Problem 88 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

A galvanic cell is set up from a zinc bar weighing \(100 \mathrm{~g}\) and \(1.0 \mathrm{~L}\) of \(1.0 \mathrm{M}\) copper sulphate solution. How long would the cell run if it is assumed to deliver a steady current of \(1.0 \mathrm{~A} ?(\mathrm{Zn}=65.4)\) (a) \(53.6 \mathrm{~h}\) (b) \(26.8 \mathrm{~h}\) (c) \(81.97 \mathrm{~h}\) (d) \(40.99 \mathrm{~h}\)

Step-by-Step Solution

Answer

The cell will run for 26.8 hours (Option b).

Step 1: Calculate the moles of zinc that will react

The zinc bar weighs 100 g. To calculate the moles of zinc, use the molar mass of zinc, which is 65.4 g/mol. Use the formula: moles of zinc = mass of zinc / molar mass of zinc.

Step 2: Calculate the total charge that can be delivered

Using the electrochemical equivalent of zinc, which is 2 (since Zn has a valency of 2), calculate the total charge using the formula: total charge (Coulombs) = moles of zinc * Faraday's constant (96485 C/mol) * valency of zinc.

Step 3: Calculate the time for the cell to run

To find the time in seconds, divide the total charge by the current delivered by the cell: time (seconds) = total charge / current. Then, convert the time from seconds to hours by dividing by 3600 seconds/hour.

Step 4: Choose the correct answer

Check the calculated time against the given options and choose the correct answer.

Key Concepts

Electrochemistry

Electrochemistry is a branch of chemistry that deals with the relationship between electricity and chemical reactions. In the context of a galvanic cell, it involves the spontaneous redox reactions that generate electrical energy. A galvanic cell is composed of two different metals connected by a salt bridge or porous membrane, with each metal immersed in an electrolyte solution containing its ions. One metal acts as the anode (oxidation occurs), and the other as the cathode (reduction occurs), enabling the flow of electrons through an external circuit, thus producing electric current.

Understanding how a galvanic cell works is essential because it is the basic principle behind batteries and various forms of corrosion prevention. When analyzing problems related to galvanic cells, one must consider aspects such as electrode potentials, molarity of solutions, and the flow of electrons to predict the outcome of the electrochemical reactions and the cell's operation time.

Galvanic Cell Operation Time

The operation time of a galvanic cell is an estimation of how long the cell can produce a steady electric current before one of the reactants is completely consumed. In practical problems, this can be calculated by determining the amount of charge the cell can deliver and dividing it by the rate at which the cell delivers that charge, which is the current.

In our textbook example, we first calculate the moles of zinc that will take part in the reaction. Knowing the moles and using the concept of valency (the charge on the zinc ion, Zn²⁺), we can determine the total charge the cell can deliver based on the number of moles of electrons that will be transferred. By dividing this charge by the steady current (in amperes), we can calculate the galvanic cell's operation time. This kind of problem is useful in understanding how long a battery can last or how to design cells for specific durations of use.

Faraday's Laws of Electrolysis

Faraday's laws of electrolysis are fundamental to understanding electrochemistry, as they quantify the relationship between the amount of electric charge used in an electrolytic process and the amount of substance altered at the electrodes. The first law states that the mass of a substance altered at an electrode during electrolysis is directly proportional to the quantity of electricity that passes through the electrolyte. The second law says that when the same amount of electricity is passed through different electrolytes, the mass of substances altered at the electrodes is directly proportional to their equivalent weights (atomic mass divided by valency).

In the problem from our textbook, Faraday's laws can be applied to calculate the total charge needed to completely use up the zinc anode. By knowing the molar mass and valency of zinc, we can use Faraday's constant, which represents the charge of one mole of electrons, to find the amount of charge (in Coulombs) associated with the reaction of the moles of zinc. This understanding is pivotal not only for solving textbook problems but also for scaling up industrial electrochemical processes.