The overall reaction in the lead storage battery is \(\mathrm{Pb}(s)+\mathrm{PbO}_{2}(s)+2 \mathrm{H}^{+}(a q)+2 \mathrm{HSO}_{4}^{-}(a q) \longrightarrow\) \(2 \mathrm{PbSO}_{4}(s)+2 \mathrm{H}_{2} \mathrm{O}(l)\) a. For the cell reaction \(\Delta H^{\circ}=-315.9 \mathrm{~kJ}\) and \(\Delta S^{\circ}=\) \(263.5 \mathrm{~J} / \mathrm{K}\). Calculate \(\mathscr{E}^{\circ}\) at \(-20 .{ }^{\circ} \mathrm{C}\). Assume \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) do not depend on temperature. b. Calculate \(\mathscr{E}\) at \(-20 .{ }^{\circ} \mathrm{C}\) when \(\left[\mathrm{HSO}_{4}{ }^{-}\right]=\left[\mathrm{H}^{+}\right]=4.5 M\). c. Consider your answer to Exercise 71 . Why does it seem that batteries fail more often on cold days than on warm days?

A lead storage battery, also known as a lead-acid battery, is a type of rechargeable battery commonly used in cars and other vehicles. It consists of plates of lead and lead dioxide submerged in an electrolyte solution of sulfuric acid. During discharge, lead and lead dioxide react with the sulfuric acid to create lead sulfate and water, releasing electrical energy. Conversely, when the battery is being charged, electrical energy is used to reverse this reaction and restore the original components.

Lead storage batteries have been the go-to choice for automotive starter batteries because of their ability to provide high surge currents, meaning they can deliver large amounts of current in a short period. Their widespread use, affordability, and relatively large power-to-weight ratio contribute to their popularity, despite being heavier compared to newer types of batteries like lithium-ion.

In electrochemistry, Gibbs free energy (ΔG) is a thermodynamic quantity that plays a pivotal role. It is a measure of the maximum reversible work that may be performed by a thermodynamic system at a constant temperature and pressure. The change in Gibbs free energy is given by the equation ΔG = ΔH - TΔS, where ΔH is the change in enthalpy (total heat content), T is the absolute temperature, and ΔS is the change in entropy (degree of disorder or randomness).

The importance of Gibbs free energy in electrochemistry is paramount for predicting the spontaneity of electrochemical reactions. A negative ΔG indicates a spontaneous process, while a positive value suggests a non-spontaneous reaction. In the case of a lead storage battery, calculating ΔG helps predict how the battery will perform under different conditions and is especially relevant in understanding the energy changes during the charging and discharging processes.

The Nernst equation allows chemists and engineers to determine the cell potential of an electrochemical cell at any given set of conditions, not just standard-state conditions. It accounts for temperature, concentration, and partial pressure of gases if involved. The general form of the Nernst equation is: E = E° - (RT/nF)lnQ, where E is the cell potential under non-standard conditions, E° is the standard cell potential, R is the universal gas constant, T is the temperature in Kelvin, F is the Faraday constant, and Q is the reaction quotient, a measure of the concentrations of reactants and products.

Applying the Nernst equation is crucial when the conditions within the battery, like temperature and concentrations, change from the standard. This allows for a precise calculation of the battery's voltage output under various conditions, which is essential for the practical use of batteries in real-life applications.

The term cell potential, denoted as E, refers to the measure of the potential difference between the electrodes of an electrochemical cell. For a lead storage battery, the standard cell potential (E°) is a fixed value that represents the voltage of the cell under standard conditions (concentrations of 1 M, pressure of 1 atm, and temperature of 298 K).

The actual cell potential (E) can differ from E° due to changes in concentration and temperature, as explained by the Nernst equation. The cell potential indicates whether a cell will operate spontaneously and provides insight into how much voltage the battery can deliver in practical use. Understanding the cell potential is essential in designing circuits and systems that depend on batteries; it ensures that the voltage supplied matches the requirements of the electrical device being powered.

Temperature can significantly affect a battery's performance, particularly evident in automotive lead storage batteries. As temperature decreases, the chemical reaction rates inside the battery slow down, resulting in a diminishing capacity and lowered voltage output. This means that on cold days, a battery might not be able to provide enough power to start a car. Additionally, lower temperatures can increase the internal resistance of a battery, further reducing its efficiency.

Conversely, at higher temperatures, chemical reactions occur more quickly, potentially leading to an increased rate of self-discharge and a shorter overall lifespan. Manufacturers typically specify an optimal temperature range for their batteries to ensure they deliver the desired performance. Understanding these temperature effects is crucial for maintaining battery health and ensuring reliability, especially in regions that experience extreme temperature variations.