Problem 189 - A Complete Resource Book in Chemistry for JEE Main

Question

For a spontaneous reaction the \(\Delta \mathrm{G}\), equilibrium constant \((\mathrm{K})\) and \(E_{\text {cell }}^{\circ}\) will be respectively \([\mathbf{2 0 0 5}]\) (a) \(-\mathrm{ve},>1,+\mathrm{ve}\) (b) \(+\mathrm{ve},>1,-\mathrm{ve}\) (c) \(-\mathrm{ve},<1,-\mathrm{ve}\) (d) \(-\) ve, \(>1,-\) ve

Step-by-Step Solution

Answer

Option (a) \(-\mathrm{ve}, >1, +\mathrm{ve}\) is correct for a spontaneous reaction.

Step 1: Understand the relationship between ΔG and spontaneity

For a reaction to be spontaneous, the Gibbs free energy change \( \Delta G \) must be negative. This is because a negative \( \Delta G \) indicates that the process can occur without the input of additional energy.

Step 2: Apply the Nernst equation for standard conditions

The Gibbs free energy change \( \Delta G \) and the standard cell potential \( E_{\text{cell}}^{\circ} \) are related by the equation:\[ \Delta G^{\circ} = -nFE_{\text{cell}}^{\circ} \]where \( n \) is the number of moles of electrons transferred in the reaction and \( F \) is the Faraday constant. For a spontaneous reaction, \( \Delta G^{\circ} \) is negative, which implies \( E_{\text{cell}}^{\circ} \) must be positive.

Step 3: Understand the relationship with equilibrium constant (K)

The relationship between \( \Delta G^{\circ} \) and the equilibrium constant \( K \) is given by:\[ \Delta G^{\circ} = -RT \ln K \]where \( R \) is the universal gas constant and \( T \) is the temperature in Kelvin. For a spontaneous reaction, since \( \Delta G^{\circ} \) is negative, \( \ln K \) must be positive. This implies \( K > 1 \) for spontaneity.

Step 4: Analyze the given options

Now that we know a spontaneous reaction requires \( \Delta G^{\circ} \) to be \(-\), \( K > 1 \), and \( E_{\text{cell}}^{\circ} \) to be \(+\), compare these requirements with the provided options.* Option (a) matches these conditions: \(-\mathrm{ve}, >1, +\mathrm{ve}\). * Other options do not match all conditions.

Key Concepts

Gibbs Free Energy

Gibbs Free Energy, denoted as \( \Delta G \), is a critical concept in chemistry that helps determine the spontaneity of a reaction. For a reaction to happen spontaneously—meaning it can occur without any additional energy input—the Gibbs Free Energy change must be negative. This negative \( \Delta G \) indicates that the system releases energy, making the process energetically favorable. The equation used to calculate Gibbs Free Energy under standard conditions is \( \Delta G^{\circ} = -nFE_{\text{cell}}^{\circ} \). Here, \( n \) represents the number of moles of electrons exchanged in the reaction, and \( F \) is the Faraday constant. When \( \Delta G^{\circ} \) is negative, it suggests that the corresponding \( E_{\text{cell}}^{\circ} \) is positive. This positive \( E_{\text{cell}}^{\circ} \) affirms that the reaction is indeed spontaneous under standard conditions.

Standard Cell Potential

The Standard Cell Potential, represented by \( E_{\text{cell}}^{\circ} \), measures the potential difference between two electrodes under standard conditions. It provides a direct indication of the driving force behind an electrochemical reaction.A positive \( E_{\text{cell}}^{\circ} \) suggests that the electrochemical cell operates spontaneously. This is because a positive cell potential corresponds to a negative \( \Delta G^{\circ} \), as per the relation: \( \Delta G^{\circ} = -nFE_{\text{cell}}^{\circ} \).In electrochemical cells, the standard cell potential can be determined using the standard reduction potentials of the cell's electrodes. A large positive \( E_{\text{cell}}^{\circ} \) typically indicates a strong spontaneous reaction, meaning the oxidizing agent and reducing agent interact energetically favorably.

Equilibrium Constant

The Equilibrium Constant, \( K \), provides insight into the position of equilibrium for a chemical reaction. It is closely related to Gibbs Free Energy through the equation: \( \Delta G^{\circ} = -RT \ln K \), where \( R \) is the gas constant, and \( T \) is the temperature in Kelvin.For a spontaneous reaction, \( \Delta G^{\circ} \) is negative, which makes \( \ln K \) positive. This leads us to the conclusion that \( K > 1 \). When \( K \) is greater than one, it implies that at equilibrium, the concentration of products is greater than that of reactants, indicating the forward reaction is favored.

Nernst Equation

The Nernst Equation is a fundamental relation in electrochemistry, extending our understanding of electrochemical reactions beyond standard conditions. It allows us to calculate the cell potential under any set of conditions, based on the concentrations or pressures of the reactants and products.The Nernst Equation is expressed as: \[ E = E_{\text{cell}}^{\circ} - \frac{RT}{nF} \ln Q \]where \( E \) is the cell potential under non-standard conditions, \( R \) is the gas constant, \( T \) is the temperature in Kelvin, \( F \) is the Faraday constant, and \( Q \) is the reaction quotient.This equation shows how deviations from standard conditions, such as concentration changes, affect the cell potential. It is key to understanding how cells perform in real-world scenarios, highlighting that even a spontaneous reaction can have its potential reduced by unfavorable conditions.