Problem 16 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following is correct equation? (a) \(\Delta U=\Delta Q-W\) (b) \(\Delta \mathrm{W}=\Delta \mathrm{U}+\Delta \mathrm{Q}\) (c) \(\Delta \mathrm{U}=\Delta \mathrm{W}+\Delta \mathrm{Q}\) (d) none of these

Step-by-Step Solution

Answer

Option (a) is the correct equation, as it aligns with the First Law of Thermodynamics.

Step 1: Understand the symbols

The symbols used in the equations refer to quantities in thermodynamics. \( \Delta U \) is the change in internal energy of a system, \( \Delta Q \) is the heat added to the system, and \( \Delta W \) is the work done by the system.

Step 2: Recall the First Law of Thermodynamics

The First Law of Thermodynamics states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. Mathematically, it can be expressed as \( \Delta U = \Delta Q - \Delta W \).

Step 3: Analyze each option

Option (a), \( \Delta U = \Delta Q - W \), matches the First Law of Thermodynamics' expression: \( \Delta U = \Delta Q - \Delta W \), assuming \( W \) is interpreted as work done by the system in appropriate units. Option (b) and (c) do not match this expression.

Step 4: Verify by elimination

Since option (a) correctly aligns with the established formula from the First Law of Thermodynamics, options (b) and (c) can be eliminated as incorrect interpretations. Option (d), "none of these," is unnecessary as (a) is already correct.

Key Concepts

Thermodynamics Symbols

In thermodynamics, we use specific symbols to represent different concepts in the equation, helping us describe processes within a system. Here are the key symbols:

• \( \Delta U \) denotes the change in internal energy of the system. This change is critical as it tells us how energy within the system is altered by various processes.
• \( \Delta Q \) stands for the heat added to the system. Heat transfer is a vital factor affecting the energy of the system.
• \( \Delta W \) indicates the work done by the system. This is the useful energy output from various processes, like expanding gas pushing a piston.

Understanding these symbols is crucial because they form the foundation for analyzing any thermodynamic process, enabling us to apply the First Law of Thermodynamics effectively.
Knowing these notations allows students to break down complex problems into simple, understandable parts, making them easier to solve.

Internal Energy

Internal energy is a core concept in thermodynamics that refers to the total energy contained within a system. This includes all forms of molecular energy—like kinetic and potential energy of the particles within the system. Changes in internal energy, marked as \( \Delta U \), inform us about how the energy state of the system shifts.
Internal energy can be influenced by two primary factors:

• Heat transfer: Adding or removing heat (\( \Delta Q \)) changes the internal energy.
• Work done: When a system does work (\( \Delta W \)), its internal energy is altered.

Understanding how internal energy varies allows us to predict how systems will react to changes in their surroundings. The complexity of a system’s internal energy is what makes thermodynamics a fascinating yet challenging field to explore.

Heat Transfer

Heat transfer refers to the movement of thermal energy from one body or system to another. In thermodynamics, we signify this with \( \Delta Q \). This transfer occurs because of a temperature difference between the systems, following nature's tendency to reach equilibrium.
There are three main methods of heat transfer:

• Conduction: Direct transfer of heat through a substance, like a metal rod heating up at one end.
• Convection: Transfer of heat by fluid motion, such as warm air rising and spreading heat.
• Radiation: Transfer through electromagnetic waves, like the warmth felt from the sun.

The role of heat transfer in thermodynamics is vital. It changes the internal energy of a system, thus influencing its state. Understanding these principles allows us to design efficient heating or cooling systems, improving everything from home energy efficiency to industrial processes.

Work Done by System

Work done by a system, denoted as \( \Delta W \), is another key concept. It represents the energy transferred from the system to its surroundings. For example, a system can do work on another system by causing physical changes, like moving a piston.
The work done by the system depends on the process:

• In an expansion process, for instance, gas inside a piston does work as it pushes the piston out, converting internal energy into work.
• The opposite is true for a compression process where work is done on the gas.

Understanding how and when work is done helps predict how a system will perform. It is essential in fields like engineering and environmental science, where energy efficiency and resource use are critical factors.