Problem 172 - A Complete Resource Book in Chemistry for JEE Main

Question

For a particular reversible reaction at temperature \(\mathrm{T}\), \(\Delta \mathrm{H}\) and \(\Delta \mathrm{S}\) were found to be both +ve. If \(\mathrm{T}_{\mathrm{e}}\) is the temperature at equilibrium, the reaction would be spontaneous when (a) \(\mathrm{T}_{n}>\mathrm{T}\) (b) \(\mathrm{T}>\mathrm{T}_{\mathrm{e}}\) (c) \(T_{e}\) is 5 times \(\mathrm{T}\) (d) \(\mathrm{T}=\mathrm{T}_{\mathrm{e}}\)

Step-by-Step Solution

Answer

The reaction is spontaneous when (b) \( \mathrm{T} > \mathrm{T}_{\mathrm{e}} \).

Step 1: Understanding Gibbs Free Energy

To determine when the reaction is spontaneous, use the Gibbs free energy equation: \( \Delta G = \Delta H - T\Delta S \). A reaction is spontaneous when \( \Delta G < 0 \).

Step 2: Relationship Between Temperature and Spontaneity

Given \( \Delta H > 0 \) and \( \Delta S > 0 \), for \( \Delta G \) to be negative (spontaneous), \( T \Delta S \) must be greater than \( \Delta H \). This occurs when \( T > \frac{\Delta H}{\Delta S} \), meaning the reaction is spontaneous at higher temperatures.

Step 3: Equilibrium Temperature Condition

At equilibrium, \( \Delta G = 0 \), implying \( \mathrm{T}_e = \frac{\Delta H}{\Delta S} \). The reaction will be spontaneous when \( T > \mathrm{T}_e \).

Step 4: Analyzing the Options

The question choices relate to the temperature's relationship with \( T_e \). The condition for the reaction to be spontaneous is effectively \( T > T_e \).

Step 5: Final Selection

Identify the correct answer. The correct choice based on our equivalence \( T > T_e \) is option (b): \( \mathrm{T} > \mathrm{T}_{\mathrm{e}} \).

Key Concepts

Spontaneity of Reactions

A chemical reaction's spontaneity is determined by whether or not it can proceed without any external input. The concept is deeply tied to Gibbs free energy, denoted as \( \Delta G \). When \( \Delta G < 0 \), a reaction is considered spontaneous, meaning it can naturally progress.
\( \Delta G \) itself is calculated using the formula:\[ \Delta G = \Delta H - T\Delta S \]Here:

• \( \Delta H \) represents the change in enthalpy, or heat content, of the system.
• \( T \) is the absolute temperature at which the reaction occurs.
• \( \Delta S \) signifies the change in entropy, or disorder, within the system.

When both enthalpy \( \Delta H \) and entropy \( \Delta S \) changes are positive, the reaction's spontaneity is temperature-dependent. As temperature increases, the term \( T\Delta S \) grows, potentially overcoming \( \Delta H \) to make \( \Delta G \) negative, leading to spontaneity.

Enthalpy and Entropy

Understanding the concepts of enthalpy and entropy is essential in explaining how reactions proceed.
Enthalpy, \( \Delta H \), relates to the energy absorbed or released as a reaction occurs. When \( \Delta H > 0 \), or positive, the reaction absorbs heat and is referred to as endothermic.
Entropy, \( \Delta S \), measures a system’s disorder. A positive \( \Delta S \) signifies an increase in disorder, which is often favorable for spontaneity.

• If a reaction results in more freedom for molecules to move (e.g., a solid to a gas), it usually implies an increase in entropy.

These changes interact in Gibbs free energy calculations, influencing the direction and spontaneity of the process. When both \( \Delta H > 0 \) and \( \Delta S > 0 \), higher temperatures make the \( T\Delta S \) term more significant, which can outweigh \( \Delta H \) and lead to a negative \( \Delta G \), driving spontaneity.

Equilibrium Temperature

The equilibrium temperature, \( T_e \), is a critical point at which a reaction's forward and reverse proceedings are balanced, resulting in no overall change. At this temperature, \( \Delta G = 0 \), meaning neither direction is favored.
For reactions with positive \( \Delta H \) and \( \Delta S \), the equilibrium temperature can be found from the equation:\[ T_e = \frac{\Delta H}{\Delta S} \]This setup reveals when the reaction shifts from being non-spontaneous to spontaneous as temperature changes.

• If \( T > T_e \), the reaction is spontaneous because the entropy-driven component \( T\Delta S \) exceeds \( \Delta H \), producing a negative \( \Delta G \).
• Conversely, if \( T < T_e \), the reaction is non-spontaneous as \( \Delta H \) prevails, making \( \Delta G \) positive.

This concept is vital for predicting reaction behavior in varying thermal conditions and determining the appropriate settings for achieving desired chemical outcomes.