Problem 84 - A Complete Resource Book in Chemistry for JEE Main

Question

The standard emf of a galvanic cell involving cell reaction with \(\mathrm{n}=2\) is found to be \(0.295 \mathrm{~V}\) at \(25^{\circ} \mathrm{C}\). The equilibrium constant of the reaction would be (Given \(\left.F=96500 \mathrm{C} \mathrm{mol}^{-1} ; \mathrm{R}=8.314 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\right)\) (a) \(2.0 \times 10^{11}\) (b) \(4.0 \times 10^{12}\) (c) \(1.0 \times 10^{2}\) (d) \(1.0 \times 10^{10}\)

Step-by-Step Solution

Answer

The equilibrium constant of the reaction is approximately \(1.0 \times 10^{10}\), option (d).

Step 1: Write the Nernst Equation for Standard Conditions

The Nernst equation for standard conditions is given by \( E^0 = \frac{RT}{nF} \ln K \), where \( E^0 \) is the standard emf of the cell, \( R \) is the gas constant, \( T \) is the temperature in Kelvin, \( n \) is the number of moles of electrons transferred, \( F \) is Faraday's constant, and \( K \) is the equilibrium constant.

Step 2: Convert Temperature to Kelvin

Given that the temperature is \(25^{\circ} C\), convert it to Kelvin by adding 273.15. Thus, \( T = 25 + 273.15 = 298.15 \text{ K} \).

Step 3: Rearrange Nernst Equation to Solve for K

Rearrange the equation \( E^0 = \frac{RT}{nF} \ln K \) to solve for \( K \). This yields \( K = e^{\left(\frac{nFE^0}{RT}\right)} \).

Step 4: Substitute Values into Equation

Substitute the given values into the rearranged equation: \( n = 2 \), \( F = 96500 \text{ C mol}^{-1} \), \( E^0 = 0.295 \text{ V} \), \( R = 8.314 \text{ J K}^{-1}\text{ mol}^{-1} \), and \( T = 298.15 \text{ K} \). Compute \( K = e^{\left(\frac{2 \times 96500 \times 0.295}{8.314 \times 298.15}\right)} \).

Step 5: Calculate the Exponential Term

First, compute the numerator: \( 2 \times 96500 \times 0.295 = 56935 \). Then, compute the denominator: \( 8.314 \times 298.15 = 2477.3 \). Now, divide the two to get \( \frac{56935}{2477.3} \approx 22.985 \).

Step 6: Compute the Exponential Function

Calculate \( e^{22.985} \) using a calculator, which approximately equals \( 1.0 \times 10^{10} \).

Step 7: Identify the Correct Answer

From the calculation in Step 6, \( K \approx 1.0 \times 10^{10} \), which corresponds to option (d).

Key Concepts

Nernst Equation

In electrochemistry, the Nernst Equation is essential for understanding the behavior of electrochemical cells under non-standard conditions. The equation links the standard electrode potential of a galvanic cell to its equilibrium constant and concentration of reactants and products at any temperature.

The standard form of the Nernst Equation is given by:

• \[E = E^0 - \frac{RT}{nF} \ln Q\]

Where:

• \(E\) is the emf under non-standard conditions.
• \(E^0\) is the standard emf of the cell.
• \(R\) is the universal gas constant.
• \(T\) is the temperature in Kelvin.
• \(n\) is the number of moles of electrons exchanged.
• \(F\) is Faraday's constant, equivalent to the charge of one mole of electrons.
• \(Q\) is the reaction quotient.

In a standard condition scenario, the Nernst equation simplifies to directly relate \(E^0\) to the equilibrium constant \(K\), reflecting how far the reaction goes towards completion at equilibrium. For reactions at 25°C (298.15 K), understanding the interplay between \(E^0\) and \(K\) helps predict reaction spontaneity with a straightforward calculation.

Equilibrium Constant

The equilibrium constant, denoted as \(K\), is a crucial concept in both chemical and electrochemical reactions. It provides insight into the balance point of a reaction, where the forward and reverse reactions occur at the same rate, thus indicating the extent of a reaction.

For electrochemical cells, \(K\) quantifies the ratio of concentrations of products to reactants at equilibrium, adjusted for their stoichiometric coefficients. It's directly related to the electromotive force (emf) in a galvanic cell under standard conditions.

• The relation: \[E^0 = \frac{RT}{nF} \ln K\], bridges electrochemical cell potentials to equilibrium constants.
• A higher \(K\) value typically signifies that a reaction heavily favors products, indicating high spontaneity.
• In the problem, a calculated \(K = 1.0 \times 10^{10}\) reflects a reaction that strongly favors the formation of products.

Understanding \(K\) empowers chemists to discern the feasibility and completeness of a reaction, as well as predict the behavior of similar reactions under specific conditions.

Galvanic Cell

Galvanic cells, or voltaic cells, are fundamental components of electrochemistry. They convert chemical energy into electrical energy through spontaneous redox reactions, which are composed of two primary half-cells.

In a galvanic cell:

• The anode undergoes oxidation, losing electrons, while the cathode undergoes reduction, gaining electrons.
• These reactions are physically separated, forcing electrons to travel through an external circuit – creating an electric current useful for doing work.
• The potential difference (emf) across these electrodes during the reaction is influenced by the concentration of reactants and can be predicted using the Nernst Equation.

Galvanic cells are characterized by a standard cell potential (\(E^0\)), which indicates the maximum potential difference when the concentrations are at standard conditions. The presence of a positive cell potential signifies a spontaneous chemical reaction, which is crucial for practical applications such as batteries and corrosion prevention. By examining terms like emf, stoichiometry, and standard conditions, students grasp a better understanding of how galvanic cells operate efficiently.