Average Translation Energy from

1
Fred J. Grieman
Average Translation Energy from Kinetic Theory
of Gases
http//www.elfyourself.com/?id1199330323
2
Kinetic Theory of Gases

• Small particles in large volume
• Constant, random motion
• No force between molecules except on collision
• Elastic collisions between molecules and walls
• (No energy transfer)

Ideal gas law PV nRT (N/No)RT
3

• Pressure Force (molecular collisions)
• Area (with walls)
• Force
  collision with wall changes momentum
• 
  m (mass) u
  (velocity)
• Imagine container c (z)
  Volume V abc
• b
  (y)
• a (x)

4
Consider 1 dimension x
mux wall area bc
A -mux a x
P F/A F ?mux/?t for one
collision ?mux mux (-mux) 2 mux ?t
distance btwn collisions/velocity
2a/ux
F 2 mux / (2a / ux ) mux2 / a For N
molecules with average average velocity ux F
N mux2/a P F/A N mux2/a /bc P N
mux2/V Velocity vector u uxi uyj uzk
u2 ux2 uy2 uz2
? ? ? ?
5
u2 ux2 uy2 uz2 ux uy uz So u2
3ux2 or ux2 ?u2 Finally, P N mux2/V
? N mu2/V
6
PV ? N mu2 Ek N(½mu2) PV ?Ek nRT
ideal gas law Ek (3/2) nRT ? Ek (per mole)
(3/2) RT For 1 molecule
ltegtt ?(RT)/NA ?(kBT) with kB R/NA
Then ltegtt ? ß-1 ?(kBT) So ß (kBT)-1
!!!!!!!!!! Put ß in everywhere nj (N/q) gj
e-ej / kBT q ?j gj e-ej / kBT qt
(2pmkBT/h2)?V
7
Energy E(per mole) NAltegt
NA-(1/q)(?q/?(1/kBT) d(kBT)-1 - (kB-1T-2 dT
) NAkBT2 (1/q)(?q/ T
RT2 ?lnq/?T Translational
Energy Et RT2 ?lnqt/?T RT2
?ln(2pmkBT/h2)?V/?TV RT2 ?lnT? /?TV
?ln(constants) /?TV RT2(?T-1) ?RT
?kBNAT as before