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HSSLIVE Plus One Physics Chapter 13: Kinetic Theory Notes

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Kinetic Theory connects macroscopic properties of gases to the microscopic motion of their constituent particles. Students explore how statistical mechanics explains pressure, temperature, and internal energy in terms of molecular motion and collisions. This chapter derives the ideal gas law from kinetic theory principles, introduces concepts of mean free path and equipartition of energy, and helps students understand molecular phenomena from gas diffusion to specific heat capacities.

Chapter 13: Kinetic Theory

1. Molecular Nature of Matter

The kinetic theory of matter is based on the idea that all matter consists of tiny particles (atoms or molecules) in constant, random motion. This theory successfully explains many macroscopic properties of matter in terms of microscopic molecular behavior.

Key postulates of kinetic theory include:

  • Matter consists of tiny particles (atoms/molecules)
  • These particles are in constant, random motion
  • Particles interact through collisions which are perfectly elastic
  • The average kinetic energy of particles is proportional to absolute temperature
  • Between collisions, particles move in straight lines

The state of matter (solid, liquid, or gas) depends on the strength of intermolecular forces relative to the kinetic energy of molecules. In solids, molecules vibrate about fixed positions; in liquids, they can move but remain close; in gases, they move freely throughout the container.

2. Behavior of Gases – Gas Laws

Experimental observations of gases led to several empirical laws that describe their behavior:

Boyle’s Law (1662): At constant temperature, the pressure of a given mass of gas is inversely proportional to its volume.

  • PV = constant (at constant T and mass)
  • This occurs because when volume decreases, molecules hit the walls more frequently, increasing pressure

Charles’ Law (1787): At constant pressure, the volume of a given mass of gas is directly proportional to its absolute temperature.

  • V ∝ T (at constant P and mass)
  • The increased temperature increases molecular speeds, resulting in greater expansion against the constant pressure

Gay-Lussac’s Law: At constant volume, the pressure of a given mass of gas is directly proportional to its absolute temperature.

  • P ∝ T (at constant V and mass)
  • Higher temperature increases molecular speeds and collision force, increasing pressure

Avogadro’s Law: Equal volumes of all gases under the same conditions of temperature and pressure contain the same number of molecules.

  • V ∝ n (at constant P and T)
  • Where n is the number of moles

3. Ideal Gas Equation

The combination of these gas laws leads to the ideal gas equation:

  • PV = nRT
  • Where R is the universal gas constant (8.314 J/mol·K)

Alternatively:

  • PV = NkT
  • Where N is the number of molecules, k is Boltzmann constant (1.38 × 10⁻²³ J/K)

The relationship between R and k is:

  • R = N_A × k
  • Where N_A is Avogadro’s number (6.022 × 10²³ molecules/mol)

An ideal gas perfectly follows the ideal gas equation. Real gases approach ideal behavior at high temperatures and low pressures, conditions under which intermolecular forces become negligible and molecular volume is insignificant compared to container volume.

4. Kinetic Theory of Gases – Assumptions

The kinetic theory makes several simplifying assumptions:

  1. Gas consists of a large number of molecules, each having mass m, moving randomly
  2. Volume of molecules is negligible compared to the container volume
  3. Molecules exert no forces on each other except during collisions (no intermolecular forces)
  4. Collisions between molecules are perfectly elastic
  5. Molecules obey Newton’s laws of motion
  6. The time of collision is negligible compared to time between collisions
  7. There is a distribution of molecular speeds (Maxwell-Boltzmann distribution)

These assumptions are most valid for monatomic gases at low pressures and high temperatures.

5. Pressure of Ideal Gas

According to kinetic theory, gas pressure results from molecular collisions with container walls. The mathematical derivation gives:

  • P = (1/3)ρ(c²)_avg = (1/3)nm(c²)_avg
  • Where ρ is mass density, n is number density, m is molecular mass, and (c²)_avg is mean squared velocity

This equation connects the microscopic property (molecular speed) with the macroscopic property (pressure).

6. Kinetic Interpretation of Temperature

Comparing the kinetic theory expression for pressure with the ideal gas equation:

  • (1/3)nm(c²)_avg = nkT
  • This yields: (1/2)m(c²)_avg = (3/2)kT

This means:

  • Average kinetic energy per molecule = (3/2)kT
  • Average kinetic energy per mole = (3/2)RT

The root mean square speed of gas molecules is:

  • c_rms = √(c²)_avg = √(3kT/m) = √(3RT/M)
  • Where M is molar mass

For example, at room temperature (300 K):

  • Hydrogen molecules move at about 1,920 m/s
  • Oxygen molecules move at about 480 m/s
  • This explains why lighter gases diffuse faster than heavier ones

7. Specific Heat Capacity of Gases

The equipartition theorem states that each degree of freedom contributes (1/2)kT energy per molecule or (1/2)RT per mole.

For monatomic gases (e.g., He, Ne, Ar):

  • 3 translational degrees of freedom
  • Energy per mole = (3/2)RT
  • Therefore, C_v = (3/2)R and C_p = (5/2)R
  • γ = C_p/C_v = 5/3 ≈ 1.67

For diatomic gases (e.g., H₂, O₂, N₂) at room temperature:

  • 3 translational + 2 rotational degrees of freedom
  • Energy per mole = (5/2)RT
  • Therefore, C_v = (5/2)R and C_p = (7/2)R
  • γ = C_p/C_v = 7/5 = 1.4

At high temperatures, vibrational modes become active, adding more degrees of freedom.

8. Mean Free Path

Mean free path (λ) is the average distance traveled by a molecule between successive collisions:

  • λ = 1/(√2πd²n)
  • Where d is molecular diameter and n is number density

For air at STP, λ ≈ 6 × 10⁻⁸ m, which is about 100 times the molecular diameter. As pressure decreases, mean free path increases, explaining why vacuum tubes work (electrons can travel farther without collisions).

The frequency of collisions:

  • f = v_avg/λ
  • Where v_avg is average molecular speed

This concept is important in understanding gas viscosity, thermal conductivity, and diffusion, all of which increase with temperature but are independent of pressure (as long as mean free path is much smaller than container dimensions).

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