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 kJ/mol (option d).

Step 1: Write down Gibbs Free Energy Formula

The Gibbs free energy change \( \Delta G \) is given by 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: Verify Given Information

We are given that \( \Delta H = -11.5 \times 10^3 \) J/mol and \( \Delta S = -105 \) J/K/mol. The temperature \( T \) is given as 300 K.

Step 3: Substitute Values Into Formula

Substitute the given values into the Gibbs free energy formula: \[ \Delta G = (-11.5 \times 10^3) - (300 \times (-105)) \]

Step 4: Calculate Temperature and Entropy Product

Calculate \( T \Delta S \): \[ T \Delta S = 300 \times (-105) = -31500 \text{ J/mol} \]

Step 5: Substitute Temperature and Entropy Product Back Into Equation

Substitute \( T \Delta S = -31500 \text{ J/mol} \) back into the Gibbs free energy formula: \[ \Delta G = -11500 - (-31500) \]

Step 6: Simplify Final Expression

Calculate \( \Delta G \): \[ \Delta G = -11500 + 31500 = 20000 \] J/mol. Convert to kJ by dividing by 1000: \( 20 \text{ kJ/mol} \).

Key Concepts

Enthalpy Change

Enthalpy change, represented as \( \Delta H \), is a central concept in understanding energy changes during chemical reactions. It refers to the total heat content of a system and indicates how much heat is absorbed or released during a reaction at constant pressure.

There are a few important points to remember about enthalpy change:

• It is measured in joules per mole (J/mol).
• A negative \( \Delta H \) signifies an exothermic reaction, where heat is released to the surroundings.
• A positive \( \Delta H \) signifies an endothermic reaction, where heat is absorbed from the surroundings.

In our example, the enthalpy change is given as \(-11.5 \times 10^3\) J/mol. This negative sign indicates that the reaction releases energy in the form of heat, making it exothermic. Understanding this helps predict how a reaction might affect its surroundings and the overall energy balance.

Entropy Change

Entropy change, represented as \( \Delta S \), describes the disorder or randomness within a system. In essence, it measures the dispersal of energy within a chemical reaction and is crucial for predicting reaction spontaneity.

Key aspects of entropy include:

• It is measured in joules per Kelvin per mole \( \text{(J/K/mol)} \).
• A positive \( \Delta S \) indicates increased disorder, a natural tendency aligned with the second law of thermodynamics.
• A negative \( \Delta S \) indicates decreased disorder, often meaning the reaction results in a more ordered state.

In the given exercise, \( \Delta S \) is \(-105 \text{ J/K/mol} \), implying that the reaction proceeds towards greater order. This can occur, for example, by forming a more structured product from less ordered reactants. Entropy change influences the overall Gibbs free energy and, consequently, the directionality and feasibility of reactions.

Thermodynamics

Thermodynamics is the branch of physical science concerning heat and its relationship with energy and work. It provides the governing principles for understanding chemical processes and predicting the course of reactions.

There are several important laws and concepts in thermodynamics:

• The first law of thermodynamics, also known as the law of energy conservation, states that energy cannot be created or destroyed, only converted from one form to another.
• The second law of thermodynamics states that in any energy exchange, if no energy enters or leaves the system, the potential energy of the state will always be less than that of the initial state, often described in terms of increasing entropy.
• The third law asserts that as a perfect crystal approaches absolute zero, its entropy approaches zero, providing a reference for entropy measurements.

Thermodynamics provides the mathematical framework for predicting the heat and work involved in chemical reactions. It is instrumental in calculating entropy and enthalpy changes, enabling us to understand how viable a reaction is energetically.

Gibbs Free Energy formula

The Gibbs Free Energy formula integrates both enthalpy and entropy changes to determine the spontaneity of a reaction. It ensures an understanding of both total energy content and entropy, central elements of thermodynamics.

The formula is given as:
\[ \Delta G = \Delta H - T \Delta S \]

• \( \Delta G \) stands for the change in Gibbs free energy.
• \( \Delta H \) is the enthalpy change.
• \( T \) is the temperature in Kelvin.
• \( \Delta S \) is the entropy change.

With this equation, a reaction with a negative \( \Delta G \) is considered spontaneous, favoring product formation, while a positive \( \Delta G \) implies that the reaction is non-spontaneous under the given conditions. In our exercise, substituting the given values yields a \( \Delta G \) of \( 20 \text{ kJ/mol} \), indicating the reaction is spontaneous at this temperature. Understanding such calculations helps predict and manipulate reactions based on desired outcomes and conditions.