Problem 2 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

Five moles of an ideal gas at 1 bar and \(298 \mathrm{~K}\) is expanded into vacuum to double the volume. The work done is :(a) \(C_{V}\left(T_{2}-T_{1}\right)\) (b) \(-\mathrm{RT}\left(\mathrm{V}_{2}-\mathrm{V}_{1}\right)\) (c) \(-\mathrm{RT} \ln \mathrm{V}_{1} / \mathrm{V}_{1}\) (d) zero

Step-by-Step Solution

Answer

The work done is zero. (Option d)

Step 1: Interpret the Problem

An ideal gas is initially at a pressure of 1 bar and a temperature of 298 K. We know that it's undergoing an expansion into a vacuum, which means it is a free expansion.

Step 2: Identify Type of Process

Expansion into a vacuum is an example of free expansion. In thermodynamics, free expansion is a type of irreversible process in which a gas expands into an insulated evacuated chamber without performing work on its surroundings because there's no external pressure resisting the expansion.

Step 3: Determine the Work Done

In free expansion against vacuum, the external pressure is zero since there's no resistance. The work done by an expanding gas is calculated as \( W = - P_{ ext{ext}}(V_2 - V_1) \), where \( P_{ ext{ext}} \) is the external pressure. In this case, \( P_{ ext{ext}} = 0 \).

Step 4: Evaluate the Given Options

The options given are different expressions for work done or energy changes. As we have already determined that the work done is zero due to the nature of free expansion, only option (d) aligns with our solution.

Key Concepts

Ideal Gas

The ideal gas is a fundamental concept in thermodynamics. It is a theoretical gas composed of a large number of randomly moving particles, and it follows several simplifying assumptions. These assumptions include:

• No intermolecular forces between the particles.
• Collisions between particles are perfectly elastic, meaning they conserve energy.
• The volume occupied by the gas particles themselves is negligible compared to the total volume of the gas.

These simplifications allow for the derivation of the ideal gas law, expressed as \[ PV = nRT \] where:

• \( P \) is the pressure,
• \( V \) is the volume,
• \( n \) is the number of moles,
• \( R \) is the ideal gas constant,
• \( T \) is the temperature in kelvins.

The ideal gas law helps predict how changing one of these variables can impact the others under ideal conditions. Nevertheless, it's important to note that real gases only approximate these behaviors under high temperature and low pressure.

Free Expansion

Free expansion is a specific type of thermodynamic process where an ideal gas expands into a vacuum without opposing external pressure. Hence, no work is done by the gas. This occurs because the external pressure, \( P_{\text{ext}} \), is zero, resulting in the following equation:\[ W = - P_{\text{ext}} (V_2 - V_1) \] Since \( P_{\text{ext}} = 0 \), the work done \( W \) is also zero.Despite the absence of work, free expansion is an important concept because it illustrates how gases behave when unconstrained by external forces. It is characteristic of an irreversible process because the system does not return to its initial state without intervention. Furthermore, in free expansion, internal energy remains constant, so any change in energy solely reflects the temperature changes due to a redistribution within the gas itself. This highlights the uniqueness of free expansion compared to other processes where work can be performed.

Work Done

Work done in thermodynamic processes is a measure of energy transfer due to volume changes. Generally, it is calculated using the formula:\[ W = - P_{\text{ext}} (V_2 - V_1) \] Where:

• \( W \) is the work done,
• \( P_{\text{ext}} \) is the external pressure,
• \( V_2 \) and \( V_1 \) are the final and initial volumes, respectively.

In the context of free expansion, such as in the problem outlined, the external pressure is zero, leading to zero work done. Including free expansion in discussions about work done helps clarify that not all expansion processes result in work, especially in vacuum conditions. In practical terms, work is only done when a force is externally applied, resulting in displacement. Recognizing situations where no work is performed can help identify whether internal energy or temperature changes occur through other means.