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