Average Kinetic Energy of a Average Kinetic Energy of a MoleculeMolecule

Thermal Physics Lesson 7Thermal Physics Lesson 7

Learning Objectives Derive an expression to calculate the

average kinetic energy of a molecule.

Mean kinetic energy

Nccccm N )...( 22

322

212

1

Nccccc N

rms

223

22

212 ...

Nmcmcmcmc N

2212

3212

2212

121 ...

221

rmsmc

The mean kinetic energy of a molecule is the total kinetic energy of all the molecules/total number of molecules.

The root mean square speed is defined as:

…and so the mean kinetic energy of one molecule is gvien by…

Mean Kinetic EnergynRTpV 2

31

rmsNmcpV

2

31

rmsNmcnRT

2

31

rmsAmcnNnRT

2

31

rmsA

mcTNR

2

31

rmsmckT

Note we have two equations with pV on the left hand side.

(N=nNA)

The Boltzmann constant, k, is defined as R/NA.

Mean Kinetic Energy

kTmcrms 23

21 2

2

3123

23

rmsmckT

nRTmcrms 23

21 2

Multiply both sides by 3/2.

Mean kinetic energy of a molecule of an ideal gas.

Kinetic energy of one mole of gas(because R=NAk)

Kinetic energy of n moles of gas.

RTmcrms 23

21 2