Problem 148 - A Complete Resource Book in Chemistry for JEE Main

Question

At \(25^{\circ} \mathrm{C}\), the standard emf of a cell having reaction involving two electron exchange is found to be \(0.295 \mathrm{~V} .\) The equilibrium constant of the reaction is approximately (a) \(9.50 \times 10^{9}\) (b) \(1 \times 10^{10}\) (c) 10 (d) \(9.51 \times 10^{\top}\)

Step-by-Step Solution

Answer

The equilibrium constant is approximately \(9.50 \times 10^9\), corresponding to option (a).

Step 1: Understand the Connection between EMF and Equilibrium Constant

The Nernst equation relates the electromotive force (EMF) of a cell to the equilibrium constant of the reaction. At standard conditions and when the temperature is 298 K (or 25°C), the Nernst equation simplifies to: \[ E_{cell} = \frac{RT}{nF} \ln K \]where \(E_{cell}\) is the standard EMF of the cell, \(R\) is the universal gas constant (8.314 J/mol·K), \(T\) is the temperature in Kelvin, \(n\) is the number of electrons exchanged (which is 2), \(F\) is Faraday's constant (96485 C/mol), and \(K\) is the equilibrium constant.

Step 2: Insert Known Values into Nernst Equation

Convert temperature to Kelvin: \[ T = 25^{\circ} \text{C} + 273.15 = 298.15 \text{ K} \]Given \( E_{cell} = 0.295 \text{ V} \) (or 0.295 J/C since 1 V = 1 J/C), we can insert this, along with the known values, into the rearranged Nernst equation:\[ \ln K = \frac{nFE_{cell}}{RT} \]

Step 3: Calculate \(\ln K\)

Plug in the values:\[ \ln K = \frac{2 \times 96485 \times 0.295}{8.314 \times 298.15} \]Compute each step:- Numerator: \(2 \times 96485 \times 0.295 = 56903.85\)- Denominator: \(8.314 \times 298.15 = 2478.8971\)Thus, \[ \ln K = \frac{56903.85}{2478.8971} \approx 22.949 \]

Step 4: Exponentiate to Find \(K\)

Use the relationship \( K = e^{\ln K} \) to find \( K \):\[ K = e^{22.949} \approx 9.5 \times 10^9 \]

Step 5: Identify the Closest Match Among the Given Options

Comparing our calculated equilibrium constant \(9.5 \times 10^9\) to the given options, the closest is option (a) \(9.50 \times 10^9\).

Key Concepts

Electromotive Force (EMF)

The Electromotive Force (EMF) is a crucial concept in electrochemistry. It represents the voltage developed by any source of electrical energy, such as a battery or dynamo. Think of EMF as the driving force that pushes charge around a circuit, similar to how pressure pushes water through a pipe. It's important to note that EMF isn't affected by the current flowing in the circuit.

In the context of galvanic cells, EMF is derived from the potential difference between the anode and the cathode at standard conditions. The formula to express this is given by:

• \(E = E^{\circ} - \frac{RT}{nF}\ln Q \)

At standard conditions, this simplifies because the reaction quotient \(Q\) is one, making the formula:

• \( E = E^{\circ} \)

Understanding the EMF of a cell helps us predict how much electrical work can be done by the chemical reaction.

Equilibrium Constant

The equilibrium constant (\(K\)) is a fundamental concept that describes the ratio of product concentrations to reactant concentrations at equilibrium. For a reversible chemical reaction, equilibrium is the state at which the rate of the forward reaction equals the rate of the reverse reaction. This constant is crucial because it helps us determine the position of equilibrium and how much product will form at equilibrium.

In the context of the Nernst equation used in electrochemistry, the equilibrium constant is linked to the EMF of a cell. The Nernst equation relates \(E_{cell}\) to \(K\), allowing us to ultimately find the equilibrium constant through this relationship:

• \( \ln K = \frac{nFE_{cell}}{RT} \)
• \( K = e^{\ln K} \)

Understanding how to derive \(K\) provides valuable insight into the feasibility and extent of electrochemical reactions.

Standard conditions

Standard conditions in chemistry are a set of specific conditions under which measurements are made to ensure consistency between data. They include a temperature of 298 K ( or 25°C), a pressure of 1 atmosphere, and solutions of 1 M concentration. These conditions are important in electrochemistry because they provide a reference point for measuring EMF and other thermodynamic quantities.

Under standard conditions, many electrochemical equations simplify. For instance, the Nernst equation is used in a simpler form when conditions are standard:

• \(E = E^{\circ}\)

The "standard" tag standardizes results, making data and experimental outcomes comparable across different studies. This facilitates more straightforward analysis and clearer communication within the scientific community.