KineticKinetic PropertiesProperties(see Chapter 2 in Shaw, pp. 21-45)(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)

1

2

Fg

FbFv

g

gFgF

ggF

)( 12

1

2

VV

Vm

net

b

g2>1 sedimentation2<1 creaming

dtdxg

dtdxg

dtdxF

fmor

fV

fv

2

1

12

1

)(

Now we need to find an expression for f...

Gravitation and Sedimentation: Stokes’ Law

•Independent of shape

•No solvation (which changes the density)

dtdx

Stokes’ Law

sRf 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)

9)(2

6)(34

)(

122

123

12

gdtdx

dtdxg

dtdxg

s

ss

R

RR

fV

Effects of Non-Sphericity & Solvation

dtdxg fm

2

11 •absorbs solvent

•m increases•measured f increases

SolvationSolvation

Non-sphericityNon-sphericity

sRf 6

dry•absorbs solvent•Rs increases•measured f increases

ideal particleof radius Rs •sphere excluded by

tumbling ellipsoid of same volume is larger •Rs increases•measured f increases

Consider quantitatively

oo ff

ff

ff *

*

f

*ff

off *

of

*f

The actual measured friction factor

The ideal friction factor: unsolvatedsphere given by Stokes’ law asMinimum possible value of f

friction factor for spherical particlehaving same volume as solvated oneof mass m

Ratio measuring increase due toasymmetry

Ratio measuring increase due tosolvation

sR6

3/1

1

21*

mm

ff b

o

Analyses also exist for the asymmetrycontribution but are complex.

*ff

Sedimentation allows for unambiguous particlemass determination, and upper limits on sizeand shape.

bmmass ofbound solvent

Furthermore, if intrinsic viscositymeasurements are also performedwe can determine unambiguouslyparticle hydration and axis ratio

Brownian Motion and DiffusionBrownian Motion and Diffusion

•All suspended particles have kinetic energy 1/2mv2 = 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).

2/12Dtx Average distance travelled by a particle:

kTDf

txcDAm d

ddd

2

2

dd

dd

xcD

tc

Diffusion - tendency for particles to movefrom regions of high concentration to regions of low concentration.

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

Adm

cx

kTDf •No assumptions!•Any particle shape or size.•D and f determined experimentally

Stokes-Einstein equation

2/1

3

66

6

As

Ass

s

NRRTtx

NRRT

RkTD

Rf

•Assumes spheres•No solvation•Original use:--finding Avogadro’snumber!

Note the two are complementary:measurement of diffusion coefficientgives a friction factor with NOassumptions: can determine particle masses

gDdtdxkT

m21 /1

Competition between sedimentationand diffusion

Note tables 2.1 and 2.2 in the text

ParticleRadius (m)

after1 hour

Sedimentationrate

10-9 1.23 mm 8 nm/hr10-8 390 m 0.8m/hr10-7 123 m 80 m/hr10-6 39m 8 mm/hr10-5 8.6m 0.8 m/hr

x

At particle sizes ca. 10-7 m radius(0.1 m) the sedimentation is perturbedto a significant step by Brownian motion:i.e particles of this size don’t sediment.

Spheres of 2 = 2.0 g/cm3 in water at 20oC

Experimental MethodsExperimental MethodsDiffusion Constants:Free boundary method

•Must thermostat (no convection effects)•Must remove any mechanical vibration

Dtx

o eDtc

dxdc 4

2/1

2

4

x

c dc/dx

0

Porous Plug Method

lccAD

dtdm )( 21