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

choppedspleenMécanique

21 févr. 2014 (il y a 3 années et 3 mois)

47 vue(s)

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