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2.1.6. Molecular weight averages: M
n
, M
w
and M
z
What is the average molecular weight of the distribution discussed above? Or
to simplify further: Assume you have:
One molecule with M = 1000, two molecules with M = 1500, and one molecule
with M = 2000. Total 4 molecules. The ‘normal ‘ average would simply be:
M
ave
= (1000 + 1500 + 1500 + 2000)/4 = 1500
The ‘normal’ average is in fact identical to what is in polymer science defined
as the number average molecular weight, with symbol M
n
. The classical
definition is:
M
n
=
N
i
M
i
i
∑
N
i
i
∑
=
n
i
M
i
i
∑
n
i
i
∑
N
i
is as before the number of molecules, where as n
i
is the number of moles.
Avogadro’s number cancels in the formula. It is straightforward to determine
M
n
in terms of masses (m
i
) or concentrations (c
i
):
c
i
=
w
i
V
w
i
=
N
i
M
i
n
i
=
N
i
N
i
i
∑
M
n
=
N
i
M
i
i
∑
N
i
i
∑
=
w
i
i
∑
w
i
M
i
i
∑
=
c
i
i
∑
c
i
M
i
i
∑
The latter formula is very useful in practical situations where distributions are
given on a mass or concentration basis.
If we replace mole fractions in the expression for M
n
by weight fractions we
obtain another type of average called the weight average molecular weight,
with symbol M
w
: