Kinetic Theory Of Gases (Copy)

1. Basic Assumptions of the Kinetic Theory of Gases

The kinetic theory of gases provides a microscopic model to explain the macroscopic behaviour of an ideal gas.

Assumptions:

1. A gas consists of a large number of molecules that are in random, continuous motion.
2. The volume of individual gas molecules is negligible compared to the total volume occupied by the gas.
3. All collisions between gas molecules and between molecules and the container walls are perfectly elastic.
4. The time spent in collisions is negligible compared to the time between collisions.
5. No intermolecular forces act between the gas molecules, except during collisions.
6. Newton’s laws of motion apply to molecular interactions.
7. The molecules move in straight lines between collisions.

These assumptions apply to ideal gases and help derive the relationships connecting pressure, volume, and molecular motion.

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

2. Pressure in Terms of Molecular Motion

How Pressure Is Caused:

• Pressure in a gas arises due to collisions of molecules with the walls of the container.
• Each collision exerts a tiny force on the wall; the combined effect of many collisions leads to macroscopic pressure.
• Molecules collide more often and with greater momentum at higher temperatures or in a smaller volume.

Derivation of the Kinetic Theory Equation

We derive the pressure formula by considering a cubic box of length L with N identical molecules of mass m, moving randomly in three dimensions.

One-dimensional model (x-direction):

• For one molecule moving with velocity cₓ, change in momentum upon collision with wall = –2mcₓ
• Time between collisions with same wall = 2L / cₓ
• Number of collisions per second = cₓ / (2L)
• Force from one molecule on wall:
  F = Δp / Δt = (2mcₓ) × (cₓ / 2L) = mcₓ² / L
• For N molecules:
  Total force ∝ sum of cₓ² over all molecules
• Average of squared speed in x-direction:
  <cₓ²> = (1/N) Σ cₓ²

Final Result in 3D:

• In 3D, by symmetry:
  <c²> = <cₓ²> + <cᵧ²> + <c_z²> = 3<cₓ²>

  So,
  <cₓ²> = (1/3)<c²>

• Substituting into pressure formula:

  p = (Nm<cₓ²>) / V = (1/3) Nm<c²> / V

• Multiply both sides by V:

  pV = (1/3) Nm<c²>

  where:
  p = pressure (Pa)
  V = volume (m³)
  N = number of molecules
  m = mass of one molecule (kg)
  <c²> = mean square speed (m²/s²)

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

3. Root Mean Square (r.m.s.) Speed

• The root mean square speed (cᵣₘₛ) is defined as:

  cᵣₘₛ = √<c²>

• This is the square root of the mean of the squares of the molecular speeds.
• It is not the same as the average speed, and it’s more representative of a molecule’s energy.

4. Comparing pV = (1/3) Nm<c²> with pV = NkT

We now compare the kinetic theory result:

pV = (1/3) Nm<c²>

with the ideal gas equation:

pV = NkT

Equating both expressions:

(1/3) Nm<c²> = NkT

Cancel N on both sides:

(1/3) m<c²> = kT

Rearranging:

(1/2) m<c²> = (3/2) kT

Average Kinetic Energy per Molecule

• The average translational kinetic energy of a gas molecule is:

  Eₖ = (1/2) m<c²> = (3/2) kT

  where:
  Eₖ = average kinetic energy per molecule (J)
  m = mass of one molecule (kg)
  <c²> = mean square speed
  k = Boltzmann constant = 1.38 × 10⁻²³ J/K
  T = absolute temperature (K)

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Key Interpretations

• Kinetic energy is directly proportional to temperature:
  • If temperature doubles, average kinetic energy doubles.
• This provides a molecular-level explanation for temperature.
• Also proves that:
  • All ideal gases at the same temperature have molecules with the same average kinetic energy, regardless of mass.

Summary of Key Equations

Conceptual Comparisons