Chapter 9: Reaction Mechanisms, Pathways, Bioreactions and Bioreactors
Example 9.5: Calculating Xn for Stoichiometric Inbalance
Case A: \( r = 0.99 \), \( p = 1 \)
\(\bar{X_n} = \frac{r+1}{r-1} = \frac{0.99+1}{1-0.99} = \frac{1.99}{0.01} = 199\)
Case B: \( r = 1 \), \( p = 0.99 \)
\(\bar{X_n} = \frac{r+1}{r+1-2rp} = \frac{1+1}{1+1-2(0.99)} = \frac{1}{1-p} = \frac{1}{0.01} = 100\)
Case C: \( p = 0.99 \), \( r = 0.99 \)
\(\bar{X_n} = \frac{r+1}{r+1-2rp} = \frac{1.99}{1.99-2(0.99)(0.99)} = 66.78\)
Case D: \( r = 1 \), \( p = 0.999 \)
\(\bar{X_n} = \frac{1}{1-p} = \frac{1}{0.001} = 1000\)
Case E: \( r = 0.99 \), \( p = 0.999 \)
\(\bar{X_n} = \frac{r+1}{r+1-2rp} = \frac{1.99}{1.99-2(0.99)(0.999)} = \frac{1.99}{0.01198} = 166.1\)
Summary
| r | p | Xn |
|---|---|---|
| 1 | 0.99 | 100 |
| 1 | 0.999 | 1000 |
| 0.99 | 1 | 199 |
| 0.99 | 0.99 | 66.8 |
| 0.99 | 0.999 | 166.1 |
Only a 1% imbalance reduces the number average chain length by almost an order of magnitude