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L₂ can further react to form lactolyl–lactolyllactic acid, L₃

      (2.5)

The structures of the species are shown in Figures 2.2 and 2.3. The step‐wise condensation oligomerization presents a significant challenge for performing separations and for modeling vapor–liquid equilibrium of aqueous solutions. The oligomerization can be represented by a succession of simultaneous equilibrium reactions with the same equilibrium constant.

(2.6)

      (2.7)

      where L₄ is the tetramer, and L _{j − 1} and L _j are generic notation for oligomers of the lengths j − 1 and j, respectively. While alternative combinations of reactions can be written, such as two L₂ molecules forming L₄, chemical equilibrium can be obtained with the most convenient combination of independent reactions. Other expressions of the equilibria can be obtained by combinations of the independent network. For generalization, addition of a monomer to a chain provides a series of reactions where each equilibrium constant is expected to be approximately the same numerical value.

      The equilibrium constant can be expressed in terms of concentration or mole fraction equivalently because the reaction has no net change in moles. Assuming that the equilibrium constant K ₂ for forming L₂, is the same for each condensation reaction K _j for forming L _j , as suggested by Flory [22]:

(2.8)

      where x _i represents a mole fraction (e.g., represents the mole fraction of species L _j ), n _i represents a number of moles, and the square brackets indicate a concentration.

The equilibrium has been studied by Watson [23], Montgomery [24], and Ueda and Terajima [25] and Bezzi [26], but the basis of the concentrations is not always clear. The distribution was studied by Witzke [1] who suggested an equilibrium constant value of approximately 0.25. Vu et al. [27] performed calibrated HPLC analysis and titrations and found that a value of 0.2023 was suitable, which is used for the calculated equilibria below. The mole fractions, moles, or concentrations appearing in Equation 2.8 are those existing at equilibrium and not those used to prepare a solution.

FIGURE 2.2 Chemical structure of lactic acid and lactoyllactic acid.

FIGURE 2.3 Chemical reaction between lactic acid and lactoyllactic acid.

      The unreacted form of lactic acid (L₁) is called the monomer. The concentration resulting from the conversion of all lactic acid to monomer is called the apparent concentration or sometimes the superficial or formal concentration. In the work of Vu et al. [27], the percent equivalent monomer lactic acid, %EMLA _j , denotes the percentage of apparent lactic acid represented by a particular oligomer. For example, the %EMLA₂ equals 10% when 10% of the apparent lactic acid molecules are dimers. The weight fraction is commonly referred to as the apparent (also known as superficial or formal) weight fraction. Using a superscript i to indicate the initial moles, for a solution composed of moles of water (mass and lactic acid (all L_1, mass ), the apparent mole fraction of lactic acid is and the apparent weight fraction is .

2.5 EQUILIBRIUM DISTRIBUTION OF OLIGOMERS

      The equilibrium and size distributions can be readily calculated using infinite series when the equilibrium constant is assumed to be independent of the oligomer length. Oligomer distribution is obtained by rearranging the material balances in terms of the equilibrium constant and subsequently empirically determining the equilibrium constant that represents the total titratable acid. The distribution is then verified against the smaller oligomers that are measurable. The equilibrium constant (Equation 2.8) can be rearranged as:

(2.9)

      where p is a lumped variable including the lactic acid monomer, free water, and equilibrium constant.

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