17
Molar mass:
There are different ways to report a molar mass like (a) Number average
molar mass, (b) mass average molar mass, (c) Viscosity average molar
mass, (d) Z- Average molar mass
Number Average Molar Mass
Colligative properties (like vapor pressure lowering, freezing point
depression, osmotic pressure etc) depend on the number of molecules
and not on their size.
Mn = ΣxiMi =
ΣNiMi
ΣNi
i
(Eq 1)
where, x i = Number fraction = mole fraction =
Ni
ΣNi
Mass Average Molar Mass
Mass average molar mass depends on size (mass/volume) as well as
number of molecules. Properties like diffusion, sedimentation, light
scattering depends both on size and mass of the molecules.
ΣNiMi2
Mw = ΣyiMi = ΣwiMi =
(Eq 2))
ΣNiMi
Σwi
i
i
w
where, y i = mass or weight fraction = i
Σwi
i
wi = NiMi
Z-Average Molar Mass
The molar mass depends both on size and mass of the molecules.
However, the contribution of the mass of the particle are weighted
further in this type of molar mass. Mz can be determined by
ultracentrifugation technique.
Mz =
ΣNiMi3
ΣNiMi2
i
(Eq 3)
18
Viscosity Average Molar Mass
Molar mass obtained from viscosity measurements is known as
Viscosity Average Molar Mass.
Mv =
ΣNiMi1.75 to 2
(Eq 4)
ΣNiMi0.75 to 1
i
If a =1, Mv = ΣyiMi = Mw; if a = -1 (not possible) then it can be shown
that Mv = Mn. It has been reported that 0.5< a <1 corresponds to 0.75
< β <1.
ΣNiMiβ+1
General Equation, M β =
ΣNiMiβ
i
Mn ; β = 0; Μw ; β = 1; Mv: 0.75<β <1 ; Mz: β = 2
Mn < Mv < Mw < Mz
[η] = KMva
=KΣyiMia
Mv = (ΣyiMia)1/a
These equations provide the basis for molecular weight determination
by GPC. It is possible to obtain polymer samples in which the
distribution of molecular weight is very narrow. In these especial
circumstances the two averages Mn and Mw are approximately equal and
hence the ratio Mw/Mn approximates to unity. Generally, however, Mw
is larger than Mn and the ratio Mw/Mn can be used to give some
indication of the Polydispersity of the sample (i.e.) the distribution of
molecular weights within the sample.
Mn
Mv
Mw
Mz
Mass fr.
yi
Molar mass
19
If a sample contains 95% and 5% molecules by weight having a molar
mass of 10,000 and 100, respectively, then using above equations the
Mw and Mn can be caculated to be 9505 and 1681, respectively. The
value of Mn, gives an inaccurate impression of the molar mass whereas
the Mw is a better indicator of the molar mass in this example. While in
experiments related to measurement of colligative properties one has to
use a value of 1681, the experiment like light scattering will involve the
molar mass of 9505.
PI =Mw/Mn
Polydispersity, an indicator of spread in the molar mass can be
expressed as above.
For polydisperse samples, molar mass determined from colligative
properties, light scattering, and appropriate data treatment of
ultracentrifugation are referred to as absolute molar mass while those
determined from GPC and viscometry are referred to as relative
molecular weights. An absolute molar mass is determined by relating
an experimental parameter with molar mass in an equation, whereas
GPC and viscometry require calibration employing polymers of known
molar mass determined by an absolute method. Typical techniques for
molecular weight determination are given in the Table.
*
109
106
*
O
108
O
O
+
O
105
O
O
107
O
log M
O
+
log[η].M
O
O
104
O
O
x
O
106
O
103
25
30 35
40
45
50
Elution volume (cm 3)
Fig. 2. Molecular weight of monodisperse
polystyrene standards as a function
of elution volume in Tetrahydrofuran.
x
O
105
18
22
24
26
28
Elution volume (cm3)
30
32
Fig. 3. Universal calibration in gel-permeation
chromatography for a variety of polymers in
THF
20
Table. Typical Molecular Weight Determination Methodsa
Method
Light scattering
Type of mol.
wt. average
Mw
Applicable wt.
range
to infinity
Membrane osmometry
Mn
2x104-2x106
Vapor phase osmometry
Mn
to 40,000
102 to infinity
Electron & X-ray microscopy
Mn,w,z
Isopiestic method (isothermal
distillation)
Ebulliometry (boiling pt
elevation)
Cryoscopy (melting pt
depreesion)
End-group analysis
Mn
to 20,000
Mn
to 40,000
Mn
to 50,000
Mn
to 20,000
Osmodialysis
Mn
500 to 25,000
Mz
Mz,w
Mw
to infinity
to infinity
to infinity
to infinity
Centrifugation:
Sedimentation equilibrium
Archibald modification
Trautman’s method
Sedimentation velocity
M
b
Other information
Can also give shape
Shape and distribution
in the early portions of effluent solution. The effluent passes through
one compartment of a differential refractometer cell and then into a
syphon before going to waste. The estimation of the mass of the
species eluting over a particular period of time is the purpose of the
differential refractometer.
The differential refractometer consists of a divided cell which is
carefully thermostated. Effluent from the sample set of columns passes
through one compartment and effluent from a referenc