Molecular Weight Distribution Molecular Weight Distribution SummarySummary

2

12wwz

21

2nnw

n

w

3ii

4ii

1z

2ii

3ii

z

iiii

2ii

i

iiw

i

ii

iin

MMM)mass.(D.S

MMM)number.(D.S

20204.1MM)P(sityPolydisper

Mn

MnM

Mn

MnM

MwMnMn

mMmM

Mw1

nMnM

Colligative Properties and MColligative Properties and Mnn For an ideal solution (dilute): V = 0 and H = 0

i0ii xlnRT

Standard chemical potential of pure component i.

Mole fraction of i.

For the solvent in a polymer solution:

2011

011 x1lnRTxlnRT

...x31x

21xx1ln 3

22222

Thus:

...x

31x

21xRT 3

2222

011

Also:

01

12

21

22

012

2

V1n;

Mcn;

nnX

MVcx

Molar volume of solvent.

Molecular weight of solute.

So for ideal solutions (V=0 and H=0):

...c

M3Vc

M2V

McRTV 3

23

3o12

22

o12o

1011

This is known as a virial equation and the property (1-1

o) relates to the colligative properties osmotic pressure, boiling pointelevation, freezing point depression and pressure lowering.

For osmotic pressure, it can be shown that:

where is the osmotic pressure.

So,

o1

011 V

...c

M3Vc

M2V1

MRT

c222

3o1

2

o1

2

For real solutions (V and H not 0):

where A2 and A3 are second and third virial coefficients.

These equations are also written as:

Or:

...cAcA1MRT

c22322

n2

...cc1cc

22322

0c2

...CcBcMRT

c222

n2

Colligative Properties and MColligative Properties and Mnn

Osmotic Pressure:

Boiling Point Elevation:

Freezing Point Depression:

Vapor Pressure Lowering:

n0c MRT

c

nvap

o1b

0c

bMH

VRTcT

nf

o1fp

0c

fp

MHVRT

cT

n

o1

o1

0c MVP

cP

Membrane Osmometry(pp. 37-44 in Shaw)

semi-permeable membrane!watch your units!!

Recall...

Assume A2, A3 … are equal to 0.

Q: What would be the osmotic pressure for a10 g/L solution of Polystyrene of 200,000 g/mol in toluene at 25C?

n0c MRT

c

Answer:

At 25C: RT ~ 23 L x atm

C ~ 10 g/L x 1 mol/200,000 g = 5 x 10-5 M

nMRT

c

P ~ 10-3 atm

P ~ 14 mm toluene

Modern osmometers can measure with accuracy of ~ 0.2 mm so the error here is ~ 0.15%. For this same sample:

Tb ~ 1 x 10-4 CP ~ 2 x 10-4 mm Hg (2.7 x 10-3 Pa)

Neither which can be measured with great accuracy!

c

RTc

2

1M

slope of A2

Practical considerations:

•Solvent•low viscosity, equilibrates fast

•Membrane•strength (i.e. 1M NaCl =0.42 atm)•no leakage or attack by solvent•does it pass what you want? mechanism?•possible membranes

–cellophane–animal membranes–polyurethane

•Measurement•Capillary effects•Temperature effects

•Impurities•Depends on number of particles, high MW sample can be contaminated by relatively (by mass) small amounts of low MW impurity.

Practical limits-- M=104-106

Two types of osmometers available:(1) Static -

(2) Dynamic – Pressure is adjusted on one side of membrane to cancel the solvent flow. (1 transparency from the Web)

Raw osmometry data are in cm (mm) of solvent:

= g h

Recall:

...cA1MRT

c 22n2

o1Av

22

2VN

21

A

So called Flory-Huggins parameter

What kind of results are possible?

/c2

c2

lower molecular weight in the same solvent

/c2

c2

better solvent

so called theta () solvent

Is this possible?

GPC,LS,UCMUsemol

g000,000,1M

VPO&ryEbulliometmol

g000,30M

osmometryUsemol

g000,10M

w

n

n

n

In Summary:In Summary:

Upper limit depends on smallest pressure that can be measured.

Lower limit depends on permeability of the membrane.

Donnan Equilibria

Problems occur when a solution of diffusible and non-diffusible ions are introduced in an osmometer (say in the case of measuring on proteins).

Unrealistically low molecular weights are determined due to the excess of diffusible ions on the side of the macromolecule.

Working with moderately low protein concentrations and moderately high salt concentrations will solve this difficulty.

Donnan Equilibria

Excess of ions in compartment 1 overcompartment 2 (text)

baaexcess

2

2

babx

xbxbxxa

2

))(()(2

Na+ = aPr- = a

Na+ = a + xPr- = aCl- = x

Na+ = bCl- = b

Na+ = b - xCl- = b - x

1 2

BeforeEqub’m

AtEqub’m

2270 m3/day, at P = 400 psi

P>

Note: P is not transmitted through membrane!

Reverse Osmosis -Desalination