Problem 116 - Chemistry

Question

The concentration of sulfuric acid in the lead-storage battery of an automobile over a period of time has decreased from 38.0 percent by mass (density \(=1.29 \mathrm{~g} / \mathrm{mL}\) ) to 26.0 percent by mass ( \(1.19 \mathrm{~g} / \mathrm{mL}\) ). Assume the volume of the acid remains constant at \(724 \mathrm{~mL}\). (a) Calculate the total charge in coulombs supplied by the battery. (b) How long (in hours) will it take to recharge the battery back to the original sulfuric acid concentration using a current of \(22.4 \mathrm{~A}\) ?

Step-by-Step Solution

Answer

The total charge supplied is 257,948.358 C, and recharging time is approximately 3.2 hours.

Step 1: Calculate Initial Mass of Acid

To find the initial mass of sulfuric acid, we first need to calculate the mass of the solution given the initial concentration. The density of the solution is given as 1.29 g/mL, and the volume is 724 mL. Thus, the total mass of the solution is:\[\text{Mass} = \text{Density} \times \text{Volume} = 1.29 \text{ g/mL} \times 724 \text{ mL} = 934.56 \text{ g}\]The sulfuric acid concentration is 38.0%, so the mass of sulfuric acid is:\[\text{Mass of } \text{H}_2\text{SO}_4 = 0.38 \times 934.56 \text{ g} = 355.1328 \text{ g}\]

Step 2: Calculate Final Mass of Acid

Using the same method as in Step 1, compute the final mass of sulfuric acid after the concentration change. The final density is 1.19 g/mL, so the total mass of the solution is:\[\text{Mass} = 1.19 \text{ g/mL} \times 724 \text{ mL} = 861.56 \text{ g}\]The sulfuric acid concentration is 26.0%, so:\[\text{Mass of } \text{H}_2\text{SO}_4 = 0.26 \times 861.56 \text{ g} = 224.0056 \text{ g}\]

Step 3: Calculate Change in Mass of Acid

The next step is to determine the change in mass of sulfuric acid:\[\Delta m_{\text{H}_2\text{SO}_4} = 355.1328 \text{ g} - 224.0056 \text{ g} = 131.1272 \text{ g}\]

Step 4: Convert Mass Change to Charge

Now, convert the change in mass of sulfuric acid to electric charge. From the balanced equation of sulfuric acid acting in a battery, each mole of \(\text{H}_2\text{SO}_4\) involves 2 moles of electrons:\[\text{Molar mass of } \text{H}_2\text{SO}_4 = 98.08 \text{ g/mol}\]\[\text{Moles of } \text{H}_2\text{SO}_4 = \frac{131.1272 \text{ g}}{98.08 \text{ g/mol}} = 1.3374 \text{ mol}\]\[\text{Total moles of electrons} = 2 \times 1.3374 = 2.6748 \text{ mol}\]The charge is obtained using the Faraday's constant (F = 96485 C/mol):\[Q = 2.6748 \text{ mol} \times 96485 \text{ C/mol} = 257948.358 \text{ C}\]

Step 5: Calculate Time to Recharge

With the charge calculated, find the time required to recharge the battery using a 22.4 A current:\[\text{Time} = \frac{Q}{\text{Current}} = \frac{257948.358 \text{ C}}{22.4 \text{ A}} = 11518.228 \text{ seconds}\]Convert the time from seconds to hours:\[\text{Time in hours} = \frac{11518.228 \text{ seconds}}{3600 \text{ seconds/hour}} \approx 3.2 \text{ hours}\]

Key Concepts

Lead-Acid Battery

The lead-acid battery is one of the oldest and most well-known types of rechargeable batteries. It's commonly used in vehicles such as cars and motorcycles due to its reliable performance and cost-effectiveness. A lead-acid battery comprises multiple cells, each containing lead dioxide ( PbO_2 ), lead ( Pb ), and an electrolyte made of a sulfuric acid ( H_2SO_4 ) and water mixture.

During discharge, the lead dioxide reacts with the sulfuric acid and lead to form lead sulfate, while releasing electrical energy. The chemical process can be simplified in terms of the changes at the electrodes:

• At the anode (negative plate), lead ( Pb ) reacts to form lead sulfate ( PbSO_4 ).
• At the cathode (positive plate), lead dioxide reacts with sulfuric acid to form more lead sulfate.

Recharging reverses this process and converts lead sulfate back into lead and lead dioxide, restoring the battery's potential. This cyclical reaction between the lead compounds and sulfuric acid is key to the battery's ability to store and release energy.

Sulfuric Acid Concentration

Sulfuric acid ( H_2SO_4 ) concentration in a lead-acid battery plays a critical role in determining the battery's state of charge. Higher sulfuric acid concentration indicates a fully charged battery, whereas a lower concentration suggests discharge.

The concentration of sulfuric acid is measured by mass percentage. In the discussed exercise, the initial concentration was 38.0%, decreasing to 26.0%. This decrease reflects the consumption of sulfuric acid as the battery discharges.

In practical terms, the battery's density measurement is often used to estimate its concentration. Battery manufacturers provide charts correlating specific gravity (density) with H_2SO_4 concentration, making it easier to monitor charge levels. Hence, understanding how sulfuric acid concentration changes inform us about the battery's health and charge status.

Faraday's Constant

Faraday's constant is a fundamental value in electrochemistry. It represents the charge of one mole of electrons and is approximately 96485 coulombs per mole. This constant is crucial in calculating the electric charge assessed during chemical reactions involving electron transfer.

In a lead-acid battery, the electrochemical reactions involve the transfer of electrons between the lead compounds and sulfuric acid. Knowing Faraday's constant allows us to convert moles of electrons into charge (in coulombs), which is essential for determining how much energy the battery can supply or store.

In the context of the exercise, the change in sulfuric acid mass was used to calculate the number of moles of H_2SO_4 . This value was then converted to moles of electrons, and Faraday's constant was employed to find the total charge supplied by the battery.

Charge Calculation

Charge calculation is vital in understanding how much electrical energy a battery can deliver over time. In the discussed problem, calculating the charge involved multiple steps, starting from determining the change in mass of sulfuric acid to finding the number of moles, and finally calculating the charge transferred using Faraday's constant.

Here's how it can be simplified:

• The initial and final masses of sulfuric acid are calculated using their respective concentrations and densities.
• The difference in these masses indicates the extent of discharge.
• Convert the mass difference into moles using sulfuric acid's molar mass (98.08 g/mol).
• Each mole of H_2SO_4 corresponds to two moles of electrons as per the reaction stoichiometry.
• Finally, apply the Faraday's constant to convert moles of electrons into total charge in coulombs.

Understanding this process helps in predicting battery performance and planning its recharge strategies, ensuring efficient use of its energy storage capacity.