Question

For a lead storage battery: (a) Sketch one cell that shows the anode, cathode, electrolyte, direction of electron and ion flow, and sign of the electrodes. (b) Write the anode, cathode, and overall cell reactions. (c) Calculate the equilibrium constant for the cell reaction \(\left(E^{\circ}=1.924 \mathrm{~V}\right)\) (d) What is the cell voltage when the cell reaction reaches equilibrium?

Short answer

At equilibrium, the cell voltage is 0 V.

Step-by-step solution

Sketching the Battery Cell

To sketch a lead storage battery cell, identify the anode as lead (Pb) and the cathode as lead dioxide \((\text{PbO}_2)\). The electrolyte is a solution of sulfuric acid \((\text{H}_2\text{SO}_4)\). Electrons flow from the anode to the cathode through the external circuit. Lead acts as the negative electrode, whereas lead dioxide acts as the positive electrode. Cations (\(\text{H}^+\)) move towards the cathode while anions (\(\text{SO}_4^{2-}\)) move towards the anode in the electrolyte.

Writing Reactions

Identify the half-reactions at each electrode. At the anode, the reaction is: \[ \text{Pb} + \text{HSO}_4^- \rightarrow \text{PbSO}_4 + \text{H}^+ + 2e^- \]. At the cathode, the reaction is: \[ \text{PbO}_2 + \text{HSO}_4^- + 3\text{H}^+ + 2e^- \rightarrow \text{PbSO}_4 + 2\text{H}_2\text{O} \].The overall cell reaction combines these two: \[ \text{Pb} + \text{PbO}_2 + 2\text{H}_2\text{SO}_4 \rightarrow 2\text{PbSO}_4 + 2\text{H}_2\text{O} \].

Calculate Equilibrium Constant

Use the Nernst Equation: \( E^\circ = \frac{RT}{nF} \ln K \) to find the equilibrium constant. Given, \( E^\circ = 1.924 \text{ V} \), \( n = 2 \). Assume standard conditions (\( T = 298 \text{ K} \)).\[\ln K = \frac{nFE^\circ}{RT} = \frac{2 \times 96485 \times 1.924}{8.314 \times 298}\]Calculate \( K \) by solving for \( \ln K \), then take the exponential.

Cell Voltage at Equilibrium

A cell reaches equilibrium when the cell voltage (\(E\)) drops to zero, because the electromotive force required to drive the spontaneous reaction is no longer present, hence \(E = 0\) at equilibrium.

Key concepts

Lead storage battery

The lead storage battery, commonly used in automobiles, is a type of rechargeable battery. It consists of several cells, each containing an anode made of lead (Pb) and a cathode made of lead dioxide (PbO₂). The electrolyte is a sulfuric acid ( H₂SO₄) solution. This setup allows the battery to store and deliver electrical energy efficiently.

When the battery discharges, chemical reactions drive the flow of electrons from the anode to the cathode through an external circuit. During charging, this process is reversed. This reversibility is what makes lead storage batteries rechargeable.

Electrode reactions

In a lead storage battery, the chemical reactions at the electrodes are crucial for its operation. At the anode, the lead (Pb) undergoes oxidation, transforming into lead sulfate (PbSO₄) and releasing electrons:

\[ \text{Pb} + \text{HSO}_4^- \rightarrow \text{PbSO}_4 + \text{H}^+ + 2e^- \].

At the cathode, lead dioxide (PbO₂) undergoes reduction by accepting electrons, forming lead sulfate and water:

\[ \text{PbO}_2 + \text{HSO}_4^- + 3\text{H}^+ + 2e^- \rightarrow \text{PbSO}_4 + 2\text{H}_2\text{O} \].

The overall cell reaction combining both electrodes is:

\[ \text{Pb} + \text{PbO}_2 + 2\text{H}_2\text{SO}_4 \rightarrow 2\text{PbSO}_4 + 2\text{H}_2\text{O} \].

Nernst equation

The Nernst equation helps us calculate the cell potential at any given condition, not just standard conditions. It accounts for concentrations of the reactants and products. For the lead storage battery, the equation can be expressed as:

\[ E = E^\circ - \frac{RT}{nF} \ln Q \],

where \( E \) is the cell potential, \( E^\circ \) is the standard cell potential (1.924 V here), \( R \) is the universal gas constant, \( T \) is temperature in Kelvin, \( n \) is the number of moles of electrons exchanged, and \( F \) is Faraday's constant. \( Q \) is the reaction quotient, reflecting the ratio of concentrations of products to reactants at any given point in time.

This equation is essential for understanding how the cell voltage changes under different conditions.

Equilibrium constant

The equilibrium constant (K) of a cell reaction provides important insight into the favorability of reactions under equilibrium conditions. To find it for a lead storage battery, we use the relationship to the Nernst equation:

\[ E^\circ = \frac{RT}{nF} \ln K \].

Replacing known values, we can solve for \( \ln K \) and hence \( K \), using:

\[ \ln K = \frac{nFE^\circ}{RT} = \frac{2 \times 96485 \times 1.924}{8.314 \times 298} \].

By calculating, \( K \) gives an understanding of how the concentrations of chemicals will be distributed at equilibrium compared to initial conditions.

Cell voltage

The cell voltage is a measure of the electric potential difference between the two electrodes. Initially, it's determined by the standard cell potential (1.924 V for a lead storage battery). This voltage indicates the ability of the battery to drive an electric current.

When the reaction approaches equilibrium, the cell voltage approaches zero. This happens because the forward and reverse reactions occur at the same rate, canceling out the driving force necessary for electron flow.

Understanding cell voltage over time is crucial for estimating battery life and efficiency.