Molecular weight of polymers
Phase change in polymer mixing : The Θ state
M.W. of polymers
a. Some natural polymer (monodisperse) :
All polymer molecules have same
molecular weights.
b. Synthetic polymers (polydisperse) :
The molecular weights of polymers are
distributed
“ temp.”
MW
• Random walk chain
The unique temp.
at which the
attractions and
repulsions of a
polymer in a
solution cancel
each other
c. Mechanical properties are influenced by molecular weight
- much lower molecular weight ; poor mechanical property
- much higher molecular weight ; too tough to process
- optimum molecular weight ; 105 -106 for vinyl polymer
15,000 - 20,000 for polar functional group containing
polymer (polyamide)
• Swelling by excluded
volume = Contraction
by solvent
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Molecular weight of polymers
Molecular weight of polymers
M.W. & properties of polymers :
Polymers are synthesized for selected physical properties
M.W. of PE
Physical
Properties
Too low
OLIGOMER/
POLYMER
Desired
Range
POLYMER
Number of Molecules
Polymer mixing
Viscosity
Too
High
POLYMER
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40
Molecular weight of polymers
Molecular weight of polymers
M.W. of polymers
Definition of average molecular weight
Unlike small molecules, polymers are typically a mixture of differently sized
molecules. Only an average molecular weight can be defined.
• Number average M.W. (Mn): Total weight of all chains divided by # of
chains
• Weight average M.W. (Mw): Weighted average. Always larger than Mn
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Molecular weight of polymers
Molecular weight of polymers
Definition of average molecular weight
What the weight mean
Mn: This gives you the true average weight
Let's say you had the following polymer sample:
1,000,000 Dalton X 2 chains
700,000 Dalton X 5 chains
400,000 Dalton X 10 chains
100,000 Dalton X 4 chains
50,000 Dalton X 2 chain
Chain
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M.W. of chain
# of chain
1 Dalton = 1 g/mole
Group weight
i
Mi
Ni
Ni x Mi(=Wi)
1
1,000,000
2
2,000,000
2
700,000
5
3,500,000
3
400,000
10
4,000,000
4
100,000
4
400,000
5
50,000
2
100,000
Σ
2,250,000
23
10,000,000
Mn
= Σ (Ni x Mi) / ΣNi
= 10,000,000/23
= 435,000
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Molecular weight of polymers
Molecular weight of polymers
What the weight mean
Weight average MW
Mn: This gives you the true average weight
Mw: Since most of the polymer mass is in the heavier fractions, this gives the
average molecular weight of the most abundant polymer fraction by mass.
Let's say you had the following polymer sample:
1,000,000 Dalton X 2 chains
700,000 Dalton X 5 chains
Mn = Σ (Ni x Mi) / Σni
400,000 Dalton X 10 chains
=
(
100,000 Dalton X 4 chains
50,000 Dalton X 2 chain
Chain
M.W. of chain
# of chain
# fraction
i
Mi
Ni
xi
xi x Mi
1
1,000,000
2
2/23
2,000,000/23
2
700,000
5
5/23
3,500,000/23
3
400,000
10
10/23
4,000,000/23
4
100,000
4
4/23
400,000/23
5
50,000
2
2/23
100,000/23
23
23/23 = 1
10,000,000/23
Σ
Chain
) / 23
Mn = Σ (Ni x Mi) / ΣNi
= Σ xi x Mi
= 10,000,000/23
≈ 435,000
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M.W. of chain # of chain Group
Weight
i
Mi
Ni
Wi=Ni x Mi
1
1,000,000
2
2,000,000 2.000 x 1012
2
700,000
5
3,500,000 2.450 x 1012
3
400,000
10
4,000,000 1.600 x 1012
4
100,000
4
400,000 0.040 x 1012
5
50,000
2
100,000 0.005 x 1012
23
10,000,000 6.095 x 1012
Σ
Mw = Σ (Wi x Mi) / ΣWi
= 6.095 Χ1012 / 10,000,000
= 609,500
cf. Mn = Σ (Ni x Mi) / ΣNi = 435,000
Molecular weight of polymers
Molecular weight of polymers
Weight average MW
Definition of average molecular weight
Mw: Since most of the polymer mass is in the heavier fractions, this gives the
average molecular weight of the most abundant polymer fraction by mass.
Chain
M.W. of chain # of chain Group
Weight
Mn=
Mi
Ni
Wi=Ni x Mi
wi
wi x Mi
1
1,000,000
2
2,000,000
0.20
200,000
2
700,000
5
3,500,000
0.35
245,000
3
400,000
10
4,000,000
0.40
160,000
4
100,000
4
400,000
0.04
4,000
5
50,000
2
100,000
0.01
500
23
10,000,000
1
609,500
Σ
Mw = Σ (Wi x Mi) / ΣWi
= Σ wi x Mi
= 609,500
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a. number average molecular weight ( Mn )
Weight
fraction
i
Wi x Mi
 Ni Mi
 Ni
(obtained by using colligative property and end group analysis)
b. weight average molecular weight ( Mw)
Mw=
WiMi
Wi
(obtained by light scattering method)
some properties(freezing points, vapor pressure, and osmotic pressure of
dilute solutions), are related directly to Mn, whereas other properties(light
scattering, sedimentation, and diffusion constants) are related directly to Mw)
cf. Mn= Σ xi x Mi = 435,000
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Molecular weight of polymers
Molecular weight of polymers
Number average MW
Weight average MW
What would be the average M.W. of
[1 Elephant (10,000 lb) + 4 mosquitos(1 lb each)]?
→ Could the value represent the group properly?
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Molecular weight of polymers
50
Molecular weight of polymers
Number average MW
Number average MW
Average DP
Number average MW
where,
MW of repeating unit = 100
DP = Degree of Polymerization (중합도)
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Molecular weight of polymers
Molecular weight of polymers
Weight average MW
Average molecular weight
ex1. Example of molecular weight calculation
a. 9 moles, molecular weight (Mw) = 30,000
5 moles, molecular weight ( Mw) = 50,000
Mn=
(9 mol x 30,000 g/mol) + (5 mol x 50,000 g/mol)
= 37,000 g/mol
9 mol + 5 mol
9 mol(30,000 g/mol)2 + 5 mol(50,000 g/mol)2
Mw =
9 mol(30,000 g/mol) + 5 mol(50,000 g/mol)
= 40,000 g/mol
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Molecular weight of polymers
Molecular weight of polymers
Average molecular weight
Average molecular weight
c. z average molecular weight ( MZ )
ex2. Example of molecular weight calculation
b. 9 grams, molecular weight (Mw) = 30,000
MZ=
5 grams, molecular weight ( Mw) = 50,000
Mn =
Mw =
(ultracentrifugation)
9g+5g
(9 g/30,000 g/mol) + (5 g/50,000 g/mol)
(9 g/30,000 g/mol) + (5 g/50,000 g/mol)
9g+5g
NiMi3
NiMi2
d. general equation of average molecular weight :
= 35,000 g/mol
 N M a+1
M =  N iM ia
i i
= 37,000 g/mol
( a=0 , Mn
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a=1 , Mw
a=2 , Mz
)
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Molecular weight of polymers
Molecular weight distribution
Polydisperse
Monodisperse
Mn ≤ Mw ≤ Mz
size
size
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Molecular weight of polymers
Molecular weight of polymers
Average molecular weight
Average molecular weight

