Kinetic Theory of Gases (KTG) - Notes for Quick Revision 📚

The Kinetic Theory of Gases (KTG) provides a microscopic understanding of the behavior of gases by considering their molecular motion. It bridges the gap between the macroscopic properties of gases, such as pressure and temperature, and their microscopic interactions.

This article presents detailed notes on the Kinetic Theory of Gases (KTG), covering assumptions, derivations, equations, and applications.

Introduction to Kinetic Theory of Gases

Definition: Kinetic Theory of Gases explains the macroscopic properties of gases (pressure, temperature, etc.) based on the motion of gas molecules and their interactions.

Key Idea: The behavior of a gas can be understood by studying the motion of its particles and the forces they exert during collisions.

Assumptions of Kinetic Theory

The Kinetic Theory is based on several simplifying assumptions:

Gas Molecules:

Gases consist of a large number of small, identical, and perfectly elastic particles (molecules).

Random Motion:

Gas molecules are in constant random motion.

Negligible Volume:

The volume of individual molecules is negligible compared to the total volume of the gas.

Elastic Collisions:

Collisions between molecules and with the container walls are perfectly elastic (no loss of kinetic energy).

Negligible Intermolecular Forces:

Molecules exert no attractive or repulsive forces on one another, except during collisions.

Duration of Collisions:

The time spent in collisions is negligible compared to the time between collisions.

Pressure:

Pressure is caused by the collisions of gas molecules with the walls of the container.

Temperature:

The average kinetic energy of gas molecules is directly proportional to the absolute temperature.

Key Quantities in KTG

1. Pressure of an Ideal Gas

The pressure exerted by a gas is derived from the momentum transfer during molecular collisions with the walls of the container.

Formula: $$P=\frac{1}{3}\rho v_{\text{rms}}^2$$ Where:

• $P$: Pressure of the gas.
• $\rho$: Density of the gas ($\rho =\frac{m}{V}$).
• $v_{\text{rms}}$: Root mean square velocity.

2. Root Mean Square Velocity ($v_{\text{rms}}$)

The root mean square velocity represents the effective speed of gas molecules.

Formula: $$v_{\text{rms}}=\sqrt{\frac{3k_{B}T}{m}}$$ Where:

• $k_B$: Boltzmann constant ($1.38\times 10^{-23}\text{J/K}$).
• $T$: Absolute temperature ($\text{K}$).
• $m$: Mass of a single molecule.

3. Average Kinetic Energy

The kinetic energy of gas molecules is directly proportional to the temperature.

Formula: $$KE=\frac{3}{2}{k}_{B}T$$ Where:

• $KE$: Average kinetic energy per molecule.

4. Ideal Gas Equation

The macroscopic behavior of gases is described by the ideal gas law:

$$PV = nRT$$

Where:

• $P$: Pressure.
• $V$: Volume.
• $n$: Number of moles.
• $R$: Universal gas constant ($8.31 \, \text{J/mol·K}$).
• $T$: Absolute temperature.

Distribution of Molecular Speeds

The distribution of molecular speeds in a gas is described by the Maxwell-Boltzmann Distribution.

Key Speeds:

Most Probable Speed ($v_{\text{mp}}$):

• The speed at which the maximum number of molecules move.
• Formula: $$v_{\text{mp}}=\sqrt{\frac{2k_{B}T}{m}}$$

Average Speed ($v_{\text{avg}}$):

• The mean speed of all molecules.
• Formula: $$v_{\text{avg}}=\sqrt{\frac{8k_{B}T}{\pi m}}$$

Root Mean Square Speed ($v_{\text{rms}}$):

• Formula: $$v_{\text{rms}}=\sqrt{\frac{3k_{B}T}{m}}$$

Degrees of Freedom

The degree of freedom ($f$) refers to the number of independent ways in which a gas molecule can move or store energy.

Degrees of Freedom for Different Gases:

Monoatomic Gas:

• $f=3$ (translational motion only).

Diatomic Gas:

• $f=5$ (3 translational + 2 rotational).

Polyatomic Gas:

• $f=6$ (3 translational + 3 rotational).

Law of Equipartition of Energy

Statement: The total energy of a molecule is equally distributed among all its degrees of freedom, with each degree of freedom having an average energy of:

$$\frac{1}{2}{k}_{B}T$$

Formula for Total Energy:

$$E=\frac{f}{2}nRT$$

Where:

• $f$: Degrees of freedom.

Heat Capacities of Gases

1. Molar Heat Capacity at Constant Volume ($C_V$):

   $$C_V = \frac{f}{2} R$$

2. Molar Heat Capacity at Constant Pressure ($C_P$):

   $$C_P = \frac{f + 2}{2} R$$

3. Relation Between $C_P$ and $C_V$:

   $$C_P - C_V = R$$

4. Adiabatic Index ($\gamma$):

   $$\gamma = \frac{C_P}{C_V}$$

Applications of Kinetic Theory

Explaining Gas Laws:

Provides a molecular basis for Boyle’s law, Charles’s law, and Avogadro’s law.

Speed and Energy Distribution:

Useful in studying chemical reactions and thermodynamic processes.

Atmospheric Science:

Describes the behavior of gases in the atmosphere.

Engineering:

Applies to the design of engines, turbines, and refrigeration systems.

Limitations of KTG

1. Ignores intermolecular forces (not accurate for high-pressure gases).
2. Assumes point-sized particles, which is unrealistic for real gases.
3. Fails to account for quantum effects in extreme conditions.

FAQs About KTG

What is the significance of KTG?

KTG provides a microscopic explanation for the macroscopic behavior of gases, bridging thermodynamics and molecular physics.

How does temperature relate to kinetic energy?

Temperature is directly proportional to the average kinetic energy of gas molecules.

What is the Maxwell-Boltzmann distribution?

It describes the probability of finding gas molecules with specific speeds at a given temperature.

Why is $v_{\text{rms}}$ greater than $v_{\text{avg}}$ and $v_{\text{mp}}$?

Because $v_{\text{rms}}$ includes the square root of the mean of squared speeds, giving more weight to higher speeds.

The Kinetic Theory of Gases (KTG) is a fundamental topic that provides a deep understanding of the behavior of gases at the molecular level. By mastering its principles, assumptions, and formulas, students can confidently tackle problems in thermodynamics, fluid mechanics, and beyond. These comprehensive notes serve as a valuable resource for academic and competitive exam preparation.

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