Problem 180 - A Complete Resource Book in Chemistry for JEE Main

Question

For a cell reaction involving two electrons, the standard emf of the cell is found to be \(0.295 \mathrm{~V}\) at \(25^{\circ} \mathrm{C}\). The equilibrium constant of the reaction at \(25^{\circ} \mathrm{C}\) will be [2003] (a) \(1 \times 10^{-10}\) (b) \(29.5 \times 10^{-2}\) (c) 10 (d) \(1 \times 10^{10}\)

Step-by-Step Solution

Answer

The equilibrium constant is approximately \(1 \times 10^{10}\) (option d).

Step 1: Understand the Nernst Equation and Relation to Equilibrium Constant

The Nernst equation relates the cell potential at any condition to the standard cell potential. At equilibrium, the cell potential difference is zero; thus, it's related as:\[E^\circ_{cell} = \frac{RT}{nF} \ln(K_{eq})\]Where:- \(E^\circ_{cell}\) is the standard cell potential (0.295 V)- \(R\) is the universal gas constant (8.314 J/mol·K)- \(T\) is the temperature in Kelvin (298 K)- \(n\) is the number of moles of electrons transferred in the reaction (2 in this case)- \(F\) is Faraday's constant (96485 C/mol)- \(K_{eq}\) is the equilibrium constant.

Step 2: Rearrange the Nernst Equation

The equation \(E^\circ_{cell} = \frac{RT}{nF} \ln(K_{eq})\) can be rearranged to solve for the equilibrium constant \(K_{eq}\):\[\ln(K_{eq}) = \frac{nFE^\circ_{cell}}{RT}\]We will use this rearranged form to calculate \(K_{eq}\).

Step 3: Input Known Values into the Equation

Substitute the known values into the equation:\[\ln(K_{eq}) = \frac{2 \times 96485 \times 0.295}{8.314 \times 298}\]Calculate the numerical values from the above equation to find \(\ln(K_{eq})\).

Step 4: Calculate the Result

First, calculate the numerator: \(2 \times 96485 \times 0.295 = 56995.7\).Then calculate the denominator:\(8.314 \times 298 = 2477.772\).Finally, calculate \(\ln(K_{eq})\):\[\ln(K_{eq}) = \frac{56995.7}{2477.772} \approx 23.0\]

Step 5: Find the Equilibrium Constant

Solve for \(K_{eq}\) by taking the exponential of both sides:\[K_{eq} = e^{23.0} \approx 1 \times 10^{10}\]Thus, the equilibrium constant is approximately \(1 \times 10^{10}\).

Key Concepts

Nernst Equation

The Nernst equation is a fundamental equation in electrochemistry that helps us understand how the cell potential changes with different conditions. It specifically shows how the potential (voltage) of an electrochemical cell depends on the concentrations of reactants and products, temperature, and the number of electrons transferred in the cell reaction. This equation is particularly useful to calculate the cell potential when the reaction is not at standard conditions (1 M concentration, 1 atm pressure, 25°C).

• The Nernst equation is written as: \[ E_{cell} = E^ ext{°}_{cell} - \frac{RT}{nF} \ln(Q) \] where - \(E_{cell}\) is the cell potential at specific conditions, - \(E^ ext{°}_{cell}\) is the standard cell potential, - \(R\) is the universal gas constant, - \(T\) is temperature in Kelvin, - \(n\) is the number of moles of electrons exchanged, - \(F\) is Faraday's constant, and - \(Q\) is the reaction quotient.
• At equilibrium, the cell potential drops to zero, implying the concentrations reach a balance point. Thus, the equation can be modified to involve the equilibrium constant \(K_{eq}\), as shown: \[ E^ ext{°}_{cell} = \frac{RT}{nF} \ln(K_{eq}) \]

Standard EMF

Standard emf (electromotive force) of a cell is a measure of its ability to drive an electric current through a circuit under standard conditions. It is the potential difference between the electrodes of the electrochemical cell when no current is flowing through the cell.

• Standard conditions generally mean concentrations of 1 M for all solutions, pressures of 1 atm for all gases, and a temperature of 25°C (298 K).
• It is crucial because it provides a baseline reference point, so scientists can predict how changes in the cell, like changes in concentration or temperature, will affect its potential.
• According to our exercise example, the standard emf is given as 0.295 V. This value is used to calculate other properties of the electrochemical cell, such as the equilibrium constant.

Cell Reaction

Cell reactions in electrochemistry refer to the reactions occurring at the electrodes of an electrochemical cell. These reactions involve the transfer of electrons between chemical species, which is the principle behind generating electricity in cells.

• Each cell has two half-reactions: an oxidation reaction occurring at the anode and a reduction reaction occurring at the cathode.
• The net cell reaction is the sum of these half-reactions, which results in the overall transfer of electrons and generates electric current.
• The net cell reaction provides information on the number of electrons ( \(n\)) involved, which is critical for using the Nernst equation accurately. For our example, the cell reaction involves two electrons.
• Understanding these reactions helps in predicting the direction of electron flow, how much energy can be harnessed, and the efficiency of the cell.

Faraday's Constant

Faraday's constant is a key component in electrochemistry, used in calculations involving the Nernst equation and many other electrochemical applications. It represents the amount of electric charge carried by one mole of electrons.

• Faraday's constant ( \(F\)) has a value of approximately 96485 Coulombs per mole of electrons.
• It provides a vital link between the macroscopic world (moles) and the microscopic world (electrons), facilitating conversions between these units during calculations.
• In our exercise, when determining the equilibrium constant, utilizing Faraday's constant was crucial in calculating the cell potential and ultimately solving for the equilibrium constant \(K_{eq}\) through the Nernst equation.

Overall, Faraday's constant is indispensable for accurate computations in electrochemical contexts.