Problem 70 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

For the galvanic cell. [1992 - 4 Marks] \(\mathrm{Ag}|\mathrm{AgCl}(\mathrm{s}), \mathrm{KCl}(0.2 \mathrm{M}) \| \mathrm{KBr}(0.001 \mathrm{M}), \mathrm{AgBr}(\mathrm{s})| \mathrm{Ag}\) Calculate the EMF generated and assign correct polarity to each electrode for a spontaneous process after taking into account the cell reaction at \(25^{\circ} \mathrm{C}\). \(\left[K_{s p}(\mathrm{AgCl})=2.8 \times 10^{-10} ; K_{s p}(\mathrm{AgBr})=3.3 \times 10^{-13}\right]\)

Step-by-Step Solution

Answer

EMF is the difference between potentials of both half-cells. Assign polarity using potential values.

Step 1: Identify Cell Components

The given galvanic cell is composed of two half-cells: \( \mathrm{Ag} | \mathrm{AgCl} | \mathrm{KCl} \) and \( \mathrm{KBr} | \mathrm{AgBr} | \mathrm{Ag} \). Each of these half-cells contains a silver electrode in contact with its respective sparingly soluble salt.

Step 2: Write Half Reactions and their Potentials

For each half-cell, write the half-reaction. The half-reaction for \( \mathrm{AgCl}/\mathrm{Ag} \) is: \[ \text{AgCl(s) + e}^- \rightarrow \text{Ag(s) + Cl}^- \]. For \( \mathrm{AgBr}/\mathrm{Ag} \), the half-reaction is: \[ \text{AgBr(s) + e}^- \rightarrow \text{Ag(s) + Br}^- \]. Calculate their standard electrode potentials using the Nernst Equation \( E = E^0 - \frac{RT}{nF} \ln Q \), where \( Q \) is the reaction quotient.

Step 3: Calculate Reaction Quotients for Each Half-Cell

The reaction quotient \( Q \) for each half-cell can be calculated using the given concentrations and solubility products. For the \( \text{AgCl} \) half-cell, \( Q = [\text{Cl}^-] = 0.2 \). For \( \text{AgBr} \) half-cell, \( Q = [\text{Br}^-] = 0.001 \).

Step 4: Use Nernst Equation for Non-standard Conditions

Using the Nernst equation, calculate the electrode potentials for each half-cell under non-standard conditions. For \( \mathrm{AgCl} \), calculate:\[ E_{\text{AgCl}} = E^0_{\text{AgCl}} - \frac{RT}{F} \ln(0.2) \].For \( \mathrm{AgBr} \), calculate:\[ E_{\text{AgBr}} = E^0_{\text{AgBr}} - \frac{RT}{F} \ln(0.001) \].Here, \( \frac{RT}{F} \approx 0.0591 \text{ V at 25°C} \).

Step 5: Calculate Cell EMF and Determine Polarity

The EMF of the cell \( E_\text{cell} \) is the difference in potential between the two half-cells. \[ E_\text{cell} = E_{\text{AgCl}} - E_{\text{AgBr}} \].Determine which electrode is positive by noting that electrons flow from lower to higher potential. Calculate the actual potentials and use this to assign anode and cathode.

Key Concepts

Galvanic Cell

A galvanic cell is a device that converts chemical energy into electrical energy through spontaneous redox reactions. In the cell given in the exercise, we have two half-cells involving silver electrodes and their corresponding sparingly soluble salts of silver chloride (AgCl) and silver bromide (AgBr). The idea is to harness the energy from these reactions to generate electricity.

A galvanic cell is made of two electrodes: the anode and the cathode. The anode is where oxidation occurs, losing electrons, while the cathode is where reduction occurs, gaining electrons. In this specific cell setup, silver acts as the solid electrode immersed in its respective solutions with AgCl and AgBr. These solid salts lead to a saturated solution, providing the ions necessary for the electrochemical reaction.

The electrodes are connected externally, allowing electrons to flow and internally by a salt bridge, which ensures electrical neutrality in the solutions by allowing ions to migrate to either side of the cell. In our exercise, the KCl and KBr act as the electrolyte solutions, providing Cl⁻ and Br⁻ ions respectively.

Nernst Equation

The Nernst Equation is a powerful tool in electrochemistry that allows us to calculate the cell potential under non-standard conditions. For the given galvanic cell, it helps us determine the electrode potential when concentrations differ from standard conditions, typically 1 M.

Mathematically, the Nernst Equation is expressed as:
\[ E = E^0 - \frac{RT}{nF} \ln Q \]
where:

• \( E \) is the electrode potential.
• \( E^0 \) is the standard electrode potential.
• \( R \) is the universal gas constant (8.314 J/mol*K).
• \( T \) is the temperature in Kelvin.
• \( n \) is the number of moles of electrons transferred.
• \( F \) is Faraday's constant (96485 C/mol).
• \( Q \) is the reaction quotient, calculated as the concentrations of products over reactants.

Using this equation, you can adjust the potential based on the real conditions of the cell. This is done by considering the concentrations of Cl⁻ and Br⁻ ions, which are given as 0.2 M and 0.001 M for our AgCl and AgBr half-cells respectively. This adjustment is crucial for computing the cell's potential in practical scenarios.

Standard Electrode Potentials

Standard electrode potentials, often denoted as \( E^0 \), provide a reference voltage for electrodes under standard conditions, which include a concentration of 1 M for ions involved in the half-reaction, a pressure of 1 atm for gases, and a temperature of 25°C or 298 K. They are commonly measured against a standard hydrogen electrode (SHE), which is assigned a potential of 0 V.

When we consider different elements and compounds, standard potentials enable us to predict the direction and feasibility of redox reactions. In our galvanic cell example, both AgCl and AgBr have their own standard potentials, which can be found in electrochemical tables. These values are essential starting points for calculating the non-standard electrode potentials via the Nernst Equation.
The greater \( E^0 \) indicates a stronger tendency to gain electrons and undergo reduction. For the silver electrodes in our problem, the potential difference between the two half-cells defines the electromotive force (EMF) or voltage of the cell. The calculated EMF offers insight into which electrode acts as the cathode and which as the anode, which is determined by the flow of electrons from the half-cell with lower to higher potential.