5.6 Ideal Gas Equation

• When calculating for gases, assume that the gas is an ideal gas
• The three gas laws (explained below) can be combined to create one equation in terms of pressure, volume, temperature and amount of gas. 

The Boltzmann Constant, k

• The Boltzmann constant k is used in the ideal gas equation and is defined as:

• Where:
  • R = molar gas constant
  • N_A = Avogadro’s constant

• Boltzmann’s constant therefore has a value of

• The Boltzmann constant relates the properties of microscopic particles (e.g. kinetic energy of gas molecules) to their macroscopic properties (e.g. temperature)
  • This is why the units are J K^-1

• Its value is very small because the increase in kinetic energy of a molecule is very small for every incremental increase in temperature

The Gas Laws

• The ideal gas laws are the experimental relationships between pressure (P), volume (V) and temperature (T) of an ideal gas
• The mass and the number of molecules of the gas is assumed to be constant for each of these quantities

Boyle’s Law

• If the temperature T of an ideal gas is constant, then Boyle’s Law is given by:

• This means the pressure is inversely proportional to the volume of a gas

Pressure increases when a gas is compressed

• The relationship between the pressure and volume for a fixed mass of gas at constant temperature can also be written as:

P₁V₁ = P₂V₂

• Where:
  • P₁ = initial pressure (Pa)
  • P₂ = final pressure (Pa)
  • V₁ = initial volume (m³)
  • V₂ = final volume (m³)

Charles's Law

• If the pressure P of an ideal gas is constant, then Charles’s law is given by:

V ∝ T

• This means the volume is proportional to the temperature of a gas
  • The relationship between the volume and thermodynamic temperature for a fixed mass of gas at constant pressure can also be written as:

• • Where:
    • V₁ = initial volume (m³)
    • V₂ = final volume (m³)
    • T₁ = initial temperature (K)
    • T₂ = final temperature (K)

Pressure Law

• If the volume V of an ideal gas is constant, the Pressure law is given by:

P ∝ T

• This means the pressure is proportional to the temperature
  • The relationship between the pressure and thermodynamic temperature for a fixed mass of gas at constant volume can also be written as:

• Where:
  • P₁ = initial pressure (Pa)
  • P₂ = final pressure (Pa)
  • T₁ = initial temperature (K)
  • T₂ = final temperature (K)

Step 1: Write down the ideal gas equation

Since the number of moles (n) is required, use the equation:

Step 2: Rearrange the equation for the number of moles n

Step 3: Substitute in values