Macroscopic versus microscopic
Fick’s laws describe time
evolution macroscopically
Physical Chemistry

Lecture 2
Random walks; microscopic
theory of diffusion


Diffusion can be understood
in terms of a microscopic
process – the random walk



Probability of event



Each event is considered
equally likely unless other
information is known
Can only discuss likelihood
of an event, p
n = number of possible
results
Only applies to a “large
sample”
Examples


Coin flips (n=2)
Throws of dice (n=6)
For events that are not
equally likely, one must
know (by some external
means) the probability, p, of
each event
p 
Uses probability concepts
Implies randomness in a
system with many
molecules
Time development occurs
in a “natural” way
Random motion of a single molecule is the
underlying driving force described by diffusion.
Probability of a sequence
Random events

Do not specifically consider
a single molecule (although
we discussed them in that
manner)
Focus on the tendency to
eliminate density gradients
Can be explained by ad hoc
appeal to kinetic theory
1
n
Probability of a sequence,
P, is product of the
probabilities of the
elementary events that
make up each sequence
When the probabilities of
the elementary events are
the same, this equation
simplifies
Example: tossing a coin
four times or tossing four
coins at the same time
 Since ph = pt = ½, each
P 

p1 p2 p3 p4 
pn
sequence has a
likelihood of happening
P = (½)4

16 possible sequences,
so the probability of any
one sequence is 1/16 (=
(½)4)
1
Probability of a configuration
P' 
 Pi
s
e
c
n
e
u
q
e
s
e
t
a
i
r
p
o
r
p
p
a
l
l
a
r
e
v
o
Often concerned only ︵
with the number of
elementary outcomes in  nappropriate sequences P
a sequence, not order
(if Pi  P for all sequences)
Example of tossing a
coin: probability of
getting 3 heads when
tossing a coin four
times


︶
Four different sequences
have three heads
P ‘ = 4(1/6) = 1/4
One-dimensional random walk
Particle hops from site
to site, starting at zero

Only one step per hop
Probability of hopping in
either direction is ½ for
each step
Determine probability
that, after m steps, the
particle is at position q
after m steps

Just like evaluating the
probability of a sequence
Numbers of configurations
Counting number of specific
configurations can be
difficult if there is a large
number of configurations
Need an expression for
number of sequences having
a certain property to use in
probability calculation
Mathematically equivalent to
counting the number of ways
of putting balls in boxes, 1
ball in each box
Example: 3 heads and 1 tails
 Nconfig = 4!/(3! 1!) = 4
nboxes !
nh !nt !
nh  nt !

nh ! nt !
N config (nh , nt ) 
Mathematics of random walks
Probability has two factors
P ' (q; m)  P N config
m
1
   N config (n, p)
2
Number of ways of having p positive and n
negative steps to end up at q after m steps
N config (n, p) 
m!

n! p!
m!
mq mq

!
!
 2  2 
These equations give the probability of being
at q after m steps in a random walk
2
Calculation of averages in a
one-dimensional random walk
Use the probability, P ’, to get averages of
functions of the distance in m steps
f (q ) 
m
1
 P' (q, m) f (q)   2 
q  m
Examples:
q
 0
q2
 m
m m
m!
 m  q m  q f (q )
qm
(
)!(
)!
2
2
Example random walk
Movement of He in a given time
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Technically only correct for P ' ( x, t ) dx
either even or odd q, but
we “smooth” the probability
over many steps
Gaussian function
Normalized probability
distribution function
Same form as the result of
solving Fick’s law
Equivalence defines the
average jump distance, x 0
2
q2
exp(
)
2m
2 m
 x2 
1
exp 
 dx
4Dt
 4 Dt 

