Chemistry Reference
In-Depth Information
Table 2.4
Properties of low density polyethylene [
48
,
49
]
DP
Softening temperature (
C)
Physical state 25
C
Molecular weight
1
28
169 (mp)
Gas
6
170
12 (mp)
Liquid
35
1,000
37
Grease
140
4,000
93
Wax
250
7,000
98
Hard wax
430
12,000
104
Plastic
750
21,000
110
Plastic
1,350
38,000
112
Plastic
X
X
o ¼
o
i
¼
M
i
N
i
By dividing the total weight of the molecules by their total number we have the
number average
molecular weight,
.
.
X
X
M
i
N
i
X
M
n
¼ o
N
i
¼
N
i
Another way to express a molecular weight average is as a
weight average
. Each molecule in such
an average contributes according to the ratio of its particular weight to that of the total,
.
X
N
i
X
2
M
w
¼
M
i
N
i
The above can be illustrated quite readily by imagining that a sample of a polymer consists of five
molecules of molecular weights of 2,4,6,8, and 10, respectively. To calculate the number average
molecular weight all the weights of the individual molecules are added. The sum is then divided by
the total number of molecules in the sample (in this case 5):
M
n
¼
2
=
5
þ
4
=
5
þ
6
=
5
þ
8
=
5
þ
10
=
5
¼
6
To calculate the weight average molecular weight of the above sample, the squares of each
individual weight are divided by the total sum of their molecular weights, in this case it is 30:
M
w
¼
2
2
=
30
þ
4
2
=
30
þ
6
2
=
30
þ
8
2
=
30
þ
10
2
=
30
¼
7
:
33
M
w
is more sensitive to the higher molecular weight species, while
M
n
is sensitive to the lower ones.
This can be seen by imagining that equal weights of two different sizes of molecules are combined,
M
1
¼
10,000 and
M
2
¼
100,000. The combination would consist of ten molecules of
M
1
and one
M
2
. The weight average molecular weight of this mixture is (10
8
/2
10
5
+10
10
/2
molecule of
10
5
)
55,000 while the number average molecular weight is only 18,182. If, however, the
mixture consists of an equal number of these molecules, then the weight average molecular weight
is 92,000 and the number average molecular weight is 55,000.
The ratio of theweight averagemolecular weight to the number averagemolecular weight is important
because it affects the properties of polymers. This ratio is called the
molecular weight distribution
:
¼
Molecular weight distribution
¼ M
w
=M
n
When all macromolecular species are of the same size, the number average molecular weight is
equal to the weight average molecular weight. On the other hand, the greater the distribution of
