Physics 213: Lecture 4, Pg 1
213 Midterm coming up
213 Midterm coming up…
…
Monday Nov. 10, 7 pm (5:15pm)
Covers:
Lectures 1-7+ superficial 8
HW 1-4
Discussion 1-4
Labs 1-2
Review Session
Sun. Nov. 9, 3-5 PM Lincoln Hall Theater
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Physics 213: Lecture 4 - Particle Diffusion and Statistical Processes - Prof. Paul G. Kwia, Study notes of Physics
A part of the lecture notes for physics 213, covering topics such as particle diffusion, brownian motion, and statistical processes. It includes information on the random walk problem, mean free path, and the diffusion constant.
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Physics 213: Lecture 4, Pg 1
213 Midterm coming up
213 Midterm coming up
Monday Nov. 10,
7 pm (5:15pm)
Covers:
Lectures 1-7+ superficial 8HW 1-4Discussion 1-4Labs 1-
Review Session
Sun. Nov. 9, 3-5 PM Lincoln Hall Theater
Physics 213: Lecture 4, Pg 2
Particle Diffusion
Particle Diffusion
Counting and Probability
Counting and Probability
The meaning of equilibrium
The meaning of equilibrium
Two
Two
cell box and the concept of
cell box and the concept of
microstates
microstates
Other binomial systems: coin flip, spins
Other binomial systems: coin flip, spins
Statistical Processes Statistical Processes
Lecture 4
Reference for Lecture 4:
Elements
Ch 5
Reference for Lecture 5:
Elements
Ch 6
http://intro.chem.okstate.edu/1314F00/Laboratory/GLP.htm
Physics 213: Lecture 4, Pg 4
Random Walk Problem:
Diffusion of molecules
Random Walk Problem:
Diffusion of molecules
Picture can also apply to:
impurity atoms in an electronic device
defects in a crystalsound waves carrying heat in solid!
Gas molecules bounce aroundrandomly, colliding with othermolecules and the walls.
How far on average does a single molecule go in time t
http://intro.chem.okstate.edu/1314F00/Laboratory/GLP.htm
OKState Demo
Physics 213: Lecture 4, Pg 5
Rough, Simplified Picture of Particle Diffusion
Rough, Simplified Picture of Particle Diffusion
Assume the particles have average speed
v
As we indicated before, there will be a distribution of
speeds
.
Here,
v
represents an average.
A particle travels a distance
d
in a straight line, then scatters
off another particle and travels in a new, random
direction. l
Each step takes an average time:
ℓ v
How far does the particle get after a time t?
After a
time t
, how many steps will the particle have taken?
τ
=
t
M
:
nswer
A
Simpler question:
Mean free path
= average
distance between collisions
ℓ
Physics 213: Lecture 4, Pg 7
Random Walk in 1
Random Walk in 1
D
D
(a slightly simpler case)(a slightly simpler case)
The net displacement
after
M steps
is:
=
M
1
i
i s x 0 s x
M
1
i
i
=
After many such walks with M steps, the mean displacement
is :
(by symmetry)
The mean square
displacement is:
The
mean square
grows
linearly
with
M, the number of steps
Width
of the distribution:
2
1/ 2
(1-D, assumes fixed
,
for steps)
t
x
rms
x
x
x
x
M
τ
τ
ℓ
Width grows
as the square root of the number of steps
For each step s
i
(random sign)
x
ℓ
x x ℓ x ℓ −
(Assume
= constant)
x
2
x
j
i
2 i
j
i
2
M ) s s ( ) s ( ) s ( ) s ( x
∑
∑
∑
∑
Physics 213: Lecture 4, Pg 8
Width
Width
and standard deviation (
and standard deviation (
s.d
s.d
10 steps, 10 steps,
n n
L L
= # steps left= # steps left
+/-
s.d
.
0
2
4
6
8
10
0.30 0.25 0.20 0.15 0.10 0.05 0.
