Problem 161 - A Complete Resource Book in Chemistry for JEE Main

Question

An ideal gas is allowed to expand both reversibly and irreversibly in an isolated system. If \(\mathrm{T}_{\mathrm{i}}\) is the initial temperature and \(\mathrm{T}_{\mathrm{f}}\) is the final temperature, which of the following statements is correct? [2006] (a) \(\left(\mathrm{T}_{\mathrm{f}}\right)_{\text {irrev }}>\left(\mathrm{T}_{\mathrm{i}}\right)_{\mathrm{rev}}\) (b) \(\mathrm{T}_{\mathrm{f}}>\mathrm{T}_{\mathrm{i}}\) for reversible process but \(\mathrm{T}_{\mathrm{f}}=\mathrm{T}_{\mathrm{i}}\) for irreversible process (c) \(\left(\mathrm{T}_{\mathrm{f}}\right)_{\text {irrev }}=\left(\mathrm{T}_{\mathrm{i}}\right)_{\mathrm{rev}}\) (d) \(\mathrm{T}_{\mathrm{f}}=\mathrm{T}_{\mathrm{i}}\) for both reversible and irreversible processes

Step-by-Step Solution

Answer

(d) \(\mathrm{T}_{\mathrm{f}}=\mathrm{T}_{\mathrm{i}}\) for both reversible and irreversible processes.

Step 1: Understanding Isolated System

An isolated system does not exchange heat or matter with its surroundings. This means that for any process occurring within the system, the change in internal energy (9E) is zero since there is no heat (9Q = 0) and no work (9W = 0) exchanged with the environment.

Step 2: Applying the First Law of Thermodynamics

The First Law of Thermodynamics states that the change in internal energy 9E of a system is equal to the heat added to the system minus the work done by the system: \[ 9E = Q - W \]However, in an isolated system, 9E = 0, Q = 0, and W = 0. Therefore, the internal energy does not change, and thus the temperature remains constant for both reversible and irreversible processes.

Step 3: Analyzing Reversible vs Irreversible Expansion

In an isolated system, since no energy is exchanged, the final temperature (Tf) of the gas must equal its initial temperature (Ti), regardless of whether the process is reversible or irreversible. This absence of change in energy and heat leads to no temperature change.

Step 4: Selecting the Correct Statement

Based on our analysis, we know that the temperature final and initial conditions in isolated systems do not change. Therefore, statement (d), "\(\mathrm{T}_{\mathrm{f}}=\mathrm{T}_{\mathrm{i}}\) for both reversible and irreversible processes," is the correct statement.

Key Concepts

Isolated System

An isolated system is a fundamental concept in thermodynamics. It describes a system that does not exchange energy or matter with its surroundings. Imagine putting a perfect lid on a pot, preventing any heat from entering or exiting. In this closed environment, anything that happens inside the pot stays inside the pot. This means that:

• There is no heat (Q = 0) transfer, meaning no energy can flow in or out.
• No work (W = 0) is done by or on the system, which means it cannot change its volume by interacting with the surroundings.
• Internal energy (E = 0) remains constant since it is not influenced by outside factors.

This notion helps us analyze processes purely based on internal changes. Since there's no exchange of energy, whatever state the system starts in, in terms of overall energy, it will remain in the same state.

First Law of Thermodynamics

The First Law of Thermodynamics is like the law of conservation of energy, but specifically applied to thermodynamics. It states:\[ E = Q - W \]This equation means the change in internal energy (E) of a system is determined by the heat added to the system (Q) minus the work done by the system (W). Here's a breakdown:

• When heat is added (Q > 0), it increases the system's internal energy unless it's entirely used to do work.
• When the system does work (W > 0), it loses energy as some of it is used to perform work on its surroundings.

In an isolated system, since both heat and work are zero, the First Law simplifies to no change in internal energy (E = 0). This simplicity means that, regardless of what happens inside the system, the total internal energy does not change.

Reversible vs Irreversible Processes

Understanding reversible and irreversible processes is crucial in thermodynamics. They describe two different kinds of changes a system can undergo:

• Reversible Process: This is an idealized process that happens so slowly that the system is always in equilibrium. In practice, it means every state of the system can be reversed without any net change in the system and surroundings.
• Irreversible Process: This occurs when the process moves the system out of equilibrium and cannot be undone by reversing the steps. Most real-world processes are irreversible due to factors like friction or rapid changes.

In an isolated system, even though these processes differ conceptually (one is ideal, the other real), neither will change the system's temperature by themselves. Since no external energy can be exchanged, the initial and final temperatures (T) will remain the same for both processes.