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209
0.0E+00
5.0E-05
1.0E-04
0
50 100 150 200 250
t
1/M
Mw
Mn
0.0E+00
1.0E+05
2.0E+05
0
50 100 150 200 250
t
M
Mw
Mn
Also note
α
= 1/DP
n
= M
0
/M
n
. For early stages of the degradation, i.e.
α
< 1/50
(M
n
> 10.000) the following approximation (simplification) holds:
ln 1
−
α
(
)
≈ −
α
= −
kt
α
=
kt
1
M
n
=
k
M
0
t
This equation is strictly valid only if we started with an indefinitely long
molecule. By starting at a given molecular weight (M
n,0
), and a corresponding
α
0
, the equation becomes:
α
=
α
0
+
kt
1
M
n
=
1
M
n
,0
+
k
M
0
t
This is a very important equation. It shows that the fraction of broken linkages
(
α
) increases linearly with time
(as long as
α
remains small),
and consequently, the
inverse
if the molecular weight also
increases linearly with time. It
follows that M
n
decreases
hyperbolically towards zero.
Since M
w
/M
n
= 2 (for random
depolymerisation) we also
obtain expressions for M
w
:
1
M
w
=
1
M
w
,0
+
k
2
M
0
t
We easily find the rate constant
(k) from monitoring the
decrease in molecular weight.
A plot of 1/M
n
versus time
should give a straight line with slope equal to k/M
0
. Alternatively, a plot of
1/M
w
versus time gives k/2M
0
as slope. This is shown in the figures above.
Note again that a consequence of the random degradation is the hyperbolic,
not linear, decrease in M
w
and M
n
with time.