# Lecture - Sed. & B.M. Chapter 2 - Jmdsdf

Mechanics

Feb 21, 2014 (7 years and 5 months ago)

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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

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:

--

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
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

(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