Problem 149 - A Complete Resource Book in Chemistry for JEE Main

Question

A heat engine absorbs heat \(\mathrm{Q}_{1}\) from a source at tem perature \(\mathrm{T}_{1}\) and heat \(\mathrm{Q}_{2}\) from a source at temperature \(\mathrm{T}_{2}\). Work done is found to be \(\mathrm{J}\left(\mathrm{Q}_{1}+\mathrm{Q}_{2}\right)\). This is in accordance with: (a) first law of thermodynamics (b) second law of thermodynamics (c) joules equivalent law (d) none of these

Step-by-Step Solution

Answer

This is according to the first law of thermodynamics (a).

Step 1: Understanding the First Law of Thermodynamics

The first law of thermodynamics, also known as the law of energy conservation, states that the energy absorbed by a system as heat, minus the energy lost by the system as work done, equals the change in its internal energy. The equation can be expressed as \( \Delta U = Q - W \), where \( \Delta U \) is the change in internal energy, \( Q \) is the heat absorbed, and \( W \) is the work done by the system.

Step 2: Analyzing the Given Information

In the provided problem, a heat engine absorbs heat \( Q_1 \) and \( Q_2 \) from two sources. The total heat absorbed by the system is therefore \( Q_1 + Q_2 \). The work done by the heat engine is given as \( J(Q_1 + Q_2) \). There is no mention of internal energy change, so we assume it remains constant.

Step 3: Matching with Thermodynamics Laws

According to the first law of thermodynamics, the work done \( W \) should be \( Q_1 + Q_2 - \Delta U \). In the given expression \( J(Q_1 + Q_2) = Q_1 + Q_2 \), we assume that the internal energy change is zero \( \Delta U = 0 \), aligning with the first law by letting \( J = 1 \).

Step 4: Applying Joule's Equivalent

In the context of the problem statement, the term \( J \) is analogous to the Joule's equivalent factor where \( J = 1 \) aligns it with work-energy correspondence showing complete work extraction from absorbed heat within the system as per the first law.

Key Concepts

Heat Engines

A heat engine is a fascinating device that turns heat energy into mechanical work, performing what seems like a magic trick. The fundamental concept behind a heat engine is based on a simple but powerful idea: it absorbs heat energy from a high-temperature source and converts a portion of it into useful work while releasing the remaining energy to a low-temperature sink. Heat engines operate in cycles, going through a sequence of processes that returns them to their initial state. In each cycle:

• They absorb heat (\(Q_1\)) from a high-temperature reservoir.
• Some of this energy is converted to work (\(W\)).
• The remaining heat is expelled (\(Q_2\)) to a lower-temperature reservoir.

Understanding how a heat engine works empowers us to appreciate many technologies, from car engines to power plants. While engines are not 100% efficient, they are instrumental in harnessing energy efficiently from fuel sources.

Work-Energy Principle

The work-energy principle is a pivotal concept in physics, linking the work done by forces to the energy changes in a system. Specifically, it suggests that the work done by the sum of all forces acting on an object equals the change in its kinetic energy. Let's break down this principle:

• Consider a system starting with a certain amount of kinetic energy.
• Upon applying forces, work is done, changing the system’s kinetic energy.
• The amount of work done (\(W\)) corresponds to the difference between the final and initial kinetic energy.

In the context of a heat engine, the work-energy principle is embodied in how the engine converts absorbed heat into mechanical work. The resulting work, which may be thought of as the energy ‘output’, can be used to drive machines and perform tasks. This principle is a testament to how energy transformations govern mechanical systems.

Energy Conservation

Energy conservation is a fundamental principle governing many natural and technological systems, dictated by the first law of thermodynamics. It stipulates that energy within a closed system remains constant, only changing form from one type to another. This underscores the idea that energy cannot be created or destroyed, merely distributed or transformed. In applications such as heat engines:

• Energy is absorbed as heat transferred from higher temperature sources.
• A portion of this energy transforms into work, with some inevitably lost to surroundings as waste heat.
• The total energy involved in these processes equals the initial input energy.

This principle assures us that even though the forms of energy may change, the total energy budget balances out. It provides the foundation for analyzing how efficiently systems like engines or electricity generation plants operate, thus helping us develop better technology and systems that leverage the existing energy resources efficiently.