Different MW distributions give different properties even if their
average MW are same
e. Polydispersity index : width of distribution
Polydispersity Index (PI) = Mw / Mn ≥ 1
Ex.1 MW 30000, 60000, 90000을 같은 몰수 포함하는 가상적 고분자
시료의 Mw , Mn 및 다분산 지수? (1.17)
monodisperse
Conventional
polymer
– a measure of the breadth of the molecular weight
Ex.2 MW 30000, 60000, 90000을 같은 무게 포함하는 가상적 고분자
시료의 Mw , Mn 및 다분산 지수? (1.22)
– PI = 1 indicates Mw = Mn, i.e. all molecules have equal length
(monodisperse)
– PI = 1 is possible for natural proteins whereas synthetic polymers
have 1.5 < PI < 5
– At best PI = 1.1 can be attained with special techniques
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Molecular weight of polymers
Molecular weight of polymers
M.W. analysis
Ex3. Example of molecular weight calculation
a. Absolute method :
End group analysis (말단기 적정법) [Mn]
Colligative properties (총괄성) [Mn]
Membrane Osmometry (MO, 삼투압법)
Ebulliometry (끓는점 오름), Cryscopy (어는점 내림), VPO(증기상삼투압법)
Ultra-centrifugation (초원심분리법) [Mw, Mz]
Light Scattering (광산란법) [Mw]
MALDI-TOF [분자량 분포]
b. Relative method : solution viscosity
Viscometry (점도법) [Mv ~ Mw]
c. Fractionation method : GPC
Gel Permeation Chromatography (크기배제 크로마토그래피) [분자량 분포]
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Molecular weight of polymers
Molecular weight of polymers
End group analysis
End group analysis
A. Basic principles
MWtPolymer= DP x MWtMonomer + MWtEnds
B. End-group must have detectable species
a) The structures of the end groups must be different from
that of the bulk repeating units
(e.g., CH3 vs. CH2 in an ideal polyethylene)
a. vinyl polymer : -CH=CH2
b. ester polymer : -COOH, -OH
c. amide and urethane polymer : -NH2, -NCO
b) ∴ If you detect the concentration of the end group and
know the total amount of sample present you can
calculate the Mn
→ need to have either a perfectly linear polymer (i.e., two
end groups per chain) or need to know information about the
amount of branching
d. radioactive isotopes or UV, IR, NMR detectable functional group
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Molecular weight of polymers
Molecular weight of polymers
End group analysis
End group analysis
D. Strengths
a) The requisite instruments are in any department
b) can be quite quick
c) Sometimes this information comes out “free” during polymer
structural studies
C. Requirement for end group analysis
- The method cannot be applied to branched polymers.
- In a linear polymer there are twice as many end of the chain
and groups as polymer molecules.
E. Weaknesses
a) does not give MW distribution information
b) need to know information about the structure
i) identity and number of end groups in each polymer molecule
c) limited to relatively low MW for sensitivity reasons
i) 5,000 - 10,000 is typical MW range
ii) Can be high with some detections types
- radioactive labeling of end groups
- fluorescent labeling of end groups
- If having different end group, the number of detected end group
is average molecular weight.
- End group analysis could be applied for polymerization
mechanism identified
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Molecular weight of polymers
Molecular weight of polymers
End group analysis
End group analysis
Epoxy Resin
Ex. 2. 양쪽 말단기가 모두 –COOH인 2g의 PET를 정량하는데 0.1M KOH 5ml가 소모되었다.
Excess HCl
O
O
OH
OH
Mn =?
Cl
(8,000)
→ Residual Excess HCl을 NaOH 로 titration
→ (초기 HCl의 양 – NaOH에 의해 적정된 잔류 HCl 양)/2
= mole수 → MW
Ex. 1
HOOC
H
O
C
C
H 4
H
N
H
C
H 6
H O
N C
H
C
H 4
O
C
H
N
n
H
C
H 6
NH2
MW = A + B x n + C
A
B
C
1g을 취하여 COOH기 정량 한 결과 0.00005mol이면
→ 1g= 0.00005mol 이므로 1 mol = (1/0.00005)g = MW
∴ 1/0.00005 = A + B x n + C → n ≒ 87
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Molecular weight of polymers
Molecular weight of polymers
Colligative properties
Osmosis
Properties of solutions that depend on the number of molecules present
and not on the kind of molecules are called colligative properties.
These properties include boiling point elevation, freezing point depression,
and osmotic pressure.
Because colligative properties depend on the number of molecules, we
expect, and will show, that colligative property experiments give a
number average molecular weight.
nB
m
c


