Ideal Gas Equation (Cambridge International A-Level Physics)

Ideal gas equation

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

$pV=nRT$

• 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:

$pV=NkT$

• 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°C 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:

$pV=nRT$

Step 2: Rearrange for the number of moles n

$n=\frac{pV}{RT}$

Step 3: Substitute in values

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

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

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

Exam Hint 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

Spec point — 15.2.3

• 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:

$k=\frac{8.31}{6.02\times10^{23}}=1.38\times10^{-23}JK^{-1}$

• The Boltzmann constant has the units J K^-1 because

  • 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