Problem 11 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

A gas undergoes change from state \(\mathrm{A}\) to state \(\mathrm{B}\). In this process, the heat absorbed and work done by the gas is \(5 \mathrm{~J}\) and \(8 \mathrm{~J}\), respectively. Now gas is brought back to \(\mathrm{A}\) by another process during which \(3 \mathrm{~J}\) of heat is evolved. In this reverse process of \(\mathrm{B}\) to \(\mathrm{A}\) :(a) \(10 \mathrm{~J}\) of the work will be done by the gas. (b) \(6 \mathrm{~J}\) of the work will be done by the gas. (c) \(10 \mathrm{~J}\) of the work will be done by the surrounding on gas. (d) \(6 \mathrm{~J}\) of the work will be done by the surrounding on gas.

Step-by-Step Solution

Answer

Option (d) is correct: 6 J of work will be done by the surroundings on the gas.

Step 1: Understand the First Law of Thermodynamics

The first law of thermodynamics is given by the equation \( \Delta U = Q - W \), where \( \Delta U \) is the change in internal energy, \( Q \) is the heat absorbed by the system, and \( W \) is the work done by the system. We will apply this principle to the processes described.

Step 2: Calculate Change in Internal Energy from A to B

For the process A to B, the heat absorbed \( Q_{AB} = 5 \text{ J} \) and the work done by the gas \( W_{AB} = 8 \text{ J} \). Using the first law, we get: \[ \Delta U_{AB} = Q_{AB} - W_{AB} = 5 \text{ J} - 8 \text{ J} = -3 \text{ J} \]

Step 3: Calculate Change in Internal Energy from B to A

Since the process B to A returns the gas to its original state, the change in internal energy \( \Delta U_{BA} = -\Delta U_{AB} = 3 \text{ J} \).

Step 4: Determine Work Done in the Reverse Process

For the reverse process (B to A), the heat evolved \( Q_{BA} = -3 \text{ J} \) (since it's released). Using the first law again:\[ 3 \text{ J} = -3 \text{ J} - W_{BA} \]Rearranging, we get \( W_{BA} = -6 \text{ J} \).

Step 5: Interpret the Sign of Work Done

A negative \( W_{BA} = -6 \text{ J} \) implies that 6 J of work is being done on the gas by the surroundings. Thus, option (d) is correct: \( 6 \text{ J} \) of work will be done by the surroundings on the gas.

Key Concepts

Internal Energy

In thermodynamics, internal energy represents the total energy contained within a system. This includes the kinetic energy of the molecules due to their motion and potential energy due to molecular interactions. It is a key concept in understanding how energy is transferred and transformed within a system.
When a system changes from one state to another, its internal energy changes. The change in internal energy (\( \Delta U \)) depends on two factors:

• The heat (\( Q \)) added or removed from the system
• The work (\( W \)) done by or on the system

For example, in the transition from state A to state B in our gas problem, the internal energy changes by \(-3 \text{ J}\). This is calculated using the equation of the first law of thermodynamics:\( \Delta U = Q - W \), where \( Q \) is the heat absorbed and \( W \) is the work done by the system. So, even if the system appears unchanged externally, like returning to state A in the problem, understanding \( \Delta U \) is crucial in quantifying the internal changes.

Heat Transfer

Heat transfer is the movement of thermal energy from one part of a system to another or between systems. It occurs because of temperature differences and can happen in three ways: conduction, convection, and radiation. In thermodynamic processes, we often talk about the amount of heat absorbed or released during a transformation.
For our gas transitioning from state A to state B, \(5 \text{ J}\) of heat is absorbed. This energy intake can be seen as an increase in molecular motion, raising the system's temperature or altering its phase. Conversely, when the gas returns from B to A, \(3 \text{ J}\) of heat is released, indicating a decrease in molecular energy.
Understanding heat transfer allows us to control and utilize energy, ensuring that the desired work can be performed efficiently. It’s fundamental in designing engines, refrigerators, and many other systems that rely on thermal energy exchange.

Work Done

Work in thermodynamics is the energy transferred when a force acts upon an object to move it over a distance. In the context of gases, work is usually done when the gas expands or compresses within a chamber. The first law of thermodynamics expresses this as one of the ways energy is exchanged in a system.During the transition of our gas from state A to B, \(8 \text{ J}\) of work is done by the gas. This work could manifest as the expansion of the gas, pushing against a piston, for example. Conversely, in the reverse process from B to A, \(6 \text{ J}\) of work is done on the gas by the surroundings.

• A positive work value indicates work done by the system on its surroundings.
• A negative work value implies work done on the system by its surroundings.

In our problem, the negative sign for the work done during the reverse process clearly shows work is done on the gas. Understanding the direction of work is essential because it helps us design systems where energy efficiency is maximized, such as in engines and power plants.