A LevelPhysicsCambridge (CIE)Revision Notes15. Ideal GasesEquation of StateIdeal Gas Equation

Ideal Gas Equation (Cambridge (CIE) A Level Physics): Revision Note

Exam code: 9702

Written by: Ashika

Reviewed by: Caroline Carroll

Updated on 2 July 2026

Ideal gas equation

The equation of state for an ideal gas (or the ideal gas equation) can be expressed as:

$p V = n R T$

Where:
- p = pressure (Pa)
- V = volume (m³)
- n = amount of substance / number of moles (mol)
- R = molar gas constant (8.31 J K^-1 mol^-1)
- T = temperature (K)

The ideal gas equation can also be written in the form:

$p V = N k T$

Where:
- N = number of molecules
- k = Boltzmann constant (1.38 × 10^-23 J K^–1)
Remember that the number of moles and the number of molecules are related by Avogadro's constant

An ideal gas is therefore defined as:
A gas which obeys the equation of state pV = nRT at all pressures, volumes and temperatures

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 amount of gas in the cylinder, in moles.

Answer:

Step 1: Write down the ideal gas equation

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

$p V = n R T$

Step 2: Rearrange for the number of moles n

$n = \frac{p V}{R T}$

Step 3: Substitute in values

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

$T = 15 ° C + 273.15 = 288.15 K$

$n = \frac{(4.5 \times 10^{7}) \times (8.3 \times 10^{- 3})}{8.31 \times 288.15} = 155.98 = 160 moles$

Examiner Tips and Tricks

Don’t worry about remembering the values of R and k, they will both be given in the equation sheet in your exam.

The Boltzmann constant

The Boltzmann constant, k, is used in the ideal gas equation and is defined by the equation:

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

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 has the units J K^-1because
- it relates the properties of microscopic particles, e.g. kinetic energy of gas molecules
- to their macroscopic properties, e.g. temperature

The Boltzmann constant is very small because the increase in kinetic energy of a molecule is very small for every incremental increase in temperature

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