Chapter 9: Reaction Mechanisms, Pathways, Bioreactions and Bioreactors
Example 9.3: Calculate Xnfor Monomers with Different End Groups
10 molecules of ARB initially
MWR = 100 , Meg ~ 18, we neglect MWeg wrt MWR.
At time t we have 5 polymer molecules with the following distribution
|
MW* |
Species |
By product |
|
||
|
100 |
ARB |
||||
|
100 |
ARB |
||||
|
200 |
AR2B |
+ AB |
|||
|
300 |
AR3B |
+ 2AB |
|||
|
300 |
AR3B |
+ 2AB |
*Neglecting Meg
p = fraction of functional groups of A that have reacted.
There are 5 molecules of polymer that remain: 2 of ARB, 1 of AR2B, and 2 of AR3B.
\( p = \frac{\text{Functional Groups Initially} - \text{Functional Groups Remaining}}{\text{Functional Groups Initially}} = \frac{10 - 5}{10} = 0.5 \)
We are giving to calculate \( \overline{X_n} \) by three different methods, all of which yield the same result.
\( \overline{X_n} = \frac{1}{1 - p} = \frac{1}{1 - 0.5} = 2 \)
or
\( \overline{X_n} = \frac{N_0}{N} = \frac{10 \text{ polymer molecules initially}}{5 \text{ polymer molecules remaining}} = 2 \)
or
\( \overline{X_n} = \frac{M_{A0}}{M_A} = \frac{10 \text{ "A" functional units}}{5 \text{ "A" functional units}} = 2 \)
\( \overline{M_n} = \frac{2 \times 100 + 1 \times 200 + 2 \times 300}{5 \text{ molecules}} = \frac{1000}{5} = 200 \)
or
\( \overline{M_n} = \overline{X_n} (MW_s) = 2 (100) = 200 \)