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target="_blank" rel="nofollow" href="#fb3_img_img_3a8d2ede-5196-51d1-91c9-5652ffc85492.png" alt="StartLayout 1st Row upper T i t r a t a b l e a c i d i t y equals sigma-summation n Subscript normal upper L Sub Subscript j Baseline equals n Subscript normal upper L 1 Baseline left-parenthesis 1 plus p plus p squared plus p cubed plus ellipsis right-parenthesis 2nd Row equals n Subscript normal upper L 1 Baseline slash left-parenthesis 1 minus p right-parenthesis equals n Subscript normal upper L Superscript i Baseline left-parenthesis 1 minus p right-parenthesis EndLayout"/>

Because water is formed by polymerization and can be removed from the solution by evaporation, the material balance constraint for water is for equilibrium moles, n _W ≥ 0, via Equation 2.17 and not the initial moles, n Subscript normal upper W Superscript i Baseline greater-than-or-equal-to 0 . Thus, negative values of are feasible and the apparent weight fractions of water are negative at high degrees of oligomerization. Vu et al. [27] have regressed K to fit titratable acidity and found a value of K = 0.2023. Using K = 0.2023, solutions of up to an apparent lactic acid concentration of 125 wt% lactic acid are theoretically feasible with an apparent water concentration of −25 wt%. The value of p and the titratable acidity are shown in Figure 2.4.

Several important relations can be developed. The distribution of oligomer lengths is given by the Flory‐Schulz distribution [22]. The %EMLA_j for species j is (note that a typographical error in equation 19 of Vu et al. [27] omits the 100):

(2.22) percent-sign upper E upper M upper L upper A Subscript j Baseline equals 100 j p Superscript j minus 1 Baseline left-parenthesis 1 minus p right-parenthesis squared

Other useful results are:

(2.23) upper N u m b e r a v e r a g e o l i g o m e r l e n g t h equals 1 slash left-parenthesis 1 minus p right-parenthesis

(2.24) upper T r u e w t percent-sign w a t e r equals 100 plus left-parenthesis a p p a r e n t w t percent-sign upper L upper A right-parenthesis left-parenthesis 0.2 p minus 1 right-parenthesis

Schematic illustration of the percent equivalent monomer lactic acid for L1 through L4.

FIGURE 2.5 The percent equivalent monomer lactic acid for L₁ through L₄. Open symbols are from Montgomery [24] and Ueda and Terajima [25]. Solid symbols are measured by Vu et al. [27].

(2.25) upper T r u e w t percent-sign normal upper L Subscript j Baseline equals left-parenthesis 0.8 j plus 0.2 right-parenthesis left-parenthesis a p p a r e n t w t percent-sign upper L upper A right-parenthesis p Superscript j minus 1 Baseline left-parenthesis 1 minus p right-parenthesis squared

The %EMLA for the short oligomers are shown in Figure 2.5. The equilibrium constant is only weakly temperature dependent. Esterification reactions are commonly nearly thermoneutral. Recently, Feng et al. [29] performed potentiometric titration at temperatures between 8 and 100°C. They found that differences in the titratable acidity were only a couple percent mostly in the range of 60–80 apparent wt% lactic acid; minor differences were observed at other concentrations.

2.6 VAPOR–LIQUID EQUILIBRIUM

Vapor–liquid equilibria for aqueous solutions of lactic acid have been measured by Sanz et al. [30] and Vu [31]. Sanz et al. used gas chromatography to measure compositions of both phases. They reported only L₂ and L₃ oligomers of lactic acid. Both flame ionization and thermal conductivity detectors were used, which enabled quantification of the water. Details of the calculations using the peak areas were not provided. Due to the low volatility of oligomers, error in the overall lactic acid concentrations of the liquid phase is likely. Analysis of the data indicates that p equals x Subscript normal upper L 2 Baseline slash x Subscript normal upper L 1 is nearly constant across the reported compositions, though p should change as shown in Figure 2.6, indicating that the reported mole fractions are not representative of reaction equilibria.

TABLE 2.4 Liquid Phase Mole Fractions of Lactic Acid + Water VLE at 101.33 kPa

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T (K)	True x _water	True	True	True	True	Apparent
378.25	0.79	0.19	0.015	0.0012	0.0001	0.22
379.25	0.8	0.18	0.014	0.0011	0.000087	0.21
380.25	0.73	0.24	0.024	0.0023	0.00023	0.30
380.75	0.72	0.25	0.025	0.0025	0.00025	0.31
381.75	0.69	0.28	0.03	0.0033	0.00037	0.35
381.85	0.71	0.26	0.027	0.0027	0.0003	0.32
383.35	0.68	0.28