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T3 · Kinetics / RatePolymer ChemistryGATE 2027 Syllabus (XE5)
Carothers Equation (Step-Growth Polymerization)
Calculates the number-average degree of polymerization (Xn) for step-growth condensation polymerizations as a function of extent of reaction (p) and average monomer functionality (f_avg).
Mathematical Canonical Relation
Plain Formula: Xn = 2 / (2 - p * favg)
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p
Range: 0.9 – 0.999 fraction (0.9-0.999)
favg
Range: 1.8 – 3.5 dimensionless
Calculated Engineering Result:
100 repeat units
Engineering Verification:Optimal engineering polymer molecular weight range.
When & Where to Apply
Use when modeling step-growth condensation systems (polyesters, polyamides, polyurethane) to predict molecular weight growth and gelation onset.
Common Engineering Pitfalls
- Confusing extent of reaction p with percentage (must be decimal 0.99, not 99%)
- Applying to chain-growth addition polymerization systems
- Ignoring stoichiometric imbalance (r < 1.0 requires modified Carothers form)
Physical Assumptions & Validity Limits
- Equal reactivity of functional groups regardless of chain length
- No intramolecular cyclization or side reactions
- Stoichiometric balance between reactive groups
Variable Symbols, SI Units & Dimensional Specifications
| Symbol | Variable Meaning | SI Unit | Typical Industrial Range |
|---|---|---|---|
| X_n | Number average degree of polymerization | dimensionless | — |
| p | Extent of reaction (fraction conversion) | dimensionless | 0.99 |
| f_{avg} | Average functionality of monomers | dimensionless | 2.0 |
Worked Industrial Numerical Example: Nylon 6,6 Polymerization at 99% Conversion
Problem Statement:
For a bifunctional step-growth monomer system (f_avg = 2.0) producing Nylon 6,6, calculate the degree of polymerization Xn when 99% of carboxyl groups have reacted (p = 0.99).
Step-by-Step Numerical Solution:
- Identify Carothers Equation: Xn = 2 / (2 - p * f_avg)
- Substitute values: Xn = 2 / (2 - 0.99 * 2.0)
- Calculate denominator: 2 - 1.98 = 0.02
- Calculate Xn: 2 / 0.02 = 100
Final Answer: Xn = 100
💡 Engineering Interpretation: At 99% conversion, average polymer chain length is 100 repeat units (Mn approx 22,600 g/mol), illustrating why step-growth requires >99% conversion for useful mechanical properties.
Authoritative Technical Reference & Source Provenance
Principles of Polymerization by George Odian (Wiley-Interscience, 2004), 4th Edition, Page 40-45.
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