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Chapter Twelve: Kinetic Theory

12.1 Introduction

• Historical Context: Boyle’s law (1661) and contributions by Newton, Maxwell, and Boltzmann.

• Kinetic Theory: Describes gases as collections of rapidly moving particles; effective in neglecting inter-atomic forces due to their insignificance in gases.

• Significance: Connects molecular motion with pressure, temperature, and properties of gases, consistent with gas laws and Avogadro’s hypothesis.

12.2 Molecular Nature of Matter

• Atomic Hypothesis: Proposed by Richard Feynman; asserts matter consists of atoms in perpetual movement, attracting at distance and repelling upon contact.

• Historical Perspectives: Concepts of atomism found in ancient Indian and Greek philosophies (Kanada and Democritus).

• John Dalton's Atom Theory:

  • Introduced to explain laws of definite and multiple proportions in compounds.

  • Key laws: fixed mass proportions in compounds, fixed ratios in compounds formed by the same elements.

• Gasses and Molecular Theory:

  • Atoms constitute molecules which make up matter; recent advancements like electron microscopy visualize these structures.

  • Atomic Size: Approximately 1 Å (10^-10 m).

12.3 Behaviour of Gases

• Ideal Gas Behaviour: Easily studied due to large inter-atomic distances and negligible interactions.

• Relationships and Laws:

  • At low pressures and high temperatures, gases follow: (PV = KT).

  • Boltzmann constant (k_B) is universal for all gases under fixed conditions, leading to Avogadro's hypothesis.

  • Gas Properties:

    • Ideal gas defined with equations: (PV = \mu RT) and individual contributions leading to Dalton's law of partial pressures.

12.4 Kinetic Theory of an Ideal Gas

• Gas Molecules: In constant random motion, only experiencing interactions during collisions.

• Pressure Derivation: Collision mechanics yield pressure relationships based on molecular number density.

• Kinetic Energy:

  • Average translational kinetic energy relationships with temperature established.

  • (PV = (2/3)E), connecting ideal gas behaviours.

• Temperature Interpretation:

  • The average kinetic energy is proportional to absolute temperature, independent of gas type.

12.5 Law of Equipartition of Energy

• Degrees of Freedom: Various degrees of freedom (translational, rotational, vibrational) contribute equally to the energy in thermal equilibrium.

• Average Energy Contributions:

  • Each translational mode contributes (1/2 k_B T).

  • Each vibrational mode contributes to total energy by 2 times the translational contribution.

12.6 Specific Heat Capacity

• Monatomic Gases: Specific heat capacity at constant volume (C_v = (3/2)R), at constant pressure (C_p = (5/2)R).

• Diatomic and Polyatomic Gases:

  • Increasing degrees of freedom lead to higher specific heat capacities.

12.7 Mean Free Path

• Definition: The average distance a molecule travels between collisions; impacted by density and size of molecules.

• Estimations: Provides insights into the behavior of gases and quantitative properties related to gas dynamics.

Summary

1. Ideal Gas Relation: (PV = \mu RT = k_B NT).

2. Kinetic Theory Insights: Relationships between pressure, volume, temperature, and internal energy established through molecular interpretations.

3. Energy Distribution: Law of equipartition allows understanding of heat capacities across gases.

4. Mean Free Path Insights: Essential for understanding collision dynamics in gases.

Points to Ponder

• Fluid Pressure Dynamics: Understanding pressure within fluids versus at walls.

• Molecular Motion Dynamics: Differences in interatomic distances and mean free paths between gases, solids, and liquids.