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: Understanding the First Law of Thermodynamics

The first law of thermodynamics is also known as the law of conservation of energy. It states that energy cannot be created or destroyed, only transferred or converted. In mathematical terms, it can be expressed as \(\Delta U = Q - W\), 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.

Step 2: Interpreting the Given Problem in Terms of Energy Conservation

In the given problem, the heat absorbed from the sources is \(Q_{1} + Q_{2}\), and the work done, \(W\), by the engine is \(J(Q_{1}+Q_{2})\). This means all absorbed heat from the different sources is being converted into work, suggesting a direct relation between heat absorbed and work done.

Step 3: Matching with Laws of Thermodynamics

According to the first law of thermodynamics, the internal energy change should account for the difference between heat added and work done. Since the work done is precisely equivalent to the heat added (as given by \(W = J(Q_{1} + Q_{2})\)), this suggests compliance with the first law where the energy is conserved as the system exchanges heat and does work.

Step 4: Conclusion

Considering the work done equals the sum of the heat absorbed, which aligns with energy conservation, the described situation is an example of the first law of thermodynamics.

Key Concepts

Conservation of Energy

The conservation of energy principle, also known as the first law of thermodynamics, is a fundamental concept in physics. It tells us that energy cannot simply vanish or appear out of nothing; it can only change forms or move from one place to another. Imagine using a battery to power a toy car. The stored chemical energy in the battery turns into kinetic energy that moves the car. No energy is lost; it just changes from chemical to mechanical form.

In formal terms, the first law of thermodynamics is expressed by the equation \( \Delta U = Q - W \). Here, \( \Delta U \) is the change in internal energy of a system, \( Q \) is the heat added to the system, and \( W \) is the work done by the system. This equation illustrates that the heat put into a system can do two things: increase the system's internal energy or be used to perform work.

In the exercise in question, heat absorbed \( (Q_1 + Q_2) \) from the surroundings is turned into work \( J(Q_1 + Q_2) \). This perfectly demonstrates the conservation of energy, as the energy in the form of heat equals the energy in the form of work done. This concept reinforces that energy within an isolated system stays balanced.

Heat Engines

Heat engines are devices that convert thermal energy, or heat, into mechanical work. They are pivotal in many practical applications, such as cars, power plants, and factories. The basic idea of a heat engine is to take in heat from a hot source, convert part of it to do work, and then expel the remaining heat to a cold sink.

In our specific task, the heat engine absorbs heat \( Q_1 \) and \( Q_2 \) from two separate sources at different temperatures \( T_1 \) and \( T_2 \). The engine uses this heat to do work signified by \( J(Q_1 + Q_2) \). The way heat engines operate showcases practical applications of thermodynamics, particularly energy transformation, and efficiency. They illustrate how heat, a form of energy, is partially converted into work.

The effectiveness of a heat engine can be evaluated using its efficiency, which is the ratio of work output to heat input. Although the system described here executes perfect energy conversion without losses, real-world engines often lose some energy as waste heat.

Thermodynamic Processes

Thermodynamic processes are the paths through which a system changes from one state to another by interacting with its surroundings. These processes can include exchanging heat, doing work, or changing internal energy. Various processes can be identified, such as isothermal, adiabatic, isobaric, and isochoric.

Within the exercise example, you encounter thermodynamic processes through a specific kind of interaction: a heat interchange between heat sources and the engine. Here, two kinds of heat are absorbed \( Q_1 \) and \( Q_2 \), and the way these exchanges lead to work production highlights specific thermodynamic paths.

Understanding these processes helps us learn how energy transfers occur in practical scenarios. For instance, during an isothermal process, the temperature remains constant, whereas an adiabatic process involves no heat exchange. Each type of process aligns with a different aspect of the first law of thermodynamics in action, helping bridge theoretical concepts with real-world energy dynamics.