Problem 104 - Chemistry The Central Science In Si Units

Question

(a) Write the reactions for the discharge and charge of a nickel-cadmium (nicad) rechargeable hattery. (b) Given the following reduction potentials, calculate the standard emf of the cell: $$ \begin{array}{r} \mathrm{Cd}(\mathrm{OH})_{2}(s)+2 \mathrm{e}^{-} \longrightarrow \mathrm{Cd}(s)+2 \mathrm{OH}^{-}(a q) \\ \qquad \begin{aligned} E_{\mathrm{red}}^{\circ} &=-0.76 \mathrm{~V} \end{aligned} \\ \mathrm{NiO}(\mathrm{OH})(s)+\mathrm{H}_{2} \mathrm{O}(l)+\mathrm{e}^{-} \longrightarrow \mathrm{Ni}(\mathrm{OH})_{2}(s)+\mathrm{OH}^{-}(a q) \\ E_{\mathrm{red}}^{\circ}=+0.49 \mathrm{~V} \end{array} $$ (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.

Step-by-Step Solution

Answer

(a) Discharge reactions: Cd oxidized, Ni(OH)3 reduced; Charge reactions: reverse. (b) Standard EMF: 1.25 V. (c) Real cell differs due to non-standard conditions. (d) Large equilibrium constant, indicating spontaneous reaction.

Step 1: Write Discharge Reactions

For the discharge reaction of the nickel-cadmium battery, the overall cell reaction is the sum of cadmium oxidation and nickel(III) hydroxide reduction. The oxidation reaction is \( \mathrm{Cd}(s) + 2\mathrm{OH}^- (aq) \rightarrow \mathrm{Cd(OH)}_2 (s) + 2e^- \), and the reduction reaction is \( \mathrm{NiO(OH)}(s) + \mathrm{H}_2 \mathrm{O}(l) + e^- \rightarrow \mathrm{Ni(OH)}_2 (s) + \mathrm{OH}^- (aq) \).

Step 2: Write Charge Reactions

For the charge part of the cycle, these reactions reverse. Cadmium(II) hydroxide is reduced: \( \mathrm{Cd(OH)}_2(s) + 2e^- \rightarrow \mathrm{Cd}(s) + 2\mathrm{OH}^-(aq) \), and nickel(II) hydroxide is oxidized: \( \mathrm{Ni(OH)}_2(s) + \mathrm{OH}^-(aq) \rightarrow \mathrm{NiO(OH)}(s) + \mathrm{H}_2 \mathrm{O}(l) + e^- \).

Step 3: Calculate Standard EMF

The standard EMF of the cell is calculated from the standard reduction potentials. EMF is given by \( E^{\circ}_{\text{cell}} = E^{\circ}_{\text{cathode}} - E^{\circ}_{\text{anode}} \). Using \( E^{\circ}_{\text{cathode}} = 0.49 \text{ V} \) and \( E^{\circ}_{\text{anode}} = -0.76 \text{ V} \), we get \( E^{\circ}_{\text{cell}} = 0.49 - (-0.76) = 1.25 \text{ V} \).

Step 4: Explain EMF Difference

The difference between the calculated EMF of 1.25 V and the typical EMF of 1.30 V is due to non-standard conditions in real batteries. These can include deviations from standard concentration, temperature, or activity differences.

Step 5: Calculate Equilibrium Constant

The equilibrium constant \( K \) is related to the EMF by the Nernst equation for equilibrium \( E = \frac{RT}{nF} \ln K \). Rearranging gives \( \ln K = \frac{nFE}{RT} \). With \( n = 2 \), \( F = 96485 \text{ C/mol} \), \( E = 1.30 \text{ V} \), and \( T = 298 \text{ K} \), we solve for \( K \). This gives \( K \) as a large value, indicating a product-favored reaction.

Key Concepts

Nickel-Cadmium Battery

Nickel-cadmium batteries, commonly referred to as nicad batteries, are a type of rechargeable battery that features cadmium as the anode and nickel (III) hydroxide as the cathode. During discharge, the cadmium undergoes oxidation, while the nickel (III) hydroxide is reduced. Discharge reactions involve converting chemical energy into electrical energy, which powers devices. When charging, these reactions reverse to restore the battery’s original state by using electrical energy to drive the chemical reactions backward.

These batteries are popular due to their capability to deliver a consistent voltage until they are nearly depleted. They are also capable of handling high discharge rates and enduring many charge-discharge cycles, which makes them ideal for use in portable electronic devices and tools. However, they must be stored properly to avoid the 'memory effect', where the battery loses its maximum energy capacity when repeatedly recharged after being only partially discharged.

Standard Reduction Potentials

Standard reduction potentials are a measure of the tendency of a chemical species to acquire electrons and thereby be reduced. They are measured in volts under standard conditions, which include 1 M concentration, 1 atm pressure, and a temperature of 25°C (298 K).

In the context of a nickel-cadmium battery, each half-reaction has its own standard reduction potential. For cadmium, the reduction potential is -0.76 V, which makes it more likely to lose electrons (oxidize) than something with a more positive potential. For nickel (III) hydroxide, the standard potential is +0.49 V, indicating it is more likely to accept electrons (reduce) compared to cadmium.

To calculate the standard cell potential, you use the formula:
\[ E^{\circ}_{\text{cell}} = E^{\circ}_{\text{cathode}} - E^{\circ}_{\text{anode}} \]
This gives you the EMF under standard conditions, providing insight into the driving forces behind the cell's reactions.

Equilibrium Constant

The equilibrium constant, denoted as \( K \), is a fundamental aspect of chemistry that gives an indication of the relative concentrations of products and reactants for a chemical reaction at equilibrium. A high equilibrium constant, typically much greater than 1, implies that the products are favored in the reaction. Conversely, a low \( K \) suggests reactants are favored.

In electrochemical cells, \( K \) can be related to the cell's potential using the Nernst equation. At standard conditions, any change in the EMF of the cell affects the value of \( K \). In the case of the nickel-cadmium battery with an EMF of 1.30 V, this indicates a strong propensity for the forward reaction, thus a large \( K \) value. This is why the nicad battery is so efficient at producing energy - it shows a strong preference for the formation of products.

Nernst Equation

The Nernst equation provides a quantitative relationship between the cell potential and the conditions of the reaction, namely the concentrations of reactants and products. It extends the concept of standard cell potential to account for changes in concentration or pressure, making it extremely versatile for practical applications.

The equation is given by:
\[ E = E^{\circ} - \frac{RT}{nF} \ln Q \]
where:

• \( E \) is the cell potential under specific conditions.
• \( E^{\circ} \) 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 Faraday's constant.
• \( Q \) is the reaction quotient.

The Nernst equation allows us to calculate the equilibrium constant by setting the equation such that the cell potential \( E \) is zero (as it would be at equilibrium), thereby giving insight into the equilibrium position of the electrochemical reaction. It’s particularly useful for understanding the behavior of batteries under non-standard conditions, such as in real-world applications.