Problem 67 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The standard reduction potential of the \(\mathrm{Ag}^{+} / \mathrm{Ag}\) electrode at \(298 \mathrm{~K}\) is \(0.799 \mathrm{~V}\). Given that for AgI, \(K_{s p}=8.7 \times 10^{-17}\), evaluate the potential of the \(\mathrm{Ag}^{+} / \mathrm{Ag}\) electrode in a saturated solution of AgI. Also calculate the standard reduction potential of the \(\mathrm{I}^{-} / \mathrm{AgI} / \mathrm{Ag}\) electrode.

Step-by-Step Solution

Answer

The Ag⁺/Ag electrode potential is 1.283 V and the standard reduction potential of the I⁻/AgI/Ag electrode is 2.275 V in the saturated AgI solution.

Step 1: Calculate the concentration of Ag⁺ in the saturated solution

The solubility product (\(K_{sp}\)) for \(\text{AgI}\) is given by \(K_{sp} = [\text{Ag}^+][\text{I}^-]\). In a saturated solution of \(\text{AgI}\), \([\text{Ag}^+] = [\text{I}^-]\). Thus, \(\text{Ag}^+\) concentration can be found by solving \([\text{Ag}^+]^2 = K_{sp}\).\\([\text{Ag}^+] = \sqrt{8.7 \times 10^{-17}} = 9.33 \times 10^{-9} \text{ M}\).

Step 2: Use the Nernst equation to find the electrode potential

To find the potential of the \(\text{Ag}^+/\text{Ag}\) electrode in the saturated solution of \(\text{AgI}\), use the Nernst equation: \[ E = E^0 - \frac{RT}{nF} \ln \frac{1}{[\text{Ag}^+]} \]Given \(E^0 = 0.799 \text{ V}\), \(R = 8.314 \ \text{J/mol K}\), \(T = 298 \text{ K}\), \(F = 96485 \ \text{C/mol}\), \(n = 1\). \[ E = 0.799 - \frac{(8.314 \times 298)}{96485} \ln \left(\frac{1}{9.33 \times 10^{-9}}\right) \]\[E = 0.799 - 0.05916 \times (-8.179) = 0.799 + 0.484 = 1.283 \text{ V}.\]

Step 3: Calculate the standard potential of the I⁻/AgI/Ag electrode

The electrode potential for \(\text{I}^-/\text{AgI}/\text{Ag}\) is the reverse of the dissolution process of \(\text{AgI}\), so consider the equilibrium \(\text{Ag}^+ + \text{I}^- \rightleftharpoons \text{AgI}\). For this reaction, \[E^0 = -\frac{RT}{nF} \ln K_{sp}\]Using \(K_{sp} = 8.7 \times 10^{-17}\), \[ E^0 = -\frac{(8.314 \times 298)}{96485} \ln(8.7 \times 10^{-17}) \]\[E^0 = -0.05916 \times (-38.509) = 2.275 \text{ V}.\]

Step 4: Relate electrodes to a conventional cell

In the saturated solution of \(\text{AgI}\), the electrode potential for the total cell, including \(\text{Ag}^+/\text{Ag}\) and \(\text{I}^-/\text{AgI}/\text{Ag}\) can be seen as a series of two reactions:- Silver electrode: \(\text{Ag}^+ + e^- \rightarrow \text{Ag}\)- Iodide electrode: \(\text{I}^- + \text{AgI} \rightarrow \text{Ag} + e^-\)The overall cell potential is a direct sum of potential changes given by individual half-reactions through this linkage.

Key Concepts

Nernst Equation

The Nernst Equation is a fundamental formula in electrochemistry used to calculate the electrode potential in non-standard conditions. It adjusts the standard electrode potential to reflect the influence of ion concentrations on the potential. Simply put, it tells us how the potential change when the concentrations of reactants and products are not at standard conditions.

The Nernst Equation is written 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 \( (8.314 \ \text{J/mol K}) \).
• \( T \) is the temperature in Kelvin.
• \( n \) is the number of moles of electrons transferred.
• \( F \) is Faraday's constant \( (96485 \ \text{C/mol}) \).
• \( Q \) is the reaction quotient, representing the ratio of the concentrations of products to reactants.

Understanding the Nernst Equation is crucial because it allows us to predict how the voltage of an electrochemical cell changes with concentration. In the example provided, it is used to find the potential of the Ag⁺/Ag electrode when the concentration of Ag⁺ is changed due to the presence of AgI in solution.

Standard Reduction Potential

Standard Reduction Potentials are key values that indicate the tendency of a chemical species to gain electrons, hence being reduced. These values are measured under standard conditions: 1 M concentration, 1 atm pressure, and 25 °C (298 K).

Every half-reaction has its own standard potential, denoted as \( E^0 \). A positive \( E^0 \) means a strong tendency to gain electrons (strong oxidizing agent), and a negative \( E^0 \) indicates a lesser tendency (strong reducing agent).

In the example, the \( E^0 \) for the Ag⁺/Ag electrode is given as 0.799 V, which is quite positive, suggesting silver ions readily accept electrons to form metallic silver under standard conditions.

Comparing different standard reduction potentials helps predict the direction of electron flow in electrochemical cells, determining which species will be oxidized and which will be reduced.

Solubility Product Constant (Ksp)

The Solubility Product Constant, abbreviated as \( K_{sp} \), is a measure of the solubility of a sparingly soluble compound. It represents the product of the concentrations of the ions each raised to the power of their stoichiometric coefficients.

For silver iodide (AgI), this means:
\[ K_{sp} = [\text{Ag}^+][\text{I}^-] \]
Since AgI is sparingly soluble, it dissolves in water only slightly until the solution becomes saturated. At saturation, Ag⁺ and I^- are in equilibrium with solid AgI, and their concentrations allow us to calculate K_sp.

In the exercise, given K_sp for AgI is 8.7 x 10^-17, indicating very low solubility, which impacts the concentrations of the ions in a saturated solution. These concentrations are used in the Nernst Equation to find the potential of the Ag⁺/Ag electrode.

Electrochemical Equilibrium

Electrochemical Equilibrium refers to the state in which there are no net macroscopic changes in reactant or product concentrations, as the forward and reverse reactions occur at the same rate. In terms of electron transfer reactions, it implies that the rates at which oxidation and reduction occur are equal.

At equilibrium, the potential difference between electrodes becomes zero, and the system reaches its lowest energy state for the given conditions. At this point, the Nernst Equation can be used to describe the relationship between the cell potential and the concentrations of the electroactive species.

Understanding this concept is crucial when dealing with the I^-/AgI/Ag electrode system as showcased in the exercise. Here, the equilibrium involved in the dissolution of AgI suggests how the potential difference changes. This potential variation ensures that despite changes in ion concentration, the system maintains equilibrium.