Kinetic
Properties
(see Chapter 2 in Shaw, pp. 21

45)
•
Sedimentation and
Creaming: Stokes’ Law
•
Brownian Motion and
Diffusion
•
Osmotic Pressure
Next lecture
:
•
Experimental Methods
•
Centrifugal Sedimentation
(Chapter 2)
•
Light Scattering (Chapter 3)
r
1
r
2
F
g
F
b
F
v
g
g
F
g
F
g
g
F
)
(
1
2
1
2
r
r
r
r
V
V
V
m
net
b
g
r
2
>
r
1
sedimentation
r
2
<
r
1
creaming
dt
dx
g
dt
dx
g
dt
dx
F
f
m
or
f
V
f
v
2
1
1
2
1
)
(
r
r
r
r
Now we need to find an expression for
f
...
Gravitation and Sedimentation:
Stokes’ Law
•
Independent of shape
•
No solvation (which
changes the density)
dt
dx
Stokes’ Law
s
R
f
6
Assumptions:
•
Spherical particles, (no solvation)
•
Particle size much larger than size of
particles making up the medium
(i.e.much larger than solvent molecules)
•
Infinitely dilute solution
•
Particles travelling slowly (no turbulence)
r
r
r
r
r
r
9
)
(
2
6
)
(
3
4
)
(
1
2
2
1
2
3
1
2
g
dt
dx
dt
dx
g
dt
dx
g
s
s
s
R
R
R
f
V
Effects of Non

Sphericity & Solvation
dt
dx
g
f
m
2
1
1
r
r
•
absorbs solvent
•
m increases
•
measured f increases
Solvation
Non

sphericity
s
R
f
6
dry
•
absorbs solvent
•
R
s
increases
•
measured f increases
ideal particle
of radius R
s
•
sphere excluded by
tumbling ellipsoid of
same volume is larger
•
R
s
increases
•
measured f increases
Consider quantitatively
o
o
f
f
f
f
f
f
*
*
f
*
f
f
o
f
f
*
o
f
*
f
The actual measured friction factor
The ideal friction factor: unsolvated
sphere given by Stokes’ law as
Minimum possible value of f
friction factor for spherical particle
having same volume as solvated one
of mass m
Ratio measuring increase due to
asymmetry
Ratio measuring increase due to
solvation
s
R
6
3
/
1
1
2
1
*
r
r
m
m
f
f
b
o
Analyses also exist for the asymmetry
contribution but are complex.
*
f
f
Sedimentation allows for unambiguous particle
mass determination, and upper limits on size
and shape.
b
m
mass of
bound solvent
Furthermore, if intrinsic viscosity
measurements are also performed
we can determine unambiguously
particle hydration and axis ratio
Brownian Motion and Diffusion
•
All suspended particles have kinetic
energy 1/2mv
2
= 3/2kT.
•
Smaller the particle, the faster is moves.
•
Moving particles trace out a complex and
random path in solution as they hit other
particles or walls

Brownian motion
(Robert Brown, 1828).
2
/
1
2
Dt
x
Average distance travelled by a particle:
kT
Df
t
x
c
DA
m
d
d
d
d
2
2
d
d
d
d
x
c
D
t
c
Diffusion

tendency for particles to move
from regions of high concentration to
regions of low concentration.
D
S > 0, second law of thermodymanics
Two laws govern diffusion:
From these laws, we may derive (text)
Einstein’s law of diffusion (pp.27

29)
Fick’s first law
Fick’s second law
A
dm
c
x
kT
Df
•
No assumptions!
•
Any particle shape or size.
•
D and f determined
experimentally
Stokes

Einstein equation
2
/
1
3
6
6
6
A
s
A
s
s
s
N
R
RTt
x
N
R
RT
R
kT
D
R
f
•
Assumes spheres
•
No solvation
•
Original use:

finding Avogadro’s
number!
Note the two are complementary:
measurement of diffusion coefficient
gives a friction factor with NO
assumptions: can determine particle masses
g
D
dt
dx
kT
m
2
1
/
1
r
r
Competition between sedimentation
and diffusion
Note tables 2.1 and 2.2 in the text
Particle
Radius (m)
after
1 hour
Sedimentation
rate
10
9
1.23 mm
8
nm/hr
10
8
390
m
0.8
m/hr
10
7
123
m
80
m/hr
10
6
39
m
8 mm/hr
10
5
8.6
m
0.8 m/hr
x
At particle sizes ca. 10

7
m radius
(0.1
m) the sedimentation is perturbed
to a significant step by Brownian motion:
i.e particles of this size don’t sediment.
Spheres of
r
2
= 2.0 g/cm
3
in water at 20
o
C
Experimental Methods
Diffusion Constants:
Free boundary method
•
Must thermostat (no convection effects)
•
Must remove any mechanical vibration
Dt
x
o
e
Dt
c
dx
dc
4
2
/
1
2
4
x
c
dc/dx
0
Porous Plug Method
l
c
c
AD
dt
dm
)
(
2
1
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