Question

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 \(\left[\mathrm{HSO}_{4}^{-}\right]=\left[\mathrm{H}^{+}\right]=4.5 \mathrm{M}\) c. Consider your answer to Exercise \(71 .\) Why does it seem that batteries fail more often on cold days than on warm days?

Short answer

The standard cell potential at -20°C is approximately 1.478 V, and the cell potential with the given concentrations at -20°C is also approximately 1.478 V. Lower temperatures lead to decreased cell potential, resulting in slower battery discharge rates and weaker overall performance. Batteries are more likely to fail on cold days compared to warmer days due to this decrease in cell potential, which affects their ability to provide the required energy to power devices.

Step-by-step solution

Calculate \(\mathscr{E}^{\circ}\) at -20 °C using Gibbs free energy equation

We are given that \(\Delta H^{\circ} = -315.9\,\text{kJ}\) and \(\Delta S^{\circ} = 263.5\,\text{J/K}\). First, we need to convert the temperature from Celsius to Kelvin. We know that at \(T = -20\,^{\circ}\text{C}\), the temperature in Kelvin is: \[T = (-20+273.15)\,\text{K} = 253.15\,\text{K}\] The relationship between the Gibbs free energy (\(\Delta G^{\circ}\)), enthalpy change (\(\Delta H^{\circ}\)), and entropy change (\(\Delta S^{\circ}\)) is given by the equation: \[\Delta G^{\circ} = \Delta H^{\circ} - T\Delta S^{\circ}\] Substitute the given values and temperature to find the Gibbs free energy change: \[\Delta G^{\circ} = (-315.9 \times 10^{3}\,\text{J}) - (253.15\,\text{K})(263.5\,\text{J/K})\] Calculate the value of \(\Delta G^{\circ}\): \[\Delta G^{\circ} \approx -285196.725\,\text{J}\] Next, we will use the relationship between the Gibbs free energy change and the standard cell potential \(\mathscr{E}^{\circ}\), given by: \[\Delta G^{\circ} = -nF\mathscr{E}^{\circ}\] Where \(n\) is the number of moles of electrons transferred, and \(F\) is the Faraday constant, approximately equal to \(96485\,\text{C/mol}\). For the given overall reaction, the number of electrons transferred, \(n\), is 2. Now, solve for the standard cell potential (\(\mathscr{E}^{\circ}\)): \[\mathscr{E}^{\circ} = -\frac{\Delta G^{\circ}}{nF}\] Plug in the values: \[\mathscr{E}^{\circ} \approx \frac{-(-285196.725\,\text{J})}{(2)(96485\,\text{C/mol})}\] Calculate the value of \(\mathscr{E}^{\circ}\): \[\mathscr{E}^{\circ} \approx 1.478\,\text{V}\] Thus, the standard cell potential at -20° C is approximately 1.478 V.

Calculate \(\mathscr{E}\) at -20 °C with given concentrations

Now, we will calculate the cell potential \(\mathscr{E}\) at -20° C when \([\mathrm{HSO}_{4}^{-}] = [\mathrm{H}^{+}] = 4.5\,\text{M}\). Because the reaction involves entropy change, the Nernst equation will be used to calculate the cell potential: \[\mathscr{E} = \mathscr{E}^{\circ} - \frac{RT}{nF} \ln(Q)\] Where \(Q\) is the reaction quotient. For the given reaction, since there are 2 moles of products and 2 moles of reactants: \(Q = \frac{[\mathrm{H}^{+}]^2}{[\mathrm{HSO}_{4}^{-}]^2}\) Since \([\mathrm{H}^{+}] = [\mathrm{HSO}_{4}^{-}] = 4.5\,\text{M}\), the reaction quotient, \(Q\), is equal to 1. Now, substitute the values and solve for \(\mathscr{E}\): \[\mathscr{E} = 1.478\,\text{V} - \frac{(8.314\,\text{J/(K*mol}))(253.15\,\text{K})}{(2)(96485\,\text{C/mol})} \ln(1)\] Since \(\ln(1) = 0\), the equation simplifies to: \[\mathscr{E} = 1.478\,\text{V}\] Thus, the cell potential at -20 °C with the given concentrations is approximately 1.478 V.

