• 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 ? 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:

• 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: