Problem 93 - A Complete Resource Book in Chemistry for JEE Main

Question

The direct conversion of \(\mathrm{A}\) to \(\mathrm{B}\) is difficult, hence it is carried out by the following path: $$ \begin{aligned} &\text { Given }\\\ &\Delta \mathrm{S}(\mathrm{A} \longrightarrow \mathrm{C})=50 \mathrm{e} . \mathrm{u} .\\\ &\Delta \mathrm{S}(\mathrm{C} \longrightarrow \mathrm{D})=30 \mathrm{e} . \mathrm{u}\\\ &\Delta \mathrm{S}(\mathrm{B} \longrightarrow \mathrm{D})=20 \mathrm{e} . \mathrm{u} \end{aligned} $$ where e.u. is entropy unit then \(\Delta \mathrm{S}(\mathrm{A} \longrightarrow \mathrm{B})\) is (a) \(+100\) e.u. (b) \(+60\) e.u. (c) \(-100\) e.u. (d) \(-60\) e.u.

Step-by-Step Solution

Answer

The change in entropy \( \Delta S(A \rightarrow B) \) is \(+60\) e.u., corresponding to option (b).

Step 1: Understanding the Problem

The problem requires computing the change in entropy \( \Delta S \) for the reaction \( A \rightarrow B \). This computation is made possible by given changes in entropy for intermediate paths using the state function property of entropy.

Step 2: Analyzing the Pathways

Given is a multistep path: \( A \rightarrow C \rightarrow D \rightarrow B \). We know the entropy changes for these segments: \( \Delta S(A \rightarrow C) = 50 \) e.u., \( \Delta S(C \rightarrow D) = 30 \) e.u., and \( \Delta S(B \rightarrow D) = 20 \) e.u.

Step 3: Writing the Entropy Equation

Use the relationship: \[ \Delta S(A \rightarrow B) = \Delta S(A \rightarrow C) + \Delta S(C \rightarrow D) - \Delta S(B \rightarrow D) \]. The subtraction for \( B \rightarrow D \) occurs because the path described is the reverse of our pathway from \( A \) to \( B \).

Step 4: Calculating \( \Delta S(A \rightarrow B) \)

Substitute the known values into the equation: \[ \Delta S(A \rightarrow B) = 50 \text{ e.u.} + 30 \text{ e.u.} - 20 \text{ e.u.} \].

Step 5: Solving the Equation

Perform the calculation: \[ \Delta S(A \rightarrow B) = 50 + 30 - 20 = 60 \text{ e.u.} \] Therefore, \( \Delta S(A \rightarrow B) = 60 \text{ e.u.} \).

Step 6: Identifying the Correct Option

The calculated \( \Delta S(A \rightarrow B) \) of 60 e.u. matches option (b). Therefore, the correct answer is (b) \(+60\) e.u.

Key Concepts

State Function

A state function is a crucial concept in thermodynamics. It refers to a property of a system that depends only on its current state, not on the process used to reach that state. This means that it does not matter how the system arrived at the current state, whether quickly, slowly, directly, or indirectly; the state function remains unchanged.
For example, entropy, energy, and pressure are all state functions. If you know the initial and final states of a system, you can calculate the change in a state function by subtracting the initial value from the final value. This is why in the problem of converting \( A \) to \( B \) through an indirect pathway, we can use the concept of state function to determine the entropy change \( \Delta S(A \rightarrow B) \). The actual path between the states does not affect the calculation of the state function value.

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat, work, temperature, and energy. It explains how energy is transferred between systems and the environment, following specific laws or principles. These laws describe how energy changes within a process and help predict how energy will behave under different conditions.
In our exercise, thermodynamics allows us to understand how entropy, a thermodynamic quantity, changes during the transition of substances from one state to another. It provides the framework to calculate entropy changes along different paths without having to carry out each step in practice. By applying the laws of thermodynamics, we can determine the feasibility and spontaneity of reactions like the transformation of \( A \) to \( B \).
Essentially, thermodynamics helps us understand and predict changes in state functions such as entropy, which measures the degree of disorder or randomness in a system.

Entropy Change

Entropy change \( \Delta S \) is a measure of how much a process increases the disorder of a system. In simple terms, it tells us how energy spreads out within a system during a process. For our initial setup, it involves moving from state \( A \) to state \( B \) indirectly via several steps, each with its own entropy change.

• \( \Delta S(A \rightarrow C) \) represents the entropy gain from \( A \) to \( C \).
• \( \Delta S(C \rightarrow D) \) shows the additional increase in entropy from \( C \) to \( D \).
• \( \Delta S(B \rightarrow D) \) is subtracted because moving from \( B \) to \( D \) is the reverse of moving from \( D \) to \( B \).

Using these changes, we can calculate \( \Delta S(A \rightarrow B) \) by adding the increases in entropy and subtracting the decrease associated with the reverse pathway. The process provides valuable insights into energy distribution, process efficiency, and sustainability.