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)
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:
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):
Recognizing Equation 2.13, we insert it into Equation 2.16 to obtain:
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)
For a given K and apparent moles
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)