Download Boltzmann Limit Translational Energy QM States

School of Aerospace Engineering
Boltzmann Limit
•
•
When can we assume that the Boltzmann limit is “accurate”
(i.e., gi >> Ni or many more available quantum states then
particles)?
Number of particles
P.G. @ SATP ~ 1019 molecules/cm3 (from pV=NkT )
–
•
Degeneracies of different energy modes (motions)
1) electronic: magnetic quantum number (g=2l+1) and spin
(g=2s+1)
– gel < O(10)
2) vibration: harmonic oscillator has no degeneracy
– gvib = 1
3) rotation: rigid rotor (2J+1)
– grot < O(100)
•
So how can gi >> Ni ?
–
translational energy?
AE/ME 6765
Boltzmann Limit-1
Copyright © 2009 by Jerry M. Seitzman. All rights reserved.
School of Aerospace Engineering
Translational Energy QM States
• Consider particle in a “box” (cube)
– recall Q.M. solution of Sch. eq’n. for (kinetic)
energy of single translating particle
h2 1
L
ε x ,nx = nx2
or nx = 8mε x ,nx
2
h
8m L
L =1 cm
• At SATP for air in a “small” box (1cm), how many QM
states have energy less than the average KE of a molecule
in x-direction (=1/2 k T… will show this later)
⇒ nx , KEavg ,x ≈ 2 mkT
nx ,avgKE ≈ 4.2 ×108
L
h
J
Boltzmann's Constant
K
h = 6.626 × 10 −34 J s
TSATP = 298K , mN 2 ≅ 4.7 ×10 − 26 kg
k = 1.38 ×10 − 23
• But this was just 1-D of cube
# States with KE < KEavg = nx , KEavg n y , KEavg nz , KEavg = nx3, KEavg ≈ 7 × 10 25
Boltzmann Limit-2
Copyright © 2009 by Jerry M. Seitzman. All rights reserved.
AE/ME 6765
1
School of Aerospace Engineering
Boltzmann Limit Validity
• So in our 1 cm box, if all our molecules were to be found in
just these states
# States " accessible" ≈ 10 26
1019/cm3
– actual number must be greater since some molecules
must be in even higher energy states for the average
to be correct
• Number of particles in 1 cm box is
L =1 cm
~1019
# QM States that could be populated 10 26
> 19 = 107
# Particles
10
⇒each state is sparsely populated ;
Boltzmann limit usually
reasonable assumption
unless densities are very
high (or T’s very low)
• For translations, we could “lump” energy states having
energies between εi and Δεi into one energy level (large box)
with approximately same εi and degeneracy gi Extended
Degeneracies
Boltzmann Limit-3
Copyright © 2009 by Jerry M. Seitzman. All rights reserved.
AE/ME 6765
2