Lecture 23 — The Maxwell Distribution Lecture 23 — The Maxwell Distribution Chapter 7, Monday March 3Chapter 7, Monday March 3rdrd
•Recap: density of states in k-space
•Single-particle density of states in energy
•Maxwell distribution of speeds in a classical gas
•Using the distribution function
Reading: Reading: Assigned problems, Assigned problems, Ch. 7Ch. 7: 6, 8, 12, 14, : 6, 8, 12, 14, 1616
Homework 7 due next FridayHomework 7 due next FridayExam 2 on Wed. after spring breakExam 2 on Wed. after spring breakReview – Monday after spring break (in Review – Monday after spring break (in
class)class)
LxLy
Lz
1 2 3sin sin sinix y z
n x n y n zA
L L L
2 2 2 2 21 2 32 2 22ix y z
n n n
m L L L
Particle (standing wave) in a boxParticle (standing wave) in a box
3/ 2
22B
x y z
mk TZ L L L
/i Bk T
i i
en Np N
Z
Boltzmann probability:
kkyy
kkxx
kkzz
Density of states in Density of states in kk-space-space
1
2
3
xx
yy
zz
nk
L
nk
L
nk
L
dVk
k
k
k
k kkyy
kkxx
kkzz
Convert to spherical coordinatesConvert to spherical coordinates
dk
2 2 2x y zk k k k
2 sinkdV k dk d d
In 1D and 2DIn 1D and 2D
k
kddk
k
dk
1D1D
2D2D
Maxwell speed distributionMaxwell speed distribution
23 3
/ 222 32
Bmu k TDmn u N u e
n(u)
u
The molecular speed distribution The molecular speed distribution functionfunction
1/ 2 1/ 2 1/ 22 8 3
m rms
kT kT kTu u u
m m m
3/ 2
2 24 exp / 22
mn u N u mu kT
kT
n u
N
/ mu u /rms mu u / mu u
