Problem 73 - A Complete Resource Book in Chemistry for JEE Main

Question

For a reaction at \(300 \mathrm{~K}\), enthalpy and entropy changes are \(-11.5 \times 10^{3} \mathrm{~J} \mathrm{~mol}^{-1}\) and \(-105 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\) respectively. What is the change in Gibbs free energy? (a) \(25 \mathrm{~kJ} \mathrm{~mol}^{-1}\) (b) \(30 \mathrm{~kJ} \mathrm{~mol}^{-1}\) (c) \(15 \mathrm{~kJ} \mathrm{~mol}^{-1}\) (d) \(20 \mathrm{~kJ} \mathrm{~mol}^{-1}\)

Step-by-Step Solution

Answer

The change in Gibbs free energy is \(20 \text{ kJ mol}^{-1}\) (Option d).

Step 1: Understand the Gibbs free energy formula

The change in Gibbs free energy (\( \Delta G \)) for a reaction can be calculated using the formula: \( \Delta G = \Delta H - T \Delta S \), where \( \Delta H \) is the change in enthalpy, \( T \) is the temperature in Kelvin, and \( \Delta S \) is the change in entropy.

Step 2: Plug in the given values

Substitute the given values into the Gibbs free energy formula.\( \Delta H = -11.5 \times 10^3 \text{ J mol}^{-1} \), \( T = 300 \text{ K} \), and \( \Delta S = -105 \text{ J K}^{-1} \text{ mol}^{-1} \).

Step 3: Convert to consistent units

Ensure that all quantities are in the correct units before calculation. Since \( \Delta H \) is given in \( \text{J mol}^{-1} \) and \( \Delta S \) is in \( \text{J K}^{-1} \text{ mol}^{-1} \), they are already consistent for substitution into the formula.

Step 4: Perform the calculation

Calculate \( T \Delta S \): \( 300 \text{ K} \times (-105 \text{ J K}^{-1} \text{ mol}^{-1}) = -31500 \text{ J mol}^{-1} \). Now, calculate \( \Delta G = -11500 \text{ J mol}^{-1} - (-31500 \text{ J mol}^{-1}) \). This simplifies to \( \Delta G = -11500 + 31500 = 20000 \text{ J mol}^{-1} \).

Step 5: Convert the result to kilojoules

Since the options are given in kilojoules per mole, convert \( 20000 \text{ J mol}^{-1} \) to kilojoules by dividing by 1000, which results in \( 20 \text{ kJ mol}^{-1} \).

Key Concepts

Enthalpy Change

In the realm of thermodynamics and chemical reactions, enthalpy change (\( \Delta H \) ) plays a pivotal role. At its core, enthalpy change indicates the heat absorbed or released during a chemical or physical process at constant pressure. When a chemical reaction occurs, bonds between atoms break and new ones form, resulting in the absorption or release of energy. The sign of the enthalpy change provides insight into the nature of the reaction:

• Exothermic Reactions: When \( \Delta H \) is negative, the reaction releases heat, warming up the surroundings.
• Endothermic Reactions: In contrast, if \( \Delta H \) is positive, the reaction absorbs heat from the surroundings, cooling them down.

In our example exercise, \( \Delta H \) is -11,500 J/mol. This indicates an exothermic reaction, meaning energy is released into the environment.

Entropy Change

Entropy is a measure of disorder or randomness within a system, denoted by \( \Delta S \). In chemical reactions, entropy change reflects the degree to which disorder or randomness changes.
- **Increase in Entropy:** A positive \( \Delta S \) means the products are more disordered than the reactants.- **Decrease in Entropy:** A negative \( \Delta S \) indicates the products are more ordered than the reactants.
For the Gibbs free energy calculation exercise, the entropy change is given as \( -105 \text{ J K}^{-1} ext{ mol}^{-1} \). This negative value suggests that the system becomes more ordered during the reaction. Such a decrease in entropy is common in reactions resulting in fewer gas molecules from many, or in crystallization processes.

Thermodynamics

Thermodynamics is the branch of physical science concerned with heat and its relation to energy and work. It provides guiding principles for predicting the direction of energy flow in chemical processes, encapsulated by Gibbs free energy.

• First Law of Thermodynamics: Energy cannot be created or destroyed; it can only be transferred or changed from one form to another.
• Second Law of Thermodynamics: The total entropy of an isolated system can never decrease over time.
• Third Law of Thermodynamics: As temperature approaches absolute zero, the entropy of a system approaches a constant minimum.

With these principles, thermodynamics helps us understand not just whether a reaction is feasible, but also the energy transformations involved.

Chemical Reaction

A chemical reaction involves the transformation of reactants into products, accompanied by energy changes due to breaking and forming of bonds. Each reaction consists of fundamental aspects that determine its spontaneity and feasibility.
- **Reactants** start off with certain energy and configuration.- **Products** form with potentially different energy level releases or absorptions.
Chemical reactions can either proceed spontaneously or require an external push. The Gibbs free energy, calculated from the enthalpy and entropy changes, helps predict this spontaneity. In the example, the computed \( \Delta G \) of 20 kJ/mol shows the energy change of this specific reaction at 300 K. Knowing \( \Delta G \) allows chemists to understand whether a reaction will occur spontaneously under constant temperature and pressure conditions.