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:

$k = \frac{R}{N_{A}}$

Where:
- R = molar gas constant
- N_A = Avogadro’s constant
Boltzmann’s constant therefore has a value of

$k = \frac{8.31}{6.02 \times 10^{23}} = 1.38 \times 10^{- 23} J K^{- 1}$

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:

Boyle's Law_2

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

Gas Volumes at Low Temperatures & High Pressures

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:

Charles's Law

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

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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:

Pressure Law

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

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Worked example

A storage cylinder of an ideal gas has a volume of 8.3 × 10³ cm³. The gas is at a temperature of 15^oC and a pressure of 4.5 × 10⁷ Pa. Calculate the number of molecules of gas in the cylinder.

Step 1: Write down the ideal gas equation

$p V = N k T$

Step 2: Rearrange the equation for the number of molecules, N

$N = \frac{p V}{k T}$

Step 3: Substitute in values

$V = 8.3 \times 10^{3} c m^{3} = (8.3 \times 10^{3}) \times 10^{- 6} = 8.3 \times 10^{- 3} m^{3}$

$T = 15 {}^{o}C + 273.15 = 288.15 K$

$N = \frac{(4.5 \times 10^{7}) \times (8.3 \times 10^{- 3})}{(1.38 \times 10^{- 23}) \times 288.15} = 9.4 \times 10^{25} molecules (2 s f)$

Examiner Tip

After you solve a problem using any of the gas laws (or all of them combined), always check whether your final result makes physical sense - e.g. if you are asked to calculate the final pressure of a fixed mass of gas being heated at constant volume, your result must be greater than the initial pressure given in the problem (since Gay- Lussac's law states that pressure and absolute temperature are directly proportional at constant volume).

Ideal Gas Equation (Edexcel International A Level Physics)

Revision Note