V MV M
nB ; mole number of solute(B)
Osmosis is the spontaneous net movement
of solvent molecules through a partially
permeable membrane into a region of
higher solute concentration, in the direction
that tends to equalize the solute
concentrations on the two sides.
V ; volume of solvent
m ; mass of solute
M ; Molecular weight of solute
What we want to know
c ; concentration of solution (g/l)
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Molecular weight of polymers
Molecular weight of polymers
Osmosis
Osmosis : Practical aspect of osmotic pressure
Net movement of solvent is from the less concentrated
(hypotonic) to the more concentrated (hypertonic) solution,
which tends to reduce the difference in concentrations.

c
V  nRT


c
n
c
RT  RT
V
M


RT
M
we expect polymer solutions to deviate from ideal behavior and
thus the osmotic pressure expression will need to be corrected.
n(mol)
c(g/l)

V(L)
M(g/mol)
In the limit of zero concentration, the solution will eventually
become ideal.
RT
M
We can therefore take a series of measurements and extrapolate
back to zero concentration to get the ideal result. In other words
This effect can be countered by increasing the pressure of the
hypertonic solution, with respect to the hypotonic.
The osmotic pressure is defined to be the pressure required
to maintain an equilibrium, with no net movement of solvent.

lim
c
c
0
71

RT
Mn
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Molecular weight of polymers
Molecular weight of polymers
Osmosis : Practical aspect of osmotic pressure
Osmosis

The question which remains is “how do we extrapolate?”
A common approach in thermodynamics is to use a virial expansion.
We thus write c as a sum of many terms:
c

RT
Mn
 RTA2 c
The *virial expansion can be applied
to measurements of osmotic
pressure in order to determine
accurate molar masses and the
nature of molecular interactions.

RT
Mn
← Ignoring higher order terms,
 RTA2 c
Slope = RTA2
Slope =
Plotting : π/c vs c
Intercept =
limitation of : 50,000～2,000,000
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Molecular weight of polymers
Molecular weight of polymers
Osmosis
Osmosis
온도 27℃에서 여러 농도의 고분자 용액의 삼투압을 측정하여
RTA2 c
 
 c 
lim
c
 
0

c

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RT
Mn
=298,900
 dyne / cm 2 


 g / cm 3 


임을 구하였다. Mn=?
 RTA2 c
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