1
Phys. 369, Fall 2009
Homework #9
Solutions
1. (14 points) a) Equation (7) of Einstein’s paper gives the diffusion coefficient for spheres of radius
P
diffusing through a fluid of viscosity
k.
As written by Einstein, it reads
1
6
RT
D
N
kP
π
=
.
(1)
Reading back through the paper, we see that
R
is the gas constant,
T
the temperature,
N
Avogadro’s
number,
k
the viscosity of the fluid, and
P
the radius of the particle. Translating into the notation of
Tipler, this equation reads
1
6
A
RT
D
N
a
πη
=
(
viscosity
η
=
and
sphere radius.
a
=
)
(2)
b) The probability distribution function is given by eq. (10) of Einstein’s paper
2
4
1
( , )
4
x
Dt
e
f x t
D
t
π
−
=
(3)
Here, I have set the number of particles to
1,
n
=
so that the distribution function is normalized to 1 rather
than to the number of particles
n
as in Einstein’s paper.
_____________________________________________________________________________
You will receive full credit for your solution to this part if you just copy down this result from Einstein’s
paper as shown above. However, you should study and understand the following solution to the diffusion
equation.
The problem we are trying to solve is the following. We suppose that the position of a particle has been
accurately measured at time
0,
t
=
and we set the origin of the spatial coordinate,
0
x
=
, to be equal to this
measured position. Therefore the distribution function must be
( ,0)
( )
f
x
x
δ
=
at time
0.
t
=
For later
times the particle distribution function
( , )
f
x t
must obey the diffusion equation
2
2
f
f
D
t
x
∂
∂
=
∂
∂
(4)
As time evolves, the particle is buffeted by many collisions and therefore undergoes a random walk
process. It follows that the distribution function at later times
t
must be a normalized Gaussian, and that
the standard deviation
σ
of the distribution should grow as the square root of the time. (See the solution to
problem 3.) So, we can say that
f
must be of the form
2
2
2
2
2
( )
2
1/2
1
1
( , )
2
( )
2
x
x
t
A t
f x t
e
e
t
At
σ
πσ
π
−
−
=
=
(5)

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*