Domain: Polymer chemistry

Consider the synthesis of \(nylon-6,6\) from \(hexamethylenediamine\) and \(adipic~acid\), the mole ratio of the diacid to diamine being 0.99. Estimate the degree of polymerization of the species which has the maximum yield by weight at the completion of the condensation.

Grading rubric:
criterion:  Correctly states the general equation for the weight fraction as $$\mathbf{w_x = rac{(2z-1) r^{(z-1)} (1-r^2)}{(1+r)}}$$ 
weight: 10
criterion:   Correctly identifies \(r\) as the stoichiometric probability factor
weight: 10
criterion:  Correctly identifies \(z\) as the segment parameter related to degree of polymerization as $$x = 2z - 1$$
weight: 10
criterion:  Correctly applies the condition for maximum weight fraction by setting the derivative to zero as $$rac{dw_x}{dz} = rac{(1 - r)^2}{(1 + r)} r^{(z-1)} [2 + 2z\ln r - \ln r] = 0.$$
weight: 10
criterion:  Correctly solves the stationarity condition to obtain $$z = rac{\ln r - 2}{2 \ln r}.$$
weight: 10
criterion:  Correctly substitutes the relationship \(x = 2z - 1\) to obtain $$x = rac{-2}{\ln r}.$$ 
weight: 10
criterion:  Correctly applies the approximate formula for the degree of polymerization at maximum weight fraction using the stoichiometric ratio \(r\) as $$\mathbf{x_{max} \approx rac{2}{-\ln r}}$$ 
weight: 10
criterion:  Correctly substitutes the stoichiometric ratio \(r = 0.99\) to evaluate \(\ln r\) numerically
weight: 10
criterion:  Correctly calculates the final result as the degree of polymerization at maximum weight fraction $$\mathbf{x_{max} = 199}.$$
weight: 20

Answer:
199