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208
5.3. RANDOM DEPOLYMERISATION OF
LINEAR
(UNBRANCHED) POLYMERS: CHANGES
IN M
W
AND M
N
5.3.1. Basic equations
for a pseudo
first order
reaction
For a polymer with n monomers (n-1
≈
n linkages), where all linkages are
cleaved with equal rate (or same probability) the following equation applies:
−
dn
dt
=
kn
k is a
pseudo
42
first order rate constant, and the reaction is first order with
respect to n. By integration from t = 0 to t we obtain:
ln
n
n
0
⎛
⎝⎜
⎞
⎠⎟ = −
kt
where n
0
is the number of linkages at t = 0 (start of the degradation). By
defining n’ as the number of cleaved linkages:
n' = n
0
- n
⇔
n = n
0
- n'
we obtain:
ln
n
0
−
n
'
n
0
⎛
⎝⎜
⎞
⎠⎟ = −
kt
⇔
ln 1
−
n
'
n
0
⎛
⎝⎜
⎞
⎠⎟ = −
kt
n'/n
0
is the
degree of scission
(fraction of broken linkages) and has the symbol
α
. Hence:
ln 1
−
α
(
)
= −
kt
42 The term pseudo indicates that other factors such as pH, buffer, temperature etc are kept konstant (but influence k if changed)