Question

In the lead storage battery, the anode reaction is \(\mathrm{Pb}(\mathrm{s})+\mathrm{HSO}_{4}^{-}+\mathrm{H}_{2} \mathrm{O}\) \(\rightarrow \mathrm{PbSO}_{4}(\mathrm{~s})+\mathrm{H}_{3} \mathrm{O}^{+}+2 \mathrm{e}^{-} .\) How many grams of \(\mathrm{Pb}\) will be used up to deliver \(1 \mathrm{~A}\) for \(100 \mathrm{~h}\) ? \((\mathrm{Pb}=208)\) (a) \(776 \mathrm{~g}\) (b) \(388 \mathrm{~g}\) (c) \(194 \mathrm{~g}\) (d) \(0.1 \mathrm{~g}\)

Short answer

388 g

Step-by-step solution

Determine the total charge delivered

First, calculate the total charge delivered by the battery using the relationship: Charge (Q) = Current (I) × Time (t). Here, I = 1 A and t = 100 hours = 100 × 3600 seconds, because the charge is measured in coulombs (C) and 1 A = 1 C/s.

Calculate the number of moles of electrons

Using the total charge from Step 1, calculate the number of moles of electrons (n) that have been transferred. Use Faraday's constant, which is the charge of one mole of electrons, approximately 96,485 C/mol. The number of moles of electrons is given by n = Q / F, where Q is the total charge and F is Faraday's constant.

Relate moles of electrons to moles of Pb

Based on the chemical reaction, two moles of electrons are needed to consume one mole of Pb. Set up the ratio of moles of electrons to moles of Pb using the stoichiometry from the reaction.

Calculate the mass of Pb used

Calculate the mass of Pb used by multiplying the number of moles of Pb calculated in Step 3 by the atomic mass of lead (Pb), which is given as 208 g/mol. The mass in grams will be the product of the moles of Pb and the atomic mass of Pb.

Compare the calculated mass with the options provided

Once the mass of Pb is found, compare it with the given options to find the correct answer.

Key concepts

Understanding Electrochemistry

Electrochemistry is the branch of chemistry that deals with the relationship between electrical energy and chemical changes. It's at the heart of various types of batteries, where chemical reactions produce electricity. In the context of the lead storage battery, which is a type of rechargeable battery, electrochemistry explains how the conversion between chemical energy and electrical energy occurs.

In the anode reaction for a lead storage battery, lead (Pb) reacts with the sulfate ion (HSO₄⁻) in the presence of water (H₂O) to form lead sulfate (PbSO₄) and hydronium ions (H₃O⁺), releasing electrons in the process. These free electrons are what travel through the circuit to provide electrical power. Understanding this anode reaction is crucial for working through problems dealing with the operation and efficiency of the battery.

Faraday's Constant and its Significance

Faraday's constant is an essential value used in electrochemistry to quantify the charge on a mole of electrons. Mathematically, it is approximately 96,485 coulombs per mole (C/mol).

When you are dealing with a battery delivering a current, it is necessary to know Faraday's constant because it allows the conversion from electrical charge to moles of electrons. The total charge Q delivered by a battery can be calculated using the current (I) multiplied by the time (t). Once Q is known, dividing by Faraday's constant (F) gives the number of moles of electrons transferred. This is a crucial step in determining the quantity of reactants or products in an electrochemical reaction.

For instance, if a lead storage battery delivers 1 ampere (A) for 100 hours, this constant allows us to calculate exactly how many moles of electrons are involved in the corresponding chemical reactions within the battery.

The Role of Stoichiometry in Electrochemistry

Stoichiometry is a section of chemistry that involves calculating the relative quantities of reactants and products in chemical reactions. In electrochemistry, it helps us relate the amount of charge transferred through the circuit to the amount of chemical reaction occurring at the electrodes.

Each chemical equation provides the stoichiometric ratio, which in the anode reaction of a lead storage battery shows that each atom of lead reacts with two electrons. When applying stoichiometry to the battery's operation, it enables us to calculate how much lead will be consumed when a certain quantity of electric charge is transferred. By knowing the number of moles of electrons (from the total charge and Faraday's constant), and the stoichiometry of the reaction, we can determine the number of moles of lead consumed. Multiply this number by the atomic mass of lead to obtain the mass in grams, which yields the answer to the original question.

Stoichiometry is thus the bridge connecting the electrons flowing through the circuit to the chemical changes happening inside the battery and is indispensable for accurately solving problems in electrochemistry and battery technology.