Class 11 Physics: Kinetic Theory of Gases Notes

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1. Introduction

• The Kinetic Theory of Gases explains the macroscopic properties of gases (pressure, temperature, volume) in terms of the microscopic motion of molecules.
• Developed to connect molecular motion with gas laws.

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2. Assumptions of Kinetic Theory (Ideal Gas)

1. Gas consists of a large number of identical molecules moving in random directions.
2. Molecules occupy negligible volume compared to the container volume.
3. No intermolecular forces except during collisions.
4. Collisions with walls or other molecules are perfectly elastic.
5. Average kinetic energy of molecules is proportional to temperature (in Kelvin).

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3. Pressure of a Gas

• Gas molecules collide with the walls of the container, exerting pressure (P).
• For N molecules of mass m in volume V:

$$P = \frac{1}{3} \frac{N m \overline{v^2}}{V}$$P=31​VNmv2​

where $\overline{v^2}$v2 = mean square speed of molecules.

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4. Mean, RMS, and Most Probable Speeds

For an ideal gas:

1. Root Mean Square Speed (v_rms)

$$v_{rms} = \sqrt{\overline{v^2}} = \sqrt{\frac{3RT}{M}}$$vrms​=v2​=M3RT​​

2. Average (Mean) Speed (v_avg)

$$v_{avg} = \sqrt{\frac{8RT}{\pi M}}$$vavg​=πM8RT​​

3. Most Probable Speed (v_mp)

$$v_{mp} = \sqrt{\frac{2RT}{M}}$$vmp​=M2RT​​

• R = universal gas constant, M = molar mass (kg/mol), T = temperature in Kelvin

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5. Kinetic Energy of Gas Molecules

• Translational kinetic energy per molecule:

$$E_{k} = \frac{1}{2} m \overline{v^2} = \frac{3}{2} k_B T$$Ek​=21​mv2=23​kB​T

where $k_B$kB​ = Boltzmann constant

• Total internal energy of n moles of ideal gas:

$$U = \frac{3}{2} nRT$$U=23​nRT

• For monoatomic gases, all energy is translational.
• For diatomic gases, rotational energy adds $\frac{1}{2}RT$21​RT per degree of freedom.

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6. Gas Laws Derived from Kinetic Theory

1. Ideal Gas Equation:

$$PV = nRT$$PV=nRT

Derived from $P = \frac{1}{3} \frac{N m \overline{v^2}}{V}$P=31​VNmv2​

2. Relation with kinetic energy:

$$\overline{E_k} = \frac{3}{2} k_B T$$Ek​​=23​kB​T

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7. Important Points

• Temperature measures average kinetic energy of gas molecules.
• Pressure arises from molecular collisions with walls.
• Kinetic theory is valid only for ideal gases.
• Molecular speeds follow the Maxwell-Boltzmann distribution.