Poly(lactic acid). Группа авторов

(2.10)

The variable p represents the probability of bond formation. Although the value of p depends on concentration, the quantity is the same for all oligomers at each concentration. Vu et al. [27] used the variable r, which is equivalent to the variable p used here. Using recursion, we recognize that Equation 2.9 can be written as:

      (2.11)

(2.12)

      Each oligomer of length j contains j lactic acid molecules, so the apparent moles of lactic acid are given by the balance found by the closed form of the sum:

(2.13)

(2.14)

Equation 2.14 can be inserted into Equation 2.12 to give the Flory‐Schulz distribution:

      (2.15)

      The water in an equilibrated solution is the sum of the apparent water plus the water from the condensation reaction. Each step during the condensation releases a water molecule, so an oligomer of length j releases (j − 1) moles of water (n _W):

(2.16)

Recognizing Equation 2.13, we insert it into Equation 2.16 to obtain:

(2.17)

Inserting Equations 2.14 and 2.17 into Equation 2.10, we develop a relation between the apparent number of moles and K that can be solved to find p

      (2.18)

      (2.19)

(2.20)

For a given K and apparent moles and Equation 2.20 provides a value of p. Then Equations 2.12 and 2.14 can be used to find the equilibrium moles of lactic oligomers and monomer, while Equation 2.17 provides the equilibrium moles of water. The titratable acidity is a measure of the true number of oligomers in solution because each oligomer has one free carboxylic acid group. The titratable acidity is

FIGURE 2.4 Left axis—total titratable acidity tabulated by Holten [28] from various workers (♢) and measurements by Vu et al. [27] (▪) compared to the model. Right axis—value of p for the model as a function of apparent wt% using K = 0.2023.

      (2.21)