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

Question

A heat engine absorbs heat \(Q_{1}\) 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

(a) first law of thermodynamics

Step 1: Understand the First Law of Thermodynamics

The first law of thermodynamics is a statement of energy conservation that can be represented by the equation \( \Delta U = Q - W \), where \( \Delta U \) is the change in internal energy, \( Q \) is the heat added to the system, and \( W \) is the work done by the system. For a cycle where the system returns to its initial state, \( \Delta U = 0 \), and therefore \( Q = W \). This implies that the total heat absorbed (from all sources) equals the total work done in a full cycle.

Step 2: Consider the Given Problem

In the problem, the engine absorbs heat from two sources: \( Q_1 \) at temperature \( T_1 \), and \( Q_2 \) at temperature \( T_2 \). The work done is \( J(Q_1 + Q_2) \). This suggests that the total heat absorbed \( Q_1 + Q_2 \) is converted into work, keeping with the energy conservation principle.

Step 3: Relate the Given Information to the First Law

According to the problem, the work done is expressed as \( J(Q_1 + Q_2) \). This expression is compatible with the first law since the heat absorbed (\( Q_1 + Q_2 \)) appears to be entirely converted into work, assuming \( J \) is a constant representing the conversion factor, similar to \( W = JQ \).

Step 4: Evaluate Other Options

Evaluate if the statement aligns with any other laws: The second law of thermodynamics primarily concerns entropy and efficiency, not direct energy balance like the first law. Joule's equivalent law refers to the conversion of heat into mechanical energy, similar to the given expression, but the phrasing 'first law' more directly ties to the conservation of energy in a cycle. Options (b) and (d) do not apply.

Key Concepts

Heat Engine

A heat engine is an essential concept in thermodynamics, typically used to illustrate how energy can be converted from heat to work. Imagine a machine designed to transform heat energy, extracted from a high-temperature reservoir, into mechanical energy or work. Heat engines operate on cycles, absorbing heat energy from a hot source, performing work, and then releasing some of the energy as waste to a cooler sink. The most commonly referenced example of a heat engine might be a car engine, where heat from fuel combustion is converted into motion. Key characteristics of a heat engine include:

• Heat input from a high-temperature source
• Work output from the system
• Heat rejection to a low-temperature sink

Understanding how a heat engine functions provides insight into the broader concept of thermodynamic cycles, particularly because they are perfect illustrations of the first law of thermodynamics applied in real-world situations.

Heat Energy Conversion

Heat energy conversion involves changing thermal energy into other forms of energy, typically mechanical work, as seen in heat engines. When a system absorbs heat energy, it can utilize this input to perform mechanical functions, thereby transforming one form of energy into another. This is the essence of operation in devices like heat engines. In the conversion process, we identify two primary endpoints:

• Converting absorbed heat into useful work output
• Releasing excess heat as waste to the surroundings

Through this conversion, it is possible to perform tasks, from powering machinery to generating electrical energy, representing a fundamental aspect of energy systems. The efficiency of this conversion process is determined by how effectively the heat input can be transformed into useful work, minimizing the energy lost to the surroundings.

Energy Conservation

Energy conservation is founded upon the principle that energy cannot be created or destroyed, only transformed from one form to another. This is crucial in thermodynamics and known as the first law of thermodynamics.When applied to a heat engine, energy conservation dictates that the total energy within the system remains constant. The engine absorbs a set amount of heat energy, some of which is converted into work, while the remainder is expelled as waste heat.The first law of thermodynamics can be mathematically expressed as:\[\Delta U = Q - W\]Where:

• \( \Delta U \) is the change in internal energy
• \( Q \) is the heat added to the system
• \( W \) is the work done by the system

In cycles like those in heat engines, where the system returns to its original state, \( \Delta U \) becomes zero, effectively summarizing the conservation principle as \( Q = W \). This means that for total heat absorbed in one complete cycle, an equivalent amount of work is performed.

Work Done

Work done in the context of thermodynamics refers to the energy transferred when a force is applied over a distance. In a heat engine, work done is the desired output from the conversion of heat energy. Within a cycle, the work done is directly related to the heat absorbed by the system. If we consider a simplified version of the first law for a complete cycle, we see that all absorbed heat energy can be translated into an equivalent amount of work performed by the heat engine. The calculation and understanding of work done involve:

• Evaluating the energy transferred to the system as heat
• Understanding the fraction of this energy converted into work
• Recognizing energy not converted remains as heat expelled to the environment

Work done is central to determining the performance and efficiency of a heat engine, directly tying into energy conservation laws. The precision of how heat translates to mechanical work informs the design and application of efficient thermal systems.