A LevelPhysicsCIERevision Notes15. Ideal Gases15.2 Equation of StateIdeal Gas Equation

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First exams 2025

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Ideal Gas Equation (CIE A Level Physics)

Revision Note

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Ideal gas equation

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

p V space equals space 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 space equals space 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 15oC 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 space equals space n R T

Step 2: Rearrange for the number of moles n

n space equals fraction numerator space p V over denominator R T end fraction

Step 3: Substitute in values

V space equals space 8.3 space cross times space 10 cubed space cm cubed space equals space open parentheses 8.3 space cross times space 10 cubed close parentheses space cross times space 10 to the power of negative 6 end exponent space equals space 8.3 space cross times space 10 to the power of negative 3 end exponent space straight m cubed space

T space equals space 15 degree straight C space plus thin space 273.15 space equals space 288.15 space straight K

n space equals space fraction numerator open parentheses 4.5 space cross times space 10 to the power of 7 close parentheses space cross times space open parentheses 8.3 space cross times space 10 to the power of negative 3 end exponent close parentheses space over denominator 8.31 space cross times space 288.15 end fraction space equals space 155.98 space equals space 160 space moles

Examiner Tip

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 space equals space fraction numerator space R over denominator N subscript A end fraction

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

  • Boltzmann’s constant therefore has a value of:

k space equals space fraction numerator 8.31 space over denominator 6.02 space cross times space 10 to the power of 23 end fraction space equals space 1.38 space cross times space 10 to the power of negative 23 end exponent space straight J space straight K to the power of negative 1 end exponent

  • 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

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