# 5.8 Average Molecular Kinetic Energy

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

Ek Nm(c)2 NkT

• 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 • Substituting this into the average molecular kinetic energy equation means it can also be written as: #### 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.  #### Exam Tip

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

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