Problem 150 - A Complete Resource Book in Chemistry for JEE Main

Question

The correct relationship between free energy change in a reaction and the corresponding equilibrium constant \(K_{c}\) is \([\mathbf{2 0 0 3}]\) (a) \(\Delta \mathrm{G}=\mathrm{RT} \operatorname{In} \mathrm{K}\) (b) \(-\Delta \mathrm{G}=\mathrm{RT} \operatorname{In} \mathrm{K}\) (c) \(\Delta G^{\circ}=R T\) In \(K\) (d) \(-\Delta G^{\circ}=\mathrm{RT} \operatorname{In} \mathrm{K}_{\mathrm{c}}\)

Step-by-Step Solution

Answer

The correct relationship is option (d): \(-\Delta G^{\circ} = RT \ln K_c\).

Step 1: Understanding the Formula

We need to find the correct relationship between the standard Gibbs free energy change \(\Delta G^{\circ}\) and the equilibrium constant \(K_c\). For a chemical reaction at equilibrium, the relationship between these quantities is represented by the formula: \(\Delta G^{\circ} = -RT \ln K_c\), where \(R\) is the universal gas constant and \(T\) is the temperature in Kelvin.

Step 2: Analyzing the Options

Examine each option:(a) \(\Delta G = RT \ln K\) - This does not match the established formula since the sign is incorrect.(b) \(-\Delta G = RT \ln K\) - This also doesn't match because the use of \(-\Delta G\) should relate to \(\Delta G^{\circ}\), not just \(\Delta G\). (c) \(\Delta G^{\circ} = RT \ln K\) - While this uses \(\Delta G^{\circ}\), the sign is missing, which makes it incorrect.(d) \(-\Delta G^{\circ} = RT \ln K_c\) - This matches the correct formula with proper use of \(-\) and inclusion of \(^{\circ}\) and \(K_c\).

Step 3: Selecting the Correct Option

Among the options provided, option (d) \(-\Delta G^{\circ} = RT \ln K_c\) is correct. It aligns with the formula used to relate standard Gibbs free energy change and equilibrium constant for a reaction occurring under standard conditions.

Key Concepts

Equilibrium Constant

The equilibrium constant, often represented as \( K_c \), is a crucial concept in understanding how reactions reach a state of balance. When a reaction is at equilibrium, the rate of the forward reaction equals the rate of the reverse reaction.
This balance is quantitatively described by \( K_c \), which is the ratio of the concentration of products to reactants, each raised to the power of their respective coefficients in the chemical equation.

• If \( K_c > 1 \), the products are favored at equilibrium.
• If \( K_c < 1 \), the reactants are favored at equilibrium.

This constant helps predict how the concentration of chemicals in a system change under different conditions, and is integral in calculating the Gibbs free energy change for reactions.

Chemical Thermodynamics

Chemical thermodynamics studies energy transformations within chemical reactions. One key focus is Gibbs free energy, a measure of the available energy for performing work when temperature and pressure are uniform throughout the system.
The formula \( \Delta G = \Delta H - T \Delta S \) can be used, where:

• \( \Delta H \) represents the change in enthalpy (heat absorbed or released).
• \( T \) is the temperature in Kelvin.
• \( \Delta S \) is the change in entropy (degree of disorder or randomness).

Knowing whether \( \Delta G \) is positive or negative indicates if a process is non-spontaneous or spontaneous, respectively. In terms of equilibrium, \( \Delta G = 0 \) signifies that a system is in equilibrium—no net change occurs.

Standard Conditions

In chemical reactions, standard conditions provide a reference point to ensure consistency when comparing data.
These conditions usually include:

• Temperature at 298 K (25°C).
• Pressure at 1 bar or 1 atmosphere.
• All solutes at 1 mol/L concentration.

Under standard conditions, the Gibbs free energy change, noted as \( \Delta G^{\circ} \), is used to determine reaction feasibility. It offers a benchmark from which researchers can determine \( \Delta G \) under non-standard conditions, helping to predict whether a reaction will occur naturally.

Reaction Equilibrium

Reaction equilibrium occurs when the forward and reverse reactions proceed at the same rate. At this point, the concentrations of reactants and products remain constant over time.
Several factors can affect equilibrium, including:

• Temperature: Changing temperature can shift equilibrium, favoring either exothermic (heat-releasing) or endothermic (heat-absorbing) reactions, depending on the nature of the reaction.
• Pressure and Volume: These can influence gaseous equilibria, according to Le Chatelier's Principle.

Understanding reaction equilibrium allows chemists to manipulate conditions to favor desired products and optimize industrial processes.