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A LevelPhysicsCIERevision Notes15. Ideal Gases15.3 Kinetic Theory of GasesAverage Kinetic Energy of a Molecule

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First exams 2025

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Average Kinetic Energy of a Molecule (CIE A Level Physics)

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Ashika

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Ashika

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Average kinetic energy of a molecule

  • An important property of molecules in a gas is their average translational kinetic energy
  • For a gas, translational kinetic energy is defined as:

The kinetic energy of the molecules as determined by their temperature

  • The average translational kinetic energy of one particle in a gas can be deduced from the ideal gas equations relating pressure, volume, temperature and speed

Deducing the equation for translational kinetic energy

p V thin space equals space N k T

p V thin space equals space 1 third N m less than c squared greater than

  • The left-hand sides of the equations are equal to pV, therefore, they can be equated:

1 third N m less than c squared greater than space equals space N k T

  • Simplify the equation by:
    • dividing both sides by N
    • multiplying both sides by 3

m less than c squared greater than space equals space 3 k T

kinetic energy = 1 half m v squared

  • In translational kinetic energy instead of v2 for the velocity of one particle, <c2> is the average speed of all molecules
  • Multiplying both sides by ½ gives the equation for the average translational kinetic energy of one molecule of an ideal gas:

E subscript k space equals space 1 half m less than c squared greater than space equals space 3 over 2 k T

  • Where:
    • EK = kinetic energy of a molecule (J)
    • m = mass of one molecule (kg)
    • <c2> = mean square speed of a molecule (m2 s-2)
    • k = Boltzmann constant
    • T = temperature of the gas (K)

  • To calculate the average translational kinetic energy for all the molecules in a gas, multiply the kinetic energy by the number of molecules in the gas, N:

E subscript k minus t o t a l end subscript equals space 3 over 2 N k T

Worked example

Helium can be treated like an ideal gas. Helium molecules have a root-mean-square (r.m.s) speed of 730 m s-1 at a temperature of 45 oC.

Calculate the r.m.s speed of the molecules at a temperature of 80 oC.

Answer:

Step 1: Write down the equation for the average translational kinetic energy

E subscript k space equals space 1 half m less than c squared greater than space equals space 3 over 2 k T

Step 2: Find the relation between cr.m.s and temperature T

  • Since m and k are constant, <c2> is directly proportional to T

less than c squared greater than space proportional to space T

  • Therefore, the relation between cr.m.s and T is:

c subscript r. m. s end subscript space equals space square root of less than c squared greater than end root space proportional to space square root of T

Step 3: Write the equation in full

c subscript r. m. s end subscript space equals space a square root of T

  • where a is the constant of proportionality

Step 4: Calculate the constant of proportionality from values given by rearranging for a

T = 45 oC + 273.15 = 318.15 K

a space equals space fraction numerator c subscript r. m. s end subscript space over denominator square root of T end fraction space equals fraction numerator space 730 over denominator square root of 318.15 end root end fraction space equals space 40.92...

Step 5: Calculate cr.m.s  at 80 oC by substituting the value of a and new value of T

T space equals space 80 degree straight C space plus space 273.15 space equals space 353.15 space straight K

c subscript r. m. s end subscript space equals space fraction numerator 730 over denominator square root of 318.15 end root end fraction space cross times space square root of 353.15 end root space equals space 769 space equals space 770 space straight m space straight s to the power of negative 1 end exponent

Examiner Tip

Keep in mind this particular equation for kinetic energy is only for one molecule in the gas. If you want to find the kinetic energy for all the molecules, remember to multiply by N, the total number of molecules.You can remember the equation through the rhyme ‘Average K.E is three-halves kT’.

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    Ashika

    Author: Ashika

    Expertise: Physics Project Lead

    Ashika graduated with a first-class Physics degree from Manchester University and, having worked as a software engineer, focused on Physics education, creating engaging content to help students across all levels. Now an experienced GCSE and A Level Physics and Maths tutor, Ashika helps to grow and improve our Physics resources.