Physics XI - Chapter 13: Kinetic Theory
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- Kinetic Theory: Explains macroscopic properties of matter using molecular motion.
- Basic Assumptions: Large number of molecules, random motion, elastic collisions, negligible volume.
- Ideal Gas: Perfectly obeys gas laws; no intermolecular forces except during collisions.
- Real Gas: Deviates from ideal behavior at high pressure and low temperature.
- Pressure Formula: P = (1/3)ρc² = (1/3)(mn)c² = (1/3)(M/V)c².
- RMS Speed: c_rms = √(3P/ρ) = √(3RT/M) = √(3kT/m).
- Average Speed: c_avg = √(8RT/πM) = √(8kT/πm).
- Most Probable Speed: c_mp = √(2RT/M) = √(2kT/m).
- Speed Ratio: c_rms : c_avg : c_mp = √3 : √(8/π) : √2 ≈ 1.73 : 1.60 : 1.41.
- Kinetic Energy: Average KE per molecule = (3/2)kT (for monatomic gas).
- Temperature: Measure of average kinetic energy of molecules.
- Boltzmann Constant: k = R/N_A = 1.38 × 10⁻²³ J/K.
- Avogadro's Number: N_A = 6.022 × 10²³ molecules/mol.
- Gas Constant: R = 8.314 J/mol·K = N_A × k.
- Mean Free Path: Average distance between collisions; λ = 1/(√2 πd²n).
- Collision Frequency: Number of collisions per second; z = √2 πd²n c_avg.
- Molecular Speeds: Maxwell-Boltzmann distribution describes speed distribution.
- Degrees of Freedom: Independent ways molecules can store energy.
- Monatomic Gas: 3 translational degrees (f=3); C_v = (3/2)R, C_p = (5/2)R.
- Diatomic Gas: 5 degrees (3 trans + 2 rot); C_v = (5/2)R, C_p = (7/2)R.
- Triatomic Gas: 6 degrees (3 trans + 3 rot); C_v = 3R, C_p = 4R.
- Law of Equipartition: Energy (1/2)kT per degree of freedom per molecule.
- Specific Heat Ratio: γ = C_p/C_v = 1 + 2/f.
- Brownian Motion: Random motion of particles in fluid due to molecular collisions.
- Van der Waals Equation: (P + a/V²)(V - b) = RT (correction for real gases).
- Boyle Temperature: Temperature where real gas behaves most ideally.
- Vapor Pressure: Pressure exerted by vapor in equilibrium with liquid.
- Critical Temperature: Maximum temperature for gas liquefaction by pressure alone.
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Chapter Summary
Kinetic Theory bridges the microscopic world of molecules with the macroscopic world we observe daily, providing a beautiful explanation of why gases behave the way they do. This chapter takes us deep into the molecular realm, revealing how the chaotic dance of countless invisible particles gives rise to the pressure, temperature, and other properties we measure in the laboratory.
We begin with the fundamental postulates of kinetic theory - that gases consist of numerous molecules in random motion, undergoing perfectly elastic collisions, with negligible volume compared to the container. These simple assumptions lead us to derive the profound relationship between pressure and molecular motion: P = (1/3)ρc², connecting the macroscopic pressure we measure with the microscopic molecular speeds.
The concept of temperature takes on new meaning as we discover it's directly proportional to the average kinetic energy of molecules. The law of equipartition of energy reveals how thermal energy distributes equally among all available degrees of freedom - (1/2)kT for each translational, rotational, and vibrational mode. This explains why different gases have different specific heats and why the ratio γ = C_p/C_v varies with molecular complexity.
We explore the Maxwell-Boltzmann distribution of molecular speeds, understanding why in any gas sample, molecules move with a range of speeds described by three characteristic values: most probable, average, and root mean square speeds. The concepts of mean free path and collision frequency help us understand gas transport properties like diffusion and viscosity.
Finally, we examine how real gases deviate from ideal behavior through the van der Waals equation, accounting for finite molecular size and intermolecular attractions. From the random walk of Brownian motion to the critical phenomena of phase transitions, kinetic theory provides the microscopic foundation for understanding the thermal world around us, connecting atomic-scale events with everyday observations in a truly remarkable way.