Probability (n
L
steps left, out of N total)
n
L
Physics 213: Lecture 4, Pg 10
The diffusion constant
The diffusion constant
2
x
t
D
For a particle randomly scattering in 3 dimensions
, the mean
square displacement along the x direction is:
with the definition*:
where
= mean free path
and
v = average particle speed
ℓ
The x-width (s.d.) of the
distribution expands as t
1/
.
Dt
x
x
rms
2
2
=
<
=
2
2
2
2
r
x
y
z
6
t
D
<
= <
<
- <
=
Radial expansion:
2
1 3
D
v
3
=
=
τ
ℓ
ℓ
(3-dimensions with average
scattering length,
)
ℓ
For a particle randomly scattering in 3 dimensions
, the mean
square displacement along
any
direction is:
(The particle has more possibilities to diffuse if we consider x, y, z.)
- The numerical coefficients in general depend on the
distribution
of
distances and time intervals. For Phys. 213 we’ll use the form above.
Physics 213: Lecture 4, Pg 11
Consider impurity atoms diffusing from the surface of analuminum film into an interface with a semiconductor.
Exercise 1: Impurity diffusion in semiconductors
Exercise 1: Impurity diffusion in semiconductors
- If the Al is 10
m thick, approximately how long will
it take before many impurities have diffused through it?
- Approximately what is the impurity diffusion constant,
D
Al Si
Assume that each impurity makes a random step
of
m about once every 10 seconds.
ℓ
x
Physics 213: Lecture 4, Pg 13
Act 1
Act 1
If we make the dimensions of the device twice as big, how
much longer will it last?
a) x ½b) x 0.71c) x 1.41d) x 2e) x 4
Physics 213: Lecture 4, Pg 14
Act 1
Act 1
1: Solution
1: Solution
If we make the dimensions of the device twice as big, how
much longer will it last?
a) x ½b) x 0.71c) x 1.41d) x 2e) x 4
The time to diffuse varies as the
square
of the thickness L: L
2
/D.
Physics 213: Lecture 4, Pg 16
Exercise 2: Isotope separation--Solution
You have the task of separating two isotopes of Uranium:
235
U and
238
U. Your lab partner suggests the following:
Put a gas containing both of them at one end of a longtube containing a gas through which they will diffuse.Which will get to the far end first?
(A)
235
U
(B)
238
U
(C)
Neither
The diffusion time t ~
ℓ
2
/3D.
From equipartition:Therefore t ~ 1/D ~ 1/v ~
√
m
the heavier isotope takes slightly longer.
(This is the technique first used in the Manhattan Project. It was thenfound that centrifuges speed up the process.)
1 3
D
v
=
ℓ
2
1
3
mv =
kT
v= 3kT/m
2
2
⇒
Physics 213: Lecture 4, Pg 17
ACT 1-2: Lifetime of batteries
Batteries can lose their charge when the separated
chemicals (ions) within them diffuse together. If youwant to preserve the life of the batteries when youaren’t using them, you should…
(A)
Refrigerate them
(B)
Slightly heat them
Physics 213: Lecture 4, Pg 19
FYI: Batteries
FYI: Batteries
Does putting batteries in the freezer or refrigerator make
them last longer?
It depends on which type of batteries and at whattemperature you normally store them.
Alkaline batteries stored at ~20˚C (room temp) dischargeat about 2%/year. However, at 38˚C (100˚F) the rateincreases to 25%/year.
NiMH and Nicad batteries, start to lose power when storedfor only a few days at room temperature. But they willretain a 90% charge for several months if you keep themin the freezer after they are fully charged.
If you do decide
to store your charged NiMH cells in the freezer orrefrigerator, make sure you keep them in tightly sealedbags so they stay dry. And you should also let themreturn to room temperature before using them.
Physics 213: Lecture 1, Pg 20
v
v = 0
End with v = 0
mgh = U
therm
½ mv
2
= U
therm
Thermal energy in block
converted to c.o.m. KE
Thermal energy
Kinetic energy
Potential energy
Have you ever seen this happen?
(when you weren’t asleep or on medication)
h = U
therm
/mg
U
therm
0
Which stage never happens?