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 \(\mathrm{K}_{c}\) is (a) \(\Delta \mathrm{G}=\mathrm{RT} \operatorname{In} \mathrm{K}_{\mathrm{c}}\) (b) \(-\Delta \mathrm{G}=\mathrm{RT}\) In \(\mathrm{K}\) (c) \(\Delta \mathrm{G}^{\circ}=\mathrm{RT}\) In \(\mathrm{K}_{\mathrm{c}}\) (d) \(-\Delta \mathrm{G}^{\circ}=\mathrm{RT}\) In \(\mathrm{K}_{c}\)

Step-by-Step Solution

Answer

(d) \(-\Delta G^\circ = RT \ln K_c\)

Step 1: Understanding the Concepts

The problem revolves around the relationship between free energy change (94G) and equilibrium constant (94K_c). The equation connecting them is:\[\Delta G^\circ = -RT \ln K_c\] where \(R\) is the universal gas constant, \(T\) is the temperature in Kelvin, and \(K_c\) is the equilibrium constant.

Step 2: Identifying the Correct Option

Based on the formula \(\Delta G^\circ = -RT \ln K_c\) from Step 1, we need to find the option that correctly expresses this relationship. The equation shows \(-\Delta G^\circ\) connected with \(RT \ln K_c\).

Key Concepts

Understanding the Equilibrium Constant

The equilibrium constant, denoted as \(K_c\), is a fundamental concept in chemistry that helps quantify the balance point of a chemical reaction at a given temperature. When a reaction reaches equilibrium, the rate of the forward reaction equals the rate of the reverse reaction.
This dynamic equilibrium means that concentrations of the reactants and products remain constant over time.

The value of \(K_c\) is derived from the concentrations of products and reactants at equilibrium. For a generic chemical reaction \(aA + bB \leftrightarrow cC + dD\), the equilibrium constant is expressed as: \[ K_c = \frac{[C]^c [D]^d}{[A]^a [B]^b} \] Here, \([A]\), \([B]\), \([C]\), and \([D]\) are the molar concentrations of the respective species. If \(K_c\) is larger, the reaction favors product formation at equilibrium. Conversely, a smaller \(K_c\) suggests that reactants are more favored.

This constant is crucial in determining how shifts in conditions, like temperature changes, might affect the concentrations of the substances involved in a reaction. Being able to determine and understand \(K_c\) is essential for predicting the outcomes of reactions in various conditions.

Thermodynamics and Free Energy

Thermodynamics is the study of energy transformations within chemical processes. One of the key parameters in thermodynamics is Gibbs Free Energy, \(\Delta G\), which indicates the amount of energy available to do work at constant temperature and pressure.
Importantly, the sign and magnitude of \(\Delta G\) determine whether reactions occur spontaneously.

When \(\Delta G\) is negative, the process is exergonic and can occur spontaneously. Conversely, a positive \(\Delta G\) implies an endergonic process, needing external energy input. There is also the standard free energy change, \(\Delta G^\circ\), which is a measure under standard conditions: 1 bar pressure, 1 M concentration, and a specified temperature (usually 298 K).

The relationship between \(\Delta G^\circ\) and the equilibrium constant is essential for understanding how energetic factors influence the positions of equilibria in chemical reactions. It is connected through the equation: \[ \Delta G^\circ = -RT \ln K_c \] This foundational equation bridges thermodynamics and equilibrium, providing deep insight into whether a reaction's products or reactants are more stable under given conditions.

Chemical Kinetics and Reaction Dynamics

Chemical kinetics investigates the rates of chemical reactions and the steps through which they occur. Unlike equilibrium, which tells us about the final state of a chemical reaction, kinetics helps us understand how fast a reaction takes place and what influences this rate.

The reaction rate is affected by several factors:

• Concentration of reactants: Higher concentrations usually increase the speed of reaction as there are more particles available to collide.
• Temperature: Higher temperatures increase kinetic energy, causing particles to move faster and collide more frequently.
• Catalysts: Substances that accelerate reactions without being consumed by providing an alternative pathway with lower activation energy.

It's crucial to distinguish between kinetics and thermodynamics. Kinetics tells us about the rate and mechanism (steps) of a reaction, whereas thermodynamics tells us about the direction and feasibility.

Thus, a reaction could be thermodynamically favorable (negative \(\Delta G\)) but kinetically slow if the activation energy barrier is high. Understanding both kinetics and equilibrium allows chemists to predict how reactions will behave in real-world conditions.