Calculator Inputs
Formula Used
For ideal stepwise polymerization, the number average degree of polymerization is calculated by the Carothers relationship:
DPn = (1 + r) / (1 + r - 2rp)
Here, p is the fractional conversion. The value r is the stoichiometric ratio of the deficient functional group to the excess functional group.
When r = 1, the equation becomes:
DPn = 1 / (1 - p)
The number average molar mass is estimated as:
Mn = DPn × effective repeat mass + end group mass
Effective repeat mass equals feed unit molar mass minus condensate loss.
How to Use This Calculator
- Enter the extent of reaction as a percentage.
- Enter the stoichiometric ratio, usually from functional group balance.
- Add feed unit molar mass and condensate loss.
- Set end group mass if needed.
- Enter target DPn or target Mn for planning.
- Add batch mass and loss values for retained mass estimates.
- Press the calculate button.
- Download CSV or PDF for reporting.
Example Data Table
| Case | Conversion (%) | r | Feed Unit MW | Loss | Expected DPn | Use Case |
|---|---|---|---|---|---|---|
| Balanced polyester | 95 | 1.000 | 194 | 18 | 20.00 | General condensation estimate |
| High conversion nylon | 99 | 1.000 | 226 | 18 | 100.00 | Target high chain length |
| Imbalanced system | 98 | 0.970 | 210 | 18 | 39.28 | Functional group mismatch |
| Lower conversion trial | 90 | 1.000 | 180 | 18 | 10.00 | Early process screening |
Article: Degree of Polymerization in Stepwise Reactions
Why Chain Growth Depends on Conversion
Stepwise polymerization builds chains through repeated reactions between functional groups. Small molecules, dimers, oligomers, and long chains can all react. This makes the process very different from chain polymerization. A high molecular size appears only when conversion becomes very high. At 90 percent conversion, a balanced system gives a DPn near 10. At 99 percent conversion, the same system gives a DPn near 100. This steep change explains why drying, purity, and reaction completion matter.
Role of Stoichiometric Balance
The ratio r describes functional group balance. It compares the smaller functional group amount with the larger amount. A value of one means perfect balance. Values below one show excess of one functional group. Even small imbalance can limit final chain length. This limit remains important even when conversion is almost complete. Therefore, accurate weighing and purity correction are essential.
Molar Mass Planning
The calculator also estimates number average molar mass. It uses effective repeat mass and optional end group allowance. Condensation systems often release water, alcohol, acid, or another byproduct. That loss reduces the repeat unit mass. The result helps compare a planned reaction with a target material grade. It can also show whether a target molar mass is realistic.
Practical Interpretation
Treat the output as an ideal calculation. Real reactions may show side reactions, cyclization, viscosity limits, or incomplete mixing. Some systems also need catalysts or vacuum removal of condensate. Use the calculated DPn as a design guide. Then compare it with lab data such as titration, GPC, viscosity, or end group analysis. This combined approach gives better process control.
FAQs
1. What is degree of polymerization?
It is the average number of repeat units in a polymer chain. Higher values usually mean longer chains and higher molar mass.
2. What does DPn mean?
DPn means number average degree of polymerization. It is based on the total number of polymer molecules and repeat units.
3. Why is conversion important?
Stepwise reactions need very high conversion. Chain length rises slowly at first, then increases sharply near complete reaction.
4. What is the stoichiometric ratio r?
It is the smaller functional group amount divided by the larger amount. Perfect balance gives r equal to one.
5. Can imbalance reduce polymer size?
Yes. Functional group imbalance limits final chain length, even when conversion is very high.
6. What is condensate loss?
It is the mass of small byproduct removed per repeat unit. Water loss in polyester formation is a common example.
7. Is the molar mass result exact?
No. It is an ideal estimate. Real data may differ because of side reactions, impurities, or measurement conditions.
8. When should I use target DPn?
Use it during process planning. It shows the conversion needed to reach a desired average chain length.