Edexcel International A Level Physics

Revision Notes

5.8 Average Molecular Kinetic Energy

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Average Molecular Kinetic Energy

  • An important property of molecules in a gas is their average kinetic energy
  • This can be deduced from the ideal gas equations relating pressure, volume, temperature and speed
  • Recall the ideal gas equation in terms of number of molecules:

pV = NkT

  • Also, recall the equation linking pressure and mean square speed of the molecules:

Kinetic Theory Final Equation_2

  • The left-hand side of both equations are equal to pV
  • This means the right-hand sides of both equations are also equal:

Equating Kinetic Energy Equations

  • N will cancel out on both sides and multiplying by 3 on both sides too obtains the equation:

m(crms)2 = 3kT

  • Recall the familiar kinetic energy equation from mechanics:

Average Kinetic Energy of a Molecule equation 3

  • Instead of v2 for the velocity of one particle, (crms)2 is the average speed of all molecules

  • Multiplying both sides of the equation by ½ obtains the average molecular kinetic energy of the molecules of an ideal gas:

Average Molecular Kinetic Energy Equation

  • Where:
    • Ek = kinetic energy of a molecule (J)
    • m = mass of one molecule (kg)
    •  (crms)2 = mean square speed of a molecule (m2 s-2)
    • k = Boltzmann constant
    • T = temperature of the gas (K)


  • Note: this is the average kinetic energy for only one molecule of the gas

  • To find the average kinetic energy for many molecules of the gas, multiply both sides of the equation by the number of molecules N to obtain: 

Ek1 halfNm(c)23 over 2NkT

  • A key feature of this equation is that the mean kinetic energy of an ideal gas molecule is proportional to its thermodynamic temperature

Ek ∝ T

  • The Boltzmann constant k can be replaced with

Boltzmann Constant Equation_2

  • Substituting this into the average molecular kinetic energy equation means it can also be written as:Average Kinetic Energy R NA Equation

Worked example

Helium can be treated as an ideal gas. Helium molecules have a root-mean-square (r.m.s.) speed of 720 m s-1 at a temperature of 45 °C. Calculate the r.m.s. speed of the molecules at a temperature of 80 °C.

Kinetic Energy Molecule Worked Example (1)Kinetic Energy Molecule Worked Example (2)_2

Exam Tip

You can remember the equation through the rhyme ‘Average K.E is three-halves kT’.

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