Ideal Gases (Copy)

A2 Level Physics – Section 15: Ideal Gases (Detailed Notes)

15.1 The Mole

1. Amount of Substance as SI Base Quantity

• Amount of substance is a fundamental SI base quantity.
• Unit: mole (mol)

2. Definition of Mole and Avogadro Constant

• 1 mole = amount of substance containing as many particles as there are atoms in 12 g of carbon-12.
• Avogadro constant (Nₐ) = 6.022 × 10²³ mol⁻¹
  • Number of atoms, molecules, or particles in 1 mole.

15.2 Equation of State for Ideal Gases

1. Ideal Gas Definition

• An ideal gas obeys:
  pV ∝ T
  • p = pressure (Pa)
  • V = volume (m³)
  • T = thermodynamic temperature (K)
  • Valid at low pressure and high temperature

2. Ideal Gas Equations

a. Molar form:
pV = nRT

• p = pressure (Pa)
• V = volume (m³)
• n = number of moles
• R = universal gas constant = 8.31 J/mol·K
• T = temperature (K)

b. Molecular form:
pV = NkT

• N = total number of molecules
• k = Boltzmann constant = 1.38 × 10⁻²³ J/K
• T = temperature (K)

3. Relationship Between k and R

• k = R / Nₐ
  • R = 8.31 J/mol·K
  • Nₐ = 6.022 × 10²³ mol⁻¹
  • k = 1.38 × 10⁻²³ J/K

15.3 Kinetic Theory of Gases

1. Assumptions of Kinetic Theory

1. Gas contains a large number of identical molecules.
2. Molecules are in random, rapid motion.
3. Molecules obey Newton’s laws of motion.
4. Collisions are elastic (no energy loss).
5. Volume of molecules ≪ volume of container.
6. No intermolecular forces except during collisions.
7. Time of collisions ≪ time between collisions.

2. Molecular Basis of Pressure and Derivation

• Pressure arises due to momentum transfer when molecules collide with container walls.
• Derivation (from Newton’s 2nd law, using a 1D model extended to 3D):

pV = (1/3)·N·m·<c²>

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

3. Root-Mean-Square Speed (cₙₘₛ)

• cₙₘₛ = √<c²>
  • Represents the effective speed of molecules in kinetic theory

4. Molecular Kinetic Energy and Temperature

Compare:

• pV = (1/3)·Nm·<c²>
• pV = NkT

From this, deduce:

(1/2)·m·<c²> = (3/2)·k·T

• Therefore, average translational kinetic energy of one molecule:

Eₖ(avg) = (3/2)·k·T

• Proportional to temperature (T in K)
• Applies to translational motion only

Key Units:

• Pressure (p): Pa
• Volume (V): m³
• Temperature (T): K
• Number of moles (n): mol
• Number of molecules (N): unitless
• Boltzmann constant (k): J/K
• Gas constant (R): J/mol·K
• Energy: J