What Is Kinetic Theory of Gases?

Important Concepts from Chapter 2

Why Is HC Verma’s Chapter Important?
- Features of HC Verma’s Solutions:
Example Problems from HC Verma
- Problem 1: Deriving Pressure Equation
- Problem 2: Average Kinetic Energy
Practical Applications of Kinetic Theory
Tips for Mastering This Chapter
FAQs on Hc Verma Solutions Class 12 Chapter 2 Kinetic Theory Of Gases

Hc Verma Solutions Class 12 Chapter 2 Kinetic Theory Of Gases

The Kinetic Theory of Gases is one of the most fascinating chapters in physics, especially for students preparing for competitive exams like JEE and NEET. This chapter helps us understand the behavior of gases at the molecular level and explains the concepts behind pressure, temperature, and the energy of gas molecules.

In this article, we will explore the key concepts from Chapter 2 of HC Verma’s Concepts of Physics (Part 2) and how its solutions can aid students in understanding the subject better. The explanations are simple to make the topic accessible to everyone.

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Hc Verma Solutions Class 12 Chapter 2 Kinetic Theory Of Gases

What Is Kinetic Theory of Gases?

The Kinetic Theory of Gases explains the properties of gases by considering them as a collection of small particles (molecules) in constant, random motion. It connects the macroscopic properties of gases, like pressure and temperature, to the microscopic behavior of gas molecules.

Key Assumptions of Kinetic Theory:

Molecules Are Point Masses: Gas molecules are considered tiny particles with mass but no volume.
Random Motion: Molecules move randomly in all directions.
Elastic Collisions: When gas molecules collide with each other or the walls of the container, no energy is lost.
No Intermolecular Forces: There are no attractive or repulsive forces between molecules, except during collisions.
Large Number of Molecules: The number of gas molecules is very large, allowing statistical methods to describe their behavior.

Important Concepts from Chapter 2

1. Pressure of Gas

The pressure exerted by a gas on the walls of its container is due to the collisions of gas molecules. Using the kinetic theory, the pressure $P$ $P$ can be derived as:

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P = \frac{1}{3} \rho v^2

$P$ $=$ $31$ $$ $ρv$ $2$

Here:

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$\rho$ $ρ$ is the density of the gas.
$v^2$ $v$ $2$ is the mean square velocity of the gas molecules.

This equation helps us understand that pressure is directly related to the kinetic energy of the molecules.

2. Temperature and Kinetic Energy

The temperature of a gas is directly proportional to the average kinetic energy of its molecules. This relationship is given by:

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KE = \frac{3}{2} k_B T

$KE$ $=$ $23$ $$ $k$ $B$ $$ $T$

Here:

$KE$ $KE$ is the average kinetic energy.
$k_B$ $k$ $B$ $$ is Boltzmann’s constant.
$T$ $T$ is the absolute temperature in Kelvin.

This equation shows that as temperature increases, the molecules move faster, increasing their kinetic energy.

3. Ideal Gas Law

The kinetic theory of gases supports the ideal gas equation:

PV = nRT

$PV$ $=$ $nRT$

Here:

$P$ $P$ is the pressure,
$V$ $V$ is the volume,
$n$ $n$ is the number of moles,
$R$ $R$ is the universal gas constant, and
$T$ $T$ is the temperature.

This law connects pressure, volume, and temperature, making it a fundamental equation in thermodynamics.

4. Degrees of Freedom

Degrees of freedom refer to the independent ways in which a gas molecule can move. For a monoatomic gas (like helium), there are 3 translational degrees of freedom. For diatomic and polyatomic gases, there are additional rotational and vibrational degrees of freedom.

The concept of degrees of freedom is crucial for understanding the distribution of energy in a gas.

Why Is HC Verma’s Chapter Important?

HC Verma’s Concepts of Physics is renowned for its clear explanations and practical approach to problem-solving. Chapter 2 provides a thorough understanding of the kinetic theory of gases through well-structured theory and carefully designed numerical problems.

Features of HC Verma’s Solutions:

Step-by-Step Explanations: The solutions break down complex problems into simple steps.
Application of Formulas: It demonstrates how to apply theoretical formulas to solve real-world problems.
Conceptual Questions: Many questions test your understanding of the core concepts rather than just calculations.

By practicing the problems from this chapter, students gain confidence in applying the kinetic theory to different scenarios.

Example Problems from HC Verma

Let’s go through a few example problems from Chapter 2 to illustrate how HC Verma’s solutions are structured.

