During a period of discharge of a lead-acid battery, \(300 \mathrm{~g}\) of \(\mathrm{PbO}_{2}(s)\) from the cathode is converted into \(\mathrm{PbSO}_{4}(s)\). (a) What mass of \(\mathrm{Pb}(s)\) is oxidized at the anode during this same period? (b) How many coulombs of electrical charge are transferred from \(\mathrm{Pb}\) to \(\mathrm{PbO}_{2}\) ?

The overall reaction in a lead-acid battery during discharge can be written as: \(Pb(s) + PbO_2(s) + 2H_2SO_4(aq) \rightarrow 2PbSO_4(s) + 2H_2O(l)\) Here, we can see that 1 mole of \(Pb(s)\) is converted to 1 mole of \(PbSO_4(s)\) during the reaction.

We are given that 300g of \(PbO_2\) is converted to \(PbSO_4\). To find the moles of \(PbO_2\), we will use its molar mass: Molar mass of \(PbO_2 = 207.2 + (16.0 \times 2) = 239.2 \mathrm{~g/mol}\) Moles of \(PbO_2 = \dfrac{300 \mathrm{~g}}{239.2 \mathrm{~g/mol}} \approx 1.255 \mathrm{~mol}\)

Since 1 mole of \(Pb(s)\) is converted to 1 mole of \(PbSO_4(s)\), the moles of \(Pb\) oxidized are equal to the moles of \(PbO_2\) converted: Moles of oxidized \(Pb\) = 1.255 mol To find the mass of oxidized \(Pb\), we will multiply the moles by its molar mass: Molar mass of \(Pb = 207.2 \mathrm{~g/mol}\) Mass of oxidized \(Pb\) = \(1.255 \mathrm{~mol} \times 207.2 \mathrm{~g/mol} \approx 259.5 \mathrm{~g}\)

In the balanced chemical equation, the oxidation state of \(Pb\) in \(Pb(s)\) is 0 and in \(PbO_2\) is +4. Therefore, 2 electrons are transferred from \(Pb\) to \(PbO_2\) per atom. Knowing the moles of \(Pb\) that react, we can find the total charge transferred: Total \(Pb\) atoms = \(1.255 \mathrm{~mol} \times 6.022 \times 10^{23} \mathrm{~atoms/mol} \approx 7.553 \times 10^{23} \mathrm{~atoms}\) Charge transferred per atom = 2 electrons = \(2 \times 1.602 \times 10^{-19} \mathrm{~C}\) Total charge transferred = \((7.553 \times 10^{23} \mathrm{~atoms}) \times (2 \times 1.602 \times 10^{-19} \mathrm{~C/atom}) = 2.42 \times 10^5 \mathrm{~C}\) (a) The mass of \(Pb(s)\) oxidized at the anode during this period is approximately 259.5 g. (b) The total charge transferred from \(Pb\) to \(PbO_2\) during this period is approximately 2.42 x 10^5 C.