Fick’s Law

x0


m

z t
1.6 cm
1 minute
12.1 cm
1 hour
93.9 cm
1 day
460 cm
1 week
1220 cm
In one minute, a molecule samples a reasonable
fraction of the environment in that flask.
Gaussian function
Occurs in many different experiments having
random processes
x2
1
P ( x,  ) 
exp( 2 )
2
2  2
Shape is determine by the standard deviation,


x  qx0
x2
x
1 second
Typical flask is of the order of 10 cm in
diameter.
Small-step-size, large-stepnumber random walk
P ' ( q, m) 
TIME
Distance moved in one direction
x 
The average position does not appear to
change with number of steps, but the square
of the distance traveled does.
It can be shown that P’ may
be approximated well by a
continuous function, when
there is a large number of
very small steps.
T = 298.15 K
P = 1 bar

Large , wide function
Small , narrow function
Random noise is gaussian
2 Dt
3
Three-dimensional diffusion
Assume diffusion in the three spatial
dimensions is uncorrelated
P ( x, y, z; t )dxdydz  P( x, t ) P( y, t ) P( z , t )dxdydz

 x2  y2  z2 
1
 dxdydz
exp 
3/ 2
4 Dt
4Dt 


In spherical co-ordinates it simplifies
and depends only on r
P ( r ,  ,  ; t ) d 
 r2  2
1
 r sin drdd
exp 
3/ 2
4Dt 
 4 Dt 
Diffusion in liquids
Average distance
traveled in a liquid
between collisions
much smaller than in a
dilute gas


Mean free path for gas
Molecular diameter for
liquid
Collisions happen on a
short time scale


Typically 1-10 ps
between collisions for
liquids
Typically 10-100 ps
between collisions for
gases
Average distance in 3D diffusion
Average over the 3D
distribution


Assumes isotropic
diffusion
Sum of three terms
r2
 x2  y2  z 2
 2 Dt  2 Dt  2 Dt
 6 Dt
 r 3D
Compare to 1D and 2D
diffusion

The average distance
traveled in a fixed time is
larger for movement in
higher-dimensional space

x 
 r 2D

r2
6 Dt
2 Dt

4 Dt
Stokes-Einstein equation
A moving particle in a fluid feels
a drag force resisting the
movement of the particle

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Proportional to the speed, with
a coefficient, f
A retarding force
Results from collisions with
molecules of the fluid
For a sphere, Stokes showed
the relationship of the
coefficient, f, to viscosity, ,
and the sphere’s radius, r
Einstein connected the
coefficient, f, to the meansquare displacement
The combination of the two
theoretical results gives the
Stokes-Einstein equation


Relates size and viscosity to the
diffusion coefficient
Knowing any two of D, , and r,
one can calculate the third
Fviscous
Fviscous
 f
f
 f
dx
dt
dx
dt
 6r
 x 2 (t )  
D
T

2kTt
f
k
6r
4
Stick, slip, and shape
Original SEE derived
under “stick” conditions

Appropriate for large
molecules in smallmolecule solvent
For small molecules,
can use the “slip”
condition

D 
Appropriate for small
molecules
For ellipsoidal
molecules, must correct
for shape by introducing
a factor, (Pe)
D 
kT
6 r
kT
4 r
Rotational diffusion
( stick )
( slip )
1 kT
( Pe) 6 re
1
kT

( Pe) 5.451 2 / 3V 1/ 3
D 
Rotational motion
opposed by frictional
forces
Treated similarly to
translational diffusion
Result for a sphere
experiencing stick
conditions
Translational and
rotational diffusion
coefficients have
different units
Drot

kT
8r 3
Dtrans

kT
6r
Summary
Use of probability theory leads to a prediction for
random movement of particles (Brownian motion)

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Apparent change in a system is the result of random events
at the microscopic level
No coherent external forces
Results of random-walk theory are equivalent to the
macroscopic theory (Fick’s laws) which postulate a
force related to concentration gradients
Connects macroscopic parameter to theory of the
microscopic structure and dynamics of matter

Stokes-Einstein relationship between diffusion coefficient
and averages over the random motion of particles
Rotational motion also hindered by viscous drag and
treated as diffusive
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