Problem 1: Deriving Pressure Equation

Question: Derive the expression for pressure exerted by an ideal gas using the assumptions of the kinetic theory.

Solution:

Consider a cubic container of side $l$ $l$ with $N$ $N$ molecules moving in random directions.
Using the assumptions of the kinetic theory, calculate the momentum change due to collisions with the walls.
By summing the contributions of all molecules and averaging, derive the pressure equation:

P = \frac{1}{3} \rho v^2

$P$ $=$ $31$ $$ $ρv$ $2$

This problem reinforces the relationship between microscopic motion and macroscopic pressure.

Problem 2: Average Kinetic Energy

Question: Calculate the average kinetic energy of one mole of a gas at 300 K.

Solution:

Use the formula for average kinetic energy per molecule: $KE = \frac{3}{2} k_B T$ $KE$ $=$ $23$ $$ $k$ $B$ $$ $T$
Multiply by Avogadro’s number $N_A$ $N$ $A$ $$ to find the energy for one mole.
Substitute $k_B = 1.38 \times 10^{-23} \, \text{J/K}$ $k$ $B$ $$ $=$ $1.38$ $\times$ $10$ $-23$ $J/K$ and $T = 300 \, \text{K}$ $T$ $=$ $300$ $K$ .

The solution shows how molecular behavior connects to measurable quantities like temperature.

Practical Applications of Kinetic Theory

The concepts from this chapter are not just theoretical; they have real-world applications:

Weather Forecasting: Understanding pressure and temperature variations in the atmosphere.
Refrigeration: Principles of gases under different pressure and temperature conditions.
Aerospace: Designing aircraft and rockets that encounter gases at varying temperatures and pressures.

Tips for Mastering This Chapter

Understand the Basics: Focus on the assumptions of the kinetic theory and their implications.
Practice Numericals: Solve all the exercises from HC Verma’s book to strengthen your problem-solving skills.
Visualize the Concepts: Try to imagine how gas molecules move and collide to connect theory with reality.
Memorize Key Formulas: Remember important equations like the ideal gas law, kinetic energy, and pressure formula.

FAQs on Hc Verma Solutions Class 12 Chapter 2 Kinetic Theory Of Gases

What is the Kinetic Theory of Gases?

The Kinetic Theory of Gases explains the behavior of gases by considering them as a collection of tiny particles (molecules) in constant random motion. It provides a molecular-level understanding of macroscopic properties like pressure, temperature, and volume.

What are the main assumptions of the Kinetic Theory of Gases?

The key assumptions of the Kinetic Theory are:

Gas molecules are small, point-like masses with negligible volume.
Molecules are in constant, random motion.
Collisions between molecules and container walls are perfectly elastic (no energy is lost).
There are no intermolecular forces except during collisions.
The number of molecules is large, allowing statistical treatment.

What is the relationship between temperature and kinetic energy in gases?

The average kinetic energy ( $KE$ $KE$ ) of gas molecules is directly proportional to the absolute temperature ( $T$ $T$ ) of the gas. The relationship is given by:

KE = \frac{3}{2} k_B T

$KE$ $=$ $23$ $$ $k$ $B$ $$ $T$

where $k_B$ $k$ $B$ $$ is Boltzmann’s constant. This means that as the temperature increases, the molecules move faster, increasing their kinetic energy.

How does the Kinetic Theory explain gas pressure?

Gas pressure is caused by collisions of gas molecules with the walls of the container. The theory relates pressure ( $P$ $P$ ) to the density ( $\rho$ $ρ$ ) and the mean square velocity ( $v^2$ $v$ $2$ ) of the gas molecules:

P = \frac{1}{3} \rho v^2

$P$ $=$ $31$ $$ $ρv$ $2$

This shows that pressure depends on the motion of the molecules and their energy.

What is the ideal gas equation, and how is it connected to Kinetic Theory?

The ideal gas equation is:

PV = nRT

$PV$ $=$ $nRT$

where $P$ $P$ is pressure, $V$ $V$ is volume, $n$ $n$ is the number of moles, $R$ $R$ is the gas constant, and $T$ $T$ is temperature. The Kinetic Theory provides a molecular explanation for this equation, linking the macroscopic properties of gases to the motion and energy of gas molecules.

Why are intermolecular forces neglected in the Kinetic Theory of Gases?

Intermolecular forces are neglected to simplify the model and because their effect is negligible in ideal gases, especially at high temperatures and low pressures. This assumption works well for ideal gases but may not hold for real gases, where deviations from ideal behavior occur.