Explain the effect of temperature on battery performance

In step 1, we have seen that colder temperatures result in a lower cell potential. A lower cell potential corresponds to a slower battery discharge rate and weaker overall performance. This is because the entropy change equation, \(\Delta G^{\circ} = \Delta H^{\circ} - T\Delta S^{\circ}\), shows that as the temperature decreases, the Gibbs free energy change increases, leading to a decrease in cell potential. This decrease in cell potential means that the battery is less capable of providing the required energy to start or power devices. Thus, batteries tend to fail more often on cold days compared to warmer days, as lower temperatures affect their overall performance.

Key concepts

Gibbs Free Energy

Gibbs Free Energy is essential in understanding how chemical reactions occur and evolve. It helps us know whether a reaction is spontaneous. The equation \( \Delta G^{\circ} = \Delta H^{\circ} - T\Delta S^{\circ} \) is key here. It expresses the relationship between the change in enthalpy \(\Delta H^{\circ}\), change in entropy \(\Delta S^{\circ}\), and temperature \(T\).

- **\(\Delta H^{\circ}\)** represents the heat involved in the reaction. Negative means the reaction releases heat, while positive means it absorbs heat.- **\(\Delta S^{\circ}\)** indicates the level of disorder or randomness in the system. - **Temperature \(T\)** affects reactions significantly, as seen in the equation.

In electrochemistry, and particularly in the case of batteries, the Gibbs Free Energy change can predict how efficient a battery's reaction will be. By using this fundamental concept, we can make calculations about battery performance under different temperatures.

Nernst Equation

The Nernst Equation is crucial for finding the cell potential under non-standard conditions. It relates the standard electrode potential to the actual cell potential, which varies with ion concentrations. The equation is \(\mathscr{E} = \mathscr{E}^{\circ} - \frac{RT}{nF} \ln(Q)\). Here's what it means:

- \(\mathscr{E}\) is the actual cell potential.- \(\mathscr{E}^{\circ}\) is the standard cell potential.- **\(R\)** is the universal gas constant, \(8.314 \text{ J/(K•mol)}\).- **\(T\)** is the temperature in Kelvin.- **\(n\)** is the number of moles of electrons exchanged in the reaction.- **\(F\)** is the Faraday constant, approximately \(96485 \text{ C/mol}\).- **\(Q\)** is the reaction quotient, which varies depending on reactant and product concentrations.

By using the Nernst Equation, you can determine how changes in concentration or temperature affect the battery's ability to function.

Battery Performance

Battery performance is influenced by multiple factors, including cell potential, temperature, and internal resistance. A battery's efficacy in different environmental conditions, such as cold weather, can vary widely. This concept helps explain why lower temperatures affect batteries more negatively:

- **Temperature:** Cold temperatures increase internal resistance and reduce the cell potential, leading to lower chemical reaction rates. This results in sluggish performance or "battery drain." - **Cell Potential:** A lower cell potential can lead to slower discharge and weaker power output. At very low temperatures, batteries sometimes can't supply the necessary power for devices. - **Entropy and Enthalpy Changes:** These thermodynamic factors, influencing Gibbs Free Energy, can alter how effective or fast a battery can operate.

Understanding these elements ensures efficient battery use and longevity, even in challenging conditions.

Cell Potential

Cell potential, also known as electromotive force (EMF), is a measure of a battery's ability to produce electrical current. It results from the difference in potential energy between reactants in a battery's electrochemical cell. Cell potential is represented by \(\mathscr{E}^{\circ}\) under standard conditions. Its importance includes:

- **Driving Current:** The greater the cell potential, the more electrical current the battery can drive through a circuit. - **Energy Supply:** A high cell potential means the battery can give out more energy before being "exhausted."- **Influence of Temperature and Concentrations:** As seen with the Nernst Equation, the cell potential can change due to temperature shifts or concentration variances in the battery's components.

A battery must maintain its cell potential to effectively function, making it vital for everything from small electronics to electric vehicles.