9.9 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(c_rms)² = 3kT

• Recall the familiar kinetic energy equation from mechanics:

• Instead of v² for the velocity of one particle, (c_rms)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:
  • E_k = kinetic energy of a molecule (J)
  • m = mass of one molecule (kg)
  •  (c_rms)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: 

E_k = 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

E_k ∝ 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: