building blocks of various types of batteries, including the nickel-cadmium (NiCad) rechargeable batteries. At the heart of these cells is a redox reaction, a chemical process in which oxidation and reduction take place, transferring electrons from one substance to another. In the context of a NiCad battery, these cells consist of two half-cells: the cadmium electrode, which undergoes oxidation during charging and reduction during discharging, and the nickel electrode, which undergoes the opposite.

Each half-cell contains an electrode submerged in an electrolyte, where the ions in the solution facilitate the flow of electric charge between the electrode and the solution. When connected in a circuit, electrons flow from the negative electrode (anode) to the positive electrode (cathode) providing electric power to external devices. The unique chemistry of NiCad batteries allows these cells to be recharged multiple times, making them a valuable power source for various applications.

Standard Reduction Potentials

The standard reduction potentials are a measure of the tendency of a chemical species to be reduced. In other words, it quantifies how likely it is that an element or compound gains electrons. These potentials are measured under standard conditions, which are 1 molar concentration, a pressure of 1 atmosphere, and a temperature of 25℃ (298.15 K).

The standard reduction potentials for the half-reactions in the NiCad battery are significant for calculating the standard electromotive force (emf) of the cell. They help us understand the direction of the electron flow and the voltage we can expect the battery to generate. Specifically, in the step-by-step solution, the nickel reaction has a positive potential, indicating it's more likely to gain electrons (reduction), while the cadmium reaction has a negative potential, indicating it's more likely to lose electrons (oxidation).

By comparing the standard reduction potentials, we are essentially gauging which half-reaction will act as the cathode (higher potential) and which will be the anode (lower potential). The difference between these potentials helps to calculate the standard emf of the cell.

Nernst Equation

The Nernst Equation is a fundamental relation in electrochemistry that describes the relationship between the electromotive force (emf) of a cell and the concentration (or activity) of the ionic species involved in the reaction. The equation provides a way to calculate the emf of an electrochemical cell under non-standard conditions by considering temperature, the number of moles of electrons transferred in the reaction (n), and the concentrations of the reactants and products.

For a reaction at equilibrium or under non-standard conditions, the Nernst equation is represented as:
\[ E_\text{cell} = E_\text{cell}^\circ - \frac{RT}{nF} \ln K \]
Where \(R\) is the gas constant, \(T\) is the temperature in Kelvins, \(F\) is Faraday's constant, \(E_\text{cell}\) is the cell potential under non-standard conditions, \(E_\text{cell}^\circ\) is the standard cell potential, and \(K\) is the equilibrium constant of the reaction.

This equation is pivotal for understanding how changes in conditions affect the performance of a battery and is therefore instrumental for calculating the actual emf of a NiCad battery during operation, as the conditions within a working battery are seldom the same as the standard conditions under which the standard reduction potentials were measured.

Equilibrium Constant Calculation

The equlibirum constant, denoted as \(K\), quantifies the extent of a chemical reaction at equilibrium. It can be determined using the Nernst equation, which connects the emf of a cell to the concentrations of the reactants and products. When the reaction reaches equilibrium, the emf of the cell is zero, and the equation simplifies to a relationship between the standard emf and the equilibrium constant.

For a typical NiCad battery, as mentioned in the solution, one can calculate the equilibrium constant using the measured emf of the cell, rather than the standard emf. This approach often yields a more realistic equilibrium constant representing the actual conditions of the battery's operation, taking into account the fact that the cell is not at the standard conditions implied in the standard reduction potentials.

The calculated equilibrium constant is a dimensionless number that indicates the direction in which the reaction naturally tends to occur. A high equilibrium constant (\(K \gg 1\)) suggests that the products are favored at equilibrium, which in the case of a NiCad battery, means that the discharge process is strongly favored, and the battery can efficiently store and deliver electrical energy.