Kinetic Theory Of Gases Notes

Introduction

The Kinetic Theory of Gases is a fundamental concept in physics that explains the macroscopic properties of gases by considering their molecular composition and motion. It provides a bridge between the microscopic world of atoms and molecules and the macroscopic world of observable quantities like pressure, temperature, and volume. This theory is crucial for understanding various gas laws and is a significant part of the NEET Physics syllabus.

Basic Assumptions of Kinetic Theory

The kinetic theory of gases is based on several key assumptions:

1. Large Number of Molecules: A gas consists of a large number of molecules moving in random directions with different speeds.
2. Negligible Volume: The volume of individual gas molecules is negligible compared to the volume of the container.
3. No Intermolecular Forces: Except during collisions, there are no attractive or repulsive forces between the molecules.
4. Elastic Collisions: Collisions between gas molecules and between molecules and the walls of the container are perfectly elastic.
5. Time of Collisions: The time duration of collisions between molecules is negligible compared to the time between collisions.

Molecular Speeds and Distribution

Root Mean Square Speed

The root mean square (rms) speed is a measure of the average speed of gas molecules. It is given by:

$$ v_{\text{rms}} = \sqrt{\frac{3k_BT}{m}} $$

where:

• $k_B$ is the Boltzmann constant ($1.38 \times 10^{-23} , \text{J/K}$)
• $T$ is the absolute temperature
• $m$ is the mass of a gas molecule

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution describes the distribution of speeds among molecules in a gas. The probability $f(v)$ that a molecule has a speed $v$ is given by:

$$ f(v) = 4\pi \left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2 e^{-\frac{mv^2}{2k_BT}} $$

This distribution shows that most molecules have speeds around a certain value, with fewer molecules having very high or very low speeds.

Pressure of an Ideal Gas

The pressure exerted by a gas is due to collisions of gas molecules with the walls of the container. According to the kinetic theory, the pressure $P$ is given by:

$$ P = \frac{1}{3} \frac{N m \langle v^2 \rangle}{V} $$

where:

• $N$ is the number of molecules
• $m$ is the mass of one molecule
• $\langle v^2 \rangle$ is the mean square speed
• $V$ is the volume of the container

Using the ideal gas law $PV = nRT$, we can relate the macroscopic and microscopic perspectives.

Temperature and Kinetic Energy

The temperature of a gas is directly related to the average kinetic energy of its molecules. The average kinetic energy per molecule is given by:

$$ \langle E_k \rangle = \frac{3}{2} k_B T $$

For one mole of gas, the total kinetic energy is:

$$ E_k = \frac{3}{2} nRT $$

Degrees of Freedom and Equipartition of Energy

The degrees of freedom of a gas molecule refer to the number of independent ways in which it can store energy. For a monatomic gas (like helium), there are 3 translational degrees of freedom. For diatomic and polyatomic gases, there are additional rotational and vibrational degrees of freedom.

According to the equipartition theorem, each degree of freedom contributes $\frac{1}{2} k_B T$ to the energy.

Real Gases and Deviations from Ideal Behavior

Real gases deviate from ideal behavior due to intermolecular forces and the finite volume of gas molecules. The Van der Waals equation accounts for these deviations:

$$ \left( P + \frac{a}{V^2} \right) (V - b) = nRT $$

where:

• $a$ accounts for intermolecular attractions
• $b$ accounts for the finite volume of gas molecules

Conclusion

The Kinetic Theory of Gases is a powerful framework that connects the microscopic world of molecules to the macroscopic properties of gases. Understanding this theory is crucial for mastering the behavior of gases in various conditions, which is essential for NEET Physics.