Problem 1 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

For the given cell; \(\mathrm{Cu}(\mathrm{s})\left|\mathrm{Cu}^{2+}\left(\mathrm{C}_{1} \mathrm{M}\right) \| \mathrm{Cu}^{2+}\left(\mathrm{C}_{2} \mathrm{M}\right)\right| \mathrm{Cu}(\mathrm{s})\) change in Gibbs energy \((\Delta \mathrm{G})\) is negative, if : (a) \(\mathrm{C}_{1}=\mathrm{C}_{2}\) (b) \(\mathrm{C}_{2}=\frac{\mathrm{C}_{1}}{\sqrt{2}}\) (c) \(\mathrm{C}_{1}=2 \mathrm{C}_{2}\) (d) \(\mathrm{C}_{2}=\sqrt{2} \mathrm{C}_{1}\)

Step-by-Step Solution

Answer

Option (d) \( C_2 = \sqrt{2}C_1 \) leads to negative \( \Delta G \).

Step 1: Understanding the Reaction

The cell notation represents a concentration cell, where the same element in different concentrations is creating a potential difference. The reaction involves copper ions migrating from the higher concentration ( C_1 or C_2) to the lower concentration, leading to the generation of electric current.

Step 2: Calculating EMF of the Cell

For a concentration cell, the electromotive force (EMF) can be calculated using the Nernst equation: \( E_{cell} = \frac{RT}{nF} \ln \frac{C_{2}}{C_{1}} \) where \( R \) is the universal gas constant, \( T \) is the temperature, \( n \) is the number of moles of electrons exchanged (\( n = 2 \) for copper), and \( F \) is Faraday's constant.

Step 3: Relating EMF to Gibbs Free Energy

The change in Gibbs energy \( (\Delta G) \) is related to the cell potential \( E_{cell} \) by the equation: \( \Delta G = -nFE_{cell} \). A negative \( \Delta G \) indicates a spontaneous reaction, meaning \( E_{cell} > 0 \).

Step 4: Analyzing Each Option

We analyze each of the options given to determine when \( \frac{C_{2}}{C_{1}} > 1 \) since this implies \( E_{cell} > 0 \):- (a) \( C_1 = C_2 \): Here, \( \frac{C_{2}}{C_{1}} = 1 \), so \( E_{cell} = 0 \), \( \Delta G = 0 \).- (b) \( C_2 = \frac{C_1}{\sqrt{2}} \): Here, \( \frac{C_{2}}{C_{1}} = \frac{1}{\sqrt{2}} < 1 \), so \( E_{cell} < 0 \), \( \Delta G > 0 \).- (c) \( C_1 = 2C_2 \): Here, \( \frac{C_{2}}{C_{1}} = \frac{1}{2} < 1 \), so \( E_{cell} < 0 \), \( \Delta G > 0 \).- (d) \( C_2 = \sqrt{2}C_1 \): Here, \( \frac{C_{2}}{C_{1}} = \sqrt{2} > 1 \), so \( E_{cell} > 0 \), \( \Delta G < 0 \).

Step 5: Conclusion

The Gibbs energy change is negative (reaction is spontaneous) when the concentration \( \frac{C_{2}}{C_{1}} > 1 \). This condition is met only for option (d): \( C_2 = \sqrt{2}C_1 \).

Key Concepts

Concentration Cell

A concentration cell is a type of electrochemical cell that generates electrical energy from the difference in concentration of ions in two half-cells. In simpler terms, imagine you have two containers filled with the same kind of solution but with different concentrations of ions. Typically, the ions will move from the area of higher concentration to the area of lower concentration, driving an electric current as they do so. This movement happens because systems tend to move towards equilibrium, where concentrations are balanced. The energy created by this movement can be thought of as a power source, much like a battery. The concentration cell containing copper ions at different concentrations, sums up this concept well and is used to understand how concentration differences can do work in the context of electrochemistry.

Nernst Equation

The Nernst Equation is crucial for calculating the electromotive force (EMF) or potential of an electrochemical cell under non-standard conditions. It adjusts the standard cell potential to account for different concentrations of ions involved in the reaction. For a concentration cell, the equation simplifies to:

• \( E_{cell} = \frac{RT}{nF} \ln \frac{C_{2}}{C_{1}} \)

Where:

• \( R \) is the universal gas constant
• \( T \) is the temperature in Kelvin
• \( n \) is the number of electrons transferred in the reaction (usually 2 for copper reactions)
• \( F \) is Faraday's constant, which converts chemical potential to electrical energy.

In simpler words, the Nernst Equation helps us determine how much voltage we can get from our concentration cell given the concentrations of ions involved.

Gibbs Free Energy

Gibbs Free Energy, symbolized as \( \Delta G \), is a measure of the maximum reversible work that a thermodynamic system can perform at constant temperature and pressure. In electrochemistry, it tells us whether a reaction will occur spontaneously.

• The relationship between Gibbs Free Energy and the EMF of a cell is given by: \( \Delta G = -nFE_{cell} \)

A negative \( \Delta G \) indicates that the process is spontaneous, meaning it can proceed without any external energy input. For electrochemical cells, if the cell potential \( E_{cell} \) is greater than zero, \( \Delta G \) is negative, implying a spontaneous reaction. Hence, understanding \( \Delta G \) in the context of concentration cells helps us know when they are capable of generating electricity on their own.

Spontaneous Reaction

A spontaneous reaction in the context of electrochemistry is a reaction that occurs on its own without needing an outside energy source to drive it. One of the indicators of spontaneity in electrochemical reactions is the sign of Gibbs Free Energy, \( \Delta G \). If \( \Delta G \) is negative, the reaction is considered spontaneous. In simpler terms, it's like a toy car that moves on its own without being pushed. Applying this concept to concentration cells, we observe that when the concentration of ions on one side of the cell is sufficiently higher than on the other, the ions will move naturally to equalize concentrations, generating an electric current. This occurs when \( \frac{C_{2}}{C_{1}} > 1 \), leading to a positive EMF and hence a negative \( \Delta G \), indicating a spontaneous reaction.