Engineering & Science
Carothers Equation: Step-Growth Polymerization Degree (Xn, Xw) & Stoichiometric Ratio (r)
Calculate number-average degree of polymerization Xn = (1 + r) / (1 + r - 2*r*p), weight-average Xw = (1 + p) / (1 - p), and Flory polydispersity PDI = 1 + p in nylon and polyester synthesis.
See why 99% conversion and exact 1:1 stoichiometry are mandatory in step-growth polymers!
Synthesizing high-molecular-weight Nylon-6,6 or PET via step-growth polycondensation requires extreme functional-group conversion p AND razor-sharp stoichiometric balance r = NA0 / NB0 (with r <= 1)! Wallace Carothers proved that number-average degree of polymerization obeys Xn = (1 + r) / (1 + r - 2*r*p). At exact equimolar feed (r = 1.0) and 99% conversion (p = 0.99), Xn = 2 / (2 - 1.98) = 100 repeat units, with Flory weight-average Xw = (1 + p) / (1 - p) = 199 and polydispersity index PDI = Xw / Xn = 1 + p = 1.99!
Now watch what happens if just 2% excess diol or diamine slips into the reactor (r = 0.98) at the same p = 0.99 conversion: Xn = 1.98 / (1.98 - 1.9404) = 50.0—cutting molecular weight in HALF! Even at 100% conversion (p = 1.0), r = 0.98 caps chain length permanently at (1 + r)/(1 - r) = 99!
| Reactor Feed Condition | Stoichiometry r & Conversion p | Number-Avg Xn | Equimolar Weight-Avg Xw |
|---|---|---|---|
| Balanced Nylon-6,6 Salt | r = 1.00, p = 0.99 | Xn = 100.0 units | Xw = 199.0 (PDI = 1.99) |
| 2% Monomer Imbalance / Capper | r = 0.98, p = 0.99 | Xn = 50.0 units (Halved!) | Capped by excess end-groups |
| Invalid r > 1 or Gel/Complete p >= 1 | r > 1 or (1+r-2rp) <= 0 | Guarded Input Error | "Invalid Carothers Inputs" |
General Carothers Number-Average Degree of Polymerization Xn(r, p)
Evaluates Xn = (1 + r) / (1 + r - 2*r*p) for stoichiometric ratio 0 < r <= 1 and conversion 0 <= p < 1.
=IF(OR(B2 <= 0, B2 > 1, B3 < 0, B3 >= 1), "Invalid Carothers Inputs", ROUND((1 + B2) / (1 + B2 - (2 * B2 * B3)), 2))Equimolar Weight-Average Degree of Polymerization Xw = (1 + p) / (1 - p)
Computes Flory's weight-average chain length Xw for equimolar linear step-growth polymerization.
=IF(OR(B3 < 0, B3 >= 1), "Invalid Carothers Inputs", ROUND((1 + B3) / (1 - B3), 2))Try the example
Use the original values or change them to test your understanding.