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T2 · ThermodynamicsPolymer ChemistryGATE 2027 Syllabus (XE5)
Flory-Fox Equation (Tg Dependence on Molecular Weight)
Relates the glass transition temperature (Tg) of a linear polymer to its number-average molecular weight (Mn) based on chain-end free volume theory.
Mathematical Canonical Relation
Plain Formula: Tg = Tg_infinity - (Kff / Mn)
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Input your production or test values below for instant, verified output.
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Tg_inf
Range: -50 – 350 °C
Kff
Range: 20000 – 400000 K·g/mol
Mn
Range: 2000 – 150000 g/mol
Calculated Engineering Result:
92.8 °C (366 K)
Engineering Verification:Depression due to chain-end free volume is 7.2 °C.
When & Where to Apply
Use when estimating the glass transition temperature of oligomers or lower molecular weight polymers before reaching the high-MW asymptotic plateau Tg,infinity.
Common Engineering Pitfalls
- Using weight-average Mw instead of number-average Mn
- Forgetting that temperatures in theoretical free volume equations must be in Kelvin before converting to Celsius
Physical Assumptions & Validity Limits
- Chain ends possess higher free volume than chain middle segments
- Linear homopolymer chains without long-chain branching
- Equilibrium free volume state at transition
Variable Symbols, SI Units & Dimensional Specifications
| Symbol | Variable Meaning | SI Unit | Typical Industrial Range |
|---|---|---|---|
| T_g | Glass transition temperature at given Mn | °C (or K) | — |
| T_{g,infty} | Asymptotic Tg at infinite molecular weight | °C | 100 |
| K_{ff} | Flory-Fox constant for polymer | K·g/mol | 180000 |
| \overline{M}_n | Number-average molecular weight | g/mol | 25000 |
Worked Industrial Numerical Example: Polystyrene Tg Prediction at Mn = 25,000 g/mol
Problem Statement:
For Polystyrene, Tg,infinity is 100°C (373 K) and Kff is 1.8 × 10^5 K·g/mol. Calculate Tg for a polystyrene sample with Mn = 25,000 g/mol.
Step-by-Step Numerical Solution:
- Identify formula: Tg = Tg_infinity - (Kff / Mn)
- Calculate depression: 180,000 / 25,000 = 7.2°C
- Calculate Tg: 100°C - 7.2°C = 92.8°C
Final Answer: Tg = 92.8°C
💡 Engineering Interpretation: The presence of chain ends lowers Tg by 7.2°C compared to high-MW polystyrene, indicating reduced thermal stability for short-chain oligomeric grades.
Authoritative Technical Reference & Source Provenance
Principles of Polymer Chemistry by Paul J. Flory (Cornell University Press, 1953), 1st Edition, Page 347-350.
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