Equation Of State (Copy)

What Is an Ideal Gas?

• An ideal gas is a theoretical model of a gas whose behavior follows a simple relationship between pressure, volume, and temperature.
• Assumes particles:
  • Have negligible volume
  • Experience no intermolecular forces
  • Undergo perfectly elastic collisions
  • Move in random motion
• Real gases behave approximately like ideal gases at:
  • Low pressure
  • High temperature

Basic Relationship

• For an ideal gas, the relationship between pressure (p), volume (V), and temperature (T) is:

  pV ∝ T (at constant amount of gas)

• That is, pressure × volume is directly proportional to thermodynamic temperature (T in kelvin).
• This relationship becomes an equation when proportionality is replaced by constants.

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

Ideal Gas Equation: pV = nRT

• The equation of state for an ideal gas is:

  pV = nRT

  where:
  p = pressure (Pa)
  V = volume (m³)
  n = number of moles (mol)
  R = molar gas constant = 8.314 J/mol·K
  T = thermodynamic temperature (K)

Units Must Be SI:

Alternate Form: pV = NkT

• You can also express the ideal gas law in terms of the number of particles (N) rather than moles (n):

  pV = NkT

  where:
  N = total number of molecules
  k = Boltzmann constant = 1.38 × 10⁻²³ J/K

Relationship Between k and R

• Since n = N / Nₐ and R is the molar constant, while k is per particle:

  k = R / Nₐ

  where:
  R = molar gas constant = 8.314 J/mol·K
  Nₐ = Avogadro’s constant = 6.022 × 10²³ mol⁻¹
  ⇒ k = 1.38 × 10⁻²³ J/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

Example Conversions and Usage

Example 1:

A gas has volume 0.02 m³, pressure 1.5 × 10⁵ Pa, and temperature 300 K. How many moles of gas are present?

Use pV = nRT
n = pV / RT = (1.5 × 10⁵ × 0.02) / (8.314 × 300)
n ≈ 1.20 mol

Example 2:

A container has 5.0 × 10²³ molecules of gas at 400 K in a volume of 0.01 m³. What is the pressure?

Use pV = NkT
p = NkT / V = (5.0 × 10²³ × 1.38 × 10⁻²³ × 400) / 0.01
p ≈ 2.76 × 10⁵ Pa

Ideal Gas Equation Variations

Real vs Ideal Gases

• At high pressure or low temperature, real gases deviate from ideal gas law.

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

Summary Table